Download What’s “SUPER” about SUPERCODUCTORS?

From last lecture ….
“… at each new level of complexity, entirely new properties
appear, and the understanding of this behavior requires research
which I think is as fundamental in its nature as any other”
Philip W. Anderson 1972
Si-crystal
semiconductor
MgB2 superconductor
NaxCoO2
superconductor
2 atoms
3 atoms
1 atom
La2-xSrxCuO4
superconductor
4 atoms
DNA giant molecule
Many atoms
Where could we find superfluidity?
• Helium - 3 atoms are fermions
 particles with half integer spin.
Helium
p
n
He - 3
p
1 millionth of a
centimetre
p
n
He - 4
n
p
• Helium - 4 atoms are bosons 
particles with integer spin.
1938
Kapitza and Allen discover superfluidity in
He-4
Superfluids flow without resistance
Normal fluid
Superfluid
For T < 2.4Κ – gravity ...
If the bottle containing helium
rotates for a while and then
stops, helium will continue to
rotate for ever – there is no
internal friction (for as long
as He is at T = -269 C or lower
1938 Pyotr L. Kapitsa discovered the superfluidity of liquid Helium 4
Nobel Prize in 1978
1941-47 Lev Landau formulated the theory of quantum Bose
liquid - 4He superfluid liquid. 1956-58 he further formulated the
theory of quantum Fermi liquid.
Nobel Prize in 1962
Early 1970s David M. Lee, Douglas D. Osheroff,
and Robert C. Richardson discovered the
superfluidity of liquid Helium 3.
Nobel Prize in 1996
Anthony Leggett first formulated the theory of superfluidity in liquid 3He in
1965.
Nobel Prize in 2003
Χαμηλές θερμοκρασίες
Διάστημα:
3000 χιλιοστά από το
απόλυτο μηδέν (-273.15 C)
5 χιλιοστά από το απόλυτο μηδέν
LOW-TC Superconductors
Lead (Pb)
7.196 K
Mercury (Hg)
4.15 K
Aluminum (Al)
1.175 K
Gallium (Ga)
1.083 K
Molybdenum (Mo) 0.915 K
Zinc (Zn)
0.85 K
Zirconium (Zr)
0.61 K
Americium (Am) 0.60 K
Cadmium (Cd)
0.517 K
Ruthenium (Ru) 0.49 K
Titanium (Ti)
0.40 K
Uranium (U)
0.20 K
Hafnium (Hf)
0.128 K
Iridium (Ir)
0.1125 K
Beryllium (Be)
0.023 K
Tungsten (W)
0.0154 K
Platinum (Pt)*
0.0019 K
Rhodium (Rh)
0.000325 K
Conductors vs. Insulators
FREE ELECTRONS
metals
No free electrons to carry
the current
wood
plastics
The foam balls (containing small magnets) organise themselves based on
the laws of minimum energy. This arrangement mimics the crystal lattice of a
solid material.
IONS (+)
What is Resistance?
ELECTRONS (-)
VOLTAGE
DIFFERENCE
ELECTRIC
FIELD
Electrical Resistance
RESISTANCE is caused by electrons colliding with:
• Thermal vibrations (phonons) of the ionic lattice
• Lattice defects
• Impurities
Cations
Electrons
I
V
Vs
copper
Liquid helium 4.2K (-269 ºC)
Impurities
R
273K = 0ºC
77K
Ro
T
V
I
Vs
Hg
Liquid helium 4.2K (-269 ºC)
Onnes (1911)
Low -Tc Superconductivity
Heike Kamerlingh Onnes
(1911)
LOW-TC Superconductors
Lead (Pb)
7.196 K
Mercury (Hg)
4.15 K
Aluminum (Al)
1.175 K
Gallium (Ga)
1.083 K
Molybdenum (Mo) 0.915 K
Zinc (Zn)
0.85 K
Zirconium (Zr)
0.61 K
Americium (Am) 0.60 K
Cadmium (Cd)
0.517 K
Ruthenium (Ru) 0.49 K
Titanium (Ti)
0.40 K
Uranium (U)
0.20 K
Hafnium (Hf)
0.128 K
Iridium (Ir)
0.1125 K
Beryllium (Be)
0.023 K
Tungsten (W)
0.0154 K
Platinum (Pt)*
0.0019 K
Rhodium (Rh)
0.000325 K
BCS Theory
No collisions
Zero resistance
John Bardeen
Leon Cooper
John Schrieffer
(1957)
Meissner Effect
• 1933 – Walther Meissner
and Robert Ochsenfeld
• T<Tc: external magnetic
field is perfectly expelled
from the interior of a
superconductor
The energy gap and BardeenCooper-Schrieffer theory
