# Elementary framework for cold field emission from quantum-confined, non-planar emitters

## Abstract

For suitably small field emitters, the effects of quantum confinement at the emitter tip may have a significant impact on the emitter performance and total emitted current density (ECD). Since the geometry of a quantum system uniquely determines the magnitude and distribution of its energy levels, a framework for deriving ECD equations from cold field electron emitters of arbitrary geometry and dimensionality is developed. In the interest of obtaining semi-analytical ECD equations, the framework is recast in terms of plane wave solutions to the Schrödinger equation via the use of the Jeffreys-Wentzel-Kramers-Brillouin approximation. To demonstrate the framework's consistency with our previous work and its capabilities in treating emitters with non-planar geometries, ECD equations were derived for the normally unconfined cylindrical nanowire (CNW) and normally confined (NC) CNW emitter geometries. As a function of the emitter radius, the NC CNW emitter ECD profile displayed a strong dependence on the Fermi energy and had an average ECD that exceeded the Fowler-Nordheim equation for typical values of the Fermi energy due to closely spaced, singly degenerate energy levels (excluding electron spin), comparatively large electron supply values, and the lack of a transverse, zero-point energy. Such characteristics suggest that emitters with non-planar geometries maymore »

- Authors:

- Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA and Microsystems Technology Laboratories, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)

- Publication Date:

- OSTI Identifier:
- 22402996

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Applied Physics

- Additional Journal Information:
- Journal Volume: 117; Journal Issue: 17; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-8979

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONFINEMENT; CURRENT DENSITY; CYLINDRICAL CONFIGURATION; ELECTRONS; ELECTROSTATICS; ENERGY LEVELS; FIELD EMISSION; INHOUR EQUATION; MATHEMATICAL SOLUTIONS; QUANTUM SYSTEMS; SCHROEDINGER EQUATION; SPIN; WAVE PROPAGATION; WKB APPROXIMATION

### Citation Formats

```
Patterson, A. A., E-mail: apatters@mit.edu, and Akinwande, A. I.
```*Elementary framework for cold field emission from quantum-confined, non-planar emitters*. United States: N. p., 2015.
Web. doi:10.1063/1.4919827.

```
Patterson, A. A., E-mail: apatters@mit.edu, & Akinwande, A. I.
```*Elementary framework for cold field emission from quantum-confined, non-planar emitters*. United States. doi:10.1063/1.4919827.

```
Patterson, A. A., E-mail: apatters@mit.edu, and Akinwande, A. I. Thu .
"Elementary framework for cold field emission from quantum-confined, non-planar emitters". United States. doi:10.1063/1.4919827.
```

```
@article{osti_22402996,
```

title = {Elementary framework for cold field emission from quantum-confined, non-planar emitters},

author = {Patterson, A. A., E-mail: apatters@mit.edu and Akinwande, A. I.},

abstractNote = {For suitably small field emitters, the effects of quantum confinement at the emitter tip may have a significant impact on the emitter performance and total emitted current density (ECD). Since the geometry of a quantum system uniquely determines the magnitude and distribution of its energy levels, a framework for deriving ECD equations from cold field electron emitters of arbitrary geometry and dimensionality is developed. In the interest of obtaining semi-analytical ECD equations, the framework is recast in terms of plane wave solutions to the Schrödinger equation via the use of the Jeffreys-Wentzel-Kramers-Brillouin approximation. To demonstrate the framework's consistency with our previous work and its capabilities in treating emitters with non-planar geometries, ECD equations were derived for the normally unconfined cylindrical nanowire (CNW) and normally confined (NC) CNW emitter geometries. As a function of the emitter radius, the NC CNW emitter ECD profile displayed a strong dependence on the Fermi energy and had an average ECD that exceeded the Fowler-Nordheim equation for typical values of the Fermi energy due to closely spaced, singly degenerate energy levels (excluding electron spin), comparatively large electron supply values, and the lack of a transverse, zero-point energy. Such characteristics suggest that emitters with non-planar geometries may be ideal for emission from both an electron supply and electrostatics perspective.},

doi = {10.1063/1.4919827},

journal = {Journal of Applied Physics},

issn = {0021-8979},

number = 17,

volume = 117,

place = {United States},

year = {2015},

month = {5}

}