Coupling the High-Throughput Property Map to Machine Learning for

Article Cite This: Chem. Mater. XXXX, XXX, XXX−XXX

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Coupling the High-Throughput Property Map to Machine Learning for Predicting Lattice Thermal Conductivity Rinkle Juneja,† George Yumnam,† Swanti Satsangi, and Abhishek K. Singh* Materials Research Centre, Indian Institute of Science, Bangalore 560012, India

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S Supporting Information *

ABSTRACT: Low thermal conductivity materials are crucial for applications such as thermoelectric conversion of waste heat to useful energy and thermal barrier coatings. On the other hand, high thermal conductivity materials are necessary for cooling electronic devices. However, search for such materials via explicit evaluation of thermal conductivity either experimentally or computationally is very challenging. Here, we carried out high-throughput ab initio calculations, on a dataset containing 195 binary, ternary, and quaternary compounds. The lattice thermal conductivity κl values of 120 dynamically stable and nonmetallic compounds are calculated, which span over 3 orders of magnitude. Among these, 11 ultrahigh and 15 ultralow κl materials are identified. An analysis of generated property map of this dataset reveals a strong dependence of κl on simple descriptors, namely, maximum phonon frequency, integrated Grüneisen parameter up to 3 THz, average atomic mass, and volume of the unit cell. Using these descriptors, a Gaussian process regression-based machine learning (ML) model is developed. The model predicts log-scaled κl with a very small root mean square error of ∼0.21. Comparatively, the Slack model, which uses more involved parameters, severely overestimates κl. The superior performance of our ML model can ensure a reliable and accelerated search for multitude of low and high thermal conductivity materials.

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INTRODUCTION Lattice thermal conductivity (κl) of materials is an important physical property, which is of great relevance to diverse applications. Materials with ultralow κl are required for thermoelectric conversion of waste heat to useful energy1,2 and also for thermal barrier coatings.3 High κl materials are essential for cooling of virtually all electronic devices.4−6 Rapid screening of materials with the desired thermal conductivity is often hampered by complexities involved in the experimental estimation of κl. Theoretically, κl can be calculated using density functional theory (DFT) in combination with linearized phonon Boltzmann transport equation (BTE). Linearized BTE is generally solved by considering the three phonon scattering, assuming the very small contribution of higher order phonon interactions. These assumptions work reasonably well in most of the cases;7 however, in strongly anharmonic systems or at very high temperatures, the higher order interactions may play an important role. In such cases, linearized BTE at most represent an upper limit to κl. Having said that, enormous increase in the computational cost and time with system size limits its usage to only simpler systems. Recently, the ab initio high-throughput screening is becoming a burgeoning area of research. These methods are utilized to narrow down the large search space for identification and prediction of compounds with desired properties. Generally, a high-throughput approach involves filtering-out materials, which satisfies simultaneously a minimum set of required criterion in terms of physical and chemical properties. For example, by defining physically © XXXX American Chemical Society

meaningful conditions on the properties, the high-throughput approach has been successful for the identification of stable structures, light-absorbing materials, materials for photoelectrochemical cells, battery materials, electrocatalytic materials, full-Heusler materials, high power factor materials, low lattice thermal conductivity materials, and transparent conductors.7−19 The high-throughput prediction for κl is mainly focused on low lattice thermal conductivity materials belonging to a particular class and the descriptor set used in the prediction model is very huge and complex.7,17,18 In this work, we carried out high-throughput screening in a systematic manner to construct a property map on a dataset of binary, ternary, and quaternary 195 compounds. From the property map, the optimized, nonmetallic, and dynamically stable 120 compounds are filtered-out, for which κl is calculated using first-principles and phonon BTE. The values of κl span 3 orders of magnitude and we identified 11 ultrahigh and 15 ultralow κl materials. Additionally, we utilize the patterns and trends in the generated property map to couple the high-throughput approach to machine learning (ML) for the development of κl prediction model. We found that maximum phonon frequency, integrated Grünesien parameter up to 3 THz, average atomic mass, and volume of unit cell show reasonable correlation with κl. Using these highthroughput proposed descriptors, a Gaussian process regresReceived: March 15, 2019 Revised: June 26, 2019 Published: June 27, 2019 A

