Machine Learning-Aided Design of Materials with Target Elastic

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Machine Learning Aided Design of Materials with Target Elastic Properties Shuming Zeng, Geng Li, Yinchang Zhao, Ruirui Wang, and Jun Ni J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01045 • Publication Date (Web): 06 Feb 2019 Downloaded from http://pubs.acs.org on February 11, 2019

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The Journal of Physical Chemistry

Machine Learning Aided Design of Materials with Target Elastic Properties Shuming Zeng,† Geng Li,† Yinchang Zhao,¶ Ruirui Wang,† and Jun Ni∗,† State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China, Collaborative Innovation Center of Quantum Matter, Beijing 100084, People’s Republic of China, and Department of Physics, Yantai University, Yantai 264005, People’s Republic of China E-mail: [email protected]

∗ To

whom correspondence should be addressed Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China ‡ Collaborative Innovation Center of Quantum Matter, Beijing 100084, People’s Republic of China ¶ Department of Physics, Yantai University, Yantai 264005, People’s Republic of China † State

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Abstract A set of universal descriptors which combines atomic properties with crystal fingerprint are presented to build interpretable models for elastic property prediction. Using the wellperformed model, 100 materials with large predicted elastic moduli are screened out and then validated by the first-principles calculations. When perform projection analysis, we find compounds with large and small elastic moduli are clearly divided into two parts by the average value of volume and atomization enthalpy (∆Hatomic ), and the relation between them is given by two discriminant equations, suggesting that compounds composed elements with large ∆Hatomic are potential large elastic moduli materials. Following this rule, we design several new stable materials like ReTcB4 and ReB which have high elastic moduli. This method is valuable for high-throughput screening and materials design.

Introduction Elastic moduli reflect the ability of a solid to resist external forces, which is important for understanding the interatomic bonding nature. Besides, knowledge of the bulk and shear modulus is useful for estimating the mechanical and thermal properties. 1–5 In order to adapt to high-throughput screening of materials , intense efforts have been made to establish database of elastic properties based on first-principles computational methods. 6–8 Up to now, there are more than 5,000 unique compounds have been calculated via an automated, integrated workflow, and this number is growing steadily. 7,8 However, materials that have been explored are still very small fraction even in the inorganic crystal structure database which contains about 60,000 entries. 9 Hindered by the expensive computational requirements, it seems to be impossible to calculate all the theoretical possible materials since the total number is a googol (10100 ). 10 To speed up screening, more efficient methods beyond first-principles calculations should be developed. Recently, Machine learning (ML) techniques have emerged as a powerful tool in materials science. By constructing specific ML models, researchers have achieved notable success in addressing various problems. 11–13 For elastic moduli prediction, ML models based on gradient boosting de2


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cision tree (GBDT), 14 statistical learning (SL), 15 supporting vector machine, 16 and crystal graph based network 17,18 have appeared recently. In spite of the difference of approaches, they all show good predictive performance. However, most of them focus on predicting the properties, not designing new materials. In this paper, we construct deep neural network (DNN) model combined with atomic property encoded fingerprint (APRF) technique to predict the mechanical properties of inorganic materials. Aided by the well-trained model, 100 materials with high predicted elastic moduli are screened out from 18,493 stable inorganic compounds and then validated by the firstprinciples calculations. After a detailed projection analysis of the descriptors, we find the volume per atom (Vpa ) and the average of atomization enthalpy [avg(∆Hatomic )] are important for elastic moduli, and the relation between them can be given by two discriminant equations, which guides us to design several new stable materials, such as ReTcB4 and ReB, of which the elastic moduli are large.

