topologyparameterdirection

ARTICLEReceived15Sep2016|Accepted13Dec2016|Published17Feb2017LearningfromdatatodesignfunctionalmaterialswithoutinversionsymmetryPrasannaV.
Balachandran1,JoshuaYoung2,TurabLookman1&JamesM.
Rondinelli3Acceleratingthesearchforfunctionalmaterialsisachallengingproblem.
Herewedevelopaninformatics-guidedabinitioapproachtoacceleratethedesignanddiscoveryofnoncentrosymmetricmaterials.
Theworkowintegratesgrouptheory,informaticsanddensity-functionaltheorytouncoverdesignguidelinesforpredictingnoncentrosymmetriccompounds,whichweapplytolayeredRuddlesden-Popperoxides.
Grouptheoryidentieshowcongurationsofoxygenoctahedralrotationpatterns,orderedcationarrangementsandtheirinterplaybreakinversionsymmetry,whileinformaticstoolslearnfromavailabledatatoselectcandidatecompositionsthatfullthegroup-theoreticalpostulates.
Ourkeyoutcomeistheidenticationof242compositionsafterscreeningB3,200thatshowpotentialfornoncentrosymmetricstructures,a25-foldincreaseintheprojectednumberofknownnoncentrosymmetricRuddlesden-Popperoxides.
Wevalidateourpredictionsfor19compoundsusingphononcalculations,amongwhich17havenoncentrosymmetricgroundstatesincludingtwopotentialmultiferroics.
Ourapproachenablesrationaldesignofmaterialswithtargetedcrystalsymmetriesandfunctionalities.
DOI:10.
1038/ncomms14282OPEN1TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NewMexico87545,USA.
2DepartmentofMaterialsScienceandEngineering,DrexelUniversity,Philadelphia,Pennsylvania19104,USA.
3DepartmentofMaterialsScienceandEngineering,NorthwesternUniversity,Evanston,Illinois60208,USA.
CorrespondenceandrequestsformaterialsshouldbeaddressedtoP.
V.
B.
(email:pbalachandran@lanl.
gov)ortoJ.
M.
R.
(email:jrondinelli@northwestern.
edu).
NATURECOMMUNICATIONS|8:14282|DOI:10.
1038/ncomms14282|www.
nature.
com/naturecommunications1Noncentrosymmetric(NCS)oxideceramicsthatbreakallimproperrotationsandcentresofsymmetryarechallengingtodiscover.
Materialswithpolar,piezoelectric,chiralandthoseexhibitingcirculardichroism(collectivelyreferredtoasNCSmaterials)aredenedbytheabsenceofinversionsymmetryandarepresenteverywhere—intheformoforganicaminoacids,sugarsandotherbiologicalmolecules1.
InorganicNCSmaterialscontainingoxideanionsarealsonotuncommon2.
Quartzcrystalswithahelicalarrangementofcorner-connectedSiO4tetrahedralunitsmaintainthepunctualityofourmechanicaltimepieces3.
Atinorganiccrystallinesurfaces,chiralityplaysacrucialroleincorrosionprocesses,heterogeneouscatalysisandthedelityofenantioselective-basedproductionorseparationofindustrialsolvents,plasticsandpharmaceuticaldrugs4.
Pb(Zr,Ti)O3,BaTiO3andBiFeO3aresomeofthearchetypalpolaroxidesthathaveimpactedmanycriticaltechnologies5.
Ofteninorganicpolarandchiralbasicbuildingunits(BBUs)areselectedandassembledtogether,butacentricorganizationofBBUswithinaunitcellaredifculttopredictduetothecomplexinterplayofchemistryandstructure.
Inthecontextofinorganicoxides,whichisthefocusofthiswork,thedesignofNCSmaterialshasreliedmainlyonBBUswithmetalcentresthathaved0electroniccongurationsorlone-paircations,wheretheacentricityarisesfromanelectronicoriginduetothepseudo-orsecond-orderJahn–Teller(SOJT)effect6,7.
Amajorityofinorganicoxides,however,stronglypreferclose-packedarrangementsofionsandhighlysymmetriccationcoordinationenvironments(forexample,octahedra).
Thisismainlyduetothedominantelectrostaticeffectsthatareoptimizedbyfavouringlike–unlikeinteractions(thatis,positiveandnegativedipolesalignequallyandoppositely),whichstabilizeatomicarrangementswithinversionsymmetry8.
Infact,thepresenceofBBUswithd0metalcentresaloneisnotasufcientconditionfordesigningNCSmaterials.
Forexample,theperovskiteSrTiO3isaquantumparaelectricorincipientferroelectric9,whereastheisoelectroniclayeredRuddlesden-Popper(RP)Sr2TiO4isacentricdielectric10.
Hence,itisthecomplexinterplaybetweenstructureandchemistrythatdeterminestheformationofNCSinorganicoxides.
Alternativetopseudo-JTorSOJTeffects,the'trilinearcoupling','hybridimproper'or'geometricferroelectricity'mechanism,wheretwononpolarlatticedistortions(octahedralrotationsortilting)coupletoapolarlatticemode,havealsobeenshowntobreaktheinversionsymmetrywithinterestingtechnologicalconsequences11,12.
Eveninthiscase,noapriorirulesexistthatguidethedesignofnewhybridimproperferroelectricmaterials,unlessexhaustivecalculationsarecarriedouttomapthechemicalandenergylandscapethatsubsequentlyinformexperiments12.
Asaresult,NCSinorganicoxidesarechallengingtodiscover.
Althoughhigh-throughputrstprinciples-basedmethodshaveshownpromiseinthedesignofNCShalf-Heusleralloys13,exhaustivecalculationsformorecomplexcrystalstructureswithnumerouspolymorphs(suchastheRPs)andthousandsofunexploredchemicalcompositionshavenot(yet)beendemonstrated.
Thisispartlybecausethepotentialenergysurfaceofcomplexoxidesisdifculttonavigate.
Phononinstabilitiesathigh-symmetrypointsawayfromtheG-pointintheirreducibleBrillouinzonescausetheprimitiveunitcelltomultiplyseveralfold,resultinginlargesystemsizesandvastnumbersofuniqueatomicarrangements.
Itischallengingtorigorouslyevaluatetheenergeticsofallstructuresinahigh-throughputmanner.
Furthermore,chemistrieswithpartiallylledd(and/orf)orbitalsandtheexistenceofenergeticallycompetinggroundstatescomplicatethestructurepredictionprocess.
Asaresult,novelapproachesaredesiredtoguidetherstprinciplescalculationsinaneffectivemanner.
Materialsinformatics,agrowingeldattheintersectionsofmanyscienticdisciplinesincludingdataandinformationscience,statistics,machinelearning(ML)andoptimization,hasthepotentialtoaccomplishthisobjective14.
Herewedevelopapredictivedata-drivencomputationalframeworkthatunitesappliedgrouptheory,informaticstechniquesandabinitioelectronicstructurecalculationsfordesigningnovelNCSmaterials.
Weapplyittothetwo-dimensionaln1RPstructurefamily(Fig.
1a),forwhichtodatefewcompositionsexistinNCScrystalclasses15–17.
Nonetheless,thechemicalsearchspaceis(Fig.
1b).
Weuseinformatics-basedmethodstoscreenthechemicalspaceanddownselect242compositionsthatshowgreaterpromiseforNCSaA-siteBO6A-siteelementsB-siteelementsBO6BO6bFigure1|Octahedralconnectivityofn1RPoxidesandthechemicalsearchspace.
(a)Then1RPphasehasasinglelayerofoctahedrathatareconnectedintwodimensions,shownwithinbrackets,whereasthereisnoconnectivityinthethirddimension.
(b)Periodictableshowingthepotential30A-siteand19B-siteelementsthatoccupythen1RPphase.
Inprinciple,therearemorethan19B-siteelementswhenwealsoconsiderthemultiplevalencestatesofcertainelements(forexample,Mn,Fe,Co,Niandsoon).
Thisdenesthechemicalspaceforourinformaticsapproach.
ARTICLENATURECOMMUNICATIONS|DOI:10.
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ThepotentialfordiscoveringnovelNCSn1RPcompoundshaskeyimplicationsintechnologicalapplicationsthatrequireabroadrangeoffunctionalitiesincludinghigh-temperaturepiezoelectricity,tunablebandgaps,improperferroelectricity,multiferroicityandthermoelectricity.
WefocusindetailonthedesignofNaRSnO4stannatesandNaRRuO4ruthenates(whereRLa,Pr,Nd,GdorY)thatwerepredictedtohaveNCSgroundstatestructuresfrominformaticsandsubsequentlyvalidatedbydensity-functionaltheory(DFT).
Forthestannates,whicharecandidatematerialsforsensorsandtransparentconductingoxides18,wendtwoenergeticallycompetingNCSgroundstatephases:P421m(piezo-active)andP21212(chiralandpiezo-active).
