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Acta Materialia

journalhomepage:www.elsevier.com/locate/actamat

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Presence of a purely tetragonal phase in ultrathin BiFeO ₃ ﬁlms:

Thermodynamics and phase-ﬁeld simulations

Yang Zhang

^a^,^b

, Fei Xue

^b^,^∗

, Zuhuang Chen

^c

, Jun-Ming Liu

^a^,^d

, Long-Qing Chen

^b

aLaboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

bDepartment of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

cSchool of Materials Science and Engineering, Harbin Institute of Technology, Shenzhen 518055, China

dInstitute for Advanced Materials and Laboratory of Quantum Engineering and Materials, South China Normal University, Guangzhou 510 0 06, China

a rt i c l e i nf o

Article history:

Received 18 July 2019 Revised 22 October 2019 Accepted 31 October 2019 Available online 8 November 2019 Keywords:

Ferroelectric Thin ﬁlms

Phase-ﬁeld simulation Morphotropic phase boundary

a b s t ra c t

ThestabilityofapurelytetragonalphaserelativetothenominalrhombohedralphaseinultrathinBiFeO₃ films is investigated usingthermodynamics and phase-field simulations. The thermodynamic analysis demonstratesthe possible presenceofapurely tetragonalstate primarilydueto theinterfacial effect fromtheconstraintoftheadjacentlayeralthoughthebuilt-inpotentialandcompressivein-planestrain alsoplayarole.Phase-field simulationsofthecorresponding ultrathinfilmsrevealthe coexistence of tetragonalandrhombohedralphasesatcertainfilmthicknessarisingfromstrainphaseseparation.Itis shownthatthepiezoelectriccoefficientd₃₃ofthetwo-phasemixtureisupto200%higherthanthatof therhombohedralsinglephase.

1. Introduction

Ferroelectric (FE)thinfilmshavebeenextensivelystudied due to their wide rangeof potential applications such as mechanical sensorsandactuators,non-volatilememories,high-frequencyelec- tronic components, etc. [1–5]. To explore device miniaturization andpossibleintriguing physics,there havebeen increasing inter- estsinultrathinfilmswiththicknesslessthantensofnanometers [6–9].Naturally,thereductionofdimensionsgivesrisetothesize effectwhichmayalterthedomainstructuresandperformancesof ultrathinfilmswhicharedramaticallydifferentfromthoseoftheir bulk counterparts [10]. For instance, it was previously believed that there exists a so-called“critical thickness” below whichthe spontaneouspolarizationwassuppressed[11].Thermodynamically, the size effect originates fromthe competitions among the bulk freeenergy,electrostaticenergy,elasticenergy,andsurfaceenergy [12–14].Inparticular, itisshownthatthedepolarizationfield in- ducedby uncompensated bound chargeson the top andbottom surfacesoffilms largelycontrolsthecriticalthicknessforthefer- roelectricityinultrathinFEfilms[15].

However, subsequent theoretical and experimental investiga- tions indicated that the ferroelectricity could persist with the thicknessdown toseveralnanometersorevenone-unit-cellscale intheultrathinFEﬁlmsorsuperlattices[16–19].Ingeneral, there

∗ Corresponding author.

E-mail address: xuefei5376@gmail.com (F. Xue).

are two conventional approaches to eliminating the influence of thedepolarizationfieldandthusstabilizingtheout-of-planespon- taneous polarization. Oneis to form109°or180°stripe domains withpolarizationspointingupanddownperiodically[20–22].The alternating positive and negative bound charges on the surfaces can reduce the electrostatic energy and thus can drastically de- creaseoreveneliminate thecritical thickness.Forexample,Fong etal.[16]in2004 observedferroelectricity withsuch180°stripe domains in three-unit-cell-thick (1.2nm) epitaxial PbTiO3 films grownon(001)SrTiO₃substrates,andin2009,similarresultswere found by Tenne et al. [18] in strained epitaxial BaTiO₃ films as thin as 1.6nm grown also on (001)SrTiO₃ substrates. Neverthe- less, many nanoscale electronic devices nowadays such as memory applicationsrequiresingledomainstructures withstableand switchablepolarizations.Then,theotherapproachistogrowultra- thinfilmsonconductingbottomelectrodelayersandexposetheir surfaces tothe ion-rich environment inorder to compensate the boundcharges.In2006,Fongetal.[17]observedmonodomainpo- larization in c-axis epitaxialultrathin PbTiO₃ films grown on the conductingSrRuO₃ layersby themetalorganicchemicalvapor de- position.Therefore,theboundchargesonbothtopandbottomin- terfaces werecompensated by freecharges, andhencethe depo- larization fieldcould be eliminated.Inaddition tothecompensa- tion by free charges,further measurements and characterizations showedthat thepropertiesof ultrathinFEfilms are alsoaffected by other interactions such as in-plane epitaxial strains, applied electric fields, dislocations, and the interfacial energy [23–27].

