Solution Of Boltzmann Transport Equation Research Articles

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

First-principles calculations on the intrinsic resistivity of realistic metals: a case study of monolayer V2N.

The Ziman resistivity formula is extensively employed to calculate the intrinsic resistivity of realistic metals using recent works of first-principles calculations. Owing to the approximation, Allen's generalization, which relates the solution of the Boltzmann transport equation (BTE) for metals to the transport electron-phonon (e-ph) spectral function, the applicability of the Ziman resistivity formula still needs to be discussed. In this work, we perform first-principles calculations of the intrinsic resistivity of the V2N monolayer, a kind of MXene material, by employing the Ziman resistivity formula and the iterative solution of BTE. We find that in the wide temperature range of 50-1400 K, the intrinsic resistivity of the V2N monolayer obtained by means of the two approaches, has the same order of magnitude. However the Ziman resistivity formula fails to correctly describe the Fermi level dependence of the intrinsic resistivity of the V2N monolayer. The underlying reason is that the band edge of the V2N and hence Van Hove singularity (VHS), is near the shifted Fermi level. We suggest a modified Ziman resistivity formula which is valid even if there is a band edge at the Fermi level.

Atomistic nonlinear carrier dynamics in Ge

An atomistic technique to successfully demonstrate the ultrafast carrier dynamics in Ge photoconductive samples is reported here. The technique is validated against the experimental findings and with the Drude conductivities. The impact of the various different scattering mechanisms is used to calibrate the experimental results. It is observed that the total scattering rate is not a constant parameter as contrast to Drude model which uses constant scattering rate as the fitting parameter to demonstrate the ultrafast carrier dynamics, but strongly dependent on the applied peak THz field strength. It also contradicts with the relaxation time approximation (RTA) method which uses scattering rate chosen on the empirical basis as the fitting parameter to demonstrate the ultrafast carrier dynamics. On the other hand the limitations and challenges offered by various types of density functional theories (DFT) pose lot of challenges. In current manuscript various types of scattering mechanisms i.e. acoustic, intervalley, Coulomb and impact ionization on the behavior of carrier conductivity are studied in details. The proposed technique has shown capability to extract low and high frequency conductivities accurately which is impossible through the Drude model or DFT based theories. It is observed that the free carrier absorption coefficient depends on the refractive index of the material at low THz frequencies. The solution of Boltzmann transport equation through Monte Carlo technique provides valuable insights for better understanding of ultrafast carrier transportation mechanism. The free carrier absorption spectra are found to be in good agreement with the experimental results at various THz field strengths.

Analysis of the electron energy distribution function and its transport coefficient in SF6-CO2 applied gas mixtuers

The Boltzmann transport equation is used to calculate the electron energy distribution function (EEDF) and the transport coefficient in pure sulfur hexafluoride (SF6) and pure Carbon dioxide (CO2) and their mixtures. The electron swarm parameters are evaluated in the range of These parameters namely are: The Diffusion coefficient of electrons and mean electron energy. The motion of electrons in plasma gas (SF6) and mixing it with (CO2) under an applied uniform electric field is simulated by using the numerical solution ofBoltzmann's transport equation technique. The numerical solutions are utilized within the international computer code kinema-Elendif and written in Fortran 77 language software. The calculated distribution function is found to be remarked non-Maxwillian that has energy variations which reflect the import electron-molecule energy exchange processes.

First-principles study of the drift and Hall mobilities of perovskite BaSnO3

Barium stannate (${\mathrm{BaSnO}}_{3}$) is an emerging perovskite oxide with promising properties for many potential applications. One of the most outstanding properties is the high-room-temperature electron mobility. The extraordinary superior electronic transport performance has attracted great attention but the research is limited by semiempirical assumptions. In this work, the first-principles calculations are employed to study the electronic transport of ${\mathrm{BaSnO}}_{3}$ by solving the Boltzmann transport equation (BTE) with full information on electron, phonon, and electron-phonon interactions taken into account. The mode-resolved analyses of scattering rates show that the longitudinal optical (LO) phonons dominate the scattering processes not only at low but also at high carrier concentrations. As a consequence, the energy relaxation time approximation significantly underestimates the solution of BTE due to the divergent coupling coefficient of electrons and LO phonons. The drift and Hall mobilities are calculated in the framework of BTE in the presence of electric and magnetic fields. At low concentration limits, the room-temperature drift and Hall mobilities are 357 and $418\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, respectively, while the Hall factor is temperature dependent with values between 1.06 and 1.17 in the temperature range of 100 K to 800 K. In the doping system with carrier concentration above $1\ifmmode\times\else\texttimes\fi{}{10}^{19}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$, the upper limit of drift mobility is $377\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ at room temperature, and the Hall factor becomes smaller than 1, which cannot be revealed by the formula under the isotropic parabolic assumption at all, indicating the importance of magnetic transport calculation from ab initio based BTE.

