A unified understanding of minimum lattice thermal conductivity

Significance

Lattice thermal conductivity, a fundamental property of materials, plays a vital role in various technological applications, including thermal energy conversion and management. Understanding its lower limit, known as the minimum thermal conductivity (${\kappa }_{\mathrm{L}}^{\min }$), is crucial but challenging, especially considering that the conventional phonon gas picture can break down in this limit. Here we propose a ${\kappa }_{\mathrm{L}}^{\min }$ model that unifies different kinds of heat carriers and apply it to thousands of inorganic compounds. We find a universal lower bound to ${\kappa }_{\mathrm{L}}^{\min }$ and unveil deep insights into its connection with the famous Cahill–Watson–Pohl model. Furthermore, we construct advanced machine learning models to enable accurate prediction of ${\kappa }_{\mathrm{L}}^{\min }$ on a large scale, thus opening the pathway for rational design of materials with ${\kappa }_{\mathrm{L}}^{\min }$.

Abstract

We propose a first-principles model of minimum lattice thermal conductivity (${\kappa }_{\mathrm{L}}^{\min }$) based on a unified theoretical treatment of thermal transport in crystals and glasses. We apply this model to thousands of inorganic compounds and find a universal behavior of ${\kappa }_{\mathrm{L}}^{\min }$ in crystals in the high-temperature limit: The isotropically averaged ${\kappa }_{\mathrm{L}}^{\min }$ is independent of structural complexity and bounded within a range from ∼0.1 to ∼2.6 W/(m K), in striking contrast to the conventional phonon gas model which predicts no lower bound. We unveil the underlying physics by showing that for a given parent compound, ${\kappa }_{\mathrm{L}}^{\min }$ is bounded from below by a value that is approximately insensitive to disorder, but the relative importance of different heat transport channels (phonon gas versus diffuson) depends strongly on the degree of disorder. Moreover, we propose that the diffuson-dominated ${\kappa }_{\mathrm{L}}^{\min }$ in complex and disordered compounds might be effectively approximated by the phonon gas model for an ordered compound by averaging out disorder and applying phonon unfolding. With these insights, we further bridge the knowledge gap between our model and the well-known Cahill–Watson–Pohl (CWP) model, rationalizing the successes and limitations of the CWP model in the absence of heat transfer mediated by diffusons. Finally, we construct graph network and random forest machine learning models to extend our predictions to all compounds within the Inorganic Crystal Structure Database (ICSD), which were validated against thermoelectric materials possessing experimentally measured ultralow κ_L. Our work offers a unified understanding of ${\kappa }_{\mathrm{L}}^{\min }$, which can guide the rational engineering of materials to achieve ${\kappa }_{\mathrm{L}}^{\min }$.

Knowing the lower limit to the lattice thermal conductivity of crystals is of fundamental interest and technological importance, particularly relevant to thermal energy conversion and management applications (1). The minimum lattice thermal conductivity (${\kappa }_{\mathrm{L}}^{\min }$), a concept first proposed by Slack (2), sets the upper limit of thermoelectric conversion efficiency for a given electronic transport profile (3–6) and provides the maximum thermal insulation for a substrate used in thermal barrier coatings (7, 8). The fundamental physics of heat transport in crystals at the lower limit of thermal conductivity continues to be an active research topic (2, 7, 9, 10).

Motivated by the observation that the experimentally accessible ${\kappa }_{\mathrm{L}}^{\min }$ is often found in amorphous solids, the initial attempts to understand ${\kappa }_{\mathrm{L}}^{\min }$ (11, 12) relied on a model proposed by Einstein (13), who assumed that the mechanism of heat transport in crystals was a random walk of the thermal energy between neighboring atoms vibrating with random phases. Einstein’s random walk model was later generalized by Cahill, Watson, and Pohl (9) (referred to as the CWP model) to incorporate the Debye model of vibrations by adopting a wavelength-dependent mean free path (MFP). An earlier approach by Slack (2) as well as the more recent study by Clarke (7) also assumed that phonons are the dominant heat carriers but used different assumptions about the phonon MFP. The CWP, Slack, and Clarke theories of ${\kappa }_{\mathrm{L}}^{\min }$ are all based on the conventional phonon gas model (PGM) and the Boltzmann gas kinetic equation (14), where the primary heat carriers are propagating phonons (10). On the other hand, recognizing the potential failure of the PGM in the disordered regime, Allen and Feldman (15, 16) developed the diffuson theory of heat transport in disordered solids. The Allen–Feldman theory is based on the Kubo–Greenwood formula in the harmonic approximation and the primary heat carriers are diffusons, which are described by the off-diagonal terms of the heat flux operator. Recently, a phenomenological ${\kappa }_{\mathrm{L}}^{\min }$ model was developed by Agne, Hanus, and Snyder (10) based on the diffuson theory, aiming to resolve the overestimation of ${\kappa }_{\mathrm{L}}^{\min }$ from the CWP model in comparison to experiments.

Despite significant advances, a comprehensive understanding of ${\kappa }_{\mathrm{L}}^{\min }$ is still elusive. A crucial yet missing piece of the puzzle is a first-principles–based theory that accurately describes ${\kappa }_{\mathrm{L}}^{\min }$ over the whole spectrum of materials between the prototypical classes of simple crystals and disordered amorphous solids. The development of such an advanced model would shed light on crucial open questions. For instance, the heat transfer pathways mediated by localized diffusons and propagating phonons are often classified into independent channels (17–21). Is there a unified physical picture of the two different kinds of heat carriers when approaching ${\kappa }_{\mathrm{L}}^{\min }$? Why does the CWP model work remarkably well for solids with strong disorder, despite retaining the PGM nature and neglecting all optical phonons (9, 11, 12)? There are recent experimental discoveries of crystalline compounds with ultralow and glass-like lattice thermal conductivity κ_L that resembles ${\kappa }_{\mathrm{L}}^{\min }$ (22–26). What is the general behavior of ${\kappa }_{\mathrm{L}}^{\min }$ in crystals, i.e., how does it vary with structural complexity and atomic composition? Ultimately, is there a physical lower or upper bound to ${\kappa }_{\mathrm{L}}^{\min }$?

To answer these questions, we propose a first-principles model of ${\kappa }_{\mathrm{L}}^{\min }$ that is based on the recently developed unified theory of thermal transport in crystals and glasses (27). By taking advantage of existing databases for phonons and our own first-principles density functional theory (DFT) calculations, we compute and chart ${\kappa }_{\mathrm{L}}^{\min }$ of 2,576 inorganic compounds in the high-temperature limit, covering wide ranges of chemical compositions, structural complexity, and space-group symmetries. Our numerical results suggest the existence of a lower bound to the isotropically averaged ${\kappa }_{\mathrm{L}}^{\min }$ around 0.1 W/(m K) in the high-temperature limit (600 K and above) that is independent of structural complexity, in striking contrast to the predictions of PGM-based theories. We explain this unusual behavior by demonstrating that the total ${\kappa }_{\mathrm{L}}^{\min }$ is relatively insensitive to various kinds of atomic and structural disorder that lead to increased structural complexity, whereas the latter can significantly affect the relative importance of different heat transport channels (phonon gas versus diffuson). With these insights, we rationalize the CWP model against the conventional PGM by pointing out that the former can in fact approximately account for contributions to heat transfer from diffusons when approaching ${\kappa }_{\mathrm{L}}^{\min }$. This finding further inspires us to provide a better analytical approximation to ${\kappa }_{\mathrm{L}}^{\min }$. Finally, we construct machine learning models amenable to large-scale predictions, which are validated against thermoelectric materials with ultralow and glasslike lattice thermal conductivity, as well as amorphous solids.

