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PHYSICAL REVIEW B
VOLUME 62, NUMBER 22
1 DECEMBER 2000-II
Tilt modulus and angle-dependent flux lattice melting in the lowest Landau level approximation
G. Mohler and D. Stroud
Department of Physics, Ohio State University, Columbus, Ohio 43210
共Received 23 May 2000; revised manuscript received 7 August 2000兲
For a clean high-T c superconductor, we analyze the Lawrence-Doniach free energy in a tilted magnetic field
within the lowest Landau level approximation. The free energy maps onto that of a strictly c-axis field, but with
a reduced interlayer coupling. We use this result to calculate the tilt modulus C 44 of a vortex lattice and vortex
liquid. The vortex contribution to C 44 can be expressed in terms of the squared c-axis Josephson plasmon
frequency ␻ 2pl . The transverse component of the field has very little effect on the position of the melting curve.
I. INTRODUCTION
II. LOWEST LANDAU LEVEL APPROXIMATION
WITH TILTED FIELD
This paper is concerned with the tilt modulus C 44 of the
vortex system in a high-T c or otherwise layered superconductor. C 44 is an elastic constant which measures the free
energy cost of applying a small transverse field in addition to
a field applied parallel to the c axis. It is a relevant parameter
in many physical processes, such as collective pinning,1 the
‘‘peak effect,’’1,2 and the transition to the Bose glass state.3
Previous investigations have shown that C 44 is very strongly
dependent on the wave vector k of the transverse field.4,5 In
addition, it is finite in the flux liquid as well as the flux solid
phase and is strongly affected by different kinds of
disorder.1,6,7
A number of authors have analyzed C 44 in various approximations. Several groups4,5,8 have calculated C 44(k) in a
flux lattice as a function of the wave vector k of the transverse magnetic field. Other authors have considered C 44 in
the flux liquid state,7,9 and in the presence of disorder in both
the solid and liquid phases.6,7
In this paper, we calculate C 44 for a layered superconductor at high fields. We describe the superconductor using a
Lawrence-Doniach10 free energy functional, and we evaluate
fluctuations in the lowest Landau level 共LLL兲 approximation,
appropriate for strong c-axis magnetic fields.11 This LLL approach accounts well for the position of the flux lattice melting curve in the magnetic-field/temperature plane,11,12 as
well as the value of the magnetization in both solid and
liquid phases,13 both in Bi2 Sr2 CaCu2 O8⫹ ␦ and the less anisotropic YBa2 Cu3 O7⫺x . To obtain C 44 , we generalize the
LLL approximation to apply to fields with a finite ab component. For a defect-free system in the LLL approximation,
the tilted field free energy can be exactly mapped onto the
usual free energy for a field B储 c, but with a weaker interlayer coupling which depends on tilt angle.
As a byproduct of this transformation, we can also analyze the behavior of the flux lattice melting curve in a magnetic field tilted at an angle to the c axis. We find that this
curve is little affected by the presence of a transverse magnetic field component at fields where the LLL approximation
is applicable, consistent with available 共but limited兲
experiment.
At fixed external magnetic field H⫽H z ẑ⫹H x x̂, the
Lawrence-Doniach Gibbs free energy in a form which incorporates the field energy is10
0163-1829/2000/62共22兲/14665共4兲/$15.00
PRB 62
G 关 ␺ ,A兴 ⫽d
兺n
冕 再
d 2 r a 共 T 兲 兩 ␺ n 共 r兲 兩 2
⫹t 兩 ␺ n 共 r兲 e ⫺i(2 ␲ /⌽ 0 )A z d ⫺ ␺ n⫺1 共 r兲 兩 2
⫹
1
2m ab
⫹V
冏冉
冊 冏
2
q
b
⫺iបⵜ⬜ ⫺ A⬜ ␺ n 共 r兲 ⫹ 兩 ␺ n 共 r兲 兩 4
c
2
冎
兩 B⫺H兩 2
8␲
⬅G̃ 关 ␺ ,A兴 ⫹V
兩 B⫺H兩 2
.
