Time-dependent quantum transport: An efficient method based on Liouville-von-Neumann equation for single-electron density matrix

Abstract

Basing on our hierarchical equations of motion for time-dependent quantum transport [X. Zheng, G. H. Chen, Y. Mo, S. K. Koo, H. Tian, C. Y. Yam, and Y. J. Yan, J. Chem. Phys. 133, 114101 (2010)], we develop an efficient and accurate numerical algorithm to solve the Liouville-von-Neumann equation. We solve the real-time evolution of the reduced single-electron density matrix at the tight-binding level. Calculations are carried out to simulate the transient current through a linear chain of atoms, with each represented by a single orbital. The self-energy matrix is expanded in terms of multiple Lorentzian functions, and the Fermi distribution function is evaluated via the Padè spectrum decomposition. This Lorentzian-Padè decomposition scheme is employed to simulate the transient current. With sufficient Lorentzian functions used to fit the self-energy matrices, we show that the lead spectral function and the dynamics response can be treated accurately. Compared to the conventional master equation approaches, our method is much more efficient as the computational time scales cubically with the system size and linearly with the simulation time. As a result, the simulations of the transient currents through systems containing up to one hundred of atoms have been carried out. As density functional theory is also an effective one-particle theory, the Lorentzian-Padè decomposition scheme developed here can be generalized for first-principles simulation of realistic systems.

Hierarchical equations of motion for an impurity solver in dynamical mean-field theory

“…For achieving the optimal efficiency of HEOM, a multiLorentzian decomposition scheme [61,62] for the hybridization function (ω) is adopted. In this way, the hybridization function is spanned by a set of Lorentzian functions (ω) ≡…”

Time-dependent density functional theory for open systems with a positivity-preserving decomposition scheme for environment spectral functions

“…(2) and (8) into Eqs. (6) and (7), we have the following exponential expansion for the equilibrium self-energies: 17,18,30 …”

“…A formally rigorous and often more efficient TDDFT-OS formalism has been developed, 17 which is based on a combined PLD scheme for resolving the memory content of self-energies 17,30 In the combined PLD scheme, Λ α (ϵ) is expanded by Lorentzian functions, see Eq. (2), and the Fermi function f α (ϵ) is cast into a Padé form and then expanded as a sum of poles, 32…”

“…One way to reduce the computational cost is to introduce a set of basis functions to decompose Λ α (ϵ). For instance, in our previous works, 17,18,30,31 we have used Lorentzian functions for the decomposition of Λ α (ϵ),…”

Time-dependent density functional theory quantum transport simulation in non-orthogonal basis

“…10,22 Beyond the WBL, the RSDM based HEOM have been applied to simulate time-dependent quantum transport in tightbinding model system. 23,24 However, in the derivation of RSDM based HEOM, a localized orthonormal basis set is assumed. In practical first principles calculation, atomic orbital basis, which are the eigenfunctions of the single-electron Hamiltonian of individual atoms, are commonly used due to their localized nature and clear chemical meaning.…”