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Physical Review A

Transfer-matrix formulation of field-assisted tunneling

C. Pérez del Valle, R. Lefèbvre, Osman Atabek

Published May 1, 1999• 30 citations
Publisher

Abstract

A transfer-matrix technique is formulated to treat the scattering of a particle incident on a piecewise constant potential and interacting with an oscillatory field. A study of tunneling through a single barrier shows in the transmission probabilities, for absorption or emission of quanta from the field, replicas below the barrier of the overbarrier field-free transmissivity, with its resonant structure. In the case of resonant tunneling through a double barrier, an appropriate choice of parameters gives evidence for a change in the regime, with inhibition of transmission at a resonance energy of the intermediate well when the period of the field approaches the lifetime of the resonant state. The technique can be extended to treat scattering of a particle by more general potentials in the presence of an oscillatory perturbation. As an illustrative example, the transmissivity as a function of a bias potential in a double-barrier structure is calculated.

DOI
10.1103/physreva.59.3701

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A transfer-matrix technique is formulated to treat the scattering of a particle incident on a piecewise constant potential and interacting with an oscillatory field. A study of tunneling through a single barrier shows in the transmission probabilities, for absorption or emission of quanta from the field, replicas below the barrier of the overbarrier field-free transmissivity, with its resonant structure. In the case of resonant tunneling through a double barrier, an appropriate choice of parameters gives evidence for a change in the regime, with inhibition of transmission at a resonance energy of the intermediate well when the period of the field approaches the lifetime of the resonant state. The technique can be extended to treat scattering of a particle by more general potentials in the presence of an oscillatory perturbation. As an illustrative example, the transmissivity as a function of a bias potential in a double-barrier structure is calculated.
We examine the transmissivity through a double-barrier potential with two resonances present in the intermediate well. The particle interacts with an oscillating field. Satellites are produced when the particle needs to emit or absorb one or more quanta of the field to reach the resonant energy which favours transmissivity. We show that as the frequency of the field goes from an off-resonance to the resonant situation, where there is matching with the energy separation of the well levels, the discontinuities responsible for the energy exchanges are first the most external ones, but finally the most internal ones. We also conclude that, even in the presence of an oscillating field, the propagation within the barriers is well accounted for by free waves.
A simle numerical method of calculating transmission coefficients across arbitrary potentials is presented by using the combination of a transfer matrix procedure and the solution of the wave functions in arbitrary potentials.This method is applied to a parabolic potential and a double barrier potential in an electric field.The transmission coefficients calculated are in agreement with those obtained using a multistep potential approximation.The wave functions at the resonant states are also given for the two examples.
Proton transfer plays a key role in the applications of advanced energy materials as well as in the functionalities of biological systems. In this work, based on the transfer matrix method, we study the quantum effects of proton transfer in a series of one-dimensional (1D) model potentials and numerically calculate the quantum probability of transferring across single and double barriers (wells). In the case of single barriers, when the incident energies of protons are above the barrier height, the quantum oscillations in the transmission coefficients depend on the geometric shape of the barriers. It is found that atomic resonant tunneling (ART) not only presents in the rectangular single well and rectangular double barriers as expected, but also exists in the other types of potential wells and double barriers. For hetero-structured double barriers, there is no resonant tunneling in the classical forbidden zone, i.e., in the case when the incident energy (E i) is lower than the barrier height (E b). Furthermore, we have provided generalized analysis on the characteristics of transmission coefficients of hetero-structured rectangular double barriers.
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 tunnelling is a phenomenon in which a particle is able to traverse a potential barrier, even if said particle possesses less energy than the barrier. This process is highly important in many modern fields of physics, such as nuclear physics and semiconductor technology. In this thesis, this phenomenon was presented through analytical study, as well as the derivation of the transmission coefficient in one and three dimensions. Of even greater interest is the phenomenon of resonant tunnelling, where particles tunnel through two or more potential barriers, leading to the formation of quantum wells and quasi-bound energy levels that result in heightened transmission probabilities at certain energies. The transfer matrix method was used in order to find an expression for the resonant tunnelling transmission coefficient in double barrier systems. In addition, a programme was developed in Python in order to investigate resonant tunnelling in different barrier systems. The programme demonstrates how the probability of transmission fluctuates with the energy of the incoming particle, as well as how it is influenced by changes to the framework of the barrier system. Furthermore, effects caused by the inclusion of asymmetry into the system are studied. Finally, the process of spin-dependent tunnelling and the related Rashba spin-orbit coupling were discussed as an extension of the resonant tunnelling phenomenon.
Based on the transmission formula derived from the analytical transfer matrix method,two cases of the transmission probabilities of the position-dependent-mass particle through the free potential field are calculated.Combined with the point canonical transformation,we conclude that the transmission probabilities are closely connected to the structure of their effective potential.If the effective potential is a well,the transmission resonance peaks lies exactly at the eigen-energies.If the effective potential is a barrier,the transmission probabilities are very low at the below-barrier energies and oscillating around 1 at the beyond-barrier energies.