ab initio Study of Strain-Induced Ferroelectricity in SrTiO3

inSrTiO3

TakatoshiHashimoto1,2∗,TakeshiNishimatsu1,

HiroshiMizuseki1,YoshiyukiKawazoe1,AtsushiSasaki2,andYoshiakiIkeda21

InstituteforMaterialsResearch(IMR),TohokuUniversity,Sendai980-8577,Japan2

NECTOKINCorporation,Sendai982-8510,Japan

February2,2008

Abstract

Valleylinesontotal-energysurfacesforthezone-centerdistortionsoffree-standingandin-planestrainedSrTiO3areinvestigatedwithanewlydevelopedfirst-principlesstructureoptimizationtechnique[Jpn.J.Appl.Phys.43(2004),6785].Theresultsofnumericalcalculationsconfirmedthattheferroelectricityisinduced,andtheCurietemperatureisincreased,byapplyingbiaxialcompres-siveortensilestrains.Alongthedistortion,strongnonlinearcouplingbetweenthesoft-andhard-modesisdemonstrated.

Keywords

densityfunctionaltheory,localdensityapproximation,biaxialstrain,potentialsurface,Slatermode,Lastmode

1Introduction

Perovskitestructurestrontiumtitanate,SrTiO3,isanextremelyimportantma-terial.Becauseofitswiderangeofphysicalproperties,suchassemiconductiv-ity,superconductivity,incipientferroelectricityandcatalyticactivity,SrTiO3isexpectedtobeusedforavarietyoftechnologicalapplications.BulkSrTiO3

crystallizesinthecubiccentrosymmetric(O1

roomtemperature.Atomicfractionalcoordinatesh)structure,andisparaelectricatareSr(0,0,0),Ti(11

1112,2,2,0,2,

∗Corresponding

author.E-mailaddress:hasitaka@imr.edu

1

SrOIzTiOIIyxOIIIFigure1:Cubic(O1

h)crystalstructureofperovskiteoxideSrTiO3.cubictoanantiferrodistortive(AFD)tetragonal(D18

4h)structure,whichisas-sociatedwithzone-boundaryphononcondensationattheRpoint.[3,4,5,6,7]ThistransitionisdrivenbyTiO6octahedralrotationalmodeinstabilities;therotationangleislessthan2◦[8].Atlowertemperatures,thedielectriccon-stantexhibitsCurie-Weisslawlikebehavior,butdoesnotexhibitdivergence.Thedielectricconstantsaturatestoavalueof∼2×104under10K[9,10],butnotransition,e.g.toaferroelectricstate,actuallyoccurswithdecreasingtemperature.Becauseofitsfailuretoexhibitaferroelectricphasetransition,SrTiO3isgenerallyregardedasanincipientferroelectric,inwhichquantumfluctuationssuppressferroelectricity.TheobservedtemperaturedependenceofSrTiO3’sdielectricconstantiswelldescribedbyBarrett’sformula[11],

ε=

M2T1

coth

󰀁

T1

straininducesferroelectricityinSrTiO3hasnotbeenclarified.WeinvestigatetherelationshipbetweenSrTiO3-ferroelectricityandin-planebiaxialstrains,compressiveandtensile,byanalyzingvalleylinesoftotal-energysurfaceswithabinitiostructureoptimizationtechniquethatwehaddeveloped[16].Com-paredtotheconventionalsoft-mode-onlystructureoptimizationtechnique,ourapproachhastheadvantagesthatitcan:(1)accuratelyestimatetotalenergyasafunctionoftheamplitudeofatomicdisplacements,(2)correctlyinvesti-gatenonlinearcouplingbetweensoft-modeatomicdisplacements,hard-modedisplacements,andlatticedeformations.

Inthenextsection,webrieflyexplainourstructureoptimizationtechniqueandthenumericalmethodsusedinthisstudy.Resultsofcalculationsareshownin§3.In§4,wesummarizethepaper.

