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arXiv:1704.08890 · Physics Letters A

Transmission singularities in resonant electron tunneling through double complex potential barrier

Nikola Opačak, Vitomir Milanović, J. Radovanović

Published Sep 12, 2017• 2 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2017.09.012

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The comprehension of the tunnel effect, the passage of matter waves and particles across a high potential barrier, was the most astounding of all discoveries in quantum mechanics. Research on tunnelling in semiconductors and superconductors, as well as the development of scanning tunnelling microscopy, garnered five Nobel prizes in physics. In this study, the resonant tunnelling effect of electron tunnelling through a one-dimensional double-potential barrier was found and verified by solving the time-independent Schrödinger equation and calculating the transmittance rate of the electron tunnelling through the two potential barriers by using the transmission matrix technique performed by MATLAB. According to the analysis, the transmittance rate was one under certain specific conditions (under some specific incident energy values or some specific distance between two potential barriers), which was called as resonant tunnelling. However, in this study, square potential wells were analyzed, which can be considered merely as ideal models in real life, and therefore all results are idealized. For further investigations, researchers can use more realistic parameters to set up potential wells and analyze real-life problems.
The exact description of tunneling is given for a smooth symmetric double barrier which is constructed with the help of both parabolic and inverted parabolic potentials. The analytical expression for transmission coefficient is found. The resonant tunneling condition is obtained. The dependence of transmission coefficient on incident particle energy is presented for different values of double barrier parameters. It is established that the number of resonances increases with growing the width of barriers and the distance between barriers. The continuous wave functions are expressed in terms of the confluent hypergeometric functions. The real and imaginary components of wave functions are shown at the resonance values of energy. The proposed smooth parabolic potential extends a very limited list of exactly solvable models that describe tunneling through double barriers. The variable shape of the considered double barrier gives the supplementary possibilities to simulate tunneling processes.
A transfer-matrix technique is formulated to treat the scattering of a particle incident on a piecewise constant potential and interacting with an oscillatory field. A study of tunneling through a single barrier shows in the transmission probabilities, for absorption or emission of quanta from the field, replicas below the barrier of the overbarrier field-free transmissivity, with its resonant structure. In the case of resonant tunneling through a double barrier, an appropriate choice of parameters gives evidence for a change in the regime, with inhibition of transmission at a resonance energy of the intermediate well when the period of the field approaches the lifetime of the resonant state. The technique can be extended to treat scattering of a particle by more general potentials in the presence of an oscillatory perturbation. As an illustrative example, the transmissivity as a function of a bias potential in a double-barrier structure is calculated.