Nonequilibrium Green Functions Simulations Research Articles

Classical and quantum theory of fluctuations for many‐particle systems out of equilibrium

AbstractCorrelated classical and quantum many‐particle systems out of equilibrium are of high interest in many fields, including dense plasmas, correlated solids, and ultracold atoms. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. While for classical systems, in principle, exact simulations are possible via molecular dynamics, this is not the case for quantum systems. Alternatively, one can use many‐particle approaches such as hydrodynamics, kinetic theory, or nonequilibrium Green functions (NEGF). However, NEGF exhibit a very unfavorable cubic scaling of the CPU time with the number of time steps. An alternative is the G1–G2 scheme [N. Schlünzen et al., Phys. Rev. Lett. 124, 076601 (2020)] which allows for NEGF simulations with time linear scaling, however, at the cost of large memory consumption. The reason is the need to store the two‐particle correlation function. This problem can be overcome for a number of approximations by reformulating the kinetic equations in terms of fluctuations – an approach that was developed, for classical systems, by Yu.L. Klimontovich [JETP 33, 982 (1957)]. Here, we present an overview of his ideas and extend them to quantum systems. In particular, we demonstrate that this quantum fluctuations approach can reproduce the nonequilibrium GW approximation [E. Schroedter et al., Cond. Matt. Phys. 25, 23401 (2022)] promising high accuracy at low computational cost which arises from an effective semiclassical stochastic sampling procedure. We also demonstrate how to extend the approach to the two‐time exchange‐correlation functions and the density response properties. [E. Schroedter et al., Phys. Rev. B 108, 205109 (2023)]. The results are equivalent to the Bethe–Salpeter equation for the two‐time exchange‐correlation function when the generalized Kadanoff‐Baym ansatz with Hartree‐Fock propagators is applied [E. Schroedter and M. Bonitz, phys. stat. sol. (b) 2024, 2300564].

Super-Suppression of Long-Wavelength Phonons in Constricted Nanoporous Geometries.

In a typical semiconductor material, the majority of the heat is carried by long-wavelength, long-mean-free-path phonons. Nanostructuring strategies to reduce thermal conductivity, a promising direction in the field of thermoelectrics, place scattering centers of size and spatial separation comparable to the mean free paths of the dominant phonons to selectively scatter them. The resultant thermal conductivity is in most cases well predicted using Matthiessen's rule. In general, however, long-wavelength phonons are not as effectively scattered as the rest of the phonon spectrum. In this work, using large-scale molecular-dynamics simulations, non-equilibrium Green's function simulations, and Monte Carlo simulations, we show that specific nanoporous geometries that create narrow constrictions in the passage of phonons lead to anticorrelated heat currents in the phonon spectrum. This effect results in super-suppression of long-wavelength phonons due to heat trapping and reductions in the thermal conductivity to values well below those predicted by Matthiessen's rule.

Accelerating Nonequilibrium Green Functions Simulations: The G1–G2 Scheme and Beyond

The theory of nonequilibrium Green functions (NEGF) has seen a rapid development over the recent three decades. Applications include diverse correlated many‐body systems in and out of equilibrium. Very good agreement with experiments and available exact theoretical results could be demonstrated if the proper selfenergy approximations were used. However, full two‐time NEGF simulations are computationally costly, as they suffer from a cubic scaling of the computation time with the simulation duration. Recently the G1–G2 scheme that exactly reformulates the generalized Kadanoff–Baym ansatz with Hartree–Fock propagators (HF‐GKBA) into time‐local equations is introduced, which achieves time‐linear scaling and allows for a dramatic speedup and extension of the simulations (Schluenzen et al. Phys. Rev. Lett. 2020, 124, 076601). Remarkably, this scaling is achieved quickly, and also for high‐level selfenergies, including the nonequilibrium GW and T‐matrix approximations (Joost et al. Phys. Rev. B 2020, 101, 245101). Even the dynamically screened ladder approximation is now feasible (Joost et al. Phys. Rev. B 2022, 105, 165155), and also applications to electron‐boson systems are demonstrated. Herein, an overview on recent results that are achieved with the G1–G2 scheme is presented. Problems and open questions are discussed and further ideas of how to overcome the current limitations of the scheme and present are presented. The G1–G2 scheme is illustrated by presenting applying it to the excitation dynamics of Hubbard clusters, to optical excitation of graphene, and to charge transfer during stopping of ions by correlated materials.

