Superconductivity

1911: discovery of superconductivity

Discovered by Kamerlingh Onnes

in 1911 during first low temperature
measurements to liquefy helium
Whilst measuring the resistivity of
“pure” Hg he noticed that the electrical
resistance dropped to zero at 4.2K
In 1912 he found that the resistive state
is restored in a magnetic field or at high
transport currents
1913
 1933: Meissner-Ochsenfeld effect

Ideal conductor! Ideal diamagnetic!

Superconductivity shows diamagnetic properties

  T 
2

H C (T )  H 0 (0) 1    
  TC  
Types of Superconductors
Superconductivity in alloys
 Isotope Effect

It has been observed that the critical temperature of superconductors varies

with the isotopic mass. In mercury Tc varies from 4.185 to 4.146 K as the
average atomic mass M varies from 199.5 to 203.4 atomic mass units.
Experimental results within each series of isotopes may be fitted with a
relation :
M  Tc  const.
1935: London Brother’s Theory
(F. London & H. London)
Macroscopic Theory of Superconductivity

Two Fluid Model

N  ns  nn (1)

ns: Super-Electrons

nn: Normal-Electrons
If E be the electric field, the equation of motion of the super-electrons

dvs dvs eE ( F  qE ) (2)

m   eE or 
dt dt m

The super-current density: J s   ns e v s (3)

dJ s dvs ns e 2 First London

Differentiating:   ns e  E Equation
dt dt m (4)

Result J s  const if E 0 (5)

Also from Maxwell’s Electromagnetic Equations

B B
 E   or  0  E  0 or B  const (6)
t t
Above solution is contradiction to Meissner Effect

B  B ( r , t )  B ( x, y , z , t )
London proposed some modifications to avoid the contradiction. Taking
the Curl of First London Equation (Eqn 4)

dJ s ns e 2 ns e 2 dB
    E    (7) On integration w.r.t time
dt m m dt

2
Constant of integration is
ns e zero as B = 0 and
 Js   B (8) Curl Js = 0 at t=0
m
Second London Equation
From another Maxwell’s equations:

  B  0 J s (9)

    B    0   J s  Taking the Curl on both sides, we get

 0 ns e 2
  B    B  
2 From Second London
B (10) Equation (8)
m
 0 ns e 2 1
 B  
2
B or  B  2
B 0 (11)
m 2
12
 m  since B  0
where    
2 
  0 ns e  is called penetration depth
From Equation (11), which is homogeneous second order differential eqn

1 which has the following solutions in one

 B ( x) 
2
B ( x)  0 dimension
2

B ( x)  B (0) exp( x  ) or H ( x)  H (0) exp( x /  )

(12)

Since B  0 H
Magnetic field inside the
superconductor decays
exponentially
From Equation 12,

B (0)
B ( x)  B (0) exp( x /  )  B ( x)  when x  
e

Thus the penetration depth is defined as the depth from the surface at
which the magnetic flux density falls 1/e of its initial value at the surface

In a thin film the Meissner Effect is incomplete when the thickness

of the film is less than penetration depth.

Thin film
 Thermodynamics of Superconductors
The transition between the normal and the superconducting states is
thermodynamically reversible. Thus we may apply thermodynamics to the
transition and we thereby obtain an expression for entropy difference between
normal and superconducting states in terms of critical field Hc curve versus T.

We treat type-I superconductor with a complete Meissner Effect (B=0). We can see
that the critical field Hc is a quantitative measure of the free energy difference
between superconducting and normal states.

The stabilization free energy of the superconducting state with respect to the
normal state can be determined by calorimetric or magnetic measurements. In
calorimetric measurement the heat capacity is measured as a function of
temperature for the superconductor and for normal conductor, which means the
superconductor in magnetic field larger than Hc. From the difference of heat
capacities we can calculate the stabilization free energy of the superconducting
state.
In the magnetic field, the stabilization free energy is found from the value of
applied magnetic field that will destroy the superconducting state, at constant
temperature. Consider a work done on a superconductor is brought reversibly
from infinity to a point r in the magnetic filed:

Ba In superconducting state:
Ba
W    M .dBa B  Ba   0 M  0  M  
0
0

Work done or Free energy

dF   M .dBa

1
dFs   Ba dBa
0
On integrating we obtain the increase in free energy density of superconductor

Fs ( Ba )  Fs (0)  Ba2 2  0  Fs ( Ba )  Fs (0)  Ba2 2 0

At critical field, normal and superconducting phases coexist, so

free energies of superconducting and normal state are equal

FN (T , Bac )  FS (T , Bac )

In case of normal non-magnetic material, we may neglect the susceptibility of a

metal in normal state (M=0). The energy of the normal metal is independent of
field.

