Phonon Boltzmann Equation Research Articles

Influence of polymorphism on the lattice thermal conductivity of Ga2O3

In this paper, the lattice thermal conductivity of Ga2O3 in its β, α, ɛ(κ), and γ phase is systematically investigated based on the first principles calculation and iterative approaches to solve the phonon Boltzmann equation. The results indicate that the crystal microstructure of Ga2O3 has a significant effect on the lattice thermal conductivity. In addition, the results also find that γ-Ga2O3 has an ultralow lattice thermal conductivity within the temperature range from 50 to 700 K. As for γ-Ga2O3, the obtained lattice thermal conductivity at room temperature (300 K) is 0.1189 W/(m K) along the [100] and [010] directions, and 0.1159 W/(m K) along the [001] direction. The lattice thermal conductivity exhibits the following order: γ-Ga2O3 ≪ ɛ(κ)-Ga2O3 < α-Ga2O3 < β-Ga2O3. The disruptive effect of Ga3+ cation vacancies on the spinel structure's symmetry is responsible for the ultralow lattice thermal conductivity observed in γ-Ga2O3. This disruption increases the complexity of the lattice and hampers the propagation and scattering of phonons. Another contributing factor is the presence of weak chemical bonding, which intensifies the oscillation of Ga atoms. The results of this study have significant implications for further investigating the factors influencing the thermal conductivity of Ga2O3 and developing thermoelectric materials.

On-the-fly training of polynomial machine learning potentials in computing lattice thermal conductivity.

The application of first-principles calculations for predicting lattice thermal conductivity (LTC) in crystalline materials, in conjunction with the linearized phonon Boltzmann equation, has gained increasing popularity. In this calculation, the determination of force constants through first-principles calculations is critical for accurate LTC predictions. For material exploration, performing first-principles LTC calculations in a high-throughput manner is now expected, although it requires significant computational resources. To reduce computational demands, we integrated polynomial machine learning potentials on-the-fly during the first-principles LTC calculations. This paper presents a systematic approach to first-principles LTC calculations. We designed and optimized an efficient workflow that integrates multiple modular software packages. We applied this approach to calculate LTCs for 103 compounds of wurtzite, zinc blende, and rocksalt types to evaluate the performance of the polynomial machine learning potentials in LTC calculations. We demonstrate a significant reduction in the computational resources required for the LTC predictions.

Thermal transport in thermoelectric materials of SnSSe and SnS[formula omitted]: A non-equilibrium Monte-Carlo simulation of Boltzmann transport equation

In the present work, thermal transport and the energy conversation in two thermoelectrically efficient candidates of Janus SnSSe and SnS2 are investigated within the non-equilibrium Monte Carlo simulation of the phonon Boltzmann equation. The phonon analysis has been performed to determine the contributed phonon in heat transport. The results present that the dominant participating phonons are the ones from the longitudinal acoustic (LA) branch while the least belongs to the transverse acoustic (TA) modes. Both materials reached the very high maximum temperature in response to the implied wasted heat. This is attributed to the low presence of the critical TA phonons which are on one hand fast enough to leave the hotspot but on the other hand have frequencies commensurable to ZA phonons’ frequency. Also, the temperature profile achieved during the heating and cooling of the materials is studied. It is obtained that the heat propagation through the SnS2 is, at first, swifter, which results in a temperature gradient through the whole material which is less than that of the SnSSe. As the time passes, the heat transfer that is directly related to the material thermal conductivity, slows down. So, the behavior of the SnS2 and SnSSe, in case of the heat propagation status, becomes similar. Moreover, the behavior of the thermoelectric figure of merit (zT), the efficiency (η), and the generated voltage have been figured out. It is stated that the higher zT and η do not guarantee a larger generated Seebeck voltage. This is true, while the generated Seebeck voltage is related to the temperature difference between the heated and the cold junction. Accordingly, how far the temperature of matter rises in response to the implied wasted heat is directly related to the obtained voltage. Mainly, it is presented that the maximum temperature that a material achieves, alongside the temperature gradient and the material property Seebeck coefficient, are essential in introducing thermoelectrically efficient materials with reasonable thermal to electrical energy conversion, useful for thermal engineering.