Methods Descriptors have been demonstrated to be a key factor in influencing the performance. 19 Ideally, the selected descriptors should be associated with the target property and remain invariant under the symmetry transformation of the crystal structure. Besides, the feature vector of different compounds should have the same dimension. To satisfy those conditions, we construct a set of descriptors that can be divided into three parts: (i) crystal related properties (qc ), including space group, point group, lattice type, volume per atom (Vpa ) and density (ρ ); (ii) atom related properties (qa ), including atomic number, Mendeleev number, 20 period and group in the periodic table, atomic mass, the number of valence electrons, the orbital exponent of Slater-type orbitals, 21 global hardness, 22 polarizability, electrophilicity indices, 23 van der Waals radii, covalent radii (rcov ), absolute radii, 21 electron affinity, electronegativity, first ionization energy, boiling point, melting point (Tmp ), 24 molar volume, thermal conductivity, 25 atomization enthalpy (∆Hatomic ), fusion enthalpy, vaporization enthalpy, 24–26 binding energy and atomic density. To ensure that compounds

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with different number of atoms have the same feature size, for each attribute qc , its average value avg(qc ) is taken as a descriptor; (iii) crystal structure fingerprints (q f ) which are used to describe the local environment of the atom. The detailed construction of the fingerprints is given below.

Figure 1: (Color online) Workflow of the APRF based DNN model. The left is the schematic construction of feature vector, and the right is the ML algorithm. K and G represent bulk and shear modulus, respectively. Fingerprint technology is widely used for crystal structure prediction, in which quantifying the dissimilarities between structures is important. For structure-property modeling, we propose APRF which is schematically illustrated in Fig.1. Considering a crystal containing N atoms in the unit cell, the first step is to calculate the N × N distance matrix D . The element di j represents the distance between the atom i and the atom j, taking full consideration of the lattice periodicity. Then the corresponding adjacency matrix A is constructed according to the neighbor relation of the atoms. This matrix describes the local environment of each atom with element ai j = 1 if the atom i is closest to the atom j, and ai j = 0 otherwise. Finally, the information of the atom is encoded to the above two matrices. It is natural to represent atom using atomic attributes mentioned in part 4


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(ii), thence APRF can be defined as

qf =

N N a ϕ (q , q ) 1 i j ij ∑ ∑ d2 N 2 i=1,i̸ ij = j j=1

(1)

where the coefficient 1/N 2 is introduced to guarantee the invariance of q f when expanding the unit cell. ϕ (qi , q j ) is the interaction term, and in this paper, it is defined as ϕ (qi , q j ) = |qi − q j |. Combining all three part features, we constructed a total of 55 descriptors to uniquely describe each crystal, and a detailed list of them can be seen in Table SI. To bridge the gap between crystal structure and its physical properties, simulation methods based on ML algorithms are confirmed to be dramatically powerful when the exact mathematical form is unknown. In our models, the relationship between the target moduli and the descriptors are established by the DNN algorithm. DNN is a kind of neural network that contains one input layer, one output layer and multiple hidden layers. In Fig.S1, we plot the mean absolute error (MAE) as a function of the number of hidden layers on the test set. Clearly, two hidden layers network is much better than one hidden layer network, and is good enough in predicting both the shear and bulk modulus. The number of neurons in each hidden layer is optimized by minimizing the MAE.

Results and discussion Mechanical properties data for inorganic materials are extracted from Material Project repository. 7,27 There are 6,975 materials which have the calculated elastic moduli in this database, but some of them are unreasonable. Thus, we remove materials with negative elastic moduli and materials that have the elastic moduli larger than 500 GPa, and finally obtain 5,518 cleaned data. These data are randomly split into two independent data sets. The first set which contains 4,500 materials is used to build and assess the ML models through five-fold cross validation method. By minimizing MAE, the optimized parameters are determined to construct two models, one for bulk modulus (KV RH ) and the other for shear modulus ( GV RH ). 7 The five-fold cross validation results for these two DNN regression models are presented in Fig.S2 and summarized in Table SII. The 5



statistical metrics MAE and root mean square errors (RMSE) are used to characterize the overall errors. Roughly speaking, the absolute errors of the predictions for GV RH and KV RH are nearly the same, which can be reflected by RMSE (about 14 GPa) and MAE (about 10 GPa). For both the bulk and shear moduli, over 65% of the materials are predicted within the error of 10 GPa, and over 88% of materials within 20 GPa.

