Is electron phonon interaction important away from fermi surface?
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In weak coupling superconductor, the effective electron phonon interaction can be written as
Heff=12∑q,k1,k2,σ1,σ2Vk1,qC†k1+q,σ1C†k2−q,σ2Ck2,σ2Ck1,σ1Heff=12∑q,k1,k2,σ1,σ2Vk1,qCk1+q,σ1†Ck2−q,σ2†Ck2,σ2Ck1,σ1
where
Vk1,q=|Dq|22ωq(Ek1+q−Ek1)2−(ωq)2Vk1,q=|Dq|22ωq(Ek1+q−Ek1)2−(ωq)2
When |Ek1+q−Ek1|<ωq|Ek1+q−Ek1|<ωq, there is atractive potential, and electrons form pairs. I don't understand that this interaction doesn't necessarily demand k1k1 or k1+qk1+q to be near the Fermi surface. It only requiers |Ek1+q−Ek1|<ωq|Ek1+q−Ek1|<ωq. What about electron states under and far away from the Fermi surface? This interaction should be smaller quickly away from the Fermi surface. But I don't see this. Could anybody give an explanation?
And more when considering electron phonon interaction people usually only consider electron states scattered by phonon near fermi surface as above. Again I don't understand why. For example, self energy correction in metals at zero tempreture caused by electron phonon interaction
∑(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]
where M(q)M(q) is electron phonon coupling function
Heff=12∑q,k1,k2,σ1,σ2Vk1,qC†k1+q,σ1C†k2−q,σ2Ck2,σ2Ck1,σ1Heff=12∑q,k1,k2,σ1,σ2Vk1,qCk1+q,σ1†Ck2−q,σ2†Ck2,σ2Ck1,σ1
where
Vk1,q=|Dq|22ωq(Ek1+q−Ek1)2−(ωq)2Vk1,q=|Dq|22ωq(Ek1+q−Ek1)2−(ωq)2
When |Ek1+q−Ek1|<ωq|Ek1+q−Ek1|<ωq, there is atractive potential, and electrons form pairs. I don't understand that this interaction doesn't necessarily demand k1k1 or k1+qk1+q to be near the Fermi surface. It only requiers |Ek1+q−Ek1|<ωq|Ek1+q−Ek1|<ωq. What about electron states under and far away from the Fermi surface? This interaction should be smaller quickly away from the Fermi surface. But I don't see this. Could anybody give an explanation?
And more when considering electron phonon interaction people usually only consider electron states scattered by phonon near fermi surface as above. Again I don't understand why. For example, self energy correction in metals at zero tempreture caused by electron phonon interaction
∑(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]
where M(q)M(q) is electron phonon coupling function
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so this is an electron-electron interaction. It might be an obvious statement, but doesn't it come from the fact that there is no phase space to scatter electrons above or below the Fermi surface ? You can probably see it when you compute quantities such as susceptibilities, free energy... Your c-operators will give electronic Green's functions, when doing the Matsubara frequency sum you get occupation factors that forbid scattering away from the Fermi surface
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