Linearized Boltzmann Transport Equation Research Articles

Extension of Mott Formula in the Linearized Boltzmann Transport Equation to the Study of Thermoelectric Power of Electron in Metal

The focus of this paper is to compute and study thermoelectric power of electron in metals using Mott formula based on free electron theory to get an insight on the effect of strain on electron thermoelectric power in metals. Electron density parameter of strained metals and Poisson ratio of metals is obtained and used in this work unlike some researchers that assumed a value for Poisson ratio of all metals. This assumption make the electron density parameter of strained and unstrained metals to be equal and then reduce the computation to only one input parameter. The experimental and simulated value are in good agreement. This agreement is due to the usefulness of free electron theory in theoretical forecast of some properties of metals. The Poisson ratio is involved in simulation. Result obtained revealed that thermoelectric power of electron in metals goes up as the electron density parameter rises due to high-order contributions from electron scattering theory. Electron thermoelectric power in metals increases as the temperature increases due to electron thermal excitation from their mean position. The electron thermoelectric power rises as strain rises for metals.

Correction: Quantitative Evaluation of Seebeck Coefficient Using Linearized Boltzmann Transport Equation for Fe2VAl-Based Compounds

Correction: Quantitative Evaluation of Seebeck Coefficient Using Linearized Boltzmann Transport Equation for Fe2VAl-Based Compounds

Linear Boltzmann equation solver for voxel-level dosimetry in radiopharmaceutical therapy: Comparison with Monte Carlo and kernel convolution.

With recent interest in patient-specific dosimetry for radiopharmaceutical therapy (RPT) and selective internal radiation therapy (SIRT), an increasing number of voxel-based algorithms are being evaluated. Monte Carlo (MC) radiation transport, generally considered to be the most accurate among different methods for voxel-level absorbed dose estimation, can be computationally inefficient for routine clinical use. This work demonstrates a recently implemented grid-based linear Boltzmann transport equation (LBTE) solver for fast and accurate voxel-based dosimetry in RPT and SIRT and benchmarks it against MC. A deterministic LBTE solver (Acuros MRT) was implemented within a commercial RPT dosimetry package (Velocity 4.1). The LBTE is directly discretized using an adaptive mesh refined grid and then the coupled photon-electron radiation transport is iteratively solved inside specified volumes to estimate radiation doses from both photons and charged particles in heterogeneous media. To evaluate the performance of the LBTE solver for RPT and SIRT applications, 177 Lu SPECT/CT, 90 Y PET/CT, and 131 I SPECT/CT images of phantoms and patients were used. Multiple lesions (2-1052mL) and normal organs were delineated for each study. Voxel dosimetry was performed with the LBTE solver, dose voxel kernel (DVK) convolution with density correction, and a validated in-house MC code using the same time-integrated activity and density maps as input to the different dose engines. The resulting dose maps, difference maps, and dose-volume-histogram (DVH) metrics were compared, to assess the voxel-level agreement. Evaluation of mean absorbed dose included comparison with structure-level estimates from OLINDA. In the phantom inserts/compartments, the LBTE solver versus MC and DVK convolution demonstrated good agreement with mean absorbed dose and DVH metrics agreeing to within 5% except for the D90 and D70 metrics of a very low activity concentration insert of 90 Y where the agreement was within 15%. In the patient studies (five patients imaged after 177 Lu DOTATATE RPT, five after 90 Y SIRT, and two after 131 I radioimmunotherapy), in general, there was better agreement between the LBTE solver and MC than between LBTE solver and DVK convolution for mean absorbed dose and voxel-level evaluations. Across all patients for all three radionuclides, for soft tissue structures (kidney, liver, lesions), the mean absorbed dose estimates from the LBTE solver were in good agreement with those from MC (median difference<1%, maximum 9%) and those from DVK (median difference<5%, maximum 9%). The LBTE and OLINDA estimates for mean absorbed dose in kidneys and liver agreed to within 10%, but differences for lesions were larger with a maximum 14% for 177 Lu, 23% for 90 Y, and 26% for 131 I. For bone regions, the agreement in mean absorbed doses between LBTE and both MC and DVK were similar (median<11%, max 11%) while for lung the agreement between LBTE and MC (median<1%, max 8%) was substantially better than between LBTE and DVK (median<16%, max 33%). Voxel level estimates for soft tissue structures also showed good agreement between the LBTE solver and both MC and DVK with a median difference<5% (maximum<13%) for the DVH metrics with all three radionuclides. The largest difference in DVH metrics was for the D90 and D70 metric in lung and bone where the uptake was low. Here, the difference between LBTE and MC had a median value<14% (maximum 23%) for bone and<4% (maximum 37%) for lung, while the corresponding differences between LBTE and DVK were<23% (maximum 31%) and<67% (maximum 313%), respectively. For a typical patient with a matrix size of 166×166×129 (voxel size 3×3×3 mm3 ), voxel dosimetry using the LBTE solver was as fast as ∼2 min on a desktop computer. Having established good agreement between the LBTE solver and MC for RPT and SIRT applications, the LBTE solver is a viable option for voxel dosimetry that can be faster than MC. Further analysis is being performed to encompass the broad range of radionuclides and conditions encountered clinically.

