Electron Self-Energy and Temperature-Dependent Effective Masses in Semiconductors:n-Type Ge and Si

The electronic self-energy due to interaction with acoustic phonons is evaluated as a function of the electron propagation vector k, and a relation is established connecting the Sommerfeld-Bethe interaction constant with the energy band separation and effective masses. For nondegenerate prolate ellipsoidal energy surfaces of revolution, the self-energy depends linearly on the temperature $T$ at high temperatures and quadratically on $T$ at low temperatures, this behavior being substantiated by the experimental results of Macfarlane and Roberts. The temperature dependence of the principal effective masses $m_i\mathrm{}\left(T\right)\mathrm{}$ at high temperatures is given by ($i=l or t$) $\frac{m_i\mathrm{}\left(0\right)\mathrm{}}{m_i\mathrm{}\left(T\right)\mathrm{}}=1+\left(\frac{128\pi }{9\rho h^{3}s{\Theta }_D}\right){m_i}^2\mathrm{}\left(0\right)\mathrm{}{\alpha }_i\mathrm{}\left(E\right)\mathrm{}{〈C^2〉}_{\mathrm{Av}}T$ thus indicating a decrease in effective mass with rising temperatures. The result does not explain the deviation from the $T^{-\frac{3}{2}}$ law for the lattice mobility as observed by Morin and Maita. The percentage decrease at room temperature for each of the electron effective masses amounts to less than 1%. These results do not account fully for the possible change determined by Lax and Mavroides.