first-principles modeling of thermal transport in

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International Journal of Thermophysics (2020) 41:9 https://doi.org/10.1007/s10765-019-2583-4

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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

AbstractThermal 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 first-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 findings on heat transfer mechanisms informed from first principles, including phonon–electron scattering, higher-order phonon–phonon scattering, and the effect of external electric field 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

http://orcid.org/0000-0001-9474-6396

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International Journal of Thermophysics (2020) 41:9

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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 effects 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 first principles, or ab initio, can well compensate the weaknesses of experimental techniques and classical molecular dynamics simulations.

In the heat transfer community, the terms “first 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 first principles approaches commonly used by the heat transfer community and discuss notable studies and findings 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 first-principles approaches, i.e., Boltz-mann transport equation, atomistic Green’s function, and simplified theories such as the diffuse mismatch model and acoustic mismatch model. In Sect. 2, we will discuss nota-ble achievements on first-principles understandings and predictions of thermal trans-port properties of materials; we also provide our thoughts on possible opportunities for using first-principles thermal transport modeling to advance specific fields 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, first-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 find 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 first-principles modeling approaches, which, to our knowledge, never appeared in previous reviews.

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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 quantified 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]

in which the summation runs over all the phonon modes � in the first Brillouin zone, c is the specific heat, v is the phonon group velocity, and � is the phonon relaxa-tion time or lifetime. Specifically, 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]. Specifi-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):

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 distribution n0

BE . Assuming the deviation depends linearly on the temperature gradi-

ent as n� = n0BE

+(−�n0

BE∕�T

)⇀

F� ⋅⇀

∇�, one can obtain

where �0� is the relaxation time of phonon mode � and ��⃗Δ𝜆 quantifies the deviation of

the population of mode � from the equilibrium distribution (i.e., the value in RTA-BTE). ��⃗Δ𝜆 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 ��⃗Δ𝜆 = 0 for all phonon modes, which is merely RTA-BTE. Notably, the inverse of �0

� ,

(1)�L =∑

�

c�v2��,

(2)v⃗𝜆 ⋅ ∇⃗T𝜕n𝜆𝜕T

=𝜕n𝜆𝜕t

||||scatter

,

(3)F⃗𝜆 = 𝜏0𝜆

(v⃗𝜆 + Δ⃗𝜆

),


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or the scattering rate �0� 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 first-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]

where the first term on the right-hand side describes the annihilation of a phonon by combination—i.e., two lower energy phonons combine into a single, higher-energy phonon—while the second term denotes the creation of a phonon by splitting, i.e., one phonon splits into two lower energy phonons. In this equation, N denotes the number of q points of the discrete grid for sampling the first Brillouin zone of pho-nons, ℏ denotes the reduced Planck’s constant, and � is the angular frequency of phonon mode. Notably, in RTA-BTE, n is usually set to be the Bose–Einstein dis-tribution (for a realistic, quantum system) or the classical n(�) = kBT∕ℏ� distribu-tion (for comparing with classical molecular dynamics simulations). In contrast, n usually deviates from the equilibrium phonon distribution functions in PBTE, as discussed in Sect. 1.2.1, even though the Bose–Einstein distribution or classical dis-tribution functions are usually used as the initial function for n in the iterative solu-tion process. In Eq. 4, the summation �± runs over phonon modes enforcing the conservation of quasi-lattice momentum: q2 = q ± q1 + Q , in which Q is the recipro-cal lattice vector with Q = 0 denoting normal process and Q ≠ 0 denoting Umklapp process. Thus, the Umklapp process is usually called momentum breaking. In addi-tion, � in Eq. 4 is the Dirac delta function, which can be treated by applying the tet-rahedron method [26] or smearing methods, for example, by replacing the � function with Gaussian [24] or Lorentzian functions [28].

Finally, the transition probability matrices in Eq. 4 are calculated as [29, 30]

(4)

�pp

�=

ℏ�

4N

+∑

�1�2

2n1 − n2

��1�2

|||V+��1�2

|||2

�(� + �1 − �2

)+

ℏ�

8N

−∑

�1�2

2n1 + n2 + 1

��1�2

|||V−��1�2

|||2

�(� − �1 − �2

),

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where m is the atomic mass and ev,q is the normalized eigenvector of the phonon mode � . In Eq. 5, l1 , l2 , and l3 run over atomic indices ( l1 runs over only the atoms in the center unit cell, which contains Nb basis atoms) and �1 , �2 , and �3 represent Car-tesian coordinates. The third-order partial derivative in Eq. 5 is the third-order anharmonic interatomic force constant obtained from DFT (which makes the calcu-lation first-principles) or lattice dynamics using empirical interatomic potentials, in which E is the total energy of the whole system and r�1

l1 denotes the �1 component of

the displacement of atom l1.As a final remark of this subsection, we note that conventionally the second- and

third-order force constants are obtained through the so-called quasi-harmonic approxi-mation [31], using zero-temperature DFT calculations. In this way, it is implicitly assumed that the potential energy surface and the corresponding force environment do not depend on temperature, which is generally acceptable for materials with rela-tively weak lattice anharmonicity. However, as pointed out by Hellman et al., harmonic and anharmonic potential energy surfaces, and thus the corresponding harmonic and anharmonic force constants, indeed depend on temperature [32–34]. A rigorous con-sideration of the finite temperature effect on these properties is critical for materials with strong lattice anharmonicity. In this regard, Hellman et al. developed a series of approaches—commonly referred to as the temperature-dependent effective potential method (TDEP)—for extracting various orders of force constants from DFT while accounting for the finite temperature effect reasonably well. We refer interested readers to the theoretical papers by Hellman et al. for details [32–34].

1.2.3 Phonon–Electron Scattering

The phonon–electron scattering rate can be computed from Fermi’s golden rule as [19]

where g is electron–phonon matrix element, f is the Fermi–Dirac distribution func-tion for electrons, � is electron wavevector, i and j are electron band indices, � is electron energy, and � is phonon frequency. The so-called electron–phonon matrix element in the above equation can be calculated using DFT as

which describes the transition of an electron at a Bloch state ik⃗ into another state at ik + q by a phonon at state � = (p, q) . In this equation, � is a ground-state Bloch wavefunction and E is the energy of the system (i.e., the self-consistent Kohn–Sham energy calculated in DFT), which depends on atomic positions. �E� denotes the

(5)V±��1�2

=

Nb�

l1

N,N�

l2l3

3,3,3�

�1�2�3

�3E

�r�1l1�r

�2l2�r

�3l3

e�1�(l1)e

�1j2,±q2

(l1)e�3j2,−q2

(l3)√ml1

ml2ml3

,

(6)

�pe

�=

2�

ℏ

∑

k,i,j

|||g�jk+q,ik

|||

2[fik(1 − fjk + q)n��(�ik − �jk + q + ℏ��) − fik(1 − fjk−q)(n� + 1)�(�ik − �jk−q + ℏ��)

],

(7)g�jk + q,ik

=

√ℏ

2��

⟨�jk + q

||�E�

||�jk

⟩,


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first-order derivative of the Kohn–Sham energy with respect to phonon displace-ment. In fact, Eq. 6 can be rewritten as

under the relaxation time approximation, eliminating the phonon distribution func-tion n . Since the energy of phonons is much smaller than that of electrons, Eq. 8 can be approximated as [35]

where �f∕�� is a window that peaks at the Fermi level, i.e., the so-called Fermi window. An important property of this function is that it broadens as temperature increases, while the value of its integral always remains as unity. In other words, the phonon–electron scattering rate is essentially a weighted average of the elec-tron–phonon scattering matrix element for electron states near the Fermi surface. This leads to straightforward strategies for tuning the electron–phonon coupling in materials, which will be discussed in Sect. 2.4.