Results and Discussion

Recent theoretical advances toward a unified theory of thermal transport in crystals and glasses have enabled a consistent treatment of different heat carriers such as propagating phonons and localized diffusons (27, 29). Herein, we construct a ${\kappa }_{\mathrm{L}}^{\min }$ model based on the unified theory of Simoncelli, Marzari, and Mauri (27, 30) by adopting the approximation that each phonon mode’s lifetime is equal to one-half of its vibrational period, τ_q^s = π/ω_q^s. This assumption follows the original proposal by Einstein (13) which was later adopted by the CWP model (9). We choose not to use the minimum MFP criteria (e.g., by assuming that MFP is equal to the smallest atomic spacing) because for optical phonon modes with vanishing group velocities, it will lead to infinite phonon lifetimes, which is unphysical and tends to diminish their off-diagonal contribution to thermal conductivity (see below). Using the formula for thermal conductivity from ref. 27, we arrive at the following expression:

$$\begin{array}{l}{\kappa }_{\text{L}}^{\min }=\frac{\pi {ħ}^2}{k_{\text{B}}{T}^{2}V{N}_q}{\displaystyle \sum _q{\displaystyle \sum _{s,s^{\prime }}\frac{{({\omega }_q^s+{\omega }_q^{s^{\prime }})}^2}{2}{v}_q^{s,s^{\prime }}\otimes v_q^{s^{\prime },s}}}\\ ·\frac{{\omega }_q^s{n}_q^s(n_q^s+1)+{\omega }_q^{s^{\prime }}{n}_q^{s^{\prime }}(n_q^{s^{\prime }}+1)}{4{\pi }^2{({\omega }_q^{s^{\prime }}-{\omega }_q^s)}^2+{({\omega }_q^s+{\omega }_q^{s^{\prime }})}^2}\end{array}$$ [1]

where ℏ, k_B, T, V, and N_q are, respectively, the reduced Planck constant, the Boltzmann constant, the absolute temperature, the volume of the unit cell, and the total number of sampled phonon wave vectors. Phonon modes are denoted by the wave vector q and mode index s. The key quantities entering Eq. 1 are phonon mode-resolved frequencies ω_q^s, population numbers n_q^s, and the generalized group velocity tensors v_q^{s, s′}, with the latter calculated as (16, 27)

$$\begin{array}{l}{v}_q^{s,s^{\prime }}=\frac{i}{{\omega }_q^s+{\omega }_q^{s^{\prime }}}{\displaystyle \sum _{\alpha ,\beta }{\displaystyle \sum _{m,p,q}{e}_q^s(\alpha ,p)D_{\beta \alpha }^{pq}(0,m)}}\\ \times (R_m+R_{pq})e^{iq·R_m}{e}_q^{s^{\prime }}(\beta ,q).\end{array}$$ [2]

Here, e, D, and R denote the polarization vector, the dynamical matrix, and the lattice vector, respectively. α/β, m, and p/q are indices labeling the Cartesian coordinate, the unit cell, and the atoms within the unit cell.

Physically, we can further decompose ${\kappa }_{\mathrm{L}}^{\min }$ into two parts: the diagonal part with s = s′, which corresponds to the PGM, denoted as PGM-${\kappa }_{\mathrm{L}}^{\min }$, and the off-diagonal part, which accounts for the diffuson channel, denoted as OD-${\kappa }_{\mathrm{L}}^{\min }$. The total ${\kappa }_{\mathrm{L}}^{\min }$ will be referred to as PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$. Immediately, we see that the evaluation of Eq. 1 only requires the knowledge of the harmonic phonon dispersion, making it amenable to large-scale DFT calculations. We applied Eq. 1 to a selected set of 2,576 inorganic compounds, taking advantage of the tabulated harmonic force constants within the phonon database (phononDB) generated by Togo et al. (31, 32) using crystalline structures from the Materials Project (MP) (33–35).

Temperature enters Eq. 1 both explicitly in the prefactor and implicitly via the phonon occupation numbers n_q^s. Because ${\kappa }_{\mathrm{L}}^{\min }$ increases monotonically with increasing temperature up to the Debye temperature as a result of enhanced heat capacity (9), and ${\kappa }_{\mathrm{L}}^{\min }$ in crystals is usually approached at high temperatures, we focus on exploring the high-temperature regime. We performed calculations at T = 300, 600, and 900 K. We find that ${\kappa }_{\mathrm{L}}^{\min }$ of most compounds calculated at 600 K exhibit considerable increment over those calculated at 300 K, whereas ${\kappa }_{\mathrm{L}}^{\min }$ varies by less than 5% between T = 600 and 900 K for approximately 95% of the studied compounds, with the largest decrease being only 14%. Considering the larger availability of experimental measurements of κ_L at 600 K than 900 K, we will focus on the T = 600 K results in the following discussion. For anisotropic crystals, the components of ${\kappa }_{\mathrm{L}}^{\min }$ are further averaged along the three principal crystallographic axes. We refer the readers to Materials and Methods for a more detailed discussion of the structure types, chemical compositions, and criteria for numerically converging ${\kappa }_{\mathrm{L}}^{\min }$.

The calculated values of PGM-${\kappa }_{\mathrm{L}}^{\min }$, OD-${\kappa }_{\mathrm{L}}^{\min }$, and PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ as functions of the number of atomic sites in the primitive cell, N_a, are plotted in Fig. 1. We associate the value of N_a with the degree of structural complexity of a compound. We see from Fig. 1A that PGM-${\kappa }_{\mathrm{L}}^{\min }$ covers a large range of values, spanning from approximately 0.01 to 1 W/(m K) in the high-temperature limit (600 K and above). There is a clear trend that PGM-${\kappa }_{\mathrm{L}}^{\min }$ decreases monotonically with increasing N_a, which can be clearly seen from the log-log plot in the Inset of Fig. 1A). In contrast, the total PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ in Fig. 1B exhibits a much narrower spread from approximately 0.1 to 2.6 W/(m K), independent of N_a. The lower [0.1 W/(m K)] and the upper [2.6 W/(m K)] bounds were determined by analyzing the frequency distribution histogram of all calculated ${\kappa }_{\mathrm{L}}^{\min }$, as detailed in SI Appendix. There are only four compounds within our calculations having values of PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ outside these bounds. The largest deviation from the lower bound is found in crystalline P6(3)/mmc-Ar, which has a value of 0.043 W/(m K), and all the other compounds have values larger than 0.081 W/(m K). Meanwhile, for the upper bound, the largest PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ is associated with P4(2)/mnm-TiO₂, with a value of 2.64 W/(m K). The comparison of PGM-${\kappa }_{\mathrm{L}}^{\min }$ and PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ reveals the increasing importance of the off-diagonal diffuson term OD-${\kappa }_{\mathrm{L}}^{\min }$ with increasing structural complexity N_a. This is further illustrated by the ratio of OD-${\kappa }_{\mathrm{L}}^{\min }$ to PGM-${\kappa }_{\mathrm{L}}^{\min }$ in Fig. 1C. We see that OD-${\kappa }_{\mathrm{L}}^{\min }$ quickly surpasses PGM-${\kappa }_{\mathrm{L}}^{\min }$ above N_a ∼ 10 and OD-${\kappa }_{\mathrm{L}}^{\min }$ becomes an order of magnitude higher than PGM-${\kappa }_{\mathrm{L}}^{\min }$ above N_a ∼ 60. We also observe a group of compounds (e.g., P3m1-ZnS) with ratios of OD-${\kappa }_{\mathrm{L}}^{\min }$ to PGM-${\kappa }_{\mathrm{L}}^{\min }$ significantly smaller than the majority trendline. Our analysis of their structures reveals that these compounds display extreme crystalline anisotropy, i.e., the lattice constant of the primitive cell along one axis is exceedingly large, and thus, N_a is large. However, for these compounds, the ratios of OD-${\kappa }_{\mathrm{L}}^{\min }$ to PGM-${\kappa }_{\mathrm{L}}^{\min }$ are reduced along the other two axes with much smaller lattice constants, thus giving rise to overall reduced ratios after averaging over the three spatial directions.

