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arXiv:2205.12588 · Journal of Physics Communications

The Dirac impenetrable barrier in the limit point of the Klein energy zone

Salvatore De Vincenzo

Published Feb 1, 2023• 0 citations

Abstract

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.

DOI: 10.1088/2399-6528/acb8ff

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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.

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.

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.

We study the solutions for a one-dimensional electrostatic potential in the Dirac equation when the incoming wave packet exhibits the Klein paradox (pair production). With a barrier potential we demonstrate the existence of multiple reflections (and transmissions). The antiparticle solutions which are necessarily localized within the barrier region create new pairs with each reflection at the potential walls. Consequently we encounter a new ``paradox'' for the barrier because successive outgoing wave amplitudes grow geometrically.

Based on the Dirac equation, the behavior of relativistic electrons which tunnel a potential barrier of height V 0 for incoming energies between V 0 and V 0 + m is studied by using the wave packet formalism. The choice of this incoming energy zone guarantees that only electrons participate in the tunneling process. In the opaque limit, as shown in a previous analysis, the transit velocity is proportional to the barrier width and for relativistic potential tends to a constant value greater than c . In this paper, a new numerical study shows a very surprising result: superluminal transmissions are also evident for thin barriers.

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.

We obtain the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein–Gordon) wave equation regime when the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation originated from the study based on non-relativistic dynamics of tunneling is reevaluated. The treatment of the problem suggests revealing insights into condensed-matter experiments using electrostatic barriers in single- and bi-layer graphenes, for which the accelerated tunneling effect deserves a more careful investigation.