Figure 2: (Color online) Comparison of the DFT calculated results and the ML models predicted values on the test set which contains 1,018 independent materials. In order to further verify the performance of the model, the remaining independent 1,018 materials are treated as test set, and the test results are shown in Fig.2, with the relevant statistics summarized in Table SII. For both the shear and bulk moduli predictions, the MAE increased less than 2 GPa, which is better than the GBDT model (the MAE increased about 3 GPa), 14 indicating that the APRF based DNN models have strong generalization ability. In addition, we note that two different models based on SL 15 and crystal graph convolutional neural networks (CGCNN) 17 are developed to predict elastic properties using the same database. By using the same number of training data, our model outperforms the other two on test set, with RMSE of 0.123 log(GPa) for GV RH and 0.079 log(GPa) for KV RH , while the RMSEs of SL model are 0.138 log(GPa) and 0.075 log(GPa), and the RMSEs of CGCNN model are 0.127 log(GPa) and 0.105 log(GPa). Despite the differences in ML algorithms, we attribute the major reason for the better perfor6


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mance of our models to the construction of the the crystal features. The crystal features constructed in this paper are divided into three groups: avg(qa ), qc and q f , which contain the composition information (the average of atomic properties), global crystal properties (such as space group) and local crystal properties, respectively. These three part features provide complete descriptions of a crystal. The SL model only contains the composition information, while the crystal graph related models such as CGCNN only contains the local information of crystal. Although GBDT model uses 2,494 features to describe a crystal, these features do not satisfy the crystal symmetry. For example, when one expands the unit cell, it makes a very different prediction. In addition, the huge amount of parameters can lead to model overfitting, because the training data is only a few thousand. In the DNN models, we just use 55 crystal features. As mentioned above, until now, only a small fraction of materials have been explored for the elastic properties, due to the limitations in computing power. Even in the Material Project database, there are still 18,493 stable inorganic compounds of which the DFT calculated elastic moduli have not been provided. Applying our trained models to these materials (the crystal structures of these materials are extracted from the Material Project and are directly fed to the models), we obtained the elastic moduli of them. After that, we picked out 50 systems (no more than 25 atoms in the unit cell) with large GV RH and 50 systems with large KV RH , respectively. The predictions for the selected materials are then validated by the DFT calculations using projector-augmented waves (PAW) 28,29 and the generalized gradient corrected Perdew-Burke-Ernzerhof (PBE) 30 exchangecorrelation functional as implemented in the VASP code, 31,32 with the relevant results are presented in Table SV and Table SVI. After carefully examining the top 15 compounds (see Table I), we find four materials with large elastic moduli (TcB2 , OsB, Ta5 B6 , ReN) which are not screened out by SL model. In addition, the top 15 systems screened by SL model are all re-screened out(except for materials included in the training set), and for these systems, our models show better predictive performance with the MAEs reduced by approximately 15 GPa and 20 GPa for KV RH and GV RH , respectively. Although the DNN model shows excellent performance in terms of elastic moduli prediction, it

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Table 1: Top 15 compounds with large predicted elastic moduli screened from 18,493 stable inorganic materials. The DFT validated results are also presented. MP-ID mp-1019317 mp-10142 mp-2850 mp-867141 mp-997617 mp-28629 mp-1010 mp-15671 mp-1097 mp-260 mp-974435 mp-974437 mp-867264 mp-1019052 mp-974430

System GV RH (GPa) GV RH (GPa) KV RH (GPa) KV RH (GPa) (DNN) (DFT) (DNN) (DFT) TcB2 202.07 241.98 296.21 303.55 Ta3 B4 199.06 203.37 295.40 292.57 OsB2 195.65 172.63 331.09 313.93 ReOs3 193.06 242.97 375.31 390.07 OsB 193.03 211.37 355.51 360.04 Ta5 B6 192.34 208.74 296.55 290.37 MnB4 191.13 236.80 253.80 276.70 Re3 B 190.45 206.80 358.75 264.53 TaB 186.70 221.38 287.72 286.32 CrB 184.02 221.29 265.09 298.16 Re3 N 181.16 185.27 393.59 384.12 Re2 C 178.75 256.48 359.85 401.17 Re3 Os 182.34 181.96 354.58 371.28 ReN 169.63 173.57 347.08 374.33 Re3 Ir 172.10 198.11 340.17 358.09