Ab initioinvestigation of the lattice dynamics and thermophysical properties of BCC vanadium and niobium.

In the present work, we have performed the phonon dispersion calculations of body-centered cubic vanadium (V) and niobium (Nb) with the supercell approach using different supercell size. Using DFT method, the calculated phonon spectra of V and Nb are found to be in a good agreement with the available experimental data. Our calculated results show a 'dip'-like feature in the longitudinal acoustic phonon mode along the Γ-H high symmetric path for both transition metals in the case of supercell size4×4×4. However, in supercell size2×2×2and3×3×3, the 'dip'-like feature is not clearly visible. In addition to this, thermodynamical properties are also computed, which compare well with the experimental data. Apart from this, the phonon lifetime due to electron-phonon interactions (τephph) and phonon-phonon interactions (PPIs) (τphph) are calculated. The effect of PPIs is studied by computing the average phonon lifetime for all acoustic branches. The value ofτephphof V (Nb) is found to be 23.16 (24.70)×10-15s at 100 K, which gets decreased to 1.51 (1.85)×10-15s at 1000 K. Theτphphof V (Nb) is found to be 8.59 (18.09)×10-12and 0.83 (1.76)×10-12s at 100 and 1000 K, respectively. Nextly, the lattice thermal conductivity is computed using linearized phonon Boltzmann equation. The present work suggests that studying the variation of phonon dispersion with supercell size is crucial for understanding the phonon properties of solids accurately.

Implementation of nonlocal non-Fourier heat transfer for semiconductor nanostructures

The study of heat transport in micro/nanoscale structures due to their application, especially in Nanoelectronics, is a matter of interest. In other words, the precise simulation of the temperature distribution inside the transistors is consequential in designing and building more reliable devices reaching lower maximum temperatures during the operation. The present study constitutes a framework for micro/nanoscale heat transport study, which leads to calculating accurate temperature/heat flux profiles with low computational cost. The newly non-dimensional parameter γ, presenting the strength of the nonlocality, is utilized through the nonlocal DPL modeling (NDPL). Alongside calculating the nonlocality coefficient, the factors appearing in DPL, including the temperature jump and phase lagging ratio, are revisited. The factor γ is found to have a linear relationship with Knudsen (Kn) number, being 3.5 for Kn=10 and 0.035 for Kn=0.1. Although the nonlocality is bold for the large Knudsen numbers, it also plays a vital role for low Knudsen number structures, especially at earlier times. Further, It is obtained that intruding γ is critical for obtaining accurate temperature and heat flux distributions that are very close to the practical results of the Phonon Boltzmann equation.

Analysis of GAAFET’s transient heat transport process based on phonon hydrodynamic equations

Compared to the classical Fourier’s law, the phonon hydrodynamic model has demonstrated significant advantages in describing ultrafast phonon heat transport at the nanoscale. The gate-all-around field-effect transistor (GAAFET) greatly optimizes its electrical performance through its three-dimensional channel design, but its nanoscale characteristics also lead to challenges such as self-heating and localized overheating. Therefore, it is of great significance to study the internal heat transport mechanism of GAAFET devices to obtain the thermal process and heat distribution characteristics. Based on this, this paper conducts theoretical and numerical simulation analyses on the phonon heat transfer characteristics within nanoscale GAAFET devices. Firstly, based on the phonon Boltzmann equation, the phonon hydrodynamic model and boundary conditions are rigorously derived, establishing a numerical solution method based on finite elements. For the novel GAAFET devices, the effects of factors such as surface roughness, channel length, channel radius, gate dielectric, and interface thermal resistance on their heat transfer characteristics are analyzed. The research results indicate that the larger the surface roughness, the smaller the channel length and the channel radius, the larger the interface thermal resistance leads to the higher hot spot peak temperature. The non-Fourier heat analysis method based on the phonon hydrodynamic model and temperature jump condition within the continuous medium framework constructed in this paper can accurately predict the non-Fourier phonon heat conduction process inside GAAFET and reveal the mechanisms of resistive scattering and phonon/interface scattering. This work provides important theoretical support for further optimizing the thermal reliability design of GAAFET, improving its thermal stability, and operational performance.

