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Physical review. B, Solid state

Quantum Rate Theory for Solids. III. N -Dimensional Tunneling Effects

Jerome H. Weiner, Yehuda Partom

Published Feb 15, 1970• 39 citations
Publisher

Abstract

A theory for rate processes in solids is presented which includes the effects of quantum statistics, tunneling, and the $N$-dimensional character of the problem. It utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets to describe the conditions of thermal equilibrium in an $N$-dimensional potential well. An exact solution is found for the motion of Gaussian wave packets on a second-degree $N$-dimensional potential surface. This solution is valid also when the initial principal directions of the wave packet differ from those of the potential surface so that the problem is not separable. It is used to compute the tunneling probability for a Gaussian wave packet on a second-degree $N$-dimensional potential surface containing a saddle point. This probability depends only on the wave-packet characteristics in the saddle-point direction, and this facilitates its incorporation in the rate expression. Since the initial characteristics of the wave packets of the ensemble are determined by the potential-well character, and their subsequent tunneling probability is determined by the saddle-point character, the theory includes the effect of a difference in orientation between the two sets of principal directions. For an artificial two-dimensional problem formulated specifically to study this aspect of the process, an orientation difference is found to produce large effects only at low temperatures where tunneling is important.

DOI
10.1103/physrevb.1.1533

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A theory for rate processes in solids is presented which includes the effects of quantum statistics, tunneling, and the NNN-dimensional character of the problem. It utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets to describe the conditions of thermal equilibrium in an NNN-dimensional potential well. An exact solution is found for the motion of Gaussian wave packets on a second-degree NNN-dimensional potential surface. This solution is valid also when the initial principal directions of the wave packet differ from those of the potential surface so that the problem is not separable. It is used to compute the tunneling probability for a Gaussian wave packet on a second-degree NNN-dimensional potential surface containing a saddle point. This probability depends only on the wave-packet characteristics in the saddle-point direction, and this facilitates its incorporation in the rate expression. Since the initial characteristics of the wave packets of the ensemble are determined by the potential-well character, and their subsequent tunneling probability is determined by the saddle-point character, the theory includes the effect of a difference in orientation between the two sets of principal directions. For an artificial two-dimensional problem formulated specifically to study this aspect of the process, an orientation difference is found to produce large effects only at low temperatures where tunneling is important.
The effects of tunneling are included in a one-dimensional rate theory which utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets (coherent states). An approximate tunneling formula for such wave packets is developed for a piecewise quadratic potential representing a well followed by a barrier; it is based on an exact solution for a Gaussian wave packet moving on a parabolic barrier. A new method for the numerical solution of the time-dependent Schr\"odinger equation is presented and used to check this approximate tunneling formula. The addition of tunneling to the effect of quantum statistics gives a rate formula which approximates the classical result at high-temperature levels and approaches a constant rate determined by tunneling at very low-temperature levels.
A model for interstitial diffusion is studied in which the impurity atom is regarded as a Brownian particle constrained to move in a single direction and subject to a periodic potential. A quantum mechanical theory is developed for the rate at which atoms leave a given potential well. The theory makes use of an ensemble of minimum-uncertainty wave packets to describe thermal equilibrium in the well and computes the tunneling probability through the potential barrier on the basis of the Schr\"odinger-Langevin equation in order to incorporate the effect of an interacting heat bath. The result of the interactions is to reduce the departure rate at a given temperature level, but the qualitative nature of the rate expression remains unchanged and predicts Arrhenius plots with straight-line behavior at high-temperature levels and curvature due to tunneling at low-temperature levels. The dynamics of the wave packet after it leaves a well is studied by computer solution of the Schr\"odinger-Langevin equation and it is found that the persistence of motion decreases with increasing strength of interaction with the heat bath.
Quantum tunneling effects of Gaussian wave packet in one- and two-dimensional double well potentials are investigated using quantum Liouville equation for time evolution of Wigner distribution in phase space. It is shown that a trajectory-based solution of this problem can be constructed by the entangled trajectory ensemble simulating the evolving quantum state. Quantum effects arise in this approach as a breakdown of the statistical independence of the trajectory ensemble. © 2014 Wiley Periodicals, Inc.
For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as 〈qi〉〈{q}_{i}〉〈qi​〉, 〈pi〉〈{p}_{i}〉〈pi​〉 and 〈qipj〉〈{q}_{i}{p}_{j}〉〈qi​pj​〉, 〈pipj〉〈{p}_{i}{p}_{j}〉〈pi​pj​〉, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.
A behavior of quantum states (superposition of two lowest eigenstates, Gaussian wave packet) in phase space is studied for one and two dimensional double well potential. Two dimensional potential is constructed from double well potential coupled linearly and quadratically to harmonic potential. Quantum trajectories are compared with classical ones. Preferable tunneling path in phase space is found. An influence of energy of initial Gaussian wave packet and trajectory initial condition on tunneling process is analyzed.. Coupling effect on trajectories and preferable tunneling path for two dimensional potential is studied.