Boundary conditions, energy spectrum, and optical transitions of electrons in bounded narrow gap crystals

Abstract

The problem of calculating the energy spectrum of the electron in a bounded crystal is formulated in terms of effective wave functions which are the envelopes of the complete electron wave function. Boundary conditions for the envelopes on crystal surface are obtained in the simple cases (the effective-mass and two-band approximation) from the requirement for the effective Hamiltonian to be hermitian. Boundary conditions involve one or a few free parameters which characterize the properties of a surface and can be calculated only within the scope of microscopic theory. The developed approach makes possible the common description and the investigation in external fields of both surface (Tamm) states in a semi-infinite narrow gap crystal and size-quantization states in a film or in a space charge layer on a semiconductor surface. The influence of boundary conditions on optical transitions in a film and on the electron spectrum in an inversion layer is discussed.