WecalculatetheirelectronicbandgapsintheP421mcrystalsymmetryusinghybridexchange-correlationfunctionals,ndingopticaltransparencyinthevisiblelightregime.
WealsocomputetheirpiezoelectricresponsesthatshowadependenceonR-cationsize.
Insharpcontrast,theNCSNaRRuO4aremagneticwithmetallic,half-metallicorinsulatingelectronicstructures.
Theirgroundstateisdeterminedtobeeitherpiezo-activewithP421msymmetrywhenRLa,PrandNdorpolarwithPca21symmetrywhenRGdandY.
Moreover,thereisatransitionfromferromagneticmetallic(RLa)orhalf-metallic(RPr,Nd)toantiferromagneticinsulating(RGd,Y)characterasafunctionofR-cationsize.
Therefore,thesebulkruthenatesarepredictedtobelongtotheintriguingclassofNCSmetals19,20andhalf-metalswithpiezo-activesymmetriesorantiferromagneticinsulatorswithpolarsymmetry(thatis,multiferroics).
Last,wealsotestourpredictionsforanadditionalninenewcompoundswithdifferentcationsoccupyingtheB-sublatticeoftheRPstructure(showninFig.
1a).
Amongthem,sevenwerevalidatedtohaveanNCSgroundstatestructure—NaLaZrO4,NaLaHfO4,KBaNbO4,NaLaIrO4,NaCaTaO4,SrYGaO4andSrLaInO4.
Theseresultsestablishourcomputationalframeworkasapowerfultoolforcrystalsymmetryclassication,structure-basedpropertydesignandcontrol.
ResultsApproach.
OursearchforNCSoxidesreliesonamultifacetedtheoreticalapproach,whichreformulatesthediscoveryobjectiveintoidentifyingstructure—chemistryinterrelationships(asshowninFig.
2).
Thedesignstrategyfocusesonthreekeycriteriaobtainedbysubdividingthedesignprocessintouniqueobjectiveswithspecictasks:Structural:Howcantheatomicstructure,orcongurationofoxygenoctahedraBBUs,bedesignedtosupportthedesiredinteractionChemical:WhichcombinationsofchemistrieswillpromotethatstructuralcongurationStability:IstheproposedcompositiontheglobalgroundstateFollowingclassicationlearningfrominformaticsandevaluationoftheenergeticstabilityfromrstprinciplesmethods,thenaldesignreliesonresponseoptimizationbyleveragingadditionaldegreesoffreedomtofurtherpromotethetargetedbehaviour.
Someofthestrategiesincludesearchingformicro-scopicmechanismsandexternalconditions(suchasepitaxialstrain)toenergeticallystabilizethosegeometries.
WenotethatthispaperisasignicantadvancementfromtheearlierworkofBalachandranetal.
15wheretheemphasiswasonenumeratingsymmetryguidelines.
Grouptheory.
Inanearlierwork,Balachandranetal.
15formulatedsymmetryguidelinesforexploringanddesigningNCSphasesinthen1RPstructuresbasedongrouptheory.
Therefore,wediscussonlythekeyresultshere.
Startingfromthecentrosymmetric(CS)aristotypestructure(showninFig.
3a),varioussymmetry-allowedcooperativeatomicdisplacements(alsoreferredtoas'shufes')wereenumeratedthattransformthearistotypeCSstructuretoaNCSstructureoflowersymmetry.
Particularly,thefocuswasonCS-NCSphasetransitionsthataresecondorderorweaklyrstorder,wherethesymmetry-loweringdistortionsarisefrom(i)non-polaroctahedraldistortions(tiltingorrotations)duetophononsofteningatthezoneboundariesintheBZoftheI4/mmmspacegroup,(ii)A/A0cationordering,(iii)theinterplaybetweentwoormoreoctahedraldistortionsand(iv)theinterplaybetweenoctahedraldistortionsandA/A0cationordering.
ThenecessitytosearchforalternativeroutestobreakinginversionsymmetrywasmotivatedbythefactthatNCSphasesareseldomseeninn1RPs,whichhasbeenexplainedbythedisconnectedoctahedrallayersdestroyingthecoherencyrequiredforcooperativeoff-centringdisplacements,andthusferroelectricity21.
Balachandranetal.
15foundthreeimportantsymmetryguidelines(givenintherowsofTable1)forliftingparityinthen1RPstructures.
NotethatallinvolveA/A0cationordering(Fig.
3b)thattransformasirreduciblerepresentation(irrep)M3andcouplewithoctahedralrotationsortilting(asshowninFig.
3c–e).
Thestructuralattributesmaybesatisedbyanyofthefollowingapproaches:Route1:Out-of-phaseoctahedraltiltingthattransformasirrepX3withorderparameterdirection(OPD)(Z1,Z1),whichonsuperpositionwithirrepM3(Z1)wouldyieldapiezoelectric(P421m)spacegroup(Fig.
3c).
Route2:Out-of-phasetiltingthattransformasirrepX3withOPD(Z1,Z2)onsuperpositionwithM3(Z1)wouldyieldachiral(P21212)andpiezo-activespacegroup(Fig.
3d).
Route3:CoupledirrepX2"X3withOPD(0,Z1;Z2,0)whensuperposedwithirrepM3(Z1)wouldyieldapolar(Pca21)spacegroup(Fig.
3e),wherethematrixelementsofX2andX3irrepsaccommodateatomicdisplacementsthatcorrespondtoJahn-Teller-likedistortionsandout-of-phasetilting,respectively.
NotethatthereisanothertypeofA/A0cationordering,transformingasirrepG3,whichliftsinversionsolelyfromtheordering(werefertoitasthetrivialcase).
However,wedonotconsiderG3A/A0cationorderinghere.
Therefore,thekeymaterialsdesignquestionis:WhatcombinationsofchemicalelementsfromthevastchemicalspacewouldstabilizetheseNCSphasesWeaddressthisquestionusingmaterialsinformatics.
Materialsinformatics.
InFig.
4,weshowthefrequencyofoccurrenceofexperimentallyknowncrystalsymmetriesinthebulkn1RPs.
WereportonlythelowtemperaturecrystalsymmetriesinFig.
4anddonotexplicitlyconsidertemperaturedependenceofthecrystalstructuresinourinformaticsanalysis.
Ourdenitionoflowtemperatureincludesexperimentallyobservedstructuresr300K.
SomeRPcompoundsalsoundergostructuraltransformationatamuchlowertemperature(forexample,La2NiO4(ref.
22)).
Undersuchcircumstances,wetakethelowertemperaturecrystalstructuretobeourlabelforinformatics.
Thissimplicationwasnecessarybecause0KDFTcalculationsareusedtovalidatetheinformatics-basedpredictions.
Balachandranetal.
15showedthatasthetemperatureincreases,thepropensityforforminghigh-symmetryphasesalsoincreases.
Weanticipatethoseresultstoholdhere.
OurliteraturesurveyshowsthatB45%ofthecompositionsareundistorted(denotedasfinFig.
4).
Similarly,therearealsoasignicantnumberofcompositionsthatundergosymmetry-loweringdistortions,albeitpreservingthespatialinversionNATURECOMMUNICATIONS|DOI:10.
1038/ncomms14282ARTICLENATURECOMMUNICATIONS|8:14282|DOI:10.
1038/ncomms14282|www.
nature.
com/naturecommunications3symmetry.
OneofthekeyobservationsfromFig.
4isthatthereareonlyninecompoundswithNCSspacegroupsthatconformwithourchemicalsearchspace(Fig.
1b).
Intheliterature,thefamilyofcation-orderedNaRTiO4andLiRTiO4(foundonlyrecently),whereRLa,Nd,Dy,Gd,Sm,Ho,EuandY,havebeenexperimentallyshown16,17tohavethepiezoelectricP421mspacegroup[X3"M3(Z1,Z1;Z1)].
ThenominalelectroniccongurationofTi4inthesecompoundsisd0.
ThecouplingbetweenTiO6octahedraltilting(thattransformasirrepX3(Z1,Z1)asshowninFig.
3c)andLi/RorNa/Rcationordering(thattransformasirrepM3(Z1)asshowninFig.
3b)liftstheinversionsymmetry—inaccordancewithRoute1.
Theonlyotherexperimentallyknownpolarn1RPoxideistheA-andB-site-ordered(LaSr)(Li0.
5Ru0.
5)O4compound,whichisreportedintheNCSImm2spacegroup23.
Inthiscompound,acombinationofA-siteandB-sitecationorderingworkinconcerttolifttheinversionsymmetry.