https://doi.org/10.1016/j.actamat.2019.10.054

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Therefore,itisimportanttofullyunderstandtheinterplaybetween thesefactorsforpotentialapplications.

AmongFEmaterials,bismuthferriteBiFeO₃(BFO)isparticularly interesting because of its room-temperature multiferroicity [28]. BulkBFOpossessestherhombohedral(R)perovskitestructurewith spacegroupR3candFEorderbelowT_C∼1100K,antiferromagnetic orderbelowT_N∼650K,andantiferrodistortiveorderbelow1200K [28,29].Moreover, strainedBFOfilms canformasupertetragonal (super T) phase under large compressive strain, an orthorhombic (O) phase under tensile strain, and several transient monoclinic phases [30–33]. Apart from the temperature control and strain engineering, many researches pointed out that the structures of substrates or electrodesthat films were directlygrown on could also play an essential role in the domain patterns of BFO,espe- ciallyinultrathinfilms[19,34–36].Hanetal.[35]andGengetal.

[34]foundthecoexistenceofRandOphasesinultrathinBFOfilms driven by the interfacialoxygen octahedral coupling. The strain- inducedmorphotropicphaseboundary(MPB)couldprovidealarge piezoelectric response. Recently, Wang et al. [19] observed stable andswitchable out-of-plane spontaneous polarization inBFO films withevenone unit-cellthicknessgrownonnear-tetragonal- SrRuO₃-buffered(001)SrTiO₃substrate.Interestingly,theBFOfilms exist in a purely T phase because of multiple factors including thechargescreeningonbothsurfaces,sizableionicpolarizationof theoxideelectrodeattheinterface,andthecrystallographicstruc- ture ofthe electrode. The ferroelectricity atthe two-dimensional limitmakesBFOapotentialcandidateforhigh-densitynon-volatile memories. Inspiredby those findings,we theoreticallyinvestigate theformationandstabilityofthepurelyTphaseinultrathinBFO films.

In this work, we report the purely T phase in ultrathin BFO ﬁlms usingthermodynamic analyses andphase-ﬁeld simulations.

Section 2 introduces the model based on the Ginzburg-Landau- Devonshire theory,containing the phenomenological descriptions of all the free energy terms and the coefficients needed. In Section3,thermodynamicanalysesbasedonthesingle-domainas- sumptionare presented.Then Section 4describesthe phase-field simulation results of multidomain structures, which demonstrate thecoexistenceofthepurelyTandR-likephasesandtheenhance- mentofpiezoelectriccoefficients.Section5istheconclusion.

2. Model

Fig.1isthesketchofthesystemthatwestudy.Toproducethe Tphase,we assumethat thebound chargesattwointerfacesare fullycompensatedandconsider theeffectsofthebuilt-in electric ﬁeldandthein-planeepitaxialstrain.

2.1. Phenomenologicaldescription

Basedon theGinzburg-Landau-Devonshiretheory,weconsider both thepolarization P andoxygenoctahedral tilt(OOT)

θ

^as^or-

derparameterstodescribethefreeenergytermsofBFO.Following previousworks[37–41],thetotalfreeenergyfortheultrathinﬁlm systemcanbewrittenas

=

V

fbulk+felast+fgrad+felec

dV+

S

fsurfdS, (1)

where f_bulk, f_elast,f_grad,and f_elec are the volume densities ofbulk free energy,elastic energy, gradient energy, andelectrostatic energy,respectively,whilef_surfisthearealdensityofsurfaceenergy.

