The relative permittivity of a crystal is a fundamental property that links microscopic chemical bonding to macroscopic electromagnetic response. Multiple models, including analytical, numerical, and statistical descriptions, have been made to understand and predict dielectric behavior. Analytical models are often limited to a particular type of compound, whereas machine learning (ML) models often lack interpretability. Here, we combine supervised ML, density functional perturbation theory, and analysis based on game theory to predict and explain the physical trends in optical dielectric constants of crystals. Two ML models, support vector regression and deep neural networks, were trained on a dataset of 1364 dielectric constants. Analysis of Shapley additive explanations of the ML models reveals that they recover correlations described by textbook Clausius–Mossotti and Penn models, which gives confidence in their ability to describe physical behavior, while providing superior predictive power.
REFERENCES
1.
D. R.
Penn
, “Wave-number-dependent dielectric function of semiconductors
,” Phys. Rev.
128
, 2093
(1962
).2.
R. D.
Shannon
and R. X.
Fischer
, “Empirical electronic polarizabilities in oxides, hydroxides, oxyfluorides, and oxychlorides
,” Phys. Rev. B
73
, 235111
(2006
).3.
R. D.
Shannon
and R. X.
Fischer
, “Empirical electronic polarizabilities of ions for the prediction and interpretation of refractive indices: Oxides and oxysalts
,” Am. Mineral.
101
, 2288
–2300
(2016
).4.
K.
Yim
, Y.
Yong
, J.
Lee
, K.
Lee
, H.-H.
Nahm
, J.
Yoo
, C.
Lee
, C.
Seong Hwang
, and S.
Han
, “Novel high-κ dielectrics for next-generation electronic devices screened by automated ab initio calculations
,” NPG Asia Mater.
7
, e190
(2015
).5.
F.
Naccarato
, F.
Ricci
, J.
Suntivich
, G.
Hautier
, L.
Wirtz
, and G.-M.
Rignanese
, “Searching for materials with high refractive index and wide band gap: A first-principles high-throughput study
,” Phys. Rev. Mater.
3
, 044602
(2019
).6.
T.
Luty
, “On the effective molecular polarizability in molecular crystals
,” Chem. Phys. Lett.
44
, 335
–338
(1976
).7.
K. J.
Miller
and J.
Savchik
, “A new empirical method to calculate average molecular polarizabilities
,” J. Am. Chem. Soc.
101
, 7206
–7213
(1979
).8.
K. J.
Miller
, “Calculation of the molecular polarizability tensor
,” J. Am. Chem. Soc.
112
, 8543
–8551
(1990
).9.
J. R.
Tessman
, A. H.
Kahn
, and W.
Shockley
, “Electronic polarizabilities of ions in crystals
,” Phys. Rev.
92
, 890
(1953
).10.
K.
Urano
and M.
Inoue
, “Clausius–Mossotti formula for anisotropic dielectrics
,” J. Chem. Phys.
66
, 791
–794
(1977
).11.
N. F.
Mott
and R. W.
Gurney
, Electronic Processes in Ionic Crystals
, 2nd ed. (Oxford University Press
, London
, 1950
).12.
A. S.
Korotkov
and V. V.
Atuchin
, “Prediction of refractive index of inorganic compound by chemical formula
,” Opt. Commun.
281
, 2132
–2138
(2008
).13.
S. T.
Pantelides
, “Mechanisms that determine the electronic dielectric constants of ionic crystals
,” Phys. Rev. Lett.
35
, 250
(1975
).14.
H.
Coker
, “Empirical free-ion polarizabilities of the alkali metal, alkaline earth metal, and halide ions
,” J. Phys. Chem.
80
, 2078
–2084
(1976
).15.
P. W.
Fowler
and N. C.
Pyper
, “In-crystal ionic polarizabilities derived by combining experimental and ab initio results
,” Proc. R. Soc. London, A
398
, 377
–393
(1985
).16.
A. R.
Ruffa
, “Theory of the electronic polarizabilities of ions in crystals: Application to the alkali halide crystals
,” Phys. Rev.
130
, 1412
(1963
).17.
J. N.
Wilson
and R. M.
Curtis
, “Dipole polarizabilities of ions in alkali halide crystals
,” J. Phys. Chem.
74
, 187
–196
(1970
).18.
P.
Jemmer
, P. W.
Fowler
, M.
Wilson
, and P. A.
Madden
, “Environmental effects on anion polarizability: Variation with lattice parameter and coordination number
,” J. Phys. Chem. A
102
, 8377
–8385
(1998
).19.
V.
Dimitrov
and S.
Sakka
, “Electronic oxide polarizability and optical basicity of simple oxides. I
,” J. Appl. Phys.
79
, 1736
–1740
(1996
).20.
J. C.
Phillips
, “A posteriori theory of covalent bonding
,” Phys. Rev. Lett.
19
, 415
(1967
).21.