Fig. 1.

Contributions to minimum lattice thermal conductivity from the phonon gas model and the off-diagonal terms based on Eq. 1. (A) Calculated minimum lattice thermal conductivity (${\kappa }_{\mathrm{L}}^{\min }$) as a function of the number of atomic sites (N_a) in a primitive cell using the phonon gas model (denoted as PGM-${\kappa }_{\mathrm{L}}^{\min }$), i.e., the diagonal part of Eq. 1. The Inset is plotted with logarithmic scales and the dashed line is plotted following the scaling of $N_{\mathrm{a}}^{-2/3}$ obtained from Slack’s model (28), showing the monotonic decay of PGM-${\kappa }_{\mathrm{L}}^{\min }$ with increasing N_a. (B) The same as (A) but with additional red disks showing the calculated ${\kappa }_{\mathrm{L}}^{\min }$ accounting for both the PGM and off-diagonal (OD) contributions, denoted as PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$. The gray dashed lines indicate values of 0.1 and 2.6 W/(m K) for the lower and the upper bounds of PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$, respectively. (C) The ratio of OD-${\kappa }_{\mathrm{L}}^{\min }$ and PGM-${\kappa }_{\mathrm{L}}^{\min }$ as a function of N_a. The Inset is displayed with logarithmic scales, showing the increasing trend of OD-${\kappa }_{\mathrm{L}}^{\min }$/PGM-${\kappa }_{\mathrm{L}}^{\min }$. All results were obtained at 600 K.

The monotonic decay of PGM-${\kappa }_{\mathrm{L}}^{\min }$ as a function of N_a is not unexpected. As already reflected in the Slack model (28, 36), κ_L based on the PGM is proportional to $N_{\mathrm{a}}^{-2/3}$ provided that acoustic phonon modes with high group velocities dominate κ_L. The monotonic decay of PGM-${\kappa }_{\mathrm{L}}^{\min }$ can be attributed to the fact that the increase in N_a leads to a reduced Brillouin zone that effectively folds back the high-energy acoustic modes into the first Brillouin zone as optical modes, resulting in both suppressed spectral weight and reduced group velocity. It is worth noting that strategies based on the above argument have been successfully used to search for complex materials with intrinsically low lattice thermal conductivity (26, 37). However, our results show that these strategies may be no longer effective when phonons are strongly scattered by anharmonicity or disorder when the heat transport is dominated by the OD terms beyond the PGM. This is especially true for systems with κ_L approaching ${\kappa }_{\mathrm{L}}^{\min }$. In this scenario, the breakdown of the PGM is due to the crossover in the heat transport from propagating phonons to localized diffusons, as initially proposed by Allen and Feldman (15, 16). As a result, the total PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ provides a much more reasonable estimation of ${\kappa }_{\mathrm{L}}^{\min }$ compared to the phonon term PGM-${\kappa }_{\mathrm{L}}^{\min }$ alone. Importantly, despite such a crossover from PGM-${\kappa }_{\mathrm{L}}^{\min }$ to OD-${\kappa }_{\mathrm{L}}^{\min }$, the bounds of the total PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ remain independent of N_a. This suggests an effective interconversion between the two fundamentally different heat transfer channels, which might be induced by variations in chemical composition, atomic disorder, lattice distortion, structural complexity (N_a), and other factors. This invites the following two questions: i) What are the key factors that determine the interconversion between PGM-${\kappa }_{\mathrm{L}}^{\min }$ and OD-${\kappa }_{\mathrm{L}}^{\min }$ across various compounds? and ii) Is it possible to revert such a interconversion, for example, via approximating PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ of compounds with large N_a using only PGM-${\kappa }_{\mathrm{L}}^{\min }$?

Answering the above two questions will help establish a unified understanding of ${\kappa }_{\mathrm{L}}^{\min }$. The challenge arises from the fact that the crossover in the heat transport from phonons to diffusons is observed in a large set of compounds with diverse characteristics. To reveal and disentangle the impacts of chemical composition, interatomic interaction, and structural complexity on ${\kappa }_{\mathrm{L}}^{\min }$, we use a simple elemental structure, i.e., face-centered cubic aluminium (fcc-Al), as a base model and then manually generate perturbations on this model to mimic the variations in structures and chemistries among different compounds. Specifically, random perturbations are applied on the atomic mass, interatomic force constants (IFCs), and atomic position. We also vary the number of atoms (N_a) in the unit cell by repeating the primitive cell of fcc-Al. As a consequence, we can use these models to approximate a variety of situations ranging from ordered to disordered by varying the strength of perturbations and N_a. Note that we strictly enforce physical constraints when applying these perturbations, such as the translational invariance of the crystal.

Fig. 2A summarizes the calculated total ${\kappa }_{\mathrm{L}}^{\min }$ and its decompositions (PGM and OD) when weak perturbations are applied, respectively, to atomic masses, IFCs, and atomic positions in unit cells with increasing N_a. First, we observe that neither the total ${\kappa }_{\mathrm{L}}^{\min }$ nor its decomposition change by simply increasing N_a without introducing disorder. This is as expected because nothing has been changed physically and merely a larger unit cell is used as the primitive cell for the perfect fcc-Al crystal. In contrast, when small perturbations are applied, we notice different trends in PGM-${\kappa }_{\mathrm{L}}^{\min }$ and OD-${\kappa }_{\mathrm{L}}^{\min }$ across the unit cells with increased N_a: i) When N_a = 1, perturbations lead to only small changes in both PGM-${\kappa }_{\mathrm{L}}^{\min }$ and OD-${\kappa }_{\mathrm{L}}^{\min }$, and the PGM-${\kappa }_{\mathrm{L}}^{\min }$ dominates over the OD-${\kappa }_{\mathrm{L}}^{\min }$, as expected for a simple crystal; ii) when N_a = 8, perturbations start to convert PGM-${\kappa }_{\mathrm{L}}^{\min }$ to OD-${\kappa }_{\mathrm{L}}^{\min }$, with the former still larger than the latter; and iii) when N_a=64, the values of PGM-${\kappa }_{\mathrm{L}}^{\min }$ and OD-${\kappa }_{\mathrm{L}}^{\min }$ are nearly exchanged, and OD-${\kappa }_{\mathrm{L}}^{\min }$ becomes the major contribution to the total ${\kappa }_{\mathrm{L}}^{\min }$. Surprisingly, the total ${\kappa }_{\mathrm{L}}^{\min }$ largely remains constant with both varying perturbations and N_a.

Fig. 2.

