Download The flux-line lattice in superconductors

Rep. Pmg. Phys. 58 (1995) 1465-1594. Printed in the UK
The flux-line lattice in superconductors
Emst Helmut Brandt
Max-Planck-lnstitutf i i Metallforschung, h t i t u t fi Physik, D-70506Stuttgart Germany
Abstract
Magnetic flux can penetrate a type-II superconductor in the form of Abrikosov vortices
(also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux
&, = h/Ze. These tiny vortices of supercurrent tend to arrange themselves in a hiangular
flux-lie lattice (FLL), which is more or less perturbed by material inhomogeneities that pin
the flux lines, and in high-T, superconductors (HTSCS) also by thermal fluctuations. Many
properties of the KL are well described by the phenomenological Ginzburg-Landau theory
or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb
alloys and HTSCs the Fu is very soft mainly because of the large magnetic penetration depth
A. The shear modulus of the FLL IS c66 l/A2, and the tilt modulus C u ( k ) (1 k2Az)-’
is dispersive and becomes very small for short distortion wavelengths 2n/k << A. This
softness is enhanced further by the pronounced anisotropy and layered structure of HTScs,
which strongly increases the penetration depth for currents along the c axis of these (nearly
uniaxial) crystals and may even cause a decoupling of two-dimensional vortex lattices in the
Cu-0 layers. Thermal fluctuations and softening may ‘melt’ the FLL and cause thermally
activated depinning of the flux lines or of the two-dimensional ‘pancake vortices’ in the
layers. Various phase transitions are predicted for the Fu in layered HTScs. Although large
pinning forces and high critical currents have been achieved, the small depinning energy
so far prevents the application of HTSCSas conductors at high temperatures except in cases
when the applied current and the surrounding magnetic field are small.
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This review was received in June 1995.
00344885/95/111465t130$59.50
@ 1995 IOP Publishing Lld
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Contents
1. Introduction
1.1. Abrikosov’s prediction of the flux-line lattice
1.2. Observation of the flux-line lattice
1.3. Ideal flux-line lattice from Ginzburg-Landau theory
1.4. Ideal flux-line lattice from microscopic BCS theory
1.5. High-T, superconductors
2. Arbitrary arrangements of flux lines
2.1. Ginzburg-Landau and London theories
2.2. Straight parallel flux lines
2.3. Curved flux lines
3. Anisotropic and layered superconductors
3.1. Anisotropic London theory
3.2. Arrangements of parallel or curved flux lines
3.3. Layered superconductors
3.4. The Lawrence-Doniach model
3.5. Josephson currents and helical instability
4. Elasticity of the flux-line lattice
4.1. Elastic moduli and elastic matrix
4.2. Nonlocal elasticity
4.3. KL elasticity in anisotropic superconductors
4.4. Line tension of an isolated flux line
4.5. Examples and elastic pinning
5. Thermal fluctuations and melting of the flux-line lattice
5.1. Thermal fluctuations of the flux-line positions
5.2. Time scale of the fluctuations
5.3. Simple melting criteria
5.4. Monte Carlo simulations and properties of line liquids
5.5. Flux-line liquid
5.6. First-order melting transition
5.7. Neutrons, muons and FLL melting
6. Fluctuations and phase transitions in layered superconductors
6.1, 2D melting and the Kosterlitz-Thouless transition
6.2. Evaporation of flux lines into point vortices
6.3. Decoupling of the layers
6.4. Measurements of decoupling
6.5. Vortex fluctuations modify the magnetization
6.6. Quantum fluctuations of vortices
7. Flux motion
7.1. Usual flux flow
7.2. Longitudinal currents and helical instability
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TheJux-line lattice in superconductors
7.3. Cross-flow of two components of magnetic flux
7.4. Dissipation in layered superconductors
7.5. Flux-flow Hall effect
7.6. Thermo-electric and thermo-magnetic effects
7.7. Flux-flow noise
7.8. Microscopic theories of vortex motion
8. Pinning of flux lines
8.1. Pinning of flux lines by material inhomogeneities
8.2. Theories of pinning
8.3. Bean’s critical state model in longitudinal geometry
8.4. Extension to transverse geometry
8.5. Surface and edge barriers for flux penetration
8.6. Flux dynamics in thin strips, disks, and rectangles
8.7. Statistical summation of pinning forces
9. Thermally activated depinning of flux lines
9.1. Pinning force versus pinning energy
9.2. Linear defects as pinning centres. The ‘Bose glass’
9.3. The Kim-Anderson model
9.4. Theory of collective creep
9.5. Vortex glass scaling
9.6. Linear a.c. response
9.7. Flux creep and activation energy
9.8. Universality of flux creep
9.9. Depinning by tunnelling
10. Concluding remarks
Acknowledgments
References
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EHBmndt
1. Introduction
1.1. Abrikosov's prediction of theflux-line lonice
Numerous metals, alloys and compounds become superconducting when they are cooled
below a transition temperature T,. This critical temperature ranges from T, c 1 K for AI, Zn,
Ti, U, W and T, = 4.15K for Hg (the superconductor discovered first, Kamerlingh Onnes
1911), over T, = 9.2K for Nb (the elemental metal with the highest T,)and Tc r+ 23 K for
Nb3Ge (the highest value from 1973 to 1986, see the overview by Hulm and Matthias 1980)
to the large Tcvalues of the high-T, superconductors (HTSCS) discovered by Bednorz and
Muller (1986), e.g. YBaZCu307-8 (YBCO, 6 (( 1, Wu et al 1987) with T, m 92.5K and
Bi2Sr,Ca+l13O,o+s (BSCCO) with T, = 1 2 0 K and Hg-compounds which under pressure
have reached Tc r+ 164 K (Gao et al 1994, Kim et af 1995~).The superconducting state is
characterized by the vanishing electric resistivity p(T) of the material and by the complete
expulsion of magnetic flux, irrespective of whether the magnetic field B, was applied
before or after cooling the superconductor below T, (Meissner and Ochsenfeld 1933). The
existence of this Meissner effect proves that the superconducting state is a thermodynamic
state, which uniquely depends on the applied field and temperature but not on previous
history. As opposed to this, an ideal conductor expels the magnetic flux of a suddenly
switched on field B,, but also 'freezes' in its interior the magnetic flux which has been there
before the conductivity became ideal.
...........
. :,/-.........
...\.
.:+m
'r:
......
.,--.-.--/.
.
....
4 .....
-.....l.n.)
~.
;y.7*'0. v .
...
.....
....
..........
@
.I*.
FLL with 20 dcfectr
YIPW
dislocation
Figure 1. The flux-line lattice (U) in typeU superconductors. kfi FLL with two-dimensional
defects (simple edge dislocation, slacking fault connecting two parrial dislocations, vacancy.
interstitial). Righl. FU. with a saew dislocation (broken curve).
It was soon realized that some superconductors do not exhibit expulsion of flux, but
the applied field partly penetrates and the magnetization of the specimen depends on the
magnetic history in a complicated way. Early theories tried to explain this by a 'sponge
like' nature of the material, which could trap flux in microscopic current loops that may
become normally conducting when the circulating current exceeds some critical value. The
true explanation of partial flux penetration was given in a pioneering work by Abrikosov
(1957). Alexei Abrikosov, then a student of Lew Landau in Moscow, discovered a
periodic solution of the phenomenological theory of superconductivity conceived a few
years earlier by Ginzburg and Landau (GL) (1950). Abrikosov interpreted his solution as a
lattice of parallel flux lines, now also called flux tubes, fluxons, or Abrikosov vortex lines
(figure 1). These flux lines thread the specimen, each carrying a quantum of magnetic flux
&, = h/2e = 2.07 x 10-15Tmz. At the centre of a flux line the superconducting order
parameter @ ( r )(the complex GL function) vanishes. The line Jr = 0 is surrounded by a
tube of radius r+ the vortex core, within which lJrl is suppressed from its superconducting
value I J r ] = 1 that it attains in the Meissner state (figure 2). The vortex core is surrounded
by a circulating supercurrent J ( r ) which generales the magnetic field B(T)of the flux line.
e,
Theflux-line lattice in superconductors
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In bulk specimens, the vortex current and field are confined to a flux tube of radius A; at large
distances r >> A, current and field of an isolated vortex decay as exp(-r/A). In thin films
of thickness d << A, the current and magnetic field of a vortex extend to the larger distance
Atr,, = 2A2/d, the circulating current and the parallel field at large distances r >> Aglm
decrease only as l/r2 and the perpendicular field as l/r3, and the vortex core has a larger
radius
(I2Afi1,f~)'/~
(Pearl 1964, 1966, Fetter and Hohenberg 1967). The coherence
length e and magnetic penetration depth A of the GL theory depend on the temperature T
and diverge at T, as (1 - T/Tc)-1/2.
Figure 2. The periodic magnetic field B ( x . y ) (1eJl and right) and order parameter I*l2(x. y )
(right) of the ideal flux-line lattice The Run-line positions correspond to the maxima in B ( r , y )
and ems in
Top. Low induction B << B,z. Bottom High induction B % Bc2. The
broken zig-zag line shows the exotic field profile in clem superconductors at zero temperature
(section 1.3).
The ratio K = A/$ is the GL parameter of the superconductor. Within GL theory, which
was conceived for temperatures close to the transition temperature Tc, K is independent of
T . The flux-line lattice (EL) (Abrikosov's periodic solution) exists only in materials with
K > 1 / d , called type-II superconductors as opposed to t y p e 1 superconductors, which
have K c l / d . Type-I superconductors in a parallel field E, < B,(T) are in the Meissner
state, i.e. flux penetrates only into a thin surface layer of depth A(T), and at B > BJT)
they become normally conducting. Here BJ!') is the thermodynamic critical field. TypelI
superconductors in a parallel applied field E, .= E,, ( T ) B,(T) are in the Meissner state
B = 0; in the field range Bel ( T ) c: E, < B,z(T) magnetic flux penetrates partly in the form
of flux lines (mixed state or Shubnikov phase 0 < B c Bo);and at B. > Bc*(T) 2 B J T )
the material is normally conducting and thus B = 8,. E,, and BCz are the lower and upper
critical fields. Within GL theory, all three critical fields vanish for T + Tc as T, - T . In
<
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E H Brandt
this paper the notation Ba = poH& B, = mH,,B,I = p ~ H ~ ~ ; a nB,zd = poH,z will be
used.
N=Q
.. .
.. .
Applied magnetic field
Figure 3. Reversible (ideal, pin-free) magnetidon cwes of long cylinders or strips in a
parallel magnetic field E. (demagnetization factor N = 0, lejV and of a sphcre (N =
right). Top T y p l superconductor with Meissner state for E. < (1 - N)B,. intermediate state
for (1 - N)B, 6 E,, < E,, and normal state for B. > Ec. The broken c w e indicates the
enhancement of the penemtion field when the wall energy of the normal and superconducting
domains in the intermediate state is accounled for (Kromiillerand Riedel1976). Middle.TypeIU1 suprumducton with Ginzburg-Landau parameter x close to I f & for N = 0 exhibit
a jump of height Bo in t h e i induction and magnetiwtion at E. = &I due to an a m f i v e
interaction between flux lies. For N > 0, this jump is sh-etched over a finite range of E.,
allowing one to observe an intermediate mixed state with domains of Meisrner phase and FU
phme. BolIomType-IYZsuperconductorswithu>> I/& Meismerphasefar B.
(l-N)E<,,
mixed state (U) for (1 - N)&I Ea En. and n o d state for E, > B,z.
4,
< <
When the superconductor is not a long specimen in a parallel field, demagnetization
effects come into play. For ellipsoidal specimens with homogeneous magnetization the
demagnetizing field is accounted for by a demagnetization factor N with 0
N
1,
figure 3. If N > 0, flux penetration starts earlier, namely, into t y p e 4 superconductors
at BLl = (1 N)Bcl in the form of a FLL, and into type-I superconductors at E: =
(1 - N ) B , in the form of normally conducting lamellae; this 'intermediate state' is described
by Landau and Lifshitz (1960), see also Hubert (1967), Fortini and Paumier (1972),
and the review by Livingston and DeSorbo (1969). From GL theory the wall energy
between normal and superconducting domains is positive (negative) for type-I (type-II)
superconductors. Therefore, at B, = B, the homogeneous Meissner state is unstable in
type4 superconductors and tends to split into normal and superconducting domains in the
finest possible way; this means a FU. appears with normal cores of radius % 5. The field
of first penetration of flux is thus (1 - N)B,1 < (1 - N ) B , for type-II superconductors and,
following Kronmiiller and Riedel (1976). Bp = I(1 - N)*E: KZ]'/' > (1 - N ) B , for
type-I superconductors with K proportional to the wall energy. Superconductivitydisappears
<
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<
the flux-line lattice in superconductors
1471
when B, reaches the critical field B,z (typen) or B, (type-I), irrespective of demagnetization
effects since the magnetization vanishes at this transition. Within GL theory these critical
fields are
The order parameter l+(r)lZand microscopic field B ( r ) of an isolated flux line for K
are approximately given by (Clem 1975a, Hao er a1 1991)
I*(r)Iz
M
1 / ~ 2+5 ’ / 4
>> 1
( 1.2)
+
B ( r ) M -Ko[(rz 2 5 2 ) ” 2 / ~ 1
( 1.3)
2nh2
with r = (xz + y2)II2 and B llz; Ko(x) is a modified Bessel function with the limits - ln(x)
(x << 1) and (n/Zx)’/’exp(-x) (x > 1). The field (1.3) exactly minimizes the GL free
energy if the variational ansatz (1.2) is inserted. The maximum field occurs in the vortex
core, B,, = B(0) M (h/2nh2)lnK m 2BCl (still for K >> 1). From (1.3) one obtains the
current density circulating in the vortex J ( r ) = &‘lB’(r)l = (4oo/2nhZjd(r/hi)K1(?/A)
with i = (r2+ 2..$2)’/2. Inserting for the modified Bessel function K I(x) the approximation
K l ( x ) % I/x valid for x << 1, one obtains the maximum current density J- = J(r =
m h/(4&nh2epo) = (27/32)’12J~where JO = @ 0 / ( 3 & a h ~ ~ piso )the ‘depairing
current density’, i.e. the maximum supercurrent density which can flow within the GL theory
in planar geometry (see, e.g. Tinkham 1975). Thus, for K >> 1 the field in the flux-line
centre is twice.the lower critical field, and the maximum vortex current is the depairing
current.
-&x)
1.2. Observation of the flux-line lattice
The first evidence for the existence of Abrikosov’s FLL in Nb came from a weak Bragg
peak seen in small-angle neutron-scattering (SANS) experiments by Cribier et a1 (1964),
which later were improved considerably (Weber et a1 1973, 1974, Thorel et al 1973, Lynn
et a1 1994). High-resolution pictures of the i%L were obtained by Essmann and Trauble
(1967, 1971) by a decoration method using ferromagnetic microcrystallites generated by
evaporating a wire of Fe or Ni in a He atmosphere of a few Torr pressure. This ‘magnetic
smoke’ settles on the surface of the superconductor at the points where flux lines end and
where the local magnetic field is thus peaked. The resulting pattern is then observed in an
electron microscope; some such pictures are shown in figure 4.
This Bitter decoration of the FLL works also with the new high-Tc superconductors
(Gammel et al 1987, 1992a,b, Vinnikov et a1 1988, Dolan et a1 1989a,b, Bolle et a1
1991, Grigorieva et a1 1993, Dai et a1 1994a. b, Durin et al 1995), see also the review by
Grigorieva (1994). Yao et al (1994) decorate the KL on both sides of a BSCCO crystal,
and Marchetti and Nelson (1995) show how to extract bulk properties of the FLL from such
double-sided decoration. Marchevsky eta1 (1995) observe droplets of FLL in NbSe2 crystals
by decoration. Small-angle neutron-scattering (SANS) from the FLL in HTSCS was performed
with YBCO by Forgan et a1 (1990) and Yethiraj et a1 (1993) and with BSCCO by Cubitt
er al (1993b) to see if the FLL melts (see section 5.7). Keimer et a1 (1994) investigate
the symmetry of the FLL in anisotropic YBCO by SANS. Kleiman e? a1 (1992) by SANS
observe the FLL in the heavy-fermion superconductor UPt3. The 3D flux distribution inside
the superconductor can also be probed by neutron depolarization (Weber 1974, Roest and
Rekveldt 1993). A further method, which probes the field distribution of the FLL in the bulk,
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E H Brandt
.
.
Figure 4. The flux-line lattice observed at l
k surface of lype-I1 superconducton in M electron
microscope after decoration wilh'Fe micmrystlllites. ( 0 ) Pbln alloy (Tc = 7.5K.Y = 2) at
T = 1.2K in a remanent magnetic field of 70G. yielding a flux-line spacing o = 0 . 6 ~ m .
is muon spin rotation (pSR) (Brandt and Seeger 1986, Piimpin et al 1988, 1990, Herlach
et al 1990, Harshman et al 1989, 1991, 1993, Brandt 1988a,b, 1991c, Lee et al 1993,
1995a,b Cubitt et a! 1993a, Weber et al 1993, Sonier et al 1994, Alves er af 1994, Song
1995, Riseman e t n l 1995), see also sections 1.5 and 5.7.
Very high resolution of the E L in NbSez (T, = 7.2K) was achieved by a scanning
tunnelling microscope (STM), which measures the spatially varying density of states of the
quasiparticles (Hess et al 1990, 1992, 1994, Renner et a1 1991, Behler et al 1994a,b,
Maggio-Aprile et al 1995). Using a STM Hasegawa and Kitazawa (1990) performed
spectroscopy with a spatial resolution of single atoms in the CuOz and (BiO)z layers of
BSCCO. Flux lines were imaged also in a magnetic force microscope by Hug er al (1994)
and Moser et al (1995), see the appropriate theories by Reittu and Laiho (1992). Wadas
The f l u d i n e lofficein superconductors
1473
Figun: 4 (continued). ( b ) High-purity Nb disks 1 nun thick, 4 mm diameter, of different
crystallographic orientations [I101 and [Oll], P T = 1.2K and R. = 8MlG (a,, = 1400G).
Due to demagnetizing effects and the small x % 0.70, round islands of Meissner phase are
surmunded by a regular EL (‘intermediate mixed state’). ( e ) High-purify Nb foil 0.16 mm thick
at T = 1.2 K and R. = 173 G . Round islands of FU. embedded in a Meissner phase.
et al (1992). and Coffey (1995). Hyun et al (1989) detected the motion and pinning of a
single flux line in a thin film by measuring the diffraction pattern from a Josephson junction
formed by superconducting cross ships. Kruithof ef al (1991) in NbMo near Bc* saw fluxline motion by its coupling to a ZD electron gas, which is formed at the interface between Si
and SiO. Bending er nl (1990) and Stoddart et al (1993) achieved singlefluxon resolufion by
microscopic Hall probes. Mannhardt ef nl (1987) observe trapped flux quanta by scanning
E H Bran&
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{d)
Figure 4 (continued). (d) Square disk 5 x 5 x I mm3 of highpurity polycrystalline Nb at
T = 1.2K and B, = 1100G. As B. is increased. magnetic flux penetrates fmm the edges in
form of fingers (upper left) which are composed of FLL shown enlarged in the right pichlre.
The rectangular cmss section of the disk causes an edge barrier (section 8.5). As scan as this
edge barrier is overcome, single flux lines or droplets of FLL (lower right) pull apaR fmm these
fingers and jump to t
k centre, filling the disk with flux from the centre. (Reproduced courtesy
of U Essmann.)
electron microscopy, see also the reviews on this method by Huebener (1984) and Gross and
Koelle (1994). Harada eral (1993) observed the mL time-resolved in thin films of BSCCO
in a field-emission transmission electron microscope by Lorentz microscopy (outside the
focal plane) following a suggestion by Suzuki and Seeger (1971) and early experiments
by Lischke and Rodewald (1974), see also Bonevich er al (1994a) and the nice work on
electron holography of flux lines by Hasegawa eral (1991) and Bonevich et al (1994b).
The flux density at the specimen surface also may be observed by a magneto-optical
method which uses the Faraday effect in a thin layer of EuSEuF2 (Kirchner 1969, Huebener
er al 1990, Habermeier and Kronmiiller 1977, Briill et al 1991) and EuSe (Schuster et al
1991, see also thereview paper by Koblischka and Wijngaarden (1995), or in a high-contrast
magneto-optical garnet plate (Dorosinskii et al 1992, Indenbom er al 1994~)covering the
surface of the superconductor. Magneto-optics was used, e.g. for comparing the penetrating
flux pattern with microscopic pictures of the same polycrystalline surface in order to identify
the pinning defects (Forkl et nl 1991a, Koblischka er al 1993, 1994), for determining the
critical current density from the measured flux density profiles (Forkl etal 1991b. Theuss er
nl 1992a), for visualizing flux motion in a YBCO film covering a step edge in the substrate
(a so called step-edge junction used for SQUIDS, Schuster e l al 1993b). for obtaining ultra-fast
time-resolved pictures of thermal instabilities in the flux-line lattice in thin films (Leiderer
er al 1993), for investigating the flux penetration into partly irradiated platelets with a
weak pinning central zone (Schuster et nl 1994d) and the anisotropy of perpendicular flux
penetration induced by a strong in-plane magnetic field (Indenbom et al 1994~).and for
visualizing pinning of flux lines at twin boundaries (Vlasko-Vlasov er al 1994a). The flux
distribution and motion at the surface may also be observed, more quantitatively but with
Thef l u d i n e lattice in superconducrors
1475
lower spatial resolution, by scanning Hall probes (e.g. Briill et a1 1991, Brawner and Ong
1993, Brawner et al 1993, Xing et a1 1994) or Hall-sensor arrays (e.g. Tamegai er al 1992,
Zeldov er a1 1994b).
1.3. Idealflux-line lattice from Ginzburg-Landau theorj
When the applied field B. is increased, more and more flux lines penetrate, and the average
induction B = ( B ( r ) )= $on (n = flux-line density) grows. At Bo = B = BCzthe vortex
cores overlap completely such that @ vanishes and superconductivity disappears. At low
inductions B c O.2Bc2 the order parameter is a product of terms of the f m (1.2), or of
the more practical approximation l@(r)I2
= 1-exp(-rz/Y2), with r replaced by lr -ril
(Brandt and Evetts 1992). The magnetic field is then a linear superposition of terms (1.3)
which are centred at the flux-line positions ri (figure 2, top). This useful approximation
applies to arbitrary arrangements of parallel flux lines as long as the vortex cores do not
overlap; this means that all distances ITi -r,l should exceed = 55. Far from the centre of an
isolated vortex a more accurate perturbation calculation gives 1 - I@(r)l exp(-&/C)
for K 5- f i and 1 7 I@(r)l exp(-2r/h) for K < f i (Gamer 1971). At high inductions
B > O.5Bcz, the GL theory yields for a periodic FLL with a r b i t r q symmetry
-
-
B ( T ) = Ba
- (60/4zhz)l@(r)lz.
(1.5)
+
Here the sum is over all reciprocal lattice vectors K,, = (2z/xly2)(myz; -mx2 n x l ) of
the periodic FLL with flux-line positions R, = (mxl nx2; nyz) ( m , n = integer). The
unit-cell area x1y2 yields the mean induction B = q5&1yz). The first Brillouin zone has
the area zk;, with kiz = 4zB/40. For general lattice symmetry the Fourier coefficients
aK and the Abrikosov parameter ,SA are (Brandt 1974)
+
aK = ( - l y + m n + n
exp(-K;,x1~2/8z)
BA = E a & .
(1.6)
K
From @(O) = 0 follows C Ka x = 0, or with U K =~ 1, CKZo
aK = - 1. In particular, for
the triangular FLL with spacing a = (2&/fiB)''z one has x , =a,x2 = a/2. y2 = &a/2,
B = 2&0/&a2, a x = (-I)"exp(-zu/&)
with U = mz mn n2 = Rmn / R I O =
K,,/Klo. This yields BA = 1.1595953 (the reciprocal-lattice s u m converges very rapidly)
and the useful relationships Kto = 2n/yz, K:, = 16irz/3a2= 8nZB/&,.
For the square
lattice one has BA = 1.1803406. The free energy F ( B ) per unit volume and the negative
magnetization M = B. - B >, 0, where B. = &aF/aB is the applied field, which is in
equilibrium with the induction B, are (still for B > 0.5Bcz)
+
+
This yields for the periodic field B ( r ) = C KBK cos Kr the Fourier coefficients BK+ =
M a K , B K =~(BO.))= B, and for the averaged order parameter (l@12) = (4iriZ/@o)M.
Klein and Pottinger (1991) rederive Abrikosov's periodic FLL solution in a simple way using
the virial theorem discovered for the GL equations by Doria et al (1989).
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E H Brandt
Bc2 the bulk superconductivity vanishes, but a thin surface layer of
At fields B.
thickness EJ remains superconducting as long as the parallel field is smaller than a third
critical field Bcs = 1.695Bc2 near Tc (Saint-James and DeGennes 1963, Abrikosov 1964)
or Bc3 EJ 2Bc2 near T = 0 (Kulik 1968). For recent work on Bc3 see Suzuki et a1
(1995) and Troy and Dorsey (1995). A very interesting situation results when a field
Bo > Bcz is applied at an angle to the surface of the superconductor. Then a lattice of
tilted vortices offinite length appears, which was confirmed by the classical experiment of
Monceau er al (1975). The theory of these ‘Kulik vortices’ (Kulik 1968) is quite intricate
and still not complete, although ’only’ the eigenvalue equation (linearized GL equation)
[-iV - (Zrr/&)AlZ@
= E @ with boundary condition [-iV - (2n/&)A]+= 0 has to be
solved, where A is the vector potential of the constant (but oblique!) applied field, see the
reviews by Thompson (1975) and Minenko and Kulik (1979).
At intermediare inductions 0.2 c B/Bd < 0.5 the GL equations have to be solved
numerically. This has been achieved by a Ritz variational method using finite Fourier series
as trial functions for the real and periodic functions I@(r)12
and B(T) (Brandt 1972a),
thereby extending early analytical calculations by Kleiner et a1 (1964) and Eilenberger
(1964) to the entire range of inductions 0 B B,z. These computations show that the
above expressions derived for low and high fields are excellent approximations.
< <
1.4. Ideal j?ux-line lattice from microscopic BCS theory
The same variational method was used to compute the ideal FLL in pure superconductors in
the entire range of B and T from the microscopic theory of superconductivity of Bardeen,
Cooper and Schrieffer (BCS) (1957). reformulated by Gor’kov (1959a.b) to allow for a
spatially varying order parameter. This computation (Brandt 1976a. b) extended ideas of
Brandt et a1 (1967) and Delrieu (1972, 1974) (originally conceived to calculate the electron
Green function near Bc2) by introducing as small parameters the Fourier coefficients of the
function V ( T ,r’) = A ( P )exp[i(Zrr/@o) A(Z)dZ] A*(?-’). where A is the gap function
and A the vector potential. For a periodic FLL, V ( r ,T’) is periodic in the variable T r‘,
and for B -+ Bc2 (i.e. when the Abrikosov solution for A applies) it is a simple Gaussian,
V ( r ,r’)= (1A12) exp(-lr - r’12rrB/&); this means that all higher Fourier coefficients of
V vanish at Bc2 and are very small in the large range B 2 O.5BC2 (Brandt 1974). However,
this elegant numerical method appears to be restricted to pure superconductors with infinite
electron mean free path 1.
For superconductors with arbitrary purity the FLL was computed from microscopic theory
in the quasiclassical formulation by Eilenberger (1968). which expresses the free energy in
terms of energy-integrated Green functions; see also Usadel (1970) for the limit of impure
superconductors (with numerical solutions by Watts-Tobin et a1 1974 and Rammer 1988)
and the carefull discussion of this quasiclassical approximation by Serene and Rainer (1983).
The computations by Pesch and Kramer (1974), Kramer and Pesch (1974). Watts-Tobin et
U / (1974), Rammer e l af (1987), and Rammer (1988) used a circular cell method, which
approximates the hexagonal Wigner-Seitz cell of the triangular FU by a circle, and the
periodic solutions by rotationally symmetric ones with zero slope at the circular boundary
(Ihle 1971). Improved computations account for the periodicity of the E L (Klein 1987,
Rammer 1991a,b), see also the calculation of the density of states near Bc2 by Pesch (1975)
and recent work by Golubov and Hartmann (1994).
Remarkably, the circular cell approximation not only yields good results at low
inductions B << Bcz, but within GL theory it even yields the exact value for B,z (Ihle 1971).
and from BSC theory a good approximate B,z (Larkin and Ovchinnikov 1995). From the
JS’
+
..
~
I477
Thefiu-line luttice in superconductors
BCS weak-coupling microscopic theory the upper critical field Bcz(T) was calculated for all
<
temperatures 0 T < T, by Helfand and Werthamer (1966) and Eilenberger (1966) and
from strong-coupling theory by Riek et a1 (1991). Zwicknagl and Fulde (1981) calculate
B,z(T) for antiferromagnetic superconductors, e.g. the Chevrel phase compound rare-earthMO&. For layered superconductors Bc2(T) was obtained by Klemm er al(1975). Takezawa
er al (1993). and recently by Ovchinnikov and Kresin (1993, who showed that in the
presence of magnetic impurities B,z(T) shows upward curvature at low T and is strongly
enhanced, such an enhancement is observed by Cooper er a! (1995). Bahcall(1995) obtains
corrections to the semiclassical B,z(T) which can also explain this positivecurvature near T,.
For non-equivalent layers &(T) was calculated by Koyama and Tachiki (1993). Spiv& and
Zhou (1995) consider B d T ) in disordered superconductors; in mesoscopic samples they
find multiple re-entrant superconducting transitions, but the T, of bulk samples remains
finite at any B such that Bc2 diverges for T + 0.
t = O.G
t=O.C
Figure 5. The penodic magnetic field of a clean superconductor near &2. equation (1.9), for
reduced tempemare f = TIT, = 0.99.0.6.0.3.0. Note the conical maxima and minima at
1 = 0. AI I > 0.6 (t
0.6) the saddle point lies between two (three)flux lines.
Calculations near BCz yield the Maki parameters K I ( T / T ~and
) KZ(T/T,)(Maki 1964)
defined by K I =B,*/&B, (1.1) and K Z = ()2.32dM/dBJ~~+00.5)'~2
(I.@, see Auer and
Ullmaier (1973) for a careful experimental analysis. Both ratios K , / K and K Z / K equal unity
at T = Tc and become z 1 at lower T. In the clean limit, K Z / K diverges at T = 0 since
the magnetization becomes a non-analytic function at Bo + BCz and T -+ 0, namely,
M x / Inx with x = (1 - Ba/BCz)m 1 - b (Delrieu 1974). For impure superconductors
one has K I Fi: K Z 4 K at all temperatures.
The full T- and B-dependences of the vortex structure in HTsCs were calculated from
BCS theory by Rammer (1991a), and the vortex structure in pure Nb from strong-coupling
theory by Rammer (1991b). The local density of states of the quasiparticles in the vortex
core (Caroli er al 1964, Bergk and Tewordt 1970). which can be measured by scanning-
-
1478
E H Brandt
tunnelling-microscopy (Hess eta1 1990, 1992, 1994), was calculated by Klein (1990). Ullah
et al (1990). Gygi and SchIiiter (1991). Zhu et a1 (1995) (accounting for the atomic crystal
field), Berkovits and Shapiro (1995) (influence of impurity scattering), and by Schopohl
and Maki (1995) (for a d-wave superconductor). For more work on the flux line in d-wave
(or combined ‘stid’-wave) superconductors see Volovik (1993) and Soininen et al (1994).
Electronic transport properties like ultrasonic absorption and thermal conductivity of the
FLL (Vnen et a1 1971) were calculated by Pesch et a1 (1974) and Klimesch and Pesch
(1978). see also Takayama and Maki (1973) and the review on the properties of type-I1
superconductors near B a by Maki (1969a).
These computations show that for impure superconductors (with electron mean free path
I smaller than the BCS coherence length t o ) the GL .result is typically a good approximation.
However, as was shown by Delrieu (1972) in a brilliant experimental and theoretical paper,
for clean superconductors near Bc2 the local relation (1.5) between B ( r ) and I @ ( T ) ~does
~
not hold. The correct nonlocal relation (with a kernel of range hvp/nksT, u p = Fermi
velocity) yields more slowly decreasing Fourier coefficients BK at T c T, (see also Brandt
1973a, 1974, Brandt and Seeger 1986, Brandt and Essmann 1987). In particular, at T << T,,
B M B,z. and 1 >> t o = hvF/nA(T=O) one has BK (-l)”/K3. This Fourier series
yields a rather exotic magnetic field B ( r ) with sharp conical maxima at the flux-line centres
and conical minima between two flux lines, and (for the triangular FLL) saddle points with
three-fold point symmetry in the middle between three flux lines, figure 5. The minima and
saddle points interchange their positions at TIT, 0.6. Explicitly, the magnetic field of
clean superconductors near B,z(T) has the form (Brandt 1974)
-
B(r) = B,
+ [MI C b K C O S K P = B + [MI
K
bK(c) = -(-I)”I”/I~
v
= m2 + mn
+
K#o
n2
bKCOSKT
p = (n’J2/3’J4)
J;
with the reciprocal FLL vectors K,,,,, from (1.4), p 2 = K i n ~ ~ y for
~ / general
8 ~ FLL
symmetry, IMI = B. B the negative magnetization, and U F the Fermi velocity. From (1.9)
one obtains the limiting expressions bK FS IMI(-1)”(33/4/2n)/u3/2 for T -+ 0 (c -+ 0)
and b~ = IMI(-l)”exp(-un/&)
for T + T, (c + 00 since B -+ 0). It is interesting to
note that the limit K + 0 yields the magnetization. Though derived for B ir BCz,the field
(1.9) approximately applies also to lower inductions B as long as the Abrikosqv solution
for. @(T) is good. The parameter c(T, E ) is explicitly obtained from
-
m
c ( T , B ) = c(t)/&
K ( r ) r [ l -e-2’c(r)’]d~ = - h t
(1.10)
with t = T / T c and b = B/B,z(T). From (1.10) one finds to a relative accuracy of 0.0012
c ( t ) = t ( l -t2)-1/2(1.959+0.182t+0.224t2).
(1.11)
<
From c ( t ) (1.10) the upper critical field of clean superconductors for 0 T c T, is obtained
as B a ( T ) = 2 & , t 2 / [ n ~ ~ c ( twith
)2]
= hvF/2RkgTc; these Bc2 values coincide with the
result of Helfand and Werthamer (1966). For arbitrary purity the bK are derived by Delrieu
(1974), Brandt (1974). and Pesch (1975).
cc
The flwc-line latrice in superconductors
1479
The strange field profile (1.9) has been observed in pure Nb by nuclear magnetic
resonance (NMR) (Delrieu 1972) and by muon spin rotation ( ~ s R(Herlach
)
et al 1990).
It is an interesting question whether this zig-zag profile of E ( T ) may be observed in other
superconductors. The type-I1 superconducting elemental metals are Nb, V, La and Tc.
Single crystals of V with the required high punty became available only quite recently and
have been studied comprehensively by Moser er a1 (1982). The HTSCS appear to be in the
clean limit since their coherence length ( is so small, smaller than the electron mean free
path I ; but by the same token their upper critical field Ec2 l / t z (1.1) is extremely large
at T << Tc such that the experimental observation of a possible exotic zig-zag field profile
is dificult.
Clean Nb has the remarkable property that its theoretical magnetization curve obtained
from the generalized GL theory of Neumann and Tewordt (1966) (an expansion in powers
of T, - T ) by Hubert (1972) and from the BCS theory by Ovchinnikov and Brandt (1975)
and Brandt (1976a,b), is not monotonic but has an S-shape if T is not close to T,
(figure 3, middle). This S-shape means that the penetrating flux lines attract each other.
The resulting equilibrium distance a0 of the flux lines, and a jump in B(B.) to a finite
induction BO % @o/at obtained by a Maxwell construction from the theoretical S-shape,
were observed in clean Nb and in Pb-TI and Ta-N alloys with K values slightly above
l / f i by decoration and magnetization experiments, see Sarma (1968), Essmann (1971),
Gageloh (1970). Krageloh et al (1971), Auer and Ullmaier (1973), and the review by
Brandt and Essmann (1987). Decoration revealed both Meissner phase islands surrounded
by nt (figure 4(b)), and FLL islands embedded in Meissner phase (figure 4(c)). The phase
diagram of clean superconductors in the K - f plane (r = T/T,) computed from the BCS theory
by Klein (1987) ahd discussed by Weber et ai (1987), near K = 1 / f i with increasing t
shows the transition from type-I to usual type4 (‘type-U-1’) superconductors via a region
where the flux lines attract each other (‘type-II-2’ superconductors). Klein et al (1987) give
an elegant and surprisingly accurate interpolation method which constructs the free energy
and magnetization (also the S-shape) of type-II superconductors using only properties at the
upper and lower critical fields.
A further rather unexpected result of microscopic theory is that for pure superconductors
at low induction B the vortex core gets rather narrow; possibly the gap function IA(r)l even
attains a vertical slope at r = 0 when I -+ CO, T --f 0, and E -+ 0. This shrinking ofthe
vorfex core was seen in computations (Gamer and Pesch 1974, Brandt 1976a,b) and was
investigated analytically by Ovchinnikov (1977). see also Rammer er al (1987).
-
1.5. High-T, superconducrors
With the discovery of high-Tc superconductivity, the interest in the FLL rose again. At
present it is not clear whether Delrieu’s (1972) exotic field distribution near BCz might be
observed in these rather clean materials; in HTscs
is very small (20-30A) such that
I >> 60 is typical (Harris et al 1994, Matsuda et al 1994, Guinea and Pogorelov 1995) (see
section 7.8), but by the same token Bc2(0) = @0/27r# is very large (% 100T),rendering
such experiments difficult, and possibly quantum fluctuations smear the cusps in E(?-). In
TI-based HTSCS the field distribution of the FLL and the vortex motion can be observed by
NMR of the zo5Tlnuclear spin (Mehring er a1 1990, Mehring 1990, Brom and Alloul 1991,
Song ef al 1992, 1993, 1994, Carretta 1992, 1993, Bulaevskii er a1 1993b, 1995a, Moonen
and Brom 1995, Song 1995). Bontemps et al (1991) measured the electron paramagnetic
resonance (EPR) linewidth of a free-radical layer on YBCO. Steegmans etal (1993) studied
the FLLby a NMR decoration technique putting a silicone layer (thinner than the flux-
1480
E H Brandt
line spacing) on grains of Nb& powder. Suh er a1 (1993). Rigamonti et a1 (1994) and
Recchia er a1 (1995) detected the thermal motion of the flux lines by nuclear spin echo
decay measurements and De Soto ef a1 (1993) observed that intrinsic pinning of the FLL in
organic superconductors enhances the N M R spin-lattice relaxation.
Besides the field distribution, there are numerous other exciting problems concerning the
FLL in HTSCs, e.g. the smcture and softness of the mL in these highly anisotropic materials,
the appearance of weakly interacting two-dimensional (ZD) vortex disks (‘pancake vortices’,
Clem 1991) in the superconducting Cu-0 layers, and of coreless vortex lines (‘Josephson
strings’) running between these layers. Furthermore, the thermal fluctuations of the 3D or 2D
vortices may lead to a melted (shear-stiffness-free) Fw in a large region of the B-T plane.
Particularly fascinating are recent ideas concerning the statistics of pinned vortices at finite
temperatures, namely, thermally activated depinning and collective flux creep, a vortex-glass
transition (Fisher 1989, Fisher et ai 1991) near which the resistivity and magnetic response
obey scaling laws, and the analogy with a Bose glass (Nelson and Vinokur 1992, 1993)
when linear pins are introduced by irradiation with heavy ions, see section 9.2. All these
points will be discussed in the following sections. More details on these theoretical concepts
may be found in recent review papers by Blatter er al (1994a) and Fischer (199%).
2. Arbitrary arrangements of flux Lines
2.1. Ginzburg-hdau and London theories
The GL theory (Ginzburg and Landau 1950) extends Landau’s theory of second-order
phase transitions to a spatially varying complex order parameter @ ( r ) .The resulting
gradient term is made gauge-invariant by combining it with the vector potential A ( r )
where V x A ( T )= B(r)= p ~ H ( ris) the local magnetic field. The two GL equations are
obtained by minimization of the GL free energy functional F(@, A ) with respect to @ and
A, i.e. from SF/S@ = 0 and S F j S A = 0. where in reduced units
A)$-’+ B’].
-
In (2.1) the length unit is the penetration depth A,
K = h/f is the GL parameter, @ is in
units of the BCS order parameter 90 (PO
- T) such that 1@1 = 1 in the Meissner
state and [@I= 0 in the normal conducting state, B is in units Z/ZB,, and A in units
&B,h. If desired, the electric field E = -aA/ar and the time dependence may be
introduced by the assumption a$/& = -ySF/S@ with y = constant (Schmid 1966, Vecris
and Pelcovits 1991, Dorsey 1992, Troy and Dorsey 1993). An extension to anisotropic
non-cubic superconductors is possible by multiplication of the gradient term in (2.1) by an
‘effective mass tensor’ flakanaka 1975, Kogan and Clem 1981). For the H,z-anisotropy in
cubic superconductors (like Nb)see Weber etni (1981) and Sauenopf et al (1982).
The phenomenological GL theory is one of the most elegant and powerful concepts in
physics, which was applied not only to superconductivity (see the textbooks of DeGennes
1966, Saint-James er a! 1969, Znkham 1975, Abrikosov 1988, Buckel 1990, Klemm 1996)
but also to other phase transitions, to nonlinear dynamics, to dissipative systems with selforganizing pattern formation, and even to cosmology (melting of a lattice of ‘cosmic strings’,
Nielsen and Olesen 1979; vortex lattice in electmweak theory, Ambjarn and Olesen 1989).
Introducing the modulus f(r)= ]@Iand phase 4(r)of the CL function @ = fe” one
finds that only this modulus and the gauge-invariant ‘supervelocity’ Q = V@/K - A, or
Q = (&/Zx)V4 - A in physical units, enter the GL theory, but not $ and A separately.
1481
Theflux-line lattice in superconductors
Thus, completely equivalent to (2.1) is the functional
F(f,&) = &OH: /d% [ - f 2
+ if4 + (vf)'/~'+ f Z Q 2 + (V x .I')&
(2.la)
Since only real and gaugeinvariant quantities enter in (2.la). any calculation based on the
theory (2.1) should contain only gauge-invariant equations, even when approximations
are made. For a straight vortex with n quanta of flux centred at r = (x2 y')'/' = 0 one
finds at r <( 5 the behaviour f ( r ) * r" and Q ( r ) = n / ( u ) . In a particular gauge one
has at r << 6 for this vortex q
( x iy)" and A % constant In the impractical gauge
with real q the vector potential A would diverge as I / ( K I T - r,l) at all vortex positions
r y= ( x u ; y,,); this real gauge would thus depend on the vortexpositions. For a FLL with n ,
flux quanta in the vth flw line, a convenient gauge is the choice (Brandt 1969b. 1977a,b)
GL
-
+
+
which allows for smooth non-singular functions g ( x , y) and A @ ,y) as exact solutions
or trial functions. In particular, for periodic positions (xu, yu) with n, = 1, the choice
g ( x . y) = constant in (2.2) describes the ideal FLL (1.4) near Bcz. Approximate expressions
for @ ( x , y) and B ( x , y) for arbitrary arrangements of straight or curved flux lines and valid
at all inductions B are given by Brandt (1977b).
Using the general GL function (2.2) Brandt (1969b) calculated the energy of structural
defects in the 2D FLL from the linearized GL theory which is valid near E,'. This 'lowest
Landau level method' (which yields formula (2.2)) was recently extended by Dodgson and
Moore (1995) to obtain the energy of 3D defects in the FLL, e.g. screw dislocations and
twisted flux lines (two-, three- and six-line braids). These authors correctly obtain that the
length of such defects diverges near Bc2 as a / ( l - b)'/' (b = E/&', a' % & / E ) ; this
finding coincides with the result of elasticity theory, a(C44/C66)''2 if one notes that near
E', the shear and tilt moduli of the FLL (section 4.1) behave as c66 (1 - b)* (4.4) and
c e ( k # 0)
1 - b (4.6~). The 3D structural defects of the FLL are thus stretched along
B when B approaches the upper critical field Bc2, similarly as the deformations of the FLL
and the superconducting nuclei caused by point pins (Brandt 1975) are stretched.
If the vortex cores do not overlap appreciably, one may put I@(r)l= 1 outside the
vortex core. The condensation energy is then constant (poH~(l@12
- $l@14) = poH:/2)
and the magnetic energy density H 2 and kinetic energy of the supercurrent f2Q2
yield
the free energy functional of the London theory, with B ( r ) = p o H ( r )
-
-
-
-
F{B)=
s
-h'V'B(r)
+ B ( T )= $0
d 3 r [ B 2+ h 2 ( V x B)'].
(2.3)
pa
Minimizing FIB] with respect to the local induction B(r) using V B ( r ) = 0 and adding
appropriate singularities along the positions ri of the vortex cores, one obtains the modified
London equation
i
1
dri 83(r -Ti)
(2.4)
where the line integral is taken along the ith flux line, see (2.8) below. The three-dimensional
(3D)delta function & ( r ) in (2.4) is the London approximation to a peaked function of finite
width ES 2fie, which can be obtained from the isolated-vortex solution (1.2), (1.3) of the
GL theory. The assumption of non-interacting vortex cores means that the B(r)-dependence
of this peaked r.h.s. of (2.4) is disregarded, resulting in an inhomogeneous linear differential
equation. The energy of vacancies and interstitials in the C L was calculated from London
1482
E H Brad
theory by Brand! (1969a.b) and Frey et a1 (1994). see also the nice London calculations
of the stability of the twisted pair and the twisted triplet in the FLL by Schonenberger et a1
(1995).
2.2. Straight parallelflw: lines
A good approximation to the GL interaction energy per unit length of an arbitrary
arrangement of straight parallel flux lines with flux #o is (Brand! 1977% 1986a)
(2.5)
1' = A / ( ~ @ ~ 2 ) c
1 ~h2/ ( l - b)'lz
:'= 5/[2(1 - b)]1'2
b = E/Ec2
(2.6)
where Ko(r) is the MacDonald function, cf equation (1.3), and i.'and 6' are an effective
penetration depth and an effective coherence length and B = I(B(r))[
is the induction
averaged over a few flux-line spacings, The first term in (2.5) is the repulsive magnetic
interaction (overlap of magnetic vortex fields) and the second term the condensation-energy
attraction (overlap of the vortex cores, see also Kramer 1971). The expression (2.5) with
good accuracy reproduces the dispersive elastic moduli c11 and c66 of the FLL (section 4.2)
in the entire induction range 0 B c Ec2 and entire k-space, and it reduces to the London
result for E << Bcz. At E << Be2 the vortex self-energy per unit length #oOH,l (with
Ifcl = B , , / p 0 ) has to be added to Fint(2.5) to obtain the total energy F. At E FIL Bc2 the
self-energy contribution is not defined but it may be chosen such as to reproduce equation
(1.7) in the case of a periodic FLL. The fact that for B/Ec2 z 0.25 the total energy of a
flux-line arrangement cannot be written as a sum over pair interactions plus a (structureindependent) self-energy, is the reason for the failure of Labusch's (1966) elegant method
to extract the shear modulus C66 of the LL from the magnetization curve M(H.), which
yields a monotonically increasing c&(E) rather than the correct C66 B(Bc2 (4.4).
The magnetic field of arbitrary vortex arrangements for K >> 1 and E << Bc2 is a linear
superposition of isolated vortex fields (1.3)
<
-
Lef. The line integrals yielding the interaction energy and selfencrgy of curved flux
lines, equatbns (2.8) and (3.3). Midd/e. U elarlically distorted by a point pinning force f.
equation (4.17). Righi. Thermal fluctuation of flux lines.
Figure 6.
Theflus-line lattice in superconductors
1483
2.3. Curvedflux lines
The structuredependent part of the GL free energy of an arrangement of arbitrarily curved
flux lines is to a good approximation (Brandt 1977b, 1986a, Miesenback 1984)
(2.8)
with rij = Ivi - rjl. Parametrizing the vortex lines as ri = q ( z ) = [ x i ( z ) :yi(z); z ] one
has dri = ( x i ; y;; 1) dr and may write the line integrals in (2.8) as (figure 6, left)
(2.9)
As in (2.3, the first term in (2.8) yields the magnetic repulsion and the second term the
core attraction of the interacting vortex segments dri. For straight flux lines, (2.8) reduces
to (2.5); for small flux-line displacements (2.8) reproduces the nonlocal linear elastic energy
of section 4; and for B << BCz it yields the London result derived from (2.4) with a GL core
cutoff.
In contrast to (2.9, the general expression (2.8) also contains the terms i = j , which
yield the self-energy of arbitrarily curved flux lines. In these self-energy terms the distance
rij of the line elements has to be cut off, e.g. by replacing the rij with (r;
2f2)’/2 for
K >> 1 and B <
< BCz. The self-energy of a curved flux line of total length L is approximately
L+ok!,1 = L(~~/4KA2CLo)(lnK+0.5),
cf equation (1.1). In the same approximation the local
magnetic field of this vortex arrangement is obtained from (2.4) by Fourier transformation
+
(2.10)
The general field (2.10) (and also (3.4) below) satisfies d i v B = 0. The London limits
$ -+ 0 of (2.8) and (2.10) were given by Goodmann (1966) and Fetter (1967). see also the
review on the FLL by Fetter and Hohenberg (1969).
3. Anisotropic and layemd superconductors
3.1. Anisotropic London theory
The HTSCS are anisotropic oxides which to a good approximation exhibit uniaxial crystal
symmetry. In YBCO a weak penetration-depth anisotropy in thea-b plane of 1.11-1.15 was
observed in decoration experiments by Dolan etal (1989b). Within the anisotropic extension
of London theory,uniaxial mscs are characterized by two magnetic penetration depths for
currents in the n-b plane, A.b, and along the c axis, A,, and by two coherence lengths
and &. The anisotropy ratio r = h&.b
= &,/& is r % 5 for YBazCu307-a (YBCO),
r > 150for Bi2Sr2CaCu20z (Bi22lZ) (Okuda et al 1991, Martinez e t a l 1992), and r FI: 90
The
for TlzBazCaCuzO, (Gray et a1 1990). One defines the GL parameter K = A,&,.
anisotropic London theory is obtained from anisotropic GL theory in the limit 6 -+ 0. These
theories yield the anisotropic critical fields B,I and Bc2 as functions of the angle 0 between
E and the c axis, E,,(@) FI: (4o/41rA~,r)@)ln~ (Balatskii et al 1986, see Klemm 1993,
1995 and Sudbg et a1 1993 for impoved expressions) and BCz(@) = [40/2~5,2~~(@)]
where E(@) = (cos2@ r-2sin20)1/2 (Bulaevskii 1990). For a FLL in thermodynamic
+
1484
E H Brundr
equilibrium, at large E >> E,, the angle B between Ha and the c axis practically coincides
with 0 since B FJ p0H.. In general one has B < 0 (Kogan 1988, Bulaevskii 1991). For
the isolated flux line (B + 0) one has €(e) c 1/@) and thus B , I ( ~ )~ iBcI(0)~(O)
:
and
B~~(e) Ecl(o)/E(e).
The modified London equation for the magnetic induction B(r) in a uniaxial
superconductor containing arbitrarily arranged straight or curved flux lines with one quantum
of flux 40 follows by minimizing the free energy F ( B } and adding appropriate singularities
' J [ B 2+
F ( B )= 2PO
(V x B ) A ( V x B)]d3r
The tensor A has the components Amp = A&p +A2c,cp where c, are the Cartesian
components of the unit vector i:along the crystaline c axis, (or, b) denote (x. y, z ) , A , = A:b,
and 1 1 2 = A-.:A.!,
For isotropic superconductors one has 111 = A2, 112 = 0, h,p = A22S,,
and the 1.h.s. of (3.1) becomes B - A'VB (equation (2.4)) since d i v B = 0. The integral
in (3.2) is along the position of the ith flux line as in (2.4) and (2.8).
Since the discovery of HTSCs, numerous authors have calculated properties of the ideal
FLL from anisotropic London theory: The energy and symmetry of the FLL (Campbell et
al 1988, Sudbg 1992, Daemen et ai 1992); the local magnetic field (Barford and Gunn
1988, Thiemann et al 1988, Grishin et al 1990); the magnetic stray field of a straight
flux line parallel to the surface (Kogan et ai 1992) and tilted to the surface (Kogan er
ai 1993b); the isolated vortex (Klemm 1988, 1990, 1993, 1995, Sudbg et a1 1993); the
elastic shear moduli (Kogan and Campbell 1989, Grishin et a1 1992, Barford and Harrison
1994); magnetization curves (Kogan er d 1988, Hao et d 1991); and the magnetization
in an inclined field (Kogan 1988, Buzdin and Simonov 1991, Hao 1993). which causes a
mechanical torque from which the anisotropy ratio r can be obtained (Farrell et al 1988,
Farrell 1994, Martinez et al 1992). The modification of the flux-line interaction near a flat
surface was calculated for isotropic superconductors by Brandt (1981a) and for anisotropic
superconductors by Marchetti (1992). Hao and Clem (1992, 1993) show that for Ha>-> H,I
the angledependent thermodynamic and electromagnetic properties of HTSCS with a FLL
depend only on the reduced field h = Ha/Hcz(e, 4). A further useful scaling approach
from isotropic to anisotropic results is given by Blatter, Geshkenbein, and Larkin (1992).
As clarified in a Reply by Blatter ef ai (1993a), such scaling rules are not unique, but the
relevant issue in constructing a scaling theory is the consisiency requirement on the set of
scaling rules.
The FLL in anisotropic HTSCSexhibits some unusual features: (a) A field reversal and
attraction of vortices occurring when B is at an oblique angle may lead to the appearance
of vortex chains (Grishin era1 1990, Buzdin and Simonov 1990, Kogan et ai 1990, Buzdin
et al 1991), which were observed in YBCO at B = 25G by Gammel et ai (1992a).
Vortex chains embedded in a perturbed E L were observed by decoration at the surface
of inclined crystals of Bi2SrzCaCuaOa+a at E = 35G (Bolle et al 1991, Grigorieva et ai
1995) and with YBa2(Cul-,A1,)3Ch~ at B = 10G (Grigorieva et a1 1993). These chains
may possibly be explained by the coexistence oftwo interpenetrating vonex lanices parallel
and perpendicular to the a-b plane, which may exhibit lower energy than a uniformly
tilted EL. The existence of this combined &trice in layered superconductors was discussed
(Theodora!& 1990, Huse 1992, Bulaevskii er ai 1 9 9 2 ~Daemen et ai 1993b, Feinberg and
Ettouhami 1993a, Feinberg 1994) and calculated explicitly (Benkraouda and Ledvij 1995),
the flux-line lattice in superconductors
1485
and it was found also within anisotropic London theory (Preosti and Muzikar 1993, Dona
and de Oliveira 1994b). (b) The magnetic forces between straight vortices in general are
no longer central forces (Kogan 1990). (c) Shear instabilities of the I%L with B in or
near the a-b planes were predicted by Ivlev et a1 (1990, 1991a). These are related to the
metastable states or frustration-induced disorder of flux lines in layered superconductors
(Levitov 1991, Watson and Canrigbt 1993). (d) Ivlev and Campbell (1993) show that fluxline chains between twin boundaries are unstable and buckle. (e) Multiple BCl values in
oblique field were predicted by Sudb0 et a1 (1993).
Flux lines in nonuniaxially anisotropic superconductors were calculated from London
theory by Schopohl and Baratoff (1988) and by Burlachkov (1989). Properties of the
anisotropic FLL were calculated from the GL theory, e.g. by Takanaka (1975), Kogan
and Clem (1981), Petzinger and Warren (1990). and Petzinger and Tuttle (1993). Using
microscopic Eliashberg theory, Teichler (1975) and Fischer and Teichler (1976) explained
the square FLL which was observed by decoration in some materials (Obst 1971, Obst and
Brandt 1978). as due to atomic lattice anisolmpy, see also the book by Weber (1977).
3.2. Arrangements of parallel or cuwedjux lines
The general solution of (3.1) and (3.2) for the induction B(r) and the free energy F { r j )
for arbitrary vortex arrangements explicitly read (Brandt 1990c, 1991%h, 1992a)
(3.3)
(3.4)
g(k, 4 ) = &q2
+ @(k2 - 4’)
= (Aik2 -I-ANl2)/(r2Ka)
(3.7)
where q = k: x & 2 is the unit vector along the c axis, A I = A*: and A2 = A: - AZb 2 0.
The integrals in (3.3) and (3.4) are along the ith and jth Bux lines (figure 6, left) and the
integration in (3.5) is over the infinite k-space. The tensor function fa’(r) (3.5)enters both
the interaction potential between the line elements of the flux lines in (3.3) and the source
field generated by a vortex segment dri in (3.4). Due to the tensorial character of fwp this
source field in general is not parallel to dri.
The factor exp(-2g) in (3.6) provides an elliptical cutoff at large k or small T and can
be derived from GL theory with anisotropic coherence length (ec = &,b/r). This cutoff,
also derived by Klemm (1993, 1993, originates from the finite vortex-core radius in the GL
theory and means that the 3D delta function &(r)in (3.1) is replaced by a 3D Gaussian of
width fi&,along a and b and width
along the c axis. Note that this GL cutoff exhibits
the correct anisotropy; choosing a cutoff with a different symmetry may lead to artificial
helical or tilt-wave instabilities of the flux lines at very short wavelengths (Sardella and
Moore 1993). Tachiki et a1 (1991) mention another trap which yields an erroneous spiral
instability. As shown by Cameiro era1 (1992), in the energy integral (3.3) (but not in the
1486
E H Brandt
magnetic field (3.4)) the tensorial potential (3.6) formally may be replaced by a diagonal
potential since the r.h.s. of (3.1) is divergence-free because vortex lines cannot end inside
the bulk. Kogan et d (1995, consider vortex-vortex interaction via material strains.
For isotmpic superconductors one has .lab = Ac = h, thus A2 = 0 and fup = Sap
exp(-r/A)/(4ar.12); this means the source field is now spherically symmetric, and (3.3),
(3.4) reduce to (2.8), (2.10) (with different cutoff). In the isotropic case the magnetic
interaction between two line elements contains the scalar product dry dv,” 6,p = dvj .dr, =
dridrj cos4 where 4 is the angle between these vortex segments; the repulsive interaction
thus vanishes at 4 = $7 and even becomes attractive when 4 >
For anisotropic
superconductors with r >> I and symmetric tilt of the two flux lines about the c axis, this
magnetic attraction begins at larger angles 4 = 2 0 close to K , i.e. when the flux lines are
nearly antiparallel and almost in the a-b plane. ?he spontaneous tilting of the vortices,
which prefer to be in the a 4 plane and antiparallel to each other, facilitates that vortices
cut each other (Sudbp, and Brandt 1991c).
1..
3.3. Layered superconductors
The HTSCS oxides consist of superconducting Cu-0 layers (or multiple Cu-0 layers) of
spacing s and thickness d <s which interact with each other by weak Josephson coupling
(Lawrence and Doniach 1971, Klemm et al 1975, Bulaevskii 1972, Deutscher and EntinWohlman 1978), see the reviews by Bulaevskii (I990) and Artemenko and Latyshev (1992)
and the book by Klemm (1996). This layered structure causes two novel phenomena:
pancake vortices and Josephson strings, figure 7.