Metal Clusters Effectively Adjust the Local Environment of Polymeric Carbon Nitride for Bifunctional Overall Water Splitting.

Compared with single-atom catalysts, clusters not only possess more metal-loadings and stability but also provide flexible active sites to break the linear scaling relationship of multistep reactions. However, exploring precise structure-activity relationships and the synergistic effect between clusters and nanosheets is still in its infancy. Here, based on first-principles and nonequilibrium Green's function simulation, the C2N-supported Fe and Co tetrahedral clusters exhibit remarkable bifunctional catalytic performance with a very low overpotential of hydrogen (0.12 and 0.07 V) /oxygen (0.20 and 0.55 V) evolution reactions (HER/OER), respectively. The C2N-regulated Fe and Co clusters have suitable d-band centers around the Fermi surface for HER. In turn, the Fe and Co clusters activate the subadjacent dual-carbon sites for OER. Simultaneously, the cluster enhances the electronic conductivity of C2N, and the initial current only needs ultralow bias voltage around 0.1-0.4 V. The desired metal cluster regulation strategy offers cost-effective potential for advancing clean energy technology.

RSDFT-NEGF transport simulations in realistic nanoscale transistors

The paper presents a device simulator for computing transport characteristics from first principles. The developed computer program effectively performs large-scale parallel calculation of quasi-one-dimensional quantum transport in realistic nanoscale devices with thousands of atoms in the cross section area of the device channel. Our simulator is based on the real-space Kohn–Sham Hamiltonian in the density functional theory and improved numerical algorithms for reducing computational burden in non-equilibrium Green’s function (NEGF) method. Several computational improvements have been introduced in constructing a reduced quantum transport model from the original Kohn-Sham Hamiltonian and implementing the R-matrix computational scheme in the NEGF simulations.

Accelerating nonequilibrium Green function simulations with embedding self-energies

Real-time nonequilibrium Green functions (NEGFs) have been very successfully used to simulate the dynamics of correlated many-particle systems far from equilibrium. However, NEGF simulations are computationally expensive since the effort scales cubically with the simulation duration. Recently, we introduced the G1--G2 scheme that allows for a dramatic reduction to time-linear scaling [N. Schl\"unzen et al., Phys. Rev. Lett. 124, 076601 (2020); J.-P. Joost et al., Phys. Rev. B 101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the computational effort with the system size. In many situations where the system of interest is coupled to a bath, to electric contacts, or to similar macroscopic systems for which a microscopic resolution of the electronic properties is not necessary, efficient simplifications are possible. This is achieved by the introduction of an embedding self-energy---a concept that has been successful in standard NEGF simulations. Here, we demonstrate how the embedding concept can be introduced into the G1--G2 scheme, allowing us to drastically accelerate NEGF embedding simulations. The approach is compatible with all advanced self-energies that can be represented by the G1--G2 scheme [as described in J.-P. Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memoryless structure of the equations and their time-linear scaling. As a numerical illustration, we investigate the charge transfer between a Hubbard nanocluster and an additional site which is of relevance for the neutralization of ions in matter.

Full quantum simulation of Shockley–Read–Hall recombination in p-i-n and tunnel diodes

We have included Shockley–Read–Hall (SRH) generation/recombination in Non-equilibrium Green’s function (NEGF) calculations via a multiphonon relaxation model. The model has been used to study how the presence of defects affects the current–voltage characteristics of GaAs p-i-n diodes, and of an InGaAs tunnel diode. Regarding p-i-n diodes, we show that SRH generation/recombination is responsible for ideality factors approaching the theoretical value of two in the forward bias regime, while in reverse bias the recombination current density varies slowly with the applied voltage. In all the considered cases, the defects located in the center of the active region proved to be the most effective in allowing trap-assisted tunneling from the valence to the conduction band. Finally, the inclusion of SRH recombination in NEGF simulations of Esaki tunnel diodes permits to predict a realistic degradation of the peak-to-valley current ratio.