Impacts of disorder and structural complexity on minimum lattice thermal conductivity. (A) Calculated total ${\kappa }_{\mathrm{L}}^{\min }$ (PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$) and the corresponding decomposed contributions (PGM-${\kappa }_{\mathrm{L}}^{\min }$ and OD-${\kappa }_{\mathrm{L}}^{\min }$) of a model system (fcc-Al) as functions of the number of atomic sites (N_a) in the designed primitive cell and the presence of various kinds of disorder. We artificially create three kinds of disorder in the perfect crystal of fcc-Al by applying small random perturbations (within 5% of their original values) to atomic mass, interatomic force constants (IFCs), and atomic position (POS), denoted in blue, orange, and yellow, respectively (with dashed line as a guide to the eye). (B) Calculated PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ (green disks), PGM-${\kappa }_{\mathrm{L}}^{\min }$ (blue triangles), and OD-${\kappa }_{\mathrm{L}}^{\min }$ (orange squares) of fcc-Al as a function of the strength of mass disorder, realized by applying random percentagewise mass perturbation. For example, 20% random mass perturbation means a random mass ranging from −20% to 20% of the original atomic mass of the element is added to each atom. The left and right panel show the results obtained for an 8-atom and 64-atom primitive cell, respectively. The shaded areas indicate the uncertainty obtained by averaging many independent random mass perturbations. (C) Phonon dispersions of the perfect fcc-Al (Left) and an 8-atom primitive cell with 20% random mass perturbations (Right). (D) Phonon group velocities of the perfect fcc-Al (Upper) and an 8-atom primitive cell with 20% random mass perturbations (Lower). The ${\kappa }_{\mathrm{L}}^{\min }$ shown in panels A and B were all calculated at 300 K.

The above results reveal that the total ${\kappa }_{\mathrm{L}}^{\min }$ is not sensitive to small perturbations of atomic masses, positions, and IFCs of a given base crystal structure. In the other words, the capability of lattice heat transfer of a given system that approaches maximum phonon scattering (τ_q^s ≈ π/ω_q^s) can neither be reduced nor enhanced significantly with small adjustments in atomic compositions, crystal structure, and interatomic interactions. This is in contrast to the case of weak phonon scattering in some materials (e.g., diamond and BN), wherein small mass perturbations such as isotope scattering can lead to a considerable reduction in κ_L (38). Conversely, the relative contributions of PGM and OD terms are very sensitive to small perturbations, especially for increasingly large unit cells. The significantly decreased PGM contribution can be mostly attributed to the reduced (diagonal) phonon group velocities. As shown in Fig. 2 C and D, the introduction of mass disorder strongly breaks the energy degeneracy of phonon bands and suppresses the (diagonal) group velocities, whereas phonon frequencies change by much less. This results in an increased OD contribution due to the lifting of degeneracy, which makes the otherwise vanishing off-diagonal velocities appreciable (39). We can also infer from Fig. 2A that the PGM-${\kappa }_{\mathrm{L}}^{\min }$ tends toward zero when N_a approaches infinity while OD-${\kappa }_{\mathrm{L}}^{\min }$ approaches the total ${\kappa }_{\mathrm{L}}^{\min }$. Interestingly, such a disorder-induced interconversion between PGM-${\kappa }_{\mathrm{L}}^{\min }$ and OD-${\kappa }_{\mathrm{L}}^{\min }$, when approached from an inverse perspective, might be leveraged to estimate the total ${\kappa }_{\mathrm{L}}^{\min }$ without explicitly computing the OD contributions. That is, we may approximate the total ${\kappa }_{\mathrm{L}}^{\min }$ of some very complex materials (large N_a) wherein OD-${\kappa }_{\mathrm{L}}^{\min }$ dominates by computing only the PGM-${\kappa }_{\mathrm{L}}^{\min }$ of a simplified model (e.g., N_a = 1). The latter could be obtained by averaging the atomic masses, positions, and IFCs of the complex structure. ${\kappa }_{\mathrm{L}}^{\min }$ based on such an approximation can be easily calculated using macroscopic properties such as materials’ density and elastic properties.

We have so far qualitatively answered the two questions raised earlier by means of showing the impacts of weak disorder and structural complexity on ${\kappa }_{\mathrm{L}}^{\min }$ in an idealized model. However, the assumption of weak disorder makes the above picture applicable to only a small group of compounds which share similar structures, chemical compositions, and interatomic interactions. To better describe complex materials with large structural and compositional fluctuations, we move on to investigate models with increasingly strong disorder. We show in Fig. 2B the calculated total and decomposed ${\kappa }_{\mathrm{L}}^{\min }$ as a function of the strength of random mass disorder (our tests show that perturbing IFCs have similar effects). We see from Fig. 2B that PGM-${\kappa }_{\mathrm{L}}^{\min }$ decreases sharply with the onset of the disorder and then continues to decrease in a slower manner with increasing disorder. In contrast, OD-${\kappa }_{\mathrm{L}}^{\min }$ increases first and then decreases with enhanced disorder, while its percent contribution keeps increasing (see SI Appendix). The resulting total ${\kappa }_{\mathrm{L}}^{\min }$ displays an overall reduced value. By comparing the left and right panels in Fig. 2B, we also observe that a larger unit cell with similar strength of disorder tends to shift the maximum of OD-${\kappa }_{\mathrm{L}}^{\min }$ toward the weaker disorder regime and make OD-${\kappa }_{\mathrm{L}}^{\min }$ more important.

Overall, we see that, in the full range of disorder from weak to strong, the total ${\kappa }_{\mathrm{L}}^{\min }$ of complex unit cells might be approximated by computing only PGM-${\kappa }_{\mathrm{L}}^{\min }$ for the perfectly ordered crystal, despite the latter displaying certain underestimation. Surprisingly, the presence of strong disorder might accidentally lead to a better agreement between the two, as can be inferred by comparing OD-${\kappa }_{\mathrm{L}}^{\min }$ without any disorder and the total ${\kappa }_{\mathrm{L}}^{\min }$ with strong disorder in Fig. 2B. These results help explain why we find that the total PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ in Fig. 1B lies in a relatively narrow range, which is bounded both from above and below over a variety of compounds. It can be interpreted as follows: i) Complex compounds are nothing but variants of simple compounds with additional compositional and structural disorder; ii) despite potentially dissimilar amounts of structural complexity, different compounds share close values of ${\kappa }_{\mathrm{L}}^{\min }$ if their averaged structural properties, such as mass, position, and interatomic interaction, are similar; and iii) it is likely that the averaged structural properties depend weakly on structural complexity for compounds with similar chemistries.

We apply these insights to bridge the knowledge gap between our ${\kappa }_{\mathrm{L}}^{\min }$ model and the CWP model. This is motivated by the missing connection between the CWP model and Allen and Feldman’s theory of diffusons, a question initially raised by Cahill and Pohl (12). Considering the PGM nature of the CWP model, we first compare the PGM-${\kappa }_{\mathrm{L}}^{\min }$ calculated using our model to those from the CWP model (denoted as CWP-${\kappa }_{\mathrm{L}}^{\min }$). To eliminate the uncertainty caused by an anisotropic average of ${\kappa }_{\mathrm{L}}^{\min }$, we only selected compounds with cubic symmetry, as shown in Fig. 3A. In contrast to the expected failure of the CWP model due to the lack of OD contribution, we find that CWP-${\kappa }_{\mathrm{L}}^{\min }$ deviates significantly from PGM-${\kappa }_{\mathrm{L}}^{\min }$ and displays much larger values, which seems to overcome the underestimation inherent in PGM-${\kappa }_{\mathrm{L}}^{\min }$. The latter is further confirmed by plotting CWP-${\kappa }_{\mathrm{L}}^{\min }$ against PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ in Fig. 3B. Overall, we find that CWP-${\kappa }_{\mathrm{L}}^{\min }$ is comparable to PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$, although the latter tends to show smaller values statistically.