DOI: 10.1021/acs.chemmater.9b01046 Chem. Mater. XXXX, XXX, XXX−XXX

Article

Chemistry of Materials sion (GPR)-based ML model is developed for the log-scale κl, which gives a very small rmse of 0.21. On the other hand, the widely used Slack model severely overestimates κl. The ML model provides a simpler and faster way to reliably screen materials having desired κl.

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METHODOLOGY All the calculations were performed using the first-principles DFT,20,21 as implemented in the Vienna Ab initio Simulation Package (VASP).22,23 The electronic exchange and correlation potential was approximated by the Perdew−Burke−Ernzerhof generalized gradient approximation.24 The ion−electron interactions were represented by the projector augmented wave (PAW) potentials.25,26 The plane wave energy cutoff was set to 1.3 times the default value in the PAW pseudopotential. A strict energy convergence criterion of 10−8 eV was used. The phonon dispersion and Grüneisen parameter were calculated using the supercell approach, as implemented in the PHONOPY.27 The lattice thermal conductivity was calculated by solving the phonon BTE using PHONO3PY.28,29 For developing the ML-based model, GPR was used. GPR is a nonparametric approach, where inferences are made about the relationship between input (X) and target output (y).30,31 The inference of the model is based on the posterior distribution over weights w of the model, which according to Bayes’ rule is given by p (w | y , X ) =

Figure 1. Schematic showing the high-throughput screening of compounds for lattice thermal conductivity evaluation. The compounds consist of combination of elements from alkali, alkaline, transition metal, lanthanides, and anions groups. Optimized, nonmetallic, and dynamically stable compounds are screened for κl calculations.

p ( y | X , w ) p (w ) p (y | X )

where the normalizing constant p(y|X) = ∫ p(y|X,w)p(w)dw is independent of the weights and is also known as marginal likelihood. These probabilities are assumed to follow a Gaussian distribution defined by a mean and covariance function. Besides the model weight parameters, the covariance function has its own set of statistical parameters, namely, variance (σ) and length scale (l), also known as hyperparameters θ.32 The hyperparameters are optimized by maximizing the marginal likelihood. Here, we used the automatic relevance determination (ARD) Matern 5/2 covariance function, which for the given data points xi and xj and hyperparameter vector θ is given by31 i k(xi , xj|θ ) = σf 2jjj1 + k

where r is defined as r =

5r + d

all the 60 binary and 85 ternary nonmetallic compounds were included for further analysis, while only 50 quaternary nonmetallic compounds were chosen. The phonon dispersion for the 195 filtered nonmetallic compounds is calculated to check their dynamical stabilities. These compounds are polar; therefore, non-analytical correction is added to the dynamical matrix for the phonon calculations. Out of 195, 120 compounds do not show any imaginary modes in the phonon dispersion and, hence, are dynamically stable. The κl values are calculated for these nonmetallic dynamically stable compounds and are shown in Figures S1−S5. To benchmark the reliability of calculations of lattice thermal conductivity, we have compared it with the experimental room temperature values, which is shown in Table S1 of Supporting Information. The calculated κl agrees reasonably well with experimental values with a mean absolute error of 0.25. The calculated κl values span 3 orders of magnitude with several of compounds having ultralow and ultrahigh κl. There are 15 compounds having κl less than 1 W/mK at T = 300 K, as reported in Table 1. Except Ba2BiAu (Fm3̅m), TlBr (Pm3̅m), and TlCl (Pm3̅m), the remaining high-throughput screened 12 ultralow κl compounds are being reported for the first time to the best of our knowledge. Although different aspects such as the type of elements, the bonding nature between constituent elements, and type of crystal structure will determine the exact underlying lattice dynamics, there are similarities in the harmonic and anharmonic properties of these ultralow compounds. The phonon frequencies for these ultralow materials span mostly in the low frequency range. The magnitude of Grüneisen parameter for the low-lying phonon modes is very high. Most of these screened ultralow compounds contain earth-abundant and nontoxic elements and, hence, can be explored for thermoelectric applications. In

