Numerically exact, time-dependent study of correlated electron transport in model molecular junctions

Effective modeling of open quantum systems by low-rank discretization of structured environments

The accurate description of the interaction of a quantum system with its environment is a challenging problem ubiquitous across all areas of physics and lies at the foundation of quantum mechanics theory. Here, we pioneer a new strategy to create discrete low-rank models of the system-environment interaction, by exploiting the frequency and time domain information encoded in the fluctuation-dissipation relation connecting the system-bath correlation function and the spectral density. We demonstrate the effectiveness of our methodology by combining it with tensor-network methodologies and simulating the quantum dynamics of complex excitonic systems in a highly structured bosonic environment. The new modeling framework sets the basis for a leap in the analysis of open quantum systems, providing controlled accuracy at significantly reduced computational costs, with benefits in all connected research areas.

A multi-layer multi-configurational time-dependent Hartree approach to lattice models beyond one dimension

The multi-layer multi-configurational time-dependent Hartree (MCTDH) approach is an efficient method to study quantum dynamics in real and imaginary time. The present work explores its potential to describe quantum fluids. The multi-layer MCTDH approach in second quantization representation is used to study lattice models beyond one dimension at finite temperatures. A scheme to map the lattice sites onto the MCTDH tree representation for multi-dimensional lattice models is proposed. A statistical sampling scheme previously used in MCTDH calculations is adapted to facilitate an efficient description of the thermal ensemble. As example, a two-dimensional hard-core Bose-Hubbard model is studied considering up to 64 × 64 lattice sites. The single particle function basis set size required to obtain converged results is found to not increase with the lattice size. The numerical results properly simulate the finite temperature Berezinskii-Kosterlitz-Thouless phase transition.

Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems

Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.

Time-Dependent Density Matrix Renormalization Group Method for Quantum Transport with Phonon Coupling in Molecular Junction

Quantum transport in molecular junctions has attracted great attention. The charge motion in a molecular junction can cause geometric deformation, leading to strong electron phonon coupling, which was often overlooked. We have formulated a nearly exact method to assess the time-dependent current and occupation number in the molecular junction modeled by the electron-phonon coupled bridge state using the time-dependent density matrix renormalization group (TD-DMRG) method. The oscillation period and amplitude of the current are found to be dependent on the electron phonon coupling strength and energy level alignment with the electrodes. In an attempt to better understand these phenomena, we have devised a new approximation that explains the bistability phenomenon and the behavior of steady currents in the strong electron-phonon coupling regime. Comparisons have been made with the multilayer-multiconfiguration time-dependent Hartree (ML-MCTDH) method and the analytical result in the purely electronic limit. Furthermore, we explore the entropy of different orderings, extending to the electron phonon model problems. Regarding finite temperature, the thermal Bogoliubov transformation of both fermions and bosons is used and compared with imaginary time evolution results.

A thermofield-based multilayer multiconfigurational time-dependent Hartree approach to non-adiabatic quantum dynamics at finite temperature

We introduce a thermofield-based formulation of the multilayer multiconfigurational time-dependent Hartree (MCTDH) method to study finite temperature effects on non-adiabatic quantum dynamics from a non-stochastic, wave function perspective. Our approach is based on the formal equivalence of bosonic many-body theory at zero temperature with a doubled number of degrees of freedom and the thermal quasi-particle representation of bosonic thermofield dynamics (TFD). This equivalence allows for a transfer of bosonic many-body MCTDH as introduced by Wang and Thoss to the finite temperature framework of thermal quasi-particle TFD. As an application, we study temperature effects on the ultrafast internal conversion dynamics in pyrazine. We show that finite temperature effects can be efficiently accounted for in the construction of multilayer expansions of thermofield states in the framework presented herein. Furthermore, we find our results to agree well with existing studies on the pyrazine model based on the ρMCTDH method.

