First-principles investigation of ferroelectricity in epitaxially strained Pb 2TiO 4

Craig J. Fennie and Karin M. RabeDepartment of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA

sReceived 10 January 2005; published 28 March 2005d

The structure and polarization of the as-yet hypothetical Ruddlesden–Popper compound Pb2TiO4 are inves-tigated within density-functional theory. Zone center phonons of the high-symmetry K2NiF4-type referencestructure, space group I4/mmm, were calculated. At the theoretical ground-state lattice constants, there is oneunstable infrared-active phonon. This phonon freezes in to give the I2mm ferroelectric state. As a function ofepitaxial strain, two additional ferroelectric phases are found, with space groups I4mm and F2mm at compres-sive and tensile strains, respectively.DOI: 10.1103/PhysRevB.71.100102 PACS numberssd: 77.84.Dy, 77.55.1f, 81.05.Zx

The occurrence of ferroelectricity in the ABO3 perovskitestructure has been known since the 1950’s.1 Recently, first-principles density-functionalsDFTd methods have proved in-valuable in elucidating the observed behavior of known per-ovskite oxide ferroelectrics, antiferroelectrics, and quantumparaelectrics.2,3 Examples include the alkaline-earthtitanates,4–6 the alkali metal tantalates and niobates,7,8 andthe lead-based perovskites.4,9 There is also a growing interestin applying these methods to the design of ferroelectricsbased on the perovskite structure.10,11

Another path to new materials leads beyond the perov-skite structure. The Ruddlesden–PoppersRPd family of com-pounds are closely related to the perovskites.12 They can beviewed as a stacking of AO-terminated ABO3 perovskitef001g slabs of thickness equal ton primitive lattice constants.Adjacent slabs are shifted relative to one another alongf110gby a/2, giving the homologous series An+1BnO3n+1. The n=1 structure is shown in Fig. 1.

Given the structural similarity to the perovskites, it seemssurprising that there have been no confirmed cases of ferro-electricity in the RP family of compounds.13 Bulk phases ofRP titanates have been reported only for some members ofthe Srn+1TinO3n+1 and Can+1TinO3n+1 series.14 It should notbe surprising that neither the strontium nor the calcium RPseries of compounds appear to display ferroelectricity giventhat the end memberssn=`d are SrTiO3 and CaTiO3, respec-tively, neither of which themselves are ferroelectric. Still, thefact that only a few RP titanates have so far been synthesizedin bulk does not preclude the possibility that a metastable RPferroelectric phase could be produced by an appropriate syn-thetic process. In order to identify such a system, it is con-venient to use first-principles DFT methods, for example, toinvestigate an as-yet hypothetical first member of a RP seriesswhere the effects of the structural modification would bemost severed whose end member is a perovskite ferroelectric.

In this paper, we show that Pb2TiO4, an n=1 RP com-pound based on ferroelectric endmember PbTiO3, is a prom-ising candidate for a high-polarization ferroelectric. Regard-ing the presence of Pb2TiO4 sor highern compoundsd in thebulk phase diagram, reports are conflicting and no structuralinformation is available.15–17However, as discussed above, itmay be possible to produce a metastable form using modernepitaxial growth techniques. Using first-principles DFT cal-culations, we compute the ground-state structure and polar-

ization, finding that Pb2TiO4 is a ferroelectric with a polar-ization comparable to PbTiO3. Furthermore, the direction ofthe polarization can be changed by an applied epitaxialstrain.

