A study of Block electrons in external fields by way of an evolution operator

An evolution operator appropriate for studying the behaviour of Bloch electrons in the presence of both an electric field and an optical field simultaneously, is developed. This evolution operator is seen to be capable of providing a unified approach for understanding various features related to Wannier-Stark ladders. By applying this evolution operator, the authors have investigated (i) the nature of the energy spectrum of Bloch electrons in the presence of an electric field alone and (ii) the absorption of electromagnetic radiation by Bloch electrons in the presence of an electric field. The principal results obtained are the following: (a) Wannier-Stark ladders would exist even in the presence of Zener tunnelling provided it is retained only up to first order. The effect of Zener tunnelling higher than the first order is to produce a broadening of the discrete Stark levels; numerical analysis relevant to this aspect is carried out with respect to a one-dimensional GaAs crystal. (b) A formula is derived for the coefficient of absorption of electromagnetic radiation by Bloch electrons in the presence of an electric field, by taking into account the k dependence of the momentum matrix element and using a nearly-free-electron model.