Electrons in crystals in a finite-range electric field

The problem of an electron in a finite-range constant electric field is treated; energy levels and eigenfunctions are calculated for the "empty crystal" and for a Mathieu-type model crystal. It is shown that the addition of boundary conditions may change the solution drastically. The influence of different boundary conditions on the eigenvalues and eigenfunctions is discussed and it is found that the former are relatively insensitive while the latter are sensitive to a change in boundary conditions. The result for the eigenvalues is shown to be consistent with an extension of the Born-Ledderman theorem to electronic states in finite crystals. The effective-mass approximation is shown to hold for this model even for moderate fields and use is made of it to explain the complex behavior of the wave functions near the bands' edges. All in all, a clear detailed picture, although limited in scope, is presented of properties of electrons in crystals under the influence of external electric fields.