2

Method

2.1

structureoptimizationtechnique

Toinvestigatethetotal-energysurfacesofperovskiteoxides,weimproveKing-SmithandVanderbilt’sscheme[17]andredefinetheamplitudeofatomicdis-placementsuαas

uα=

󰀄

u2x+u2y+u2zunder

theconditionthatvτ

thetotalαandthestraincomponentsηi(i=1,...,6;Voigtnota-tion)minimizeenergyforeachuusinganabinitionorm-conserving

pseudopotentialmethodandgeometricoptimization.TosimulateD1

4hepitaxialSrTiO3thinfilms,asillustratedinFig.2(a),underasubstrateinducedbiaxialin-planestrain,onlythecaxis(i.e.,η3)ispermittedtorelaxbutotherparame-tersa=b,α=β=γ=90◦(i.e.,η1,η2,η4,η5,andη6)arefixedforalldirectionsofpolarizations,[001](Fig.2(b)),[110](Fig.2(c)),and[100](Fig.2(d)),tosat-isfysymmetryconstraints.Totalenergyminimizationundertheconstant-uαconstraint(i.e.,onthespheresurfacewitharadiusuα)isperformediteratively.Fullydetailedformalismofthisstructureoptimizationtechniqueisgiveninref.[16].Wecontinueiterativeoptimizationofatomicandlatticestructureun-tildifferencesintotalenergiesremainlessthan10−7Hartreefortwosuccessiveiterations.

2.2CalculationMethods

Forallabinitiocalculations,weusetheABINITpackage[18]withadaptingitforourstructureoptimizationtechniqueinitssourcecodeofSrc

(a)[001]cab[010][100]Substrate(b)(c)(d)[001][100][010][100]Figure2:(a)ExaggeratedillustrationofepitaxiallygrownSrTiO3onsubstrate.Latticeconstantsa,b(=a),andcandcrystallographicdirections[100],[010],and[001]areindicated.(b)Patternofatomicdisplacementsaccompanying[001]tetragonaldistortionunderbiaxialcompressivestrainisindicatedbyarrowsin(0¯10)projection.(c)Thatof[110]monoclinicdistortionunderbiaxialtensilestrainin(100)projection.(d)Thatof[100]monoclinicdistortionunderbiaxialtensilestrainin(100)projection.

4

ofkineticenergiesofplanewavesjustbelowtheenergycutoffisintroduced[20]asimplementedintheABINIT.This“energycutoffsmearing”techniqueiseffi-cientnotonlyforconstant-pressuremoleculardynamicsbutalsoforcomparingtotalenergiesindifferentvolumes.Withthistechnique,onedoesnothavetoachieveconvergencebetweenconstant-number-of-plane-wavescalculationsandconstant-energy-cutoffcalculationswithrespecttothebasissetand,therefore,doesnothavetoadoptasuperfluouslylargeenergycutoff.Thepseudopoten-tialsincludeO2sand2p,Ti3s,3p,3dand4s,Sr4s,4pand5s,asvalenceelectrons.Blochwavefunctionsaresampledonan8×8×8gridofk-pointsinthefirstBrillouinzone.Theexchange-correlationenergyistreatedwithinthelocaldensityapproximation(LDA).Astheparametrizedcorrelationenergy,weuseTeter’srationalpolynomialparameterization,[21]whichreproducestheresultsobtainedbyCeperleyandAlder.[22]Theelectronicstatesarecalculatedbytheiterativeschemetoreachatoleranceofconvergencethatrequiresdifferencesofforcestobelessthan5×10−7Hartree/Bohrfortwosuccessiveiterations.