Figure 7. Top left The magnetic field lines of a poinf vortex (pancake vonex) in a stack of
superconducting layers indicated by horizontal lines. Top rtghr. Stack of point vonices forming
a straight and a distorted flux line which threads the superconducting Cu-0 layers. Bottom le,?.
Josephson currents (vertical arrows) between, and supercurrents (horizontal arrows) inside the
Cu-0 layers, eirrulating around a Josephson vortex whose core is marked by a dot. Bottom
right. Kinked flux lines in a layered superconductor consist of point vortices connected by
Josephson strings.
the flux-line lattice in superconductors
1487
3.3.1. Pancake vortices. When B, is nearly along the c axis the flux lines formally consist
of a stack of 2D point vortices, or pancake vortices (Efetov 1979, Artemenko and Kruglov
1990, Guinea 1990, Clem 1991, Buzdin and Feinberg 1990, Fischer 1991, Genenko 1992)
which have their singularity (the zero of $ and infinity of V4) only in one layer. The field
of a single point vortex with centre at rj = 0 is confined to a layer of thickness % 2Aab. Its
rotationally symmetric z-component and its in-layer radial component are
B,(r) = (~4014xl;~r)exP(-r/Aab)
(3.8)
BI(T) = (s@0z/4n&r.~)[e x ~ ( - l z l / l ~ ) / l z l exp(-r/Adr]
(3.9)
where TI = ( x : y ) and T = (x; y ; z ) . The field (3.8). (3.9) satisfies d i v B = 0 as it
should. In a vortex line, the radial components of the point-vortex fields cancel ( i f s <<A)
and only the z-components survive. A point vortex in plane z = zn contributes a Rux
4 ( z , ) = (4~s/2l.b)exp(-lz,l/h.b)
to the flux through the plane z = 0. This means the
flux of one point vortex is << 40, but the sum of all @(z.) along a stack yields,&. The
self-energy of a single point vortex diverges logarithmically with the layer extension, but
this fact does not disturb us since the energy of a stack of such point vortices, a vortex line,
has an energy per unit length and also diverges for an infinitely long vortex. Remarkably.
the interaction between point vortices is exactly logarithmic up to arbitrarily large distances,
in contrast to the interaction of point vortices in an isolated film (Pearl 1964. 1966). which
is logarithmic only up to distances As],,, = 2h&/d where d is the film thickness. Point
vortices in the same layer repel each other, and those in different layers attract each other.
This is the reason why a regular lattice of straight flux lines has the lowest energy. The
pancake vortex in a superconducting layer squeezed between two infinite superconducting
half spaces is calculated in detail by Ivanchenko et a1 (1992). Pudikov (1993) shows that
the logarithmic interaction of the pancake-vortices applies for any number of layers N > 1,
and that the interaction is softer near the specimen surface. Theodorakis and Ettouhami
(1995) calculate the undulating or ‘sausaging’ vortex along z when T,(z) varies periodically
to model a layered system with a finite order parameter between the layers. Brandt et a1
(1996) show that a distorted stack of pancake vortices feels a long-range attraction to a
planar surface..
3.3.2. Josephson strings. When the applied field is nearly along the a-b plane the vortex
core prefers to run between the C u 4 layers (Schimmele et al 1988, Tachiki and Takahashi
1989, 1991, Tachiki e?a1 1991). When the coupling between the layers is weak, the vortex
lines along the a-b plane are so called Josephson vortices or Josephson swings. These
have no core in the usual sense of a vanishing order parameter $ since @ is assumed to
be zero in the space between the layers anyway. The circulating vortex current thus has
to tunnel across the isolating space between the layers. The width of the Josephson core
is AJ = i
3(the Josephson length, Barone and Patemo 1982) and its thickness is s. Its
aspect ratio AJ/s = r = &&/$c is thus the same as for the core of a London vortex in
the a 4 plane with width t a b and thickness 6- The vortex structure and anisotropy of the
penetration depth in various superconductin~normalmultilayers was calculated by Krasnov
e t a l (1993).
3.3.3. Lock-in transition and oscillation. When the angle 0 between the c axis and the
applied field Ha is close to f n , the flux lines form steps or kinks consisting of point vortices
separated by Josephson vortices (Ivlev et al 1991b), see the reviews by Feinberg (1992,
1994), the work on kink walls by Koshelev (1993), and computer simulations by Machida
1488
E H Brundl
and Kaburaki (1995). If it is energetically favourable and not impeded by pinning, the
flux lines in this case will spontaneously switch into the a-b plane. This lock-in transition
(Feinberg and Villard 1990, Theodora!& 1990, Feinberg and Ettouhami 1993b, Koshelev
1993) was observed in HTSCs by Steinmeyer et al (1994a,b), Vulcanescu et al (1994),
and Janossy et a1 (1995) in mechanical torque experiments, and in an organic layered
superconductor by Mansky et a1 (1993, 1995) from ac. susceptibility. It means that the
perpendicular component of the applied field should only penetrate above some critical
angle, for Ix/2 - 01 (1 - N)B,i,,JB,, where N is the demagnetization factor. The
anisotropic magnetic moment of layered HTSCS and the mechanical torque were calculated
by Bulaevskii (1991) and measured, e.g. by Fane11 ef a1 (1989) and Iye etal (1992). In a
nice experiment Oussena et al (1994b) in untwinned YBCO single crystals with B in the
a-6 plane observe lock-in oscillations as a function of B which are caused by matching
of the Fu with the Cu-0 layers. In a nice theory Koshelev and Vinokur (1995) calculate
corrections to the resistivity when the flux lines form a small angle with the layers, see also
section 8.2.1.
>
3.3.4. Motion of Josephson vortices. The viscous drag force on moving Josephson vortices
is calculated by Clem and'coffey (1990). extending the BardeenStephen (1965) model.
Gurevich (1992a, 1993. 1995a), Aliev and S i n (1993), and Mints and Snapiro (1994, 1995)
find nonlocal electrodynamics and nonlinear viscous motion of vortices moving along planar
defects: Vortices.in long Josephson contacts (characterized by a coupling strength or critical
current density Jc) are usually described by the sine-Gordon equation for the phase difference
@ ( x , t ) ; but when @ varies over lengths shorter than the London penetration depth A, then
a nonlocal description is required, which has as solutions Josephson vortices (section 3.3.2)
or Abrikosov vortices with Josephson core (Gurevich 1995a). depending on the value of
J,. In the static case the equations describing this nonlocal Josephson electrodynamics turn
out to be similar to the equations describing crystal-lattice dislocations in a Peierls potential
(Seeger 1955), allowing one to use results of dislocation theory to obtain periodic solutions
which describe vortex structures in high& Josephson contacts (Alfimov and Silin 1994)
3.4. The Lawrence-Doniach m d e l
A very useful phenomenological theory of layered superconductors is the LawrenceDoniach
(LD) theory (Lawrence and Doniach 1971), which contains the anisotropic GL and London
theories as limiting cases when the coherence length & ( T ) exceeds the layer spacing s.
The LD model defines for each superconducting layer a two-dimensional (ZD) GL parameter
qn(.x,y ) (n = layer index, zn = ns) and replaces the gradient along z IIc by a finite difference.
The LD free energy functional reads
(3.10)
(3.11)
Here l'i = ( x ; y ) , (Y a T, - T and B are the usual GL coefficients, m and M are
the effective masses of Cooper pairs moving in the a 4 plane or along c, B = rotA,
1489
Thejk-line lanice in superconductors
2e/h = 2rr/&,, A, = i A , and the integral in (3.11) is along a shaight line. One
has t:, = h2/2ma, c2 = h2/2Ma, A:b = m@/4poe2a, and A$ = M@/4poe2aY,
thus
A;/A:,
= c2b/.$z = M/m = r2 >> 1 for HTSCs. For cc >> s the difference in (3.10)
may be replaced by the gradient (a/& - 2ieA,p)Q(x, y, z ) , and the anisotropic GL
theory is recovered. In the opposite limit, cc << s, (3.10) describes weakly Josephson
coupled superconducting layers. An extension of the LD model to non-vanishing order
parameter between the layers (i.e. to alternating superconducting and normal layers coupled
by proximity effect) was recently given by Feinberg et al (1994), see also Abrikosov et a1
(1989).
The LD theory can be derived from a tight binding formulation of the BCS theory near
Tc (Klemm et ai 1975, Bulaevskii 1990), but like many phenomenological theories the
LD theory is more general than the microscopic theory from which it may be derived. In
particular, if $!"(x, y) = constant x exb[i$.(x, y)] is assumed, the resulting London-type
LD theory applies to all temperatures 0 < T < Tc (Feigel'man et al 1990)
(3.12)
M r d = 6"-#"+I - 1,.
The magnetic field of an arbitrary mangement of point vortices at 3D positions T, is in the
limit of zero Josephson coupling, corresponding to the limit Ac -+ M
where k = ( k l ; kz), k l = ( k x ;ky), 1: = E, and I have introduced the cutoff from (3.61,
(3.7) (note that k: = q2 since q = k x E in (3.6)). Even for finite Ac the field (3.13) is
a good approximation when Bo is along z or close to it and may be used to calculate the
field variance of arbitrary arrangements of point vortices (Brandt 1991~).The free energy
of a system of point-vortices at positions r, = (xu; yw;2,) is composed of their magnetic
interaction energy FM and the Josephson coupling energy FJ caused by the difference 8,
(3.12) of the phases 4" of
in adjacent layers, F = FM F J , with
+
(3.15)
In general, F, is not easily calculated since the gauge-invariant phase differences have to
be determined by minimizing FM F J . The vortex lattice with B parallel to the layers
was calculated analytically (Bulaevskii and Clem 1991) and numerically (Ichioka 1995).
Solutions of the LD theory for'a single straight flux line and for a periodic FLL for all
orientations of B have been obtained by Bulaevskii et al (1992c), Feinberg (1992, 1994),
and Klemm (1995, 1996). The energy of a distorted vortex line was calculated by Bulaevskii
eta! (1992d); the interaction between two pairs of point vortices (four-point interaction)
entering that energy, corresponds to the interaction of two vortex line elements d r in the
London energy, equations (2.8) and (3.3), see figure 6, left. Phase transitions caused by
vortices and by the layered structure of HTSCs are discussed in section 6.
+
1490
E H Brundt
3.5. Josephson currents and helical instability
Within the LD model the current flowing perpendicular to the layers is a Josephson current,
which tunnels across the insulating layer and which is determined by the phase difference
6. (3.12) between neighbouring superconducting layers. From the LD energy (3.12) follows
the Josephson relation (Josephson 1962) for this current density
Jk,
Y ) = Jmsin 6.(x,
Y)
Jm = 4 o / ( ~ s A ~ w ~ )
(3.16)
where Jm is the critical current density along the c axis. Thermal fluctuations of &(PA)
reduce the loss-free current which can flow along c (sections 6.3 and 7.4) (Glazmann
and Koshelev 1990. 1991b. Genenko and Medvedev 1992, Daemen et a1 1993a, Ioffe et
al 1993). Considering the depression of the order parameter l@(r)I2
by the current Jn,
Genenko et a1 (1993) obtain a generally non-sinusoidal relation between this current and
the phase difference 8..
Intrinsic Josephson effects between the Cu-0 planes in BSCCO single crystals were
observed by Kleiner et a1 (1992, 1994) and Muller (1994) in intricate experiments. Ling
et a1 (1995a) observe even two types of (intra- and inter-unit-cell) Josephson junctions in
YBCO. Fraunhofer oscillations as a function of Hoin mscs with a width in the a-b plane
not exceeding the Josephson length h, = rs were predicted by Bulaevskii et a1 (1992b).
Interestingly, a critical current density of the same order as the decoupling current density
Jm (3.16) results from anisotropic London theory if one considers the helical instability of
a flux line (or a stack of point vortices) in the presence of a longitudinal current of density
Jll, cf section 7.2. By a spontaneous helical distortion u(z) = uo(&coskz Gsinkz) a flux
line gains an energy from Jll which may exceed the elastic energ: I f this distortion. The
total energy per unit length of the helix is (u32)[ k z P ( k ) -kJu&,] where P ( k ) is the fluxline tension, see (4.13) below. An unpinned flux line (Clem 1977) or Fu (Brandt 198Oc,
1981b) is thus unstable with respect to a helical deformation with long pitch length 2 r / k at
arbitrarily small Jll > P(O)k/&,, but even weak pinning may suppress this ‘inflation’ of a
helix. However, due to the dispersion of P ( k ) (nonlocal elasticity) a second instability
occurs at the shortest possible pitch length 2 , i.e. at k = r / s , where s is the layer
spacing. With P ( k ) = 4:/(4ap&)
(putting In(r/kt0b) = ln(s/rtc) = 1 in (4.14))
this helical instability of a stack of point vortices occurs at a longitudinal current density
J,l, = &/(4sA:&3) c .Id. A longitudinal current of the critical value Jm thus, in zero
magnetic field, suppresses superconductivity; but in the presence of a longitudinal field or
of vortices, this current ‘blows up’ the flux line into independent point vortices. During
this instability, the point vortices in adjacent layers are tom apart by the Josephson string
which connects them and which deforms into a circle with growing radius.
+
4. Elasticity of the flux-line Lattice
4.1. Elastic moduli and elastic matrix
From the general expressions (2.5) and (2.8) for the energy of a distorted E L , or from
equation (3.14) for the magnetic energy of point vortex arrangements, in principle one
can calculate arbitrarily large vortex displacements caused by pinning forces, by structural
defects in the U, by field gradients or currents, by temperature gradients, or by thermal
fluctuations. However, in many cases, the distortion is so small that it may be calculated
from linear elasticity theory. The linear elastic energy F e of~ the n L is expressed most
conveniently in k-space. First, the displacements of the flux lines ui(z) = q ( z ) - Rj =
The fludine lattice in superconductors
1491
(qX;
qY;
0) from their ideal positions &. = (Xi;Yj:z ) are expressed by their Fourier
components
and then the most general quadratic form in the u(k) = (U,; u s ; 0) is written down
with (a,p ) = ( x , y). The k-integrals in (4.1) and (4.2) are over the first Brillouin zone
(BZ)of the FLL and over -5-I
k, 6
The coefficient Q m ~ ( kin) (4.2). called the
elastic mahix, is real, symmetric, and periodic in k-space. Within continuum theory of this
is related to the elastic moduli C I I for uniaxial compression,
uniaxial elastic medium,
ca for shear, and CM for tilt, by (figure 8)
<
e-'.
+
+
or in Compact form, Qap
(CII - C66)kkp
&+q[(k: f k;)C66 f k:Ca
ar(k)]. For
completeness in (4.3) the Lahusch parameter OIL. is added, which describes the elastic
interaction of the FLL with the pinning potential caused by material inhomogeneities
(section 8.2). A k-independent cq would mean that all flux lines are pinned individually
(Labusch 3969b). For weak collective pinning (Larkin and Ovchmnikov 1979) a~(k)
should
decrease when k i > R;' or k, > L;' where R, and L,
(cM/c~~)'/*R,are the radius
and length of the coherent (short-range order) regions of the pinned FLL (section 8.7).
p
1l.i
R?
elastic matrix element
r/a
elastic moduli
h.
1j.i
1.
Figure 8. Elastic matrix of the FU defined by (4.2) (left) and elastic moduli (4.3) (right)
from isotropic CL theory The depicted matrix elements give the elastic energy of pure tilt,
compression, and shear waves with wavelength k / t . The parabola (broken line) gives the
local approximation to the compressional and tilt energies valid fork K A-'.
1492
E H Brandi
For uniform distortions the elastic moduli of the triangular FLL are (Labusch 1969a,
Brandt 1969a,b, 1976~.1986)
C I I -C66
CM
= B2a2F/aB2 = (B’/po)aB,/aB
= BaF/aB = BB,/po
(4.4)
- - b)’(1 -0.58b+0.29bZ)
(
is the modulus for isotropic compression and BJpO
cs m (B@o/16~A’po) 1 -
-
(1
$2)
with b = B/Bc2. Here C I I c66
=
aF/aB is the applied field which is in equilibrium with the FLL at induction B. The last two
factors in c 6 show that the shear modulus vanishes when K = I/& (in this special case all
flux-line arrangements have the same energy), when A + 00 (strongly overlapping vortex
fields), and when B + BCz (strongly overlapping vortex cores). In addition C66 vanishes
at the melting transition of the LL (Sgik and Stroud 1994b, Franz and Teitel 199%.1995).
For B (< &/4xA2 Fi: B,I/InK one has cI1= 3C.56 exp(-a/@ (the Cauchy relation for
nearest neighbour interaction by central forces) and CM = BH,I >> CII. For B >> &,/47rA2
one has C66 << C I IFi: CM Fi: B*/po (incompressible solid). The shear modulus C66 of the 2D
RL in thin isotropic films was calculated by Conen and Schmid (1974), the bulk c66(B. T)
is given by Brandt (1976c), and the elastic properties of anisotropic films were calculated
by Martynovich (1993).
In uniaxially anisotropic superconductors the free energy F(B, 0 ) depends explicitly
on the angle 0 between B and the c axis, and thus on the tilt angle 01 if the tilt is not in
the a-b plane. One then has for uniform tilt
-
+
cM = d2F/dar2 = B B , / ~ ~a2F/aa2.
(4.5)
The first term in (4.5) originates from the compression of the field lines when BL is held
constant during the tilt, B(a) = BJcosa cs Bz(l (u2/2). In the special case where the
equilibrium direction z is along the crystalline c axis one has 01 = 0. In HTSCs F ( O )
decreases with increasing angle 0 since the flux lines prefer to run along the a-b plane.
In this case the second term i n (4.5) is negative. For large anisotropy and low flux density
B << B,I with BIlc, the second term nearly compensates the first (isotropic) term such
that cu becomes very small (section 4.3). However, for larger B >> B,I the modulus for
uniform tilt is almost isotropic since the free energy of the FLL is F .% B2/po m B2jp.0
(cf equations (1.7) and (3.1)), which does not depend on any material parameter and is thus
isotropic. The anisotropic part of F is proportional to the magnetization M = B - B”,
which is a small correction when B >> BcI. The magnetization in HTSCs strongly depends
on 0 such that the mechanical torque exerted by an applied field is sharply peaked when
Bu is close to the a-b plane (Kogan 1988,Bulaevskii 1991, Farrell era1 1988, Okuda 1991,
Martinez et a[ 1992,Iye er 4l 1992, Doria and de Oliveira 1994a).
+
4.2. Nonlocal elasricity
The continuum approximation (4.3) for the elastic matrix in general applies if k: =
k: k,’ << k& = 4nB/&; as k l approaches ksz, the quadratic k-dependence ceases to hold
since Op,ohas to be periodic in k-space, Oa8(k)= %p(k+ K)where K are the vectors of
the reciprocal FLL, cf (1.4) and figure 8. However, for the FLL the direct calculation of the
elastic matrix from GL and London theories reveals that the quadratic k-dependence holds
only in the smaller region k <<A-’, which typically means a very small central part of the
BZ and a small fraction of the range 0 < lk,l < 5-l (Brandt 1977a). Within the continuum
approximation one may account for this ‘elastic nonlocality’ by introducing k-dependent
+
the flux-line lum'ce
in superconductors
1493
(dispersive) elastic moduli in (4.3). For isotropic superconductors at E > B,I one finds for
>> 1, b = E / & << 1 (London limit), and k = (k:+ k; +k:)'/' << ksz (continuum limit)
K
and a k-independent c s (4.4). The physical reason for the strong dispersion of c11 and c#
is that the range 1. of the flux-line interaction is typically much larger than the flux-line
spacing a. This means that the mL is much softer for short wavelengths 2n/k c 2rrA of
compressional and tilt distortions than it is for uniform compression or tilt.
The elastic matrix of the FLL was first calculated from GL theory by solving for @(r)
and B ( r ) up to terms linear in arbitrarily chosen displacements ui(z)(Brandt 1977a,b).
Remarkably, it turns out that the correct expansion parameter near B,z for periodic ui ( z )
is not the spatial average (]@Iz)
M (1 - b ) but the parameter ([@[*)/(kzAz), which diverges
fork 4 0, i.e. for uniform compression or tilt. This may be seen from the approximate GL
result for the tilt and compressional moduli valid for k'A' < A Z / p = %'(I - b )
From this argument it follows that truncating the infinite series ( 4 . 6 ~ )(or similar
expressions) can give qualitatively wong results and that it is not always allowable to
put B(r) = constant in such calculations even when K is very large and B is close to
Ec2. For example, the elastic moduli ( 4 . 6 ~ )would then diverge as k-z in the limit k + 0
as is evident from the term n = 1. Only when the infinite summation is performed does
one obtain the unity in the denominator in (4.6u), which has the meaning of a 'mass' or
'self-energy' that renormalizes the spectrum of excitations. This unity originates from the
spatial variation of B(r) as it becomes obvious from London results like (4.6).
This somewhat unexpected finding partly explains erroneous results which were obtained
for the fluctuating FLL, namely, that there exists a 'massless mode' of the FLL, that the
long-wavelength thermal fluctuations diverge, and that long-range order is destroyed in
an infinite PLL (Eilenberger 1967, Maki and Takayama 1971, Houghton and Moore 1988,
Moore 1989, Ikeda er ul 1990, 1991a, 1992), see also later discussion by Houghton er al
(1990), Moore (1992), and Ikeda (1992b). In such analytical calculations B(r) = constant
was used explicitly, corresponding to the limit h -+ 03 and to the neglection of effects
of magnetic screening. Interestingly, even when the diverging m n g tilt modulus CM =
E'([$r[*)/(~k'A') is used, the correct average fluctuation (U*} (section 5.1) is obtained
(Moore 1989) since the main contribution to (U') (5.3) comes from large k FT kaZ.
It should be remarked here that the l i t 1. + 03 (or K -+ 03, or E(r) = constant) does
not only mean vanishing tilt stiffness c#(k # 0) + 0 and compressibility cll(k # 0) -+ 0,
but also vanishing shear modulus c s -+ 0 since all three moduli are A-'. This exact result
might appear counter-intuitive since large A means a long-range interaction between the flux
lines, which sometimes is erroneously believed to lead to a stiff EL. A further little-known
consequence of the academic limit h + 03 is that the magnetic flux 4 of each flux line
vanishes since it is compensated by the flux of its image flux line if A exceeds the distance x
of the flux line to the surface; for a planar surface one has @ ( x ) = $o[ 1-exp(-x/A) 1, thus
the flux of a vortsx vanishes when the vortex approaches the surface. Therefore, the limit
A -+ 03 has to be taken such that A still stays smaller than the size of the superconductor.
But what about vortices in supemuid 4He? The interaction between vortex lines in
a neutral superfluid is not screened, i.e. formally one has h = 03 for 4He. In spite
of this, the triangular vortex lattice in 4He has a h i r e shear modulus as shown by
Tkachenko (1969). see also the stability considerations by Baym (1995) and the intricate
-
1494
E H Brandt
observation of vortex arrangements by trapped ions (Yarmchuck et a1 1979 and Yarmchuck
and Packard 1982). This seeming paradox is solved by noting that the neutral supemuid
4He formally corresponds to the limit of zero electron charge e, i.e. the absence of magnetic
screening. In this limit both the flux quantum & = h/2e and the London penetration depth
A' = ( m / 2 e 2 ~ o n ~ )(in
' / 2SI units with 2m, Ze, and nc the mass, charge, and density of the
Cooper pairs, see Tilley and Tilley 1974) diverge. Therefore, in the shear modulus per flux
line, C66/n = &/(161rA~/~o)(4.4) in the London limit, the factors e-2 in the numerator
and denominator cancel and C66 is independent of e.
While the GL calculations of Fu.elasticity are rather intricate, and the subsequent
derivations from the microscopic BCS-Gor'kov theory,by Larkin and Ovchinnikov (1979)
and from the anisotropic GL theory by Houghton et a1 (1989) are even more complicated, it
is much simpler to obtain the elastic matrix from London theory: At not too iarge inductions
B < 0 2 5 B d and for not too small K > q, the London and GL results practically coincide:
the main correction from GL theory is that at larger B the London penetration depth A (or
the anisotropic Amb and A,) have to be multiplied by a factor ( ~ ~ ( r ) 1 2 %
) - (1
1~
-zb)-'/',
cf equation ( 4 . 6 ~ ) ;the resulting increase of A accounts for the reduced screening near the
upper critical field Bc2. Another correction from the GL theory is a cutoff in the interaction
(3.6), which originates from the finite vortex core and enters the vortex self-energy. The
approximation (4.6) considers only the magnetic interaction between the flux lines. When
k increases above % k ~ z / 3then c11 becomes smaller than (4.6) and c~ ,becomes larger
than (4.6), cf figure 8. The compression modulus C I I has an additional factor (1 +k2tR)-2
with
= t 2 / ( 2 - 2b), which comes from the overlapping vortex cores and is important
only at large b = B/Bc2 > 0.3; c11is reduced further by geometric dispersion such that at
kl w kez. kz = 0, one has c )I (k) % c s ; this is so since in a hexagonal lattice compressional
waves with k = Kxo are identical to shear waves with k = K I Iand thus must have the
same energy. The tilt modulus c a has an additional term which originates from the selfenergy of the flux lines, see (4.10) and section 4.3; this line-tension term has only weak,
logarithmic dispersion and exceeds the interaction term (4.6) when k2 > kiz/ In K .
The elastic nonlocality (4.6) has been questioned by Matsushita (1989, 1994), who also
claims that the general expressions for B(r) (2.7) and (2.10) violate d i v B = 0. While it is
easy to show that the B(r) (2.7), (2.10). (3.4), (3.8). (3.9), (3.13) do satisfy d i v B = 0, one
can trace back the claim of a local tilt modulus to the confusion of Abrikosovjllx lines and
magnetic field lines. When thefield lines are shifted by small displacements U(T) the new
induction is B V x (Ux B),which is a local relationship. However, when thejur lines
are shifted then the resulting B follows from the nonlocal expression (2.10). In particular,
for small periodic u ( r )one obtains thus the new induction B'+ V x (U x B)/(I+k2hZ)
(Brandt 1977%b). This fact leads also to the dispersion of the compressional and tilt moduli.
+
4.3. FLL elasticity in anisotropic superconductors
Within London theory, the elastic matrix of the FLL in uniaxially anisotropic superconductors
is obtained straightforwardly by expanding the free energy F[Ri U;(Z)}(3.3) up to
quadratic terms in the flux-line displacements u~(z).
The general result may be written
as an infinite sum over all reciprocal lattice vectors K ,which guarantees periodicity in k
space. Choosing Bllz one gets for arbitrary orientation of the c axis (Brandt 1977% Sudbg
and Brandt 1991a, b, Nieber and Kronmiiller 1993b)
+
The pux-line lanice in superconductors
1495
+
k%(k) = k:fap(k)
kakgfdk) - 2kzkpfdk)
(4.8)
with fap(k) given by (3.6). The last term -2R,kp&.(k) in (4.8) was first obtained by
Sardella (1991, 1992). This term influences the compression and shear moduli of the FLL
when B is not along the c axis, but it does not enter the tilt moduli. Taking the limit
k l < K I Gin (4.7)one obtains the continuum approximation as follows. The term K = 0
yields the dispersive compression and tilt moduli while the nondispersive shear moduli are
obtained by integrating over all terms K 0. For general orientation of Bllz with respect to
the c axis, there are several moduli for compression and tilt or for combined compression and
tilt (Ovchinnikov and Julev 1991, Sudb0 and Brandt 1991b, Sardella 1992, Schonenberger
erul 1993) and several shear moduli (Kogan and Campbell 1989). Since the FLL orientation
couples to the crystal lattice anisotropy, there also appears a 'rotation modulus', see Kogan
and Campbell (1989) and the extension to films of a biaxial superconductor by Mwtynovich
(1993). The situation simplifies when B is along the c axis. The FLL is then a uniaxial
elastic medium like in isotropic superconductors, and the elastic matrix resumes the form
(4.3) but now with the k-dependent moduli
+
(4.9)
(4.10)
C66 M
B@O/U~~A:&I).
(4.12)
These London IeSUlts apply to b B/Bcz < 0.25 and K = A\ob/&,b > 2; in (4.11) ko * kez
is a cutoff. For the extension to b z 0.5 the shear modulus c66 (4.4) should be used with A
replaced by hab. the A,: and A: in (4.9) and (4.10) should be divided by 1-b, and the second
term in the brackets in (4.10) should be replaced by (1 - i ~ ) / ( h =
~ (1
~ ~ b)&,/(4nBA:).
)
Note that the factors 1 - b provide vanishing of all moduli as B + Bc2. An interesting
and useful scaling argument, which fork >> A-' allows one to obtain the anisotropic elastic
energy (and other properties) of the E L from the isob.opic results, is given by Blatter et ai
(1992, 1993a).
The compression modulus c11 (4.9) and the first term in the tilt modulus (4.10) originate
from the vortex-vortex interaction (the term K = 0 in (4.7)). The second term in (4.10)
originates mainly from the interaction of each vortex line with itself and may thus be called
the isolated-vortex contribution (Glazman and Koshelev 1991a, Brandt and Sudb0 1991,
Fiscber 1991). Due to the k,-dependence of this term the exact tilt modulus even of isotropic
superconductors depends on the wavevector k rather than on k = lkl. This correction term
dominates for large k, namely, when k: z ki2/ln(rr) or when k: > k ~ 2 ~ 2 / I n ( r / ~ o b k r ) ,
and for small B .
-
4.4. tine tension of an isolutedflwc line
In the limit B + 0 the second term in c a (4.10) yields the line tension P = ca&/B of
an isolated flux line oriented along the c axis
(4.13)
1496
E H Brundt
Due to the factor I / rz= h:/AEb >> 1 in (4.13) the Line tension is very small compared to
the self-energy J = &H,1 of this flux line
J
(&/4n~&~)ln6
PfkJ
RS
( ~ ~ / 4 n P o h ~ ) I n ( r / k , ~ ~ b(4.14)
).
The P(k,) (4.14) applies to large x = hob/f,,b >> 1 and not to long tilt wavelength,
k&b >-> 1. If k2h.b << 1, the bracket in (4.13) becomes r-’in(rK)
f . In particular, in
the limit r --f CO the line tension of a straight flux line (k2 = 0) is P = &(87rp012b).
This result may also be obtained from Clem’s (1991) stack of pancake vortices, which has
a line energy
+
(4.15)
if it runs at an angle 0 with respect to the c axis. In general, the line tension of a straight
flux line is related to its self-energy by (Brandt and Sudb0 1991)
P =J
+ azJ/aa2
(4.16)
cf equation (4.5). A similar relation holds for the l i e energy and line tension of dislocations
in anisotropic crystals (Seeger 1955, 1958). Using the approximate expression for the
sin’@)”’
self-energy of a flux line (Balatskii er a1 1986) J ( 0 ) sx J(0)(cosz Q r2
one obtains for Q = 0 (flux l i e along c ) P(0) = J ( 0 ) / r z since the two terms in
(4.15) almost compensate each other; for 0 = 47r (tlux Line in the a-b plane) one
has the self-energy J ( $ T ) = J ( O ) / r and the line tension for tilt out of the a-b plane
Pout($n)= J ( f n ) r 2= P(0)r3. For tilt within the a-b plane the flux l i e s do not feel the
anisotropy and &us Pln(in)= J($n).
These arguments show that the line tension of flux lines, and the tilt modulus of the FLL,
are strongly reduced by the material misotropy if B is along the crystalline c axis and if the
tilt is not uniform, e.g. if tilt or curvature occurs only over a finite length L c 2RA.b, which
means that values lk,l
L-’dominate (Brandt 1992~).Note that the elastic nonlocality IS
crucial for this reduction.
For layered superconductors this means that very M e energy is required ro displace
one pancake vortex from the stack, since for this distortion the characteristic wavevector
k, % z / s is very small (s < h& is the layer spacing, figure 7). One can show that the above
elasticity theory derived from anisotropic CL and London theory approximately also applies
to layered superconductors. In a more accurate formulation (Glazman and Koshelev 1991),
k: in (4.3) and in (4.8x4.14) should be replaced by Qz= 2(1 -cosq)/s2 and all integrals
f”, dz f(k:) by J”, dq f
In general one may say that the anisotropic London theory
gives good results even for layered superconductors if one excludes resulting lengths which
are less than the layer spacing s and replaces these by s.
+
<
(ez).
4.5. Examples and elustic pinning
As discussed in the previous section, the flux l i e displacements obtained by the correct
nonlocal elasticity theory typically are much larger than estimates using the usual local
theory of elasticity. This enhancement is mainly due to the dispersion of the tilt modulus
ca(k) or line tension P = c ~ @ o / BThree
.
examples shall illustrate this.
4.5.1. Elastic response to a poinr force. The maximum elastic displacement u(0) caused
in the FLL by a point-pinning force f acting on the flux-line segment at T = 0 (figure 6,
middle) is easily obtained in k space (Schmucker and Brandt 1977). The force balance
the flux-line lattice in superconductors
1497
equation f,(k) = SF/Gu,(k) = @,p(k)up(k) is solved by inverting the elastic matrix,
yielding U&)
= @$(k) fp(k):Noting that for a point force f(k) =constant, one gets in
the continuum approximation (4.3)
$5
-
(4dB~~)'/'p~oh,bi,/(t b)3/2.
(4.17)
The second term in the first line of (4.17) may be disregarded since C I I > C66. The three
factors in the middle line of (4.17) have the following interpretation. The first factor is
the result of local elasticity, which assumes a dispersion-free CM = c ~ ( 0ss) BZjpo. The
~ kazhob/(l - 6)''' ss ( B hK/BcI)'/Z is the
second factor [cu(O)/cM(kL = k ~ z ) ] l /$5
correction caused by elastic nonlocality. The third factor hc/h,b = r is the correction due
to anisotropy. Both correction factors are typically much larger than unity. Therefore, the
elastic nonlocality and the anisotropy enhance the pinning-caused elastic displacements of
the KL.
4.5.2. Thermalfluctuations of the PLL. The thermal fluctuations of the flux-line positions
(figure 6, right) are given by the same integral (4.17), (U') ss ZkBTu(0)/f, and are thus
enhanced by nonlocality and anisotropy in the same way as is the elastic response to a point
force. These thermal fluctuations will be discussed in section 5.1.
4.5.3. Elastic energy for general non-ungoim distortions. If the tilt wavelength of the
flux lines is less than 2zh,b, i.e. for k,h,b > 1, the elastic energy (4.2) with (4.9t(4.12)
inserted takes an instructive form. For each Fourier component U exp(ikR,) of the general
displacement field one obtains an energy density
~ 6 kz;6$.b:
= 2b = 2B/Bcz. In (4.18)
where k&h:b = 2 b ~ '% BInK/B,, $5 ~ ~ 1 4 and
the first term (= 1) originates from compression (cl]) and tilt (CM)and presents a global
parabolic potential caused at the position of a flux-line segment by all other flux lines.
The second term ($ comes from the shear distortion (C6.5) and may be disregarded for
estimates. The third and fourth terms originate from the interaction of a flux-line segment
with the other segments of the same flux line. The third term describes the coherence of
a flux line ('Josephson coupling of the segments'); this term (- k:) is a line tension with
weak logarithmic dispersion, a four-point interaction in the language of pancake vortices
(Bulaevskii et i l 1992.d). The fourth term stems from the magnetic forces between the
flux-line segments of one flux line; for small k, < A;; this term is k: and thus is a line
tension, but for larger kz z h,-d the k,-dependence becomes logarithmic and thus this term
is more like a local parabolic potential, which has no dispersion. These two terms can be
smaller or larger than unity, depending on B and on the characteristic value of kz.
The last term (= 6/2b) in the energy density (4.18) originates from elastic pinning of
the FLL. Here the pinning strength E and the Labusch parameter CY' in (4.3) were estimated
i)
-
1498
E H Brand
as follows. The maximum possible pinning energy Up,,, per cm is achieved when all fluxline cores are centred in long holes of radius rp > &.?.ab. The gain in condensation energy
is then Up.- M MOH:
2 z & = ~p,3(4z@&~).The same estimate follows if one cuts off
the magnetic self-energy (4.14) of a flux line at the radius rp of the long pinning cylinder;
this inner cutoff replaces the factor l n =
~ ln(h,b/cob) by In(h,b/r,) and yields the energy
/ ~ =(cf
b ) equation (1.1) and
difference (pinning energy) Up,-. ss ( ~ ~ / 4 n p ~ h \ . : b ) [ l n ( rtp0.51
Brandt (1992~)).Realistic pinning by spatial variation of material parameters is weaker; it
has a pinning energy per cm Up = (&4zp0i2) . E with an unknown parameter E
1,
typically E << 1. The Labusch parameter OLLin the model of direct summation (section 8.7)
is estimated as the curvature rz
of this potential times the density B / @ o of the flux
lines, OLI. M (&4zp&,)~5-~B/@,3. From the elastic pinning energy density u2ff~/2one
obtains the last term c/2b in the brackets in (4.18). This term can be smaller or larger than
unity, depending on the induction B = bB,z and on the pinning strength E.
Summarizing the discussion of (4.18), one may say that for non-uniform distortion each
flux-line segment behaves as if it were sitting in a parabolic potential and experiencing a
slightly dispersive line tension, since the elastic energy has the form ( C I c2kz)s2. The
parabolic potential is caused by the magnetic interaction with other flux lines (first term)
and with the same flux line (fourth term), and by pinning (fifth term). The line tension
comes from the coherence of the flux line (third term), which is absent in the limit of very
large anisotropy or for decoupled layers. Depending on B, kz and on the pinning strength
E. any of these terms may dominate.
From this picture one can easily reproduce the displacement (4.17) caused by a point
pinning force. Including elastic pinning, one finds that the response (4.17) is multiplied
by a factor (1 ~ / 2 b ) - ' / ~which
,
reduces the displacement (and the thermal fluctuations,
section 5.1) when the pinning strength E exceeds the reduced induction b = B / B C 2 .
<
+
+
5. Thermal fluctuations and melting of the flux-line lattice
5.1. Thermalfluctuations of theflux-line positions
The thermal fluctuations of the FLL in HTSCS can become quite large due to the higher
temperature and the elastic softness which is caused by the large magnetic penetration
depth (elastic nonlocality) and the pronounced anisotropy or layered structure. The mean
square thermal displacement of the vortex positions can be calculated within linear elasticity
theory (section 4) by ascribing to each elastic mode of the FLL (with discrete k-vectors in
a finite volume v) the average energy $BT. With (4.2) one has
i ( % ~ ( k@&)
)
up(k))= k ~ T / 2
thus
(u&)
U&))
= ksT @$(k)
(5.1)
where @$(k) is the reciprocal elastic matrix. From the Fourier coefficients (5.1). the
thermal average (uz) = ( l ~ , ( z ) 1 ~ is
) obtained as a sum over all (lu(k)I2)(F'arseval's
theorem). Letting V -+ 03 this sum becomes an integral over the Brillouin zone and over
all kz
With the continuum approximation (4.3) for @up, which for consistency requires one to
replace the hexagonal tint BZ by a circle of radius kBz = (4zB/&)'P = (2b)'I2/5, and
1499
The fludine lattice in superconductors
omitting the compression modes since C I I >> c66. one gets explicitly for Bllc
(5.3)
This is the same integral (4.17) as that which determines the flux-line displacement at
the position of a point force. The main contribution to this integral comes from large
values kL w kBz due to the factor kl in the numerator. The tilt modulus (4.10) is thus
strongly reduced with respect to its values for uniform strain (k = 0). The dispersion
(nonlocality) of c ~ ( k enhances
)
the fluctuation (U’) pear Bc2 approximately by a factor
kszAc/2(1 - b)”’
[bKz/2(l -b)I”’F = [BhK/4BC1(1- B/B,z)]’/’r >> 1 (Houghton
ef al 1989, Feigel’man and Vinohr 1990, Brandt 1989% Moore 1989, 1992). The result
may be written as the (U’) of local elasticity theory, obtained first by Nelson and Seung
(1989), times typically large correction factors originating from elastic nonlocality and
from anisotropy, times a reduction factor from local pinning, cf (4.17) and the discussion
below (4.1 8).
A good approximation which slightly overestimates (u2) (5.3) is the replacement of
k r by ksz in the tilt modulus c u ( k ) (4.10); the dominating contribution to cu then
comes from the second term in (4.10). Noting that kiz = 43rB/& = %/.$ab and
ln(.$;’/rZkiz) 4 ln(1/2b) (for K >> 1). one has the interpolation formulae c a ( k l =
k~z)/C*(o) (1 - b ) h [ 2 + (1/2b)1’2]/(2bK2r2) and C66 b(l - 6)’ (4.4); note that the
replacement of A2 by h“ = Az/(l - b ) in the low-field C66 b/AZ would yield the wrong
high-field limit C66 b(1 -b). One thus obtains for 0 < b < 1, K >> 1, and Bllc the thermal
fluctuations (2)
,(ksr/4?~)[ksz- (ffL/C6)”z][C66C~(kL
=kBZ)]-1’2, or explicitly
--
(u2)
[
kBT( 1 6 R k ? A ~ br2)112 x (1
B&
- b)3 In (2 + &)]-1’2
[ I’/‘):(-
x 1-
.
(5.4)
In the following the pinning correction 26/b = Q / C & ~
= 4 o r & p O / ~ ~ <c 1 will be
disregarded. The general result (5.4) is conveniently expressed in terms of the Ginzburg
number Gi = f [ k ~ T , r ~ o / 4 n ~ ~ ~ ( O ) (9.1),
B ~ ( Owhich
) ] ’ determines the relative width of
the fluctuation regime of a superconductor near B&), Gi = 1 - Ty/T, (cf section 9.1)
where a2 = 2 B / f i & was used. The fluctuation (U’) is thus strongly enhanced by
nonlocality and anisotropy. It is further enhanced when one goes beyond the continuum
approximiition, e.g. by inserting in (5.2) the correct (periodic) elastic matrix from the London
theory (4.8) or from the GL theory (Brandt 1977% 1989a). (U’) is enbancedeven further by
nonlinear elasticity and plasticity, i.e. by the creation and motion of dislocations and other
structural defects in the FLL (Brandt 1986, Marchetti and Nelson 1990, Koshelev 1993).
5.2. lime scale of thefluctuations
Time scale and frequency speclrum of thermal fluctuations of the FLL are determined by the
elastic restoring force and the viscous motion of the flux lines. A KL moving with velocity
w exerts a drag force qu per unit volume on the atomic lattice. The viscosity q (per unit
volume) is related to the flux-flow resistivity p p
(B/Bc2)pn and to the normal-state
resistivity p. (equal to p of the superconductor at B = Bc2(T)) by (cf section 7.1)
q = B’/PFF w B B d p n .
(5.6)
1500
EHBrMdt
The FLL has two types of elastic eigenmodes, which are strongly overdamped (i.e. not
oscillating) since the viscous drag force by far exceeds a possible inertial force (up to
very high frequencies >> 10” s-l); the small flux-line mass per cm, mp‘ % (2/rr3)m.kp
(me is the electron mass and kp the Fermi wavevector), was &timated by Suhl (1965)
and Kuprianov and Likharev (1975), see also Ha0 and Clem (1991b) and Blatter and
Geshkenbein (1993) (vortex mass in anisotropic superconductors), Coffey and Clem (1991b)
(in layered superconductors), and Simhek (1992) and Coffey (1994a) (contribution of
atomic-latticedistortions to the vortex mass). The elastic eigenmodes of the EL are obtained
by diagonalization of the elastic matrix @=J, yielding a compressional mode ul(k)and a
shear mode uz(k)(Brandt 1989b. 1991b). After excitation these eigenmodes relax with
exp(-rlt) and
exp(-rzr), respectively. In the continuum
exponential time laws
approximation (4.3) one has
-
-
ri (IC)
= [Cii(k)k:
+ ca(k)k:
I/v
%
ri
(5.7)
+
rz(k) = [C66k:
ca(k)k: J/q rlkt/kz
(5.8)
with rl = B2kz/p0q EJ b (1 - b) p,,/po hZ = 2n B( 1 - b) pJ&, po K 2 . The power spectra
P ( w ) of these viscous thermal fluctuations are Lorentzian
P ~ Ok),= n-1rl,z(k)/[~2
+ r~,z(wz~.
(5.9)
This Lorentzian specbum should describe the high-frequency depinning noise in the electric
conductivity of typeII superconductors since each ‘plucking’ of a vortex triggers a viscoelastic relaxation process, see section 7.7 and the review on flux noise by Clem (1981b).
These two overdamped elastic modes (Brandt 1989b) recently were termed ‘vortons’ by
Bulaevskii and Maley (1993) and Bulaevskii e t n l (1994%~).
Due to the dispersion of the moduli CII (4.9) and c a (4.10), the compressional mode
is nearly dispersion free, rl(k) EJ Fl.
ul(k) for short wavelengths k > kh = (1 - b)’h.;;
The relaxation rate or fluctuation frequency rl has thus a very high density of states.
At long wavelengths k c kh the compressional relaxation is a dilfusion process since
r l ( k ) EJ (BZ/poq)k2; the diffusivity is B2/poq = m / h (Brandt 1990b, see also
section 7.1). The relaxation of mere shear strains is always diffusive, r&) cx k: if
k, = 0, and much slower than the compressional mode, rz(k) << rl since C66 << cll(k).
5.3. Simple melting criteria
In analogy to crystal lattices, a ‘melting temperature’ T ,of the three-dimensional (3D)FLL
has been estimated from the Lindemann criterion for the thermal fluctuations, (uz)i/z= c ~ a ,
where cL EJ 0.1 - -0.2 and U is the dux-line spacing. With (5.4). T, = tmTc is implicitly
given by (see also Houghton etnl 1989, Brandt 1989% 199Oc, 1991%Blatter et al 1994a)
(5.10)
This is still a general (for K >> 1, Bllc), though slightly large, estimate of the fluctuations
since kL EJ ksz was assumed in cu(k). To obtain explicit expressions one may assume
B d T ) FIJ Bcz(0)(I - t ) and A.b(T) EJ h.~,(O)/(l - t)l/’, which means afemperature
independent K = hnb/cab. Writing B = bB,z(T) = LBcz(0), 1 - b = 1 b/(l - t ) =
(1 - t - &)/(1 t), and 2, = Tm/Tcas above, one obtains
-
-
c;
%
3Gi b
3 G i Lt:(l-t,)
r,
--
r 2 (1-b)3 I-tm
iTz
(l-tm-L)3’
(5.11)
The JPux-linelanice in superconductors
1501
For low inductions b << 1 the explicit expression for the melting field B,(T) reads
This means B,,(T) c( (Tc- T)’ near Tc and B,(T) c( 1 / T 2 far below T,. The thus
defined melting temperature T:D(B) is rather low. As stated above, softening of the FLL
by structural defects or by anharmonicity decreases T
D
: even more. For TiD % Tcone has
(5.13)
Comparing this uith the 2D melting temperature of the FLL in thin films of thickness d ,
Tio = a 2 d c 6 & r k ~= constant(B) (6.2)one finds that the 3D and 2D melting temperatures
formally coincide for a film of thickness d = ~ ( C ~ / O . ~ O ) ~ A thus
~ ~ / dA ~=, a/r for
CL = 0.1, see also Markiewicz (1990a). This length is of the same order as the length
a / n r over which the flux lines curve to hit the surface at a right angle (Hocquet et al
1992, Plaqais etal 1993, 1994, Brandt 1993a). Note., however, that the melting thickness
depends also on the structure-dependent shear modulus, d2/a2F3 C ~ ( ~ R ) ’ C M=~Bz)/c%.
(~L
As a matter of principle, it is not clear whether the Lindemann criterion applies to line
liquids; it was shown to fail if the lines are confined to planes, e.g. to the space in between
the Cu-0 planes in HTScs (Mikheev and Kolomeisky 1991). Interestingly, the Lindemann
criterion has been applied also to the melting of a lattice of ‘cosmic strings’ in cosmoIogy
(Nielsen and Olesen 1979) or ‘colour magnetic flux tubes’ in field theory (Ambj0m and
Olesen 1989). Cosmic strings have a microscopic radius of %
cm and are thousands
of light years long, see the review by Hindmarsh and Kibble (1995). They are believed to
form a line lattice which should melt due to quantum fluctuations (section 6.6).
Other possible melting criteria consider the thermal fluctuation of the vortex distances
IT^ - T ~ +I,I of the vortex separation Ixj - x i + , I (nearest neighbour direction along x ) , or of
the shear strain y (Brandt 1989b). These fluctuations are related by
1
(U~)%~(IT~-T~+I~
2
2
&a2
- a ) ~ ( I x i - ~ ~ + ~ ~ ~ - a ~ ) ~ (5.14)
- ( y ~ ) .
4R
The melting temperatures resulting from these criteria differ only by numerical factors of
order unity from the Lindemann result T , ( B ) (5.13). At the Lindemann temperature T,,
one has large shear-strain fluctuations ( y 2 ) 1 / 2% 2 . 7 ~ ~In . metals the critical flow strain
at which plastic deformation sets in is ye % (Seeger 1958), and a similar flow strain (or
flow stress C66yc) was assumed for the ZD FLL (Schmucker 1977). Nice measuremens of
the flow stress of the 2~ vortex lattice in specially prepared weak-pinning channels were
performed by Pmymboom et a1 (1988) and Pastoriza and Kes (1995). Equating ( y 2 ) 1 / 2to
this small critical shear swain would yield a much smaller T,,;but probably local thermal
fluctuations of the shear strain will not usually produce permanent structural defects in the
FLL. After a short time of local entanglement the flux lines will straighten again and global
melting should not occur at such low temperatures. Similar arguments apply to the distance
criterion. Even when the vortices touch locally, they might merge for a short time and then
separate and stretch again since the barrier for vortex cutting is small (Brandt et al 1979,
Wagenleithner 1982, S u d b ~and Brandt 1991, Moore and Wilkin 1994, Carraro and Fisher
1995, Schonenberger et a1 1995). A periodic KLis indeed observed up to rather high T
by Lorentz electron microscopy (Harada et a1 1993).
As for the vanishing of the shear modulus c66 of the FLL, this melting criterion is obvious
but not trivial since the shear stiffness in principle depends on the time and length scales
&
1502
E H Brandt
and on pinning. If one shears a lorge Fu. uniformly it will creep by thermal activation of
dislocations, which are always present and can move. In particular, screw dislocations do
not see a Peierls potential and thus could move easily along the flux lines if there were no
pinning. The shear and tilt creep rates thus depend crucially on pinning. Marchetti and
Nelson (1990) show that a hexatic (i.e. with short-range order) flux-line liquid, believed to
lie in between the line lattice and normal line liquid states, can be described by a continuum
elasticity theory in which one includes free dislocation loops; this yields an effective shear
modulus caa(kl, kz = 0) k:. Thus, C66 remains finite for k l > 0 but vanishes for k l = 0
as expected for a liquid. Interestingly, when a weak random pinning potential is switched
on, the shear modulus of the FLL is reduced (Brechet er al 1990). The influence of pinning
and thermal depinning on FLL melting is discussed by Markiewicz (1990b). A universality
of B,z(T) and of the melting line of the FLL in clean HTSCs in terms of the Ginzburg number
Gi is discussed by Mikitik (1995).
-
5.4. Monte Curio simulations und properties of line liquids
Numerous theories on a possible melting transition of the 3D FLL have been published since
1990. The melting transition of a 3D lattice of point vortices was investigated by Sengupta
et a/ (1991). Jackson and Das (1994) and Menon and Dasgupta (1994) using a density
functional method based on the smcture function of liquids and solids. Ryu et al (1992)
performed.Monte Carlo (MC) simulations on this point-vortex system and found two melting
lines where first the translational and then the bound-angle order parameters drop to zero.
Xing and TeSanoviC (1990) in their MC simulation found a melting transition very close to
the H,,(T)-line; however, due to the large distance and weak interaction between flux lines
near Hcl, this transition should be suppressed by even extremely weak pinning. Ma and
Chui (1991, 1992) discussed dislocation-mediated melting of the 3D FLL and performed MC
simulations of a small system of flux lines interacting by the correct 3D potential (3.3) with
periodic boundary conditions.
Probably the most realistic MC study of the 3D FLL in anisotropic London superconductors
uses the correct interaction (3.3) and discretizes the space by cubic cells into which vorticity
can enter from all six sides (Carneiro et nl 1993, Carneiro 1992, 1994, 1995, Chen and
Teitel 1995); in this way a melting line B ( T ) is obtained; for weak anisotropy, entangled
and disentangled vortex-liquid phases are observed. but for strong anisotropy, a nearly 2D
point-vortex liquid results. Other MC simulations of the 3D FLL consider a 3D xy-model
with the Hamiltonian (Ebner and Stroud 1985)
H
-
Jij
COS(@^
- @j - A i j )
,
A.. - -l
j
Adl
(5.15)
where Jij are nearest-neighbour exchange interactions on a cubic or triangular lattice, @i
are phase angles, and V x A = H = constant is assumed. With this model Hetzel et al
(1992) found a first-order melting transition with a hysteresis. Recent analytical solutions
of the anisotropic 3D xy-model were obtained by Choi and Lee (1995) and Shenoy and
Chattopadhyay (1995). Li and Teitel (1991, 1992, 1993, 1994) in their simulations find two
sharp phase transitions: from an ordered FLL to a disentangled flux-line liquid and then into
a normal-conducting phase with much entangling, cutting, and loop excitations. Hikami er
ul (1991) investigated the perturbation series for the 3D GL free energy in a random pinning
potential at high fields near Hc2, see also Fujita and Hikami (1995). Blatter and Ivlev
(1993, 1994) showed that quantum fluctuations modify the theoretical melting line B ( T )
as observed by de Lima and Andrade (1995). Quantum fluctuations of the
were also
The flux-line lattice in superconductors
1503
calculated by Bulaevskii and Maley (1993) and Bulaevskii et a1 (1994a,c), cf section 6.6.
An excellent realistic 3D Mc study of YBCO was performed by h i k and Stroud
(1993, 1994%b, 1995). These authors start from the CL free energy and insert a spatially
varying critical temperature Tc(r), which describes the layered structure (CuOz double
layers) and pinning in the layers, and then use an expansion in terms of lowest-Landaulevel basic functions in a system of 8 x 8 flux lines and 28 double layers. To check
for a melting transition SSik and Stroud (1993, 1994a) consider the ‘helicity modulus
tensor’, defined as the second derivative of the fluctuating free energy F with respect to the
supervelocity Q = (@&n)V@- A (section U ) , a kind of superRuid density tensor; note
that S F J S Q = J is the current density and, with (2.1a), SZF/61QI2 I@[’. In the pin-free
case, the tensor S2F/8Q,8Q~ is found to have zero components parallel to the layers but
a finite component along the c axis; with pinning, all diagonal elements are finite. Without
pinning and with weak pinning, this tensor vanishes at the same melting temperature where
the structure factor of the FLL looses its Bragg peak and also the shear modulus of the FLL
is found to vanish ( h i k and Stroud 1994b). The nearly approximation-free (apart from the
periodic boundary conditions) 3D computations of S a i k and Stroud (1993, 1994a,b) thus
indicate a sharp melting transition in YBCO.
-
5.5. Flux-line liquid
The entanglement of flux lines in a liquid state has been discussed by Nelson (1988,
1989), Marchetti and Nelson (1990, 1991) (hexatic line liquid and hydrodynamic approach),
Marchetti (1991), and with pinning included by Nelson and Le Doussal (1990) and Marchetti
and Nelson (1993) (entanglement near a surface, slab geometry). A kinetic theory of fluxline hydrodynamics was developed by Radzihovsky and Frey (1993). Glyde et a1 (1992a. b)
studied the stability and dynamics of the FLL and its dependence on film thickness, and Jkeda
et a1 (1995) the fluctuations of the vortex liquid.
In order to take advantage of Nelson’s (1989) analogy of his model FLL with an
interacting 2D Bose superfluid, these theories use a local flux-line tension and a simplified
ZD interaction between the flux lines suggested by Nelson (1989). Ko[lri(z) - r , ( z ) l / h ] ,
cf (2.5), rather than the full 3D interaction (2.8) or (3.3). This qUaSi-2D interaction is good for
long tilt wavelengths b / k , > &A, whereas the non-dispersive weak line tension (reduced
due to anisotropy, cf section 4.3) is valid for short tilt wavelengths < 2nA (Brandt 1992~).
Nelson’s model interaction yields elastic moduli of the FLL which have the form of (4.9) to
(4.12) but with k, = 0 inserted. Fortunately, this simplification does not change the resulting
thermal fluctuations of the FLL too much, since in (2)(5.3) the main contribution comes
from large k l kBz and from small kz, because the isolated-vortex-term (4.11) dominates
k ~ z this
; dominance was stated fust by Glazman and Koshelev
in cqq (4.10) when k l
(1991a). Note, however, that the quasi-ZD model interaction between flux lines is always
repulsive, while the correct 3D London interaction [the first term in (2.8). or (2.8) with
E + 01 becomes attractive when the angle between the flux lines exceeds 90”. Recently
Feigel’man er a1 (1993a) argued that the FLL with the correct 3D London interaction indeed
can be mapped onto a boson model, from which they conclude that the flux-line liquid
state is a genuine thermodynamic state occumng between the FLL state and the normal
conducting state. Fleshler et a1 (1995) measure the resistivity of the vortex liquid in YBCO
crystals with and without electron irradiation.
Comparing entangled flux lines with polymeres, Obukhov and Rubinstein (1990)
obtained astronomically large relaxation times for the disentanglement of a flux-line liquid.
These times probably do not apply since the barrier for vortex cutting is rather low (Sudb0
-
1504
E H Brandt
and Brandt 1991~.Moore and W i k n 1994, Carraro and Fisher 1995). Schonenberger et a1
(1995) recently have investigated vortex entanglement and cutting in detail for a lattice and
liquid of London flux lines; they find that the twisted (by 180") vortex pair is unstable when
embedded in a FL.L or liquid, but the twisted (by 120") vortex triplet presents a metastable,
confined configuration of high energy. Chen and Teitel (1994, 1995) consider the 'helicity
modulus' of a fluctuating elastic EL and find that at the FLL melting transition there remains
superconducting coherence (zero resistivity) parallel to the applied magnetic field, provided
the finite-wavelength shear modulus c & ~ ) stays finite for k~ > 0. Cameiro (1995) finds
that the melting and entanglement transitions in nearly isotronic superconductors occur at
the same temperature within the resolution of the computer simulations. Statistical pinning
of flux-line liquids and of defective FLL is considered bS. Vhokur er a1 (1990a, 1991b) and
Koshelev (1992a), note also the analogies of strongly pinned flux lines with a 'vortex glass'
(section 9.5) and a 'bose glass' (section 9.2).
5.6. First-order melting transition
So far the experimental evidence for a sharp melting transition is not compelling but quite
likely. hobably the strongest evidence for a first-order melting (or other) transition in
BSCCO crystals is an abrupt jump in the local induction B of small amplitude 0.3 G,
measured by Hall probes when Ha or T is swept (Zeldov el a1 1995a, Majer et a[ 1995).
Very recently a sharp dip in the differential resistivity of clean, untwinned YBCO crystals
was observed at a field H ( T ) and was interpreted as melting of the moving FLL (Safar er al
1995). First indication of FLL melting in vibrating YBCO crystals by Gammel etal (1988)
can also be explained by thermally activated depinning (Bmndt et al 1989). cf section IO.