ThebulkfreeenergydensitytermisdescribedbyLandaupoly- nomials extended up to the eighth order with respect to order parameters. For the elastic energy term, the total strain tensor ɛ containscontributions fromthe electrostriction,rotostriction,and elastic strain,respectively [42].See Supplementary MaterialA for

Fig. 1. (Color online) Sketch of the system. The system is divided into four layers, i.e., air, BFO, electrode, and substrate. The bound charges at interfaces of the BFO layer are assumed to be fully compensated, and the built-in electric ﬁeld and in- plane epitaxial strain are considered in the BFO layer.

detailed expressions for the bulk free energy andelastic energy terms.

The gradient energyterm isintroduced by thespatial deriva- tivesofPand

θ

^(SeeSupplementaryMaterialA)as

fgrad

∇

^P,

∇ θ

=1

2gi jklPi,jPk,l+1

2

v

i jkl

θ

i,j

θ

k,l, (2)

⁼^−P^·^E⁻¹₂^E^T

⁰

κ

^E, ⁽³⁾

whereEisthetotalelectricﬁeld,

0isthevacuumdielectriccon- stant,and

κ

^is^the^rank-2^background^dielectric^constant^tensor.

^P

g44

P₁²+P₂²

+g11P₃²

+ 1 2

δ

θ

v

44

θ

1²+

θ

2²

δ

P and

δ

_θ ^indi-

catethestrongersurfaceeffect.

2.2.Coeﬃcients

All the temperature-independent coefficients in the bulk and elasticenergyterms[Eqs.(S1)–(S7)andEqs.(S9)–(S12)]aretaken fromfirst-principlecalculationsattemperatureT=0Kreportedin literature[41].Onlya₁andb₁aretemperaturedependent,andthe valuesofthetransitiontemperaturesarefromliterature[40].Co- efficientsofthebulkandelasticenergytermsarelistedinTableSI (seeSupplementaryMaterialB).

Tothebestof ourknowledge,the gradientenergycoeﬃcients forBFOin Eqs.(2) and(S13) are not knowndirectly eitherfrom

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^is ^then ^the ^difference ⁱⁿ ^the

work function step [37], h_f is the thickness of films, and nz= (⁰,0,1)^Tîs^theûnit^vector^from^bottom^to^top.^Therefore,^theêlec- trostaticenergywithoutanexternalappliedfieldisgivenby

f_elec=

δϕ

ft =

δϕ

h_fP3+a^∗₁

P₁²+P₂²

+a^∗₃P₃²+a^∗₁₁

P₁⁴+P₂⁴

+a^∗₃₃P₃⁴+a^∗₁₂P₁²P₂²+a^∗₁₃

P₁²P₃²+P₂²P₃²

+b^∗₁

θ

3²+t₁₂^∗

2²

+t44P1P2

θ

1

θ

2

⁺

C11+C12−2C₁₂² C11

ε

s²

+sixth-andeighth-orderterms, (8)

wherethecoefficientswithsuperscript^∗signsindicatetherenor- malizedcoefficients,andthesixth-andeighth-ordertermsremain the same as inEqs. (S2), (S3), and (S5)-(S7). See Supplementary MaterialDforthedetails oftheserenormalizedcoefficients. Note thata^∗₁,a^∗₃,b^∗₁,andb^∗₃arefunctionsofthefilmthicknessh_f.

From Eqs. (8) and (S18)-(S20), five parameters besides the filmthicknessh_f haveinfluences ontheenergyprofiles. Theyare temperature T, in-plane substrate strain ɛs, built-in potential

δϕ

^,

effective extrapolation length of polarization

δ

^effP , and effective extrapolation length of OOT

δ

_θ^eff^. ^Their ^roles ^can ^be illustrated with energy contours. Fig. 2(b) gives the energy contours at T=298Kunder thestrain

ε

s=−2% (effective energyexpression f_bulk+f_elast). Compared with Fig. 2(a), the four minima change fromthe Rphase to themonoclinic (M_A,or R-like)phase dueto the compressive strains. Then, adding an external electric ﬁeld alongthe[001]directionwithE^app=5MV/cm(describingtheef- fectofthebuilt-inpotential)breaks thesymmetryalong[001],as showninFig.2(c)(effectiveenergyexpression f_bulk+f_elast+f_elec).