5 2yz r zzexp( − 5 r ) 3 {

∑m = 1

(xim − xjm)2 σm 2

and σf, d, and σm

are the standard deviation, number of predictors in the data, and length scale for each predictor m, respectively. The hyperparameters for the covariance functions were optimized using the Broyden−Fletcher−Goldfarb−Shanno algorithm.32

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RESULTS AND DISCUSSION A dataset of 2162 compounds is obtained from the Materials Project.33 These compounds include 322 binary, 306 ternary, and 1524 quaternary systems with elements belonging to alkali, alkaline earth, transition metals, lanthanides, and anions, as highlighted in the periodic table of Figure 1. The lattice parameters of these compounds are optimized. Because κl has predominant contribution to total thermal conductivity in nonmetallic systems, the metallic cases are dropped from the dataset. In order to keep the computational cost manageable, B


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Chemistry of Materials

hybridization of acoustic and optical modes, which lowers the κl. The phonon modes of the high κl compounds span over a much larger frequency range, which leads to reduction in the hybridization of acoustic and optical modes. Furthermore, some of these compounds also have phononic gaps, which prohibit the phonon−phonon scattering in the gapped regions, thereby resulting in even higher κl. These effects are illustrated in Figure 2 for two representative ultralow and ultrahigh κl

Table 1. Ultralow Lattice Thermal Conductivity of 15 Compounds at T = 300 K compound

κl (W/mK) at T = 300 K

CsK2Sb (Fm3̅m) TlI (Pm3̅m) Ba2BiAu (Fm3̅m) SrTePd (F4̅3m) Ba2SbAu (Fm3̅m) TlBr (Pm3̅m) Cs2Se (Fm3̅m) PbI2 (R3̅m) LiFeP (I4mm) TlCl (Pm3̅m) TlCl (Fm3̅m) Ba2AgSb (Fm3̅m) PbI2 (P3̅m1) TlBr (Fm3̅m) LaCoTe (F4̅3m)

0.13 0.15 0.21 0.22 0.27 0.34 0.50 0.51 0.65 0.65 0.69 0.72 0.84 0.87 0.98

addition to this, the high-throughput screening has resulted in 11 compounds with κl greater than 200 W/mK at T = 300 K, as reported in Table 2. Except BeTe (F4̅3m), BeSe (F4̅3m), BP Table 2. Ultrahigh Lattice Thermal Conductivity of 11 Compounds at T = 300 K compound

κl (W/mK) at T = 300 K

BiB (F4̅3m) BeTe (F4̅3m) SnC (F4̅3m) BP (P63mc) B2AsP (P4̅m2) SiSn (F4̅3m) BeSe (F4̅3m) SiC (F4̅3m) BP (F4̅3m) BSb (F4̅3m) GeC (F4̅3m)

214.85 234.01 310.86 409.57 414.83 421.28 423.54 447.18 456.58 546.45 908.27

Figure 2. κl as a function of temperature, phonon dispersion, and mode Grünesien parameters for ultralow (a,b) and ultrahigh (c,d) κl compounds.

