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Physics Letters A

Some special features of relativistic tunnelling through multi-barrier systems with δ-function barriers

Chandana Roy

Published Jun 1, 1994• 7 citations
Publisher
DOI
10.1016/0375-9601(94)90015-9

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In this paper, we examine transmission through one-dimensional potential barriers that are piecewise constant. The transfer matrix approach is adopted, and a new formula is derived for multiplying long matrix sequences that not only leads to an elegant representation of the wave function but also results in much faster computation than earlier methods. The proposed method covers a broad spectrum of potentials, of which multi-barrier systems are special cases. The procedure is illustrated with a finite lattice of nonuniform rectangular barriers—non-uniformity being a novel feature, as the uniform case has been solved exactly by Griffiths and Steinke. For the nonuniform multi-barrier problem, the intervening wells strongly influence the transmission probability. Surprisingly, we find that the wells act “individually,” i.e., their influence is a function only of their width and is independent of their exact locations in a multi-barrier system. This finding leads to an observation that we have termed the “alias effect.” The exact solutions are supplemented with asymptotic formulas.
We present a general theory of the quantum-mechanical tunneling through an inhomogeneous planar delta barrier. The delta barrier means that the potential energy is proportional to the Dirac delta function in the direction perpendicular to the plane of the barrier. The inhomogeneity of the delta barrier means that the delta function is multiplied by a non-negative function, different from a constant, of lateral coordinates. We assume that this function, being the barrier strength, may be arbitrary. To exemplify our theory, we consider delta barriers that are made inhomogeneous by embedding equal circular windows in the barrier plane, assuming that the barriers are homogeneous both inside and outside the windows. (The value of the barrier strength is taken higher by the side of the windows than inside the windows.) With the inhomogeneous delta barriers of this kind, we show how the tunneling theory is related to the theory of diffraction and scattering. Although our general solution of the problem is new in the context of the tunneling theory, it is essentially based on a method which was used by Kirchhoff in the 19th century in the theory of waves.
The analytic nature of the transmission coefficient for a δ-function barrier makes it a useful tool to examine a variety of technologically important applications, such as photoemission from semiconductors with an alkali coating, the examination of tunneling times for wave packets incident on a barrier, and for parameterizing tunneling through the narrow barrier of a normal-superconducting point contact. The analytic model of a δ-function barrier inside a confining well is extended to the finite height and width rectangular barrier (a delta-function sequence). Methods to exactly evaluate the eigenstates are given and their dependencies are examined. The time evolution of a superposition of the lowest eigenstates is considered for barriers having comparable Gamow tunneling factors so as to quantify the impact of barrier height and shape on time evolution in a simple and exact system and, therefore, serve as a proxy for tunneling time. Last, density profiles and associated quantum potentials are examined for coupled wells to show changes induced by weaker and wider barriers.