Inadditiontothesecompounds,Pb2TiO4,Ca2IrO4,Sn2SnO4,cation-orderedLaANiO4(ASr,CaandBa)LaSrAlO4andLaSrMnO4havealsobeentheoreticallypredictedtohaveNCSstructures15,24–28;however,theseresultshavenot020103DensityfunctionaltheoryMaterialsinformaticsAppliedgrouptheoryStructuralcriterionChemicalcriterionEnergeticstabilityMaterialsdiscoveryBuildBBUdistortionmodesubspaceEnumeratemaximalandminimalspacegroupsStructurallibrariesBayesiananalysisRecursivepartitioningPrincipalcomponentanalysisComputeenergiesofstructuresEvaluatepropertiesanddescriptorsIdentifycandidatestructuresandgeometriesExtract,quantifyandanalysechemicaldependenceofstructuralmodestorecommendspecificchemistriesValidationofpredictedchemistriesandthedeterminationofgroundstatesstructuresSuggestionsforexperimentalsynthesisandcharacterizationFigure2|Predictivematerialsdiscoveryframework.
Synergisticintegrationofappliedgrouptheory,materialsinformaticsandabinitioelectronicstructurecalculationsfordesigningnovelfunctionalmaterials.
AppliedGroupTheorydeterminesthegeometricrules,uncoversthecrystallographicsymmetryrestrictionsandthensubsequentlyshowshowtoliftthemtoachieveNCSstructuresforagivencrystalstructuretopology.
Materialsinformaticsusesthedatafromexperiments,features(suchasorbitalradii)thatcapturethechemicaltrendsintheconstructeddatasetandstatisticalinferencetoolstoextractQCSRthatguidesselectionofchemicalcompositions.
DFTcalculationsvalidatethepredictionsfrommaterialsinformatics.
Wethenrecommendthevalidatedchemicalcompositionsforexperimentalsynthesisandcharacterization,eventuallyleadingtoitsdiscovery.
Experimentallysynthesizedcompositionsaugmentthetrainingsetforasecondmaterialsinformaticsiterationandtheprocessrepeatsuntildesiredmaterialsarediscovered14.
Inthispaper,wefocusoncomputationaltasks2and3(boxed).
(I4/mmm)M–3(1)(P4/nmm)AabAA′zxcdeX–3(1,1)(P42/ncm)X–3(1,2)(Pccn)X–2(0,1)ababyzxyzxyzxX–3(2,0)X–2X–3(0,1;2,0)(Pbca)Figure3|A/A0cationorderingandoctahedraltiltinginthen1RPsforNCSmaterialsdesign.
(a)High-symmetryaristotypestructure(f,I4/mmm).
(b)OneoftheA/A0cationorderingschemes(irrep:M3(Z1);spacegroup(s.
g.
):P4/nmm).
(c)Out-of-phaseoctahedraltilting(oxygendisplacementsindicatedusingarrows)(irrep:X3(Z1,Z1);s.
g.
:P42/ncm)andlatticeconstantsaandbareofequallength.
(d)Out-of-phaseoctahedraltilting(irrep:X3(Z1,Z2);s.
g.
:Pccn)andlatticeconstantaab.
(e)Coupleddistortions(irrep:X2"X3(0,Z1;Z2,0);s.
g.
:Pbca),whereX2(0,Z1)andX3(Z2,0)representJahn–Teller-likeout-of-planecompressionandout-of-phaseoctahedraltilting,respectively.
ARTICLENATURECOMMUNICATIONS|DOI:10.
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Recently,themetastableCa2IrO4wasepitaxiallygrownonaYAlO3substrateinthen1RPphaseusingpulsedlaserdeposition29.
However,theauthorsdidnotreportitscrystalsymmetry.
Therefore,wedonotconsiderthesechemistriesinourinformaticsanalysis.
Inthefamilyofn1RPswithrelativelysimplestoichiome-triessuchasAA0BO4,whereAandA0aretwochemicalspecies(similarordissimilar)occupyingtheA-siteandBisacationwith6-foldoctahedralcoordination,thereareB3,200potentialchemicalcompositionsthatsatisfycrystalchemistryandstoichiometricguidelines(forexample,chargeneutrality),andthereforeare,inprinciple,amenableforexperimentalsynthesis.
However,only3%havebeenexperimentallysynthesized,andamongthese,onlyninehaveNCSphases.
Theobjectiveofourinformaticsanalysisistoutilizestatisticalinferenceandmachinelearning(ML)methodsforestablishingquantitativechemistry-symmetryrelationships(QCSR)ofknownmaterialsinFig.
4.
TheseQCSRs,inturn,serveasaguidetorapidlyscreenthevastchemicalspaceandidentifynew,previouslyunexploredcompositionsthatfavourthedistortionsgivenintheTable1.
Dataset.
InourMLapproach,webuildadatasetofexperimentallyknownmaterialsthatincludesbothCSandNCSstructures.
EventhoughourcomputationaldesignfocusesonAA0BO4stoichiometries,ourtrainingdatasetincludesRPcompositionsthatdeviatefromtheAA0BO4stoichiometry(seedatasetintheSupplementaryInformation).
Wedescribeeachn1RPcompositionuniquelyintermsofitscrystalsymmetryorirrep(referredtoas'classlabel'intheMLjargon)andasetoffeatures.
WeuseWaber–CromerorbitalradiiasfeaturesforML30.
Orbitalradiianddistortionmodeshavebeenutilizedinthepastforpredictingstructuresandformabilitiesofcomplexoxides31,32.
OurMLobjectiveistobuildaclassicationmodelthatpredictscrystalsymmetriesorirreplabelsfromorbitalradii.
All83experimentallyknownRPchemicalcompositions(afterremoving(LaSr)(Li0.
5Ru0.
5)O4,becausewedonotconsidertheelementLiinourchemicalspace,seeFig.
1b)werewritteninthesimpliedA2BO4stoichiometricform,wheretheA-andB-sitescanhavetwoormoreelementswithpartialsiteoccupancies.
Weusedatotalof12and10orbitalradiifeaturestodescribetheA-andB-sites,respectively.
IfthereweretwoormoreelementsoccupyingeithertheA-orB-sites,thenlinearcombinationsweightedbytheirrelativestoichiometricproportionswereusedtobuildthefeatures.
Weconstructedtwodatasetsforclassicationlearningthatuses:(i)spacegroupsasclasslabels(anobviouschoice)and(ii)irrepscorrespondingtooctahedraltilting,rotations,orlackthereofasclasslabels.
Here,wefocusmainlyontheMLresultsfromthelatterdataset(case(ii))thatusesirrepsasclasslabels,whichallowsustoelegantlyisolateoctahedralrotationsortiltingfromcationordering.
Asaresult,wecangrouporcombinetwospacegroupsunderthesamelabel.
Forexample,wecombinecompositionswiththeI4/mmmandP4/nmmspacegrouptogether(underthelabel,f),becauseinbothcasestherearenooctahedralrotationsortilting.
OneofthekeydifferencesbetweenI4/mmmandP4/nmmisthatinP4/nmmtheA-siteWyckofforbitissplitintotwouniquecrystallographicsites15.
Similarly,wecancombinespacegroupsP421mandP42/ncmintoasingleirrep,X3(Z1,Z1).
Suchdatatransformationreducesthenumberofuniqueclasslabelsfrom9to7(seeinsetinFig.
4)forclassicationlearning.
ThemaindisadvantagewithsuchgroupingisthatourQCSRmodelnowcannotdistinguishbetweenorderedanddisorderedstructures.
ThisshouldnotaffectourNCSmaterialsdesigngoalbecauseofadvancementsinthenonequilibriumsynthesisandprocessingoftheseoxides.
Recently,therehavebeenexperimentaldemonstrationsoflayer-by-layergrowthofA/A0cation-orderedn1RPsusingmolecularbeamepitaxywithunprecedentedcontrol33.
WealsotestedthepredictivepowerofourMLmodelsbyintentionallyleavingout14compoundsduringtraining(whichreducesthesizeofourtrainingsetfrom83to69compounds).
Oneofourinformaticsgoalsistovalidatewhetherourclassicationlearningcanidentifythelabelscorrectlyfortheleftoutcompounds,beforeusingthemformakingnewpredictions.
Evenafterreducingthenumberofuniqueclasslabelsfrom9to6(sincethereisonlyonechemicalcompositionwithirrepG3,whichwedonotconsiderforML),wemuststilladdresstheproblemofclassimbalance,wheresomeirrepclasslabelsarefoundmorefrequentlythanothers.
ThiskindofclassimbalanceisproblematicforML.
Totesttheimplicationsofclassimbalance,wetrainedadecisiontreeclassicationmodelusingtheimbalanceddatasetandfoundthatcompositionswithspacegroupPccnorX3(Z1,Z2)were100%misclassied.
AsshowninTable1andFig.
3,PccnorX3(Z1,Z2)isoneofthedesiredclasslabelsfordesigningNCSmaterials.
