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Doklady of the National Academy of Sciences of Belarus

ТУННЕЛИРОВАНИЕ ЧЕРЕЗ ГЛАДКИЙ ПАРАБОЛИЧЕСКИЙ ДВОЙНОЙ БАРЬЕР

Ágnes Baran, V. V. Kudryashov

Published Oct 5, 2017• 0 citations
OpenAlex

Abstract

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.

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ТУННЕЛИРОВАНИЕ ЧЕРЕЗ ГЛАДКИЙ ПАРАБОЛИЧЕСКИЙ БАРЬЕР КОНЕЧНОЙ ВЫСОТЫ

V. V. Kudryashov, Ágnes Baran

The smooth barrier of finite height and variable shape is constructed by means of joining the central inverted parabolic potential and two side parabolic potentials. The problem of tunneling through this barrier is solved exactly. The dependence of the transmission coefficient on energy is presented. The real and imaginary components of wave functions are shown.

Doklady of the National Academy of Sciences of Belarus•Jan 6, 2017•0 citations

Smooth double barriers in quantum mechanics

Avik Dutt, Sayan Kar

Quantum mechanical tunneling across smooth double barrier potentials is modeled using Gaussian functions and is analyzed numerically and by using the WKB approximation. The transmission probability, resonances as a function of incident particle energy, and their dependence on the barrier parameters are obtained for various cases. We also discuss the tunneling time, for which we obtain generalizations of the known results for rectangular barriers.

arXiv:1008.1640•American Journal of Physics•Nov 18, 2010•27 citations

Resonant Delay in Tunnelling through Smooth Double and Triple Barriers

L. V. Chebotarev

The energy dependence of the delay time associated with tunnelling through smooth (symmetric) double and triple barriers is investigated. Sharp resonances in the delay are found to mirror resonances in the transmission rate. In the strong-localization limit, particles with off-resonant energies do not perceive the inner potential relief and tunnel through the given double, or triple, barrier in much the same way as they would tunnel through a simple, unstructured barrier. Verification of the derived relations through a high-precision numerical solution of the Schrödinger equation confirms the quality of the obtained expressions in wide energy ranges up to the barrier top.

physica status solidi (b)•Jul 1, 1998•2 citations

Analysis of One-dimensional Double-potential Barrier Structures Resonant Tunnelling Based on Transmission Matrices

Zihan Zhang

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.

Highlights in Science Engineering and Technology•Jun 11, 2024•0 citations

Transmission coefficient in double barrier quantum tunnelling effect

Zhongzheng Duan, Wenxi Luo, Xiaohan Xu

At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.

Theoretical and Natural Science•Dec 19, 2023•0 citations

Single-point potentials with total resonant tunneling

Alexander V. Zolotaryuk

Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double-barrier structure.

arXiv:1303.4162•Physical Review A•May 15, 2013•15 citations
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.
The energy dependence of the delay time associated with tunnelling through smooth (symmetric) double and triple barriers is investigated. Sharp resonances in the delay are found to mirror resonances in the transmission rate. In the strong-localization limit, particles with off-resonant energies do not perceive the inner potential relief and tunnel through the given double, or triple, barrier in much the same way as they would tunnel through a simple, unstructured barrier. Verification of the derived relations through a high-precision numerical solution of the Schrödinger equation confirms the quality of the obtained expressions in wide energy ranges up to the barrier top.
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.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double-barrier structure.