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arXiv:0706.2266 · Physical Review A

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

P. Christillin, Emilio d’Emilio

Published Oct 4, 2007• 10 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.76.042104

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quant-pharXiv:2609.15287•Sep 15, 2026
The transmission of fermions of mass mmm and energy EEE through an electrostatic potential barrier of rectangular shape (i.e., supporting an infinite electric field), of height U>E+mc2U>E+m{c}^{2}U>E+mc2---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.
Dirac particles can undergo perfect transmission through a sufficiently high potential barrier in the Klein zone. Although the perfect Klein tunneling (often referred to as the Klein paradox) is similar to the non-relativistic resonant transmission which occurs only when the kinetic energy exceeds the barrier, the underlying mechanism is believed to be fundamentally distinct. In this work, we show that for the relativistic double-barrier model the perfect-transmission curve can pass continuously from the above-barrier zone to the Klein zone. Additionally, in the Klein zone, perfect transmission occurs even for subcritical barrier heights, supported by both bound-state analysis and wave-packet dynamics. These findings suggest a connection between perfect Klein tunneling and resonant transmission, and provide new insights into the physical nature of the Klein paradox.
Abstract We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height V 0 with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e. E → V 0 − mc 2, for a given V 0. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, V 0 → E + mc 2, for a given relativistic energy E( < V 0), is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when V 0 → E + mc 2 does not seem to be compatible with the existence of the impenetrable barrier.
We present a new resolution of the Klein paradox by considering the problem of electron incidence on a step potential barrier using modified Dirac equations (MDE). Contrary to the generally accepted interpretation, which is based on the Dirac equation (DE) and predicts electron tunneling through an infinitely high barrier, our calculations show that tunneling is impossible if the electron kinetic energy is less than the barrier height. We propose to investigate MDE as an alternative DE relativistic quantum mechanical equation for spin ½ particles․
The early papers by Klein, Sauter and Hund which investigate scattering off a high step potential in the context of the Dirac equation are discussed to derive the 'paradox' first obtained by Klein. The explanation of this effect in terms of electron-positron production is re-assessed. It is shown that a potential well or barrier in the Dirac equation can become supercritical and emit positrons or electrons spontaneously if the potential is strong enough. If the well or barrier is wide enough, a seemingly constant current is emitted. This phenomenon is transient whereas the tunnelling first calculated by Klein is time-independent. It is shown that tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when the barrier is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and is not necessarily connected to particle emission. The Coulomb potential is investigated and it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, as Z increases the Coulomb barrier should become increasingly transparent to positrons. This is an example of Klein tunnelling. Phenomena akin to supercritical positron emission may be studied experimentally in superfluid 3 He.
Particle penetration through a square potential barrier is studied with the Dirac equation and relativistic tunnelling occurs in an overcritical potential. Relations between this phenomenon and the Klein paradox are discussed. However, the relativistic correction to mesoscopic conduction, the Landauer formula, is negligible and relativistic tunnelling would not occur in solid-state physics owing to the barrier height never reaching the overcritical region. Relativistic tunnelling is essentially a high-energy phenomenon.
The Lorentz covariance of relativistic Dirac equations serves as a fundamental principle underlying the laws of electromagnetism across different inertial frames. Exploiting the covariance, we obtain the general solutions for Dirac fermions under both the in-plane electric E\boldsymbol{E}E and perpendicular magnetic B\boldsymbol{B}B fields, which reduce to either a magnetic or electric field in the inertial frame with drift velocity along the E×B\boldsymbol{E}\times\boldsymbol{B}E×B direction. This dichotomy defines the magnetic and electric regimes, separated by the critical field ratio E/B=vFE/B=v_\text{F}E/B=vF​ with vFv_\text{F}vF​ denoting the Fermi velocity of Dirac fermions. Using these solutions, we revisit Klein tunneling through a heterojunction with generalized electromagnetic potentials. In the magnetic regime, the transmission exhibits oscillations governed by the Fabry-Pérot interference. In the electric regime, perfect transmission occurs at normal incidence in the drifted frame. The interference phase is further analyzed in terms of the solid angles on the Bloch sphere, providing a geometric interpretation of Klein tunneling. Finally, we briefly discuss the relation between the tilting of Dirac cones and the in-plane electric field, establishing the correspondence of the undertilted and overtilted cases to the magnetic and electric regimes, respectively. Our findings reveal the manipulation of Klein tunneling by electromagnetic fields, offering a theoretical basis for designing novel electronic devices.