Peierls-Boltzmann Transport Equation Research Articles

Unravelling ultralow thermal conductivity in perovskite Cs2AgBiBr6: dominant wave-like phonon tunnelling and strong anharmonicity

Understanding the lattice dynamics and heat transport physics in the lead-free halide double perovskites remains an outstanding challenge due to their lattice dynamical instability and strong anharmonicity. In this work, we investigate the microscopic mechanisms of anharmonic lattice dynamics and thermal transport in lead-free halide double perovskite Cs2AgBiBr6 from first principles. We combine self-consistent phonon calculations with bubble diagram correction and a unified theory of lattice thermal transport that considers both the particle-like phonon propagation and wave-like tunnelling of phonons. An ultra-low thermal conductivity at room temperature (~0.21 Wm−1K−1) is predicted with weak temperature dependence( ~ T−0.34), in sharp contrast to the conventional ~T−1 dependence. Particularly, the vibrational properties of Cs2AgBiBr6 are featured by strong anharmonicity and wave-like tunnelling of phonons. Anharmonic phonon renormalization from both the cubic and quartic anharmonicities are found essential in precisely predicting the phase transition temperature in Cs2AgBiBr6 while the negative phonon energy shifts induced by cubic anharmonicity has a significant influence on particle-like phonon propagation. Further, the contribution of the wave-like tunnelling to the total thermal conductivity surpasses that of the particle-like propagation above around 310 K, indicating the breakdown of the phonon gas picture conventionally used in the Peierls-Boltzmann Transport Equation. Importantly, further including four-phonon scatterings is required in achieving the dominance of wave-like tunnelling, as compared to the dominant particle-like propagation channel when considering only three-phonon scatterings. Our work highlights the importance of lattice anharmonicity and wave-like tunnelling of phonons in the thermal transport in lead-free halide double perovskites.

Intrinsically low lattice thermal conductivity in layered Mn3Si2Te6

The ferrimagnetic nodal-line semiconductor Mn3Si2Te6 has recently received much attention due to its colossal angular magnetoresistance (Seo et al 2021 Nature 599 581). The magnetic and electronic properties of Mn3Si2Te6 have been extensively studied. Meanwhile, a recent experiment showed that Mn3Si2Te6 has a low in-plane lattice thermal conductivity, which implies its potential applications in thermoelectricity. Here, we have investigated phonon dispersion and lattice thermal conductivity of Mn3Si2Te6 by the first-principles calculations and the Peierls–Boltzmann transport equation. It is found that the lattice thermal conductivities of Mn3Si2Te6 are quite low, which are 1.33 and 0.96 Wm−1K−1 along the a and c axes at 300 K, respectively. A significant contribution (>90%) to the thermal conductivity comes from the acoustic phonons and low-frequency optical phonons linked to the vibration of Te atoms. Meanwhile, it is found that such low thermal conductivities of Mn3Si2Te6 are a consequence of the low group velocities and relatively short phonon lifetimes, which are intrinsically derived from the quite complex crystal structure, heavy Te atoms, and relatively weak chemical bonding. Our work not only explains the origin of the intrinsically low thermal conductivity of Mn3Si2Te6 but also could be helpful to the study on the thermal conductivity of other similar layered magnetic materials.

Thermal resistance from non-equilibrium phonons at Si–Ge interface

As nanostructured devices become prevalent, interfaces often play an important role in thermal transport phenomena. However, interfacial thermal transport remains poorly understood due to complex physics across a wide range of length scales from atomistic to microscale. Past studies on interfacial thermal resistance have focused on interface-phonon scattering at the atomistic scale but overlooked the complex interplay of phonon-interface and phonon-phonon scattering at microscale. Here, we use the Peierls-Boltzmann transport equation to show that the resistance from the phonon-phonon scattering of non-equilibrium phonons near a Si–Ge interface is much larger than that directly caused by the interface scattering. We report that non-equilibrium in phonon distribution leads to significant entropy generation and thermal resistance upon three-phonon scattering by the Boltzmann's H-theorem. The physical origin of non-equilibrium phonons in Ge is explained with the mismatches of phonon dispersion, density-of-states, and group velocity, which serve as general guidance for estimating the non-equilibrium effect on interfacial thermal resistance. Our study bridges a gap between atomistic scale and less studied microscale phenomena, providing comprehensive understanding of overall interfacial thermal transport and the significant role of phonon-phonon scattering.

