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Journal of Sichuan University

Research on tunneling time of delta complex potential

Xiao Qu

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

First,resolving the Schroinger equation for the boundary condition,we got the tunneling time of one and double delta complex potentials.Then,we also analyzed the effection of the related aspects to the phase time by using the Matlab and Origin.The research we got that,the phase time under the two conditions,was different for the effection of the real and imaginary part,as well as the projectile energy. Reserching the signal delta complex potentials,we got the maximum,while the double delta complex potentials having the minimum.Lastly,the two different potentials were simulated and analyzed.

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•Jan 1, 2002•0 citations
First,resolving the Schroinger equation for the boundary condition,we got the tunneling time of one and double delta complex potentials.Then,we also analyzed the effection of the related aspects to the phase time by using the Matlab and Origin.The research we got that,the phase time under the two conditions,was different for the effection of the real and imaginary part,as well as the projectile energy. Reserching the signal delta complex potentials,we got the maximum,while the double delta complex potentials having the minimum.Lastly,the two different potentials were simulated and analyzed.
The comprehension of the tunnel effect, the passage of matter waves and particles across a high potential barrier, was the most astounding of all discoveries in quantum mechanics. Research on tunnelling in semiconductors and superconductors, as well as the development of scanning tunnelling microscopy, garnered five Nobel prizes in physics. In this study, the resonant tunnelling effect of electron tunnelling through a one-dimensional double-potential barrier was found and verified by solving the time-independent Schrödinger equation and calculating the transmittance rate of the electron tunnelling through the two potential barriers by using the transmission matrix technique performed by MATLAB. According to the analysis, the transmittance rate was one under certain specific conditions (under some specific incident energy values or some specific distance between two potential barriers), which was called as resonant tunnelling. However, in this study, square potential wells were analyzed, which can be considered merely as ideal models in real life, and therefore all results are idealized. For further investigations, researchers can use more realistic parameters to set up potential wells and analyze real-life problems.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
The electron tunnelling phase time τp and dwell time τD through an associated delta potential barrier U(x) = ξδ(x) are calculated and both are in the order of 10−17−10−16s. The results show that the dependence of the phase time on the delta barrier parameter ξ can be described by the characteristic length lc = 2/meξ and the characteristic energy Ec = meξ2/2 of the delta barrier, where me is the electron mass, lc and Ec are assumed to be the effective width and height of the delta barrier with lcEc = ξ, respectively. It is found that τD reaches its maximum and τD = τp as the energy of the tunnelling electron is equal to Ec/2, i.e. as lc = DB, DB is de Broglie wave length of the electron.
Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.