Current through a multi-lead junction caused by applied bias with arbitrary time-dependence

TL;DR

This paper derives a comprehensive analytical formula for the time-dependent current in multi-terminal nanojunctions under arbitrary biases using NEGF, capturing both transient and steady-state behaviors with numerical validation.

Contribution

It provides a closed-form, partition-free NEGF-based expression for current in multi-lead systems with arbitrary time-dependent biases, extending previous results to transient regimes.

Findings

Analytical formula reduces to known results in special limits.

Numerical simulations show current 'ringing' oscillations at specific frequencies.

Dependence of oscillations on system parameters like bias amplitude and temperature.

Abstract

We apply the Nonequilibrium Green's Function (NEGF) formalism to the problem of a multi-terminal nanojunction subject to an arbitrary time-dependent bias. In particular, we show that taking a generic one-particle system Hamiltonian within the wide band limit approximation (WBLA), it is possible to obtain a closed analytical expression for the current in each lead. Our formula reduces to the well-known result of Jauho et. al. [doi:10.1103/PhysRevB.50.5528] in the limit where the switch-on time is taken to the remote past, and to the result of Tuovinen et. al. [doi:10.1088/1742-6596/427/1/012014] when the bias is maintained at a constant value after the switch-on. As we use a partition-free approach, our formula contains both the long-time current and transient effects due to the sudden switch-on of the bias. Numerical calculations performed for the simple case of a single-level quantum dot coupled to two leads are performed for a sinusoidally-varying bias. At certain frequencies of the driving bias, we observe `ringing' oscillations of the current, whose dependence on the dot level, level width, oscillation amplitude and temperature is also investigated.