Rattling-induced suppression of thermal transport in cubic In2O3 with Sn-Ga diatomic defect

We present the first principles study of cubic In2O3 with a diatomic defect composed of a Sn atom substituting the In atom at the b-site and a Ga atom embedded in the nearest c-site (structural vacancy) with lattice positions according to the Wyckoff notations. Structural, electronic, phononic and thermal properties were investigated within density functional theory formalism. The lattice anharmonicity effects were taken into account for all possible three-phonon scattering processes. The phonon transport was considered within the Peierls-Boltzmann transport equation with relaxation time approximation. In the relaxed lattice, a strong rearrangement of the initial positions of the atoms in the defect vicinity was revealed, which primarily manifests itself in the displacement of the Sn atom toward another interstitial site. Thus, a cage is formed around the defect by 12 O and 12 In atoms. The calculations of elastic constants and mean square displacements of cage region atoms showed the rattling-like behavior of the Sn atom. Bader charge analysis and electron localization function allowed a deeper understanding and explanation of such behavior. Phonon energy spectra as compared to In2O3 and In2O3:(Sn) demonstrated flattening of phonon branches with spatial localization of phonon modes. They also revealed a decrease in average group velocities of phonons, including those of acoustic type, the presence of avoided-crossing features in the low energy range, and an increase of available phase space for three-phonon scattering. Accounting for all these vibrational features due to defect atoms resulted in a thermal conductivity drop at room temperature by more than seven times compared to In2O3.

Anharmonicity-induced strong temperature-dependent thermal conductivity in CuInX2 ( X=Se , Te)

The lattice thermal $\mathrm{conductivity}({\ensuremath{\kappa}}_{\mathrm{L}}$) of most single crystals usually follows ${T}^{\ensuremath{-}1}$ temperature dependent relation at high temperature region. Here, we investigate the microscopic mechanism of the significant deviation of ${\ensuremath{\kappa}}_{\mathrm{L}}\ensuremath{\sim}{T}^{\ensuremath{-}1}$ rule in Cu-based chalcopyrite compounds $\mathrm{CuIn}{X}_{2}$($X$=Se, Te) by performing first-principles calculations and solving Peierls-Boltzmann transport equation with phonon anharmonic renormalization. The temperature sensitive second-order force constants lead to the dramatically softened optical phonon bands, which contributes to the enlarged phonon scattering phase space and consequent enhanced scattering rate. The strong temperature dependent thermal conductivity in $\mathrm{CuIn}{X}_{2}$($X$=Se, Te) can be well reproduced when calculating thermal conductivity with renormalization force constants. The motion of Cu contributes the most to the large anharmonicity at high temperature, which makes the anharmonic renormalization non-negligible for estimating accurate lattice dynamic and thermal conductivity. Our work will deepen the understanding of thermal transport property of Cu-based chalcopyrite, which is helpful for designing better thermoelectric or photovoltaic devices.

BTE-Barna: An extension of almaBTE for thermal simulation of devices based on 2D materials

We present BTE-Barna (Boltzmann Transport Equation - Beyond the RTA for NAnosystems), a software package that extends the Monte Carlo (MC) module of the almaBTE solver of the Peierls-Boltzmann transport equation for phonons (PBTE) to work with nanosystems based on 2D materials with complex geometries. To properly capture how the phonon occupations evolve in momentum space as a result of scattering, we have supplemented the relaxation-time approximation with an implementation of the propagator for the full linearized version of the PBTE. The code can now find solutions for finite and extended devices under the effect of a thermal gradient, with isothermal reservoirs or with an arbitrary initial temperature distribution in space and time, writing out the temperature and heat flux distributions as well as their spectral decompositions. Besides the full deviational MC solver, a number of useful approximations for highly symmetric devices are also included. Program summaryProgram Title::BTE-BarnaCPC Library link to program files:https://doi.org/10.17632/t58z289tpc.1Developer's repository link:https://github.com/sousaw/BTE-BarnaCode Ocean capsule:https://codeocean.com/capsule/4171806Licensing provisions: Apache-2.0Programming language: C++Nature of problem: Calculation of temperature profiles, thermal flux and/or effective thermal conductivities for nanosystems based on 2D materials.Solution method: For highly symmetric systems confined along some direction(s), the linearized phonon Boltzmann transport equation is solved iteratively by partially suppressing the phonon lifetimes due to boundaries. For more complex geometries, an energy-based deviational Monte Carlo method including off-diagonal terms in the linearized scattering operator arising from the phonon-phonon interaction is used.Additional comments including restrictions and unusual features: Depends on the following external libraries: boost, Eigen, HDF5, MessagePack, MPI, oneTBB, spglib, and almaBTE.

