Linearized Boltzmann Transport Equation Research Articles

Phonon transport properties of two dimensional group-III nitrides (BN, AlN, and GaN)

In this work, the lattice thermal conductivity of two-dimensional honeycomb structures of BN, AlN, and GaN studied using first-principles density functional theory and solving the linearized Boltzmann transport equation by relaxation time approximation. Investigation of the influence of the phonon modes on the thermal conductivity revealed that ZA, and LA + TA + ZO modes have the most and TO + LO have the least contribution to the thermal conductivity of BN whereas in AlN, the contribution of ZA decreases while the effect of TO + LO increases. In GaN, the TO + LO branches Play the major role on the thermal conductivity. The observed differences in the lattice thermal conductivity of these monolayers were explained by considering the variations of their frequency-dependent phonon lifetimes, group velocities, and heat capacities. In BN monolayer, the group velocity and phonon lifetime of the acoustic phonons found to be higher whereas the heat capacity of it's LO and TO branches was lower than the monolayers of AlN and GaN and therefore, the contribution of the acoustical phonon branches to thermal conductivity has become more important. However in GaN, the LO and TO phonons revealed the highest phonon lifetime and specific heat resulting the dominance of the contribution of these phonon modes to the total thermal conductivity.

Low lattice thermal conductivity of pentadiamond

The lattice thermal conductivity of carbon materials is particularly interesting because it can vary within a range spanning five orders of magnitude depending on the atomic configuration. Herein, we systematically study the lattice thermal conductivity and phonon transport properties of pentadiamond, a new three-dimensional carbon allotrope consisting of pentagonal carbon rings. Based on first-principles calculations and an iterative solution to the linearized Boltzmann transport equation, the intrinsic lattice thermal conductivity (kl) is found to be 490.88 W/mK at room temperature, much lower than 2664.93 W/mK of diamond. A detailed analysis of both harmonic and anharmonic properties reveals that the low kl of pentadiamond essentially originates from its large phonon phase space, short phonon relaxation time resulting from strong overlap between the acoustic and low-lying optical phonon branches, and the low phonon group velocity. The distinct thermal transport behavior exhibited in pentadiamond further shows the diversity and complexity in lattice thermal conductivity of carbon allotropes.

Assessment of empirical interatomic potential to predict thermal conductivity in ThO2 and UO2

Computing vibrational properties of crystals in the presence of complex defects often necessitates the use of (semi-)empirical potentials, which are typically not well characterized for perfect crystals. Here we explore the efficacy of a commonly used embedded-atomempirical interatomic potential for the U x Th1−x O2 system, to compute phonon dispersion, lifetime, and branch specific thermal conductivity. Our approach for ThO2 involves using lattice dynamics and the linearized Boltzmann transport equation to calculate phonon transport properties based on second and third order force constants derived from the empirical potential and from first-principles calculations. For UO2, to circumvent the accuracy issues associated with first-principles treatments of strong electronic correlations, we compare results derived from the empirical interatomic potential to previous experimental results. It is found that the empirical potential can reasonably capture the dispersion of acoustic branches, but exhibits significant discrepancies for the optical branches, leading to overestimation of phonon lifetime and thermal conductivity. The branch specific conductivity also differs significantly with either first-principles based results (ThO2) or experimental measurements (UO2). These findings suggest that the empirical potential needs to be further optimized for robust prediction of thermal conductivity both in perfect crystals and in the presence of complex defects.

An integrated experimental and computational investigation of defect and microstructural effects on thermal transport in thorium dioxide