Recently, first-order transitions were indicated by a sharp jump with hysteresis in the
resistivity of very weak-pinning untwinned YBCO single crystals by Safar er al (1992b)
and Kwok et al (1992, 1994b). Charalambous et al (1993) find that the FLL superheats
at low current densities, but supercooling is negligible. Stronger pinning suppressed the
observed hysteresis. Geshkenbein et al (1993) explain the observed hysteresis by vortex
glass transitions of the first order. In general, a hysteresis may also be caused by dynamic
processes or disorder.
It might be argued that even weak pinning may cause a similar hysteretic jump in the
current-voltage curves. For example, a sharp Wansition from weak ZD to strong 3D pinning
in films of amorphous Nb,Ge and MoxSi, with an abrupt jump of the critical current density
& ( B ) by a factor > 10 has been observed by Wordenweber and Kes (1985,1986). However,
hysteresis was not observed in that experiment. Farrell e? al'(1991, 1995) in a sensitive
torsional oscillator experiment on a very clean untwinned YBCO crystal see no hysteresis
and no jump in the magnetization against temperature. Jiang et a1 (1995) in untwinned
YBCO crystals observe a decreasing hysteresis with increasing angle between the applied
field and the c axis and propose a current-induced nonequilibrium effect as the possible
origin of this hysteresis. Worthington et a1 (1992) measured resistivity C U N ~ S over large
ranges of current density, magnetic field, and temperature in various YBCO samples. finding
evidence for hvo phase transitions at the same field, separated by a phase of 'vortex slush'
and interpreted as disorder-dependent melting and a vortex-glass transition (section 9.4).
The influence of twin-boundaries on FLL melting in YBCO is measured by Fleshler et id
(1993). Kwok et al (1994a) in a YBCO single crystal containing only two twin planes
observe a pronounced peak in the critical current density J J T ) in fields between 0.3 and
1.5T for Hall& which is indicative of %L softening prior to melting.
Melting transitions were even observed in conventional superconductors. Berghuis er
the flux-line iattice in superconductors
1505
al (1990) in thin films of amorphous Nb3Ge observed the dislocation-mediated 2D melting
predicted by Feigel'man ef a1 (1990) and Vinokur ef a1 (1990b). Similar resistivity data by
Schmidt er af (1993) suggest 20 melting of the ELin thin Nb films.
All these experiments actually measure resistivities and dissipation and thus require
at least some pinning of the flux lines by material inhomogeneities (section 8.2). In a
completely pin-free superconductor, the dissipation comes from uniform free flux flow,
which is not expected to exhibit any discontinuity or extremum at the proposed melting
transition. Besides this, dissipation by uniform flux flow is very small at low frequencies
o/Zz < ~O'S-', much smaller than the hysteresis losses caused by even extremely weak
pinning. These hysteresis losses in principle also originate from flux-line motion, but
here from the highly non-uniform jerky motion triggered by depinning processes. A
different dissipation mechanism, which seemingly is independent of pinning, was suggested
by Marchetti and Nelson (1990, 1991), the plastic deformation or viscous shearing of an
entangled EL. However, shear deformation of the FLL can only be excited by very local
forces acting on one flux line or on one flux-line row (Brandt 1978). Such local forces
are exerted by linear or point pins, by planar pins, or by the specimen surface, but not
by applied magnetic fields or cnrrents since these act over the penetration depth A, which
typically is larger than the flux-line spacing. Therefore, all these dissipation mechanisms
are caused by pinning. Since pinning in mscs strongly depends on temperature (due to
thermally activated depinning, section 9), resistivity experiments in principle will not give
unique evidence for a melting transition of the KL. However, introducing weak-pinned
channels of 12 pm width in BSCCO, Pastoriza and JSes (1995) observe a sudden increase
of the shear viscosity of the vortex liquid upon decreasing the temperature, thus supporting
the melting scenario.
5.7. Neutrons, muom and mL melting
In contrast to resistivity measurements, small-angle neutron-scattering (SANS) and muon
spin rotation (pSR) experiments in principle can provide direct information on the smcture
of the EL. Transverse rotation experiments with positive muons measure the probability
P ( E ) to find a magnetic induction value E inside the specimen. The field distribution
P(E') = (S[ B ( r ) - E ' ] ) ((. ..) denotes spatial averaging) of a perfect F'LL exhibits van
Hove singularities (Seeger 1979, Brandt and Seeger 1986). which will be smeared by the
random distortions of the E L caused by pinning (Piimpin et of 1988, 1990, Brandt 1988a. b.
1991~.Herlach et ai 1990, Harshman et ai 1991, 1993, Weber et ai 1993, Riseman er a1
1995) or by thermal fluctuation (Song et al 1993, Song 1995, Schneider et nl 1995) or
flux diffusion (Inui and Harshman 1993). Using ~ S Cubitt
R
et ~f (1993a) measure the
angular dependence of the field distribution in BSCCO and find evidence for the presence
of weakly coupled point vortices. The same conclusion is drawn by Harshman ef al (1993)
from the very narrow pSR line observed in BSCCO at low T as a consequence of the
pinning-caused reduction of the correlation between the 2D vortex lattices in adjacent layers.
Complete longitudinal disorder reduces the field variance U = ( [ B ( r ) (B)]')
(the line
width squared) from the value U M = 0.0609+&,2, for the ideal triangular FLL along the
c axis (Brandt 1988a) to the much smaller value 020 = 1.4(s/a)'&~ where s is the layer
spacing and a the flux-line spacing. (There is a misprint in uzo of Harshman et a1 (1993)).
In contrast, pinning-caused shear or compressional disorder within the 2~ vortex lattices, or
in the 3D EL, enhances the variance, and a completely random disorder also in the layers
has the variance umd = (E&,s/8zA~a)'/2.which increases with B (Brandt 1988a,b, 1 9 9 1 ~
Harshman ef a1 1993).
1506
E H Brandt
Recent SANS experiments by Cubitt et a1 (1993b) and pSR studies by Lee et a1 (1993,
1995a,b) gave evidence for FLL melting in BSCCO when T was increased, and for a
crossover from ZD to 3D behaviour (decouphg of layers, section 6.3) when Ha was
increased. With increasing temperature the Bragg peaks vanished at about the same T ( B )
value where the pSR line abruptly became narrower and changed its shape (lost its highfield tail). Using Jmentz microscopy Harada et a1 (1993) saw that in BSCCO films the FLL
averaged over the exposure time of 30s stays regular even slightly above the irreversibility
line obtained from d.c. magnetization measurements; this finding shows that the mL did not
melt at the irreversibility temperature. In their SANS experiments on Nb, Lynn et a1 (1994)
interpret the increase in transverse width of the Bragg peaks as melting of the FLL in Nb.
However, as argued by Forgan et al (1995), this observation more likely reflects a slight
loss of orientational perfection of the weakly pinned FLL, which is caused by the softening
of the FLL (peak effect, Pippard 1969, see section 9.8).
6. Fluctuations and phase transitions in layered superconductors
Due to their layered smcture, HTSCs may exhibit two-dimensional character. When the
temperature is lowered from T,, a transition from 3D to ZD superconductivity is expected
when the coherence length along the c axis, Ec(T) (Tc - T)-”’, becomes smaller than
the layer spacing s (Lawrence and Doniach 1971). Various phase transitions have been
predicted for this quasi ZD system.
-
6.I .
2D melting and the Kosterlitz-Thouless transition
An infinite pin-free ZD vortex lattice is always unstable because its free energy can be
reduced by thermal nucleation of edge dislocations even when T + 0. But even weak
pinning stabilizes a 2D lattice. This can be seen from the estimate of 2D fluctuations
cf (5.3) with
JT.$,
dk, = l / d inserted. Thus, for a large elastic pinning restoring force
>> kZZC66 = B2/(4A2wo) one h a (u2)61, k ~ T k g ~ / ( 4 a C d i =
d ) kBTB/($offLd) %
kBT/(ocLda2),but for small LYL the fluctuations diverge logarithmically with decreasing 01’
or d for films of infinite size.
As argued by Hubermann and Doniach (1979) and Fisher (1980). in the 2D FLL of a
thin film of thickness d and radius r , spontaneous nucleation of edge dislocations in the
FLL of spacing a COSIS the energy U = (da2c66/41r)In(R/a) and enhances the entropy
by S = kBln(R/a); here R 2 / a Z is the number of possible positions of the dislocation
cores. The free energy F = U - TS can, therefore, be lowered by spontaneous creation of
dislocations, leading to dislocation-mediated melting at T = T
:’ = a2dc66/(4nks) (5.13).
To obtain melting in a stack of films or layers, as in HTSCS, one has to replace d by the
layer spacing s. Inserting the shear modulus C66 (4.12), one explicitly gets
‘a
Note that T;D does not depend on B.
Earlier, a topological phase transition in a 2D superfluid was predicted by Berezinskii
(1970) and Kosterlitz and Thouless (1973) (BKT) and elaborated by Halperin and
The jhx-line lattice in superconductors
1507
Nelson (1979). At T r 0, vortex-antivortex pairs nucleate spontaneously in a single
superconducting layer or film even in zero applied field. These pairs dissociate at a
temperature
Formula (6.3) exceeds (6.2) by a factor of 8x. Accounting for the temperature dependence
of the penetration depth Xob(T) one obtains (Feigel’man et al 1990)
One may say that the 2D melting (6.2) is also a Kosterlitz-Thouless transition, with the
dissociating objects being not vortex-antivortex pairs but dislocation pairs with opposite
Burgers vectors. The 2D melting temperature is lower than TBKTsince the binding ‘energy
of dislocation pairs is lower than that of vortex pairs by a factor of 1/(8n).
A Kosterlitz-Thouless transition in layered HTSCS was predicted by Stamp er a1 (1988).
In a nice work, Blatter etal (1991b) discuss that in the presence of a large field parallel to the
layers the KL may undergo a transition to a smectic state and this would lead to a vanishing
inter-layer shear stiffness. The changed elastic properties of this FLL then would modify
the Kosterlitz-Thouless type behaviour and would lead to an algebraic current-voltage law
down to zero current density.
Clear evidence for a BKT transition is observed in current-voltage curves of thin YBCO
films by Ying and Kwok (1990). Current-voltage curves by Gorlova and Latyshev (1990),
originally interpreted as evidence for a BKT transition, are shown by Minnhagen (1991) to
obey Coulomb gas scaling associated with 2D vortex-pair fluctuations, see also Jensen and
Minnhagen (1991), Minnhagen and Olsson (1992), and the review by Minnhagen (1987).
A renormalization group theory of vortex-string pairing in thin BSCCO films yields a BKT
transition (Ota etal 1994). The effect of a magnetic field on the BKT transition is investigated
by Martynovich (1994), who finds that the transition can become first-order at high fields.
Fischer (1995b) shows that point pinning practically does not change the BKT transition, see
also Fischer (1991, 1992a). Yazdani et af (1993) observe that in an amorphous MoGe film
the BKT melting is not strongly influenced by pinning when the order in the vortex lattice
is sufficiently large, but Hsu and Kapitulnik (1992) in detailed studies of ultrathin clean
monocrystalline Nb films see a strong influence of pinning on the BKT transition. Chen et
al (1993a, b) observe 20 and 3~ flux unbinding in YBCO thin films at small fields.
Melting of the 2D lattice of flux lines in a film with the interaction (2.5) was simulated
by Brass and Jensen (1989). Using the solution for the complex GL-function q(x. y )
(2.2) which describes a ZD FLL with arbitrary flux-line positions near Hc* (Brandt 1969b),
TeSanoviC (1991), TeSanoviC and Xing (1991) and TeSanoviC et a1 (1992) obtained analytical
expressions for the partition function and melting line of 2D superconductors near the upper
critical field Hcz. For new ideas on the critical behaviour at low fields see TeSanovit
(1995). In 2D MC simulations using periodic boundary conditions O’Neill and Moore (1992,
1993) and Lee and Moore (1994) found no evidence for a liquid-to-solid transition. Hikami
et a1 (1991) and Wilkin and Moore (1993) developed a perturbation series for the free
energy from GL theory, hut their Pad&-Bore1 resummation still does not converge to the
exact low-temperature limit. In other 2D MC simulations Kat0 and Nagaosa (1993) and Hu
and McDonald (1993, 1995) obtained evidence for a first-order transition in the positional
correlation function of a 2D superconductor. In the London limit and in extensive 2D MC
simulations, Franz and Teitel (1994, 1995) find a weak first-order melting transition with a
1508
E H Brnndt
discontinuous vanishing of the shear modulus. Yates et a1 (1995) in their ZD MC simulations
using periodic hexagonal boundary conditions find a melting transition of the FLL.
6.2. Evaporation offlux lines into point vom’ces
Clem (1991) and Bulaevskii et a1 (1991) predicted that a thermally fluctuating vortex line
(figure 6, right) ‘evaporates’ into independent point vortices (figure 7) at a temperature
Tsvap
which coincides with the BKT transition temperature (6.3). At this temperature the
mean square radius ( r Z ) of the vortex line diverges. Hackenbroich and Scheidl (1991)
and Scheidl and Hackenbroich (1992) considered this problem in detail and confirmed the
identity Tmp = TBKTeven when the virtual excitation of point-vortex pairs is accounted for
and even in the case of nonequivalent layers.
t
Figurr 9. Six-terminal experiment on platelw of a layered high-T, superconductor in a
perpendicular magnetic field 2nd with applied current I. From top to bottom: at low temperatures
the flux lines (stacks of Josephson-coupled point vomces) are mherent and parallel: the resistivity
along the Rux lines is thus zro. and lherefore the top (Vi)and m o m (V2) voltage drops are
equal if the current is not too large (left) (after Busch eta1 1992 and L6pez et d 1994). with
inmasing temperature the Josephson coupling between the superconducting layers is reduced by
thermal flucNations; the flux tines will then break npa17 if the current is large enough (middle,
the arrow denotes the vortex velocity U). The resistivity perpendicular to the layers thus becomes
larger and larger, and V2 decreases. At high t e m p a ” the current flows mainly near the top
surface. Eonom In a flux transformer two hlms are separated by a very thin insulating layer
and a perpendicular field generales flux lines which thread both film.When a current is applied
to the upper film, the upper flux lines will move and drag the flux lines in the l o w film, see the
field lines of hva flux lines depicted in the middle figure. The moving flux lines also generate
a voltage drop also in the lower film, though no current is applied to this film. This dragging
of one N by the drifting periodic field of anolhe~Fu.ewes when the film separation exceeds
the flu-line spacing divided by 2z or when the velocity becomes loa large.
6.3. Decoupling of the layers
One expects that the ZD pancake vortices in neighbouring layers of distance s decouple
when the typical shear energy of the E L ik&,s starts to exceed the tilt energy ik:cw(ki)
the flux-line lattice in superconductors
I509
at k, = x / s (at the shortest possible shear and tilt wavelengths). With the elastic moduli
from (4.10)44.12) this estimate gives a decoupling field BZDabove which the layers are
decoupled and the ZD lattices of pancake vortices are uncorrelated
(6.5)
B~ = (a&/ r2s2)In(rs/a~,,,).
The thermal fluctuations of the point-vortex lattice in the superconducting layers lead
to a misalignment of the point vortices in adjacent layers (figure 9, middle). With perfect
alignment, i.e. if the vortex lines are straight and perpendicular to the layers, the phase
difference between neighbouring layers (3.12) is & ( r l )= 0. With increasing temperature,
the space and time average (8,”)increases until at (8;s F5 1 the phase coherence between
the layers is destroyed and the layers thus decouple. Glazman and Koshelev(l99la, b) find
for this decoupling transition T = TEf sx ~ ~ ~ ( 4 n ) - ” ~ / ( ~ 0 kIn~ terms
h ~ ~ofh the
~ ) 2D
.
melting transition temperature (6.2) one may write this as
T E a TF(B,,/B)‘12
with
B~~= 64n40/r2s2.
(6.6)
Thus, at sufficiently large fields B > B,, decoupling of the layers should occur at T
below TiD. However, at present it is not clear if the decoupling is a true phase transition,
set the review by Blatter et al (1994a) for a more detailed discussion and the recent
careful renormalization group analysis of coupled ZD layers by Pierson (1994, 1995). By
Monte Carlo simulations Sgik and Stroud (1995) find a smooth crossover from 3D to 2D
behaviour characterized by a continuous change of the helicity modulus Q (section 5.4) in
the c direction, and of the shear modulus c s in the a-b plane, when the effective. inter-layer
coupling was decreased.
A different decoupling transition, driven by topological excitations, has been suggested
by Feigel’man et a1 (1990) (FGL).The excited objects, called ‘quartets’, consist of four 2D
edge dislocations and correspond to a pair of dislocation loops in a 3D vortex lattice. With
increasing T, the density of these quartets increases, and a decoupling transition is expected
to occur at
TL:L % TiD/ln(B/Bm).
(6.6~)
Daemen et al (1993a) (DB) have followed the decoupling idea further and suggest
that due to these phase fluctuations the Josephson coupling between neighbouring layers
is renormalized to a smaller value as T increases. As consequences, the effective
penetration depth hc,e# and anisotropy ratio res = hc,eff/h& increase and the maximum
possible current density along the c axis .Ii,- = Jo(cosS.(X, y ) ) = (@o/ZnpOsA&) o(
Joexp[-B/B:;(T)]
decreases and vanishes at a decoupling field
However, this self-consistent treatment of the decoupling is probably not allowed. Formula
(6.7) at low fields B < BZDcoincides with the ‘second melting line’ of Glazman and
Koshelev (1991a,b), but at higher B it does not apply since c66 was neglected in the
calculation of the phase-difference.
Several authors find decoupling (3D-2D crossover) in zero applied field from xy-models
using Monte Carlo simulations (Pierson et a1 1992, Minnhagen and Olsson 1991, Weber
and Jensen 1991) or a renormalization group treatment (Chattopadhyay and Shenoy 1994,
Shenoy and Chattopadhyay 1995, Pierson 1994, 1995, Friesen 1995, Timm 1995). The
exact vortex-antivortex interaction energy within the anisotropic xy-model is calculated by
1510
E H Brandf
Choi and Lee (1995). Genenko and Medvedev (1992) predict a suppression of the critical
current by ZD vortex fluctuations and an avalanche-like generation of free vortices,
Apart from thermal fluctuations, decoupling of the superconducting layers may also be
caused by random pinning of the point-vortices. This problem has been considered in detail
by Koshelev et a1 (1995). who find decreasing effective coupling as the pinning strength
or the magnetic field increase. However, above a 'decoupling field' this decrease comes
to a halt and the coupling strength saturates to a value which is proportional to the square
of the original Josephson coupling. These authors also give an explicit expression for the
point-vortex wandering on the tail of the field distribution, which can be measured by muonspin rotation. As shown by Brandt (1991~).while in conventional superconductors random
pinning distorts the FLL such that the field dismbution becomes broader (Brandt 1988a, b),
in layered mscs pinning can narmw this distribution since the flux lines (as stacks of point
vortices) become broader and overlap more strongly such that the magnetic field variance
is reduced, cf section 5.7. The resulting strong narrowing of the ~ S line
R was observed in
BSCCO by Harshman et ~l (1993).
A further decoupling problem occurs when the field Ha is applied in the a-b plane.
Decoupling of the superconducting Cu-O layers then is equivalent to the melting of
the lattice of Josephson vortices between the layers (Efetov 1979). It was shown both
for low fields (Mikheev and Kolomeisky 1991) and high fields (Korshunov and Larkin
1992) that such a decoupling does not occur before total desmction of superconductivity.
Thermal nucleation of vortex rings between the Cu-O planes, or crossing these planes, is
calculated with a renormalization group technique by Korshunov (1990, 1993). Horovitz
(1991. 1992 1993, 1994). and Carneiro (1992). The importance of vortex rings in layered
superconductors was also stressed by Friedel (1989) and Doniach (1990).
6.4. Measurements of decoupling
Decoupling and depinning transitions have been studied in artificially grown superlattices of
YBa&u3O#rBazCu307 (Triscone etal 1990,1994, Brunner etal 1991, Fischer etal 1992,
Li et ai 1992a, Fu et a1 1993b, Obara et a1 1995, Suzuki er ~l 1995b) where the coupling
between the superconducting layers can be gradually decreased to zero by increasing the
thickness of the Praseodymium layers. Neerinck et ~l (1991) investigate GePb superlattices.
Decoupling in BSCCO crystals is observed by Hellerquist et al (1994) and in a nice work
by Pastoriza et al (1994) who find a drop of the resistivity in the c-direction by five orders
of magnitude in a temperature interval of 0.1 Kat B. = 360G.White etal (1991, 1994) and
Steel et a1 (1993) observe decoupling in synthetic MoGdGe multilayers, see also Urbach
etal (1994). In micrometre-thick such layers with various anisotropies Klein et al (1995)
observe a strong decrease of the critical current with increasing anisotropy. Korevaar et
al (1994) in NbGdGe multilayers with varying Ge thickness observe a crossover from ZD
melting in the full sample to 2D melting in single layers at high fields, but a 3D melting at
intermediate fields. A decoupling transition is found in ultrathin and multilayered BSCCO by
Raffy etal (1991, 1994) and in fully oxidized (and thus extremely anisotropic) YBa2Cu307
by Gao etal (1993). Mikheenko et al (1993) show that decoupling of the Cu-O layers can
be induced by an in-plane current density whose critical value &(T) is found to vanish at
T B ~ A~ similar
.
current-induced decoupling in BSCCO films is observed by Balestrino et
ai (1995). A review on thermal softening of the interplane Josephson coupling in HTSCS is
given by Kopelevich et al (1994).
A nice six-terminal method applies a current between two contacts on the top surface
of a HTSC platelet in a perpendicular field Hllc and measures the 'primary' and 'secondary'
Theflux-line lattice in superconductors
1511
voltage drops on the top and bottom by two further pairs of contacts (figure 9) (Busch
et al 1992, Safar et a1 1992c, 1994, Wan et ai 1994, Ldpez et a1 1994. de la CNZ e t
al 1994a, b, Eltsev el a1 1994, Eltsev and Rapp 1995). Above some temperature (or at
higher current densities) the bottom voltage drop falls below that of the top surface since
the vortex lattices in the Cu-0 layers decouple. This effect thus has to do with decoupling
and with currents along the c axis (Bricefio et al 1991, 1993, Charalambous et a1 1992.
Rodriguez et a1 1993), but also with flux-line cutting and with the helical instability of
flux lines in a longitudinal current (section 7.2). It is thus certainly not an exactly linear
effect. A nice phenomenological explanation is given by L6pe.z et a1 (1994). The 2D FL.L in
one layer drags the 2D FLL in the adjacent layer; when the coupling force decreases below
the pinning force on the 2D FLL, the adjacent lattices slip and decouple. This decoupling
was calculated for two layers by Yu and Stroud (1995) and for N layers by Uprety and
Domhguez (1995). A similar dragging of magnetically coupled flux-line lattices in two
films separated by an insulating layer was observed by Giaever (1966) and by Deltour and
Tinkham (1968) and was termed the ‘flux transformer effect’ since a voltage drop can appear
at one film when the current is applied to the other film. This effect was investigated in
detail by Ekin et al (1974), Ekin and Clem (1975), and Clem (1974b, 1975b). Glazman
and Fogel’ (1984b) consider the coupling between two films with solid and melted flux-line
lattices. An interesting related ‘cross-talk‘ effect, possibly also caused by moving vortices,
is observed in metal/insulator/metal trilayers near the superconducting transition of one of
the layers (Shimshoni 1995).
In my opinion it does not seem appropriate to ascribe the coincidence between the
top and bottom voltage drop to a nonlocal resistivity as suggested by Safar el al (1994).
Busch et al (1992) and Eltsev and Rapp (1995) successfully explain their six-terminal
measurements on YBCO crystals by a local anisotropic conductor. A nonlocal resistivity
in principle could follow, e.g. from a shear viscosity ?hear
of an entangled flux-line liquid
(Marchetti and Nelson 1990, 1991) in combination with the viscous drag force exerted by
the atomic lattice, usually also termed ‘viscosity’ q = B Z / p (5.6).
~ ~ The characteristic
length of this nonlocality [oloc
= (qJhear/q)l/2 (Huse and Majumdar 1993) is not known at
present; possibly it is smaller than the flux-line spacing a and thus does not play a role.
Nonlocal electrodynamics on length scale t ( T ) in clean superconductors is discussed by
Pippard (1953). Brandt (1972b, 1973b), Mou etal (1995) and Kogan and Gurevich (1995),
see also Feigel’man and Ioffe (1995). and Blum and Moore (1995). However, note that
any lattice is ‘nonlocal’ on the length scale of the lattice spacing a if it is described by
a continuum theory: the elastic moduli become dispersive when the wavevector k of the
distortion approaches the radius of the Brioullin zone kBZ ~3 r/a, since the elastic energy
,fa lattice has to be periodic in k-space, see figure 8. For example, in the simple model of
d ID lattice with nearest-neighbour interaction, the factor k2 in the elastic energy is replaced
by the periodic function [2 sin(ak/2) /aI2, which Formally yields a dispersive compression
modulus cl,(k) = ~ ~ ~ ( 0 ) [ 2 s i n ( a k / 2 ) j kThis
a ] ~ .natural ‘geometric’ nonlocality of a lattice
should not be confused with the nonlocality discussed in section 4.2, which is due to the large
range A >> n / n of the forces between flux lines. Note also that the top and bottom voltage
drops must exactly coincide when the flux lines are coherent and aligned, since the voltage
drop along flux lines is zero (the space between stationary vortex cores is ideal conducting).
The linear a.c. resistivity of a coherent FLL is indeed found to be local (section 9.5), and
Busch etal (1992) could explain their six-terminal measurements by an anisotropic local
d.c. conductivity. This anisotropic conductivity depends on the material anisotropy and
also on the direction of theflux lines, which varies with the measuring current due to the
self-field of the current. Furthermore, with pinned flux lines, a weak current typically will
1512
E H Brandt
not Row through the bulk but along the surface due to the Meissner effect. These facts
complicate the interpretation of six-terminal measurements.
6.5. Vortexfluctuations modifr the magnetization
The thermal fluctuation of the vortex lines gives an entropy contribution to the free energy
and thus to the magnetization M = BF/BH.. Bulaevskii et ai (1992e, 1994b) predicted
that all magnetization curves M(H,, T) should cross at one point if M is plotted against
the temperature T with the applied field H, as a parameter. At this crossing point one has
T = T" % T, and M ( H , , T') = M * = ( c m k ~ T ' / & ) ( 4 l r / ~with
~ ) c, m 1. This result was
also obtained from GL theory for H,,% HCz by TeSanoviC etal(1992). In a nice overview on
the fluctuation magnetization of 2D superconductors Koshelev (1994b) derived the'correction
factor as c; = 0.346 573. . .. This relation was experimentally confirmed, e.g. by Kes et
at (1991), Li er al (1992b, 1993), Zuo ef al (1993). Kogan et al (1993a), Thompson et al
(1993), Jin el al (1994), Wahl et al (1995), and Genoud et al (1995). It also agrees with
earlier data M(H,, T) of Welp et al (1991). which depend only on one scaling variable
[T - T c ( H ) ] / ( T H ) Z /as3 explained by 3D GL fluctuation theory, see also the comment by
Salamon and Shi (1992). Buzdin and Feinberg (1994a) calculate the angle dependence of
the fluctuation magnetization and specific heat for anisotropic superconductors, and Baraduc
et al (1995) extend this to include the 2D-3D crossover regime in YBCO. The influence of
thermal fluctuations on the vortex-caused 205TL.
NMR frequency shift is discussed by Song
et al (1993) and Bulaevskii et al (1993b), see also Song (1995). Kogan etal (1993a) show
that thermal fluctuations of vortices in layered superconductors also influence the penetration
depth h,b(T), the GL parameter K ( T ) ,and the upper critical field H&(T) extracted from
magnetization data.
Martinez et al (1994) find experimental evidence that both quantum and thermal
fluctuations modify the reversible magnetization of a BSCCO crystal. Blamer etat (3993b)
consider the fluctuations of vortices in HTSCS and in thin films accounting also for the
fluctuations of the amplitude of the order parameter q: they argue that a spontaneous
creation of flux lines cannot occur in a 3D superconductor.
6.6. Quanrunifluctiiations of vortices
The possible melting of the vortex lattice in HTSCs due to quantum fluctuations was
considered by Narahari Achar (1991) and by Blatter and lvlev (1993. 1994); a melting
line with quantum correction was observed in YBCO by de Andrade and de Lima
(1995). A related effect, the macroscopic tunnelling of vortices out of pins, is discussed
in section 9.9. A particularly elegant formulation of the quantum theory of superfluid
undamped vortices was given by Fetter (1967) and is extended to undamped point vortices
in layered superconductors by Fetter (1994). see also the calculations of helicon modes in
HTSC by Blatter and Ivlev (1995). Following Blatter and Ivlev (1993, 1994) and Blatter et
al (1994b), the combined thermal and quantum fluctuations (U') are given by a sum over
Matsubara frequencies on= 2nnkT/h (n = 0,1,2, .. .) with the term n = 0 corresponding
to the pure thermal fluctuations. The cutoff of this sum crucially enters the resulting (U').
Bulaevskii and Maley (1993) calculate the low-temperature specific heat of the vortex
lattice, introducing the term 'vorton' for the viscously relaxing elastic modes considered in
section 5.2. In subsequent work Bulaevskii etal (1994a, b,c) consider the effect of quantum
fluctuations on the reversible magnetization in HTSCS starting from the classical Langevin
equation (Schmid 1982) and adding the vorton contribution obtained bom Caldeira and
The flux-linelattice in superconductors
1513
Leggett’s (1981) action for a damped oscillator. The Hall effect of moving vortices yields
a force term av x Z which is linear in the velocity v l i e the viscous drag force qv, but
which does not dissipate energy. This Hall term was considered in quantum fluctuations
(Fetter 1967, 1994, Blatter and Ivlev 1994, Morais-Smith et al 1994, 1995, Horovitz 1995)
and quantum depinning (Feigel’man et al 19934 Ao and Thouless 1994, Stephen 1994,
Sonin and Horovitz 1995, Chudnovsky 1995).
7. Flux motion
7.1. Usualfiuxfipow
An electric current density J flowing through the superconductor exerts a Lorentz force
density B x J on the flux-line lattice which causes the vortices to move with mean velocity
v. This vortex drift generates an electric field E = B x v where B is the flux density or
magnetic induction in the sample, see the review by Kim and Stephen (1969). An interesting
discussion of the voltage generated along normal conductors and superconductors is given
by Kaplunenko et al (1985), who show that the ‘Faraday voltage’ a@/& induced by the
changing magnetic flux 0 can be compensated by the ‘Josephson voltage’ (h/2e)a$/at
caused by the changing phase difference q5 of the order parameter @ at the contacts,
see also Clem (1981a.b). Moving flux lines dissipate energy by two effects which give
approximately equal contributions:
(a) By dipolar eddy currents that surround each moving flux line and have to pass through
the vortex core, which in the model of Bardeen and Stephen (1965) is approximated by a
normal conducting cylinder.
(b) By the retarded recovery of the order parameter @(T) at places where the vortex
core (a region of suppressed 1@1) has passed by (linkham 1964).
In the solution of this flux-flow problem by the time-dependent Ginzburg-Landau theory
(”DGL).
achieved for B % BCz and for curved flux lines by Thompson and Hu (1971, 1974)
and for B << Be2 by Hu and Thompson (1972), both dissipation effects are combined.
Applying the TDGL theory to HTscs, Ullah and Dorsey (1990, 1991), Vecris and Pelcovits
(1991), and Troy and Dorsey (1993) consider the influence of fluctuations on the transport
properties of type-I1 superconductors and on the moving FLL;Dorsey (1992) and Kopnin
eta1 (1993) obtain the Hall effect during flux flow, and Ivlev eta1 (1990), Hao and Clem
(1991a), Golosovsky eral (1991, 1992, 1993, 1994). Hao and Hu (1993), and Krivoruchko
(1993) give the flux-flow resistivity of anisotropic superconductors. The Bardeen-Stephen
model is generalized to layered superconductors by Clem and Coffey (1990). For reviews on
moving vortices see Larkin and Ovchinnikov (1986), Gor’kov and Kopnin (1975). Shmidt
and Mkrtchyan (1974). and Tinkham’s book (1975).
As observed by neutron scattering (Simon and Thorel 1971, Schelten eral 1975, Yaron
et al 1994, 1995) and NMR (Delrieu 1973), during flux flow the FLL can become virtually
perfect over the entire specimen, in spite of pinning; see also the NMR experiments on rf
penetration and flux flow by Carretta (1993) and recent decoration experiments by Duarte
et al (1995). This means the FLL flows past the random pinning potential averaging out its
effect such that the FLL correlation length increases. This ‘shaking’ and ordering of a FLL
flowing across pins, and maximum disorder just above J, when depinning starts, is also seen
in computer simulations (Brandt, unpublished) and is calculated explicitly by Koshelev and
Vinokur (1994), see also Shi and Berlinsky (1991) and Bhattacharya and Higgins (1993,
1994, 1995).
In thin superconducting films in the presence of a driving current and an d field, the
1514
E H Brandt
differential resistivity dVldI exhibits peaks caused by the interaction of the moving periodic
FLL with the pins; this effect allows the determination of the shear modulus of the FLL (Fiory
1971, 1973). Fiory’s nice experiments were explained by Larkin and Ovchinnikov (1973)
and Schmid and Hauger (1973); see also the recent theory by Muller and Schmidt (1995).
Recent new such experiments on the ‘washboard frequency’ during flux flow by Harris et
al (1995a) show that the peaks in dV/dl vanish rather abruptly at a ‘melting field’ B,(T)
of the FLL, e.g. B, = 3.5T at T = 86K, cf section 5.3, Mathieu et al (1993) investigate
the dissipation by short vortices or ‘flux spots’ in a thin surface sheath above H d . Tilted
vortices in thin films in an oblique field are considered in a brilliant theoretical paper by
Thompson (197.5). and are reviewed by Minenko and Kulik (1979). In @in films with a
current the curvature of the flux lines can be much less than the curvature of the magnetic
field lines (Brandt 1993a).
Since at low B the dissipation of the vortices is additive and since at the upper critical
field Bcz(T) the flux-flow resistivity pm has to reach the normal conductivity pn,one expects
p ~ p ( BT)
, % pnB/Bcz(T). This linear relationship was observed by Kim et al (1995) in
a NbTa alloy at low T << T,,while p w ( B ) became strongly concave as T -+ T,. Raffy
et ol (1991) carefully measured the flux-flow resistivity of thin BSCCO films. Recently
) experiments using short
Kunchur et d (1993, 1994) nicely confirmed the linear p p ~ ( B in
pulses of extremely high current density J, which provides spatially constant J and ’free
flux flow’, i.e. absence of pinning effects, but still does not heat the sample, such that even
the depairing current density JO = ($00/3&~~~/lo) can be reached.
At high flux-flow velocities in YBCO Doettinger eta1 (1994, 1995)observe an instability
and a sudden jump to a state with higher resistivity, which was predicted by Larkin and
Ovchinnikov (1975) as caused by the nonequilibrium distribution of the quasiparticles,
see also Guinea and Pogorelov (1995). This electronic instability should not be confused
with the pinning-caused thermomagnetic instability predicted as the solution of a system of
nonlinear differential equations by Maksimov (1988); this explosive instability occurs during
flux jumps (see, e.g. McHenry et ol 1992, Gerber et al 1993, and Muller and Andrikidis
1994) and may lead to the nucleation and dendritic fractal growth of normal spots, which
was observed by Leiderer eta1 (1993) in YBCO films and explained by Maksimov (1994).
Magnetic instabilities in typen superconductors were also investigated by Wipf (1967) and
Ivanchenko et al (1979) and were reviewed by Mints and Rakhmanov (1981), Gurevich
and Mints (1987) and Wipf (1991).
7.2. Longitudinul currents and helical instability
When the current flows exactly parallel to the flux lines it does not exert a Lorentz force
and thus should flow without dissipation. Flux-line arrangements which satisfy B x J = 0
(or B x rot B = 0 in the bulk) are called ‘force-free’, see Walmsley (1972) and Campbell
and Evetts (1972) and references therein. For example, a twisted FLL with B, = B cos(kz)
and By = B sin(kz) exhibits rot B = -kB and thus is a force-free configuration. In real,
finitesize superconductors such force-free configurations develop an instability near the
surface since there the condition BllJ is violated. Longitudinal currents are limited by
the onset of a helical instability of the single flux line (Clem 1977) or of the FLL in the
bulk or near a planar surface (Brandt 1980c, 1981b), or of the macroscopic current path
similarly to the way a wire carrying a current between the poles of a magnet will bend into
an arc (Campbell 1980). This instability (see also section 3.5)defines a longitudinal critical
current density Jell, which has been measured, e.g. by Irie et af (1989). see also D’Anna er al
(1994b.c). The onset of spontaneous helical deformation of the flux lines is delayed by the
The fludine lattice in superconductors
-
1515
finite shear modulus C66 of the FLL or by elastic pinning (Labusch parameter 01'): for zero
~
pinning one has Jcn (c~~/cII)''', and for weak pinning one has jcn ( a r ~ / c 1 1 ) ' /(Brandt
1981b). The low power means that even very weak pinning can stabilize longitudinal
currents. Khalfin er al (1994) considered a layered superconductor with a vortex line driven
by a longitudinal a.c. current. Genenko (1993, 1994, 1995) calculates the relaxation of
vortex rings in a cylinder, the irreversible right-handed vortex-helix entry into a current
carrying cylindrical wire in a longitudinal field, and the left-handed helical instability near
the surface.
the FLL deforms spontaneously and the flux-line helices grow until they
For J >
cut other flux lines or hit the specimen surface. The resulting dissipative, oscillating, and
possibly chaotic state, with spiral structure at the surface of a cylindrical specimen, has
been the subject of intricate experimental (Timms and Walmsley 1976, Ezaki and Irie 1976,
Walmsley and Timms 1977, Cave and Evetts 1978, Blamire and Evetts 1985, LeBlanc er
al 1993) and theoretical (Clem 1980, 1981a, 1992, Brandt 1980b, 1982, Perez-Gonzales
and Clem 1985, 1991, Matsushita and Irie 1985) investigation, which recently was resumed
by Marsh (1994) and Kriimer (1995). The basic problem with this resistive state, say in a
cylinder with longitudinal current. is that a continuous migration of helical flux lines from
the surface leads to a pile-up of longitudinal flux in the centre of the specimen, since there
the azimuthal flux component of a helix vanishes (vortex rings contract to a point) but the
longitudinal component remains. The diverging longitudinal flux cannot be removed from
the centre simply by outgoing helices since these will cut and reconnect with the incoming
helices such that the transport of azimuthal flux comes to a halt and thus the observed
longitudinal voltage drop cannot be explained. An intricate double-cutting model which
solves this problem was suggested by Clem (1981a). From these problems it is clear that
there are situations where the magnetic flux, carried by continuous Abrikosov flux lines,
resists cutting and reconnection, in contrast to the imaginary magnetic fields lines in a
normal metal which can easily go through each other.
4
-.
7.3. Cross-~%wof two componenrs of magneticj7u.x
In a long strip the diffusion (or flow) of transverse flux slows down when longitudinal flux
is present (Brandt 1992b). Similarly, in a strip with pinning the penetration of transverse
flux is delayed when a longitudinal field is applied. This induced anisotropy is discussed
in a nice paper by Jndenbom et al (1994c), see also Andre et a1 (1994). The resulting
enhancement of the critical current by this 'cross-flow' is measured by LeBlanc et al (1993)
and Park and Kouvel (1993). This enhancement is closely related to the stabilization of
longitudinal currents and the delay of helical instability by pinning (Brandt 1981b). For a
long strip containing a parallel flux density Bll sz p&,b, the obstruction of the penetration
of a perpendicular flux density BI from the long edges can be understood in three different
ways (figure IO):
(a) The application of a small perpendicular field H a l tilts the flux lines near the edges.
Penetration of BI means that this tilt has to penetrate, but for long strips even small tilt
causes large flux-line displacements, which are prevented by even weak pinning. The flux
lines thus stay parallel to the strip, and BL cannot penetrate if the flux lines are coherent.
(b) 3efore B l has penetrated, the application of HaIinduces a Meissner screening
current along the four edges of the strip and over its entire surface. At the shorr edges, this
screening current is perpendicular to E,, and thus tilts the flux lines, which means that B l
penetrates from there. However, at the long edges, the screening current flows parallel to
the flux lines and does not exert a force or torque on these until helical instability sets in,
IS16
E H Brandt
Bo:D
BpB,
((11
Ibl
IcI
Figure 10. Visualization of the penelmtion of a small dditional perpendicular magnetic field
into an isotropic type-ll superconducting strip in a large longitudinal field. ('11 kji. If no
longitudinal magnetic held is applied. the flux of the perpendicular held will penetrate from the
long edges. taking the shortest way. Middle. With a longitudinal flux present. the penetrmion
of a perpendicular flux proceeds from the short egdes along the strip. kf~The stream lines oi
the screening c u ~ e n t sindued w the two Ral surfaces of the sVip when B perpendicular field
is applied. These surface cumnts exert a Lorentz force only on the ends of the longitudinal
flux lines, where the U-fuming current and the Rux lines are at a right angle. N e v the long
edges the cument is p d l e l to the flux lines and thus does not exert a force on them. ( b ) The
tilt of the longitudinal vortices penetrates from the edges, driven by Lhe U.tuming screening
current. After some time (in an Ohmic strip with flux diffusion) or when the perpendicular held
is increased (in the Bean model with highly non-linear resistivity) the Lilt of the flux lines. and
thus the perpendiculv Rux, have penetrated completely. ( e ) The penetration of an additional
perpendicular held imm the long edges M be described as the cutting and reconnection of
the flux lines belonging to the longitudinal and perpendicular flux components The resulting
slightly tilled flux lines are then pushed into the strip by the screening current flowing near the
long edges; hoeever, for small tilt angle the driving force is small compared 10 the pinning
force. and thus this type of flux penetntian is suppressed.
which can be stabilized by even weak pinning.
(c) In a more local picture, the penetrating BL has to cross the large B f , which means
that flux lines have to cut each other at an angle of 90'. After each culling process the new
flux-line ends reconnect. The resulting flux lines are thus tilted and will prevent further
penetration of perpendicular flux if they are pinned and coherent.
These three pictures show that the presence of weakly pinned longitudinal coherent
flux lines causes a high threshold for the penetration of perpendicular flux, but if the flux
lines consist of almost uncoupled point vortices and of Josephson strings as in BSCCO,
this threshold will be low. At 7' = 0 the flux penetration (or initial magnetization) will be
highly nonlinear. At higher temperature, when thermally activated pinning occurs, the flux
penetration may be described as a linear anisotropic diffusion of flux (Brandt 1992b).
The cross-flow problem may also be illustrated with the simpler example of aslab in the
y-z plane containing a parallel flux density B, (Campbell and Evetts 1972). A penetrating
B, will tilt the flux lines at the surface, but even weak pinning can prevent this. In fact,
as discussed by Brandt and Indenbom (1993), in the above example of a long strip with
pinning, the penetration of perpendicular flux will start at the $at sugaces. From there a
Bean critical state penetrates from both sides as with the slab. When the fronts of antiparallel
flux meet at the midplane of the strip, the flux lines will close first at the long edges and
EL will then penetrate from there as discussed in section 8.4.
Angle (dea)
Perpendicular F i e l d (li
Figure 11. (n)~eresistivityofaBSCCOplatelet(lx2x0.2mm3. 7 =74,36K)asafunction
of the angle 8 of the magnetic field B, wifh respect to the a-b plane for (from top to bottom)
E, = 0.1.0.3.0.5, I . 1.5.2, 2.5. 3, 3.5.4,4.5.5,5.5.6.6.5 and 7 T . The current is applied in
the n-b plane parallel to the axis of rotation. (b)These curves collapse into one single curve
when plolted against the perpendicular field companenl B,sinB (afrer lye n d 1992).
7.4. Dissipation in layered superconductors
In HTSCs the flux-flowresistivity is highly anisotropic, see, e.g. the beautiful angle-dependent
data of Iye et al (1990a,b, 1991a,b, 1992) (figure 11) and the careful comparison of the
resistivity in the two cases B I J and BllJ by Kwok et a! (1990). Most remarkably, if
both B and J are in the a 4 plane, the resistivity in BSCCO is found to be independent of
the angle between B and J , in contradiction to the prediction of the flux-flow picture (lye
er a1 1990a,b, Raffy et a[ 1991). This puzzling observation can be understood in various
ways: by the relevance of only the induction component perpendicular to the layers (Kes
et nl 1990); by vortex segments hopping across the layers, thereby nucleating pancakeantipancake pairs which are driven apart by the Lorentz force (Iye et al 1990b); by the
correct angular scaling of anisotropic resistivity (Blatter et al 1992, Iye et a1 1992, Fu et
al 1993a); or by the motion of vortex kinks (Ando er al 1991, Iye et al 1991b). Hao et
a1 (1995) show that this Lorentz-force independence of the resistivity in BSCCO (observed
to a lesser extent in the less anisotropic YBCO, but not at all in isotropic MoCe films) is
just a consequence of the large material anisotropy when one accounts for the fact that the
electric field induced by vortex motion is always perpendicular to B.Note thaionly the
perpendicular part JI of the current density J = JA+ J , = &JB) B x ( J x B) causes
dissipation, while the parallel current density Jll is dissipation-free or may cause a helical
instability (Clem 1977, Brandt 198Oc, 1981b), which distorts the RL.
A further explanation of the Lorentz-force independence of the resistivity near T,,(Ikeda
1992a) is based on the renormalized fluctuation theory of the order parameter developed
by Ikeda et al.(1991b). These fluctuations also modify the conductivity (Aslamazov and
Larkin 1968, Schmidt 1968, Maki and Thompson 1989, Dorsey 1991, Ullah and Dorsey
1990, 1991, Dorin eta! 1993). the magnetoresistance (Hikami and Larkin 1988). and the
+
1518
E H Brundf
diamagnetism (Hopfenghtner et al 1991, Koshelev 1994b) just above and below T,,and
they broaden the resistive transition in multilayer HTSCs (Koyama and Tachiki 1992),,see
also section 6.5. For a comprehensive review on the theory and experiments of fluctuation
effects in superconductors see Tlnkham and Skocpol (1975). Lang et al (1995) recently
gave a detailed study of the normal-state magnetotransport in BSCCO thin films with regard
to contributions of the superconducting fluctuations.
Current transport along the c axis (see also section 3.5) was investigated in YBCO for
Ellc and B I c by Cbaralambous et al (1992). for Bllc by Hettinger et al (1995), and
in BSCCO by Briceno etal (1991. 1993) and.Rodriguez et a1 (1993). ?he pronounced
maximum in the c axis resistivity at a temperature below T, was interpreted and well fitted
by Briceiio eta1 (1993) using the Ambegaokar and Halperin (1969) resistivity of weak links
at finite temperature; the same model was found to fit the resistivity of various HTSCS very
well (Soulen et a1 1994); see also Kosbelev (1996) and the explanations of the magneticfield-caused broadening of the resistive transition by Tinkham (1988) and Inui etal (1989),
and the phenomenology of flux motion in HTSCs by Doniach (1990). Balestrino etal (1993)
fit the enhanced c axis resistivity in thin BSCCO films to the fluctuation model of Ioffe et
a1 (1993). For recent work on the c axis resistivity of BSCCO see Zhao et a1 (1995), Luo
etal (1995), and Cho era1 (1995).
Woo er a1 (1989) and Kim et a1 (1990b) in TIzBaZCaCuzO,, and Kim et a1 (1990a) in
granular NbN films, observe a Lorentz-force independent resistance tail in magnetic fields
near T, and explain this quantitatively by Josephson coupling between grains using the
Josephson critical current derived by Ambegaokar and Baratoff (1963). Recent experiments
on fluctuations in TI-based HTSCS were performed by Ding et a1 (1995) and Wahl et al
(1995). Another dissipation mechanism in terms of ?D vortex-antivortex pair breaking is
suggested by Ando ef a1 (1991). In "/AN
multilayer films Gray et a1 (1991) find a
crossover between flux pinning and Josephson coupling in the current-voltage curve as H,
is increased. Iye et al (199Ob) carry out a comparative study of the dissipation processes
in HTSCs. A comprehensive review on Lorentz-force independent dissipation in HTSCs is
given by Kadowaki eta1 (1994).
7.5. FluxlffowHall effect
A much debated feature of HTSCS and of some conventional superconductors (see Hagen et
a1 (1993) for a compilation) is that near T, the Hall effect observed during flux flow changes
sign as a function of B or T . This sign reversal offhe Hull mgle is not yet fully understood
though numerous explanations were suggested, e.g. large thermomagnetic effects (Freimuth
et ul 1991). flux flow and pinning (Wang and Ting 1992, Wang er a1 1994), opposing
drift of quasiparticles (Feme11 1992). complex coefficient in the timedependent GL equation
(Dorsey 1992), bound vortex-antivortex pairs (Jensen et a1 1992). the energy derivative of
the density of states of quasiparticles at the Fermi energy may have both signs (Kopnin
er a1 1993). segments of Josephson strings between the Cu-0 layers (Harris ef al 1993,
1994; see the criticizing comments by Geshkenbein and Larkin 1994 and Wang and Zhang
1995), renomalized tight-binding model (Hopfeng2rtner et a1 1993, and unpublished 1992;
Hopfengwner notes that the sign reversal occurs at a temperature where the longitudinal
resistivity is dominated by fluctuations of the order parameter p), fluctuations (Aronov and
Rapoport 1992). vortex friction caused by Andreev reflection of electrons at the vortex
core boundary (Meilikhov and Farzetdinova 1993). unusual Seebeck effect (Chen and Yang
1994), motion of vacancies in the FLL (A0 1995), and the difference in the electron density
inside and outside the vortex core (Feigel'man et a1 1994, 1995), and charged vortices
The flux-line lattice in superconductors
1519
(Khomskii and Freimuth 1995). See also the recent calculations of the Hall effect in dirty
type4 superconductors by h k i n and Ovchinnikov (1995) and Kopnin and Lopatin (1995)
and the microscopic analysis by van Otterlo et a1 (1995).
Samoilov (1993) points out a universal scaling behaviour of the thermally activated
Hall resistivity. Wnokur et al (1993) prove the universal scaling law pxy p:z for the
resistivity tensor in HTSCS, which is confirmed, e.g. by the experiments of WBltgens et a1
(1993b) and Samoilov et al (1994). Pinning Of flux lines influences the Hall signal (Chien
et nl 1991): the thermally activated nature of flux flow renormalizes the effective viscosity
q but leaves the Hall parameter 01 unchanged (Vinokur et ul 1993). Duect evidence of
the independence of the Hall resistivity on the disorder caused by heavy-ion irradiation of
both YBCO and T12BazCaCu20g (with negative and positive sign of the Hall effect in the
pinned region, respectively) is observed by Samoilov et al (1995). Including the Hall effect
into collective pinning theory, Liu et a1 (1995) also find the scaling pxy p:, but no sign
reversal, suggesting that the sign reversal is not a pinning effect. Indeed, the sign reversal
of pxy was found to be even more pronounced at very high current densities where pinning
is unimportant (Kunchur et al 1994). Budhani et a1 (1993) find suppression of the sign
anomaly by strong pinning at columnar defects.
A double sign change (explained by Feigel’man et a1 (1994, 1995)) is observed in
YBa2Cu408 by Schoenes et a/ (1993), and a sign change in MoGe/Ge. multilayers by
Graybeal etal (1994) and in amorphous isotropic MosSi by Smith etal (1994). Bhattacharya
e t a l (1994) observe a sharp minimum in pry just below HCz. Ginsberg and Manson (1995)
in twin-free YBCO crystals observe a Hall conductivity a,, = -c,sz/Hu
cz?H; which
has the dependence on field H,, and reduced temperature r = 1 - T / Z predicted by
Dorsey (1992), Troy and Dorsey (1993). and Kopnin et al (1993). Harris et a1 (1995b)
corroborate the additivity of the Hall conductivities of quasiparticles (U;,
H ) and of
vortices (oi, 1/H) with u;y > 0 and with o& =- 0 in LaSrCuO, but aA. < 0 in untwinned
YBCO, resulting in a sign reversal of the total a,, in YBCO but not in LaSrCuO. So far in
all these experiments and theories it remains open why one of the contributions to the Hall
conductivity is negative. A mechanism for an even Hall effect in ceramic or inhomogeneous
superconductors is proposed by Meilikhov (1993). Horovitz (1995) predicts a quantized Hall
conductance when the vortex dynamics is dominated by the Magnus force.
-
-
+
-
-
7.6. Thermo-electricand therm-magnetic effects
Another interesting group of phenomena which for brevity cannot be discussed here in detail,
are the thermo-electric (Seebeck and Nernst) effects and the thermo-magnetic (Peltier and
Ettingshausen) effects, i.e. the generation of a temperature gradient by an electric current
or vice versa, and the dependences on the applied magnetic field, see, e.g. Huebener et
a1 (1991, 1993), Hohn et al (1991), Fischer (1992b), Oussena et al (1992), Ri et a1
(1993), Zeuner et al (1994), and Kuklov et a1 (1994). The Peltier effect is discussed
by Logvenov et al (1991). Meilikhov and Farzetdinova (1994) present a unified theory of
the galvanomagnetic and thermomagnetic phenomena based on a phenomenological threefluid model (superconducting and normal quasiparticles plus flux lines) in analogy to the
two-fluid model of superconductors by Gorter and Casimir (1934). The thermo effects
originate from the transporr of entropy by moving flux lines (Maki 1969b. 1982, Palstra
et a1 1990a, Huebener et a/ 1990, Kober er al 1991% Golubov and Logvenov 1995). A
comprehensive comparative study of Nernst, Seebeck and Hall effects was recently given
by Ri et a/ (1994), see also the review paper on superconductors in a temperature gadient
by Huebener (1995).
1520
E H Brandt
7.7. Fluxlpow noise
The motion of flux lines causes both magnetic and voltage noise originating from pinning in
the bulk and at surfaces. Measurements and theories of the flux-flow noise were reviewed
by Clem (1981b). Ferrari et a1 (1991, 1994) and Wellstood et,al (1993) find that noise in
thin YBCO films can be explained by individual vortices hopping between two potential
wells in a confined region and that flux noise is suppressed by a very weak current. Lee
et nl (1995a, b) observe correlated motion of vortices in YBCO and BSCCO films up to
30pm thick by measuring the noise at opposing surfaces; depending on temperature, the
flux lines move as rigid rods, or they are pinned at more than one pin along their length,
thus destroying the correlation. Marley et al (1995) and D’Anna et al (1995) measure an
increase of flux-flow noise near the peak effect (section 9.8) as an indication of a dynamic
transition in the KL. Marx eral (1995) observe and discuss l/f noise in a BSCCO bicrystal
grain-boundary Josephson junction, and Kim et ai (1995a) observe voltage noise in YBCO
films. For interesting ideas related to l/f noise and to noise during self-organized criticality
see Jensen (1990) and Wang et al (1990). Ashkenazy etal (1994, 1995a) give theories of
random telegraph noise and (Ashkenazy et a1 1995b) of flux noise in layered and anisotropic
superconductors. Tang et al (1994) find the noise of a single driven flux line by computer
simulafion, see also the comment by Brandt (1995b) and new work by Denniston and Tang
(1995). Houlrik er al (1994) and Jonsson and Minnhagen (1994) calculate the flux-noise
spectrum for ZD superconductors from a xy-like model with time-dependent GL dynamics.
7.8. Microscopic theories of vortex motion
The flux-flow resistivity, the Hall effect, and the effects of a temperature gradient on flux
flow have to do with the forces on moving vortex lines in superconductors (Huebener and
Brandt 1992). This complex problem is still not fully understood. A nonlocal generalization
of the model of Bardeen and Stephen (1965) to IOW-K superconductors was given by Bardeen
and Sherman (1975) and Bardeen (1978). The equation of motion of a vortex in the
presence of scattering was derived by Kopnin and Kravtsov (1976). Kopnin and Salomaa
(1991), and Kopnin and Lopatin (1995). The modem concept of the Berry phase was
applied to reproduce the Magnus force on a moving vortex (Ao and Thouless 1993, Gaitan
1995, Ao and Zhu 1995). The classical theory of vortex motion by NoziBres and Vinen
(1966) and Vinen and Wmen (1967) recently was confirmed quantitatively by Hofmann
and Kiimmel (1993) who show that half of the Magnus force on the vortex originates from
the electrostatic field derived from Bernoulli’s theorem, and the other half from Andreev
scattering of the quasiparticles inside the vortex core. During Andreev scattering at an
interface or at inhomogeneities of the order parameter, a Cooper pair dissociates into an
electron and a hole, and one of these two particles is reflected and the other one passes; see
Kiimmel et al (1991) for a review on Andreev scattering in weak links. Andreev scattering
at the vortex core also changes the thermal conductivity and ulhasonic absorption in the
presence of a FLL (Vinen et al 1971, Forgan 1973). Accounting for the excitations in the
vortex core (Caroli et al 1964, Klein 1990, Ullah et ai 1990), Hsu (1993) and Choi er al
(1994) calculate the frequency-dependent flux-flow conductivity up to infrared frequencies,
cf also section 9.6. For further ideas on moving vortices see the reviews by Gor’kov and
Kopnin (1975) and Larkin and Ovchinnikov (1986), and the work by Gal’perin and Sonin
(1976). Ao and Thouless (1993, 1994), Niu et a1 (1994). and Gaitan (1995). The analogy
between the vortex dynamics in superfluids and superconductors is only partial. see Vinen
(1969) and Sonin and Krusius (1994) for overviews and the book by Tilley and Tilley
The flux-linelanice in superconductors
1521
(1974). Recently Domfnguez et al (1995) showed that the Magnus force on the flux lines
results in the acoustic Faraday effect, i.e. the velocity of ultrasound propagating along the
magnetic field depends on the polarization.
Another interesting problem is the low-temperature viscosity q of flux lines. Matsuda
et a1 (1994) in YBCO measure extremely large q (or low ~ F =
F B Z / q ) . which suggests
a large Hall angle and extremely clean materials ( 1 >3 f ) , see also Harris et al (1994).
This vortex viscosity also enters the theories of the low-temperature specific heat of the FLL
(Bulaevskii and Maley 1993, Fetter 1994). which thus may be used to measure q. Guinea
and Pogorelov (1995) find that mscs with their short coherence length are in the ’ultraclean’ limit, with electronic levels in the vortex core separated by energies comparable to
the energy gap, resulting in a large vortex viscosity.
8. Pinning of E u lines
8.i. Pinning offlux lines by material inhomogeneities
In real superconductors at small current densities J -= Jc the flux lines are pinned by
inhomogeneities in the material, for example by dislocations, vacancies, interstitials, grain
boundaries, twin planes, precipitates, irradiation-caused defects (e.g. by fast neutrons, Weber
1991, Sauerzopf et a1 1995), by planar dislocation networks (e.g. competing with oxygen
vacancies in BSCCO, Yang et a1 1993), or by a rough surface, which causes spatial variation
of the flux-line length and energy. HTSCS and some organic superconductors exhibit intrinsic
pinning by their layered structure (Tachiki and Takahashi 1989). Only with pinned flux
lines does a superconductor exhibit zero resistivity. When J exceeds a critical value Jc,
the’ vortices move and dissipate energy. Pinning of flux lines was predicted by Anderson
(1962) and is described in a comprehensive review by Campbell and Evetts (1972), see
also the books by Ullmaier (1975) and Huebener (1979), the reviews by Campbell (1991,
1992a. b), Zhukov and Moshchalkov (1991), Malozemoff (1991). Senoussi (1992), Pan
(1993) and McHenry and Sutton (1994), and the general articles on high-field high-cument
superconductors by Hulm and Matthias (1980), on critical currents and magnet applications
of HTSCs by Larbalestier (1991), and on pinning centres in HTSC by Raveau (1992) and the
comparison of pinning parameters between low-T, superconductors and YBCO by Kiipfer
er al (1995). Pinning has several consequences:
(i) The current-voltage curve of a superconductor in a magnetic field is highly nonlinear.
