A simple scheme capable of treating transport in molecular junctions in the language of many-body states is presented. An ansatz in Liouville space similar to generalized Kadanoff-Baym approximation is introduced in order to reduce exact equation-of-motion for Hubbard operator to quantum master equation (QME)-like expression. A dressing with effective Liouville space propagation similar to standard diagrammatic dressing approach is proposed. The scheme is compared to standard QME approach, and its applicability to transport calculations is discussed within numerical examples.
We propose a self-consistent generalized quantum master equation (GQME) to describe electron transport through molecular junctions. In a previous study [M.Esposito and M.Galperin. Phys. Rev. B 79, 205303 (2009)], we derived a time-nonlocal GQME to cure the lack of broadening effects in Redfield theory. To do so, the free evolution used in the Born-Markov approximation to close the Redfield equation was replaced by a standard Redfield evolution. In the present paper, we propose a backward Redfield evolution leading to a time-local GQME which allows for a self-consistent procedure of the GQME ge…
Based on our recent work on quantum transport [Li et al., Phys. Rev. B 71, 205304 (2005)], where the calculation of transport current by means of quantum master equation was presented, in this paper we show how an efficient calculation can be performed for the transport noise spectrum. Compared to the longstanding classical rate equation or the recently proposed quantum trajectory method, the approach presented in this paper combines their respective advantages, i.e., it enables us to tackle both the many-body Coulomb interactionand quantum coherence on equal footing and under a wide range of …
Time-dependent currents in molecular junctions can be caused by structural fluctuations or interaction with external fields. In this publication, we demonstrate how the hierarchical quantum master equation approach can be used to study time-dependent transport in a molecular junction. This reduced density matrix methodology provides a numerically exact solution to the transport problem including time-dependent energy levels, molecule-lead coupling strengths and transitions between electronic states of the molecular bridge. Based on a representative model, the influence of a time-dependent mole…
In this review article, we present a non-equilibrium quantum transport theory for transient electron dynamics in nanodevices based on exact master equation derived with the path integral method in the fermion coherent-state representation. Applying the exact master equation to nanodevices, we also establish the connection of the reduced density matrix and the transient quantum transport current with the Keldysh nonequilibrium Green functions. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation…
We present a theoretical approach to solve Markovian master equation for quantum transport with stochastic telegraph noise. Considering probabilities as functionals of a random telegraph process we use the Novikov's functional method to convert stochastic master equation to a set of deterministic differential equations. The equations are then solved in the Laplace space and the expression for the probability vector averaged over ensemble of realisations of stochastic process is obtained. We apply the theory to study manifestations of telegraph noise in transport properties of molecular junctio…
Electrons in the active region of a nanostructure constitute an open many-body quantum system, interacting with contacts, phonons, and photons. We review the basic premises of the open system theory, focusing on the common approximations that lead to Markovian and non-Markovian master equations for the reduced statistical operator. We highlight recent progress on the use of master equations in quantum transport, and discuss the limitations and potential new directions of this approach.