1.2.4 Atomistic Green’s Function

The atomistic Green’s function (AGF) approach has been extensively used to study phonon transport across small structures and interfaces over the last decade [16, 36, 37]. It can rigorously capture phonon-defect scattering mechanisms. In particular, simulations using DFT-based AGF methods have predicted approximately one order of magnitude stronger phonon-vacancy scattering in various materials [38, 39] com-pared to those predicted using empirical models, of which more details and refer-ences can be found in Lindsay et al.’s [15] review paper. To date, most studies using AGF adopted the harmonic version, which neglects anharmonic scatterings. In gen-eral, harmonic AGF is sufficiently accurate for modeling phonon transport at low temperatures, in materials with relatively weak anharmonicity, and in short struc-tures [36, 40].

Here, we provide a brief introduction to the simplest version of harmonic AGF, while readers interested in advanced versions are referred to Refs. [41–43] for details. Specifically, Ref. [41] reports Mingo’s pioneering efforts to incorporate anharmonic phonon scattering mechanisms in AGF, through which he found sig-nificant anharmonic effects down to even 0 K for atomic constrictions. However, one obvious “disadvantage” of this anharmonic extension is that it requires substan-tially more effort to implement than the basic harmonic version. Ref. [42] reports the use of harmonic AGF for dimensionally mismatched interfaces, i.e., between 2D graphene and 1D graphene nanoribbon, while Ref. [43] reports Sadasivam et al.’s efforts to implement electron–phonon scattering mechanisms to AGF through the phenomenological Büttiker probe approach.

(8)�pe

�=

1

�ep

�

=2�

ℏ

∑

k,i,j

|||g�jk + q,ik

|||

2[fik − fik + q

]× �(�ik − �jk + q + ℏ��),

(9)�pe

�≈ 2��

∑

k,i,j

|||g�jk + q,ik

|||

2 �f (�ik, T)

��× �(�ik − �jk + q + ℏ��),

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In the framework of AGF, the dynamical response of the lattice system under a small perturbation is governed by [36],

where ω is the angular frequency of phonon, H is the harmonic matrix, G is the Green’s function, and I is the identity matrix. It is worth noting that the diagonal ele-ments of the harmonic matrix used in AGF are �ii = −

∑

i≠j�2E∕�ui�uj , while those

used in conventional lattice dynamics are defined as �ii = �2E∕�u2i , where E is the

energy of the system, u is the spatial displacement from the equilibrium bond length, and i and j are the atom indices [36]. It is worth noting that if the harmonic matrices are computed from DFT, the corresponding AGF approach can be viewed as first principles.

The phonon transmission function across the interfacial region can be calculated as

where �d,d is the Green’s function of the interfacial region, �1 = i(�1 − �+

1

) , and

�2 = i(�2 − �+

2

) . The superscript + represents conjugate transpose. �1 and �2 are

the self-energy matrices of the left and right leads, which can be computed as

and

respectively. In Eqs. 12 and 13, the H’s are the harmonic matrices between the leads and the interfacial region with 1 denoting the left lead, 2 the right lead, and d the interfacial region. In the above equations, �1 and �2 are the uncoupled Green’s func-tions of the left and right lead, respectively, which can be calculated as

and

in which �1,1 and �2,2 are the harmonic matrices of the left lead and the right lead, respectively. In Eqs. 13 and 14, �i is a small imaginary number, which represents the broadening of phonon energy and helps solve the AGF numerically.

1.2.5 Acoustic Mismatch Model and Diffuse Mismatch Model

The acoustic mismatch model (AMM) and diffuse mismatch model (DMM) in various variations have been widely used for estimating phonon conduction across solid–solid interfaces [44–51]. Considering an interface formed by two materials 1 and 2, the heat flux from material 1 to 2 can be expressed as [36]

(10)(�2� −�)� = �,

(11)�(�) = Trace[�1Gd,d�2G

+d,d

],

(12)�1 = �d,1�1�1,d

(13)�2 = �d,2�2�d,2,

(14)�1 =[(ω + δi)2� −�1,1

]

(15)�2 =[(ω + δi)2� −�2,2

],


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where p denotes Phonon Branch Index (polarization), ℏ is the reduced Planck’s con-stant, � is phonon frequency, vg(�) is phonon group velocity, no

BE is the Bose–Ein-

stein distribution function, T is temperature, �(�) is transmission probability, � is wave vector, � is incidence angle, and � is the azimuthal angle. Assuming an infini-tesimal temperature drop across the interface, the interfacial thermal conductance G can thus be obtained from Eq. 16 as [36]

The phonon band structures, or phonon dispersion relations, of the two materials are the essential input (i.e., the � , vg , and p of phonons) to Eqs. 16 and 17. If these parameters are obtained from DFT-based calculations, we can call such calculations first principles.

The final step for obtaining G through Eq. 17 is to find the phonon transmission function �(�) . AMM is one of the earliest theories for predicting �(�) , in which phonons are assumed to scatter at the interface in a specular way like acoustic waves (as illustrated in Fig. 1). In AMM, the transmission function can be computed as [36, 52, 53]

where Z1 and Z2 are defined as the product of density of mass and phonon group velocity. Typically, AMM only works well for smooth interfaces of which the rough-ness is much smaller than the critical phonon wavelength. Therefore, AMM usu-ally works well at low temperatures, at which phonons are concentrated in the long-wavelength regime. At high temperatures, phonon reflection and transmission at the interface possess more diffuse features. Thus, Swartz and Pohl proposed DMM [53] under the assumption that the direction of phonon propagation is completely inde-pendent of its initial direction after the phonon gets scattered at the interface, as illustrated in Fig. 1. Accordingly, the transmission function for DMM can be written as [36, 53]

in which D(�) is the phonon density of states. It is worth mentioning that Eq. 19 assumes elastic scattering at the interface, i.e., a phonon in material 1 can only trans-mit into a phonon with the same frequency in material 2, regardless of the mode.

(16)

q1→2 =1

8π3

∑

p∫

2�

0 ∫�∕2

0 ∫∞

0

ℏ�vg(�)noBE(�, T1)�(�)k

2dk sin � cos �d�d�,

(17)

G =1

8π3

∑

p∫

2�

0 ∫�∕2

0 ∫∞

0

ℏ�vg(�)�no

BE(�, T1)

�T�(�)k2dk sin � cos �d�d�.

(18)�AMM =4Z1Z2 cos �1 cos �2

(Z1 cos �2 + Z2 cos �1)2,

(19)�DMM =

∑

p2

D2(�)vg2

∑

p1

D1(�)vg1 +∑

p2

D2(�)vg2,

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This has been found to be untrue due to inelastic phonon-interface scatterings [54] and interfacial mode conversion [55].

2 Achievements and Opportunities

In this section, we will provide a brief review of notable discoveries made through first-principles thermal transport modeling. Moreover, we will share our thoughts on possible opportunities for using these approaches to further enhance certain critical areas in modern science and technology. Specifically, we have selected nine important topics—namely, two-dimensional materials, superhard materials, higher-order phonon scattering processes, electron–phonon interactions, thermo-electric materials, effect of external electric fields, polymers, coherent phonons, and machine learning—which represent the areas that have benefited most from first-principles thermal modeling or could benefit thermal property prediction significantly in the future.