Withthesecalculationmethods,thecalculatedequilibriumlatticeconstantforcubicSrTiO3is7.27Bohr,whichis∼1.3%lessthantheexperimentalvalueof7.36Bohrobtainedbylinearlyextrapolatingthelatticeconstantathightemperaturetozerotemperature(seeFig.1inref.[23]).Thisunderestima-tioniscommonlyconsideredtobearesultoferrorsfromLDA.Propertiesofferroelectrics,especiallytotal-energysurfaces,areverysensitivetothelatticeconstant[24].Semi-empiricalconstraintsonlatticeconstantsincalculationsofperovskiteoxideshavebeencommonlyusedtoacquireagreementsbetweentheexperimentallyobservedvaluesandcalculatedresults.Althoughartificialconstraintscanbeintroducedinourstructureoptimizationtechnique,wedonotemploysuchsemi-empiricalconstraints.Nevertheless,webelievethatcal-culationsusingLDAclarifysometrendsofdisplacivetransitionsinstrainedSrTiO3.

TheBerry-phasemethod[25,26]isusedtoevaluatespontaneouspolariza-tions.

ForsimplicityandtoemphasizetheoriginofferroelectricityinstrainedSrTiO3,weneglecttheAFDinstabilitiesanduseasingleunitcelltocalcu-latetotal-energysurfaces.WebelievethatthisisareasonableapproximationbecausetheeffectofAFDondielectricresponseisnegligiblysmall.[27]

3

3.1

ResultsandDiscussion

Free-standingSrTiO3

Beforeexaminingin-plainstrainedSrTiO3,weevaluatedthetotal-energysur-faceoffree-standingSrTiO3thatisdistortedfromcubictotetragonalandhas[001]polarization.ShowninFig.3(a)isthetotalenergyasafunctionoftheamplitudeofatomicdisplacementsuz.M.Itohetal.foundthatferroelectricitywasinducedinSrTiO3bytheisotopeexchangeof18Ofor16O.[10]ThisisotopeexchangeexperimentandtheincipientferroelectricitywithT0=38K>0inBarrett’sformulasuggestthatthetotalenergyvsuzcurvemighthavedoublewellstructure,butourcalculatedresultdoesnot.ThisisnotjustanartifactofthesensitivityofLDAtotal-energysurfacestolatticeconstants,asdescriedinrefs.[24]and[28],rater,asweshowinnextsubsection§3.2,thesystemisvery

5

closetohavingadoublewell.

Calculatedlatticeconstantsaandcarewellfittedbyquadraticfunctionsin

Sr

Fig.3(b).ItissurprisingthattheamplitudeofSr-displacement,vz,issmaller

Ti

thanthatofTi,vz,forthesoft-modeeigenvectorcalculatedbythefrozen

SrTi

phononmethodatuz=0,butvzbecomeslargerthanvzforuz>0.07asshowninFig.3(c)and(d).

3.2

Effectsofin-planestrainsonSrTiO3

TotalenergiesofSrTiO3asfunctionsoftheamplitudeofatomicdisplacementsuarecalculatedforthetetragonal[001]distortionunderin-planebiaxialcompres-sivestrains(Fig.4(a))andthemonoclinic[110](Fig.4(b))and[100](Fig.4(c))distortionsunderin-planebiaxialtensilestrains.Itisconfirmedthat,forthebi-axialtensilestrainedSrTiO3,the[110]directionofferroelectricpolardistortionisenergeticallymorepreferablethanthatof[100]ascomparedinFig.5.Energygain,thedeferencebetweentotalenergyatu=0anditsminimumvalue,growsupaccordingtobothcompressiveandtensilebiaxialstrains,thoughitiszerointhezero-straina=a0case.Thevalueofu,atwhichthesystemexhibitsminimumtotalenergyandequilibriumstructure,alsogrowsupaccordingtothebiaxialstrains.InFig.6,weshowcalculatedspontaneouspolarizationsfortheequilibriumstructuresunderbiaxialstrainsa=0.95a0∼1.05a0.Underthecompressivestrain,spontaneouspolarizationappearsabove1%andincreaseslinearlyasafunctionofstrain.Underthetensilestrain,spontaneouspolariza-tionappearsalittlebelow1%andincreasesmoregraduallywithstrain.Thesecalculatedresultsrepresentthatthetransitiontemperatureandthespontaneouspolarizationforferroelectricorderingarepredictedtobemonotonicallyincreas-ingfunctionsofbiaxialin-planestrains.Inourcalculatedresults,theenergygainbecomesbarelytheorderofroomtemperature(300K≈26meV)at4%ofcompressiveortensilestrains.The4%ofcompressiveortensilestrainsareim-practicalevenbythethermalnon-equilibriumthin-filmgrowthtechniques,andlargerthanthe1%oftensilestrainunderwhichferroelectricitywasexperimen-tallyfound,[13]thoughitisdifficulttocompareexperimentsandLDA-basedcalculations.Ontheotherhand,Haenietal.couldnotobservetheferroelec-tricityinthe0.9%compressivestrainedSrTiO3thinfilmona(La,Sr)(Al,Ta)O3substrate.[13]Itisalsodifficulttoconcludethat0.9%compressivestrainislessthanthecriticalstrainrequiredtoinduceferroelectricityorthat0.9%isenoughtoinducetheferroelectricity,butasurfaceeffect(oranyothereffects)suppressesit.

