Predicting band gaps and band-edge positions of oxide perovskites using DFT and machine learning
Wei Li, ZigengWang, Xia Xiao, Zhiqiang Zhang, Anderson Janotti1, Rajasekaran Sanguthevar, Bharat Medasani
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Predicting band gaps and band-edge positions of oxideperovskites using DFT and machine learning
Wei Li *, Zigeng Wang *, Xia Xiao , Zhiqiang Zhang , Anderson Janotti , , Rajasekaran Sanguthevar & Bharat Medasani , Department of Materials Science and Engineering, University of Delaware, Newark, DE 19716,USA ComputerScience andEngineeringDepartment,UniversityofConnecticut,CT 06269,USA Department ofPhysics,UniversityofDelaware, Newark,DE 19716,USA Delaware EnergyInstitute,UniversityofDelaware, Newark, DE 19702,USA Princeton PlasmaPhysicsLaboratory,Princeton,NJ 08540* These authors contributed equally to this work.
Density functional theory within the local or semilocal density approximations (DFT-LDA/GGA)has become a workhorse in electronic structure theory of solids, being extremely fast and re-liable for energetics and structural properties, yet remaining highly inaccurate for predictingband gaps of semiconductors and insulators. Accurate prediction of band gaps using first-principles methods is time consuming, requiring hybrid functionals, quasi-particle GW, orquantum Monte Carlo methods. Efficiently correcting DFT-LDA/GGA band gaps and un-veilling the main chemical and structural factors involved in this correction is desirable fordiscovering novel materials in high-throughput calculations. In this direction, we use DFTand machine learning techniques to correct band gaps and band-edge positions of a repre- entative subset of ABO perovskite oxides. Relying on results of HSE06 hybrid functionalcalculations as target values of band gaps, we find a systematic band gap correction of ∼ ∼ ∼ The band gap and band-edge positions (i.e., ionization energy and electron affinity) are basic prop-erties of semiconductors and insulators, and often dictate the suitability of materials for deviceapplications. Their prediction, based on first-principles methods, is key to novel materials dis-covery. DFT calculations
1, 2 based on LDA or GGA are often used to predict stable crystalstructures, with lattice parameters within 1-2% of the experimental values
5, 6 . These calculationsare extremely fast and scalable, permitting the study of the energetic and structural properties ofthousands of materials with relatively modest computing resources and in relatively short times,playing a central role in current materials discovery research efforts based on high-throughputcomputation. However, when standard LDA or GGA functionals are employed, band gaps (Eg)predicted by DFT are severely underestimated in comparison to experimental values. Predicting E g of semiconductors and insulators requires going beyond LDA or GGA approximations in DFT,making the calculations much more involved and computationally expensive.Methods that accurately predict band gaps are very expensive with respect to both compu-tational resources and wall time. The simplest approach is to mix Fock exchange with GGA ex-2 e α a (a) (b) (c) (d)(e) Figure 1: Crystal structures of ABO perovskite prototypes and selected A and B atoms.Crystal structure of (a) P m ¯3 m cubic, (b) I m mm tetragonal, (c) P nma orthohombic, and(d) R ¯3 c rhombohedral strutures of ABO perovskites. Green, blue and red spheres repre-sent of A, B and O atoms, respectively. The apical and equatorials B-O-B bond angles, α a and α e are indicated in (e). The A and B atoms selected for this study are indicated inthe Periodic Table in the lower panel. , partially correcting the self-interaction error in DFT-LDA/GGA,giving band gaps very close to the experimental values for many materials . This increases thecomputation time by ten fold compared to DFT-LDA/GGA calculations. More formally rigorousapproaches would be to use the Greens function quasi-particle GW or the wavefunction-basedquantum Monte Carlo method, yet at the expense of at least an extra order of magnitude incomputational time. As a result, these are not generally amenable to high-throughput computa-tional approaches, posing a stringent obstacle to novel materials discovery.Machine learning (ML) techniques have emerged as powerful tools in materials science re-search, with applications in a variety of directions, such as prediction and classification of crystalstructures and building predictive models of various materials properties . Recent effortsalso include predicting band gaps, however with limited accuracy . A straightforward directionwould be to predict band gaps using the DFT-GGA band structures available in AFLOW database as a training set for machine learning