Four-Phonon Scattering Effect and Two-Channel Thermal Transport in Two-Dimensional Paraelectric SnSe.

In recent years, increased attention has been paid to the study of four-phonon interactions and diffusion transport in three-dimensional (3D) thermoelectric materials because they play an essential role in understanding the thermal transport process. In this work, we study four-phonon scattering and diffusion transport in two-dimensional (2D) thermoelectric materials using the paraelectric phase of 2D SnSe as an example. The inherent soft phonon modes are treated by the self-consistent phonon theory, which considers the temperature-induced renormalization of the phonons. Based on density functional theory and the Peierls-Boltzmann transport equation for phonons, we show that the four-phonon interactions can reduce the thermal conductivity of the 2D SnSe sheet by nearly 40% due to the collapse of soft optical modes, and the contribution of diffusion transport to the total thermal conductivity accounts for 14% at a high temperature of 800 K due to the short phonon mean free path approaching the Ioffe-Regel limit, suggesting the two-channel transport in this system. The results are further confirmed by using the machine learning-assisted molecular dynamics simulations. This work provides a new insight into the physical mechanisms for thermal transport in 2D systems with strong anharmonic effects.

Reexamination of hydrodynamic phonon transport in thin graphite

The recent hydrodynamic phonon transport theory for graphitic materials has been supported by the measurements of the second sound at temperatures up to about 100 K. When boundary scattering becomes comparable to momentum-conserving normal phonon scattering processes that are responsible for phonon hydrodynamics, Poiseuille phonon flow phenomena can emerge to give rise to unique size-dependent thermal conductivity in thin graphite. Here, we examine the thickness range for the Poiseuille phonon flow to become observable in thin graphite with the use of both deviational Monte Carlo simulation of the Peierls-Boltzmann transport equation and four-probe thermal transport measurements. As the basal-plane thermal conductivity calculated by prior first-principles theories saturates to the graphite value when the thickness is increased to five graphene layers, the phonon dispersion of graphite is used in the current calculations of thin graphite of micrometer thickness and a 23-layer thick ultrathin graphite (UTG) sample. The calculations show that diffuse surface scattering by surface defects can lead to Poiseuille phonon flow at 50 K in thin graphite with the thickness close to several micrometers but not in the 65 μm thin graphite and 23-layer UTG, where phonon scattering by the top and bottom surfaces become, respectively, much less and more frequent than the normal processes. In addition, the calculation results with the bulk graphite dispersion and diffuse surface scattering show decreased basal-plane thermal conductivity with decreasing thickness, opposite to recent thermocouple measurements of thin graphite samples. In comparison, the calculation results reveal that partially diffuse surface defect scattering can yield the four-probe measurement results of UTG samples, which are prepared here with an improved process to minimize surface contamination.

Observation of second sound in graphite over 200\u2009K

Second sound refers to the phenomenon of heat propagation as temperature waves in the phonon hydrodynamic transport regime. We directly observe second sound in graphite at temperatures of over 200 K using a sub-picosecond transient grating technique. The experimentally determined dispersion relation of the thermal-wave velocity increases with decreasing grating period, consistent with first-principles-based solution of the Peierls-Boltzmann transport equation. Through simulation, we reveal this increase as a result of thermal zero sound—the thermal waves due to ballistic phonons. Our experimental findings are well explained with the interplay among three groups of phonons: ballistic, diffusive, and hydrodynamic phonons. Our ab initio calculations further predict a large isotope effect on the properties of thermal waves and the existence of second sound at room temperature in isotopically pure graphite.

Novel insights into lattice thermal transport in nanocrystalline Mg3Sb2 from first principles: the crucial role of higher-order phonon scattering.