compounds. The variation of ωmax with log-scaled κl at T = 300 K is shown in Figure 3a, which shows that κl increases with ωmax. Therefore, the maximum of the phonon frequency (ωmax) could be a reasonable descriptor to broadly capture the variation in κl. Because the Grüneisen parameter can shed some light on anharmonicity in a material, we next analyze the correlation between mode Grüneisen parameter and κl. The Grüneisen parameters are shown for 116 compounds in Figures S11−S15. The Grüneisen parameters for two ultralow and ultrahigh κl materials are shown in Figure 2. Although the expected inverse dependence of magnitude of Grü neisen parameter on κl roughly holds (Figure 2), it was not sufficient to explain the observed differences in thermal conductivity. Interestingly though, the spread of the Grüneisen parameter over the frequency range shows distinct variations for compounds having different thermal conductivity. For example, the low κl materials are having large spread in the lower frequency range as compared to the high κl. This spread can be quantified by calculating the area under the Grüneisen parameter versus frequency curve. However, integrating the Grüneisen parameter over the entire frequency range may not give a better correlation with κl. The low-lying modes dominate the

(F4̅3m), and BSb (F4̅3m), the remaining seven compounds are reported for the first time. Recently, boron-based compounds have attracted a lot of interest because of their very high thermal conductivity at room temperature.34−36 The highthroughput screening has resulted in three additional boronbased ultrahigh κl compounds BiB (F4̅3m), BP (P63mc), and B2AsP (P4̅m2). The ultrahigh κl compounds also have few similarities in their properties, such as the presence of large phononic gaps, low magnitude of Grüneisen parameter, and so on. These new ultrahigh κl compounds can be utilized in power control systems, thermal barrier coatings, security systems, and cooling of electronic devices. With the increase in the size of the system, the explicit evaluation of κl becomes computationally very demanding. Therefore, the models which can predict the κl with reasonable accuracy and time are of utmost importance for rapid screening of materials space. To develop such a model, we generated an extensive property map, which includes harmonic and quasiharmonic properties such as phonon dispersion and Grüneisen parameter. The phonon spectra of all the compounds are shown in Figures S6−S10. The phonon modes of the compounds having low κl are lying mostly in the lower frequency range. In these compounds, there is a strong C


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Figure 3. Variation of log-scaled κl with (a) maximum of phonon frequency (ωmax), (b) inverse of log-scaled integrated Grüneisen parameter up to 3 THz (ln(1/γc=3)), (c) average atomic mass M, and (d) volume of cell V.

correlation among these features is very less. Further, we calculated the correlation of these descriptors with the κl, to check the relevance of these descriptors. As shown in Figure 4b, ωmax has highest correlation with κl, followed by γc=3, V, and M. Various GPR models are developed using different covariance functions. The four descriptors ωmax, γc=3, M, and V to the GPR model are a four-dimensional (4D) array. This dataset is split randomly 1000 times in 90:10 train/test ratio. Because each descriptor has different dimensions, they are standardized by subtracting the mean and dividing by the standard deviation for the training data during each splitting. The performance of the developed prediction model is evaluated by statistical regression metrics such as root mean square error (rmse) and coefficient of determination (R2). The R2 determines the proportion of variability in the predicted values captured by input features. The best prediction model for the log-scaled κl was obtained by using ARD Matern 5/2 covariance function, which gives the train/test rmse of 0.20/ 0.21 and the R2 of 0.99/0.99. To ensure no over-fitting, we checked the learning curve for the developed model, which is shown in Figure S16. The train and test rmse converge around 0.21 as a function of increase in the training data, thereby implying no over-fitting. The ML-predicted versus DFT κl is plotted in Figure 5a. The difference between DFT and ML κl (residual) is shown in Figure 5b. For most of the compounds, the residual is less than 0.1. To further check the expected

anharmonicity, and hence, the integration up to a cut-off frequency may have a better correlation with κl. To determine a suitable cut-off frequency, we calculate the integrated Grüneisen parameter for different cut-off frequencies and checked the Pearson correlation coefficient with the κl (PCC(κl)). As shown in Figure 4a, the integrated Grüneisen

Figure 4. (a) Pearson correlation coefficient for κl (PCC(κl)) with integrated Grüneisen parameter at different cut-off frequencies, (b) Pearson correlation map of the high-throughput four descriptors with each other, and PCC(κl) with the four descriptors.