Table 2: The space group, E f , Vpa , avg(∆Hatomic ) and the elastic moduli of the new designed materials. System

Space Group

OsTcB4 NbTcB4 ReTcB4 TaTcB4 ReB ReB4

187 187 187 187 187 71

Ef Vpa avg(∆Hatomic ) (eV/atom) (Å3 ) (kJmol−1 ) -0.361 9.05 617.00 -0.358 9.71 607.67 -0.438 9.11 614.83 -0.382 9.70 615.83 -0.283 10.73 669.50 -0.059 8.47 605.60

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GV RH (GPa) 239.44 165.38 264.39 194.59 233.08 199.45

KV RH (GPa) 324.28 267.61 326.73 278.59 353.35 292.24

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does not guide us in designing new materials. To achieve this goal, we turn to model interpretability, which enables us to understand the underlying physics. However, it is notoriously difficult to know how the DNN model works due to its black-box nature. Therefore, we propose an intuitive method to characterize the relative importance of descriptors. By continuously changing one descriptor xk [from -20% to 20%, increase 5% each time, totally 8 times (l)] and keeping others invariant, we obtained a series of corresponding MAEs (δk,i ). The importance of descriptor xk can be defined as p(xk ) =

1 l−1 |δk,i+1 − δk,i | . ∑ l − 1 i=1 δk,i

(2)

Larger p(xk ) means more important of xk . It is should be emphasized that p(xk ) can only be a reference for importance assessment, because of the complex nonlinear relations between descriptors and crystal properties. Utilizing p(xx), we filter out descriptors that have significant impact on elastic moduli prediction, and the top five are Vpa , ρ , avg(Tmp ), avg(∆Hatomic ) and avg(rcov ). This finding indicates the fact that elastic properties are strongly correlated with bond strength, since these descriptors are bond-length related. 8 Besides, when projected the whole data set onto the dual space of descriptors, the elastic moduli are clearly partitioned into two regions in avg(∆Hatomic )Vpa space, as shown in Fig.3. For almost all the large elastic moduli materials, Vpa is less than 20 (Å3 ), and the avg(∆Hatomic ) is greater than 370 (kJmol−1 ). Crystals with small elastic moduli are located above the dashed lines, while large elastic moduli materials are under the lines. The dashed lines for GV RH and KV RH are given by the discriminant equations Dg = 0 and Dk = 0, respectively, which are defined as: Dg = Vpa − 0.041 × avg(∆Hatomic ) − 2.39,

(3)

Dk = Vpa − 0.062 × avg(∆Hatomic ) + 4.91.

(4)

For Dg > 0 and Dg < 0, the average values of GV RH are 79 GPa and 31 GPa, respectively. For Dk > 0 and Dk < 0, the average values of KV RH are 165 GPa and 67 GPa, respectively. The patterns obtained by ML models and the DFT calculations are almost the same, which means that the model captures the main physics of elastic moduli. 9



