Steady state conductance in a double quantum dot array: The nonequilibrium equation-of-motion Green function approach

Abstract

We study steady state transport through a double quantum dot array using the equation-ofmotion approach to the nonequilibrium Green functions formalism. This popular technique relies on uncontrolled approximations to obtain a closure for a hierarchy of equations, however its accuracy is questioned. We focus on 4 different closures, 2 of which were previously proposed in the context of the single quantum dot system (Anderson impurity model) and were extended to the double quantum dot array, and develop 2 new closures. Results for the differential conductance are compared to those attained by a master equation approach known to be accurate for weak system-leads couplings and high temperatures. While all 4 closures provide an accurate description of the Coulomb blockade and other transport properties in the single quantum dot case, they differ in the case of the double quantum dot array, where only one of the developed closures provides satisfactory results. This is rationalized by comparing the poles of the Green functions to the exact many-particle energy differences for the isolate system. Our analysis provides means to extend the equation-of-motion technique to more elaborate models of large bridge systems with strong electronic interactions.

A non-equilibrium equation-of-motion approach to quantum transport utilizing projection operators

“…EOM formulated on the Keldysh contour has been used in the literature to describe transport in junctions by us 47,48 and others. 44,45,52,53 Two main weaknesses of the method are: 1. The necessity to make uncontrollable approximations in truncating the (in general infinite) chain of EOMs and 2.…”

“…We show that proper symmetrization, as discussed in Refs. 42,52,53, is built into the scheme. We also show explicitly that projection (orthogonalization) is crucial for proper description of a system coupled to the bath.…”

“…It resolves the symmetry-breaking problem in decoupling chain of Green function EOMs in the sense of left vs. right EOM incompatibility. The latter problem is relevant for EOM formulations for both usual 42,52,53 and Hubbard 37-40 Green functions, and may lead to complications (for example, non-Hermiticity of the density matrix). In the presented formulation Hermiticity and positive definiteness of the density matrix is guaranteed by the canonical from of Eq.…”

“…This is an example of the built-in symmetrization property of the formulation, which otherwise should be obtained within a stepby-step approach. 52,53 Note also that in the Coulomb blockade regime the EOM for the canonical Green function G (0) A1,A1 (τ, τ ′ ) can be solved directly (without a selfconsistent procedure). This is reminiscent of the similar observation in Ref.…”

“…[44][45][46][47][48][49][50][51][52][53] EOM permits working with correlation functions of any operators to produce (in general) an infinite chain of equations of motion. The main drawback of the method is the necessity to make an uncontrolled approximation to close the chain of equations.…”