Solution of the time dependent Schr\"odinger equation leading to Fowler-Nordheim field emission

TL;DR

This paper solves the time-dependent Schrödinger equation for electron emission under an external electric field, showing the wavefunction approaches Fowler-Nordheim tunneling with specific decay behavior and femtosecond time scales.

Contribution

It provides an exact solution to the time-dependent Schrödinger equation for field emission, revealing the asymptotic approach to Fowler-Nordheim tunneling and associated decay dynamics.

Findings

Wavefunction approaches Fowler-Nordheim form as t→∞

Deviation decays as t^{-3/2}

Relevant time scales are femtoseconds

Abstract

We solve the time-dependent Schr\"odinger 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.