This paper presents an $ab$ $initio$ study of hydrogen dynamics inside nanometer-size voids in $a$-Si within the framework of the density-functional theory for a varying hydrogen load of 10 to 30 H atoms/void at the low and high temperature of 400 K and 700 K, respectively. Using the local density approximation and its generalized-gradient counterpart, the dynamics of hydrogen atoms inside the voids are examined with an emphasis on the diffusion of H atoms/molecules, and the resulting nanostructural changes of the void surfaces. The results from simulations suggest that the microstructure of the hydrogen distribution on the void surfaces and the morphology of the voids are characterized by the presence of a significant number of monohydride Si-H bonds, along with a few dihydride Si-H$_2$ configurations. The study also reveals that a considerable number of (about 10--45 at.%) total H atoms inside voids can appear as H$_2$ molecules for a hydrogen load of 10--30 H atoms/void. The approximate shape of the voids is addressed from a knowledge of the positions of the void-surface atoms using the convex-hull approximation and the Gaussian broadening of the pseudo-atomic surfaces of Si and H atoms.
We use empirical molecular dynamics technique to study the low-energy vibrations in a large 4096 atom model for pure amorphous silicon and a set of models with voids of different size based on it. Numerical vibrational eigenvalues and eigenvectors for our models are obtained by exact diagonalization of their dynamical matrices. Our calculations show that localized low-energy vibrational excitations of rather complex structure are present in amorphous silicon models with voids. According to their spatial localization patterns we make an attempt to classify these excitations as modes associated …
We present a computational study of the void-induced microstructure in amorphous silicon ($\it a$-Si) by generating ultra-large models of $\it a$-Si with a void-volume fraction of 0.3$\%$, as observed in small-angle x-ray scattering (SAXS) experiments. The relationship between the morphology of voids and the intensity of scattering in SAXS has been studied by computing the latter from the Fourier transform of the reduced pair-correlation function and the atomic-form factor of amorphous silicon. The effect of low-temperature ($\le$ 600 K) annealing on the scattering intensities and the microstr…
The paper presents an $ab$ $initio$ study of temperature-induced nanostructural evolution of hydrogen-rich voids in amorphous silicon. By using large $a$-Si models, obtained from classical molecular-dynamics simulations, with a realistic void-volume density of 0.2%, the dynamics of Si and H atoms on the surface of the nanometer-size cavities were studied and their effects on the shape and size of the voids were examined using first-principles density-functional simulations. The results from $ab$ $initio$ calculations were compared with those obtained from using the modified Stillinger-Weber po…
We present a computational study of small-angle X-ray scattering (SAXS) in amorphous silicon ($a$-Si) with particular emphasis on the morphology and microstructure of voids. The relationship between the scattering intensity in SAXS and the three-dimensional structure of nanoscale inhomogeneities or voids is addressed by generating ultra-large high-quality $a$-Si networks with 0.1-0.3% volume concentration of voids, as observed in experiments using SAXS and positron annihilation spectroscopy. A systematic study of the variation of the scattering intensity in the small-angle scattering region wi…
A disorder-order transition from amorphous carbon (aC) to layered amorphous graphene (LAG) has been predicted using ab-initio methods. Amorphous carbon at densities close to the graphitic density show a strong proclivity to layer in NVT simulations near 3000K . Our calculations have shown that graphitization is strongly dependent on the simulation temperature and is not observed under 2500K. The origin of such a disorder-order transition has been studied using various structural analyses. Each layer of LAG is a layer of amorphous graphene (aG) including pentagons and heptagons in addition to h…
Apart from its unique and exciting electronic properties, many sensor based applications of graphene are purely based on its mechanical and structural properties. Here we report a numerical and analytical study of a void in amorphous (small domain polycrystalline) graphene, and show that the energetics of a void is a balance between the line tension cost versus the increased area gain. Using the concepts of classical nucleation theory, we show that the critical radius of a void formed in amorphous graphene at constant pressure is simply the ratio of line tension at the void and the applied pre…
We discuss a novel approach that allows to obtain effective potentials from ab initio trajectories. Our method consists in fitting the weighted radial distribution functions obtained from the ab initio data with the ones obtained from simulations with the effective potential, and using the parameters of the latter as fitting variables. As a case study, we consider the example of amorphous silica, a material that is highly relevant in the field of glass science as well as in geology. Our approach is able to obtain an effective potential that gives a better description with respect to structural…