At T = 0 or for conventional superconductors, one has the electric field E = 0 for J c Jc
and E = ~ F FforJ J >> Jc. For J slightly above Je various concave or convex shapes of
E ( J ) are observed, depending on the type of pinning and on the geometly of the sample.
Often a good approximation is (with, e.g. p = 1 or p = 2)
E ( J ) = 2pFF[1 - ( J C / J ) ~ ] ” ~ (J
.IC).
(8.1)
(ii) The magnetization curve M ( H , ) is irreversible and performs a hysteresis loop when
the applied magnetic field H, = Bo/@,, is cycled. When Ha is increased or decreased
the magnetic flux enters or exits until a crifica6 slope is reached as with a pile of sand,
for example, a maximum and nearly constant gradient of B = [BI.More generally, in
this critical state the current density J = IJI attains its maximum value J,. Averaged
over a few flux-line spacings, the current density of the FLL in one-dimensional problems is
J = ( B H / B B ) V x B X pO1Vx B,where H ( E ) B / p o is the (reversiblejmagnetic field
H, which would be in equilibrium with the induction E . In general one has J = V x H
with He = BF/BB, (a = x , y , z ) where F is the free energy of the superconductor
-
1522
E H Brandt
(Josephson 1966).
(iii) In general, the current density in t y p e n superconductors can have three different
origins: (a) su$ace currents (Meissner currents) within the penetration depth A; (b) a
gradient of the flux-line density; or (c) a curvature of the flux lines (or field lines). The
latter two contributions may be seen by writing V x B = V B x h BV x B where
B = B/B. In bulk samples typically the gradient term dominates, J &‘VB, but in thin
films the current is carried almost entirely by the curvature of the flux lines. Since in simple
geometries a deformation of the FLL is caused mainly by pinning, it is sometimes said that
‘the current flows in regions where the FLL is pinned’. However, due to a finite lower
critical field H c l . the current density in type-U superconductors may considerably exceed
the critical current density J,. required to move the vortices. Such ‘overcritical current’
flows not only near the specimen surface (as a magnetization current or as a Meissner
but also inside the specimen at the flux front when
current, which remains also above HCl),
HcI > 0 (Campbell and Evetts 1972, Ullmaier 1975). Recently an excess vblume current
was observed in the form of a ‘current string’ in YBCO and BSCCO platelets by Indenbom
er al (1993. 1995).
(iv) A kind of opposite situation, with a finite driving force on the flux lines in spite
of zero average current, occurs in the ‘flux transformer’ (section 6.4 and figure 9) when
the flux lines in an isolated film are dragged by the moving periodic field caused by the
maving flux lines in a very close-by parallel film io which a current is applied. In general,
the driving force is given by the average of the current density at the vortex cores, which
may deviate from the global average (transport) current density. This deviation is largest
at low inductions B < B,I where the relative spatial variation of the periodic B ( r ) is
not small. A further ‘current-free’ driving force on flux lines is exerted by a temperature
gradient (section 7.6). see the review paper by Huebener (1995). In all these cases, flux
flow induces a voltage drop and can be suppressed by pinning.
(v) Pinned flux lines exert a force on the atomic lattice which leads to magnetomechanical effects, e.g. to ‘suprastriction’ (Kronmiiller 1970) or ‘giant magnetostriction’
(Ikuta er al 1993, 1994) and to a change of the velocity and damping of ultrasound
(Shapira and Neuringer 1967, Forgan and Cough 1968, Liu and Narahari Achar 1990,
Pankert er al 1990, Lemmens et al 1991, Gutlyanskii 1994, Zavaritskii 1993, Dominguez
er a1 1995). The pinning force can enhance the frequency of flexural or tilt vibrations
of superconductors while depinning will attenuate these vibrations, see section 10 and the
reviews by Esquinazi (1991) and Ziese et al (1994a). Another mechanical effect of pinning
is a strong ‘internal friction’ which enables superconductors to levitate above (or below) an
asymmetric permanent magnet motionless, without oscillation or rotation, figure 12, since
part of the magnetic field lines is pinned inside the specimen in form of flux lines (Moon
et al 1988, Peters et al 1988, Huang er al 1988, Brandt 1988c, 1989c, 1990a, Kitaguchi
er al 1989, Murakami et al 1990). Levitation forces were calculated by various authors
(Nemoshkalenko er 4l 1990, 1992, Yang et al 1992, Huang er al 1993, Barowski et al
1993, Chan et al 1994, Xu et al 1995, Haley 1995, Coffer 1995).
+
8.2. Theories of pinning
Theories of pinning deal with several different groups of problems.
8.2,I . Models for the nmgnetization. The distribution of flux and current inside the
superconductor and the (more easily measurable) total magnetic moment or nonlinear
a s . response were calculated for various models of the reversible magnetization curve
The flux-line lattice in superconductors
-
1523
Figure 12. Various c o n f i g d o n s of superconducting levitation. Clockwise from tap left.
The levitation of a type-l superconductor above a magnet usually is unstable. A type11
superconductor can levitate above, and be suspended below, a magnet in a stable position
and orientation since part of the magnetic Rux penetrates in form of flux lines which are pinned.
A ring magnet magnetized along its axis, exhibits WOpoints on its axiC where the magnetic
field is zero Any diamagnet or superconductor is attracted to these points and. if light enough,
can levitate there (Kitaguchi et a1 1989). Levitation of a magnet above a Rat typeII (type-I)
superconductor is stable (unstable). The levitation above an appropriately curved 'watch glass'
type-I superconductor can be stable.
-=
E(&), e.g. the model B = 0 for H,
H,,,B = po(H, - cH,I) for H, > H,I
with c zz (Clem and Hao 1993, Indenbom er al 1995), or models of the irreversible
magnetization or field dependence of the critical current, e.g. the Kim-Bean model
& ( E ) = Jc(0)/(l+IBI/B~)'~2
(Kim etal 1963, Cooley and Grishin 1995) or the AndersonKim model & ( E ) = Jc(0)/(l
IBI/Bo) (Anderson and Kim 1964. Lee and Kao 1995)
or other models (Yamafuji el uf 1988, Chen and Goldfarb 1989, Bhagwat and Chaddah
1992, Yamamoto et a1 1993, Gugan and Stoppard 1993, Gamory and T f i s 1993, Shatz
et a1 1993). In addition, these results depend on the shape of the specimen and on the
orientation of the homogeneous applied field, cf section 8.4. They are further modified by
the assumptions of inhomogeneous or anisotropic pinning and of a surface barrier (Clem
1979) (section 8.5).
4
+
8.2.2. Granular superconductors. Various effects are responsible for the low J, of
polycrystalline HTSCs (Mannhardt and Tsuei 1989). The granular structure of ceramic HTSCS
was modelled as a percolative network of inter-grain Josephson junctions (weak links)
(Deutscher et al 1974, Clem et d 1987, Clem 1988, Peterson and Ekin 1988, Trnkham
and Lobb 1989). The almost constant & ( B ) at very low fields, and the rapid decrease
at fields between 10G and 30G, have been well fitted (Kugel et a2 1991, Schuster et al
1992b) by assuming that the current through each junction is defined by its own Fraunhofer
diffraction pattern whose oscillations are smeared out after averaging the junction lengths
and orientations. Pereyra and Kunold (1995) observe quantum interference in the critical
current of polycrystalline YBCO. In a nice work Schindler et al (1992) measured a single
grain boundary in DyBalCu30x by microcontacts and find that it acts as a Josephson
junction. Dong and Kwok (1993) observe J,(H) oscillations in bitextured YBCO films
caused by the weak-link character of 45" grain boundaries. Amrein et al (1995) measure
the orientation dependence of .Ic across the grain boundary in a BSCCO bicrystal and find
a similar exponential decrease (for angles > 45") as Dimos et al (1988) saw in an Y3CO
bicrystal.
,
1524
I
E H Brandt
Fisher et al (1990) and Gilchrist (1990a,b) explain critical current and magnetic
hysteresis of granular YBCO as due to Josephson junctions with randomly distributed
vortices and with a Bean-Livingston surface banier, see also Kugel and Rakhmanov
(1992). Nikolsky (1989) discusses the possibility that the intergrain junctions arc nontunnelling but of metallic kind, at which Andreev reflection of the quasipaicles occurs.
DobrosavljeviE-Grujit and RadoviC (1993) and ProkiC et al (1995) calculate critical currents
of superconductor-normal metal-superconductor junctions; see Deutscher and DeGennes
(1969) and Deutscher (1991) for reviews on the related proximity effect and recent
experiments on the proximity effect in HTSC edge-junctions (Antognazza et al 1995), in
Nb/insulator/Nbjunctions (Camerlingo eta/ 1995) and in Cu- and Ag- wiresby Courtois et
a / (1995). The role of grain boundaries in the high-field high-current superconductors NbTi
and Nb3Sn was comprehensively studied by Dew-Hughes (1987). Evetts and Glowacki
(1988) consider the flux trapped in the grains of ceramic YBCO. Inter- and intra-grain
critical currents in KTSCS were measured by Kupfer et al (1988) and Zhukov et al (1992).
The competition between the demagnetization factors of the sample and of the grains was
investigated by Senoussi et al (1990) and Yaron er al (1992). Schuster er al (1992b)
investigate H,,(T) and J J T ) of YBCO with various grain sizes. Isaac etal (1995) measure
relaxation of the persistent current and energy barrier U ( j ) (section 9.7) in a ring of grainaligned YBCO. For further work on ceramic HTSCs see, e.g. Dersch and Blatter (1988),
Rhyner and Blatter (1989), Stucki et al (1991) (low-field critical state), Muller (1989,
1990, 1991), Murakami (1990). Gilchrist and Konczykowski (1990). Senoussi etal (1991),
Halbritter (1992, 1993), Kugel and Rakhmanov (1992), Jung er a! (1993, 1994), Castro
and Rinderer 1994, Lee and Kao 1995, and the review on magnetic a s . susceptibility by
Goldfarb et a1 (1991).
8.2.3. Elementary pinning forces. Most pinning forces act on the vortex core and
thus typically vary spatially over the coherence length 6. Pinning may be caused by
spatial fluctuations of Tc(r)(Larkin 1970) or of the mean free-path of the electrons 1(r),
corresponding to fluctuations of the prefactors of 1$12 or of l(-iV/K -A)$[' in the GL free
energy (2.l), respectively. Griessen et a1 (1994) find evidence that in YBCO films pinning
is caused mainly by fluctuations of the mean free path. The elementary pinning force
between a flux line and a given material defect was estimated from GL theory by numerous
authors (e.g. Seeger and Kronmuller 1968, Labusch 1968, Kammerer 1969, Kronmiiller
and Riedel 1970, Schneider and Kronmiiller 1976, Fihnle and Kronmiiller 1978, F h l e
1977, 1979, 1981, Shehata 1981, Ovchinnikov 1980, 1982, 1983), see also the reviews
by Kronmiiller (1974) and Gamer (1978). Trapping of flux lines at cylindrical holes of
radius rp was calculated by Mkrtchyan and Schmidt (1971), Buzdin (1993), Buzdin and
Feinberg (1994b) (pointing out the analogy with an electrostatic problem when h + CO),
and Khalfin and Shapiro (1993); cylindrical holes can pin multiquanta vortices with the
number of flux quanta n , up to a saturation number n, GZ. r p / g ( T ) ;when more flux lines
are pinned (n, > np), the hole acts as a repulsive centre, see also Cooley and Grishin (1995).
Anisotropic pinning by a network of planar defects is calculated and discussed in detail by
Gurevich and Cooley (1994). From microscopic BCS theory pinning forces were calculated
by Thuneberg et al (1984) and Thuneberg (1984, 1986, 1989), see also Muzikar (1991).
The most comprehensive and correct discussion of the microscopic theory of pinning forces
is given by Thuneberg (1989), who also points out that some previous estimates of pinning
forces from GL theory were wrong since they used reduced units, which in inhomogeneous
superconductors should vary spatially. The main result from microscopic theory is that
the flux-line lattice in superconductors
1525
the pinning energy of a small pin of diameter D << t o (the coherence length at T = 0)
is enhanced by a factor :o/D >> 1 (or / I D for impure superconductors if D < l << CO)
with respect to the GL estimate U p % p0H:D3 due to the ‘shadow effect’ caused by
the scattered Cooper pairs. KuliC and Rys (1989), Blatter et al (1991~). Marchetti and
Vinokur (1995) and Larkin et al (1995) calculate pinning by parallel twin planes, which
occur in YBCO crystals, and Svensmark and Falikov (1990) consider equilibrium flux-line
configurations in materials with twin boundaries. Shapoval (1995) considers the possibility
of localized superconductivity at a twin boundary due to the modification of the electron
spectrum.
Pinning by twin boundaries in YBCO crystals was measured, e.g. by Swartzendruber
etal (1990), see also Roitburd et al (1990). Flux lines pinned near a saw-tooth twin were
observed in decoration experiments by Gammel et a/ (1992b), and the flux pileup at a twin
plane was seen magneto-optically by Vlasko-Vlasov et a/ (1994a) and Welp et al (1994).
Oussena et al (1995) observe that if bulk pinning is strong, flux lines can move along the
twin planes (channelling), but only if the angle between pins and flux lines is less than a
lock-in angle of 3.5”. Random arrays of point pins have been studied in detail, e.g. by
van Dover (1990). The elastic response of the pinned FLL (the Labusch parameter a‘) has
been measured by vibrating reed techniques (Esquinazi et al 1986, Gupta et d 1993, Kober
et al 1991b, Gupta et a/ 1993, Ziese and Esquinazi 1995, Ziese et a/ 1994a) and torque
magnetometry (Fame11 e t a / 1990). Doyle et a/ (1993, 1995), Tomlinson et al (1993), and
Seow eta! (1995) measure the linear and nonlinear force4isplacement response of pinned
flux lines by a directed transport current, which gives more information than the circulating
currents flowing in the usual magnetic measurements.
8.2.4. Attractive and repulsive pins. Typical pins athact flux lines. For example, the
maximum possible (negative) pinning energy which a cylindrical normal inclusion or hole
with radius rp z t can exert on a parallel flux-line core is per unit length Up,,,,- .%
($~/4ap0h~)[ln(r,/<) 0.51 (cf section 4.5 and Brandt 1980a, 1992c); from this energy
the maximum pinning force is estimated as fp,,
% Up,,,,&.
A normal cylinder or hole
attracts flux lines since a flux line saves core energy when it is centred at the defect.
However, at high inductions close to ECz the same cylinder repels the flux lines. This
is so since near the surface of a normal inclusion or hole the order parameter l*I2 is
enhanced (Ovchinnikov 1980) by the same reason which also leads to the surface sheath
of thickness % f at a planar surface in the field range Ec2 E
Ec3 (SaintJames and
DeGennes 1963, DeGennes 1966), see also section 1.3. The (strongly overlapping) cores
of the flux lines h y to avoid the region of enhanced 1$1’ around the pin similarly as they
are repelled from the surface sheath. Another type of repulsive pins are inclusions of a
material with higher T, than the mahix; these enhance the order parameter locally (Brandt
and Essmann 1995). Such inclusions of various shapes have been considered within GL
thecry by Brandt (1975), and within BCS theory by Thuneberg (1986), see also Zwicknagl
may stretch along B to far
and Wilkins (1984). Remarkably, the region of enhanced
outside the pin. For example, a small spherical inclusion of radius < 6 above Be’ creates
a cigar-shaped superconducting nucleus of radius = 5 and length % t / ( E / E , z
I)”’,
which diverges as E approaches Ecz from above (Brandt 1975). Below Ecz this nucleus
qualitatively survives and repels the surrounding flux lines. A repulsive pinning force is also
exerted by a pin that has trapped a flux l i e which now repels other flux lines magnetically.
Repulsive pins typically are less effective in pinning the FLL than are attractive pins, since
the flux lines may Bow around such pins. This effect was seen in early simulations of flux-
+
< <
-
E H Brandl
1526
line pinning (Brandt 1983%b). However, if the pins are very dense such that their distance
becomes less than t vthe notions repulsive and attractive lose their sense and the pinning
potential becomes a 2D random function in each plane perpendicular to the flux lines.
8.2.5. Statistical summation of pinning forces. Numerous theories try to sum the random
pinning forces that act on the flux lines, e.g. the theory of collective pinning discussed
in section 8.7. Early summation theories should be understood as just parametrizing the
measured critical current density J,(B, T ) , since the assumptions of a theory remain either
unclear or are not fulfilled in experiments. For example, the scaling law by Kramer (1973).
J,(B, T ) bP(1 - b)q where b = E/E&)
and p and q are constants depending on
the pinning mechanism, typically fits data very well. For p = 1 or and q = 2 the
Kramer law follows from the assumption that the FLL flows plastically around large pins.
see also Wordenweber (1992); therefore, a softer FLL yields weaker pinning and one ha5
J, c66(BI T) when the flow stress of the E L is proportional to its shear modulus c66 (4.4).
As opposcd to this idea, the statistical theories of weak pinning yield stronger pinning for
softer FLL, e.g. Jc l/c66, section 8.7.
N
-
i
-
8.2.6. Thermally activated depinning. After the discovery of HTscs, the collective pinning
theory, which applies at zero temperature, was extended to include thermally activated
depinning, section 9. Various phase diagrams of the FU in the B-T plane were suggested
by combining this collective creep theory (section 9.4), or the vortex-glass (section 9.5) and
Bose-glass (section 9.2) pictures, with the melting criteria and with the phase transitions of
layered superconductors discussed in section 6, see, e.g. the general article by Bishop er n/
(1992) and the comprehensive review by Blatter etal (1994a). Transitions from 3D pinning
of vortex lines to ZD pinning of point vortices in BSCCO were observed by many authors.
e.g. by Gray er a1 (1992), Almasan et al (1992), Marcon et al (1992, 1994) and Sarti er a/
(1994). see also section 6.4.
8.2.7. Anisotropic pinning. The critical current density Jc in layered HTSCS is highly
anisotropic (Barone et a1 1990, Pokrovsky et a1 1992, Blatter and Geshkenbein 1993.
Balents and Nelson 1994) since flux lines are pinned much more strongly when they are
parallel to the Cu-0 layers (Tachiki and Takahashi 1989, Tachiki etal 1994, Schimmele rta/
1988, Senoussi etal 1993, Nieber and Kronmiiller 1993a, Koshelev and Vinokur 1995). In
YBCO planar twin-boundaries cause anisotropic pinning (Kwok eta1 1990, Liu et al 1991,
Crabtree er a1 1991, Hergt et a/ 1991, Fleshler et al 1993. Vlasko-Vlasov et al 1994a).
Anisotropic pinning by linear defects (e.g. Crabtree et al 1994) is discussed in section 9.2.
An anisotropic depinning length was observed by Doyle etal (1993). Nice angle-dependent
J,(O) with sharp peaks were measured in HTSCS by Roas et a1 (1990b). Schmitt er a1
(1991). Kwok et al (1991), Labdi er a/ (1992). Schalk e t a [ (1992), Theuss el al (1994).
Kraus et al (1994b) (figure 13). and Solovjov et al (1994), and in NbMb2.r multilayers by
Noshima et a1 (1993). The anomaly in the resistivity observed by Kwok et al (1991) for B
nearly parallel to the a-b plane, was recently explained by Koshelev and Vinokur (1995).
Anisotropic Jc was also observed by magneto-optics by Schuster et a1 (1993a). A similar
angular dependence of pinning and flux flow in thin type4 superconducting Sn films was
investigated by Deltour and Tinkham (1966, 1967), see also Tinkham (1963). Thin films
of type-I superconductors behave like type-U superconductors, e.g. a Pb film of thickness
< 0.16pm (Dolan 1974). Gurevich (1990a. 1992b) shows that uniform current flow in
HTSCS with anisotropic J, may become unstable and that current domain walls and macro
The pur-line lanice in superconductors
.
5
"EU
YBa2CW0r-a
1.0
1527
BizSrzCaCu*Os+.
0.8
0
v
4
.s' 0.6
.-3
.A
5
-0
0.4
Y
c
2!
L
= 0.2
-m
U
U
'j
.-
U
0
0
90
180
270 360 0
90
field orientation 0 (")
180
270
360
Figure 13. The critical cumnt densily I,. as a function of the angle 0 between the applied field
and the e ais. ( a ) YBCO film irradiated by 340 MeV Xe ions at an angle Oe = 60' until a
density of linea defects was reached that corresponds to a magnetic induction of Bq = 1.7T if
there is one flux line per defect. ( b ) BSCCO film irradiated by 0.9GeV pb ions with QQ = 30'
and Bq = 2. I T. Note the sharp maximum of J, when B I c (intrinsic pinning by the ab planes). and in YBCO also when B is along the linear defeas caused by the irradiaion. In
BSCCO lhir latter m i m u m is absent. indicating that point vorrices are pinned independently
at T = 40 K and B = 1 T (after b u s ef al 1994a).
vortex patterns may occur. A similar macroturbulence was observed by Vlasko-Vlasov et
a1 (1994b). which may also be related to the 'current string' observed and explained as a
consequence of finite H,,by Indenbom et nl (1995).
8.2.8. Periodic pinning struczures. Besides layered superconductors such as NbSez
(Monceau et al 1975) and HTSCS (Ivlev et nl 1991a, Levitov 1991, Watson and Canright
1993), artificial periodic pinning structures have been investigated extensively, e.g. films
with spatially modulated composition (Raffy er a1 1972, 1974), films with periodically
varying thickness (Martinoli 1978, Martinoli et a1 1978). NbTi-alloy wires containing a
hexagonal may of cylindrical pins (Cooley et nl 1994), and a lattice of holes (Fiory et
al 1978, Baert et al 1995, Metlushko et al 1994a, Bezryadin and Pannetier 1995). For
recent reviews see Lykov (1993) and Moshchalkov er al (1994b). Very recently Cooley
and Grishin (1995) showed that in the critical state in superconductors with periodic pins,
the flux density forms a terrace structure, causing stratification of the transport current into
the terrace edges where the flux-density gradient is large; the magnetization thus changes
abruptly at rational or periodic fields when a new terrace appears inside the superconductor.
This effect might explain the periodic magnetization observed by Metlushko et nl (1994a)
and Cooley el a/ (1994), and the rational fractions observed in YBCO crystals with field
parallel to the Cu-0 planes (Raffy et a1 1972, 1974) and in multilayer films in parallel
field (Brongersma et a1 1993, Hiinneckes et a1 1994). A further interesting behaviour is the
exhibition of micro-networks of Josephson junctions (Theran er al 1993, 1994, Korshunov
1994) and of superconducting wires (Fink e t a f 1985, 1988, 1991, Moshchalkov eta! 1993,
Itzler et al 1994), which may contain vortices as seen by decoration (Runge and Pannetier
1993) and which exhibit a resistive transition at a critical current density (Giroud et nl 1992,
Fink and Haley 1991), see Pannetier (1991) for a review.
E H Brandt
1528
8.2.9. Conductorsandinstubilities.
Some theories or models are more application oriented.
For example, Campbell (1982) and Khalmakhelidze and Mints (1991) calculate the losses
in multifilamentary superconducting wires and Wpf (1967, 1991). Mints and Rakhmanov
(1981) and Gurevich and Mints (1987) consider the thermal stability of the critical state in
such wires. The limits to the critical current in BSCCO tapes were estimated by Bulaevskii
et a1 (1992a) within the ‘brick-wall model’, which assumes that the current flows i n the
a-b planes until a structural defect interrupts this path and forces the current to flow in the
c direction into the next ‘brick‘; the current is thus limited by the conductivity between the
layers, i.e. .by ideal c axis grain boundaries. A bktter description of BSCCO tapes is the
‘railway switch model’ by Hensel et al (1993, 1995), which considers realistic small-angle
c axis grain boundaries as current limiters. A further theory of inter- and intra-cell weak
links in BSCCO is given by Iga et a1 (1995). see also the magneto-optical imaging of flux
penetration and current flow in silver-sheathed BPbSCCO by Pashitski eta1 (1995) and the
special impurity phases of Ullrich et a1 (1993).
8.3. Bean’s critical state model in longitudinal geometry
The critical state of a pinned FLL is approximately described by the ‘Bean model’, which
in its original version (Bean 1964, 1970) assumes a B-independent critical current density
J, and disregards demagnetizing effects that become important in flat superconductors in
a perpendicular magnetic field. It further assumes zero reversible magnetization, namely,
B(H,) = &offu and zero lower critical field H,, = 0. When finite H,, is considered,
the penetrating flux front performs a vertical jump to E = 0 (Campbell and Evetts 1972,
Ullmaier 1975, Clem and Hao 1993, Indenbom et a1 1995). In flat specimens of finite
thickness the consideration of a finite HC,leads to the appearance of a ‘current string’
in the centre of the specimen as predicted and observed by Indenbom et a1 (1995). The
original Bean model further disregards the nonlocal relationship between the flux density
B ( r ) and the flux-line density n ( r ) FZ B(T)/@o(Brandt 1991~);this nonlocality originates
from the finite penetration depth h and inwoduces discontinuities in the flux-line density in
the Bean critical state as shown by Voloshin et a1 (1994) and Fisher et a1 (1995).
For long cylinders or slabs in longitudinal applied field & ( t ) along z , the Bean model
predicts that in regions where magnetic flux has penetrated, the flux density B ( x , y, t ) is
piecewise linear with slope lVB[ = poJ,, and in the central flux-free zone one has B = 0
and J = 0, figure 14. The detailed pattern of B depends on the magnetic history. Explicitly,
for a, slab of thickness 2a (-a x a), when H. is increased from zero, the Bean model
yields J ( x ) = 0, H ( x ) = B(x)/po = Ofor 1x1 b, and J ( x ) = J,x/lxl, H ( x ) = J,(lxl-b)
forb 1x1 a. The negative magnetic moment M per unit area of the slab is for H, H,
< <
< <
<
<
and for H, 2 H, it saturates to M = J,a2. Here Hc = J,a is the field at which full
penetration is achieved and a - b = aH,/H, = H J J , is the penetration depth of the flux
front. Interestingly, the longitudinal currents and the transverse currents of the U-turn at
the far-away ends of the slab give identical contributions to M , thus cancelling the factor
in the general definition of M. Note also that in this longitudinal geometry the magnetic
moment coincides with the local magnetization H ( x ) - Ha integrated over the sample
cross section. The correctness of (8.2) is easily checked by considering the Meissner state
(H,,
<< Hc,b + a , ideal shielding) in which M = Mideal = %Ha equals the specimen
width (volume per unit area) times the expelled field. The deviation of M from the ideal
4
the flux-line lattice in superconductors
Meissner value initially is M - 2aH.
(b = 0) and thus M = M,, = a2J,.
-
H,". For 1H.I
1529
> H, one has complete penetration
Figure 14. Bean model in a cylinder or slab in a parallel magnetic field H.. Top left, Flux lines
pinned by point pins, ihe LorenQ force of density BJc acts to the right (schematic). Top righr.
Negative magnetizarion against applied field. The virgin curve, from equation (8.2). saturates
when full penetration is reached at Ha = H,. The hysteresis loop follows from equation (8.3).
Middle. Profiles of the penetrating (leff) and leaving (righf) flux. The numben correspond to
the numbers in the magnetization curve. Bollom. The current density during penetration and
exit of flux and penetration of reverse flux.
When Ha is cycled with amplitude 0 c Ho < Hc = &a, the flux profile is as shown in
figure 11. The magnetic moment for Ha decreasing from +Ho to -Ho is then
M i ( % , Ho, J c ) = ( H , Z + 4 H , H c - 2 H , H o - H i ) / 2 J c .
(8.3)
The corresponding branch M + for H, increasing from -Ha to +Ho follows from symmetry,
M?(H,, Ho, J,) = - M $ ( - H a , Ho. J,). The dissipated power P ( H 0 ) per unit area of the
slab is the frequency v times the area of the hysteresis loop, P = vpo $ M dH,. Introducing
a function f ( x ) with f ( x < 1) = fx', f ( x I ) = x - ;, one gets for all amplitudes Ho
the dissipation P(Ho) = 4 a v p o H : f ( H o / H c ) . Exact solutions of the Bean model for the
penetration of a rotating field are given by Gilchrist (1972, 1990a, 1994b).
In general, the flux penetration obeys a nonlinear diffusion equation, which for slabs or
a half space has been solved numerically (van der Beek and Kes 1991b, van der Beek et
al 1992b, Rhyner 1993, Bryksin and Dorogovtsev 1993, Calzona eta1 1993, Gilchrist and
van der Beek 1994). Defining h ( x , y , t ) = H ( x . y , t ) - H , ( t ) one has for specimens with
general cross section in parallel field the boundary condition h = 0 at the surface and the
equation of motion (Brandt 199%) (see also (8.18) below)
>
i ( x , y, f) = V . (DVhJ - & ( t )
(8.4)
where D = E ( J ) / p o J is a nonlinearfilu difusivity with E ( J ) the current-voltage law of
the superconductor. This diffusivity in general depends on space and time.
1530
E H Brandt
Remarkably, even in the fully penetrated critical state where the current density
J = IVHl is practically constant, the electric,field E = p ( J ) J in general is not constant;
E is a linear function of space satisfying V x E = -w&
=constant. For a rectangular
specimen cross section both in longitudinal and transverse geometries) the magnitude
E exhibits a strange zig-zag profile which is independent of the detailed form of E ( J )
as long as'this is sufficiently nonlinear (sharply bent at J = J,) (Brandt 199%). In
particular, for a logarithmic activation energy U ( J ) = UoIn(Jz/J) (section 9.7) one gets
for E exp[-U(J)/k~T] the power-law E ( J ) = E , ( J / J J ' with n = Uo/ksT >> 1 and
E, = E(&). Assuming an integer odd 'exponent n , equation (8.4) becomes particularly
simple and reads for slabs and cylinders, respectively
-
h(x, t ) = c[ (h')"]' - & ( t )
h(r, t ) = (c/r)[r(h')"]'- &(t)
(8.4~)
where c = E,/J;po. The equations for the current density J = h' are even simpler but
do not contain the driving force H.(t), j ( x , t ) = c(J")", j ( r , t ) = c [ (rJn)'/r ]'. The
nonlinear diffusion equations (8.4) or (8.40) are easily integrated on a personal computer,
e.g. starting with H.(t) = h ( x , y , t ) = 0 and then increasing H n ( t ) . The numerical
integration of ( 8 . 4 ~ is
) facilitated noting that h(x, t) and h(r, t ) are even functions of x
or r which may be represented by Fourier series, and that one of the derivatives acts on an
even function and one on an odd function. In the limit n -f CO one obtains the Bean model;
finite n allows for flux creep. Rhyner (1993) studied such power-law superconductors in
detail and obtained scaling laws for the dependence of their ac. losses on frequency and
amplitude. Corrections to the critical state theory of the surface impedance were derived
by Fisher et al (1993) due to the finite resistivity and by Voloshin etal (1994) and Fisher
et al (1995) due to the nonlocal relation between the vortex density n , and the flux density
B.
The nonlinear diffusion equations (8.4) and (8.4a) are closely related to the concept
of 'self-organized criticality' (Bak et ai 1988, Jensen 1990, Pla and Non 1991, Ling et al
1991, Richardson erai 1994, DeGennes 1966) and to Gurevich's idea of the universality
of flux creep (section 9.8), or to problems with logarithmic time scale (Geshkenbein et al
1991, Vinokur et al 1991a, van der Beek et al 1993). From the universal electric field
during creep E ( r ,t ) = F ( r ) / t (Gurevich and Kiipfer 1993, Gurevich and Brandt 1994)
it follows that (8.4) is a linear da$Tusion equation in the logarithmic time T = I n t since
D E l/t. This universal behaviour is independent of the explicit forms of E ( J . B )
or J,(B) and of the specimen shape, and it applies even to transverse geometry where the
diffusion equation becomes nonlocal (section 9.8).
--
8.4. Extension to transverse geometry
In typical magnetization experiments the above longitudinal geometry is not realized.
Measurements are often performed on films or flat monocrystalline platelets in perpendicular
magnetic field in order to get a larger magnetization signal. In this perpendicular or
transveme geometry the original Bean model does not apply and has to be modified. The
critical state of circular disks was computed by Frankel (1979), D2umling and Jxbalestier
(1989), Conner and Malozemoff (1991), Theuss et al (1992a.b). Hochmut and Lorenz
(1994), and Knorpp et al (1994), see also the reviews on flux pinning by Huebener (1979)
and Senoussi (1992).
The problem of flux penetration in transverse geometry was recently solved analytically
for thin circular disks (Mikheenko and Kuzovlev 1993, Zhu etal 1993) and for thin long
strips (Brandt er al 1993a, b). For strips carrying a transport current in a magnetic field see
The flux-lineluffice in superconductors
1531
Brandt and Indenbom (1993) and Zeldov er al (1994a). These theories, like Bean's, still
assume constant J, and B,I = 0. They consider the sheet current J s ( x .y ) = I J ( x . y. z ) dz
(the current density integrated over the specimen thickness d ) and the perpendicular
component Hz of the magnetic field at the flat surface, related to the flux density BL by
B, = fi0Hz. For the ship with width 2a (-a x a) in the ideal Meissner state the sheet
current along y
< <
J:&d(x) = 2.xHa/(aZ - x * ) ' / ~
(8.5)
diverges at the ship edges in the limit d + 0. Inside the ship this sheet current exactly
compensates the constant applied field H.. Thus one has HZ= 0 for 1x1 < a, and outside
the strip one finds in the plane z = 0
H,'dw'(lxI
> a ) = IxlH./(x2
- a2p2.
(8.5~)
The negative magnetic moment per unit length of the ship M = J:=xJ.(x) dr in this ideal
shielding state is Mi&d = nn2H,. In general, the perpendicular field H,(x) generated in
the film plane z = 0 by the sheet current J s ( x ) follows from the London equation (2.4) by
integration over the film thickness d using the flux-line density n ( x ) = wH,/q50(Larkin
and Ovchinnikov 1971)
If a >> hfilm = 2hz/d applies, the gradient term in (8.6) may be disregarded and (8.6)
reduces to Ampire's law. Note that (8.6) is a nonlocal relationship, whereas for longitudinal
geometry (for d + CO) the local relation J ( x ) = H ' ( x ) holds.
Remarkably, both unknown functions J d x ) and H z ( x ) can be determined from one
equation (8.6) if one adds the Bean-condition [Js(x)l < J,d (constant bulk pinning). A
different such condition, which also allows for a unique solution, is the assumption of zero
bulk pinning but a finite edge barrier (section 8.5), which means I&(lxl = a)[ =
for thin films with d < A one may assume Jdse = Jod with JO the depairing current
(Maximov and Elistratov 1995). For films which exhibit both bulk pinning and an edge
barrier, solutions of (8.6) were obtained by Zeldov er al (1994b,c) and Khaykovich et al
(1994). The main effect of the disregarded gradient term ;hm,J'(x) in (8.6) (or of the finite
thickness d if d > A) is to remove the infinity of H z ( x ) at the edges x = &a (Ivanchenko
er al 1979). Flux penetration into thin circular disks with d << h is calculated by Fetter
(1980) and Borovitskaya and Genkin (1995).
When the sheet current is limited to the constant critical value J,d as in the Bean model,
then, when Ha is increased from zero the shielding current near the specimen edges saturates
to J,(x z b) = J,d and J,(x < -b) = -J,d since flux starts to penetrate in the form of
flux lines. One thus has H,(x) # 0 for 1x1 > b and Hz(x) 5 0 for -b < x < b < a. The
current distribution J ( x ) of this twdimensional problem with field-independent J, can be
obtained by conformal mapping as shown by Norris (1970) for the similar problem of a strip
with transport current. From the condition that J>,(x)is continuous, or H ( x ) finite inside
the specimen, one finds the flux-front position b = a/cosh(H,/H,) where H, = J,d/n is
a critical field. With the abbreviation c = (1 - bz/a2)1/2
= tanh(H,/H,) the sheet current
J,, the magnetic field HZ in the plane z = 0, the negative magnetic moment M , and the
penetrated flux @ = 2p0 J; H ( x ) d x for H,,increased from zero take the form (cf figure 15)
1532
E H Brundt
\\\\IJ,
1
0
I\\\\
!
,
-a
-a
0
a
X +
I
Figure 15. Modified Bean model for a superconducting strip in P perpendicular magnetic field
(section 8.4) which increases (top and middle) and then decreases (bottom). Top left, The
satumtion of the cunent density 3 to its maximum allowed value Jc near the rjght ship edge
x = a . Top middle. Cunent density J ( x ) (8.7) for fields Q / H C = 0.5. I , 1.5.2.2.5 with
Hc = J , d / x . Topright. Penetrationdeptha-b=a-a/cosh(H,/H,). Mi&k Perpendicular
magnetic field H ( x ) (8.8) for HJH, = 0.5.1.1.5.2,2.5, Bonom. Perpendicular magnetic
field H ( x ) (left) and current density J ( x ) (right) when the applied field is decreased E"
H./H, = 1.5 (upper curves) via 1. 0.5, 0 (dashed curves. remanent state), -0.5, -1 to -1.5
(lower curve). The corresponding curves for inereasing Ha are obtained by changing the signs
of H ( x ) and J ( x ) .
Hz(x)=
lo
- b*)'/*/c 1x11
H, tanh-'[clxl/(x2 - b2)'/2]
H, tanh-'[(x2
1x1 < b
b c 1x1 < U
1x1 > a
M(H,) = Jcdazc = Jcda2tanh(H,/H,)
(8.9)
@(Ha)= 2poHca In(a/b) = 2p,3Hca1ncosh(H,/Hc).
In the limit Jc -+
(8.8)
(8.10)
these equations reproduce the ideal-shielding results (8.5), (8.6),
@,deal = 0.
For weak penetration (Ha << H,) one gets
b = U - aH:/2H:,
M = naZH.(l - H,2/3H:), and the flux @ = poaH:/Hc.
For almost complete penetrution (Ha >> Hc) one gets b = 2aexp(-H,/Hc) << U ,
M = Jcduz[l - 2exp(-2H./Hc)I, Js(lxl < b) = (2JCd/n)sin-'(x/b), Hz(b < 1x1 c
4 2 ) = H,cosh-'Ix/bl, Hz(2b e 1x1 < 00) = Ha Hcln(la2/x2 - I[-'/'), and
@ = 2poa(H. - 0.69Hc).
Thus in transverse geometry full penetration is realized only at large fields Ha %
Hcln(4u/d) obtained from the condition b c d/2. This estimate is confirmed by
computations for finite thickness d (Fork1 and Kronmiiller 1994). Note also the vertical
slopes of HL(x)and J ( x ) at 1x1 = b and the logarithmic infinity of H,(x) at 1x1 -+ a. The
Mihd
00
= nu2H., and
+
The fludine lattice in superconductors
1533
analytical solution H,(x) (8.8) thus proves a vem’cal slope of the penetrating flux front as
opposed to the finite and constant slope in the original longitudinal Bean model.
In a disk of radius a, the sheet current J ( r ) is also given by the equation (8.7) for
the strip, but with x replaced by r and H, = Jc/x by Hc = JJ2 everywhere. Analytical
solutions for the perpendicular field H,(rY at the disk surface are not available, but the
negative magnetic moment M = x
J(r)dr of the disk is
I,“?
with
S ( X )= 2.x
[
COS-’
+-I
1
coshx
sinhlxl
coshzx
(8.11)
and H, = JJ2. For Ha < H, (ideal shielding) one hai M = 8H,a3/3 and for H, >> H,
(full penetration) M = nJca3/3since S(0) = 1 and S(r > 1) = n/(4x). The magnetization
curves of strips, disk, and squares in the Bean model are very similar when normalized to
the same initial slope and same saturation value; the magnetization curve M(H.) of the
strip (8.9) then exceeds that of the disk (8.11) only by O.OlIM(w), and the magnetization
of the square (Brandt, unpublished) differs from the disk magnetization only by less than
O.OOZM(w) along the entire curve.
From these virgin solutions one may obtain the solutions for arbitrary magnetic history
Ha@). For example, if H&) is cycred between the values +Hoand -Ho,one has in the
half period with decreasing H.
MJ.(H,) = M(H0) - 2M(Ho/2
- Ha/2).
(8.12)
At H., = -Ho the original virgin state is reached again but with J , H, M , and 4
having changed sign. In the half period with increasing Ha one has J + ( y ,Ha, Jc) =
-J+(Y* -H<,
J c ) . H T ( Y ,H a , J c ) = -HL(Y,-Ha, J c ) . Mt(Ha,Jc ) -- -M+(-H L?I J c ) and
%(Ha, Jc)
= -@+(-Ha,
J,). When Ha oscillates with frequency U and amplitude Ho the dissipated
power per unit length of the strip is P = up0 $ M(H,) dH, (frequency times area of the
hysteresis loop) yielding
7
P(H0) = 4v~oaZJ,Xo[(2/h)Incoshh- tanhh]
(8.13)
with h = Ho/H,. At low amplitudes HO < H, this gives P = (2nupoa2/3H:) H
.: and
for large amplitudes HO> Hc (full penetration) one has P = 4up0a~J,(Ho- 1.386Hc).
The energy loss is thus initially very small, P H;.
The nonlinear ax. susceptibility for disks is obtained from (8.1 1) and (8.12) by Clem
and Sanchez (1994). Magnetization curves of disks of varying thickness are measured by
Oussena et a1 (1994a). Irreversible magnetization curves and nonlinear ac.susceptibilities
of the pinned Fu.for various geometries are compared by Bhagwat and Chaddah (1992).
Stoppard and Gugan (1993, and in a particularly nice short overview by Gilchrist (1994a).
Gilchrist and Dombre (1994) calculate the nonlinear ax. response and harmonics of a slab
with power-law resistivity, see also Rhyner (1993). Gugan (1994) presents a model which
extends the superposition (8.12) to general induction-dependent J,(B). see also the ideas on
critical currents in disks and films by Caplin et a1 (1992), Atsarkin eta1 (1994), and Porter
eta1 (1994). Rogacki et a1 (1995) confirm the magnetization curve (8.9) by the vibrating
reed technique. The field profile (8.8) is confirmed, e.g. by magneto-optics (Schuster et a1
1994d). by Hall probes (Zeldov et a1 1994b), and by an eleckon-spin resonance microprobe
(Khasanov et al 1995), see also section 1.2. Interesting extensions of the magnetostatics of
thin superconducting strips are given by Zeldov et al (1994b) ( i u e n c e of edge barrier)
and Koshelev and Larkin (1995) (explanation of the ‘paramagnetic Meissner effect’).
-
E H Brandt
1534
8.5. Surjiuce and edge barriers for pia penetration
-
Interestingly, all low-field results contain one power of HO or H, more in perpendicular
geometry than in longitudinal geometry, namely, the hysteresis losses are P Hi (- Hi),
the initial magnetization change is M - M,ad H,”(- H:), and the penetration depth is
a -b If,’ (- If,). This means that perpendicular flux penetrates into flat superconductors
with bulk pinning delayed as if there were a surface barrier. with penetration depth
a - b If:. This pinning-caused surface barrier in lransverse field (or better, edge barrier)
was investigated magneto-optically by Indenbom et nl (1993). The flux penetration over
an artificial surface barrier in single crystal BSCCO platelets which were irradiated near
the edges to enhance pinning, was observed magneto-optically by Schuster et nl (1994b)
and with an array of Hall sensors by Khaykovich et al (1994), and in both cases it was
successfully compared with theory.
-
-
Flgure 16. Visualiwion of the edge banier for flux penetration into a thin type-11
supcrconducting strip or disk with rectangular cmss secfion (left insert). Plotted is the hysteresis
loop of the negative mametimion. The points A, B, and C on this curve correspond to the
patterns of penetrating flux lies shown in the (right insen). At point B the edge barrier is
overcome e d y and flux penetrates directly to Ihe middle of the superconductor (chain line).
A similar surface or edge barrier, which does not require pinning, is related to the
rectangular (or in general, non-elliptic) cross section of flat superconductors in transverse
fields (Grover et al 1990, 1992, Indenbom et al 1994a,b,d, Indenbom and Brandt 1994,
Zeldov et al 1994b,c, Doyle and Labusch 1995, Benkraouda and Clem 1996), figure 16.
Solutions of the London theory for a vortex penetrating into thin films of constant thickness
d were given for strips by Larkin and OvchiMikov (1971), Likharev (1971, 1979), and
Kuprianov and Likharev (1974), for disks by Fetter (1980). and for an infinite film by
Ivanchenko et al (1979), see also Kogan (1994). A similar edge-shape barrier delays flux
penetration also with type4 superconductors, which do not contain flux lines but domains of
normal and superconducting phase; the magnetic irreversibility caused by thii edge-barrier
is investigated in detail by Provost et ol (1974). Fortini and Paumier (l976), Fortini et a1
(1980). and was rediscovered by Grover et al (1990, 1992). Superconductors of ellipsoidal
shape do not exhibit this type of surface barrier and have thus been used to determine the
lower critical field Hcl (L.iang er al 1994). Exact analytical solution for the current and
field distribution in the limit of large width a >>
= 2A2/d were obtained for thin long
strips with a surface barrier for homogeneous (or zero) pinning by Zeldov et al (19944 c)
and Maksimov and Elistratov (1995). and for inhomogeneous pinning by Khaykovich et al
(1994).
These important and often overlooked geometric surface barriers should be compared
with the microscopic surface banier’of Bean and Livingston (1964) for the penetration of
straight flux lines through a perfectly flat parallel surface. The Bean-Livingston barrier
The flw(-line lanice in superconductors
-
1535
-
results from the superposition of the attractive image force KlY2x/A) and the repulsive
force H,exp(-x/A) exerted by the Meissner current on a flux line at a distance x from
the surface. The resulting potential has a maximum at a distance = (avortex-core radius)
from the planar surface. In HTSCS,flux lines or point vortices can overcome this microscopic
barrier by thermal activation, section 9, which reduces the field of first penetration (Buzdin
and Feinberg 1992, Burlachkov 1993, Koshelev 1992b, 1994% Mints and Snapiro 1993,
Kogan 1994, Burlachkov et al 1994). In contrast, the geometric barrier cannot be overcome
by thermal activation and is not sensitive to surface roughness. Extending previous work by
Temovskil and Shehata (1972) and Clem (1974a), Burlachkov (1993) very carefully analyses
the thermally activated nucleation of half circular flux-line loops at a planar surface and the
penetration of flux lines into a superconductor which already contains flux lines. Burlachkov
and Vmokur (1994) apply these ideas to thin strips with transport current and find the selfconsistent current distribution between flux entry and flux exit over a barrier at the two
edges. An observed asymmetric current-voltage curve in films in parallel field is explained
by a Bean-Livingston surface barrier and inhomogeneous bulk pinning (Jiang et a1 1994).
The onset of vortices at the surface of strips and wires with cross section << A' is calculated
from timedependent GL theory by Aranson et a1 (1995), and the nucleation and instability
of vortex helices at the surface of a cylindrical wire by Genenko (1994, 1995).
At least seven different surface or edge barriers for the penetration of transverse flux
into thin superconductors may be distinguished:
(i) The Bean-Livingston barrier.
(ii) Clem's Gibbs free energy barrier for penetration of flux tabes or flux lines into
superconducting strips of elliptical cross section (Clem et a1 1973, Buck et a1 1981).
(iii) Likharev's (1971) barrier for the penetration of a perpendicular vortex into a thin
film of width 2a and thickness d << A. The barrier width is w Afilm = U2/d and the barrier
height is of the order of HCl(A/d)*(Indenbom unpublished). Depending on the parameter
y = da/(2nA2) = a/(?rAfilm),Likharev (1971) finds that vortex penetration over the batrier
becomes energetically favourable at Ha HA,where MHA = ($0y/4a2)In(Az/td) =
($0/4naAfi1,)1n(hfi1~/~)
for y >> 1 and @LOHA
= ( h / 2 . a 2 ) I n ( a / 0 for Y << 1.
(iv) The edge shape barrier caused by the rectangular edges and by the constant thickness
of the sample as compared to the ellipsoid (Provost et a1 1974, Indenbom et al 1994%b, d,
Doyle and Labusch 1995, Labusch and Doyle 1995). This barrier can be reduced by
rounding the sharp rectangular edges.
(v) A barrier caused in the Bean model by the consideration of the nonlocal relationship
between the flux density E(r) and the flux-line density n ( r ) (Fisher ef al 1995).
(vi) Surface pinning by the different structure or composition of the materia! near the
edge (see, e.g. Flippen et a1 1995).
(vii) The apparent barrier caused in the presence of bulk pinning with J,d = constant
by the slow initial increase of the flux penetration depth a - b H
.' (Brandt et a[ 1993),
see the discussion following equation (8.10).
Notice that the barriers (i)-(iv) delay only the penetration of flux but not the exit
of flux; the hysteresis loop of the irreversible magnetization thus becomes asymmetric.
This is so since these four pin-free surface barriers are essentially caused by the shielding
current, which flows along the edge in increasing field Ha, but which vanishes when in
decreasing field the magnetization goes through zero. Notice also that only (i) and (v)
are real snrface barriers while the other barriers are edgeshape barriers and depend on the
profile of the entire specimen. Interestingly, in type-I superconductors in the intermediate
state, which occurs for demagnetization factors N > 0, the positive wall energy of the
domains (e.g. lamellae) also enhances the field of flux penetration into an ellipsoid above
>
-
1536
E H Brandt
the value (1 - N ) B , which would hold for zero wall energy, see figure 3 and the discussion
preceding equation (1.1). This effect is largest in perpendicular geometry (1 - N << 1). but
in contrast to edge or surface barriers, it occurs also in ellipsoidal specimens where it yields
a reversible magnetization curve if domain-wall pinning is absent.
StejiC er a1 (1994) give detailed measurements and calculations of critical currents in
thin films in parallel and oblique field, see also Brongersma er a1 (1993) and Mawatari and
Yamafuji (1994). Sharp matching peaks in the attenuation of thin vibrating YBCO films
in parallel field were observed by Hiinneckes et al (1994). In SQUID magnetometers Sun
et a1 (1994) observe the threshold field caused by bulk pinning (section 8.4) and by the
edge barrier. Surface barriers in weak-pinning HTSC single crystals in perpendicular field
were seen in magnetization measurements, e.g. by Kopylov er a1 (1990). Chikumoto et
al (1992). Konczykowski er a1 (1991), Chen et al 1993a,b, and Xu et ai (1993), and by
magneto-optics as a surface step of B by Dorosinskii et a1 (1994). It is now believed that
the observed surface barrier is not the microscopic Bean-Livingston barrier but a geometric
barrier. Kugel and Rakhmanov (1992) and Sun et a1 (1995) study the flux penetration into
granular HTSCS accounting for the Bean-Livingston barrier of the grains.
Fhbrega er ai (19944 1995) measured the magnetic relaxation of Cebased HTSCs near
the first flux penetration, finding a strong dependence on the applied field, which indicates
thermal overcoming of surface or geometric barriers. Careful measurements in weak fields
parallel to the a 4 plane by Nakamura et a1 (1993) in BSCCO, Zuo el a1 (1994%b) in Ndand TI-based HTSCS, and Hussey et a1 (1994) in TI-based HTscs, yield abrupt jumps of the
magnetization and a large barrier for flux penetration, but nearly no barrier for flux exit;
Dauding et a1 (1994) measure different magnetization curves of YBCO platelets in parallel
and perpendicular field. These findings were explained, respectively, by a Bean-Livingston
barrier, by decoupling of the layers (section 6.3), by a lock-in transition (section 3.3.3). or by
material anisotropy, but they may also be caused by the edge-shape barrier (Indenbom and
Brandt 1994). In YBCO films of thickness d , Liu er a1 (1993) measured an irreversibility
field Birr
d’/’, see also the newer measurements and discussion by Neminsky er a/
(1994). As argued by Grover et al (1990, 1992). Indenbom et a1 (1994a,b,d), Zeldov er
al (1994b,c, 1995b) and Majer et a1 (1995). the measured ‘irreversibility line’ in HTSC in
transverse geomehy may well be due to the edge-shape barrier, or at least a section of this
line in the B-T plane.
-
8.6. Flux dynnmics in thin strips, dish, and rectangles
The flux profiles observed magneto-optically during penetration and exit of flux by Schuster
et al (1994b.d. 1995a) (see also Koblischka and Wijngaarden 1995) were quantitatively
explained by computer simulations based on a one-dimensional integral equation for the
sheet current J s ( r , t ) in thin strips (Brandt 19936, 1994a) and disks (Brandt 1994b).
The only material properties entering these computations are the reversible magnetization
(usually B = ~ Q Hand Hcl = 0 is assumed) and a nonlinear resistivity p ( J ) = E / J or sheet
resistivity ps = p / d = E / J , , which may depend on the position explicitly (e.g. because
of inhomogeneous pinning or varying thickness d ) or implicitly via B(r) and J S ( r ) . For
homogeneous pinning, the above quasistatic results (8.5) to (8.13) for strips and disks in a
transverse field Hu(t) are reproduced by inserting a sharply bent model resistivity, e.g. a
power law p (J/Jc)”-’ with tt >> 1 (Evetts and Glowacki 1988, Ries etal 1993, Rhyner
1993, Gilchrist and van der Beek 1994). and the desired history H,(t) with H.(O) = 0,
and then integrating over time (Brandt 1994d), figure 17. The same equation for the sheet
current can be used to compute flux creep in transverse geometry (Gurevich and Brandt
-
Theflux-line lanice in superconductors
1537
Figure 17. Flux penetration into a disk with radius a and current-voltage power-law E ( J ) =
( J / J c ) ' 9 E , when the applied field is increased from zero to Ho = 0.1,0.2.0.3,0.4.0.5,0.7, 1
in units of the maximum sheet current Ja = Jmd. This smooth (non-steplike) E ( J ) smears
the v d d slops of the sheet current J ( r ) d and field H ( r ) (both in units Jso) at the flux
fmnt as compared 10 (he Bean model. Top, Constant Jc = J,o. Mid& Field dependent
J J H ) = J d J ( O . 8 t IHI). Bonom. Engshaped disk with a hole of ndius 0 . k . for consmt
Jq = Jm, Flux penetration into the ring is complete at H. 0.6Jo.
>
1994) (section 9.8) and the linear response (section 9.6.2) of HTSCs in the TAFF regime
(where p is Ohmic) or in the vortex-glass regime with arbitrary complex and dispersive a.c.
resistivity p(w) (Kotzler er ai 1994b, Brandt 1994~).
This method has recently been extended to describe the time- and space-dependence of
the sheet current J,(x, y) in thin squares and rectangles with linear or nonlinear resistivity
in transverse magnetic field (Brandt 1995a,b). Superconducting (or nonlinearly conducting)
squares or rectangles during penetration or exit of flux exhibit sharp diagonal lines which
mark the positions where the sheet current bends sharply since it has to flow parallel to
the edges and has to keep constant modulus Js = IJsI = J,d. These discontinuity lines
(Campbell and Evetts 1972, Gyorgy et a1 1989, Zhukov and Moshchalkov 1991, Briill
er al 1991, Vlasko-Vlasov et al 1992, Schuster er al 1994a. b, c, Grant er al 1994, Xing
et a1 1994) are clearly seen as logarithmic infinities in the perpendicular field H,(x, y)
(figures 18-20). In the fully penetrated critical state with J = J, everywhere in a rectangle
of extension -a
x < 0 , -b
y 6 b, b a, the current stream lines are concentric
rectangles, and the BiotSavart law yields
<
<
>
H 2 ( X . y ) = H ~ + ; iJcd
;;[f(x,)~)+f(-X,y)+f(X,
f ( x , y ) = &In
&P + a
&Q - a
+
+b - x
+b - x
-Y)+f(-X*-Y)]
(8.14)
-y
(P + y - b)(y - b + a ) ( P + x - a ) x
+In)
-y
(y - b)(Q+ y - b + a ) ( x - a)(Q + x )
+ -
with P = [(a - x)'
(b - y ) z ] ' / zand Q = [ x z (b a - y)2]'/2, figure 18. Note the
star-like penetration pattern, the flux concentration in the middle of the sample edges, and
the logarithmic infinities near the diagonal discontinuity'lines (Iyl - 1x1 = b) and near the
1538
E H Brandl
Figure 18. Critical state of a rectangular typn superconductor film with side ratio b f a = 1.35
in a perpendicular field, calculated from equation (8.14). kJi.The l o d magnethation p ( x . y)
(top) and the current stream lines. which a
x lines E(%. y) = constant (bottom). Right. The
flux density or magnetic fied H(x. y ) , Note the sharp maxima of H at lhe edges and the sharp
minima of H at the discontinuity lines of the cumnt. The thicker line m k s the locus where
the self-field (8.16) of the circulating cumnu is zero and thus H b , y ) = Ha, On lhis 'neutral
line' H(x. y ) stays constant during flux creep.
edges (1x1 = a or IyI = b), cf (8.8) forb + 0.
The determination of the timbdependent sheet current J&, y . I ) and perpendicular field
component U&, y. t ) for thin planar conductors or superconductors of arbitrary shape
in a time dependent applied field i H , ( t ) is an intricate problem which can be solved
as follows. First, one has to express the sheet current by a scalar function g ( x , y ) as
Js(x. y) = -E x V g ( x . y) = V x i g ( x , y ) . This substitution guarantees that div J, = 0
and that the current flows along the specimen boundary if one puts g ( x , y) =constant = 0
there. For rectangular specimens g ( x , y ) is conveniently expressed as a 2D Fourier series
in which each term vanishes at the edges. The contour lines g ( x . y) = constant coincide
with the current stream lines. The physical meaning of g ( x , y ) is the local mgnefiuuion
or density of tiny current loops. %e integral of g ( x , y ) over the specimen area (or the
appropriately weighted sum over all Fourier coefficients) yields the total magnetic moment
m=
4s~
x J(r)d
Tx
Js(T)dZr
2
s
g(T)d2r.
(8.15)
Next one determines the integral kernel K ( T , T') (T = x , y ) whicb relates the perpendicular
field H&, y) in the specimen plane z = 0 to g(x', y') by (for H. = 0)
s
s
H,(T)= K ( T , T ' ) ~ ( T ' ) ~ % ' g(r) = K - ] ( T , T') H,(r')dzr'
(8.16)
the flux-line lattice in superconductors
I539
Figure 19. Penemtion of perpendicular magnetic flux into a rectangular film with side ralio
bla = 1.35,computedfromequations(8.18)and(8.16)usingagridof 18~24independentpoints
on a personal computer. "he film is characterized by B = pa H and a highly nonlinear currentvoltage curve, E ( J ) = ( J / J J m E , , (Ec = J, = I ) which models a type-II superconductor
at low temperatures. Depicted are the current stream l i e s (top row) and the lines of callstant
magnetic field H ( x . y ) (boltom row): the thicker line marks H(x. y ) = H.. The applied fields
am,f"left to right, Ha = 0.24.0.38.0.61 and 1.16 in units of &d. At H. = 1.16 Eiux
has penemei almost campletely. and a critical state is reached which loolrs as calculated from
(8.14) and depicted in figure 18.
where the integrals are over the specimen area. This is a nontrivizl problem, since when
one performs the limit of zero thickness in the Biot-Savart law, the kernel becomes highly
singular, K = -1/4nlr - #I3: this form of the kernel only can be used if r lies outside
the specimen: but inside the specimen area (where T = r' can occur) one has to perform
part of the integration analytically to obtain a well behaved kernel (Brandt 1992d, Schuster
et ai 199%). or numerically for a small but finite height z above the specimen (Roth et a1
1989, Xing et a1 1994). Explicitly one has
The inverse kernel K-' in (8.16) may be obtained by Fourier transform or by performing
the integration on a grid with positions Ti = ( x i , yi), weights w i , the tables Hi = H,(ri) and
gi = g ( r i ) , and the matrix Kij = K ( r i , rj) w,. The integrals (8.16) then are approximated
by the sums Hi =
Kijg, and gi = E.