Therefore,theclass-imbalanceproblemmustbeaddressed.
Anumberofmethodshavebeendevelopedinthecomputerscienceandarticialintelligenceliteraturetoovercometheclass-imbalanceproblem34,35.
Someoftheminclude:oversampling(thatis,randomlyduplicatinginstancesoftheunder-represented403530FrequencyofoccurrenceFrequencyofoccurrence252015105040TargetNCSΓ–3NCS35302520151050P4I4/mmmP4/nmmPbcaPbcmPccnImm2P42/ncmCmcaI41/acdP421mX+3(0,1)X+3(1,1)X+3(1,2)X+2X+3–Figure4|DistributionofexperimentallyknownRPoxides.
Oursurveyresultedinatotalof84compounds,whichwenoterepresentsonlyasmallfractionoftheoverallcombinationsofhypotheticallyfeasiblechemistries.
ExceptfortheninecompoundsindicatedinspacegroupsP421mandImm2,therearenootherexperimentalreportsofNCSphasesinn1RPoxides.
Inset:Thespacegroupsaretransformedintotheircorrespondingirreduciblerepresentations(irreps)andA/A0cationorderingisnotexplicitlyconsidered.
Thesymbolfdenotesnooctahedralrotationortilting.
IrrepsthatwetargetforNCSmaterialsdesignareindicatedusingthedottedrectangleintheinset.
Table1|Irreps,OPDs,SGsandmoderepresentationofdistortedstructuresarisingfromrotationalmodes(X2andX3)andA-sitecationordering(M3).
IrrepsOPDSGMRX3"M3(Z1,Z1;Z1)P421mRotationACOX3"M3(Z1,Z2;Z1)P21212RotationACOX2"X3"M3(0,Z1;Z2,0;Z1)Pca21RotationsACOACO,A-sitecationordering;MR,moderepresentation;OPD,orderparameterdirection;SG,spacegroup;",coupleddistortions.
NATURECOMMUNICATIONS|DOI:10.
1038/ncomms14282ARTICLENATURECOMMUNICATIONS|8:14282|DOI:10.
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Inthiswork,weutilizeanoversamplingschemereferredtoassyntheticminorityclassoversamplingtechnique(orSMOTE)34,inwhichtheunder-representedclasslabelsareoversampledbycreating'synthetic'examplesofextraorctitioustrainingdatapointsfromtheoriginalimbalanceddata.
Itisbasedonak-nearest-neighbouranalysisandoneofitsmainadvantages(relativetootheralgorithms)isthattheextradatapoints,inprinciple,informstheMLmodelstocreatelargerandlessspecicdecisionregions.
AdditionaldetailsaboutthealgorithmaredescribedintheMethodssection.
WetookthedatasetthatcontainedirrepsasclasslabelsandappliedSMOTEtoconstructsyntheticdatapointsforthetwoirreplabels,P4andX3(Z1,Z2).
Wecreatedatotalofthreeandsixsyntheticdatapointsfortheunder-representedP4andX3(Z1,Z2)labels,respectively.
Ourtrainingdatasetsizenowincreasedto78compounds(69originally9fromSMOTE)forclassicationlearning.
Weconrmedusingprincipalcomponentanalysis(PCA)thatSMOTEdidnotaffectourdatamanifold(SupplementaryFig.
1).
Datapreprocessing.
OurNCSmaterialsdesignisinitiatedbyexhaustivelyenumerating,atrst,allpossibleAA0BO4combinationsthatsatisfycrystalchemistryandstoichiometricrules(forexample,chargeneutrality).
Asnotedbefore,weuseWaber–Cromerorbitalradiiasfeatures.
Wethenaugmentthisexhaustivedatasetwiththe78n1RPs.
Notethatatthispoint,wedonotincludetheirrepclasslabelsinourdataset.
Now,wehaveatotalof3,253chemicalcompositionsand22orbitalradiifeatures.
Weautoscaledthedata(normalizedtozeromeanandunitvariance)andappliedPCA,whichconstructslinearcombinationsofweightedcontributionsoforbitalradii(seeSupplementaryFigs2and3).
Inarecentwork,Balachandranetal.
36showedthatinadatasetcontainingorbitalradiiasfeatures,PCAremovesredundancyofinformation,reducesdatadimensionalityandconstructsphysicallymeaningfullinearcombinationsoforbitalradii(seeSupplementaryNote1).
Inaddition,principalcomponents(PCs)arealsoindependentofoneanother(assumingGaussianorNormaldistribution).
AfterPCA,wereducedthedimensionalityofourdatasetfrom22orbitalradiifeaturesto8PCs,whichtogethercapture490%oftotalvarianceinthedataset.
Wethenidentifyandisolate78chemicalcompositionsforwhichtheirreplabelsareexperimentallyknown;werefertothisdatasetasthetrainingset.
Theremainingcompositionsarereferredtoasthe'virtualset'deningthevastchemicalsearchspaceyettobeexploredfornewNCSmaterialsdesign.
Classicationlearning.
WeutilizedtheJ48decisiontreeclassicationlearningalgorithm,asimplementedinWEKA,forestablishingQCSR37,38.
ThereasonsforchoosingtheJ48algorithmarediscussedintheMethodssection.
Weconstructedvebootstrappedsamplesof78compositionseachfromtheoriginaltrainingset.
Wethentrainedthedecisiontreealgorithmusingthevebootstrappedsamplesandconstructedvedecisiontreemodels(SupplementaryFigs4–8).
Theclassicationaccuraciesforthevedecisiontreemodelswereevaluatedonthetrainingdatasetandby10-foldcross-validation.
TheresultsaregiveninSupplementaryTable1andSupplementaryNote2.
Theaverageclassicationaccuracyfromthevebootstrappeddecisiontreesusingthe10-foldcross-validationisB80%.
TheseresultsindicatethatmoreaccurateQCSRmodelscouldpotentiallybeformulatedeitherthroughalternativefeatureselectionmethods39orbyutilizingother(kernel-based)MLalgorithms(whichwedonotaddresshere).
Furthermore,wealsotestedourdecisiontreestodeterminewhethertheycouldcorrectlyidentifytheirreplabelsfor14compounds,whichwereintentionallyheldoutduringthetrainingprocess.
ResultsaregiveninTable2.
OurensembleofdecisiontreescorrectlylabelledwithZ60%accuracy(exceptforYSrCrO4andCa2CrO4)12outof14compoundsintheindependenttestset,givingcondenceinourclassicationlearning.
Usingthevebootstrappeddecisiontrees,wescreenedatotalof3,175compositionsinthevirtualsetandltered242newcompositionsthatshowedpotentialforNCSgroundstatestructures.
Atthisstage,weretainedonlythosecompositionsthatwereidentiedtobeNCS,thatis,belongingtoeitherX3(Z1,Z1),X3(Z1,Z2)orX2"X3(0,Z1;Z2,0),byatleastthreeoutofthevedecisiontrees.
Wethencreatedadditionallterstoremovedatapointsthatcontained(i)toxicelements,suchasPb,HgandCd,(ii)compositionswherebothAandA0siteswereoccupiedbythesameelementand(iii)compositionswithAorA0siteelementsthatwerenotpartoftheoriginaltrainingdataset(forexample,Cs,Rb,Tl,AgandMg).
Wenotethatsomedisagreementisexpectedbetweenourpredictionsandexperiments(orcalculations),particularlywhenconcernedwiththetransitionmetalelementswhosevalencestatefallswithinthestrongelectroncorrelationsregime(forexample,Table2|Acomparisonbetweenexperimentalandpredictedirrepstoindependentlyvalidatetheclassicationmodels.
RPoxidesExperimentalirrepPredictedirrepPredictionaccuracy(in%)CaSrRuO4(ref.
74)P4P460LaSrFeO4(ref.
75)ff100LaSrCoO4(ref.
76)ff100NdSrCoO4(ref.
76)ff100GdSrCoO4(ref.
76)ff100LaSrCrO4(ref.
77)ff100YCaCrO4(ref.
77X3(Z0,Z1)X3(Z0,Z1)80YSrCrO4(ref.
77)X3(Z1,Z2)f0SmCaCrO4(ref.
78)X3(Z0,Z1)X3(Z0,Z1)100LaCaFeO4(ref.
79)X3(Z0,Z1)X3(Z0,Z1)80Ca2CrO4(ref.
80)P4P4andX3(Z0,Z1)40NaDyTiO4(ref.
16)X3(Z1,Z1)X3(Z1,Z1)100NaSmTiO4(ref.
16)X3(Z1,Z1)X3(Z1,Z1)100NaHoTiO4(ref.
16)X3(Z1,Z1)X3(Z1,Z1)100Predictionaccuracy(in%)istheratioofthenumberoftreesthatcorrectlypredictedtheirreplabeltothetotalnumberoftrees(5)usedforprediction.
