Noncanonical Wannier-Stark ladders and surface state quantization in finite crystals subjected to a homogeneous electric field

In the one-particle single band approximation, which is the basis of the original Wannier result, commonly referred to as the Wannier-Stark ladder (WSL), we have extended the concept by predicting the existence of noncanonical WSLs which are a set of evenly spaced levels (in the middle of the tilted band) with noncanonical level spacing equal to the Plank constant times (1-2m#prime#/m)-1 times Bloch oscillation frequency. To observe a particular WSL, the certain voltage must be applied. The latter is related to the numbers m=3,4,hor-ellipsis and m#prime#=1,2,hor-ellipsis<m/2. We also show that, if the electrostatic energy due to applied voltage is larger than the zero-field band width, the quantization of surface localized states smoothly changes from the Airy type (at the spectrum edges) to the Wannier-Stark type with a pronounced energy interval in between, where the level spacing doubles that of canonical WSL. Analytical results are derived within the exactly solvable model of finite tilted tight-binding band. Their experimental implications and further-to-go directions are addressed to dielectric crystalline layers and superlattices, whose thickness (length) admits the direct tunneling