Green’s function methods for single molecule junctions

Floquet non-equilibrium Green's function and Floquet quantum master equation for electronic transport: The role of electron-electron interactions and spin current with circular light

The non-equilibrium Green's function (NEGF) and quantum master equation (QME) are two main classes of approaches for electronic transport. We discuss various Floquet variances of these formalisms for transport properties of a quantum dot driven via interaction with an external periodic field. We first derived two versions of the Floquet NEGF. We also explore an ansatz of the Floquet NEGF formalism for the interacting systems. In addition, we derived two versions of Floquet QME in the weak interaction regime. With each method, we elaborate on the evaluation of the expectation values of the number and current operators. We examined these methods for transport through a two-level system that is subject to periodic driving. The numerical results of all four methods show good agreement for non-interacting systems in the weak regime. Furthermore, we have observed that circular light can introduce spin current. We expect these Floquet quantum transport methods to be useful in studying molecular junctions exposed to light.

Emergence of Boltzmann Subspaces in Open Quantum Systems Far from Equilibrium

Single molecule junctions are important examples of complex out-of-equilibrium many-body quantum systems. We identify a nontrivial clustering of steady state populations into distinctive subspaces with Boltzmann-like statistics, which persist far from equilibrium. Such Boltzmann subspaces significantly reduce the information needed to describe the steady state, enabling modeling of high-dimensional systems that are otherwise beyond the reach of current computations. The emergence of Boltzmann subspaces is demonstrated analytically and numerically for fermionic transport systems of increasing complexity.

Assessing the importance of multireference correlation in predicting reversed conductance decay

In a classical electronic resistor, conductance decays as the device length increases according to Ohm's Law. While most molecular series display a comparable exponential decay in conductance with increasing molecular length, a class of single-molecule device series exists where conductance instead increases with molecular/device length, a phenomenon called reversed conductance decay. While reversals of conductance decay have been repeatedly theoretically predicted, they have been far more difficult to demonstrate experimentally. Previous studies have suggested that theoretical multi-reference(static) correlation errors may be a major cause of this discrepancy, yet most single-molecule transport methods are unable to treat multireference correlation. Using our unique multireference transport method based on non-equilibrium Green's function and multiconfigurational pair-density functional theory (NEGF-MCPDFT), we examined a previously predicted case of reversed conductance decay in systems of linear chains of phenyl rings with varying lengths and electrode designs. We compare our NEGF-MCPDFT results to those of non-multireference NEGF methods to quantify the exact role of static correlation in conductance decay reversals and clarify their relative importance to geometric and electrode design/coupling considerations.

Shaping Electronic Flows with Strongly Correlated Physics

Nonequilibrium quantum transport is of central importance in nanotechnology. Its description requires the understanding of strong electronic correlations that couple atomic-scale phenomena to the nanoscale. So far, research in correlated transport has focused predominantly on few-channel transport, precluding the investigation of cross-scale effects. Recent theoretical advances enable the solution of models that capture the interplay between quantum correlations and confinement beyond a few channels. This problem is the focus of this study. We consider an atomic impurity embedded in a metallic nanosheet spanning two leads, showing that transport is significantly altered by tuning only the phase of a single local hopping parameter. Furthermore─depending on this phase─correlations reshape the electronic flow throughout the sheet, either funneling it through the impurity or scattering it away from a much larger region. This demonstrates the potential for quantum correlations to bridge length scales in the design of nanoelectronic devices and sensors.