Advanced nuclear reactor concepts aim to use fuels that must withstand unprecedented temperature and radiation extremes. In these fuels, thermal energy transport under irradiation is directly related to fuel longevity, reactor safety, and is arguably one of the most important performance metrics. Here we provide a comprehensive, first-principles-informed treatment of phonon mediated thermal transport in a defect-bearing actinide oxide with direct comparison to experimental measurements. Pristine and proton irradiated thorium dioxide was chosen as a model system to treat the complexity of thermal transport in the presence of lattice defects. A thermal transport model is implemented using the linearized Boltzmann transport equation (LBTE) with input from first principles calculations and defect evolution models. Density functional theory is used to calculate phonon dispersion in thorium dioxide and used as an input to calculate both intrinsic and extrinsic, defect-induced relaxation times. In addition, a defect evolution model is benchmarked using microstructure characterization of as-irradiated thorium dioxide using a combination of electron microscopy and optical spectroscopy. The output of the LBTE is compared directly to mesoscopic measurements of thermal conductivity on length scales commensurate with defect accumulation. Parametric measurements of conductivity with irradiation dose and temperature suggest a saturation in the reduction of thermal conductivity with increasing defect generation, which is partially captured in our defect evolution model and LBTE framework. This comprehensive, atomistic- to meso-scale treatment provides the necessary basis to investigate thermal transport under irradiation in more complex systems that exhibit strong electron correlation.

MCBTE: A variance-reduced Monte Carlo solution of the linearized Boltzmann transport equation for phonons

MCBTE solves the linearized Boltzmann transport equation for phonons in three-dimensions using a variance-reduced Monte Carlo solution approach. The algorithm is suited for both transient and steady-state analysis of thermal transport in structured materials with size features in the nanometer to hundreds of microns range. The code is portable and integrated with both first-principles density functional theory calculations and empirical relations for the input of phonon frequency, group velocity, and mean free path required for calculating the thermal properties. The program outputs space- and time-resolved temperature and heat flux for the transient study. For the steady-state simulations, the frequency-resolved contribution of phonons to temperature and heat flux is written to the output files, thus allowing the study of cumulative thermal conductivity as a function of phonon frequency or mean free path. We provide several illustrative examples, including ballistic and quasi-ballistic thermal transport, the thermal conductivity of thin films and periodic nanostructures, to demonstrate the functionality and to benchmark our code against available theoretical/analytical/computational results from the literature. Moreover, we parallelize the code using the Matlab Distributed Computing Server, providing near-linear scaling with the number of processors. Program summaryProgram Title:MCBTECPC Library link to program files:https://doi.org/10.17632/rzn86w7t3p.1Developer's repository link:https://github.com/abhipath90/MCBTECode Ocean capsule:https://codeocean.com/capsule/7196257Licensing provisions: GPLv3Programming language: MATLABNature of problem: Calculation of time- and space-dependent temperature and heat flux profiles, and frequency-resolved effective thermal conductivity in structured systems where heat is carried by phonons.Solution method: Solution of linearized Boltzmann transport equation for phonons, variance-reduced Monte Carlo approach.Additional comments including restrictions and unusual features: Runtime about 1 to 10 hours on a personal computer.

Thermal conductivity ofCaF2at high pressure

We study the thermal transport properties of three CaF$_{2}$ polymorphs up to a pressure of 30 GPa using first-principle calculations and an interatomic potential based on machine learning. The lattice thermal conductivity $\kappa$ is computed by iteratively solving the linearized Boltzmann transport equation (BTE) and by taking into account three-phonon scattering. Overall, $\kappa$ increases nearly linearly with pressure, and we show that the recently discovered $\delta$-phase with $P\bar{6}2m$ symmetry and the previously known $\gamma$-CaF$_{2}$ high-pressure phase have significantly lower lattice thermal conductivities than the ambient-thermodynamic cubic fluorite ($Fm\bar{3}m$) structure. We argue that the lower $\kappa$ of these two high-pressure phases stems mainly due to a lower contribution of acoustic modes to $\kappa$ as a result of their small group velocities. We further show that the phonon mean free paths are very short for the $P\bar{6}2m$ and $Pnma$ structures at high temperatures, and resort to the Cahill-Pohl model to assess the lower limit of thermal conductivity in these domains.

