Discovery of low thermal conductivity compounds with first-principles anharmoniclattice dynamics calculations and Bayesian optimization

Atsuto Seko,1, 2, ∗ Atsushi Togo,2, ∗ Hiroyuki Hayashi,1 Koji Tsuda,3 Laurent Chaput,4 and Isao Tanaka1, 2, 5, †

1Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan2Center for Elements Strategy Initiative for Structure Materials (ESISM), Kyoto University, Kyoto 606-8501, Japan

3Department of Computational Biology, Graduate School of Frontier Sciences,The University of Tokyo, Kashiwa 277-8561, Japan

4Institut Jean Lamour, UMR CNRS 7198, Universite de Lorraine,Boulevard des Aiguillettes, BP 70239, 54506 Vandoeuvre Les Nancy Cedex, France

5Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan(Dated: June 23, 2015)

Compounds of low lattice thermal conductivity (LTC) are essential for seeking thermoelectricmaterials with high conversion efficiency. Some strategies have been used to decrease LTC. However,such trials have yielded successes only within a limited exploration space. Here we report the virtualscreening of a library containing 54,779 compounds. Our strategy is to search the library throughBayesian optimization using for the initial data the LTC obtained from first-principles anharmoniclattice dynamics calculations for a set of 101 compounds. We discovered 221 materials with verylow LTC. Two of them have even an electronic band gap < 1 eV, what makes them exceptionalcandidates for thermoelectric applications. In addition to those newly discovered thermoelectricmaterials, the present strategy is believed to be powerful for many other applications in whichchemistry of materials are required to be optimized.

Thermoelectric generators are essential for utilizingotherwise waste heat. Because of the technological im-portance, researchers have been seeking materials withhigh conversion efficiency for decades[1–4]. Compoundsof low lattice thermal conductivity (LTC) are essentialfor this purpose. Different strategies have been usedto decrease LTC. Recently, high throughput screening(HTS) of materials using materials database constructedby first principles calculations has been recognized asan efficient tool for accelerated materials discovery[5–9].Thanks to the recent progress of computational powerand techniques, a large set of first principles calculationscan be performed with the accuracy comparable to ex-periments. This is a straightforward strategy when bothof the following conditions are satisfied: 1) the targetphysical property can be accurately computed by firstprinciples methods. 2) The exploration space is well de-fined and not too large to compute the target physicalproperty exhaustively in the space.

In order to evaluate LTC with the accuracy compara-ble to experimental data, however, we need to developa method that is far beyond the ordinary density func-tional theory (DFT) calculations. Since we need to treatmultiple interactions among phonons, or anharmonic lat-tice dynamics, the computational cost is many orders ofmagnitudes higher than the ordinary DFT calculations.Such expensive calculations are practically possible onlyfor a small number of simple compounds. HTS of a largeDFT database of LTC is not a realistic approach un-less the exploration space is narrowly confined. In theyear 2014, Carrete and coworkers concentrated their ef-forts to search low LTC materials within half-Heuslercompounds[10]. They made HTS of wide variety of half-

Heusler compounds by examination of thermodynamicalstability via DFT results. Then LTC was estimated ei-ther by full first principles calculations or by a machine-learning algorithm for a selected small number of com-pounds. HTS of low LTC using a quasiharmonic Debyemodel was also reported in 2014[11]. Efficient predictionof LTC through compressive sensing of lattice dynam-ics was recently demonstrated[12]. Development of suchnew methods would bring accelerated discovery of newmaterials in the future.

In the present study, we do not want to restrict theexploration space by empirical knowledge, for example,by crystal structure. We firstly evaluated LTC of 101compounds with three prototype structures, i.e., rock-salt, zincblende and wurtzite-type structures, by first-principles anharmonic lattice dynamics calculations andsolving Boltzmann transport equation with the single-mode relaxation-time approximation[13, 14]. Then theresults are used to construct a model for making “virtualscreening” of many compounds in a library with a diver-sity of structures and chemical compositions employingBayesian optimization procedure. For the Bayesian opti-mization, predictors are determined by kriging method tofind the lowest LTC compound among the 101 first prin-ciples data. The highly ranked compounds are suppliedto first principles LTC calculations to verify the result ofthe screening.

