Boltzmann Transport Equation Approach Research Articles

Unleashing the power of artificial intelligence in phonon thermal transport: Current challenges and prospects

The discovery of advanced thermal materials with exceptional phonon properties drives technological advancements, impacting innovations from electronics to superconductors. Understanding the intricate relationship between composition, structure, and phonon thermal transport properties is crucial for speeding up such discovery. Exploring innovative materials involves navigating vast design spaces and considering chemical and structural factors on multiple scales and modalities. Artificial intelligence (AI) is transforming science and engineering and poised to transform discovery and innovation. This era offers a unique opportunity to establish a new paradigm for the discovery of advanced materials by leveraging databases, simulations, and accumulated knowledge, venturing into experimental frontiers, and incorporating cutting-edge AI technologies. In this perspective, first, the general approach of density functional theory (DFT) coupled with phonon Boltzmann transport equation (BTE) for predicting comprehensive phonon properties will be reviewed. Then, to circumvent the extremely computationally demanding DFT + BTE approach, some early studies and progress of deploying AI/machine learning (ML) models to phonon thermal transport in the context of structure–phonon property relationship prediction will be presented, and their limitations will also be discussed. Finally, a summary of current challenges and an outlook of future trends will be given. Further development of incorporating AI/ML algorithms for phonon thermal transport could range from phonon database construction to universal machine learning potential training, to inverse design of materials with target phonon properties and to extend ML models beyond traditional phonons.

Transport theory up to second-order scattering for reaction-diffusion-phonon systems with applications to active transport in catalysis, explosions, and biological membranes.

A Boltzmann transport equationapproach is developed for reaction-diffusion systems which incorporates phonon transport in addition to the traditional approach. Scattering processes up to second order are taken into account. Two forces emerge from this analysis when a spatial gradient exists, one force on reactants and products, the other force on phonons. The forces are equal and opposite and have the tendency for separation of the phonons away from the reactants and products. These forces are capable of creating the types of instabilities that can lead to the formation of Turing patterns. The existence of these forces allows for exergonic conversion where not all of the released energy from reactions and diffusion becomes heat. When applied to homogeneous catalysis, one finds that reactants and products are pushed toward regions of greater catalytic activity. In the realm of high-energy explosions, calculations show that reactants and products can be accelerated laterally to the direction of a TNT reaction front up to speeds near 1000 m/s. This acceleration is in opposition to diffusion and represents active transport. Calculations also show that active transport observed in biological systems such as bacteria, mitochondria, and chloroplasts may be explained by this second-order transport theory. Using reasonable values for key parameters, calculations show that up to one-third of the available chemical energy can be converted toward pumping protons uphill to a potential of 50 mV.

Revisiting thermal transport in single-layer graphene: On the applicability of thermal snapshot interatomic force constant extraction methodology for layered materials

The thermal conductivities of single-/bi-layer graphene and bulk-graphite are obtained using the Boltzmann transport equation (BTE) framework by accounting for three-phonon and four-phonon scatterings. For single-layer graphene, the thermal conductivity and interatomic force constants obtained using temperature-independent finite-difference, and temperature-dependent molecular dynamics-based approaches agree with each other. The use of the thermal snapshot approach to get temperature-dependent force constants results in a non-physical description of interatomic distances for single-layer graphene. The predicted thermal conductivity at room temperature using finite-difference based force constants is 800 W/m K, which is a severe under-prediction of experimentally measured values. For bi-layer graphene and bulk graphite, the thermal snapshot methodology is applicable and thermal conductivity changes by 25% and 5% with temperature-dependent force constants. The effect of four-phonon scattering is less than 10% on the predicted thermal conductivity of bi-layer graphene and graphite, and the obtained thermal conductivities using thermal snapshot methodology are in agreement with the literature. The limitation in the prediction of thermal conductivity of single-layer graphene via the BTE approach stems from non-accountability of temperature-dependence in finite-difference based force constants and non-physical description of interatomic bonds in thermal snapshot based force constants extraction for planar 2-atoms unitcell of single layer graphene.

