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Journal of Mathematical Physics

Tunneling through inhomogeneous delta barriers as diffraction and scattering

Viktor Bezák

Published Nov 1, 2007• 5 citations
Publisher

Abstract

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.

DOI
10.1063/1.2806498

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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.
Abstract Having examined free particles and particles that are confined in space by a potential energy term, we now consider the impact of a disturbance in the flat energy landscape for a free particle. By disturbance we means some kind of fixed “obstacle” which is either a positive (repelling) or negative (attractive) potential. We are interested in determining the impact on the free particle. Continuing to work mostly in one dimension, the particle described by a plane wave corresponding to momentum moving in the positive direction (a positive k−vector in the x−direction), we study elastic scattering. In one dimension, this means that we determine the probability that the particle is transmitted (continuing in the forward direction) or reflected (now moving in the backward direction.) We will also determine the nature of the solution inside the potential and in the case that the potential energy maximum is greater than the kinetic energy of the particle, we will show that the particle tunnels through the barrier. Interestingly, when we have two barriers, we can find conditions where the probability that the particle is transmitted is unity. This is the result of resonance, a feature of the wave-like nature of the particle’s wave function.
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.
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.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
A two-dimensional scattering potential represents the quantum extension of a diffractive lattice: a Dirac delta function with a modulated permeability along the y -axis. This model does not have an explicit classical analogue and quantum effects such as tunneling and diffraction play an important role. An analytical solution for the one-harmonic case is found. For the general case of an arbitrary number of harmonics a simple criterion is derived for the range of parameters where quantum chaos is permitted (but does not necessarily occur). The statistical properties of the S -matrix for the given model have been investigated. The deviations from the usual predictions for irregular scattering in the random matrix theory (RMT) framework have been found and are discussed.
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.