Driven Liouville von Neumann Equation in Lindblad Form

Electron Dynamics in Open Quantum Systems: The Driven Liouville-von Neumann Methodology within Time-Dependent Density Functional Theory

A first-principles approach to describe electron dynamics in open quantum systems driven far from equilibrium via external time-dependent stimuli is introduced. Within this approach, the driven Liouville-von Neumann methodology is used to impose open boundary conditions on finite model systems whose dynamics is described using time-dependent density functional theory. As a proof of concept, the developed methodology is applied to simple spin-compensated model systems, including a hydrogen chain and a graphitic molecular junction. Good agreement between steady-state total currents obtained via direct propagation and those obtained from the self-consistent solution of the corresponding Sylvester equation indicates the validity of the implementation. The capability of the new computational approach to analyze, from first principles, non-equilibrium dynamics of open quantum systems in terms of temporally and spatially resolved current densities is demonstrated. Future extensions of the approach toward the description of dynamical magnetization and decoherence effects are briefly discussed.

Performance of reservoir discretizations in quantum transport simulations

Quantum transport simulations often use explicit, yet finite, electronic reservoirs. These should converge to the correct continuum limit, albeit with a trade-off between discretization and computational cost. Here, we study this interplay for extended reservoir simulations, where relaxation maintains a bias or temperature drop across the system. Our analysis begins in the non-interacting limit, where we parameterize different discretizations to compare them on an even footing. For many-body systems, we develop a method to estimate the relaxation that best approximates the continuum by controlling virtual transitions in Kramers turnover for the current. While some discretizations are more efficient for calculating currents, there is little benefit with regard to the overall state of the system. Any gains become marginal for many-body, tensor network simulations, where the relative performance of discretizations varies when sweeping other numerical controls. These results indicate that typical reservoir discretizations have little impact on numerical costs for certain computational tools. The choice of a relaxation parameter is nonetheless crucial, and the method we develop provides a reliable estimate of the optimal relaxation for finite reservoirs.

Analytic expressions for the steady-state current with finite extended reservoirs

Open-system simulations of quantum transport provide a platform for the study of true steady states, Floquet states, and the role of temperature, time dynamics, and fluctuations, among other physical processes. They are rapidly gaining traction, especially techniques that revolve around "extended reservoirs," a collection of a finite number of degrees of freedom with relaxation that maintains a bias or temperature gradient, and have appeared under various guises (e.g., the extended or mesoscopic reservoir, auxiliary master equation, and driven Liouville-von Neumann approaches). Yet, there are still a number of open questions regarding the behavior and convergence of these techniques. Here, we derive general analytical solutions, and associated asymptotic analyses, for the steady-state current driven by finite reservoirs with proportional coupling to the system/junction. In doing so, we present a simplified and unified derivation of the non-interacting and many-body steady-state currents through arbitrary junctions, including outside of proportional coupling. We conjecture that the analytic solution for proportional coupling is the most general of its form for isomodal relaxation (i.e., relaxing proportional coupling will remove the ability to find compact, general analytical expressions for finite reservoirs). These results should be of broad utility in diagnosing the behavior and implementation of extended reservoir and related approaches, including the convergence to the Landauer limit (for non-interacting systems) and the Meir-Wingreen formula (for many-body systems).

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.

Open System Tensor Networks and Kramers' Crossover for Quantum Transport

Tensor networks are a powerful tool for many-body ground states with limited entanglement. These methods can nonetheless fail for certain time-dependent processes-such as quantum transport or quenches-where entanglement growth is linear in time. Matrix-product-state decompositions of the resulting out-of-equilibrium states require a bond dimension that grows exponentially, imposing a hard limit on simulation timescales. However, in the case of transport, if the reservoir modes of a closed system are arranged according to their scattering structure, the entanglement growth can be made logarithmic. Here, we apply this ansatz to open systems via extended reservoirs that have explicit relaxation. This enables transport calculations that can access steady states, time dynamics and noise, and periodic driving (e.g., Floquet states). We demonstrate the approach by calculating the transport characteristics of an open, interacting system. These results open a path to scalable and numerically systematic many-body transport calculations with tensor networks.

Unified theory of plasmon-induced resonance energy transfer and hot electron injection processes for enhanced photocurrent efficiency

Plasmons in metal nanoparticles (MNPs) promise to enhance solar energy conversion in semiconductors. Two essential mechanisms of enhancement in the near-field regime are hot electron injection (HEI) and plasmon-induced resonance energy transfer (PIRET). Individual studies of both mechanisms indicate that the PIRET efficiency is limited by the short lifetime of the plasmon, whereas the hot electrons result from the plasmon decay. The development of a unified theory of the coupled HEI and PIRET processes is fundamentally interesting and necessary for making reliable predictions but is complicated by the multiple interactions between various components that participate in the enhancement process. In this paper, we use the model-Hamiltonian approach to develop a combined theoretical framework including both PIRET and HEI. The coupled dynamics as well as the time evolution of hot electron energy distribution are studied. The theory further predicts an interference-induced asymmetry in the spectral dependence of PIRET, which can be used to distinguish it from HEI. As the relative contributions of PIRET and HEI strongly depend on the size of the MNPs, this presents itself as a simple route to control the strength of their contributions. The results presented here can further guide future applications of plasmonic solar energy harvesting.

