Superconductivity

Superconductivity III:

Theoretical Understanding

Physics 355

Superelectrons

•
Two Fluid Model


s
d
m v qE
dt

s s
j n ev

2
s
n e
j E
m

London Phenomenological Approach

•
Ohm’s Law

•
Magnetic Vector Potential

•
Maxwell IV

•
London Equation

j E


0
B j

 
2
0 L
1
j A

 


2
0
2
L
2
2
L
1
1
B B j B
B B



      
 
Net Result

London Penetration Depth

L
0
/
( )
x
B x B e



The penetration depth for pure metals is in the range of 10
-
100 nm.

2
0
L
2
mc
nq



Coherence Length

•
Another characteristic length that is
independent of the London penetration depth
is the coherence length

.

•
It is a measure of the distance within which
the SC electron concentration doesn’t change
under a spatially varying magnetic field.

The effects of lattice vibrations

The localised deformations of the lattice caused by the electrons are subject to
the same “spring constants” that cause coherent lattice vibrations, so their
characteristic frequencies will be similar to the phonon frequencies in the lattice

The Coulomb repulsion term is effectively instantaneous

If an electron is scattered from state k to k’
by a phonon, conservation of momentum
requires that the phonon momentum must
be
q = p
1



p
1
’

The characteristic frequency of the
phonon must then be the phonon
frequency

q
,

p
1


p
2


p
2

p
1

q

The electrons can be seen as interacting by
emitting and absorbing a “virtual phonon”, with a
lifetime of

=2

/


determined by the uncertainty
principle and conservation of energy

Lecture 12

The attractive potential

It can be shown that such electron
-
ion interactions modify the screened
Coulomb repulsion, leading to a potential of the form

2
2
2 2 2 2
( ) 1
( )
q
o s q
e
V q
q k

  
 
 
 
 
 
 
 
Clearly if

<

q


this (much simplified) potential is always negative.

2
2 2 2 2
o
1
1
( ) 1
s q
e
q k
  
 
 
 
 
 
 
 
This shows that the phonon mediated interaction is of the same order of
magnitude as the Coulomb interaction

The maximum phonon frequency is defined by the Debye energy
ħ

D

=k
B

D
,

where

D


is the Debye temperature (~100
-
500K)

The cut
-
off energy in Cooper’s attractive potential can therefore be
identified with the phonon cut
-
off energy
ħ

D

2
2 2 exp
( )
F D
F
E E
D E V

 

 
 
 
Lecture 12

The maximum (BCS) transition temperature

D(E
F
)V

is known as the electron
-
phonon coupling constant:

( )/2
ep F
D E V



ep

can be estimated from band structure calculations and from estimates of
the frequency dependent Fourier transform of the interaction potential, i.e.
V(q,

)

evaluated at the Debye momentum.

Typically

ep

~ 0.33

For
Al

calculated

ep

~ 0.23

measured

ep

~ 0.175

For
Nb

calculated

ep

~ 0. 35

measured

ep

~ 0.32
















ep
D
c
B
1
exp
2
T
k
75
.
1

In terms of the gap energy we can write

which implies a maximum possible T
c

of 25K

!

Lecture 12

Bardeen Cooper Schreiffer
Theory

In principle we should now proceed to a full treatment of BCS Theory

However, the extension of Cooper’s treatment of a single electron pair
to an N
-
electron problem (involving second quantisation) is a little too
detailed for this course

Physical Review,
108
, 1175 (1957)

Lecture 12