approaches. However, this would have limited use consider-ing that the predicted band gaps would still be severely underestimated. Or one could use DFT+U for band gaps, with computational costs similar to those of DFT-LDA/GGA; the problem is whatvalue of U to choose and the justification of applying U to dispersive valence and conductionbands. An interesting approach involves graph convolutional neural networks (CGCNN) basedon atomic connections in the crystal structure after being trained using DFT band gaps . How-ever, this method was also trained and aimed at DFT-GGA band gaps. Recently, reports on auto-mated, high-throughput calculations of band gaps based on hybrid functional have appeared in theliterature , pointing toward more reliable predictions of band gaps, yet the nature and size of4he band-gap corrections from the DFT-GGA values have not been discussed or analyzed.In this work we developed machine learning models for mapping band gaps computed withDFT-GGA into band gaps with higher accuracy HSE06 hybrid functional. We chose perovskiteoxides as example to demonstrate the applicability of our approach. An interesting feature ofABO perovskite semiconductors and insulators is the dependence of their band gaps on the metalelements A and B as well as on rotations and tilting of the BO octahedra. Here we restrictedthe scope of perovskite materials to those for which the valence band is derived from oxygen 2 p orbitals and the conduction band is derived from A or B valence orbitals, as indicated in Fig. 1.We did not consider perovskites where the valence and conduction bands are determined by tran-sition metal d orbitals and the gap associated with spin-splitting of d bands or d - d transitions. Weexplicitly included octahedral tilting and rotations leading to tetragonal, orthorhombic, and rhom-bohedral crystal structures as shown in Fig. 1. Using a high throughput approach , we calculatedthe band structures of the perovksites with PBEsol and HSE06 functionals. We analyzed the map-ping of valence band maximum (VBM) and conduction band minimum (CBM) between PBEsoland HSE06 functionals by employing different machine learning models. Our combined DFT-MLmodel predicts E g within an error of 0.16 eV to that of HSE computed E g , and reveals the mainatomic and structural factors that determine the correction to the VBM, CBM, and consequently E g predicted at GGA level. 5 a) PBEsol band gap (eV) H SE band gap ( e V ) Eg correctionCBM correctionVBM correction
HSE band gap (eV) E ne r g y ( e V ) (b) (c) PBEsol HSECBMVBM 0.5 eV1.0 eV Figure 2: Correction of the band gap of ABO perovskites based on HSE06 and DFT-GGAPBEsol calculations. (a) HSE06 vs PBEsol band gaps, (b) the band-gap correction ( ∆ E g ,light green), correction of the valence-band maximum ∆VBM (light blue) and conduction-band minimum ∆CBM (dark red) vs HSE06 band gap. (c) schematic of the correction ofthe band-edge of the positions. The dashed line in (a), placed to guide the eye, has slopeequal 1 and crosses the vertical axis at 1.5 eV. esults and Discussion We selected 118 oxide perovskites ABO , and for each we considered four crystal structures,with symmetries P m ¯3 m (cubic), I /mmm (tetragonal), P nma (orthorhombic) and P /mmc (rhombohedral), as shown in Fig. 1, totaling 472 structures. The selected A and B atoms, alsoindicated in the Periodic Table in Fig. 1, are: A = Li, Na, K, Rb, Cs, Cu, Ag, Au, Be, Mg, Ca,Sr, Ba, Pb, Zn, Cd, Sn, Sc,Y, La, or Bi, and B = P, As, Sb, V, Nb, Ta, Si, Ge, Sn, Ti, Zr, Hf, Al,Ga, In, or Tl, such that the considered compounds satisfy valence(A) + valence(B)=6. A data set ofDFT-GGA band gaps was constructed using this set of materials.The four crystal structures for all ABO compounds were first optimized with the DFT-GGAPBEsol functional. Then their electronic structures were calculated using PBEsol and HSE06. Inthis way, since the average electrostatic potential is used as reference for the Kohn-Sham bandenergies, and does not depend on exchange and correlation, we can directly compare the PBEsoland HSE06 band structures, extracting the corrections for VBM and CBM, and the band gap (i.e., ∆VBM , ∆CBM , and ∆ E g ). We note that for all compounds studied here, the VBM for thecubic structure occurs at the R point (0.5, 0.5, 0.5) and the CBM occurs at the Γ point in thecubic Brillouin zone, characterizing an indirect R- Γ fundamental band gap. For the tetragonal,orthorhombic, and rhombohedral structures, both VBM and CBM occur at Γ , characterizing adirect Γ - Γ fundamental band gap.The calculated HSE06 band gaps vs PBEsol band gaps are shown in Fig. 2(a). First, wenote that the DFT-GGA underestimates the band gap with respect to HSE06 by ∼ compounds,the Cu d orbitals mix with the O 2 p orbitals, pushing the