Superconductivity III: Theoretical Understanding

1. Superconductivity III:Theoretical Understanding Physics 355

2. Superelectrons • Two Fluid Model

3. Net Result London Phenomenological Approach • Ohm’s Law • Magnetic Vector Potential • Maxwell IV • London Equation

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

5. 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.

7. 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 p1 p2 q p1 p2 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 = p1 p1’ The characteristic frequency of the phonon must then be the phonon frequency q, Lecture 12

8. It can be shown that such electron-ion interactions modify the screened Coulomb repulsion, leading to a potential of the form Clearly if <q this (much simplified) potential is always negative. The attractive potential 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 =kB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 Lecture 12

9. D(EF)V is known as the electron-phonon coupling constant: In terms of the gap energy we can write The maximum (BCS) transition temperature 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 which implies a maximum possible Tc of 25K ! Lecture 12

11. 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