Computational prediction of high thermoelectric performance in As2Se3 by engineering out-of-equilibrium defects.

We employed first-principles calculations to investigate the thermoelectric transport properties of the compound As2Se3. Early experiments and calculations have indicated that these properties are controlled by a kind of native defect called antisites. Our calculations using the linearized Boltzmann transport equation within the relaxation time approximation show good agreement with the experiments for defect concentrations of the order of 1019 cm-3. Based on our total energy calculations, we estimated the equilibrium concentration of antisite defects to be about 1014 cm-3. These results suggest that the large concentration of defects in the experiments is due to kinetic and/or off-stoichiometry effects and in principle it could be lowered, yielding relaxation times similar to those found in other chalcogenide compounds. In this case, for relaxation time higher than 10 fs, we obtained high thermoelectric figures of merit of 3 for the p-type material and 2 for the n-type one.

Comprehensive understanding of electron mobility and superior performance in sub-10 nm DG ML tetrahex-GeC2 n-type MOSFETs.

In this study, we have investigated the electron mobility of monolayered (ML) tetrahex-GeC2 by solving the linearized Boltzmann transport equation (BTE) with the normalized full-band relaxation time approximation (RTA) using density functional theory (DFT). Contrary to what the deformation potential theory (DPT) suggested, the ZA acoustic mode was determined to be the most restrictive for electron mobility, not the LA mode. The electron mobility at 300 K is 803 cm2 (V s)-1, exceeding the 400 cm2 (V s)-1 of MoS2 which was calculated using the same method and measured experimentally. The ab initio quantum transport simulations were performed to assess the performance limits of sub-10 nm DG ML tetrahex-GeC2 n-type MOSFETs, including gate lengths (Lg) of 3 nm, 5 nm, 7 nm, and 9 nm, with the underlap (UL) effect considered for the first two. For both high-performance (HP) and low-power (LP) applications, their on-state currents (Ion) can meet the requirements of similar nodes in the ITRS 2013. In particular, the Ion is more remarkable for HP applications than that of the extensively studied MoS2. For LP applications, the Ion values at Lg of 7 and 9 nm surpass those of arsenene, known for having the largest Ion among 2D semiconductors. Subthreshold swings (SSs) as low as 69/53 mV dec-1 at an Lg of 9 nm were observed for HP/LP applications, and 73 mV dec-1 at an Lg of 5 nm for LP applications, indicating the excellent gate control capability. Moreover, the delay time τ and power dissipation (PDP) at Lg values of 3 nm, 5 nm, 7 nm, and 9 nm are all below the upper limits of the ITRS 2013 HP/LP proximity nodes and are comparable to or lower than those of typical 2D semiconductors. The sub-10 nm DG ML tetrahex-GeC2 n-type MOSFETs can be down-scaled to 9 nm and 5 nm for HP and LP applications, respectively, displaying desirable Ion, delay time τ, and PDP in the ballistic limit, making them a potential choice for sub-10 nm transistors.