J. C.
Phillips
, “Dielectric definition of electronegativity
,” Phys. Rev. Lett.
20
, 550
(1968
).22.
J. A.
Van Vechten
, “Quantum dielectric theory of electronegativity in covalent systems. I. Electronic dielectric constant
,” Phys. Rev.
182
, 891
(1969
).23.
J. A.
Van Vechten
, “Quantum dielectric theory of electronegativity in covalent systems. II. Ionization potentials and interband transition energies
,” Phys. Rev.
187
, 1007
(1969
).24.
J. H.
Gladstone
and T. P.
Dale
, “XIV. Researches on the refraction, dispersion, and sensitiveness of liquids
,” Philos. Trans. R. Soc. London
153
, 317
–343
(1863
).25.
R. D.
Allen
, “A new equation relating index of refraction and specific gravity
,” Am. Mineral.
41
, 245
–257
(1956
).26.
O. L.
Anderson
and E.
Schreiber
, “The relation between refractive index and density of minerals related to the earth’s mantle
,” J. Geophys. Res.
70
, 1463
–1471
, (1965
).27.
R. A.
Eggleton
, “Gladstone-dale constants for the major elements in silicates; coordination number, polarizability, and the Lorentz–Lorentz relation
,” Can. Mineral.
29
, 525
–532
(1991
).28.
F. D.
Bloss
, M.
Gunter
, S.-C.
Su
, and H.
Wolfe
, “Gladstone-dale constants; a new approach
,” Can. Mineral.
21
, 93
–99
(1983
).29.
P. W.
Fowler
, R. W.
Munn
, and P.
Tole
, “Polarisability of the oxide ion in crystalline BeO
,” Chem. Phys. Lett.
176
, 439
–445
(1991
).30.
J. A.
Mandarino
, “The Gladstone Dale compatibility of minerals and its use in selecting mineral species for further study
,” Can. Mineral.
45
, 1307
–1324
(2007
).31.
K. T.
Butler
, D. W.
Davies
, H.
Cartwright
, O.
Isayev
, and A.
Walsh
, “Machine learning for molecular and materials science
,” Nature
559
, 547
–555
(2018
).32.
J. J.
de Pablo
, N. E.
Jackson
, M. A.
Webb
, L.-Q.
Chen
, J. E.
Moore
, D.
Morgan
, R.
Jacobs
, T.
Pollock
, D. G.
Schlom
, E. S.
Toberer
et al., “New frontiers for the materials genome initiative
,” npj Comput. Mater.
5
, 41
(2019
).33.
A.
Dunn
, Q.
Wang
, A.
Ganose
, D.
Dopp
, and A.
Jain
, “Benchmarking materials property prediction methods: The matbench test set and automatminer reference algorithm
,” arXiv:2005.00707 (2020
).34.
M.
Lee
, Y.
Youn
, K.
Yim
, and S.
Han
, “High-throughput ab initio calculations on dielectric constant and band gap of non-oxide dielectrics
,” Sci. Rep.
8
, 14794
(2018
).35.
Y.
Umeda
, H.
Hayashi
, H.
Moriwake
, and I.
Tanaka
, “Prediction of dielectric constants using a combination of first principles calculations and machine learning
,” Jpn. J. Appl. Phys., Part 2
58
, SLLC01
(2019
).36.
Y.
Noda
, M.
Otake
, and M.
Nakayama
, “Descriptors for dielectric constants of perovskite-type oxides by materials informatics with first-principles density functional theory
,” Sci. Techol. Adv. Mater.
21
, 92
–99
(2020
).37.
I.
Petousis
, D.
Mrdjenovich
, E.
Ballouz
, M.
Liu
, D.
Winston
, W.
Chen
, T.
Graf
, T. D.
Schladt
, K. A.
Persson
, and F. B.
Prinz
, “High-throughput screening of inorganic compounds for the discovery of novel dielectric and optical materials
,” Sci. Data
4
, 160134
(2017
).38.
G.
Petretto
, S.
Dwaraknath
, H. P.
Miranda
, D.
Winston
, M.
Giantomassi
, M. J.
Van Setten
, X.
Gonze
, K. A.
Persson
, G.
Hautier
, and G.-M.
Rignanese
, “High-throughput density-functional perturbation theory phonons for inorganic materials
,” Sci. Data
5
, 180065
(2018
).39.
A.
Jain
, S. P.
Ong
, G.
Hautier
, W.
Chen
, W. D.
Richards
, S.
Dacek
, S.
Cholia
, D.
Gunter
, D.
Skinner
, G.
Ceder
et al., “Commentary: The materials project: A materials genome approach to accelerating materials innovation
,” APL Mater.
1
, 011002
(2013
).40.
S. P.
Ong
, S.
Cholia
, A.
Jain
, M.
Brafman
, D.
Gunter
, G.
Ceder
, and K. A.