Fig. 3.

Comparisons of the minimum lattice thermal conductivity based on Eq. 1 with the CWP model for cubic crystals at 600 K. (A) Comparisons of the phonon gas model of ${\kappa }_{\mathrm{L}}^{\min }$ (PGM-${\kappa }_{\mathrm{L}}^{\min }$) with the CWP model (CWP-${\kappa }_{\mathrm{L}}^{\min }$). (B) Comparisons of the unified minimum thermal transport model that combines both phonon gas model and the off-diagonal terms (PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$) with the CWP model. (C) The linear correlation of the PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ with an approximated formula of the CWP model in the high-temperature limit, i.e., 0.5k_Bn^2/3(Y/ρ)^1/2, wherein n, Y, and ρ are the number density, Young’s modulus, and mass density, respectively. The dashed lines in (A–C) denote the deviations from the diagonal within a factor of two.

The above comparison reveals that the CWP model seems to work remarkably well, which is quite surprising because i) the CWP model is relatively simple and only needs sound velocity and number density as inputs (see Eq. 3 in ref. 9) and ii) the CWP model does not at all account for the OD contribution, which is dominated by optical phonon modes (27). We attribute the success of the CWP model to the fact that the CWP model could be a good approximation to the sophisticated PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ model based on Eq. 1 by averaging out disorder and phonon unfolding, in the same spirit of our model detailed earlier. This is achieved by neglecting the details of both chemical composition and atomic arrangement using a single parameter, i.e., the number density n, to describe the structure in the CWP model. In such a simplified picture with only one atom in an averaged unit cell, only three acoustic branches arise, which induces an effective back conversion from the OD contribution to the PGM contribution, thus making the PGM dominant again. This picture is supported by Fig. 1C, which shows that PGM-${\kappa }_{\mathrm{L}}^{\min }$ tends to dominate over OD-${\kappa }_{\mathrm{L}}^{\min }$ in simple crystals. Importantly, the above analysis implies that it is inappropriate to use the CWP model as a second heat transport channel on top of the PGM model as adopted in recent studies (18, 40), which will result in a double-counting of the PGM contribution.

The interpretation of the CWP model based on the structure averaging and phonon unfolding picture offers additional insights into its potential limitations or uncertainties. On the one hand, neglecting structural details might lead to a lack of additional flattening of phonon dispersions caused by disorder or symmetry breaking, thus giving rise to an overestimated ${\kappa }_{\mathrm{L}}^{\min }$. On the other hand, OD contributions from the three acoustic branches are still missing, which leads to a general underestimation of ${\kappa }_{\mathrm{L}}^{\min }$. Considering these two competing factors, there is no clear answer on the net effect. However, through our numerical experiment presented in Fig. 3B, we see that CWP-${\kappa }_{\mathrm{L}}^{\min }$ is likely to overestimate ${\kappa }_{\mathrm{L}}^{\min }$, thus indicating that the dominant uncertainty probably comes from the first factor. This might explain why experiments tend to find lower lattice thermal conductivities than the values predicted by the CWP model (10).

To establish a quantitative measure of the extent that the CWP model overestimates ${\kappa }_{\mathrm{L}}^{\min }$ compared to our model, in Fig. 3C, we numerically fit our calculated PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ based on the relation of ${\kappa }_{\mathrm{L}}^{\min }\propto k_{\mathrm{B}}{n}^{2/3}{(Y/\rho )}^{1/2}$ suggested by Clarke (7), wherein n is the number density, Y is Young’s modulus, and ρ is the mass density. We find that PGM+OD-${\kappa }_{\mathrm{L}}^{\min }\approx 0.5k_{\mathrm{B}}{n}^{2/3}{(Y/\rho )}^{1/2}$, which is considerably smaller than that from the CWP model in the high-temperature limit (9), i.e., ${\kappa }_{\mathrm{L}}^{\min }\approx 1.1k_{\mathrm{B}}{n}^{2/3}{(Y/\rho )}^{1/2}$ if sound velocity is approximated as v_s = (2v_T + v_L)/3 ≈ 0.94(Y/ρ)^1/2 (7, 10). Our analysis unambiguously demonstrates that the CWP model tends to overestimate ${\kappa }_{\mathrm{L}}^{\min }$ intrinsically, thus offering a different perspective from the proposal by Agne et al. (10), who propose to mitigate such an overestimation by resorting to an alternative heat transfer through diffusons. We note that our conclusion only applies to the general behavior of the CWP model but may not work for a specific compound, as indicated by the large spread of data points along the diagonal in Fig. 3C. Furthermore, we see from the relation of ${\kappa }_{\mathrm{L}}^{\min }\propto k_{\mathrm{B}}{n}^{2/3}{(Y/\rho )}^{1/2}$ that the bound to ${\kappa }_{\mathrm{L}}^{\min }$ of crystals can be attributed to the competing parameters of n, Y, and ρ, all of which might have physical bounds for crystals.

With our ${\kappa }_{\mathrm{L}}^{\min }$ model established, we wanted to investigate the distribution of ${\kappa }_{\mathrm{L}}^{\min }$ across all experimentally known compounds in the Inorganic Crystal Structure Database (ICSD) (45). Due to the high computational cost of calculating phonon properties from first principles, it would be unfeasible to calculate them all, making it necessary to use a more efficient model. We decided on two complementary approaches to machine-learn ${\kappa }_{\mathrm{L}}^{\min }$: MatErials Graph Network (MEGNet) (46)—a state-of-the-art crystal graph convolutional neural network—and a random forest—a more interpretable approach (as seen in Fig. 4A). MEGNet directly uses crystal structure information to construct crystal graphs as features. In contrast, the random forest requires featurization. For the random forest, we used a variety of features generated by Matminer (47) using the Magpie (48) preset including the mean, standard deviation, minimum, maximum, mode, and range of elemental properties such as atomic number (N), mass (m), electronegativity (EN), Mendeleev number (N_m), melting temperature (T_m), column, row, covalent radius (R_covalent), number of electrons and unfilled slots in the s, p, d, and f valence orbitals, bandgap (E_gGS), magnetic moment (M_GS), and volume per atom in the ground state (V_GS). We also included some simple structural features such as the space group, and the mean and standard deviations of both bond length (L_B) and angle (θ_B), totaling 141 features for each compound. We refer the readers to Materials and Methods for more details on training machine learning models.

Fig. 4.

Machine learning models of minimum lattice thermal conductivity. (A) Depiction of the two machine learning models, and the error on the test set for (B) MEGNet and (C) the random forest. (D) Feature importance from the random forest. (E) MEGNet latent graph features reduced to two dimensions through t-SNE. Black points show the training set and other points show the ICSD dataset, colored by their estimated ${\kappa }_{\mathrm{L}}^{\min }$. (F) Kernel density estimate of the MAE distribution for ${\kappa }_{\mathrm{L}}^{\min }$ for both models on the test set. (G) Kernel density estimate of ${\kappa }_{\mathrm{L}}^{\min }$ for both models on the ICSD dataset. (H) ${\kappa }_{\mathrm{L}}^{\min }$ of seven selected compounds from experiments [TlInTe₂ at 600 K (41), Tl₃VSe₄ at 300 K (18), CsAg₅Te₃ at 600 K (25), Cu₂Se at 600 K (23), AgTlTe at 600 K (22), BiCuSeO at 600 K (42), Cu₃SbSe₃ at 600 K (43), and Sr₈Ga₁₆Ge₃₀ at 300 K (44)], direct calculation using our PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ model, MEGNet, and the random forest at the corresponding temperatures. All ${\kappa }_{\mathrm{L}}^{\min }$ were isotropically averaged.

