International Journal of Thermophysics (2020) 41:9 https://doi.org/10.1007/s10765-019-2583-4 First‑principles Modeling of Thermal Transport in Materials: Achievements, Opportunities, and Challenges Tengfei Ma1 · Pranay Chakraborty1 · Xixi Guo1 · Lei Cao1 · Yan Wang1 Received: 9 August 2019 / Accepted: 30 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract Thermal transport properties have attracted extensive research attentions over the past decades. First-principles-based approaches have proved to be very useful for predicting the thermal transport properties of materials and revealing the phonon and electron scattering or propagation mechanisms in materials and devices. In this review, we provide a concise but inclusive discussion on state-of-the-art frst-prin- ciples thermal modeling methods and notable achievements by these methods over the last decade. A wide range of materials are covered in this review, including two- dimensional materials, superhard materials, metamaterials, and polymers. We also cover the very recent important fndings on heat transfer mechanisms informed from frst principles, including phonon–electron scattering, higher-order phonon–phonon scattering, and the efect of external electric feld on thermal transport. Finally, we discuss the challenges and limitations of state-of-the-art approaches and provide an outlook toward future developments in this area. Keywords Density functional theory · Electron · First principles · Phonon · Thermal transport 1 Introduction 1.1 Overview Thermal transport is of fundamental importance in a number of modern technologi- cal drivers, for instance, transistors, optoelectronics, photovoltaics, and thermoelectric energy conversion [1–3]. The materials used in these applications vary in size, lattice structure, constituting elements, type and number of defects, etc., imposing great chal- lenges for optimal thermal design of the devices. Moreover, it is usually challenging to accurately measure thermal transport properties such as thermal conductivity and * Yan Wang [email protected] 1 Department of Mechanical Engineering, University of Nevada, Reno, NV 89557, USA Vol.:(0123456789)1 3 9 Page 2 of 37 International Journal of Thermophysics (2020) 41:9 interfacial thermal resistance directly, as the current thermal measurement techniques are limited by the size, dimension, shape, surface roughness, range of thermal con- ductivity, and electrical properties (conductive or insulating) of the samples [4–8]. In terms of modeling, the widely used classical molecular dynamics approach can capture phonon transport naturally, and it could be enhanced to include electron efects phe- nomenologically through the two-temperature model [9, 10]. However, its accuracy is never guaranteed for thermal modeling, because the empirical interatomic potentials were rarely developed to ensure the accuracy in its second- and third-order force con- stants, which are essential for accurately modeling phonon scatterings. Furthermore, there are no empirical interatomic potentials for most of the materials that have been discovered and almost none for those that are to be discovered. Thus, thermal modeling approaches based on frst principles, or ab initio, can well compensate the weaknesses of experimental techniques and classical molecular dynamics simulations. In the heat transfer community, the terms “frst principles” or “ab initio” are usu- ally used interchangeably to refer to a density-functional theory (DFT) based approach, which takes inputs from the powerful DFT method that solves the Schrödinger equa- tion approximately—but usually accurately enough. In this review, we aim to provide a brief overview of the frst principles approaches commonly used by the heat transfer community and discuss notable studies and fndings enabled by these approaches. This review is organized as follows. In Sect. 1, we will provide a brief review of the theories and methodologies of several well-established frst-principles approaches, i.e., Boltz- mann transport equation, atomistic Green’s function, and simplifed theories such as the difuse mismatch model and acoustic mismatch model. In Sect. 2, we will discuss nota- ble achievements on frst-principles understandings and predictions of thermal trans- port properties of materials; we also provide our thoughts on possible opportunities for using frst-principles thermal transport modeling to advance specifc felds of science and technology. In Sect. 3, we will discuss the challenges and limitations of the state- of-the-art approaches; if they can be overcome, frst-principles thermal modeling can play a more important role in material discovery and device designs. Finally, we will conclude this review with an outlook to this research area. It is worth noting that there already exist several excellent reviews covering thermal modeling approaches or computational studies on thermal materials [11–17]. To distin- guish from them, we place an emphasis on discussing notable achievements and oppor- tunities for several important topics. The readers might fnd that the topics are interre- lated to each other, while each of them represents a burgeoning or hot topic. Moreover, Sect. 3 is dedicated to discussing the challenges and limitations of state-of-the-art frst- principles modeling approaches, which, to our knowledge, never appeared in previous reviews. 