FN ( Bac )  FN (0)
At critical value of Bac of the applied magnetic field, the energies are
equal in normal and superconducting state:

FN ( Bac )  FS ( Bac )  FS (0)  Bac2 2 0

0
FN (0)  FS (0)  H c2 [ Bac   0 H c ; in SI unit ]
2
Stabilization free energy of the superconductor state is

F  FN (0)  FS (0)  0 H c2 2

Free Energy and Entropy in superconducting transition

dH c
F  U  TS  S   dF dT  S    0 H c
dT
 Entropy of Superconductors
dH c
S N  S S  S   0 H c
dT
S N  S S  S  0; dH c dT  0

Superconducting state is more ordered state and hence less entropy

 Specific heat of superconductors
From entropy relation

dH c dH c
S N  S S   0 H c  TS N  TS S  QN  QS    0TH c
dT dT

dS N dS S d  dH c  dS N dS S d  dH c 
   0  c
H   T  T    0T  c
H 
dT dT dT  dT  dT dT dT  dT 

2
 dH c  d 2Hc
C N  C S    0T     0TH c
 dT  dT 2
Rutgers Equation
2
 dH c 
C N  CS T T    0T   at T  Tc ; H c  0
c
 dT 
At transition temperature, there is discontinuity in
specific heat. As the lattice specific heat is same in
superconducting and normal state. The discontinuity is
due to electronic specific heat only.
 Superconducting Gap

(C N  CS )T Tc  A exp(2); where   1 k B T

Superconducting band gap exist at

Fermi energy 2=1.65 meV (for Hg)

Superconducting energy gap changes with

temperature
 BCS Theory of Superconductivity
(Bardeen, Cooper and Schrieffer 1957)

1. The electron in the superconducting state form bound pairs

(Cooper pairs). The interaction between electron and lattice,
through lattice distortion leads to bound electron pair .
2. An electron with wave vector k 
is coupled with another electron with wave vector  k 

3. Energy gap corresponds to the binding energy and the

coherence length is the maximum length at which these
electrons are coupled.

4. The penetration depth and the coherence length emerges as a

natural consequences of the BCS theory. The London equation
is obtained for magnetic fields that vary slowly in space. Thus
Meissner effect is obtained in a natural way.

5. Magnetic flux through a superconducting ring is quantized and

the effective unit of charge is 2e rather than e. Thus flux
quantization in terms of pair charge is a consequence of the
theory.
 BCS Ground State

The filled Fermi sea is the ground state of a Fermi gas of non-interacting
electrons. This state allows arbitrarily small excitations – we can form an excited
state by taking an electron from the Fermi surface and raising it just above the
Fermi surface

BCS Theory shows that with an appropriate attractive interaction between

electrons the new ground state is superconducting and is separated by a finite
energy Eg from its lowest excited state.
The formation of BCS ground state is suggested by above Figure. The
BCS state in (b) contains admixture of one-electron orbitals from above
the Fermi energy εF. At first site, the BCS state appears to have a higher
energy than the Fermi state: the comparison of (b) and (a) shows that
the kinetic energy of BCS state is higher than that of Fermi state. But
the attractive potential energy of BCS state acts to lower the total
energy of BCS state with respect to Fermi state.
The central feature of the of the BCS state is that the one particle
orbitals are occupied in pairs: if an orbital with wave vector k with
spin up is occupied, then orbital with wave vector –k with spin
down is also occupied. If one is empty the other is also empty. These
pairs, called Cooper Pairs, are bosons with spin zero.

The superconducting transition temperature is much lower and

takes place when the electron pairs berak-up into two fermions.
The model of superconductor composed of non-interacting
bosons can not be taken absolutely literally, for there are 106
electrons in the volume occupied by a single Cooper pair.
 Flux quantization in a superconducting ring

We prove that the total magnetic flux that passes through a

superconducting ring may assume only quantized values, integral
multiples of the flux quantum

2  c / q where q  2e

Flux quantization is the beautiful example

of a long range quantum effect in which
the coherence of the superconducting
state extend over a ring or solenoid
Let  (r ) be particle probability amplitude

   n number of electron pairs

We can write   n1 2 e i ( r ) ;    n1 2 e  i ( r )

From Hamilton’s Equations of Mechanics, the velocity of the particle is

1 q  1 q 
v  p  A     i   A
m c  m c 

Particle flux n q 
 v     A 


m c 
 nq  q 
The electric current density is j  q v     A 


m c 
Quantization of magnetic flux through a ring is a dramatic
consequence of the above equation. Let us take a closed path C
through the interior of the superconducting material, well away from
the surface. The Meissner effect tells us that B and j both are zero in
the interior . From the above equation,

q
c  qA or   A
c
Taking the integration along the closed loop:

