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arXiv:1004.3892 · Advances in Mathematical Physics

Relativistic Double Barrier Problem with Three Transmission Resonance Regions

A. D. Alhaidari, H. Bahlouli, A. Jellal

Published Jan 1, 2012• 8 citations
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Abstract

We obtain exact scattering solutions of the Dirac equation in 1 + 1 dimensions for a double square barrier vector potential. The potential bottom between the two barriers is chosen to be higher than 2mc2, whereas the top of the barriers is at least 2mc2above the bottom. The relativistic version of the conventional double barrier transmission resonances is obtained for energies within ±mc2from the height of the barriers. However, due to our judicious choice of potential configuration we also find two more (subbarrier) transmission resonance regions below the conventional one. Both are located within the two Klein energy zones and characterized by resonances that are broader than the conventional ones. The design of our double barrier so as to enable us to establish these two new subbarrier transmission resonance regions is our main finding.

DOI
10.1155/2012/762908

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We obtain exact scattering solutions of the Dirac equation in 1 + 1 dimensions for a double square barrier vector potential. The potential bottom between the two barriers is chosen to be higher than 2mc2, whereas the top of the barriers is at least 2mc2above the bottom. The relativistic version of the conventional double barrier transmission resonances is obtained for energies within ±mc2from the height of the barriers. However, due to our judicious choice of potential configuration we also find two more (subbarrier) transmission resonance regions below the conventional one. Both are located within the two Klein energy zones and characterized by resonances that are broader than the conventional ones. The design of our double barrier so as to enable us to establish these two new subbarrier transmission resonance regions is our main finding.
We obtain exact scattering solutions of the Dirac equation in 1 + 1 dimensions for a double square barrier vector potential. The potential bottom between the two barriers is chosen to be higher than 2 mc 2 , whereas the top of the barriers is at least 2 mc 2 above the bottom. The relativistic version of the conventional double barrier transmission resonances is obtained for energies within ± mc 2 from the height of the barriers. However, due to our judicious choice of potential configuration we also find two more (subbarrier) transmission resonance regions below the conventional one. Both are located within the two Klein energy zones and characterized by resonances that are broader than the conventional ones. The design of our double barrier so as to enable us to establish these two new subbarrier transmission resonance regions is our main finding.
Dirac particles can undergo perfect transmission through a sufficiently high potential barrier in the Klein zone. Although the perfect Klein tunneling (often referred to as the Klein paradox) is similar to the non-relativistic resonant transmission which occurs only when the kinetic energy exceeds the barrier, the underlying mechanism is believed to be fundamentally distinct. In this work, we show that for the relativistic double-barrier model the perfect-transmission curve can pass continuously from the above-barrier zone to the Klein zone. Additionally, in the Klein zone, perfect transmission occurs even for subcritical barrier heights, supported by both bound-state analysis and wave-packet dynamics. These findings suggest a connection between perfect Klein tunneling and resonant transmission, and provide new insights into the physical nature of the Klein paradox.
We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.
Based on the Dirac equation, the behavior of relativistic electrons which tunnel a potential barrier of height V 0 for incoming energies between V 0 and V 0 + m is studied by using the wave packet formalism. The choice of this incoming energy zone guarantees that only electrons participate in the tunneling process. In the opaque limit, as shown in a previous analysis, the transit velocity is proportional to the barrier width and for relativistic potential tends to a constant value greater than c . In this paper, a new numerical study shows a very surprising result: superluminal transmissions are also evident for thin barriers.
The two-component approach to the one-dimensional Dirac equation is applied to the Woods–Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E = − m ) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero momentum when the corresponding well supports a half-bound state at E = − m is demonstrated.
We have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential. We have used Dirac equation in the presence of the scalar and vector potentials in the quaternionic formalism of relativistic quantum mechanics to study the problem. The wave functions of different regions have been derived. Then, using the reflection coefficient, transmission coefficient, and the continuity equation, the scattering problem has been investigated in detail. It has been shown that we have faced some fluctuations in the reflection and transmission coefficients.