Quantum Chemical Studies on Possible Molecular Devices Based on Electric Field-Induced Intramolecular Charge Transfer

We report quantum chemical studies on possible molecular devices working based on electric field-induced intramolecular charge transfer (EFIMCT). In the case of donor-acceptor (DA)-type molecular systems, intramolecular charge transfer (IMCT) can be induced by applying the external electric field to molecular systems along the charge transport direction, providing a possible switching mechanism which does not depend upon the electron-phonon coupling effect and is different from the negative differential resistance mechanism observed in the well-known NO2-substituted phenylene ethynylene oligomers. When the EFIMCT proceeds, the molecular systems have strong static electron correlation effects, where the standard nonequilibrium Green's function-density functional theory (DFT) approach cannot be applied to the molecular junction. As a first step toward practical switching devices, we do quantum chemical studies on the EFIMCT in such molecular systems as an isolated molecule, instead of using the electrode-junction-electrode open quantum system model. A prototype molecule P1 is designed as a tentative candidate molecule where the EFIMCT can proceed. The complete active space self-consistent field (CASSCF) molecular orbital calculations on P1 indicate that the EFIMCT can proceed at the external electric field intensity of 0.003 au, corresponding to about 2.25 V bias voltage. This calculated result strongly suggests that the development of this type of switching devices working at practically low bias voltage is feasible if the molecular system is properly designed. Broken symmetry unrestricted Hartree-Fock and spin-polarized Kohn-Sham DFT calculations also qualitatively reproduce the CASSCF results on P1, to some extent, indicating that these approaches can be employed for rough estimations on the EFIMCT such as the first screening of a large quantity of candidate molecules for this type of molecular devices. The possibility of molecular memory devices based on the EFIMCT is also discussed by analyzing the ground and excited potential energy surface model. Remaining challenges to develop practical molecular devices are discussed.

Control and Enhancement of Single-Molecule Electroluminescence through Strong Light-Matter Coupling

The energetic positions of molecular electronic states at molecule/electrode interfaces are crucial factors for determining the transport and optoelectronic properties of molecular junctions. Strong light-matter coupling offers a potential for manipulating these factors, enabling a boost in the efficiency and versatility of these junctions. Here, we investigate electroluminescence from single-molecule junctions in which the molecule is strongly coupled with the vacuum electromagnetic field in a plasmonic nanocavity. We demonstrate an improvement in the electroluminescence efficiency by employing the strong light-matter coupling in conjunction with the characteristic feature of single-molecule junctions to selectively control the formation of the lowest-energy excited state. The mechanism of efficiency improvement is discussed based on the energetic position and composition of the formed polaritonic states. Our findings indicate the possibility to manipulate optoelectronic conversion in molecular junctions by strong light-matter coupling and contribute to providing design principles for developing efficient molecular optoelectronic devices.

A derivation of the conditions under which bosonic operators exactly capture fermionic structure and dynamics

The dynamics of many-body fermionic systems are important in problems ranging from catalytic reactions at electrochemical surfaces to transport through nanojunctions and offer a prime target for quantum computing applications. Here, we derive the set of conditions under which fermionic operators can be exactly replaced by bosonic operators that render the problem amenable to a large toolbox of dynamical methods while still capturing the correct dynamics of n-body operators. Importantly, our analysis offers a simple guide on how one can exploit these simple maps to calculate nonequilibrium and equilibrium single- and multi-time correlation functions essential in describing transport and spectroscopy. We use this to rigorously analyze and delineate the applicability of simple yet effective Cartesian maps that have been shown to correctly capture the correct fermionic dynamics in select models of nanoscopic transport. We illustrate our analytical results with exact simulations of the resonant level model. Our work provides new insights as to when one can leverage the simplicity of bosonic maps to simulate the dynamics of many-electron systems, especially those where an atomistic representation of nuclear interactions becomes essential.

One-Shot GW Transport Calculations: A Charge-Conserving Solution

Transport measurements are a common method of characterizing small systems in chemistry and physics. When interactions are negligible, the current through submicrometer structures can be obtained using the Landauer formula. Meir and Wingreen derived an exact expression for the current in the presence of interactions. This powerful tool requires knowledge of the exact Green's function. Alternatively, self-consistent approximations for the Green's function are frequently sufficient for calculating the current while crucially satisfying all conservation laws. We provide here yet another alternative, circumventing the high computational cost of these methods. We present expressions for the electric and thermal currents in which the lowest-order self-energy is summed to all orders (one-shot GW approximation). We account for both self-energy and vertex corrections such that current is conserved. Our formulas for the currents capture important features due to interactions and, hence, provide a powerful tool for cases in which the exact solution cannot be found.