Thermal conductivity and lattice dynamics of thermoelectric oxychalcogenide BiCuTeO

Recently, BiCuTeO is considered as one of the promising thermoelectric materials due to its ultra-low thermal conductivity. For this reason, the thermoelectric characteristic of this material has been studied to evaluate the lattice thermal conductivity (𝜅𝜅𝐿𝐿) from firstprinciples calculations which are based on solving linearized Boltzmann transport equations (LBTE) through the relaxation time (RTA) approximation. These calculations are used to predict the behavior of phonons in order to understand the origin of the ultralow thermal conductivity of BiCuTeO. The lattice thermal conductivity of BiCuTeO is reproduced with high accuracy. Our calculations predict that BiCuTeO announces a strong anharmonicity, which is the cause of the very low value of the thermal conductivity. This results in very high group speeds. Moreover, the calculations of the elastic properties, dielectric constants, phonon group velocities, lifetimes, and Grüneisen parameters shows that the lattice thermal conductivity exhibits an obvious anisotropy.

First principle study of electronic and optical properties of WS2(1-x)Se2x obtained by isoelectronic Se substitution on S-site of monolayer WS2

Two-dimensional semiconducting transition metal dichalcogenides have recently grabbed attention among the researchers, due to their extraordinary electrical, optical and thermal properties. Also, they have shown their suitability for application in digital electronics, photovoltaic cells, Thermoelectric generators and so on. Still, it is desired to precisely control the material properties to expand their scope of application and revamp the device performance. The substitutional doping technique is one of the widely explored processes, which is conventionally used to modulate the material properties. In the present work, the substitutional doping of selenium at the sulfur site within the WS2 sheet, is used indigenously to improve the electrical and optical response of the material. In doing so, electrical and optical properties of such doped materials are studied by employing density functional theory and linearized Boltzmann transport equation by considering the relaxation time approximation. The study has shown an almost linear variation of electronic band gap energy with the doping percentages. Also, the electron and hole mobility were found to be tailored due to the doping. Notably, hole mobility has shown a noticeable improvement at or near 60 percent of selenium doping. In the last section of the current work, the optical properties are extracted for the doped structures. The WS2 monolayer has shown the highest absorption peak near 3 eV. But for the doped materials, absorption peaks are relatively smaller and left shifted in the energy axis due to their low band gap energies.

Physics-Informed Neural Network with Fourier Features for Radiation Transport in Heterogeneous Media

Typical machine learning (ML) methods are difficult to apply to radiation transport due to the large computational cost associated with simulating problems to create training data. Physics-informed Neural Networks (PiNNs) are a ML method that train a neural network with the residual of a governing equation as the loss function. This allows PiNNs to be trained in a low-data regime in the absence of (experimental or synthetic) data. PiNNs also are trained on points sampled within the phase-space volume of the problem, which means they are not required to be evaluated on a mesh, providing a distinct advantage in solving the linear Boltzmann transport equation, which is difficult to discretize. We have applied PiNNs to solve the streaming and interaction terms of the linear Boltzmann transport equation to create an accurate ML model that is wrapped inside a traditional source iteration process. We present an application of Fourier Features to PiNNs that yields good performance on heterogeneous problems. We also introduce a sampling method based on heuristics that improves the performance of PiNN simulations. The results are presented in a suite of one-dimensional radiation transport problems where PiNNs show very good agreement when compared to fine-mesh answers from traditional discretization techniques.

From Green-Kubo to the full Boltzmann kinetic approach to heat transport in crystals and glasses

We show that vertex corrections to the quasi-harmonic Green-Kubo theory of heat transport in insulators naturally lead to a generalisation of the expression for the conductivity that could be derived from the linearized Boltzmann equation, when the effects of the full scattering matrix are accounted for. Our results, which are obtained from the Mori-Zwanzig memory-function formalism, provide a fully ab initio derivation of the linearized Boltzmann transport equation and establish a connection between two recently proposed unified approaches to heat transport in insulating crystals and glasses.