P05.14 Dosimetric Impact of Using the Acuros XB for Planning of SBRT of Centrally Located NSCLC

Stereotactic Body Radiation Treatment (SBRT) of lung cancer is applied to a heterogeneous density gradient. Dose modeling is highly dependent on the accuracy of the calculation algorithm. This study aims to assess the dosimetric implications calculated by Acuros External Beam (AXB) instead of the Anisotropic Analytical Algorithm (AAA) in SBRT of centrally located-Non small cell lung cancer (CL-NSCLC). Twelve IMRT plans for SBRT of CL-NSCLC were investigated retrospectively. The prescribed scheme was 50 Gy in 5 fractions every 2 days. The plans originally calculated by AAA aimed to cover 95% PTV by varying prescription isodose lines at 70 to 90%. These plans were recalculated by AXB (Dose to medium) and verified by the benchmark Monte Carlo (MC) method. Dosimetric parameters e.g. V20Gy Lungs, D95%(PTV), D98%(GTV), dose gradient (the ratio of 50% prescription isodose volume to the PTV, R50%), and doses to organs at risk (spinal cord, heart, esophagus, trachea, bronchus, and major vessels) were compared. AAA has generally overestimated the PTV dose compared to AXB and MC. The mean differences for D95%(PTV) and D98%(GTV) is 3.1% and 2.8% higher than that of AXB, respectively. The differences were significant with p-value<0.05. They were further verified by MC method, the results agreed with that of AXB. The agreement between AXB and MC in all PTV parameters were within 1.6% and the differences were not significant. No significant differences were also noted on V20Gy Lungs between the three algorithms but three cases were observed with maximum point dose to the lung tissue 1cm surrounding the PTV increased over 11% after the recalculation of AXB. Moreover, two cases were reported exceeding in heart dose and three cases were found degrading from none to minor deviation in dose gradient R50%. AXB has been investigated intensively and found to perform superiorly to AAA especially in calculating the rapid change in heterogeneous density gradient e.g. in regions of re-buildup in soft tissue after passed through lung/air. Although AXB and MC have two completely different way in solving Linear Boltzmann Transport Equation (LBTE), the agreement between AXB and MC was widely acceptable, AXB provides an accurate but faster alternative to MC method. In this study, a significant under-dosage in PTV and large dose difference in modeling tissue/air interface were observed after remodeled with AXB and MC. Since the PTV of SBRT for NSCLC is relatively small, the PTV missed in AAA may lead to a reduction in treatment outcome. Hence, AXB should be used in preference to AAA to model SBRT for centrally located NSCLC.

Metavalent bonding induced abnormal phonon transport in diamondlike structures: Beyond conventional theory

A phenomenon appears in a few examples of the chalcopyrites (space group $I$-42 $d$) where heavier atoms do not necessarily lead to lower lattice thermal conductivity, in contradiction with Keyes expression that formulates an inverse relation of thermal conductivity with mean atomic mass. Herewith, the thermal conductivity of $\mathrm{CuIn}{\mathrm{Se}}_{2}, \mathrm{CuIn}{\mathrm{Te}}_{2}, \mathrm{AgIn}{\mathrm{Se}}_{2}$, and $\mathrm{AgIn}{\mathrm{Te}}_{2}$ was calculated and compared at room temperature from the linearized Boltzmann transport equation using ab initio density functional theory. $\mathrm{CuIn}{\mathrm{Se}}_{2}$ and $\mathrm{AgIn}{\mathrm{Se}}_{2}$ solids exhibit lower lattice thermal conductivity than that of $\mathrm{CuIn}{\mathrm{Te}}_{2}$ and $\mathrm{AgIn}{\mathrm{Te}}_{2}$, respectively, despite the fact that Te atoms are significantly heavier than Se. A comparison between dispersion relation, the Gr\"uneisen parameter, and projected density of states leads to the conclusion that anharmonic transverse acoustic modes in the form of anomalous vibrations of Cu and Ag cause the lower values of the thermal conductivity. By analyzing the electronic structure, the compounds under study fit perfectly into a recently defined region of the metavalent bonding well known for its pronounced anharmonicity. The insight gained from the current results deepens our understanding of the unusual heat transfer phenomenon related to the metavalent bonding and sheds light on design and discovery of thermally functional materials that break the prediction by the conventional theory.

On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

The paper considers a class of linear Boltzmann transport equations which models charged particle transport, for example in dose calculation of radiation therapy. The equation is an approximation of the exact transport equation containing hyper-singular integrals in its collision terms. The paper confines to the global case where the spatial domain G is the whole space Existence results of solutions for the due initial value problem are formulated by applying variational methods. In addition, some regularity results of solutions are verified in scales of relevant anisotropic mixed-norm Sobolev spaces.

