Arbitrary Potential Barrier Research Articles

Penetration of arbitrary double potential barriers with probability unity: Implications for testing the existence of a minimum length

Quantum tunneling across double potential barriers is studied. With the assumption that the real space is a continuum, it is rigorously proved that large barriers of arbitrary shapes can be penetrated by low-energy particles with a probability of unity, i.e., realization of resonant tunneling (RT), by simply tuning the interbarrier spacing. The results are demonstrated by tunneling of electrons and protons, in which resonant and sequential tunneling are distinguished. The critical dependence of tunneling probabilities on the barrier positions not only demonstrates the crucial role of phase factors but also points to the possibility of ultrahigh accuracy measurements near resonance. By contrast, the existence of a nonzero minimum length puts upper bounds on the barrier size and particle mass, beyond which effective RT ceases. A scheme is suggested for dealing with the practical difficulties arising from the delocalization of particle position due to the uncertainty principle. This work opens a possible avenue for experimental tests of the existence of a minimum length based on atomic systems. Published by the American Physical Society 2024

Tunnelling of Hermite-Gaussian wavepackets

We examine the tunnelling process of incident Hermite-Gaussian wavepackets which feature several nodes and peaks. An analytic formula for the transmitted wave function is developed which applies to arbitrary incident wave functions and potential barriers with known transmission amplitudes. This formula permits the accurate numerical calculation of the transmitted wave packet, even when the transmission probability is extremely small. For square potential barriers, the shape of the transmitted packet, its relationship to the shape of the incident packet, the momentum of the transmitted packet, and the duration of the tunnelling process are investigated. The shape of the reconstructed transmitted wave packet is also studied for the smooth repulsive Pöschl-Teller potential.

Calculation a New Transmission Coefficient of Tunneling for an Arbitrary Potential Barrier and Application to Alpha Decay

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 known to be. This event is called tunneling. Tunneling is a probability, and it is measured by a transition 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 this paper, 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.

Numerical simulation of tunneling through arbitrary potential barriers applied on MIM and MIIM rectenna diodes

With the continuous miniaturization of electronic devices, quantum-mechanical effects such as tunneling become more effective in many device applications. In this paper, a numerical simulation tool is developed under a MATLAB environment to calculate the tunneling probability and current through an arbitrary potential barrier comparing three different numerical techniques: the finite difference method, transfer matrix method, and transmission line method. For benchmarking, the tool is applied to many case studies such as the rectangular single barrier, rectangular double barrier, and continuous bell-shaped potential barrier, each compared to analytical solutions and giving the dependence of the error on the number of mesh points. In addition, a thorough study of the J − V characteristics of MIM and MIIM diodes, used as rectifiers for rectenna solar cells, is presented and simulations are compared to experimental results showing satisfactory agreement. On the undergraduate level, the tool provides a deeper insight for students to compare numerical techniques used to solve various tunneling problems and helps students to choose a suitable technique for a certain application.

Andreev reflection enhancement in semiconductor-superconductor structures

We develop a theoretical approach for modeling a wide range of semiconductor-superconductor structures with arbitrary potential barriers and a spatially dependent superconducting order parameter. We demonstrate asymmetry in the conductance spectrum as a result of a Schottky barrier shape. We further show that the Andreev reflection process can be significantly enhanced through resonant tunneling with appropriate barrier configuration, which can incorporate the Schottky barrier as a contributing component of the device. Moreover, we show that resonant tunneling can be achieved in superlattice structures as well. These theoretically demonstrated effects along with our modeling approach enable much more efficient Cooper pair injection into semiconductor-superconductor structures, including superconducting optoelectronic devices.

