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Вісник Національного технічного університету України Київський політехнічний інститут. Серія: Радіотехніка. Радіоапаратобудування

Порівняння традиційного та імпедансного методів моделювання квантово-розмірних структур

Є. А. Нелін, M. V. Vodolazka

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

Introduction. Comparative analysis of traditional and impedance approaches in modeling asymmetric potential quantum-mechanical barrier is fulfilled. Features of waves tunneling. The effect of waves tunneling is considered. The features of quantum-mechanical wave tunneling through asymmetric potential barrier are illustrated. Traditional approach. Quantum-mechanical wave transmission through asymmetric potential barrier on the basis of quantum-mechanical approach is explored. The expression for reflec-tion coefficient is obtained. Impedance approach. The expression of the reflection coefficient using the impedance meth-od, which is based on the concept of quantum-mechanical impedance and the theory of transmission lines is obtained. The identity of expressions received by two approaches is demonstrated. Tunnling regime. The expressions for transmission coefficient for tunneling regime in exact and approximate forms are obtained. Applicability of the approximate formula is presented. Matched overbarrier transmission. The conditions of reflectionless resonant and nonresonant overbarrier transmission are obtained. Dependences of transmission coefficient for asymmet-ric barrier that illustrate unmatched and matched overbarrier transmission are presented. Conclusions. Impedance method significantly simplifies modeling of quantum-mechanical structures in comparison with the traditional method of solving quantum-mechanical prob-lems.

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Вісник Національного технічного університету України Київський політехнічний інститут. Серія: Радіотехніка. Радіоапаратобудування•Jan 1, 2015•0 citations

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Introduction. Comparative analysis of traditional and impedance approaches in modeling asymmetric potential quantum-mechanical barrier is fulfilled. Features of waves tunneling. The effect of waves tunneling is considered. The features of quantum-mechanical wave tunneling through asymmetric potential barrier are illustrated. Traditional approach. Quantum-mechanical wave transmission through asymmetric potential barrier on the basis of quantum-mechanical approach is explored. The expression for reflec-tion coefficient is obtained. Impedance approach. The expression of the reflection coefficient using the impedance meth-od, which is based on the concept of quantum-mechanical impedance and the theory of transmission lines is obtained. The identity of expressions received by two approaches is demonstrated. Tunnling regime. The expressions for transmission coefficient for tunneling regime in exact and approximate forms are obtained. Applicability of the approximate formula is presented. Matched overbarrier transmission. The conditions of reflectionless resonant and nonresonant overbarrier transmission are obtained. Dependences of transmission coefficient for asymmet-ric barrier that illustrate unmatched and matched overbarrier transmission are presented. Conclusions. Impedance method significantly simplifies modeling of quantum-mechanical structures in comparison with the traditional method of solving quantum-mechanical prob-lems.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
The quantum tunneling effect, which is based on the quantum theory that microscopic particles behave in waves through barriers higher than their own energy, has important applications in practical technology. Microscopes such as scanning tunneling microscopes, which use quantum tunneling to generate tunnel currents, have a resolution of up to the desired level. Moreover, it can be used in biology to observe molecular structures such as macromolecules and biofilms. In this paper, the transmission coefficients of one-dimensional asymmetric unilateral barrier and asymmetric two-sided barrier under general arbitrary boundary conditions are strictly derived using the conventional method of solving the steady state Schrodinger equation by quantum mechanics and matrix. Meanwhile, the transmission coefficients of symmetric two-sided barrier are numerically studied. From the above series of methods, it is deduced that the basic working principle of resonant tunneling diode and other semiconductor electronic devices is exactly the tunnel effect. Thus, it is found that quantum tunneling effect plays an important role in life.
Introduction. In this paper the impedance models double well structures (DWS) which is based on δ-inhomogeneities and rectangular potential are developed. The models of double well potential which is based on δ-functions and rectangular depending allow to obtain analytical solutions and to investigate important features of DWS. Analogy of double well structure and coupled oscillatory circuits. The analogy of DWS and coupled oscillatory circuits are considered. It is presented comparison the Schrodinger equation for the wave function and the equation for the current circuit without loss. It is shown that DWS as the system with two states performs the function of logical operations similar bits in classical computer. Models based on the impedance δ-inhomogeneities. Two models of DWS based on the impedance δ-inhomogeneities: δ-barrier in the potential well and two δ-wells are developed. The analytical expressions for input impedance and eigenvalues are received and investigated. It is shown that characteristics of DWS with finite size and δ-inhomogeneities agree well. Eigenvalues of asymmetric double well structure. The most general case of asymmetric DWS with a rectangular potential is considered. The eigenvalues of such a structure on the basis of a generalized model of barrier structures are found. Dependences of eigenvalues of symmetric and asymmetric DWS are presented. Conclusions. Impedance models allow to obtain the analytical solutions with substantial generalization problems in comparison with known traditionally solved problems. By analysis of input impedance characteristics conditions for the eigenvalues DWS placed between environments with different wave impedance are received.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
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.
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.