K-'
H, where K;' is the inverse matrix of Ki,
I
GI
(Brandt 1994a. b, Xing et al 1994).
As the last step the equation of motion for g ( x , y . t ) is obtained from the (3D) induction
law V x E = -E (the dot denotes a p t ) and from the material laws B = poH and E = p J
valid inside the sample where J = J,/d = -2 x V g ( x , y)/d. In order to obtain an equation
of motion for the function g ( x , y , t). it is important to note that the required z-component
xi
E H Brandt
1540
Figure 20. Distribution of perpendicular magnetic flux in a square YBCO film (d = 800nm.
T = 20K) and in a rectangular YBCO single crystal (d = 80wm, T = 65K) viwalized
magneto-optically by a ferrimagnetic gamet indicator. For the squan the perpendicular field B.
is applied in the sequence ( a ) 54mT. ( b ) 92mT.(c) 151 mT and ( d ) -24mT. In (a)and ( b )
the flux penetration sfans in a star-like pmtem; in (e) full penemtion is almast reached and the
flux density exhibits sharp ridges along lhe diagonals of the s q m : in ( d ) the penetration of
reversed flux is Seen at the edges. For the rectangle E, is (e) 149 mT and (f) 213 mT. the black
line indicates the sample edge. At almost complete penetnuion (f). cumnt discontinuity lines
with the shape of a double fork BR seen as sharp negative ridges of the flux density (reproduced
coutesy of Th Schuster).
Bz = i B = -i (Vx E ) = -(.i.x V)E = -ai: aE/ay+a aE/ax does not depend on the
(unknown) derivative aE/az. With the sheet resistivity ps = p/d one may write inside the
sample E = p J = p,J, = - p s ( i x V)g and thus B, = (2 x V ) ( p , i x Vg) = V(p, Vg).
Using (8.16) with an applied field 2 H&) added, Bz = PO&
po J K ( r , 7’) b(r’)d’r’,
one obtains the equation of motion for g ( x , y . t ) in the form (Brandt 1995a. h)
+
j ( ~t ),=
/
K-’(T, r’)[f(r’,
t ) - fi&)]d*r’
with
f ( r ,f) = V(p, Vg)/po.
(8.18)
This general equation, which applies also when ps = p / d depends on r , J , and H I ,
is easily integrated over time on a personal computer. Some results obtained from
(8.18) for flux penetration into a superconducting rectangle are depicted in figure 19.
Since p/po has the meaning of a diffusivity, (8.18) describes nonlocal (and in general
nonlinear) diffusion of the magnetization g ( x , y , t ) . In the limit of a long ship of a
London superconductor with (formally) p = iopoA’ (cf equation (9.10) below), (8.18)
reduces to the static (time-independent) equation (8.6). If the resistivity is anisotropic
( E , = p,,J,, Er = p J Y ) then (8.18) still applies but with a modified function f ( r .t ) =
V,[(pYy/d)V,g]
VY[(px,/d)Vyg]. Interestingly, similar distributions of magnetic flux as
+
The flux-line lattice in superconductors
1541
in figures 18 and 19 are obtained for a square network of Josephson junctions in an external
magnetic field (Reinel et al 1995).
8.7. Statistical summation of pinning forces
The statistical summation amounts to the calculation of the average pinning force density
Jc B exerted by random pinning forces, or more precisely, by the forces exerted by a random
pinning potential. These forces thus depend on the flux-line positions, and the problem has to
be solved self-consistently. If the FLL were ideally soft. each pin could exert its maximum
force and the direct summation would apply, yielding a pinning force per unit volume
J,B = npfp were n p is the number density of pins and fp is the elementary pinning force.
In the opposite limit, if the FLL were ideally stiff, there would be no correlation between
the positions of the flux lines and pins, and thus the pinning forces would be truly random
and average to zero. Obviously, the elmtic or plastic distortion of the FLL by the pins is
crucid for the pinning summation. A softening of the shear stiffness of the FLL may lead
to a steep increase of J,(B) near the upper critical field B,z where C66 (Bc2 - B)’ (4.4)
decreases (Pippard 1969).
It huns out that the ‘dilute limit’, considered in early summation theories by Yamafuji
and Irie (1967) and Labusch (1969b) and discussed by Campbell and Evetts (1972). does
not exist (Brandt 198Oa). For random pins with force fp and number density n p this limit
yields J,B
n pf:/cef for fp >> f , and J,B = 0 for fp
fk.Here c , is
~ an effective
elastic modulus of the FLL (C66 or ( C S C ~ ) ~ / *for linear or point pins, respectively) and
f , r is a threshold force which is an artifact of the non-existing dilute limit: In an infinite
system, even for weak dilute pins the correct J,B is never proportional to the number of
pins since each pin ‘feels’ the FU.distortions caused at its position by the other pins, even
when these are far away. A single pin or a few pins in fact do exhibit this threshold
effect, but with increasing number of pins the threshold decreases. One may see this by
formally combining N random pins of strength fp into one ‘superpin’ which has the larger
force N i l 2 f pthat will exceed the threshold if N is large. A summafion over point pins by
considering ‘bundelling’ of flux lines was given by Schmucker and Kronmiiller (1974). see
also Lowell (1972), Campbell (1978). Matsushita and Yamafuji (1979) and Takics (1982).
The existence of a finite average pinning force requires the occurrence of elastic
instabilities in the ELwhen the FLL moves past the pins and the flux lines are plucked.
Without such instabilities the total pinning force would be a unique smooth periodic function
F(X) of the (uniformly driven) centre of gravity coordinate X = ut, with zero average
( F ( X ) ) = 0 (Brandt 1983a, b). The time-averaged pinning force (friction) would thus
be zero. Instabilities of the E L in the presence of a few pins were calculated explicitly
by Ovchinnikov (1982, 1983) and for planar pins by Brandt (1977~).see also Gurevich
and Cooley (1994) for a detailed consideration of planar pinning and Martinoli (1978) for
periodic pins.
The first correct statistical summation was presented by Larkin and Ovchinnikov (LO)
(1979). after they had arrived at similar results earlier (Larkin and Ovchinnikov 1973) by a
more complicated calculation using Feynman graphs and disregarding the important elastic
nonlocality of the FLL. For similar summation theories see Kerchner (1981, 1983), and for
pinning of charge-density waves see Lee and Rice (1979) and the detailed theory by Fisher
(1985). A general scaling law for pinning summation in arbitrary dimension was suggested
by Hilzinger (1977) and applied to collective pinning of flux lines and of crystal-lattice
dislocations (‘solution hardening’) by Brandt (1986~).A renormalization group argument
for this scaling law is given by Fisher (1983).
-
-
<
1542
E H Brundt
The LO 'theory of collective pinning' applies to weak random pins in an elastic KL
without dislocations where it prkdicts for 3D and 2D pinning in a film of thickness d
J,B = ( 1 . 5 ' / 2 / 3 2 n z ) W ' / ( r ~ ~ ~ 6 c(3D)
~)
(8.19)
J,B
(8.20)
izf
(8n)-'/'W/(rPdc,)
(20).
fi,i)pins
Here W = np(
is the average pinning force squared, with nF the volume density of
pins and fp,j the actual force exerted by the ith pin on the nL, and rp izf ( is the range of
the pinning forces. A detailed calculation of W and rp as functions of B/Bc2 for various
types of random pins was performed from GL theory by Brandt (1986d). The main idea of
LO is that the random pins destroy the long-range order of the FLL (Larkin 1970) such that
with 3 0 pinning the short-range order only is preserved in a cigar-shaped volume of radius
Rse = 8 n r , ' c 2 c 2 f / W and length(a1ong B ) L, = ( c ~ / c , # R ~ , = 8nrzc44c66/'W >> R3e.
0.
This correlated volume V, is defined as the volume within which the pinning-caused meansquare vortex displacement g(r) = (IzL(T) - u(0)12)is smaller than the pinning force range
squared, g(r) r,'. This condition yields an elliptical volume V, = ( 4 n / 3 ) R ~ , L CFor
.
films of thickness d -= L, and radius R , pinning becomes 2D and the correlated volume is a
cylinder of height d and radius R k = rpc&nd/(W h ~ ( R / R k ) ) l ' /thus
~ , V, = nR&d. Lo
then assume that within the volume V, the pinning forces are uncorrelated and therefore their
squares add. This idea yields for the average pinning force the estimate I,B = (W/V,)'n,
or explicitly (8.19) and (8.20). In the above LO expressions corrections from nonlocal
elasticity are omitted. With increasing pinning strength these nonlocal corrections (Brandt,
1986b, Wordenweber and Kes 1987) become important when R3c A. When pinning is
so strong that R3e becomes comparable to the flux-line spacing a, the collective pinning of
bundles goes over to the collective pinning of individual vortices, which was considered
by numerous authors, see below. When pinning becomes even stronger (or the EL softer)
such that L, is reduced to rp or t >then the collective pinning result goes over into the direct
summation, J,B izf (npW)'/' = n p ( f i , i ) $ For dilute pins, with spacing n;I3 exceeding
the flux-line spacing ( q 5 0 / B ) ' / ~this
, may be written as J,B izf npfp since all pins exert
their maximum force &. For dense pins, not all pins are active and one gets in the direct
summation limit J,B
(n'f3B/q5~)fF.
The 20 LO result (8.20) is borne out quantitatively by 2D computer simulations which
artificially suppress plastic deformation by assuming a spring potential between nearest
neighbours and obtain a J,B which is izf 1.1 times the LO expression (8.20) (Brandt,
unpublished). A nice experimental verification of (8.20) was obtained with weak pinning
films of amorphous MoGe by Kes and Tsuei (1981, 1983), White et a1 (1993), and Garten
e t a / (1995). The 3D LO result (8.19) does not always apply to real superconductors since
in an infinite 3D FLL even weak pins may induce plastic deformation, which softens the
EL and enhances J, (Mullok and Evens 1985). But recently the 3D LO result (8.19) was
confirmed by small-angle neutron scattering from 2H-NbSez single crystals (Yaron et a1
1994). Very recently Giamarchi and Le Doussal (1994, 1995) by a powerful 'functional
variational method' confirmed the 3D and 2D Lo results for J,B and found the remarkable
result that the 2D and 3D pinned FLL (or any elastic medium in a random pinning potential)
does not have to contain dislocations to lower its strain and energy.
The LO collective pinning theory was extended to films of finite thickness by Brandt
(1986b) and M'ordenweber and Kes (1987). Nice physical ideas on the pinned KL are
given by Bhattacharya and Higgins (1993, 1994, 1995) and by Yamafuji et a( (1994, 1995).
Computer simulations of the pinned FLL were performed by Brandt (1983a, b), Jensen et d
(1989, 1990), Shi and Berlinsky (1991), Pla and Non (1991). Kat0 eta1 (1991). Enomoto
<
<
Theflux-line lanice in superconductors
1543
et of (1993), Machida and Kaburaki (1993, 1995), Reefman and Brom (1993), Wang et
d
(1995). Probert and Rae (1995), and particularly nicely by Koshelev (1992~).Koshelev and
Vinokur (1994) introduce the notion of ‘dynamic melting’ of the FLL by defining an effective
temperature of the FLL caused by its jerky motion over random pins during flux flow, which
like thermal fluctuations may be described by a h g e v i n equation. Wang et al (1995)
compute the complex resistivity of the randomly pinned FLL by computer simulations.
The long-range distortion of the pinned FLL, which crucially enters the pinning
summation (Larkin 1970) and which was also discussed in the context of decoration
experiments (Crier et a1 1991, Chudnovsky 1990), was derived from scaling arguments.
by Feigel’man er a[ (1989), who found for these transverse fluctuations power laws of the
type g ( r ) = ( ( u ( ~
- )u(0)l2) rc, where depends on the dimensionalities d and n of
the elastic manifold and of the displacement field U (e.g. d = 3, II = 2 for the K L ; d = 1,
n = 2 for a single flux line), see Blatter et nf (1994a) for detailed discussion. Recently
this distortion was calculated by a Gaussian variational method using the replica method
(Bouchaud et al 1992), a replica density functional approach (Menon and Dasgupta 1994),
and a random phase model (Batmuui and Hwa 1994).
In my opinion the fundamental problem of an elastic medium in a random potential
in thermal equilibrium, was solved definitively by Giamarchi and Le Doussal (1994, 1995,
1996) by the Gaussian variational method and the renormalization group; Giamarchi and Le
Doussal find that previous theories strongly overestimate the pinning-caused displacements
of the FLL and that the correlation function g ( r ) = (lu(r)- u(0)12)at large distances
(where gl/* exceeds the lattice spacing a ) increases only slowly with the logarithm of r,
since a pinning centre essentially displaces only the nearest lattice point (or flux line). For
more work on the random field xy-model as a model for vortex pinning see Le Doussal
and Giamarchi (1995). By computer simulations k i k et al (1995) show that in a 2D
superconductor the vortex-liquid transition is not accompanied by a simultaneous divergence
of the correlation length of the phase of the order parameter, and that even in the solid FLL
phase the phaseangle coherence-length remains short ranged, of the order of the magnetic
penetration depth A.
The problem of a single vortex line depinning from a random potential has been studied
extensively, see e.g. Huse et af (1985), Kardar (1985, 1987), Brandt (1986~).Ivlev and
Mel’nikov (1987), Nattermann et al (1992), Parisi (1992), Dong et af (1993), Er@$ and
Kardar (1994), Mikheev et al (1995), and the simulation of depinning noise of one moving
flux line by Tang eta1 (1994) commented by Brandt (1995b). A detailed discussion is given
in the review paper by Blatter er a[ (1994a). The softening of the randomly pinned EL
and the fluctuation of the elastic moduli are calculated by a replica method by Ni and Gu
(1994). Collective pinning of Josephson vortices in a long disordered Josephson junction
(e.g. between the superconducting Cu-0 planes in HTSCS) was calculated by Vinokur and
Koshelev (1990) and in a comprehensive nice work by Fehrenbacher et ai (1992). see also
the recent experiments by Itzler and Tinkham (1995), the theory by Balents and Simon
(1995), and the book on Josephson junctions and their critical currents by Barone and
Patemo (1982).
-
9. Thermally activated dephning of flux lines
9.1. Pinning force versus pinning energy
The ideal pinning picture of section 8 strictly spoken applies only at zero temperature. At
T > 0, thermally activated depinning of flux l i e s causes a non-vanishing resistivity even
1544
E H Brandt
below Jc (Anderson 1962, Kim and Anderson 1964). In conventional superconductors this
effect is observed only close to the transition temperature Tc as flux creep (Beasley et a1
1969, Rossel etal 1990, Berghuis and Kes 1990, Suenaga et& 1991, Svedlindh etal (1991).
Schmidt et al 1993). Flux creep occurs in the critical state after the applied magnetic field is
increased or decreased and then held constant. The field gradient and the persistent currents
and magnetization then slowly decrease with an approximately logarithmic time law. As
mentioned in section 8.6 and section 9.6 below, flux creep is caused by a highly nonlinear,
current-dependent Rux-flow resistivity p ( J ) , which may be modelled, e.g. by p a ( J / J J
(n >> 1). p a exp(J/J,), or p s exp[-(Jz/ J)']. InitiaUy, the persistent currents in a ring
feel a large p. but as the current decays, p decreases rapidly and so does the decay rate
-.f(t) 0: p ( J ) .
In mscs thermal depinning is observed in a large temperature interval below T,. This
'giant flux creep' (Dew-Hughes 1988, Yeshurun and Malozemoff 1988, Kes et a1 1989,
Inui et a1 1989) leads to reversible behaviour of HTSB
above an 'irreversibility line' or
'depinning Line' in the B-T plane, see, e.g. Yazyi etal (1991), Schilling etal (1992, 1993),
and Deak et a1 -(1994). Thermal depinning occurs mainly because: (a) the superconducting
coherence length (evortex core radius) is small; (b) the magnetic penetration depth 1 is
large; and (c) these materials are seongly anisoeopic (section 3); here the notation ='tab,
1 = A&,, and K = ho&b is used. All three properties decrease the pinning energy but
tend to increase the pinning force.
Small 6 means that the elementary pinning energy Up of small pins (e.g. oxygen
vacancies or clusters thereof) is small, of the order of (B:/p0)t3 = (U$/8nZfi&2)(, The
elementary pinningforce U,,/e, however, is independent of 5 in this estimate and is thus
not necessarily small in WSCS,cf section 8.2.3.
Large 1 means that the stiffness of the F'LL with respect to shear deformation and to shortwavelength tilt (nonlocal elasticity, section 4.2) is small. Therefore, the flux lines can better
adjust to the randomly positioned pins. According to the statistical summation theories for
random pins (Labusch 19694 Campbell and Evetts 1972, Larkin and Ovchmnikov 1979),
this flexibility increases the average pinning force density. Therefore, the naive argument
that a soft F'LL or a flux-line liquid with vanishing shear stiffness cannot be pinned since it
may flow around the pins, does not apply to the realistic situation where there are many more
pins than flux lines. However, in HTSCs the decrease of the pinning energy enhances the rate
of thermally activated depinning and thus effectively reduces pinning at finite temperature.
Large material anisotropy, like the large penetrations depth effectively softens the FLL
and increases the average pinning force but decreases the pinning energy. This will be
particularly clear with the columnar pins discussed in section 9.2.
A usefd~quantity which shows in which superconductors thermal fluctuations and
thermal depinning become important, is the Ginzburg number Gi (section 511). Ginzburg
(1960) showed that in phase transitions of the second order the temperature range Tf T
Tc where the relative fluctuations of the order parameter @ are large, P@/@l*(Tf)e 1 is
determined by a constant Gi = 1 - T,/T,. For anisotropic superconductors this Ginzburg
number is (in SI units)
e
< <
This means that fluctuations are large when Tc, b and the anisotropy r are large, and when
( is small.
The flux-line lattice in superconductors
1545
9.2. Linear defects as pinning centres. The 'Base glass'
Long columnar pins of damaged atomic lattice were generated 6y high-energy heavy-ion
irradiation perpendicular to the Cu-0 planes in YBCO (Xoas etal 1990%Civale etal 1991,
Budhani et nl 1992, Konczykowski et al 1993, Klein et al 1993%b, Schuster et al 1993c,
1 9 9 4 ~1995b,c, Leghissa et al 1993) or BSCCO (Gerhauser et al 1992, Leghissa et al
1992, Thompson et al 1992, Schuster et al 1992a, Hardy et al (1993), Klein et al 19934 c,
Kummeth et al 1994, Kraus er al 1994a,b). Clearly these linear pins are most effective
when the flux lines are parallel to the pins, since then large sections of the flux line are
pinned simultaneously, which means a large pinning energy and a reduction of thermally
activated depinning. However, this desired effect is observed only in the less anisotropic
YBCO. In the very anisotropic BSCCO the flux lines have very low line tension (section 4.3)
and thus easily break into short segments or point vortices which then depin individually
with very small activation energy (Gerhsuser et al 1992, Brandt 1992~).As a consequence,
BSCCO tapes are good superconductors only at low temperatures, say at T = 4 K, where
they may he used to build coils for extremely large magnetic fields. In principle, if B could
be kept strictly parallel to the layers, large Jc and weak thermal depinning could be achieved
even at T = I1 K, but this geometric condition is difficult to meet.
Figure 21. Depinning of a flux line f"linear defects visualized as svaight cylinders. Leji By
thermal activation and supported by the applied cumnt, a pinned flux line can extend a parabolic
Imp (nucleus) towards a neighbouring linear defect. This loop is then inflated by the current.
and the flux line gradually slips to the neighbouring defect. Kinks in flux lines pinned at two
or more linear defects are driven along the defect by the c m n t , allowing the flux line 10 move
in the direction of the Lorenh force. Right. Depinning of a flux line from two linear pins
in a layered superconductor. When the Josephson coupling between the layers is weak (the
anisotropy large). individual pint vortices can depin and be uapped by a neighbluing defect.
The decrease of the pinning energy with increasing anisotropy r = hc/h,,b and
the eventual depinning of independent point vortices from linear pins oriented along
the c axis can be understood as follows. The flux-lie tension (4.14) is P %
( ~ ~ / 4 n ~ o h ; 4 b r 2 ) I n ( r Lwhere
/ ~ ~ b )L, = l / k z <
is the typicd length of the
deformation; the pinning energy per unit length of a hear defect with radius rp >
is Up = E ( @ ~ / ~ Z @ & ~ ) with t < 0.5 -k ln(rp/&,b)
% 1; the equals sign applies for
cylindrical holes, cf equation (1.1) and section 4.5. Now consider the depinning nucleus
when a flux line switches from one line pin to a neighbouring line pin at a distance a,,
and back to the original line pin, figure 21. The energy U and minimum width 2L of this
double kink is obtained by minimizing the sum of the depinning energy ZLU, and the tilt
energy (P/2) (u'(z)'dz
where u(z) is the flux-line displacement. The result is a trapezoidal
1546
E H Brundt
nucleus with (Brandt 1992~)
-
L = ( P / ~ U ) ~ ~=~ [U2, 1 n ( ~ r / ~ ~ ~ ) / ~ ] ' f ~Ua =
, / (8pu,)L/za,
r
=~ L U , I /
r.
(9.2)
Both the width 2L and energy U of the depinning nucleus decrease with increasing
anisotropy r as 1/r. When L has shrunk to the layer separation s at r % 2up/s, the
anisotropic London theory ceases to be valid, and L saturates to L = s. A similar picture
results when the distance up between the linear pins is so large, or the applied current density
J so high, that depinning is triggered by the current. The critical nucleus for depinning
is now a parabola of height U,/(&J), width 2L = ( S U p P ) 1 ~ 2 / ( & o J ) I / r, and energy
U = $ LU, = $(2U~P)'/*/(&oJ) I/ r. The general expression for arbitrary a, and J is
given by Brandt (1992~).Note that the pinningforce per unit length Fp % U p / & b and the
critical current density Jc = Fp/@o= €&0/(4Rpo&?~~) do not depend on the anisotropy
in this geometry.
Pinning of flux lines by randomly positioned parael columnar pins is an interesting
statistical problem (Lyuksyutov 1992). Nelson and Vinokur (1992, 1993) show that in the
model of a simplified flux-line interaction this system is equivalent to a 'Bose glass', which
has similar properties to the 'vortex glass' of section 9.5; see also Tauber et a1 (1995) and
Ciamarchi and Le Doussal (1996). While Woltgens et a1 (1993a) do not find Bose-glass
behaviour in thin films, Krusin-Elbaum etal (1994) and Jiang etal (1994) in YBCO crystals
with columnar defects observe Bose-glass melting, and Konczykowski (1994) in YBCO and
BSCCO, Konczykowski etal (1995) in BSCCO crystals, Miu etal (1995) in BSCCO films,
and Budhani et a1 (1994) in a TI-HTSCwith columnar defects find evidence for Bose-glass
behaviour. In a nice paper Krusin-Elbaum er al (1994). with YBCO crystals bombarded
by 1 GeV Au ions, confirm various predictions of the Bose-glass melting transition; this
transition is assumed to occur along the irreversibility line Hi,(T)
(1
determined
from the maximum of the dissipative component x" of the a.c. susceptibility at 1 MHz, see
Krusin-Elbaum eta! (1991) and Civale et al (1992) for a discussion of this a.c. method. In
this experiment, with increasing defect density the effective power exponent 01 grows from
= to % 2, in quantitative agreement with the predicted (Nelson and Vinokur 1992, 1993)
shift of the Bose-glass transition field to higher temperatures. Van der Beek et al (1995)
investigate the vortex dynamics in BSCCO with columnar defects using an at.-shielding
method and find good agreement with the Bose-glass theory. Zech et nl (1995) measure
in detail the flux-line correlation and irreversibility line in BSCCO with columnar defects
aligned at 0" or 45" to the c-axis.
The state when flux lines are locked-in at the columnar defects was observed by d.c.
measurements (Budhani etal 1992) and by microwave absorption (Lofland etal 1995). Ryu
et al (1993) performed Monte Carlo simulations of this system. The influence of additional
('competing') disorder by point defects is calculated by Hwa ei al (1993b) and Balents
and Kardar (1994). The effect of non-parallel ('splayed') columnar pins is considered by
Hwa etal (1993a). This splay of columnar defects reduces the vortex motion (Civale et af
1994). Thermal depinning and quantum depinning from a single l i e defect is considered by
KuliC eta! (1992). Krwer and Kulii (1993, 1994). Khalfin and Shapiro (1993), Sonin and
Horovitz (1995), Sonin (1995), Chudnovsky e t a l (1995) and Krwer (1995). The phase
diagram of a thin superconducting film with a hole in an axial magnetic field is presented
by Bezryadin et af (1995) as a model for a columnar pin. Thermal depinning from a single
twin pIane is calculated by Khalfin etal (1995). The low-temperaturedynamics of flux lines
pinned by parallel twin planes is found to be dominated by Mott variablerange hopping of
superkinks (Marchetti and Vinokur 1994, 1995).
-
-
-
4
the flux-line lattice in superconductors
1547
By a decoration method Leghissa et 01 (1993) observe the destruction of translational
order in the K L by these linear defects. Doyle et a1 (1995) measure the effect of columnar
defects on the elasticity of the E L . Dai et a1 (1994b) in BSCCO simultaneously observe
columnar defects by etching and flux l i e s by .decoration and thus can see where the
flux lines are pinned. By scanning tunnelling microscopy Behler et ai (1994a) see the
columnar defects and the pinned flux lines on the same NbSe2 sample. In YBCO films
irradiated at various angles by heavy ions Holzapfel ef al (1993) observe sharp peaks in
the angle-dependent critical current J,@) when B is parallel to the columnar pins, see
also Klein et al (1993a,c). In a similar HTsc with oblique and crossed linear defects
Schuster et al (1994c, 1995c) observe anisotropic pinning by magneto-optics. Becker et a1
(1995) measure magnetic relaxation in DyBCO and interpret these data by a distribution
of activation energies. Klein et a1 (1993b.c) in BSCCO with radiation-induced columnar
defects inclined at 45" with respect to the c axis observe that at T = 40K pinning is isotropic
and thus the point vortices are independent, but at T = 60K anisompic pinning indicates
that the pinned objects have line character and thus the point vortices are aligned and form
flux lines. Kraus et ai (1994a) give a review on such tailored linear defects. Lowndes et
01 (1995) introduce columnar defects by growing YBCO films on miscut, mosaic LaAlO
substrates, and observe strong asymmetric pinning.
9.3. The Kim-Anderson model
In kscs at sufficiently high temperatures, a linear (Ohmic) resistivity p is observed at small
currenf densities J << Jc due to thermally assisted flux flow (TAW, Kes et ai 1989). Both
effects, flux creep at J X J, and TAFF at J << J,, are limiting cases of the general expression
of Anderson (1962) and Anderson and Kim (1964) for the electric field E ( B , T,J ) caused
by thermally activated flux jumps out of pinning centres
E ( J ) = 2pJC exp(-U/ksT)sinh(JU/J,kBT).
(9.3)
In (9.3) we introduce the phenomenological parameters Jc(B) (critical CUrrent density
at T = 0), p,(B, T) (resistivity at J = Jc). and U ( B , T) (activation energy for flux
jumps). The physical idea behind (9.3) is that the Lorentz force density J x B acting on
the FLL increases the rate of thermally activated jumps of flux lines or flux-line bundles
along the force, uoexp[-(U - w ) / k ~ T ] ,and reduces the jump rate for backward jumps,
uoexp[-(U+ W)/ksT]. Here U ( B , T)isan activationenergy, W = J B V l theenergy gain
during a jump, V the jumping volume, 1 the jump width, and uo is an attempt frequency.
AI1 these quantities depend on the microscopic model, which is still controversial, but by
defining a critical current density J, = J U / W = U / B V I , only measurable quantities enter.
Subtracting the two jump rates to give an effective rate U and then writing the drift velocity
U = ul and the electric field E = U E = p J one arrives at 19.3).
For large currents J % Jc one has W % ZI >b kBT and thus E 0: exp(J/JI) with
J 1 = J,kBT/U. For small currents J << 51 one may linearize the sinh(W/kBT) in (9.3) and
obtain Ohmic behaviour with a thermally activated linear resistivity prm m exp(-U/kBT).
Combining this with the usual flux-flow resistivity pFF valid at J > J, (Qiu and Tachiki
1993) or with the square-root result ( p = 2 in (8.1)) for a particle moving viscously across
a one-dimensional sinusoidal potential (Schmid and Hauger 1973) one gets (figure 22)
P = (2P,U/ksT)exP(-u/ksT) = Pcm
P = pcexpI(J/Jc
p =
- ~ ) U / ~ B aT exp(JIJ1)
I
- J;/Jz)'/'
ppp 0
p.B/B,z(T)
for J
<< JI
for J
~3
for J
>> J,
(TAFF)
(9.44
Jc (flux creep)
(9.4b)
(flux flow).
(9.4c)
EH B r d t
1548
T
0.7
0.8
0.9
1.0
Figure 22. Left Electric field E ( 3 ) induced by moving flux lines which are driven by acurrent
density 1. cf equations (9.4aH9.4~).The inset shows a flux line or flux-line bundle hopping
over a periodic pinning potential which is tilted by the Lonnfz f o r e exerted on the flux,lines by
the current. Thermally activated hopping becomes more likely al high tempemure T and high
induction B . Right. irreversibility lines (&pinning lines) in the B-T plane s e p m e the region
where free flux flow or Ulermally activated flux flow (TAR) DC~UTS(boh with linear E ( 3 ) )from
the flux-creep region (with highly nonlinear E ( 3 ) ) .
The linear TAFF regime (9.4~)is observed if B and T are sufficiently large (Palstra et al
1990b. Shi etal 1991, Worthiigton etal 1991, 1992, Seng eta1 1992, van der Beek et al
1992a, Wen and Zhao 1994). At lower T and B nonlinear resistivity is observed (Koch et al
1989, Gammel et a1 1991, Safar et al 1992%Yeh et a1 1992 1993a,b, Dekker et a[ 1992a.
Worthington er a1 1992, Seng et al 1992, van der Beek et al 1992% Sandvold and Rossel
1992, White et al 1993, 1994, Woltgens et al 1993a, b, Sun et al 1993, Jiang et al 1994,
Wagner etal 1,994,1995, Deak eta1 1994, Heinzel e r a l 1994), which often can be fitted by
the vortex-glass expressions, section 9.5. It appears that in the TAFF regime the KL is in a
‘liquid’ state (Feigel’man and Vnokur 1990, Chhavarty et al 1990, Vinoh et al 1990%
1991b. Koshelev 1992a). i.e. it has no shear stiffness; therefore, elastic deformations of the
TW. at different points are not correlated. This assumption leads to an activation energy U
which does not depend on the current density. Another situation where an extended KimAnderson theory fits the measured E ( J ) data well is when there are two types of pinning
centres (Adamopoulos et al 1995).
9.4. Theory of collective creep
Theories of collective creep go beyond the Anderson model (9.3) and predict that the
thermally jumping volume V of the FLL depends on the current density J and becomes
infinitely large for J -+ 0. Therefore, the activation energy also diverges, e.g. as
U cx V cx l / J ‘ with (Y > 0. As a consequence, the thermally activated resistivity
p ( J , 2’) exp[-U(J)/kBT] becomes truly zero for J + 0. This result follows if weak
random pinning acts on a
that is treated as an ideally elastic medium which may be
isotropic (Feigel’man er al 1989, Natterman 1990, Fischer and Nattermann 1991, Feigel’man
et al 1991), anisotropic (Pokrovsky et al 1992, Blatter and Geshkenbein 1993). or layered
(Koshelev and Kes 1993). Qualitatively the same result is obtained by the vortex-glass
picture, which starts from complete disorder, section 9.5. Interestingly, all three moduli
for compression, shear, and tilt of the FLL enter into theories of collective creep, whereas
the collective pinning theory of Larkin and Ovchinnikov (1979) (valid at zero temperature)
involves only the shear and tilt moduli. This is so because the activated (jumping) volume
of the KL depends also on the compression caused by the driving current, but the correlated
volume V, of the pinning-caused distortions of the EL (section 8.7) depends only on the
combination c i P c;’ M cGp ( p = 1 or 2 for dimensionality D = 2 or 3).
An activation energy which diverges when J -+ 0 is also obtained in theories that
-
+
the flux-line lattice in superconductors
1549
assume the &pinning proceeds via a kink mechanism which allows the flux lines to climb out
from the space between the Cu-0 layers (Ivlev and Kopnin 1990%b, 1991, Chakravarty et
a1 1990) or from columnar pins (Nelson and Vinokur 1992, 1993, Brandt 1992~.Lyuksyutov
1992, K r a e r and KuliC 1993, 1994) (section 9.2).
9.5. Vortex glms scaling
Similar current-voltage curves as in the collective creep theory follow from the ‘vortexglass’ picture (Fisher 1989, Fisher etaf 1991). The hasic idea of the glassy behaviour is
that if there is a second-order phase transition to this vortex glass similar to that in theories
of spin glasses, then both a characteristic correlation length 6, (the size of the jumping
volume) and the relaxation time r, of the fluctuations of the glassy order pahmeter diverge
at the glass-transition temperature T,
r,(T, B ) N T,(B)ll - T/T,I-”z.
(9.5)
Cg(T,B ) = C,(B)ll - T/T,I-”
The vortex-glass picture predicts scaling laws, e.g. the electric field should scale as
Et:-’ = f*(Jti-’) where z % 4, D is the spatial dimension, and f&) are scaling
functions for the regions above and below T,. For x + 0 one has f + ( x ) = constant
At Tgr a power-law current-voltage curve is expected,
and f - [ x ) --f exp(-x-”).
E a J ( Z + 1 ) / ( D - l ) , thus
r -
(9.7)
In the theory of collective creep there is no explicit glass temperature, but the picture is
similar since collective creep occurs only below a ‘melting temperature’ T, above which
the FU.looses its elastic stiffness. Thus T, has a similar meaning to T,. A vortex glass
state should not occur in ZD flux-line lattices (Feigel’man et a1 1990, Vinokur et al 1990b,
Dekker et al 1992b, Bob! and Young 1995). More details may be found in the reviews by
Blatter etol (1994a) and Fischer (1995a).
Experiments which measure the magnetization decay or the voltage drop with high
sensitivity, appear to confirm this scaling law in various HTXS in an appropriate range
of B and T. For example, by plotting T-dependent creep rates b? = dM/df in reduced
form, (1jJ)Il - T/T,I-Y(z-’)b? against Jll - T/T,l-*”, van der Beek et al (1992a) in
BSCCO measured T, = 13.3K, z = 5.8 & 1, and U = 1.7 & 0.15, figure 23. Vortexglass scaling of p ( J , T) was observed, e.g. in YBCO films (Koch et a1 1989, Olsson et
ai 1991, Dekker et a1 1992h Woltgens et a1 1993% 1995). in YBCO crystals Cyeh et a1
1992, 1993% Charalambous er nl 1995), in BSCCO films (Yamasaki et a1 1994), and in
TlzBalCaCu2Os films (Li et al 1995). Very detailed data p ( J ) for three YBCO samples
with different pinning (without and with irradiation by protons or Au ions) are presented
by Worthington et a1 (1992) for various inductions B with T as parameter. In the sample
with intermediate pinning two transitions are seen in p ( J ) , a ‘melting transition’ at T,
(e.g. T, % 91.5K at B = 0 2 T , J = l @ A n r 2 ) and a ‘glass transition’ at T , E 90K
(above case) or T, = 84.92 K (strong pinning sample, very sharp transition at B = 4 T,
J
4 x 10s Am-*). The FLL phase in between these two transitions was termed ‘vortex
slush’. Krusin-Elhaum et a1 (1992) in YBCO crystals observe a crossover between singlevortex and collectivepinning regimes by plotting lines of &(E, T) = constant, and Li
et a1 (1994) observe a vortex-glass-to-liquid transition in a BSCCO/Ag tape. Wagner et
a1 (1995) see a vortex-liquid-vortex-glass transition in BSCCO films. Ando er a1 (1992,
1993) in narrow YBCO ships observe a size effect on the vortex-glass transition, see also
<
1550
;;-ik
E H Brandt
-10
0
5
10
l!
log IJ/TI T-T9?"I
16'
10'
1Os
CurrmtdwsityJIA m?
Figure 23. Lq? Electric held E and reSiStivity p = E j J against "rent density J in a log-log
plot for a clean YBCO u y s t a l in a held of 1 T for various tempemhms. Note the abrupt change
of uuvanu'e in a narrow tempera" interval of 0.1 K (=produced colutesy T K Wonhiogton).
Right. Vortex-glass scaling of the resistivity p ( 3 ) of a thick YBCO hlm. Plotted in this way.
data taken at various temperatures wUapse into one curve (after Dekker CI 01 1992b).
the observation of collective pinning and fluctuation effects of strongly pinned flux lines
in ultrathin NbTi wires by Ling et al (1995b). Metlushko e t a l (1994b) identify different
pinning regimes in BSCCO single crystals, and Moshchalkov et a1 (1994a) see a crossover
from point pinning to columnar pinning when BSCCO crystals are irradiated, these data
may also be explained by the crossover of two B ( T ) lines corresponding, respectively,
to Fu melting or depinning and to the vanishing of the edge-shape barrier (section 8.5)
as suggested by Indenbom et al (1994a.b). Kim et d (1995b) find two different scaling
behaviours of the critical current in thin BSCCO films below and above To M 25 K. A nice
comprehensive study of the E-J-B surface of mscs was performed by Caplin et al (1994)
and Cohen et a1 (1994).
A different method of checking the vortex-glass scaling predictions is to measure the
linear a.c. response. This has been achieved over the remarkable frequency range from
~
by Kotzler et ~l
3mHz to 5OMHz (over ten orders of magnitude) on the S Q specimen
(1994a,b), see also Olsson eta1 (1991). Reed etal (1994, 1995), Wu etal (1993b, 1995),
and Ando et QZ (1994). The complex resistivity of a circular YBCO film was obtained
by measuring the a.c. susceptibility and then applying an inversion scheme based on the
integral equation for the sheet current in circular disks (Brandt 1993a.b. 1994~.see also
section 8.6). which yields the magnetic susceptibility x ( w ) as a function of the complex
resistivity p(w) (section 9.5). This contact-free method allows a very precise determination
of both the modulus and the phase of p(w). Within the glass model, the magnitude and
phase angle of the linear complex conductivity u ( w ) = I ( p ( o ) are predicted to have the
following forms, characteristic for a second-order phase transition (Fisher et al 1991, Dorsey
1991)
IU(w)l
= (Tg/eg)&(wr)
~ - ' ( U ' ' / U ' ) = p*(OS)
(9.8)
Thej w - l i n e lattice in superconductors
1551
where & ( x ) and P*(x) are homogeneous scaling functions above and below T8, which
describe the change from pure Ohmic behaviour (U“ = 0) for T >> T, to pure screening
(U’ = 0) at T <
< T8. While their limiting behaviours are known, the detailed shapes of these
scaling functions have not yet been worked ont. With their contact-free method Kotzler
et ai (1994b) found that in YBCO films the phase angle of u(o) monotonically i n c m e s
with o at temperatures T > T, and decreases at T < T,, but at T = T8 the phase remains
frequency-independent over ten orders of magnitude in w af a non-trivial value (1 - 1/z)n/2
as predicted by Dorsey (1991) and Dorsey eta1 (1992), with z = 5.5 f0.5, figure 24. This
constancy of the phase angle follows from (9.5) and (9.8) by applying the Kramers-Kronig
relation between magnitude and phase angle of the linear conductivity u(o).Physically it
means that at T8 there coexist islands of ideally pinned ‘glassy’ mL and of mobile EL.
For these data both the modulus and the phase of U ( @ ) scde perfectly over more than 14
orders of magnitude in the scaled frequency ors if v = 1.7 Y 0.1 is used, figure 25.
76
78
80
82
84
TEMPERATURE ( K )
86
Figure 24. kji. Temperahm dependence of the real and imaginary pans of the magnetic
susceptibility x of a YBCO film measured at fixed frequencies in a.c. and d.c. fields pardel to
the c axis and perpendicular to the film. (a) B. = 0.4T and (b) B. = 4T. Righr. Frequency
dependence of (a) the phase angle and (b) the modulus of the dynamic conductivity U@)
evaluated from the dynamic susceptibility x(o) by invelfing equation (9.14) for the YBCO film
data at 8. = 0.4T. Note thaf at T = 88.9 K, interpreted as the glass temperature T,,the phase
angle has a frequency-independent non-hivial value (&er Kotzler et a1 1994b).
9.6. Linear ax. response
9.6.1.Thermally activatedfluxfiw. Within the TAFF model ( 9 . 4 ~ the
) linear complex a.c.
resistivity p ( o ) and ac. susceptibility x(o) of the FLL are obtained as follows. Consider
1552
E H Brandt
I
I
I
I
Figure 25. Dynamical (frequency) scaling of the linear conductivity o ( w ) of Ihe YBCO film
of figure 24 at B. = 0.4T and B,, = 4T. Both the phase angle ( n ) and modulus (b) of a scale
well and exhibit a transition from pure Ohmic to full screening behaviour (after K6tzler et a1
1994b3.
a superconducting half space in which a constant external field generates a uniform FLL
with induction B. If the flux lines are rigidly pinned, an additional applied small a.c.
magnetic field H&) = Hoexp(iwt) will penetrate exponentially to the London depth A.
If the flux lines are pinned elastically with force constant OIL. (Labusch parameter) such
that a small uniform displacement U causes a restoring force density -OIL.& then H&)
penetrates exponentially to the Campbell penetration depth AC = ( B 2 / ~ o o r r ) 1 ’(Campbell
z
1971, Campbell and Evetts 1972). More precisely, the penetration depth is now (A2+A;)’D
(Brandt 1991d). If the flux lines are parallel to the planar surface this penetration proceeds
by compressional waves in the FLL; if they are perpendicular to the surface the penetration
occurs by tilt waves. In both cases the same penetration depth results since the moduli for
long-wavelength compression and tilt are approximately equal, C I I= c a = B Z / p o (4.6).
The viscous drag force density on the FLL. -q au f a t , may be accounted for by replacing O I L
by OIL iwq. This means that viscous damping dominates at frequencies o >>
where
TO = OIL. At lower frequencies the a.c. response of the superconductor with elastically
+
c’
1553
the flux-line lam‘ce in superconductors
pinned flux lines is that of the Meissner state with increased penetration depth.
At finite temperatures one has to account for thermal depinning. This was achieved in
three different ways, which all yield essentially the same result: a model of periodic pins was
considered by Coffey and Clem (1991a); a relaxing Labusch parameter was introduced by
Brandt (1991d); a visco-elastic force-balance equation was solved following a suggestion by
A M Campbell (Brandt 1992a). If the E L is displaced homogeneously and then stopped, the
elastic restoring force relanes due to thermally activated (or tunnelling-caused) depinning.
This means the Labusch parameter decreases with time. It ‘seems natural‘ to assume
) a~(O)exp(-t/r). This assumption in fact reproduces the
an exponential decay, L Y L ( ~=
results obtained by the two other methods, but not all experimental results confirm this
TAFF picture, see section 9.5 (Kotzler el a1 1994b) and section 9.6.3 (Behr et al 1992),
and also Koshelev and Vinokur (1991) and van der Beek et a1 (1993). The relaxation time
may be expressed as 5 = B2/(prApFaL) % ZoeXp(-U/kBT) with prm = pmeXp(U/kBT)
the TAFF resistivity from (9.4~)and 50 = q / c q = B’/(~F@’L)with ~ F F= B Z / q (5.6) the
flux-flow resistivity. Formally this relaxation replaces ( Y L by (rLiw/(l +iwr). The resulting
complex a.c. penetration depth A,,&)
is thus
In the second version of (9.9) s = soexp(U/kT) >> was assumed. From A,, the complex
surface impedance is obtained as 2, = iofioA,,(w). Notice that the result (9.9) does not
specify the physical nature of the creep and thus in principle applies also to quantum creep
where r = roexp(S~/h)with SE a Euclidian action, section 9.9.
From the complex ax. penetration depth (9.9) and the Maxwell equations one finds that
for the half space the electric field E ( x ) has the same spatial dependence exp(-x/kaC)
as the current densify J(n). Thus the ratio E ( x ) / J ( x ) = poe is spatially constant. This
means that the usual local relation E = paJ applies, although the penetration of J and E
is mediated by coherent flux lines, see the discussion in section 6.4. With (9.9) the complex
a.c. resistivity p,,(w) = E / J = iop&,
becomes
-
p,,,(o)
G=
iwpok’t p r d l + iwr)/(l+ iosoo).
(9.10)
For low and high frequencies, this resistivity is real and independent of w ; the flux motion
in this m e is diffusive with diffusivity D = poC/po. The a.c. penetration depth is then
the usual skin depth 8 = (2D/o)’/’ as in normal conducting metals. For intermediate
frequencies, pOcis imaginary and proportional to w; the surface currents are then nearly
loss free due to strong elastic pinning, and the screening. in this case is similar to the
Meissner state but with a larger penetration depth (A2 A:)’”.
From A,, (9.9) or paC(9.10) one in principle obtains the linear complex susceptibility of
superconductors of arbitrary shape. For example, for a slab with 1x1 < a and for a cylinder
with radius a one has the penetrating parallel magnetic a.c. fields
+
where I&) is a modified Bessel function. Averaging HIover the specimen cross section
one gets the linear complex ax. susceptibility ~ ( w =) 1 ~ ( w =) (Hl)/Ho
for slabs and
cylinders (Clem et al 1976) in a parallel a.c. field,
+
(9.12)
E H Brandt
1554
and I , @ ) = dIo/du. In the Ohmic (diffusive) cases at low and high frequencies, the known
skin-effect behaviour results, with a complex argument U = (1 +i)(od2/8D)'/z containing
the flux diffusivities D = &~m/@ = @ / r for w << l / r , and D = m / p o = A:/%
for
11%< w. Explicitly one then gets for the Ohmic slab (Kes et al 1989, Geshkenbein et al
1991, Brandt 1991e, 1992b)
p(o) = p'
- ip" =
(si& U
+ sin U) - i(sinh U - sin U)
U (cosh U
+ cos.U)
(9.13)
with U = (od2/2D)'/' = d/6, figure 26. 'Ihis gives p(o) = 1 for o = 0 and
p(w) (l-i)/ufor\ul > 1. Thedissipativepartp"(o) hasamaximump~~.=0.41723at
v = 2.2542 corresponding to oy = 1.030 % 1 with rd = d2/n2D, or to d/26 = 1.127 w 1
with 6 = ( ~ D / O ) ' /the
~ skin depth.
*
1 .o
0.8
0.6
0.4
0.2
I
0.0
1
2
3
4
5
U+
+
x ( w ) = d - iw" of a sVip of width
2, thickness d << a, and Ohmic resistiviy p in a longitudinal a c . field (full curve, ~111
(9.13)) and PerpendicULv ac. field (broken curve. PLI (Bmdt 1993h 1994a)). The time scales
rl = d2mLa/16pand Q = ndwo/Zap are chosen such thal the slopes of p[ and py coincide at
o = 0. These Ohmic p ( o ) apply to normal conducting malerials and to superconductorsin the
free nux-flow or thermally assisted flux-flow states.
Figure 26. Complex a.c. permeability p(o) = I
9.6.2. General susceptibilifies. The linear susceptibilities of rectangular bars in a parallel
a.c. field, and of thin strips and disks in a perpendicular a.c. field for arbitrary complex
resistivity pae are given by Brandt (1994b,c) (figure 27), see also Dorsey (1995). In general,
these linear permeabilities p and susceptibilities x = p - 1 may be expressed as infinite
sums over first-order poles in the complex o plane (Brandt 1994~)
(9.14)
The variable w in (9.14) is proportional to io/p,,(o); for a slab or cylinder in a longitudinal
a.c. field one has w = uz = (a/A,,)2 = ioa2po/p.c. and for the slab and disk in a
perpendicular field, w = ad/(2nA;J = iwadpo/(2npa,). The real and positive constants
A. and c,, (n = 0, .. . , m) follow from an eigenvalue problem, namely, a differential or
integral equation describing local or nonlocal diffusion of longitudinal flux or of a sheet
current in a parallel or perpendicular geometry, cf equations (8.4) and (8.18). The pole
The f l u l i n e lanice in superconductors
1555
1
0.1
............
0.01
I.,.-
0.1
' ' """'
'
'
1
. , , , ,,., , ,
10
,
,.,,
100
UT0
Figure 27. The a.c. permeabilities &(a) = p' - is" in various longitudinal and t " v w
geometries for Ohmic resistivity plotted on a log-log scale. The time unit is the relaxation
time ZQ of the slowest relaxation mode in each of these geometries, e.g. ro = d 2 / D r 2
(TO = 0.24924ad/D)for ships with cross section 2a x d and with flux diffusivity D = plwg
in longitudinal (perpendicular) a.c. held. For details see Brand1 (1994b). At large oy,one
has in the longih~dinal a.c. field /I' a p"
lf&
and in the perpendicular a.c. held
p In(consm1 x w ) / a , thus p'
110 and $'= (21~)
In(cnnstan1 x w)p''.
-
-
-
positions An are the eigenvalues and the amplitudes c, are squares of integrals over the
eigenfunctions (matrix elements). For example, for the strip this eigenvalue problem reads
(9.14u)
+
yielding A0 = 0.63852, A, %z A0 n, and c, = 0.5092,0.159,0.091,0.062,. , ., obtained
from c. = 4A:b: where b, = f , ( x ) x d x . In particular, for an Ohmic strip with length
2a, thickness d , and resistivity p. the lowest eigenvalue A0 determines the relaxation time
SO = adpO/(ZzpAo) with which the sheet current decays exponentially after a jump in the
applied magnetic field and which yields the position of the maximum in the dissipative part
of the ax. susceptibility x ( w ) = x' - ix", x,& = && = 0.4488 at or0 = 1,108 %z I.
For practical purposes the infinite sums (9.14) may be approximated by their first few
terms. Tables of the c. and A, for the slab, cylinder, strip and disk are given by Brandt
(1994~);the coefficients for the square and for rectangles will be publishd. Brandt and
Gurevich (1996) and Gurevich and Brandt (1996) find that the small-amplitude linear a s .
response of superconducting strips and disks during flux creep is given also by expressions
(9.14) but with w = iot, where t is the time elapsed after creep has started. By inverting
the formulae (9.14), Kotzler et af (1994b) extracted the linear complex a s . conductivity of
YBCO and BSCCO disks in parallel and perpendicular d.c. fields from ax. susceptibility
measurements; they confirmed the predictions of vortex-glass scaling (section 9.5) in three
of these geometries, but for BSCCO with the d.c. field along the c axis this scaling did not
work.
Skvortsov and Geshkenbein (1994) obtain the linear susceptibility of anisotropic
ellipsoids. The linear screening by an infinite type-I1 superconducting film between two
ax. coils was calculated by Hebard and Fiory (1982), Fiory et al (1988), Jeanneret et
al (1989). Clem and Coffey (1992). Pippard (1994), Klupsch (1995) and Gilchrist and
Brandt (1996). Nonlinear screening is considered by Nurgaliev (1995) and by Gilchrist and
Jb
1556
E H Brandt
Konczykowsk (1990, 1993), who showed that screening of coils by a superconductor can
be described by a series of inductive loops with given mutual- and self-inductancesand with
nonlinear resistivity; see Gilchrist and van der Beek (1994) for applications and Fhbrega ef
ai (1994a) for experiments.
9.6.3. Algebraic relaxation In &c. experiments on ceramic BSCCO by Behr et ai (1992)
in the large frequency range 2Hz < 0/2n 4 2MHz both the real and imaginary parts of
the susceptibility p(o) (9.12) for cylinders below T = % 70K could be fitted very well
by replacing the above exponential decay of the Labusch parameter, q , ( t )= 011. exp(-t/r)
yieldingcYL(w) = ocLios/(l+iwr), by an algebraic decay of the form q ( t ) = orL/(l+f/t)l
yielding for U T << 1
LYL(O)= C$L(iWT)@
r(1
- p)
kit = h$/[(ioT)@I-(l - p ) ]
(9.15)
where r(l - 6) is Euler's gamma function. Interestingly, in these experiments the
effect of temperature on the dynamics is entirely determined by that of the exponent
p ( T ) ss 1/(1 U / k B T ) , see equation (9.16) below, while T % 4 x 10-'*s (for B = 1T)
is nearly independent of T . Moreover, p ( T ) overlaps with exponents determined from
the algebraic decay of the time dependent magnetization of this sample at the same field
(Spirgatis er ai 1992). Behr et al (1992) find a depinning limit (p = 1) for T( w 70K
4 T Tc % 90 K and a glass transition with p % 0.07 at Tg % 24 K.
+
<
9.6.4.Further work More microscopic theories of the linear a s . susceptibility of the EL
in layered superconductors are given by Koshelev and Vinokur (1991). Ivanchenko (1993),
and Artemenko et ai (1994). see also Hsu (1993). Tachiki et ai (1994), and Bulaevskii
et ai (199%) for the low-frequency magneto-optical properties. Sonin and Traito (1994)
consider the surface impedance of the arbitrarily inclined E L also taking into account a
possible surface barrier (section 8.5) and the thin surface layer over which the flux lines
bend in order to hit the surface at a right angle (Hocquet et al 1992, Mathieu et ai 1993,
Plaqais et ai 1993, 1994, Brandt 1993a). Parks et a1 (1995) measure vortex dynamics in
the Terahertz domain by phase-sensitive spectroscopy and find disagreement with the usual
flux-flow picture. Anomalous vortex mobility vanishing logarithmically in the limit of
small frequencies in an ideal 2D superconductor made from a hiangular array of Josephson
junctions, was observed by Th6ron eta1 (1993, 1994) and is explained by Beck (1994) and
Korshunov (1994), see also the MC simulations by Yu and Stroud (1994).
A discussion of the linear and nonlinear ax. response of HTSCS is given by van der Beek
eta1 (1993), see also Matsushita etal (1991) and the discussion below equation (8.13). Note
that a nonlinear frequency-dependent (and time-independent) resistivity p = E / J cannot
be defined since in a nonlinear conductor the time dependences of E and J in general are
different and do not cancel.
Numerous research groups investigate pinning and flux flow by microwave absorption,
see, e.g. Blazey et ai (1988), Portis etal (1988), Shridar er ai (1989), Giura et al (1992),
Kessler et 01 (1994), Cough and Exon (1994), Blackstead et ai (1994), Lofland er ai
(1995). Colosovsky et al (1991, 1992, 1993, 1994. 1993, Wu et al (1995) and the
reviews by Pie1 and Muller (1991). Portis (1992) and ,Dumas et 01 (1994). Unexpected
sharp magnetoabsorption resonances with cyclotronic and anticyclotronic polarisation were
observed in BSCCO at 30 and 47MHz in fields up to 7 T by Tsui et nl (1994) and
are explained by Kopnin et ai (1995). Several groups perform microwave precision
mwurements of the penetration depth A(T) in the absence of flux lines in order to check
microscopic theories of superconductivity (e.g. s- or d-wave pairing), see, e.g. Anlage and
Thejlux-line lattice in superconductors
1557
Wu (1992), Hardy er aE (1993). Wu et a[ (1993a), Klein et af (1993d), Lee et a[ (1994).
Kamal et a1 (1994) and Xu et a1 (1995). By mutual inductance of coaxial coils, A(T)
was measured, e.g. by Fiory et a1 (1988), Jeanneret et al (1989), Ulm et a1 (1995). see
also section 9.6.2. However, caution has to be taken in interpreting the measured A ( T ) as
indicative of s-wave or d-wave pairing, since A(T) may be influenced by the presence of
vortices, by fluctuations of the phase 4 of the order parameter (Coffey 1994b. Roddick and
Stroud 1995). or by the layered structure (Klemm and Liu 1995, Adrian et ol 1995).
Careful measurements of &(T) and A&")
in a suspension of aligned ultrafine
YBCO particles smaller than hob were performed by Neminsky and Nikolaev (1993).
who found a power-law of the form Aob(T)/Aab(0) % A,(T)/A,(O) % 1 0.34(T/Tc)Z.5,
which they interpret as due to a T-dependent inelastic scattering length lin(T) inserted in
A(T)/A(O) = 1+.$/2li,. Buzdin etal (1994) calculate and measure the quasi-20 fluctuation
contribution to the anisotropic penetration depth in YBCO and find excellent agreement.
+
9.7.Flux creep and activation energy
The decay of screening currents, or of the magnetization, in principle is completely
determined by the geometry and by the resistivity p(T, B , J). In superconducting rings and
hollow cylinders in an axially oriented magnetic field one simply has M E J a p ( T , B , J).
Within the Kim-Anderson model (9.3) the decay of persistent currents in this geometry
was calculated analytically for all times (Zhukov 1992, Schnack etal 1992). Twelve orders
of magnitude of the electric field E = 10-13-10-1Vm-1 were measured by Sandvold
and Rossel (1992) combining current-voltage curves withhighly sensitive measurements
of decaying currents in rings of YBCO films of 3 mm diameter, 0.1 mm width, and 200nm
thickness. In a nice work Kunchur et al (1990) show that by the superposition of decaying
flux distributions, usual flux creep can lead to a memory effect, which had been observed
by Rossel er a[ (1989) but was interpreted as glass-like behaviour. Dislocation-mediated
('plastic') flux creep is considered by van der Beek and Kes (1991a), andMee etal (1991)
investigate creep in a granular HTSC with a distribution of coupling strengths.
The current dependent activation energy U ( J ) may be extracted from creep experiments
by the method of Maley et 01 (1990), figure 28, see e.g. van der Beek et a1 (1992a), Kung
et al (1993). Sengupta et a1 (1993), and Rui et al (1995) (for a Hg-based HTSC). The KimAnderson model (9.3) originally means an effective activation energy E Jc - J, cf equation
(9.4b), and actually corresponds to a zig-zag shaped pinning potential, which has a barrier
height proportional to the current-caused tilt of the total potential. As shown by Beasley et
al (1969), a more realistic rounded potential yields U cx (Jc - J)3/2. Collective creep theory
yields U a l / J u with 01 > 0 depending on E and on the pinning strength. For single-vortex
pinning one predicts 01 = f; 01 crosses over from to 1 to as the vortex bundles increase
from small (a < R I , RII% L, e A) to intermediate (a e RI < A < RI % L,) to large
(A < R I , RI% L,) size. Here RI,RN,and L, denote the transverse or parallel (to the flux
motion) and L, the longitudinal (to the field) dimensions of the bundle (section 8.7), and a
is the flux-line spacing (Feigel'man et a1 1989, Nattermann 1990, Fisher and Nattermann
1991, Blatter et ai 1994a). For the more general (uniaxial) aniso&opy situation, the result
is more complicated and may be found in Blatter et al (1994a). These various exponents
a are nicely seen in proton irradiated YBCO by Thompson et al (1994). Dekker et a1
(1992a) find 01 = 0.19 at low, and 01 = 0.94 at high current densities. Many experiments
suggest a logarithmic dependence U
In(Jz/J) (Ries et a1 1993, Gj0lmesli et al 1995),
corresponding to the limit 01 + 0 or to a logarithmic interaction between flux lines (Zeldov
et a1 1990) or point vortices (Zhukov et al 1994). For U In(Jz/J) the creep rate can be
2
-
-
1558
E H Brand?