Classical master equations and broadened classical master equations: Some analytical results

Some analytical results for the steady-state properties of the single-molecule tunneling junction are obtained with the use of the broadened classical master equations and classical master equations. The case of the one electronic level of the bridge molecule coupled to a single classical harmonic oscillator is considered within the spin-less model. Based on these equations, we establish some relations between different average values of interest, considering the large bias limit and the limit of the weak electron-oscillator coupling. We derive the analytical expressions for a number of characteristic properties of the tunneling junction in these limiting cases, compare our results with those obtained by the numerically exact calculations, and find that our expressions work very well. In the diabatic regime, the approximate solutions of the classical master equations are suggested, which permit us to introduce the effective temperature T_eff and perform rather simple calculations of the average vibrational excitations N and the tunnel current I. It is shown that in the adiabatic regime, the properties of the tunneling junction depend essentially on the effective temperature T_{eff ad}. We obtain the analytical expressions for T_{eff ad} using different approaches for the treatment of the adiabatic regime. For both the diabatic and adiabatic regimes, we calculate T_eff, T_{eff ad}, N, and I, compare our results with those available in the literature, and confirm well agreement. The dependence of N and I on the reorganization energy and the position of the electronic level of the bridge molecule is discussed.

Non-adiabatic quantum dynamics without potential energy surfaces based on second-quantized electrons: Application within the framework of the MCTDH method

A first principles quantum formalism to describe the non-adiabatic dynamics of electrons and nuclei based on a second quantization representation (SQR) of the electronic motion combined with the usual representation of the nuclear coordinates is introduced. This procedure circumvents the introduction of potential energy surfaces and non-adiabatic couplings, providing an alternative to the Born-Oppenheimer approximation. An important feature of the molecular Hamiltonian in the mixed first quantized representation for the nuclei and the SQR representation for the electrons is that all degrees of freedom, nuclear positions and electronic occupations, are distinguishable. This makes the approach compatible with various tensor decomposition Ansätze for the propagation of the nuclear-electronic wavefunction. Here, we describe the application of this formalism within the multi-configuration time-dependent Hartree framework and its multilayer generalization, corresponding to Tucker and hierarchical Tucker tensor decompositions of the wavefunction, respectively. The approach is applied to the calculation of the photodissociation cross section of the HeH⁺ molecule under extreme ultraviolet irradiation, which features non-adiabatic effects and quantum interferences between the two possible fragmentation channels, He + H⁺ and He⁺ + H. These calculations are compared with the usual description based on ab initio potential energy surfaces and non-adiabatic coupling matrix elements, which fully agree. The proof-of-principle calculations serve to illustrate the advantages and drawbacks of this formalism, which are discussed in detail, as well as possible ways to overcome them. We close with an outlook of possible application domains where the formalism might outperform the usual approach, for example, in situations that combine a strong static correlation of the electrons with non-adiabatic electronic-nuclear effects.

Non-adiabatic effects of nuclear motion in quantum transport of electrons: A self-consistent Keldysh-Langevin study

The molecular junction geometry is modeled in terms of nuclear degrees of freedom that are embedded in a stochastic quantum environment of non-equilibrium electrons. The time-evolution of the molecular geometry is governed via a mean force, a frictional force, and a stochastic force, forces arising from many electrons tunneling across the junction for a given nuclear vibration. Conversely, the current-driven nuclear dynamics feed back to the electronic current, which can be captured according to the extended expressions for the current that have explicit dependences on classical nuclear velocities and accelerations. Current-induced nuclear forces and the non-adiabatic electric current are computed using non-equilibrium Green's functions via a timescale separation solution of Keldysh-Kadanoff-Baym equations in the Wigner space. Applying the theory to molecular junctions demonstrated that non-adiabatic corrections play an important role when nuclear motion is considered non-equilibrium and, in particular, showed that non-equilibrium and equilibrium descriptions of nuclear motion produce significantly different current characteristics. It is observed that non-equilibrium descriptions generally produce heightened conductance profiles relative to the equilibrium descriptions and provide evidence that the effective temperature is an effective measure of the steady-state characteristics. Finally, we observe that the non-equilibrium descriptions of nuclear motion can give rise to the Landauer blowtorch effect via the emergence of multi-minima potential energy surfaces in conjunction with non-uniform temperature profiles. The Landauer blowtorch effect and its impact on the current characteristics, waiting times, and the Fano factor are explored for an effective adiabatic potential that morphs between a single, double, and triple potential as a function of voltage.