One mayobservetheappearance ofa purelyTphase inFig. 2(c) withthe spontaneous polarization P₃∼1.1C/m².In fact, inaddition tothe stableR/R-like phases, theenergyproﬁles in Fig.2(a) and (b) also include the existence of such T metastable phases according to our numerical results, but the energy difference

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Table 1

Default parameters used in this work.

Parameter Value

T 298 K

ɛ s −2 . 0%

δϕ −1 . 0 V

δ^effP 10 ˚A

δ^eff_θ 0.1 ˚A

betweentheRandTphasesaretoohightoobservetheTphases.

Next,thesurfaceeffectonthephasestabilitiesisstudied.Fig.2(d) and(e)istheenergycontourswiththesamevaluesofparameters asFig.2(c)exceptthevalueofh_f.Thedefaultvaluesofthesefive parametersarelistedinTable1.Allthefollowingresultsarebased onthedefaultvaluesunlessnotedotherwise.Forafilmwiththick- nessh_f=3.5nm,asshowninFig.2(d),thereexistboththeR-like phaseandthepurelyTphase,whiletheformerisstable.Forh_f= 2.8nm[Fig.2(e)],however,theTphasebecomesthestatewiththe lowestenergyalthoughtheR-likephaseismetastable.Finally,the R-likephasebecomesunstableandonlytheTphaseremainswith sufficientlysmallh_f,e.g.,h_f=2.0nminFig.2(f).Sincethecontri- butionsofthesurfaceenergyandelectrostaticenergyareinversely proportional tothethicknessh_f [Eqs.(S16)and(7)],thetendency in Fig. 2(d)–(f) indicates that the surface effect and the built-in potentialhelptostabilizethepurelyTphasewithdecreasingfilm thickness. It should be noted that the above T phase has only onenonzeropolarization component(out-of-planeP₃),withthree zeroOOTcomponents,inagreementwiththeexperimentalresults [19].

Fig.2(d)–(f)istheresultsforthreetypical valuesofh_f.Tofur- ther understandthe behaviorsofthe purely TandR-like phases, itisnecessarytocomputethetotalenergiesofthetwophasesas functionsoffilm thickness.Fig. 3(a)givesthecurves oftotalfree energy densities ft withrespect to thefilm thickness h_f forboth the purelyTandR-likephases. One mayclearlyobserve thatthe phase transition betweentwo phases occursat h^critical_f ∼ 3 nm. For h_f<h^critical_f , the total free energy ofthe T phase isthe lowest one, andthe corresponding region can be referred as “Stable T”; the region with h_f>h^critical_f , on the other hand, can be referred as“Stable R” sincethe R-like phaseis stable, asshownin the insert of Fig. 3(a). For h_f→∞,f_elec and f_surf can be ignored, sotheenergydensitiesofthepurelyTandR-likephasesconverge tosolutionofBFOfilmswithmechanicalfilmboundaryconditions [Fig.2(c)],whicharerepresentedbytheblueandreddashedlines inFig.3(a),respectively.Ourrecentexperimentalresultsalsoshow that the BFO films grown on SrRuO₃-buffered DyScO₃ substrate exhibit a “criticalthickness” around 2nmbelow whichtheR-like phasetransitstothepurelyTphase.

To further compare the contribution from each energy term, the curvesofenergydifferencesbetweenthepurely Tphase and the R-like phase versus ﬁlm thickness are plotted in Fig. 3(b).

For four energy terms andtotal free energydensity f_i (i.e., f_bulk, f_elast,f_elec, f_surf^Vol,andf_t), theenergydifference isdeﬁned as f_i=

f_i^purely^T−f^R-like

i ,where f_i^purely^Tand f^R-like

i aretheenergydensity ofthepurelyTphaseandtheR-likephase,respectively.Itisobvi- ousthatif f_i<0,suchenergyfavorstheformationofthepurelyT phase,andviceversa.FromFig.3(b),wecouldclearlyﬁndthatthe elastic,electrostatic,andsurfaceenergytermsallfavorthepurely T phase.As fortheir hierarchy, their valuesaround the regionof the phase transition ( ft=0) are compared. Since f_surf^Vol is the lowest at the range h_f=2.5 ∼ 10 nm, it is the surface energy that dominatesthe formationofthepurely Tphase.At thesame time,theelasticandelectrostaticenergiesalsomakecontributions tothestabilityofthepurelyTphase.