AllexperimentallyreportedcompoundshavedisorderedA-sitearrangement.
InCa2CrO4,ourclassierpredictswith40%condencethatbothP4andX3(Z0,Z1)labelsareequallylikelyandexperimentally,P4isobserved.
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com/naturecommunicationsTi3,Cr3,V3,Mn3andsoon),mainlybecausetherewereveryfewinstancesofchemicalcompositionswiththesetransitionmetalcationsinourtrainingset.
Ourrenedresults,afterscreeningthroughvariousltersandremovingchemicalcompositionsthatcouldfallinthestronglycorrelatedregime,includedatotalof242newchemicalcompositionsthatshowpromiseforNCSstructures.
ThefollowingoctahedralB-sitecationsinthevirtualsetarepredictedtohaveNCSstructuresinthen1RPoxides:Ga3,In3,Ti4,Zr4,Ru4,Sn4,Hf4,Ir4,Nb5andTa5.
WecouldalsoexcludeIn3,becauseoftheexperimentaldifcultiesinformingn1RPstructuresusingequilibriumsynthesisandprocessingtechniques40(althoughwedonotprecludestabilizingIn-basedn1RPsusingnon-equilibriummethods).
ThechemicalcompositionsforallpredictedNCSmaterialsarelistedinTable3.
AdditionaldetailscanbefoundinSupplementaryTable2,SupplementaryNote3andthedatasetscanbedownloadedfromref.
41.
Tosummarize,usinginformaticsweidentied242newn1RPchemicalcompositionswithpotentialforNCScrystalstructures,whichsignicantlyexpandsthechemicalspaceofNCSn1RPoxides(B25-foldincrease).
Density-functionaltheory.
Onthebasisofthegrouptheoryandmaterialsinformaticsanalysis,werstvalidateourpredictionsbyassessingtheenergeticstabilitycomponent(Task3inFig.
2)fortendownselectedNaRSnO4andNaRRuO4compounds,whereRisarare-earthelement(RLa,Pr,Nd,GdandY)usingDFTcalculations.
Inourcalculations,Na1andR3cationswereorderedinaccordancewiththeirreplabelM3(Z1),asshowninFig.
3b.
Tothebestofourknowledge,nopreviousexperimentalortheoreticaldataexistsforeitherNaRSnO4orNaRRuO4com-pounds.
Inaddition,stannateshaveimplicationsinthedesignoftransparentconductingoxides18andruthenatesarepotentialmaterialsforinvestigatingmetal–insulatortransitions42.
WechooseespeciallyNaRSnO4andNaRRuO4forvalidation,motivated(albeitnaively)bytheadaptivedesignparadigm14,wheretheobjectiveistoiterativelyimprovethepredictionsoftheclassicationmodel.
Typically,theimprovementsaremadebychoosingchemicalcompositionsforexperimentthatshowpromisingcharacteristics(suchasNCScrystalclassesasdiscussedhere),yethavelargeuncertainties.
Here,NaRSnO4andNaRRuO4satisfythesecriteria,becausethepredictionsfromthevedecisiontreeswereX2"X3(NCS),X3(Z1,Z2)(NCS),X3(0,Z1)(CS),X2"X3(NCS)andX3(Z1,Z2)(NCS),correspondingtoPca21(polar),P21212(chiral),Pbcm(centrosymmetric),Pca21(polar)andP21212(chiral)spacegroups,respectively.
Fouroutofthevedecisiontreespredictthesecompoundstohaveachiralorpolarstructure,makingthempromisingNCScandidates,yettheirreplabelsorspacegroupsaredifferent,indicatinguncertainty.
Furthermore,withstannatesthenominalelectroniccongurationofSn4(4d10)isdifferentfromthatofSOJT-cationTi4(3d0),therebypresentinganinterestingcaseforcomparisonbetweenthetwoB-siteoctahedralcations.
TheShannonionicradiiforSn4andTi4inthesix-foldcoordinationare0.
69and0.
605,respectively43,makingtheirionicsizeswithinthehard-spheremodelalsodifferent.
Similarly,ruthenates(withRuinnominally4ionicstate)havepartiallylled4delectronswithfourelectronsoccupyingthet2gorbitalmanifoldandarequitedistinctfromthe3d0titanates.
Stannates.
WeperformedfullstructuralrelaxationsforNaRSnO4(whereRLa,Pr,Nd,GdandY)withinthegeneralizedgradientapproximation(cf.
Methods).
ThephonondispersionsaregiveninSupplementaryFig.
9,fromwhichweidentifyacommonsetofsixcandidatecrystalsymmetriesfrom'freezingin'theimaginaryphononmodesofthehigh-symmetryparaelectricreferencephase(P4/nmm)fordeterminingthegroundstatestructure.
TheyincludePmn21,Pc,P421m,P42m,I42mandPnma.
Inadditiontothesesixcrystalsymmetries,wealsoconsideredthreemoresymmetries,namelyP21212,PbcmandPca21,asrecommendedbyMLtounambiguouslyconrmthegroundstate.
Therefore,intotal,weconsideredninedistortedcandidatestructures.
ThetotalenergydatafromDFTcalculationsisgiveninTable4,whichshowsthatallstannatesexhibitastrongenergeticcompetitionbetweentheNCSpiezoelectricallyactiveP421m[X3(Z1,Z1)]andchiralP21212symmetries[X3(Z1,Z2)].
Wendthatthetotalenergydifferenceiso0.
1meVperf.
u.
(Table4)betweenthetwoNCSphases.
Acloserexaminationofthetwoconvergedcrystalstructuresrevealedthattheydiffermainlyinthein-planelatticeparameters(inP421mab,whereasinP21212aabandthisisshowninFig.
3c,d,respectively).
Furthermore,inP21212thein-planelatticeconstantTable3|Fulllistof242predictedAA0BO4RPcompoundsfromclassicationlearningthatshowpropensitytowardsNCSstructures.
B-cation[A;A0cationcombinations]Ga3[ASr;A0Y,Er,TmandYb][ABa;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu]In3[ACa;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][ASr;A0Y,Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][ABa;A0YandBi]Ti4[ANa;A0Bi,Ce,Pm,Tm,YbandLu]Zr4[ANa;A0Y,Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][AK;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][ACa,Sr;A0Ba]Ru4[ANa;A0Y,Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][AK;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][ACa,Sr;A0Ba]Sn4[ANa;A0Y,Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][AK;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][ACa;A0Ba]Hf4[ANa;A0Y,Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][AK;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][ACa;A0Ba]Ir4[ANa;A0Y,Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu][AK;A0Bi,La,Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,YbandLu]Nb5[ANa;A0Ca,SrandBa][AK;A0CaandBa]Ta5[ANa;A0Ca,SrandBa][AK;A0CaandBa]NCS,noncentrosymmetric;RP,Ruddlesden-Popper.
NATURECOMMUNICATIONS|DOI:10.
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com/naturecommunications7awasfoundtobenotequaltobonlyinthefourthorfthdecimalpoint.
Therefore,weassignthegroundstatestructuretobeNCSP421mspacegroupforthestannates.
WeconcludefromourDFTcalculationsthattheRPstannatesareNCS,ingoodagreementwiththeinsightsfromMLandtheinversionsymmetryisbrokenduetothecoupledactionofSnO6oxygenoctahedraltiltingandNa/Rcationordering(Route1).
Wethencomputedthebandgaps(Eg)foreachofthecompoundsusingtheHSEsolexchange-correlationfunctional(whichoftenmoreaccuratelyreproducesexperimentalresults44)andfoundthemtobeintherange4.
3to4.
5eV(Table5),similartoBa2SnO4(Eg4.
41eV)18.
TheamountofexactexchangeusedinthecalculationswastunedusingtheknownexperimentalbandgapofBaSnO3(ref.
45).
Wenextcomputedthepiezoelectricstraincoefcients(dij)foreachcompoundinP421mspacegroup(Fig.
5);thedijresponseismarginallysmallerthanthatreportedforthetitanates16,butfollowsthesametrend(increasingwithdecreasingatomicradius,uptoRGdandthendecreases).
Ruthenates.
AllDFTcalculationswereperformedusingthespin-polarizedDFTUmethod,whereaneffectiveHubbard-Uof1.
5eVwasusedtotreatthecorrelatedRu4delectrons(cf.
Methods).
ThephonondispersionsaregiveninSupplementaryFig.
10andshowsomesimilaritieswiththestannates.
Weexploredatotalofninedistortedcrystalsymmetriestodeterminethegroundstate(sixfromphononcalculationsandthreefromML).
ThetotalenergiesfromDFTUforNaRRuO4indifferentcrystalsymmetriesandferromagneticspinorderaregiveninTable4;thegroundstateisdeterminedtobeNCSforNaLaRuO4,NaPrRuO4andNaNdRuO4withtwocompetingstructures,P21212andP421m.