Josephson-like currents in graphene for arbitrary time-dependent potential barriers

From the exact solution of the Dirac-Weyl equation we find unusual currents j y running in y-direction parallel to a time-dependent scalar potential barrier W(x, t) placed upon a monolayer of graphene, even for vanishing momentum component p y . In their sine-like dependence on the phase difference of wave functions, describing left and right moving Dirac fermions, these currents resemble Josephson currents in superconductors, including the occurance of Shapiro steps at certain frequencies of potential oscillations. The Josephson-like currents are calculated for several specific time-dependent barriers. A novel type of resonance is discovered when, accounting for the Fermi velocity, temporal and spatial frequencies match.

Calculation of penetration probability across an arbitrary potential barrier in fusion reactions

The penetration probability across an arbitrary nucleus-nucleus potential barrier is studied by using transfer matrix approach, including continuous variations of effective mass. The calculated penetration probabilities for reaction ${}^{16}$O+${}^{208}$Pb are closed to those of WKB and modified Numerov calculations based on the Bass potential. With the transfer matrix approach, the influence of the effective mass on the penetration probability are simultaneously investigated. The calculations imply that the effective mass is an important quantity for studying the hindrance of fusion cross sections of some reactions at deep sub-barrier energies. The fusion cross sections of heavy-ion fusion reactions and the resonance phenomena due to double-hump barriers are also studied with this approach.

A transfer matrix approach to electron transport in graphene through arbitrary electric and magnetic potential barriers

A transfer matrix method is presented for solving the scattering problem for the quasi-one-dimensional massless Dirac equation applied to graphene in the presence of an arbitrary inhomogeneous electric and perpendicular magnetic field. It is shown that parabolic cylindrical functions, which have previously been used in the literature, become inaccurate at high incident energies and low magnetic fields. A series expansion technique is presented to circumvent this problem. An alternate method using asymptotic expressions is also discussed and the relative merits of the two methods are compared.

Matrix method for simulating the tunneling transfer

We develop an efficient approach for computer simulation of stationary scattering and tunneling transfer across an arbitrary one-dimensional potential barrier. New algorithms and programs were worked out and tested and the convergence of the method in question was examined.

Absence of Anomalous Tunneling of Bogoliubov Excitations for Arbitrary Potential Barrier under the Critical Condensate Current

We derive the exact solution of low energy limit of Bogoliubov equations for excitations of Bose-Einstein condensate in the presence of arbitrary potential barrier and maximum current of condensate. Using this solution, we give the explicit expression for the transmission coefficient against the potential barrier, which shows partial transmission in the low energy limit. The wavefunctions of excitations in the low energy limit do not coincide with that of the condensate. The absence of the perfect transmission in the critical current state originates from local enhancement of density fluctuations around the potential barrier.

Explicit expression for the reflection and transmission probabilities through an arbitrary potential barrier

Without solving the Schrödinger equation, exact and general expressions of the reflection and transmission probabilities for a quantum particle through an arbitrary potential barrier are presented by using the analytical transfer matrix method. It is seen that in addition to the parameters of the ambience and both the start and the final point of the barrier, the unique dependence on the formulae is the total phase shift accumulated by the mainwaves and the subwaves.

Universal potential-barrier penetration by initially confined wave packets

The dynamics of an initially sharp-boundary wavepacket in the presence of an arbitrary potential barrier are investigated. It is shown that the penetration through the barrier is universal in the sense that it depends only on the values of the wavefunction and its derivatives at the boundary. The dependence on the derivatives vanishes at long distances from the barrier, where the dynamics are governed solely by the initial value of the wavefunction at the boundary.

Quantum-Classical Transition in Dissipative Double-Well Systems – A Numerical Study by a New Monte Carlo Scheme –

Thermodynamics of dissipative quantum systems with double-well potentials is studied by the path-integral Monte Carlo (PIMC) method without truncation to the two-state model. For efficient simulation at low temperatures, we develop a new scheme of local update based on approximate decomposition of the Boltzmann weight to the Gaussian distributions. Localization transition induced by ohmic dissipation is clarified numerically for arbitrary potential barriers.