Ideal models of complex materials must satisfy all available information about the system. Generally, this information consists of experimental data, information implicit to sophisticated interatomic interactions and potentially other {\it a priori} information. By jointly imposing first-principles or tight-binding information in conjunction with experimental data, we have developed a method: Experimentally Constrained Molecular Relaxation (ECMR) that uses {\it all} of the information available. We apply the method to model medium range order in amorphous silicon using Fluctuation Electron mic…
Using empirical potential molecular dynamics we compute dynamical matrix eigenvalues and eigenvectors for a 4096 atom model of amorphous silicon and a set of models with voids of different size based on it. This information is then employed to study the localization properties of the low-energy vibrational states, calculate the specific heat C(T) and examine the low-temperature properties of our models usually attributed to the presence of tunneling states in amorphous silicon. The results of our calculations for C(T) and "excess specific heat bulge" in the C(T)/T^3 vs. T graph for voidless a-…
Polymeric nitrogen with single bonds can be created from the molecular form at high pressure and due to large energy difference between triple and single bonds it is interesting as energetic material. Its structure and properties are, however, still not well understood. We studied amorphous nitrogen by \textit{ab initio} simulations, employing molecular dynamics and evolutionary algorithms. Amorphous nitrogen was prepared at pressure of 120 GPa by quenching from hot liquid, by pressure-induced amorphization of molecular crystal and by evolutionary search. All three amorphous forms were found t…
In this paper, we present ab initio computer models of Cu-doped amorphous Ta2O5 , a promising candidate for Conducting Bridge Random Access Memory (CBRAM) memory devices, and study the structural, electronic, charge transport and vibrational properties based on plane-wave density functional methods. We offer an atomistic picture of the process of phase segregation/separation between Cu and Ta2O5 subnetworks. Electronic calculations show that the models are conducting with extended Kohn-Sham orbitals around the Fermi level. In addition to that, we also characterize the electronic transport usin…
We present an ab initio calculation of the DC conductivity of amorphous silicon and hydrogenated amorphous silicon. The Kubo-Greenwood formula is used to obtain the DC conductivity, by thermal averaging over extended dynamical simulation. Its application to disordered solids is discussed. The conductivity is computed for a wide range of temperatures and doping is explored in a naive way by shifting the Fermi level. We observed the Meyer-Neldel rule for the electrical conductivity with E_MNR = 0.06 eV and a temperature coefficient of resistance, TCR ~ -2.0% K^-1 for a-Si:H. In general, experime…
Use of amorphous silicon ($a$-Si) and hydrogenated amorphous silicon ($a$-Si:H) in photovoltaics has been limited by light-induced degradation (the Staebler-Wronski effect) and low hole mobilities, and voids have been implicated in both problems. Accurately modeling the void microstructure is critical to theoretically understanding the cause of these issues. Previous methods of modeling voids have involved removing atoms according to an {\it a priori} idea of void structure and/or using computationally expensive molecular dynamics. We propose a new fast and unbiased approach based on the estab…
We have studied the photoinduced transformation of small rings (3-membered) in amorphous silica by Car-Parrinello simulations. The process of ring opening leading to the formation of a couple of paramagnetic centers, namely an E' and a non-bridging-oxygen hole center (NBOHC), has been proposed experimentally to occur in silica exposed to F2 laser irradiation (at 7.9 eV). By using a new scheme for the simulation of rare events in ab-initio molecular dynamics (Iannuzzi, Laio and Parrinello, Phys. Rev. Lett. 90, 238303 (2003)), we have identified the transformation path for the opening of a 3-mem…
The understanding of dynamic failure in amorphous materials via the propagation of free boundaries like cracks and voids must go beyond elasticity theory, since plasticity intervenes in a crucial and poorly understood manner near the moving free boundary. In this Letter we focus on failure via a cavitation instability in a radially-symmetric stressed material, set up the free boundary dynamics taking both elasticity and visco-plasticity into account, using the recently proposed athermal Shear Transformation Zone theory. We demonstrate the existence (in amorphous systems) of fast cavitation mod…
We report results of ab initio constant-pressure molecular dynamics simulations of sulfur compression leading to structural transition and pressure-induced amorphization. Starting from the orthorhombic S-I phase composed of S$_8$ ring molecules we find at room temperature and pressure of 20 GPa a transformation to monoclinic phase where half of the molecules develop a different conformation. Upon further compression, the monoclinic phase undergoes pressure-induced amorphization into an amorphous phase, in agreement with experiments. We study the dynamics of the amorphization transition and inv…