Effect of four-phonon interaction on phonon thermal conductivity and mean-free-path spectrum of high-temperature phase SnSe

The phonon thermal conductivity and mean-free-path (MFP) spectrum of high-temperature phase SnSe (β-SnSe) are studied using the Boltzmann transport equation and ab initio approaches. The particle picture for phonon transport in β-SnSe is revisited, and the imaginary phonon frequencies caused by the ground-state within conventional density-functional theory are resolved. We show that between 800 and 950 K, the in-plane and cross-plane thermal conductivity has an average decrease of 38% and 19%, respectively, when four-phonon scatterings are considered. This large suppression of phonon transport stems mainly from the strong redistribution scattering process. With both the phonon and electron MFP spectra revealed, a characteristic length of 10 nm is suggested to reduce the in-plane and cross-plane thermal conductivity by 18% and 52%, respectively, via nanostructure engineering without sacrificing the power factor.

Anharmonic effects on lattice dynamics and thermal transport of two-dimensional InTe monolayer

The lattice thermal conductivity (κl) plays a key role in the performance of thermoelectric (TE) materials, where the lower values lead to higher figure of merit values. Two-dimensional (2D) group III-VI monolayers such as InTe are promising materials for TE energy generation owing to their low κl that lead to high TE figure of merit values. In this work, we investigate the influence of the lattice anharmonicity on the lattice thermal conductivity of InTe monolayer. The thermodynamic parameters are calculated by using the self-consistent phonon (SCP) theory. The κl value of the InTe monolayer is obtained to be 0.30 Wm−1 K−1 by using the standard Boltzmann transport equation (BTE) approach, while it is 3.58 Wm−1 K−1 by using SCP + BTE approach. These results confirm the importance of the anharmonic effects on the κl value, where it was found to be significantly higher (91%) using the SCP + BTE approach than that obtained using the standard BTE approach.

Revisiting Mott's model for the thermopower of PdxAg1−x alloys with support of density functional theory

Electronic properties of palladium-silver alloys have been studied for almost a century as a prototype of a transition-metal alloy with entangled $s$ and $d$ bands. We review the literature starting from Mott's model of 1935 and discuss from a quantitative point of view a series of common assumptions stemming from that seminal paper. We also show that spectral functions obtained by unfolding the band structures of a sample of supercells can be used for the ab initio computation of the thermopower of such systems within a Boltzmann transport equation approach.

The effect of finite-temperature and anharmonic lattice dynamics on the thermal conductivity of ZrS2 monolayer: self-consistent phonon calculations

Two-dimensional (2D) ZrS2 monolayer (ML) has emerged as a promising candidate for thermoelectric (TE) device applications due to its high TE figure of merit, which is mainly contributed by its inherently low lattice thermal conductivity (LTC). This work investigates the effect of the lattice anharmonicity driven by the temperature-dependent phonon dispersions on the thermal transport of ZrS2 ML. The calculations are based on the self-consistent phonon (SCP) theory to calculate the thermodynamic parameters along with the LTC. The higher-order (quartic) force constants were extracted by using an efficient compressive sensing lattice dynamics technique, which estimates the necessary data based on the emerging machine learning program as an alternative of computationally expensive density functional theory calculations. Resolve of the degeneracy and hardening of the vibrational frequencies of low-energy optical modes were predicted upon including the quartic anharmonicity. As compared to the conventional Boltzmann transport equation (BTE) approach, the LTC of the optimized ZrS2 ML unit cell within SCP + BTE approach is found to be significantly enhanced (e.g., by 21% at 300 K). This enhancement is due to the relatively lower value of phonon linewidth contributed by the anharmonic frequency renormalization included in the SCP theory. Mainly, the conventional BTE approach neglects the temperature dependence of the phonon frequencies due to the consideration of harmonic lattice dynamics and treats the normal process of three-phonon scattering incorrectly due to the use of quasi-particle lifetimes. These limitations are addressed in this work within the SCP + BTE approach, which signifies the validity and accuracy of this approach.