parameter up to 3 THz (γc=3) has the maximum correlation with κl, as compared to other high and low cut-off frequencies, including the full frequency range. In addition to it, because the low frequency phonon modes predominantly contribute to κl, the anharmonicity in the low frequency regime is expected to be fairly well represented by Grüneisen parameter up to a low cut-off frequency. Hence, we defined an integrated Grüneisen parameter up to 3 THz as a descriptor for κl, without explicitly calculating the expensive third order force constants. Figure 3b shows the variation of log-scaled κl at T = 300 K against inverse log-scaled integrated Grüneisen parameter (1/ γc=3). Similar to ωmax, κl increases with 1/γc=3. Furthermore, the magnitude of κl also depends on the average atomic mass of constituent elements (M) and volume of the system (V), we plotted log-scaled κl at T = 300 K as a function of these simple descriptors. As shown in Figure 3c,d, overall κl decreases with M and V. Hence, maximum phonon frequency, integrated Grüneisen parameter up to 3 THz, average atomic mass, and volume of cell could broadly capture the trends of κl in a given material. To check the capability of these descriptors emerging from the analysis of high-throughput property map, for prediction model development, we employed the data-driven ML methods, which have been successful in predicting various resource extensive properties such as band gaps, bandedges.37,38 First, to ensure the linear independence for these four descriptors (ωmax, γc=3, M, and V), the Pearson correlation is calculated. As shown by the heat map in Figure 4b, the

Figure 5. Scatter plot of DFT-calculated log-scaled κl vs (a) ML model-predicted κl, and (c) Slack model-predicted κl. Panels (b,d) show the difference between true and predicted response for both cases. D


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ical operations iteratively. The feature spaces ϕ1, ϕ2, and ϕ3 consist of 70, 10 860, and 350 767 452 features, respectively. The accuracy of the models increases with complexity and dimensionality, as described in the Supporting Information. The best solution for the proposed descriptors for room temperature log-scaled κl using 4D descriptor from SISSO is

applicability of the model, we tested it on few independent compounds, which were not the part of initial dataset. The comparison between DFT and model κl values is shown in Table S2 in Supporting Information. The model works fine for these completely unseen data with a mean absolute error of 0.74. The error bar is slightly higher as there are less unseen data points, and for some cases DFT κl is not close to ML κl. However, the model still successfully predicts the correct order of magnitude for these new compounds. Furthermore, the performance of the ML model is compared with the widely used Slack model. This model uses quasiharmonic based, yet involved quantities, such as Debye temperature θD, average Grüneisen parameter γ, average atomic mass M, volume of unit cell V, and number of atoms in the unit cell n.39 At temperatures near θD, the Slack model κl is given by39 κl = A

ln(κl)4D = 0.488((ln(M )ωmax1/3) − (γc = 31/4))

ij yz |(M − V )| jj zz ln(ωmax ) jj zz zz − 0.419jjj jj (M + ωmax ) + (ωmax − V ) zzz j z k {

(

ij (ω |(M − V )|) max − 0.023jjjj j (M − V ) − γ c=3 k

γ Tn

2/3

where the constant A is determined as 2.43 × 10 1−

0.514 γ

+

yz zz + 1.614 zz {

It has a very high correlation of 0.93 with the DFT κl and a fairly reasonable R2 of 0.88 with rmse of 0.66. Hence, the proposed four descriptors provide a less expensive route to estimate the order of magnitude of lattice thermal conductivity. Furthermore, these models were also developed for the Slack descriptors (see Supporting Information). For the low dimensions and less complex formulas, the SISSO models using proposed descriptors have lower rmse compared to models developed using Slack descriptors.