Within the hierarchical quantum master equation (HQME) framework, an approach is presented, which allows a numerically exact description of nonequilibrium charge transport in nanosystems with strong electronic-vibrational coupling. The method is applied to a generic model of vibrationally coupled transport considering a broad spectrum of parameters ranging from the nonadiabatic to the adiabatic regime and including both resonant and off-resonant transport. We show that nonequilibrium effects are important in all these regimes. In particular in the off-resonant transport regime, the inelastic c…
In this paper, the exact transient quantum transport of non-interacting nanostructures is investigated in the presence of initial system-lead correlations and initial lead-lead correlations for a device system coupled to general electronic leads. The exact master equation incorporating with initial correlations is derived through the extended quantum Langevin equation. The effects of the initial correlations are manifested through the time-dependent fluctuations contained explicitly in the exact master equation. The transient transport current incorporating with initial correlations is obtaine…
We present a hierarchical quantum master equation (HQME) approach, which allows the numerically exact evaluation of higher-order current cumulants in the framework of full counting statistics for nonequilibrium charge transport in nanosystems. The novel methodology is exemplarily applied to a model of vibrationally coupled electron transport in a molecular nanojunction. We investigate the influence of cotunneling on avalanche-like transport, in particular in the nonresonant transport regime, where we find that inelastic cotunneling acts as trigger process for resonant avalanches. In this regim…
In this work a practical scheme is developed for the first-principles study of time-dependent quantum transport. The basic idea is to combine the transport master-equation with the well-known time-dependent density functional theory. The key ingredients of this paper include: (i) the partitioning-free initial condition and the consideration of the time-dependent bias voltages which base our treatment on the Runge-Gross existence theorem; (ii) the non-Markovian master equation for the reduced (many-body) central system (i.e. the device); and (iii) the construction of Kohn-Sham master equation f…
We investigate to which extent a many-body Bloch-Redfield master equation description of quantum transport is consistent with the exact generalized equilibrium conditions known as exchange fluctuation theorems. Thereby we identify a class of master equations for which this is the case. Beyond this class, we find deviations which exhibit characteristic scaling laws as functions of the dot-lead tunneling, the inter-dot tunneling, and the temperature. These deviations are accompanied by an increase of lead energy fluctuations inherent in the Bloch-Redfield equation beyond rotating-wave approximat…
This paper presents a comprehensive review of the wave-function approach for derivation of the number-resolved Master equations, used for description of transport and measurement in mesoscopic systems. The review contains important amendments, clarifying subtle points in derivation of the Master equations and their validity. This completes the earlier works on the subject. It is demonstrated that the derivation does not assume weak coupling with the environment and reservoirs, but needs only high bias condition. This condition is very essential for validity of the Markovian Master equations, w…
Time-resolved electron transport in nano-devices is described by means of a time-nonlocal quantum master equation for the reduced density operator. Our formulation allows for arbitrary time dependences of any device or contact parameter. The quantum master equation and the related expression for the electron current through the device are derived in fourth order of the coupling to the contacts. It is shown that a consistent sum up to infinite orders induces level broadening in the device. To facilitate a numerical propagation of the equations we propose to use auxiliary density operators. An e…
We construct a particle-number(n)-resolved master equation (ME) approach under the self-consistent Born approximation (SCBA) for quantum transport through mesoscopic systems. The formulation is essentially non-Markovian and incorporates the interlay of the multi-tunneling processes and many-body correlations. The proposed n-SCBA-ME goes completely beyond the scope of the Born-Markov master equation, being applicable to transport under small bias voltage, in non-Markovian regime and with strong Coulomb correlations. For steady state, it can recover not only the exact result of noninteracting tr…
Based on an exact formulation, we present a master equation approach to transport through Majorana zero modes (MZMs). Within the master equation treatment, the occupation dynamics of the regular fermion associated with the MZMs holds a quite different picture from the BdG S-matrix scattering process, in which the "positive" and "negative" energy states are employed, while the master equation treatment does not involve them at all. Via careful analysis for the structure of the rates and the rate processes governed by the master equation, we reveal the intrinsic connection between both approache…
For quantum transport through mesoscopic system, a quantum master equation approach is developed in terms of compact expressions for the transport current and the reduced density matrix of the system. The present work is an extension of Gurvitz's approach for quantum transport and quantum measurement, namely, to finite temperature and arbitrary bias voltage. Our derivation starts from a second-order cummulant expansion of the tunneling Hamiltonian, then follows conditional average over the electrode reservoir states. As a consequence, in the usual weak tunneling regime, the established formali…
