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arXiv:1304.1561 · Journal of Physics A Mathematical and Theoretical

A study of transit times in Dirac tunneling

Stefano De Leo

Published Apr 3, 2013• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/46/15/155306

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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.
We solve numerically the time-dependent Dirac equation for a quantum wave packet tunneling through a potential barrier. We analyze the spatial probability distribution of the transmitted wave packet in the context of the possibility of effectively superluminal peak and front velocities of the electron during tunneling. Both the Dirac and Schr\"odinger theories predict superluminal tunneling speeds. However, in contrast to the Dirac theory the Schr\"odinger equation allows a possible violation of causality. Based on an analysis of the tunneling process in full temporal and spatial resolution, we introduce an instantaneous tunneling speed that can be computed inside the potential barrier.
We have analyzed the influence of the wave packet size in the relativistic tunnelling time τ and its uncertainty �τ when it traverses a given potential barrier. The analytical expressions obtained for both magnitudes confirm that the size of the incident pulse has a significant effect on the tunnelling process. This effect is greater for short pulses, compared with the length of the barrier. For the evanescent zone, we have derived an analytical expression for τ with a good limit of validity. This expression constitutes a value tool to calculate the relativistic tunnelling time as a function of the incident wave packet with a good limit of validity. Superluminal propagation is found in this region but with a large value of the uncertainty �τ compared with the tunnelling time itself. We can conclude that the probability of superluminal propagation is practically negligible in the evanescent region. In respect to the Klein zone, we have derived an analytical expression for τ that depends on the size of the incident
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.
In the case of tunneling of relativistic particles, differently from the nonrelativistic case, a limit of "transparent" barrier can also lead to an apparent "superluminal" behavior when considering the phase time. In this limit, the restricting condition of "opaque" barrier of the nonrelativistic case is avoided, nevertheless, the very thin width of a single barrier to obtain this ?transparent? limit can result in a problem itself, for probing the effect. A combination of two successive transparent barriers can show an apparent "superluminal" behavior along a macroscopic arbitrary distance "L". Two solutions for energy E above and below the potential square barrier V are found, for both solutions there the apparent superluminal behavior is possible above a threshold of free travelling group velocity (energy) and dependent on the ratio barriers length free path as function of the ratio group velocity - speed of light.
This paper investigates the problem of a relativistic Dirac half-integer spin free particle tunneling through a rectangular quantum-mechanical barrier. If the energy difference between the barrier and the particle is positive, and the barrier width is large enough, there is proof that the tunneling is always superluminal. For antiparticle states, the tunneling may be either subluminal or superluminal instead, depending on the barrier width. These results derive from studying the tunneling time in terms of phase time. For particle states these are always negatives while for antiparticle states they are always positives, whatever the height and width of the barrier. The scattering also leads to an anomalous distortion of the Dirac spinor that tends to disappear as the particle velocity approaches the speed of light. Moreover, the phase time tends to zero, increasing the potential barrier both for particle and antiparticle states. This agrees with the interpretation of quantum tunneling that the Heisenberg uncertainty principle provides. This study’s results are innovative with respect to those available in the literature. Moreover, they show that the superluminal behaviour of particles occurs in those processes with high-energy confinement.