We review our previous work on the dynamic structure factor S(k,omega) of liquid Ge (l-Ge) at temperature T = 1250 K, and of amorphous Ge (a-Ge) at T = 300 K, using ab initio molecular dynamics [Phys. Rev. B67, 104205 (2003)]. The electronic energy is computed using density-functional theory, primarily in the generalized gradient approximation, together with a plane wave representation of the wave functions and ultra-soft pseudopotentials. We use a 64-atom cell with periodic boundary conditions, and calculate averages over runs of up to 16 ps. The calculated liquid S(k,omega) agrees qualitativ…
Since the pioneering efforts of Duwez and coworkers in 1965, when a solid amorphous phase of Pd-Si ($a$-PdSi) was obtained in the vicinity of the eutectic concentration, much work has been done. However, some points related to the atomic structures remain to be systematized. In this work, 8 amorphous Pd$_{100-c}$Si$_{c}$ alloys (c = 2.5, 5, 10, 13.34, 15, 17.5, 20, and 22 at %) were generated by molecular dynamics $ab$ $initio$ simulations; the short-range structure is analyzed using several correlation functions, like Pair Distribution Functions, reduced Pair Distribution Functions, Plane Ang…
Boron nitride (BN) is a structurally versatile insulator since it can be found in several crystalline structures with interesting mechanical and electrical properties, making this material attractive for technological applications. Seeking to improve its features, experimental and simulational studies for the amorphous phase (a-BN) have been carried out by some groups, focusing on the electrical and structural properties, pressure-induced phase transformations, and hydrogenated a-BN. In this work two amorphous structures are computationally generated and studied using ab initio Molecular Dynam…
The nanostructure of hydrogenated amorphous silicon (a Si:H) is studied by a combination of small-angle X-ray (SAXS) and neutron scattering (SANS) with a spatial resolution of 0.8 nm. The a-Si:H materials were deposited using a range of widely varied conditions and are representative for this class of materials. We identify two different phases which are embedded in the a-Si:H matrix and quantified both according to their scattering cross-sections. First, 1.2 nm sized voids (multivacancies with more than 10 missing atoms) which form a superlattice with 1.6 nm void-to-void distance are detected…
Realistic models of amorphous ZrO2 are generated in a ``melt-and-quench'' fashion using ab-initio molecular dynamics in a plane-wave pseudopotential formulation of density-functional theory. The structural properties of the resulting amorphous models are analyzed, with special attention to coordination statistics. The vibrational and dielectric properties of one of these models are then investigated from first principles using linear-response methods. The electronic dielectric constant and Born effective charges are found to be very similar to those of the crystalline phases. Encouragingly, th…
By employing $ab$ $initio$ molecular dynamics simulations at constant pressure, we investigated behavior of amorphous carbon dioxide between 0-100 GPa and 200-500 K and found several new amorphous forms. We focused on evolution of the high-pressure polymeric amorphous form known as a-carbonia on its way down to zero pressure, where it eventually converts into a molecular amorphous solid. During decompression, two nonmolecular amorphous forms with different proportion of three and four-coordinated carbons and two mixed molecular-nonmolecular forms were observed. Transformation from a-carbonia t…
We calculate the dynamic structure factor S(k,omega) of liquid Ge (l-Ge) at temperature T = 1250 K, and of amorphous Ge (a-Ge) at T = 300 K, using ab initio molecular dynamics. The electronic energy is computed using density-functional theory, primarily in the generalized gradient approximation, together with a plane wave representation of the wave functions and ultra-soft pseudopotentials. We use a 64-atom cell with periodic boundary conditions, and calculate averages over runs of up to 16 ps. The calculated liquid S(k,omega) agrees qualitatively with that obtained by Hosokawa et al, using in…
Density functional theory (DFT) calculations are carried out to study the structure and electronic structure of amorphous zinc oxide (a-ZnO). The models were prepared by the "melt-quench" method. The models are chemically ordered with some coordination defects. The effect of trivalent dopants in the structure and electronic properties of a-ZnO is investigated. Models of a-X_0.375 Z_0.625 O (X=Al, Ga and In) were also prepared by the "melt- quench" method. The trivalent dopants reduce the four-fold Zn and O, thereby introducing some coordination defects in the network. The dopants prefer to bon…
Void swelling can develop in materials under persistent irradiation when non-equilibrium vacancy and self-interstitial populations migrate under sufficiently asymmetric interaction biases. In conventional metals, the propensity is determined to first approximation by comparing point-defect relaxation strains. We thus present DFT-based calculations of structures and formation energies and volumes of point defects in the $α$ and the $δ$-phases of plutonium. Our results show that lattice defects in $δ$-Pu, in contrast to most fcc metals, have surprisingly small formation volumes. Equally unexpect…