-....
Y
2
---.. -:-=:.,
100 :
0
0
~
"
~
"
2 io9
'
'
4
'
~
io9
-. ,.. ,
'
"
~ ~
6 IO9
'
'
~
~
a io9
j ( Am" )
Figure 28. The current-density dependence ofthe activation energy U ( J ) extracted from several
experimenls on lhe s a BSCCO single cryswl 3t different lemperatures (5-21 K) using the
method of Mdey et ai (1990). Plotred in this way, J
!a curves nearly fall on one fine. and a
nearly temperature-independent U ( J ) is obtained (courtesy of C I van der Beek. see also van
der Beek el nl 1992a).
-
calculated analytically (Mnokur eta! 1991a) and a power-law current-voltage curve E J"
holds which leads to the nonlinear diffusion equations ( 8 . 4 ~ )(Rhyner 1993, Cilchrist and
van der Beek 1994) and to self-organized criticality.
for J << J, with the KimCombining the collective creep result U ( J ) = Uc(Jc/J)u
Anderson formula U ( J ) = U,(1 - J / J c ) for J % J, one obtains for the decaying current
density in rings and cylinders the interpolation formula
where to is an integration constant. Formula (9.16) is confirmed nicely, e.g. with YBCO
by Thompson et a1 (1991). From (9.16) one obtains a temperature dependent J ( T ) and the
normalized creep rate S (Malozemoff and Fisher 1990, Feigel'man et al 1991)
(9.18)
Experiments on HTSCs by Senoussi et al (1988) yield TO % 10K and U,/kB ZZ 100IOOOK. Note that the decay rate S (9.18) decreases with increasing time and saturates at
kBT z U,/[olln(t/to)] to a value I/[a In(?/to)]. Numerical solutions to the relaxation rates
for various dependencies U ( J )are given by Schnack et af (1992). Caplin etal (1995) show
that the often measured value S = 0.03 reflects the experimental conditions rather than being
a characteristic feature of the underlying vortex dynamics. McElfresh et al (1995) measure
and explain the influence of a transport current on flux creep in YBCO films.
Based on the concept of self-organized criticality or avalanche dynamics (Bak et af 1988,
Tang and Bak 1988, Tang 1993, Ling et al 1991, Field et al 1995, Bonabeau and Lederer
1995), Wang andShi (1993) derived anexpression J ( t ) = J,--J,[(ksT/Uo)In(l+B?/s)ll/B
which is similar to (9.16) and for B = 1 coincides with the Kim-Anderson result, see also
~
'
the flux-line lattice in superconductors
1559
the measurements of creep by micro Hall probes by Koziol et a1 (1993). Okomel'kov and
Genkin (1995) by a microscopic pinning model show that the creep rate increases when
the interaction between vortices is increased; this finding agrees with the result of statistical
summation in section 8.7. Good fits to relaxation rates are also achieved by fitting a spectrum
of activation energies (Hagen and Griessen 1989, Griessen 1990, Gurevich 1990b. Niel and
Evetts 1991, Martin and Hebard 1991, Theuss eta1 1992a,b, Theuss 1993). The success
of theories which assume only one activation energy U suggests that at given values of E ,
J and T essentially one effective U of an entire specbum determines the physical process
under consideration.
9.8. Universality offlux creep
Gurevich and Kiipfer (1993) recently p o i n h out a new simplicity in the flux-creep problem.
When the applied field H,(t) is increased or decreased with constant ramp rate fiu and then
held constant for times t
0, then after some transition time the electric field becomes
universal taking the form E ( r ,t ) = F ( r ) / ( t + s )with F(r) a space-dependent function and
5 cx l/lfial. T h universal behaviour holds if the current-voltage law E ( J ) is sufficiently
nonlinear, namely, if one has aE/aJ = E/JI with a constant J1. This condition is exact if
E ( J ) = Eoexp(J/J1) (Anderson's creep (9.3b)) and it holds to a good approximation for
a wide class of E ( J ) curves, e.g. for E c( J" with n >> 1 (it was checked numerically that
practically the same function F(r) results for all n 2 3 (Gurevich and Brandt 1994)) or for
the vortex-glass and, collective creep models which yield
E ( J ) = E,exp[U(J)/keT]
U ( J ) = [ ( J c / J ) " - 1JUo
(9.19)
>
AfW a sufficiently long ramp period the superconductor has reached the critical state
corresponding to its E ( J ) and to the geometry; in this saturated state E ( r ) is a linear
function of T since V x E = -@OH,,
see also Zhukov et a1 (1994). When ramping is
stopped at t = 0, then the linear E ( r ) starts to curve and to decrease until, after a transient
period of duration = t (Gurevich et a1 1991, Gurevich 1995b), the universal behaviour
E ff l / t is reached. In particular, for a slab of width 20 (1x1 < a ) in a parallel field
Gurevich and Kupfer (1993) obtain
(9.21)
This universality also applies to transverse geometry, figure 29; for a strip of width 2a
(1x1 < a ) and thickness d <<a in a perpendicular field Gurevich and Brandt (1994) obtain
(9.22)
f ( u ) = (1 u)ln(l + U ) - (1 - u)ln(l - U ) - 2u lnlul
and a similar expression for the circular disk.
The current density J ( r ,t ) , magnetic field H ( r ,f ) , and magnetization M(r) during
creep are obtained by inserting the universal E ( r , t ) into the specific relation E ( J ) ,
e.g. J ( x , t ) a E(x, t)"" for E cx J" or J ( x , f ) = Jc - J I In[ E,/E(x, t ) ] for the vortex
glass and collective creep models (9.,20). The latter yield in both geometries
+
M(t) =
I
M(0) - M I In(1
Mc
+ t/rd
- MI Wtlto)
(0 < t
< b)
(t >> to)
(9.23)
(9.24)
1560
E H Brandt
2
1
-
0
0
1
7)
Figure 29. Upper cumex Universality of the electric field E(x.1) and E(,, I) approached
during flucreep in strips of half width a (1x1
a) and disks of radius U (r G a). We
show the normalired field E(q,t)/E(I,t) against q = x j a = rjo obtained numerically for
power-law resistivities E(J) = (J/Jc)"Ec for n = 1.3. and W. Thc asymptotic curve n = m
meavy curve) (9.22) is nearly reached at n 9. Lower NWLI. Relaxation (creep) of E ( x . I)
for a ship with n = 25 al times I = 0.0.003,0.01,0.03,0.1,0.3,
1 and 3 in units of time
II = udlcoJ&rE, sz nto (cf (9.24). Jc = Jln) (aRer Gurevich and Brandt 1994).
<
where for parallel geometry to = 0.66Jla/lHgl, M I= a2J1, to = 8a2fi~Jl/e3E,,and for
perpendicular geometry TO = 0.41Jld/lHal, M I= a2J1,to = adp&/4Ec.
In a nice work Cai et a1 (1994) applied these results to extract the unrelaxed irreversible
magnetization from combined magnetization and flux-creep measurements, finding that the
anomalous 'fish tail' or 'peak effect' observed by many groups in the critical current density
and thus in the irreversible magnetization and torque, is a static rather than a dynamic
effect; this is debated by other authors, see, e.g. Schnack et al (1992), Yeshurun et a1
(1994). van Dalen et a1 (1995) and J i a et a1 (1995). A method to extract the 'real'
critical current (not influenced by creep) is developed by Scbnack et a1 (1993) and Wen et
a1 (1995). The influence of the viscous motion of flux lines on the magnetic hysteresis is
considered by Takircs and Gomory (1990) and by Wst (1992). As discussed above, creep
does always occur in superconductors which are not in magnetic equilibrium, in particular
during variation of the applied field; this fact is described in a unified picture by treating
the superconductor as a nonlinear conductor, cf equations (8.4) and (8.18). Therefore, in
HTSCs the measured J, will always depend on the time and length scales of the experiment
as confirmed nicely by the ax. experiments of Fhbrega et al (1994a). For more work on
the 'fish-tail effect' see Ling and Budnick (1991), Kohayashi et a1 (1993), Klupsch et a1
(1993). Gordeev etal (1994). Solovjev e t a l (1994). Perkins et a1 (1995), and D'Anna et
al (1995). A careful analysis of the fish-tail effect in YBCO and TmBCO with various
oxygen content was recently performed by Zhukov et al (1995b). An explanation of the
peak effect in terms of plastic flow of the FLLis given by Bhattacharya and Higgins (1994),
while Larkin et a1 (1995) calculate a sharp peak effect for twinned YBCO crystals; see also
the flux-line lattice in superconductors
1561
the experiments on t
his by Zhukov et al (1995a).
An interesting difference between flux creep in these two geometries is that the universal
E ( x ) is monotonic in parallel geometry but has a maximum in perpendicular geometry at
x = a / d for the strip and x = 0.652 for the disk. In perpendicular geometry after the
increase of & ( t ) is stopped, the creeping magnetic field H ( T , t) increases near the centre
as expected, but it decreases near the edges because of demagnetizing effects. The regions
of increasing and decreasing H are separated by a 'neutral line' at which H ( r , t) = H,
is constant during creep. For the strip two neutral lies run at 1x1 = a/& where E ( x ) is
maximum. For the disk a neueal circle appears at r = 0.876a where r E ( r ) is maximum.
For squares and rectangles the neutral line, defined by H,(x,y) = Ha in (8.14), has a
star-like shape (figures 18, 19) which was observed by magneto-optics by Schuster et a1
(1995a). For sample shapes differing from a circular disk or infinite strip, flux motion is a
genuine ZD problem and the creep-solution for the electric field E(x, y, t ) does not factorize
into time and space functions. As a consequence, the neutral line slightly changes its shape
during a long creep time, but probably this effect is difficult to observe.
A universality of a different type is discussed by Geshkenbein et a l (1991). In the
critical state, i.e. during flux creep, the response to a small pembation SB(x, t) << B is
described by a diffusion equation in which the time is replaced by the logarithm of the time,
see also section 8.3. As a consequence, in the case of complete penetration the relaxation
is universal and does not depend on the magnetic field. The absorption of a small a.c. field
6 B cos(ot) is then proportional to the inverse square root of the waiting time (creep
time) t , ~ " ( w ) I / a . This universality of the linear response during flux creep applies
also to perpendicular geometry (Brandt and Gurevich 1996, Gurevich and Brandt 1996).
-
-
I
Fwre 30. Flux creep in a disk with nonlinw current-voltage law E ( J ) = ( J / J , ) 1 9 E c after
the increase af H, was stopped at I = 0. Shown is the relaxation of the sheet current Js(r,I)
and magnetic field H ( r , f ) both in units .
I
&
at times f = 0.01, I , 100, lod. IO6, IO8, and
10I2 in units of 11 = adfloJ,/ZnE,. Insel. The magnetization M ( 1 ) at f >> ti is approximately
M ( I ) / M ( o ) sz ( i l / i ) l / l B = I - fg In(t/tl).
1562
E H Brandt
9.9. Depinning by tunnelling
In numerous experiments the creep rate plotted agajnst T appears to tend to a finite value
at zero temperature. Flux creep observed at low temperatures by Lensink et a1 (1989),
Mota eta1 (1989, 1991), Griessen etal (1990), Fruchter etal (1991), Lairson etal (1991).
Liu er al (1992), Prost et af (1993). and Garcla et a1 (1994). and in ultrathin Nb wires
by Ling et nl (1995b), in principle may be explained by usual thermal activation from
smooth shallow pinning wells (Griessen 1991), but more likely it indicates ‘quantum creep’
or ‘quantum tunnelling’ of vortices out of the pins as suggested first by Glazman and Fogel’
(1984a) and Mitin (1987). The thermal depinning rate a exp(-ll/keT) is now replaced
by the tunnelling rate a exp(-SE/fz) where SE is the Euclidean action of the considered
tunnelling process; see Seidler er al (1995) for a recent measurement of SE. Rogacki et
a1 (1995) measure the low-temperature (T 2 0.1 K) creep rate in DyBCO crystas by the
vibrating reed technique.
Tunnelling of vortices differs from tunnelling of particles by the smallness of the inertial
mass of the vortex (Suhl 1965, Kuprianov and Likharev 1975, Hao and Clem 19916, Coffey
and Clem 1991b, Simgnek 1992, Coffey 1994a, Blatter and Geshkenbein 1993, Blatter et
a1 1994a. b). This means that the vortex motion is overdamped and there are no oscillations
or resonances. This overdamped tunnelling out of pins is calculated by Ivlev et al (1991~)
and Blatter et a1 (1991a) and recently by Gaber and Achar (1995). Griessen et al (1991)
show that the dissipative quantum tunnelling theory of Caldeira and Leggett (1981) with
the usual vortex viscosity q inserted reproduces the overdamped results. A very general
theory of collective creep and of vortex-mass dominated quantum creep in anisotropic and
layered superconductors is given by Blatter and Geshkenbein (1993) and a short review on
flux tunnelling by Fruchter et a1 (1994). Since their coherence length is so small, HTSCS
may belong to a class of very clean type-Il superconductors in which quantum tunnelling
of vortices is governed by the Hall term (Feigel’man er a1 1993b. Garcla et al 1994, Ao
and Thouless 1994, Boulaevski er al 1995b). Macroscopic tunnelling of a vortex out of
a SQUID (a small superconducting loop with two Josephson junctions) was calculated in
the underdamped case (Ivlev and Ovchinnikov 1987) and in the overdamped limit (MoraisSmith el a1 1994).
10. Concluding remarks
This review has not presented general phase diagrams for the Fu in HTSCS since these
might suggest a universality which is not generally valid. Lines in the B-T plane which
separate ‘phases’, e.g. the ideal n L or elastically distorted FLL from the plastically distorted
or amorphous or liquid n ~are, not always sharp and in general depend on their definition
and on the pinning strength and purity of the material, and also on the size and shape of
the specimen and on the time scale or frequency of the experiment.
For example, an irreversibility line or depinning line may be defined as the line Tdp(B)
at which the imaginary (dissipative) part of the linear a.c. permeability ~ ‘ ‘ ( w )(9.13) of
a HTSC in the TAFF regime is maximum for a given frequency (Kes et nl 1989, van der
Beek er a1 1992a, Seng et a1 1992, Karkut et a1 1993). This condition coincides with
the maximum attenuation of a vibrating HTSC. Superconductors performing tilt vibrations
in a magnetic field have been used extensively to study elastic pinning (Esquinazi 1991,
Brandt et nl 1986a,b, Esquinazi er a1 1986, Brandt 1987) and depinning lines (Gammel
ef a1 1988, Kober et nl 1991b, Brandt 1992a. Gupta et a1 1993, Ziese et al 1994a,b,c.
Drobinin et a1 1994, Shi et a1 1994). A typically sharp maximum in the attenuation occurs
the flux-line lattice in superconductors
1563
when the a.c. (or vibrational) frequency w/2n coincides with a reciprocal relaxation time
(urd = l), or the skin depths equals half the specimen thickness (d/26 = l), or when the
thermally activated resistivity p ~ m ( ET)
, = pmexp[LI(B, T)/keT] goes through the value
pd, = wd2p&r2. From these three equivalent conditions it is clear that such a depinning
line in the B-T plane depends on the frequency and on the size of the specimen, at least
logarithmically. In certain geometries the attenuation should even exhibit two peaks, which
may belong to two different relaxation modes (Brandt 1992b. Ziese et a1 1994~).but other
reasons for the occurrence of two or more peaks are also possible (Durin et a/ 1991, Xu et
al 1992, Arribkre et a/ 1993, Jos.6 and Ramiirez-Santiago 1993, D’Anna et a1 1992, 1994a,
Giapintzakis et a1 1994).
Such peaks, therefore, do not necessarily indicate a melting (or other) transition in the
FLL (Brandt et a1 1989, Kleimann et a/ 1989, Geshkenbein et a/ 1991, Steel and Graybeal
1992, Ziese et a/ 1994a,b, c). Moreover, an observed ‘melting line’ may also correspond to
the vanishing of a geometric edge-shape barrier (section 8.5) as shown by Indenbom et a1
(1994a, b) and Grover eta/ (1990). In extremely weak pinning YBCO specimens, however,
the observed sharp peak in the attenuation of slow torsional oscillations probably indicates
a melting transition in the FLL (Fame11 et U! 1991, Beck et a1 1992). see the discussion in
section 5.6. Notice also that the glass &ansition detected in YBCO and BSCCO by Kotzler
er al (1994b) by measuring the complex ax. susceptibility x of a disk over ten orders of
magnitude of the frequency, is not seen directly in the raw data x ( B , T, w) (figure 24) but
is evident in the phase angle of the complex resistivity p ( E , T, w) evaluated from these
data, section 9.5.
Several topical features of the KL, which give further insight into the properties of the
pinned or moving K L , for brevity could not be discussed here in detail, e.g. the Hall effect
(section 7.3, the thermo-effects (section 7.6), and the flux noise (section 7.7) reviewed by
Clem (1981b). Another such topical problem is the enhancement of pinning by an applied
electric field (Mannhardt er a[ 1991, Mannhardt 1992, Burlachkov et nl 1993, Jenks and
Testardi 1993, Lysenko et a/ 1994, Frey et a1 1995, Ghinovker et a1 1995). Also omitted
from this review were the rich properties of magnetic superconductors (Krey 1972, 1973),
which may contain self-induced and spiral flux lines, see the reviews by Tachiki (1982)
and Fulde and Keller (1982). of heavy-fermion superconductors like UPt3 (Kleiman et a1
1992). and of possible ‘exotic’ superconductors with more-component order-parameter or
with d-wave (or %+id’-wave) pairing; see, e.g. Abrikosov (1995). Recent theories of the
vortex structure in d-wave superconductors are given by Ren el a/ (1995). Berlinsky et a/
(1995), and Schopohl and Maki (1995).
This overview was intended to show that the discovery of high-T, superconductivity has
revived the interest in the flux-line lattice. The intriguing electromagnetic properties of the
HTSCS inspired numerous novel ideas related to the layered structure and short coherence
length of these cuprates, which both favour large thermal fluctuationsand thermal depinning
of the FLL. At the same time, old classical work on the FLL was rediscovered and improved
or adapted to these highly anisotropic and fluctuating superconductors. This review only
could present a selection from the wast amount of work performed in a very active field of
research.
Acknowledgments
During the time in which this review was written many helpful conversations with colleagues
contributed to the clarification of the numerous issues. I particularly want to thank Misha
Indenbom for numerous suggestions and Alehander Gurevich, Alexei Koshelev, Anatolii
1564
E H Brandl
Larkin, Yuni Ovchinnikov, Gianni Blatter, Dima Geshkenbein, Alexei Abrikosov, Lev
Bulaevskii, Harald Weber, Uwe Essmann, Lothar Schimmele, Thomas Schuster, Michael
Koblischka, Holger Kuhn, Alexander Forkl, Horst Theuss, Helmut Kronmiiller, J i g e n
Kotzler, Asle Sudbg, John Clem, Ted Forgan, Jan Evetts, Archie Campbell, Peter Kes,
Joe Enen, Colin Gough, Mike Moore, Boris Ivlev, Denis Feinberg, John Gilchrist, Ronald
Griessen, Helene Raffy, Sadok Senoussi, Ian Campbell, Kazuo Kadowaki, Yoichi Ando,
Richard Klemm, George Crabtree, Valeri Viokur, Vladimir Kogan, Marty Maley, Igor
Maksimov, Aleksander Grishin, Aleksander Zhukov, Victor Moshchalkov, Miodrag KuliC,
Gertrud Zwicknagl, Roland Zeyher, Roman Mints, Eli Zeldov, Terry Doyle, Edouard Sonin,
Yurii Genenko, Klim Kugel, Leonid Fisher, Georg Saemann-lschenko, Roland Busch,
Milltino Leghissa, Flonan Steinmeyer, Rudolph Huebener, Rudolf Gross, Konrad Fischer,
Pablo Esquinazi, Dierk Rainer, Werner Pekh and B%d Thrane. Thanks are also due to Uwe
Essmann, Yasuhiro Iye, Martino Leghissa, Thomas Schuster, Gerd Nakielski and Kees van
der Beek for figures 4, 11, 13, 20, 24, 25 and 28. Part of this work was performed at the
Institute for Scientific Interchange in Torino, Italy; this is gratefully acknowledged.
References
Abrikosov A A 1957 zh. Ehp,Teor. Fk. 32 1442 @ngl Transl. 1957 SOY.Phys,-JETP 5 1174)
-I*
U.Ekrp. Teor. Fir. 47 720 (Engl. Transl. 1965 Sov. Phys.-JETP 20 480)
-1988
Fundamentals ofthe 7hcory of Merals (Amsterdam: Nonh-Holland)
-1995
Phys. Rev. B 51 11955; I995 Pkysica 7A4C 243
Abrikosov A A, Budin A I, KuliC M Land Kuptsov D A 1989 Supercond Sci. Zcknol. 1 260
Adamopoulos N, Soylu B. Yan Y and Evetts J E 1995 Physica 242C 68
Adrian S D, Reeves M E, Wolf S A and Krezin V 2 1995 Phys. Rev. B 51 6800
Alfimov G Land Silin V P 1994 U.Ehp. TED?.F k 106 671 (Engl. T m l . 1994 So". Phys.-JETP 79 369)
Aliev Yu M and Silin V P 1993 Phyr. Lett. 177A 259
Almasan C C, de An&
M C, Dalichaouch Y. Neumeiec J 1. Seaman C L. Maple M E, Guertio R P. Kuric M
V and Garland J C 1992 Phys. Rev. B 69 3812
Alves S. Boek#pa C, Halim C, Lam J C, Whang E,Cooke D W and Leon M 1994 Phys. Rev. B 49 12396
Ambegaokar V and Baramff A 1963 Phys. Rev. Lett 10 486 (erratum 11 104)
Ambeganhr V and Halperin B I 1969 Phys. Rev. bit. 22 1364
Ambj0m J and Olesen P 1989 Phys. Lett. 218B 67
Amrein T,Schnltz L, Kabiur B and Urban K 1995 Phys. Rev. B 51 6792
Anderson P W 1962 Phys. Rev. Lett. 9 309
Anderson P W and Kim Y B 1964 Rev. Mod. Phys. 36 39
Ando Y,Kubota H,Sat0 Y and Terasaki I 1994 Phys. Rev. B 50 9680
Ando Y, Kubota H and Tanaka S 1992 Phys. RN. Len. 69 2851
Phys. Rev. B 48 7716
-1993
Ando Y, Motohira N, Kitazawa K, Takeya J and Akita S 1991 Phys. Rev. Lett 67 2737
And& M-0. D'Ama G. lndenbom M V and Benoit W 1994 Physica Z35-4OC 2627
Anlage S M and Wu D-H 1992 J. Supereond 5 395
Antognazza L, Berkowitz S J. Geballe T H and Char K 1995 Phys. Rev. B 51 8560
Ao P 1995 1. Supercond 8 1
A0 P and Thouless D J 1993 Phys. Rev. Lett. 70 2158
Phys. Rev. Len. 72 132
-1994
A0 P and Zhu X-M 1995 Phys. Rev. Lett. 74 4718
Aranson I, Gitterman M and Shapio B Ya 1995 Phys. Rev. B 51 3092
Amnov A G and Rapport A B 1992 Mod. Phys Len. B 6 1083
Anibere A, Pastorira H, Goffman M F, de la Cruz F, Mitzi D B and Kapitulnik A 1993 Phys. Rev. B 48 7486
Artemenko S N, -er
C and Wonneberger W 1994 Physica 22OC 101
Artcmenko S Nand Kruglov A N 1990 Phys. Lett. 143A 485
Artemenko S N and Lalyshev Yu I 1992 Mod Phyr. Lett, B 6 367
Ashkenay V D, J u g G. Khalfin I B and Shapim B Ya 1994 Phys. Rev. B 50 I3679
-1995a
Phys. Rev. B 51 1236
The flux-line lattice in superconductors
1565
Ashkenazy V D,lung G,and Shapim B Ya 1995b Phys. Rev. B 51 9052.91 18
Aslamazov L G and Larkin A I 1968 Phys. k f r . MA 238
Atsarkin V A, Vasneva G A and Zybtsev S G 1994 Supercond Sci. Technol. 1423
Auer I and Ullmaier H 1973 Phys. Rev. 7 136
Baal M. Metlushko V V, Jonckheere R, Moshchalkov V V and Bruynsenede Y 1995 Eumphys. Len. 29 157;
1995 Phys. Rev. Len. 74 3269
Bahcall S R 1995 Phys. Rev. Left, 75 1376
Bak P, Tang C and Wiesenfeld K 1988 Phys. Rev. A 38 364
Balatskil A V. Burlachkov L I and Gor’kw L P 1986 Ur Ehp. %or. Fiz. 90 1478 (Engl Transl. 1986 Sov.
Phys.-JETP 63 866)
Balents L and Kardw M 1994 Phys. Rev. B 49 13030
Balents L and Nelson D R 1994 Phys. Rev. Len. 73 2618
Balents L and Simon S H 1995 Phys. Rev. Len. 51 6515
Balestrino G, Crisan A, Livanov D V, Milani 9, Montuori M and Varlamov A A 1995 Pkys. Rev. B 51 9100
Balestrino G, Marinelli M, Milani E, Varlamov A A and Yu L 1993 Pkys. Rev. B 47 6037
Bamduc C. Buzdin A, Henry I-Y, Brison I-P and Puech L 1995 Physica 248C 138
Bardeen J 1918 Phys. Rev. B 17 I472
Bardeea~J. C o o p r L N and ScMeffer I R 1957 Phys. Rev. 108 1115
Bardeen J and Sherman R D 1975 Phys. Rev. B 12 2634
Bardm I and Stephen M I 1965 Phys. Rev. A 140 1197
Barford W and Gunn M 1988 Physica 156C 515
Barford W and Harrison M 1994 Phys. Rev. B 50 13748
Barone A, Larldn A I and Ovchinnikov Yu N 1990 J. Supercond. 3 155
Barone A and Paterno G 1982 Physics nnd Applicatiom ofthe Josephson Effect (New York Wiley)
Barowski H. S d e r K M and Schoepe W 1993 J. LoH1 Temp. Phys. 93 85
BaUouni G G and Hwa T 1994 Phys. Rev. Leu. 72 4133
Baym G 1995 Phys. Rev. B 51 I1697
Bean C P 1964 Rev. Mod Phys. 36 31
Bean C P 1970 J. AppL Phys. 41 2482
Bean C P and Livingston I D 1964 Phys. Rev. Len. 12 14
Beasley M R, Labusch R and Webb W W 1969 Phys. Rev. 181 682
Beck H 1994 Phys. Rev. B 49 6153
Beck R G, Farrell D E. Rice J P, Ginsberg D M and Kogan V G 1992 Phys. Rev. Len. 68 1594
Becker Th, Them H. Schuster Th, Kronmiiller H, Kraus M and Saemann-lschenko G 1995 Physica 245C 273
Bednorz I G and Mliller K A 1986 2 Phys. B 64 189
Behler S, Jess P, Baratoff A and GUntherodt H-J 1994a Pkysica W W O C 2703
Behler S, Pan S H, Jess P. Baratoff A, GUntherodt H-I, My F, Wuth 0 and Wiesner I 1994b Phys. Rev. Len. 72
1750
Behr R. Klirzler J, Spirgatis A and Ziese M 1992 Physica 191A 464
Bending S J, van Kliaing K and Ploog K 1990 Phys. Rev. B 42 9859
Benkraouda M and Clem J R 1996 preprint
Benkraouda M and Ledvij M 1995 Phys. Rev. B 51 6123
Berezinskii V L 1970 UI.Ehp. Teor. Fir. 59 901 (Engl. Transl. 19711 Sov. Phys.-JETP 34 610)
Berghuis P and Kes P H 1990 Physica 16568 1169
Berghuis P. van der Slot A L F and Kes P H 1990 Phys. Rev. Len. 65 2583
Be@ W and Tewordt L 1970 2 Phys. WO 178
Berkovits R and Shapiro B Ya 1995 Phys. Rev. Len. 70 2611
Berlinsky A I, Fetter A L,Franz M, Kallin C and Soininen P I 1995 Phys. Rev. Leff.75 2200
Bevyadin A, Buzdin A and Pannetier B 1995 Phys. Rev. B S 3718
Bevyadin A and Pannetier B I995 1. Low Temp. Phys. 98 251
Bhagwat K V and Chaddah P 1992 Physica 19062 444
Bhattaeharya S and Higgins M I 1993 Phys. Rev. Len. 70 2617
-1994
Phys. Rev. B 49 10005
-1995
Phys. Rev. B 52 64
Bhattacharya S, Higeins M I and R&shnan
T V 1994 Phys. Rev. Leff.73 1699
Bishop D 1. Gammel P L. Huse D A and Murray C A 1992 Science 255 165
Blackstead H A, Pulling D B, Horwitz J S and Chrisey D B 1994 Phys. Rev. B 49 15335
Blamire M G and Evens I E 1985 Appl. Phys. Len. 46 1181
1566
E H Brandt
Blatter G, Feigel'man M V. Geshkenbein V B. Larkin A I and Vmokur V M 1994a Rev. Mod. P h y . 66 1125-388
Blmtcr G and Geshltenbein V B 1993 Phys. Rev. B 47 130
Blaner G. Geshkenbein V B and L A n A I 1992 Phys. Rev. LelI. 68 875
-1993a
Phys. Rev. Left C 71 302
Blaner G. Geshkenbein V B and Wnoku V M 1991a Phys. Rev. Len. 66 3297
Blatter G and Ivlev B 1993 Phys. Rev. Len. 70 2621
Blaner G and lvlev B 1994 Phys. Rev. B 50 10272
Blatter G and lvlev B 1995 Phys. Rev. B 52 4588
Blaner G, lvlev B, Kagan Yu, Theunissen M. Volotkin Ya and Kes P 1994b Phys RCX B 50 13013
Blatter G, lvlev B and Nordborg H 1993b Phys. Rev. B 48 10448
Blaner G , lvlev B and Rhyner J 1991b Phys. Rev. Leff. 66 2392
Blaner G , Rhyner J and Wnokur V M 1991c Phyr Rev. B 43 7826
Blazey K W. P o d A M, Muller K A and Holeborg F 1988 Ewophys. Lelf. 462 1988
Blum T and Moore M A 1995 Phys. Rev. B 51 15359
Bokil H S and Young A P 1995 Phys. Rev. Len. 74 3021
Bolle C A , Gammel P L,Crier D G . Munay C A , Bishop D J, Mitzi D B, and KapiNlnik A 1991 Phys. Rev. Leu.
66 112
Bonevich J E, Hamda K. Kasai H. Matsuda T, Yoshida T,Pozn G and Tonomua A 1994a Phyx Rev. B 49 6800
Bonevich J E, H
d K, Kasai H.Matsuda T. Yoshida T and Tonomura A 1994b Phyr. Rev. B 50 567
Bontemps N. Davidov D. Monod P and Even R 1991 Phys. Rev. B 43 I1512
Boroviukaya E S and Genldn V M 1995 Physica C (in print)
Bouchaud J-P. Mdzard M and Yedida S 1992 Phys, Rev. B 46 14686
B m d t E H 1969a Phys. Star. Sol. B 35 1027.36 371
-1969b
Phys. Slur. Sol. B 36 381, 393
-1972a
Phys. Star. Sol. B 51 345
-1972b
Phys. Len. 39A 193,227
-1973a
Phys. Lefl. 43A 539
-1973b
Phys, Sfaf. Sol. B 57 277, 465
-1974
Phys. Sfat. Sol. B 64 257,461.65 469
-1975
Phys. Star. Sol. B 71 277
-1976a
Phys. Sfor.Sol. B 77 105
-1976b
J. Low Temp. Phys. 24 409
-1976~ Phys. Slat. Sol. B 77 551
-1977a
J. Low Temp. Phys. 26 709.735
-1977b
J , Low Temp. Phys. 28 263,291
-1977~ Phys. SlQf. Sol. B 84 269.637
-1978
Phil Mag B 37 293
-1980a
Phys. Len. 77A 484
-1980b
J. Low Temp. Phys. 39 41
-1980~ Phys. Lelr. 79A 207
-1981a
J. Low Temp. Phys. 42 557
-1981b
J. Low Temp. Phys. 44 33. 59
Phys. REV.B 25 5756
-1982
-1983a
Phys. Rev. Lem. 50 1599
-1983b
J. Low Temp. Phys 53 41. 71
-1986a
Phys, Ree B 34 6514
-1986b
J . Low Temp. Phys. 64 375
-1986~
Plrys. Rev. Leu. 56 1381
-1986d
Phys. Rev. Len 57 1347
-1987
J. Physique C 8 31
-1988a
Phys. Rev. B 37 2349
J. Low Temp. Phys. 73 355
-1988b
-1988~
Appl. Phys. Leu. 53 1554
-1989a
Phys. Rev. Len. 63 1106
-1989b
Physicu 162-4C 1167
-1989~ Science 243 349
-1990a
Am J. Phys, 58 43
-1990b
Z Phys, B 80 167
The &dine lanice in superconductors
1567
-1990~
Physica 1 6 M C 1129
-1991a
Physica 1698 91
J. Mood Phys. B 5 751
-1991bln1.
-1991c
Phys. Rev. Lett. 66 3213
-1991d
Phys. Rev. Lett. 67 2219
-1991e
Physicn 18S9C 270
-1992a
Physic= 195C I
-1992b
Phys. Rev. Lett. 68 3769
-1992c
Phys, Rev. Len 69 1105; 1992 Europhys Lett. I8 635
-1992d
Phys. Rev. B 46 8628
-1993a
Phys, Rev. B 48 6699
-1993b
Phys. Rev. Lea 71 2821
-1994a
Phys. Rev. B 49 9024
-1994b
Phys. Re,'. B 50 4034
-1994~ Phys. Rev. B 50 13833
-1994d
Physica 235-40C 2939
Phys. Rev. Lett. 74 3025
-1995a
-1995b
Phys. Rei,. Len, C 74 3499
-199%
Phys. Rev. B (submitted)
Brandt E H, Clem I R and Walmsley D C 1979 J. Low Temp. Phys. 37 43
Brandt E H, Esquinazi P and Neckel H 1986a J . Low Temp. Phys. 63 187
Brandt E H, Esquinazi P, Neckel H and Weiss G 1986b Phys. Ret lean. 56 89
Brandt E H , Esquinazi P and Weiss G 1989 Phys. Rev. Lett. C 62 2330
Brandt E H and Essmann U 1975 fhys. Lett. %A 45
-1987
Phys. Star Sol. B 144 13
Brandt E H and Eveus 3 1992 Concise Encyclopedia of Magnetic and Superconducting Materials ed I Eveus
(Oxford: Pergamon) p 150
Bmdr E H and Gurevich A 1996 Phys. Rev. Len. 76 (submitted)
Bmndt E H and lndenbom M 1993 Phys. Rev. B 48 12893
B m d t E H.lndenbom M and Forkl A 1993 Europhys. Lett 22 735
Brandl E H. hlinls R G and Shapiro I E 1996 Phys. Rev. Lett. 76 (submitted)
Brandt E H and Seeger A 1986 Adv. Phys. 35 189
Bmndl E Hand Sudtw A 1991 Physica 18OC 426
Brandt U, Pesch W and Tewordt 1967 2 Phys. 201 209
Brass A and Jensen H I 1989 Phys. Rev. B 39 9587
Brawner D A and Ong N P 1993 1. Appl. Phys. 73 3890
Brawner D A, Ong N P and Wang 2 2 1993 Phys. Rev. B 47 1156.48 13188
Brechet Y J M, Doupl B. Jensen H J and Shi A-C 1990 Phys. Rev. B 42 21 16
Brice130 G. Crommie M F and Zed A 1991 Phys Rev. Lett. 66 2164
-1993
PhyAica 204C 389
Brom H B and Alloul H 1991 Physien 177C 297
Brongenma S H, Verweij E. Koema N I. de Groat D G,Griessen R and lvlev B I 1993 Phys. Rev. Len. 71 2319
BriiU P, KirchgsSsner D and Leiderer P 1991 Physicn 182C 339
Brunner 0. Antognazza L. Triscone J-M, Midville L and Fisher 0 1991 Phys. Rev. Len. 67 1354
Bryksin V V and Dorogovtsev S N 1993 Physics 215C 173
Buck W. Selig K-P and Parisi J 1981 J. Low Temp. Phys. 45 21
Buckel W 1990 Superconductiviry (New York 'VCM
Budhani R C, Holstein W L and Suenaga M 1994 Phys. Rev. Lett. 72 566
Budhani R C, Liou S H and Cai Z X 1993 Phys. Rev. Lett. 71 621
Budhani R C. Suenaga M and Liou S H 1992 Phys. Rev. Lett 69 3816
Bulaevskii L N 1972 2.Eksp. Tear. Fir.64 2241 (Engl. Transl. 1973 Sou. Phys.-JETP 37 1133)
-1990
Inl. J. Mod Phys. B 4 1849
-1991
Phys. Rev. B 44 910
Bulaevskii L N, Cho J H, Maley M P, Li Q, Suenaga M and Ledvij M 1994a Phys, Rev. B 50 3507
Bulaevskii L N and Clem I R 1991 Phys. Rev. B 44 10234
Bulaevskii L N, Clem J R. Glazman L and Malazemoff A P 1992a Phys. Rev. B 45 2545
Bulaevskii L N,Clem I R, and Glazman L 1992b Phys. Rev. B 45 2545
Bulaevskii L N, Daemen L L,Maley M P and Coulter J Y 1993a Phys. Rev. B 48 13798
1568
E H Brondt
Bulaevskii L N. Ha"l P C and Vinokur V M 1995a Pkys. Rev. B 5l 15355
Bulaevskii L N, Kolesnikov N N. Schegolev I F and Waselev 0 M 1993b Phys. Rev. let^ 71 1891
Bulaevskii L N. Larkin A 1. Maley M P and Vinokw V M 1995b Pkys. Rev. B 52 9205
Bulaevskii L N. Ledvij M and Kogan V G 1992c Pkys. REV.B 46 366
-1992d
Phys, Rev. B 46 11807
-199%
Phys. Rev. Lett. 68 3773
Bulaevskii L N and Maley M P 1993 Pkys. Rev. Len. 71 3541
Bulaevskii L N. Maley M P and Cho I H l994b Physica W U O C 87
Bulaevskii L N. Maley M P and Schegolev I F 1994c Physicn 197B 506
Bulaevskii L N. Maley M P and 'Tachiki 199% Phys. Rev. Len 74 801
Bulaevskii L N, Meshkov S V and Feinberg D 1991 Pkys. REV.B 43 3728
Burlachkov L 1989 Eumphys. Lea 8 673
-1993
Phys. Rev. B 47 8056
Burlachkov L, Khalfin I B and Shapiro B Ya 1993 Phys. Rev. B 48 I156
Burlachkov Land Vinokur V M 1994 Pkysica 194-6C 1819
Burlachkov L. Geshkenbein V B, Koshelev A B,Larldn A I and Vinokur V M 1994 Phys. Rev. B SO I6770
Busch Q Ries G, Werthner H. Kreiselmeyer G and Saemann-lschenko G 1992 Phys. Rev. k i t . 69 522
Buzdin A I 1993 Phys. Rev. B 47 I1416
Buzdin A I and Feinbetg D I990 J. Physique 51 1971
-1992
Phys. Lett. 165A 281, 167 89
-1994a
Physica 220C 74
-1994b
Pkysica W U O C 2755
Buzdin A I. Krotov S S and Kuptsov D A 1991 Physieo 175C 42
Buzdin A I,Neminsky A, Nikolaev P and Bmduc C 1994 Pkysica 227C 365
Buzdin A I and Simonov A Yu 1990 Pkysica lMTC 421
-1991
Physica 175C 143
Cai X Y,Gurevich A, Larbalestier D C. Kelley R J, Onellion M, Beger H and Margaritondo G 1994 Phys. Rev. B
50 I6774
Caldeira A 0 and Leggen A J 1981 Phys. Rev. Lett. 46 211
Calzona V, Cimberle M R, Ferdeghini C Grass0 G, lorio A, Licci F, Putti M. Rirmto C and Siri A S 1993
Supepcreond Sci 7crhnoL 6 46
Camerlingo C. Monaco R. Ruggiero B, Russo M and Testa G I995 Phys, Rev. B 51 6493
Campbell A M 1971 3, Phys. C: SolidStafePkys. 4 3186
-1978
Phil. Mag. B 37 149, 169
-1980
Helveticn Physiea A m 53 404
-1982
C,yopzics 82 3
-1991
IEEETrans. Magn 27 1660
Campbell L J. Dona M M and Kogan V G 1988 Phys. Rev. B 38 2439
Campbell A M and Eveas J E 1972 Adv. Phys. 72 199
Caplin A D,Angadi M A, Laverty I R and de O l i v e h A L 1992 Supercond. Sci. Tcchnol. 5 S186
Caplin A D.&hen L F. Perkin G K and Zhukov A A 1994 Supeaond Sci. Tcchnol. 7 412
Caplin A D. Perkin G K and &hen L F 1995 Super" Sci. Technol. 8 366
Cameiro G I992 Phys. Rev. B 45 2391, 2403
-1994
Phys. Rev. B 50 6982
Phys. Rev. Len. 75 521
-1995
Cameim G, Cavalcanti R and Gamer A 1993 Phys. Rev. B 47 5263
Cameiro G, Dona M M and de Andrade S C B 1992 Pkysica 203C 167
Caroli C DeGennes P G and Matriwn J 1964 Phys Le& 9 307
h
o C and Fisher D S 1995 Phys. Rev. B SI 534
Carrena P 1992 Phys. Rev. B 45 5760
Pkys. Rev. B 48 528
-1993
Caslro H and Rinderer L 1994 Physka W M C 2915
Cave I R and Evefts 1 E 1978 Phil. Mag. B 37 I l l
Chakravw S,Ivlev B I and Ovchinnikov Yu N 1990 Phys Rei.. Len. 64 3187, 1990 Phys, Rev. B 42 2143
Chan W C,Jwo D S,Lin Y F and Huang Y 1994 Pky&za DOC 349
Charalambous M, Chaussy J and Lejay P 1992 Phys. Rev. B 45 5091
Charalambous M, Chaussy J. Lejay P and Vmokur V 1993 Phys. Ret,. Lett. 71 436
Charalambous M. Koch R H, Masselink T, Doany T.Field C and Holtzberg F 1995 Phys. Rev. Lett. 75 2578
Thefludine lattice in superconductors
1569
Chauopadhyay B and Shenoy S R 1994 Phys. Rev. Lett. 72 4W
Chen D-Xand Goldfarb R B 1989 1. Appl. Phys. 66 2489
Chen D-X, Coldfah R B, Cross R W and Sanchez A 1993a Phys. Rev. B 48 6426
Chen I L and Yang T I 1994 Phys. Rev. B 50 4064
Chen S Q. Skocpol W I. DeObaldia E, OMalley M and Mankiewich P M 1993b Phys. Rev. B 47 2936
Chen T and Teitel S 1994 Phys. Rev. Lea 72 2085
-1995
Phys. Rev. Len.74 2792
Chien T R, J i g T W, Ong N P and Wang Z Z 1991 Phys: Rev. Lett. 66 3075
Chiimoto N, Konczykowski M, Motohiro N and Malozemoff A P 1992 Phys. Rev. Lett. 69 1260
Cho 1 H, Maley M P, S a f a ~H. Coulter I Y and Bulaevskii L N I995 Pkys. Reu. Letr. 75 (submitted)
Choi E-J, Lihn H-T S, Drew H D and Hsu T C 1994 Phys. Rev. B 49 13271
Choi M-S and Lee S-I 1995 Phys. Rev. 8 51 6680
cbudnovsky E 1990 Phys. Rev. Lett. 65 3040
-1995
Phys. Rev. B 51 1181, 15351
Chudnovsky E, Ferrera A and Vlknkin A 1995 Phys. Rev. 8 51 1181
Civale L, Krusin-Elbaum L. Thompson J R, Wheeler R, Manvick A D, Khk M A, Sun Y R. Holtzberg F and
Feild C 1994 Phys. Rev. B 50 4102
Civale L, Manvick A D. Wonhington T K, K3k M A. Thompson J R. Krusin.Elbaum L. Sun Y , Clem J R and
Holtzberg F 1991 Phys. Rev. B 67 648
Civale L. Wonhington T K. Krusin-Elbaum L, Sun Y and H o l m F I992 Magnefic Suxepribility of
Superconductors nnd Other Spin System ed R A Hein. T L F m v i l l a and D H Liebeenberg (New York:
Plenum)
Clem J R 1974a Pmc. I31h Conz Low Temp. Pkys. ed K D Timmerhaus. W J OSullivan and E F " e l ,
vol. 3
(New York Plenum) p 102
-1974b
Phys. Rev. B 9 898
J. Low Temp.'Phys. 18 427
-1975a
Phys Rev. B 12 1742
-1975b
Phys. Rev. Lett 24 1425
-1977
-1979
J. Appl. Phys. 50 3518
-1980
J. Low Temp Pkys. 38 353
Physica 107B 453
-198la
-1981b
Phys. Rep. 75 I
-1988
Physica 15S5C 50
-1991
Phys. Rev. B 43 7837
-1992
Concise Encyclopedia of Magnetic and Superconducting Marerinls ed J Evetts (Oxford Pergamon)
p 612
Clem J R, Bumble B, Raider S I, Gallagher W J and Shih Y C 1987 Phys. Rev. B 35 6637
Clem J R and Coffey M W 1990 Pkys. Rev. B 42 6209
-1992
Phys. Rev. B 46 14662
Clem J R and Han Z 1993 Phys. Rev. B 48 13774
Clem J Q Huebener R P and Gallus D E 1973 J. Low Temp. Phyx 12 449
Clem J R, Kerchner H R and SekuIa T S 1976 Phys. Rev. B 14 1893
Clem 1 Rand Sanchez A 1994 Phys. Rev. B 50 9355
Coffey M W 1994a Phys. Rev. B 49 9774
-1994b
Physica W M C 1961
Pkys. Rev. B 52 R9851
-1995
Coffey M W and Clem J R 1991a Phys. Rev. Lett. 61 386
Phys. Rev. B 44 6903
-1991b
Cohen L F, Zhukov A A, Perkins G, Jensen H I, Klestov S A, Vorokova V I, Abell S, Kiipfer H, Wolf T and
Caplin A D 1994 Physic= WOC 1
Conen E and Schmid A 1974 1. Low Temp. Phys. 17 331
Comer L W and Malozemoff A P 1991 Phys Rev. B 43 402
Cooley L D and Grishin A M 1995 Phys. Rev. Lett 74 2788
Cooky L D,Lee P I and Larbalestier D C 1994 Appl. Phys. Len. 64 1298
Cooper J R, Lam J W and Wade I M 1995 Phys. Rev. B 51 6179
Counois H. Gandit P H and Pannetier B 1995 Pkys. Rev. B 52 1162
Crabfree G W, Kwok W K, Welp U, Downey J, Fleshler S and K G Vandervoort 1991 Physica 18S9C 282
Crabfree G W. Pendrich I. Kwok W K and Glagola B G 1994 Physica 23540C 2543
1570
E H Brandt
Cnbier D, Jacrot B, Rao L M and Famou B 1964 Phys. Len. 9 106
de la CNZ F. Pastonza H, L6pez D, Go&"
M F, Arribh A and Nieva G 1994a Physicn W4-4OC 83
de la Cruz F. Mpez D and Nieva G 1994b Phil. Mag. B 70 773
Cubiu R, Forgan E M, Warden M. lee S L, Zimmermann P. KeUer H. SaviC I M. W e a P, Zech D, Kes P H.
Li T W, Menovsky A A and Tamawski 2 1993a Physic0 213C 126
Cubit1 R. Forgan E M, Yang G. Lee S L, Paul D McK, Mook H A, Yethiraj M, Kef P H. Li T W. Menovsky A A,
Tamawski Z and Morlensen K 1993b Nnrure 365 407
D'Anna C. And* M - 0 and Benoit W 1994a Europhys. Len. 25 539
DAnna C. Andrb M-0, lndenbom M V and Benoit W 1994b Physica WOC 115
DAnna G Benoit W, Sadowski W and Walker E 1992 Eumphys. Len.20 167
D'Anna G. Gammel P L, Safar H, Alen G B and Bishop D J 1995 Phys. Rev. Len. in press
D ' A ~ MG, Indenbom M V. Andre M-0. Benoit W and Walker E 1994c Europhys. Len. 25 225
D m " L L. Campbell L 1 and KO@ V G 1992 Phys. R#v. B 46 3631
Daemen L L, Bulaevskii L N, Maley M P and Coulter J Y 1993a Phys, Rev. Len 70 1167: 1993 Phys. Rev. B 47
11291
Dmnen L L,Campbell L J. Simonov A Yu and Kogan V G 1993b Phys. Rev. Len. 70 2948
Dai H, Liu 1 and Lieber C M 1994a Phys. Rev. Left 72 748
Dai H, Yoon S, Liu J, Budhani C and Lieber C M 1994b Science 265 1552
Dbmling M and Lahalestier D C 1989 Phys, Rev. B 40 9350
Dkmling M, Walker E and Wikiger R 1994 Phys, Rev. B 50 13024
Deak I, McElfresh M, Clem J R, Ha0 Zh, Konczykowski M, Muenchausen R. Foltyn S and Dye R 1994 Phys.
Rev. B 49 6270
de Andrade R and de Lima 0 F 1995 Phys. Rei: B 51 9383
DeGennes P G 1966 Superconductivify ofMeIols nnd Alloys (New York Benjamin)
Demer C, Eidellolh W and Koch R H 1992a Phys. Rev. Len. 68 3347
Dekker C. Waltgens F I M,Koch R H, Hussey B W and Gupta A 1992b Phys. Rev. Left. 69 2711
de Lima 0 F and Andrade R Physica 248C 353
Dclrieu J M I972 J. Low Temp. Phys. 6 197
-1973
3, Phys. F: Mer. Phys. 3 893
Sfare Thesis Universit.6 Paris-Sud
-1974
Deltour R and Tinkham M 1966 Phys. Len 23 183
-1967
Phys. Rev. Len. 19 125
-1968
Phys. Res, 174 418
Dennlston C and Tang C 1995 Phys, Rev. B 51 8457
Dersch H and Blaner G 1988 Phys, Rev. B 38 11391
De Soto S M, Slichter C P, Wang H H, Geker U and Williams 1 M 1993 Pkys. Rev. Len. 70 2956
Deutscher G 1991 Physica 185-9C 216
Deutscher G and DeGennes P G 1969 SuperconduefivifyvoI 11, ed R D Parks (New York Marcel Dekker) p 1005
Deutscher G and Entin.Wohlman 0 1978 Phys. Rev. B 17 1249
Deutscher G, lmry Y and Gunther L 1974 Phys. Rev. B 10 4598
Dew-Hughes D 1987 PhiL Mag. B 55 459
DewHughes D 1988 Cryogenic8 28 674
Dimos D, Chaudhari P, Mannhardt 1 and LeGoues F K 1988 Phys. Rev. B 61 219
Ding S Y, Wang G Q, Yao X X, Peng H T, Peng Q Y and ulou S H 1995 Phys. Rev. B 51 9107
DobrosavljeviC-GrujiC L and RadoviC Z 1993 Supercond Sci. Technol. 6 537
Dodgson M J W and Moore M A 1995 Phys. Rev. B 51 11887
Doettinger S G. Anders S , Huebener R P. Haemel Hand Ovchinnikov Yu N 1995 Eumphys, Len. 30 549
Dwttinpr S C, Huebener R P. Gerdemann R. Kiihk A, Anden S,TrXuble T G and Villegier 1 C 1994 Phys. Rev.
Lelf. 73 1691
Dolan G J 1974 I. Low Temp. Phys. 15 133
Dolan G J, Chandrashekar G V. Dmger T Q Feild C and Holtzberg F 1989a Phys. Rev. Len. 62 827
Dolan C J, Holtzberg F. Feild C and Dinger T R 1989b Phys. Rev. Len. 62 827
Dornfnguez D, Bulaevskii L N, lvlev B, Maley M and Bishop A R 1995 Phys. Rev. L d t 74 2579; 1995 Phys.
Rev. B 51 15649
Dong M. Marchetti M C, Middleton A A and V i n o h V M 1993 Phys. Rev. Len. 70 662
Dong S Y and Kwolt H S 1993 Phys. Rq. B 48 648&---Doniach S 1990 High-Tempemlure Supemonduerivify ed K S Bedell, D Coffey, D E Meleer, D Pines and
1 R Schriefier (New York: Addison-Wesley) p 406
--
the flux-line lattice in superconductors
1571
Dona M M, Gubemafis J E and Rainer D 1989 Phys. Rev. B 39 9573
Dona M M and de Oliveira I G 1994a Phys. Rev. B 49 6205
-1994b
Physica 235-40C 2577
Donn V V, Klem R A, Varlamov A A, Buzdin A I and Livanov D V 1993 Phys. Rev. B 48 12951
Dorosinskii L A, lndenbom M V, Nikitenko V I, Ossip'yan Yu A, Polyanskii A A and Vlasko-Vlasov V K 1992
Pkysicn 203C 149
Dorosioskii L A, Nildtenko V I and Polyanskii A A 1994 Phys. Rev. B 50 501
Doney A T 1991 Phys. Rev. B 43 7575
-1992
Phys. Rev. B 46 8316
-1995
Phys. Rev. B 51 15329
Dorsey A T, Huang M and Fisher M P A 1992 Phys. Rev. B 45 523
Doyle R A, Campbell A M and Somekh R E 1993 Phys. Rev. Lcn. 71 4241
Doyle R A, S a w W S, Johnson J D, Campbell A M, Berghuis P,Somekh R E, Eveus J E, Wuth G and Wiesner J
1995 Phys. Rev. B 51 12763
Doyle T B and Labusch R 1995 Pme. Int. C?yogenics Marerials Conf: 1994. Crilicol Slate in Superconductors.
October 1994, Honolulu
Drobinin A V, Tsipenyuk Yu M and Kamentsev K E 1994 (preprint for LTZO,unpublished)
Duarte A, Femandez Righi E, Bolle C A. de la Cruz F. Gammel P L. Oglesby C S, Batlogg B and Bishop D J
1995 Phys. Rev. Lelt. (submitted)
Dumas J, Nemimky A. Thrane B P and Schlenker C 1994 Molecular Phys. Rep. 8 51
D u r h C A, Gammel P L, Miller R E and Bishop D J 1995 Phys. Rev. B 52 75
Du& C, Yayi J, de la CNZ F, Bishop D J, Mitn D B and Kapituld A 1991 Phys. Rev. B 44 7737
Ebner C and Straud D 1985 Phys. Rev. B 31 165
Efetov K B 1979 W .Ekrp. Tear. Fir. 76 1781 @ngl. Transl. 1979 Sov. Phys.-JETP 49 905)
EilenteEer G 1964 2 Phys. 180 32
-1966
Phys. Rev. 153 584
-1967
Phys. Rev. 164 628
-1968
2 Phys. 214 195
E h J W and Clem J R 1975 Phys. Rev. B 12 1753
Ekin J W. Serin B and Clem J R 1974 Phys, Rev. B 9 912
Eltsev Yu, Holm W and Rapp 0 1994 Phys. Rev. B 49 12333; 1994 Phyrica 235-lOC 2605
Eltsev Yu and Rapp 0 1995 Phys. Rev. B 51 9419; 1995 Phys. Rev. Lett C 75 2446
Enomoto Y,Kasufni K and Maekawa S 1993 Physica 215C 51
D and Kardar M 1994 Phys. Rev. Lctt. 73 1703
E s q u i n ~P 1991 J. Low Temp. Phys. 85 139
Esquinazi P. Neckel H,Brandt E H and Weiss G 1986 J. Low Temp. Phys. 64 1
Essmann U 1971 Superconducliviry ed F Chilton (Amsterdam: North Holland) p 38
Essmann U and Tfiuble H 1967 Phys. Lett. 24A 526
-1971
Sei. Am. 224 75
Evetts J E and Glowacki B A 1988 C i y p t i c s 28 641
Ewki T and Me F 1976 J. Phys. Soc. Japan 40 682
Fabrega L, Fontcubem J. Civale t and Pili01 1994a Phys. Rev. B 50 1199
Fabrega L, Fontcubem J. Madnez B and Pinol 1994b Phys. Rev. B 50 3256
Fabrrga L, Fontcubem J and Pili01 1995 Phy.sica 245C 325
F a d e M 1977 Phys. Slat. Sol. B 83 433
-1979
Phys. Slat. Sol. B 92 K121
-1981
1. Low Temp. Phys. 46 3; 1981 IEEETrans. M q n . Mag-17 1707
F a d e M and Kronmiiller H 1978 J. Nuel. Mater. 72 249
Farrell D E 1994 Physical Properlies of High TemperalureSuperconducforsN e d D M Ginsberg (Singapore: World
Scientific)
Farrell D E, Bonham S, Foster J, Chang Y C, Jiang P 2,Vandervoort K G, Lam D J and Kogan V G 1989 Phys.
Rev. Left. 63 182
Fanell D E, Kwok W K,Welp U, Fendrich J and Crabrree G W 1995 Phys. Rev. B 51 9148
F m l l D E. Rice J P and Ginsberg D M 1991 Phys. Rev. Left. 67 I165
Farrell D E, Rice J P, Ginsberg D M and Liu Z D 1990 Phys. Rev. Lett 64 1573
Farrell D E, Williams C M, Wolf S A, Bansal N P and Kogan V G 1988 Phys. Rev. Len. 61 2805
Fehrenbacher R, Geshkenbein V B and Blatter G 1992 Phys. Rev. B 45 5450
Feigel'man M V, Geshkenbein V B, loffe L B and Larkin A I 1993a Phys. Rev. B 48 16641
1572
E H Brandt
Feigel'man M V, Geshkentcin V B and Larkin A I 1990 Physua 167C 177
Feigel'man M V. Geshkenbein V B, Larkin A Iand Lmit S 1993b Pis'ma Zh,E h p . Teor. Fir. 57 699 @ngl.