Lattice thermal conductivity of β-, α- and κ- Ga2O3: a first-principles computational study

The thermal transport properties of Ga2O3 in different phases remain inadequately explored. We employ first-principles calculations and the phonon Boltzmann equation to systematically study the lattice thermal conductivity of β-, α- and κ-Ga2O3. Our results reveal that κ-Ga2O3 exhibits pronounced phonon anharmonicity due to its complex polyhedral configurations and weak bonding, resulting in significantly lower lattice thermal conductivity compared to β- and α-Ga2O3. This work provides critical knowledge of the fundamental phonon thermal transport properties of different-phase Ga2O3, as well as helpful guidance for the thermal design of Ga2O3-based high-power devices.

An investigation into the reliability of newly proposed MoSi2N4/WSi2N4 field-effect transistors: A Monte Carlo study

Recently, the two dimensional complex MA2Z4 structures have been suggested as suitable replacements for silicon channels in field-effect transistors (FETs). Specifically, two materials of MoSi2N4 and WSi2N4 due to their very desirable electrical and thermal properties are noticed. On the other hand, the reliability of transistors, which is determined by the maximum temperature they obtain during the performance, specifies the usefulness of the newly proposed channels for thermal management solution. In this work, the FETs, including MoSi2N4 and WSi2N4 channels, are investigated using Monte Carlo simulation of the phonon Boltzmann equation. In particular, the phonon analysis has been carried out to investigate the peak temperature rise. Our calculations confirm that MoSi2N4 and WSi2N4 present lower maximum temperature than the previously suggested candidate, the blue phosphorene (BP) which itself reaches a shallow temperature. Concretely, the phonon exploration shows that the competition between the dominant heat carrier velocity, and its related frequency settles the maximum temperature value. The material WSi2N4 with much more phonons in TA mode, with almost high velocity and relatively low-frequency, shows adequate thermal condition, and its peak temperature is very low, say 110 K, less than that of BP. The material MoSi2N4 attains the maximum temperature of only 10 K less than BP peak temperature. This behavior attributes to the dominant LA phonons which are fast but also have high frequency and consequently make the temperature get larger than that of the WSi2N4. In summary, WSi2N4, with very low peak temperature, and in the next step MoSi2N4, both with beneficial electrical/thermal properties, are suggested as very suitable candidates for producing more reliable FETs, fulfilling the thermal management.

Calculation of thermal conductivity of UO2±0.25 solving phonon Boltzmann equation

Calculation of thermal conductivity of UO2±0.25 solving phonon Boltzmann equation

Coupled Monte Carlo–Deterministic simulation of thermal behavior of graphene on substrate

The purpose of this paper is to study thermal effects in a graphene sheet on substrate. The temperature evolution of the phonon branches and of the crystal lattice is obtained when also the presence of the substrate is taken into account. The numerical strategy is based on the coupling of stochastic and deterministic numerical methods; Boltzmann equation for charge transport is treated by using the Direct Simulation Monte Carlo approach, whose results are treated as source terms for the phonon Boltzmann equation which is solved deterministically. The results are useful also for applications as correct design of graphene-based prototypes, when heating effects in the presence of a substrate are relevant.