Chiral-Induced Spin Selectivity and Non-equilibrium Spin Accumulation in Molecules and Interfaces: A First-Principles Study

Electrons moving through chiral molecules are selected according to their spin orientation and the helicity of the molecule, an effect known as chiral-induced spin selectivity (CISS). The underlying physical mechanism is not yet completely understood. To help elucidate this mechanism, a non-equilibrium Green's function method, combined with a Landauer approach and density functional theory, is applied to carbon helices contacted by gold electrodes, resulting in spin polarization of transmitted electrons. Spin polarization is also observed in the non-equilibrium electronic structure of the junctions. While this spin polarization is small, its sign changes with the direction of the current and with the handedness of the molecule. While these calculations were performed with a pure exchange-correlation functional, previous studies suggest that computationally more expensive hybrid functionals may lead to considerably larger spin polarization in the electronic structure. Thus, non-equilibrium spin polarization could be a key component in understanding the CISS mechanism.

A Green's Function Approach for Determining Surface Induced Broadening and Shifting of Molecular Energy Levels

Upon adsorption of a molecule onto a surface, the molecular energy levels (MELs) broaden and change their alignment. This phenomenon directly affects electron transfer across the interface and is, therefore, a fundamental observable that influences electrochemical device performance. Here, we propose a rigorous parameter-free framework, built upon the theoretical construct of Green's functions, for studying the interface between a molecule and a bulk surface and its effect on MELs. The method extends beyond the usual wide-band limit approximation, and its generality allows its use with any level of electronic structure theory. We demonstrate its ability to predict the broadening and shifting of MELs as a function of intramolecular coupling, molecule/surface coupling, and the surface density of states for a molecule with two MELs adsorbed on a one-dimensional model metal surface. The new approach could help provide guidelines for the design and experimental characterization of electrochemical devices with optimal electron transport.

Gate-controlled electron quantum interference logic

Inspired by using the wave nature of electrons for electron quantum optics, we propose a new type of electron quantum interference structure, where single-electron waves are coherently injected into a gate-controlled, two-dimensional waveguide and exit through one or more output channels. The gate-controlled interference effects lead to specific current levels in the output channels, which can be used to realize logic gate operations, e.g., NAND or NOR gates. The operating principle is shown by coherent, dynamic Wigner quantum electron transport simulations. A discussion of classical simulations (Boltzmann) allows to outline the underlying process of interference. Contrary to other electron control approaches used for advanced information processing, no magnetic or photonic mechanisms are involved.

Long-range charge transport in homogeneous and alternating-rigidity chains

We study the interplay of intrinsic-electronic and environmental factors in long-range charge transport across molecular chains with up to N ∼ 80 monomers. We describe the molecular electronic structure of the chain with a tight-binding Hamiltonian. Thermal effects in the form of electron decoherence and inelastic scattering are incorporated with the Landauer–Büttiker probe method. In short chains of up to ten units, we observe the crossover between coherent (tunneling, ballistic) motion and thermally-assisted conduction, with thermal effects enhancing the current beyond the quantum coherent limit. We further show that unconventional (nonmonotonic with size) transport behavior emerges when monomer-to-monomer electronic coupling is made large. In long chains, we identify a different behavior, with thermal effects suppressing the conductance below the coherent-ballistic limit. With the goal to identify a minimal model for molecular chains displaying unconventional and effective long-range transport, we simulate a modular polymer with alternating regions of high and low rigidity. Simulations show that, surprisingly, while charge correlations are significantly affected by structuring environmental conditions, reflecting charge delocalization, the electrical resistance displays an averaging effect, and it is not sensitive to this patterning. We conclude by arguing that efficient long-range charge transport requires engineering both internal electronic parameters and environmental conditions.