The authors apply the generalized master equation to analyze time-dependent transport through a finite quantum wire with an embedded subsystem. The parabolic quantum wire and the leads with several subbands are described by a continuous model. We use an approach originally developed for a tight-binding description selecting the relevant states for transport around the bias-window defined around the values of the chemical potential in the left and right leads in order to capture the effects of the nontrivial geometry of the system in the transport. We observe a partial current reflection as a m…
Beyond the second-order Born approximation, we develop an improved master equation approach to quantum transport by virtue of a self-consistent Born approximation. The basic idea is replacing the free Green's function in the tunneling self-energy by an effective reduced propagator under the Born approximation. We found that the effect of this simple improvement is remarkable, for instance, it can not only recover the exact result of noninteracting transport under arbitrary voltages, but also predict the challenging nonequilibrium Kondo effect. In addition to having an elegant structure, the ap…
For an open quantum system, we investigate the pumped current induced by a slow modulation of control parameters on the basis of the quantum master equation and full counting statistics. We find that the average and the cumulant generating function of the pumped quantity are characterized by the geometrical Berry-phase-like quantities in the parameter space, which is associated with the generator of the master equation. From our formulation, we can discuss the geometrical pumping under the control of the chemical potentials and temperatures of reservoirs. We demonstrate the formulation by spin…
We demonstrate that with a stepwise introduction of complexity to a model of an electron system embedded in a photonic cavity and a carefully controlled stepwise truncation of the ensuing many-body space it is possible to describe the time-dependent transport of electrons through the system with a non-Markovian generalized quantum master equation. We show how this approach retains effects of an external magnetic field and the geometry of an anisotropic electronic system. The Coulomb interaction between the electrons and the full electromagnetic coupling between the electrons and the photons ar…
We discuss the use of super-fermion formalism to represent and solve quantum kinetic equations for the electron transport problem. Starting with the Lindblad master equation for the molecule connected to two metal electrodes, we convert the problem of finding the nonequilibrium steady state to the many-body problem with non-Hermitian Liouvillian in super-Fock space. We transform the Liouvillian to the normal ordered form, introduce nonequilibrium quasiparticles by a set of canonical nonunitary transformations and develop general many-body theory for the electron transport through the interacti…
We use a generalized Master equation (GME) formalism to describe the non-equilibrium time-dependent transport through a short quantum wire connected to semi-infinite biased leads. The contact strength between the leads and the wire are modulated by out-of-phase time-dependent functions which simulate a turnstile device. One lead is fixed at one end of the sample whereas the other lead has a variable placement. The system is described by a lattice model. We find that the currents in both leads depend on the placement of the second lead. In the rather small bias regime we obtain transient curren…
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 re…
The interplay between interference effects and electron-electron interactions in electron transport through an interacting double quantum dot system is investigated using a hierarchical quantum master equation approach which becomes exact if carried to infinite order and converges well if the temperature is not too low. Decoherence due to electron-electron interactions is found to give rise to pronounced negative differential resistance, enhanced broadening of structures in current-voltage characteristics and an inversion of the electronic population. Dependence on gate voltage is shown to be …
Quantum transport through single molecules is very sensitive to the strength of the molecule-electrode contact. When a molecular junction weakly coupled to external electrodes, charging effects do play an important role (Coulomb blockade regime). In this regime, the non-equilibrium Green function is usually substituted with master equation approaches, which prevents the density functional theory from describing Coulomb blockade in non-equilibrium case. Last year, we proposed an Ansatz to combine the non-equilibrium Green function technique with the equation of motion method. With help of it, C…
We report an observation of charge conservation breaking in a model study of electronic current noise of transport through a dissipative double quantum dot within generalized master equation formalism. We study the current noise through a double quantum dot coupled to two electronic leads in the high bias limit and a dissipative heat bath in the weak coupling limit. Our calculations are based on the solution of a Markovian generalized master equation. Zero-frequency component of the current noise calculated within the system, i.e., between the two dots, via the quantum regression theorem exhib…
