Emergence of Landauer transport from quantum dynamics: A model Hamiltonian approach

Abstract

The Landauer expression for computing current-voltage characteristics in nanoscale devices is efficient but not suited to transient phenomena and a time-dependent current because it is applicable only when the charge carriers transition into a steady flux after an external perturbation. In this article, we construct a very general expression for time-dependent current in an electrode-molecule-electrode arrangement. Utilizing a model Hamiltonian (consisting of the subsystem energy levels and their electronic coupling terms), we propagate the Schrödinger wave function equation to numerically compute the time-dependent population in the individual subsystems. The current in each electrode (defined in terms of the rate of change of the corresponding population) has two components, one due to the charges originating from the same electrode and the other due to the charges initially residing at the other electrode. We derive an analytical expression for the first component and illustrate that it agrees reasonably with its numerical counterpart at early times. Exploiting the unitary evolution of a wavefunction, we construct a more general Landauer style formula and illustrate the emergence of Landauer transport from our simulations without the assumption of time-independent charge flow. Our generalized Landauer formula is valid at all times for models beyond the wide-band limit, non-uniform electrode density of states and for time and energy-dependent electronic coupling between the subsystems. Subsequently, we investigate the ingredients in our model that regulate the onset time scale of this steady state. We compare the performance of our general current expression with the Landauer current for time-dependent electronic coupling. Finally, we comment on the applicability of the Landauer formula to compute hot-electron current arising upon plasmon decoherence.

Numerical Simulation of Non-Equilibrium Stationary Currents Through Nano-scale One-Dimensional Chains

“…2 a), we show the current across the sample's center as a function of time, for both initial setups, with or without disorder. In the partitioned setup, the results are similar to the ones obtained earlier for a multi-level system by Pal et al [9], where a quasisteady state Landauer current is found after a short transient. The Landauer current was obtained by writing the transmission function in Eq.…”

Landauer transport as a quasisteady state on finite chains under unitary quantum dynamics

“…The quasisteady-state lasts until a recurrence time t r ≈ 2v −1 F L l , where current inversions start happening. Besides being related to the inverse spacing of the energy levels in the system 21 , this recurrence time may also be interpreted as the time taken by a Fermi-level electron to leave the sample and return to it, by traveling back and forth inside a lead. This conclusion was seen to be independent of the central sample's features, as long as the leads are much larger than it.…”

“…( 11), in the eigenbasis of the unperturbed Hamiltonian. Thus, we obtain (21) where O αβ (t) = Ψ α |O| Ψ β and |Ψ α is an eigenstate of H 0 with energy ε α . Within linear response theory, we write the reduced density matrix as…”

“…In more recent years, a significant effort was devoted to the study of time-dependent transport and transient dynamics in mesoscopic systems attached to infinite leads 11,[15][16][17][18][19][20] . In the work of Pal et al 21 , this assumption was relaxed and time-dependent transport through a quantum-dot connected to two systems with a quasicontinuum spectrum (discrete, but dense), which take the role of finite leads, was considered in the partitioned approach. It was found that even in this case, after transients died-out, a steady-state transport regime across the dot emerges.…”