First-principles DFT calculations were performed withinthe local density approximation as implemented in the Vi-ennaab initio Simulations PackagesVASPd.18 A plane-wavebasis set and projector-augmented wave potentials wereemployed.19 The electronic wave functions were expanded inplane waves up to a kinetic energy cutoff of 500 eV. Inte-grals over the Brillouin zone were approximated by sums ona 63636 G-centered mesh ofk points. Polarization wascalculated using the modern theory of polarization as imple-mented inVASP.20

We approach the problem of searching for possibleferroelectric states by first calculating the properties ofthe RP high-symmetry reference structure, space group:I4/mmm, the structure one expects for the paraelectricphase. We performed full optimization of the lattice param-eters sa=3.857 Å, c=12.70 Åd and internal coordinatesszPb=0.3507,zOII

=0.1562d. The residual Hellmann–Feynmanforces were less than 2 meV/Å. Next, we calculated thezone-center phonons of this reference system by computingthe dynamical matrix atq=0 using the direct method inwhich each ion was moved by approximately 0.01 Å. A simi-lar approach could be used to identify possible zone-boundary instabilities, not considered here. We then froze inthe real-space eigendisplacements of selected unstable modesand performed full relaxations in the space group determinedby the symmetry breaking mode. Finally, we compute thepolarization.

For Pb2TiO4 in the I4/mmm high-symmetry referencestructure, there are three infraredsIRd-active modes thattransform according to the irreducible representationA2u andfour IR modes that transform according toEu. The one-dimensional A2u modes involve atomic distortions alongf001g while in the two-dimensionalEu modes, atoms move inthe plane perpendicular tof001g. Our calculations reveal thatat the ground-state structural parameters, I4/mmm Pb2TiO4has one phonon with an imaginary frequencysv=96i cm−1d, indicative of an instability. This unstable pho-non is IR active of Eu symmetry type. Therefore,Pb2TiO4 inthe RP structure does indeed display a ferroelectric instabil-ity. The real-space eigendisplacements of this unstable ferro-

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electric mode consists of Pb and Ti atoms moving against anonrigid oxygen octahedronswith larger displacements ofthe apical oxygens in the PbO layerd. As for PbTiO3,

2 thecharacter of the ferroelectric instability in Pb2TiO4 involvessignificant Pb displacements moving against oxygen in thePb–O planes. This involvement of theA-site cation in bothPbTiO3 and Pb2TiO4 differs from non-Pb-based compoundsse.g., BaTiO3d and has been attributed to the Pb2+ 6s2 lonepair. It is in fact this lone-pair physics that stabilizes PbTiO3in the tetragonal phase4 and may have a role in facilitatingthe ferroelectric distortion in Pb2TiO4.

21

The key role of Pb is further emphasized by comparisonwith Ba2TiO4. A compound at this composition, bariumorthotitanate, has been identified in the bulk phase diagram.It crystallizes not in the RP structure but rather in the mono-

clinic distorted-K2SO4 structuresspace group: P21/nd.22 Acalculation for the structure and zone-center phonons of I4/mmm Ba2TiO4 in the hypothetical RP structure, exactlyanalogous to that for Pb2TiO4, shows no evidence for anyferroelectric instability, even for varying epitaxial strain.

Returning to Pb2TiO4, we search for the ferroelectricground state by freezing-in the real-space eigendisplacementpattern of the unstable Eu mode, and performing full relax-ations of all ions and lattice constants consistent with theresultant space group. Since this Eu mode is doubly degen-erate, any linear combination of the degenerate modes polar-ized alongf100g and f010g is an equivalent eigendisplace-ment. We considered two linear combinations; one polarizedalongf100g, a second polarized alongf110g. Freezing-in theEu mode polarized alongf100g results in a body-centeredorthorhombic space group, I2mm. For the distortion alongf110g, the resulting space group is face-centered orthorhom-bic F2mm. Since this F2mm structure is slightly higher inenergys20 meV/formula unitd than the I2mm structure, wenow consider only the latterswe will revisit F2mm belowwhen discussing the effect of epitaxial straind. Our calculatedstructural parameters of Pb2TiO4 in this orthorhombic spacegroup are displayed in Table I. We imposed the convention,oiDxi =0, whereDxi is the deviation alongf100g from thecentrosymmetric position of ioni. It can be seen that therelaxed structure can in fact be related to the high-symmetryRP structure by small displacements of Pb and Ti ions mov-ing against the nonrigid oxygen octahedra, consistent withthe freezing-in of the Eu phonon instability of the high-symmetry structure as proposed. Finally, we calculate thespontaneous polarizationPs. We find thatPs=68 mC/cm2,along f100g as required by symmetry. ThereforePb2TiO4 inthe RP structure is a ferroelectric with a spontaneous polar-ization comparable to that of PbTiO3.

