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

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+mc(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.