2.1 Two‑Dimensional Materials

The advent of various two-dimensional (2D) materials since the first production of graphene in 2004 [56] has triggered extensive researches on their thermal [57–61], electrical [62–65], photonic [65–67], chemical [63], and mechanical [68–70] proper-ties. By 2017, approximately 700 2D materials have been theoretically predicted to be stable, even though the majority of them are yet to be synthesized [71]. Thermal transport in many 2D materials have been investigated using first-principles methods before they can be actually produced or measured in labs [72–87]. Here, we pro-vide a brief review of notable studies in this area, with a focus on studies reporting

Fig. 1 Schematics of the specular phonon-interface scattering assumption for the acoustic mismatch model (AMM) and the diffuse phonon-interface scattering assumption for the diffuse mismatch model (DMM)


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unique thermal transport properties of 2D materials. Readers interested in details of computational studies on graphene are referred to Ref. [16].

Lindsay et al. [88] pioneered the use of RTA-BTE and PBTE for thermal trans-port calculation. They emphasized that using the proper selection rule for phonon scattering is important for accurately estimating the intrinsic lattice thermal conduc-tivity of materials. The 2D nature of graphene leads to a unique selection rule of phonon scattering: out-of-plane phonons only scatter in pairs owing to the mirror symmetry of graphene. This selection rule is also valid for graphene nanoribbon (GNR). Specifically, narrower GNRs have less phonon modes. Moreover, optical phonon modes are pushed to higher frequencies in narrower GNRs. Consequently, GNRs can have much reduced phonon scattering rates and their intrinsic thermal conductivity can even be higher than bulk graphene, as shown in Fig. 2. However, boundary scattering [89], edge-roughness scattering [90], and phonon localization [91] in GNRs can indeed strongly hinder phonon transport in these narrow struc-tures, rendering their thermal conductivity much lower than bulk graphene, as pre-dicted by molecular dynamics simulations [89, 91].

The structural similarity between graphene and carbon nanotube raises an interesting and fundamental question regarding the connection between their phonon transport properties. In this regards, Lindsay et al. [88] have established plausible phonon transport and scattering theories bridging the thermal transport properties of single-walled carbon nanotube (SWCNT) with those of single-layer graphene (like an SWCNT with infinitely large diameter), as shown in Fig. 2. Notably, a reduced diameter removes three-phonon scattering channels in SWC-NTs, leading to a higher thermal conductivity. On the other hand, as the diameter decreases, increasing curvature intensifies phonon scatterings due to the violation of the aforementioned selection rule enforced by the mirror reflection symmetry of 2D graphene, leading to reduced thermal conductivity. These two competing mechanisms lead to a non-monotonic thermal conductivity-diameter relation for SWCNT, as later confirmed by Qiu et al.’s [92] molecular dynamics simulations.

Fig. 2 Thermal conductiv-ity (normalized by the value of bulk graphene) of various SWCNTs as a function of tube diameter [88]. The red circles, blue squares, and green triangles denote zigzag, armchair, and chiral SWCNTs, respectively. The open black squares are for GNRs of which the width equals the perimeter of the cor-responding SWCNTs. Figure is reproduced with permission

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In a recent work by Feng and Ruan, four-phonon scattering process—in which four phonons participate in a phonon scattering event rather than the three-pho-non process considered in earlier studies—was claimed to be even more sig-nificant than three-phonon scattering for flexural phonons [93]. This is because the reflection symmetry imposes more restrictions on three-phonon processes involving flexural acoustic (ZA) phonons than four-phonon ones. For exam-ple, the three-phonon process including three ZA phonons is forbidden by the reflection symmetry while four-ZA scattering processes ZA + ZA↔ZA + ZA and ZA↔ZA + ZA + ZA are allowed, as discussed in Ref. [93]. As a result, including four-phonon processes leads to reduced values of predicted thermal conductivi-ties of graphene, compared to calculations only considering three-phonon pro-cesses. Moreover, the contribution of ZA modes to the overall thermal conductiv-ity of graphene is significantly reduced by four-phonon processes.

In fact, the above notable findings for phonon behaviors in graphene can usu-ally be generalized to other 2D, flat atomic layers like hexagonal boron nitride. However, for non-planar 2D materials in which atoms are not strictly in the same plane, such as silicene, phosphorene, and MXenes, it is obvious that phonon–pho-non and phonon–electron scatterings do not need to satisfy certain selection rules enforced by mirror symmetry. This difference could lead to some unique proper-ties of buckled 2D layers. For instance, Xie et al. [85] found through both RTA-BTE and PBTE that the thermal conductivity of silicene can increase dramati-cally with tensile strain, owing to the large increase in acoustic phonon lifetime. The mechanism is straightforward: silicene turns flatter at larger tensile strains, rendering the aforementioned mirror symmetry-enforced selection rule more effective by suppressing the scattering processes involving an odd number of flex-ural acoustic phonon modes, thereby leading to a higher thermal conductivity.

The electron contribution to thermal conductivity was relatively less investi-gated for 2D materials. He et al. [94] predicted high electronic and lattice thermal conductance of 2D hydrogen boride, i.e., hydrogenated graphene-like borophene, through the AGF approach. Specifically, its lattice part is found to be compara-ble to that of graphene, while the electronic part is almost 10 times of that of graphene, rendering hydrogen boride much more thermally and electrically con-ductive. With the wide deployment of first-principles approaches for calculating thermal and electrical properties of materials, we expect them to play a more sig-nificant role in discovering and designing new 2D materials with desired trans-port properties.

2.2 Superhard Materials

Superhard materials are a category of unique materials characterized by ultrahigh hardness, typically exceeding 40 GPa in the Vickers hardness test. These include, but not limited to, diamond, boron carbide, cubic boron nitride, and tungsten boride. They are being intensely investigated for various applications such as polishing and cutting tools, wear-resistant and protective coatings, and body armors. Plus, super-hard materials (like diamond) are being widely used in thermal applications owing


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to their high thermal conductivity, which primarily arises from the strong covalent bonds.

Ward et al. [30] pioneered first-principles PBTE calculations of lattice thermal transport in diamond, after which Broido et al. [95] analyzed the pressure depend-ence of thermal conductivity in this material. Notably, Broido et al. found a sig-nificant pressure dependence of thermal conductivity in cubic diamond, as shown in Fig. 3. As revealed in their work, optical phonons are involved in approximately 80 % of the three-phonon scattering processes for acoustic phonons in diamond at zero pressure. Furthermore, a compressive hydrostatic pressure elevates the overall pho-non frequencies and significantly reduces the three-phonon scattering rates, leading to a dramatic increase in thermal conductivity. This work revealed the importance of optical phonons to thermal transport in materials, which was usually overlooked previously.