The two models achieve similar accuracies on our test set, with MEGNet slightly outperforming the random forest (Fig. 4B, C, and F), and both also predict similar distributions for ${\kappa }_{\mathrm{L}}^{\min }$ on the ICSD data set (Fig. 4G). The distribution is bounded on the low end near 0.1 W/(m K) and a maximum around 4 W/(m K). Only ∼100 of the ∼36,000 structures show MEGNet-predicted ${\kappa }_{\mathrm{L}}^{\min }$ values above 2.64 W/(m K) (the maximum observed in the calculated training set) and are almost entirely various allotropes of Be, C, and Co or carbides and nitrides, all having strong bonding and relatively low densities. Part of the difference in predicted values between the models arises from a limitation of the random forest, as it is restricted to interpolating between the closest data points in our training set, preventing it from predicting smaller or larger values than those it was trained on. On the other hand, MEGNet is capable of extrapolation and finds more compounds with higher values of ${\kappa }_{\mathrm{L}}^{\min }$ when compared to the random forest and finds a smoother distribution for intermediate values, as shown in Fig. 4G.

The random forest model provides advantages in the form of faster computation, interpretable feature importance, and easily visualized features. From the random forest, we extracted the 15 most important features and have plotted them in Fig. 4D. Our most information-dense features are related to bonding strength (L_B, θ_B, EN, N_valence) and factors concerning the atomic properties (m, N, ρ, R_covalent, V_GS). In some cases, the mean properties are more useful, such as for bond length (L_B), bond angle (θ_B), and valence electron counts (N_valence), while in other cases, the differences between atomic properties carry more information (σ_m, σ_N, σ_Row, σ_EN). This is not unexpected, as phonon transport is sensitive to both similarities and differences within compounds. For example, we tend to expect shorter bonds to be stronger which results in higher characteristic frequencies, while mass difference among atoms could significantly alter the phonon spectrum. Notably, the most information-dense features being related to bonding strength are consistent with the positive correlation with Y in the relation of ${\kappa }_{\mathrm{L}}^{\min }\propto k_{\mathrm{B}}{n}^{2/3}{(Y/\rho )}^{1/2}$.

Finally, we performed dimensionality reduction using t-SNE (t-distributed stochastic neighbor embedding) with PCA (principal component analysis) initialization in order to project MEGNet’s latent graph features into two dimensions for visualization, as seen in Fig. 4E. The distribution of the training set and the ICSD dataset are considered together, and the overlap generally tells us how well the training set covers the ICSD dataset, indicating potential transferability of the model. The points containing ICSD materials with low and high predicted ${\kappa }_{\mathrm{L}}^{\min }$ are segregated in the latent space, allowing the model to clearly separate them from each other. Additionally, the training set points span the extent of the axes and their distribution varies smoothly, with the exception of less density near some outer edges and a small cluster toward the top center of the figure. Nevertheless, the coverage is especially dense in the area of low predicted ${\kappa }_{\mathrm{L}}^{\min }$ which is most relevant for thermoelectric or thermal barrier coating materials, suggesting that our training set sufficiently covers the space, but coverage could be improved by increasing the size of our training set in the future.

We further validate these predictions against experimental measurements and direct calculations using Eq. 1. Since ${\kappa }_{\mathrm{L}}^{\min }$ is relatively rare in ordered crystals, we deliberately choose thermoelectric materials exhibiting ulralow κ_L, which presumably best represent ${\kappa }_{\mathrm{L}}^{\min }$, as shown in Fig. 4H. We find that the experimental values are either higher or comparable to PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$, consistent with the definition of ${\kappa }_{\mathrm{L}}^{\min }$. It is noteworthy that compounds including CsAg₅Te₃ (25), Cu₃SbSe₃ (43), and BiCuSeO (42) have κ_L already close to ${\kappa }_{\mathrm{L}}^{\min }$ from our model, suggesting that further improvement in thermoelectric efficiency should be focused on optimizing their electronic transport properties. We note that CsAg₅Te₃ and Cu₃SbSe₃ indeed resemble amorphous solids and display nearly temperature-independent and glasslike κ_L (25, 43). However, for compounds such as Tl₃VSe₄ (18) and TlInTe₂ (41), there is still room for further reducing κ_L, although their κ_L are already very low. Also, we find it interesting to see a large gap between our calculated ${\kappa }_{\mathrm{L}}^{\min }$ and the experimentally measured κ_L in Sr₈Ga₁₆Ge₃₀ (44, 49), a typical electron-crystal phonon-glass material. This finding implies that although κ_L of a material might exhibit a temperature dependence similar to that of glasses, κ_L may still not achieve the minimum.

Overall, the machine-learned ${\kappa }_{\mathrm{L}}^{\min }$ from both MEGNet and the random forest agree very well with our direct calculations. This is rather encouraging because most of the compounds are not contained in the training dataset for the machine learning models except for TlInTe₂. The largest deviation between experiment, machine learning models, and direct calculation is found in Cu₂Se. The discrepancy might be attributed to the following factors: i) We used an ordered structure (β-Cu₂Se) to compute ${\kappa }_{\mathrm{L}}^{\min }$, while in the experimental structure the copper ions are highly disordered around the Se sublattice (23); ii) β-Cu₂Se often exhibits off-stoichiometry with copper vacancies, resulting in relatively large variations in the experimentally measured values of κ_L in the range of 0.4 to 0.6 W/(m K) among different samples (23). Therefore, despite the disordered nature of β-Cu₂Se, our models, including both machine learning and direct calculation, seem to provide reasonable estimations using ordered structures.

The above results on Cu₂Se indicate that κ_L of amorphous compounds might be approximated with a reasonable accuracy using ordered crystals. To confirm this hypothesis, we further apply our model to amorphous silica (a-SiO₂). We calculated PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ for two ordered phases of SiO₂ (α-quartz and α-cristobalite) that are free of lattice instabilities, which are compared to the CWP model and the experimental measurement (9) in Fig. 5A. We see that PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ of both α-quartz and α-cristobalite agree well with experiment, displaying overall improvement over the CWP model, particularly at high temperatures. This implies that specific atomic arrangement might not be very relevant when approaching ${\kappa }_{\mathrm{L}}^{\min }$, although it is responsible for subtle differences, which are probably more profound at low temperatures.

Fig. 5.

Prediction of lattice thermal conductivity in amorphous solids using machine learning models of minimum lattice thermal conductivity. (A) Comparison of temperature-dependent κ_L of amorphous silica (a-SiO₂) between the experiment (gray disks) (9) and theoretical ${\kappa }_{\mathrm{L}}^{\min }$ (solid/dashed lines). Results from the CWP model are indicated by solid magenta lines. The dashed lines display the calculated PGM+OD-${\kappa }_{\mathrm{L}}^{\min }$ for two ordered phases of SiO₂, namely α-quartz and α-cristobalite. (B) Predicted ${\kappa }_{\mathrm{L}}^{\min }$ (orange disks) at 600 K using MEGNet for 317 structures of ordered SiO₂ from Materials Project (33) as a function of the number of atomic sites in a primitive cell. The upper, middle, and lower dashed lines denote the values from the experiment, mean of MEGNet predictions, and the CWP model, respectively. (C) Comparison of κ_L at 300 K of six amorphous compounds between experiments (9), our MEGNet model, and the CWP model (9).

