First principles investigation of ferroelectricity in epitaxially strained Pb$_2$TiO$_4$

0.16 MB·English

by  Craig J. Fennie

First principles investigation of ferroelectricity in epitaxially strained Pb TiO
2 4
Craig J. Fennie and Karin M. Rabe
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019
(Dated: February 2, 2008)
Thestructureandpolarizationoftheas-yethypotheticalRuddlesden-PoppercompoundPb2TiO4
are investigated within density-functional theory. Zone center phonons of the high-symmetry
5
0 K2NiF4-typereferencestructure,spacegroupI4/mmm,werecalculated. Atthetheoreticalground-
statelattice constants, thereisoneunstableinfrared-active phonon. Thisphonon freezes in togive
0
the I2mm ferroelectric state. As a function of epitaxial strain, two additional ferroelectric phases
2
are found, with space groups I4mm and F2mm at compressive and tensile strains, respectively.
n
a PACSnumbers: 77.84.Dy,77.55.+f,81.05.Zx
J

6
The occurrence of ferroelectricity in the ABO per- flicting and no structuralinformation is available.15,16,17
3
] ovskite structure has been known since the 1950’s.1 Re- However, as discussed above, it may be possible to pro-
i
c cently, first-principles density functional methods have duce a metastable form using modern epitaxial growth
s
provedinvaluableinelucidatingtheobservedbehaviorof techniques. Using first-principles DFT calculations, we
-
l known perovskite oxide ferroelectrics, anti-ferroelectrics, compute the ground-state structure and polarization,
r
t and quantum paraelectrics.2,3 Examples include the finding that Pb TiO is a ferroelectric with a polariza-
m 2 4
alkaline-earth titanates,4,5,6 the alkali metal tantalates tion comparable to PbTiO . Furthermore, the direction
3
t. andniobates,7,8 andthelead-basedperovskites.4,9 There ofthepolarizationcanbechangedbyanappliedepitaxial
a is also a growing interest in applying these methods to strain.
m
the design of new ferroelectrics based on the perovskite First-principles DFT calculations were performed
- structure.10,11 within the local density approximation as implemented
d
n Another path to new materials leads beyond the per- intheViennaabinitioSimulationsPackage(VASP).18 A
o ovskite structure. The Ruddlesden-Popper (RP) family plane wave basis set and projector-augmented wave po-
c of compounds are closely related to the perovskites.12 tentials were employed.19 The electronic wavefunctions
[ They canbe viewed as a stacking AO-terminatedABO3 wereexpandedinplanewavesuptoakineticenergycut-
1 perovskite [001] slabs of thickness equal to n primitive off of 500 eV. Integrals over the Brillouin zone were ap-
v lattice constants. Adjacent slabs are shifted relative to proximated by sums on a 6×6×6 Γ-centered mesh of
1 one another along [110] by a/2, giving the homologous k-points. Polarization was calculated using the modern
2 series An+1BnO3n+1. The n=1 structure is shown in theory of polarization as implemented in VASP.20
1
Fig. 1. We approach the problem of searching for possible
1
0 Given the structural similarity to the perovskites, it ferroelectric states by first calculating the properties of
5 seemssurprisingthattherehavebeennoconfirmedcases the RP high-symmetry reference structure, space group:
0 offerroelectricityintheRPfamilyofcompounds.13 Bulk I4/mmm, the structure one expects for the paraelectric
/ phases ofRP titanates have beenreportedonly for some phase. We performed full optimization of the lattice pa-
t
a members of the Srn+1TinO3n+1 and Can+1TinO3n+1 se- rameters(a=3.857˚A,c=12.70˚A)andinternalcoordinates
m ries.14 It shouldnot be surprisingthat neither the stron- (zPb=0.3507, zOII=0.1562). The residual Hellmann-
- tium nor the calcium RP series of compounds appear Feynman forces were less than 2 meV/˚A. Next we cal-
d
to display ferroelectricity given that the end members culated the zone-center phonons of this reference system
n
(n=∞) are SrTiO and CaTiO , respectively, neither of
o 3 3
which themselves are ferroelectric. Still, the fact that
c
: many RP titanates are not thermodynamically stable
v
does not preclude the possibility that a metastable RP
i
X ferroelectric phase could be produced by an appropriate
r synthetic process. In order to identify such a system,
a
it is convenient to use first-principles density functional
(DFT) methods, for example, to investigate an as-yet
hypothetical first member of a RP series (where the ef-
fectsofthestructuralmodificationwouldbemostsevere)
whose end member is a perovskite ferroelectric.