Corrosiveness of Li2TiO3 ceramic tritium breeder to structural RAFM steel: Effects on microstructure and thermal conductivity

Structural steel will be in long-term contact with ceramic tritium breeder at a high temperature in a solid tritium breeding blanket. The maximum temperature in the tritium breeding zone would reach nearly 1173 K. At this high temperature, a large amount of lithium oxide would evaporate and deposit on the structural materials, causing the overall corrosion of the structural materials. Corrosiveness of Li2TiO3 ceramic tritium breeder to reduced activation ferritic-martensitic steel CLF-1 was investigated in this study. The experiment was carried out at 1123 K under He + 0.1 vol% H2 atmosphere for up to 400 h. The formation of non-uniform corrosion layers with a Fe-rich outer layer and a Cr-rich inner layer were found on the surface of CLF-1 steel. Secondary ion mass spectrometry (SIMS) test at contact and non-contact areas were performed to demonstrate evaporation of lithium compound. The corrosion layers were characterized by scanning electron microscopy (SEM) and energy dispersive X-ray (EDX) spectroscopy. The non-contact area with tritium breeding pebbles were corroded indicating a large amount of lithium oxide on the non-contact zone, resulting in serious overall corrosion of structural materials. LiFe5O8 was found as the main phase on the surface of the corroded samples by X-ray diffraction (XRD). The thermal conductivity of LiFe5O8 (∼0.8 W/m/K, 300 K) calculated by first principles phonon Boltzmann equation (PBE) is significantly lower than the original CLF-1 (∼23 W/m/K, 300 K). Almost no change of thermal conductivity for experimental results was found after corrosion, which is due to non-uniform corrosion layer presenting thinner layer at some areas, as well as the hole, crack and peeling of corrosion layer indicated by SEM results. However, in the long-term service of blanket structural material, it is predicted that a much thicker corrosion layer may form on the surface of RAFM steels thus bring a risk to the heat transfer of blanket.

Physical Insights on the Thermoelectric Performance of Cs2SnBr6 with Ultralow Lattice Thermal Conductivity.

This study has investigated the microscopic mechanisms of ultralow lattice thermal conductivity by the first-principles density functional theory. By solving the phonon Boltzmann equation iteratively, we find that the thermal conductivity of the lattice is abnormally low and that glass like heat transfer behavior occurs. Therefore, in addition to the contribution about the particle-like propagation to heat transport, the off-diagonal elements of the heat-flux operator through wave-like interbranch tunneling of phonon modes are also considered. The results provided new insights into the minimum thermal conductivity (κL) for Cs2SnBr6 (0.17 W m-1 K-1 at 450 K). It was also found that polar optical phonon scattering severely affects carrier lifetime. In addition, an impressive thermoelectric figure of merit of 0.55 at 450 K for Cs2SnBr6 was obtained in the case of doping p-type carriers. The study helps us understand the ultralow κL in complex crystals with strong anharmonicity and find that Cs2SnBr6 is a new and promising thermoelectric material.

Hydrodynamic heat transport in dielectric crystals in the collective limit and the drifting/driftless velocity conundrum

We apply a recently developed method for solving the linearized phonon Boltzmann equation to study the hydrodynamic thermal transport in dielectrics in the collective limit, i.e., when normal collisions dominate resistive ones. The method recovers Guyer and Krumhansl results for a single Debye branch and extends them to general dispersion relations and branches. Specifically, we obtain explicit microscopic expressions for the phonon distribution and for the transport coefficients in this limit. We find that the phonon distribution differs from the commonly used displaced distribution in two terms: one accounting for viscous flow and another one which allows us to solve a long-standing issue on drifting and driftless second-sound velocities. Thus, the new method allows us to generalize previous results and fill some gaps on fundamental aspects of the collective limit through a simple mathematical formalism. We compare the hydrodynamic framework with previous models and discuss its limitations.