L.H .S .     dl 2 s   2  1 s is integer and ψ is single-valued

C
 q q q q
R.H .S .   A.dl   (curl A ).d   B.d  
c C c C c C c
q
L.H .S .  R.H .S . ;  2  1  2 s  
c
  2c q s   0 s ; s  1,2,3...
 0  2c 2e  2.0678 10 7 gauss cm 2

This unit of flux is called fluxon or fluxoid

 Josephson Tunneling
Under suitable conditions we observe remarkable effects associated with
the tunneling of superconducting electron pairs from a superconductor
through a layer of an insulator into another superconductor. Such a
junction is called a week link. The effect of pair tunnelling includes
DC Josephson Effect: A dc current flows
across the junction in the absence of any
electric or magnetic filed.

AC Josephson Effect: A dc voltage applied

across the junction causes rf current
oscillations across the junction.

Macroscopic long-range Quantum Interference: A dc magnetic field

applied through a superconducting circuit containing two junctions
causes the maximum super-current to show interference effects as a
function of magnetic field intensity. This effect can be utilized in
sensitive magnetometers (SQUID)
 DC Josephson Effect

 1  n11 2 ei
1
 2  n12 2 ei 2

1 2

The time - dependent Schrodinger equation gives

i / t  H
i 1 / t  T 2 (1)
i 2 / t  T 1

Here ħT represents the effect of electron-pair coupling or transfer

interaction across the insulator; T has the dimension of rate or frequency.
It is a measure of leakage of ψ1 into ψ2 and ψ2 into ψ1.
Let  1  n11 2 ei
1
and  2 n12 2 ei 2
Using above in eqn 1 gives:

 1 1 1 2 i1 n1 
 n1 e  i 1 1  iT 2
t 2 t t
 2 1 1 2 i 2 n2  2
 n2 e  i 2  iT 1
t 2 t t
On further solving, above equations yield:

1 n1 1
 in1  iT (n1n2 )1 2 ei ; where    2  1
2 t t
1 n2  2
 in2  iT (n1n2 )1 2 e i
2 t t
Separating the above equation into real and imaginary parts:
n1 n2
 2T (n1n2 ) sin  ;
12
 2T (n1n2 )1 2 sin 
t t
12 12
1  n2   2  n1 
 T   cos  ;  T   cos 
t  n1  t  n2 
For identical superconductors 1 & 2, n1  n 2 We get:
 n2 n
 2  1   0; &   1 ;   const.
t t t
Current density:
 n
(n2 )  2e 2  4eT n1n2  sin   J 0 sin( 2  1 )
12
J q
t t
J0 is the maximum zero-voltage current that could pass through the
junction
AC Josephson Effect: When a dc voltage is applied across the junction
we get AC current through the junction: We can say that a pair (2e)has a
potential energy – eV on one side and + eV on the other side: The time
dependent equation of motion becomes:

i  1 t  T 2  eV 1 ; i  2 t  T 1  eV 2 ;

Putting,  1  n11 2 ei
1
and  2 n12 2 ei 2 in above equations :

Separating real and imaginary parts, we get

n1 n2
 2T (n1n2 ) sin  ;
12
 2T (n1n2 )1 2 sin 
t t
12 12
1 eV  n2   2 eV  n1 
  T   cos  ;   T   cos 
t   n1  t   n2 
For identical superconductors 1 & 2, n1  n 2 We get:
 n2 n
 2  1    t   2eV  ; &   1 ;   const.
t t t
Integrating above equation
 (t )   (0)  (2eVt )
J  J 0 sin (0)  (2eVt  )

The current oscillates with the frequency   2eV 

A dc voltage of 1 V produces a frequency of 483.6 MHz. A photon of

energy ħ = 2eV is emitted or absorbed when an electron pair crosses
the barrier. By measuring voltage and frequency it is possible to
obtain very precise value of e/ħ.
 Macroscopic Quantum Interference
We consider two Josephson junctions
in parallel as shown in the figure. No
voltage is applied.
Let the phase difference between points
1 & 2 be a & b for junctions a & b. In
absence of magnetic field, these two
phases must be equal.

Let the magnetic field be applied normal to the parallel circuit as shown in the
figure. The phase difference of two paths is given by:

 2  1   b   a  (2e / c) or

e e
b  0   ; &  a  0  
c c
Total Current:
  e   e 
J  J a  J b  J 0 sin   0     sin   0   
  c   c 
J  2 J 0 sin  0 cose c ; sin( A  B)  sin( A  B )  2 sin A cos B

Where flux    B.d

C
 Superconducting Quantum Interference Device (SQUID)

Magnetism at low temperature

Measurements of small magnetic fields (5×10−18 Tesla)

Small quantities of samples