−8

A=

yz zz zz z {

ij i V + ωmax yz zz − 0.013jjj (|M − (V /ωmax )|) − jjj j k ln M { k

MθD3V1/3 2

)

0.228 γ2

The θD was calculated by fitting the phonon density of states at frequencies from 0 to 1/4 of the maximum phonon frequency.27 θD is given by θD = ℏωD/kB, where ωD = 9n/a1/3, is the Debye frequency, and kB and a are Boltzmann constant and fitting parameter, respectively. The κl was calculated for all the 120 compounds, using the Slack model. The κl calculated using the Slack model does not agree very well with the calculated first-principles values. The κl calculated using the Slack model is severely overestimated for all the compounds, as shown in Figure 5c. It shows very poor variability in data and hence very less correlation with first principles κl. The corresponding residual plot is shown in Figure 5d. The difference between first-principles and the κl calculated using the Slack model is at least 2 W/mK, and for some of the compounds, it reaches up to 8 W/mK. The residual for the κl calculated using the Slack model is 1 order of magnitude larger as compared to ML model. This highlights the superiority of the ML model in predicting the κl, with a high degree of accuracy. Furthermore, to check the potential of proposed descriptors, we developed a ML prediction model with Slack descriptors namely M, V, n, θD, and γ. By training the GPR model on 90:10 train/test over 1000 random trials, the rmse of train/test for best model was 0.29/0.30 for log-scaled κl, which is higher than the earlier proposed ML model. The Slack−MLpredicted versus DFT κl and the difference between DFT and Slack−ML κl are shown in Figure S17. Hence, the proposed descriptors, in addition to being simpler than Slack descriptors, result in better ML prediction models. Having realized the predictive capability of proposed descriptors, we tried to find a suitable solution using different combinations of these descriptors by applying various algebraic and function operations, by employing the sure independence screening and sparsifying operator (SISSO)40 method. Using the sparsifying l0 constraint and screening procedure, SISSO provides the best combination of descriptors having the largest correlation with the target output. Starting from the descriptors space ϕ0 having four descriptors, enormous number of features are generated through SISSO by applying different mathemat-

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CONCLUSIONS In summary, we coupled the high-throughput property map to the ML approach to develop a prediction model for lattice thermal conductivity. Starting with the generation of dataset for 2162 binary, ternary, and quaternary compounds, we applied constraints to screen out 120 nonmetallic and dynamically stable compounds and calculated their κl. The values of κl of the compounds in the dataset vary by 3 orders of magnitude. An analysis of the property map reveals a reasonable correlation of κl with maximum phonon frequency, integrated Grüneisen parameter up to 3 THz, average atomic mass, and volume of the unit cell. These parameters are used as descriptors in GPR to develop a prediction model for κl. The model shows excellent accuracy with an rmse of ∼0.21 and captures the large variability in κl. Furthermore, our model outperforms widely used Slack model by 1 order of magnitude in terms of accuracy and can accelerate the search for low and high κl materials for a wide range of applications. Our approach of finding the descriptors using the high-throughput property map can be generalized to develop robust predictive models for other physical and chemical properties of interest, which can help to accelerate the discovery of novel functional materials.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.9b01046. Calculated lattice thermal conductivity, phonon dispersion, Grü neisen parameter of 116 compounds, learning curve, GPR model using Slack descriptors, E