A Theoretical Investigation of the Stability, Electronic, Optical, and Thermoelectric Properties of BaLiP

The density functional theory and post density functional theory techniques are used to investigate the properties of BaLiP for potential applications in photovoltaic and thermoelectric (TE) applications. Density functional calculations of the cohesive energy, phonon‐dispersion, and elastic constant studies establish structural, dynamical, and mechanical stability of the compound. Density functional calculations show that BaLiP is a direct bandgap semiconductor while optical properties at the Bethe–Salpeter equation level of approximation suggest that BaLiP is anisotropic with the optical bandgaps of and eV for in‐plane and out‐of‐plane absorption, respectively. A maximum absorption coefficient of for in‐plane and for out‐of‐plane absorption in the visible range at and eV is found, respectively. Lattice thermal conductivity is computed using a Boltzmann transport equation approach. The calculations reveal that the average lattice thermal conductivity is at room temperature. The figure of merit, which is the main criterion for estimating the TE potential of a material, is estimated to have a maximum of at 1000 K at a hole carrier concentration of . This study shows that BaLiP has some promise as a candidate for applications in electronics, optoelectronics, and TE applications.

Large lattice thermal conductivity, interplay between phonon-phonon, phonon-electron, and phonon-isotope scatterings, and electrical transport in molybdenum from first principles

We describe an ab initio phonon Boltzmann transport equation (BTE) approach accounting for phonon-electron scattering in addition to the well-established phonon-phonon and isotope scatterings. The phonon BTE is linearized and can be exactly solved beyond the relaxation time approximation (RTA). We use this approach to study the lattice thermal conductivity (${\ensuremath{\kappa}}_{\text{ph}}$) of molybdenum (Mo). ${\ensuremath{\kappa}}_{\text{ph}}$ of Mo is found to possess several anomalous features: (1) like in another group VI element tungsten (W), ${\ensuremath{\kappa}}_{\text{ph}}$, with a large value of 37 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature, follows weak temperature dependence due to interplay between phonon-phonon (ph-ph), phonon-electron (ph-el), and phonon-isotope (isotope) scatterings; and (2) compared with W, though Mo is much lighter in mass, Mo has a smaller ${\ensuremath{\kappa}}_{\text{ph}}$. This is attributed to weaker interatomic bonding, larger isotope mixture, and larger density of states at Fermi level in Mo. In isotopically pure samples, ${\ensuremath{\kappa}}_{\text{ph}}$ increases from 37 to 48 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature. Considering the similarity of the phonon dispersion, our work suggests that chromium should also have a large ${\ensuremath{\kappa}}_{\text{ph}}$, which, rather than the complexity of the electronic band structure argued in the literature, accounts for the significant deviation of measured Lorenz number $L$ from the expected Sommerfeld value. The electrical conductivity ($\ensuremath{\sigma}$) and electronic thermal conductivity (${\ensuremath{\kappa}}_{\text{e}}$) of Mo are also calculated by using an ab initio electron BTE approach. $\ensuremath{\sigma}$ and the total thermal conductivity ($\ensuremath{\kappa}$) agree with the experimental data reasonably. These results demonstrate that the ab initio calculations can quantify the lattice and electronic contributions to $\ensuremath{\kappa}$. We also look into the cumulative $\ensuremath{\sigma}$ and ${\ensuremath{\kappa}}_{\text{ph}}$ with respect to electron and phonon mean free paths (MFPs), respectively, in order to reveal the size effect in Mo. The MFPs of electrons contributing to conductivity range from 5 to 22 nm, whereas the MFPs of phonons primarily distribute between 5 and 73 nm with more than 80% contribution to ${\ensuremath{\kappa}}_{\text{ph}}$. This suggests that a reduced Lorenz number can be observed in Mo nanostructures when the relevant size goes below $\ensuremath{\sim}70$ nm.

Thermal conductivity of silicon at elevated temperature: Role of four-phonon scattering and electronic heat conduction

While using first-principles-based Boltzmann transport equation approach to predict the thermal conductivity of crystalline semiconductor materials has been a routine, the validity of the approach is seldom tested for high-temperature conditions. Most previous studies only focused on the phononic contribution, and neglected the electronic part. In this paper, we present a Boltzmann transport equation study on high-temperature thermal conduction in bulk silicon by considering the effects of both phonons and electrons. Both polar and bipolar contributions to the electronic thermal conductivity are calculated from first-principles. More than 25% of heat is shown to be conducted by electrons at 1500 K. The computed total thermal conductivity of silicon faithfully reproduces the measured data. The approach presented in this paper is expected to be applied to other high-temperature functional materials, and the results could serve as benchmarks and help to explain the high-temperature phonon and electron transport phenomena.