Computational procedure of LTC is described in de-tail elsewhere[13]. LTCs were calculated from phononlifetimes, group velocities, and mode-heat capacities solv-ing the phonon Boltzmann transport equation within therelaxation time approximation. The phonon propertieswere calculated from the force constants. We employed

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first-principles calculation to obtain second-order forceconstants (FC2) and third-order force constants (FC3)with the supercell and finite displacement approaches.Phonopy code was used for these phonon calculations[14].Finite displacements of 0.03 A were systematically in-troduced to perfect supercells to fill up all elements offorce constant tensor elements among atoms in the su-percells. The Brillouin zone integration for the phononlifetime calculation was performed by the linear tetrahe-dron method.

For the first principles calculations, we employedthe plane-wave basis projector augmented wave (PAW)method[15] in the framework of DFT and the generalizedgradient approximation of the Perdew-Burke-Ernzerhof(PBE) form[16] as implemented in the VASP code[17–19]. Much more attention for the convergence of DFTcalculations should be paid in the phonon calculationsas compared to the ordinary first principles calculationswith respect to the k-point mesh, plane wave energy cut-off and tolerances of energy, residual force and stress.The size of the supercell was chosen by observing theconvergence of phonon properties by changing the su-percell size. Low LTC crystals are generally more an-harmonic and the atomic interaction range is consideredrelatively large. LTC calculations of the highly rankedcompounds required larger supercells than those for or-dinary crystals with smaller anharmonicity. The planewave energy cutoff was chosen to be at least 20% higherthan the recommended values in the PAW dataset. To-tal energies were minimized until the energy convergencesbecame less than 10−8 eV.

Results of first principles LTC of 101 compounds areshown with crystalline volume per atom, V , and den-sity, ρ, in Figs. 1 (a) and (b). Among 101 com-pounds, PbSe with the rocksalt structure show the lowestLTC, 0.9 W/mK (@300 K). It is in the similar trend asthe recent report showing low LTC for lead- and tin-chalcogenides[20–23]. The computed results are com-pared with available experimental data in Fig. 1 (c).Satisfactory agreements between experimental and com-puted results are evident in Fig. 1 (c), demonstratingthe usefulness of the first principles LTC data for furtherstudies. A phenomenological relationship has been pro-posed that log κL is proportional to log V [24]. Althoughqualitative correlation can be seen between our LTC andV , it is difficult to predict LTC quantitatively, hence todiscover new compounds with low LTC, only from thephenomenological relationship. It can be noted that thedependence on V is remarkably different between rocksalttype and zincblende or wurtzite type compounds, whilezincblende and wurtzite type compounds show similarLTC when the chemical compositions are the same.

The 101 first principles LTC data are then used tomake a model for the prediction of LTC of compoundswithin a library on the basis of the Bayesian optimization.For the purpose of the prediction, it is preferable to select

“good” predictors. Our rule of thumb is as follows: 1)whenever experts’ knowledge is available as a physical orphenomenological rule, it should be examined as the firststep. 2) Predictors may be better included in a library orthose easily made by combining the physical quantitiesin a library. Alternatively, the predictors may be easilycomputed by DFT calculations. 3) High efficiency for theBayesian optimization procedure needs to be examined.

On the basis of these ideas, we firstly determine pre-dictors for the Bayesian optimization procedure to findthe lowest LTC compound among the 101 first princi-ples LTC data. We adopt kriging method based on theGaussian process regression (GPR)[25, 26] of LTC sim-ply using two physical quantities, V and ρ, as predic-tors. These quantities are available in most of the experi-mental or computational crystal structure database, suchas ICSD[27], Atomwork[28], Materials Project Database(MPD)[29], and aflowlib[30]. Although a phenomenolog-ical relationship has been proposed between log κL andV [24], the correlation between them is not so high. Thecorrelation between log κL and ρ is even worse.

We start from an observed data set of 5 compoundsthat is randomly chosen from 101 compounds. In thekriging, a compound with maximum probability of im-provement among the remaining data is searched, namelya compound with the highest Z-score derived from GPR.The compound is included into the observed data setand then another compound with maximum probabilityof improvement is searched. Both the kriging and ran-dom searches are repeated fifty times and the averagenumber of observed compounds required for finding thecompound with the lowest LTC is examined.