The effect of three-phonon scattering phase space on thermal transport can be manifested in other ways. Chakraborty et al. [96] revealed through their first-prin-ciples PBTE calculations that lattice structure plays a vital role in determining the lattice thermal conductivity of superhard diamonds and boron nitrides, namely, cubic diamond, hexagonal diamond (lonsdaleite), cubic boron nitride, and hexago-nal boron nitride. They found that the hexagonal structures have lower thermal con-ductivity than their cubic counterparts, despite the fact that both phases have almost the same bond energy and bond length. As revealed through their phonon analysis, the lower thermal conductivity of hexagonal phases stems from the smaller acous-tic-optical phonon bandgap and, correspondingly, enlarged three-phonon scatter-ing phase space, which causes stronger phonon scatterings, as shown in Fig. 4. In Sects. 2.3 and 2.8, we will discuss another manifestation of the phase space effect on phonon transport, i.e., high lattice thermal conductivity resulting from limited phase space in materials with large phonon bandgaps.

Fig. 3 (a) Phonon density of states (DOS) at different pressures: 0 Pa and 125 GPa. (b) Thermal conduc-tivity of isotopically pure cubic diamond and that of naturally occurring one as a function of pressure. Both figures are reproduced with permission [95]

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2.3 Boron Arsenide: The Importance of Higher‑Order Phonon Scatterings

Due to the restriction of computational power and the seemingly sufficient agree-ment between measured and calculated values of lattice thermal conductivity for a wide variety of materials, prior researches have mainly focused on three-phonon scattering processes for thermal conductivity prediction and neglected higher-order processes like four-phonon scattering. Since the derivation of a rigorous formalism (referred to as Feng-Ruan formalism in this paper) for evaluating four-phonon scat-tering rates [97], the effect of four-phonon scattering has been found to be non-neg-ligible in materials with strong lattice anharmonicity [98, 99], with restricted three-phonon scattering phase space [93, 99], or at higher temperatures [97].

One notable material is boron arsenide (BAs). In 2013, Lindsay et al. [100] pre-dicted through first-principles PBTE calculations that BAs could have an ultrahigh room temperature thermal conductivity over 2000 W·m-K−1, comparable to that of diamond and graphite. Its high thermal conductivity stems from the unconvention-ally large acoustic-optical phonon bandgap, which severely restricts three-phonon scattering phase space, as illustrated in Fig. 5(a). In addition, the bunching of the three acoustic branches is also necessary for the reduced phonon scattering in BAs. In 2017, Ruan and coworkers found that four-phonon scattering is important in determining the thermal conductivity of BAs [99]. In particular, four-phonon pro-cesses reduce the predicted room temperature thermal conductivity from 2200 W·m-K−1 (three-phonon only) to 1400 W·m-K−1 (three-phonon plus four-phonon), as shown in Fig. 5(b) [101]. The reduction at higher temperatures is even more signifi-cant, e.g., 60 % at 1000 K.

In 2018, three independent papers published in Science [101–103] reported the successful synthesis of BAs along with a measured high thermal conductiv-ity around 1000–1300 W·m-K−1 at room temperature. The around 30 % variation among experimental values in the three studies [101–103] may arise from quality of samples. Nonetheless, they all unanimously demonstrate reasonable agreement with first-principles predictions, which further confirms the capability of first-prin-ciples approaches for predicting the thermal conductivity of materials, especially newly or to-be discovered ones. Currently, there are still intense research efforts on

Fig. 4 Three-phonon scattering rates at 300 K for longitudinal acoustic phonons in the x–y plane of the first Brillouin zone of (a) cubic diamond and (b) hexagonal diamond. (c) Comparison between the differ-ential volume of the three-phonon scattering phase space for cubic diamond and hexagonal diamond. All figures are reproduced with permission [96]


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investigating thermal transport in BAs and similar materials, for the sake of apply-ing them to thermal management of modern electronic and photonic devices. For readers interested in more details of recent progress in the research on BAs—includ-ing synthesis, theoretical predictions, and experimental measurements—we refer the readers to the short review by Tian and Ren [104].

2.4 Electron–Phonon Interactions: Metals, Semiconductors, and Metal‑Nonmetal Interfaces

As discussed in Sect. 1.2.3, phonon–electron scattering rate is strongly affected by the number of electron states available at and near the Fermi level. This is evidenced by Wang et al.’s [35] first-principles calculations for various metals. As shown in Fig. 6, platinum has much higher electron density of states at the Fermi level than gold. As a result, the phonon–electron scattering rate in plati-num is significantly higher than that in gold. This indeed provides a reliable and fast approach for identifying materials with possibly strong phonon–electron coupling. As shown by Jain et al. [105] and Wang et al. [35], phonon–electron scattering is generally strong in metals with large electron density of states at the Fermi level, for example, platinum, nickel, and aluminum. Vice versa, pho-non–electron scattering is weak in metals with small density of states at the Fermi level. It is worth noting that electronic heat capacity and electron–phonon scattering rate (i.e., the rate at which electrons are scattered by phonons) follow the same rule: higher density of states at the Fermi level leads to higher elec-tronic heat capacity and stronger electron–phonon scattering, which is directly governed by the Fermi window. Detailed analysis and discussions can be found in Lin et al.’s [106] classic paper on first-principles calculations of electron–phonon coupling and electronic heat capacity of various metals.

Fig. 5 (a) Phonon dispersion relations of BA [99]. (b) Comparison between measured thermal conductiv-ity of BAs and first-principles predictions including or excluding four-phonon scattering process [101]. All panels are reproduced with permission [99, 101]

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Excitingly, the same conclusion applies to semiconductors, of which the Fermi level can be tuned (relative to the band structure) much more conveniently by gat-ing or doping than metals. Prior to Jain et al. and Wang et al.’s studies on metals, Liao et al. [107] found significant phonon–electron scattering in heavily-doped sili-con based on first-principles calculations. Specifically, they predicted an up to 45 % (for p-type silicon at around 1021 cm−3) reduction in the room temperature lattice thermal conductivity of silicon as the carrier (electron or hole) density rises above 1019 cm−3, a doping range readily achieved in thermoelectric materials. Later on, Liao et al. predicted the feasibility of using electrostatic gating to tune phonon–elec-tron scattering and thus lattice thermal conductivity of two-dimensional materials, such as silicene and phosphorene, through first-principles calculations [108].

In addition to the aforementioned studies on phonon–electron scattering in ele-mental metals and semiconductors, there are several recent studies on exploring its effect in more complicated systems. For instance, Li et al. [101] found strong pho-non–electron scattering, even comparable to phonon–phonon scattering, in group V transition metal carbides, including vanadium carbide, niobium carbide, and tanta-lum carbide. All these metals are characterized by nested Fermi surfaces and large frequency gaps between acoustic and optic phonons, leading to relatively stronger phonon–electron scattering effect on lattice thermal transport. Similarly, Tong and Bao found that phonon–electron scattering reduces the lattice thermal conductivity

Fig. 6 Electron density of states (a, c) and phonon–electron (p–e) scattering rates (b, d) for platinum and gold obtained from first principles. All panels are reproduced with permission [35]


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of NiAl and Ni3Al by 55 % and 75 %, respectively, at 100 K [109]. However, this effect decreases quickly at higher temperatures, owing to the fast increase in anhar-monic phonon–phonon scatterings. It is worth noting that both elemental Ni and Al have rather strong phonon–electron scattering, as predicted in Wang et al.’s [35] ear-lier study. Therefore, one might expect to see similarly strong effect in their alloys, though theoretical proofs on this are needed.

