Question

Why electron phonon interaction happends near fermi surface?

Answer 1

Usually the energy correction to electron by electron phonon interaction in metals at zero tempreture has the form(Gerald D.Mahan Many-Particle Physics, Third Edition, Sec. 7.4)

∑(k,u)=∫d3q(2π)3M(q)2[1−nf(ξ(k+q))u−ϵ(k+q)−ω(q)+iδ+nf(ξ(k+q))u−ϵ(k+q)+ω(q)+iδ]∑(k,u)=∫d3q(2π)3M(q)2[1−nf(ξ(k+q))u−ϵ(k+q)−ω(q)+iδ+nf(ξ(k+q))u−ϵ(k+q)+ω(q)+iδ]

The phonon energy is typically much smaller compared to Fermi energy. The author said that the self energy has appreciable value only for uu within a Debye energy of the Fermi surface. I can't see this in this formula. And I have use the electron phonon coupling constant(here M(q)) in Solid State Physics(Ashcroft/Mermin, Chapter 26 Phonons in Metals, P-523)

g(q)2=1V4πe2q2+k2012ωqg(q)2=1V4πe2q2+k0212ωq

to calculate this self energy(self consistently). I am not able to get the typically electron phonon self energy contribution to electron in Ashcroft/Mermin's book. 

Answer 2

Hola mate !! ✌️✌️

here's your answer...⬇️⬇️

The electron and photon interaction happens near fermi surface . this is so because the condition for this type of interactions are satisfied at Fermion level . hence can be carried out effectively.

hope it helps !! ❤️❤️