Fig. 3. (Color online) (a) Curves of the energy density f t with respect to the ﬁlm thickness h ffor the purely T and R-like phases. Blue and red dashed lines indicate the case of h f→ ∞ for the T and R-like phases, respectively. Insert: zoom-in curves of the grey rectangular area. A dot-dashed line separates the regions of “Stable T”

and “Stable R”. (b) Curves of the differences of the total free energy density and each energy term between the purely T phase and the R-like phase with respect to the ﬁlm thickness h f. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Anotherinteresting phenomenon in Fig. 3(a)is that there ex- istsa peak inthe curve ofenergydensityf_t for theR-like phase ath_f∼4.4nm,whereastheenergycurveforthepurelyTphaseis monotonicallyincreasing.Itcanbeexplainedbythesubtlebalance andcompetitionbetweentwothickness-relatedenergyterms,i.e., theelectrostaticenergyf_elec andthesurfaceenergy f_surf^Vol (SeeSup- plementaryMaterialEforthedetails).Itshouldbementionedthat ifone variessome oftheparametersinTable1,differenttenden- ciesmaybeobtained.However,itislargelythebalanceoff_elecand f_surf^VolthatdeterminesthebehaviorsofftforboththepurelyTphase andtheR-likephase.

DifferentparametersinTable1playdifferentrolesinthecriti- calthicknessh^critical_f betweenthepurelyTandR-likephases.Toex- ploretheinfluenceofeachparameteronthecriticalthicknessand phase transition, the phase diagrams of certain parameters with respecttothefilmthicknessareillustratedinFig.4.Foreachsub- figure,theother fourparameters areassumedtotake thedefault valuesinTable1,andthedashedwhitelineindicatestheposition of the default value of the controlled parameter. From Fig. 4(a), one can find that the temperature T has a minor impact on the phasetransition.Thecriticalthicknessrisesslowlywithincreasing temperature,showing thathighertemperaturemayfavorthe for- mationoftheTphaseinBFO.Nevertheless,thein-planesubstrate strainandthebuilt-inpotential aremoreimportant.Asshownin Fig. 4(b)and (c), it is evident that the greater compressive sub- stratestrainandlargerabsolutevalue ofthebuilt-inpotentialen- large the region of “StableT”. As for the extrapolation length of

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Fig. 4. (Color online) Phase diagrams of ultrathin BFO films with respect to film thickness h fand temperature T (a), in-plane substrate strain ɛ s(b), effective built-in potential δϕ(c), effective extrapolation length of polarization δPêff(d), and effective extrapolation length of OOT δêff_θ (e). White lines indicate the value of the default parameters in Table 1 .

polarization inFig. 4(d), theseparation line oftwo phases (solid blackline)isverticalaroundthedefaultvalue,indicatingthatthe surfaceeffectofthepolarizationhaslittleinﬂuenceonthestability oftheTphase.Itisnotablethatforthesmallervaluesof

δ

P^effbe- yondthescopeinFig.4(d),thesymmetryofthephasesbecomes complicated,whichwillnot bediscussed inthispaper.Moreover, theextrapolationlength ofOOTplays asigniﬁcantroleintheR/T transition,asshowninFig.4(e).Since

δ

_θ^eff^measures^the^strength^of

thesurfaceenergyforOOT,onecansaythatitistheinterfacialef- fectofOOTthatfavorsthepurelyTphasethemostintheultrathin BFOﬁlms.Notethattheinterfacialeffectsfromthetopandbottom interfacescannotbedistinguishedinthethermodynamicanalyses.