1. Existence of condensate.
2. Weak attractive electronphonon interaction leads to
the formation of bound pairs
of electrons, occupying states
with equal and opposite
momentum and spin.
3. Pairs have spatial
extension of order  .
The key point is the existence of energy gap between
ground state and quasi-particle excitations of the system.
E g (0)  2(0)  3.528kTc
The electron-electron attraction of the Cooper pairs caused
the electrons near the Fermi level to be redistributed above
or below the Fermi level. Because the number of electrons
remains constant, the energy densities increase around the
Fermi level resulting in the formation of an energy gap.
Essential details:
F. and H. London (1935) proposed a simple theory to describe the electrodynamics
of superconductors. They assumed that superconductivity is generated by
superelectrons, which are not scattered by either impurities or lattice vibrations,
thus are not contributing to the resistivity. They started from the equation of
E
motion of a free electron in an applied electrical field
m vs  eE
where
vs
(1.1)
is the velocity of the superelectrons and m and -e are their mass and
charge, respectively. Hence the supercurrent density is given by
J   n s evs
here
ns
(1.2)
is the density of superelectrons. Substituting (1.2) into (1.1), they derived,
the so-called first London equation
E
m
J .
ns e 2
(1.3)
Taking the curl of both sides of (1.3), and using Maxwell's third equation
(Faraday's law), they obtained
B
m
Ñ J .
ns e 2
(1.4)
Equation (1.4) can be integrated with respect to time and obtain
B  B0  
m
Ñ   J  J0 
ns e 2
(1.5)
where B0 andJ 0 , related by Ñ  B0   0 J 0 , are the magnetic field and current density
at t  0 , respectively. However, according to the Meissner effect (Meissner and
Ochsenfeld 1933) the magnetic flux inside a superconductor is completely expelled,
irrespective of whether the magnetic field was applied before or after cooling below
Tc
, i.e.
B0  0
B
. Therefore (1.5) leads to the postulated second London equation
m
Ñ J .
2
ns e
(1.6)
The field distribution within a superconductor is calculated from (1.6) in combination
with Maxwell's fourth equation   B  0 J to obtain
B
L2
(1.7)
m
0 e 2 ns
(1.8)
2 B 
where
L 
is called the London penetration depth. Equation (1.7) implies that the magnetic field is
exponentially screened from the interior of a sample within a distanceL (typically
L
 0.1 m ). Therefore, if the sample size is much larger than
be effectively screened.
, the whole specimen will
The Ginzburg-Landau theory
Ginzburg and
Landau (1950) introduced a
complex
pseudo-wave
function
 ( r )   ( r ) exp(i ) as a superconducting order parameter. The theory assumes that the
local density of superconducting carriers is given by
ns*  ( r ) .
2
(1.9)
Therefore, the order parameter( r ) is zero above Tc and increases continuously as the
temperature decreases. For small amplitudes and slow variation in space of  ( r ) , the
free energy density f can be expanded in series of the form
f = fn +    r 
2
1
1
B2
4
2
+   r  +
- i Ñ - 2 e A   
2
4m
2 0
(1.10)
A is the vector potential which is
where f n is the free energy density in the normal state,
related to the local magnetic induction B by the formula Ñ  A  B . In equation (1.10) it
is assumed that the superconducting carriers are electron pairs (Cooper pairs) with mass
and charge equal to 2m and 2e (e  0) , respectively (Bardeen, Cooper and Schrieffer
1957).