Quantum transport with electronic relaxation in electrodes: Landauer-type formulas derived from the driven Liouville-von Neumann approach

We derive the exact steady-state solutions for the simplest model systems of resonant tunneling and tunneling with destructive quantum interference from the driven Liouville-von Neumann (DLvN) approach. Under the finite-state lead condition (the two electrodes have finite states), we analyze the asymptotic behavior of the steady-state current in the two limits of electronic relaxation. Under the infinite-state lead condition, the steady-state solutions of the two model systems can be cast as Landauer-type current formulas. According to the formulas, we show that the transmission functions near the resonant peak and the antiresonant dip can be significantly influenced by electronic relaxation in the electrodes. Moreover, under intermediate and strong electronic relaxation conditions, we analytically show that the steady-state current of the DLvN approach dramatically deviates from the Landauer current when destructive quantum interference occurs. In the regime of zero electronic relaxation, our results are reduced to the Landauer formula, indicating that the DLvN approach is equivalent to the Landauer approach when the leads have infinite states without any electronic relaxation.

Pathways for charge transport through material interfaces

Modeling charge transport across material interfaces is important for understanding the limitations of electronic devices such as transistors, electrochemical cells, sensors, and batteries. However, modeling the entire structure and full dimensionality of an interface can be computationally demanding. In this study, we investigate the validity of an efficient reduced one-dimensional Hamiltonian for calculating charge transport along interfaces by comparing to a two-dimensional model that accounts for additional charge transport pathways. We find that the one-dimensional model successfully predicts the qualitative trend of charge transmission probability among Pt/Fe₂O₃ and Ag/Fe₂O₃ interfaces. However, the two-dimensional model provides additional information on possible pathways that are not perpendicular to the interface direction. These charge transport pathways are directed along the lowest potential energy profile of the interface that correlates with the crystal structure of the constituting materials. However, the two-dimensional paths are longer and take more scattering time. Therefore, the one-dimensional model may hold sufficient information for qualitative estimation of charge transport through some material interfaces.

Extending the hierarchical quantum master equation approach to low temperatures and realistic band structures

The hierarchical quantum master equation (HQME) approach is an accurate method to describe quantum transport in interacting nanosystems. It generalizes perturbative master equation approaches by including higher-order contributions as well as non-Markovian memory and allows for the systematic convergence to the numerically exact result. As the HQME method relies on a decomposition of the bath correlation function in terms of exponentials, however, its application to systems at low temperatures coupled to baths with complexer band structures has been a challenge. In this publication, we outline an extension of the HQME approach, which uses re-summation over poles and can be applied to calculate transient currents at a numerical cost that is independent of temperature and band structure of the baths. We demonstrate the performance of the extended HQME approach for noninteracting tight-binding model systems of increasing complexity as well as for the spinless Anderson-Holstein model.

Path integral molecular dynamics for exact quantum statistics of multi-electronic-state systems

An exact approach to compute physical properties for general multi-electronic-state (MES) systems in thermal equilibrium is presented. The approach is extended from our recent progress on path integral molecular dynamics (PIMD), Liu et al. [J. Chem. Phys. 145, 024103 (2016)] and Zhang et al. [J. Chem. Phys. 147, 034109 (2017)], for quantum statistical mechanics when a single potential energy surface is involved. We first define an effective potential function that is numerically favorable for MES-PIMD and then derive corresponding estimators in MES-PIMD for evaluating various physical properties. Its application to several representative one-dimensional and multi-dimensional models demonstrates that MES-PIMD in principle offers a practical tool in either of the diabatic and adiabatic representations for studying exact quantum statistics of complex/large MES systems when the Born-Oppenheimer approximation, Condon approximation, and harmonic bath approximation are broken.

The Fate of Atomic Spin in Atomic Scattering off Surfaces

We explore model electron dynamics of an atom scattering off a surface within the time-dependent complete active space self-consistent field (TD-CASSCF) approximation. We focus especially on the scattering of a hydrogen atom and its resulting spin dynamics starting from an initially spin-polarized state. Our results reveal competing electronic time scales that are governed by the electronic structure of the surface as well as the character of the atom. The time scales and nonadiabaticity of the dynamics are reported on by the final spin polarization of the scattered atom, which may be probed in future experiments.