VBM to higher energies. In the case ofSn-B-O and Pb-B-O , the VBM has large contributions from Sn and Pb s valence orbitals, whichalso pushes the VBM to higher energies. In all the cases where the valence band is mostly derivedfrom O 2 p orbitals, the ∼ ∆VBM and ∆CBM , i.e., the amount the VBM and CBM inHSE06 differ from the VBM and CBM in DFT-GGA are shown in Fig. 2(b). Contrary to commonwisdom, where it is often assumed that to correct the DFT-GGA band gap only an upward shiftof the CBM is necessary, we find that about 2/3 of the gap correction comes from shifting downthe VBM and only about 1/3 of the correction comes from shifting the conduction band upward.This is attributed to large self-interaction correction of the O 2 p -derived valence bands in thesematerials. Again, the outliers, where the VBM is corrected by a lesser amount, correspond tocompounds containing Cu, Sn, or Pb in the A site. It is also interesting to note the correction inthe VBM derived from O 2 p is larger than the correction of CBM derived from d orbitals, such asin SrTiO and similar compounds, despite the rather flat nature of their conduction bands that arederived from the quite localized transition metal d orbitals. Finally, we also note that the band-gapcorrection ∆ E g is slightly larger than 1.5 eV for compounds with larger band gaps, approaching2 eV, and this is traced back to the correction of the CBM which approaches 1 eV for compoundswith E g & ∆VBM , ∆CBM , and ∆ E g correc-tions to atomic and structural properties of the compounds. The atomic properties as input to themachine learning models include electronegativity, ionization energy, valence-orbital energies, andatomic radius of both A and B atoms. Structural properties include octahedral tilting and rotationsthat are characterized by the apical α a and equatorial α e angles corresponding to B-O-B angles par-allel and perpendicular to the c axis. We employed three machine learning models, which are thelinear ridge regressor (LRR), kernel ridge regressor (KRR), and the gradient boosted decision tree(GBDT) regressor, as implemented in Scikit-Learn Toolbox . We used a regularization strengthof . to both LRR and KRR models. We used polynomial kernel in KRR with maximum order . For the GBDT model, we set the maximum tree depth to with base estimators.The prediction performance of the LRR, KRR, and GBDT models can be seen in Table. 1.In these models we use two third of the data as training set. We also use mean absolute error(MAE) to measure the performance in predicting ∆VBM , ∆CBM , and ∆ E g . Among the threemodels, GBDT gives the highest prediction accuracy with low variance; the KRR model performsbetter than LRR. Note that we obtain lower MAE than previous models
26, 31, 39, 49, 50 , likely to thebetter quality or more uniformity of our training dataset. The results indicate that there exists acomplex nonlinear relation between the input properties and the target results, explaining why thepure linear model LRR performs poorly. Note that all the three ML models predict ∆VBM withsimilar performance, indicating that the VBM correction has a more linear relation with the inputproperties than the CBM and E g corrections. 9 lectronegativity_Aequatorial angleapical angle s orbital energy_B p orbital energy_Batomic number_Belectronegativity_B s orbital energy_Aionization energy_Aionization energy_B p orbital energy_Aatomic radius_Batomic radius_Aatomic number_Aoxidation state_Aoxidation state_B . . . . . . . . electronegativity_Bionization energy_Bequatorial angle p orbital energy_Aoxidation state_Bapical angle p orbital energy_Batomic number_A s orbital energy_Batomic number_B s orbital energy_Aatomic radius_Aelectronegativity_Aatomic radius_Bionization energy_Aoxidation state_A . . . . . . . .
175 0 . P r ope r t i e s Feature importance Feature importanceelectronegativity_A p orbital energy_Aequatorial angleionization energy_Bapical angle s orbital energy_Belectronegativity_Batomic number_Aionization energy_Aoxidation state_B s orbital energy_A p orbital energy_Batomic number_Boxidation state_Aatomic radius_Batomic radius_A . . . . .
200 0 . (a) Δ VBMFeature importance (b) Δ CBM (c) Δ E g Figure 3: Feature importance in the gradient boosted decision tree (GBDT) model fordetermining the band-gap ( ∆ E g ) and band-edge corrections ( ∆VBM , ∆CBM ) of ABO perovskites.Table 1: Mean absolute error (MAE) used to evaluate the performance of the linear ridgeregressor (LRR), kernel ridge regressor (KRR), and the gradient boosted decision tree(GBDT) models in predicting the corrections of the valence-band maximum ( ∆VBM ),conduction-band minimum ( ∆CBM ) and band gap ( ∆ E g ) of oxide perovskites in DFT-GGA PBEsol compared to the HSE06 values.LRR KRR GBDT ∆VBM 0 . ± .