Tmsl. 1993 SOY.Pkys.-JETPLett. 57 711)
Feigel'man M V, GesMnbein V 8, Larkin A Iand Vmokur V M 1989 Phys. Rev. Lett. 63 WO3
-1994
Physica 7.3MOC 3127
Phys. Rev. Lett. 75 (submitted)
-1995
Feigel'man M V, Geshkenbein V B and Vinokur V M 1991 Phys, Rev. B 43 6263
Feigel'man M V and Vinokur V M 1990 P h y . REI..B 41 8986
Feintcrg D 1992 Pkysica 194C 126
-1994
J. Physique Ill 4 169
Feinberg D and Enouhami A M 1993a Phys. Scr. T 49 159
Int. J. Mod. Phys. B 7 2085
-1993b
Feinberg D. Theodonkis S and Enouhami A M 1994 Pkys. Re,*. B 49 6285
Feintcrg D and Villard C 1990 Mod. Phys. Len. B 4 9
Fendrich I A, Kwok W K. Giapintzakis J. van der Beek C J. Vinokur V M. Fleshler S. Welp U, Viswanathan H K
and Cnbuee 0 W 1995 Phys. Rev, Lett. 74 1210
Fermi M J. Johnson M, Wellstood F C. Kingston 1 1. Shaw T J, and Clarke J 1994 J, ,!.ow Temp. Phys. 94 I5
Ferrari M J. Wellstwd F C. Kingston J J and Clarke I 1991 Phys. Rev. Lett. 67 1346
Ferrell R A 1992 Pkys. Rev. 68 2524
Fetter A L 1967 Phys. Reu. 163 390
-1980
Phys. Rev. B 22 1200
-1994
Phys. Rev. B 50 13695
Fetter A L and Hohenberg P C 1967 Phys. RCY.159 330
-1969
Superconductivity VOI11. ed R D Parks (New York Marcel Dekker) p 817
Field S. Win J, Nori F and Ling Xinsheng 1995 Pkys. Rev. Lett. 74 1206
Fink H 1. Buisson 0 and Pannetier B 1991 Phys. Rev. B 43 10144
Fink H J and Haley S B 1991 Phys. Rev. Len. 64 216
Fink H I, Lopez A and Maynard R 1985 Pkys. REV.B 25 5237
Fink H J. Rodriguez D and Lopez A 1988 Phys. Rev. B 38 8767
Fiory A T 1971 Phys. Rev. Lett. 27 501
-1973
Pkys. Rev. B 7 I881
Fiory A T. Hebad A F, Mankiewich P M and Howard R E 1988 Appl. Pkys. Lett. 52 2165
Fiory A T,Hebard A F and Somekh S 1978 Appl. Phys. Len. 32 73
Fischer K and Teichler H 1976 Phys. Lett. 58A 402
Fischer K H 1991 Physicn 178C 161
Physka 193C 401
-1992a
Physic0 2OOC 23
-1992b
-1993
Physica 2lOC 179
-1995a
Superconductivity Review 1 153-206
Fischer K H 1995b Influence of disorder on KT transition Preprint
Fischer K H and Natter" T 1991 Phys. Rev. B 43 10372
Fischer 0. Bmnner 0.Antognana L,Mieville Land Tiscone L M 1992 Phys. Scr. T 42 46
Fisher D S 1980 Phys. Rev. B 22 I190
-1992
Phenomenology and Application of High.Emperatwe Superconductors ed K S Bedell, M Inui.
D E Meltzer. I R Schrieffer and S Doniach (New York: Addison-Wesley) p 287
Fisher D S, Fisher M P A and Huse D A 1991 Pkys, Rev. B 43 130
Fwher L M. Il'in N V. Podlevskikh N A, and zlkharchenko S I1990 SolidStotc Commun. 72 687
Fisher L M, Il'in N V, Volmhin I F. Makarov N M. Yampol'skii V A, Perez Rodriguez F and Snyder R L 1993
Physica 206C 195
Fisher L M. Voloshin 1 F. Gorbachev V S, Savel'ev S E and Yampol'skii V A 1995 Physica 245C 231
Fisher M P A 1989 Phys. Rev, Lett. 62 1415
Fleshler S. Kwok W-K. Welp U, Vinokur V M. Smith M K. Downey J and Crabtree G W 1993 Phys Rev. B 47
14448
Flippen R B. Askew T R. Fendrich J A and van der Beek C J 1995 Phys. Rev. Len. (submitted)
Forgan E M 1973 J. Phys. F: Met. Phys. 3 1429
Forgan E M, Cubin R, Yethiraj M, Christen D K, Paul D M, Lee S L and G a e l P 1995 Phys Rev. Len. C 74
1697
Forgan E M and Cough C E 1968 Phys. Lett. 26 602
Thefludine [onice in superconductors
1573
Forgan E M, Paul D McK, Mook H A , T i i n s P A, KeUer H, Sutton S and J S Abell 1990 Norure 343 735
Forkl A 1993 Phys. Scr. T 49 148
Forkl A, Dragon T and KronmUllcr H 1991a J. Apgl. Phys. 67 3047
Forkl A, H a h m e i e r H-U, Leibold B, Dragon T and Krvnmtiller H 195'1b Physim IBDC 155
Forkl A and Kmnmiiller H 1994 Physica u 8 C 1
Fortini A, Haire A and Paumier E 1980 Fhys. Rev. B 21 5065
Fortini A and Faumier E 1972 Phys. Rev. B 5 1850
-1976
Phys. Rev. B 14 55
F d e J D 1 1979 3. AppL Phys. 50 5402
Franz M and Teitel S 1994 Phys. Rev. Len. 73 480
Phys Rev. B 51 6551
-1995
Freimuth A, Hohn C and Galffy M 1991 Phys. Rev. B 61 10396
Frey E, Nelson D R and Fisher D S 1994 Phys. Rev. B 49 9723
Frey T, Mannhardt 1, Bednorz J G and Williams E J 1995 Phys. Rev. B 51 3257
Friedel J 1989 3. Phys. Cond. Mar. 1 7757
Friesen M 1995 Phys. Rev. B 51 632, 12786
Fruchter A, Malozemoff A P. Camptell I A, Sanchez I, Konczykowski M, Griessen R and Holtzberg F 1991 Phys.
Rev. B 43 8709
Fruchter A, Rost D and Campbell I A 1994 Physica 235-4OC 249
Fu C M. Moshchalkov V V, Boon W, Temst K, BmynSeraede Y,Jakob 0, Hahn T and Adrian H 1993a Physicn
205c 111
Fu C M, Moshchalkov V V, Rosseel E, Baert M, B w o W and BNynseraede Y 1993b Phydca u)6C 110
Fulde P and Keller J 1982 Superconductivity in Temnry Compounds 11: Superconductivity m d Mqmlism
ed M B Maple and 0 Fischer (Berlin: Springer) p 249
Fujila A and Hikami S 1995 Phys. Rev. E 51 16259
Gaber M W and Achar B N N 1995 Phys. Rev. B 52 1314
Gaitan F 1995 Phys. Rev. B 51 9061
Gal'perin Yu M and Sonin E B 1976 Fir.Tverd. Teln IS 3034 (Engl. Transl. 1976 So". Phys. SolidStnre 18 1768)
G q m e l P L,Bishop D I. Dolan G J. Kwo J R,M m y C A. Schneemeyer L F and Wacs& I V 1987 Phys. Rev.
Len. 59 2592
Gammel P L, Bishop D I. Rice J P and Ginsberg D M 1992a Phys. Rev. Len. 68 3343
Gammel P L, D u b C A, Bishop D J. Rice J P,Kogan V G, M v i j M, Simonov A Yo.Rice J P and Ginsberg D M
1992b Phys. Rev. Len. 69 3808
Gammel P L.Schneemeyer L F. Waszczak I V and Bishop D J 1988 Phys. Rev. Len. 61 1666
Gammel P L. Schneemeyer L F and Bishop D J 1991 Phys. RLY.Len. 66 953
Gao L. Xue Y Y.Chen F, Xiong Q, Meng R L, Ramirez D. Chu C W, Eggert J H and Mao H K 1994 Phys.
Rev. B 50 4260
Gao 2 X, Osquiguil, Maenhoudt M, Wuyts 8, Libbrecht S and Biuynseraede Y 1993 Phys, Rev. Len. 66 953
Garcia A, Zhang X X. Tejada 1, Manzel M and Bruchlos H 1994 Phys. Rev. B 50 9439
Garten F. White W Rand Beasley M R 1995 Phys. Rev. B 51 1318
Genenko Yu A 1992 Sverkkprovodims~'5 1402 ([email protected] m l . 1992 Superconductivity 5 1376)
-1993
Physicn 215C 343
P i s ' m U . %p. Teor. Fiz 59 807 (Engl. Transl. 1994 Sov. Phys.-JLTP Left. 59 841): 1994 Phys.
-1994
Rev. B 49 6950
-1995
Phys. Rev. B 51 3686
Genenko Yu A and Medvedev Yu V 1992 Phys. Len. 167A 427
Genenko Yu A, Medvedev Yu V and Shuster G V 1993 PiP'ma U . E k p . Eor, Fir. 57 705 (Engl. Transl. 1993
Sov. Phys.-JETPLett 57 717)
Genoud I-Y, Triswne G, Juncd A, Tsuakamoto T and Miiller I 1995 Physica 242C 143
Gerber A. Li J N. Tarnawski Z Franse I I M and Menovsky A A 1993 Phys. Rev. B 47 6047
GerMuser W, Ries G, NeumUUer H-W,Schmidt W. Eibl 0, Saemnn-lschenko G and Klaumilnzer S 1992 Phys.
Rev. Len. 68 879
Geshkenbein V B, Feigel'man M V and Vinokur V M 1991 Physic0 1 S 9 C 2511
Geshkenbein V B, loffe L B and Larkin A I 1993 Phys. Rev. B 48 9917
Geshkenbein V B and Larkin A I 1994 Phys. Rev. Len. C 73 609
Geshkenbein V B, Vinokur V M and Fehrenbacher R 1991 Phys. Rev. B 43 3748
Ghinovker M. Sandomirsky V E and Shapiro B Ya 1995 Phys. Rev. B 51 8404
Giaever I 1966 Phys Rev. Len. 16 460
1574
E H Brad
GiDnwrchi T and Le Doussal P 1994 Phys. Rev. kif. 72 1530
-1995
Phys. Rev. B 52 1242
-1996
Preprinl
Giapinwris I. Neiman R L,Ginsberg D M and Kirk M A 1994 Phys. Rev. B 50 16001
Gilchrist I 1972 J. Phys, D: Appl. Phys. 5 2252
-1990a
Supercond. Sci Technol. 3 93
-1990b
Solid Stafe C o m w 75 225
-1994a
Physico 219C 67
-1994b
Supercond. Sci. TechnoL 7 849
Gilchrist J and Brandt E H 1996 unpublished
Gilchrist J and Dombre T 1994 Phys. Rev. B 49 1466
Gilchrist I and Konczykowski M 1990 Physica 168C 123
-1993
Physicn C 212 43
Gilchrist J and van der Beek C J 1994 Physic= C 231 147
Ginsberg D M and Manson J T 1995 Phys. Ret'. B 51 515
Ginzburg V L 19M) Fiz T v e d Tela 2 2031 (Engl. Transl. 1961 Sov. Phys. Sol. Sfofe 2 1824)
Ginzburg V L and Landau L D 1950 Zh.Ehp, TEOT.
Fiz 20 1064 (End.Transl.1965 Men of Physics: L D Landau
ed D ter Haar (New York: Pergmon) val 1 pp 138-57)
Giroud M, Buisson 0, Wang Y Y and Pannetier B 1992 J. Low Temp. Phys. 87 683
Giura M. Marcon'R, Fastampa R and Silva E 1992 Phys. Rev. B 45 7387
Gjalmesli S, Fossheim K. Sun Y R and Schwartz 1 1995 Phys. Rev. B 52 10447
Glazman L and Fogel' N Ya 1984a Fir. NW Temp. 10 95 (Engl. Trans1 1984 So". J. LOW Temp. Phys. 10 51)
-1984b
Fiz. Tvud. Tela 26 800 (Engl. Tmsl. 1984 Son Phys. Sol. Sfafc26 482)
Glzman L and Koshelev A 1990 Zh. Ekp. Teor. Fir. 97 1371 (Engl. Transl. 1990 Sov. Phys.-JETP 70 774)
Phys. Rev. B 43 2835
-1991a
-1991b
Physicn 173C 180
Glyde H R, Lesage F and Findeisen P 1992a Phys. Rev. B 46 9175
Glyde H R, Moleko L K and Findeisen P 1992b Pkys. R N . B 45 2409
Goldfarb R B, Lelental M. Thompson C A 1991 Mopefie Suscepfibiliry of Superconductors andother Spin Syslems
ed R A Hein, T L Fmcavilla and D H Liebenberg (New York: Plenum) p 49
Golosovsky M, Ginodman V. Shaltiel D. Gerhkser W and Fischer P 1993 Phyr. Rev. B 47 9010
Golosovsky M, Naveh Y and Davidov D 1991 Physica 18OC 164
-1992
Phys. Rev. B 45 7495
Golosovsky M, Snortland H J and Beasley M R 1995 Phys. Rev. B 51 6162
Golosovsky M, Tsindlekht M, Chayet H and Davidov D 1994 Phys. Rei. B 50 470 (em"
51 12062)
Golubov A A and Harrmann U 1994 Phys. Rev. Len. 72 3602
Golubov A A and Logvenov G Yu 1995 Phys. Rev. B 51 3696
Oilm6ry F and Takafs S 1993 Physic= 2l2C 297
Goodman B B 1966 Rep. Prop. Phys. 29 445
Gordeev S N, Jahn W, Zhukov A A, Kiipfer H and Wolf T 1994 Phys. Rev. B 49 15420
Gor'kov L P 1959aZh. Ekp, Teor. Fiz 36 1918 (Engl. Transl. 1959 Sov. Phys.-JETP 9 1364)
Zh Eksp. Teor. Fiz 37 833 (Engl. T m l . 1960 Sov. Phys.JETP 10 593)
-1959b
Gor'kov L P and Kopnin N B 1975 Usp. Fiz N m k 116 413 (End.Transl. 1976 SOY.Phys.-Uspechi 18 496)
Gorlova I G and btyshev Yu I 1990 Pis'- Zh. Eksp. Teor. Fk 51 197 (Engl.T m l . 1990 Sov. Phys.-JETP
Len. 51 224)
Gorter C J and Casimir H B G 1934 2 Tech Phys. 15 539
Gough C E and Exon N I 1994 Phys. Rev. B 50 488
Grant P D. Denhoff M W. Xing W, Brown P,Govorkov S, Lmin IC,Heinrich B, Zhou H,Fife A A and Cragg A R
1994 Physi.cn U9C 289
Gray K E, Kampwirth R T and Farrell D E 1990 Phys. Rev. B 41 819
Gray K E, Kampwirth R T, Miller D I. Murduck J M. Hampshire D. Heaog R and Weber H W 1991 Physica
174C 340
Gray K F, Kim D H. Veal B W, Seidler G T, Rosenbaum T F and Parrell D E 1992 Phys. Rev. B 45 10071
Graybeal I M, Luo J and White W R 1994 Phys. Rev. B 49 12923
Grier D G, Murray C A, Bolle C A. G m e l P L, Bishop D 1. Mitzi D B and Kapilulnik A 1991 Phys. Rev. Left
66 2270
Griwen R 1990 Phys. REV.Left. 64 1674
-1991
Pkysica 172C 441
The fludine laltice in superconductors
1575
Griessen R. Lensink J G and Schnack H G 1991 Physic4 185-9C 337
Griessen R. Lensink J 0, Schrbder T A M and Dam B 1990 C~~a&wnics
30 536
Griessen R. Wen H-H, van Dalen A J J , Dam B, Rector J, Schnack H 0, Libbrecht S, Osquiguil E and
Bruynseraede Y 1994 Phys. Rev. Len. 72 1910
Grigorieva I V 1994 Supercond Sci.Technol. 7 161
Grigorieva I V, Steeds J W. B W s h n a n G and Paul D M 1995 Phys. Rev. B 51 3765
Grigorieva 1 V, Steeds J Wand Sasaki K 1993 Phys. Rev. B 48 16865
Grishin A M, Matqnovich A Yu and Yampol'skiy S V 1990 W .Eksp. nor. Fir. 91 1930 CEngl. Transl. 1990 SOP.
Phys.-JETP 70 1089)
-1992
M E k p . Teor. Fk. 101 649 W g l . Transl. 1992 Sov. Phys.-JETP 74 345)
Gmss R and Koelle D 1994 Rep, Pmg. Phyr. 57 651-741
Gmver A K. Kumar R, Chaddah P. Subramanian C K and Sankannarayanan Y 1990 Physic0 170C 431
Grover A K, Ramakrishnan S, Kumar R. Paulose P L, Chandra G, Mdik S K and Chaddab P 1992 Physica 192C
372
Gugan D 1994 Physica w3C 165
Gugan D and Stoppard 0 1993 Physica 213C 109
Guinea F 1990 Phys. Rev. B 42 6244
Guinea F and Pogorelov Yu 1995 Phys. Rev. Lett. 74 462
Gupta A Kopelevich Y,Ziese M, E q u i n e P. Fisher P. Schulz F I and Bran" H P 1993 Phys. Rev. B 48 6359
Gurevich A 199Oa Phys. Rev. Len. 65 3197
-1990b
Phys. Rev. B 42 4852
Phye Rev. B 46 3187. E14329
-1992a
Phys. Rev. B 46 3638: 1992 Supercond. Sci Technol. 5 383
-1992b
-1993
Phys. Rev. B 48 12857
Physica 243C 191
-199%
-1995b
Int. 3. Mod Phys. B 9 1045-65
Gurevich A and Brandt E H 1994 Phys. Rev. Lett. 73 178
Gurevich A and Brandt E H 1996 (unpublished)
Gurevich A and Cooley L D 1994 Phys. Rev. B 50 13563
Gurevich A and Kupfer H 1993 Phys. Rev. B 48 6477
Gurevich A, Kupfer H, Runtsch B and Meier-Hirmer R 1991 Phyz Rev. B 44 12090
Gurevich A V and Mints R G 1987 Rev. Mod. Phys. 59 941
Gutlyanskii E D 1994 Pis'mo M Ekp. Teor. Fiz, 59 459 (engl. Transl. 1994 Sou. Phys.-IETP Lett. 59 459)
Gygi F and Schliiter M 1991 Phys. Rev. B 43 7609
Gyorgy E M, van Dover R B, Jackson K A, Schneemeyer L F and Waszczak J V 1989 Appl. Phys. LetI. 55 283
Habermeier H-U and Kromniiller H 1977 AppL Phys. 12 297
Hackenbroich G and Scheidl S 1991 Physicn ISlC 163
Hagen C Wand Griessen R 1989 Phys. Rev. Len. 62 2857
Hagen S I, Smith A W, Rajeswari M, Peng 1 L, Li Y Z Grme R L, Mao S N, Xi X X, Bhanacharya S, Li Q
and Lobb C J 1993 Phys. Rev. B 47 1064
Hdbritter J 1992 Phys. Rev. B 46 14861; 1992 1. Appl. Phys. 71 339
-1993
Phys. Rev. B 48 9735
Haley S B 1995 Phys. Rev. Len. 74 3261
Halperin B I and Nelson D R 1919 Phys. Rev. B 19 2457
Ha0 2 1993 Phys. Rev. B 48 16822
Hao Z and Clem J R 1991a IEEE Tram Magn. 27 1086
Phys. Rev. B 43 1622
-1991b
-1992
Phys. Rev. B 46 5853
Phys. Rev. Le& C 71 301
-1993
Hao Z Clem J R. McBlfresh M W, Civale L, Malozemoff A P and Holtzberg F 1991 Phys. Rev. B 43 2844
Hao Z and Hu G R 1993 Phvs. Rev. B 48 16818
Ha0 Z Hu C-R and Ting C S 1995 Phys. Rev. B 51 9387
Harada K. Matsuda T, Kasai H. Bonevich J E Yoshida T,Kawabe U and Tonomura A 1993 Phys. Rev. Lett. 71
3371
Hardy V, Simon ch, Provost J and Groult D 1993 Physica 205C 371
Hardy W N, Bonn D A, Morgan D C Liang R and Zhang K 1993 Phys. Rev. Len. 70 3999
Harris 1 M, Ong N P,Gagnon R and Taillefer J 199% Phys. Rev. Len. 74 3684
Harris 1 M, Ong N P, Matl P, Gagnon R Taillefer I, Kimura T and Kitazawa K 1995b Phys. Rev. Len. 74 3684
1576
E H Brand:
Harris J M,Ong N P and Yan Y F 1993 Phys. Rev. Lea 71 1455
Harris J M,Yan Y F. Tsui 0 K C Matsuda Y and Ong N P 1994 Phys, Rev. Lett. 73 171I
Harsh~nanD R. Brandt E H, Fiory A T , Inui M, M k i B, Schneemeyer L F and Waszcrak 1 V 1993 Phyr. Rev. B
47 2905
Harshman D R, Kleiman R N. lnui M,Espinosa G P, Mitzi B, KapiNlnik A, Pfiz T and Williams D L 1991 Phys.
Rev. k l l . 67 3152
Harshman D R, Schneemeyer L F, Waszczak J V, Aeppli G, Cava R I. Batlogg 8, Rupp L W, Ansaldo E J and
WlliamS D L 1989 Phys. Rev. B 39 851
Hasegawa S and Kitazawa K 1990 Jap,J, Appl. Phys. 29 434
Hasegawa S, Matsuda T, h d o I, Osakak N, Igarashi M, Kobayashi T,Naito M,Tonomura A and Aoki R 1991
Phys. Rev. B 43 7631
Hebard A F and Piory A T 1982 Physica 109-1OB 1637
Heinzel Ch, H&g J. Yiao 2 Land Ziemann P 1994 Physicu 2 3 M C 2679
Helfand E and Werthamer N R 1966 Phys. Rev. 147 288
Hellerquist M C,Ryu S,Lombardo L W and KapiNlnik A 1994 Physica 23OC 170
Hensel B. Grass0 G and FlUkiger R 1995 Phys. Rev. B 51 15456
Hensel B, Grivel J-C. Jeremie A, Perin A, Pollini A and Fliikiger R 1993 Physica 205C 329
Hergt R. Andrii W and Fischer K 1991 Physica 185-9C 2197
He:lach D, Majer G. Rosenkranz I, S c h m b M, Schwan W. Seeger A, Temp1 W. Bmdt E H, Essmann U.
Fiirderer K and Gldisch M 1990 H y p e e e Inleroclionr 63-5 41
Hess H F, Murray C A and Was%& I Y 1992 Phys, Rev. Len. 69 2138
Phys. Rev. B 50 16528
-1994
Hess H F, Robinson R B and Waszczak J V 1990 Phys. Rev. Lett 64 2711
Hettinger I D, Gray K E, Veal B W. Panlikas A P. Kostic P. Washbum B R Tonjes W C and Flewelling A C
1995 Phys. Rev. Lelt. 74 4726
Hefzel R E. SudbB A and Huse D A 1992 Phys. Rev. Len. 69 518
Hikami S. Fujita A and Larkln A I 1991 Phys. Rev. B 44 10400
Hikami S and Larkin A I 1988 Mod. Phys Len. B 2 693
Hilzinger H R 1977 Phil. Mag. 36 225
Hindmarsh B and Kibble T W B 1995 Rep. Prog. Phys. 58 477
Hochmut H and Larenr M 1994 Physica WOC 209
Hocquet T, Mathihieu P and Simon Y 1992 Phys. Rev. B 46 1061
H o h a n n S and Kiimmel R 1993 Phys. Rev. Lelt. 70 1319
Hohn C. Calm M. Dascoulidou A, Freimuth A. Soltner H and Poppe U 1991 Z Phys. B 85 161
Holzapfel B. Kreiselmeyer G, Kraus M. Saemann-lscknko G, Bouffard S. Klaumiinzer S and Schultz L 1993
Phys Rev. B 48 600
HopfengZmer R. H e m 1 Band Saemann-lschenko G 1991 Phys, Rev. B 44 741
Hopfengiirtner R. Leghissa M. Kreiselmeyer G. Holzapfel B. Schmitt P and Saemann-lschenko G 1993 Phys.
Rev. B 47 5992
Horovitz B 1991 Phys. Rev. Lett. 67 378
Phys. Rev. B 45 12632
-1992
Phys. Rev. B 47 5947.5964
-1993
P h y . Rev. Leu C 72 1569 Wan’93
-1994
Phys. Rev, B 51 3989
-1995
Houghton A and Moore M A 1988 Phyr. Rev. Len 60 1 3 7
Houghton A. Pelcovits R A and Sudb0 A 1989 Phys. Rev. B 40 6163
Pkys. Rev. B 42 906
-1990
Houlrik J. Jonsson A and Minnhagen P 1994 Pkys. Rev. B 50 3953
Hsu J W P and Kapitulnik A 1992 Phys. Rev. B 45 4819
Hsu T C 1993 Physica Z1X 305
Hu C-Rand Thompson R S 1972 Phys. Rev. B 6 1 I O
Hu I and MacDonald A H 1993 Phys. Rev. Lclt. 71 432
Phys. Rev, B 52 1286
-1995
Huang C Y. Shapira Y. McNiff E 1, Peters P N, SchwarCz B B. Wu M K,Shull R D and Chiang C K 1988 Mod.
Pkys. Len. B 2 869. 1027
Huang Y, Lin Y F. Wang M J and Wu M K 1993 J. Appl, Phys. 73 3580
Hubemann B A and Donisch S 1979 Phys, Rev. Len. 43 950
Huben A 1967 Phys. Slat. Sol. 2A 669
The$ux-line lattice in superconductors
1577
-1972
Phys. Stat. Sol. B 53 147
Huebener R P 1979 Magneric F h Structure$ in Supemonducfors (Berlin: Springer)
-1984
Rep. Pro& Phys. 47 175
Supercond. Sci. Technol. 8 189
-1995
Huebener R P and Brandt E H 1992 Physic= 202C 321
Huebener R P, Kober F, Ri H-C, Knorr K, Tsuei C C, Chi C C and S c h e u e m n M R 1991 Physica l8lC 345
Huebener R P, Ri H C , Gross R and Gollnik F 1993 Physicn 209C 27
Huebener R P,Ustinov A and Kaplunenko V 1990 Phys. Rev. B 42 4831
Hug H J, Moser A, Parashikov I, Stiefel B, Fritz 0, Guntherodt H-J and Thomas H 1994 Physica W W O C 2695
Hulm J K and Matthias B T 1980 Science 208 881
HUnneckes C, Bohn H 0. Schilling W and Schulz H 1994 Phys. Rev. Len, 72 2271
Huse D A 1992 Phys. Rev. B 46 12230
Huse D A, Henley C L and Flsher D S 1985 Phys. Rev. Len. 55 2924
Huse D A and Majumdar S N 1993 Phys. Rev Lett. 71 2473
Hussey N E, Canington A, Cooper J R and Sinclair D C 1994 Physicn 23WOC 271 1
Hwa T. Le Doussal P. Nelson D and Vinokur V M 1993a Phys. Rev. B 48 I167
Hwa T, Nelson D and Vinokur V M 1993b Phys. Rev. B 48 I167
Hyun 0 E, Clem J R and Finnemore D K 1989 Phys. Rev. B 40 175
lchioka M 1995 Phys. Rev. B 51 9423
Iga F,Grover A K, Yamaguchi Y, Nishihara Y. Goyal N and Bhat S V 1995 Phys. Rev. B 51 8521
lhle D 1971 Phys. Sfof. Sol. B 47 423
lkeda R 1992a Physic0 ZDlC 386
Phys. Rev. B 46 14842
-1992b
lkeeh R. Ohmi T and Tsuneto T 1990 1. Phys. Soc. Japan 59 1740
Physico 185-9C 1563
-19912
-1991b
J Phys. Soc. Japan 60 10.51
J, Phys. Soc. Japan 61 254
-1992
Ikuta H. Hirota N. Kishio K and Kitmwa K 1994 Physico 235-40C 237
Ikuta H, Hirota N, Nakayama Y, Kishio K and Kitvawa K 1993 Phys. Rev. Len. 70 2166
lndenbom M V and Brandt E H 1994 P h y . Rev. Lett. C 73 1731
lndenbom M V, D’Anna G D. Andk M-0, Benoit W, Kronmiiller H. Li T W and Kes P H 1994a Critical Currenfs
in Superconductors ed H W Weber (Singapore: World Scientific) p 327
lndenbom M V, DAnna G D, Andr6 M-0, Kabanov V V and Benoit W 1994b Physica 2 3 5 4 0 C 201
lndenbom M V, Forkl A, Habermeier H-U and Kronmiiller H 1993 J. Alloys Comp. 195 499
lndenbom M V, Forkl A, Ludescher B. Kronmiiller H, Habermeier H-U, Leibold B, D’Anna G. Li T W, Kes P H
and Menovsky A A 1994c Physica 226C 325
lndenbom M V. Kronmiiller H,Li T W,Kes P H and Memvsky A A 1994d Pkpica 222C 203
lndenbom M V, Schuster Th. Kuhn H. KronmGller H, Li T W and Menovsky A A 1995 Phys, Rev. B 51 15484
lnui M and Harshman D R 1993 Phys. Rev. B 47 12205
lnui M, Littlewood P B and Coppersmith S N 1989 Phys. Rev. Left. 63 2421
laffe L B. Larkin A I, W a m o v A A and Yu L 1993 Phys. Rev. B 47 8936
Me F. Matsushita T,Otabe S,Malsuno T and Yamafuji K 1989 Cryogenics 29 317
Isaac I, Jung 1. Murakami M, Tanaka S, Mohammed M A-K and Friedrich L 1995 Phys. Rev. B 51 11806
ltzler M A. Danner G M, Bojko R and Chaikin P M 1994 Phys. Rev. B 49 6815
ltzler M A and Tinkham M 1995 Phys. Rev. B 51 435,941 I
lvanchenko Yu M 1993 Phys. Rev. B 48 15966
lvanchenko Yu M. Belevtsov L V, Genenko Yu A and Medvedev Yu V 1992 Phy8ico 193C 291
lvanchenko Yu M. Khirnyr V F and Mikheenko P N 1979 Zh. Ekrp. Teor. FU. 77 952 (Engl. Transl. 1979 Sou.
Phys.-JETP 50 479)
Ivlev B I and Campbell L J 1993 Phys. Rev. B 47 14514
Ivlev B I and Kopnin N B 1990a J. Low Temp.Phys. 80 161
-1990b
Phys. Rev, Le//. 64 1828; 1990 Phys. Rev, B 42 10052
Phys. Rev. B 44 2747
-1991
Ivlev B 1. Kopnin N B and Pokrovsky V L 1990 J. Low Temp. Phys. 80 187
Ivlev B I. Kopnin N B and Salomaa M M 1991a Phys. Rev. B 43 2896
Ivlev B I and Mel’nikov V I 1987 Phys. Rev. B 36 6889
lvlev B I and Ovchinnikov Yu N 1987 Zh. a s p . Teor. F ~ L93 668 (Engl. Transl. 1987 Sov. Phys.-JETP 64 378)
Ivlev B 1. Ovchinnikov Yu N and Pokrovsky V L 1991b Mod. Phys. Len B 5 73
1578
E H Brand1
fvlev B I, Ovchinnikov Yu N and Thompson R S 1991c Phys. Rei.. B 44 7023
lye Y, Fukushima A. Tamegai T. T w h i m T an6 Bando Y 1991a Physicn lSS-9C 297
lye Y, Nakamuta S. Tame& T,T8rashima T.Yamamolo K and Bando Y 1994a Phydca 1ffiC 62
lye Y,Oguro I, Tamgai T. D a m W R. Motohira N and I m w a K 1992 Physim 199C 154
lye Y. Twashjma T and Bando Y 1991b Phydca 184C 362
lye Y , Wat3nabe A, NaRmura S, Tame@ T,
T, ~ a ” 0 t DK and Bando Y 19% Physica 167C 2%
Jackson D J C and Das M P 1994 Solid S f a e Conutrrm. 20 475’
Janossy B. de Gfaaf A, Kes P H. Knpylov V N and Togonidze T G 1995 Pkysica 246C 217
Jemeret B, Gavilano J h Raane G A. teemann Ch and Martinoti P 1989 AppL P h p t e i r 55 2336
le& W G and Tesrtardi L R 1993 Phys. Reu. B 48 12993
J e n m H J 1990 P h p Rev. Lea 64 3103
lensen H 3, Brass A, Shi A-Ch and Berlinski A J 1990 Phys. Rev. B 41 6394
Jenen H 1. Bmhet Y and B m A 1989 J. Low Temp Phys. 74 293
Jenan H J and Minnhagcn P 1991 Phyz Rev. &U. 66 1630
Jenren H J, Minnhagen P, Sonin E and Weber H 1992 Eumphys. kif.28 463
Jiang W. Yeh N-C, Rced D S. fiplani U. Beam D A, Konczykowski M. Tombrello T A and Holrzberg F 1994
Phys. Rex &U. 72 550
Jiang W,Yeh N-C, Reed D S. fiptani U. and Holubcrg F 1995 Phys. Rev. k i t , 14 1438
Jiang X, COCUOIIYPI. Hagen S J and Lobb C J I994 Phys. Rev. 8 49 9244
J w R. Schilling A and On H R 1994 Phsr. Rev. B 49 9218
l m M. Koblirch!a. M Rand van Walen A J I 1995 Phyj. Rev. B 52 (submilted)
Janssan A and MinnMgen P 1994 Phys. Rev, Lm 73 3576
Jose 1 V and Ramim.Santiago G 1993 3. Phy. A: Mah. Cen 26 U35
Josephson B W 1962 Phys Len. 1 251
-1966
Pkys. Rev. 152 21 I
lung 1. Isaac 1 and Mohammed M A-K 1993 Phyj. Rev. B 48 75526
Juug I,M o h w d M A.K. Is& I and Friednch L 1994 Phys. Rev. €3 49 12188
Kadowaki K Songliu Y and Kitazawa K I994 &p”.
Sei Tehwl. 7 519
Kamal S. Bonn D A. Golldenteld N, Hirschfeld P J. Gang R and Hardy W N 1994 Pkys. Rev. h i . 73 1845
Krrmerlingh Onnes H 1911 Leiden CO”. 12Ob, 122b, 124~
Kammerer U 1969 2.Phyr. 227 I2.S
Kaplunenko V K, MoskVin S I and Shmidi V V 1985 Fiz. N k k Tcmp 11 846 (Engl. T m l . 1985 SOS.1. Low
Temp. Phys. 11 4 W
Eudw M 1985 Phys. Rav. Len. 55 2923
-1987
Nurl. Php. 8 290 582
Karkut M G,Slaski ht, Heill L K. Sagdahl L T snd Fossheim K 1993 Physicsic0 215C 19
Kato Y,Enomoto Y and Maekawa S 1991 Phys. Rev. B 44 6916
Kat0 Y md Nagaaw. N 1993 Phy. Rev. B 47 293248 7383
Keimer E, Shih W Y.Erwin R W.Lynn J W,Dogan Fand &ay 1 A 1994 Phys. Rev. Lerc 73 3459
KC"^ R 1981J, hW
i h p . p h p . 46 zas
-1983
J. Low Tenp Phys;.50 337
Kes P H 1992a PhmomenoIogy M d Applican‘on of High-Temproture Supmonducfars 4 K S Bedell. M Inui.
D E Meluer. J R Scfvieffcr and S Doniach (New York: Addison-Wesley) p 390
-19926
Concise Encyclopedia of Magn<ficand Suflreonduciing Marcrials ed J E v e n (Oxford: Peeamon)
p 00. I63
Ker P H,A m S, w den Berg 1. van der Beek C J, Mydosh 1 A 1989 S u p e m d . Sci. Techno[ 1 242
Kcs P H,A m J. V i a o h V Man& vao dc; Beek C J 1990 Phys. Rev. Lcrt. 64 1063
Kes P H. van der Beek C 1. Maley M P, McHenry M E. Huse D A, Menken M 1 V and Menovsky A A 1991
Phy. Ret,. Mi. 67 2383
Kes P H and Tsuei C C 1981 Phys. Rev. &if. 47 1930
Kcs P H and Tsuei C C 1983 Phys. Rcv. B 28 5126
Kessler C. Netendnhl B. Petipd D-N.DulEic A, Habemier H-U and Mehring M 1994 Physicd 219C 233
KhaIfin I B. Dorosinskii L A and Shapim B Ya 1995 Phys. Rev. B 51 1245
KhaI6n 1 8, Malomad B A and Shapiro B Ya 1994 Phys. Rev. B 50 9351
Khalfin I Band Shapim B Ya 1993 Physica ZOYC 359
KhaImakhclidm Yu G and Mints R G 1991 Cryogenics 29 10.41
Khannov R 1. Talanov Yu 1, V&dw
Yu M and Teitelbaum 0 B 1995 Physic0 SU: 333
Khaykovich 8. Zeldov E Konczykowsb M, hbjer W. L d n A 1 and Clem I R 1994 Physiea U54W 2757
The pux-iine lattice in superconductors
1519
Khomskii D I and Freimuth A 1995 Phys. Rev. Len. 75 1384
Kim D H, Gray K, Kampwirth R T, and McKay D M 1990a Phys. Rev. B 41 I1642
Kim D H.Gray K, Kampwifth R T. Woo K C. McKay D M and Stein 1 199% Phys. Rev. B 42 6249
Kim D H. Kang W N, Kim Y H. Park J H, Lee J 1. Yi G H. Hahn T S and Choi S S 199% Physicn 246C 235
Kim S F, Li 2 2 and Raffy H 1995b Physicn Z44C 78
Kim Y B, Hempstead C F and Stmad A R 1963 Phys. Rev. 129 528
-1965
Phys. Rev. A U9 1163
Kim Y 6 and Stephen M J 1969 Supeperconducrivily VOIn, ed R D Parks (New York Marcel Dekkw) p 1107
Kim Y C, Thompson 1 R, Ossandon I G, Christen D K and Pacanthaman M 199% Phys. Rev. B 51 11767
Kitaguchi H, Takada I, Oda K, Osaka A, Miura Y, Tomii Y, Mazaki H and Takano M 1989 Physicn 157C 267
Kleiman R N. Gammel P L, Schneemeyer L F, Waszczak I V and Bishop D I 1989 Phys. Rev. 62 R2331
Kleiman R N, Broholm C, Aeppli G, Bucher E, Stiicheli N, Bishop D J, Clausen K N, Mortensen K, Pedersen J S
and Howard 6 1992 Phys. Rev. Lett 69 3120
Klein L, While W R. Beasley M R and KapitulnifA 1995 Phys. Rev. B 51 6796
Klein L, Yacoby E R, Wolfus Y, Yeshurun Y, Burlachkov L, Shapiro B Ya, Konczykowsld M and Holtzberg F
1993a Phys. Rev. B 47 12349
Klein L. Yacoby E R, Yeshurun Y,Konczykowski M, Holtzkrg F and Kishio K 1993b Physica 209C 251
Klein L. Yacoby E R, Yeshurun Y, Konczykowski M and Kishio K 1993c Phys. Rev. B 48 3523
Klein N, Tellmain N. Schulz H, Urban K, Wolf S A md b i n V Z 1993d Phys. Rev. Lett. 7 1 3355
Klein U 1987 J. Low Temp. Phys. 69 1
-1990
Phyz. Rev. B 41 4819
Klein U and Paanger 1991 Phys. Rev. B 44 7704
Klein U, Rammer J and Pesch W 1987 J. Low Temp. Phys. 66 55
Kleiner R, Miiller P. Kohlstedt H, Pedecsen N F and Sakai S 1994 Phys. Rev. B 50 3942
Kleiner R, Steinmeyer F, Kunkel G and Miiller P 1992 Phys. Rev. k t r . 68 2394
Kleiner W H. Roul L M and AuUer S W 1964 Phys. Rev. A 133 1226
KIemm R A 1988 Phys. Rev. B 38 6641
-1990
Phys. Rev. B 41 117
Phys. Rev. B 47 14630
-1993
-1995
SIAM J. Appl. Math (August 1995)
-1996
Layered Superconducfors (New York: Oxford) (in preparation)
Klemm R A and Liu S H 1995 Phys. Rev. Left. 74 2343
Klemm R A. Luther A and Beasley M R 1975 Phys. Rev. B 12 877
Klimesch P and Pesch W 1978 J. Low Temp. Phys. 32 869
Klupsch Th 1995 Physien 244C 165
Klupsch Th. H e r s R and Hiergeist R 1993 Physicn 211C 65
Knorpp R. Forkl A, Habermeier H-U and Kronmiiller H 1994 Phyico DOC 128
Kobayashi T. Nakayama Y, Kishio K, Kimura 7, Kitazawa K and Yamafuji K 1993 Appl. Phy% Len 62 1830
K o k r F, Ri HC, Gross R, Kcelle D, Huebener R P and Gupta A 1991a Phys. Rev. B 44 I1951
Kober J, Gupta A, Esquinazi P, Braun H F and Brandt E H 1991b Phys. Rev. Lea. 66 2507
Koblischka M R, Schuster Th and Kronmiiller H 1993 Physica 211C 263
-1994
Physic= 219C 20.5
Koblischka M R and Wijjngden R I 1995 Supercond. Sci. TechnoL 8 199
Koch R H, Foglielti V, Gallagher W I, Koren G, GupIa A and Fisher M P A 1989 Phys. Rev. Letr. 63 151 1
Kogan V G 1988 Phys. Rev. B 38 7049
-1990
Phys. Rev. Lett. €4 2192
-1994
Phys. Rev. B 49 15874
Kogan V G. Bulaevsldi L N, MinlnoviC P and DobrosavIjevi6-GNjit L 1995 Phys. Rev. 6 51 15344
Kogan V G and Campbell L J 1989 Phys. Rev. Left. 62 1552
Kogan V G and Clem I R 1981 Phys. Rev. B 24 2497
Kogan V G, Fang M M and Mitra S 1988 Phys. Rev. 6 38 11958
Kogan V G and Gurevich A 1995 Preprint
Kogan V G, Ledvij M and Bulaevskii L N 1992 Phys. Rev. Lett. 70 1870
Kogan V G, Ledvij M, Simonov A Yu, Cho J H and Johnston D C 1993a Pkys Ret,. Len 70 1870
Kogan V G, Nakagawa N. Thiemann S L I990 Phys. Rev. B 42 2631
Kogan V G, Simonov A Yu and Ledvij M 1993b Phys. Rev. B 48 392
Konczykowski M 1994 Physica D U O C 197
Konczykowski M, Burlachkov L I, Yeshurun Y and Holtzberg F 1991 Phyr. Rev. B 43 13707
1580
E H Brandi
Konczykowski M. Chikumoto N,Mnokur V M and Feigel'man M V 1995 Ph,vs. Rev. B 51 3951
Konczykowski M,Vinokur V M,Rullier-Albenque F, YeshuNn Y and Holtzberg F 1993 Phys. Rev. B 41 5531
Kopelevich Y, Esquinazi P and Ziese M 1994 Mod Phys. Left. B 8 1529
Kopnin N B. Ivlev B I and Kalatsky V A 1993 J. Low Temp. Phys. 90 1
Kopnin N B and Kravtsov V E 1976 Pis'ma Zh Eksp, Tmr. FU 23 631 (Engl. Transl. 1976 Sov, Phys.JETPLen.
23 578); 1916 Ur f i p . Tcor. FiC I1 1644 (Engl. Tmsl. 1976 Sov. P k p J E T P 44 861)
Kopnin N B and Lopaiin A V 1995 Phys. Rev. B 51 15291
Kopnin N B, Lapatin A V. Sonin E B and Trait0 K B 1995 Phys. Rev. Lett. 74 4527
Kopnin N B and Salomaa M M 1991 Phys. Rev. B 44 9661
Kopylov V N, Koshelev A E, Schegolev I F and Togonidze 1990 Phyk-o 170C 291
Korevaar P, Kes P H,Koshelev A E and Aarts J 1994 Phys. Rev. Leff. 12 3250
Korshunov S E 1990 Europhys. Letf. 11 757
-1993
Phys. Rev, B 48 3969
Phys. Rev. B 50 13616
-1994
Korshunov S E and Larkin A I 1992 Phys. Rev. B 46 6395
Koshelev A E 19921 Phys Rev. B 45 12936
-1992b
Phyma 191C 219
-1992c
Pk&
198C 371
Phys. Rev. B 48 I I80
-1993
-1994a
Physica 223C 276
-1994b
Phys. Rev. B 50 506 (enatum 51 12063)
-1996
Preprint
KosheleV A E, Glarman L I and k!dn A I 1995 Phys, Re,: B (submitted)
Koshelev A E and Kes P H 1993 Phys. Rev. B 48 6539
Koshelev A E and Larkin A I 1995 Phys. Rev. B 52
Koshelev A E and Viioku~V M 1991 Physicn Il3C 465
-1994
Phys, Rev. Lett. 13 3580
-1995
Phys. Rev. B 51 11678
Kosterlitz J M and Thouless D I 1973 J. Phys. C: Solid State Phys. 6 I181
Kiitzler I. K a u h n M, Nakielski G and Behr R 1994a Phys. Rev. Lett. 12 2081
Kiitzler J. Nakielski G, Baumann M, Behr R, Goerke F and Brandt E H 19946 Phys, Rev. B 50 3384
Koyama T and Tachiki M 1992 Physica 193C 163
Physica 210C 509
-1993
Koziol 2,de Chael P, Franse I J M, Tamavski 2 and Menovsky A A 1993 Physica 212C 133
Kriigeloh U 1970 Phys. Stet. Sol. 42 559
Kriigeloh U,Kumpf U and Seeger A 1971 Pmc. 12fhInt. CO@ Low TemperamrePhysics LT12 ed E Kmda (Kyoto:
Academic Press of Japan) p 473
Kramer E J 1973 J. Appl. Phyr. 44 1360
-1978
J. Nuel. Mater. 12 5
Kramer L I971 Phys. Rev. B 11 3821
Kramer L and Pesch W 1914 Z Phys. 269 59
K & w A 1995 Physica 240C 203
K r h e r A and KuliC M L 1993 Phys. Rev. B 48 9673
KrXmer A and KuliE M L 1994 Phys. Rev. B 50 9484
Krasnov V M, Pedenen N F and Golubov A A 1993 Pkysica 209C 519
Kraus M, Kreiselmeyer 0, Daniel I, Leghissa M, S a e m - l s c h e n k o G, Holrapfel B, Kummeth P, Scholz R and
Vinnikov L Ya 1994a Nuclearlwfr. Methods B 89 307
Kraus M, Leghissa M and Saemann-lschenko G 1994b Phys. BL 50 333
Krey U 1972 Int. J, Magnerirnt 3 65
-1973
Inf. 1. Magnetism 4 153
Krivoruchko V N 1993 J. Low Temp. Phys, 92 217
Kronmtlller H 1970 Phys. Stat. Sol. 40 295
-1974
Int. Discussion Meefingon F l u Piwing in Superconductors ed P Haasen and H C Freyhardt (G6ningen:
Akademie der Wissenschaften) p 1
Kronmtlller H and Riedel H 1970 Phyr. Stat. Sol. 38 403
-1976
Phys. Star. Sol. 71 581
Kmithof G H,van Son P C and Klapwijk T M 1991 Phyr. Rev. Lett. 61 2725
Kmsin-Elbaum L, Civale L Blatter G. Manvick A D, Holtzberg F and Field C 1994 Pkys. Rev. Lett. I2 1914
Thepux-line lattice in superconductors
1581
Krusin-Elbaum L,Civale L, Holtzberg F. Maloremoff A P and Field C 1991 Phys. Rev. Len 67 3156
Kmsin-Elbaum L, Civale L, Vinokur V M and H o l t h r g P 1992 Phys. Rev. Len. 69 2280
Kugel K I, Lisovskaya T Ya and Mints R G 1991 Supercanducrivily 4 2166
Kugel K I and Raldunanov A L 1992 Physica 196C 17
Kuklov A, Bulatov A and Buman J L 1994 Phys. Rev. Lerr. 72 2274
KuliC M L, KrXrner A and Schotte K D 1992 SolidSlaIe Commrm. 82 541
KuliC M L a n d Rys F S 1989 J. Low Temp. Phys. 76 167
Kulik I O 1968 Zh E k p . Teor. Fir. 55 889 (Engl. Transl. 1969 Sov. Phys.-JETP 28 461)
Klimmel R. Schiissler U, Gunsenheimer U and Plehn H 1991 Physica 1 8 S 9 C 221
Kummeth P, Smller C. Neumilller H-Wand Saemann-Ischenko G 1994 Appl. Phys. Len. 65 1302
Kunchur M N, Christen D K and Phillips J M 1993 Phys. Rev. Lell. 70 998
Kunchur M N, Christen D K, Khbunde C E and Phillips 1 M 1994 Phys. Rev. Len. 72 752.2259
Kunchur M N, Poon S J and Subramanian M A 1990 Phys. Rev. B 41 4089
Kung P I , Maley M P, MeHenry M E,Willis 10, Murakami M and Tanaka S 1993 Phys. Rev. B 48 13922
KUpfer H, Apfelstedl I, Flukiger R, Keller C, Meier-Hirmer R, Runtsch B, Tumwsld A. Wrech U and Wolf T 1988
Cryogenics 28 650
Kiipfer H. Zhukov A A, Kresse R. Meier-Hirmer R lahn W, Wolf T, MatsushitaT. Kimura K and Salama K 1995
Phys. Rev. B 52 7689
Kuprianov M Yu and Likharev K K 1974 Fiz Twrd Teh 16 2829 (Engl Transl. 1975 Sov. Phys. Solid Stair 16
1835)
-1975
2%. Eksp. Teor. Fiz. 68 1506 (Engl. Transl. 1975 SOP.
Phys.-JETP 41 755)
Kwok W K, Fendrich J, van der Beek C 1and Crabuet G W 1994a Phys. Rev. Lett 73 2614
Kwok W K, Fendrich J, Welp U, Fleshler S, Downey 1and Crabtree G W 1994b Phys. Rex Len. 72 1088. 1092
Kwok W K, Fleshler S, Welp U, Ylnokur V M, Downey J and Crabtree G W 1992 Phys. Rev. Len 69 3370
Kwok W K, Welp U, Crabtree G W. Vandervmn K. Hulscher G and Liu J Z 1990 Phys. Rev. Len 64 966
Kwok W K, Welp U, Ylnokur V M, Fleshler S, Downey 1 and Crabme G W 1991 Phys. Rev. Let[, 67 390
Labdi S. Ram H,b b o r d e 0 and Monceau P I992 Physica 197C 274
Labusch R 1966 Phys. Len. 22 9
Phys. Rev. 170 470
-1968
-1969a
t'hys. Stat. Sol. 32 439
Cryslal Latrice Defeclrs 1 I
-1969b
Labusch R and Doyle T B 1995 Pmc. Inf Cryogenics Malerids Con$ 195'4. Critical S m e in Supercondru"
(Oclober 1994, Honolulu)
Lainon M. Sun J Z, Geballe T H,Beasley M R and Bravman J C 1991 Phys. Rev. B 43 10405
Landau L D and Lifshik E M 1960 Electrodynamics of Continuous Media (New York: Pergamon) p 178
Lang W, Heine G, Kula W and Sobolewski R 1995 Phys, Rev. B 51 9180
Larbalestier D C 1991 Physics Today 44 (6) 74
Larkin A I 1970 U. E k p . Teor. Fir. 58 1466 (Engl. Transl. 1970 Sow Phys.-JETP 31 784)
Larkin A I, Marchelti M C and Vinokur V M 1995 Phys. Rev. Lett 75 2992
Larkin A I and Ovchinnikov Yu N 1971 Zh, E k p . Teor. Fic 61 1221 (Engl. Tmsl. 1972 Sov. Phys.-JETP 34
651)
-1973
U. E k p . Teor. Fiz. 65 1704 (Engl. Transl. 1974 SOP.Phys.-JETP 38 854)
-1975
Zh. E k p . Teor. Fir. 68 1915 (En& Transl. 1976 Sov. Phys.-JETP41 960)
-1979
J. Low Temp. Phys. 73 109
-1986
Nonequilibrium Supercunduoiviry ed D N Langenberg and A I Larkin (Amsterdam: Elsevier) p 493
-1995
Phys. Rev. B 51 5965
Lawrence W E and Doniach S 1971 Pmc, I21h Inl. Con$ Low Tempemrure Physics LT12 ed E Kanda (Kyoto:
Academic Press of Japan) p 361
LeBlanc M A R. Celebi S, Wang S X and Plecwhek V 1993 Phys. Rev. Len. 71 3367
Le Doussal P and Giamarchi T 1995 Phys. Rev. Lex 74 606
Lee C Y and Kao Y H 1995 Physic0 241C 167
Lee H H and Moore M A 1994 Phys. Rev. B 49 9240
Lee J Y. Paget K M, Lemberger T R. Foltyn S R and Wu X 1994 Phys. Rev. E 50 3337
Lee P A and Rice T M 1979 Phys. Rev. B 19 3970
Lee S L, Warden M, Keller H,Schneider I W. Zimmermann P, Cubill R. Forpan E M. Wylie M T, Kes P H,
Li T W, Menovsky A A and Tamavski Z 199% Phys. Rev. Len. 75 922
Lee S L, Zimmermann P, Keller H,Warden M. SaviC I M. Schauwecker R, Zech D. Cubit1 R. Forgan E M,
Kes P H,Li T W, Menovsky A A and Tamavski Z 1993 Phys. Rev. Len. 71 3862
1582
E H Brand1
Lee Th S,Missen N. Sagdahl L T,Clarke J, Clem 1 R. Char K, Eckstein J N. Fork D K, Lombard0 L,Kapilulnik A,
Schneemyer L F, Waszczak J V and van Dover R B 1995b Phys. Rev. Len.74 2796
Leghissa M. Gurevich L A . Knus M. Saemn-lschenko G and Vmikov L Ya 1993 Phys. Rev. B 48 31341
Leghissa M, Schuster 'In. Gerhauser W, KlaumIinzer S. Koblischka M R. KronmUUer H, Kuhn H. NeumUller H-W
and Saemann-lcchenko G 1992 Eumphys. Left, 19 323
Leiderer P, Eoneberg J. Briill P, Bujok V and He-inghaus S 1993 Phys. Rev. Letf. 71 2646
Lemmens P, Frirnig P, Ewen S. Pankal J. Mybach G and Comberg A 1991 PhyJica 174C 289
Lensink J G. F l i p C F J. Roobeek 1. Griessen R and Dam B 1989 Phyxica 16-C
663
Levitov L S 1991 Phys. Rev. Lett. 66 224
Li H C, Li J W. Wang R L. Yin E. Zhao Z X. Yan S L. Fang L and Si M S 1995 Physicn 246C 330
Li Qi. Kwon C, Xi X X, Ehatlacharya S,Walkenhorst A, Venkatesan T,Hagen S J. Jiang Wand Green R L 1992a
Phys. Rec Left. 69 2713
Li Qiang, Suenaga M. Hikata T and Sat0 K 1992b Phys. Rev. E 46 5857
Li Qiang, Suenaga M. Bulaevskii L N.H i k m T and Sa10 K 1993 Pkys. Rev. B 48 13865
Li Qiang, W l e s m n H J, Suenaga M, Motowidlow Land Haldar P 1994 Phys. Rev, B 50 4256
Li Y-H and Teitel S 1991 Phys. Rev. Len 66 3301
-1992
Phys. Rev. E 45 5718
-1993
Phys. Rev. E 47 359
-1994
Phys. Rev. E 49 4136
Liang R, Dosanjh P. Bonn D A. Hardy W N and Eerlinsky A J 1994 Phys. Rev. E 50 4212
Likharev K K 1971 b Vuzov Radio&. 14 909, 919 (Engl. Transl. 1971 Radiophys. Quonlum Electron 14 714,
722)
Rev. Mod. Pkys. 51 101
-1979
Ling D C. Yang G, Chen J T and Wenpr L E 199% Phys. Rev. Len. 75 201 1
Ling X S and Budnick 1 I 1991 Mognefic Suscepribiliry of Superronduefors and OfherSpin System ed R A Hein.
T L Fmcavilla and D H Liebenberg (New York: Plenum) p 377
Ling X S. MKambridge J D, R i m N D. Sleight J W, Prober D E, Motowidlo L R and Zeitlin B A 1995b Phys.
