Tunneling through molecules and quantum dots: master-equation approaches

TL;DR

This paper reviews and clarifies various master-equation formalisms used to model electronic transport through molecules and quantum dots, highlighting their equivalences, assumptions, and limitations, and introduces a diagrammatic scheme for higher-order calculations.

Contribution

It unifies different master-equation approaches, clarifies the role of approximations, and develops a diagrammatic method for systematic higher-order tunneling calculations.

Findings

Equivalent master equations by different authors are identified.

The role of the Markov approximation is clarified.

A diagrammatic scheme for higher-order tunneling processes is introduced.

Abstract

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 equations derived by Wangsness, Bloch, and Redfield and by Koenig et al. are equivalent. The roles of the large-reservoir and Markov approximations are clarified. The Markov approximation is traced back to nonzero bias voltage and temperature, whereas interactions and the corresponding rapid relaxation in the leads are shown to be irrelevant for the transport under certain conditions. It is explained why the T-matrix formalism gives incomplete results except for diagonal density operators and 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.