A quantum master equation (QME) is derived for the many-body density matrix of an open current-carrying system weakly coupled to two metal leads. The dynamics and the steady-state properties of the system for arbitrary bias are studied using projection operator techniques, which keep track of number of electrons in the system. We show that coherences between system states with different number of electrons, n, (Fock space coherences) do not contribute to the transport to second order in system-lead coupling. However, coherences between states with the same n may effect transport properties whe…
We recall theoretical studies on transient transport through interacting mesoscopic systems. It is shown that a generalized master equation (GME) written and solved in terms of many-body states provides the suitable formal framework to capture both the effects of the Coulomb interaction and electron--photon coupling due to a surrounding single-mode cavity. We outline the derivation of this equation within the Nakajima-Zwanzig formalism and point out technical problems related to its numerical implementation for more realistic systems which can neither be described by non-interacting two-level …
We study shot noise in tunneling current through a double quantum dot connected to two electric leads. We derive two master equations in the occupation-state basis and the eigenstate basis to describe the electron dynamics. The approach based on the occupation-state basis, despite widely used in many previous studies, is valid only when the interdot coupling strength is much smaller than the energy difference between the two dots. In contrast, the calculations using the eigenstate basis are valid for an arbitrary interdot coupling. We show that the master equation in the occupation-state basis…
Understanding electron transport processes in molecular wires connected between contacts is a central focus in the field of molecular electronics. Especially, the dephasing effect causing tunneling-to-hopping transition has great importance from both applicational and fundamental points of view. We analyzed coherent and incoherent electron transmission through conjugated molecular wires by means of density-functional tight-binding theory within the D'Amato-Pastawski model. Our approach can study explicitly the structure/transport relationship in molecular junctions in a dephasing environmental…
The superconducting reservoir effect on the current carrying transport of a double quantum dot in Markovian regime is investigated. For this purpose, a quantum master equation (QME) at finite temperature is derived for the many-body density matrix of an open quantum system. The dynamics and the steady-state properties of the double quantum dot system for arbitrary bias are studied. We will show that how the populations and coherencies of the system states are affected by superconducting leads. The energy parameter of system contains essentially four contributions due to dots system-electrodes …
We derive a Markovian master equation for the single-electron density matrix, applicable to quantum cascade lasers (QCLs). The equation conserves the positivity of the density matrix, includes off-diagonal elements (coherences) as well as in-plane dynamics, and accounts for electron scattering with phonons and impurities. We use the model to simulate a terahertz-frequency QCL, and compare the results with both experiment and simulation via nonequilibrium Green's functions (NEGF). We obtain very good agreement with both experiment and NEGF when the QCL is biased for optimal lasing. For the cons…
We present a novel hierarchical quantum master equation (HQME) approach which provides a numerically exact description of nonequilibrium charge transport in nanosystems with electronic-vibrational coupling. In contrast to previous work [Phys. Rev. B $\bf{94}$, 201407 (2016)], the active vibrational degrees of freedom are treated in the reservoir subspace and are integrated out. This facilitates applications to systems with very high excitation levels, for example due to current-induced heating, while properties of the vibrational degrees of freedom, such as the excitation level and other momen…
We give a detailed comparison of the hierarchical quantum master equation (HQME) method to a continuous-time quantum Monte Carlo (CT-QMC) approach, assessing the usability of these numerically exact schemes as impurity solvers in practical nonequilibrium calculations. We review the main characteristics of the methods and discuss the scaling of the associated numerical effort. We substantiate our discussion with explicit numerical results for the nonequilibrium transport properties of a single-site Anderson impurity. The numerical effort of the HQME scheme scales linearly with the simulation ti…
Transport of electrons through two-dimensional semiconductor structures on the nanoscale in the presence of perpendicular magnetic field depends on the interplay of geometry of the system, the leads, and the magnetic length. We use a generalized master equation (GME) formalism to describe the transport through the system without resorting to the Markov approximation. Coupling to the leads results in elastic and inelastic processes in the system that are described to a high order by the integro-differential equation of the GME formalism. Geometrical details of systems and leads leave their fing…
We present measurements of magnetic tunnel junctions made using a self-assembled-monolayer molecular barrier. Ni/octanethiol/Ni samples were fabricated in a nanopore geometry. The devices exhibit significant changes in resistance as the angle between the magnetic moments in the two electrodes is varied, demonstrating that low-energy electrons can traverse the molecular barrier while maintaining spin coherence. An analysis of the voltage and temperature dependence of the data suggests that the spin-coherent transport signals can be degraded by localized states in the molecular barriers.