ElecTra code: Full-band electronic transport properties of materials

This paper introduces ElecTra, an open-source code which solves the linearized Boltzmann transport equation in the relaxation time approximation for charge carriers in a full-band electronic structure of arbitrary complexity, including their energy, momentum, and band-index dependence. ElecTra stands for ‘ELECtronic TRAnsport’ and computes the electronic and thermoelectric transport coefficients electrical conductivity, Seebeck coefficient, electronic thermal conductivity, and mobility, for semiconductor materials, for both unipolar and bipolar (small bandgap) materials. The code uses computed full-bands and relevant scattering parameters as inputs and considers single crystal materials in 3D and 2D. The present version of the code (v1) considers: i) elastic scattering with acoustic phonons and inelastic scattering with non-polar optical phonons in the deformation potential approximation, ii) inelastic scattering with polar phonons, iii) scattering with ionized dopants, and iv) alloy scattering. The user is given the option of intra- and inter-band scattering considerations. The simulation output also includes relevant relaxation times and mean-free-paths. The transport quantities are computed as a function of Fermi level position, doping density, and temperature. ElecTra can interface with any DFT code which saves the electronic structure in the ‘.bxsf’ format. In this paper ElecTra is validated against ideal electronic transport situations of known analytical solutions, existing codes employing the constant relaxation time approximation, as well as experimentally well-assessed materials such as Si, Ge, SiGe, and GaAs. Program summaryProgram title: ElecTra – Electronic Transport simulation labCPC Library link to program files:https://doi.org/10.17632/ycgx2fjzb6.1Licensing provisions: GPLv3Programming Language: MATLAB®Nature of the problem: computing the electronic and thermoelectric charge transport coefficients of materials with arbitrary complex full-band electronic structures, considering the carrier energy-, momentum-, and band-dependence of the scattering rates.Solution method: Semiclassical Linearized Boltzmann transport equation, with electronic structures (DFT or analytical) as input, formed into constant-energy surfaces, with scattering rates evaluated using Fermi's Golden Rule.Additional comments including restrictions and unusual features:•Programming interface: any DFT code which saves data in the ‘.bxsf’ format.•RAM: a case study for a half-Heusler bandstructure on a 51×51×51k-mesh, 2 Gb per processor is needed•Running time: for the example above, depending on the number and complexity of the scattering mechanisms and the number of simulated Fermi levels and temperatures considered, the time needed varies from ∼ 1 hour on a desktop PC or laptop (light simulations), to 5-10 hours on an HPC with 30-45 cores (heavy simulations). Using the constant relaxation time and constant mean-free-path approximations on a desktop PC or laptop, the running time is of the order of minutes.

NaBeAs and NaBeSb: Novel Ternary Pnictides with Enhanced Thermoelectric Performance

Thermoelectrics is a clean energy technology that converts thermal energy into electrical energy and provides an environmentally friendly and viable route to produce electricity. In this work, we thoroughly explore the thermoelectric performance of two novel ternary pnictide compounds, NaBeAs and NaBeSb, using full ab initio calculations and solving linear Boltzmann transport equations. We predict that both compounds exhibit excellent dynamical and thermal stability. The combined effect of low phonon group velocity and weak bonding interaction results in intrinsic low lattice thermal conductivities of 1.41 and 0.59 W/mK for NaBeAs and NaBeSb, respectively, in the z-axis direction at 600 K. The electronic structure exhibits high conduction band degeneracy, which results in favorable electrical transport characteristics. The unique combination of very low lattice thermal conductivity and good electrical transport properties leads to excellent n-type thermoelectric properties. Furthermore, in n-type doping, the figures of merit reach 1.82 and 4.26 for NaBeAs and NaBeSb at 600 K. Our findings show that both compounds are promising n-type thermoelectric materials at medium temperatures.

Effects of phonon bandgap on phonon–phonon scattering in ultrahigh thermal conductivity θ-phase TaN

Phonon bandgap typically has a significant effect on phonon–phonon scattering process. In this work, the effects of mass modified phonon bandgap in θ-phase TaN are systemically investigated by the means of first-principles calculations with linearized Boltzmann transport equation. Through detailed calculations, we find that phonon bandgap has a significant effect on three-phonon process while exhibits a much weaker effect on four-phonon process. The reason for the ultrahigh thermal conductivity of θ-phase TaN is the long lifetime of phonons including both three-phonon and four-phonon processes, which originates from the weak phonon anharmonicity and large phonon bandgap-induced small phonon–phonon scattering phase space. This work advances the understanding of phonon bandgap effects on phonon transport.

Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators

This work presents a data-driven, projection-based parametric reduced-order model (ROM) for the neutral particle radiation transport (linear Boltzmann transport) equation. The ROM utilizes the method of snapshots with proper orthogonal decomposition. The novelty of the work is in the detailed proposal to exploit the parametrically affine transport operators to intrusively, yet efficiently, build the reduced transport operators in real time in a matrix-free manner compatible with sweep-based transport solvers. This affine-based ROM is applied to one-dimensional (1-D), two-dimensional (2-D), and 2-D multigroup transport benchmarks and is found to significantly outperform less intrusive ROMs in terms of speed for a desired accuracy level. The ROM has an 18.2 to 89.4 speedup with an error range of 0.0002% to 0.01% for the 1-D benchmark, a 1120× to 4870× speedup with an error range of 0.0009% to 0.01% for the 2-D benchmark, and a 54 600× to 399 800× speedup with an error range of 0.00022% to 0.01% for the multigroup 2-D benchmark. Even higher speedups are expected for three-dimensional multigroup transport problems.

Thermal conductivity of group-III phosphides: The special case of GaP

We calculate the lattice thermal conductivity of cubic group-III phosphides by solving the linearized Boltzmann transport equation for phonons and discuss how there is a large disagreement between the literature values for GaP. We find the presence of comparably large third-order interatomic force constants at large interaction distances that influence the thermal conductivity of GaP. When both four-phonon scattering and the contributions from the long-range interactions are included in the calculation, a thermal conductivity of $76\phantom{\rule{0.28em}{0ex}}{\mathrm{W}\phantom{\rule{0.16em}{0ex}}\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ is obtained at room temperature, in good agreement with a reported experimental value of $77\phantom{\rule{0.28em}{0ex}}{\mathrm{W}\phantom{\rule{0.16em}{0ex}}\mathrm{m}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$. Our results show the importance of carrying out convergence tests with respect to all the calculation parameters and could help explain the thermal conductivity in other systems where similar ambiguities are found.

Electron and Hole Mobility of SnO2 from Full-Band Electron–Phonon and Ionized Impurity Scattering Computations

Mobility is a key parameter for SnO2, which is extensively studied as a practical transparent oxide n-type semiconductor. In experiments, the mobility of electrons in bulk SnO2 single crystals varies from 70 to 260 cm2V−1s−1 at room temperature. Here, we calculate the mobility as limited by electron–phonon and ionized impurity scattering by coupling the Boltzmann transport equation with density functional theory electronic structures. The linearized Boltzmann transport equation is solved numerically beyond the commonly employed constant relaxation-time approximation by taking into account all energy and momentum dependencies of the scattering rates. Acoustic deformation potential and polar optical phonons are considered for electron–phonon scattering, where polar optical phonon scattering is found to be the main factor which determines the mobility of both electrons and holes at room temperature. The calculated phonon-limited electron mobility is found to be 265 cm2V−1s−1, whereas that of holes is found to be 7.6 cm2V−1s−1. We present the mobility as a function of the carrier concentration, which shows the upper mobility limit. The large difference between the mobilities of n-type and p-type SnO2 is a result of the different effective masses between electrons and holes.

Accuracy of Acuros^{text {TM}} BV as determined from GATE monte-carlo simulation

The American Association of Physicists in Medicine’s Task Group No.43 has provided a standardised dose calculation methodology that is now the international benchmark for all brachytherapy dosimetry publications and treatment planning systems. However, limitations in this methodology has seen the development of Model-Based Dose Calculation Algorithms (MBDCA). In 2009, Varian (Varian Medical Systems, Palo Alto, CA, USA) released Acuros^{text {TM}} BrachyVision (ABV) which calculates dose by explicitly solving the Linear Boltzmann Transport Equation. In this study we have assessed the accuracy of ABV dose calculations within a range of materials relevant to high dose rate brachytherapy with an iridium-192 (^{text {192}}Ir) source. Accuracy assessment has been achieved by implementing a modelled GamaMed Plus ^{text {192}}Ir source within a series of phantoms using the GEANT4 Application for Emission Tomography (GATE) to calculate dose for comparison with dose as determined by ABV. Comparisons between GATE and ABV were made using point-to-point profile comparisons and 1D gamma analysis. Source validation results yielded good agreement with published data. Spectrum and TG43U1 comparisons showed no major differences, with TG43U1 comparisons agreeing within ± 1%. Point-to-point comparisons showed large differences between GATE and ABV near the source and in low density materials. 1D gamma analysis pass criteria of 2%/1 mm and 2%/2 mm yielded passing rates ranging between 51.72–100% and 62.07–100% respectively. A critical analysis of this study’s results suggest that ABV is unable to accurately calculate doses in low density materials. Furthermore, spatial accuracy of dose near the source is within 2 mm.