Zintl phase Mg3Sb2, which has ultra-low thermal conductivity, is a promising anisotropic thermoelectric material. It is worth noting that the prediction and experiment value of lattice thermal conductivity (κ) maintain a remarkable difference, troubling the development and application. Thus, we firstly included the four-phonon scattering processes effect and performed the Peierls-Boltzmann transport equation (PBTE) combined with the first-principles lattice dynamics to study the lattice thermal transport in Mg3Sb2. The results showed that our theoretically predicted κ is consistent with the experimentally measured, breaking through the limitations of the traditional calculation methods. The prominent four-phonon scatterings decreased phonon lifetime, leading to the κ of Mg3Sb2 at 300 K from 2.45 (2.58) W m-1 K-1 to 1.94 (2.19) W m-1 K-1 along the in (cross)-plane directions, respectively, and calculation accuracy increased by 20%. This study successfully explains the lattice thermal transport behind mechanism in Mg3Sb2 and implies guidance to advance the prediction accuracy of thermoelectric materials.

GPU_PBTE: an efficient solver for three and four phonon scattering rates on graphics processing units

Lattice thermal conductivity (LTC) is a key parameter for many technological applications. Based on the Peierls–Boltzmann transport equation (PBTE), many unique phonon transport properties of various materials were revealed. Accurate calculation of LTC with PBTE, however, is a time-consuming task, especially for compounds with a complex crystal structure or taking high-order phonon scattering into consideration. Graphical processing units (GPUs) have been extensively used to accelerate scientific simulations, making it possible to use a single desktop workstation for calculations that used to require supercomputers. Due to its fundamental differences from traditional processors, GPUs are especially suited for executing a large group of similar tasks with minimal communication, but require completely different algorithm design. In this paper, we provide a new algorithm optimized for GPUs, where a two-kernel method is used to avoid divergent branching. A new open-source code, GPU_PBTE, is developed based on the proposed algorithm. As demonstrations, we investigate the thermal transport properties of silicon and silicon carbide, and find that accurate and reliable LTC can be obtained by our software. GPU_PBTE performed on NVIDIA Tesla V100 can extensively improve double precision performance, making it two to three orders of magnitude faster than our CPU version performed on Intel Xeon CPU Gold 6248 @2.5 GHz. Our work also provides an idea of accelerating calculations with other novel hardware that may come out in the future.

Importance of quadratic dispersion in acoustic flexural phonons for thermal transport of two-dimensional materials

Solutions of the Peierls-Boltzmann transport equation using inputs from density functional theory calculations have been successful in predicting the thermal conductivity in a wide range of materials. In the case of two-dimensional (2D) materials, the accuracy of this method can depend highly on the shape of the dispersion curve for flexural phonon (ZA). As a universal feature, very recent theoretical studies have shown that the ZA branch of 2D materials is quadratic. However, many prior thermal conductivity studies and conclusions are based on a ZA branch with linear components. In this paper, we systematically study the impact of the long-wavelength dispersion of the ZA branch in graphene, silicene, and $\ensuremath{\alpha}$-nitrophosphorene to highlight its role on thermal conductivity predictions. Our results show that the predicted $\ensuremath{\kappa}$ value, its convergence and anisotropy, as well as phonon lifetimes and mean free path can change substantially even with small linear to pure quadratic corrections to the shape of the long-wavelength ZA branch. Also, having a pure quadratic ZA dispersion can improve the convergence speed and reduce uncertainty in this computational framework when different exchange-correlation functionals are used in the density functional theory calculations. Our findings may provide a helpful guideline for more accurate and efficient thermal conductivity estimation in mono- and few-layer 2D materials.

Space-time dependent thermal conductivity in nonlocal thermal transport

Nonlocal thermal transport is generally described by the Peierls-Boltzmann transport equation (PBE). However, solving the PBE for a general space-time dependent problem remains a challenging task due to the high dimensionality of the integro-differential equation. In this work, we present a direct solution to the space-time dependent PBE with a linearized collision matrix using an eigendecomposition method. We show that there exists a generalized Fourier type relation that links heat flux to the local temperature, and this constitutive relation defines a thermal conductivity that depends on both time and space. Combining this approach with ab initio calculations of phonon properties, we demonstrate that the space-time dependent thermal conductivity gives rise to an oscillatory response in temperature in a transient grating geometry in high thermal conductivity materials. The present solution method allows us to extend the reach of our computational capability for heat conduction to space-time dependent nondiffusive transport regimes. This capability will not only enable a more accurate interpretation of thermal measurements that observe nonlocal thermal transport but also enhance our physical understanding of nonlocal thermal transport in high thermal conductivity materials that are promising candidates for nanoscale thermal management applications.

