Signatures of Wannier-Stark and surface states in electron tunneling and related phenomena: Electron transmission through a tilted band

Abstract

Predicted by Wannier in 1960 band states quantization in a constant electric field, $E_n$ = const $\pm n {\cal E}$, where $n =0$, 1, 2, ..., and ${\cal E}$ is proportional to the strength of electric field [this kind of spectrum is commonly referred as the Wannier-Stark ladder (WSL)], implies that the probability of tunneling through a tilted band should have ${\cal E}$ spaced peaks, at least, under the weak coupling of the band states to the source and drain electrodes. It has been shown, however, (Phys. Rev. B {\bf 63}, ..., 2001), that the appearance of the canonical WSL is preceded by WSLs with other level spacing, namely, ${\cal E}_{m'/m}/(1-2m'/m)$, where $m$ and $m'< m/2$ are positive integers specifying certain applied voltage. Here we show that canonical and noncanonical WSLs, in addition to different peak spacing in the transmission spectrum, have other pronouncedly distinctive features. As an example, for the former, the peak and valley tunneling probability decays exponentially with the increase of applied voltage. The corresponding exponents are given by the sum and difference of two Fowler-Nordheim-type exponents implying an anomalous increase of the peak-to-valley ratio. These and other signatures of extended states (es), surface localized states (sls), and Wannier-Stark (WS) states in the through tilted band transmission spectrum are discussed on the basis of (derived for the first time) nearly exact explicit expressions of es-, sls-, and WS states assisted tunneling probability.