To provide more evidence for this observation, we next examine other phases of SiO₂ using our machine learning model. Particularly, we used MEGNet instead of the random forest because the former better encodes structural information. Since our MEGNet model is trained on structures from the Materials Project (MP) (33–35), we collected all structures with the chemical formula SiO₂ available within MP, totaling 317 structures. The calculated ${\kappa }_{\mathrm{L}}^{\min }$ using MEGNet is shown in Fig. 5B. We find a relatively large spread of ${\kappa }_{\mathrm{L}}^{\min }$ when N_a is small (N_a < 25), while the spread becomes much smaller when N_a is large (N_a > 75). The mean value of ${\kappa }_{\mathrm{L}}^{\min }$ of the 317 structures is 1.45 W/(m K), close to the experimental value of 1.65 W/(m K) for the amorphous phase (9) and better than the value of 1.18 W/(m K) from the CWP model at 600 K (9). These results confirm our earlier hypothesis that κ_L of amorphous compounds can be approximated by ordered crystals. Moreover, a reliable prediction may be achieved by averaging over different structures, effectively capturing the distinct local structures in amorphous solids.

We apply our MEGNet model to other amorphous solids by averaging over ordered phases and compared to the experiments and the CWP model in Fig. 5C. Our machine learning model achieves an accuracy comparable with the CWP model. The advantage of our model is that the calculation requires very low computational cost (within a second for each compound) and simple input (only structural information) and thus is readily applicable to many other compounds on a large scale. We note that compared to the amorphous SiO₂, our predicted κ_L of amorphous Si is considerably smaller than both the CWP model and the experiment. We attribute such a discrepancy mainly to the observations reported in the literature that i) propagating modes contribute significantly more in amorphous Si than amorphous SiO₂ and ii) phonon lifetimes in amorphous Si are notably larger than π/ω (50), both of which indicate that κ_L in amorphous Si still does not reach the minimum value. We hypothesize that this might also be the case for amorphous CdGeAs₂ and Ge.

Before closing, we briefly comment on the limitations of our theoretical and machine learning models. First, both the identified lower bound to ${\kappa }_{\mathrm{L}}^{\min }$ and the constructed machine learning models are based on the isotropically averaged ${\kappa }_{\mathrm{L}}^{\min }$. Consequently, it is reasonable to question how ${\kappa }_{\mathrm{L}}^{\min }$ behaves in anisotropic structures. This anisotropy is of particular interest because κ_L as small as 0.05 W/(m K) below our isotropically averaged lower bound at room temperature has been realized in layered WSe₂ crystals (51). Therefore, we analyzed the anisotropic ${\kappa }_{\mathrm{L}}^{\min }$ for all of the calculated compounds (SI Appendix, Fig. S3). We find that there are 67 compounds in total that have ${\kappa }_{\mathrm{L}}^{\min }$ less than 0.1 W/(m K) in at least one of the three Cartesian directions. However, there are only five compounds exhibiting ${\kappa }_{\mathrm{L}}^{\min }$ less than 0.07 W/(m K) (see listed compounds and ${\kappa }_{\mathrm{L}}^{\min }$ in SI Appendix). This observation suggests that even though it may be difficult to find such materials due to their rarity, anisotropic materials, including layered crystals, have the potential to achieve a lattice thermal conductivity of less than 0.1 W/(m K), consistent with the prior work (51).

Second, we computed all ${\kappa }_{\mathrm{L}}^{\min }$ using the harmonic phonon dispersion in the absence of finite temperature-induced phonon frequency shifts; the latter are important in a range of low-thermal-conductivity materials (52, 53). As a first attempt to investigate the effects of phonon anharmonicity on ${\kappa }_{\mathrm{L}}^{\min }$, we computed ${\kappa }_{\mathrm{L}}^{\min }$ at 300 K for Tl₃VSe₄ using harmonic and anharmonic phonon dispersions, respectively. The anharmonic phonon dispersion was computed using the self-consistent phonon theory at 300 K in a previous study (52). We find that accounting for anharmonic effects only leads to a small change (about 7% decrease) in ${\kappa }_{\mathrm{L}}^{\min }$, which is much smaller than the change (about 162.5% increase) in κ_L when phonon lifetimes are limited by intrinsic three phonon scattering (52). The relative insensitivity of ${\kappa }_{\mathrm{L}}^{\min }$ to anharmonic effects can be explained by the relationship ${\kappa }_{\mathrm{L}}^{\min }\propto k_{\mathrm{B}}{n}^{2/3}{(Y/\rho )}^{1/2}\propto k_{\mathrm{B}}{n}^{2/3}{v}_s$ (7), which indicates that ${\kappa }_{\mathrm{L}}^{\min }$ is largely unchanged unless anharmonic effects lead to a sizable change in sound velocity v_s.

Third, it is not entirely justified to use one-half of the vibrational period as the minimum phonon lifetime, i.e., τ = π/ω. The adoption of τ = π/ω was first proposed by Cahill, Watson, and Pohl (9), motivated by Einstein’s thermal conductivity model (13). Specifically, Einstein’s model relies on a critical assumption that there is no coherence between the motions of neighboring atoms, which seems to be a reasonable choice for glassy materials. Einstein further derived the energy exchange between neighboring atoms by integrating over one-half of a period of oscillation, however, without an explicit explanation in his original paper (13). We hypothesize this might be motivated by the fact that it is the shortest time period that an atom returns to its original position after exchanging energy with neighboring atoms. Cahill et al. interpreted Einstein’s result to mean that each atom undergoes a thermal energy loss or gain in the span of half an oscillation period (9) and subsequently assumed that τ = π/ω. Mathematically, τ adopted in computing ${\kappa }_{\mathrm{L}}^{\min }$ can have an arbitrary value, which presumably is small in order to reduce the PGM contribution. However, physically, τ cannot be arbitrarily short because our derived equation for ${\kappa }_{\mathrm{L}}^{\min }$ relies on a critical assumption that phonons are well-behaved quasiparticles (τω ⪆ 2π). The very small τ implies strong interactions and will invalidate the quasiparticle picture, making the phonon properties (such as group velocities and frequencies) less well-defined and invalidating the usage of Eq. 1. To explore the potential uncertainty due to the adopted τ, we performed sensitivity analysis of ${\kappa }_{\mathrm{L}}^{\min }$ by varying τ for a randomly selected set of compounds (SI Appendix). We find that ${\kappa }_{\mathrm{L}}^{\min }$ decreases for most compounds when τ is decreased. Specifically, a sharp decrease in ${\kappa }_{\mathrm{L}}^{\min }$ is found when τ is reduced from a larger value to π/ω, a value we adopted for computing ${\kappa }_{\mathrm{L}}^{\min }$. When τ is less than π/ω, slower decrease or no change in ${\kappa }_{\mathrm{L}}^{\min }$ is observed, especially for lower values of ${\kappa }_{\mathrm{L}}^{\min }$. Since τ cannot be arbitrarily small as argued previously, we hypothesize that τ = π/ω might be a reasonable choice to achieve a relatively low value of ${\kappa }_{\mathrm{L}}^{\min }$ while approaching the limit of the quasiparticle picture.