First-principles calculations of inherent properties of Rb based state-of-the-art half-Heusler compounds: promising materials for renewable energy applications

In the present work, we have studied structural, electronic, optical and thermoelectric properties of Rb based state-of-the-art materials RbYZ (Y = Be, Mg, Ca, Sr and Ba; Z = P, As, Sb and Bi) having 8 valence electron count (VEC) using density functional theory followed by solution of Boltzmann transport equation with constant relaxation time approximation. The exchange and correlation potential are described by the GGA of Wu and Cohen (GGA-WC); the Becke-Johnson approach modified by Tran and Blaha (TB-mBJ) has been used to model the exchange-correlation potential. The bandgap of these materials lies in the range of 0.201 eV—2.591 eV. The various optical parameters are comparable with the state-of-the-art photovoltaic materials. Thermoelectric properties have been computed at 300 K, 600 K and 900 K. At these temperatures lattice thermal conductivity have been computed using Slack’s model. This detailed study shows that these compounds are promising for renewable energy applications.

Electronic and thermo‐physical properties of double antiperovskites X6SOA2 (X = Na, K and A = Cl, Br, I): A non‐toxic and efficient energy storage materials

AbstractWe have explored the structural, electronic and transport parameters of cubic double antiperovskite structure X6SOA2 (X = Na, K and A = Cl, Br, I) using density functional theory followed by the solution of Boltzmann transport equation with constant relaxation time approximation. The exchange and correlation potential are described by the PBE‐GGA; the Becke‐Johnson approach modified by Tran and Blaha (TB‐mBJ) has been used to model the exchange‐correlation potential. Band gap of these materials have been found in between 2.85–4.24 eV. Thermoelectric properties have been computed at 300, 600 and 900 K. It has been found that figure of merit of double antiperovskite materials approaches to unity in both n‐ and p‐type regions. Hence these materials may turn out to be potential thermoelectric candidates. As these properties of the titled compounds have been explored for the very first time, hence, this work may open a new panorama for various detailed experimental and theoretical studies in the quest for non‐toxic, environmentally safe, and efficient energy sources.

Physics-informed neural networks for solving multiscale mode-resolved phonon Boltzmann transport equation

Boltzmann transport equation (BTE) is an ideal tool to describe the multiscale phonon transport phenomena, which are critical to applications like microelectronics cooling. Numerically solving phonon BTE is extremely computationally challenging due to the high dimensionality of such problems, especially when mode-resolved properties are considered. In this work, we demonstrate the use of physics-informed neural networks (PINNs) to efficiently solve phonon BTE for multiscale thermal transport problems with the consideration of phonon dispersion and polarization. In particular, a PINN framework is devised to predict the phonon energy distribution by minimizing the residuals of governing equations and boundary conditions, without the need for any labeled training data. Moreover, geometric parameters, such as the characteristic length scale, are included as a part of the input to PINN, which enables learning BTE solutions in a parametric setting. The effectiveness of the present scheme is demonstrated by solving a number of phonon transport problems in different spatial dimensions (from 1D to 3D). Compared to existing numerical BTE solvers, the proposed method exhibits superiority in efficiency and accuracy, showing great promises for practical applications, such as the thermal design of electronic devices.

An explicit modal discontinuous Galerkin method for Boltzmann transport equation under electronic nonequilibrium conditions

The present paper deals with the development of a numerical scheme for the solution of two-dimensional Boltzmann transport equation (BTE) under electronic nonequilibrium conditions. A two-dimensional explicit modal discontinuous Galerkin (DG) method with uniform meshes is developed for solving the BTE in conjunction with the relaxation time approximation. The spatial discretization is carried out using a high-order DG method, where the polynomial solutions are represented using scaled Legendre basis functions. A third-order explicit SSP-RK scheme is applied for temporal discretization to the resulting semi-discrete ordinary differential equation. The analytic steady state solution of BTE at low electromagnetic field is considered as the validation study. The numerical scheme is applied to the description of electron transport in non-linear dynamic conductivity based on static electric field. The electrical and Hall conductivities with nonequilibrium conditions is estimated for the BTE solution in presence of the electromagnetic field. These transport coefficients are also examined with the dependence of temperature and chemical potential. It is observed that the electrical and Hall conductivity decreases in the presence of a magnetic field. Numerical results demonstrate the potential advantages of the high-order scheme in treating ultrafast nonequilibrium dynamics.