Electron transport characteristics are investigated through some molecular chains attached to two non-superconducting electrodes by the use of Green's function method. Here we do parametric calculations based on the tight-binding formulation to characterize the electron transport through such bridge systems. The transport properties are significantly influenced by (a) the length of the molecular chain and (b) the molecule-to-electrodes coupling strength and here we focus are results in these aspects. In this context we also discuss the steady state current fluctuations, the so-called shot nois…
Electron transport characteristics through molecular wires are studied by using the Green's function formalism. Parametric calculations are performed based on the tight-binding model to investigate the transport properties through the wires. The transport characteristics are significantly influenced by (a) the interference effects, (b) chemical substituent group, (c) molecule-to-electrode coupling strength and (d) the gate voltage, and, here we focus our results in these aspects. In this article we also discuss the noise power of current fluctuations. The noise power gives key information abou…
We explore multi-terminal quantum transport through a benzene molecule threaded by an Aharonov-Bohm flux $ϕ$. A simple tight-binding model is used to describe the system and all the calculations are done based on the Green's function formalism. With a brief description of two-terminal quantum transport, we present a detailed study of three-terminal transport properties through the benzene molecule to reveal the actual mechanism of electron transport. Here we numerically compute the multi-terminal conductances, reflection probabilities and current-voltage characteristics in the aspects of molec…
An important class of approaches to the description of electronic transport through molecules and quantum dots is based on the master equation. We discuss various formalisms for deriving a master equation and their interrelations. It is shown that the master equations derived by Wangsness, Bloch, and Redfield and by Koenig et al. are equivalent. The roles of the large-reservoir and Markov approximations are clarified. The Markov approximation is traced back to nonzero bias voltage and temperature, whereas interactions and the corresponding rapid relaxation in the leads are shown to be irreleva…
The effect of molecular vibrations on electronic transport is investigated with the smallest fullerene C20 bridge, utilizing the Keldysh nonequilibrium Green's function techniques combined with the tight-binding molecular-dynamics method. Large discontinuous steps appear in the differential conductance when the applied bias-voltage matches particular vibrational energies. The magnitude of the step is found to vary considerably with the vibrational mode and to depend on the local electronic states besides the strength of electron-vibration coupling. On the basis of this finding, a novel way to …
We analyze non-equilibrium current transport in molecular electronic devices, using as an example devices formed by two terphenyl dithiol molecules attached to gold electrodes. Using a first-principles based self-consistent matrix Green's function method, we show that the spatially resolved current density profiles and local electrochemical potential drops provide valuable information regarding the local field effect on molecular transport, which depend on the internal structure of the molecules and cannot be obtained from measuring the current- and conductance-voltage characteristics alone.
We discuss the conductance of a molecular bridge between mesoscopic electrodes supporting low-dimensional transport and bearing an internal structure. As an example for such nanoelectrodes we assume semi-infinite (carbon) nanotubes. In the Landauer scattering matrix approach, we show that the conductance of this hybrid is very sensitive to the geometry of the contact unlike the usual behaviour in the presence of bulk electrodes.
We discuss the description of full counting statistics in quantum transport with a nonMarkovian master equation. We focus on differences arising from whether charge is considered as a classical or a quantum degree of freedom. These differences manifest themselves in the inhomogeneous term of the master equation which describes initial correlations. We describe the influence on current and in particular, the finite-frequency shotnoise. We illustrate these ideas by studying transport through a quantum dot and give results that include both sequential and cotunneling processes. Importantly, the n…
We study coherent transport through a double quantum dot. Its two electronic leads induce electronic matter and energy transport and a phonon reservoir contributes further energy exchanges. By treating the system-lead couplings perturbatively, whereas the coupling to vibrations is treated non-perturbatively in a polaron-transformed frame, we derive a thermodynamic consistent low-dimensional master equation. When the number of phonon modes is finite, a Markovian description is only possible when these couple symmetrically to both quantum dots. For a continuum of phonon modes however, also asymm…
Understanding the electronic and phononic transport properties of junctions consisting of a scattering region such as a nanoscale matters or molecules connected to two or more electrodes is the central basis for future nano and molecular scale applications. The theoretical and mathematical techniques to treat electron and phonon transport are leading to model the physical properties of nano and molecular scale junctions. My aim in this paper is to review the theoretical and mathematical techniques to treat electron and phonon transport in nano and molecular scale junctions leading to models of…