In this paper we show that Pb TiO , an n=1 RP
2 4
compoundbasedonferroelectricend-memberPbTiO ,is
3
a promising candidate for a high-polarization ferroelec-
tric. Regarding the presence of Pb TiO (or higher n FIG. 1: (color online) Crystal structure of Ruddlesden-
2 4
compounds) in the bulk phase diagram,reports are con- Popper compound Pb2TiO4, Space group I4/mmm.
2
sistent with the resultant space group. Since this Eu
TABLE I: Crystal structure of ferroelectric Pb2TiO4, Space mode is doubly degenerate, any linear combination of
Group: I2mm, a=3.985˚A, b=3.826˚A, c=12.70˚A.
the degenerate modes polarized along [100] and [010] is
Atom Wyckoff Coordinates an equivalent eigendisplacement. We considered two lin-
earcombinations;onepolarizedalong[100],asecondpo-
Pb (4c) m 0.0616 0 0.3508
Ti (2a) 2mm 0.0256 0 0 larized along [110]. Freezing-in the Eu mode polarized
along[100]resultsinabody-centeredorthorhombicspace
OIx (2a) 2mm -0.0190+12 0 0 group, I2mm. For the distortion along [110], the result-
OIy (2b) 2mm -0.0307 21 0 ing space group is face-centered orthorhombic F2mm.
OII (4c) m -0.0496 0 0.1538 Since this F2mm structure is slightly higher in energy
(20meV/formulaunit)thantheI2mmstructure,wenow
consideronlythelatter(wewillrevistF2mmbelowwhen
discussing the effect of epitaxial strain). Our calculated
by computing the dynamical matrix at q = 0 using the
structural parameters of Pb TiO in this orthorhombic
direct method where each ion was moved by approxi- 2 4
mately 0.01˚A. We then froze in the real-space eigendis- space group are displayed in Table I. We imposed the
placementsofselectedunstablemodesandperformedfull convention,Pi∆xi=0,where∆xi is thedeviationalong
[100] from the centrosymmetric position of ion i. It can
relaxationsinthespacegroupdeterminedbythesymme-
be seen that the relaxed structure can in fact be related
trybreakingmode. Finally,wecomputethepolarization.
to the high-symmetry RP structure by small displace-
ForPb TiO intheI4/mmmhigh-symmetryreference
2 4
ments of Pb and Ti ions moving against the non-rigid
structure, there are three infrared-active (i.r.) modes
oxygen octahedra, consistent with the freezing-in of the
that transform according to the irreducible representa-
tion A2u and four i.r. modes that transform according Eu phonon instability of the high-symmetry structure as
proposed. Finally, we calculate the spontaneous polar-
atotoEmuic.diTsthoertoionnes-dailmonegns[i0o0n1a]lw(h1i-lde)inAt2huem2-oddEesu imnvoodlevse, ization Ps. We find that Ps = 68µC/cm2, along [100]
as required by symmetry. Therefore Pb TiO in the RP
atomsmoveintheplaneperpendicularto[001]. Ourcal- 2 4
structure is a ferroelectric with a spontaneous polariza-
culations reveal that at the ground-state structural pa-
tion comparable to that of PbTiO .23
rameters, I4/mmm Pb TiO has one phonon with an 3
2 4
imaginary frequency (ω=96i cm−1), indicative of an in- Epitaxy plays a dual role in our thinking about the
stability. This unstable phonon is infrared-active of Eu Pb2TiO4 system. Aswillbe discussedshortly,onepossi-
symmetry type. Therefore, Pb2TiO4 in the RP struc- ble route to synthesize thin films of Pb2TiO4 in the RP
ture does indeed display a ferroelectric instability. The structureisthroughtheuseofepitaxialstabilization.24,25
real-space eigendisplacements of this unstable ferroelec- In addition, it is becoming increasingly possible to grow
tric mode consists of Pb and Ti atoms moving against a oxidethinfilmscoherentlyonsubstrateswitharelatively
non-rigidoxygenoctahedra(withlargerdisplacementsof wide rangeoflattice constants (1-2%lattice mismatchis
the apical oxygens in the PbO layer). As for PbTiO ,2 currentlythe norm). Thisprovidesanadditionalparam-
3
the character of the ferroelectric instability in Pb TiO eter to “tune” the properties of the material to desired
2 4
involvessignificantPbdisplacementsmovingagainstoxy- valuesbyapplyinganin-plane(orepitaxial)straintothe
gen in the Pb-O planes. This involvement of the A-site thin film compared to bulk.