Particlelike Phonon Propagation Dominates Ultralow Lattice Thermal Conductivity in Crystalline Tl3VSe4

We investigate the microscopic mechanisms of ultralow lattice thermal conductivity (κ_{l}) in Tl_{3}VSe_{4} by combining a first principles density functional theory based framework of anharmonic lattice dynamics with the Peierls-Boltzmann transport equation for phonons. We include contributions of the three- and four-phonon scattering processes to the phonon lifetimes as well as the temperature dependent anharmonic renormalization of phonon energies arising from an unusually strong quartic anharmonicity in Tl_{3}VSe_{4}. In contrast to a recent report by Mukhopadhyay etal. [Science 360, 1455 (2018)SCIEAS0036-807510.1126/science.aar8072] which suggested that a significant contribution to κ_{l} arises from random walks among uncorrelated oscillators, we show that particlelike propagation of phonon excitations can successfully explain the experimentally observed ultralow κ_{l}. Our findings are further supported by explicit calculations of the off-diagonal terms of the heat current operator, which are found to be small and indicate that wavelike tunneling of heat carrying vibrations is of minor importance. Our results (i)resolve the discrepancy between the theoretical and experimental κ_{l}, (ii) offer new insights into the minimum κ_{l} achievable in Tl_{3}VSe_{4}, and (iii)highlight the importance of high order anharmonicity in low-κ_{l} systems. The methodology demonstrated here may be used to resolve the discrepancies between the experimentally measured and the theoretically calculated κ_{l} in skutterides and perovskites, as well as to understand the glasslike κ_{l} in complex crystals with strong anharmonicity, leading towards the goal of rational design of new materials.

Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment

Phonon heat conduction over length scales comparable to their mean free paths is a topic of considerable interest for basic science and thermal management technologies. Although the failure of Fourier's law beyond the diffusive regime is well understood, debate exists over the proper physical description of thermal transport in the ballistic to diffusive crossover. Here, we derive a generalized Fourier's law that links the heat flux and temperature fields, valid from ballistic to diffusive regimes and for general geometries, using the Peierls-Boltzmann transport equation within the relaxation time approximation. This generalized Fourier's law predicts that thermal conductivity not only becomes nonlocal at length scales smaller than phonon mean free paths, but also requires the inclusion of an inhomogeneous nonlocal source term that has been previously neglected. We demonstrate the validity of this generalized Fourier's law through direct comparison with time-domain thermoreflectance (TDTR) measurements in the nondiffusive regime without adjustable parameters. Furthermore, we show that interpreting experimental data without this generalized Fourier's law leads to inaccurate measurement of thermal transport properties.

Revisiting phonon-phonon scattering in single-layer graphene

Understanding the mechanisms of thermal conduction in graphene is a long-lasting research topic, due to its high thermal conductivity. Peierls-Boltzmann transport equation (PBTE) based studies have revealed many unique phonon transport properties in graphene, but most previous works only considered three-phonon scatterings and relied on interatomic force constants (IFCs) extracted at 0 K. In this paper, we explore the roles of four-phonon scatterings and the temperature dependent IFCs on phonon transport in graphene through our PBTE calculations. We demonstrate that the strength of four-phonon scatterings would be severely overestimated by using the IFCs extracted at 0 K compared with those corresponding to a finite temperature, and four-phonon scatterings are found to significantly reduce the thermal conductivity of graphene even at room temperature. In order to reproduce the prediction from molecular dynamics simulations, phonon frequency broadening has to be taken into account when determining the phonon scattering rates. Our study elucidates the phonon transport properties of graphene at finite temperatures, and could be extended to other crystalline materials.