Moreover,intheP21212symmetry,awasfoundtobenotequaltobonlyatthefourthdecimalpoint(similartothestannates).
WealsoperformedadditionalDFTUcalculationsforthetoptwolowestenergystructures(namelyP421mandPca21),wherewenowimposeantiferromagneticspinorderonthein-planeRuatoms(shownschematicallyinSupplementaryFig.
11).
ThetotalenergyresultsaregiveninTable6,fromwhichweconcludethattheNCSP421mspacegroupwithferromagneticRu4–O2–Ru4interactionsisthelikelygroundstateforthesecompounds(Route1).
InthecaseofNaGdRuO4andNaYRuO4,thegroundstatestructureisalsodeterminedtobeNCS,butinpolarPca21crystalsymmetry(seeTable4).
Furthermore,inbothNaGdRuO4andNaYRuO4,thePca21structurewithin-planeantiferromagneticRu4–O2–Ru4interactions(SupplementaryFig.
11)werefoundtobe1.
44and5.
54meVperatomlowerinenergy,respectively,thanthatfortheferromagneticstructures.
ThetotalenergydataalongwithRu-atommagneticmomentsaregiveninTable6.
Thus,wepredictNaGdRuO4andNaYRuO4tohavepolarPca21groundstatestructures(Route3)withantiferromag-neticspinorder.
WealsocalculatedtheelectronicbandstructuresforallveNaRRuO4intheirrespectivegroundstates.
TheresultsareshowninSupplementaryFig.
11.
WendthatNaLaRuO4ismetallicwithbandscrossingtheFermilevelinboththespin-upandspin-downelectronchannels.
Ontheotherhand,theNaPrRuO4andNaNdRuO4arefoundtobehalf-metals,thatis,bandscrosstheFermilevelonlyinthespin-downchannelandagapappearsforthespin-upchannel.
Moreover,thesizeofthegapincreasesastherare-earthcationsizedecreases.
ThisoccursbecausetherelativeamplitudeofRuO6octahedraltiltingalsoincreaseswithdecreasingrare-earthcationsize,impactingtheelectronicbandwidthsoftheRu-t2gorbitals.
Notethatthisisnotthersttimeferromagneticmetalsorhalf-metalsarereportedinruthenium-basedoxides46,47.
However,ourintriguingndingisthatNaLaRuO4,NaPrRuO4andNaNdRuO4RPoxidesarealsoNCSwithpiezo-activesymmetries.
Thus,thesecompoundsaddtothegrowinglistofNCSmetals19,20orhalf-metalswithunusualcoexistingproperties(brokeninversionsymmetryandmetallic-likeconduction).
Incontrast,theNCSNaGdRuO4andNaYRuO4arefoundtobeinsulatingwithagapappearinginbothspin-upandspin-downelectronchannels(seeSupplementaryFig.
11).
Wenotethatrutheniumoxideswithantiferromagneticinsulatinggroundstatesarealsonotuncommon.
Forexample,RPCa2RuO4isaTable4|ThetotalenergydifferenceandthermodynamicstabilityfordifferentknownandpredictedRPphasesfromQuantumESPRESSO63.
RPoxidesCrystalsymmetriesfromphononcalculations(DE)Machinelearning(DE)DEDP4/nmmPmn21PcP421mP42mI42mPnmaP21212PbcmPca21KnowncompositionCa2IrO4(Pbca)34Ca2IrO4(I4/mmm)156NewpredictionsStannatesNaLaSnO42.
31.
71.
702.
42.
12.
300.
90.
368.
6NaPrSnO49.
59.
39.
309.
53.
49.
503.
42.
979.
9NaNdSnO414.
714.
715.
4014.
43.
914.
705.
41.
381.
2NaGdSnO440.
234.
834.
5028.
05.
435.
2014.
610.
975.
6NaYSnO446.
837.
136.
4032.
55.
937.
6016.
611.
773.
6RuthenatesNaLaRuO45.
75.
15.
104.
92.
65.
100.
50.
472.
2NaPrRuO415.
514.
914.
9010.
94.
614.
901.
80.
778.
3NaNdRuO421.
120.
420.
4013.
85.
021.
102.
80.
353.
3NaGdRuO446.
141.
741.
71.
026.
67.
143.
21.
038.
9014.
1NaYRuO4179.
947.
647.
62.
632.
78.
849.
22.
611.
401.
3DFT,density-functionaltheory;RP,Ruddlesden-Popper;OQMD,OpenQuantumMaterialsDatabase.
ThetotalenergydifferenceDE(inunitsofmeVperatom)istakenwithrespecttothelowestenergyphase.
CrystalsymmetrywithDE0isidentiedasthegroundstatestructure.
Forallruthenates,weimposedferromagneticspinorderontheRuatom.
DEDinmeVperatomisthetotalenergydifferencecalculatedfromDFTforadecompositionreactionobtainedfromOQMD50,51.
NegativeandpositivevaluesforDEDindicatethatthecompoundisthermodynamicallystableandunstable,respectively.
CorrespondingdecompositionreactionsaregiveninSupplementaryNote4.
ForCa2IrO4,spacegroupsPbcaandI4/mmmarethetheoreticalgroundstateandhigh-symmetrystructures15,respectively.
Furthermore,instannatesstructuresinitializedwithPnmasymmetryconvergedtoP21/mwhenRLa,ProrNd.
Similarly,inruthenatesPcstructureconvergedtoP1whenRPr,GdorY.
ARTICLENATURECOMMUNICATIONS|DOI:10.
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com/naturecommunicationsknownantiferromagneticinsulatorintheCSPbcaspacegroup(Fig.
3e)atlowtemperatures48,49.
Thus,wepredictNaGdRuO4andNaYRuO4aspotentialmultiferroicswithpolarsymmetry,antiferromagneticspinorderandabandgap.
ArethesestannatesandruthenatesalsothermodynamicallystableWeaddressthisquestioninthenextsection.
Thermodynamicstability.
Weusegrandcanonicallinearprogramming50todeterminethethermodynamicstabilityforthepredictedRPstannatesandruthenates.
The'reservoir'ofstablecompoundspresentintheOpenQuantumMaterialsDatabase51werechosentodescribethetheoreticalconvexhull.
Theprocessinvolvescalculationofthetotalenergychange(DED)forachemicalreactioninvolvingreactantsthatareknowntobethermodynamicallystableandaproduct,whichisthegroundstatestructureofourpredictedRPcompounds.
CompoundswithnegativeDEDareidentiedtobethermodynamicallystable.
ItisalsoimportanttonotethatcompoundswithpositiveDED(metastable)havealsobeensynthesized.
Commonly,whenDEDiso25meVperatomabovetheconvexhull,itissuggestedthatthecompositioncouldbepotentiallysynthesizedunderappropriateexperimentalconditions52.
Toevaluatethiscriterionforourdesignproblem,werstcalculatedtheDEDforCa2IrO4thatwasrecentlyepitaxiallygrownintheRPstructure-typeusingthepulsedlaserdepositionmethod29.
ItiswellknownintheliteraturethatCa2IrO4inRPstructuretypeisametastablephase29.
OurmainmotivationistocomparetheDEDforCa2IrO4withournewlypredictedcompounds(especiallythosewithpositiveDED)andgleanadditionalinsights.
TheresultsaregiveninTable4.
TheDEDforRPCa2IrO4inthetheoreticalgroundstateandhigh-symmetrystructuresare34and156meVperatom,respectively,abovetheconvexhull,yetitwassuccessfullysynthesized.
WegivetheDEDdataforboththetheoreticalgroundstateandhigh-symmetrystructures,becauseSourietal.
29donotreportthecrystalsymmetryoftheirthinlm,andthereforethereferencepointisunclear.
HavingbenchmarkedtheDEDdataforCa2IrO4,wereturntoourpredictedNCSstannatesandruthenates.
InTable4,weprovidetheDEDdataforbothstannatesandruthenates.
TheassociateddecompositionreactionsaregivenintheSupplementaryNote4.
Twooutof10compounds—NaGdRuO4andNaYRuO4—havenegativeDED,andtherefore,weidentifythemtobethermodynamicallystableandpromisingforsynthesis.
TheremainingeightcompoundshaveDEDr82meVperatom.
Additionalpredictions.
InTable7,wereportourresultsfornineadditionalrandomlychosencompoundsthatwerepredictedtohaveNCSgroundstatestructuresfromML.
Thetotalenergydata,alongwiththedifferentcrystalsymmetriesobtainedfrombothphononcalculationsandML,aregivenintheSupplementaryTable3.
SevenoutofninecompoundsarefoundtohaveNCSgroundstatestructures,ingoodagreementwithourclassicationlearning.
