### Abstract

We test the consistency with which Simmons’ model can predict the local current density obtained for flat metal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missing factor of 1/2. Beside this technical issue, Simmons’ model relies on a mean-barrier approximation for electron transmission through the potential-energy barrier between the metals. In order to test Simmons’ expression for the local current density when the correct image potential energy is included, we compare the results of this expression with those provided by a transfer-matrix technique. We also consider the current densities provided by a numerical integration of the transmission probability obtained with the WKB approximation and Simmons’ mean-barrier approximation. The comparison between these different models shows that Simmons’ expression for the local current density actually provides results that are in good agreement with those provided by the transfer-matrix technique, for a range of conditions of practical interest. We show that Simmons’ model provides good results in the linear and field-emission regimes of current density versus voltage plots. It loses its applicability when the top of the potential-energy barrier drops below the Fermi level of the emitting metal.

Original language | English |
---|---|

Pages (from-to) | 63-77 |

Number of pages | 15 |

Journal | Jordan Journal of Physics |

Volume | 12 |

Issue number | 1 |

Publication status | Published - 1 Jan 2019 |

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### Keywords

- Field electron emission
- Mean-barrier approximation
- Metal-vacuum-metal junction
- Theory
- Transfer-matrix technique
- Transmission probability

### Cite this

*Jordan Journal of Physics*,

*12*(1), 63-77.

}

*Jordan Journal of Physics*, vol. 12, no. 1, pp. 63-77.

**Numerical testing by a transfer-matrix technique of Simmons' equation for the local current density in metal-vacuum-metal junctions.** / Mayer, Alexandre; Mousa , Marwan; Hagmann, Mark; Forbes, Richard.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Numerical testing by a transfer-matrix technique of Simmons' equation for the local current density in metal-vacuum-metal junctions

AU - Mayer, Alexandre

AU - Mousa , Marwan

AU - Hagmann, Mark

AU - Forbes, Richard

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We test the consistency with which Simmons’ model can predict the local current density obtained for flat metal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missing factor of 1/2. Beside this technical issue, Simmons’ model relies on a mean-barrier approximation for electron transmission through the potential-energy barrier between the metals. In order to test Simmons’ expression for the local current density when the correct image potential energy is included, we compare the results of this expression with those provided by a transfer-matrix technique. We also consider the current densities provided by a numerical integration of the transmission probability obtained with the WKB approximation and Simmons’ mean-barrier approximation. The comparison between these different models shows that Simmons’ expression for the local current density actually provides results that are in good agreement with those provided by the transfer-matrix technique, for a range of conditions of practical interest. We show that Simmons’ model provides good results in the linear and field-emission regimes of current density versus voltage plots. It loses its applicability when the top of the potential-energy barrier drops below the Fermi level of the emitting metal.

AB - We test the consistency with which Simmons’ model can predict the local current density obtained for flat metal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missing factor of 1/2. Beside this technical issue, Simmons’ model relies on a mean-barrier approximation for electron transmission through the potential-energy barrier between the metals. In order to test Simmons’ expression for the local current density when the correct image potential energy is included, we compare the results of this expression with those provided by a transfer-matrix technique. We also consider the current densities provided by a numerical integration of the transmission probability obtained with the WKB approximation and Simmons’ mean-barrier approximation. The comparison between these different models shows that Simmons’ expression for the local current density actually provides results that are in good agreement with those provided by the transfer-matrix technique, for a range of conditions of practical interest. We show that Simmons’ model provides good results in the linear and field-emission regimes of current density versus voltage plots. It loses its applicability when the top of the potential-energy barrier drops below the Fermi level of the emitting metal.

KW - electronic field emission

KW - transfer matrix

KW - theory

KW - junction

KW - electronic transport

KW - Field electron emission

KW - Mean-barrier approximation

KW - Metal-vacuum-metal junction

KW - Theory

KW - Transfer-matrix technique

KW - Transmission probability

UR - https://arxiv.org/abs/1812.07381

UR - http://www.scopus.com/inward/record.url?scp=85065918803&partnerID=8YFLogxK

M3 - Article

VL - 12

SP - 63

EP - 77

JO - Jordan Journal of Physics

JF - Jordan Journal of Physics

SN - 1994-7607

IS - 1

ER -