For a small range of temperatures near Tc the parameters

and
are approximately
given by
T

   0   1
 Tc 
(1.11)
  constant
(1.12)
where  0  0 is temperature independent.
If the free energy density is integrated over all space and minimised with respect to local
changes in A and  , two coupled differential equations are obtained. These govern the
equilibrium variation of A and  with position, given particular boundary conditions,
and are known, respectively as the first and second Ginzburg-Landau equations
     
2
1
2

i
Ñ

2
e
A

 0
4m
2e 2  2 A e
ie
*
*
J 
 Ñ   Ñ   
 

2m
m
m
where  is the phase of the order parameter.
2

Ñ   2e A 
(1.13)
(1.14)
The upper critical field and coherence length
For sufficiently high fields, superconductivity is destroyed and the field is uniform in the
sample. If the field is continuously reduced, at a certain field B = Bc 2 , called the upper
critical field, superconducting regions begin to nucleate spontaneously. In the regions
where the nucleation occurs, superconductivity is just beginning to appear and
therefore  is small, and equation (1.13) becomes
1
2

i
Ñ

2
e
A

     .
4m
(1.15)
Equation (1.15) is identical to the Schrödinger equation for a particle of charge 2e and
mass 2m in a uniform magnetic field. For an applied fieldB along the z-axis, the highest
solution corresponding to the upper critical field is
Bc 2 
0
2 2
and the corresponding order parameter
  x  x0  2 
i  k y y  kz z 
e
 exp  

2
2


(1.16)
(1.17)
with

4m

 (0)
1  T Tc
(1.18)
4m 0 is the
where  0  h 2e is the flux quantum, x0  k y  0 2 B , and  (0) 
value of  at T  0 . Equation (1.17) shows that is the characteristic length over
which  can vary appreciably. The parameter is called the Ginzburg-Landau
coherence length.
The penetration depth

In the case where the dimension of the sample are much greater than
B  0 inside the sample. Then 
, then
is constant, if varied the gradient term in (1.10)
would mean that the free energy increased. The constant value of
is given from
equation (1.13):
  0
2
2
 0  T 
   1   .
   Tc 
(1.19)
2
Since the order parameter is constant, i.e. Ñ   0 , equation (1.14) becomes
2e 2
2
J 
0 A .
m
(1.20)
Taking
the curl of both sides of equation (1.20), and substituting for the vector potential
B Ñ A
yields to
B
m
2

0 Ñ J .
2e 2
(1.21)
Equation (1.21) is identical to the second London equation (1.6) with a penetration depth
given by

m
2 0 e  0
2
2

 (0)
1  T Tc
(1.22)
2
where  (0)  m  2 0 e  0 is the penetration depth at zero temperature. The above
equation, in contrast to the expression (1.8) of the London penetration depth, contains
2
the temperature dependent parameter,  0 , which is defined in terms of (T ) .
The thermodynamic critical field
The existence of the Meissner effect, where the magnetic flux is completely expelled
from a type-I superconductor, implies that the superconducting state has a lower free
energy than the normal state. Therefore, the thermodynamic critical field Bc (T ) required
to destroy the superconducting state, is defined from the condition when the work done in
magnetic expulsion equals the zero field free energy difference between the normal and
superconducting states, or in term of free energies densities as
Bc2
 f  fn  fs 
20 .
(1.23)
The quantity  f , called the condensation energy density, is the energy per unit volume
released by transformation from the normal into the superconducting state. In the case of
zero applied magnetic and small variation of the order parameter , the solution (1.19)
can be substituted into (1.10), and the minimum free energy density corresponding to the
superconducting state at zero field will be given by
1 2 
fs = fn    .
2  
(1.24)
Comparing (1.24) to (1.23), and using the expressions of the penetration depth (1.22), and
Bc (T )
the coherence length (1.18), the thermodynamic critical field
can be written in the
form
Bc (T ) 
0
.
2 2
(1.25)
B
For a thin film of thickness d   in an external magnetic field
applied parallel to the
plane of the film and having the same value at both faces, the Ginzburg-Landau equations
have the solution (Tinkham 1996)
  0
2
2

d 2 B2 
1 
2 2 
24

Bc 

(1.26)
where  and Bc are the penetration depth and thermodynamic critical field of the bulk
B  Bc 2 //
material, respectively. Thus, the film becomes normal, i.e.   0 , when
,
2
given by
Bc 2 //  2 6
Here
Bc 2 //
Bc 
12 0