Even though more challenging than semiconductors, it is still possible to tune electron–phonon interaction in metals. Lanzillo et al. [110] conducted first-princi-ples calculations on electrical resistivity limited by electron–phonon scattering, i.e., the scattering of an electron mode by phonons. Interestingly, they found that a pres-sure (compressive stress) up to 2 GPa can reduce the electrical resistivity consider-ably (Fig. 7), which agrees with their experimental results. A further analysis of the Eliashberg spectral function from the density functional perturbation theory reveals that the reduction in electrical resistivity arises from weakened electron–phonon interaction, which is claimed to be caused by stiffened phonons under pressure.

As we can see from the above scarce but interesting studies on phonon–electron scattering and its effect on lattice thermal transport, this mechanism provides a new route for tuning the lattice thermal conductivity of materials, which could be useful for applications such as thermoelectric materials and thermal switch. However, tun-ing phonon–electron scattering in materials, even for semiconductors, is not always feasible. For example, in Li et al.’s [111] recent work, phonon–electron scattering was found to have negligible influence on lattice thermal transport in crystalline tin selenide (SnSe). In this regard, schemes other than tuning the location of Fermi level should also be actively explored to maximally tune the strength of phonon–electron scattering in materials. For example, one could tune the band structure of materials to increase the electron density of states at the Fermi level.

Another topic of significant fundamental and practical importance is elec-tron–phonon coupled thermal transport across metal–nonmetal interfaces, which vastly exist in electronic devices [2, 112–115]. In 2004, Majumdar and Reddy derived an electron–phonon thermal boundary resistance—caused by the thermal nonequilibrium between electrons and phonons near a metal-nonmetal interface—that adds to the conventionally known phonon–phonon thermal boundary resistance [113]. In 2013, Wang and Ruan predicted analytically through the two-temperature

Fig. 7 Electrical resistivity as a function of pressure for aluminum and copper. Figure is reproduced with permission [110]

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model that inserting a nanometer-thin (5 nm–15 nm) interlayer of metal (e.g., Ti, Ni, Cr, Al, and, not mentioned in their work, Pt) with strong electron–phonon interac-tion can reduce electron–phonon nonequilibrium near Au–Si interface, thereby lead-ing to enhanced interfacial thermal transport [116]. In Wang and Ruan’s subsequent paper in 2016, they first obtained essential phonon and/or electron properties of var-ious metals (Au as a substrate; Pt and Al as an interlayer for Au–Si interface) and Si from first-principles calculations (lattice dynamics, RTA-BTE and DMM) and then fed them to two-temperature Boltzmann transport equations [44]. Their simula-tions rigorously demonstrated how a Pt or Al interlayer of 10 nm can significantly increase the thermal transport across Au–Si interfaces and accelerate hot electron relaxation in Au thin films supported on a Si substrate.

Finally, we note the thermal transport in metal alloys, laser-processed metals and alloys, and additively manufactured metallic components are important for various critical applications (such as high-voltage–power transmission, heat exchangers, nuclear fuel rods, and thermal management in microelectronics), while their com-plex microstructures [117–123] impose great challenges and, at the same time, cre-ate exciting opportunities for first-principles based investigation of thermal transport in these materials.

2.5 Thermoelectric Materials

The advance in nanotechnology has enabled the development of various thermoelec-tric (TE) materials. However, even with decades of extensive research, the figure-of-merit ZT of even the best TE materials is still lower than the target value of 3 to 4, which is the critical number for TE devices to compete with conventional power generators and refrigerators in terms of efficiency [124]. First-principles approaches have provided a deeper understanding of phonon transport and scattering in various existing and new materials and structures [125, 126]. Herein, we focus on notable first-principles studies on thermal transport in TE materials, because a low lattice thermal conductivity is essential for achieving a higher ZT, while readers who are interested in the basic theories and recent developments of TE materials are referred to Refs. [127–130].

Based on the BTE of lattice thermal transport (Eq. 1), lattice thermal conduc-tivity is the product of phonon relaxation time � (its inverse is the scattering rate � ), group velocity, and heat capacity, among which � is mostly convenient to engi-neer for most materials and structures. As informed from the lattice dynamics of phonons, materials or structures containing one or several of the following prop-erties tend to have a low lattice thermal conductivity: containing many phonon scatters such as dislocations, impurities, alloys, grain boundaries, interfaces, etc.; possessing strong lattice anharmonicity owing to resonant bonds or other mecha-nisms; possessing complicated band structure that leads to lower phonon group velocity or larger phonon scattering phase space. In addition, there are less inves-tigated mechanisms like phonon–electron scattering as discussed in Sect. 2.4.

The earliest first-principles calculations were mostly on single materials (iso-topically pure or naturally occurring) or simple binary or ternary alloys, in which


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the mass difference between elements causes additional phonon scatterings. Since 2007, Broido and coworkers’ have conducted a series of first-principles studies on simple TE materials such as nanostructured Si and Ge (elemental materials and alloys) and more complicated ones such as MgSiSn alloys, with the earliest stud-ies using RTA-BTE while the more recent ones using PBTE [131–134]. Since the 2010s, Mingo and coworkers [135] have used the AGF approach while Chen and coworkers have used BTE approaches, similar to those by Broido et al., to investigate a variety of TE materials, including nanostructured Si/Ge, Bi/Sb, and FeSb [136, 137]. Through these studies: optical phonons were found to not only contribute to over 20 % of thermal conductivity but also scatter acoustic phon-ons significantly in PbTe and PbSe alloys [137]; long-range interaction caused by resonant bonding was found to significantly increase the lattice anharmonicity of lead chalcogenides, SnTe, Bi2Te3, Bi, Sb, and Bi-Sb alloy [136, 138]; the lat-tice and electronic thermal conductivity accumulation with respective to phonon mean-free-path and electron mean-free-path, respectively, have been quantified [139, 140], as shown in Fig. 8. In addition to materials with intrinsically low lat-tice thermal conductivity, various strategies can be applied to further reduce it. For instance, the effect of various defects such as impurities, interfaces, and grain boundaries on thermal transport has been analyzed through first-principles calcu-lations, as reviewed in Ref. [16].

2.6 Effect of External Electric Field

Strategies for actively tuning thermal transport is of great importance for a wide range of practical applications, for example, thermal switch and thermal signal pro-cessing. Most of the current approaches for tuning thermal transport, such as doping

Fig. 8 Normalized cumulative conductivity (at 300 K), i.e., electrical conductivity σ and lattice thermal conductivity �L , with respective to the corresponding electron or phonon mean-free-path for Si [139] and PbTe [140]. The normalized cumulative conductivity quantifies how material size can affect the conduc-tivity of nanomaterials, i.e., the percentage of conductivity remaining (vertical axis) for a specific size (horizontal axis) of nanomaterial due to the truncation of mean-free-path by materials’ boundaries

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and nanostructuring, will irreversibly modify the atomic structure and thus the trans-port properties of materials. Applying an external field such as electric, magnetic, or electromagnetic (e.g., light irradiation) field is a promising method to actively manipulate thermal transport in materials and, at the same time, preserve their pristine microstructures. The development of Berry-phase-based modern theory of polarization enables a rigorous inclusion of electric field effect in DFT, from which equilibrium atomic structures and force constants under specified external electric field can be obtained conveniently [141]. Several theoretical studies adopting first-principles-based PBTE have pioneered the exploration of thermal transport modula-tion via external electric field, while experimental proofs are yet to be provided.