Feasibility of streamline upwind Petrov-Galerkin angular stabilization of the linear Boltzmann transport equation with magnetic fields

To accurately model dose in a magnetic field, the Lorentz force must be included in the traditional linear Boltzmann transport equation (LBTE). Both angular and spatial stabilization are required to deterministically solve this equation. In this work, a streamline upwind Petrov-Galerkin (SUPG) method is applied to achieve angular stabilization of the LBTE with magnetic fields. The spectral radius of the angular SUPG method is evaluated using a Fourier analysis method to characterize the convergence properties. Simulations are then performed on homogeneous phantoms and two heterogeneous slab geometry phantoms containing water, bone, lung/air and water for 0.5 T parallel and 1.5 T perpendicular magnetic field configurations. Fourier analysis determined that the spectral radius of the SUPG scheme is unaffected by magnetic field strength and the SUPG free parameter, indicating that the Gauss-Seidel source iteration method is unconditionally stable and the convergence rate is not degraded with increasing magnetic field strength. 100% of simulation points passed a 3D gamma analysis at a 2%/2 mm (3%/3 mm) gamma criterion for both magnetic field configurations in the homogeneous phantom study, with the exception of the 1.5 T perpendicular magnetic field in the pure lung phantom where a 77.4% (87.0%) pass rate was achieved. Simulations in the lung slab geometry phantom resulted in 100% of points passing a 2%/2 mm gamma analysis in a 0.5 T parallel magnetic field, and 97.7% (98.8%) of points passing a 2%/2 mm (3%/3 mm) gamma criterion in a 1.5 T perpendicular magnetic field. For the air slab geometry phantom, 72.1% (79.2%) of points passed a 2%/2 mm gamma criterion in a 0.5 T parallel magnetic field and 90.3% (92.8%) passed the same gamma criterion in a 1.5 T perpendicular magnetic field. While the novel SUPG angular stabilization method shows feasibility in some cases, it was found that the accuracy of this method was degraded for very low density media such as air.

Deterministic linear Boltzmann transport equation solver for patient-specific CT dose estimation: Comparison against a Monte Carlo benchmark for realistic scanner configurations and patient models.

Epidemiological evidence suggests an increased risk of cancer related to computed tomography (CT) scans, with children exposed to greater risk. The purpose of this work is to test the reliability of a linear Boltzmann transport equation (LBTE) solver for rapid and patient-specific CT dose estimation. This includes building a flexible LBTE framework for modeling modern clinical CT scanners and to validate the resulting dose maps across a range of realistic scanner configurations and patient models. In this study, computational tools were developed for modeling CT scanners, including a bowtie filter, overrange collimation, and tube current modulation. The LBTE solver requires discretization in the spatial, angular, and spectral dimensions, which may affect the accuracy of scanner modeling. To investigate these effects, this study evaluated the LBTE dose accuracy for different discretization parameters, scanner configurations, and patient models (male, female, adults, pediatric). The method used to validate the LBTE dose maps was the Monte Carlo code Geant4, which provided ground truth dose maps. LBTE simulations were implemented on a GeForce GTX 1080 graphic unit, while Geant4 was implemented on a distributed cluster of CPUs. The agreement between Geant4 and the LBTE solver quantifies the accuracy of the LBTE, which was similar across the different protocols and phantoms. The results suggest that 18 views per rotation provides sufficient accuracy, as no significant improvement in the accuracy was observed by increasing the number of projection views. Considering this discretization, the LBTE solver average simulation time was approximately 30s. However, in the LBTE solver the phantom model was implemented with a lower voxel resolution with respect to Geant4, as it is limited by the memory of the GPU. Despite this discretization, the results showed a good agreement between the LBTE and Geant4, with root mean square error of the dose in organs of approximately 3.5% for most of the studied configurations. The LBTE solver is proposed as an alternative to Monte Carlo for patient-specific organ dose estimation. This study demonstrated accurate organ dose estimates for the rapid LBTE solver when considering realistic aspects of CT scanners and a range of phantom models. Future plans will combine the LBTE framework with deep learning autosegmentation algorithms to provide near real-time patient-specific organ dose estimation.

