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Israel Journal of Chemistry

Thermal resonant tunneling rates by a generalized flux averaging method

Maytal Caspary Toroker, Lihu Berman, Uri Peskin

Published Dec 1, 2002• 2 citations
Publisher

Abstract

Abstract The calculation of the thermal rate constant as a time integral over flux‐flux correlation functions is a challenging task when the potential energy along the reaction coordinate cannot be associated with a distinctive single barrier. In the case of resonant tunneling through a double barrier potential, the calculations may become formidable due to the population of long‐lived resonance states and the corresponding long time‐decay of the flux‐flux correlation functions. The flux averaging method was introduced recently in order to circumvent this problem in cases where the long time dynamics is due to a single resonance state with the longest lifetime in the system. In this work we generalize the method for calculations of thermal resonant‐tunneling rates in systems of many resonances, where the long time‐decay is accompanied by an internal dynamics within the quasi‐bound system. This extra complication is handled by additional averaging of flux‐flux correlation funcation over the time period of the internal dynamics. The result is an exact expression for the rate constant in terms of a linear combination of time integrals over flux‐flux correlation functions, which reaches its asymptotic time limit in a short (direct scattering) time, regardless of the long time‐decay of the flux‐flux correlation functions. This is derived for an analytic model system, and demonstrated in a numerical simulation of resonant tunneling through a double barrier potential.

DOI
10.1560/822k-5lt2-acbb-hc15

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Abstract The calculation of the thermal rate constant as a time integral over flux‐flux correlation functions is a challenging task when the potential energy along the reaction coordinate cannot be associated with a distinctive single barrier. In the case of resonant tunneling through a double barrier potential, the calculations may become formidable due to the population of long‐lived resonance states and the corresponding long time‐decay of the flux‐flux correlation functions. The flux averaging method was introduced recently in order to circumvent this problem in cases where the long time dynamics is due to a single resonance state with the longest lifetime in the system. In this work we generalize the method for calculations of thermal resonant‐tunneling rates in systems of many resonances, where the long time‐decay is accompanied by an internal dynamics within the quasi‐bound system. This extra complication is handled by additional averaging of flux‐flux correlation funcation over the time period of the internal dynamics. The result is an exact expression for the rate constant in terms of a linear combination of time integrals over flux‐flux correlation functions, which reaches its asymptotic time limit in a short (direct scattering) time, regardless of the long time‐decay of the flux‐flux correlation functions. This is derived for an analytic model system, and demonstrated in a numerical simulation of resonant tunneling through a double barrier potential.
The tunneling time of a particle through a given barrier is commonly defined in terms of “internal clocks” that effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Büttiker and Landauer ( Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a “clock independent” intrinsic time scale (the resonance lifetime) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
We study the real time dynamics of a dissipative quantum system in a metastable state which may decay by crossing a potential barrier. Starting from an initial state where the system is in thermal equilibrium on one side of the barrier, the time evolution of the density matrix is evaluated analytically in the semiclassical approximation for coordinates near the barrier top. In a region about a critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ large quantum fluctuations render the harmonic approximation of the potential insufficient and anharmonicities become essential. Accounting for non-Gaussian fluctuation modes, we show that the density matrix approaches a quasistationary state with a constant flux across the barrier. This extends our earlier results [Phys. Rev. E 51, 4267 (1995)] on the quantum generalization of the Kramers flux state to the region about Tc{\mathrm{T}}_{\mathrm{c}}Tc​. By matching the flux state onto the equilibrium state on one side of the barrier, we determine the decay rate out of the metastable state. The rate constant shows a changeover from thermally activated decay to quantum tunneling for temperatures below Tc{\mathrm{T}}_{\mathrm{c}}Tc​.
Abstract The tunneling dynamics of a particle moving in a bistable potential with fluctuating barrier is studied. For barriers fluctuating randomly in time we show by exact numerical calculation the significant effect of barrier fluctuation on the tunneling behavior of the particle. At nonzero temperatures the computed tunneling rate constant passes through a maximum when plotted against fluctuation frequency. The resonant frequency (at which the maximum appears) slowly decreases with increase in temperature and attains a constant value at higher temperature and it increases linearly with increase in barrier height of the potential. Another important observation is that in presence of barrier fluctuation the dependence of tunneling rate constant on temperature is strongly guided by the barrier fluctuation frequency. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 280–285, 2003
The tunneling time of particle through given barrier is commonly defined in terms of "internal clocks" which effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Buttiker and Landauer (Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a "clock independent" intrinsic time scale (the resonance life time) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
We consider an analytic solution of the time-dependent Schr\"odinger equation with the initial condition \ensuremath{\psi}(x,0)=exp(ikx) along -\ensuremath{\infty}0 to investigate the time evolution for x>0 of the wave function in a double-barrier resonant structure at resonance. For typical parameters of the structure we find that the single-resonance approximation is valid from a few tenths of the corresponding lifetime onward. Very short times require the contribution of many far away resonances. The buildup time along the internal region takes a few lifetimes. At birth the transmitted wave front is blurred; however, for long times it becomes well defined and moves with classical velocity yielding a delay time \ensuremath{\tau}\ensuremath{\approx}2\ensuremath{\hbar}/\ensuremath{\Gamma} as in the stationary-phase treatment.
Abstract We analyze the tunneling‐time of a spatial wave packet which traverses a resonant structure by means of the presence‐time formalism. With this method we can evaluate the finite size effects of the corresponding energy probability distribution in the tunneling‐time. We show that these effects are not negligible and reduce the tunneling‐time by 25%. An expression for the resonant tunneling‐time is derived, 3ħ/2Γr, where Γr is the width of the resonance. We confirm numerically this result for two resonant structures: a superlattice with n cells and square‐barrier periodic potential, and an asymmetrical rectangular double‐barrier potential. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)