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Learning-Driven Synthesis of Full-Heusler Compounds. Chem. Mater. 2016, 28, 7324−7331. (12) Qin, G.; Hu, M. Accelerating Evaluation of Converged Lattice Thermal Conductivity. npj Comput. Mater. 2018, 4, 3. (13) Petretto, G.; Gonze, X.; Hautier, G.; Rignanese, G.-M. Convergence and Pitfalls of Density Functional Perturbation Theory Phonons Calculations from a High-Throughput Perspective. Comput. Mater. Sci. 2018, 144, 331−337. (14) Singh, H. K.; Zhang, Z.; Opahle, I.; Ohmer, D.; Yao, Y.; Zhang, H. High-Throughput Screening of Magnetic Antiperovskites. Chem. Mater. 2018, 30, 6983−6991. (15) Burton, L. A.; Ricci, F.; Chen, W.; Rignanese, G.-M.; Hautier, G. High-Throughput Identification of Electrides from All Known Inorganic Materials. Chem. Mater. 2018, 30, 7521−7526. (16) Wang, S.; Wang, Z.; Setyawan, W.; Mingo, N.; Curtarolo, S. Assessing the Thermoelectric Properties of Sintered Compounds via High-Throughput Ab-initio Calculations. Phys. Rev. X 2011, 1, 021012. (17) Carrete, J.; Li, W.; Mingo, N.; Wang, S.; Curtarolo, S. Finding Unprecedentedly Low-Thermal-Conductivity Half-Heusler Semiconductors via High-Throughput Materials Modeling. Phys. Rev. X 2014, 4, 011019. (18) van Roekeghem, A.; Carrete, J.; Oses, C.; Curtarolo, S.; Mingo, N. High-Throughput Computation of Thermal Conductivity of HighTemperature Solid Phases: the Case of Oxide and Fluoride Perovskites. Phys. Rev. X 2016, 6, 041061. (19) Woods-Robinson, R.; Broberg, D.; Faghaninia, A.; Jain, A.; Dwaraknath, S. S.; Persson, K. A. Assessing High-Throughput Descriptors for Prediction of Transparent Conductors. Chem. Mater. 2018, 30, 8375−8389. (20) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133. (21) Sham, L. J.; Schlüter, M. Density-Functional Theory of the Energy Gap. Phys. Rev. Lett. 1983, 51, 1888. (22) Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 11169. (23) Kresse, G.; Furthmüller, J. Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15−50. (24) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865. (25) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 50, 17953. (26) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758. (27) Togo, A.; Tanaka, I. First Principles Phonon Calculations in Materials Science. Scr. Mater. 2015, 108, 1−5. (28) Chaput, L.; Togo, A.; Tanaka, I.; Hug, G. Direct Solution to the Linearized Phonon Boltzmann Equation. Phys. Rev. Lett. 2013, 110, 265506. (29) Togo, A.; Chaput, L.; Tanaka, I. Distributions of Phonon Lifetimes in Brillouin Zones. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91, 094306. (30) Murphy, K. Machine Learning: A Probabilistic Perspective; MIT Press, 2012. (31) Rasmussen, C. E. Advanced Lectures on Machine Learning; Springer, 2004; pp 63−71. (32) Williams, T. M.; Fletcher, R. Practical Methods of Optimization. Vol. 2 Constrained Optimization. Oper. Res. 1982, 33, 675−676. (33) Jain, A.; Ong, S. P.; Hautier, G.; Chen, W.; Richards, W. D.; Dacek, S.; Cholia, S.; Gunter, D.; Skinner, D.; Ceder, G.; Persson, K. a. Commentary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Mater. 2013, 1, 011002. (34) Li, S.; Zheng, Q.; Lv, Y.; Liu, X.; Wang, X.; Huang, P. Y.; Cahill, D. G.; Lv, B. High Thermal Conductivity in Cubic Boron Arsenide Crystals. Science 2018, 361, 579−581.

SISSO model using proposed descriptors and Slack descriptors, comparison table for DFT and experimental κl, and comparison table for DFT and ML-predicted κl for eight independent compounds (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Rinkle Juneja: 0000-0001-6372-1524 George Yumnam: 0000-0001-9462-7434 Abhishek K. Singh: 0000-0002-7631-6744 Author Contributions †

R.J. and G.Y. have equal contributions.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Atsuto Seko, Atsushi Togo, and Isao Tanaka from Department of Materials Science and Engineering, Kyoto University, Kyoto, Japan for the useful discussions and providing computational facilities. R.J., G.Y., and A.K.S. acknowledge Isao Tanaka and the Kyoto University for supporting their stay to carry out initial part of work. R.J. acknowledges Morgana Ribas for useful inputs. R.J. thanks DST for INSPIRE fellowship (IF150848). The authors thank the Materials Research Centre, Thematic Unit of Excellence, Materials Informatics Initiative of IISc (MI3), and Supercomputer Education and Research Centre, Indian Institute of Science, for providing computing facilities.