1 3 International Journal of Thermophysics (2020) 41:9 Page 3 of 37 9 1.2 First‑Principles Thermal Transport Modeling Methods: A Brief Overview 1.2.1 Peierls‑Boltzmann Transport Equation The ability of a material to conduct heat can be quantifed by its thermal conductivity . As depicted by the kinetic theory, the lattice thermal conductivity L , i.e., heat trans- fer by phonons, can be computed as [18, 19] = c v2 , L (1) in which the summation runs over all the phonon modes in the frst Brillouin zone, c is the specifc heat, v is the phonon group velocity, and is the phonon relaxa- tion time or lifetime. Specifcally, each phonon mode = (p, q) can be uniquely determined by its branch index p and wave vector q . Obtaining , or its inverse, the phonon scattering rate , is the most challenging task for calculating L through Eq. 1. Generally speaking, it is of acceptable accuracy to merely sum up the rates of all the scattering mechanisms based on the Matthiessen’s rule for low-L materials, in which the Umklapp process is comparable to the normal process [15]. Specif- cally, the of each phonon mode is obtained assuming all other phonons are in equi- librium. The Boltzmann transport equation (BTE) solved in this way is generally referred to as the relaxation time approximation (RTA), which will be abbreviated as RTA-BTE in the following sections. However, for high-L materials with weaker Umklapp process [20, 21], it is generally more accurate to evaluate from an exact solution to the Peierls–Boltzmann transport equation (PBTE): �n �n v⃗ ⋅ ∇⃗ T � = � , � �T �t (2) scatter ⃗ where ∇ T is the temperature gradient and n is the phonon distribution function. In practice, Eq. 2 can be solved iteratively [22] or using variational methods [23]. The term on the right-hand side of Eq. 2 sums all the phonon scattering mechanisms, including scatterings through three-phonon interactions and higher-order ones, iso- topes, grain boundaries, impurities, and other extrinsic mechanisms. As indicated by Eq. 2, these mechanisms as a sum (right-hand side) alters the phonon distribution function n (left-hand side), typically deviating from the equilibrium Bose–Einstein 0 distribution nBE . Assuming the ⇀deviation⇀ depends linearly on the temperature gradi- 0 0 ⋅ ent as n = nBE + −nBE∕T F ∇, one can obtain ⃗ 0 ⃗ F� = �� v⃗� + Δ� , (3) 0 � �⃗ where is the relaxation time of phonon mode and Δ� quantifes the deviation of the population of mode from the equilibrium distribution (i.e., the value in RTA- � �⃗ ⇀ BTE). Δ� depends on the value of F of other phonon modes if two- or three-phonon interactions are considered. Thus, Eq. 3 must be solved iteratively by considering all the phonon modes altogether. A typical iterative solution process starts by assuming � �⃗ 0 Δ� = 0 for all phonon modes, which is merely RTA-BTE. Notably, the inverse of , 1 3 9 Page 4 of 37 International Journal of Thermophysics (2020) 41:9 0 or the scattering rate under RTA, is calculated as the summation of the rates of all the phonon scattering mechanisms: for instance, intrinsic scattering mechanisms like three-phonon interactions (discussed in Sect. 1.2.2) and higher-order ones; and extrinsic scattering mechanisms such as isotopes, boundaries, and impurities. There are several open-source codes available for predicting phonon proper- ties, particularly lattice thermal conductivity, through RTA-BTE or PBTE, such as ShengBTE [24], almaBTE [25], Phono3py [26], and ALAMODE [27]. Aux- iliary tools are provided by these codes to obtain second-order and third-order force constants through DFT (which makes the calculation frst-principles) or using empirical interatomic potentials. These force constants are essential for cal- culating phonon band structures and scattering rates, as discussed below. 1.2.2 Anharmonic Phonon–Phonon Scattering Most PBTE studies so far consider the three-phonon process (i.e., anharmonic phonon scattering process involving three phonons), of which the scattering rate can be computed from Fermi’s golden rule as [19] + − n − n 2 n + n + 1 2 pp 2 1 2 + 2 1 2 − , = V + 1 − 2 + V − 1 − 2 4N 12 1