Role of the slope of realistic potential barriers in preventing relativistic tunneling in the Klein zone

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 $U>E+m{c}^2$—due to the many-body nature of the Dirac equation evidentiated by the Klein paradox—has been widely studied. Here we exploit the analytical solution, given by Sauter for the linearly rising potential step, to show that the tunneling 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 wavelength of the particle—the latter circumstance being hardly escapable in most realistic cases.