Bloch electrons in finite crystals in the presence of a uniform electric field

Taking the crystal potentials to be of the Mathieu type, we have investigated the energy spectrum and the wave functions of Bloch electrons in finite crystals in the presence of a uniform electric field. Our main purpose is to examine the various aspects related to the (Wannier) Stark ladders (SL) in finite crystals. The treatment depends on solving the Schrödinger equation numerically; Gelarkin's approach, in combination with Householder's tridiagonalization procedure and the $Q-L$ algorithm, is used for the purpose. Our investigation takes into account both the infinite-wall boundary conditions and the periodic boundary conditions, and it encompasses a wide range of parameters such as the crystal potential strength, the electric field, and the length of the crystal. The results obtained by us clarify the circumstances which would control the occurrence of SL and throw considerable light on the nature of the wave functions corresponding to the SL states.