Frequency‐ and wave‐vector‐dependent dielectric function of a model semiconductor

Closed analytical expressions for the real part ϵ 1 ( q , ω) and the imaginary part ϵ 2 ( q , ω) of the frequency‐ and wave‐vector‐dependent dielectric function are obtained including a gap E g in an appropriate manner in the spirit of the nearly‐free electron approximation, and requiring that these functions will exactly satisfy the Kramers‐Kronig relations and the f ‐sum rule. In the long wavelength limit the classical Lorentz oscillator result comes out, whereas for ω = 0 the static screening function is decisively improved with regard to earlier model calculations. The results also yield the plasmon dispersion relations ω L ( q ) and ω T ( q ) for Si, Ge, GaAs, and ZnSe in good agreement with previous detailed computations. With the average gap width E g going to zero the expressions obtained transform into the well‐known RPA (Lindhard) formulae for a free electron gas.