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Physica Scripta

Relativistic phase time in delta-potential scattering of Dirac fermions in 1D and 2D systems

Jorge Villavicencio, Alberto Hernández-Maldonado, I. K. A. ROMERO MALDONADO

Published Nov 26, 2025• 0 citations
Publisher

Abstract

Abstract We present a unified analytical study of the transmission phase time for Dirac particles scattered by delta-function potentials in one and two dimensions. For the one-dimensional case, we obtain a closed-form expression for the transmission relativistic phase time and show that it coincides with both the reflection phase time and the self-interference delay, in agreement with general predictions for symmetric, zero-width scatterers. The resulting delay alternates periodically between positive and negative values as the coupling strength increases, and converges to the known non-relativistic expression in the appropriate limit. This delay-advance pattern can be interpreted through the spectral structure of the Dirac Hamiltonian, where electron and positron bound-state sectors correlate with time advances and delays, respectively, reflecting the underlying nature of the delta interaction. In the two-dimensional case, we derive an exact expression for the transmission phase time in graphene and examine its angular and coupling dependence. The delay exhibits a periodic, butterfly-like interference pattern governed by resonant transmission and Klein tunnelling, vanishing at normal incidence and at integer values of the reduced coupling strength. These features reveal how angular interference and relativistic dynamics shape temporal signatures of quantum transport in Dirac materials. The closed-form analytical expressions for the relativistic phase time in one dimension and in two dimensions (graphene), which constitute the main results of this work. Our results provide analytic insight into phase-time phenomena in relativistic systems and provide compact benchmarks for time-resolved quantum transport in narrow-gap and Dirac-like media.

DOI
10.1088/1402-4896/ae24b7

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Abstract We present a unified analytical study of the transmission phase time for Dirac particles scattered by delta-function potentials in one and two dimensions. For the one-dimensional case, we obtain a closed-form expression for the transmission relativistic phase time and show that it coincides with both the reflection phase time and the self-interference delay, in agreement with general predictions for symmetric, zero-width scatterers. The resulting delay alternates periodically between positive and negative values as the coupling strength increases, and converges to the known non-relativistic expression in the appropriate limit. This delay-advance pattern can be interpreted through the spectral structure of the Dirac Hamiltonian, where electron and positron bound-state sectors correlate with time advances and delays, respectively, reflecting the underlying nature of the delta interaction. In the two-dimensional case, we derive an exact expression for the transmission phase time in graphene and examine its angular and coupling dependence. The delay exhibits a periodic, butterfly-like interference pattern governed by resonant transmission and Klein tunnelling, vanishing at normal incidence and at integer values of the reduced coupling strength. These features reveal how angular interference and relativistic dynamics shape temporal signatures of quantum transport in Dirac materials. The closed-form analytical expressions for the relativistic phase time in one dimension and in two dimensions (graphene), which constitute the main results of this work. Our results provide analytic insight into phase-time phenomena in relativistic systems and provide compact benchmarks for time-resolved quantum transport in narrow-gap and Dirac-like media.
For the relativistic tunneling effect described using Dirac's equation, Winful et al. [Phys. Rev. A 70, 052112 (2004)] presented the deduction of a general result that allows for the determination of the phase time (group delay) as the sum of the particle dwell time inside a potential barrier and of the self-interference delay associated with the incident and reflected wave functions interaction. In this paper, a mathematical model is derived through a construction analogous to the proposal mentioned above, but based on an alternative representation for Dirac's equation. This representation is similar to the one introduced by Ajaib [Found. Phys. 45, 1586 (2015)]. Thus, from the application of this model in the study of the tunneling effect in the absence of an external magnetic field, the influence of spin on the tunneling times is described. More specifically, the tunneling time is obtained as the sum of the dwell times inside the potential barrier for particles with spin up and spin down and the self-interaction time associated with the incident and reflected wave functions for particles with spin up.
We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the nnn--nnn--nnn, nnn--ppp--nnn, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.
In one-dimensional scattering, the portion of particles that is transmitted in general vanishes as the kinetic energy of the incident particles approaches zero. Based upon published results from the literature, the present work shows that for a potential consisting of two Dirac δ functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential. The underlying reasons are shown to be related to the structure of the bound levels of the potential. It is shown that for a potential with a finite range but of general shape the reflection coefficient in general tends to unity near threshold, which is what is usually observed. However, in the presence of a bound level at the onset of the continuum, a finite portion of the incident particles is transmitted at threshold for a potential with a general shape and, furthermore, it can be shown that if the potential V is symmetric with V(x)=V(−x) then resonance transmission is observed, i.e., the reflection coefficient vanishes at threshold.
Abstract We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components. These are non-singular short-range potential functions (not necessarily analytic like a square well) such that they are well represented by their matrix elements in a finite subset of a square integrable basis set. This set is chosen to support a tridiagonal symmetric matrix representation for the free Dirac operator. We derive the transmission and reflection coefficients. This work will be followed by another where we apply the theory to obtain scattering information for different potential coupling modes including those with evidence of Klein tunneling and of supercritical resonances.
We present an analytical framework for studying quantum tunneling through multiple Dirac delta potential barriers in one dimension. Using the transfer matrix method, we derive a closed-form expression for the total transfer matrix of a system composed of NNN equally spaced delta barriers. In a systematic manner, a compact expression is obtained for the first element of the transfer matrix, based on triangular numbers. This, in turn, allows us to compute the transmission coefficient exactly as a function of the number of barriers. The proposed method successfully reproduces well-known results for one and two barriers and efficiently captures complex interference effects for larger values of NNN, such as N=4N=4N=4.