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Physics of Particles and Nuclei Letters

Transformation-matrix method for tunnel-effect simulation

V. A. Fedirko, Sergey Vladimirovich Polyakov, Dmitry Alexeevich Zenyuk

Published Sep 1, 2011• 1 citation
Publisher

Abstract

The problem of numerically simulating steady-state scattering and the tunneling transfer of electrons was considered for a 1D potential barrier with an arbitrary shape. An effective numerical approach to solving this problem was developed on the basis of the transformation-matrix method. To test this approach, a computer program was written and applied to calculations of the tunneling processes. The convergence of the method proposed was further investigated in numerical experiments.

DOI
10.1134/s1547477111050062

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The problem of numerically simulating steady-state scattering and the tunneling transfer of electrons was considered for a 1D potential barrier with an arbitrary shape. An effective numerical approach to solving this problem was developed on the basis of the transformation-matrix method. To test this approach, a computer program was written and applied to calculations of the tunneling processes. The convergence of the method proposed was further investigated in numerical experiments.
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.
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.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
Application of the variational RRR-matrix method to one-dimensional quantum tunneling
We have applied the variational RRR-matrix method to calculate the reflection and tunneling probabilities of particles tunneling through one-dimensional potential barriers for five different types of potential profiles -- truncated linear step, truncated exponential step, truncated parabolic, bell-shaped, and Eckart. Our variational results for the transmission and reflection coefficients are compared with exact analytical results and results obtained from other numerical methods. We find that our results are in good agreement with them. We conclude that the variational RRR-matrix method is a simple, non-iterative, and effective method to solve one-dimensional quantum tunneling problems.