Communication: Master equations for electron transport: The limits of the Markovian limit

Master equations are increasingly popular for the simulation of time-dependent electronic transport in nanoscale devices. Several recent Markovian approaches use "extended reservoirs"-explicit degrees of freedom associated with the electrodes-distinguishing them from many previous classes of master equations. Starting from a Lindblad equation, we develop a common foundation for these approaches. Due to the incorporation of explicit electrode states, these methods do not require a large bias or even "true Markovianity" of the reservoirs. Nonetheless, their predictions are only physically relevant when the Markovian relaxation is weaker than the thermal broadening and when the extended reservoirs are "sufficiently large," in a sense that we quantify. These considerations hold despite complete positivity and respect for Pauli exclusion at any relaxation strength.

Origin of Plasmon Lineshape and Enhanced Hot Electron Generation in Metal Nanoparticles

Plasmon-generated hot carriers are currently being studied intensively for their role in enhancing the efficiency of photovoltaic and photocatalytic processes. Theoretical studies of the hot electrons subsystem have generated insight, but we show that a unified quantum-mechanical treatment of the plasmon and hot electrons reveals new physical phenomena. Instead of a unidirectional energy transfer process in Landau damping, back energy transfer is predicted in small metal nanoparticles (MNPs) within a model-Hamiltonian approach. As a result, the single Lorentzian plasmonic line shape is modulated by a multipeak structure, whose individual line width provides a direct way to probe the electronic dephasing. More importantly, the hot electron generation can be enhanced greatly by matching the incident energy to the peaks of the modulated line shape.

Communication: Relaxation-limited electronic currents in extended reservoir simulations

Open-system approaches are gaining traction in the simulation of charge transport in nanoscale and molecular electronic devices. In particular, "extended reservoir" simulations, where explicit reservoir degrees of freedom are present, allow for the computation of both real-time and steady-state properties but require relaxation of the extended reservoirs. The strength of this relaxation, γ, influences the conductance, giving rise to a "turnover" behavior analogous to Kramers turnover in chemical reaction rates. We derive explicit, general expressions for the weak and strong relaxation limits. For weak relaxation, the conductance increases linearly with γ and every electronic state of the total explicit system contributes to the electronic current according to its "reduced" weight in the two extended reservoir regions. Essentially, this represents two conductors in series-one at each interface with the implicit reservoirs that provide the relaxation. For strong relaxation, a "dual" expression-one with the same functional form-results, except now proportional to 1/γ and dependent on the system of interest's electronic states, reflecting that the strong relaxation is localizing electrons in the extended reservoirs. Higher order behavior (e.g., γ2 or 1/γ2) can occur when there is a gap in the frequency spectrum. Moreover, inhomogeneity in the frequency spacing can give rise to a pseudo-plateau regime. These findings yield a physically motivated approach to diagnosing numerical simulations and understanding the influence of relaxation, and we examine their occurrence in both simple models and a realistic, fluctuating graphene nanoribbon.

Electron transport in real time from first-principles

While the vast majority of calculations reported on molecular conductance have been based on the static non-equilibrium Green's function formalism combined with density functional theory (DFT), in recent years a few time-dependent approaches to transport have started to emerge. Among these, the driven Liouville-von Neumann equation [C. G. Sánchez et al., J. Chem. Phys. 124, 214708 (2006)] is a simple and appealing route relying on a tunable rate parameter, which has been explored in the context of semi-empirical methods. In the present study, we adapt this formulation to a density functional theory framework and analyze its performance. In particular, it is implemented in an efficient all-electron DFT code with Gaussian basis functions, suitable for quantum-dynamics simulations of large molecular systems. At variance with the case of the tight-binding calculations reported in the literature, we find that now the initial perturbation to drive the system out of equilibrium plays a fundamental role in the stability of the electron dynamics. The equation of motion used in previous tight-binding implementations with massive electrodes has to be modified to produce a stable and unidirectional current during time propagation in time-dependent DFT simulations using much smaller leads. Moreover, we propose a procedure to get rid of the dependence of the current-voltage curves on the rate parameter. This method is employed to obtain the current-voltage characteristic of saturated and unsaturated hydrocarbons of different lengths, with very promising prospects.