73.40.Gk The solution of the one-dimensional Dirac equation is examined for the symmetrical double deltafunction potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.
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:
The solution of the one-dimensional Schrödinger wave equation is presented for the potential-energy function that describes a double delta-barrier under the application of a constant electrical field perpendicular to it. The transfer matrix technique is employed to determine the transmission coefficient in an analytical form. Some attributes of the transmission coefficient are established. The transmission coefficient is shown to exhibit maxima and minima, the conditions for maxima and minima in the transmission coefficient are discussed. The current-voltage characteristic of the biased double delta-barrier is calculated numerically. It is found to exhibit the same oscillatory behaviour as the transmission coefficient when the voltage applied to the double delta-barrier is increased. The width of the double delta-barrier is shown to modulate the peak-to-valley ratio in the current-voltage characteristic.
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