Thermoelectric transport properties of XAgP (X = Sr and Ba) from first principles

For thermoelectric applications those materials are of interest that have significant power factor (PF) and low lattice thermal conductivity, . Here we theoretically explore of two novel materials SrAgP and BaAgP using linearized Boltzmann transport equation with a single-mode relaxation time approach. We estimate the figure of merit zT by employing ab-initio calculations based on density functional theory and semiclassical Boltzmann transport theory. It is observed that at room temperature SrAgP exhibits slightly higher lattice thermal conductivity than BaAgP, which is mainly due to the large phonon group velocity. The relaxation time derived from deformation potential theory indicates a higher p-type PF for SrAgP compared to BaAgP over the entire temperature range. This provides an estimate for the figure of merit for the two materials. The low lattice thermal conductivity and higher PF make SrAgP a more promising thermoelectric material.

Assessing the quality of relaxation-time approximations with fully automated computations of phonon-limited mobilities

The mobility of carriers, as limited by their scattering with phonons, can now routinely be obtained from first-principles electron-phonon coupling calculations. However, so far, most computations have relied on some form of simplification of the linearized Boltzmann transport equation based on either the self-energy, the momentum- or constant relaxation time approximations. Here, we develop a high-throughput infrastructure and an automatic workflow and we compute 69 phonon-limited mobilities in semiconductors. We compare the results resorting to the approximations with the exact iterative solution. We conclude that the approximate values may deviate significantly from the exact ones and are thus not reliable. Given the minimal computational overhead, our work encourages to rely on this exact iterative solution and warns on the possible inaccuracy of earlier results reported using relaxation time approximations.

Sensitivity Comparison between Monolayer Graphene and Multilayer Graphene

Graphene is an excellent piezoresistive material. The gauge factor of graphene mirrors the sensitivity of electromechanical devices. This paper mainly studies the gauge factors of different layers of graphene under different deformation conditions. Specifically, a theoretical model was combined with linearized Boltzmann transport equation, and the density function theory (DFT) to explore how the layer number of graphene affects sensitivity. The results show that monolayer graphene is slightly more sensitive than two-layer graphene, and significantly more sensitive than three-layer graphene and four- layer graphene. In particular, monolayer graphene remains highly sensitive under large deformation conditions, which gives monolayer graphene a significant advantage over other layers of graphene. Furthermore, a microelectromechanical system (MEMS) pressure sensor was proposed with monolayer graphene, and compared with previous similar sensors with multilayer graphene in terms of sensitivity.

Electron mobility in ordered β -(AlxGa1−x)2O3 alloys from first-principles

Alloying Ga2O3 with Al2O3 yields diverse structural phases with distinctive optoelectronic properties, making them promising candidates for ultrawide bandgap semiconductors in next-generation power electronics. Yet, there is a lack of sound knowledge of the carrier dynamics in the (AlxGa1−x)2O3 alloys due to their structural complexity. Herein, we focus on the ordered β-(AlxGa1−x)2O3 alloys, predict their carrier mobility, and determine the intrinsic electron mobility limit based on solving linearized Boltzmann transport equations from first principles. The predicted electron mobility for ordered β-(Al0.25Ga0.75)2O3 and β-(Al0.5Ga0.5)2O3 alloys at 300 K, respectively, is 103.6 and 80.60 cm2/V s, demonstrating excellent agreement with literature experiments. Such low electron mobility is limited by the intrinsically strong polar optical phonon (POP) scattering process. As the Al content further increases, the alloy's electron mobility further reduces mainly due to the enlarged Pauling ionicity, Fröhlich coupling constant, and POP scattering. This work provides physical insight into the carrier dynamics in ordered β-(AlxGa1−x)2O3 alloys and seeks to improve the electron mobility for potential applications in high-power electronics.