Indirect electron-phonon interaction leading to significant reduction of thermal conductivity in graphene

We investigate the effect of electron-phonon interaction (EPI) on the lattice thermal conductivity κph of graphene from first-principles calculations. By employing an iterative solution of Boltzmann transport equation (BTE), we highlight the marked effect of the indirect coupling between the flexural acoustic (ZA) phonons and electrons on the thermal conductivity in graphene. Although the ZA phonons, the dominant carriers of κph, do not interact with electrons directly due to the reflection symmetry with respect to the basal plane, their anharmonic interactions with the in-plane transverse acoustic (TA) and longitudinal acoustic (LA) phonons that can be effectively scattered by electrons have a significant effect on κph. This is originated from the dominance of normal processes over Umklapp processes in graphene. Specifically, this indirect effect can result in up to 21% reduction of κph even at room temperature, and 32% reduction in κph at 200 K. Moreover, κph does not decrease monotonically with increasing charge carrier density n. Instead, κph is minimized at ∼4.9×1014cm−2. This unusual finding is found to be strongly correlated with the electron density of states at corresponding Fermi levels. This indirect effect should also exist widely in other materials, whose intrinsic lattice thermal conductivities are dominated by normal processes. On the other hand, we also explore the electronic thermal conductivity κe varying with n. Intriguingly, at room temperature κe starts increasing dramatically at n=4.9×1014cm−2, where the corresponding Bloch-Grüneisen transition temperature ΘBG exceeds the room temperature.

An innovative technique for electronic transport model of group-III nitrides

An optimized empirical pseudopotential method (EPM) in conjunction with virtual crystal approximation (VCA) and the compositional disorder effect is used for simulation to extract the electronic material parameters of wurtzite nitride alloys to ensure excellent agreement with the experiments. The proposed direct bandgap results of group-III nitride alloys are also compared with the different density functional theories (DFT) based theoretical results. The model developed in current work, significantly improves the accuracy of calculated band gaps as compared to the ab-initio method based results. The physics of carrier transport in binary and ternary nitride materials is investigated with the help of in-house developed Monte Carlo algorithms for solution of Boltzmann transport equation (BTE) including nonlinear scattering mechanisms. Carrier–carrier scattering mechanisms defined through Coulomb-, piezoelectric-, ionized impurity-, surface roughness-scattering with acoustic and intervalley scatterings, all have been given due consideration in present model. The direct and indirect energy bandgap results have been calibrated with the experimental data and use of symmetric and asymmetric form factors associated with respective materials. The electron mobility results of each binary nitride material have been compared and contrasted with experimental results under appropriate conditions and good agreement has been found between simulated and experimental results.

High figure-of-merit in the heavy-fermion UN2 system for radioisotope thermoelectric applications

The design of uranium-based thermoelectric (TE) materials presents a novel and intriguing strategy for directly converting nuclear heat into electrical power. Using high-level first-principles approach combined with accurate solution of Boltzmann transport equation, we demonstrate that a giant n-type power factor of 13.8 mW m−1 K−2 and a peak ZT value of 2.2 can be realized in the heavy-fermion UN2 compound at 700 K. Such promising TE performance arises from the large degeneracy (N v = 14) of heavy conduction band coupled with weak electron–phonon interactions, which is in principle governed by the strong Coulomb correlation among the partially filled U-5f electrons in the face-centered cubic structure. Collectively, our theoretical work suggests that the energetic UN2 could serve as both excellent heat source and efficient power convertor, which also uncovers an underexplored area for TE research.

Maximum evaporating flux of molecular fluids from a planar liquid surface.