Toclarifythestructuraleffectsofin-planebiaxialstrain,weanalyzethedetailedbehaviorofatomicdisplacementsfor4%biaxialcompressiveandtensilestrains:

Predictedresultsforthecompressivestrain,asfunctionsofuz,areplottedinFig.7:(a)totalenergy;(b)latticeconstants;(c)and(d)atomicdisplacements.Ferroelectrictetragonalequilibrium,witha40.2meVstabilizationenergygain,ispredictedatuz=0.456Bohr.Thelatticeconstantc(uz),iswellfittedwith

τ

aquadraticfunction.Normalizedatomicdisplacementsvz/uzchangegreatlyintheinvestigatedrangeofuz,incontrastwithBaTiO3inwhichtheyremainalmostconstanttotheΓ15soft-modeeigenvectorinferroelectricdistortion[16].Intheatomicdisplacementsofthepositivecharge,wecanfindthatthedis-placementofTiisdominantfromparaelectricuz=0statetouz=0.456at

6

250.0tetra-SrTiO3 (valleyline [001])energy [meV]200.0150.0100.050.00.08.07.87.67.47.27.06.80.60.40.20.0-0.2(a)calattice constant[Bohr](b)SrTidisplacementτvz [Bohr](c)OIIIOI-0.4Sr0.80.60.4Ti0.20.0OIII(d)-0.2-0.4OI-0.6-0.80.00.10.20.30.40.50.60.70.80.9uz [Bohr]Figure3:Calculatedresultsforfree-standingSrTiO3regardingatomicdisplace-mentsoftetragonal[001]direction.(a)Calculatedtotalenergy(inmeV)asafunctionofuz(inBohr)(+’sconnectedwithsolidlines).Zerooftheenergyscaleisplacedatthetotalenergyofthecubicstructurewhenuz=0.(b)LatticeconstantsaandcinBohrasfunctionsofuzfittedbyquadraticfunctionsdrawnwithadottedlineandadashedline,respectively.(c)Atomicdisplacements

OIIIOIISrTiOI

asfunctionsofuz.(d)Normalizedatomicdisplace-,vzvz,vz,vz=vz

τ

mentsvz/uzasfunctionsofuz.TheΓ15soft-modeeigenvectorcalculatedbythefrozenphononmethodisadditionallyshownatuz=0.

