Tunneling through molecules and quantum dots: Master-equation approaches

An important class of approaches to the description of electronic transport through molecules and quantum dots is based on the master equation. We discuss various formalisms for deriving a master equation and their interrelations. It is shown that the master equation derived by König et al. [Phys. Rev. Lett. 76, 1715 (1996); Phys. Rev. B 54, 16820 (1996)] is equivalent to the Wangsness-Bloch-Redfield master equation. The roles of the large-reservoir and Markov approximations are clarified. At low temperatures, when the quasiparticle lifetime becomes large, the Markov approximation can be derived from the assumption of weak tunneling under certain conditions. Interactions in the leads are shown to be irrelevant for the transport in the case of momentum-independent tunneling. It is explained why the $T$-matrix formalism gives incomplete results except for diagonal density operators to second order in the tunneling amplitudes. The time-convolutionless master equation is adapted to tunneling problems and a diagrammatic scheme for generating arbitrary orders in the tunneling amplitudes is developed.

Figure 1

Diagram for the second term in the ME [Eq. (53)]. Time is increasing to the right. The bare propagator $e^{-i({\mathcal{L}}_{\text{dot}}+{\mathcal{L}}_{\text{leads}})(t-t^{\prime })}$ for the density operator (not occurring here) would be represented by just two lines. The full propagator $e^{-i\mathcal{L}(t-t^{\prime })}$ is shown as a hatched box. Its irreducible part $e^{-i\mathcal{Q}\mathcal{L}\mathcal{Q}(t-t^{\prime })}$ is distinguished by writing “Q” beside it. All additional projection operators are likewise indicated by “P” or “Q.” An insertion of $-i{\mathcal{L}}_{\mathrm{hyb}}$ is denoted by a heavy bar connecting the two lines. The projected density operator $\mathcal{P}\rho \left(t^{\prime }\right)$ at time $t^{\prime }$ is shown as a filled semicircle.

Figure 2

Diagram for the superoperator $\Sigma (t-t_0)$. The interpretation of symbols is given in the caption of Fig. 1. The rightmost superoperator in Eq. (63) corresponds to the lower right corner of the diagram.

Figure 3

Contributions to the TCL ME [Eq. (66)] containing (a) one and (b) two powers of the superoperator $\Sigma$. The filled semicircle denotes the density operator $\mathcal{P}\rho \left(t\right)$. In (b), no time ordering of $t^{\prime }$ and $t^{″}$ is implied.

Figure 4

General form of all terms involving tunneling in the TCL ME [Eq. (66)]. Here, the filled semicircle denotes the projected density operator $\mathcal{P}\rho$ at time $t^{\prime }$.

Figure 5

Diagrams for all terms in the TCL ME up to fourth order in the tunneling amplitudes. The dashed lines denote the pairing of lead-electron operators $a$, $a^{†}$ with the same quantum numbers.