Qin et al. [142] predicted that the thermal conductivity of the single-atomic-layer, 2D silicon—that is, silicene—can be reduced by two orders of magnitude from 19.2 W·m-K−1 to 0.091 W·m-K−1 by a 0.5 V·A−1 electric field in the cross-plane direction. The charge density is redistributed under the electric field, which leads to phonon renormalization and thus changed lattice anharmonicity. As a result, ther-mal conductivity can vary remarkably under external electric fields. Based on a dif-ferent mechanism, Liu et al. [143] observed significant modulation of the thermal conductivity of barium titanate (BTO) by external electric fields, which is caused by the amorphous-crystalline transition of this ferroelectric material. Through BTE calculations, the authors found that crystalline BTO possesses a thermal conduc-tivity 3.9 times higher than that of its amorphous counterpart without an external electric field, as shown in Fig. 9(a). Moreover, thermal conductivity of crystalline BTO can be further enhanced by 2.4 times under electric field. Later in 2019, the same research group demonstrated similar electric field effect on another ferroelec-tric material, PbTiO3, through BTE [144]. Figure 9(b) shows their calculated ther-mal conductivity at external electric fields from − 4 MV·cm−1 to 2 MV·cm−1, in which thermal conductivity increases with increasing polarization due to reduced anharmonic phonon scattering.

2.7 Polymers

Polymers have been viewed as thermal insulators with very low thermal conduc-tivity in the range of 0.1−1 W·m-K−1 for several decades since its mass produc-tion. In 2010, Shen et al.’s [145] groundbreaking paper reported a surprisingly high experimentally measured thermal conductivity of ~ 104 W·m-K−1 for ultra-drawn polyethylene (PE) nanofibers, revealing that the thermal conductivity of polymers is not inherently low.

In a 2017, study by Wang et al. [146], phonon transport in individual PE chains and bulk PE crystals was investigated using first-principles PBTE. Notably, they obtained an ultrahigh thermal conductivity of 1400 W·m-K−1 for single-chain PE while a much lower value of 237 W·m-K−1 for bulk PE crystals (along the chain direction), owing to the strong anharmonicity caused by inter-chain van der Waals interactions. Moreover, they found that high-frequency longitudinal acoustic pho-nons dominate thermal transport in PE chains, due to their large group velocity


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and low scattering rate; in contrast, transverse phonons contribute little to ther-mal conductivity because of their vanishing group velocities and short lifetimes.

In the same year as Wang et al.’s [147] work, Shulumba et al. investigated pho-non thermal transport in PE using a more sophisticated approach: first-principles PBTE with force constants obtained from the temperature-dependent effective potential method (i.e., TDEP, which can consider the finite temperature effect on potential energy surface) discussed in Sect. 1.2.2. Moreover, they further enhanced the TDEP method by incorporating the nuclear quantum effect (i.e., zero-point motion), which significantly alters the behavior of the ultralight hydro-gen atoms that widely exist in polymers. As a consequence, the predicted room temperature thermal conductivity of PE (along the chains) drops from around 250 W·m-K−1 (agreeing with Wang et al.’s value of 237 W·m-K−1) to around 160 W·m-K−1, owing to the strong zero-point motion of hydrogen atoms (as put by Shulumba et al., hydrogen atoms have zero-point kinetic energy corresponding to around 1000 K).

Finally, we note that first-principles studies of thermal transport in polymers (par-ticularly semi-crystalline and amorphous ones) and across polymer-inorganic mate-rial interfaces are still rare, while it is an important topic that deserves more research attention, especially when polymer-based 3D printing technologies are fast advanc-ing these years.

2.8 Superlattice and Coherent Phonons

Superlattice, a structure composed of alternating nanolayers of two or more mate-rials, has been extensively investigated for photoelectronic [148–150] and ther-moelectric applications [124, 151–154]. In particular, its value for thermoelectric applications arises from the ultralow lattice thermal conductivity caused by its high-interface density. So far, it is well understood that the lattice thermal transport

Fig. 9 Thermal conductivity of BTO (a) [143] and PTO (b) [144] under different external field. Both panels are reproduced with permission [143, 144]

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behavior in superlattices is a manifestation of two competing mechanisms: coherent and incoherent phonon transport [155–157].

In 2011, Garg et al. [158] reported calculations of the in-plane and cross-plane thermal conductivity of SiGe superlattices through RTA-BTE. Noting that a superla-ttice with a large period length generally contains too many atoms for DFT calcula-tions, Garg et al. alleviated this issue with a virtual crystal approximation, in which they first calculated the force constants of pure Si and Ge supercells from DFT and then used their average for the superlattice. In this way, they found that the ther-mal conductivity of short-period SiGe superlattices increases as the period length decreases, as shown in Fig. 10(a), which agrees with previous lattice dynamics and molecular dynamics results [159–161]. The increase in thermal conductivity (as period length decreases) is mainly due to increased phonon group velocities, while the phonon relaxation times stay almost the same, as displayed in Fig. 10(b), which agrees with previous lattice dynamics predictions by Hyldgaard and Mahan [160] and Tamura et al. [161]. Notably, Fig. 10(b) shows much longer phonon relaxation times in the shortest SiGe superlattice (“1 + 1”: one period has 1 unit cell thick of Si and 1 unit cell thick of Ge) than longer-period ones—which was attributed to limited phase space for phonon scatterings, particularly the absence of acoustic–acoustic-optical combination channel caused by the large band gap of the “1 + 1” superlattice, as shown in Fig. 10c and 10d. In a subsequent study [162], Garg and Chen extended this approach by including interfacial disorder scattering, through which they were able to obtain a non-monotonic “thermal conductivity-superlattice period” relation, which is commonly observed in molecular dynamics simulations and experiments of high-interface-quality superlattices [157, 159, 163–165].

Similarly, in 2018, Carrete et al. [166] adopted the virtual crystal approxima-tion and calculated thermal conductivity of InAs/GaAs superlattices using PBTE

Fig. 10 (a) Thermal conductivity of Si/Ge superlattice as a function of number of period. (b) Phonon group velocity and relaxation time under different superlattice periods. (c) Scattering rate due to different mechanisms. (d) Total scattering rate. All panels are reproduced with permission [129]


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with all force constants obtained from DFT. In addition to three-phonon scattering, they incorporated two additional types of scatterings—scattering due to isolated point defects (short-range disorder) and scattering due to an average mass distri-bution modulated along the growth direction of the superlattice (long-range disor-der)—to better model realistic superlattice samples in experiments. Their theoreti-cal predictions matched with their experimental data reasonably well, suggesting the effectiveness of first-principles approaches for predicting thermal conductivity of superlattices.

AGF is another useful approach for studying thermal transport in superlattices. In 2014, Tian et al. [167] studied phonon transport in Si/Ge superlattices using force constants extracted from DFT. They observed that interface roughness destroys the coherence of high-frequency phonons. However, small-period superlattices with rough interfaces show higher thermal conductivity than those with smooth inter-faces, which was attributed to the smoother change of vibrational density of states between layers in rough-interfaced superlattices [167, 168]. Later, Qiu et al. [169] investigated the effect of aperiodicity and interface roughness on Si/Ge superlattices using AGF, through which they found that both interface roughness and aperiodicity can reduce the transmittance of high-frequency phonon significantly. More recently, Mendoza and Chen performed AGF simulations on GaAs/AlAs superlattices with embedded ErAs particles [170]. They found that Anderson localization of phonons can lead to a thermal conductivity maximum in the thermal conductivity-device length relation.

