State Representation Approach for Atomistic Time-Dependent Transport Calculations in Molecular Junctions

Abstract

Abstract:We propose a new method for simulating electron dynamics in open quantum systems out of equilibrium, using a finite atomistic model. The proposed method is motivated by the intuitive and practical nature of the driven Liouville vonNeumann equation approach of Sánchez et al. [J. Chem. Phys., 124, 214708 (2006)]. A key ingredient of our approach is a transformation of the Hamiltonian matrix from an atomistic to a state representation of the molecular junction. This allows us to uniquely define the bias voltage across the system while maintaining a proper thermal electronic distribution within the finite lead models. Furthermore, it allows us to investigate complex molecular junctions, including non-linear setups and multi-lead configurations. A heuristic derivation of our working equation leads to explicit expressions for the damping and driving terms, which serve as appropriate electron sources and sinks that effectively "open" the finite model system.Although the method does not forbid it, in practice we find neither violation of Pauli's exclusion principles nor deviation from density matrix positivity throughout our numerical simulations of various tight-binding model systems. We believe that the new approach offers a practical and physically sound route for performing atomistic time-dependent transport calculations in realistic molecular junctions.2

Electron transport in real time from first-principles

“…In tight-binding implementations of the driven Liouvillevon Neumann equation, 18,20 the system is divided in three regions: Source, Drain, and Molecule (S, D, and M, respectively). In this context, within an atomistic or site representation for a two lead setup, the density matrix and Hamiltonian can be written as in Equations (1) and (2), respectively,…”

“…Starting from the formulation above, Hod and coworkers 20 proposed a modified working expression in which the damping contribution affects not only the pure lead elements S, D, SD, and DS but also the lead-molecule coherences SM, MS, MD, and DM,…”

“…20 The modification of the standard Liouville-von Neumann equation adding imaginary absorbing potentials of magnitude Γ to the hamiltonian elements associated with the source and drain regions leads to a damping term as appearing in Equation (5) except for the ρ 0 elements. The final form of this equation is recovered if electron injection is represented in an analogous way, by including a potential of the same magnitude but opposite sign acting on the initial charge of the lead regions, ρ 0 S and ρ 0 D .…”

“…19 Later on, Hod and collaborators implemented a modified form of the equation of motion which led to improvements in the stability and steady-state convergence of the quantum dynamics. 20 This form, which can be derived from first principles, 24 was shown to conserve density matrix positivity and N-representability. In particular, they introduced a unitary transformation from the orthogonal, tight-binding atomic orbital basis to a state representation where the new basis elements can be identified with the source, drain, or device.…”

“…This state representation redefines the bias voltage in terms of the coupling between the eigenstates of the isolated sections of the full system. 20 Very recently, these authors generalized this approach to non-orthogonal localized basis sets using an extended Hückel model. 21 Our goal in the present study is to realize a first-principles implementation of the driven Liouville-von Neumann equation discussed in the previous paragraph, suited for transport simulations in realistic molecular systems.…”