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

Alexandre Mayer, Marwan Mousa , Mark Hagmann, Richard Forbes

Research output: Contribution to journalArticle

Abstract

We test the consistency with which Simmons’ model can predict the local current density obtained for flatmetal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missingfactor of 1/2. Besides this technical issue, Simmons’ model relies on a mean-barrier approximation forelectron 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 comparethe results of this expression with those provided by a transfer-matrix technique. This technique is knownto provide numerically exact solutions of Schrodinger’s equation for this barrier model. We also considerthe current densities provided by a numerical integration of the transmission probability obtained with theWKB approximation and Simmons’ mean-barrier approximation. The comparison between these differentmodels shows that Simmons’ expression for the local current density actually provides results that are ingood agreement with those provided by the transfer-matrix technique, for a range of conditions of practicalinterest. We show that Simmons’ model provides good results in the linear and field-emission regimes ofcurrent density versus voltage plots. It loses its applicability when the top of the potential-energy barrierdrops below the Fermi level of the emitting metal.
Original languageEnglish
Pages (from-to)63-77
Number of pages10
JournalJordan Journal of Physics
Volume12
Issue number1
Publication statusPublished - 2019

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potential energy
current density
vacuum
metals
approximation
numerical integration
field emission
plots
electric potential

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    title = "Numerical testing by a transfer-matrix technique of Simmons' equation for the local current density in metal-vacuum-metal junctions",
    abstract = "We test the consistency with which Simmons’ model can predict the local current density obtained for flatmetal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missingfactor of 1/2. Besides this technical issue, Simmons’ model relies on a mean-barrier approximation forelectron 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 comparethe results of this expression with those provided by a transfer-matrix technique. This technique is knownto provide numerically exact solutions of Schrodinger’s equation for this barrier model. We also considerthe current densities provided by a numerical integration of the transmission probability obtained with theWKB approximation and Simmons’ mean-barrier approximation. The comparison between these differentmodels shows that Simmons’ expression for the local current density actually provides results that are ingood agreement with those provided by the transfer-matrix technique, for a range of conditions of practicalinterest. We show that Simmons’ model provides good results in the linear and field-emission regimes ofcurrent density versus voltage plots. It loses its applicability when the top of the potential-energy barrierdrops below the Fermi level of the emitting metal.",
    keywords = "electronic field emission, transfer matrix, theory, junction, electronic transport",
    author = "Alexandre Mayer and Marwan Mousa and Mark Hagmann and Richard Forbes",
    year = "2019",
    language = "English",
    volume = "12",
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    journal = "Jordan Journal of Physics",
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    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.

    In: Jordan Journal of Physics, Vol. 12, No. 1, 2019, p. 63-77.

    Research output: Contribution to journalArticle

    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

    Y1 - 2019

    N2 - We test the consistency with which Simmons’ model can predict the local current density obtained for flatmetal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missingfactor of 1/2. Besides this technical issue, Simmons’ model relies on a mean-barrier approximation forelectron 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 comparethe results of this expression with those provided by a transfer-matrix technique. This technique is knownto provide numerically exact solutions of Schrodinger’s equation for this barrier model. We also considerthe current densities provided by a numerical integration of the transmission probability obtained with theWKB approximation and Simmons’ mean-barrier approximation. The comparison between these differentmodels shows that Simmons’ expression for the local current density actually provides results that are ingood agreement with those provided by the transfer-matrix technique, for a range of conditions of practicalinterest. We show that Simmons’ model provides good results in the linear and field-emission regimes ofcurrent density versus voltage plots. It loses its applicability when the top of the potential-energy barrierdrops 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 flatmetal-vacuum-metal junctions. The image potential energy used in Simmons’ original papers had a missingfactor of 1/2. Besides this technical issue, Simmons’ model relies on a mean-barrier approximation forelectron 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 comparethe results of this expression with those provided by a transfer-matrix technique. This technique is knownto provide numerically exact solutions of Schrodinger’s equation for this barrier model. We also considerthe current densities provided by a numerical integration of the transmission probability obtained with theWKB approximation and Simmons’ mean-barrier approximation. The comparison between these differentmodels shows that Simmons’ expression for the local current density actually provides results that are ingood agreement with those provided by the transfer-matrix technique, for a range of conditions of practicalinterest. We show that Simmons’ model provides good results in the linear and field-emission regimes ofcurrent density versus voltage plots. It loses its applicability when the top of the potential-energy barrierdrops below the Fermi level of the emitting metal.

    KW - electronic field emission

    KW - transfer matrix

    KW - theory

    KW - junction

    KW - electronic transport

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

    M3 - Article

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    JO - Jordan Journal of Physics

    JF - Jordan Journal of Physics

    SN - 1994-7607

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    ER -