Heat transport in carbon nanotubes: Length dependence of phononic conductivity from the Boltzmann transport equation and molecular dynamics

In this article, we address lattice heat transport in single-walled carbon nanotubes (CNTs) by a quantum mechanical calculation of three-phonon scattering rates in the framework of the Boltzmann transport equation (BTE) and classical molecular dynamics (MD) simulation. Under a consistent choice of an empirical, realistic atomic interaction potential, we compare the tube length dependence of the lattice thermal conductivity (TC) at room temperature determined from an iterative solution of the BTE and from a nonequilibrium MD (NEMD) approach. Qualitatively similar trends are found in the limit of short tubes, where an extensive regime of ballistic heat transport prevailing in CNTs of lengths $L\lesssim 1\,\rm{\mu m}$ is independently confirmed. In the limit of long tubes, the BTE approach suggests a saturation of TC with tube length, whereas direct NEMD simulations of tubes extending up to $L=10\,\rm{\mu m}$ are demonstrated to be insufficient to settle the question of whether a fully diffusive heat transport regime and an intrinsic value of TC exist for CNTs. Noting that acoustic phonon lifetimes lie at the heart of a saturation of TC with tube length as per the BTE framework, we complement the quantum mechanical prediction of acoustic phonon lifetimes with an analysis of phonon modes in the framework of equilibrium MD (EMD). A normal mode analysis (NMA) with an emphasis on long wavelength acoustic modes corroborates the BTE prediction that heat transport in CNTs in the long tube limit is governed by the low attenuation rates of longitudinal and twisting phonons.

Thermal and mechanical properties of U3Si2: A combined ab-initio and molecular dynamics study

Triuranium disilicide (U3Si2) has been identified as an accident resistant alternative to uranium dioxide (UO2) based fuels. While experimental studies of U3Si2 are not new, only few studies on the thermal conductivity has been reported. Moreover, theoretical studies of the thermal properties of U3Si2 are scarce. We found that inaccuracies in experimental data were a source of discrepancies in recently reported theoretical results, as such need to be addressed. In this work, the structural, thermal, and mechanical properties of, the non-magnetic and magnetic phases, of U3Si2 were investigated using Hubbard corrected density functional theory (DFT+U) and molecular dynamics (MD) simulations. The lattice and electronic thermal properties were determined based on Boltzmann transport equation (BTE) approaches. A considerable anisotropy was determined the lattice and electronic contributions to the thermal conductivity from DFT+U/BTE results, in which the electronic contribution is predominant. A relative good agreement between the lattice contributions to the thermal conductivity, predicted from DFT+U and MD computations, was associated to similarities in the acoustic phonon modes. The present computational results significantly enhance the understanding of the thermal properties of U3Si2. The good agreement between DFT+U/BTE results with experimental data confirm the accident tolerant characteristic of U3Si2, but also further demonstrating the accuracy of the DFT(+U)/BTE approach for the investigation of thermal properties of advanced nuclear fuels.

A comparative analysis of the phonon properties in UO2 using the Boltzmann transport equation coupled with DFT + U and empirical potentials

The vibrational and thermal properties of UO2 are determined by the solution of the Boltzmann transport equation (BTE) for phonons. The BTE approach relies on interatomic force constants derived from atomic force calculations. Density functional theory (DFT) calculations provide a reliable approach to compute interatomic forces. However, the calculation of interatomic forces using DFT is computationally demanding, and thus significantly limiting its applications. In such cases, empirical potential (EP) calculations are an efficient alternative. However, EPs are not usually constructed to correctly describe lattice vibration properties. In this work, we perform a comparison of lattice vibrations and thermal properties of UO2 solving the BTE using DFT + U and EP computed forces. The experimental lattice thermal conductivity of UO2 is found to be correctly described by DFT + U, while it is only approximated by the Cooper and Potashnikov potentials. However, the heat capacity is only closely reproduced by DFT + U and the Potashnikov potential. In contrast, significant errors were obtained with the Arima and Yamada potentials in all the computed results. The discrepancies in the predicted thermal properties by interatomic potentials were determined to be related to inaccuracies in the description of phonon properties.

