A Correlated Source-Sink-Potential Model Consistent with the Meir–Wingreen Formula

Abstract

We model a molecular device as a molecule attached to a set of leads treated at the tightbinding level, with the central molecule described to any desired level of electronic structure theory. Within this model, in the absence of electron-phonon interactions, the Landauer-Büttiker part of the Meir-Wingreen formula is shown to be sufficient to describe the transmission factor of the correlated device. The key to this demonstration is to ensure that the correlation selfenergy has the same functional form as the exact correlation self-energy. This form implies that non-symmetric contributions to the Meir-Wingreen formula vanish, and hence conservation of current is achieved without the need for Green's Function self-consistency. An extension of the Source-Sink-Potential (SSP) approach gives a computational route to the calculation and interpretation of electron transmission in correlated systems. In this picture, current passes through internal molecular channels via resonance states with complex-valued energies. Each resonant state arises from one of the states in the Lehmann expansion of the one-electron Green's Function, hole conduction deriving from ionised states, and particle conduction from attached states. In the correlated device, the dependence of transmission on electron energy is determined by four structural polynomials, as it was in the tight-binding (Hückel) version of the SSP method. Hence, there are active and inert conduction channels (in the correlated case, linked to Dyson orbitals) governed by a set of selection rules that map smoothly onto the simpler picture.

A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices

“…Expression (17) has no divergences at molecular eigenvalues and has been used in our previous work [60,63,70] to derive selection rules for transmission. The numerical computation of SSP transmission does not use expressions such as (15) directly, relying instead upon a standard linear numerical solution of the SSP matrix equations that does not suffer from numerical instabilities.…”

“…that relates the energy, E, of a stream of electrons in the lead to the parameter β L p and the associated wavevector q L p (0 ≤ q L p ≤ 2π) in an infinite wire. The description of the molecule may, in principle, go beyond the Hückel model in the manner we have described previously [70], but, for simplicity, here, we derive the Hückel version of the theory.…”

“…where we have used the Jacobi relation from Equation ( 13) and the generalised SP notation in Equations ( 11) and ( 14). We derived this theory in terms of the Hückel tight-binding approximation, but it also holds for correlated molecular GFs, where the MO coefficients, U, in the definitions of SPs are replaced by the GF Dyson orbital coefficients [70]. Furthermore, the cluster expansion in Equation ( 53) holds for devices with any number of leads.…”

“…In our previous work on two-lead devices, we were able to describe 11 specific cases for conduction with leads attached to distinct vertices [60,63,70]. In particular, we identified inert MOs (or Dyson orbitals for correlated molecular calculations) that had no effect on the transmission at any electron energy, E. On the other hand, active orbitals-more correctly active shells, allowing for possible degeneracy-each produce a maximum, or a shoulder, in the transmission spectrum.…”

“…Conduction of the Hückel device can be analysed in terms of internal channels based on molecular orbitals of the central molecule, which are either active or inert [63]. It can be shown [70] that, as the level of theory is raised, a similar analysis applies to Hartree-Fock MO channels and, then at the second-order (or higher) Green's Function level, to internal channels associated with Dyson orbitals. This evolution of the channel picture is consistent with the standard Meir-Wingreen [19] and Landauer-Büttiker [71,72] expressions for elastic conduction of electrons through molecules.…”