Charge Transport Through Single Molecules : Different Theoretical Approaches I .

Two different methods which allow to calculate electric current through molecular junctions are compared. The first method is based on the scattering theory and Landauer formula, and the second on the master equation technique. Two models of the junction with different molecular bridges (with and without an internal vibrational degree of freedom) were used to compare and discuss features of both methods. The way to improve the scattering approach in the inelastic transport regime using realistic vibrational states populations instead of Boltzmann factors in the Landauer formula is proposed. Introduction Charge transport through individual molecules is promising subject of research nowadays. Work in this field is stimulated by the idea to create a new type of electronics (which is called “molecular electronics” or “moletronics”) in the future, where individual molecules will be used as building blocks for electronic circuits [Ratner and Ratner, 2002]. A whole class of experimental methods was established where a molecule (or a chain of molecules which is called a molecular bridge) can be connected between two solid electrodes [Selzer and Allara, 2006]. Such molecular junctions allow to measure electric current through the bridging molecule. Molecular bridges are complicated quantum systems with many internal degrees of freedom which may actively participate in the energy exchange during the charge transport process. There are few theoretical approaches available to treat such junctions [Galperin et al., 2007]. In this paper, we compare two of them. In the approach based on the Landauer formula [Datta, 1997] the current can be expressed through the transmission function of the bridge. Calculation scheme includes consecutive computation of self-energies of the leads and Green’s function of the bridge. Scattering theory formulas should be used afterwards to transform the Green’s function to the T-matrix and, finally, to the tunneling probability [Troisi et al., 2003]. Another possible way to calculate the current is to use the master equation (ME) theory. By solving the WangsnessBloch-Redfield ME [Timm, 2008] the reduced density matrix (RDM) of the molecular bridge can be determined. RDM contains detailed information about the bridge while the leads are treated “half rigorously” as large particle reservoirs in equilibrium in this method. The RDM can be used to calculate any observable of interest, including the current [Hartle et al., 2009]. Both methods will be discussed in the next two sections. Theory To compare methods we formulate two models of molecular junctions. In model 1 molecular bridge has only one vacant electronic level with energy ε0 available for tunneling. Interactions with internal degrees of freedom are neglected in this model and the tunneling process is elastic. In model 2 we add one active vibrational degree of freedom on the bridge which can exchange energy with electrons (Fig. 1). Such a situation is theoretically described in terms of the socalled independent boson model (see Mahan [1993] for details), where harmonic oscillator is coupled to the electronic state. This model reflects the fact that the bridge, when occupied by an electron, has a potential energy which is different from the energy of the unoccupied bridge (see Fig. 1). This is schematically depicted in Fig. 1 as two different harmonic potentials with different sets of vibrational states. In this paper we denote the energy spectra of unoccupied 175 WDS'10 Proceedings of Contributed Papers, Part III, 175–180, 2010. ISBN 978-80-7378-141-5 © MATFYZPRESS