In this work, we use the kinetic theory of gases (KTG) to develop a theoretical model to understand the role of internal motions of molecules on the maximum evaporation flux from a planar liquid surface. The kinetic theory is applied to study the evaporation of molecular fluids into a vacuum and predict the dimensionless maximum evaporation flux (J_{R,max}, i.e., the ratio of the maximum evaporation flux to the molar flux emitted from a liquid surface). The key assumptions regarding the velocity distribution function (VDF) of polyatomic molecules in the highly nonequilibrium vapor near the evaporating surface are validated by the VDF obtained directly from molecular dynamics (MD) simulations. Our KTG-based analysis shows that J_{R,max} is affected by the specific heat (c_{V,int}) associated with internal degrees of freedom of fluid molecules. When the maximum evaporation flux is reached, the isotropic evaporating vapor far from the liquid surface moves at its speed of sound regardless of whether it is a monatomic vapor or polyatomic vapor. To fundamentally understand the evaporation of a molecular fluid into a vacuum, we solve the Boltzmann transport equation (BTE) to obtain the temperature, density, and flow speed distributions in the highly nonequilibrium evaporating vapor flow. Our BTE solutions indicate that there are several universal features of the evaporating vapor when the maximum evaporation flux occurs. In particular, we find that the evaporating vapor flow speed reaches the maximum value of sqrt[1.5] times the most probable thermal speed in the vapor flow direction at the vacuum boundary, and this maximum value is independent of fluid properties. All theoretical predictions in this work are verified by the MD simulation results of the evaporation of the model liquid Ar and the model liquid n-dodecane into a vacuum, and existing experimental data.

Accelerating first-principles estimation of thermal conductivity by machine-learning interatomic potentials: A MTP/ShengBTE solution

Accurate evaluation of the thermal conductivity of a material can be a challenging task from both experimental and theoretical points of view. In particular for the nanostructured materials, the experimental measurement of thermal conductivity is associated with diverse sources of uncertainty. As a viable alternative to experiment, the combination of density functional theory (DFT) simulations and the solution of Boltzmann transport equation is currently considered as the most trusted approach to examine thermal conductivity. The main bottleneck of the aforementioned method is to acquire the anharmonic interatomic force constants using the computationally demanding DFT calculations. In this work we propose a substantially accelerated approach for the evaluation of anharmonic interatomic force constants via employing machine-learning interatomic potentials (MLIPs) trained over short ab initio molecular dynamics trajectories. The remarkable accuracy of the proposed accelerated method is confirmed by comparing the estimated thermal conductivities of several bulk and two-dimensional materials with those computed by the full-DFT approach. The MLIP-based method proposed in this study can be employed as a standard tool, which would substantially accelerate and facilitate the estimation of lattice thermal conductivity in comparison with the commonly used full-DFT solution.

Strongly Out-of-Equilibrium Simulations for Electron Boltzmann Transport Equation Using Modal Discontinuous Galerkin Approach

In this study, we present simulations of strongly out-of-equilibrium conditions for the one-dimensional electron Boltzmann transport equation (BTE) in semiconductor devices. An explicit modal discontinuous Galerkin method is employed to solve the electron BTE along with the simplest collisional model–relaxation time approximation. The electron BTE system is discretized in momentum and time, and applied to the description of the dynamics of non-linear electron transport under a strong static electric field. A third-order explicit SSP-RK scheme is adopted for the temporal discretization of the resulting semi-discrete ODE. For the steady-state electron BTE, the analytical solutions are derived at low and high electric fields which are used to validate the numerical approach. The computed solution show a good agreement with derived analytic solution in wide range of tested parameters and regimes. An extensive range of numerical simulations has also been performed to investigate the impact of exploited flow parameters on the electron BTE solutions. Ultimately, these results predict that the modal DG approach is particularly effective in treating the strongly out-of-equilibrium regimes encountered like in ultrafast transport dynamics.

Electronic Thermal Conductivity and Thermoelectric Performance of PbBi4Te7

Bismuth telluride and its related compounds are the state-of-the-art thermoelectric materials operating at room temperature. Bismuth telluride with Pb substituted, PbBi4Te7, has been found to be a new quasi-binary compound with an impressive high power factor. In this work, in the framework of density functional theory, we study the electronic thermal conductivity of the compound by employing the solution of Boltzmann Transport Equation in a constant relaxation-time approximation. The results show that the electronic thermal conductivity drastically increases with the increase of temperature and carrier concentration which have a detrimental effect on the thermoelectric performance. At a particular temperature, the competition between the thermal conductivity, the Seebeck coefficient and the electrical conductivity limits the thermoelectric figure of merit, ZT. The maximum ZT value of about 0.47 occurs at 520 K and at the carrier concentration of 5.0×1019cm-3 for n-type doping. This suggests that to maximize the thermoelectric performance of the compound, the carrier concentration must be carefully controlled and optimized whereas the best operating temperature is around 500 K.