Finally, despite encouraging results from the comparison between the calculations and the experiments, our theoretical model only accounts for harmonic heat flux (39, 54) and Eq. 1 fails to describe phonon satellite structures deviating from the quasiparticle picture (55). For the machine learning model, local structural motifs may not be well captured by ordered structures for amorphous solids, and our simple averaging scheme with equal weight for each structure is probably not optimal. We deem establishing a more reliable physical bound to phonon lifetime, including higher-order anharmonic effects on the phonon frequency shifts and the heat flux beyond the phonon quasiparticle picture, and machine learning structural motifs inherent in amorphous solids are interesting avenues of further research to improve our model.

Conclusion

In summary, we have developed a first-principles minimum lattice thermal conductivity model based on a unified theory of thermal transport in crystals and glasses. By applying such a model to thousands of crystalline compounds, we have found a universal bound to minimum lattice thermal conductivity independent of structural complexity. In striking contrast to the conventional phonon gas model, such an unusual behavior is found to be deeply rooted in the conversion of the heat transfer mechanism from the phonon gas model to the diffuson picture with the presence of disorder, while the value of the total minimum lattice thermal conductivity largely remains unchanged. With these insights, we bridge the knowledge gap between the Cahill–Watson–Pohl model and our unified model by pointing out that the former could be viewed as an approximation of the latter. We further construct machine learning models based on a graph network and random forest to enable fast and accurate prediction of minimum lattice thermal conductivity on a large scale, which is validated against thermoelectric materials with ultralow and glasslike lattice thermal conductivity. We demonstrate the applicability of our graph network model to simulate lattice thermal conductivity in amorphous solids. These findings highlight a unified understanding of the lower limit of lattice thermal transport in solids. The first-principles–based theory and machine learning model built in this work are universal and readily applicable in research relevant to thermal energy conversion and management.

Materials and Methods

Theoretical Calculation of ${\mathit{\kappa }}_{\mathrm{L}}^{\min }$.

The key ingredients for modeling ${\kappa }_{\mathrm{L}}^{\min }$ based on Eq. 1 from first principles are materials harmonic vibrational spectra, which can be explicitly calculated if materials structure and harmonic interatomic force constants are known. To construct a large database for ${\kappa }_{\mathrm{L}}^{\min }$, we used the phonon database (phononDB) generated by Togo (31), who used crystalline structures from the Materials Project (MP) (33–35) and calculated harmonic interactions using Phonopy (32). It is worth noting that these phonon calculations were performed by means of the Vienna Ab Initio Simulation Package (VASP) (56–59), which employed the projector-augmented wave (PAW) (60) method in conjunction with the revised Perdew–Burke–Ernzerhof version [PBEsol (61)] of the generalized gradient approximation (GGA) (62) for the exchange-correlation functional (63). We performed post processing of the phononDB (version of 2018-04-17) to generate harmonic force constants and downselected the compounds that are free of lattice instabilities (imaginary phonon frequencies) with supercell structures constructed from diagonal matrices [required by our ${\kappa }_{\mathrm{L}}^{\min }$ implementation within ShengBTE (64)], totaling 2,576 compounds. These compounds cover wide ranges of chemical compositions and space group symmetries, with 189, 922, 310, 660, and 495 compounds from cubic, orthorhombic, tetragonal, trigonal/hexagonal, and triclinic/monoclinic crystal systems, respectively. Other compounds and their harmonic phonon properties, including TlInTe₂, Tl₃VSe₄, CsAg₅Te₃, Cu₂Se, TlAgTe, CuBiSeO, SnSe, α-cristobalite, and α-quartz, were calculated following the similar DFT settings. For β-Cu₂Se which has imaginary phonon frequencies, we have performed anharmonic phonon renormalization at finite temperature (600 K) using self-consistent phonon theory (52, 53, 65, 66).

We implemented the ${\kappa }_{\mathrm{L}}^{\min }$ model based on Eq. 1 within ShengBTE (64). The off-diagonal group velocities were calculated following the derivations by Allen and Feldman (16) and Simoncelli et al. (27). We calculated ${\kappa }_{\mathrm{L}}^{\min }$ for these 2576 compounds at three temperatures of 300 K, 600 K, and 900 K using a constant mesh density (mesh_density = 50) following Phonopy conventions (32). The convergence is carefully monitored across different compounds to achieve good balance between accuracy and efficiency.

Machine Learning Models.

MEGNet: For the MEGNet model, we used frozen initial atom embeddings transferred from another MEGNet model (46) trained on 133,420 formation energies from Materials Project (33). Crystal graphs were constructed using 50 features per bond with a Gaussian smearing width of 0.5 Å up to a cutoff radius of 5 Å for the bond attributes, transferred atom embedding weights of size 16 for the atom attributes, and one single global attribute embedding the temperature in Kelvin. For the model hyperparameters, we used only a single MEGNet block and set the number of dense layers in the MEGNet block to n₁ = 16, n₂ = 16, and n₃ = 8, followed by 2 Set2Set passes, and the learning rate was set to 10⁻³. In order to prevent overfitting, we set weight dropout to 25% during both training and prediction and added a small L2 weight regularization parameter of 10⁻⁵. The ${\kappa }_{\mathrm{L}}^{\min }$ data was split 80%:10%:10% into a training set, validation set, and testing set. Predictions were then run over 10 trials and the average was taken as the final output. The model was run for 500 epochs with a patience of 250, and the model with the best validation performance was selected for further predictions. The MEGNet training mean absolute error (MAE) was 0.033 W/(m K), the validation MAE was 0.0406 W/(m K), and the testing MAE was 0.0479 W/(m K). Finally, we performed dimensionality reduction on the combined phononDB and ICSD dataset using t-SNE to project the latent graph features into two dimensions. We initialized the t-SNE with PCA and chose the settings of 1,000 iterations, a perplexity of 5, and a learning rate of 39,514 (n).

Random Forest: Features for the random forest were created through Matminer (47) using the Magpie preset (48), along with global symmetry features and site statistic fingerprints, totaling 141 features for each compound as detailed in the main text. The random forest was constructed using 1,000 estimators, and we verified that including more estimators did not improve the accuracy significantly. For this model, we used fivefold cross-validation on the 90% training set for parameter selection and left a holdout test set of 10%. In order to prevent overfitting, we changed the allowable maximum depth of each tree, where larger maximum depths showed better prediction accuracies at the cost of a larger gap between training and testing accuracies. A maximum depth 8 was then chosen to balance training, validation, and testing accuracy. For the random forest, the mean training MAE was 0.0375 W/(m K), mean validation MAE was 0.0458 W/(m K), and the testing MAE was 0.0570 W/(m K). Feature importance was extracted from a random forest trained on only 600 K data.

The training dataset used in both MEGNet and random forest models consists of our calculated ${\kappa }_{\mathrm{L}}^{\min }$ at three temperatures of 300 K, 600 K, and 900 K, totaling 7,728 datapoints. For the ICSD dataset, 36,199 crystal structures were pulled from the Open Quantum Materials Database (OQMD) (67, 68), and all machine learning predictions of ${\kappa }_{\mathrm{L}}^{\min }$ were then performed at 600 K unless otherwise specified.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

The codes, datasets, and machine learning models have been deposited in Github (https://github.com/yimavxia/Minikappa) (69).