8␲
共1兲
Here ␺ n is the order parameter in the n th layer, d is the
distance between layers, q⫽⫺2 兩 e 兩 , and t⫽ប 2 /(2m c d 2 ) is
the interlayer coupling energy. In a cuprate superconductor
such as Bi2 Sr2 CaCu2 O8⫹ ␦ , d represents the repeat distance
in the c direction. “⬜ is the xy component of the gradient
operator, V is the system volume, and the magnetic induction
B⫽“⫻A, where the vector potential A⫽(A⬜ ,A z ).
We choose the gauge A⫽⫺B z yx̂⫹B x yẑ and expand the
␺ n ’s as a linear combination of LLL states, to obtain ␺ n (r)
2 2 2
2
(T)/4b 2 ) 1/4兺 k c k,n e ikx⫺(y⫺kl ) /2l , where k⫽2 ␲ p/
⫽( 冑3a H
L x is the quantized momentum (p a positive integer兲,
l⫽ 冑⌽ 0 /2␲ B z is the magnetic length, a H (T)⫽ 关 a(T)
⫹2t 兴关 1⫺B z /H c2 (T) 兴 , and sample volume is V⫽L x L y L z .
In terms of measurable quantities, a(T)⫹2t⫽
⫺បqH c2 (T)/(2m ab c), where m ab is the transverse effective
mass and H c2 (T) is the upper critical field as a function of
temperature T. Substituting this expansion into Eq. 共1兲 and
integrating over x and y yields
R14 665
©2000 The American Physical Society
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R14 666
G̃⫽
G. MOHLER AND D. STROUD
2
aH
共 T 兲␲ l 2d
b
Nx
兺
k,n
再
⫺ 兩 c k,n 兩 2 ⫺
* c k,n⫺1 e ⫺i2 ␲ B x dl
⫻ 共 c k,n
⫹
3 1/4
兺
5 5/2 q ,q
1
*
c k,n c k⫹q
2
2 k/⌽
0
corresponding, at low temperatures, to a tilted Abrikosov
lattice.
t exp„⫺ 共 ␲ B x dl/⌽ 0 兲 2 …
兩 a H共 T 兲兩
⫹c.c. 兲
c*
c
e
1 ,n k⫹q 2 ,n k⫹q 1 ⫹q 2 ,n
2
2
⫺(q 1 ⫹q 2 )l 2 /2
PRB 62
冎
III. TILT MODULUS
C 44 is defined by
,
冉 冊
1 ⳵ 2G
C 44⫽
V ⳵␪2
共2兲
where N x ⫽(L x /l)( 冑3/4␲ ) . This expresses the Gibbs free
energy as a function of the expansion coefficients c k,n for a
tilted magnetic field. This expression differs from that for
B储 ẑ in only two respects:11 共i兲 there is an extra phase factor
in the interlayer coupling; and 共ii兲 the strength of the interlayer coupling is renormalized by an exponentially decaying
factor.
The phase factor in Eq. 共2兲 can be eliminated by introduc2
ing a new set of coefficients c̃ k,n ⬅c k,n e ⫺i(2 ␲ B x dl k/⌽ 0 )n , in
2
* c k,n⫺1 e ⫺i2 ␲ B x dl k/⌽ 0
Eq. 共2兲. The product term becomes c k,n
* c̃ k,n⫺1 . This transformation does not affect the free
⫽c̃ k,n
energy term which contains products of four coefficients. To
see this, note that when such a typical product is transformed,
it
picks
up
a
phase
factor
2
e i(2 ␲ B x dl n/⌽ 0 )(k⫺(k⫹q 1 )⫺(k⫹q 2 )⫹(k⫹q 1 ⫹q 2 )) , where each (k
⫹q i ) term is an integer modulo N ⌽ , N ⌽ being the number
of flux lines in the sample. The terms in the exponential can
be summed to yield zero, leaving this term unaltered. Similarly, 兩 c k,n 兩 2 ⫽ 兩 c̃ k,n 兩 2 .
