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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