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The Journal of Chemical Physics

Analysis of a dissipative resonant tunneling trap by temperature-dependent Langevin–Schrödinger equations

Uri Peskin

Published Nov 1, 2000• 17 citations
Publisher

Abstract

Resonant tunneling of electrons through a dissipative barrier is studied within a generalized spin–boson model. The model accounts for a nonlinear inelastic coupling between the electron and harmonic (bath) nuclear modes inside the barrier. In particular, we study the case in which the nuclear frequencies are similar to the decay rate of the electronic resonance states, as may appear in electron transport through narrow molecular barriers. Two cases are considered. In the case of a single responsive mode numerically exact wave-packet dynamics shows that the inelastic coupling increases the time delay of the tunneling electron in the barrier and the decay process out of the barrier is characterized by nonexponential steps. In the case of a multimode (Ohmic) bath an approximate treatment of the dynamics, based on temperature dependent nonlinear Langevin–Schrödinger equations, implies that the trapped electron wavepacket is fully solvated in the barrier due to irreversible dissipation at low temperatures. An interpretation of the solvation process in terms of the mean-field approximation is given and discussed.

DOI
10.1063/1.1313387

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Resonant tunneling of electrons through a dissipative barrier is studied within a generalized spin–boson model. The model accounts for a nonlinear inelastic coupling between the electron and harmonic (bath) nuclear modes inside the barrier. In particular, we study the case in which the nuclear frequencies are similar to the decay rate of the electronic resonance states, as may appear in electron transport through narrow molecular barriers. Two cases are considered. In the case of a single responsive mode numerically exact wave-packet dynamics shows that the inelastic coupling increases the time delay of the tunneling electron in the barrier and the decay process out of the barrier is characterized by nonexponential steps. In the case of a multimode (Ohmic) bath an approximate treatment of the dynamics, based on temperature dependent nonlinear Langevin–Schrödinger equations, implies that the trapped electron wavepacket is fully solvated in the barrier due to irreversible dissipation at low temperatures. An interpretation of the solvation process in terms of the mean-field approximation is given and discussed.
In order to better understand the trapping and response time of resonant electron wave packets in a resonant tunneling structure, we examined the time-dependent transport of a spherically symmetric Gaussian wave packet through a model double-barrier potential. Within the effective mass picture, the wave packet is projected onto the complete eigenstates of the model potential. The wave packet is initially localized to the left of the barriers and allowed to propagate through the structure. At wave packet energies corresponding to the resonant energies, the probability transmitted to the right of the barriers exhibits a steplike behavior as a function of time. A computation is made of the quantum mechanical delay time, to~tclass, where to is the packet transit time and tclass is the transit time for a classical particle. An increase in this delay time is observed when the packet energy corresponds to the energy of the resonant levels, and a comparison is made to the decay time of a particle in the first resonant state of the well. The delay time increases with increasing barrier area and decreasing packet energy width.
We have modeled the transmission of an electron wave packet through a resonant tunneling semiconductor nanostructure by solving the time-dependent Schr\"odinger equation using the finite-difference method. We have found in all cases that the passage of the electron wave packet through the tunneling barrier is accompanied by a propagation delay relative to the propagation of an undisturbed wave packet. Tunneling transport is shown to be causal, and no evidence of superluminal behavior is seen, either for resonant or for off-resonant tunneling. In the case of off-resonant tunneling, the peak of the transmitted wave packet is observed to exit a double resonant tunneling barrier before the peak of the incident wave packet enters the structure. However, these two peaks are not directly related, and the appearance of a well-formed peak in the transmitted intensity is shown to be a result of the transient behavior of the tunneling event.
A temperature-dependent Schrödinger equation is proposed for the study of quantum dynamics in systems which interact with an external bath. The derivation is based on a time-dependent self consistent field (TDSCF) approximation for a system-bath wave function. As previously shown by Miller, this approximation leads to a quantum mechanical analog to the classical generalized Langevin equation. By replacing the time-evolution of the quantum mechanical bath observables with the corresponding classical trajectories, the fluctuating force and the nonlinear friction kernel in the Langevin-Schrödinger equation become temperature-dependent where the fluctuations intensity is proportional to T. Application of the new equation to the spin-boson model shows agreement with numerically exact path integral calculations for T>0. We relate the success of the TDSCF approximation in this case to the bath-induced noise which diminishes the importance of quantum mechanical system-bath correlations.
A variational approach for the quantum dynamics of statistical mixtures is developed, which is based upon the representation of the natural states of the mixture in terms of hybrid Gaussian-based Multiconfiguration Time-Dependent Hartree (G-MCTDH) wavefunctions. The method, termed ρG-MCTDH, is combined with a treatment of dissipation and decoherence based on the nonstochastic open-system Schrödinger equations. The performance and the convergence properties of the approach are illustrated for a two-dimensional tunneling system, where the primary tunneling coordinate, represented by flexible single-particle functions, is resonantly coupled to a second harmonic mode, represented by Gaussian wave packets. The harmonic coordinate is coupled to the environment and two different processes are studied: (i) vibrational relaxation at zero temperature described by a master equation in the Lindblad form and (ii) thermalization induced by the Caldeira-Leggett master equation. In the second case, the evolution from a quantum tunneling regime to a quasistationary classical-limit distribution, driven by the heat bath, is visualized using a flux analysis.
The resonant tunneling of a particle coupled with a set of local bosons can be mapped onto the single-site model in an appropriate time scale. In this paper we investigate the process of the particle through a barrier associated with the dissipative source. The frequency and temperature-dependent tunneling probability has been evaluated explicitly in the spirit of independent boson assumption by using golden rule for transition rate. The character of sideband structure is revealed and the tunneling current with dissipation is also given.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.