Notethatsomeofthem(forexample,KBaNbO4andNaCaTaO4)havespacegroupsthatarenotseeninanyknownorreportedRPcompounds(seeFig.
4).
ThisisbecausewedidnotconstrainourDFTcalculationstoonlyknownstructuresorthosefromML,butperformedphononcalculationsandfullstructurerelaxations.
Thedecompositionenergies,DED,forallninecompoundsarealsogiveninTable7.
SixoutofninepredictedcompoundshaveeitheranegativeDED(thermodynamicallystable)orDEDr34meVperatom(thatis,stablerelativetoCa2IrO4),indicatingpromise.
ExperimentalTable5|Bandgap(EgineV)attheHSEsollevelforeachNaRSnO4compoundfromVASP69,70intheNCSP421mspacegroup.
CompoundEg(eV)NaLaSnO44.
35NaPrSnO44.
45NaNdSnO44.
42NaGdSnO44.
34NaYSnO44.
34HSE,Heyd–Scuseria–Ernzerhof;NCS,Noncentrosymmetric;VASP,ViennaabinitioSimulationPackage.
4GdYNdPrLad14,d25d3632Idijl(pC/N)1011.
051.
1rRE()1.
151.
21.
251.
3Figure5|Calculatedpiezoelectriccoefcients.
Piezoelectricstraincoefcients(yaxis)fortheP421mNaRSnO4structuresasafunctionoftherare-earthcationionicsizein,rRE(xaxis).
Therearethreesymmetry-alloweddijcomponents(d14,d25andd36)andtwoofwhichareequivalent(d14d25).
Table6|Totalenergydifference(DEinmeVperatom)withrespecttothelowestenergystructureforNaRRuO4intwoP421mandPca21structureswithbothFMandAFMspincongurations.
CompoundDElRuBP421mFMP421mAFMPca21FMPca21AFMNaLaRuO407.
30.
46.
00.
91NaPrRuO406.
80.
71.
80.
91NaNdRuO406.
70.
30.
50.
91NaGdRuO42.
58.
71.
400.
85NaYRuO48.
114.
15.
500.
84AFM,antiferromagnetic;FM,ferromagnetic.
AllcompoundsinitializedwithAFMP421mconvergedtoAFMP21212structuresindicatingevidenceofspin–latticecoupling.
ConstrainingAFMcongurationinP421mstructures(wherewexedthelatticeconstantstothatofFMP421m)onlyresultedintotalenergieshigherthanthatforAFMP21212.
StructureswithDE0representthegroundstateconguration.
mRuBistheabsolutevalueforthemagneticmomentperRu-site(inBohrmagnetons)inthecorrespondinggroundstatestructures.
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InTable3,chemistriesforall242predictedRPoxidesthatshowpotentialforNCSstructuresarelisted.
TheDFToptimizedgroundstatecrystallographicinformationlesforall19compoundscanbedownloadedfromref.
53.
Asageneralobservation,wenotethattheNCSP421mspacegroupthatwepredictfor13outof19compositionsfromDFTisalsooneofthemostcommonlyobservedexperimentalgroundstates16,17(alsoseeFig.
4)forthen1RPcompounds.
DiscussionWedevelopedacomputationalstrategybuiltonthefoundationsofappliedgrouptheory,MLandDFTtodesignNCSRPcompounds.
Intermsofthenoveltyofourinformaticsapproach,wenotethattheuseofirrepsasclasslabelsforMLisnewtomaterialsscience.
Normally,spacegroupsareutilized.
Theroleofgrouptheoryinourframeworkwastotransformthespacegroupsintoirreps.
FromusingirrepsasclasslabelsforML,wewereabletoreducethecomplexityofourclassicationproblemfrom9to6classlabels.
Evenafterreducingthecomplexity,wefoundthatourdatasetsufferedfromclassimbalance.
Toaddressthisdeciency,weappliedtheSMOTEalgorithmtogeneratesyntheticdatapointsandthenconstructedanensembleofdecisiontreesforirrepclassication.
Ourdecisiontreesidentied242newcompositions(fromscreeningB3,200compositions)thatshowpotentialforNCSgroundstate.
Wetestedourpredictionfor19compositionsusingDFT,amongwhich17werevalidatedtohaveanNCSgroundstatestructure.
Wethusndgoodagreementbetweenourinformatics-basedpredictionsandDFTgroundstatestructures.
Oneofthemajordesignoutcomesistheidenticationoftwonewmultiferroics(NaGdRuO4andNaYRuO4),whichwerealsodeterminedtobethermodynamicallystable.
ItisalsoimportanttorecognizethatnotallourMLpredictionsagreedwiththeDFTcalculations.
Forexample,KLaIrO4andBaLaGaO4werepredictedtobeNCSbutourfrozen-phononcalculationsandfullstructuralrelaxationsfromDFTindicatedisagreement(Table7).
Moreover,theinconsistenciesarefoundtobepronouncedwhenbothA/A0cationshaverelativelylargeionicsizes(forexample,K,BaorLa).
OurDFTcalculationsrevealthatthepresenceoflargeA/A0cationssignicantlyreducestheamplitudeofoctahedraltilting,whichweascribetothestericeffects.
OurMLmodelsappeartoincorrectlyclassifythemasNCS.
ThereareseveralwaystoreducesuchmisclassicationerrorsandimproveourMLpredictionaccuracies.
Welistsomeofthemhere:First,oneofthemostpromisingdirectionsistosynthesizethepredictedmaterialsanddeterminethecrystalstructureforeachcompound,whichwillallowustoaugmentourdatasetwithnewdatapointsandretrainourMLmodels.
WeanticipateourMLmodelstolearnrapidlyfromthesenewdatapointsandimprovetheirpredictionaccuracyinsubsequentiterations32.
Second,ourcurrentMLmodelsarebasedonvedecisiontreeclassiers;oneofthenaturalextensionswouldbetoconstructmorethanvebootstrappedsamplesandgenerateadditionaldecisiontrees(orapplyarandomforestalgorithmwithhundredsofclassiers)thatcould,inprinciple,reducethemisclassicationTable7|DFTaidedvalidationforninerandomlyselectedRPoxidesthatwerepredictedtohaveanNCSgroundstatestructurefromML.
RPoxidesDFTgroundstateNCSgroundstate(in%)PredictedspacegroupsfromML[irreplabel]DEDaspredictedfromMLTree1Tree2Tree3Tree4Tree5NaLaHfO4P421m(NCS)100Pca21[X2"X3]P21212[X3(Z1,Z2)]P421m[X3(Z1,Z1)]Pca21[X2"X3]P21212[X3(Z1,Z2)]17.
9NaLaZrO4P421m(NCS)80Pca21[X2"X3]P21212[X3(Z1,Z2)]Pbcm[X3(0,Z1)]Pca21[X2"X3]P21212[X3(Z1,Z2)]22.
6NaLaIrO4(FM)P421m(NCS)100Pca21[X2"X3]P21212[X3(Z1,Z2)]P421m[X3(Z1,Z1)]Pca21[X2"X3]P21212[X3(Z1,Z2)]204.
6KLaIrO4(FM)Pbcm(CS)80Pca21[X2"X3]P421m[X3(Z1,Z1)]Pca21[X2"X3]Pca21[X2"X3]Ibca[P4]135.
4KBaNbO4P21(NCS)100Pca21[X2"X3]P421m[X3(Z1,Z1)]Pca21[X2"X3]Pca21[X2"X3]Pca21[X2"X3]832NaCaTaO4Pca21(NCS)100Pca21[X2"X3]Pca21[X2"X3]P421m[X3(Z1,Z1)]Pca21[X2"X3]P21212[X3(Z1,Z2)]15.
9SrLaInO4P421m(NCS)100Pca21[X2"X3]P21212[X3(Z1,Z2)]P21212[X3(Z1,Z2)]P21212[X3(Z1,Z2)]P21212[X3(Z1,Z2)]38.
9SrYGaO4P21(NCS)80P21212[X3(Z1,Z2)]fP21212[X3(Z1,Z2)]P21212[X3(Z1,Z2)]Pca21[X2"X3]26.
4BaLaGaO4P4/nmm(CS)60Pbcm[X3(0,Z1)]P21212[X3(Z1,Z2)]Pbcm[X3(0,Z1)]P21212[X3(Z1,Z2)]P21212[X3(Z1,Z2)]51.
1CS,centrosymmetric;DFT,density-functionaltheory;FM,ferromagneticspinorderimposedontheIr-atom;ML,machinelearning;NCS,noncentrosymmetricstructures.
NotethatinavastmajorityofcompoundstheDFTenergydifferencebetweenspacegroupsP21212andP421misoftheorderoffewtenthsofmeVperatom.
AdditionaldetailsaregiveninSupplementaryTable3andSupplementaryNote4.
