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The Journal of Chemical Physics

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

Hang Xie, Feng Jiang, Heng Tian, Xiao Hong Zheng, YanHo Kwok, Shuguang Chen, ChiYung Yam, YiJing Yan, GuanHua Chen

Published Jul 27, 2012• 51 citations
Publisher

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.

DOI
10.1063/1.4737864

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Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.

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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.
We present an extension of second-principles density functional theory (SPDFT) to perform time-dependent simulations. Our approach calculates the real-time, real-space evolution of the density matrix using the Liouville-von Neumann equation of motion, enabling the determination of optical and transport properties in very large systems, containing tens of thousands of atoms, on modest computational platforms. Unlike other methods, SPDFT is applicable to a wide range of materials, including both metals and insulators. We demonstrate its capabilities by computing the spectra of SrTiO3{\mathrm{SrTiO}}_{3}SrTiO3​, diamond, and metallic lithium. While SPDFT results for SrTiO3{\mathrm{SrTiO}}_{3}SrTiO3​ closely match those obtained from DFT with linear perturbation theory, significant improvements are observed for diamond and metallic lithium. In diamond, the inclusion of electron-electron interactions during the density-matrix evolution yields spectra that more accurately reproduce Bethe-Salpeter equation results than perturbative DFT. In lithium, time-dependent SPDFT captures both interband transitions and the Drude peak, enabling detailed ab initio studies of transport properties beyond usual approximations.
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.
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.
A linear-scaling first-principles quantum mechanical method is developed to evaluate the optical responses of large molecular systems. Instead of a many-body wave function, the equation of motion is solved for the reduced single-electron density matrix in the time domain. The locality of the reduced single-electron density matrix is utilized to ensure that computational time scales linearly with system size. The two-electron Coulomb integrals are evaluated with the fast multipole method, and the calculation of exchange-correlation quadratures utilizes the locality of an exchange-correlation functional and the integral prescreening technique. As an illustration, the resulting time-dependent density-functional theory is used to calculate the absorption spectra of polyacetylene oligomers and linear alkanes. The linear-scaling of computational time versus the system size is clearly demonstrated.
A method for the numerical solution of the dissipative Liouville–von Neumann equation is presented. The reduced density operator is expanded in a basis of time-dependent wave functions. This guarantees that the size of the basis required in the representation of the density operator is minimal. Equations of motion for the expansion wave functions and the density matrix elements are obtained from the Dirac–Frenkel variational principle. The numerical effort of the method scales proportional to N, where N is the dimension of the relevant Hilbert space. As a first example, the dynamics of a three-mode system with vibronic coupling (with N≈50 000) coupled to a dissipative bath has been studied, modeling the S1/S2 states of pyrazine. For the cw-absorption spectrum, fast convergence with respect to the number of expansion wave functions has been obtained.
Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.