Low temperature-semiconductor band gap thermal shifts: T⁴ shifts from ordinary acoustic and T² from piezoacoustic coupling

At low temperature T, the experimental gap of silicon decreases as E_g(T)=E_g(0)-AT^4. The main reason is electron-phonon renormalization. The physics behind the T^4-power law is more complex than has been realized. Renormalization by intraband scattering requires a careful non-adiabatic treatment in order to correctly include acoustic phonons and avoid divergences from piezoacoustic phonon interactions. The result is an unexpected low T term E_g(0)+A' T^p with positive coefficient A', and power p=4 for non-piezoelectric materials, and power p=2 for piezoelectric materials. The acoustic phonons in piezoelectric semiconductors generate a piezoelectric field, modifying the electron-phonon coupling. However, at higher T, when thermal acoustic phonons of energy hbar v_s q acquire energies comparable to the electronic intermediate state (higher than the band-edge state by hbar^2 q^2 /2m*), the low q and higher q intraband contributions to T^p rapidly cancel, giving little thermal effect. But there is an additional T-dependence from interband effects of acoustic phonons. This turns out to have power law T^4 for both non-piezoelectric and piezoelectric semiconductors. This term can have either sign, but usually reduces the size of gaps as T increases. It arises after cancellation of the T^2 terms that appear separately in Debye-Waller and Fan parts of the acoustic phonon interband renormalization. The cancellation occurs because of the acoustic sum rule.