cationinbothPbTiO andPb TiO differsfromnon-Pb With this in mind we consider again the high-
3 2 4
basedcompounds(e.g. BaTiO )andhasbeenattributed symmetry,I4/mmmRPstructure andexplorethe effect
3
to the Pb2+ 6s2 lone-pair. It is in fact this lone-pair of epitaxial strain on the low-frequency infrared-active
physics that stabilizes PbTiO in the tetragonal phase4 modes. We impose epitaxial strain by constraining the
3
and may have a role in facilitating the ferroelectric dis- two basal primitive vectors of the bct lattice to an angle
tortion in Pb TiO .21 of90degreesandtoafixedequallength(i.e. correspond-
2 4
ThekeyroleofPbisfurtheremphasizedbycomparison ing to that of an implicit coherently matched square-
withBa TiO . Acompoundatthiscomposition,barium lattice substrate). In Fig. 2 we show how the phonon
2 4
orthotitanate, has been identified in the bulk phase dia- frequenciesforthelowestfrequencyEu andA2u phonons
gram. ItcrystallizesnotintheRPstructurebutratherin vary as a function of compressive epitaxial strain. We
the monoclinic distorted-K SO structure (space group: use the computed a parameteroftetragonalferroelectric
2 4
P21/n).22 Acalculationforthestructureandzone-center PbTiO323 as the reference strain (i.e. for 0% strain we
phonons of I4/mmm Ba2TiO4 in the hypothetical RP fixed the in-plane lattice constant of Pb2TiO4 to that of
structure, exactly analogous to that for Pb2TiO4, shows ferroelectricPbTiO3).26 Foreachvalueoffixedstrainwe
noevidenceforanyferroelectricinstability,evenforvary- again perform relaxation of the ions and c-axis.
ing epitaxial strain. Referred to tetragonal PbTiO , the ground-state
3
Returning to Pb TiO , we search for the ferroelectric I4/mmm structure has an in-plane strain of approxi-
2 4
ground state by freezing-in the real-space eigendisplace- mately -0.3%. For small compressive strains the phonon
ment pattern of the unstable Eu mode, and perform- instabilitiesofepitaxialPb2TiO4 areexpectedtobesim-
ing full relaxations of all ions and lattice constants con- ilar to those in the unconstrained structure. This corre-
3
71^2 90
−2m ) 80
d (c 0 70
e
ar
ency squ−71^2 2µC/cm )5600 c−phase a−phase aa−phase
qu n (
e o
phonon fr−100^2 Polarizati3400 I4/mmm
I2mm
−122^2 Eu 20 F2mm
A2u I4mm
10
−141^2 0
−6 −5 −4 −3 −2 −1 0 −6 −4 −2 0 2 4
epitaxial strain (%) epitaxial strain (%)
FIG. 2: Soft infrared-active phonon frequencies as a func- FIG.3: Polarization along[001](I4mm),[100](I2mm),and
tion of in-plane compressive strain for the lowest frequency [110] (F2mm) as a function of epitaxial strain.
Eu phonon and lowest frequency A2u phonon of the high-
symmetry,paraelectric I4/mmm reference structure.
from 34µC/cm2 at -3.3% strain to 56µC/cm2 at +0.7%
strain. Theminimumenergystructureinthea-phaseoc-
spondstothefarrightoftheFig.2. Itisevidentthatthe curs at a tensile strain of +0.55%, corresponding to the
lattice dynamics in this region of zero or small epitaxial groundstate structurediscussedabove. As the compres-
strains are dominated by the largely unstable Eu mode, sive strain increases, the energy of the I2mm structure
as previously discussed. If we now increase the compres- approaches that of the paraelectric I4/mmm structure,
sive epitaxialstrain (i.e. from right-to-left)the Eu mode as shown in Fig. 4, while remaining about 3 meV lower
stiffens while the A2u mode softens considerablyandbe- for the values of strain that we considered. This is con-
comes unstable at ≈ -2.5% strain. Fig. 2 shows that for sistent with the leveling off of the Eu phonon as shown
large compressive strains (-4 to -5%), the highly unsta- in Fig.2 andexplains why the polarizationof the I2mm
ble A2u should dominate the lattice dynamics while for structure remains nonzero for large compressive strains.
