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International Symposium on Advanced Topics in Electrical Engineering

Wavelet solution of the time independent Schrödinger equation for a rectangular potential barrier

Simona Mihaela Bibic, Emilia-Simona Malureanu

Published May 1, 2013• 3 citations
Publisher

Abstract

Electronics industry is a major user of Metal-Insulator-Metal (MIM) structures. When the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM structures, we are dealing with the Fowler-Nordheim field emission. In MIM structures we have to calculate the currents densities that go through. Unfortunately, the calculation methods currently used lead to values that get predictions of emission current density too low. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schrödinger equation. One of the oldest and most efficient methods of solving the Schrödinger equation is the WKB method. This paper proposes the wavelet method for solving the 1D Schrödinger equation, that is: determining the wave function using the harmonic multiresolution analysis.

DOI
10.1109/atee.2013.6563472

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Electronics industry is a major user of Metal-Insulator-Metal (MIM) structures. When the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM structures, we are dealing with the Fowler-Nordheim field emission. In MIM structures we have to calculate the currents densities that go through. Unfortunately, the calculation methods currently used lead to values that get predictions of emission current density too low. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schrödinger equation. One of the oldest and most efficient methods of solving the Schrödinger equation is the WKB method. This paper proposes the wavelet method for solving the 1D Schrödinger equation, that is: determining the wave function using the harmonic multiresolution analysis.
Quantum tunneling has many important applications in electronics. In thin layers structures, when the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM or MOS structures, we are dealing with the Fowler-Nordheim field emission. In these structures we have to calculate the currents densities that go through. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schrödinger equation. The calculation methods currently used lead to values that get predictions of emission current density too low. The transmission coefficient is calculated by wavelet approximation and the results are validated experimentally on a lateral MOM tunnel junction.
We have used several different methods to solve the one-dimensional, time-dependent Schrödinger equation for a sinusiodally modulated barrier. Analytical solutions are given for the case in which the time-dependent part of the potential has several different forms, but is independent of position. For more general fields, Floquet's theorem is used to write the wavefunction as a summation of components which have different energies as a result of the absorption and emission of modulation quanta. A system of coupled ordinary differential equations is obtained, which is then solved numerically using shooting methods. The examples show a resonance in which the tunneling current is markedly increased. For square barriers, this resonance occurs when the particles absorbing modulation quanta are above the barrier, and the length of the barrier is an integer multiple of one-half the de Broglie wavelength. The existence of the resonance is confirmed by asymptotic solutions for large and small frequencies. Examples suggest that it may be possible to make a microwave power amplifier by illuminating a field emitter array with an amplitude-modulated laser operating near the new resonance. © 1995 John Wiley & Sons, Inc.
Quantum tunneling is a phenomenon where particles have a probability of penetrating a potential barrier despite having total energy llower than the barrier height. This study analyzes the tunneling effect by solving the one-dimensional (1D) Schrodinger equation using the finite difference method to obtain the wave function evolution for various potential barrier configurations. The solution is implemented in a Graphical User Interface (GUI) MATLAB to facilitate analysis and visualization, allowing users to interactively adjust potential parameters, energy, and other conditions. Simualation results demonstratehow transmission probability depends on energy, height and width of potential barrier. This GUI provides an intiuitive tool for exploring quantun tunneling, making it valuable for both education and research in quantum physics.
A new numerical algorithm for tunneling is presented. This algorithm can be used to analyse an onedimensional\npotential barrier with an arbitrary profile. It is able to calculate not only the transmission coefficient\naccurately, but also the phases of the transmitted and reflected waves. Using the Schrodinger equation and plane waves,\nwe introduce a tunneling matrix to relate the transmitted and incident wave functions. Thus the transmission coefficient\nand phase can be calculated from the tunneling matrix.To test the accuracy of this algorithm,we calculate the transmission\ncoefficients and phases for a double -barrier (resonant tunneling) and a triangular barrier (field emission), and\nmake comparisous with corresponding analytical values. It shows that this algorithm is fast, accurate and efficient.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.