A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times

The phonon Boltzmann transport equation with dual relaxation times is often used to describe the heat conduction in semiconductor materials when the classical Fourier's law is no longer valid. For practical engineering designs, accurate and efficient numerical methods are highly demanded to solve the equation. At a large Knudsen number (i.e., the ratio of the phonon mean free path to a characteristic system length), steady-state solutions can be obtained via the conventional iterative scheme (CIS) within a few iterations. However, when the Knudsen number becomes small, i.e., when the phonon transport occurs in the diffusive or hydrodynamic regime, thousands of iterations are required to obtain converged results. In this work, a general synthetic iterative scheme (GSIS) is proposed to tackle the inefficiency of CIS. The key ingredient of the GSIS is that a set of macroscopic synthetic equations, which is exactly derived from the Boltzmann transport equation, is simultaneously solved with the kinetic equation to obtain the temperature and heat flux. During the iteration, the macroscopic quantities are used to evaluate the equilibria in the scattering terms of the kinetic equation, thus guiding the evolution of the phonon distribution function, while the distribution function, in turn, provides closures to the synthetic equations. The Fourier stability analysis is conducted to reveal the superiority of the GSIS over the CIS in terms of fast convergence in periodic systems. It is shown that the convergence rate of the GSIS can always be maintained under 0.2 so that only two iterations are required to reduce the iterative error by one order of magnitude. Numerical results in wall-bounded systems are presented to demonstrate further the efficiency of GSIS, where the CPU time is reduced by up to three orders of magnitude, especially in both the diffusive and hydrodynamic regimes where the Knudsen number is small.

Lattice Boltzmann scheme for hydrodynamic equation of phonon transport

In this work, a lattice Boltzmann scheme is developed for numerical solution of the phonon Boltzmann equation under Callaway's dual relaxation model in the hydrodynamic limit. Through a Chapman-Enskog expansion to the lattice Boltzmann equation with the resistive scattering term as an equivalent source term, we recover a phonon hydrodynamic equation which is reduced to the Guyer-Krumhansl heat transport equation and the Fourier's law in the limit of dominant normal scattering and dominant resistive scattering respectively. The transition of heat transport from diffusive regime to hydrodynamic regime is modeled extensively by the present numerical scheme, which produces results in good agreement with the benchmark solutions and experimental measurement. Two well-known phonon hydrodynamic phenomena including the phonon Poiseuille flow and second sound propagation are well captured by the lattice Boltzmann scheme. This work will promote the numerical modeling and deeper understanding of the non-Fourier heat transport induced by phonon normal scattering.

Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors

We present a formalism to solve the phonon Boltzmann transport equation (BTE) for finite Knudsen numbers that supplies a hydrodynamic heat transport equation similar to the Navier-Stokes equation for general semiconductors. This generalization of Fourier's law applies in general cases, from systems dominated by momentum-preserving normal collisions, as is well known, to kinetic materials dominated by resistive collisions, where it captures nonlocal effects. The key feature of our framework is that the macrostate is described in terms of the heat flux and its first derivatives. We obtain explicit expressions for the nonequilibrium phonon distribution and for the geometry-independent macroscopic parameters as a function of phonon properties that can be calculated from first principles. Ab initio model predictions are found to agree with a wide range of experiments in silicon. In contrast to approaches directly based on the BTE, the hydrodynamic equation can be solved in arbitrary geometries, thus providing a powerful tool for nanoscale heat modeling at a low computational cost.

Phonon vortex dynamics in graphene ribbon by solving Boltzmann transport equation with ab initio scattering rates

In this work, we study thermal phonon vortex in graphene ribbon by a discrete-ordinate solution of phonon Boltzmann equation under Callaway's dual relaxation model. The phonon scattering rates of normal and resistive processes are acquired from ab initio calculation without need of any empirical input parameters. The temperature, size and isotope effects on transition from phonon vortex transport to conventional Fourier's heat conduction in both simple and complex geometries are investigated. The physical mechanism for the evolution of phonon vortex is declared by wide phonon mean free path distribution of resistive processes. A hierarchical vortex series is obtained with primary, secondary and terniary vortexes in a complicated geometry. The present work provides an accurate and efficient multi-scale numerical framework for modeling hydrodynamic phonon transport in high-thermal-conductivity materials and also sheds light on the heat dissipation applications.