In terms of the new variables, the free energy is thus
1/2
G̃⫽
2
aH
共 T 兲 ␲ l 2 dN x
b
兺
k,n
再
⫺ 兩 c̃ k,n 兩 2 ⫺
* c̃ k,n⫺1 ⫹c.c 兲 ⫹
⫻ 共 c̃ k,n
⫻
兺
q ,q
1
*
c̃ k,n c̃ k⫹q
2
t⬘
兩 a H共 T 兲兩
5
t ⬘ ⫽te ⫺( ␲ B x dl/⌽ 0 )
2
2
⫺(q 1 ⫹q 2 )l 2 /2
冎
,
2
⫻
冊
兺
k,n
再
兩 B⫺H兩 2
.
8␲
⫺ 兩 c k,n 兩 2 ⫺
1/4
兺k c̃ k,n e ⫺ik(2 ␲ B dl /⌽ )n e ikx⫺(y⫺kl ) /2l ,
x
2
0
3 1/4
兺
2 5/2 q 1 ,q 2
共7兲
t exp„⫺ 共 ␲ B x dl/⌽ 0 兲 2 …
兩 a H共 T 兲兩
2 2
2
共5兲
*
c̃ k,n c̃ k⫹q
c̃ *
2
c̃
1 ,n k⫹q 2 ,n k⫹q 1 ⫹q 2 ,n
2 2 2
/l
e ⫺(q 1 ⫹q 2 )
冎
.
共8兲
Then, writing ⳵ 2 / ⳵ ␪ 2 ⫽B z2 ⳵ 2 / ⳵ B 2x , taking B z ⬃H z , and
evaluating the derivatives from Eqs. 共7兲, with the result
共4兲
is the renormalized interlayer coupling. Thus, the tilted
B-field free energy is identical in form to the free energy for
a field B储 ẑ, but with a weaker interlayer coupling. The
ground state solution for the redefined amplitudes c̃ k,n are
identical to those for the c k,n ’s in a strictly longitudinal field
but weaker B储 ẑ. However, the order parameter ␺ n picks up a
B x dependency,
4b 2
k,n
兩 B⫺H兩 2
8␲
Then, writing ⳵ / ⳵ ␪ →H z ⳵ / ⳵ H x for small H x , we find that
c
the second term in Eq. 共7兲 contributes a term C 44
2
2
⫽H z /(4 ␲ )⬃B z /(4 ␲ ) in the LLL regime, where the magnetization is small. This is the compressive contribution to the
tilt modulus.
To calculate the remaining 共vortex-related兲 contribution to
v
C 44 , which we denote C 44
, we first write G̃/k B T⫽H/T,
2
2
where T⫽bk B T/a H (T) ␲ l d and
⫹
where
冉
⬅G v ⫹V
冕兿
* dc̃ k,n e ⫺G̃/k B T ⫹V
dc̃ k,n
* c̃ k,n⫺1 ⫹c.c. 兲
⫻ 共 c̃ k,n
共3兲
␺ n 共 r兲 ⫽
G⫽⫺k B Tln
5/2
c̃ *
c̃
e
1 ,n k⫹q 2 ,n k⫹q 1 ⫹q 2 ,n
冑3a H2 共 T 兲
where ␪ is the angle between H and the c axis, and where G,
the Gibbs free energy appropriate to an experiment at constant H and T, is given by
H⫽N x
3 1/4
共6兲
,
␪ ⫽0
v
⫽
C 44
冉冑 冊 冉 冑
冑 冓兺
3
8
1/2
⫻ Bz
␲ td 3 兩 a 共 T 兲 ⫹2t 兩
k,n
⌽ 0 bL y L z
冊
共 1⫺B z /H c2 兲
冔
* c k,n⫺1 ⫹c.c. 兲 T ,
共 c k,n
共9兲
where we have dropped the tildes on the c k,n ’s because this
expression is evaluated at B x ⫽0. Equation 共9兲 can be expressed in terms of measurable quantities using the identities
a(T)⫹2t⫽បqH c2 (T)/2m ab c and b⫽2 ␲ ␬ 2 (qប/m ab c) 2 ,
where ␬ ⫽␭ ab (B z ,T)/ ␰ ab (B z ,T) is the ratio of the ab penetration depth and coherence length. We also write t
⫽ប 2 /(2m ab d 2 ␥ 2 ), where ␥ 2 is the anisotropy parameter.