Apparently, limited by the computational cost of first-principles methods (DFT-based RTA-BTE, PBTE, and AGF), the investigation of superlattices is only limited to those with small periods, which restricts a full-scale analysis of coherent phonon transport in this important metamaterial. For larger-period superlattices, fictitious force constants obtained through the virtual crystal approximation have to be used, which prevents a quantitative prediction of thermal transport properties.

2.9 Machine Learning‑Based Techniques

Machine learning (ML)-based techniques are attracting more research attentions and great progress has been achieved in artificial intelligence areas such as gameplay (DeepBlue [171] and AlphaGo [172]). Recently, extensive efforts have been devoted to implementing ML in material research. Even though ML itself cannot be counted as first principles, the use of data obtained from DFT calculations has been a pop-ular way for training ML models. Owing to its great promise as a useful tool for predicting various properties of materials, herein we briefly discuss recent ML stud-ies—using first-principles data, experimental data, or classical molecular dynamics data, or a combination of them—on thermal transport properties.

ML techniques have been applied to search for materials with low �L as thermoe-lectric materials. Carrete et al. in 2014 calculated the �L of 450 thermally stable half-Heusler compounds, using a random-forest regression model trained by 32 sets of �L ’s [173]. In particular, the training data were obtained from first-principles PBTE calculations. The parameters that are easy to obtain and empirically considered to

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correlate strongly with �L were used as descriptors, for instance, atomic number and weight, lattice constant, band gap, specific heat, and volume of phase space for three-phonon scattering. To obtain a quantitatively precise model, the authors added anharmonic interatomic force constants as descriptors to further train the ML model, demonstrating reasonable accuracy (as shown in Fig. 11) and, at the same time, much better time efficiency compared with full first-principles calculations.

In 2015, Seko et al. [174] calculated the �L of 101 compounds with first-princi-ples RTA-BTE, the less accurate but widely used predecessor of PBTE. Using these data, a Bayesian optimization model was built and applied to screen 54,779 com-pounds to find the materials with the lowest �L . 221 materials were found to possess very low �L from the initial screening. After a further filtering, two compounds with room temperature �L less than 0.5 W·m-K−1 and band gap less than 1 eV were suc-cessfully discovered, demonstrating the feasibility and power of ML for searching for target materials.

There are several recent studies exploring ML-based predictions of interfacial thermal resistance Rint between two materials. The prediction of Rint is a high-dimensional mathematical problem as Rint depends on many factors, including interface roughness, binding energy, and the presence of impurities, which alto-gether determine Rint in a complex way. ML is thus very suitable for this applica-tion due to its superior capability of dealing with black-box problems. In 2018, Yang et al. [175] trained several ML models using samples obtained from clas-sic molecular dynamics simulations to predict the Rint between single-layer gra-phene and hexagonal boron nitride, demonstrating the time efficiency of ML compared to direct molecular dynamics simulations. A more recent study by Wu et al. [176] reported the use of experimental-data-trained ML models to predict Rint . As shown in Fig. 12(a), they used three types of descriptor sets: property descriptors (e.g., temperature, heat capacity, melting point, density, and unit cell volume), compound descriptor (e.g., atomic ratio, location in the periodic table, electronegativity, ionic potential, and binding energy), and process descrip-tors (film thickness and type, i.e., whether the film is an interlayer or free on one side). These huge sets of descriptors could feed the ML model with physical, chemical, and structural information of the sample that can affect Rint but could

Fig. 11 Lattice thermal conductivity predicted from ML model (blue dots) and first principles (red plus symbols). �anh is the thermal conductivity predicted from ML model with second-order force constants. �� is the thermal conductivity calculated from BTE using reconstructed third-order force constants. Fig-ure is reproduced with permission [173]


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not be comprehensively and accurately captured by simple models like AMM and DMM. Figure 12(b) shows the nice agreement between ML predictions and experimental data.

ML has been applied to more complicated structures such as aperiodic super-lattices, or random multilayer (RML), which were found to exhibit much lower �L than periodic superlattices, because coherent phonons are localized in these dis-ordered structures [157]. The major challenge for obtaining the best RML struc-ture, one with lowest �L that is essential for thermoelectric materials and ther-mal barriers, is to find the optimal thickness distribution (i.e., number of layers for each specific thickness) and arrangement (i.e., the order of layers of various thickness). Recently, Chakraborty et al. [177] reported ML prediction of �L of RMLs, in which nonequilibrium molecular dynamics simulations on thousands of different RML structures were used to train a ML model, which turned out to predict the �L of RMLs rather accurately (error is approximately 5% to 10 %). Moreover, with the aid of ML, the authors identified several key parameters to quantify the disorder of the structure, which can in turn improve the accuracy of ML predictions.

3 Challenges

3.1 Dependence of Predicted �L on exchange–correlation functionals in DFT

DFT has proved to be a powerful tool for predicting various material properties. In particular, its use in first-principles predictions of �L has been very successful, as reviewed in Sect. 2 of this review. However, DFT does not provide a strictly accu-rate solution to the Schrödinger equation. Instead, it solves a simplified version, the Kohn–Sham equation, which is the one-electron Schrödinger-like equation of a ficti-tious system of non-interacting particles. One particular source of inaccuracy of this approach arises from the use of exchange–correlation (XC) functionals, for which

Fig. 12 (a) List of the descriptor sets used in machine learning prediction of interfacial thermal resist-ance Rint . (b) Comparison of Rint predicted from ML model and experimental data. All panels reproduced with permission [176]

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currently no one knows the exact form and thus can only use various versions of functionals as approximations. The local density approximation (LDA) and gener-alized gradient approximation (GGA) are two of the mostly used XC functionals in DFT. Specifically, LDA is one of the simplest approximations in which the XC energy depends only on electronic density at each grid point, while GGA includes the density gradients term and can generally improve the prediction. As reported in Refs. [35] and [178], different XC functionals may predict different lattice constants, force constants, and thus �L . We have checked this conclusion for graphene using first-principles PBTE for this review paper. As shown in Fig. 13, the choice of XC functionals affects the predicted values of �L for graphene significantly. This incon-sistency restricts a direct comparison between predicted �L and experimental values and for predicting the �L of new materials quantitatively, because there lack clear rules for judging which XC functional is most appropriate for a given material.

3.2 Long‑Range Interaction Necessitating Large Cutoff Radius for Force Constant Calculations

As discussed in Sect. 1.2.2, the calculation of transition probabilities and thus phonon relaxation times requires second- and third-order interatomic force con-stants (IFCs). In supercell-based DFT calculations of third-order IFCs, a cut-off distance rc is almost always introduced to reduce the computational cost by neglecting the interaction between atoms with a distance larger than rc . In pre-vious PBTE studies, an rc covering 3rd- to 7th- nearest neighbors were usually considered to be sufficient for including any significant interactions between atoms. However, it has been found that a large rc extending to 9th- or even more neighbors is needed to account for the significant long-range interactions in cer-tain materials, for instance, Bi/Sb [136], SnSe [179], graphene [179], and phos-phorene [179] (Fig. 14). A small rc almost always leads to an overestimation of �L or even a diverged calculation, especially for low-dimensional structures. Unfortunately, it seems that a rigorous convergence study with respect to the cut-off radius used in IFC calculations are missing in many previous studies using

Fig. 13 Lattice thermal conduc-tivity of single-layer graphene predicted from first-principles PBTE, for which the force constants were obtained from DFT calculations with different exchange–correlation function-als


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first-principles PBTE, which means many reported values may suffer from seri-ous errors, at least quantitatively.