Parameter-free model to estimate thermal conductivity in nanostructured materials

Achieving low thermal conductivity and good electrical properties is a crucial condition for thermal energy harvesting materials. Nanostructuring offers a very powerful tool to address both requirements: in nanostructured materials, boundaries preferentially scatter phonons compared to electrons. The computational screening for low-thermal-conductivity nanostructures is typically limited to materials with simple crystal structures, such as silicon, because of the complexity arising from modeling branch- and wave-vector-dependent nanoscale heat transport. The phonon mean-free-path (MFP) dependent Boltzmann transport equation (MFP-BTE) approach is a model that overcomes this limitation. To illustrate this, we analyze thermal transport in 75 nanoporous half-Heusler compounds for different pore sizes. Our calculations demonstrate that, in most cases, the optimization of thermal transport in nanostructures should take into account both bulk thermal properties and geometry-dependent size effects, two aspects that are typically engineered separately. To enable efficient calculations within this paradigm we derive a model, based on the ``gray'' formulation of the BTE, that can decouple the influence of the geometry and the material on the effective thermal conductivity with relatively little loss in accuracy compared to the MFP-BTE. Our study motivates the need for a holistic approach to engineering thermal transport and provides a method for high-throughput low-thermal conductivity materials discovery.

Tailoring phononic, electronic, and thermoelectric properties of orthorhombic GeSe through hydrostatic pressure

In this paper, we systematically investigate the effect of hydrostatic pressure on the phononic and electronic transport properties of orthorhombic p-type GeSe using first-principles based Boltzmann transport equation approach. It is found that the lattice thermal conductivities along the a and c directions increase with pressure, whereas it experiences a decrease along the b direction. This anomalous pressure dependent lattice thermal conductivity is attributed to the combined effect of enhanced phonon group velocity and reduced phonon lifetime. Additionally, the optical phonon branches have remarkable contributions to the total lattice thermal conductivity. The electronic transport calculations indicate that the Seebeck coefficient undergoes a sign change from p-type to n-type along the a direction under pressure, and a dramatic enhancement of the power factor is observed due to the boost of electrical conductivity. The predicted ZT values along the a, b, and c directions are 1.54, 1.09, and 1.01 at 700 K and 8 GPa, respectively, which are about 14, 7.3, and 1.9 times higher than those at zero pressure at experimental carrier concentration of ~1018 cm−3. Our study is expected to provide a guide for further optimization of the thermal and charge transport properties through hydrostatic pressure.

Thermal transport properties of single-layer black phosphorus from extensive molecular dynamics simulations

We compute the anisotropic in-plane thermal conductivity of suspended single-layer black phosphorus (SLBP) using three molecular dynamics (MD) based methods, including the equilibrium MD method, the nonequilibrium MD (NEMD) method, and the homogeneous NEMD (HNEMD) method. Two existing parameterizations of the Stillinger–Weber (SW) potential for SLBP are used. Consistent results are obtained for all the three methods and conflicting results from previous MD simulations are critically assessed. Among the three methods, the HNEMD method is the most and the NEMD method the least efficient. The thermal conductivity values from our MD simulations are about an order of magnitude larger than the most recent predictions obtained using the Boltzmann transport equation approach considering long-range interactions in density functional theory calculations, suggesting that the short-range SW potential might be inadequate for describing the phonon anharmonicity in SLBP.

Diffusive Phonons in Nongray Nanostructures

Nanostructured semiconducting materials are promising candidates for thermoelectrics (TEs) due to their potential to suppress phonon transport while preserving electrical properties. Modeling phonon-boundary scattering in complex geometries is crucial for predicting materials with high conversion efficiency. However, the simultaneous presence of ballistic and diffusive phonons challenges the development of models that are both accurate and computationally tractable. Using the recently developed first-principles Boltzmann transport equation (BTE) approach, we investigate diffusive phonons in nanomaterials with wide mean-free-path (MFP) distributions. First, we derive the short MFP limit of the suppression function, showing that it does not necessarily recover the value predicted by standard diffusive transport, challenging previous assumptions. Second, we identify a Robin type boundary condition describing diffuse surfaces within Fourier's law, extending the validity of diffusive heat transport in terms of Knudsen numbers. Finally, we use this result to develop a hybrid Fourier/BTE approach to model realistic materials, obtaining good agreement with experiments. These results provide insight on thermal transport in materials that are within experimental reach and open opportunities for large-scale screening of nanostructured TE materials.

