General theory for calculating disorder-averaged Green's function correlators within the coherent potential approximation

We report a diagrammatic method to solve the general problem of calculating configurationally averaged Green's function correlators that appear in quantum transport theory for nanostructures containing disorder. The theory treats both equilibrium and nonequilibrium quantum statistics on an equal footing. Since random impurity scattering is a problem that cannot be solved exactly in a perturbative approach, we combine our diagrammatic method with the coherent potential approximation (CPA) so that a reliable closed-form solution can be obtained. Our theory not only ensures the internal consistency of the diagrams derived at different levels of the correlators but also satisfies a set of Ward-like identities that corroborate the conserving consistency of transport calculations within the formalism. The theory is applied to calculate the quantum transport properties such as average ac conductance and transmission moments of a disordered tight-binding model, and results are numerically verified to high precision by comparing to the exact solutions obtained from enumerating all possible disorder configurations. Our formalism can be employed to predict transport properties of a wide variety of physical systems where disorder scattering is important.

Figure 1

Diagrammatic visualization of self-energies $\mathrm{\Sigma }$, $\tilde{\mathrm{\Sigma }}$ and correlators $L^{\left(2\right)}$, ${\tilde{L}}^{\left(2\right)}$. The directed thick line represents the average Green's function. Dashed lines represent disorder vertices.

Figure 2

Diagrammatic illustration of high-order kernels and their relations. Dashed lines represent disorder vertices.

Figure 3

Diagrammatic visualization of the correlators $L^{\left(2\right)}$, $L^{\left(3\right)}$, $L^{\left(4\right)}$, and auxiliary functions $A^{\left(2\right)}$ and $A^{\left(3\right)}$. Directed double lines represent the average Green's function. The same diagrammatic building blocks are marked by same colors. Internal indices are to be integrated over.

Figure 4

Diagram of the tight-binding chain model. The external leads are disorder free while the central region contains disordered sites.

Figure 5

Disorder-induced transmission moments $〈T^n\left(E\right)〉$ of a tight-binding transport junction containing four disordered sites in its central region. The numerical results are obtained with two methods: CPA and brute-force computation. The onsite energies of the two atomic species are ${\varepsilon }_{A/B}=\mp 0.25\theta$ and the disorder concentration $x$ varies from 0.01 to 0.5. The energy ($E$) is in units of the hopping probability ($\theta$).

Figure 6

Disorder-averaged ac dynamic conductance (vs response frequency $\mathrm{\Omega }$) of a one-dimensional transport junction with nine disordered sites in the central region. Each disordered site can be occupied either by species $A$ or species $B$ with an equal probability and the onsite energy of $A$ is fixed at the equilibrium chemical potential, i.e., ${\varepsilon }_A=0$, while ${\varepsilon }_B$ is adjustable. The solid (dashed) lines denote the real (imaginary) part of the conductance. The response frequency ($\mathrm{\Omega }$) is in unit of $\theta /\hslash$, where $\theta$ is the hopping probability.

Figure 7

Diagram of the closed time contour used in the Keldysh formalism. The contour begins and ends both at $-\infty$. It consists of a forward branch (denoted by $+$) and a backward branch (denoted by $-$). The complex time is denoted by $\tau$ as opposed to $t$ for the real time.