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

## Abstract

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{minus}2m{sup {prime}}/m){sup {minus}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{sup {prime}}=1,2,{hor_ellipsis}{lt}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.

- Authors:

- Publication Date:

- Sponsoring Org.:
- (US)

- OSTI Identifier:
- 40203609

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 63; Journal Issue: 23; Other Information: DOI: 10.1103/PhysRevB.63.235410; Othernumber: PRBMDO000063000023235410000001; 050123PRB; PBD: 15 Jun 2001; Journal ID: ISSN 0163-1829

- Publisher:
- The American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; DIELECTRIC MATERIALS; ELECTRIC FIELDS; ELECTROSTATICS; OSCILLATIONS; QUANTIZATION; SUPERLATTICES; THICKNESS; TUNNELING

### Citation Formats

```
Onipko, Alexander, and Malysheva, Lyuba.
```*Noncanonical Wannier-Stark ladders and surface state quantization in finite crystals subjected to a homogeneous electric field*. United States: N. p., 2001.
Web. doi:10.1103/PhysRevB.63.235410.

```
Onipko, Alexander, & Malysheva, Lyuba.
```*Noncanonical Wannier-Stark ladders and surface state quantization in finite crystals subjected to a homogeneous electric field*. United States. doi:10.1103/PhysRevB.63.235410.

```
Onipko, Alexander, and Malysheva, Lyuba. Fri .
"Noncanonical Wannier-Stark ladders and surface state quantization in finite crystals subjected to a homogeneous electric field". United States. doi:10.1103/PhysRevB.63.235410.
```

```
@article{osti_40203609,
```

title = {Noncanonical Wannier-Stark ladders and surface state quantization in finite crystals subjected to a homogeneous electric field},

author = {Onipko, Alexander and Malysheva, Lyuba},

abstractNote = {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{minus}2m{sup {prime}}/m){sup {minus}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{sup {prime}}=1,2,{hor_ellipsis}{lt}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.},

doi = {10.1103/PhysRevB.63.235410},

journal = {Physical Review B},

issn = {0163-1829},

number = 23,

volume = 63,

place = {United States},

year = {2001},

month = {6}

}