normalized displacement7

200.0150.0100.0energy [meV]SrTiO3 (out-of-plane)a = 1.00*a0a = 0.99*a0a = 0.98*a0a = 0.97*a0a = 0.96*a0a = 0.95*a0a0 = 7.266 [Bohr] SrTiO3 (in-plane, [110])a = 1.00*a0a = 1.01*a0a = 1.02*a0a = 1.03*a0a = 1.04*a0a = 1.05*a0a0 = 7.266 [Bohr] SrTiO3 (in-plane, [100])a = 1.00*a0a = 1.01*a0a = 1.02*a0a = 1.03*a0a = 1.04*a0a = 1.05*a0a0 = 7.266 [Bohr]50.00.0-50.0-100.0-150.00.00.10.20.30.40.5uz [Bohr]0.60.70.80.9 0.00.20.40.6u [Bohr]0.81.01.2 0.00.20.40.6u [Bohr]0.81.01.2(a)(b)(c)Figure4:Calculatedtotalenergies(inmeV)ofSrTiO3asfunctionsofu=󰀂

0.7spontaneous polarization [C/m2] 1.02.03.04.0biaxial compressive strain [%]5.0 0.01.02.03.0biaxial tensile strain [%]4.05.00.60.50.40.30.20.10.0-0.10.0Figure6:Calculatedspontaneouspolarizationsoftheequilibriumstructuresasafunctionofappliedbiaxialstrains.

Table1:Conventionalatomicdisplacivemodes.Slatermode,Lastmode,andtheoctahedron-deformationmode.ThetranslationalmodeandtheΓ25modearealsolisted.

Slatermode

ABOII

−1/−1/

√12√12

Lastmode√20−1/

√20−1/

20

octa.deform.mode01/√√6

−2/

√51/√1/√51/

0√2

−1/

whichthetotalenergybecomesminimum,whereasthedisplacementofSrbe-comeactivatedinsteadofTiintheregionofu>0.456,asshowninFig.7(c).TheremaybearelatednessbetweenthisSr-Ti-displacements-crossingpointandtheferroelectricequilibriumstructurewiththeminimumtotalenergy,thuswementionedthatfree-standingSrTiO3,whichhavethecrossingpointataroundu=0.07,isveryclosetohavingadoublewell.Inthenegativecharge,theatomicdisplacementofOI(≡OII)isdominantcomparedtothatofOIIIthroughouttheinvestigatedrangeofuz.Toobtainthedeeperinsightsaboutthesebehavioroftheatomicdisplacements,weanalyzethepresentlyobtainednormalizedatomicdisplacementsbydecomposingthemintoconventionalthreeΓ15modes:Slatermode,[29]Lastmode,[30]andtheoctahedron-deformationmode,aslistedinTable1.AsshowninFig.8,althoughtheSlatermodeisdominantthroughouttheentireinvestigatedrangeofatomicdisplacements,theLastmodetendtoin-creasewhiletheSlatermodedecreasesmoderatelyasfunctionsoftheamplitudeofatomicdisplacementsuz.ThisresultiscorrespondingtothesightpointedoutbySchimizu[14]andHarada[31]thattheLastmodeisalsoimportantwhiletheSlatermodeplaysakeyroleinthephasetransitionofSrTiO3.

ForSrTiO3underthe4%tensilestrain,thecalculatedresultsoftotalenergy,

9

60.040.020.00.0-20.0-40.0-60.08.07.87.67.47.27.06.80.60.40.20.0-0.2tetra-SrTiO3 (a=0.96*a0)energy [meV](a)lattice constant[Bohr]ca(b)displacementτvz [Bohr]SrTi(c)OIIIOI-0.40.8Sr0.60.4Ti0.20.0OIII(d)-0.2-0.4OI-0.6-0.80.00.10.20.30.40.50.60.70.80.9uz [Bohr]Figure7:CalculatedresultsforSrTiO3under4%biaxialcompressivestrainregardingatomicdisplacementsoftetragonal[001]direction.(a)Calculatedtotalenergy(inmeV)asafunctionofuz(inBohr)(+’sconnectedwithsolidlines).Zerooftheenergyscaleisplacedatthetotalenergyofthecubicstruc-turewhenuz=0.(b)LatticeconstantsinBohr.aisfixed.cisfittedbyquadraticfunctionsofuzanddrawnwithadashedline.(c)Atomicdisplace-SrTiOIOIIOIII

mentsvz,vz,vz=vz,vzasfunctionsofuz.(d)Normalizedatomicdis-τ

placementsvz/uzasfunctionsofuz.