Analytical formula of the transmission probabilities across arbitrary potential barriers

The analytical transfer matrix method is applied to the evaluation of the tunnelling in the semiconductor heterostructures. An analytical formula of the transmission probabilities across arbitrary shaped potential barriers which includes the variations of the effective masses has been developed in this paper. Various potential barrier structures are analysed and the results are of high precision. Tunnelling resonance is obtained for the double-barrier structure. We can correlate the energy location of the peaks of the transmission probability with the bound state energies of the included potential wells through our analytical transfer matrix method.

Entangled electron current through finite size normal-superconductor tunneling structures

We investigate theoretically the simultaneous tunneling of two electrons from a superconductor into a normal metal at low temperatures and voltages. Such an emission process is shown to be equivalent to the Andreev reflection of an incident hole. We obtain a local tunneling Hamiltonian that permits to investigate transport through interfaces of arbitrary geometry and potential barrier shapes. We prove that the bilinear momentum dependence of the low-energy tunneling matrix element translates into a real space Hamiltonian involving the normal derivatives of the electron fields in each electrode. The angular distribution of the electron current as it is emitted into the normal metal is analyzed for various experimental setups. We show that, in a full three-dimensional problem, the neglect of the momentum dependence of tunneling causes a violation of unitarity and leads to the wrong thermodynamic (broad interface) limit. More importantly for current research on quantum information devices, in the case of an interface made of two narrow tunneling contacts separated by a distance $r$, the assumption of momentum-independent hopping yields a nonlocally entangled electron current that decays with a prefactor proportional to $r^{-2}$ instead of the correct $r^{-4}$.

To the developement of the transfer matrix formalism

The present work is devoted to the development of some mathematical aspects of the transfer matrix theory. It is shown that a transfer matrix for an arbitrary potential barrier can be obtained directly by solving the corresponding differential equation. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Differential Equation for the Transfer Matrix

An electron propagating through a crystal toward an interface can either reflect or transmit. The determination of its transmission and reflection probabilities represents an actual task in such fields as nanoelectronics, magnetoelectronics, or spin electronics. Within the framework of the effective mass approximation the problem can be reduced to the tunneling of the quantum particle through one-dimensional potential barrier. The tunneling process can be described by means of the transfer matrix, which contains all the information about the energetic dependence of the transmission and reflection coefficients. In the present work the differential equation for the transfer matrix of the arbitrary potential barrier is derived. The method proposed represents an alternative way of the calculation of the transfer matrix.

Tunneling through arbitrary potential barriers and the apparent barrier height

A simple matrix formalism is presented that allows the tunneling current between two planar electrodes with an arbitrary barrier potential to be calculated conveniently. This formalism is a straightforward extension of the usual textbook example for tunneling through a rectangular barrier and can be easily implemented on a personal computer. A first principles derivation of the tunneling current based on free-electron electrodes is given that is useful for the teaching of solid-state physics. Examples showing the utility of the method are the transmission through a tunnel barrier modified by the image potential and tunneling via intermediate states (resonances). These issues are interesting topics in scanning tunneling microscopy where both phenomena are believed to contribute to the image formation.

THE CONDITIONAL TUNNELING TIME FOR REFLECTION USING THE WKB WAVE-FUNCTION

We derive an expression for the conditional time for the reflection of a wave from an arbitrary potential barrier using the WKB wavefunction in the barrier region. Our result indicates that the conditional times for transmission and reflection are equal for a symmetric barrier within the validity of the WKB approach.

Larmor precession and tunneling time of a relativistic neutral spinning particle through an arbitrary potential barrier

The Larmor precession of a relativistic neutral spin-1/2 particle in a uniform constant magnetic field confined to the region of a one-dimensional arbitrary potential barrier is investigated. The spin precession serves as a clock to measure the time spent by a quantum particle traversing a potential barrier. With the help of general spin coherent state it is explicitly shown that the precession time is equal to the dwell time.