Triple-, double-, and fractionally-spaced Wannier–Stark ladders

The concept of Wannier–Stark (WS) quantization literally refers to infinite crystals or bulk electron states. In real finite crystals, the bulk and edge or surface states always coexist. Moreover, when the surface states are already considerably localized due to the presence of a constant electric field, the WS ladder (WSL) and hence the WS localization may not yet come into play, at least, in the canonical form of WS quantization Ej=const±jϵ, j=0,1,2,… (Ej is the one-electron energy, and the parameter ϵ is associated with the electric field strength). We show that at certain voltages Vm, m=3,4,…, which are lower than V∞ needed for the WS band opening (the sub-WS regime), the mid-spectrum levels can form triple-, double-, and fractional-spaced WSLs, where El=±l[1+2/(m−2)]ϵm. It is also found that in the WS regime, the quantization of surface localized states (sls) smoothly changes from the Airy type (at the spectrum edges) to the WS type with a pronounced energy interval in between, where the level spacing doubles that of canonical WSL. Possible experimental manifestations of predicted effects are also outlined.