Numerically "exact" simulations of a quantum Carnot cycle: Analysis using thermodynamic work diagrams

We investigate the efficiency of a quantum Carnot engine based on open quantum dynamics theory. The model includes time-dependent external fields for the subsystems controlling the isothermal and isentropic processes and for the system--bath (SB) interactions controlling the transition between these processes. Numerical simulations are conducted in a nonperturbative and non-Markovian SB coupling regime using the hierarchical equations of motion under these fields at different cycle frequencies. The work applied to the total system and the heat exchanged with the baths are rigorously evaluated. In addition, by regarding quasi-static work as free energy, we compute the quantum thermodynamic variables and analyze the simulation results using thermodynamic work diagrams for the first time. Analysis of these diagrams indicates that, in the strong SB coupling region, the fields for the SB interactions are major sources of work, while in other regions, the field for the subsystem is a source of work. We find that the maximum efficiency is achieved in the quasi-static case and is determined solely by the bath temperatures, regardless of the SB coupling strength, which is a numerical manifestation of Carnot's theorem.

The laws of thermodynamics for quantum dissipative systems: A quasi-equilibrium Helmholtz energy approach

Using the quasi-equilibrium Helmholtz energy (qHE), defined as the thermodynamic work in a quasi-static process, we investigate the thermal properties of both an isothermal process and a transition process between the adiabatic and isothermal states (adiabatic transition). Here, the work is defined by the change in energy from a steady state to another state under a time-dependent perturbation. In particular, the work for a quasi-static change is regarded as thermodynamic work. We employ a system--bath model that involves time-dependent perturbations in both the system and the system--bath interaction. We conduct numerical experiments for a three-stroke heat machine (a Kelvin-Planck cycle). For this purpose, we employ the hierarchical equations of motion (HEOM) approach. These experiments involve an adiabatic transition field that describes the operation of an adiabatic wall between the system and the bath. Thermodynamic--work diagrams for external fields and their conjugate variables, similar to the P-V diagram, are introduced to analyze the work done for the system in the cycle. We find that the thermodynamic efficiency of this machine is zero because the field for the isothermal processes acts as a refrigerator, whereas that for the adiabatic wall acts as a heat engine. This is a numerical manifestation of the Kelvin-Planck statement, which states that it is impossible to derive mechanical effects from a single heat source. These HEOM simulations serve as a rigorous test of thermodynamic formulations because the second law of thermodynamics is only valid when the work involved in the operation of adiabatic wall is treated accurately.

Generalized master equation for charge transport in a molecular junction: Exact memory kernels and their high order expansion

We derive a set of generalized master equations (GMEs) to study charge transport dynamics in molecular junctions using the Nakajima-Zwanzig-Mori projection operator approach. In the new GME, time derivatives of population on each quantum state of the molecule, as well as the tunneling current, are calculated as the convolution of time non-local memory kernels with populations on all system states. The non-Markovian memory kernels are obtained by combining the hierarchical equations of motion (HEOM) method and a previous derived Dyson relation for the exact kernel. A perturbative expansion of these memory kernels is then calculated using the extended HEOM developed in our previous work [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. By using the resonant level model and the Anderson impurity model, we study properties of the exact memory kernels and analyze convergence properties of their perturbative expansions with respect to the system-bath coupling strength and the electron-electron repulsive energy. It is found that exact memory kernels calculated from HEOM exhibit short memory times and decay faster than the population and current dynamics. The high order perturbation expansion of the memory kernels can give converged results in certain parameter regimes. The Padé and Landau-Zener resummation schemes are also found to give improved results over low order perturbation theory.

