Wave-Number-Dependent Dielectric Function of Semiconductors

Expressions for the wave-number-dependent dielectric function are derived for various models of a semiconductor. The calculation is carried out for the diagonal part of the dielectric function at zero frequency. It is found that calculations based on plane wave models (such as the free electron model) give poor results for small values of the wave number due to neglect of both Bragg reflections and Umklapp processes. We use instead an isotropic version of the nearly free electron model in which dielectric function depends on only one parameter $E_g$ representing an average energy gap that can be determined from optical data. It is noted that for small wave numbers Umklapp processes give the major contribution to the dielectricfunction, where-as for large wave numbers normal processes dominate. The dielectric function is evaluated numerically for a value of $E_g$ appropriate to Si.