Tunneling Dynamics and Time Delay in Electron Transport through Time-Dependent Barriers with Finite-Bandwidth Reservoirs

TL;DR

This paper introduces an analytical approach to study electron tunneling through time-dependent barriers with finite-bandwidth reservoirs, revealing finite tunneling times in non-Markovian environments and connecting barrier modulation to measurable phase shifts.

Contribution

It provides a simple, Floquet-free analytical framework for time-dependent tunneling, relating barrier modulation to phase shifts and tunneling time, especially in finite-bandwidth reservoirs.

Findings

Finite tunneling delay in finite-bandwidth reservoirs.

Vanishing tunneling time in wide-band (Markovian) limit.

Analytical expressions for time-dependent tunneling current.

Abstract

We present a transparent and analytically tractable approach to the problem of time-dependent electron transport through tunneling barriers. Using the Single-Electron Approach, we study a model system composed of a time-dependent tunneling barrier coupled to two reservoirs of finite bandwidth. Avoiding Floquet expansions, we derive simple expressions for the time-dependent tunneling current in both adiabatic and non-adiabatic regimes. Our formulation, based on the tunneling Hamiltonian framework, relates barrier modulation to measurable phase shifts in the steady-state current, offering a physically intuitive definition of the tunneling (or traversal) time. Remarkably, in the Markovian limit (wide-band reservoirs), we recover the well-known result of vanishing tunneling time. In contrast, for finite-bandwidth leads, we predict a finite time delay given by the inverse bandwidth. Our findings provide a robust foundation for understanding tunneling dynamics in non-Markovian environments and may serve as a benchmark for experimental investigations involving tunable band structures.