Unification of nonequilibrium molecular dynamics and the mode-resolved phonon Boltzmann equation for thermal transport simulations

Nano-size confinement induces many intriguing non-Fourier heat conduction phenomena, such as nonlinear temperature gradients, temperature jumps near the contacts, and size-dependent thermal conductivity. Over the past decades, these effects have been studied and interpreted by nonequilibrium molecular dynamics (NEMD) and phonon Boltzmann transport equation (BTE) simulations separately, but no theory that unifies these two methods has ever been established. In this work, we unify these methods using a quantitative mode-level comparison and demonstrate that they are equivalent for various thermostats. We show that different thermostats result in different non-Fourier thermal transport characteristics due to the different mode-level phonon excitations inside the thermostats, which explains the different size-dependent thermal conductivities calculated using different reservoirs, even though they give the same bulk thermal conductivity. Specifically, the Langevin thermostat behaves like a thermalizing boundary in phonon BTE and provides mode-level thermal-equilibrium phonon outlets, while the Nose-Hoover chain thermostat and velocity rescaling method behave like biased reservoirs, which provide a spatially uniform heat generation and mode-level nonequilibrium phonon outlets. These findings explain why different experimental measurement methods can yield different size-dependent thermal conductivity. They also indicate that the thermal conductivity of materials can be tuned for various applications by specifically designing thermostats. The unification of NEMD and phonon BTE will largely facilitate the study of thermal transport in complex systems in the future by, e.g., replacing computationally unaffordable first-principles NEMD simulations with computationally less expensive spectral BTE simulations.

Interfacial heat transport across multilayer nanofilms in ballistic–diffusive regime

In recent years, investigations of nanoscale heat transfer have been regarded as a critical challenge for thermal engineering applications. Therefore, improving the thermal conduction inside nanodevices is required to ensure the reliability of future-generation nanoelectronics. Owing to the fast improvement in the semiconductor industry, researchers have been strongly motivated to study interfacial phonon transport to reduce the self-heating mechanism caused by ballistic thermal conduction, which induces size-dependent effective thermal conductivity. Based on the phonon Boltzmann equation (BTE), we obtain the ballistic–diffusive equation and Cattaneo–Vernotte models to explain the heat conduction in multiple interfaces of silicon and silicon dioxide nanolayers. Both the temperature-jump boundary condition and thermal boundary resistance, also referred to as Kapitza resistance, can control the interfacial heat transport in the ballistic–diffusive regime. In the present study, our analytical models are compared with the phonon BTE to validate the performance of our methodology. We find that if the thermal boundary resistance reduces, the heat transport across the interfaces can be enhanced, which is necessary to limit the heat dissipation in silicon nanolayers.

Direct simulation of second sound in graphene by solving the phonon Boltzmann equation via a multiscale scheme

The direct simulation of the dynamics of second sound in graphitic materials remains a challenging task due to lack of methodology for solving the phonon Boltzmann equation in such a stiff hydrodynamic regime. In this work, we aim to tackle this challenge by developing a multiscale numerical scheme for the transient phonon Boltzmann equation under Callaway's dual relaxation model which captures well the collective phonon kinetics. Comparing to traditional numerical methods, the present multiscale scheme is efficient, accurate and stable in all transport regimes attributed to avoiding the use of time and spatial steps smaller than the relaxation time and mean free path of phonons. The formation, propagation and composition of ballistic pulses and second sound in graphene ribbon in two classical paradigms for experimental detection are investigated via the multiscale scheme. The second sound is declared to be mainly contributed by ZA phonon modes, whereas the ballistic pulses are mainly contributed by LA and TA phonon modes. The influence of temperature, isotope abundance and ribbon size on the second sound propagation is also explored. The speed of second sound in the observation window is found to be at most 20 percentages smaller than the theoretical value in hydrodynamic limit due to the finite umklapp, isotope and edge resistive scattering. The present study will contribute to not only the solution methodology of phonon Boltzmann equation, but also the physics of transient hydrodynamic phonon transport as guidance for future experimental detection.