Solution of the time dependent Schrodinger equation leading to FowlerNordheim field emission
Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz
2018
Abstract
We solve the timedependent Schrodinger equation describing the emission of electrons from a metal surface by an external electric field \(E\), turned on at \(t=0\). Starting with a wave function \(\psi(x,0)\), representing a generalized eigenfunction when \(E=0\), we find \(\psi(x,t)\) and show that it approaches, as \(t\to\infty\), the FowlerNordheim tunneling wavefunction \(\psi_E\). The deviation of \(\psi\) from \(\psi_E\) decays asymptotically as a power law \(t^{\frac32}\). The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds. We plot the shorttime evolution of the current and density.
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This work has been presented at the following conferences:

[HF19]: Timeevolution of electron emission from a metal surface
Mathematical Physics at the Crossings, VirginiaTech, Virginia, USA, May 24 2019
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[MSU19]: Timeevolution of electron emission from a metal surface
Michigan State University, East Lansing, Michigan, USA, May 02 2019
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[GT19]: Field electron emission and the FowlerNordheim equation
GeorgiaTech, Atlanta, Georgia, USA, Feb 22 2019
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[Pr18b]: Field electron emission and the FowlerNordheim equation
Princeton University, New Jersey, USA, Nov 13 2018
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source

[Rut18]: Solution of timedependent Schrodinger equation for field emission
Rutgers University, New Jersey, USA, Sep 27 2018
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source