The role of the slope of `realistic' potential barriers in preventing relativistic tunnelling in the Klein zone

TL;DR

This paper investigates how the slope of realistic potential barriers affects relativistic tunnelling in the Klein zone, revealing that a finite electric field region exponentially suppresses tunnelling when its length exceeds the particle's Compton wavelength.

Contribution

It provides an analytical study showing the exponential suppression of tunnelling through trapezoidal barriers with finite electric field regions, extending understanding beyond idealized models.

Findings

Tunnelling rate is exponentially depressed with increasing barrier slope.

Finite electric field regions significantly suppress relativistic tunnelling.

Suppression becomes prominent when the field region exceeds the particle's Compton wavelength.

Abstract

The transmission of fermions of mass m and energy E through an electrostatic potential barrier of rectangular shape (i.e. supporting an infinite electric field), of height U> E + m - due to the many-body nature of the Dirac equation evidentiated by the Klein paradox - has been widely studied. We exploit here the analytical solution, given by Sauter for the linearly rising potential step, to show that the tunnelling rate through a more realistic trapezoidal barrier is exponentially depressed, as soon as the length of the regions supporting a finite electric field exceeds the Compton wavelenght of the particle - the latter circumstance being hardly escapable in most realistic cases.