Altshuler-Aronov effects in nonequilibrium disordered nanostructures

The Altshuler-Aronov (AA) effect is one of the most basic quantum many-body effects in the mesoscopic regime. It originates from the coexistence of disorder and electron-electron interaction. In this paper, we reformulate the Feynman diagrammatic theory of the AA effect in a real-space nonequilibrium Green's function framework, in which an effective medium technique (via coherent potential) is employed to evaluate disorder-induced vertices in the diagrams. As such the developed real-space formalism is compatible with the prevailing nanodevice simulation paradigm, leading to an effective numerical approach to calculating the AA effects on electronic structures and quantum transport properties of nanostructures. As an application, we analyze the $I\text{−}V$ characteristics and the full local density of states (DOS) profile of an Anderson-Hubbard lattice sandwiched between biased electrodes. We show how the DOS anomaly due to the AA mechanism is reshaped by the geometrical confinement and the nonequilibrium effects in a nanosystem. Our numerical findings are well understood by the analytical results we provide in this paper.

Figure 1

(a) Keldysh contour. (b) Free energy diagrams to the first order in e-e interaction (wavy line). The gray area in each diagram represents the disorder vertex induced by configuration average.

Figure 2

Demonstration of the Kadanoff-Baym scheme for deriving self-energies (a),(b). Here the Fock (exchange) diagram is used for example. The Hartree self-energy can be derived in a similar manner. (c) displays the recursive structure of the vertex correction.

Figure 3

Hartree (a) and Fock (b) diagrams for the interaction correction to the DOS of weakly disordered systems. The ladder series composed of dashed lines represents the diffuson objects. Boldface letters (e.g., R) indicate spatial coordinates on a macroscopic scale greater than the impurity mean free path.

Figure 4

Plot of the function $\mathrm{\Xi }\left(y\right)=(1-2y){(y-1)}^2{y}^2(4y^2-4y-1)$, which we invoke in Eq. (20).

Figure 5

Diagrams for the first-order interaction correction to the charge current in weakly disordered systems. Here only the Fock diagrams are shown, since those of Hartree can be obtained by simply rearranging the interaction vertex [see Fig. 3].

Figure 6

$\delta I-V$ curve generated from Eq. (27) for a disordered wire at zero temperature. $\delta I$ is the charge current correction to the first order in e-e interaction.

Figure 7

Schematic diagram of the tight-binding model. Disorder and interaction are confined within the central scattering region. The rest (leads) serves as electronic reservoirs.

Figure 8

Diagrammatic representation of the charge current correction [see Eq. (31)] to the first order in interaction. ${\mathrm{\Sigma }}_{\mathrm{AA}}$ represents the self-energy diagrams of Fig. 2. Note that the current vertex must be dressed with Fig. 2 in order to ensure charge conservation in numerical calculations.

Figure 9

Diagrammatic representation for the equations used to compute (a) $K_i^{\left(2\right)}$ and (b) $K_i^{\left(3\right)}$ under CPA. All the objects in the diagrams are local in space. The angle brackets denote disorder average over the random potential $v_i$. The stripes in (b) denote the ladder series of $K_i^{\left(2\right)}$ (see Ref. [14] for more details).

Figure 10

(a) Numerically simulated frequency dependence of ${\mathcal{P}}_x\left(\omega \right)$, i.e., the diffusive recurrence probability, at the middle of a disordered tight-binding chain ($L=40a$) sandwiched between ballistic leads. (b) Position dependence of ${\mathcal{P}}_x\left(\omega \right)$ at $\omega =0$ under a set of differing disorder strengths.

Figure 11

$\delta I\text{−}V$ relation calculated using the tight-binding chain model [see Eq. (30)] under the setting $U_0=\xi ,U_{ij}=0$. $\delta I$ is the current correction to the first order in $U_0$. (a) The system size is set at $L=40a$ while the disorder strength is varied by tuning $v$. (b) Results for another two different system sizes under a given $v$.

Figure 12

DOS corrections at the middle of the model chain. The interaction is assumed to be local. The solid lines mark the nonequilibrium results obtained under $V=0.4\xi /e$. The dashed lines mark the corresponding results obtained in equilibrium (i.e., $V=0$). (a) The system size $L$ is fixed while the disorder strength $v$ varies. (b) The other way around.

Figure 13

(a) Complete profile of the local DOS correction as a function of position and energy in a system of $L=40a$, $U_{ij}=0$, and $v=0.5\xi$. (b) Cut at $x=10a$. (c) Cut at $\varepsilon =\pm eV/2$.

Figure 14

DOS correction from the Fock contribution (see Fig. 2) at the middle of the model chain ($L=40a,v=0.5\xi$). The interaction takes the form $U_{ij}=ua/|x_i-x_j|$. Blue curve: The bare interaction $U_{ij}$ is used in the diagram. Red curve: Screening is taken into account via Eq. (5) where the polarization $P$ is dressed with vertex correction [see Eq. (6) and Fig. 2]. Green curve: The screened interaction is calculated with the bare polarization, i.e., without vertex correction. Dashed curves: corresponding equilibrium results.

Figure 15

Energy relaxation effects on the DOS correction under differing bias voltages (a) and interaction strengths (b). The dashed curves represent the energy distribution function extracted from the onsite diagonal of $\mathrm{i}{G}^{-+}$, which is computed with the self-consistent GW-CPA scheme developed in Ref. [33]. The solid curves are obtained by computing the diagrams of Fig. 2 using the self-consistent Green's functions. The result is presented for a model chain ($L=40a$) at its middle point. The bare interaction is modeled as $U_{ij}=ua/|x_i-x_j|$.