Abstract Resonant tunneling is a quantum phenomenon that has been extensively studied and applied in electronic devices. In this paper, the transmission coefficient and resonant energies of a particle passing through a linear potential barrier are derived and investigated. We propose that approximated values of resonant energies can be achieved by considering the energy levels of a particle confined in an infinite tilted potential well. It is found that approximation of high resonant energies is more accurate than that of low resonant energies.
Abstract Resonant tunneling is a quantum phenomenon that has been extensively studied and applied in electronic devices. In this paper, the transmission coefficient and resonant energies of a particle passing through a linear potential barrier are derived and investigated. We propose that approximated values of resonant energies can be achieved by considering the energy levels of a particle confined in an infinite tilted potential well. It is found that approximation of high resonant energies is more accurate than that of low resonant energies.
Quantum tunnelling is a phenomenon in which a particle is able to traverse a potential barrier, even if said particle possesses less energy than the barrier. This process is highly important in many modern fields of physics, such as nuclear physics and semiconductor technology. In this thesis, this phenomenon was presented through analytical study, as well as the derivation of the transmission coefficient in one and three dimensions. Of even greater interest is the phenomenon of resonant tunnelling, where particles tunnel through two or more potential barriers, leading to the formation of quantum wells and quasi-bound energy levels that result in heightened transmission probabilities at certain energies. The transfer matrix method was used in order to find an expression for the resonant tunnelling transmission coefficient in double barrier systems. In addition, a programme was developed in Python in order to investigate resonant tunnelling in different barrier systems. The programme demonstrates how the probability of transmission fluctuates with the energy of the incoming particle, as well as how it is influenced by changes to the framework of the barrier system. Furthermore, effects caused by the inclusion of asymmetry into the system are studied. Finally, the process of spin-dependent tunnelling and the related Rashba spin-orbit coupling were discussed as an extension of the resonant tunnelling phenomenon.
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.
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.
Abstract This paper examines tunnelling through potential well protected by double barriers as well as the motion of a wave packet encountering this type of potential. Various options for the left and right barriers’ penetration factors are taken into consideration. When one of the penetration factors has limiting value of 0.5, the transmission coefficients turns out to be independent of incoming energy of the particle and the width of potential well. When both penetration factors have a limiting value of 0.5, a scenario similar to resonance tunneling is achieved. Furthermore, by doing the wave packet analysis, we found that the width and height of potential barriers have a negligible impact on the probability of finding the particle inside the potential well in the limit of large penetration factors.
According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is well known. This event is called tunneling. Tunneling is a probability, and it is measured by a transmission coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In previous publications, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values. In this article, this general transmission coefficient is applied to calculate transmission coefficient for an arbitrary form external electrical field barrier (cold emission by the metal). The values obtained from our calculations and those of classical method (WKB) have been compared.
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.