We discuss the efficacy of evolutionary method for the purpose of structural analysis of amorphous solids. At present ab initio molecular dynamics (MD) based melt-quench technique is used and this deterministic approach has proven to be successful to study amorphous materials. We show that a stochastic approach motivated by Darwinian evolution can also be used to simulate amorphous structures. Applying this method, in conjunction with density functional theory (DFT) based electronic, ionic and cell relaxation, we re-investigate two well known amorphous semiconductors, namely silicon and indium…
Chalcogenide alloys are materials of interest for optical recording and non-volatile memories. We perform ab-initio molecular dynamics simulations aiming at shading light onto the structure of amorphous Ge2Sb2Te5 (GST), the prototypical material in this class. First principles simulations show that amorphous GST obtained by quenching from the liquid phase displays two types of short range order. One third of Ge atoms are in a tetrahedral environment while the remaining Ge, Sb and Te atoms display a defective octahedral environment, reminiscent of cubic crystalline GST.
We have extended our experimentally constrained molecular relaxation technique (P. Biswas {\it et al}, Phys. Rev. B {\bf 71} 54204 (2005)) to hydrogenated amorphous silicon: a 540-atom model with 7.4 % hydrogen and a 611-atom model with 22 % hydrogen were constructed. Starting from a random configuration, using physically relevant constraints, {\it ab initio} interactions and the experimental static structure factor, we construct realistic models of hydrogenated amorphous silicon. Our models confirm the presence of a high frequency localized band in the vibrational density of states due to Si-…
Exponential band edges have been observed in a variety of materials, both crystalline and amorphous. In this paper, we infer the structural origins of these tails in amorphous and defective crystalline Si by direct calculation with current {\it ab initio} methods. We find that exponential tails appear in relaxed models of diamond with suitable point defects. In amorphous silicon (a-Si), we find that structural filaments of short bonds and long bonds exist in the network, and that the tail states near the extreme edges of both band tails are are also filamentary, with much localization on the s…
Localized basis ab initio molecular dynamics simulation within the density functional framework has been used to generate realistic configurations of amorphous silicon carbide (a-SiC). Our approach consists of constructing a set of smart initial configurations that conform essential geometrical and structural aspects of the materials obtained from experimental data, which is subsequently driven via first-principles force-field to obtain the best solution in a reduced solution space. A combination of a priori information (primarily structural and topological) along with the ab-initio optimizati…
The PEO3:LiCF3SO3 polymer electrolyte has attracted significant research due to its enhanced stability at the lithium/polymer interface of high conductivity polymer batteries. Experimental studies have shown that, depending on the preparation conditions, both the PEO3:LiCF3SO3 crystalline complex and the PEO3:LiCF3SO3 amorphous phase can be formed. However, previous theoretical investigations focused on the short chain amorphous PEO3:LiCF3SO3 system. We report ab initio density-functional-theory calculations of crystalline PEO3:LiCF3SO3. The calculated results about the bonding configuration, …
A combination of classical molecular dynamics (MD) and ab initio Car-Parrinello molecular dynamics (CPMD) simulations is used to investigate the adsorption of water on a free amorphous silica surface. From the classical MD SiO_2 configurations with a free surface are generated which are then used as starting configurations for the CPMD.We study the reaction of a water molecule with a two-membered ring at the temperature T=300K. We show that the result of this reaction is the formation of two silanol groups on the surface. The activation energy of the reaction is estimated and it is shown that …
We use first-principles simulations to investigate the topological properties of Bi$_2$Se$_3$ thin films deposited on amorphous SiO2, Bi$_2$Se$_3$/a-SiO$_2$, which is a promising substrate for topological insulator (TI) based device applications. The Bi$_2$Se$_3$ films are bonded to a-SiO$_2$ mediated by van der Waals interactions. Upon interaction with the substrate, the Bi$_2$Se$_3$ topological surface and interface states remain present, however the degeneracy between the Dirac-like cones is broken. The energy separation between the two Dirac-like cones increases with the number of Bi$_2$Se…
We employ ab initio molecular dynamics to simulate the response of hydrogenated amorphous silicon to light exposure (Staebler-Wronski effect). We obtain improved microscopic understanding of PV operation, compute the motion of H atoms, and modes of light-induced degradation of photovoltaics. We clarify existing models of light-induced change in aSi:H and show that the Hydrogen collision model of Branz3 is correct in essentials.