Valance Band Maximum and Thermoelectric Properties of Bi\(_2\)O\(_2\)Se: First-Principles Calculations

Bi2O2Se has been known as a promising thermoelectric material with low thermal conductivity. Detail understanding of band structure is therefore important. In this report, by employing first-principles density functional theory and using primitive unit cell, the electronic band structure of Bi2O2Se is examined. The compound is found to be a narrow band gap semiconductor with very flat bands at the valence band maximum (VBM). Nevertheless, the curvature of energy surface at VBM is directional dependent. Overall, the heavy bands at VBM do not reduce drastically electrical conductivity. It is demonstrated by utilizing the solution of Boltzmann Transport Equation to compute the transport coefficients, i.e. the Seebeck coefficient, the electrical conductivity thereby the power factor and the electronic thermal conductivity. The figure of merit of the compound is also estimated and discussed. The p-type doping is suggested increasing the thermoelectric performance of the compound. All results are in good agreement with experiments and calculations reported earlier.

Strongly anisotropic thermal conductivity in planar hexagonal borophene oxide sheet

The flat hexagonal borophene oxide (B2O) has the highest Li storage capacity among existing two-dimensional materials. Thermal conductivity is an important parameter for the safety of Li-ion batteries. We investigate the lattice thermal conductivity of B2O by solving phonon Boltzmann transport equation combined with the first-principles calculations. We found that the relaxation time approximation remarkably underestimate the thermal conductivity (κ) of monolayer B2O, revealing phonon hydrodynamics characteristic. The κ of B2O from the exact solution of Boltzmann transport equation is 53 Wm−1K−1 and 130 Wm−1K−1 along armchair-direction and zigzag-direction at 300 K, respectively. B2O exhibits strong thermal transport anisotropy due to anisotropic phonon group velocity, obviously larger than that of other borophene allotropes. At room temperature, the phonon mean free path of B2O is about 231 nm and 49 nm along armchair-direction and zigzag-direction, respectively. The highly anisotropic thermal conductivity of B2O offers new possibilities for its applications in thermal management.

Modeling of carrier mobility for semispherical quantum dot infrared photodetectors (QDIPs)

Carrier mobility for quantum dot infrared photodetectors is considered as one of the critical parameters to determine many important device’s performance parameters such as the electrical conductivity, drift velocity, dark current and photocurrent. In this paper a complete theoretical model of the carrier mobility for semispherical quantum dot structures is developed. This model is based on the solution of Boltzmann transport equation all over the device. A parametric study of the effects of the QD density and the dimensions of the QD on the carrier mobility is investigated. Finally, the carrier mobility in semispherical QDIP is compared with other QD structures. The presented model is a generic model that can be applied for different semispherical QD structures.

Transport phenomena in the Knudsen layer near an evaporating surface.

Using the combination of the kinetic theory of gases (KTG), Boltzmann transport equation (BTE), and molecular dynamics (MD) simulations, we study the transport phenomena in the Knudsen layer near a planar evaporating surface. The MD simulation is first used to validate the assumption regarding the anisotropic velocity distribution of vapor molecules in the Knudsen layer. Based on this assumption, we use the KTG to formulate the temperature and density of vapor at the evaporating surface as a function of the evaporation rate and the mass accommodation coefficient (MAC), and we use these vapor properties as the boundary conditions to find the solution to the BTE for the anisotropic vapor flow in the Knudsen layer. From the study of the evaporation into a vacuum, we show the ratio of the macroscopic speed of vapor to the most probable thermal speed of vapor molecules in the flow direction will always reach the maximum value of sqrt[1.5] at the vacuum boundary. The BTE solutions predict that the maximum evaporation flux from a liquid surface at a given temperature depends on both the MAC and the distance between the evaporating surface and the vacuum boundary. From the study of the evaporation and condensation between two parallel plates, we show the BTE solutions give good predictions of transport phenomena in both the anisotropic vapor flow within the Knudsen layer and the isotropic flow out of the Knudsen layer. All the predictions from the BTE are verified by the MD simulation results.