3.3 Error Amplification from Force Calculation in DFT to �L Prediction

in PBTE

The accuracy of �L obtained through the aforementioned first-principles approaches greatly depends on the accuracy of the force constants obtained from DFT, which appear in the expressions for calculating the transition probabilities and thus phonon relaxation times (see details in Sect. 1.2.2). In DFT, which seeks for an approximate solution to the ground state of a system, various factors affect the accuracy of the forces. For instance, the exchange–correlation functionals (discussed in Sect. 3.1), cutoff radius used in IFC calculations (discussed in Sect. 3.2), cutoff energy for wavefunctions, k-point grid, and reciprocal space projection algorithms used in DFT all affect the predicted values of �L significantly. The error in force constants will propagate into the final PBTE calculations for �L , leading to significant variation of

Fig. 14 (a) Root-mean-square second-order IFCs as a function of the distance between two atoms. (b) Thermal conductivity of phosphorene as a function of cutoff radius used in third-order IFC calculations. Both panels are reproduced with permission [179]

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values of �L among studies using different setups of DFT and choice of cutoff radius for IFC calculations.

Xie et al. [180] studied the propagation and amplification of the errors from force constant calculations to the PBTE-based thermal conductivity predictions. The authors used the empirical Tersoff potential to calculate third-order IFCs with 16-digit precision. Note that the authors used empirical potentials instead of DFT, because the explicit analytical form of empirical potentials guarantees that the error only comes from numerical truncation. First, the IFCs obtained using the empirical potential are used to calculate the “actual” �L using PBTE as the benchmark value. Then, the authors intentionally introduced errors to the third-order IFCs by either truncating digits or adding random digits to the IFCs, thus producing “erroneous” �L’s. As shown in Fig. 15, when the error of IFCs lies in the typical error range for DFT-based force calculations (~ 10−5 to 10−3 eV Å−1), the calculated �L could be substantially different from the benchmark results. Moreover, the authors found that imposing translational invariance conditions, i.e., acoustic sum rule, on IFCs does not always improve the accuracy of �L , if the IFCs are not sufficiently accurate. The findings of this study thus reveal a great challenge facing the current first-principles PBTE approach: many DFT calculations may not have provided accurate enough IFCs to PBTE, and therefore, the predicted �L ’s might deviate from the accurate values substantially. This is even worse for first-principles PBTE calculations of materials with complicated unit cells, for which the high-computational cost of DFT usually restrict one to use a small cutoff radius for IFC calculations, a small cutoff energy for truncating plane waves, a coarse k-grid, etc. Moreover, as one can expect, the error could be even larger for higher-order force constants, which are deriva-tives of lower-order ones. This renders it even more challenging to predict accurate higher-order anharmonic phonon–phonon scattering rates.

Fig. 15 Error propagation and amplification for (a) silicon and (b) graphene by Xie et al. Benchmark thermal conductivity is obtained from IFCs calculated from the forces with 16-digit accuracy. “Trunca-tion” and “addition” in the legends mean the IFCs used for thermal conductivity calculation are trun-cated in digits or randomized in the last few digits to introduce random error, respectively. “Original” or “modified” mean the thermal conductivity values are obtained without or with applying the acoustic sum rule, respectively. The red, shaded region highlights the typical error range for forces obtained from DFT. Both panels are reproduced with permission [180]


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In practice, the errors in force constants predicted via DFT can be alleviated (though not completely removed) by enforcing crystal symmetries to the force con-stants, even though Xie et al. [180] found that the simple acoustic sum rule still cannot always guarantee a reasonable prediction if the raw force constants are not accurate enough. In a notable PBTE study by Lindsay et al. [133], three approaches for enforcing translational invariance (arising from crystal symmetry) conditions are tested, including the acoustic sum rule, �2 minimization, and Lagrange multipliers. As the acoustic sum rule does not enforce derivative premutation and point group symmetries of the system, it does not help improve the �L prediction as much as the other two approaches, which can guarantee all symmetry properties of the crys-tal while enforcing translational invariance. They have demonstrated that enforcing translational invariance to force constants leads to significantly improved (better match to experimental values) prediction of the �L of Si and Ge in their work [133]. Moreover, Bonini et al. [181] demonstrated the importance of enforcing rotational symmetry to force constants for predicting the �L of graphene, which should also be applicable to other 2D materials with rotational symmetry.

4 Outlook and Conclusion

First-principles-based thermal modeling is still a fast-growing area, thanks to the advent of faster computers and better algorithms, the advancement in our under-standing of heat transfer physics, and the increasing experimental data for validation. In particular, we expect to see more machine learning-based predictions of mate-rial properties, either by predicting the properties through machine learning directly or using this technique to accelerate conventional methods, e.g., DFT or molecular dynamics. Moreover, it would further enhance machine learning-based approaches by incorporating physics insights of specific materials systems or physical processes into machine learning models.

So far, most first-principles studies on thermal transport are for static/quasi-static systems in the reciprocal space, while a real-space-real-time first-principles simu-lation would be very useful for thermal designs, for instance, when one wants to find the locations of hot spots in a device. BTE could be a very useful tool in this sense, but the computational cost is formidably expensive for a spectral BTE simu-lation aimed at considering phonon scattering processes rigorously, i.e., by enforc-ing the selection rules as done in the k-space PBTE. Therefore, all the BTE studies reported so far have made a certain degree of simplifications to mitigate the high-computational cost of a full BTE solution. Moreover, BTE relies on predetermined mathematical expressions to describe carrier scatterings and drifting, which is less straightforward than molecular dynamics that can consider the transport, scatter-ing, rectification, and localization of phonons naturally [155, 157, 182–184]. With the advancement in hardware and algorithms for high-performance computing (and possibly the development of feasible quantum computing technologies), we might see more first-principles molecular dynamics (also referred to as ab initio molecular dynamics) simulations, in which forces on atoms are calculated from DFT on-the-fly. So far, very few studies [185–188] adopted this approach and the system sizes

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were very small, which is not practical for materials with long phonon mean-free-path or wavelength.

Furthermore, over the last decade, there has been a growing interest in the funda-mental physics of spin transport and spin–lattice interaction in materials [189–191], owing to its practical importance to emerging applications such as spin-thermoe-lectrics, spin-caloritronics, spintronics, and quantum computing. However, there are rather few developments [192] of first-principles methods for modeling spin–lattice and magnon–phonon interactions and their effect on thermal transport. The develop-ment of theories and computational approaches to rigorously model spin or magnon involved thermal transport processes will certainly benefit those burgeoning areas.

In summary, we have briefly discussed state-of-the-art first-principles approaches for modeling thermal transport in materials. We have provided a brief review of notable findings relevant to several important topics on thermal transport physics. Obviously, the predictive power of first-principles methods has proved to be very useful for discovering new materials and understanding thermal transport processes in existing materials, particularly those that are challenging to analyze experimen-tally. However, there are still challenges with the application and development of first-principles-based thermal modeling approaches, fundamentally buried in the inaccuracy of DFT-based force calculations, which will be propagated and even amplified in subsequent phonon/electron and thermal transport analyses.

Acknowledgments The authors would like to thank LC’s and YW’s faculty startup funds from the Uni-versity of Nevada, Reno. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund (60587-DNI10) for partial support of this research.

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