.
d
2 d
(1.27)
is known as the parallel upper critical field for a thin film. Taking into account
the temperature dependence of the coherence length, (1.27) can be written in the form
Bc 2 // 
12 0
1  T Tc .
2 d (0)
(1.28)
It is clear from (1.28) that the temperature dependence of the parallel upper critical field
is following a power law. This is in contrast to the linear dependence of equation (1.16)
of the upper critical for a bulk sample.
The Ginzburg-Landau parameter
The surface energy,  , of a superconducting-normal boundary is defined as the
difference between the Gibbs free energy per unit area between a homogeneous phase
(either all normal or all superconducting) and a mixed phase. Assuming that the
superconducting phase is located in the half-space x  0 , and the normal phase in the
other side (figure 1.1), and using the Ginzburg-Landau free energy density expression
(1.10), the surface energy is given by (e.g. Tinkham 1996)
2
4

Bc2 
B    

 1  Bc     0   dx .
2  0 


(1.29)
Here, the term to the left-hand side of the square bracket in (1.29) represents the positive
contribution to the surface energy associated with the diamagnetic screening energy. The
term on the right represents the negative contribution to the surface energy associated

with the condensation energy. Hence, it can be seen from (1.29) that the sign of is
determined from the balance of the positive magnetic expulsion and the negative
condensation energies.

Detailed numerical calculations of (1.29) show that the sign of the surface energy, ,
  
depends on the value of
, called the Ginzburg-Landau parameter. The surface
energy is positive for materials with   1
negative for materials with   1
2 , called type I superconductors, and
2 , called type II superconductors. The magnetic
behaviour of these materials is shown in figure 1.2.
Type I superconductors completely exclude magnetic flux from their interior, i.e. are in
the Meissner state, for all applied magnetic field below the thermodynamic critical field
Bc . The superconducting elements, with the exception of niobium, are all type I.
Type II superconductors allow the penetration of the magnetic flux when the applied field
exceeds a value referred to as the lower critical field,Bc1 . For increasing applied fields
above Bc1 , the magnetic field penetrates partially forming what is called a mixed state.
Eventually, when the applied field reached the value of the upper critical field
Bc 2
material becomes normal. The superconducting alloys and compounds are type II.
Type I
Bc
Type II
Bc

normal
superconducting

normal
0
B
0
superconducting
B

0

B

x
0

x

Figure 1.1: Diagram of variation of B and in a domain wall. The case refers to a type I
superconductor (positive surface energy); the case refers to a type II superconductor
(negative surface energy).
, the
Type-I
B (T)
Type-II
B (T)
c
c2
B
Normal state
B
Normal state
Mixed state
Meissner phase
B (T)
c1
Meissner phase
T
(a)
T
c
0
T
T
c
(b)
Figure 1.2: Magnetic phase diagram for (a) Type-I and (b) type-II superconductor.
The anisotropic Ginzburg-Landau theory
Anisotropic superconductors, such as NbSe2, the high temperature superconductors, and
artificially prepared superconducting multilayers, differ from isotropic materials in many of their
properties. As seen in the previous sections, the properties of isotropic superconductors are
described in term of the penetration depth which is proportional to m (equation (1.22)), and
the coherence length  proportional to 1
m (equation (1.18)), where m is the mass of the
superelectrons. The simplest way to extend the Ginzburg-Landau theory to the case of materials
with anisotropic superconducting properties is by introducing a phenomenological anisotropic
mass tensor mik instead of the isotropic m (Clem 1989). This mass tensor is diagonal, and the
diagonal elements mi (i  a, b, c ) are normalised such that  m1m2 m3 
1/ 3
 1 , where a, b, and c are
the three principal crystal directions. The coherence lengths and penetration depths along the
i direction are given by i  
mi and i   mi , respectively, with the normalisation
properties    a b c  and    a b c  , and the Ginzburg-Landau parameter is defined
1/ 3
as      i i .
1/ 3
Hence, within the mass tensor approach, an anisotropic superconductor is characterised
by two average lengths and  , and two mass ratios, for example ma / mc and mb / mc ,
the third mass being determined from the above normalisation. In this theory, the
thermodynamic critical field is similar to the isotropic case and is given by
Bc 
0
0

.
2 2i i 2 2
(1.30)
The upper critical field along principal axis i can be written as
Bc 2 // i   i 2 Bc 
 
where i
respectively.
0
2 j k

mi 
, j and k are the coherence lengths along the j and k-axis,
(1.31)
However, most of the superconducting multilayers and the high temperature
superconductors are uniaxial or almost uniaxial materials. In this case the
superconducting properties are uniquely defined by the in-plane ma  mb  mab and axial
mc effective masses, and equations (1.30) and (1.31) become
Bc 
0
2 2ab ab
Bc 2 // c 
Bc 2 // ab 