Coherent and incoherent contributions to molecular electron transport

We numerically isolate the limits of validity of the Landauer approximation to describe charge transport along molecular junctions in condensed phase environments. To do so, we contrast Landauer with exact time-dependent non-equilibrium Green’s function quantum transport computations in a two-site molecular junction subject to exponentially correlated noise. Under resonant transport conditions, we find Landauer accuracy to critically depend on intramolecular interactions. By contrast, under nonresonant conditions, the emergence of incoherent transport routes that go beyond Landauer depends on charging and discharging processes at the electrode–molecule interface. In both cases, decreasing the rate of charge exchange between the electrodes and molecule and increasing the interaction strength with the thermal environment cause Landauer to become less accurate. The results are interpreted from a time-dependent perspective where the noise prevents the junction from achieving steady-state and from a fully quantum perspective where the environment introduces dephasing in the dynamics. Using these results, we analyze why the Landauer approach is so useful to understand experiments, isolate regimes where it fails, and propose schemes to chemically manipulate the degree of transport coherence.

Orbital-resolved visualization of single-molecule photocurrent channels

Given its central role in utilizing light energy, photoinduced electron transfer (PET) from an excited molecule has been widely studied^1-6. However, even though microscopic photocurrent measurement methods^7-11 have made it possible to correlate the efficiency of the process with local features, spatial resolution has been insufficient to resolve it at the molecular level. Recent work has, however, shown that single molecules can be efficiently excited and probed when combining a scanning tunnelling microscope (STM) with localized plasmon fields driven by a tunable laser^12,13. Here we use that approach to directly visualize with atomic-scale resolution the photocurrent channels through the molecular orbitals of a single free-base phthalocyanine (FBPc) molecule, by detecting electrons from its first excited state tunnelling through the STM tip. We find that the direction and the spatial distribution of the photocurrent depend sensitively on the bias voltage, and detect counter-flowing photocurrent channels even at a voltage where the averaged photocurrent is near zero. Moreover, we see evidence of competition between PET and photoluminescence¹², and find that we can control whether the excited molecule primarily relaxes through PET or photoluminescence by positioning the STM tip with three-dimensional, atomic precision. These observations suggest that specific photocurrent channels can be promoted or suppressed by tuning the coupling to excited-state molecular orbitals, and thus provide new perspectives for improving energy-conversion efficiencies by atomic-scale electronic and geometric engineering of molecular interfaces.

Computational perspective on recent advances in quantum electronics: from electron quantum optics to nanoelectronic devices and systems

Quantum electronics has significantly evolved over the last decades. Where initially the clear focus was on light-matter interactions, nowadays approaches based on the electron's wave nature have solidified themselves as additional focus areas. This development is largely driven by continuous advances in electron quantum optics, electron based quantum information processing, electronic materials, and nanoelectronic devices and systems. The pace of research in all of these areas is astonishing and is accompanied by substantial theoretical and experimental advancements. What is particularly exciting is the fact that the computational methods, together with broadly available large-scale computing resources, have matured to such a degree so as to be essential enabling technologies themselves. These methods allow to predict, analyze, and design not only individual physical processes but also entire devices and systems, which would otherwise be very challenging or sometimes even out of reach with conventional experimental capabilities. This review is thus a testament to the increasingly towering importance of computational methods for advancing the expanding field of quantum electronics. To that end, computational aspects of a representative selection of recent research in quantum electronics are highlighted where a major focus is on the electron's wave nature. By categorizing the research into concrete technological applications, researchers and engineers will be able to use this review as a source for inspiration regarding problem-specific computational methods.

Control of Molecular Conformation and Crystal Packing of Biphenyl Derivatives

This Review presents a discussion of the conformation of biphenyl derivatives in different chemical environments. The interplay between aromatic stabilization and steric repulsion, normally considered to explain the conformation of the molecule, is contrasted with the interpretation provided by models not based on molecular orbitals. The electronic control of conformation by means of appropriate hydrogen substitution is discussed by examples taken from chemistry and molecular electronics. Supramolecular synthons involving biphenyl are critically analyzed in terms of the molecular conformation, crystal packing and intermolecular forces. Some directions for future research on the control of the conformation of biphenyls are also presented.