Phonon thermal transport in β−NX(X=P,As,Sb) monolayers: A first-principles study of the interplay between harmonic and anharmonic phonon properties

The investigation of thermal properties of recently emerged two-dimensional (2D) materials is a necessary step towards fulfilling their potential applications in nano-electronics devices. In this study, the thermal conductivity of novel \b{eta}-NX (X=P, As, Sb) monolayers are investigated using a first-principles density functional theory (DFT) study based on the full solution of the linearized Peierls-Boltzmann transport equation (PBTE). The results show that the room temperature thermal conductivities of \b{eta}-NP, \b{eta}-NAs, and \b{eta}-NSb are about 1.1, 5.5, and 34.0 times higher than those of single-element \b{eta}-P, \b{eta}-As, and \b{eta}-Sb monolayers, respectively. The phonon transport analysis reveals that higher phonon group velocities as well as phonon lifetimes are responsible for such an enhancement in the lattice thermal conductivities of \b{eta}-NX (X=P, As, Sb) binary compounds compared to single-element group-VA monolayers. We found that \b{eta}-NP has the minimum thermal conductivity among \b{eta}-NX (X=P, As, Sb) monolayers, while it has the minimum average atomic mass, which is in contrast with the common assumption that lower mass systems exhibit higher thermal conductivities. This work demonstrates the trade-off between harmonic and anharmonic phonon properties in determining the variation of the thermal conductivity among \b{eta}-NX (X=P, As, Sb) monolayers. The higher anharmonicity in \b{eta}-NP is found to be responsible for the lower thermal conductivity of this monolayer.

Thermal Resistance by Transition Between Collective and Non-Collective Phonon Flows in Graphitic Materials

ABSTRACTPhonons in graphitic materials exhibit strong normal scattering (N-scattering) compared to umklapp scattering (U-scattering). The strong N-scattering cause collective phonon flow, unlike the relatively common cases where U-scattering is dominant. If graphitic materials have finite size and contact with hot and cold reservoirs emitting phonons with non-collective distribution, N-scattering change the non-collective phonon flow to the collective phonon flow near the interface between graphitic material and a heat reservoir. We study the thermal resistance by N-scattering during the transition between non-collective and collective phonon flows. Our Monte Carlo solution of Peierls-Boltzmann transport equation shows that the N-scattering in graphitic materials reduce heat flux from the ballistic case by around 15%, 30%, and 40% at 100, 200, and 300 K, respectively. This is significantly larger than ~ 5% reduction of Debye crystal with similar Debye temperature (~2300 K). We associate the large reduction of heat flux by N-scattering with the non-linear dispersion and multiple phonon branches with different group velocities of graphitic materials.

Anharmonic stabilization and lattice heat transport in rocksalt β-GeTe

Peierls-Boltzmann transport equation, coupled with third-order anharmonic lattice dynamics calculations, has been widely used to model lattice thermal conductivity (κl) in bulk crystals. However, its application to materials with structural phase transition at relatively high temperature is fundamentally challenged by the presence of lattice instabilities (imaginary phonon modes). Additionally, its accuracy suffers from the absence of higher-than-third-order phonon scattering processes, which are important near/above the Debye temperature. In this letter, we present an effective scheme that combines temperature-induced anharmonic phonon renormalization and four-phonon scattering to resolve these two theoretical challenges. We apply this scheme to investigate the lattice dynamics and thermal transport properties of GeTe, which undergoes a second-order ferroelectric phase transition from rhombohedral α-GeTe to rocksalt β-GeTe at about 700 K. Our results on the high-temperature phase β-GeTe at 800 K confirm the stabilization of β-GeTe by temperature effects. We find that considering only three-phonon scattering leads to significantly overestimated κl of 3.8 W/mK at 800 K, whereas including four-phonon scattering reduces κl to 1.7 W/mK, a value comparable with experiments. To explore the possibility to further suppress κl, we show that alloying β-GeTe with heavy cations such as Pb and Bi can effectively reduce κl to about 1.0 W/mK, whereas particle size needs to be around 10 nm through nanostructuring to achieve a comparable reduction in κl.

Thermal conductivity of InN with point defects from first principles

We present ab initio calculations of thermal conductivity of InN with vacancies and substitutional defects using a full solution of the Peierls-Boltzmann transport equation. Our parameter-free calculations are in good agreement with experimental measurements demonstrating the predictive power of this approach. Phonon-defect scattering rates are computed from a Green's function methodology that is nonperturbative and includes interatomic force constant variance induced near the defects. Restricting calculations to first-order perturbation approaches can overestimate optic phonon scattering rates by nearly three orders of magnitude. On the other hand, neglecting the force variance weakens the scattering rates by about an order of magnitude, mostly in the low-frequency region below 2 THz. This work elucidates important properties of phonon-defect scattering in thermal transport and demonstrates the predictive power of the coupling of Peierls-Boltzmann transport, Green's function methods, and density functional theory.