Transport in molecular states language: Generalized quantum master equation approach

A simple scheme, capable of treating transport in molecular junctions in the language of many-body states, is presented. By introducing an ansatz in Liouville space, similar to the generalized Kadanoff-Baym approximation, a quantum master equation (QME)-like expression is derived starting from the exact equation of motion for Hubbard operators. Using an effective Liouville space propagation, a dressing similar to the standard diagrammatic one is proposed. The scheme is compared to the standard QME approach and its applicability to transport calculations is discussed.

Figure 1

(Color online) Current-voltage characteristic for resonant level model (43). The generalized QME (solid line, red) is compared to the standard QME (dashed line, blue) result. Note that the generalized QME result is exact for this model. See text for parameters.

Figure 2

(Color online) Conductance vs applied bias for the model of a resonant level coupled to a single vibration [Eq. (52)]. Generalized QME (solid line, red) is compared to standard QME (dashed line, blue) result. See text for parameters.

Figure 3

(Color online) Nondegenerate, ${\epsilon }_{\uparrow }\ne {\epsilon }_{\downarrow }$, quantum dot (QD) model. (a) Current-voltage characteristic compares generalized (solid line, red) and standard QME (dotted line, blue) approaches. Calculation is done for $V_g=-0.55$. (b) Conductance vs applied bias $V_{sd}$ and gate voltage $V_g$ calculated within generalized QME approach.

Figure 4

(Color online) Conductance vs applied bias $V_{sd}$ and gate voltage $V_g$ for a degenerate quantum dot (QD) coupled to a vibration. The calculation is done using the generalized QME approach. The vibration frequency is ${\omega }_0=0.1$ and the strength of the electron-vibration coupling is $M=0.1$. Other parameters are the same as in Fig. 3, except that ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=-0.5$ (degenerate case).

Figure 5

(Color online) Conductance vs applied bias $V_{sd}$ and gate voltage $V_g$ for a degenerate QD asymmetrically coupled to the contacts. The calculation is performed using the generalized QME approach. Parameters used in the calculation are the same as in Fig. 3 except that ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=-0.5$ and ${\Gamma }_{L,\sigma }=0.01$ and ${\Gamma }_{R,\sigma }=0.1$.

Figure 6

(Color online) Two-level bridge model. Comparison between the prediction of the generalized QME of the present paper and the standard QME of Ref. 21. Shown are the current (a) and the probability for the system to be unoccupied (b) vs the applied bias–generalized (solid line, red) and standard (dashed line, blue) QME considerations. (c) represents the real (solid, red and dashed, blue) and imaginary (dotted, red and dash-dotted, blue) parts of the coherences in the local eigenbasis vs the applied bias for the generalized (red) and the standard (blue) QME treatment, respectively. See text for parameters.