Carrier Scattering - Optical Phonons

Optical Phonons

Typically, phonons in the optical branches of vibrational dispersion relationships have energies on the order of or greater than kT and, therefore, the approximations ħω<q>>1 cannot be made. Yet, a reasonable route that still provides a detour from dealing with complex phonon dispersions is using the Einstein model which states that only one phonon mode exists in solids. For optical phonons, this approximation turns out to be sufficient due to very little slope variation in ω(q) and, thus, we can claim ħω(q) ≅ ħω, a constant. Consequently, N_q is also a constant (only T dependent). The last approximation, g(E')=g(E±ħω) ~ g(E), cannot be made since ħω ~ E and there is no workaround for it, but the added complexity to the sum for τ is minimal.

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The sum turns to the density of states at E' and the Bose-Einstein distribution can be taken out of the sum due to ħω(q) ≅ ħω.