The result is
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PRB 62
TILT MODULUS AND ANGLE-DEPENDENT FLUX . . .
v
C 44
⫽
1
32␲ ␬ ␥
2 2
⫻
冓兺
k,n
关 H c2 共 T 兲 ⫺B z 兴 B z
冔
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Nx
N ⌽N z
* c k,n⫺1 ⫹c.c. T .
c k,n
共10兲
Finally, the total tilt modulus is
C 44⬃
B z2
4␲
v
⫹C 44
.
共11兲
In a triangular Abrikosov lattice, for example, only N y
⫽N ␾ /N x of the N ⌽ coefficients in each layer are nonzero.
The
nonzero
coefficients
are
c 2pN x ,n ⫽(2/␤ A ) 1/2,
c (2p⫹1)N x ,n ⫽i(2/␤ A ) 1/2, where p is an integer ranging from
0 to N y and ␤ A ⫽1.159595 . . . is the Abrikosov ratio.14 Substituting these values into Eq. 共10兲, using N y /L y ⫽2N x /
(L x 冑3), and adding the compressive term, we obtain
C 44⫽
B z2
4␲
⫹
1
8 ␲ ␤ A␬ 2␥ 2
„H c2 共 T 兲 ⫺B z …B z .
共12兲
Equation 共12兲 can be compared to a previous estimate9,15
LVG
2
C 44
⫽(B z2 /4␲ ) 关 1⫹1/(4 ␲ ˜␭ ab
(B z ,T) ␥ 2 n s ) 兴 , where n s is
the superfluid number density for a two-dimensional Bose
system related to the three-dimensional flux line system by a
path integral formalism.3 Writing ␭ ab (B z ,T)⫽ ␬ ␰ ab (B z ,T),
2
(B z ,T)⬃2 ␲ „H c2 (T)⫺B z …/⌽ 0 , and approximating n s
1/␰ ab
by n 0 ⫽B z /⌽ 0 , the number of bosons 共i.e., flux lines兲 per
LVG
⬃B z2 /(4 ␲ )⫹ 关 „H c2 (T)
unit area,6,7 we obtain C 44
2 2
⫺B z …B z 兴 /(8 ␲ ␬ ␥ ) This result is in agreement with our
own result, Eq. 共12兲, except for the factor (1/␤ A )⬃0.86,
which arises from the nonuniformity of n s in the Abrikosov
lattice phase.
Our C v44 is closely connected to the so-called Josephson
plasmon frequency ␻ pl , as calculated in the same LLL
approximation.16 ␻ 2pl (B z ,T) is the squared plasma frequency
of the Josephson junction formed between adjacent ab layers, and is given by16
F 2
␻ 2pl 共 B,T 兲 ⫽ 共 ␻ M
pl 兲
⫻
冓兺
k,n
FIG. 1. Vortex contribution C v44 to the tilt modulus, plotted versus temperature for Bi2 Sr2 CaCu2 O8⫹ ␦ at B z ⫽2 T, as evaluated
using Eq. 共14兲 and ␻ 2pl from Ref. 16. C v44 is discontinuous at the
flux lattice melting temperature T M (B z ) 共indicated by arrow兲. Inset:
enlargement of C v44 near and above T M .
v
that ␻ 2pl , and hence C 44
, have discontinuities at flux lattice
melting in clean Bi2 Sr2 CaCu2 O8⫹ ␦ . 17,18
IV. FLUX LATTICE MELTING IN A TILTED
MAGNETIC FIELD
Equations 共3兲 and 共4兲 have some striking implications for
the LLL phase diagram of a clean layered superconductor.