0
(1.32)
2 2c c
0
2 ab2
0
.
2 ab c
(1.33)
(1.34)
The anisotropy ratio , which describes the degree of anisotropy of uniaxial
superconductors, is defined from the formula    mc mab   c ab   ab  c . This
12
number enters the expressions of many anisotropic quantities, such as the ones describing
the vortex matter in layered superconductors (see next chapter). The magnitude of
 depends on the different classes of superconductors, for example   3.3 for NbSe2
(Morris et al 1972),   7.7 for YBa2Cu3O7- (Farrell et al 1990), and   150 for
Bi2Sr2CaCu2O8+ (Okuda et al 1991).
In summary …
Characteristic lengths in SC
London equation:
The London equation shows that the magnetic field
exponentially decays to zero inside a SC (Meissner
effect)
Penetration depth is the characteristic length of
the fall off of a magnetic field due to surface
currents.
The Pippard coherence length:
Coherence length is a measure of the
shortest distance over which
superconductivity may be established
Ginzburg-Landau parameter:
for pure SC far from Tc temperaturedependent Ginzburg-Landau coherence
length is approximately equal to Pippard
coherence length
Magnetic properties
Dependences of critical fields on temperature.
Phase boundaries between
superconducting, mixed and normal states
of type I and II SC.
Intermediate state (SC of type I)
(Type I SC show a reversible 1st order phase transition with a latent heat when the applied
field reached Bc. At this particular field relatively thick Normal and SC domains running
parallel to the field can coexist, in what is known as the intermediate state)
A distribution of superconducting and
normal states in tin sphere
(superconducting regions are shaded)
Intermediate state of a mono-crystalline
tin foil of 29 m thickness in perpendicular
magnetic field (normal regions are dark)
Mixed state (SC of type II)
(In type II SC finely divided quantized flux vortices or flux lines enter the
material over a range of applied fields below Bc, and remain stable over a
range of applied fields, in what became known as the mixed state. If these flux
lines are pinned by lattice defects or other agencies, type II SC can carry a
large super-current: see development of useful high-field SC magnets.)
Supercurrent
Abrikosov: [1957]
Normal core
One vortex carries one
quantum of the flux:
Normal regions are
approximately 300nm
Closer packing of normal regions
occurs at higher temperatures or
higher external magnetic fields
Triangular lattice of vortex lines going out
to the surface of SC Pb0.98In0.02 foil in
perpendicular to the surface magnetic field
Vortex characteristics
• Magnetic field of a
vortex
A quantum of magnetic flux is
hc
0 
2e
Vortex state of type II superconductors
• Type II
Phase of GL pseudo-wave function changes
by 2 when going around spatial lines
where  is zero

1
||
0
Normal core
Vortex state of type II superconductors
• Type II
Phase of GL pseudo-wave function changes
for 2 when going around spatial lines
where  is zero
In type-II SC field penetrates to the bulk of
material in the form of vortices (or
magnetic flux lines, or fluxons)
Each vortex represents magnetic flux
quantum

1
||
B/Beq

0
Normal core
Critical current density
Critical current is the maximum current SC
materials can carry, above which they stop being
SCs. If too much current is pushed through a
SC, the latter will become normal, even though i
may be below its Tc. The colder you keep the SC
the higher the current it can carry.
Three critical parameters Tc, Hc and Jc define
the boundaries of the environment within which
a SC can operate.
Fig. demonstrates relationship between Tc, Hc and Jc (a criti
surface). The highest values for Hc and Jc occur at 0K, while
highest value for Tc occurs when H and J are zero.
Josephson effect
(see also hand-out)
In 1962 Josephson predicted Cooper-pairs can tunnel
through a weak link at zero voltage difference. Current
in junction (called Josephson junction – Jj) is then
equal to:
J  J c sin 1   2 
Electrical current flows between two SC
materials - even when they are separated by
a non-SC or insulator. Electrons "tunnel"
through this non-SC region, and SC current
flows.
The Meissner-Ochsenfeld Effect
Walter Meissner
Robert Ochsenfeld
(1933)
Magnet
T>T
T<TC
Superconductor
DIAMAGNETISM