Persson
, “The materials application programming interface (API): A simple, flexible and efficient API for materials data based on representational state transfer (REST) principles
,” Comput. Mater. Sci.
97
, 209
–215
(2015
).41.
D.
Davies
, K.
Butler
, A.
Jackson
, J.
Skelton
, K.
Morita
, and A.
Walsh
, “SMACT: Semiconducting materials by analogy and chemical theory
,” J. Open Source Software
4
, 1361
(2019
).42.
S. P.
Ong
, W. D.
Richards
, A.
Jain
, G.
Hautier
, M.
Kocher
, S.
Cholia
, D.
Gunter
, V. L.
Chevrier
, K. A.
Persson
, and G.
Ceder
, “Python materials genomics (pymatgen): A robust, open-source python library for materials analysis
,” Comput. Mater. Sci.
68
, 314
–319
(2013
).43.
S.
Baroni
, S.
De Gironcoli
, A.
Dal Corso
, and P.
Giannozzi
, “Phonons and related crystal properties from density-functional perturbation theory
,” Rev. Mod. Phys.
73
, 515
(2001
).44.
F.
Pedregosa
, G.
Varoquaux
, A.
Gramfort
, V.
Michel
, B.
Thirion
, O.
Grisel
, M.
Blondel
, P.
Prettenhofer
, R.
Weiss
, V.
Dubourg
, J.
Vanderplas
, A.
Passos
, D.
Cournapeau
, M.
Brucher
, M.
Perrot
, and E.
Duchesnay
, “Scikit-learn: Machine learning in Python
,” J. Mach. Learn. Res.
12
, 2825
–2830
(2011
).45.
A. E.
Teschendorff
, “Avoiding common pitfalls in machine learning omic data science
,” Nat. Mater.
18
, 422
–427
(2018
).46.
C.
Chen
, W.
Ye
, Y.
Zuo
, C.
Zheng
, and S. P.
Ong
, “Graph networks as a universal machine learning framework for molecules and crystals
,” Chem. Mater.
31
, 3564
–3572
(2019
).47.
P. E.
Blöchl
, “Projector augmented-wave method
,” Phys. Rev. B
50
, 17953
(1994
).48.
G.
Kresse
and J.
Furthmüller
, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
,” Comput. Mater. Sci.
6
, 15
–50
(1996
).49.
G.
Kresse
and J.
Furthmüller
, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
,” Phys. Rev. B
54
, 11169
(1996
).50.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Generalized gradient approximation made simple
,” Phys. Rev. Lett.
77
, 3865
(1996
).51.
S. M.
Lundberg
and S.-I.
Lee
, “A unified approach to interpreting model predictions
,” in Advances in Neural Information Processing Systems 30
, edited by I.
Guyon
, U. V.
Luxburg
, S.
Bengio
, H.
Wallach
, R.
Fergus
, S.
Vishwanathan
, and R.
Garnett
(Curran Associates, Inc.
, 2017
), pp. 4765
–4774
.52.
A.
Mannodi-Kanakkithodi
, G.
Pilania
, and R.
Ramprasad
, “Critical assessment of regression-based machine learning methods for polymer dielectrics
,” Comput. Mater. Sci.
125
, 123
–135
(2016
).53.
O.
Isayev
, C.
Oses
, C.
Toher
, E.
Gossett
, S.
Curtarolo
, and A.
Tropsha
, “Universal fragment descriptors for predicting properties of inorganic crystals
,” Nat. Commun.
8
, 15679
(2017
).54.
A.
Mansouri Tehrani
, A. O.
Oliynyk
, M.
Parry
, Z.
Rizvi
, S.
Couper
, F.
Lin
, L.
Miyagi
, T. D.
Sparks
, and J.
Brgoch
, “Machine learning directed search for ultraincompressible, superhard materials
,” J. Am. Chem. Soc.
140
, 9844
–9853
(2018
).55.
I.
Petousis
, W.
Chen
, G.
Hautier
, T.
Graf
, T. D.
Schladt
, K. A.
Persson
, and F. B.
Prinz
, “Benchmarking density functional perturbation theory to enable high-throughput screening of materials for dielectric constant and refractive index
,” Phys. Rev. B
93
, 115151
(2016
).56.
D.
Jha
, K.
Choudhary
, F.
Tavazza
, W.-k.
Liao
, A.
Choudhary
, C.
Campbell
, and A.
Agrawal
, “Enhancing materials property prediction by leveraging computational and experimental data using deep transfer learning
,” Nat. Commun.
10
, 5316
(2019
).57.
L.
Pauling
, The Nature of the Chemical Bond
, 3rd ed. (Cornell University Press
, Ithaca, NY
, 1960
).58.
J. C.
Phillips
and J. A.
Van Vechten
, “Spectroscopic analysis of cohesive energies and heats of formation of tetrahedrally coordinated semiconductors
,” Phys. Rev. B
2
, 2147
–2160
(1970
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
