Driven Liouville-von Neumann equation for time-dependent transport problems

Driven Liouville-von Neumann equation for time-dependent transport problems

Langerak, N.

Thijssen, J.M. (mentor)
Dubbeldam, J.L.A. (mentor)

Applied Sciences

QN/Quantum Nanoscience

TN-TW

2016-07-26

Electronic devices are downsizing at a rapid pace and since the 1990s, single molecules have been studied as candidates for such devices. This thesis focusses on a new approach to simulate the electric current through these molecular devices. We followed the approach of Hod [17], which is based on the time propagation of the density matrix and adds a driving term to the Liouville-von Neumann equation. We have implemented this approach into a computer code to see whether this could correctly simulate a time-dependent transport problem. The setup for the simulations is a molecular junction, which is modelled under the tight-binding approximation. For appropriate damping rates, the simulations produced a steady-state current in close agreement with the Landauer current. An optimal damping rate was found, but the steady-state current is rather stable with respect to the choice of the damping rate. To simulate a time-dependent problem, we reproduced the results of Napitu [9] who modelled a molecular pump with the more traditional Keldysh Green’s function approach. To this end, our simulations do not match the traditional approach for this time-dependent setup. The reasons for the differences are still unclear. Finally, the driven Liouville-von Neumann equation has been checked with the Lindblad conditions. These turned out to be satisfied in the limit of infinite leads with equilibrium occupation. Intuitively we do not expect the driven Liouville-von Neumann equation to be in the Lindblad form without taking a limit and we have not succeeded in doing so. Although we have successfully reproduced Hod’s results, the results for the pump are still inconclusive. The disagreement with the results obtained by Napitu [9] may be due to a difference in the model or in the method used. This leads to recommendations for future research.

http://resolver.tudelft.nl/uuid:d8043d91-2db7-4225-8849-6211ec151910

Student theses

bachelor thesis

(c) 2016 Langerak, N.