When − log κL is expressed as f , Z-score for a com-pound with predictors x∗ is defined as

Z(x∗) = [f(x∗) − fbest] /√v(x∗) (1)

where f(x∗) and v(x∗) denote the predicted value of− log κL and its prediction variance at a point expressedby predictors x∗, respectively. v(x∗) is expected tobe small for compounds near the observed data, whileit can be large for compounds far from the observeddata. fbest denotes the lowest LTC value among “ob-served” compounds, which is updated at each krigingstep. Z-score that is evaluated by dividing [f(x∗) − fbest]by the square root of the prediction variance,

√v(x∗)

tends to select candidates with maximum probability ofimprovement[31]. Here the prediction and its varianceare described using the Gaussian kernel function. There-fore, our GPR has two free parameters, i.e. variances ofGaussian kernel and prior distribution. Here, they aregiven as 20 and 0.1, respectively.

Figure 2 (a) shows the result of the kriging search incomparison to the random search of the lowest LTC com-pounds within the 101 compounds. The average num-bers of compounds required for the optimization usingthe kriging and random searches, Nave, are 11 and 55,

3

100

101

102

103

100 101 102 103

BaO

CaO

KBr

KClKF

KILiBr

LiF

LiHMgO

NaBr

NaCl

NaF

NaI PbS

PbSePbTe

RbBrRbCl

RbF

RbI

SrO

AlN

AlPAlSb

BN

BP

CdTe

GaAs

GaP

GaSb

InAs

InP

InSb

ZnS

ZnSeZnTe

AgI

AlNBeO

CdS

CuBr

CuCl

CuI

GaNSiC

ZnO

LTC

ca

lcu

latio

n (

W/m

∙K)

LTC experiment (W/m∙K)

100

101

102

103

10

Rocksalt-typeZincblende-typeWurtzite-type

BaO

BaS

BaSeBaTe

CaOCaS

CaSe

CaTeCdO

CsF

KBr

KClKF

KH

KILiBr

LiCl

LiF

LiH

LiI

MgO

NaBr

NaCl

NaFNaH

NaIPbS

PbSePbTe

RbBr

RbClRbF

RbH

RbI

SrO

AgI

AlAs

AlN

AlPAlSb

BAs

BeO

BeS

BeSe

BeTe

BN

BP

CdS

CdSe

CdTe

CuBr

CuCl

CuH

CuI

GaAs

GaN

GaP

GaSbInAs

InN

InP

InSbMgTe

SiC

ZnO

ZnS

ZnSeZnTe

20 30 40 505 60

LTC

ca

lcu

latio

n (

W/m

∙K)

Volume (Å3/atom)

100

101

102

103

1 10

Rocksalt-typeZincblende-typeWurtzite-type

5

LTC

ca

lcu

latio

n (

W/m

∙K)

Density (g/cm3)

a b

c

FIG. 1. LTC calculated from first principles for 101 compounds along with (a) volume, V , and (b) density, ρ. (c) ExperimentalLTC data are shown for comparison when experimental LTCs are available.

0 20 40 60 80 100

Number of observations

Kriging with 2 descriptors

(Nave

= 11)

Kriging with 2 descriptors

+ 34 elemental descriptors

(Nave

= 19)

Random

(Nave

= 55)

0 20 40 60 80 100

Number of observations

PbSe LiI

Kriging with 2 descriptors

(Nave

= 65)

Random

(Nave

= 50)

10

1

50

5

Min

imu

m o

f L

TC

(W

/m∙K

)

Kriging with 2 descriptors

+ 34 elemental descriptors

(Nave

= 19)

a b

FIG. 2. Lowest LTC values at each iteration in kriging searchfor finding (a) PbSe and (b) LiI. Those by random searches arealso shown for comparison. When performing a kriging searchfor finding LiI, PbSe and PbTe are intentionally omitted andthe rest of 99 compounds are used.

respectively. The compound with the lowest LTC amongthe 101 compounds, i.e., rocksalt PbSe, can be foundmuch more efficiently using the kriging technique andonly with two variables, V and ρ. However, we realizethat the kriging only with these two variables is not arobust way for finding the lowest LTC. As an example,Fig. 2 (b) shows the result of the kriging search using thedataset after intentionally removing 1st and 2nd lowestLTC compounds, i.e., rocksalt PbSe and PbTe, from the101 compounds. Then rocksalt LiI should be the rightanswer of the optimization. However, Nave is 65 for find-ing LiI using the kriging only with V and ρ, which islarger than that of the random search, Nave = 50. Thedelay of the optimization should originate from the factthat LiI is an outlier when LTC is modeled only with Vand ρ. Such outlier compounds with low LTC are difficult

to find only with V and ρ.