Phonon hydrodynamics in crystalline GeTe at low temperature

A first-principles density functional method along with the direct solution of linearized Boltzmann transport equations are employed to systematically analyze the low-temperature thermal transport in crystalline GeTe. The extensive thermal transport simulations, ranging from room temperature to cryogenic temperatures, reveal the emergence of a phonon hydrodynamic regime in GeTe at low temperature. The reduction of grain boundary scattering is found to play a crucial role along with the divergent trend of umklapp and normal scattering at low temperatures in accommodating the hydrodynamic regime. Average scattering rates for normal, umklapp, and other resistive processes are distinguished for a wide range (4-300 K) of temperatures and used for identifying various phonon transport regimes. Therefore, the variations of lattice thermal conductivity, phonon propagation length, and thermal diffusivity with temperature, related to these transport regimes (ballistic, hydrodynamic, and kinetic), have been thoroughly investigated. The modewise decomposition of lattice thermal conductivity and the distinction of thermal diffusivity according to different scattering processes reveal rich information on the dominant phonon modes and phonon scattering processes in GeTe at low temperature. Further, the kinetic-collective model is used to elucidate the hydrodynamic behavior of phonon scattering through the relative study of collective and kinetic contributions to the thermal transport properties. In this context, the Knudsen number is estimated through the characteristic nonlocal length and the grain size, which further quantifies the consistent hydrodynamic behavior of phonon thermal transport for GeTe. Finally, phonon-vacancy scattering for GeTe is realized, and vacancies are found strongly to influence the hydrodynamic window while incorporating the other resistive scattering mechanisms.

Asymptotic Derivation of the Simplified PN Equations for Nonclassical Transport with Anisotropic Scattering

In nonclassical transport, the free-path length variable s is modeled as an independent variable, and a nonclassical linear Boltzmann transport equation incorporating s has been derived. To model transport in diffusive regimes, the simplified spherical harmonic equations (SPN) have been successfully employed. To model nonclassical transport in diffusive regimes, nonclassical versions of the SPN equations are needed. Nonclassical SPN equations to model isotropic scattering have been derived, and the nonclassical SP1 equation with anisotropic scattering has been determined, but the method used to derive it cannot derive the higher order equations. This article presents a new method which will be able to derive all the nonclassical SPN equations with anisotropic scattering. This method is presented and is used to produce the first of these equations, and it is shown to be equivalent to the previously derived nonclassical SP1 equation with anisotropic scattering.

Lattice thermal conductivity of pure and doped (B, N) Graphene

In this paper, the effect of B and N doping on the phonon induced thermal conductivity of graphene has been investigated. This study is important when one has to evaluate the usefulness of electronic properties of B and N doped graphene. We have performed the calculations by employing density functional perturbation theory(DFPT) to calculate the inter-atomic forces/force constants of pristine/doped graphene. Thermal conductivity calculations have been carried out by making use of linearized Boltzmann transport equations (LBTE) under single-mode relaxation time approximation(RTA). The thermal conductivity of pristine graphene has been found to be of the order of 4000 W/mK at 100 K, which decreases gradually with an increase in temperature. The thermal conductivity decreases drastically by 96% to 190 W/mK when doped with 12.5% B and reduces by 99% to 30 W/mK with 25% B doping. When graphene is doped with N, the thermal conductivity decreases to 4 W/mK and 55 W/mK for 12.5% and 25% doping concentration, respectively. We have found that the thermal conductivity of doped graphene show less sensitivity to change in temperature. It has also been shown that the thermal conductivity of graphene can be tuned with doping and has a strong dependence on doping concentration.