normalized displacement10

1.0dten0.8seompe0.6mcoalcp0.4Slateresdid0.2octahedron-deformationLast0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9uz [Bohr]Figure8:NormalizedatomicdisplacementsoftetragonalSrTiO3aredecom-posedintothreeconventionalmodes:Slatermode(solidline),Lastmode(dashedline),andtheoctahedron-deformationmode(dottedline)asfunctionsofuzinBohr.

latticeconstantsandatomicdisplacements󰀂ofx-directionasfunctionsoftheamplitudeofatomicdisplacementsu=

200.0mono-SrTiO3 (a=1.04*a0)[C][A][B]energy [meV]150.0100.050.00.0-50.07.87.67.47.27.00.60.40.20.0-0.2-0.4-0.60.80.60.40.20.0-0.2-0.4-0.6-0.80.0ac(a)lattice constant[Bohr](b)SrTiOIIIOIOIISrdisplacementτvx [Bohr](c)normalized displacement(d)TiOIIIOIOII1.20.20.40.60.8u [Bohr]1.0Figure9:CalculatedresultsforSrTiO3under4%biaxialtensilestrainregardingatomicdisplacementsoftetragonal[110]direction.(a)Totalenergy(inmeV)asafunctionofu(inBohr)(+’sconnectedwithsolidlines).Zerooftheenergyscaleisplacedatthetotalenergyofthecubicstructurewhenu=0.(b)Lattice

τ

constantscandfixedainBohr.(c)Atomicdisplacementsvx.(d)Normalized

τ

atomicdisplacementsvx/uxasfunctionsofu.

12

[A] u = 0.0 Bohr[B] u = 0.443 Bohr[C] u = 0.870 BohrFigure10:Pseudovalencechargedensitymapsinthe(001)crosssectionsandaprojectionforSrTiO3under4%biaxialtensilestrainatthefollowingthreepointsofatomicdistortion;[A]paraelectricstateatu=0,[B]monoclinicequilibriumstateatu=0.443Bohr,and[C]largedisplacementstateatu=0.870Bohr,whichareindicatedontotal-energysurfaceshowninFig.9(a).MapsweredrawnwithVENUS.[32]

13

beregardedasatypicalioniccrystal.Theoriginoftheferroelectricitycanbeconsideredthattherigid-sphere-likeionsgointoopenspacesinducedbythebiaxialtensilestrain.

4Summary

Inthisstudy,weinvestigatedtheeffectofthein-planebiaxialcompressiveandtensilestrainsofSrTiO3ontheferroelectricdistortionbydeterminingthetotal-energysurfaceaccuratelywithourstructureoptimizationtechnique.(a)WeconfirmedthattheferroelectricdistortionisinducedstronglyandCurietemperatureisraisedbyapplyingthebiaxialcompressiveortensilestrainstoSrTiO3.(b)Fromtheresultsofthenormalizedatomicdisplacements,itisclarifiedthattheferroelectricdistortionofSrTiO3isstronglyinfluencedbytheatomicdisplacementscorrespondingtothehardmodeaswellasthesoftmode.(c)Byanalyzingtheobtainednormalizeddisplacementsfor4%biaxialcompressivestrain,weshowedthattheSlatermodeplaysadominantrolewhiletheLastmodeisalsoimportantintheferroelectricphasetransitionofSrTiO3.(d)Thebehaviorofatomicdisplacementassociatedwithferroelectricdistortionwasclarifiedbyanalyzingthechargedensitydistributionineachamplitudeofatomicdisplacementsfor4%biaxialtensilestrain.

Acknowledgements

ComputationalresourceswereprovidedbytheCenterforComputationalMate-rialsScience,InstituteforMaterialsResearch(CCMS-IMR),TohokuUniversity.WethankthestaffatCCMS-IMRfortheirconstanteffort.

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