In order to overcome the outlier problem, we add pre-dictors about constituent chemical elements. There aremany choices for such variables: They are, for exam-ple, electronegativity, atomic radius, ionization energy,etc[26]. Here, we newly introduced “elemental descrip-tors”, which is a set of binary digits representing thepresence of chemical elements. Since the 101 LTC datais composed of 34 kinds of elements, we use 34 elementaldescriptors. Results of the kriging are shown in Figs. 2(a) and (b) with 34 elemental descriptors on top of Vand ρ. In both cases, the compound of the lowest LTC isfound with Nave = 19. The use of the elemental descrip-tors is found to improve the robustness of the efficientsearch.

As described in the Supplemental Material (SM), bet-ter correlations with LTC can be found for parametersthat are obtained from phonon density of states. How-ever, we do not use such phonon parameters as predictorsin the present study, because there is no data libraryavailable for such phonon parameters for a wide rangeof compounds. Hereafter, we show results only with thepredictor set composed of 34 elemental descriptors on topof V and ρ.

Screening for low LTC compounds over compounds in alarge library is carried out using a GPR prediction model.Such a screening based on a prediction model is called“virtual screening” in biomedical communities[32]. Forthe virtual screening, we adopt all 54,779 compounds inMPD library[29, 33], which is composed of most of crystalstructure data available in ICSD[27]. On the basis of theGPR prediction model made by V , ρ and 34 elementaldescriptors for the 101 LTC data, a ranking for low LTCcompounds is made according to the Z-score of the 54,779compounds.

Figure 3 shows the distribution of Z-scores for the54,779 compounds along with V and ρ. The magni-

4

0-5-10-15-20-25

Z-score of LTC

log V

log

ρ

FIG. 3. Dependence of Z-score on constituent elements forcompounds in the MPD library. The magnitude of Z-scoreis shown by colors along with volume, V , and density, ρ, foreach element.

tude of Z-score is plotted in panels corresponding to con-stituent elements. (Transition metal and other elementsare shown in SM.) The Z-score is relative to rocksaltPbSe showing the lowest LTC among 101 compounds.Among 54,779 compounds, 221 compounds show posi-tive Z-score, which are expected to have lower LTC thanthat of rocksalt PbSe, i.e., < 0.9 W/mK (@300 K). Theyare highlighted by red dots. They are widely distributedin V −ρ space; which means it is difficult to pick them upwithout performing the Bayesian optimization with ele-mental descriptors. The Z-score is widely distributed forlight elements such as Li, N, O and F. This implies thatthe presence of such light elements by itself have littleeffects on lowering the LTC. When such light elementsform a compound with heavy elements, the compoundtends to show high Z-score. It is also noteworthy thatmany compounds composed of some light elements suchas Be and B tend to show high LTC.

Special features are recognized for Pb, Cs, I, Br andCl. Many compounds composed of these elements ex-hibit high Z-score. (The number of compounds with pos-itive Z-score is shown in SM.) Most of compounds show-ing positive Z-score have any of atomic combinations ofthese five elements. On the other hand, elements in thePeriodic table neighboring to these five elements do notshow analogous trends. For example, compounds withhigh Z-scores are rarely found for Tl and Bi, which areneighboring to Pb. This may sound odd since Bi2Te3 is afamous thermoelectric compound. This may be ascribedto our selection of the training dataset composed onlyof AB compounds with 34 elements and three kinds ofsimple crystal structures. In other words, the trainingdataset is somehow “biased”. This is unavoidable at themoment since the first-principles LTC calculations arestill too expensive to obtain sufficiently unbiased train-ing dataset with a large enough number of data to coverthe diversity of chemical composition and crystal struc-

Cs

I

Cl

Pd

K

Cd Pb

a b

Cs2[PdCl

4]I2

K2CdPb

FIG. 4. Crystal structures of K2CdPb and Cs2[PdCl4]I2predicted to show low LTC of < 0.5 W/mK (@300 K) andnarrow band gap of < 1 eV.

tures. Nevertheless, the “biased” training dataset will beverified to be useful for finding low LTC materials. Be-cause of the use of the “biased” training dataset, we maynot be able to discover all of the low LTC materials inthe library. However, we can discover at least a part ofthem.