We consider the question of the amorphization of metallic alloys by melt quenching, as predicted by molecular dynamics simulations with semi-empirical potentials. The parametrization of the potentials is discussed on the example of the ternary Cu-Ti-Zr transition metals alloy, using as reference the ab-initio simulation. The pair structure in the amorphous state is computed from a potential of the Stillinger Weber form. The transferability of the parameters during the quench is investigated using two parametrizations: from solid state data, as usual, and from a new parametrization on the liqui…
We present a novel approach for parameter-free modeling of the structural, dynamical and electronic properties of non-crystalline materials based on ab-initio Molecular Dynamics, improved signal processing technique and computer visualization. The method have been extensively tested by investigating hydrogen and silicon dynamics in hydrogenated amorphous silicon (a-Si:H). By comparing the theoretical and experimental vibrational spectra we demonstrate how to relate vibrational properties to the structural stability, bonding and hydrogen diffusion. We extracted microscopic characteristics that …
From Bulk to Surface: Structure and Dynamics of Amorphous Alumina from Deep Potential Molecular Dynamics
Princeton University（普林斯顿大学）；Shell International Exploration & Production Inc.（壳牌国际勘探与生产有限公司）；Shell India Markets Pvt., Ltd.（壳牌印度市场私人有限公司）；Shell Information Technology International Inc.（壳牌信息技术国际有限公司）
Normal mode decomposition of atomic vibrations has been used to provide microscopic under-standing of thermal transport in amorphous solids for decades. In normal mode methods, it is naturally assumed that atoms vibrate around their equilibrium positions and that individual normal modes are the fundamental vibrational excitations transporting heat. With the abundance of predictions from normal mode methods and experimental measurements now available, we care-fully analyze these calculations in amorphous silicon, a model amorphous solid. We find a number of discrepancies, suggesting that treati…
Accurate prediction of structural and mechanical properties on amorphous materials enabled through machine-learning potentials: a case study of silicon nitride
The amorphous phases of antimony oxide at three different oxidation levels have been investigated by {\it ab initio} molecular-dynamics calculations using a simulated `melt-quench' approach. The atomic and electronic structure of the amorphous phases are analyzed and compared to their crystalline counterparts. The amorphous structure of the trioxide ({\it a\,}--Sb$_2$O$_3$) resembles the $β$-phase of the crystalline form (valentinite), in agreement with previous observations. In the relaxed athermal structure, however, more senarmontite-like structural features are apparent. The phase diagram …
We perform an {\it ab initio} modeling of amorphous copper-doped alumina (a-Al$_2$O$_3$:Cu), a prospective memory material based on resistance switching, and study the structural origin of electronic conduction in this material. We generate molecular dynamics based models of a-Al$_2$O$_3$:Cu at various Cu-concentrations and study the structural, electronic and vibrational properties as a function of Cu-concentration. Cu atoms show a strong tendency to cluster in the alumina host, and metallize the system by filling the band gap uniformly for higher Cu-concentrations. We also study thermal fluc…
Lithium phosphorus oxynitride (LiPON) is an amorphous solid-state lithium ion conductor displaying exemplary cyclability against lithium metal anodes. There is no definitive explanation for this stability due to the limited understanding of the structure of LiPON. We provide a structural model of RF-sputtered LiPON via experimental and computational spectroscopic methods. Information about the short-range structure results from 1D and 2D solid-state nuclear magnetic resonance experiments investigating chemical shift anisotropy and dipolar interactions. These results are compared with first pri…