Emergence of Landauer transport from quantum dynamics: A model Hamiltonian approach

The Landauer expression for computing current-voltage characteristics in nanoscale devices is efficient but not suited to transient phenomena and a time-dependent current because it is applicable only when the charge carriers transition into a steady flux after an external perturbation. In this article, we construct a very general expression for time-dependent current in an electrode-molecule-electrode arrangement. Utilizing a model Hamiltonian (consisting of the subsystem energy levels and their electronic coupling terms), we propagate the Schrödinger wave function equation to numerically compute the time-dependent population in the individual subsystems. The current in each electrode (defined in terms of the rate of change of the corresponding population) has two components, one due to the charges originating from the same electrode and the other due to the charges initially residing at the other electrode. We derive an analytical expression for the first component and illustrate that it agrees reasonably with its numerical counterpart at early times. Exploiting the unitary evolution of a wavefunction, we construct a more general Landauer style formula and illustrate the emergence of Landauer transport from our simulations without the assumption of time-independent charge flow. Our generalized Landauer formula is valid at all times for models beyond the wide-band limit, non-uniform electrode density of states and for time and energy-dependent electronic coupling between the subsystems. Subsequently, we investigate the ingredients in our model that regulate the onset time scale of this steady state. We compare the performance of our general current expression with the Landauer current for time-dependent electronic coupling. Finally, we comment on the applicability of the Landauer formula to compute hot-electron current arising upon plasmon decoherence.

Wavepacket dynamics and the multi-configurational time-dependent Hartree approach

Multi-configurational time-dependent Hartree (MCTDH) based approaches are efficient, accurate, and versatile methods for high-dimensional quantum dynamics simulations. Applications range from detailed investigations of polyatomic reaction processes in the gas phase to high-dimensional simulations studying the dynamics of condensed phase systems described by typical solid state physics model Hamiltonians. The present article presents an overview of the different areas of application and provides a comprehensive review of the underlying theory. The concepts and guiding ideas underlying the MCTDH approach and its multi-mode and multi-layer extensions are discussed in detail. The general structure of the equations of motion is highlighted. The representation of the Hamiltonian and the correlated discrete variable representation (CDVR), which provides an efficient multi-dimensional quadrature in MCTDH calculations, are discussed. Methods which facilitate the calculation of eigenstates, the evaluation of correlation functions, and the efficient representation of thermal ensembles in MCTDH calculations are described. Different schemes for the treatment of indistinguishable particles in MCTDH calculations and recent developments towards a unified multi-layer MCTDH theory for systems including bosons and fermions are discussed.

Driven Liouville von Neumann Equation in Lindblad Form

The Driven Liouville von Neumann approach [J. Chem. Theory Comput. 2014, 10, 2927-2941] is a computationally efficient simulation method for modeling electron dynamics in molecular electronics junctions. Previous numerical simulations have shown that the method can reproduce the exact single-particle dynamics while avoiding density matrix positivity violation found in previous implementations. In this study we prove that in the limit of infinite lead models the underlying equation of motion can be cast in Lindblad form. This provides a formal justification for the numerically observed density matrix positivity conservation.

Voltage Quench Dynamics of a Kondo System

We examine the dynamics of a correlated quantum dot in the mixed valence regime. We perform numerically exact calculations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibration times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. The time-dependent current saturation temperature connects the equilibrium Kondo temperature to a substantially increased value at voltages outside of the linear response. These signatures are directly observable by experiments in the time domain.

Multilayer Multiconfiguration Time-Dependent Hartree Theory

Multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory is a rigorous and powerful method to simulate quantum dynamics in complex many-body systems. This approach extends the original MCTDH theory of Meyer, Manthe, and Cederbaum to include dynamically contracted layers in a recursive way, within which the equations of motion are determined from the Dirac-Frenkel variational principle. This paper presents the general derivation of the theory and analyzes the important features that make the ML-MCTDH method numerically efficient. Furthermore, we discuss the generalization of the theory to treat many-body identical particles (fermions or bosons) as well as calculating energy eigenstates via the improved relaxation method.

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

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-von-Neumann equation approach of Sánchez et al. [J. Chem. Phys. 2006, 124, 214708] and Subotnik et al. [J. Chem. Phys. 2009, 130, 144105]. 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 multilead 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 junction models.