WeemphasizetheimportanceofthebottomlayersbecauseinBFO films, the constraint of the bottom layers usually dominates the surfaceenergy [19,34,35]. Onthe other hand, forthe BFO layers inBFOsuperlattices,we expectthat both thetopandbottom in- terfacialeffects are important.The deterministic factor is thetilt patternin the layers that are adjacent to the BFO layers. If the oxygen octahedra within the adjacent layers havesmall or even notilts,theywouldconfinetheformationofOOTintheBFOlay- ers.Inaddition,theanisotropy ofthegradientenergycoefficients forOOT,i.e.,v₁₁<<v₄₄[44],leadstostrongerOOTcouplingalong the[100]and[010]directionsthanthatalongthe[001]direction.

Altogether,thepurelyTphasewouldbethelowest-energystateif thesurfaceeffectiscrucial,whichisthecaseofultrathinﬁlms.It isnoteworthythattheaboveanalysesarebasedontheassumption thattheadjacentelectrodesorsubstratespossessnoOOTorsmall

θ

3(sincev₁₁<<v₄₄).Usually,theOOTofthesubstrateplaysasig- nificant role, andtheOOT in theelectrode dependson its thickness and the tilt patternof the substrate [47]. Since SrTiO₃ and DyScO₃ substrateshavedifferenttilt patterns,we expectthat the two substrates provide different interfacialcouplings on the BFO films. Recently, Han et al.[35] observed an Ophase inultrathin BFOfilmsdirectlygrown onthe(110)-orientedNdGaO₃ substrate.

Interestingly, such Ophase wasfound in the BFOlayers close to thesubstrate, while thelayers closetothe surfacearestill the R phase.Thisphasetransitionalongthe[001]directionindicatesthe dominantinﬂuenceofinterfacialoxygenoctahedralcoupling.

4. Phase-ﬁeldsimulations

Sofar,weonlyfocusonthethermodynamicstabilityofasingle domaininultrathinBFOﬁlms.Real thinﬁlms areexpectedtoex- hibitcomplicateddomainpatternsduetothecompetitionsamong theshort-rangebulk,gradient,andsurfaceenergyterms,together with the long-range electric and elastic energy terms. Fig. 5(a)

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Fig. 5. (Color online) (a) Curves of the total energy density f tversus substrate strain ɛ sfor the purely T phase and the R-like phase at h f= 3 nm . The green line indicates the common tangent of two curves. (b-d) Domain patterns with εs= −2 . 4% , εs= −2 . 2% , and εs = −1 . 8% , respectively. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

showsthecurvesofthetotalfreeenergydensityftversus thein- plane strain ɛs forthe purely T phase andthe R-like phase with ﬁlmthicknessh_f=3.0nm,showingthatthephasetransitionbe- tween twophaseswouldhappenat

ε

^s ∼ −2%.The commontan- gentofthetwocurves(solidgreenline)indicatestheexistenceof two-phase mixtureswhichcan formthrough theso-called“strain phaseseparation”[48].

In order to further investigate the inhomogeneous domain structures and properties of ultrathin BFO films, we perform phase-fieldsimulationsonthesystemillustratedinFig.1.Assum- ing fullycharge-compensatedinterfacesofthefilms andwiththe introduction of the built-in potential, the short-circuit electrical boundary conditions are employed, andthe potential ofthe bot- tomlayerofBFOfilmsandwholeelectrodeissettobezero,while thepotentialofthetoplayeroffilmsequalstothebuilt-inpotential

δϕ

^.^SeeSupplementaryMaterialFforthedetailedsystemset- tingsofphase-ﬁeldsimulationsincludingthesystemsizes,bound- aryconditions,andnumericalmethods.

Thedomainstructures withdifferentin-planesubstratestrains areshowninFig.5(b)–(d).Owingtothebuilt-inpotentialthatcre- ates an electric ﬁeld pointingto the positive [001] direction, the ﬁnaldomainstructureonlycontainsfourR-likephasevariantsand one Tphasevariant.When thecompressivestrain isratherlarge, asillustratedinFig.5(b),thedomainpatternisthepurelyTphase.

For smaller compressive strain in Fig. 5(d),the domain structure isR-likemulti-domains.Underanintermediatecompressivestrain, however,thedomainpatternexhibitsthecoexistenceofthepurely TandR-likephasesasshowninFig.5(c).

