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Application of hierarchical equations of motion to time dependent quantum transport

Heng Tian

Published Jan 1, 2012• 0 citations
Publisher

Abstract

Within the exact framework established recently, which is a successful marriage\n\nbetween the time dependent density functional theory for open electronic system\n\nand quantum dissipation theory formulated in the hierarchical equations of motion,\n\nan entirely new scheme is proposed in this thesis to simulate the time-dependent\n\nquantum transport in nano-devices at both zero and finite temperature equally\n\nwithout relying on the pole structure of the Fermi distribution function. Neither\n\ndoes it depend on any non-unique parametrization of the line-width matrix, hence,\n\nthis new practical approach can be integrated with the first principles simulations\n\nseamlessly.\n\nBeyond the exact framework, a reliable method which works under the Wide-\n\nBand-Limit approximation at zero temperature is also developed. At the price of\n\nloss of some non-Markovian memory effects on the dynamics, a set of equations\n\nof motion which terminates at the first tier instead of the second tier is obtained.\n\nBenefiting from the latest advancement of numerical analysis, a hybrid fourth-order\n\nRunge-Kutta algorithm is proposed to solve this set of equations of motion which\n\ncomprises stiff ones. Based on this result, an alternative scheme is considered to\n\ndeal with the same approximation at finite temperature.\n\nAs an illustration of these new approaches, the transient current of the one\n\ndimensional tight-binding periodical chain with and without a single impurity, driven\n\nby some time alternating and/or static bias voltages, are investigated. The influence\n\nof temperature and switch-on rate of bias voltage is exemplified. Particularly, in the\n\none dimensional tight-binding chain with a single impurity which breaks its perfect\n\nperiodicity, an asymmetry between the left and right transient current is found.\n\nComparison between the results under the Wide-Band-Limit approximation and\n\nthose with the exact description is carried out.

DOI
10.5353/th_b4786944

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Within the exact framework established recently, which is a successful marriage\n\nbetween the time dependent density functional theory for open electronic system\n\nand quantum dissipation theory formulated in the hierarchical equations of motion,\n\nan entirely new scheme is proposed in this thesis to simulate the time-dependent\n\nquantum transport in nano-devices at both zero and finite temperature equally\n\nwithout relying on the pole structure of the Fermi distribution function. Neither\n\ndoes it depend on any non-unique parametrization of the line-width matrix, hence,\n\nthis new practical approach can be integrated with the first principles simulations\n\nseamlessly.\n\nBeyond the exact framework, a reliable method which works under the Wide-\n\nBand-Limit approximation at zero temperature is also developed. At the price of\n\nloss of some non-Markovian memory effects on the dynamics, a set of equations\n\nof motion which terminates at the first tier instead of the second tier is obtained.\n\nBenefiting from the latest advancement of numerical analysis, a hybrid fourth-order\n\nRunge-Kutta algorithm is proposed to solve this set of equations of motion which\n\ncomprises stiff ones. Based on this result, an alternative scheme is considered to\n\ndeal with the same approximation at finite temperature.\n\nAs an illustration of these new approaches, the transient current of the one\n\ndimensional tight-binding periodical chain with and without a single impurity, driven\n\nby some time alternating and/or static bias voltages, are investigated. The influence\n\nof temperature and switch-on rate of bias voltage is exemplified. Particularly, in the\n\none dimensional tight-binding chain with a single impurity which breaks its perfect\n\nperiodicity, an asymmetry between the left and right transient current is found.\n\nComparison between the results under the Wide-Band-Limit approximation and\n\nthose with the exact description is carried out.
A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.
A quantum dissipation theory is formulated in terms of hierarchically coupled equations of motion for an arbitrary fermionic system coupled with grand canonical fermionic bath ensembles.1 The theoretical construction starts with the second-quantization influence functional in path integral formalism, in which the fermionic creation and annihilation operators are represented by Grassmann variables. Temporal derivatives on influence functionals are then performed in a hierarchical manner, on the basis of calculus-on-path-integral algorithm. Both the multiple-frequency-dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting formalism is in principle exact, applicable to interacting systems, with arbitrary time-dependent external fields. It renders an exact tool to evaluate various transient and stationary quantum transport properties of many-electron systems. At the second-tier truncation level the present theory recovers the real-time diagrammatic formalism developed by Schön and coworkers.2 For a single-particle system, the hierarchical formalism terminates at the second tier exactly, and the Landuer-Büttiker's transport current expression is readily recovered. Numerical studies will be presented to highlight the richness of transient current through both interacting and noninteracting model systems.
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.
An accurate and efficient numerical approach is developed for the transient electronic dynamics of open quantum systems at low temperatures. The calculations are based on a formally exact hierarchical equations of motion quantum dissipation theory [J. S. Jin et al., J. Chem. Phys. 128, 234703 (2008)]. We propose a hybrid scheme that combines the Matsubara expansion technique and a frequency dispersion treatment to account for reservoir correlation functions. The new scheme not just admits various forms of reservoir spectral functions but also greatly reduces the computational cost of the resulting hierarchical equations, especially in the low temperature regime. Dynamical Kondo effects are obtained and the cotunneling induced Kondo transitions are resolved in the transient current in response to time-dependent external voltages.
We present an exact quantum dissipation theory in terms of the hierarchical equations‐of‐motion formalism for an arbitrary open electronic system. This formalism provides a theoretical tool to study various transient and stationary properties of many‐body systems far away from equilibrium. In contact with transport current, we explicitly show that how the hierarchial scheme relates to various tunneling processes, such as sequential tunneling, cotunneling, and high‐order events. The present hierarchical scheme numerically converges in general. It accounts for the combined effects of the many‐particle interaction within the system and the nature of the system‐bath coupling. In particular, for noninteracting systems, it terminates at the second tier exactly.
In this paper, we present a non-equilibrium quantum theory for transient electron dynamics in nanodevices based on the Feynman–Vernon influence functional. Applying the exact master equation for nanodevices we recently developed to the more general case in which all the constituents of a device vary in time in response to time-dependent external voltages, we obtained non-perturbatively the transient quantum transport theory in terms of the reduced density matrix. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation and decoherence being fully taken into account. For a simple illustration, we apply the theory to a single-electron transistor subjected to ac bias voltages. The non-Markovian memory structure and the nonlinear response functions describing transient electron transport are obtained.