ForKBaNbO4,thestructureinitializedwithPca21symmetryconvergedtoP21inourDFTcalculations.
DED(inmeVperatom)isthedecompositionenergyforachemicalreactiongiveninSupplementaryNote4.
NegativeandpositivevaluesforDEDindicatethatthecompoundisthermodynamicallystableandunstable,respectively.
ARTICLENATURECOMMUNICATIONS|DOI:10.
1038/ncomms1428210NATURECOMMUNICATIONS|8:14282|DOI:10.
1038/ncomms14282|www.
nature.
com/naturecommunicationserrors.
Also,exploringkernel-basedMLalgorithms,suchassupportvectormachinesandsemisupervisedlearningschemesrepresentalternativeinformatics-basedavenuestogaincondenceorreduceuncertaintiesinourpredictions.
Furthermore,wedemonstratedtheuseoftheSMOTEalgorithmforthersttimeinmaterialsdesignproblems;recently,anumberofnewalgorithms35havebeendevelopedforaddressingsimilarclass-imbalanceproblems,whichcouldalsobeexplored.
Wenotethatclass-imbalanceproblemsareubiquitousinmaterialsdesignandremainsanuncharteredterritoryinmaterialsinformatics54.
Finally,thechoiceofmorerobustfeaturescouldalsoimprovethepredictionaccuracies.
Furthercomputationaleffortsaimedatexhaustivelyevaluatingthepotentialenergysurfaceofrelatedphases55oralternatively,data-drivenapproaches56involvinginferencemodelscouldfurtherrenethepredictionsbyaddressingissuesrelatedtocompoundformabilityandorder-disordertransitions.
Notwithstandingthelimitations,ourapproachprovidesarationalframeworkforstructure-baseddesignofnovelfunctionalmaterialswithimplicationsbeyondthelayeredRPoxides.
Forinstance,ourmethodologycanbeextendedtoexploreNCSstructuresinDion–Jacobson,Aurivillius,Brownmilleriteoranycrystalfamily.
Inprinciple,ourstrategycouldalsoguidethesearchformaterialswithintriguingfunctionalitiessuchasferroaxiality57.
Thekeycomponenttorealizesuchpredictionswillbethedatabaseconstructionprocessandmoreimportantly,thenatureofavailabledata(includingfeatures)woulddeterminethetypeofquestionsthatcanbeaddressed.
IntermsofMLmethods,off-the-shelfclassicationlearningwithclass-imbalancealgorithms(suchasthosedemonstratedinthiswork)hasthepotentialtoprovideinsightsnecessaryforguidingtheacceleratedsearchofnewmaterialswithtargetedcrystalsymmetryorfunctionality.
Advancedlearningstrategies(forexample,semisupervisedlearning,algorithmsbeyondSMOTEandBayesianmethods)maybenecessary,butthechoiceanditsformulationwillhingecriticallyontheavailabledatabasesand/orpriordomainknowledge.
MethodsGrouptheory.
ThegrouptheoreticalanalysiswasperformedusingtheISO-TROPY58toolandelectronicresourcesavailablefromtheBilbaoCrystallographicServer59.
Materialsinformatics.
WeusedthefollowinginferenceandMLmethodsinthispaper:PCAfordata-dimensionalityreductionandfeatureextraction60,samplingtechniquessuchasbootstrapmethodthatconstructsmultipledatasetsfromourexperimentaldatasetviasamplingwithreplacement,decisiontreeclassicationlearning61forformulatingQCSRdesignrulesandSMOTE34torectifytheclass-imbalanceproblem.
Wechosethedecisiontreeclassicationlearnerforthefollowingreasons62:(i)theyareinterpretablemakingthemodeltransparenttodomainexperts;(ii)thesplittingcriteria(forexample,Shannonentropy)servestoaccomplishfeatureselectionwithouttheneedforusinganyadditionalMLmethods;(iii)theyarescalable;and(iv)theyhavethecapabilitytomatchthepredictionaccuraciesofstate-of-the-artMLmethods.
MLcalculationswereperformedusingRSTUDIOandWEKA.
ThedecisiontreealgorithmasimplementedinWEKAwasused.
ThedatasetwasconstructedusingtheWaber–Cromerorbitalradiiasfeatures.
Theclass-imbalanceproblemwasrectiedusingtheSMOTEalgorithm.
Whenthereisclass-imbalance,theseMLmodelscouldignorethelessfrequentlyobservedclasslabelsandgroupthemwithotherclasslabelsinthenearest-neighborhigh-dimensionaldataspacethatoccurmorefrequently.
Thisisnotdesirableforthiswork,becausethefrequencyofoccurrenceoftheNCSspacegroups,tobeginwith,arealreadyunder-represented.
TheinputtoSMOTEisourdatasetandthreeadditionalparameters:(i)theunder-representedorminorityclasslabelthatweintendtooversample,(ii)thenumberofnearestneighbours(k)and(iii)thenumberofextrasyntheticdatasamples(in%)tobecreated.
TheSMOTEalgorithmfunctionsasfollows:ittakesthedifferencebetweenthefeaturevectors(thatis,orbitalradii)oftheunder-representedirrepsanditsknearestneighboursandmultipliesthedifferencebyarandomnumberbetween0and1tocreateanewfeaturevector.
Thisnewfeaturevectorisaugmentedtotheoriginaldataset.
Asaresult,theselectionofarandomdatapointismadealongthelinesegment(asimpliedvisualrepresentationoftheprocessbasedonourdatasetisgiveninSupplementaryFig.
1).
WeusedPCAtoensurethatSMOTEdidnotaffectthemanifoldofourdataset.
WeusetheSMOTEalgorithmasimplementedinWEKA37.
Electronicstructurecalculations.
DFTcalculationsforallRPcompoundswereperformedusingtheplanewavepseudopotentialcode,QuantumESPRESSO(QE)63toobtainthetotalenergies.
Weusedultrasoftpseudopotentials64withthePBEsolexchange-correlationfunctional65takenfromthePSlibrary66.
Aplane-wavecutoffof60Rywasusedduringtheionicandelectronicrelaxationsteps.
ElectroncorrelationsinRu-4dandIr-5delectronsweretreatedusingtheHubbard-UmethodwithintheDudarevformalism67.
Spin-polarizedcalculationswithcollinearferromagneticspinorderwereimposedontheRuandIratoms.
AneffectiveHubbard-Uof1.
5eVwaschoseninbothcases.
FrozenphononcalculationswereperformedusingPHONOPYcode68thatusestheforcesfromQEasinputforcalculatingthedynamicalmatricesandinteratomicforceconstants.
Weemployedasupercellofsize222with112atomsforthefrozenphononcalculations.
AllcalculationstoobtainbandgapsandpiezoelectriccoefcientsforNaRSnO4wereperformedusingDFTasimplementedintheViennaabinitioSimulationPackage69,70.
ThecrystalstructuresweretakenfromconvergedQEcalculations.
Weusedprojectoraugmented-wavepotentials71withthePBEsolfunctional.
Thepiezoelectricandelastictensorswerecomputedwithinthedensity-functionalperturbationtheory72,73withaplane-wavecutoffof800eV.
ThedensityofstateswerecomputedrstwithPBEsol,andthenwithdifferentamountsofexactexchangeusingHSE(Heyd–Scuseria–Ernzerhof).
BycomparingtheexperimentalbandgapofBaSnO3withourcomputedvalues,weselectedtheamountofexactexchangetouse(here35%).
Dataavailability.
ThedatasetsfortheinformaticsstudyandtheDFToptimizedcrystallographicinformationlesaredepositedatgshare(refs41,53.
).
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AcknowledgementsP.
V.
B.
andT.
L.
acknowledgefundingsupportfromtheLosAlamosNationalLaboratory(LANL)LDRDno.
20140013DRonMaterialsInformaticsandtheCenterARTICLENATURECOMMUNICATIONS|DOI:10.
1038/ncomms1428212NATURECOMMUNICATIONS|8:14282|DOI:10.
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nature.
com/naturecommunicationsforNonlinearStudies(CNLS).
J.
M.
R.
andJ.
Y.
weresupportedbyNSFundergrantnos.
DMR-1454688andDMR-1420620,respectively.
TheauthorsacknowledgetheHigh-PerformanceComputingModernizationoftheDODandLANLInstitutionalComputing(IC)forcomputationalresourcesthathavecontributedtotheresearchresultsreportedherein.
AuthorcontributionsThestudywasplanned,calculationsperformedandthemanuscriptpreparedbyP.
V.
B.
,J.
Y.
,T.
L.
andJ.
M.
R.
Allauthorsdiscussedtheresults,wroteandcommentedonthemanuscript.
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V.
etal.
Learningfromdatatodesignfunctionalmaterialswithoutinversionsymmetry.
Nat.
Commun.
8,14282doi:10.
1038/ncomms14282(2017).
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