When B储 c, this phase diagram depends on only two parameters, namely g⫽a H 冑␲ l 2 d/(bk B T) and ␩ ⫽t/ 兩 a H 兩 . 12 Our results show that, even with a transverse field B x , the phase
diagram still depends on only two parameters, except that ␩
is now replaced by ␩ ⬘ ⫽ ␩ exp关⫺(␲B2x d2)/(2Bz⌽0)兴. As found
previously,11–13 this phase diagram contains a single firstorder melting line separating a triangular vortex lattice from
a vortex liquid. In the regime where the LLL approximation
is adequate, our results show that this first-order line should
persist in a tilted magnetic field 共cf. Fig. 2兲. Furthermore, in
most high-T c materials, the line is shifted very little by a
␤A
关 ␤ A N x / 共 2N ⌽ N z 兲兴
2
冔
* c k,n⫺1 ⫹c.c. 兲 T ,
共 c k,n
共13兲
F
2 2
where ␻ M
is the
pl ⫽ 冑关 „H c2 (T)⫺B…cq 兴 / 关 ⑀ 0 ␬ ␥ ប ␤ A 兴
mean-field Josephson plasmon frequency and ⑀ 0 is an interlayer dielectric constant. Combining Eqs. 共10兲 and 共13兲 gives
v
C 44
共 B z ,T 兲
⑀ 0 ␻ 2pl 共 B z ,T 兲
⫽
B zប 2
8⌽ 0 q 2
.
共14兲
v
(B z ,T) for B z ⫽2 T in clean
Figure 1 shows C 44
Bi2 Sr2 CaCu2 O8⫹ ␦ , using this relation ␻ 2pl as calculated in
v
Ref. 16. Like ␻ 2pl , C 44
has a discontinuity at the flux lattice
melting transition, and remains finite in the flux liquid state.
Experiments on ␻ 2pl are consistent with this result, indicating
FIG. 2. Phase diagram of a clean high-T c material in the LLL
approximation, including a transverse magnetic field component
B x . The parameters ␩ ⬘ and g are defined in the text. Points and
spline fit are given by Hu and MacDonald 共Ref. 12兲.
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R14 668
G. MOHLER AND D. STROUD
nonzero B x . 共For Bi2 Sr2 CaCu2 O8⫹ ␦ , t ⬘ ⬃t exp关⫺7.5
⫻10⫺7 B 2x /B z 兴 .兲
We have found no experimental melting data in tilted
magnetic fields at fields where the LLL approximation is
applicable. In BiSr2 Ca2 Cu2 O8⫹x , the low-field melting temperature has been reported19 to depend only on the c component of H, which, though obtained in a very different regime,
would be consistent with the result of our calculations. It
would be of great interest to have further tests of the LLL
predictions in the relevant high-field regime.
PRB 62
melting temperature will be little affected by the application
of an oblique magnetic field in the range where the LLL
approximation is valid, except possibly at angles nearly parallel to the layers. This prediction appears consistent with
some existing 共but low-field兲 experiments.19 We also obtain
an expression for the zero-wave-vector tilt modulus C 44 , in
good agreement with previous estimates by other means.9,15
v
Finally, the vortex contribution C 44
is proportional to the
squared Josephson plasmon frequency ␻ 2pl , as calculated in
the same LLL approximation, and remains finite in the vortex liquid as well as the vortex solid phase.
V. SUMMARY
In this paper, we have extended the LLL approximation
for high-T c superconductors to treat fields tilted at an angle
to the layer perpendicular. The resulting free energy has exactly the same form as the usual case, except that the effective interlayer coupling is reduced. For high-T c materials,
this reduction is small; hence, we predict that the flux lattice
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ACKNOWLEDGMENTS
This work was supported by the Midwest Superconductivity Consortium through Purdue University, Grant No.
DE-FG 02-90 ER45427, and by NSF Grant No. DMR9731511.
edited by E. Kanda 共Keigaku, Tokyo, 1971兲, p. 361.
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11