Investigation of heat transport across Ge/Si interface using an enhanced ballistic-diffusive model

In this paper, we investigate the heat transport across GermaniumSilicon interface using an enhanced ballistic-diffusive equation (EBDE), where we introduce the temperature jump boundary condition coupled with a thermal boundary resistance (TBR) and an effective thermal conductivity model. This paper focuses on the thermal transport of sub-23 nm Ge/Si thin films. The present model is aimed to describe the ballistic-diffusive phonon transport across Ge/Si interface. We have found that the temperature jump occurs in the interface due to phonon-boundary interactions. In addition, the interfacial heat transport is influenced by the surface roughness effect. The prediction of the suggested EBDE model are in good accordance with analytical method reported in the literature. The proposed model shows excellent agreement with the phonon Boltzmann transport equation (BTE) approach and Monte Carlo simulation (MC). Further, the analytic model for the effective thermal conductivity (ETC) is in strong agreement with experimentally based approach and also the theoretical model.

Ab-initio Study of the Electron Mobility in a Functionalized UiO-66 Metal Organic Framework

This study leverages density functional theory accompanied with Boltzmann transport equation approaches to investigate the electronic mobility as a function of inorganic substitution and functionalization in a thermally stable UiO-66 metal-organic framework (MOF). The MOFs investigated are based on Zr-UiO-66 MOF with three functionalization groups of benzene dicarboxylate (BDC), BDC functionalized with an amino group ( $$\hbox {BDC} + \hbox {NH}_2$$ ) and a nitro group ( $$\hbox {BDC} + \hbox {NO}_2$$ ). The design space of this study is bound by UiO-66(M)-R, [ $$\hbox {M}=\hbox {Zr}$$ , Ti, Hf; $$\hbox {R}=\hbox {BDC}$$ , $$\hbox {BDC}+\hbox {NO}_2$$ , $$\hbox {BDC}+\hbox {NH}_2$$ ]. The elastic modulus was not found to vary significantly over the structural modification of the design space for either functionalization or inorganic substitution. However, the electron–phonon scattering potential was found to be controllable by up to 30% through controlled inorganic substitution in the metal clusters of the MOF structure. The highest electron mobility was predicted for a UiO-66( $$\hbox {Hf}_5\hbox {Zr}_1$$ ) achieving a value of approximately $$1.4\times 10^{-3}\,\hbox {cm}^2$$ /V s. It was determined that functionalization provides a controlled method of modulating the charge density, while inorganic substitution provides a controlled method of modulating the electronic mobility. Within the proposed design space the electrical conductivity was able to be increased by approximately three times the base conductivity through a combination of inorganic substitution and functionalization.

Chirality effect on electron phonon relaxation, energy loss, and thermopower in single and bilayer graphene in BG regime

We investigate the energy dependent electron-phonon relaxation rate, energy loss rate, and phonon drag thermopower in single layer graphene (SLG) and bilayer graphene (BLG) under the Bloch-Gruneisen (BG) regime through coupling to acoustic phonons interacting via the Deformation potential in the Boltzmann transport equation approach. We find that the consideration of the chiral nature of electrons alters the temperature dependencies in two-dimensional structures of SLG and BLG from that shown by other conventional 2DEG system. Our investigations indicate that the BG analytical results are valid for temperatures far below the BG limit (∼TBG/4) which is in conformity with a recent experimental investigation for SLG [C. B. McKitterick et al., Phys. Rev. B 93, 075410 (2016)]. For temperatures above this renewed limit (∼TBG/4), there is observed a suppression in energy loss rate and thermo power in SLG, but enhancement is observed in relaxation rate and thermopower in BLG, while a suppression in the energy loss rate is observed in BLG. This strong nonmonotonic temperature dependence in SLG has also been experimentally observed within the BG limit [Q. Ma et al., Phys. Rev. Lett. 112, 247401 (2014)].