Strain engineering on the electronic properties and interface contact of graphene/GeN3 van der Waals heterostructure

Graphene-based van der Waals (vdW) heterostructures have shown great potential in electronic and optoelectronic nanodevices. Herein, we investigate the electronic property and Schottky barrier of graphene/GeN3 vdW heterostructure by first-principles calculations. It is noted that the electronic natures of graphene and GeN3 monolayers are well preserved in the heterostructure lattice due to the weak vdW interaction. Interestingly, the p-type Schottky contact in graphene/GeN3 heterostructure with a barrier height of 0.21 eV can be effectively tuned by both vertical and horizontal strains. Herein, the carrier concentration in the graphene layer reaches ∼1013 cm−2 level by strain engineering. It is noteworthy that the designed optoelectronic field-effect transistor based on graphene/GeN3 heterostructure exhibits distinguished responsivity of 0.297 AW−1 and impressive external quantum efficiency of 54.5% under illumination based on further non-equilibrium Green's function simulations. Our findings are of utmost significance for the metal–semiconductor vdW contact and corresponding applications in high-performance electronic and optoelectronic devices.

A Current Model for Ballistic/Quasi-Ballistic GNRFETs Considering Line-Edge Roughness

In this article, a physics-based analytical current model for armchair graphene nanoribbon field-effect transistors (A-GNRFETs) is developed, which is valid from ballistic to quasi-ballistic transport regimes. The roles of line-edge roughness (LER) for different roughness parameters and Fermi level of GNRs are numerically investigated. Furthermore, the effect of LER-related tunneling current has a strong impact on the drain–source current for the OFF state, and it is considered in this model. The good agreement between our proposed current model and the nonequilibrium Green function (NEGF) simulations proves its validity. The results indicate that depending on the geometrical and roughness parameters, the transport properties can be ballistic or quasi-ballistic.

Doublon Production in Correlated Materials by Multiple Ion Impacts

Recently, it was demonstrated that ions impacting a correlated finite graphene‐type honeycomb cluster can excite strong nonequilibrium states. In particular, this can lead to an enhanced population of bound pairs of electrons with opposite spin—doublons—where the doublon number can be increased further via multiple ion impacts. These predictions were made based on nonequilibrium Green functions (NEGF) simulations allowing for a time‐dependent non‐perturbative study of the energy loss of charged particles penetrating a strongly correlated system. Herein, these simulations are extended to larger clusters and longer simulation times, utilizing the recently developed G1–G2 scheme, which allows for a dramatic speedup of NEGF simulations. Furthermore, the dependence of the energy and doublon number on the time interval between ion impacts and on the impact point is investigated.

Thermoelectric Performance Enhanced by Destructive Quantum Interference in Nanoporous Carbon Nanotube Based Junctions

Aided by density functional theory with nonequilibrium Green's functions simulations, the thermoelectric (TE) properties of nanoporous carbon nanotube junctions formed by covalently binding carbon nanotubes via benzene bridges with either meta‐ or para‐connections are investigated. The results show that the TE performances of meta‐connected nanoporous carbon nanotube junctions are significantly improved due to the influence of destructive quantum interference (DQI) effect. Moreover, the TE properties of meta‐connected carbon nanotube junctions can be further improved by substitution of nitrogen atom and gate voltage. The reason is that the quantum interference patterns can be transformed from DQI to constructive quantum interference (or Fano resonance). The theoretical analysis shows that the TE figure of merit (ZT) for meta‐connected carbon nanotube junctions at room temperature is close to 0.5 near the Fermi level, ≈20 times higher than that of the perfect carbon nanotube junction. These results reveal that quantum interference effect can indeed be used to enhance the TE performance of molecular junctions.

Speed Up Quantum Transport Device Simulation on Ferroelectric Tunnel Junction With Machine Learning Methods

As the device size scales down to the nanometer regime, quantum effects play an important role in device characteristics and performance. Quantum transport device simulation based on the nonequilibrium Green's function (NEGF) has been extensively applied to simulate the nanoscale devices. The NEGF simulations, however, can be computationally expensive, especially in the presence of scattering. In this study, a machine learning (ML)-based framework is developed, targeting on replacing the computationally intensive NEGF simulations. This framework first learns a sparse representation of a quantum transport property of interest and then trains a model to describe the quantitative mapping relation between the device parameters and properties. Also, the accuracy is further improved with the application of feature engineering. As an example, a graphene-ferroelectric-metal (GFM) ferroelectric tunnel junction (FTJ) is simulated. The results show that the ML-based framework allows circumventing the NEGF calculation and simultaneously maintaining high accuracy in quantum transmissions and tunneling I-V characteristics. This ML-based framework can be applied to speed up the quantum transport device simulations and enable efficient tunneling device design.

