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physica status solidi (b)

Coefficients of Abovebarrier and Subbarrier Transmission for a Family of One-Dimensional Structured Barriers

Maria Tashkova, Stoil Donev

Published Mar 1, 1997• 1 citation
Publisher

Abstract

We use the transfer-matrix approach to solve the one-dimensional Schrödinger equation for a particular family of structured barriers and to compute in an explicit form the abovebarrier and subbarrier transmission coefficients as functions of barrier parameters. For rectangular barriers the reflectionless conditions are solved and the admissible energy expressions are obtained. For “opaque” barriers the influence of the different barrier factors on the tunneling coefficients is explicitly shown.

DOI
10.1002/1521-3951(199703)200:1<75::aid-pssb75>3.0.co;2-6

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We use the transfer-matrix approach to solve the one-dimensional Schrödinger equation for a particular family of structured barriers and to compute in an explicit form the abovebarrier and subbarrier transmission coefficients as functions of barrier parameters. For rectangular barriers the reflectionless conditions are solved and the admissible energy expressions are obtained. For “opaque” barriers the influence of the different barrier factors on the tunneling coefficients is explicitly shown.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.
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.