Figure 3: (Color online) Projection of the DFT calculated value and the prediction value in the avg(∆Hatomic )-Vpa space. The bulk and shear moduli are rescaled to the range of [0,100] and [0,200], respectively. As seen in Fig.3, large avg(∆Hatomic ) and small Vpa can lead to large elastic moduli, suggesting that materials composed of transition metal elements especially for V, Pt, Zr, Hf, Nb, Ta, Mo, W, Tc, Re, Ru, Os, Ir (∆Hatomic > 500 kJmol−1 ) and non-metallic elements B, C, N, Si (∆Hatomic > 400 kJmol−1 ) 24,26 with close-packed structure or short bond (like diamond) are more likely to have large elastic moduli. In fact, these elements are included in the top 100 compounds with large moduli except for Al2 O3 and Be3 Ni (see Table SIII and Table SIV). According to this practical heuristic rule, we can change the composition of the compounds to change its mechanical properties. For example, among the screened materials, TcB2 (two Tc atoms and four B atoms in the unit cell) has the largest shear modulus of 241.98 GPa, and its bulk modulus is 303.55 GPa. However, the ∆Hatomic of element Tc is not the largest. Therefore, we replace one Tc atom with Ta, W, Re and Os atom, and re-relax these structures by VASP. We do not replace all the Tc atoms because replacing one atom is enough to illustrate how to design new materials with large elastic moduli and XB2 (X=Ta, W, Re, Os) exist in the Material Project. After that, the formation energy (E f ) and the mechanical properties are calculated by VASP and summarized in Table II. Surprisingly,

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the shear and bulk modulus of ReTcB4 are 264.39 GPa and 326.73 GPa, respectively, which are almost 23 GPa larger than TcB2 . In addition, to obtain large elastic moudli materials, we can also specify elemental composition according to avg(∆Hatomic ) to search for materials with small Vpa . For example, with the help of the evolutionary algorithm USPEX, 33,34 we construct thousands of possible structures of Tcx Rey Bz (see Fig.S3). Among these materials, two new materials (not existing in the database), ReB and ReB4 , are picked out with low E f and small Vpa . The DFT calculations validated that these materials are indeed large elastic moduli materials (see Table II). Furthermore, the phonon spectrums calculated by Phonopy 35 indicate that ReTcB4 and ReB are dynamic stable (see Fig.4). (a)

Re B

Tc

(b)

Re B

Figure 4: (Color online) The crystal structures (left) and the phonon spectrums (right) of ReTcB4 (a) and ReB (b). All the structure files of the new designed materials are provided in the Supporting Information.

conclusions In summary, we construct a set of universal descriptors to build the DNN models. This approach shows high accuracy for elastic moduli prediction with small deviations as compared with the DFT results. By applying these models to 18,493 stable inorganic materials in the Materials Project 11



database, we have screened out 100 compounds that are potential high elastic moduli materials and validated them by the first-principles calculations. In addition, we proposed a descriptor analysis method to improve the model interpretability, and find that the strength of bond is the crucial factor to affect both the bulk and shear moduli, which is consistent with the common physics insight. Furthermore, when projected to the avg(∆Hatomic )-Vpa space, the entire data set is dramatically clearly partitioned into two parts by the discriminant equations, suggesting that compounds with elements of large ∆Hatomic and small volume are promising large elastic moduli materials. Following this rule, we design several new stable materials materials beyond the database, such as ReTcB4 and ReB, which have large elastic moudli. Our method paves the way for quick sieving and materials design for desired mechanical properties.

Acknowledgement This research was supported by the National Key Research and Development Program of China under Grants No. 2016YFB0700102, the National Natural Science Foundation of China under Grants No. 11774195 and No. 11704322; the Natural Science Foundation of Shandong Province for Doctoral Program under Grant No. ZR2017BA017.

Supporting Information Available Complete set of the descriptors (Table SI), a detailed example to construct the feature vectors (in Subsection “Feature vector construction”), the detailed hyperparameters and statistic results of the DNN models (Table SII), the MAE as a function of the number of hidden layers (Fig.S1), the comparisons between the five-fold cross validated results and the DFT results (Fig.S2), the possible crystals of Rex Tcy Bz generated by USPEX (Fig.S3), the top 100 materials with large calculated bulk (Table SIII) and shear moduli (Table SIV), and the top 50 screened materials with large predicted bulk (Table SV) and shear moduli (Table SVI) along with the DFT results. Complete data set to construct and test the models, and the structures data for the new designed materials. This material is available free of charge via the Internet at http://pubs.acs.org/. 12


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Machine Learning-Aided Design of Materials with Target Elastic

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