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arXiv:2602.23650 · Physical Review A

Perfect transmission of a Dirac particle in one-dimension double square barrier

Xu Zhang, Qiang Gu

quant-phcond-mat.mes-hallhep-thSubmitted Feb 27, 2026• Announced Mar 2, 2026• 2 citations

Abstract

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.

Journal reference: Physical Review A 113, 032208 (2026)
DOI: 10.1103/3j51-vct5
Comments: 9 pages, 5 figures

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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.

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.

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 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.

We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.

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.