# Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections

## Abstract

When particles are injected according to the Fowler-Nordheim (FN) field emission equation, the transmitted current density will transition to the space charge limited (SCL) current density, with increasing applied diode voltage. The actual transmitted current density is the so-called SCL-FN current density. In this work, Barbour's analytic solution for the SCL-FN current density is modified with consideration of injection velocity and also geometric effects, by solving the advanced FN equation with the effective field enhancement factor, the energy conservation equation with an initial velocity term, and Poisson's equation simultaneously. The solution is also extended to the relativistic regime where similar transition process is found. This solution has been verified using particle-in-cell simulation with varying diode voltage, electron injection velocity, and field enhancement factor.

- Authors:

- Department of Nuclear Engineering, University of California, Berkeley, California 94720-1730 (United States)
- NanoScience Simulation Laboratory, Department of Physics, Fu Jen Catholic University, Taipei County 24205, Taiwan (China)

- Publication Date:

- OSTI Identifier:
- 21120270

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Plasmas; Journal Volume: 15; Journal Issue: 4; Other Information: DOI: 10.1063/1.2907365; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; ANALYTICAL SOLUTION; CURRENT DENSITY; ELECTRIC POTENTIAL; ELECTRON BEAM INJECTION; ELECTRONS; FIELD EMISSION; PLASMA; POISSON EQUATION; RELATIVISTIC RANGE; SIMULATION; SPACE CHARGE; VELOCITY

### Citation Formats

```
Feng, Y., Verboncoeur, J. P., and Lin, M. C.
```*Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections*. United States: N. p., 2008.
Web. doi:10.1063/1.2907365.

```
Feng, Y., Verboncoeur, J. P., & Lin, M. C.
```*Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections*. United States. doi:10.1063/1.2907365.

```
Feng, Y., Verboncoeur, J. P., and Lin, M. C. Tue .
"Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections". United States.
doi:10.1063/1.2907365.
```

```
@article{osti_21120270,
```

title = {Solution for space charge limited field emission current densities with injection velocity and geometric effects corrections},

author = {Feng, Y. and Verboncoeur, J. P. and Lin, M. C.},

abstractNote = {When particles are injected according to the Fowler-Nordheim (FN) field emission equation, the transmitted current density will transition to the space charge limited (SCL) current density, with increasing applied diode voltage. The actual transmitted current density is the so-called SCL-FN current density. In this work, Barbour's analytic solution for the SCL-FN current density is modified with consideration of injection velocity and also geometric effects, by solving the advanced FN equation with the effective field enhancement factor, the energy conservation equation with an initial velocity term, and Poisson's equation simultaneously. The solution is also extended to the relativistic regime where similar transition process is found. This solution has been verified using particle-in-cell simulation with varying diode voltage, electron injection velocity, and field enhancement factor.},

doi = {10.1063/1.2907365},

journal = {Physics of Plasmas},

number = 4,

volume = 15,

place = {United States},

year = {Tue Apr 15 00:00:00 EDT 2008},

month = {Tue Apr 15 00:00:00 EDT 2008}

}