Transmission through an Asymmetrical Double Delta Barrier

Abstract

The solution of the Schro È dinger equation is examined for the potential that is formed of two delta functions of unequal strength. An analytical expression for the transmission coefficient is derived from the solution. The transmission coefficient is shown to exhibit relative maxima and minima. Moreover, it is proved that the transmission coefficient in its maxima is larger and in its minima is smaller than the transmission coefficient for the corresponding single delta barrier.A great deal of interest has been paid to the study of the transmission through onedimensional double-barrier structures ([1] and papers cited therein). With the recent progress of the fabrication technique, the transmission through such structures has become a physical reality. Modern computers now allow problems of the transmission through realistic potentials to be solved numerically with a relative ease. However, an analytical solution of the transmission problem is still of an instructive value, since it enables one to get an insight into phenomena which typically take place. Thus, the need for simple models of double-barrier structures has not decreased. A double-barrier structure is often modelled by a rectangular double barrier. Yamamoto [2] analytically derived a strict expression for the transmission coefficient and the resonance condition for a symmetrical rectangular double barrier. Zhao et al. [3] studied the transmission through an asymmetrical rectangular double barrier. However, that analytical expression for the transmission coefficient appears to be very intricate. There exists a more simple archetype of a double barrier structure, namely a double delta barrier. The transmission through a symmetrical double delta barrier was thoroughly studied by Galindo and Pascual [4]. In this work, the study of the transmission through an asymmetrical double delta barrier is carried out to present briefly the attributes of the transmission coefficient.For this purpose, the Schro È dinger equation is solved for a particle of mass m moving in the following potential:where dx is the Dirac delta function, g 1 and g 2 are the strengths of the two delta functions, respectively. Evidently, its solution is a wave function wx that has the form of plane waves moving from the left to the right and vice versa:

Transmission Properties of the One-Dimensional Array of Delta Potentials

“…The problem of one-dimensional delta-function potential array has been studied in previus works 13,14,15,16,17 . In reference 13 they use a field-theoretic approach to obtain the transfer matrix in terms of a propagator.…”

“…The problem of one-dimensional delta-function potential array has been studied in previus works 13,14,15,16,17 . In reference 13 they use a field-theoretic approach to obtain the transfer matrix in terms of a propagator. In references 14,15,16,17 the case of two delta-function potentials is studied using a quantum-mechanical approach. We use a similar aproach to develop a convenient way to deal with the problem, based in a transfer matrix methodology.…”

On Relativistic Transmissions through a Symmetrical Pair of Delta-Barriers or Delta-Wells

“…In other words, the transmission coefficient for certain symmetrical double delta-barriers was found to be equal to unity at all energies and thus at low, so-called, non-relativistic energies, too. However, the non-relativistic treatment of the transmissions through the symmetrical double delta-barrier that is based on the solution of the Schrö dinger equation is not at all involved with a situation of this sort [2,3]. In general, the relativistic treatment of the solid state phenomena is expected to introduce at most some small though in some cases significant corrections to the non-relativistic results.…”

“…2, 196-208 (2002) Ideal transmissions through double barrier structures as well as through double well structures have recently attracted a great deal of interest because of their applications in solid state devices. In the non-relativistic setting, an archetype of such structures is also represented by the double delta-function potential in the one-dimensional system [1][2][3]. A rectangular barrier of large height and small width affords a gross realisation of the delta-barrier.…”

On the Transmission Coefficient for the Double δ'-Function Potential

“…The transmission through a symmetrical double delta barrier was thoroughly studied by Galindo and Pascual [3]. The study of the transmission through an asymmetrical double delta barrier was carried out by the author [4]. The δ-fυπctiοn potential belongs to a family of point interactions in one--dimensional quantum mechanics [5].…”