Rev. Len. 74 805
Ling X S, Shi D and Budnick J I 1991 Physicn 185-9C 2181
Lischke E and Rcdewald W I974 Phys Snr. Sol. B 63 97
Liu C and Narahari Achar B N 1990 Physica 168C 515
Liu Ch J, Schlenker C Schuben J and Slrihker E 1993 Phys. Rev. E 48 13911
Liu J 2, Jia Y X, Shelton R N and Fluss M J 1991 Phys. Rev. Left. 66 1354
Liu Wu. Clinton T W and Lobb C J 1995 Phys Rev. E 52 7482
Liu Y, Haviland D E, Glzman L I and A M Galdman 1992 Phys. Rev. Left. 68 2224
Livingston 1 D and DeSorbo W 1969 Supexonducfiviry vol 11, ed R D Parks (New York Marcel Dekker) p 1235
LoRvld S E, EhagaI S M. Rajeswari M. Vedaman T,Kanjilal D. Senapati L and Mehta G K 1995 Phys. Rev. E
51 8489
Logvenov G Yu. Ry-ov
V V. Ustinov A V and Huebener R P 1991 Physica 175C I79
L6m2 D. Nieva G, de la Crur F. Jensen H J and OKane D 1994 Phss. Rev. B 50 9684
Lowell J 1972 1.Phys. F: Mer. Phys. 2 547
Lowndes D H. Christen D K, Klabunde C E, Wang 2 I. Kmeger D M, Bud& J D. Zhu S and Norton D P 1995
Phys. Rev. Left. 74 2355
Luo S. Yang G and Cough C E 1995 Phys. Rev. B 51 6655
Lykov A N 1993 Adv. Phys. 42 263
Lynn J W. Romv N. Griwreit T E. Zhang H and Clinton T W 1994 Phss. Rev. Letf. 72 3413
Lysenko V S,GomeniukY V, Lozovski 7,Tyagulski I P and Varukhin V N 1994 Physicn W U O C 2631
Lyuksyutov I F 1992 Europhys. Len, 20 273
Ma H and Chui S T 1991 Phys. Rev. Left. 67 505
-1992
Phys. Rev. Lelt. 68 2528
Machida M and Kabunki H 1993 Phys, Rev. Len 71 3206
-1995
Phys. Rev. Len 74 1434
Maggio-Aprile I. Renner Ch. Erb A. Walker E and Fischer 01995 Phys. Rev. Left. 75 2754
Majer D, Zeldov E and Konzcykowski M 1995 Phys. Rev. Len. 75 1166
Maki K 1964 Physics 121
-1969a
Superconducrivity vol 11. ed R D Parks (New York: Marcel Dekker) p 1035
-1969b
J. Low Temp. Phys. 1 4 5
-1982
Prog, Theor. Phys. 41 902
the flux-line lattice in superconductors
1583
Maki K and Takayam H 1971 Pmg. Theor. Phys. 46 1651
Maki K and Thompson R S 1989 Phys. Rev. B 39 2767
Maksimov I L 1988 Phys. Len. 12SA 289
-1994
PhysicaW5-4OC 3017
Maksimov I L and Elistratov A A 1995 Pis'ma Zh E k p . %or. Fk. 61 204 (End. Transl. 1995 SOY.PhysJETP
k t f . 61 208)
Maley h4 P, Willis J 0,Lessure H and McHenry M E 1990 Phys. Rev. B 42 2639
Malozemoff A P 1991 Physica 185-9C 264
Malozemoff A P and Fisher M P A 1990 Phys. Rev. B 42 6784
Mannhardt J 1992 Mod. Phys. Len. B 6 555
Mannhardl 1. Bosh I, Gross R and Huebener R P 1987 Phys. Rev. B 35 5267
Mannhardt 3, Scholm D 0. Bednon J G and Muller K A 1991 Phys Rev. kn.67 2099
Mannhardt J and Tsuei C C 1989 Z Phys. B 77 53
Mansky P A , Chaikin P M and Haddon R C 1993 Phys, Rev. Len.70 1323
Mansky P A , Danner G and Chaikin P M 1995 Phys Rev. B 52 7554
Marchetti M C 1991 Phys. Rev. B 43 8012
-1992
Physicn ZOOC 155
Marchetti M C and Nelson D R 1990 Phys. Rev. B 41 1910. 42 9938
-1991
Physica 174C 40
-1993
Phys. Rev. B 47 12214
-1995
Phys. Rev. B 52
Marchetti M C and Viokur M V 1994 Phys, Rev. Len. 72 3409
-1995
Phys. Rev. B 51 16276
Marchevsky M, Gurevlch L A, Kes P H and A m J 1995 Phys. Rev. Letf. 75 2400
Marcon R, Silva E, Fastampa R and Giura M 1992 Phys. Rev. B 46 3612
Marcon R. Silva E, Fastampa R, Giura M and Sard S 1994 Phys. Rev, B 50 13684
Markiewicz R S 1990a 3.Phys: Condew. Matter 2 4005
-1990b
Physica 171C 479
Marley A C. Higgins M J and Bhanacharya S 1995 Phys, Rev, Len. 74 3029
Manh G E 1994 Phys Rev. B 49 450,50 571
Mardn S and Hebard A F 1991 Phys Rev. B 43 6253
Martinez I C Brongersma S H, Koshelev A, Ivlev B, Kes P H,Griessen R P, de Gmot D G, Tamavski z and
Menovsky A A 1992 Phys. Rev. Len. 69 2276
Martinez J C van der Linden P I E M, Bulaevskii L N, Bmngersma S H,Koshelev A, Ferenboom I A A I,
Menovsky A A and Kes P H 1994 Phys. Rev. Lett. 72 3614
Martinoli P 1978 Phys. Rev. B 17 1175
Mardnoli P, Olsen I L and Clem 1 R 1978 1. Less Comm Met. 62 315
Marrynovich A Yu 1993 Physic0 218C 19
-1994
Physicn 227C 22
Marx A, Fafh U, Ludwig W,Gross R and Amrein T 1995 Phys. Rev. B 96735
Mathieu P, Plapis B and Simon Y 1993 Phys. Rev. B 40 7376
Matsuda Y,Ong N P, Yan Y F, Harris J M and Pete" I B 1994 Phys. Rev. B 49 4380
Matcushita T 1989 Physicn 160C 328
Physien 22OC 1172
-1994
Matsushita T aOd Me F 1985 1. Phys. Soc. Japan 54 1066
Matsushita T, Otak E S and Ni B I991 Physica 182C 95
Matsushita T aOd Yamafuji K I979 3. Phys. Soc. 3apm 47 1426
Maw& Y and Yamafuji K 1994 Phpica UBC 336
MeElfresh M, Zeldov E, Clem I R. Darwin M. De& J and Hou L 1995 Phys. Rev. B 51 911 1
McHenry M E, Lessure H S, Maley M P. Coulter I Y,Tanaka I and Kojima H 1992 Physica 190C 403
McHenry M E and Sutton R A 1994 Prog. Mat. Sci. 38 159-310
Mee C Rae A I M. Vinen W F and Gough C E 1991 Phys. Rev. B 43 2946
Mehring M 1990 Earlier and Recenf Aspects of Supercondufivily ed J G Bednon and K A Mtiller (Berlin:
Springer) p 467
Meluing M, Hentsch F, M W s c h Hj and Simon A 1990 Solidsfate Cammm 75 753
Meilikhov E A 1993 Physico 209C 566
Meilikhov E A and Farzetdinova R M 1993 Physica 21OC 473
-1994
Physlca 221C 27; 1994 J. Supercod. 7 897
1584
E H Brundt
Meissner W and Ochsenfeld R 1933 Na:um~issensckgf:en
21 787
Menon G I and Dasgupta C 1994 Phys. Rev. Len. 73 1023
Metlushko V V. Baen M, Jouckheem R, Moshcdkov V V and Bruynseraede Y 1994a Solid Sfare Commun. 91
331
Metlushko V V, Gunthemdl G, Moshchalkov V V and Bruynseraede Y 1994b Eumphys. Len.26 371
Miesenmk H M 1984 J. Low Temp. Pkys. 56 1
Mikheenko P N. Genenko Yu A. Medvedev Yu V, Usosldn A I and Chu!mnova I N 1993 Pkysicn 212C 332
Mibeenkc P N and Kuzovlev 1993 Physica 2MC 229
Mikheev L V, Dmssel B and Kardar M ‘1995Pkys. Rev. Lett 75 1170
Mikheev L V and Kolomeisky E B 1991 Phys. Rev. B 43 10431
Miitik G P 1995 Physica 245C 287
Minenko E V and Kulik I 0 1979 Fiz NizA Tern# 5 1237 (E@. Transl. 1979 Sov. 3. Low Temp. Phys. 5 583)
Minnhapn P 1991 Phys. Rev. B 44 7546
Minnhagen P and Olsson P 1991 Phys. Rm. Len 67 1039
Pkys. Rev. B 45 5722, 1992 Pkys. Scr. T 42 29
-1992
Mints R G and Rakhmanov A L 1981 Rev. Mod Phys. 53 551
Mints R G and Snapiro I B 1993 Phys. Rei,. B 47 3273; 1993 Eumphys. Le:. 21 611
-1994
Phys. Rev. B 49 6188
-1995
Phys. Rev. B 51 3054
Mitin A V 1987 zh. Eksp, Tcor. Fiz 93 590 (end. Transl. 1987 Sol*.Phys.-JETP 66 335)
Miu L, Wagner P, Hadish A, Hillmer P, Adrian H, Wiener J and Wuth G 1995 Pkys. Rev. B 51 3953
Mkrichyan G S and Schmidt V V 1971 zh. &p, Teor. Fiz 61 367 (Engl. Transl. 1972 Sov. Phys.-JETP 34 195)
Monceau P,SaintJames D and Waysand G 1975 Phys. Rev. B 12 3673
Moan F C. Weng K C and Cheng P Z 1989 J. Appl Phys. 66 5643
Moon F C, Yanoviak M M and Ware R 1988 Appl. Pkys, Leff.52 1534
Moonen J T and Brom H B 1995 Physica 244C 1, IO
Moore M A 1989 Phys. Rev. B 39 136
-1992
Pkys. Rev. B 45 7336
Moore M A and Wilkin N K 1994 Phys. Rei). B 50 10294
MoraisSmilh C, lvlev B and Blaner G 1994 Phys. Rev. B 49 4033
-1995
Pkys. Rev. B 52 10581
Moser A, Hug H J, Panshikov I, Stiefel B. FriQ 0, Thomas H, Baratoff A, Giinthemdt H-J and Chaudhari P 1995
Pkys. Rev. Len. 74 1847
Moser E, Seidl E and Weber H W 1982 J. Low Temp Pkys. 49 585
Moshchalkov V V, Cielen L. DhalM M, Van Haesendonck C and Bruynseraede Y 1993 Nnfure 361 617
Moshchalkov V V. Metlushko V V, Giintherodt G, Goncharov I N, Didyk A Yu and Bruynsemde Y 1994a Phys.
Rev. B 50 639
Moshchalkov V V. Gielen L, Baen M, Metlushko V. Neuttiens G,Strut& C. Bruyndoncx V. Qiu X, DhaUd M,
Temsr K. Potter C.lonckheere R. Stockman L. Van B e l M, Van Haesendonck C and Bruynsemde Y 1994b
Phys. Scr. T 55 168
Mota A C. Juri G, Wsani P and Pollini A 1989 Pkssicn 162-W 1152
Mota A C, Juri G, Wsani P, Pollini A, TeNvi T A d A u p k K 1991 Pkysicn 1 8 W C 343
Mou Chung-Yu.Wonis R, Doney A T and Huse D A 1995 Phys. Rev. B 51 6575
Muller K H 1989 Physicn 159C 717
Physica 168C 585
-1990
-1991
Physica 185-9C 1609
Mffller K H and Andrikidis C 1994 Pkys, Rev. B 49 1294
Muller P 1994 Physic0 W5-4OC 289
Mullen J and Schmid A 1995 Phys, Rev. Left. 75 136
Mullok S J and Evetts 1 E 1985 3. Appl. Pkys. 57 2588
Munkami M 1990 Mod. Phys. Left. B 4 163
Munkami M, Oyama T, Pujimoto H, Taguchi T. Gotoh S. Shiolwra Y, Koshizuga N and Tanaka S 1990 Jap. J.
Appl. Phys. 29 L1991
Muzikat P 1991 Phys, Rev. B 43 10201
Nakamm N, Gu G D and Koshizuka N 1993 Phys, Rev. Len. 71 915
Narahari A c h B N 1991 Solid Sfme Commm 77 689
Natterman T 1990 Phys. Rev. Len. 64 2454
Natter”
T, Stepanov S, Tang L-Hand Leshom H 1992 J. Physique I1 2 1483
Thefludine lattice in superconiiuctors
1585
Neerinck D. Temf K, BaeR M, Osquiguil Van Hasendonck C. Bmynseraede Y, Cilabea A and Schuller I K
1991 Phys. Rev. Len. 67 2577
Nelson D R 1988 Phys. Rev. Leif 60 1973
3. Stat. Phys 57 511
-1989
Nelson D R and Le Doussal 1990 Phys. Rev. B 42 10113
Nelson D R and Seung H S 1989 Phys. Rev. B 39 9153
Nelson D R and Vinokur V M 1992 Phys. Rev. Lex 60 2398
-1993
Phys. Rev. B 48 13060
Neminsky A M, Dumas I, h
e B P, Schlenker C, Karl H and SIriIzker B 1994 Phys. Rev. B 50 3307
Neminsky A M and Nikolaev P N 1993 Physica 212C 389
Nemoshkalenko V V, Ivanov M A, Nikitin B G and Pogorelov Yu G 1992 High-T, Superconductivity
ed A S Davydov and V M Loktev (Berlin: Springer) p 173-93
Nemoshkalenko V V, Ivanov M A, Nikidn B G, Pogorelov Yu 0 and Klimenko G A 1990 Solid Sfnu C o m u n .
74 637
Neumann L and Tewordt L 1966 2 Phys. 189 55,191 73
Ni Iun and Gu Binglin 1994 Phys. Rev. B 49 15276
Nieter S and Kronmiiller H 1993a Physica 21OC 188
-19936
Physica 213C 43
Niel L and Evem J 1991 Europhys. Len. 15 543
Nielsen H B and Olesen P 1979 Nud Phys. B 160 380
Nikolsky R 1989 Cryogenics 29 388
Niu Q,Ao P and Thouless D I 1994 Phys. Rev. Le!. 72 1706
Noms W T 1970 J. Phys. D: Appl. Phys 3 489
Noshima T, Kinoshita M. Nakano S and Kuwasawa Y 1993 Physica 206C 387
Nozih P and vlnen W F 1966 Phil. Mag. 14 667
Nurgaliev T 1995 Physica 249C 25
Obara H. Anderson M, Fhbrega L, Fivat P, Triscone J-M, Decroux M and Fischer 0 1995 Phys. Rev. Lett. 74
3041
Obst B 1971 Phys. Scar. Sol. 45 467
Obst B and Brandt E H 1978 Phys. Len. M A 460
Obukhov S P and Rubinstein M 1990 Phys. Rev. Left. 65 1279
Okomel'kov A V and Genkin G M 1995 Phys. Rev. B 52 9697
Okuda K. Kawamata S, Noguchi S, ltoh N and Kadowaki K 1991 3. Phys. Soc. Japm 60 3226
Olsson H K. Koch R H, Eidelloth W and Rotertawj R P 1991 Phys. Rev. ten.66 2661
O'NeiU I A and Moare M A 1992 Phys. Rev. Len.69 2582
-1993
Phys. Rev. B 48 374
Ota T, Mads I, Teras& I and Uchinokura K 1994 Phys. Rev. B 50 3363
Oussena M, Gagnon R, Wang Y and Aubin M 1992 Phys. Rev, B 46 528
Oussena M. de Groot P A I, Gagnon R and Taillefer L l994a Phys. Rev. B 49 9222
Phvs. Rev. Len. 72 3606
-1994b
Ousma M. de Groat P A I. Porter S 1, Gagnon R and Taillefer L 1995 Phys. Rev. B 51 1389
Ovchinnikov Yu N 1977 2 Phys. B 27 239
-1980
Zh. E b p . 720,. Fk I 9 1496 (Engl. Tnnsl. 1980 Sov. Phys.-JETP 52 755)
-1982.w
~ b pEO,.
. F&. 82 2020 (Engi. T-I.
1982 so". P ~ ~ ~ . - J E55
T P1162)
-1983
Ur Eksp. Eor. Fk 84 237 (Engl. Transl. 1983 SOP.Phys.-JETP SI 136)
Ovchinnikov Yu N and Brandt E H 1975 Phys. Slot. SOL 67 301
Ovchinnikov Yu N and Julev B I 1991 Phys. Rev. B 43 8024
Ovchinnikov Yu N and Kresin V Z 1995 Phys. Rev. B 52 3075
Palsw T T M. Baltlog B, Schneemyer L F and Waszczak J V 1990a Phys. Rev. Len. 64 3090
F'alstra T T M. Baltlog B, van Dover R B, Schneemeyer L F and Waszczak I V 1990b Phys. Rev. B 41 6621
Pan V M 1993 J. Alloys Comp. 195 387
PankerI 1. Marbach G. Camberg A, Lemens P, FrBnig P and EwerI S 1990 Phys. Rev. Len. 56 3052
Pannetier B 1991 Qumtum Coherence in Meroscopic System ed B Krame~(New York Plenum) p 457
Park S J and Kouvel J S 1993 Phys. Rev. B 48 13995
Park; B, Spielman S, Orenstein I, Nemeth D T,Ludwig F, Claike 1. Merchant P and Lew D J 1995 Phys. Rev.
Lett. 74 3265
Pashitski A E, Polyanskii A, Gurevich A, Panell J A and Larbalestier P C 1991 Physica 24K 133
Pastoriza H. Goffmann M F, M M r e A and de la Cmz F 1994 Phys. Rev. Left. 72 2951
1586
E H Brandt
Pastoriza H and Kes P H 1995 Preprint
Pearl 1 1964 J. Appl. Phys. Lett. 5 65
J. Appl. Phys. 31 4139
-1966
Pmyra P and Kunold A 1991 Phys. Rev. B SI 3820
Perez-Gonwlez A and Clem J R 1985 Phys. Rev. B 31 7048; 1985 I. Appl. Phys. 58 4326
Phys. Rev. 8 43 7792
-1991
Perkins G K, Cohen L F, Zhukov A A and Caplin A D 1995 Phys. Rev. B 52
Pcsch W 1975 Z Phys, B 21 263
Pesch W and Kramer L 1974 J. Low Temp Phys. 15 367
Pesch W, Watts-Tobin R and h e r L 1974 2 Phys. 269 253
Peters P N, Sisk R C Urban E W, Huang C Y and Wu M K 1988 Appl. Phys. Len. 52 2066
Petersen R L a n d Ekin J W 1988 Phys. Rev. B 37 9848
Petzinger K G and Tuttle B 1993 Phys. Rcv. B 47 2909
Petzinger K G and Warren G A 1990 Phys. Rev. B 42 2023
Pie1 H and Muller G 1991 Trans, Mag. 27 85
Pierson S W 1994 Phys. Rev. Len. 13 2496
-1995
Phys. Rev. B 51 6663; 1995 Phys. Rev. Lett. 74 '2359
Pierson S W, Valls 0 T and Bahlouii H 195'2 Phys. Rev. B 45 13035. 13076
Pippard A B 1953 Proc. R. Soc. A 216 547
Phil. Mus. 19 220
-1969
-1994
Superrond. Sci. lichnol. 7 696
Pla 0 and Non F 1991 Phys, Rev. Lett. 67 919
Plaqais B, Mathieu P and Simon Y 1993 Phys. Rev. Lett. 70 -1521
-1994
Phyx Rev. 8 49 15813
Pokrovsky V L. Lyuksyutov I and N a t t e r " T 1992 Phys. Rev. B 46 3071
Porter S J, Daniel G 1. Oussena M and de Gmot P A 1 1994 Physicu W S A I C 3099
Portis A M 1992 J. Supereond. 5 319
P o h s A M, Blazey K W and Waldner F 1988 P h y i c o 153-5C 308
Preosti G and Muzikar P 1993 Phys. Rev. B 48 8831, 9921
Probert M I J and Rae A I M 1995 Phys. Rev. Lett. I S 1835
Prokit V. Davidovif D and Dobrosavljevit-GNjiC L 1995 Phys. Rev. B 51 1270
Prost D, Fruchter L, Campbell I A, Motohira N and Konczykowski M 1993 Phys. Rev. B 47 3457
Provost I. Paumier E and Fonini A 1974 1. Phys. F: Met. Phys. 4 439
PNymboom A, Kes P H, van der Drift E and Radelaan S 1988 Phys, Rev. Lex 60 1430
Pudikov V 1993 Physicu 212C
Piimpin B, Keller H,Kiindig W. Odermau W. Patterson D, Schncider J W. Simmler H, Connell S. Miiller K A,
Bednon J G, B l m y K W, Morgenstem 1. Rossel C and Savif M 1988 Z Phys. B I2 175
Piimpin B. Keller H. Kiindig W. Odematt W. Savit I M. Schneider I W. Simmler H, Zimmermann P. Kddis E
Rusiecki S. Maeno Y and Rossel C 1990 P h y . Rev. B 42 8019
P&t L 1992 Supercond. Sci. Technol. S 497
Qiu X C and Tachiki M 1993 Physics ZOIC 255
Qiu X G,Z h o B R. Guo S Q. Z h g J L. Li L and Tachiki M 1993 Phys. Rev. B 48 16180
Radzihovsky L and Frey E 1993 Phys. Rev. B 48 10357
Raffy H. Renard J C and Guyon E 1972 SolidStote Commun. 11 1679
SolldSlate Commwr. 14 427,431
-1974
Raffy H,Labdi S. Laborde 0 and Monceau P 1991 Phys. Rex Lett. 66 2515; 1991 Pkysicu 184C 159
Raffy H. Labdi S, Li 2 2,RiR H, Kim S F. Megten S. Laborde 0 and Monceau P 1994 Physicn W5-4OC 182
Rammer I 1988 J. Low Temp Ph.w '71 323
-1991a
Physica l l 7 C 421
-1991b
Physicu 181C 99
Rammer J. Pesch W and Kramcr L 1987 Z Phys. B 68 49
Raveau B 1992 Phys. Today 45 53
Recchia C H, Penningtoo C H, Haughlin H and Lafyatis G P 1995 Phys. Rev. B 52 9746
Reed D S, Yeh N-C. Iiang W. Kriplani U. Beam D A and Holbberg F 1994 Phys. Rev. B 49 4384
Reed D S, Yeh N - C Konczykowski M, Samoilov A V and Hollzberg F 1995 Phys. Rev. B 51 16448
Reefnian D and Brom H B 1993 Physicu 213C 229: Phys. Rev. B 48 3567
Reinel D, Dictench W, Majhofer A and Wolf T 1995 Physica 24512 193
Reittu H 1 md Laiho R 1992 S w r c o r d . Sci TechnoL 5 448
The fllur-line lattice in superconductors
1587
Ren Y. Xu J-H and 'ling C S 1995 Phys. Rev. Len. 74 3680
Renner Ch, Kent A D. Niedermann Ph and Fisher 0 1991 Phys. Rev. Len 67 1650
Rhper J 1993 PhyEicn 212C 292
Rhyner I and Blauer G 1989 Phys. Rev. B 40 829
RI H-C, Gross R. Gollnik F, Beck A, Huebener R P, Wagner P and Adrian H 1994 Phys. Rev. B 50 3312
Ri H-C, Kober F, Beck A, Alff L, Gross R and Huebener R P 1993 Phys. Rev. B 47 12312
Richardson R A, Pla 0 and Non F 1994 Pkys, Rev. Lett. 72 1268
Riek C T, Schamberg K and Schopohl N 1991 J. Low Temp. Phys. 84 381
Ries G. Neumuller H-W, Busch R. Kummeth P, Leghissa M, Schmitl P and Saemann-Ischenko G 1993 J. Affnys
Comp. 72 1268
Rigamonti A, Aldrovandi S.Bona F, Carretta P. Cinlolesi F, C o d M and Rubini S 1994 Nuovo Cimenm 16 1743
Riseman T M, Brewer J H, Chow K H. Hardy W N, Kiefl R F, Kreitzman S R, Liang R. MacFariane W A,
Mendels P. Moms G D, Rammer I. Schneider 1W,Niedermayer C and Lee S L 1995 Phys. Rev. B 52 10569
Roas B, Hensel B, Henke S,Klaumiinzer S. Kabius B. Watanabe W. Saemann-lschenko G. Schultz Land Urban K
199% Ewophys. Len. 11 669
Roas B, Schultz L and Saemann-lschenko G 1990b Phys. Rev. Len 64 479
Roddick E and Stroud D 1995 Phys. Rev. Left 74 1430
Roest W and Rekveldl M Th 1993 Phys. Rev. B 40 6420
Rogacld K, Esquinazi P, Fmlhaber E and Sadowski W 1995 Physiea M6C I 2 3
Roitburd A. Swamendruber L J. Kaiser D L, Gayle F W and Bennett L H I990 Phys. Rev. Len. C 64 2962
Rossef C, Maeno Y and Morgenstem I 1989 Phys. Rev. Lett. 62 681
Rossel C, Sandvold E, Sewn1 M. Chevrel Rand Potel M 1990 Physiea 165C 233
Rolh B 1, Sepulveda N G and Wiskwo J P 1989 J. Appl. Phys. 65 361
Rui Y, Ji H L, Xu X N, Shao H M, Qin M J, Jin X, Yao X X. Rang X S. Ying B i d Zhao Z X 1995 Phys.
Rev. B 51 9161
R u ~ g eK and Pannetief B 1993 1.Physique I 3 389
Ryu S,Doniach S. Deutscher G and Kapitulnik A 1992 Phys. Rev. Len. 68 710
Ryu S. KapiNlnik A and Doniach S 1993 Phys. Rev. Len. 71 4245
Safar H, Gammel P L. Bishop D J, Mitzi D B and Kapitulnik A 199% Phys. Rev. Len. 68 2672
Safar H. Gammel P L, Huse D A. Ales G B. Bishop D 1, Lee W C, Giapiolsakis I and Ginsberg D M 1995 Phys.
Rev. B 52 621 1
Safar H, Gmmel P L. H u e D A, Bishop D J. Rice I P and Ginsberg D M 1992b Phys. Rev. Letf 6 9 824
Safar H. Gammel P L. Huse D A, Maiumdar S N. Schneemeycr L F, Bishop D I, L ~ E ZD, Nieva G and de la Cruz F
1994 Phys Rev. Len. 72 1272
Safar H. Rodriguez E, de la CNZ F, Gmmel P L. Schneemeyer L F and Bishop D 1 199% Phys. Rev. B 46 14238
Saint-James D-and DeGemes P G 1963 Phys. Len. 7 306
Saint-James D, Sanna G and Thomas E J 1969 Type-I1 Superconduefivify (New York: Benjamin)
Sslamon M B and Shi Jing 1992 Phys. Rev. Len C 69 I622
Samoilov A V 1993 Phys. Rev. Lei!. 71 617
Samoilov A V, lvanov Z G and Johansson L G 1994 Phys. Rev. B 49 3667
Samoilov A V. Lefis A, Rullier-Albenque F. Lejay P, Bouffard S, lvanov Z G and Johansson L-G 1995 Phys.
Rev. B 51 2351
Sandvold E and Rossel C 1992 Physica 19OC 309
Sardella E 1991 Phys. Rev. B 44 5209
-1992
Phys. Rev. B 45 3141
Sardella E and Moore M A 1993 Phys. Rev. B 48 9664
Sarma N V 1968 Phil. Mag. I8 171
Sartj S,Silva E, Fasfampa R, Giura M and Marcon R 1994 Phys. R m B 49 556
SGik R and Stroud D 1993 Phys. Rev. B 48 9938
-1994a
Phys. Rev. Len. 72 2462
-1994b
Phys. Rev. B 49 16074.50 3294
-1995
Phys. Rev. B 52 3696, Phys. Rev. Len.75 2582
S%lk R, Stroud D and Tebanovich 1995 Phys. Rev. B 51 3042
Sauerzopf F M. Seidl E and Weber H W 1982 J. Low Temp. Phys. 49 249
Sauempf F M, Wiesinger H P. Weber H W and Crabtree G W I995 Phys. Rev. B 51 6002
Schalk R M, Weber H W, Barber Z H. Przylupsky P and €vats J E 1992 Physica 199C 31 1
Scheldl S and Hackenbroich G 1992 Phys. Ret'. B 46 14010
Schelten I, Ullmaier H and Lippmann G 1975 Phys. Rev. B 12 1772
1588
E H Brandt
Schilling A, Jin R, Guo J D and Olt H R 1993 Phys. Rev. Len. 71 1899
Schilling 4 Ott H R and Wolf Th 1992 Phys. Rev. B 46 14253
Schimmele L, Kronmtiller H and Teichler H 1988 Phys. Stat. Sol. B 147 361
Schindler 0, Seebachet B, Kleiner R. Mffller P and Andrer K 1992 Physica 196C 1
Schmid A I966 Phys. KO&. Male& 5 3D2
-1982
J. Low Temp. Phys. 49 6G9
Schmid A and h u g e r W 1973 J. t o w 7emp. Phys. 11 667
Schmidt H 1968 2 Phys. 216 336
Schmidt M F, lsneloff N E and Goldmann A M 1993 Phys. Rev. Lett. 70 2162
SchmiII P. Kummelh P, Schultz L and Saemann-hchenko G 1991 Phys. Rev. Lett. 67 267
Schmucker R 1977 PhiL Mag. 35 431,453
Schmucker R and B m d t E H 1977 Phys. Stat. Sol. B 79 479
Schmucker R and Kronmffller H 1974 Phys. Stat. SOL B 61 181
Schnack H 0, Griessen R. Lensink J G. van der Beek C J and Kes P H 1992 Physicn 197C 337
Schnack H C. Griessen R, Lensink J G. and Wen Hi-hu 1993 Phys. Rev. B 48 13178
Schneider E and Kronmffller H 1976 Phys. Stat. SOL B 74 261
Scheider J W, S c W t h S and Meier P F 1995 Phys. Rev. B 52 3790
Schoenes I, Kaldis E and Karpinski I 1993 Phys, Rev. B 48 16869
Schanenberger A M. Geshkenbein V B and Blatter G 1993 Phys. Rev. B 48 15914
-1995
Phys, Rev. Lett. 75 1380
Schopohl N and Baratoff A 1988 Physica 153-5C 689
Schopohl N and Maki K 1995 Phys. Rev. B 52 490
Schusler Th. lndenbom M V, Koblischka M R. Kuhn H and KronmiiUer H 1994a Pkys, Rev. B 49 3443
Schuster Th. Indcnbom M V. Kuhn H, Brandl E H and Konczykowski M 1994b Phys. Rev. Lett. 73 1424
Schuster Th, lndenbom M V. Kuhn H, Kronmiiller K Leghissa M, and Kreiselmeyer G 1994~Phys, REV.B SO
9499
Sehuster Th, Koblischka M R. Kuhn H. Gliicker M. Ludewher B and Kronmlilier H 1993a J, Appl. Phys. 74 3307
Schuster Th. Koblischka M R, Kuhn H. Kronmffller H. Fried1 G, R o s B and Schultz L 1993b Appl. Phys. Lett,
62 768
Schusier Th, Koblischka M R, Kuhn H, Kronmiiller H. Leghissa M, GerMuser W, Saemann-Ischenko G, NeumUller
H-W and Klaumiinzer S 199% Phys. Rev. B 46 8496
Schusler Th. Koblischka M R, Ludescher B. Moser N and Kronmliller H 1991 Cryogenics 31 811
Schuster Th, Koblischka M R. Reininger T, Ludescher B. Henes R and Kronmiiller H 1992b Supenond Sci.
Technol. 5 614
Schusler Th. Kuhn H and Brandt E H 1995a Phys. Rev. B SI 697
Schusler Th. Kuhn H. Brandt E H. Indenbom M V. K I h r M. Miiller-Vom G. H a b e m i e r H-U. Kronmiiller H
and Forkl A 1995b Phys. Rev. B 52 10375
Schuster Th. Kuhn H, Brandt E H. Indenbom M V. Kobkchka M R and Konavkowski M 1994d Phw, Rev. B
50 16684
Schuster TIT Kuhn H,Indenbom M V, Leghissa U Kraus M and Konczykows!4 M 199% Phys Rev, B 51 16358
Schusler Th. Kuhn H. Koblhchka M R, Theuss H, Kronmtiller H, Leghissa M. b u s M and Saemann.Ischenko G
1993c Phys. Rev. B 47 373
Seeger A 1955 Theone der Citte,fehLstellen. Handbuch der Physik vol 7, pari I, ed S Fliigge (Berlin: Springer)
p 383
-1958
Kristallpla~tuiuTr.Hrmdbuch der Physik "017. pan 2, ed S Fliigge (Berlin: Springer) p 1
-1970
Metalhrs. Tram. 1 2987
-1979
Phys. Lett. 77A 259
Seeger A and KronmUller H 1968 Phys. Stat. Sol. 27 371
Seidler G T, Rosenhum T F, Beauchamp K M. Jaeger H M, Cnbtrre G W, Welp U and Vinokur V M 1995
Phys. Rev. Lett. 74 1442
Seng Ph, Gross R, Baier U, Rupp M, Koelle D.Huebener R P, Schmin P and Saemann-Ischenko G 1992 Physica
192C 403
Sengupta S, Dasgupta C, Krishnamurty H R, Menon G I, Ramakrishnan T V 1991 Phys. Rev. Lett. 67 3444
Sengupta S. Shi D, Wang Z Smith M E and Mffiinn P I 1993 Phys. Rer B 47 5165
Senoussi S 1992 1. Physique Ill 2 1041
Senoussi S, AguiUon C and Hadjoudi S 1991 Physic= 175C 215
Senoussi S, Hadjoudi S. M
a
w R and Fert A 1990 Physica 16SC 364
Senoussi S,Mosbah F,Snnhini 0 and Hammond S 1993 Physica 211C 288
-
The flux-line lattice in superconductors
1589
Senoussi S, Ou&na M. Collin G and Campkll I A 1988 Phys. Rev. B 37 9792
Seow W S. Doyle R A, Johnson J D, Kumar D, Somekh R, Walker D J C, and Campbell A M 1995 Physicu 241C
71
Serene 1Wand Rainer D 1983 Phys. Rep. 101 221
Shapira Y and Neuringer L J 1967 Phys, Rev. 154 375
Shapoval E A I995 P i s ' m W %p. nor. Fiz. 61 124 (Engl. Transl. 1995 So". Phys.-JETPLen 61 131)
Sham S, Shaulov A and Yeshurun Y 1993 Pkys. Rev. B 48 13871
Shehata L N 1981 Phys. slot SOL B 105 77
Shenoy S R and Chattopadbyay B 1995 Phyr. Rev. B 51 9129
Shi An-Chang and Berlinsky A J I991 Phys. Rei,. Len. 67 1926
Shi Donglu, Kourous H E, Xu Ming and Kim D H 1991 Phys. Rev. B 43 514
Shi X D, Chaikin P M, Ong N P and Wang Z Z 1994 Phys. Rev. B 50 13845
Shimshoni E 1995 Phys. Rev. B 51 9415
Shmidt V V and Mwchyan G S 1974 Usp. Fir. Nauk 112 459 (Engl. Transl. 1974 So". Phys. Usp. 17 170)
Shridar S, Wu D-H and Kennedy W 1989 Phys. Re$. Left 63 1873
~invinekE I992 Phyx Rev. B 46 14054
Simon Y and Thorel P 1971 Phys. Len. 35A 450
Slrvomov M A and Geshkenkin V B 1994 zlr. Ekp. nor. Fiz. 105 1379 (Engl. Transl. 1994 So". Phys.JETP
78 743)
Smith A W, Clinton T W, ' h e i C C and Lobb C I 1994 Phys. Rev. B 49 12927
Soininen P I, Kallin C and Eerlinsky A J 1994 Phys. Rev. B 50 13885
Solovjov V F, Pan V M and Freyhardt H C 1994 Phys. Rev. B 50 13724
Song Y-Q 1995 Physica 241C 187; Phys. Rev. Len. 75 2W8
Song U-Q, Halperh W P, Tonge L, Marks T I , Ledvij M, Kogan V G and Bulaevskii L N 1993 Phyr. Rev. Len.
70 3127
Song Y-Q. Lee M, Halperin W P. Tonge L M and Marks T J 1992 Phys, Rev. B 45 4945
Song Y-Q. Tripp S, Halperin W P. Tong L and Marks T I and Bulaevskii L N 1994 Phys. Rev. B SO 16570
Sonier 1E, KieR R F,Brewer J H. Bonn D A, Camlan J F, Chow K H, Dasanjh P. Hardy W N, MacFarlane W A,
Mendels P, Morris G D, Riseman T M and Schneider I W 1994 Phys. Rev. Lett 72 744
Sonin E B 1995 Physicu 2108 234
Sonin E B and Horovitz B 1995 Phys. Rev. B 51 6526
Sonin E B and Krusius M 1994 The Vmex Sture ed N Bontemps, Y Bruyuseraede. G Deutscher and A Kapitulnik
(Dordrecht: Kluwer) p 193
Sonin E B and Trait0 K B 1994 Phys. Rev. B 50 13547
Soulen R J. Francavilla T L. Fuller-Mora W W, Miller MM, Joshi C H. Carter W L, Rodenbush A I, Manlief M D
and Aired D 1994 Phys. Rev. B 50 478
Spirgatis A, Trox R. Kakler J and Bock J 1992 Cryogenies 33 138
Spivak B and Zhou F 1995 Phys. Rev. Leu. 74 2800
Stamp P C E, F m L and Ayache C 1988 Phys. Rev. B 38 2847
Steegmans A, Provoost R. Moshchalkov V V, Silverans R E,Libbrecht S, Bnekenhoudt A and Bruynseraede Y
1993 Physicu 218C 295
Steel D G and Graybed J M 1992 Phys. Rev. B 45 12643
Steel D G, White W Rand Graybeal J M 1993 Phys. Rev. Left 7 l 161
Steinmyer F, Kleiner R. MUer P, Muller H and W
inzer K 1994a Eumphys, Leu. 25 459
Steinmyer F, Kleiner R. Miier P and Winzer K 1994b Physicn 192401
StejiC G, Gureneh A, Kadyrov E, Christen D,loynt R and Larbalestier D C 1994 Phys. Rev. B 49 1274
Stephen M I 1994 Phys. Rev. Len 72 I534
Stoddart S T, Bending S I, Geim A K and Henini M 1993 Phys. Rev. Lett 71 3854
Stoppard 0 and Gugan D 1995 Physicn 241C 375
SNcki F. Rhyner I and Blamer G 1991 Physica 181C 385
Sudb# A 1992 Physicu ZOIC 369
Sudb0 A and Brandt E H 1991a Phys. Rev. Leu. 66 1781
-1991b
Phys. Rev. B 43 10482
-1991~ P b s . Rev. Len. 67 3116
Sudbg A, Brandt E H and Huse D A 1993 Phys. Rev. Leu. 71 1451
Suenaga M, Gosh A K, Xu Y and Welsh D 0 1991 Phys Rev. Len. 66 1777
Suh B I, Torgewn D R and Bona F 1993 Phys. Rev. Len. 71 3011
Suhl H 1965 Phys. Rev. Leu, 14 226
1.590
E H Brandt
Sun J Z Gallagher W J and Koch R H 1994 Phys. Rev. B 50 13664
Sun Y R. Thompson J R, Chen Y J, Christen D K and Goyal A 1993 Phys. Rev. B 47 14481
Sun Y R Thompson J R, Schwartz I. Christen D K, Kim Y C and Paranthaman M 1995 Phys. Rev. B 47 14481
Suzuki T, Iwai T and Takakanaka K 1995a Physica 242C 90
Suzuki T and Seeger A 1971 Commenrs SolidSrare Phys. 3 128, 173
Suzuki T, Triscone J-M, Beasley M R and Geballe T H 1995b Phyr. Rev, B 52 6858
Svedlindh P, Niskanen K, Norling P, Nordblad P, Lundgren L, Rossel C, Sergent M. Chevrel R and Potel M 1991
Phys. Rev. B 43 2135
Svensmark H and Falkov L M 1990 Phys. Rev. B 42 9951
Swartzendrukr L J, Roilburd A, Kaiser D L. Gayle F W and BeMet L H 1990 Phys. Rev. kif.64 483
Tachiki M 1982 Physic0 109-108 1699
Tachiki M, Koyama T and Takahashi S 1991 Physica 185-9C 303
-1994
Phys. Ret-, B 50 7065
Tachiki M and Takahashi S 1989 SolidSrare C o m n . 70 2991
-1991
Physica 169B 121
Tachiki M, Takahashi S and Sunaga K 1991 Phys. Rev. B 47 6095
Takics S 1982 Phys. Star. Sol. A 74 437
T&cs S and Gam&y F 1990 Supercond Sei. Technol. 3 94
T h & a K 1975 Phys, Star. Sol. B 68 623
Takayama H and Maki K 1973 I. Low Tcmp. Phys. 12 195
Takezawa N, Koyama T and Tachiki M 1993 Phyxica 207C 231
Tamcgai T. Krusin-Elbaum L, Civale L,Sanlhanam P, Brady M 1, Masselink W T, Holtzberg F and Feild C 1992
Phys. Rev. B 45 8201
Tang C 1993 Physica 194A 315
Tang C and Bak P 1988 P h y . Rev. Lerf. 60 2341
Tang C, Feng S and Golubovic L 1994 Phys. Rev. Len. 12 1264
T h k r U C Dai H. Nelson D and tieber C M 1995 Phvs. Rev. Lert 74 5132
Teichler H 1975 Phys. Star. Sol. B 72 211
TemovskiY F F and Shehah L N 1972 ut &n
..
Teor. Fiz 62 2297 (end.
Transl. 1912 So". Phw-IETP 35 1202)
.~
TeSanoviE Z 1991 Phys. Rev. B 44 12635
Phys. Rev. B 51 16204
-1995
TeSanoviC Z and Xing L 1991 Phys. RN. Lett 67 2729
TeSanoviC Z. Xing L, Bulaevskii L, ti Q and Suenaga M 1992 Phys. Rev, Len. 69 3563
Thwdorakis S 1990 Phys. Rev. B 42 10172
Thmdorakis S and Enhouami A M 1995 Phys. Rev. B 51 11664
T h h n R, Korshunov S E,Simond J-B, Leemann Ch and Martinoli P 1994 Phys. Rev. Len 72 562
Them" R. Simond J-B, Leemann Ch, Beck H, Martinoti P and Minnhagen P 1993 Phys. Rev. Lett. 71 1246
Theuss H 1993 Physica 208C I55
T h e w H. Becker T and KronmUller H 1994 Physica 233C 179
T h e w H. Forkl A and Kronmtiller H 1992a Physica 190C 345
Theuss H.Reininger T and Kronmiiller H 1992b 3, Appl. Phys. 72 1936
Thiemann S,RadoviE Z and Kogan V G 1988 Phys. Rev. B 39 11406
Thompson J R. Ossandon J 0, Christen D K. Chakoumakos B C, Sun Y R. Paranthaman M and Blynestad I 1993
Phys. Re\*. B 48 14031
Thompson 1 R, Sun Y R. Chrisen D K, Marwick A D and Holtzberg F 1994 Phys. Rev. B 49 13287
Thompson J R, Sun Y R. and Holkkrg F 1991 Phys. Rev. B 44 458
Thompson J R. Sun Y R, Kerchner H R, Christen D K. Sales B C Chakoumakos B C. Marwick A D. Civale L
and Thomson J 0 1992 Appl. Phys. Lett. 60 2306
Thompson R S 1915 Zh Ekrp. Teor. Fiz 69 2249 (Engl. Transl. 1976 So". Phys.JETP 42 1144)
Thompson R S and Hu C-R 1971 Phys. Rev. Len. 20 1352
Proe. l3rh h i . Con$ on Low Temp. Physics ed K D T h e h a u s . W J O'Sullivan and E F Hamme1
-1914
(New York Plenum) p 6763
Thorel P, Kahn P, Simon Y and Cribier C 1973 J. Physique 34 441
Thuneberg E V 1984 I, Low Temp. Phys. 57 415
I. Low Temp. Phys. 62 21
-1986
-1989
Cryogenics 29 236
Thuneberg E V, Kurkijwi J and Rainer D 1984 Phys. Rev. B 29 3913
Tilley D R and Taey I 1914 Supefluidig nndSuperconductirip (New York van Nostrand Reinhold)
The fludine lam'ce in superconductors
1591
Tim C 1195 Phys. Rev. B 52 9751
T
' im W E and W a l d e y D G 1976 J Phys. F: Mer. Phys. 6 21 07
Tinkham M 1963 Phys. Rev. 129 2413
-1965
Phys. Rev. Lett. 13 804
-1975
Introduction to Superconductivity (New Yok McGraw-Hill)
-1988
Phys. Rev. Len. 61 1658
Tlnkham M and Lubb C J 1989 Solid State Physics 42 91
Tlnkham M and Skocpol W 1 1975 Rep. Pmg. Phys. 38 1049
Tkachenko V K 1969 Zh E k p . Teor. Fiz. 56 1763 @ngl Transl. 1969 So". Phys.-JETP 29 945)
Tomlinson E I, Paylupsky P and Evetts J E 1993 Cryogenics 33 28
Triscone 1-M, Fischer 0,Brunner 0, Antognazza L, Kent A D and K a r h t M G 1990 Phys. R w Lerr. 64 804
Triscone 1-M, Fivat P, Andersson M, Decroux M and Fischer 0 1994 Phys. Rev. B 50 1229
Troy R I and Dorsey A T 1993 Phys. Rei,. B 47 2715
-1995
Phys. Rev. B 51 11728
Dui 0 K C, Ong N P, Matsuda Y,Yan Y F and Petersen I B 1994 Phys. Rev. Lea 73 124
Ullah S and Dorsey A T 1990 Phys. Rev. Lett. 65 2066
-1991
Phys. Rev. B 44 262
Ullah S , Dorsey A T and Buchholtz L 1 1990 Phys. Rev. B 42 9950
UUmaier H 1975 Irreversible Properties OfTypeJl Superconductors (Berlin: Springer)
UUrich M, Muller D. Heinemann K. Niel L and Freyhardt H C 1993 AppL Phys. Left 63 406
Ulm E R. Kim J-T, Lemberger T R, Foltyn S R and Wu X 1995 Phys. Rev. B 51 9193
Uprety K K and Domhguez D 1995 Phys. Rev. B 51 5955
Urbach I S, Lombard0 L W, White W R, Beasley M R and Kapitulnik A 1994 Physic= Z19C 93
Usadel K 1970 Phys. Rev. Lett. 25 560
van Dalen A J J, Koblischka M R, Griessen R, Jina M and Ravi Kumar G 1995 PhyJica 2SOC 265
van der Beek C I and Kes P H 1991a Phys. Rev. B 43 13032
-1991bPhysico
185-9C 2241
van der Beek C 1, Geshkenbein V B and Vinokur V M 1993 Phys. Rev. B 48 3393
van der Beer C 1, Kes P H, Maley M P, Menken M 1 V and Menovsky A A 1992a Physica 195C 302
van der Beek C J. Konczykowski M. Vinokur V M, Crabtree G W. Li T W and Kes P H 1995 Phys. Rev. B 51
15492; Phys. Rev Lett. 74 1214
van der Beek C 1, Nieuwenhuys G 1, Kes P H, Schnack H G and Griessen R P l992b Physica 197C 320
van Dover R B 1990 AppL Pkys. Leu. 56 2681
van Olterlo A, Feigel'man M V, Geshkenbein V B and Blalter G I995 On the Hall anomaly Preprint
Vecris G and Pelcovits R A 1991 Phys. Rev. B 44 2767
Vinen W F 1969 Superconductiviry ed R D Parks (New York Marcel Dekker) volI1. p 1167
Vinen W F, Forgan E M. Gough C E and Hood M I 1971 Physico 55 94
Vinen W F and Warren A C 1967 Pmc. Phys. Soc. London 91 409
ViMikov L Ya, Gurevich L A. Yemel'chenko G A and Ossipyan Yu A 1988 Solid. Stare Commun. 67 421
Vinokur V M, Feigel'man M V and Geshkenbein V B 1991a Phys. Rev. Lett. 67 915
Vinokur V M, F e i g e l ' m M V, Geshkenbein V B and BlatterG 1993 Phys. Rev. Lett. 71 1242
Vinokur V M, Feigel'man M V. Geshkenbein V B and Larkin A I 1990a Phys. Rev. Lett. 65 259
Wnokur V M. Geshkenbein V B, Feigel"
M V and Larkin A I 1991b ul. Ekrp. Teor. Fir. 100 1104 (E@.
Transl. 1991 SOL,.
Phys.-JETP 73 610)
Vinokur V M, Kes P H and Koshelev A E 1990h Physico 168C 28
Vinokur V M and Koshelev A E 1990 Zh. Ekp. Teor. Fir. 97 976 (Engl. Transl. 1990 So". Phys.-JETP 70 547)
Vlasko-Vlasov V K. Dorosinskii L A. Polyanskil A A, Nikitenko V I, Welp U, Veal B W and Crabuee G W 1994a
Phys. Rev. Lett. 72 3246
Vlasko-Vlasov V K, Indenbom M V, Nikitenko V I,PolyanskiT A A, Prowrov R L. Grekhov I V. Delimova L A.
Linifchuk I A. Antonov A V and Gusev M Yu 1992 Sverkhprovodimosr' 5 1637 (Engl T m s l . 1992
Superconductivio 5 1582)
Vlasko-Vlasov V K, Nikitenko V 1. Polyanskir A A, Crabfree G U', Welp G W and Veal B W 1994b Physica
2 W : 361
Voloshin I F. Fisher L M, Gorbachev V S , Savel'ev S E and Yampol'skii V A 1994 Pis'ma Zh Ekrp. Teor. Fiz.
59 55 (Engl. Transl. 1994 Sov. Phys.-JETP Left 59 55)
Volovik G E 1993 Pis'rna Zh. Eksp. Teor. Fiz 58 457 (Engl. Transl. 1993 Sov. Phys.-JETP Lett. 58 469)
Vulcanescu V. Collin G . Kojima H, Tanaka I and Fruchter L 1994 Phys. Rev. B 50
Wadas A. Fritz 0.Hug H 1 and Giintherodt H-1 1992 Z Phys. B 88 317
E H Brandt
1592
Wagenleithner P 1982 J. Low Temp. Phys. 48 25
Wagner R. Frey U, Hillmer F, and Adrian H I995 Phys. Rev. B 51 1206
Wagner R, Hillmer F, Frey U and Ad~ianH 1994 Phys, Rev. B 49 13184
Wahl A, Maignan A. Martin C, Hardy V, Provost J and Simon Ch 1995 Phys. Rev, B 51 9123
Walmsley D G 1972 J. Phys. F: Met. Phys. 2 510
Wdmsley D G and Tim- W E 1977 J. Phys. F Mer. Pkys, 7 2373
Wan Y M. Hebboul S E and Garland J C 1994 Phys. Rev, Len 72 3867
Wang L. Zhu Y, Zhao H L and Feng S 1990 Pkys. Rev. Lett 64 3094
Wang Yu-P and Zhang D-L 1995 Phys. Rev. Len,C 74 2838
Wang Z and Shi D 1993 Phyf. Rev. B 48 4208,9782, 16179
Wang Z D, Dong J and l i n g C S 1994 Phys. Rev. Lett 72 3875
Wang 2 D, Ho K M, Dong J and Zhu 1-X 1995 Phys. Rev. B S I 6119
Wang 2 D and Ting C S 1992 Phys. Rei$. B 46 284; 1992 Pkys. Rev. Len. 69 1435
Watson G I and Canright G S 1993 Phys. Rev. B 48 15950
Watts-Tobin R, Kramer Land Pes& W 1974 J. Low Temp. Phys. 17 71
Weber H and Iensen H J 1991 Phys. Rev. B 44 454
Weber H W 1974 J, Low Temp. Phys. 17 49
-1977
Anisotropy Effects in Supereonduetors (New York. Plenum)
-1991
Physica 185-9C 309
Weber H W, Bodo M. Sauempf F M. Wtesinger H and Klein U 1987 Jap. J. AppL Phys, 16 917
Weber H W. Sauerzopf F M and Seidl E 1981 Physica IWB 429
Weber H W, Schellen J and Lippmann G 1973 Phys. Stat. SOL B 57 515
J. Low Temp. Phys. 16 367
-1974
Weber H W, Seidl E. Botlo M, Laa C. Mnyerhofer E,Sauempf F M, Schalk R M,Wxesinger H P and Rammer I
1989 Physicn 161C 272
Weber M, Amato A. Gygax F N, Schenck A. Malem H. Duginov V N. Grebinnik V G. L m v A B,
Olshevsky V G, Pomjakushin V Yu. Shilw S N, Zhukov V A, Kirillov B F, Pirogov A V , Ponomarev A N.
Storchak V G, Kapusta S and Bock J 1993 Phys. RN. B 48 13022
Wellstood F C, Fenari M J, Kingston J 1, Shaw T J and Clarke J 1993 Phys. Rev. Lett, 70 89
Welp U, Fleshler S, Kwok W K, Klcmm R A. Vurokur V M, Downey J, Veal B and CrabG W I991 Phys
Rev. Letf. 67 3180
Welp U,Gardiner T. Gunlher D. Crabme G W, Vlasko-Vlasov V K and Nikitmko V 1 1994 Phys. Rev, Leu 67
3180
Wen Hai-hu. Schnack H G, Griessen R. Dam B and Rector J 1995 Physic0 241C 353
Wen Hai-hu and Zhan Zhong-xian 1994 Phys. Ret: B 50 13853
White W R, Kapilulnik A and Beasley M R 1991 Phys. Rev. Len.66 2826
Phys. Rev. Letr. 70 670
-1993
Phys. Rev. B 50 6303
-1994
Wilkin N and Moore M A 1993 Phys. Rev. B 47 957.48 3464
Wipf S L 1967 Phys. Rev. 161 404
Ctyogenics 31 936
-1991
W6ltgens P 1 M. Dekker C, Koch R H, Hussey B W and Gupta A 1995 Phys. Rev. B 52 4536
W6ltgens P J M. Dekker C. SwOste J and de Wjn H W 1993a Phys REY.B 48 16826
Waltgeens P J M. Dekker C and de Wijn H W 1993b Phys. Rev. Lett, 71 3858
Woo K C. Gray K E, Kampwilth R T. Kang J H, Stein S J. East R and McKay D. M 1989 phys. Rev. Lett. 63
1877
Wardenweber R 1992 Phys. Rev. B 46 3076
Wardenweber R and Kes P H 1985 Physinr 13% 136
Phys. Rev. B 34 494
-1986
-1987
J. Low Temp.Phys. 67 I
Worlhinmon T K. Fisher M P A. Huse D A. Toner J. Manvick A D. zabel T. Feild C A and Holtzbere F 1992
PGs. Rev. B 46 11854
Wonhinmon T K. Olsson E Nichols C S, Shaw T M and Clarke D R 1991 Phw. Rev. B 43 10538
Wu D Booth I C and Anlage S M 1995 Phys. Rev. Lcn. 75 525
Wu D H,Mao J. Mao S N, Peng I L, Xi X X, Venkatesan T.Greene R L and Anlage S 1993a Phys. Rev. Lett.
70 85
Wu H, Ong N P and Li Y Q 1993b Phys. Rev. Lett. 71 2642
Wu M K, Ashbum 1 R,Tomg C J, Hor P H, Meng R L. Cao L. Huang 2 J. Wang Y Q and Chu C W 1987 Phys.
I
6
Theflux-line lattice in superconductors
1593
Rev. Len 58 908
Xing L and TeSanoviC Z 1990 Phys. Rev. Len. 65 794
Xing W, Heinrich 8, Zhou Hu,Fife A A and Cragg A R 1994 J. Appl. Phys. 76 4244
Xu J H. Miller J H and T i g C S 1995 Phys. Rev. B 51 424
Xu M, Finnemore D K, Crabtree G W,V i a o h V M. Dabmvski B, Hinks D G and Zhang K 1993 Pkys. Rev. B
48 10630
Xu 2,Drulis H, Hou I-C De Long L E and Brill 1 W 1992 Physica 202C 256
Yamafuji K, Fujiyoshi T, Toko K, Matsuno T, Kobayashi T and Kishio K 1994 Pkystysica 226C 133
Yamafnji K, Fujiyoshi T, Toko K and Marsushifa T 1988 Physica 159C 743
Yamafuji K and lrie P 1967 Pkys. Lett. %A 387
Yamafuji K, Wakuda T, Fujiyoshi T and Punaki K 1995 Physica 242C 251
Yammato K. M d Hand Yasuoka H 1993 Pkys. Rev. B 47 915
Y m a k i H,Endo K, Kosaka S, Umeda M, Yoshida S and Kajimm K 1994 Pkys. Rev, E 50 12959
Yang G, Shang P, Sntton S D,Jones I F, Abell J S and Gough C E 1993 Phys. Rev. B 48 4054
Yang Z J. Johansen T H, Bratsberg H. Bhainagar A and Skjeltorp A T 1992 Physic0 197C 136
Yao Z,Ymn S, Dai H. Fan S and Lieber C 1994 Nmun 371 777
Yarmchuk E J, Gordon M 1V and Packard R E 1979 Phys. Rev. Lett. 43 214
Yarmchuk E J and Packard R E 1982 J. Low Temp. Pkys. 46 479
Yamn U, Gammel P L, Huse D A, K l e i m R N, Oglesby C S, Bucher E, Batlogg B. Bishop D I, Monensen K,
Clausen K N,BoUe C A and de la Cmz 1995 Pkys. Rev. Len. 73 2748
Yamn U. Gammel P L, Huse D A, Kleiman R N,Oglesby C S,Bucher E. Batlogg B. Bishop D I. Monensen K,
Clausen K N, Bolle C A and de la CNZ F 1994 Pkys. Rev. Len. 73 2748
Yamn U, Komyushin Y and Felner I 1992 Phys. Rev. B 46 14823
Yazdani A, Howald C M, White W R, Beasley M R. Kapitulnik A 1994 Phys. Rev. B 50 16117
Yazdani A, White W R, Hahn M R, Gabay M, Beasley M R, Kapitulnik A 1993 Phys. Rev. Len. 70 505
Yazyi J, Anib6re A, D u b C,de la CNZ P,Mitzi D B and Kapitulnik A 1991 Pkysien 184C 254
Yeh N-C,Iiang W, Reed D S, Kriplani U and Holtzberg F 1993a Phys. Rev. B 47 6146
Yeh N-C,Reed D S. Jiang W. Kriplani U. Holt&erg F, Gupta A, Hunt B D, Vasquez R P, Fmte M C and Bajuk L
1992 Phys. Rev. B 45 5654
Yeh N-C,Reed D S, Iiang W, Kriplani U, Tsuei C C Chi C C and Holaberg F 1993b Phys. Rev, Leu. 71 4043
Yeshurun Y. Bontemps N, Burlachkov Land Kapitulaik A I994 Phys. Rev. B 49 1548
Yeshurun Y and Malozemoff A P 1988 Phys. Rev. Len. 64 2202
Yethiraj M, Mook H A, Wignall G D. CubiU R, Forgan E M, Paul D McK and Armstrong T 1993 Phys. Rev. Len.
70 857
Ymg Q Y and Kwok H S 1990 Phys. Rev. B 4 2 2242
Yu W and Stroud D 1994 Pkys. Rev. B 50 13632
-1995
Phys. Rev. B 51 3725
Zavaritskil N V 1993 Pis'ma Zh.Eksp. %or. Fk.57 695 (Engl. Transl. 1993 Son Phys.JBTP Lett. 57 707)
Zech D,Lee S L, Keller H, Blatter G,Janossy B, Kes P H, Li T W and Menovsky A A 1995 Phys. Rev. B 52
6913
Zeldov E. Amer N M,Koren G and Gupta A 1990 Appl P h p Lett, 56 1700
Zeldov E, Clem I R, MeElfresh M and Danvin M 1994a Phys Rev, B 49 9802
Zeldov E, Larkin A L Geshkenbein V 9, Konczykowski M. Majer D,Khaykwieh B, Vinokur V M and S h U " H
1994b Phys. Rev. Lett 73 1428
Zeldov E, Larkin A I. Konczykowski M, Khaykovich B, Majer D, Geshkentein V B and Vinokur V M 19%
Physiea 235-40C 2761
Zeldov E,Majer D, Konczykowski M, Geshkenbein V B, Vinokur V M and S h a m a n H 1995a Nature 375 373
Zeldov E$ Majer D, Konczykowski M, Larkin A I, Vinokur V M, Geshkenbein V B. Chikumoto N and Shtrikman H
199% Ewopkys. Lett 30 367
Zeuner S,Prettl W. Renk K Q and Lengfellner H 1994 Phys. Rev. B 49 9080
zhao Y. Gu G D, Cnchrane J W, Russell G J, Wen J G,Nakamm N, Tajima S and Koshiznka N 1995 Phys.
Rev. B 51 3806
Zhu J. Mester J. L o c M 1 and Tumeaure J 1993 Pkysica 212C 216
Zhu Y-D,Zhang F C and Sigrist M 1995 Phys. Rev. B 51 1105
Zhukov A A 1992 Solid Sme C o m " 82 983
Zhukov A A, Komarkov D A, Karapetrov G. Gordeev S N and Antonov R I 1992 Supercond Sci'.Tmhol. 5 338
Zhukov A k Kiipfer H, Claw H. WUhl H, Klaser M and Muller-Vogt G 1995a Phys. Rev. B 52 R9871
Zhukov A A. KIkpfer H. Perkins 0, a h e n L E Caprin A D,Klestov S A, &us H, Voronkova V I, Wolf T and
1594
E H Brand
Wtihl H 1995b Phys. Rev. B 51 12704
Zhukov A A, Ktipfcr H, Rybachuk V A, P o n o m n k o L A. Munshov V A and Martynkin A Yu 1994 Physica
219C 99
Zhukov A A and Moshchakov V V 1991 Svcrkkpmvndimost'4 850 (Engl. Transl. 1991 Supercnnducriviry 4 759)
Ziese M and Esquinazi P 1995 Z Phys. B 94 265
Ziese M, Esquinazi P, and Braun H F 1994a Supercond. Sci. Technal. 7 869
Ziese M. E s q u i k P, Gupta A and Braun H F 1994b Phys. Rev. B 50 9491
Ziese M, Esquinazi P, Kupelevich Y and Sheman A B 1994c Physica Z24C 79
Zuo F, Khiuoev S. liang X, Peng J Land -ne
R L 1994a Phys. Rev. Lm. 72 1746
Zuo F, Khivoev S, Voss S and H e m m A M 1994b Phys. Rev. B 49 9252
Zuo F, VaD,Duan H M and H e m m A M 1993 Phys. Rev. B 47 8327
Z w i h a g l G E and Fulde P 1981 Z Phys. B 43 23
Zwicknagl G E and Wilkins I W 1984 Phys. Rev. L f f . 53 1276