23

Epitaxy plays a dual role in our thinking about thePb2TiO4 system. As will be discussed shortly, one possibleroute to synthesize thin films of Pb2TiO4 in the RP structureis through the use of epitaxial stabilization.24,25In addition, itis becoming increasingly possible to grow oxide thin filmscoherently on substrates with a relatively wide range of lat-tice constantss1–2% lattice mismatch is currently the normd.This provides an additional parameter to “tune” the proper-ties of the material to desired values by applying an in-planesor epitaxiald strain to the thin film compared to bulk.

With this in mind we consider again the high-symmetry,I4/mmm RP structure and explore the effect of epitaxialstrain on the low-frequency IR-active modes. We impose ep-itaxial strain by constraining the two basal primitive vectors

TABLE I. Crystal structure of ferroelectric Pb2TiO4, spacegroup: I2mm,a=3.985 Å,b=3.826 Å,c=12.70 Å.

Atom Wyckoff Coordinates

Pb s4cd m 0.0616 0 0.3508

Ti s2ad 2mm 0.0256 0 0

OIx s2ad 2mm −0.0190+12 0 0

OIy s2bd 2mm −0.0307 12 0

OII s4cd m −0.0496 0 0.1538

FIG. 1. sColor onlined Crystal structure of RP compoundPb2TiO4, space group I4/mmm.

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of the body-centered-tetragonal lattice to an angle of 90° andto a fixed equal lengthsi.e., corresponding to that of an im-plicit coherently matched square-lattice substrated. In Fig. 2,we show how the phonon frequencies for the lowest-frequency Eu and A2u phonons vary as a function of com-pressive epitaxial strain. We use the computeda parameter oftetragonal ferroelectric PbTiO3 sRef. 23d as the referencestrain si.e., for 0% strain, we fixed the in-plane lattice con-stant of Pb2TiO4 to that of ferroelectric PbTiO3d.26 For eachvalue of fixed strain, we again perform relaxation of the ionsandc axis.

Referred to tetragonal PbTiO3, the ground-state I4/mmmstructure has an in-plane strain of approximately −0.3%. Forsmall compressive strains, the phonon instabilities of epitax-ial Pb2TiO4 are expected to be similar to those in the uncon-strained structure. This corresponds to the far right of theFig. 2. It is evident that the lattice dynamics in this region ofzero or small epitaxial strains are dominated by the largelyunstable Eu mode, as previously discussed. If we now in-crease the compressive epitaxial strainsi.e., from right toleftd the Eu mode stiffens while the A2u mode softens con-siderably and becomes unstable at<−2.5% strain. Figure 2shows that for large compressive strainss−4 to −5%d, thehighly unstable A2u should dominate the lattice dynamicswhile for intermediate strain values, both an A2u mode andan Eu mode are unstable and comparable in value. This be-havior with strain can be simply understood as arising fromvolume effects. As we increase the in-plane compressivestrain, the effective volume in which the Eu modespolarizedin-plane, perpendicular to thec axisd vibrates decreases. Thisincreases the short-range repulsive forces, thereby stiffeningthe force constant.27 In contrast, the effective volume of theA2u mode spolarized parallel to thec axisd increases withincreasing compressive strain leading to a softening of theforce constant.

Next, we use these phonon instabilitiessFig. 2d as a guideto search for additional epitaxially stabilized ferroelectricstructures. At each value of strain, we first freeze-in sepa-

rately the real space eigendisplacement pattern correspond-ing to the A2u modesI4mmd, the Eu f100g modesI2mmd, theEu f110g modesF2mmd, and both the A2u and the Eu f100gmodessCmd. Then, we relax all ions and thec-axis latticeparameter while keeping the in-plane lattice parametersfixed. Finally, we calculate the total energy, Fig. 4, and thepolarization, Fig. 3, of the resultant structures as a functionof epitaxial strain.

