Time-Dependent Partition-Free Approach in Resonant Tunneling Systems

TL;DR

This paper develops a time-dependent, partition-free approach using an extended Keldysh formalism to analyze the current response in resonant tunneling systems, including both noninteracting and interacting cases, with exact and approximate solutions.

Contribution

It introduces a partition-free formalism for time-dependent tunneling systems, proving asymptotic equivalence with partitioned schemes and deriving exact formulas for steady-state currents.

Findings

Steady-state current is independent of external perturbation history.

Analytic time-dependent current in wide-band limit.

Exact steady-state current formula for interacting systems.

Abstract

An extended Keldysh formalism, well suited to properly take into account the initial correlations, is used in order to deal with the time-dependent current response of a resonant tunneling system. We use a \textit{partition-free} approach by Cini in which the whole system is in equilibrium before an external bias is switched on. No fictitious partitions are used. Besides the steady-state responses one can also calculate physical dynamical responses. In the noninteracting case we clarify under what circumstances a steady-state current develops and compare our result with the one obtained in the partitioned scheme. We prove a Theorem of asymptotic Equivalence between the two schemes for arbitrary time-dependent disturbances. We also show that the steady-state current is independent of the history of the external perturbation (Memory Loss Theorem). In the so called wide-band limit an analytic result for the time-dependent current is obtained. In the interacting case we propose an exact non-equilibrium Green function approach based on Time Dependent Density Functional Theory. The equations are no more difficult than an ordinary Mean Field treatment. We show how the scattering-state scheme by Lang follows from our formulation. An exact formula for the steady-state current of an arbitrary interacting resonant tunneling system is obtained. As an example the time-dependent current response is calculated in the Random Phase Approximation.