Realistic Modeling of MoS₂ Piezoelectric Transistor

This article presents a comprehensive analysis of the characteristics of an ${n}$ -type monolayer MoS2 piezoelectric field-effect transistor (PZFET) based on a realistic model that includes scattering by intrinsic phonons, remote phonons, and charged impurities. By employing the density-functional theory (DFT), first, we evaluated the effect of the uniaxial compressive out-of-plane strain on the band structure and carrier’s effective masses of a MoS2 monolayer. Thereafter, by solving the linearized Boltzmann transport equation (BTE), the carrier mobility, mean free path, and mean-free-time modulation under different vertical strains in the presence of various scattering mechanisms are computed. By the use of the computed properties, we assessed the characteristics of the PZFET, in which the vertical strain is induced in the monolayer MoS2 channel from the piezoelectric layer ( $\pi $ -layer). The results indicate that the studied PZFET in relation to both the ballistic and nonballistic transports is subthermionic [subthreshold swing (SS) smaller than 60 mV/decade] with a relatively large ${I}_{ \mathrm{\scriptscriptstyle ON}}/{I}_{ \mathrm{\scriptscriptstyle OFF}}$ ratio. The results based on the realistic model indicate that a monolayer MoS2 PZFET enables the scaling of the channel length and the supply voltage beyond the conventional field-effect transistors (FETs).

Thermal conductivity of amorphous and crystalline GeTe thin film at high temperature: Experimental and theoretical study

Thermal transport properties bear a pivotal role in influencing the performance of phase change memory (PCM) devices, in which the PCM operation involves fast and reversible phase change between amorphous and crystalline phases. In this paper, we present a systematic experimental and theoretical study on the thermal conductivity of GeTe at high temperatures involving fast change from amorphous to crystalline phase upon heating. Modulated photothermal radiometry (MPTR) is used to experimentally determine thermal conductivity of GeTe at high temperatures in both amorphous and crystalline phases. Thermal boundary resistances are accurately taken into account for experimental consideration. To develop a concrete understanding of the underlying physical mechanism, rigorous and in-depth theoretical exercises are carried out. For this, first-principles density functional methods and linearized Boltzmann transport equations (LBTE) are employed using both direct and relaxation time based approach (RTA) and compared with that of the phenomenological Slack model. The amorphous phase experimental data has been described using the minimal thermal conductivity model with sufficient precision. The theoretical estimation involving direct solution and RTA method are found to retrieve well the trend of the experimental thermal conductivity for crystalline GeTe at high temperatures despite being slightly overestimated and underestimated, respectively, compared to the experimental data. A rough estimate of vacancy contribution has been found to modify the direct solution in such a way that it agrees excellently with the experiment. Umklapp scattering has been determined as the significant phonon-phonon scattering process. Umklapp scattering parameter has been identified for GeTe for the whole temperature range which can uniquely determine and compare Umklapp scattering processes for different materials

A Study on Phonon-Mediated Thermal Transport and Lattice Thermal Conductivity Prediction Using First-Principles Calculations

The systematic theoretical approaches and atomistic simulation programs to predict thermal properties of crystalline nanostructured materials within first-principles framework are studied here. Recent progress in computational power has enabled an accurate and reliable way to investigate nanoscale thermal transport in crystalline materials using first-principles based calculations. Extracting a large set of anharmonic force constants with low computational effort remains a big challenge in lattice dynamics and condensed-matter physics. This paper focuses on recent progress in first-principles phonon calculations for semiconductor materials and summarizes advantages and limitations of each approach and simulation programs by comparing accuracy of numerical solutions, computational load and calculating feasibility to a wide range of crystalline materials. This work also reviews and presents the coupling model of first-principles molecular dynamic (FPMD) approach that can extract anharmonic force constants directly and solution of linearized Boltzmann transport equation to predict phonon-mediated lattice thermal conductivity of crystalline materials.

