Deformation potentials and electron-phonon scattering: Two new theorems

This paper gives two new theorems about the band deformation potential D_αβᵇ(→k,n) which expresses the shift of electron energies under external strains, ∂ε_→kn∂S_αβ, and the electron-phonon deformation potential D_αβ^e-ph(→k,n) which gives the matrix element for an electron to scatter from state (→k,n) to a nearby state (→k+→Q,n) by absorption of an acoustic phonon of branch j and amplitude →u(→Qj), namely $〈{{→}}{k}+{{→}}{Q},n|{H}_{e{-}ph}|{{→}}{k},n〉=i{u}_{{α}}({{→}}{Q}j){Q}_{{β}}{D}_{{α}{β}}^{e{-}ph}({{→}}{k},n)$. First, it is shown for a rigid ion model that ${D}_{{α}{β}}^{e{-}ph}({{→}}{k},n)$ equals ${D}_{{α}{β}}ᵇ({{→}}{k},n)+m{v}_{{{→}}{k}n{α}}{v}_{{{→}}{k}n{β}}$, where ${{{→}}{v}}_{{{→}}{k}n}$ is the electron group velocity. Second, it is shown that the Fermi surface average of ${D}_{{α}{β}}^{e{-}ph}({{→}}{k},n)$ equals the Fermi surface average of $m{v}_{{{→}}{k}n{α}}{v}_{{{→}}{k}n{β}}$. From the second theorem, it is deduced that the first theorem is probably generally valid in a metal [i.e., not restricted to a rigid-ion model) provided the band deformation potential ${D}_{{α}{β}}ᵇ({{→}}{k},n)$ is defined relative to an energy which moves with the Fermi energy under strains. The first theorem contradicts the common belief that the two deformation potentials are always the same, but preserves the usual form of the deformation-potential theorem at band edges where v_→knα vanishes.