General derivation of the Green's functions for the atomic approach of the Anderson model: application to a single electron transistor (SET)

Abstract

We consider the cumulant expansion of the periodic Anderson model (PAM) in the case of a finite electronic correlation U, employing the hybridization as perturbation, and obtain a formal expression of the exact one-electron Green's function (GF). This expression contains effective cumulants that are as difficult to calculate as the original GF, and the atomic approach consists in substituting the effective cumulants by the ones that correspond to the atomic case, namely by taking a conduction band of zeroth width and local hybridization. In a previous work (T. Lobo, M. S. Figueira, and M. E. Foglio, Nanotechnology 21, 274007 (2010)) we developed the atomic approach by considering only one variational parameter that is used to adjust the correct height of the Kondo peak by imposing the satisfaction of the Friedel sum rule. To obtain the correct width of the Kondo peak in the present work, we consider an additional variational parameter that guarantees this quantity. The two constraints now imposed on the formalism are the satisfaction of the Friedel sum rule and the correct Kondo temperature. In the first part of the work, we present a general derivation of the method for the single impurity Anderson model (SIAM), and we calculate several density of states representative of the Kondo regime for finite correlation U, including the symmetrical case. In the second part, we apply the method to study the electronic transport through a quantum dot (QD) embedded in a quantum wire (QW), which is realized experimentally by a single electron transistor (SET). We calculate the conductance of the SET and obtain a good agreement with available experimental and theoretical results. Copyright 2012 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License.

The thermoelectric figure of merit for the single electron transistor

“…We solve the Hamiltonian employing the atomic approach for the finite correlation U case. This method was developed in earlier papers [24,25], and here we present a short review of it in Section 4. The atomic approach is able to capture the low temperature physics, dominated by the Kondo effect as well as the region T !…”

“…1a). The second term describes the QD defined by a two-level structure: one localized (f-electrons) bare level E fs ¼ E QD and the local Coulomb interaction of the QD, characterized by the double occupation level at E QD þ U [24,25]. These localized levels are expressed in the representation of Hubbard operators, which are convenient for working with correlated local states and are defined in general by X p,ab ¼ jp,a><p,bj.We would like to mention that the simple two-level structure of the isolated QD, is changed when this QD is allowed to interact with the conduction electron band.…”

“…In previous papers [24,25], one of us developed a simple method, that employs the atomic solution of the periodic Anderson model as a "seed", for generating approximate Green's functions for describing the Kondo physics of strongly correlated electron systems.…”

“…Since the calculation of the exact effective cumulant M s is equivalent to obtaining the exact Green's functions [24,25], we introduce an approximation that consists of substituting the exact effective cumulant in the Green's functions with the approximated ones, obtained from the atomic solution of the model, which is associated with the transitions described by the Hubbard operators. We employ the index I x ¼ 1,2,3,4, defined in Table 1, to characterize these X operators:…”

Spin-current Seebeck effect in an interacting quantum dot: Atomic approximation for the Anderson impurity model

“…In this section, we present a compact summary of the atomic approach necessary for the reader to follow the paper; the details of the method can be found in our previous papers [5,6]. The atomic approach is a simple method, that employs the atomic solution of the periodic Anderson model (PAM) as a "seed", to generate approximate Green's functions to describe the Kondo physics of strongly correlated electron systems.…”