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REFERENCES

(1) Bell, L. E. Cooling, Heating, Generating Power, and Recovering Waste Heat with Thermoelectric Systems. Science 2008, 321, 1457− 1461. (2) Snyder, G. J.; Toberer, E. S. Complex Thermoelectric Materials. Nat. Mater. 2008, 7, 105−114. (3) Toberer, E. S.; Zevalkink, A.; Snyder, G. J. Phonon Engineering Through Crystal Chemistry. J. Mater. Chem. 2011, 21, 15843−15852. (4) Yeh, L.; Chu, R. Thermal Management of Microelectronic Equipment: Heat Transfer Theory; ASME Press, 2002. (5) Arik, M.; Becker, C. A.; Weaver, S. E.; Petroski, J. Thermal Management of LEDs: Package to System. Third International Conference on Solid State Lighting, 2004; pp 64−76. (6) Wilcox, W. R. Heat Transfer in Power Transistors. IEEE Trans. Electron Devices 1963, 10, 308−313. (7) Seko, A.; Togo, A.; Hayashi, H.; Tsuda, K.; Chaput, L.; Tanaka, I. Prediction of Low-Thermal-Conductivity Compounds with FirstPrinciples Anharmonic Lattice-Dynamics Calculations and Bayesian Optimization. Phys. Rev. Lett. 2015, 115, 205901. (8) Curtarolo, S.; Hart, G. L. W.; Nardelli, M. B.; Mingo, N.; Sanvito, S.; Levy, O. The High-Throughput Highway to Computational Materials Design. Nat. Mater. 2013, 12, 191. (9) Carrete, J.; Mingo, N.; Wang, S.; Curtarolo, S. Nanograined Half-Heusler Semiconductors as Advanced Thermoelectrics: An Ab Initio High-Throughput Statistical Study. Adv. Funct. Mater. 2014, 24, 7427−7432. (10) Toher, C.; Plata, J. J.; Levy, O.; de Jong, M.; Asta, M.; Nardelli, M. B.; Curtarolo, S. High-Throughput Computational Screening of Thermal Conductivity, Debye Temperature, and Grüneisen parameter using a Quasiharmonic Debye Model. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 174107. (11) Oliynyk, A. O.; Antono, E.; Sparks, T. D.; Ghadbeigi, L.; Gaultois, M. W.; Meredig, B.; Mar, A. High-Throughput MachineF


Article

Chemistry of Materials (35) Zheng, Q.; Polanco, C. A.; Du, M.-H.; Lindsay, L. R.; Chi, M.; Yan, J.; Sales, B. C. Antisite Pairs Suppress the Thermal Conductivity of BAs. Phys. Rev. Lett. 2018, 121, 105901. (36) Ravichandran, N. K.; Broido, D. Non-monotonic Pressure Dependence of the Thermal Conductivity of Boron Arsenide. Nat. Commun. 2019, 10, 827. (37) Rajan, A. C.; Mishra, A.; Satsangi, S.; Vaish, R.; Mizuseki, H.; Lee, K.-R.; Singh, A. K. Machine-Learning Assisted Accurate Band Gap Predictions of Functionalized MXene. Chem. Mater. 2018, 30, 4031−4038. (38) Mishra, A.; Satsangi, S.; Rajan, A. C.; Mizuseki, H.; Lee, K.-R.; Singh, A. K. Accelerated Data-Driven Accurate Positioning of the Band Edges of MXenes. J. Phys. Chem. Lett. 2019, 10, 780−785. (39) Morelli, D. T.; Slack, G. A. High Thermal Conductivity Materials; Springer, 2006; pp 37−68. (40) Ouyang, R.; Curtarolo, S.; Ahmetcik, E.; Scheffler, M.; Ghiringhelli, L. M. SISSO: A Compressed-sensing Method for Identifying the Best Low-dimensional Descriptor in an Immensity of Offered Candidates. Phys. Rev. Mater. 2018, 2, 083802.

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