Weexpectthatthestrain-inducedMPBsseparatingthepurelyT phaseandtheR-likephasewouldenhancethepiezoelectricprop- ertiesofultrathinBFOﬁlms. Thelongitudinal piezoelectriccoeﬃ- cient d₃₃ iscalculatedby thechangeofthe straincomponentɛ33

withrespecttotheexternalappliedelectricﬁeldcomponentE^app₃ , i.e., d₃₃=

∂ ε

33/

∂

Êâpp₃ ^[49–51]^. ^See Supplementary Material G for detailedproceduresofcalculatingd₃₃viaboththethermodynamic andphase-fieldsimulations.

Fig. 6. (Color online) Longitudinal piezoelectric coeﬃcient d 33as a function of in- plane strain ɛ sfrom phase-ﬁeld simulations (dots) and thermodynamic analysis for the purely T and R-like phases (lines).

The solid lines in Fig. 6 show the results of d₃₃ obtainedby the thermodynamic analysisas a function ofɛs.The valuesfrom thethermodynamicanalysisarecomparabletothevalueunderthe stress-freeconditions(∼16pC/N based onthe coefficientsinthis paper).Notethatd₃₃ oftheR-likephaseislargerthanthatofthe Tphase. The T phase only has one nonzero polarization compo- nentandallOOTcomponentsarezero.Extraappliedelectricfields alongthe samedirectionjustchange themagnitudeofthe order parameters, and thus the induced strain is small. For the R-like phase,however, its polarization directionrotatestowards the ex- ternalfield directionalong [001], so the strain ɛ33 would be increasedby a largerextent. Onecan observesimilar trends under differentfilmthicknesses,asshowninFig.S2(c).

The values of d33 from phase-ﬁeld simulations are listed by the dots in Fig. 6. Compared to the thermodynamic results, the piezoelectriccoeﬃcients underlargecompressive in-plane strains

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(

ε

^s<−2.3%) aresimilarsince bothare purelyTphaseswithonly one domain variant. Then, for d₃₃ with small in-plane strains (

ε

^s>−1.8%), the phase-fieldresults are larger than the thermo- dynamicones. This is because the thermodynamic analysis only considers one R-like domain, while the phase-field results exist fourR-likedomainvariantsandtransitionregions (domainwalls) that separate these R-like domains. The domain wall regions are more sensitive to the applied electric field, which leads to enhanced piezoelectric responses [50].Near the MPBs, the R-like and T phases coexist, resulting in a large deviation from the thermodynamic values with in-plane strain −2.3%<

ε

^s<−1.8%

inFig. 6. The enhancement ofd₃₃ in thisrange iscaused by the phasetransitionfromtheR-likephasetotheTphase.Itisnotable that the piezoelectric coeﬃcient d₃₃ is increasedby up to 200%, comparedtothatofthesingleRdomain.

5. Conclusions

We study thethermodynamic stabilities of thetetragonal and rhombohedral phases in the ultrathin BFO films using both the thermodynamicanalysisandphase-fieldsimulations.Accordingto the thermodynamic analysis, the purely T phase rather than the RorR-likephase isfound tobe thestablestate inthinnerfilms.

Thesurfaceeffect,namely,theconstraintfromtheelectrodeorthe substrate,isresponsibleforthestabilizationoftheTphase,while thebuilt-in potential and compressive substratestrain also favor theTphase.Thestabilitiesofthetwophasesaremainlycontrolled bythecompetitionsbetweenthebulkfreeenergyandsurfaceen- ergyterms. Ourphase-fieldsimulations reveal the coexistence of thepurelyTandR-likephases,andthepiezoelectriccoefficientof thetwo-phasemixtureissignificantlyenhancedcomparedtothat ofsingle-domainBFObulkcrystals.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Acknowledgments

Y.Z. gratefully acknowledges the ﬁnancial support from China ScholarshipCouncilunderNo.201706190099.NationalNaturalSci- enceFoundation of China Z.H.C. acknowledges the support from theNationalNatural ScienceFoundation ofChina underthe con- tractnumbers51802057 andU1932116.Division ofMaterialsSci- encesandEngineeringNationalScienceFoundationF.X.andL.Q.C.

acknowledgethesupport bythe NationalScienceFoundationun- derDMREFGrantDMR-1629270.

Supplementarymaterials

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.actamat.2019.10.054. References

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