Long-Range Surface-Assisted Molecule-Molecule Hybridization

Metalated phthalocyanines (Pc's) are robust and versatile molecular complexes, whose properties can be tuned by changing their functional groups and central metal atom. The electronic structure of magnesium Pc (MgPc)-structurally and electronically similar to chlorophyll-adsorbed on the Ag(100) surface is investigated by low-temperature scanning tunneling microscopy and spectroscopy, non-contact atomic force microscopy, and density functional theory. Single, isolated MgPc's exhibit a flat, fourfold rotationally symmetric morphology, with doubly degenerate, partially populated (due to surface-to-molecule electron transfer) lowest unoccupied molecular orbitals (LUMOs). In contrast, MgPc's with neighbouring molecules in proximity undergo a lift of LUMOs degeneracy, with a near-Fermi local density of states with reduced twofold rotational symmetry, indicative of a long-range attractive intermolecular interaction. The latter is assigned to a surface-mediated two-step electronic hybridization process. First, LUMOs interact with Ag(100) conduction electrons, forming hybrid molecule-surface orbitals with enhanced spatial extension. Then, these delocalized molecule-surface states further hybridize with those of neighbouring molecules. This work highlights how the electronic structure of molecular adsorbates-including orbital degeneracies and symmetries-can be significantly altered via surface-mediated intermolecular hybridization, over extended distances (beyond 3 nm), having important implications for prospects of molecule-based solid-state technologies.

Influence of the Contact Geometry and Counterions on the Current Flow and Charge Transfer in Polyoxometalate Molecular Junctions: A Density Functional Theory Study

Polyoxometalates (POMs) are promising candidates for molecular electronic applications because (1) they are inorganic molecules, which have better CMOS compatibility compared to organic molecules; (2) they are easily synthesized in a one-pot reaction from metal oxides (MO _x ) (where the metal M can be, e.g., W, V, or Mo, and x is an integer between 4 and 7); (3) POMs can self-assemble to form various shapes and configurations, and thus the chemical synthesis can be tailored for specific device performance; and (4) they are redox-active with multiple states that have a very low voltage switching between polarized states. However, a deep understanding is required if we are to make commercial molecular devices a reality. Simulation and modeling are the most time efficient and cost-effective methods to evaluate a potential device performance. Here, we use density functional theory in combination with nonequilibrium Green's function to study the transport properties of [W₁₈O₅₄(SO₃)₂]^4-, a POM cluster, in a variety of molecular junction configurations. Our calculations reveal that the transport profile not only is linked to the electronic structure of the molecule but also is influenced by contact geometry and presence of ions. More specifically, the contact geometry and the number of bonds between the POM and the electrodes determine the current flow. Hence, strong and reproducible contact between the leads and the molecule is mandatory to establish a reliable fabrication process. Moreover, although often ignored, our simulations show that the charge balancing counterions activate the conductance channels intrinsic to the molecule, leading to a dramatic increase in the computed current at low bias. Therefore, the role of these counterions cannot be ignored when molecular based devices are fabricated. In summary, this work shows that the current transport in POM junctions is determined by not only the contact geometry between the molecule and the electrode but also the presence of ions around the molecule. This significantly impacts the transport properties in such nanoscale molecular electronic devices.

Time dependent reduced density matrix functional theory at strong correlation: insights from a two-site Anderson impurity model

The one-body density matrix has recently attracted considerable attention as a promising key quantity for the description of systems out of equilibrium. Its time evolution is given in terms of the two-body density matrix, and thus the central challenge is to find approximations to the latter. An extra layer of difficulty is added when dealing with strong electron correlations. In this work, we explore precisely this regime by looking at the two-site Anderson impurity model as a case study. To address the system's dynamics, we use an adiabatic approximation based on the exact ground-state two-body density matrix. We find that this adiabatic extension does not reproduce the exact results even for a slow switch-on of the external perturbation, and we trace back this behavior to the lack of an accurate imaginary part of the adiabatic approximation to the two-body density matrix. The attempt to restore an approximate imaginary part through a Hilbert transform of the real part works well only for very short times, but quickly deteriorates for longer times, with the one-body density matrix being pushed out of its N-representability domain. Our results thus pose an important constraint on practical prescriptions to perform the time evolution of the one-body density matrix.