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Chinese Physics B

Does the Hartman effect exist in triangular barriers

Qing-Ling 青玲 Li 李, Shuang 爽 Zheng 郑, Zhi Xiao

Published Aug 23, 2023• 4 citations
Publisher

Abstract

We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.

DOI
10.1088/1674-1056/acf2fe

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We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.
This paper explores the Larmor time and compares it for various shapes of barriers such as rectangular barrier and triangular barrier, and we arrive at consistent results between the traditionally defined Larmor time and a newly defined Larmor time. Interestingly, we have found some regular sharp jumps (discontinuities) in reflection Larmor times for a given rectangular barrier. With further investigation, it was found that the wave numbers where the Larmor time jumps correlate with those where the reflection probability reaches minimal values. Based on known results for triangular barriers, this paper also presents the Larmor times for positive and negative slope trapezoidal barriers, whose treatments are similar to the positive and negative triangular barriers, respectively, though the trapezoidal barriers exhibit more oscillatory behaviors for transmission times in comparison to their triangular counterparts. Finally, we demonstrate that the transmission and reflection Larmor times for M-shaped barriers exhibit more pronounced peaks, as expected. Also interestingly, some of the sharp jumps in reflection time are associated with sharp variations in the corresponding reflection probabilities, which is similar to the correlation mentioned earlier between the sharp jumps in reflection time and reflection probabilities in wave numbers for rectangular barriers.
We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.
We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.
An electron wave packet tunneling through a barrier has a transmission (or ``group delay'') time ${\ensuremath{\tau}}_{g}$ that, for a rectangular barrier, is commonly held to become independent of the barrier width LLL as the width increases (the McColl-Hartman effect). In the present study, it is shown that first, the McColl-Hartman effect for a rectangular barrier is dependent upon LLL as the Gamow tunneling factor $\ensuremath{\theta}(k)$ vanishes, and ${\ensuremath{\tau}}_{g}$ is only independent of LLL when $\ensuremath{\theta}(k)$ is large; and second, for a triangular barrier to model field emission, although ${\ensuremath{\tau}}_{g}$ can be large for small field, it vanishes when the energy matches the barrier height.
As was shown in quant-ph/0405028, the state of a tunneling particle can be uniquely presented as a coherent superposition of two states to describe alternative sub-processes, transmission and reflection. In this paper, on the basis of the stationary wave functions for these sub-processes, we give new definitions of the dwell times for transmission and reflection. In the case of rectangular potential barriers the dwell times are obtained explicitly. In contrast with the well-known Büttiker's dwell-time, our dwell time for transmission increases exponentially, for the under-barrier tunneling, with increasing the barrier's width. By our approach the well-known Hartman effect is rather an artifact resulted from an improper interpretation of the wave-packet tunneling and experimental data, but not a real physical effect accompanying the tunneling phenomenon.