01 0 . ± .
01 0 . ± . . ± .
01 0 . ± .
01 0 . ± . E g . ± .
02 0 . ± .
01 0 . ± . p valence orbital of atom A, and the equatorial angle of the octahedralrotation are the main properties that determine ∆VBM . For ∆CBM , the main properties are theelectronegativity, ionization energy of atom B atom and the equatorial angle of the octahedral rota-tion. We have also applied LRR, KRR and GBDT models to the data by excluding the discoveredless-important features for each label and no obvious accuracy improvement is identified.For both ∆VBM and ∆CBM , the equatorial angle determines the overlap between the or-bitals of B and O in the directions parallel to the a - b plane, which in turn, affect both VBM andCBM positions. Note that the dependence on the apical angle α a is less than that on the equato-rial angle α e since the former affects the B-O orbital overlap only along the c direction. Finally,we also note that the relative importance of the electronigativity, ionization energies, and rotationangles is higher for atom A than for atom B in determining the band gap. This is attributed to thelarger contribution of the VBM correction than the CBM correction to ∆ E g . Conclusions
Using high-throughput DFT-GGA PBEsol and HSE06 calculations we determined the band gapcorrection of a representative set of oxide perovskites, finding that the HSE06-based correctionpushes down valence band by ∼ ∼ octahedra are the main factors involved in the corrections. These results serve as startingpoint and guide to developing machine-learning-based approaches applicable to the discovery ofnovel electronic materials. Methods
The first-principles calculations are based on DFT within the generalized gradient approximationof Perdew, Burke, and Ernzerhof revised for solids (PBEsol) and the projector augmented wavemethod
44, 45 as implemented in the Vienna Ab initio Simulation Packaged (VASP)
46, 47 . The wavefunctions are expanded in plane waves with cutoff energy of 650 eV. Structure optimizations areperformed using 7 × ×
7, 7 × ×
7, 7 × ×
5, and 7 × × Γ -centered k -point grid for the integra-tions over the Brillouin zones of the cubic, tetragonal, orthorhombic, and rhombohedral primitivecells, respectively. The screened hybrid functional HSE06
9, 10 is employed to compute target bandgaps, using the structural parameters found using the PBEsol functional. In tests we found thatPBEsol and HSE06 give lattice parameters that differ by less than 1%, and in good agreement withexperimental values. So we neglected the differences in the band gap calculated using the PBEsol-optimized lattice parameters and those calculated using the HSE06-optimized lattice parameters.Test calculations indicate that these differences are less than 0.1 eV.12e used different ML algorithms to build our band-gap prediction model, including thelinear ridge regressor, kernel ridge regressor, and gradient boosted decision tree from open-sourcesoftware package Scikit-Learn Toolbox . The input to the model is comprised of atomic andstructural properties, including the B-O-B apical angle α a and B-O-B equatorial angle α e . Theregression fit to the input gives the predicted band gaps. Prediction performance of the learningmodels is evaluated by the mean-absolute error. The feature importance of all the descriptors isobtained with GBDT to interpret the importance of various descriptors in the training model. Data availability
The datasets generated and/or analyse during the current study are available in the GitHub reposi-tory https://github.com/vera-weili/perovskite_ML . Acknowledgements
This work was supported by the National Science Foundation EAGER-1843025.This research was also supported by the Xtreme Science and Engineering Discovery Environment (XSEDE)facility, National Science Foundation grant number ACI-1053575, and the Information Technologies (IT) re-sources at the University of Delaware, specifically the high performance computing resources. This researchwas also supported in part by the following National Science Foundation grants: 1447711, 1743418, and1843025. AJ acknowledges support from NSF Faculty Early Career Development Program DMR-1652994.
Author Contributions
A.J, R.S and B.M conceived the research and provided guidance. W.L performedsimulations of material properties. Z.W, X.X, B.M performed ML analysis. All authors analyzed the results,wrote and revised the paper. ompeting Interests The authors declare that they have no competing financial interests.
Correspondence
Correspondence and requests for materials should be addressed to Anderson Janotti,Sanguthevar Rajasekaran, Bharat Medasani, (email: [email protected], [email protected],[email protected], [email protected]).
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