As shown in Fig. 3, RP Pb2TiO4 undergoes a series ofstructural transitions with epitaxial strain. Over the range ofslightly tensile to compressive strains, Pb2TiO4 forms in theI2mm space group. Following the convention appearing inthe literature,28 we refer to this phase as thea phase. Thepolarization in this phase varies from 34mC/cm2 at −3.3%strain to 56mC/cm2 at +0.7% strain. The minimum energystructure in thea phase occurs at a tensile strain of +0.55%,corresponding to the ground-state structure discussed above.As the compressive strain increases, the energy of the I2mmstructure approaches that of the paraelectric I4/mmm struc-ture, as shown in Fig. 4, while remaining about 3 meV lower

FIG. 2. Soft IR-active phonon frequencies as a function of in-plane compressive strain for the lowest-frequency Eu phonon andlowest-frequency A2u phonon of the high-symmetry paraelectric I4/mmm reference structure.

FIG. 3. Polarization alongf001g sI4mmd, f100g sI2mmd, andf110g sF2mmd as a function of epitaxial strain.

FIG. 4. Energysper formula unitd as a function of epitaxialstrain for space groups I4/mmmsparaelectricd, I2mm sa phased,F2mm saa phased, and I4mmsc phased.

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for the values of strain that we considered. This is consistentwith the leveling off of the Eu phonon as shown in Fig. 2 andexplains why the polarization of the I2mm structure remainsnonzero for large compressive strains. A transition from theaphase, with the polarization in plane, to a phase with thepolarization along thec axis, i.e., from I2mm to I4mm, oc-curs for large compressive strains as anticipated from thephonon instabilities of the I4/mmm structure. This occurs at< -3.3% strain. We refer to this I4mm phase as thec phase.The polarization in thec phase approaches 70mC/cm2 at−4.0% strain and continues to increase. In the range wherethe two unstable mode frequencies cross, we considered thepossibility that coupling between the two modes could leadto additional ferroelectric structures. However, relaxing thestructures in the low-symmetry space group Cm alwaysyielded one of the two higher-symmetry structuressa or cphase, depending on the value of the misfit straind. Thus, thetransition froma to c with increasingly compressive in-planestrain appears to be first order.29 Finally, for large enoughtensile strainssgreater than<0.7%d the F2mm structure be-comes lower in energy than that of the I2mm structure. Thepolarization of thisaa-phase is comparable to that of theaphase while the minimum energy structure occurs at aslightly more positive strain of +0.7% strain. The point atwhich the energy curves for thea andaa phases cross is ofparticular interest, as the in-plane polarization is nearly iso-tropic. The free rotation of the polarization might result in

some unexpected interesting physical properties.High-pressure techniques have been successful in the syn-

thesis of RP structures, e.g., polycrystalline Ba2RuO4.30 This

route seems promising to synthesize bulk RP Pb2TiO4. Fur-ther, we suggest synthesis of nonbulk phases of this materialthrough the use of epitaxial stabilization.24,25 In fact, thismethod has proven quite successful for stabilizing a varietyof oxide thin films in the RP structure. One example is that ofthe Srn+1TinO3n+1 series.31 Another example has been thelow-pressure synthesis of Ba2RuO4 in the RP structure.32 Thegrowth of thin films of Pb2TiO4 would provide a means torealize the interesting behavior of this material with epitaxialstrain.

Initially, we asked the question whether Pb2TiO4 in theRP structure would be a ferroelectric. Using first-principlesDFT calculations, we have seen that indeed it does display aferroelectric instability. We have argued that if Pb2TiO4could be made in the RP structuresbulk or thin filmsd itwould undergo a ferroelectric structural transition to theorthorhombica phase with a spontaneous polarization com-parable to that of bulk PbTiO3.

The authors would like to acknowledge many valuablediscussions with Darrell Schlom. We also thank David Singhfor suggesting possible growth techniques at APS 2004. Thiswork was supported by NSF-NIRT Grant No. DMR-0103354.

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