Ultra-low lattice thermal conductivity and giant phonon–electric field coupling in hafnium dichalcogenide monolayers

Phonons in crystalline solids are of utmost importance in governing its lattice thermal conductivity (kL). In this work, kL in hafnium (Hf) dichalcogenide monolayers has been investigated based on ab initio DFT coupled to linearized Boltzmann transport equation together with single-mode relaxation-time approximation. Ultra-low kL found in HfS2 (2.19 W m−1 K−1), HfSe2 (1.23 W m−1 K−1) and HfSSe (1.78 W m−1 K−1) monolayers at 300 K, is comparable to that of the state-of-art bulk thermoelectric materials, such as, Bi2Te3 (1.6 W m−1 K−1), PbTe (2.2 W m−1 K−1) and SnSe (2.6 W m−1 K−1). Gigantic longitudinal-transverse optical (LO-TO) splitting of up to 147.7 cm−1 is noticed at the Brillouin zone-centre (Γ-point), which is much higher than that in MoS2 single layer (∼2 cm−1). It is driven by the colossal phonon–electric field coupling arising from the domination of ionic character in the interatomic bonds and Born effective or dynamical charges as high as 7.4e on the Hf ions, which is seven times that on Mo in MoS2 single layer. Enhancement in kL occurs in HfS2 (2.19 to 4.1 W m−1 K−1), HfSe2 (1.23 to 1.7 W m−1 K−1) and HfSSe (1.78 to 2.2 W m−1 K−1) upon the incorporation of the non-analytic correction term. Furthermore, the mode Grüneisen parameter is calculated to be as high as ∼2.0, at room temperature, indicating a strong anharmonicity. Moreover, the contribution of optical phonons to kL is found to be ∼12%, which is significantly high than that in single-layer MoS2. Large atomic mass of Hf (178.5 u), small phonon group velocities (4–5 km s−1), low Debye temperature (∼166 K), low bond and elastic stiffness (Young’s modulus ∼75 N m−1), small phonon lifetimes (∼6 ps), low specific heat capacity (∼17 J K−1 mol−1) and strong anharmonicity are collectively found to be the factors responsible for such a low kL. These findings would be immensely helpful in designing thermoelectric interconnects at the nanoscale and 2D material-based energy harvesters.

Generalization of Fourier’s Law into Viscous Heat Equations

Heat conduction in dielectric crystals originates from the propagation of atomic vibrations, whose microscopic dynamics is well described by the linearized phonon Boltzmann transport equation. Recently, it was shown that thermal conductivity can be resolved exactly and in a closed form as a sum over relaxons, $\mathit{i.e.}$ collective phonon excitations that are the eigenvectors of Boltzmann equation's scattering matrix [Cepellotti and Marzari, PRX $\mathbf{6}$ (2016)]. Relaxons have a well-defined parity, and only odd relaxons contribute to the thermal conductivity. Here, we show that the complementary set of even relaxons determines another quantity --- the thermal viscosity --- that enters into the description of heat transport, and is especially relevant in the hydrodynamic regime, where dissipation of crystal momentum by Umklapp scattering phases out. We also show how the thermal conductivity and viscosity parametrize two novel viscous heat equations --- two coupled equations for the temperature and drift-velocity fields --- which represent the thermal counterpart of the Navier-Stokes equations of hydrodynamics in the linear, laminar regime. These viscous heat equations are derived from a coarse-graining of the linearized Boltzmann transport equation for phonons, and encompass both limits of Fourier's law and of second sound, taking place, respectively, in the regimes of strong or weak momentum dissipation. Last, we introduce the Fourier deviation number as a descriptor that captures the deviations from Fourier's law due to hydrodynamic effects. We showcase these findings in a test case of a complex-shaped device made of graphite, obtaining a remarkable agreement with the very recent experimental demonstration of hydrodynamic transport in this material. The present findings also suggest that hydrodynamic behavior can appear at room temperature in micrometer-sized diamond crystals.

Phonon limited mobility in 3D Dirac semimetal Cd3As2

We report the numerical calculations of phonon limited electron mobility, μ in Cd3As2 three dimensional Dirac semimetal using Boltzmann transport theory over a temperature range 20<T<300K. By employing the Ritz iteration technique, we have directly solved the 3D linearized Boltzmann transport equation to obtain the first order perturbation distribution function Φ−1(Ek) as a function of electron energy Ek and hence the mobility, μ. we have considered electrons scattering by acoustic phonons via deformation potential scattering and polar optical phonons. The μ due to the latter is dominating the former for T>80K, for electron concentration ne = 1x1018cm−3. Thiscrossover shifts to lower T, with decreasing ne. The calculations are qualitatively agreeing with the experimental results.