Nanowire gate-all-around MOSFETs modeling: ballistic transport incorporating the source-to-drain tunneling

Incorporating the source-to-drain tunneling current that is valid in all operating regions, an analytical compact model is proposed in this paper for cylindrical ballistic gate-all-around n-type metal-oxide-semiconductor field-effect transistors with ultra-short silicon channel. From taking the drain-induced barrier lowering effect into consideration, the potential distribution within the device channel has been modeled based upon a 2D analysis in our previous work. In this study, by introducing a parabolic function when modeling the potential profile in the channel direction, we found out that the source-to-drain tunneling effect in the subthreshold region could be evaluated analytically by applying Wentzel–Kramers–Brillouin approximation. Then, it is practical to estimate the ballistic drain current for all operating regions analytically with this compact model considering both the source-to-drain tunneling and thermionic transport. The resulting analytic compact model is tested against non-equilibrium Green's function simulation using SILVACO, and good accuracy is demonstrated. Finally, we perform an NMOS inverter circuit simulation using HSPICE, when introducing our model to it as a Verilog-A script.

G1-G2 scheme: Dramatic acceleration of nonequilibrium Green functions simulations within the Hartree-Fock generalized Kadanoff-Baym ansatz

The time evolution in quantum many-body systems after external excitations is attracting high interest in many fields. The theoretical modeling of these processes is challenging, and the only rigorous quantum-dynamics approach that can treat correlated fermions in two and three dimensions is nonequilibrium Green functions (NEGF). However, NEGF simulations are computationally expensive due to their $T^3$-scaling with the simulation duration $T$. Recently, $T^2$-scaling was achieved with the generalized Kadanoff--Baym ansatz (GKBA), for the second-order Born (SOA) selfenergy, which has substantially extended the scope of NEGF simulations. In a recent Letter [Schl\"unzen \textit{et al.}, Phys. Rev. Lett. \textbf{124}, 076601 (2020)] we demonstrated that GKBA-NEGF simulations can be efficiently mapped onto coupled time-local equations for the single-particle and two-particle Green functions on the time diagonal, hence the method has been called G1--G2 scheme. This allows one to perform the same simulations with order $T^1$-scaling, both for SOA and $GW$ selfenergies giving rise to a dramatic speedup. Here we present more details on the G1--G2 scheme, including derivations of the basic equations including results for a general basis, for Hubbard systems and for jellium. Also, we demonstrate how to incorporate initial correlations into the G1--G2 scheme. Further, the derivations are extended to a broader class of selfenergies, including the $T$ matrix in the particle--particle and particle--hole channels, and the dynamically screened-ladder approximation. Finally, we demonstrate that, for all selfenergies, the CPU time scaling of the G1--G2 scheme with the basis dimension, $N_b$, can be improved compared to our first report: the overhead compared to the original GKBA, is not more than an additional factor $N_b$.

Electrical Contact between an Ultrathin Topological Dirac Semimetal and a Two-Dimensional Material

Ultrathin films of topological Dirac semimetal, Na$_3$Bi, has recently been revealed as an unusual electronic materials with field-tunable topological phases. Here we investigate the electronic and transport properties of ultrathin Na$_3$Bi as an electrical contact to two-dimensional (2D) metal, i.e. graphene, and 2D semiconductor, i.e. MoS$_2$ and WS$_2$ monolayers. Using combined first-principle density functional theory and nonequilibrium Green's function simulation, we show that the electrical coupling between Na$_3$Bi bilayer thin film and graphene results in a notable interlayer charge transfer, thus inducing sizable $n$-type doping in the Na$_3$Bi/graphene heterostructures. In the case of MoS$_2$ and WS$_2$ monolayers, the lateral Schottky transport barrier is significantly lower than many commonly studied bulk metals, thus unraveling Na$_3$Bi bilayer as a high-efficiency electrical contact material for 2D semiconductors. These findings opens up an avenue of utilizing topological semimetal thin film as electrical contact to 2D materials, and further expands the family of 2D heterostructure devices into the realm of topological materials.

Achieving the Scaling Limit for Nonequilibrium Green Functions Simulations.