intermediatestrainvalues,bothanA2u mode andanEu A transition from the a-phase, with the polarization in-
modeareunstableandcomparableinvalue. Thisbehav- plane, to a phase with the polarization along the c-axis,
ior with strain can be simply understood as arising from i.e. from I2mm to I4mm, occurs for large compressive
volume effects. As we increase the in-plane compressive stains as anticipatedfrom the phononinstabilities of the
strain, the effective volume in which the Eu mode (po- I4/mmm structure. This occurs at ≈-3.3% strain. We
larizedin-plane,perpendiculartothec-axis)vibratesde- refertothisI4mmphaseasthec-phase. Thepolarization
creases. This increases the short-range repulsive forces, inthe c-phaseapproaches70µC/cm2 at-4.0%strainand
thereby stiffening the force constant.27 In contrast, the continues to increase. In the range where the two unsta-
effective volume of the A2u mode (polarized parallel to ble mode frequencies cross, we considered the possibility
the c-axis) increases with increasing compressive strain that coupling between the two modes could lead to ad-
leading to a softening of the force constant. ditional ferroelectric structures. However, relaxing the
Next we use these phonon instabilities (Fig. 2) as a structures in the low symmetry space group Cm always
guidetosearchforadditionalepitaxiallystabilizedferro- yieldedoneofthetwohighersymmetrystructures(aorc
electricstructures. Ateachvalueofstrainwefirstfreeze- phase,dependingonthevalueofthemisfitstrain). Thus
in separately the real space eigendisplacement pattern the transition from a to c with increasingly compressive
corresponding to the A2u mode (I4mm), the Eu [100] in-plane strain appears to be first-order.29 Finally, for
mode (I2mm), the Eu [110] mode (F2mm), and both large enough tensile strains (greater than ≈0.7%) the
theA2u andtheEu [100]modes(Cm). Thenwerelaxall F2mm structure becomes lower in energy than that of
ions and the c-axis lattice parameter while keeping the the I2mm structure. The polarization of this aa-phase
in-plane lattice parameters fixed. Finally we calculate is comparable to that of the a-phase while the minimum
the total energy, Fig. 4, and the polarization, Fig. 3, of energystructureoccursata slightlymorepositive strain
the resultant structures as a function of epitaxial strain. of+0.7%strain. Thepointatwhichtheenergycurvesfor
As shown in Fig. 3, RP Pb TiO undergoes a series the a and aa phases cross is of particularinterest, as the
2 4
of structural transitions with epitaxial strain. Over the in-plane polarization is nearly isotropic. The free rota-
range of slightly tensile to compressive strains, Pb TiO tion of the polarizationmight result in some unexpected
2 4
forms in the I2mm space group. Following the conven- interesting physical properties.
tion appearing in the literature,28 we refer to this phase Compounds unstable in the RP structure at atmo-
as the a-phase. The polarization in this phase varies spheric pressures have been synthesized under high-
4
0.35 bilization.24,25 Infact,thismethodhasprovenquite suc-
cessful for stabilizing a variety of oxide thin films in the
0.3
RP structure. One example is the higher order mem-
0.25 II42/mmmmm bers (n > 3) of the Srn+1TinO3n+1 series where bulk
0.2 F2mm phasesareknowntobeunstable.31 Anotherexamplehas
I4mm
V) been the low pressure synthesis of Ba2RuO4 in the RP
gy (e 0.15 structure.32 The growth of thin films of Pb2TiO4 would
er 0.1 provideameanstorealizetheinterestingbehaviorofthis
n
E
material with epitaxial strain.
0.05
Initially, we asked the question whether Pb TiO in
0 2 4
the Ruddlesden-Popper structure would be a ferroelec-
−0.05 tric. Using first-principles DFT calculations, we have
−0.1 seen that indeed it does display a ferroelectric instabil-
−6 −4 −2 0 2 4
epitaxial strain (%) ity. We have argued that if Pb TiO could be made in
2 4
the RP structure (bulk or thin films) it would undergo
FIG. 4: Energy (performula unit) as a function of epitaxial a ferroelectric structural transition to the orthorhombic
strain for space groups I4/mmm (paraelectric), I2mm (a-
a-phase with a spontaneous polarization comparable to
phase), F2mm (aa-phase),and I4mm (c-phase).
that of bulk PbTiO .
3
The authors wouldlike to acknowledgemany valuable
pressure conditions, e.g. polycrystalline Ba RuO .30 discussions with Darrell Schlom. We also thank David
2 4
This route seems promising to synthesize bulk RP Singh for suggesting possible growth techniques at APS
Pb TiO . Further, we suggest synthesis of non-bulk 2004. ThisworkwassupportedbyNSF-NIRTGrantNo.
2 4
phases of this material through the use of epitaxial sta- DMR-0103354.
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6 E. Cockayne and B. P. Burton, Phys. Rev. B 62, 3735 cutoff of 500 eV,a=3.867˚A and Ps =65µC/cm−2.
(2000). 24 D.G. Schlom, et al.,Mater. Sci. Eng. B 87, 282 (2001).
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