The quantum master equation applied to electronic transport through nanoscopic devices provides information not only on the stationary state but also on the dynamics. The dynamics is characterized by the eigenvalues of the transition-rate matrix, or generator, of the master equation. We propose to use the spectrum of these eigenvalues as a tool for the study of nanoscopic transport. We illustrate this idea by analyzing a molecular quantum dot with an electronic orbital coupled to a vibrational mode, which shows the Franck-Condon blockade if the coupling is strong. Our approach provides complem…
Based on our recently developed quantum transport theory in term of an exact master equation, the corresponding particle-number resolved ($n$-resolved) master equation and the related shot noise spectrum formalism covering the full frequency range are constructed. We demonstrate that the noise spectra of transport current through single quantum dot and double quantum dots show characteristic steps and/or peak-dips in different tunneling frequency regimes through tuning the applied bias voltage and/or gate voltage at low temperatures. The peak-dips crossing the tunneling resonant frequencies is…
Full counting statistics (FCS) for the transport through a molecular quantum dot magnet is studied theoretically in the incoherent tunneling regime. We consider a model describing a single-level quantum dot, magnetically coupled to an additional local spin, the latter representing the total molecular spin s. We also assume that the system is in the strong Coulomb blockade regime, i.e., double occupancy on the dot is forbidden. The master equation approach to FCS introduced in Ref. [12] is applied to derive a generating function yielding the FCS of charge and current. In the master equation app…
This review presents recent results on the physics of electron transport in molecular devices. The review is organized as follows. A brief description of molecular junction (MJ) technology is first given followed by an introduction to the basic physics of electron transport through MJs from DC to about 20 GHz. Then, several sections review selected results on spin-dependent transport, plasmonics, quantum interferences, thermal transport and electronic noise in ME devices.
The consistency with Onsager's theorem is examined for commonly used perturbative approaches, such as the Redfield and second-order von Neumann master equations, for thermoelectric transport through nanostructures. We study a double quantum dot, which requires coherences between states for a correct description, and we find that these perturbative approaches violate Onsager's theorem. We show that the deviations from the theorem scale with the lead-coupling strength in an order beyond the one considered systematically in the respective approach.
We study the transport properties of the Fano-Anderson model with a Lorentzian-shaped density of states in one of the electronic reservoirs. We explicitly show that the energy dependence of the density of states can cause non-Markovian effects and that the non-Markovian master equation may fail if these effects are strong. We evaluate the stationary current, the zero frequency current noise and the occupation dynamics of the resonant level by means of a quantum master equation approach within different approximation schemes and compare the results to the exact solution obtained by scattering t…
An ab initio based theoretical approach to describe nonequilibrium many-body effects in molecular transport is developed. We introduce a basis of localized molecular orbitals and formulate the many-body model in this basis. In particular, the Hubbard-Anderson Hamiltonian is derived for single-molecule junctions with intermediate coupling to the leads. As an example we consider a benzenedithiol junction with gold electrodes. An effective few-level model is obtained, from which spectral and transport properties are computed and analyzed. Electron-electron interaction crucially affects transport …
Solution of master equations by fermionic-duality: Time-dependent charge and heat currents through an interacting quantum dot proximized by a superconductor
Thermal conductance of single molecular junctions at room temperature has been measured recently using picowatt-resolution scanning probes. However, fully understanding thermal transport in a much wider temperature range is needed for the exploration of energy transfer at single-molecular limit and the development of single-molecular devices. Here, employing a semiclassical Langevin molecular dynamics method, a comparative study is performed on the thermal transport of an alkane chain between Au and graphene electrodes, respectively. We illustrate the different roles of quantum statistics and …
We explore electron transport properties in molecular wires made of heterocyclic molecules (pyrrole, furan and thiophene) by using the Green's function technique. Parametric calculations are given based on the tight-binding model to describe the electron transport in these wires. It is observed that the transport properties are significantly influenced by (a) the heteroatoms in the heterocyclic molecules and (b) the molecule-to-electrodes coupling strength. Conductance ($g$) shows sharp resonance peaks associated with the molecular energy levels in the limit of weak molecular coupling, while t…
We study qubit-mediated energy transfer between two electron reservoirs by adopting a numerically-exact influence functional path-integral method. This non-perturbative technique allows us to study the system's dynamics beyond the weak coupling limit. Our simulations for the energy current indicate that perturbative-Markovian Master equation predictions significantly deviate from exact numerical results already at intermediate coupling, $πρα_{j,j'}\gtrsim 0.4$, where $ρ$ is the metal (Fermi sea) density of states, taken as a constant, and $α_{j,j'}$ is the scattering potential energy of electr…