The dynamics of strongly correlated fermions following an external excitation reveals extremely rich collective quantum effects. Examples are fermionic atoms in optical lattices, electrons in correlated materials, and dense quantum plasmas. Presently, the only quantum-dynamics approach that rigorously describes these processes in two and three dimensions is the nonequilibrium Green functions (NEGF) method. However, NEGF simulations are computationally expensive due to their T^{3} scaling with the simulation duration T. Recently, T^{2} scaling was achieved with the generalized Kadanoff-Baym ansatz (GKBA), for second-order Born (SOA) selfenergies, which has substantially extended the scope of NEGF simulations. Here we demonstrate that GKBA-NEGF simulations can be performed with order T^{1} scaling, both for SOA and GW selfenergies, and point out the remarkable capabilities of this approach.

A computationally efficient hybrid approach based on artificial neural networks and the wavelet transform for quantum simulations of graphene nanoribbon FETs

Numerical modeling of graphene nanoribbon field-effect transistors (GNRFETs) using quantum-mechanical approaches is often associated with a heavy computational burden, indicating the urgent need for new methods to resolve this issue. A computationally efficient hybrid approach for quantum simulations of ballistic GNRFETs is developed herein. The proposed simulation method is based on solving the two-dimensional (2D) Poisson equation using both wavelet-based adaptive meshing and wavelet-based matrix compression techniques, self-consistently, with well-trained neural network models that imitate the mode-space (MS) nonequilibrium Green’s function (NEGF) solver in generating the charge density. The results obtained by applying the hybrid approach show good agreement with NEGF simulations. Numerical experiments reveal that the developed simulation method can offer a speed-up of about one order of magnitude over the conventional MS NEGF approach. The encouraging results obtained in this work indicate that the developed numerical model is particularly appropriate for incorporation into nanoelectronic device simulators to investigate future GNRFET-based circuits.

Nonequilibrium Green function simulation of coupled electron–phonon transport in one-dimensional nanostructures

Phonon thermal-conductance of a one-dimensional semiconductor nanostructure has been calculated by self-consistently solving coupled nonequilibrium Green function equations for electrons and phonons. The channel length, N, dependence of the phonon thermal-conductance change, ΔKp, due to the electron–phonon interaction is found to obey a simple relation of with a constant proportionality factor β for the calculated N range up to N = 100 (Ge is the electrical conductance). The phonon thermal-conductance change in the isotopically disordered system is found to be hardly affected by the presence of the isotope impurities.

Ion Impact Induced Ultrafast Electron Dynamics in Finite Graphene‐Type Hubbard Clusters

Strongly correlated systems of fermions have an interesting phase diagram arising from the Hubbard gap. Excitation across the gap leads to the formation of doubly occupied lattice sites (doublons) which offers interesting electronic and optical properties. Moreover, when the system is driven out of equilibrium interesting collective dynamics may arise that are related to the spatial propagation of doublons. Here, a novel mechanism that was recently proposed by the authors [Balzer et al., Phys. Rev. Lett. 121, 267602 (2018)] is verified by exact diagonalization and nonequilibrium Green functions (NEGF) simulations – fermionic doublon creation by the impact of energetic ions. The formation of a nonequilibrium steady state with homogeneous doublon distribution is reported. The effect should be particularly important for correlated finite systems, such as graphene nanoribbons, and directly observable with fermionic atoms in optical lattices. It is demonstrated that doublon formation and propagation in correlated lattice systems can be accurately simulated with NEGF. In addition to two‐time results, single‐time results within the generalized Kadanoff–Baym ansatz (GKBA) with Hartree–Fock propagators (HF‐GKBA) is presented. Finally systematic improvements of the GKBA that use correlated propagators (correlated GKBA) and a correlated initial state are discussed.

Doublon Formation by Ions Impacting a Strongly Correlated Finite Lattice System.

Strongly correlated systems of fermions have a number of exciting collective properties. Among them, the creation of a lattice that is occupied by doublons, i.e., two quantum particles with opposite spins, offers interesting electronic properties. In the past a variety of methods have been proposed to control doublon formation, both, spatially and temporally. Here, a novel mechanism is proposed and verified by exact diagonalization and nonequilibrium Green functions simulations-fermionic doublon creation by the impact of energetic ions. We report the formation of a nonequilibrium steady state with homogeneous doublon distribution. The effect should be particularly important for strongly correlated finite systems, such as graphene nanoribbons, and directly observable with fermionic atoms in optical lattices.