Superconductivity III:Theoretical Understanding

Physics 355

Superelectrons

• Two Fluid Model

sd

m v qEdt

s sj n ev

2sn ej Em

London Phenomenological Approach• Ohm’s Law

• Magnetic Vector Potential

• Maxwell IV

• London Equation

j E

0B j

20 L

1j A

20 2

L

22L

1

1

B B j B

B B

Net Result

London Penetration Depth

L0

/( ) xB x B e

The penetration depth for pure metals is in the range of 10-100 nm.

20

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 vibrationsThe 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 = p1 p1’

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

p1 p2

p2p1

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

22

2 2 2 2( ) 1

( )q

o s q

eV q

q k

Clearly if <q this (much simplified) potential is always negative.

2

2 2 2 2o

11

( ) 1s 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 ħD =kBD,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 ħD

22 2 exp

( )F DF

E ED E V

Lecture 12

The maximum (BCS) transition temperature

D(EF)V is known as the electron-phonon coupling constant:

( ) / 2ep FD 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 ~ 0.33For Al calculated ep ~ 0.23 measured ep ~ 0.175For Nb calculated ep ~ 0. 35 measured ep ~ 0.32

epDcB

1exp2Tk75.1

In terms of the gap energy we can write

which implies a maximum possible Tc 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