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Solution of the time dependent Schrodinger equation leading to Fowler-Nordheim field emission

Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz

2018

Abstract

We solve the time-dependent 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 Fowler-Nordheim 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 short-time evolution of the current and density.

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• Costin_Costin_Jauslin_Lebowitz_2018.pdf

LaTeX source:

• tarball: 18ccjl-1.1.tar.gz
• git repository: 18ccjl-git (the git repository contains detailed information about the changes in the paper as well as the source code for all previous versions).

Other releases

• arXiv preprint: arXiv:1808.00936.
• peer reviewed version, published in Journal of Applied Physics, volume 124, number 213104.

Related articles

• [CCJL19]: Exact solution of the 1D time-dependent Schrodinger equation for the emission of quasi-free electrons from a flat metal surface by a laser
  Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz, 2019
  (published in Journal of Physics A: Mathematical and Theoretical, volume 53, issue 36, number 365201, 2020)
  pdf, source

Talks

This work has been presented at the following conferences:

• [HF19]: Time-evolution of electron emission from a metal surface
  Mathematical Physics at the Crossings, VirginiaTech, Blacksburg, Virginia, USA, May 24 2019
  pdf


• [MSU19]: Time-evolution of electron emission from a metal surface
  Michigan State University, East Lansing, Michigan, USA, May 02 2019
  pdf


• [GT19]: Field electron emission and the Fowler-Nordheim equation
  GeorgiaTech, Atlanta, Georgia, USA, Feb 22 2019
  pdf


• [Pr18b]: Field electron emission and the Fowler-Nordheim equation
  Princeton University, New Jersey, USA, Nov 13 2018
  pdf


• [Rut18]: Solution of time-dependent Schrodinger equation for field emission
  Rutgers University, New Brunswick, New Jersey, USA, Sep 27 2018
  pdf