Spectroscopy of the transition-rate matrix for molecular junctions: dynamics in the Franck–Condon regime

Abstract

The quantum master equation applied to electronic transport through nanoscopic devices provides information not only on the stationary state but also on the dynamics. The dynamics is characterized by the eigenvalues of the transition-rate matrix, or generator, of the master equation. We propose to use the spectrum of these eigenvalues as a tool for the study of nanoscopic transport. We illustrate this idea by analyzing a molecular quantum dot with an electronic orbital coupled to a vibrational mode, which shows the Franck-Condon blockade if the coupling is strong. Our approach provides complementary information compared to the study of observables in the stationary state.Recent progress in nanotechnology has allowed us to fabricate transistors with a single molecule forming the active region [1][2][3][4]. The transport properties of such devices have been studied extensively both experimentally and theoretically. Theoretical approaches have been reviewed by Andergassen et al [5] and Zimbovskaya and Pederson [6].The transport properties of single-molecule devices are often dominated by strong interactions. Electron-electron interaction, which leads to Coulomb blockade [7,8] and the coupling of electrons to vibrational modes [7-10], have to be taken into account. We assume the nuclear motion to be slow on the timescale of electronic transitions, which is the case for most, but not all, single-molecule devices studied so far [11]. The electron-vibron coupling is then due to the changes of the equilibrium nuclear configuration with the electronic state, i.e. the Franck-Condon effect: even if a molecule is initially in a vibrational eigenstate, after an electronic transition it will be in a superposition of vibrational eigenstates belonging to the new electronic configuration. The probability of ending up in any one of them is determined by the Franck-Condon matrix elements between vibrational eigenstates for the old and new electronic configurations [12][13][14][15][16]. These matrix elements can be very small if the equilibrium value of the relevant normal coordinate changes strongly with the electronic state. Consequently, the rates for electrons tunneling into or out of the molecule and thus the current can be strongly suppressed-this is the Franck-Condon blockade [14][15][16][17][18][19][20][21]. The dynamics is also unusual in this regime: electrons tunnel in avalanches separated by quiet time intervals [14,15]. The avalanchetype transport is self-similar on intermediate timescales and also leads to a strong enhancement of the zero-frequency Fano factor [14][15][16].Strong interactions and states far from equilibrium render the description of molecular devices difficult in general [5,6]. We focus here on the case of a weak hybridization between the molecule and the leads. In this case, master-equation (ME) approaches [13,14,[16][17][18][19][22][23][24] or the equivalent real-time diagrammatic approach [25][26][27] can be employed. In principle, the ME takes into account all the interactions in the molecule but r...

Vibron-assisted spin relaxation at a metal/organic interface

“…Note that in this case, for the chosen initial conditions, no vibrons are excited since the FC matrix, whose elements F qq enter the transition rates of Eq. (4), is the identity matrix [35,37,38].…”

“…The molecule is assumed weakly coupled to an electrode, which acts as a charge reservoir at the chemical potential μ = 0. The total (molecule plus electrode) Hamiltonian reads [34][35][36][37][38] …”

“…[34][35][36][37][38] for details. H t is proportional to a tunneling amplitude V between the molecule and the electrode.…”

“…We treat tunneling as a perturbation and evaluate the dynamics of our system through a master equation [38,40] …”

“…D σ αα = α|d σ |α are the electronic matrix elements, while F qq are the Franck-Condon (FC) matrix elements between two states with q and q vibrons [35,37,38], which read…”