Superconductivity III: Theoretical Understanding Physics 355

Superelectrons Two Fluid Model

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

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

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, p1p1 p2p2 p2p2 p1p1 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 Clearly if  <  q this (much simplified) potential is always negative. 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 (~ K) The cut-off energy in Cooper’s attractive potential can therefore be identified with the phonon cut-off energy ħ  D Lecture 12

The maximum (BCS) transition temperature D(E F )V is known as the electron-phonon coupling constant: 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 ~ For Nb calculated ep ~ measured ep ~ 0.32 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