We report first principles ab initio density functional calculations of hydrogen dynam- ics in hydrogenated amorphous silicon. Thermal motion of the host Si atoms drives H diffusion, as we demonstrate by direct simulation and explain with simple models. Si-Si bond centers and Si ring centers are local energy minima as expected. We also describe a new mechanism for break- ing Si-H bonds to release free atomic H into the network: a fluctuation bond center detachment (FBCD) assisted diffusion. H dynamics in a-Si:H is dominated by structural fluctuations intrinsic to the amorphous phase not presen…
In 1965 Duwez \textit{et al.} reported having generated an amorphous, stable phase of palladium-silicon in the region 15 to 23 atomic percent (at. \%) silicon. These pioneering efforts have led to the development of solid materials that are now known as Bulk Metallic Glasses (BMG). In 2019 we discovered, computationally, that bulk amorphous Pd becomes magnetic, and so does porous/amorphous Pd. Puzzled by our results we undertook the study of several solid binary systems in the Pd-rich zone; in particular, the study of the glassy metallic alloy $a$-Pd$_{100-c}$Si$_{c}$, for $0 \leq c \leq 22$, …
Leadership Computing Facility, Argonne National Laboratory（阿贡国家实验室领导计算设施）；Institute for Computational Physics, Universität Stuttgart（斯图加特大学计算物理研究所）；Applied Materials Division, Argonne National Laboratory（阿贡国家实验室应用材料部）；University of Cambridge（剑桥大学）；X-ray Science Division, Argonne National Laboratory（阿贡国家实验室X射线科学部）；Computational Science Division, Argonne National Laboratory（阿贡国家实验室计算科学部）
We propose a novel active learning scheme for automatically sampling a minimum number of uncorrelated configurations for fitting the Gaussian Approximation Potential (GAP). Our active learning scheme consists of an unsupervised machine learning (ML) scheme coupled to Bayesian optimization technique that evaluates the GAP model. We apply this scheme to a Hafnium dioxide (HfO2) dataset generated from a melt-quench ab initio molecular dynamics (AIMD) protocol. Our results show that the active learning scheme, with no prior knowledge of the dataset is able to extract a configuration that reaches t…
Bismuth displays puzzling superconducting properties. In its crystalline equilibrium phase, it does not seem to superconduct at accessible low temperatures. However, in the amorphous phase it displays superconductivity at ~ 6 K. Under pressure bismuth has been found to superconduct at Tcs that go from 3.9 K to 8.5 K depending on the phase obtained. So the question is: what electronic or vibrational changes occur that explains this radical transformation in the conducting behavior of this material? In a recent publication we argue that changes in the density of electronic and vibrational states…
We present a theory for the ab-initio computation of NMR chemical shifts (sigma) in condensed matter systems, using periodic boundary conditions. Our approach can be applied to periodic systems such as crystals, surfaces, or polymers and, with a super-cell technique, to non-periodic systems such as amorphous materials, liquids, or solids with defects. We have computed the hydrogen sigma for a set of free molecules, for an ionic crystal, LiH, and for a H-bonded crystal, HF, using density functional theory in the local density approximation. The results are in excellent agreement with experiment…
Superconducting resonators used in millimeter-submillimeter astronomy would greatly benefit from deposited dielectrics with a small dielectric loss. We deposited hydrogenated amorphous silicon films using plasma-enhanced chemical vapor deposition, at substrate temperatures of 100°C, 250°C and 350°C. The measured void volume fraction, hydrogen content, microstructure parameter, and bond-angle disorder are negatively correlated with the substrate temperature. All three films have a loss tangent below $10^{-5}$ for a resonator energy of $10^5$ photons, at 120 mK and 4-7 GHz. This makes these film…
We have investigated the dynamics of Na ions in amorphous Na2Si2O5, a potential solid electrolyte material for Na-battery. We have employed quasielastic neutron scattering (QENS) technique in the amorphous Na2Si2O5 from 300 to 748 K to understand the diffusion pathways and relaxation timescales of Na atom dynamics. The microscopic analysis of the QENS data has been performed using ab-initio and classical molecular dynamics simulations (MD) to understand the Na-ion diffusion in the amorphous phase. Our experimental studies show that the traditional model, such as the Hall and Ross (H-R) model, …