Verification process for the candidates of low LTC com-pounds after the virtual screening is one of the most im-portant steps to “discover” low LTC compounds. Firstprinciples LTCs are evaluated for the top 8 compoundsafter the virtual screening. All of them are consideredto form ordered structures. LTC calculation was unsuc-cessful for Pb2RbBr5 due to the presence of imaginaryphonon modes within the supercell used in the presentstudy. Z-scores and first principles LTC of the rest of thecompounds are listed in Table 1. All of top 5 compoundsshow LTC of < 0.2 W/mK (@300 K), which are muchlower than that of the rocksalt PbSe, i.e., 0.9 W/mK(@300 K). This confirms the powerfulness of the presentGPR prediction model for efficiently discovering low LTCcompounds. Crystal structures of highly ranked com-pounds, PbRbI3, PbIBr, PbRb4Br6 and PbI2 (P63mc)are shown in SM. PbICl and PbClBr have the same crys-tal structures as PbIBr. PbI2 (R3m) and PbI2 (P63mc)are different only in their stacking sequences. All of thesecompounds contain either six-fold or eight-fold coordi-nated Pb by halogen ions, and are of stoichiometric chem-ical composition when Pb is divalent.

When such LTC materials are considered for ther-moelectric applications, properties related to electronicstructures, namely electronic contribution of the thermalconductivity, electrical conductivity and Seebeck coeffi-cient should also be optimized. Although they can betuned by elemental doping, the band gap, Eg, should bea simple measure of the electronic structure and allows todiscriminate in a simple way between materials that canbe good thermoelectrics or not. All of 221 compoundsshowing positive Z-score are listed in SM together withEg (DFT-PBE) given in the MPD library. Among themonly 19 compounds satisfy 0.1 < Eg < 1.0 eV. First prin-ciples LTCs are evaluated for them. Crystal structures

5

TABLE I. First principles LTCs and Z-scores for highlyranked compounds by the virtual screening. Band gaps byDFT-PBE are taken from MPD library[29, 33].

Ranking Z-score FormulaSpace LTC Band

group (W/mK) gap (eV)

1 1.90 PbRbI3 Pnma 0.10 2.46

2 1.76 PbIBr Pnma 0.13 2.56

3 1.56 PbRb4Br6 R3c 0.08 3.90

4 1.56 PbICl Pnma 0.18 2.72

5 1.56 PbClBr Pnma 0.09 3.44

7 1.44 PbI2 R3m 0.29 2.42

8 1.43 PbI2 P63mc 0.29 2.45

121 0.39 K2CdPb Ama2 0.45 0.18

144 0.29 Cs2[PdCl4]I2 I4/mmm 0.31 0.88

and LTC for two of them are shown in Fig. 4 and Ta-ble I. Both of K2CdPb and Cs2[PdCl4]I2 are predicted toexhibit LTC of less than 0.5 W/mK (@300 K) togetherwith band gap of smaller than 1 eV. The discovery ofsuch compounds may open a gate toward designing newthermoelectric materials with exceptionally high figure ofmerit.

In this study, we first report the theoretical LTC of 101compounds by first-principles anharmonic lattice dynam-ics calculations. Using these data, the virtual screeningof a library containing 54,779 compounds is performedby Bayesian optimization using kriging method based onthe Gaussian process regressions. 221 materials with verylow LTC are found from this screening. A final filteringof those low LTC compounds is made using the electronicband gap, which is a measure to discriminate in a simpleway between materials that can be good thermoelectricsor not. Two compounds with low LTC of < 0.5 W/mK(@300K) and narrow band gap of < 1 eV are thus dis-covered, which may open a gate toward designing newthermoelectric materials with exceptionally high figure ofmerit. The present method should be useful for searchingmaterials for many different applications in which chem-istry of materials are required to be optimized.

This work was supported by Grant-in-Aid for Scien-tific Research (A) and Grant-in-Aid for Scientific Re-search on Innovative Areas “Nano Informatics” (GrantNo. 25106005) from the Japan Society for the Promo-tion of Science (JSPS).

∗ These authors contributed equally to this work.† tanaka@cms.mtl.kyoto-u.ac.jp

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