Table Of ContentAb initio study of proper topological ferroelectricity in layered perovskite La Ti O
2 2 7
Jorge Lo´pez-P´erez and Jorge ´In˜iguez
Institut de Ci`encia de Materials de Barcelona (CSIC), Campus UAB, 08193 Bellaterra, Spain
Wepresentafirst-principlesinvestigationofferroelectricity inlayeredperovskiteoxideLa2Ti2O7
(LTO), one of the compounds with highest Curie temperature known (1770 K). Our calculations
reveal that LTO’s ferroelectric transition results from the condensation of two soft modes that
havethe same symmetry and are strongly coupled anharmonically. Further,the leading instability
mode essentially consists of rotations of the oxygen octahedra that are the basic building block
1 of the perovskite structure; remarkably, because of the particular topology of the lattice, such O6
1 rotations give raise to a spontaneous polarization in LTO. The effects discussed thus constitute an
0 exampleofhownano-structuring–providedherebythenaturallayeringofLTO–makesitpossible
2 to obtain a significant polar character in structural distortions that are typically non-polar. We
n discuss the implications of our findings as regards the design of novel multifunctional materials.
a Indeed,theobserved proper ferroelectricity driven by O6 rotations provides theideal conditions to
J obtain strong magnetoelectric effects.
1
1 PACSnumbers: 77.84.-s,61.50.Ah,75.85.+t,71.15.Mb
]
i I. INTRODUCTION
c
layer with n=4
s
- perovskite-like
l Because of their physicalappeal andtechnologicalim- planes
r
t portance, ferroelectrics and related materials have been
m the object of continued attention for decades.1–3 Bulk BO group
6
. oxides with the ideal perovskite structure – ranging
t
a from classic ferroelectric BaTiO to strong dielectric
3
m Ba1−xSrxTiO3, or from piezoelectric PbZr1−xTixO3 to
- relaxor (PbMg1/3Nb2/3O3)1−x-(PbTiO3)x – have been top side
d
especially well studied. Indeed, partly thanks to a num-
n
ber of key contributions from first-principles theory,4 we
o
now understand the fundamental atomistic origin of the
c
[ ferroelectric (FE) and response properties of the most
important members of this family. During the past
1
decade, the focus has increasingly shifted towards nano-
v
structuredmaterials,especiallyin the formofthin films.
6
6 Work on films has led to a better understanding of how y view x view z view
0 elastic (i.e., the epitaxial strain exerted by a substrate)
2 and electric (e.g., the imperfect screening of the depo-
1. larizingfieldassociatedto particularmetallic electrodes) FIG. 1: (Color on-line.) Different views of the layered per-
0 boundaryconditionsaffectthe FEstate.5 Further,ithas ovskite structure of the AnBnO3n+2 compounds for n = 4,
wherenisthenumberofperovskite-likeplaneswithinalayer.
1 beenshownthatexoticFEpropertiescanbeobtainedin
1 artificially created super-lattices, as e.g. in the recently OThnelydtehfieneAdCataormtessi(aansabxaelslsx),ayn,danBdOz6(owchtaichhedforalloawrethsheocwonn-.
v: discussed PbTiO3/SrTiO3 heterostructures.6 vention adopted in Ref. 32) are used throughout thepaper.
i The emergence of magnetoelectric multiferroics7 (i.e.,
X
materials with coupled magnetic and FE orders) has
ar contributed to refuel interest in ferroelectrics, especially ferroelectrics YMnO311,12 and Ca3Mn2O7.13
in what regards unconventional mechanisms for ferro- This is the context of our work on the layered
electricity. If we restrict ourselves to oxides with the perovskite oxide La Ti O (LTO), whose structure is
2 2 7
ideal ABO perovskite structure, ferroelectricity (usu- sketchedinFig.1. LTOisoneofthehighest-temperature
3
ally drivenby a B-site transitionmetalwith annd0 elec- ferroelectrics known, with a Curie point (T ) of approx-
C
tronic configuration) and magnetism (which requires lo- imately 1770 K,14 and is lately being considered for ap-
calized d or f electrons) seem to be mutually exclusive, plications as a high-T piezoelectric.15,16 The FE transi-
the known exceptions being very scarce.8,9 So far, the tion temperature of LTO is enormous, especially when
mostnotablewaysaroundthisproblemconsistin(i)hav- comparedwiththe T ’sofcubicperovskitetitanatesone
C
ingferroelectricitydrivenbytheA-sitecation,asinroom might have expected to be similar to LTO: most signif-
temperaturemultiferroicBiFeO ,10 and(ii)movingaway icantly, BaTiO (BTO) becomes ferroelectric at about
3 3
fromtheidealperovskitestructureandresortingtoother 400 K, and the related compound SrTiO remains (a
3
mechanisms for ferroelectricity, as e.g. in the improper quantum) paraelectric down to 0 K. In order to explain
2
the surprising behavior of LTO, we need to identify the that our results have for the design of novel magneto-
atomistic origin of its FE instability. LTO might owe electric materials. Finally, in Section IV we summarize
its very high T to the kind of mechanisms known to and present our conclusions. Whenever it is possible,
C
operate in the cubic titanates (in essence, long-range we compare our first-principles results with the (scarce)
dipole-dipole interactions that destabilize the non-polar experimental information available for LTO.
phase17,18), which might somehow be enhanced by the
peculiartopologyoftheLTOlattice. Alternatively,LTO
mightpresentsomenew formofverystrongFE instabil- II. METHODOLOGY
ity. Either way, the study of LTO will provide us with
information that might be useful for the design of new A. Theoretical approach to La2Ti2O7
ferroelectrics.
LetusalsonotethatLTOisamemberofthefamilyof We adopted the usual first-principles approach to the
oxideswithgeneralformulaLanTinO3n+2,19whosestruc- investigation of structural phase transitions of the dis-
turecanbeseenastheidealcubicperovskiteperiodically placivetype,whichisroutinelyappliedwithgreatsuccess
truncated along the [011]c direction of the cubic lattice, to FE perovskite oxides.21 Here we describe a simplified
n being the number of perovskite-like planes within one versionofsuchanapproachthatissufficientforourstudy
layer(seeFig.1). Thesestructuresarethusrelatedtothe of LTO (i.e., we will assume one-dimensional instability
well-known Ruddlesden-Popper, Aurivillius, and Dion- modes,willnotdiscusshowtodealwithcellstrains,etc.).
Jacobson families of layered perovskites,20 for which the In essence, one needs to identify a reference equilibrium
truncation direction is [001]c and which include very fa- phase of high symmetry (HS) – which corresponds to
mous members such as (La,Ba)2CuO4, the parent com- the high-temperature PE phase of the compound – and
poundofhigh-TC superconductors. Thebasicfeaturesof study its stability against all possible structural distor-
the electronic structure of the LanTinO3n+2 compounds tions. Moreprecisely,onewritestheenergyofthecrystal
arecontrolledbyn: TheLaandOatomscanbeassumed as the following Taylor series:
tobe intheir mostcommonionizationstates–i.e., La3+
andO2− –,whichimpliesthattheTicationswillpresent E = E0 + 1 K u u +O(u3), (1)
mn m n
a positive chargeof 3+4/n. Accordingly, the number of 2
Xm,n
3delectronsintheTiatomswillbe1−4/n,whichallows
us to move quasi-continuously from the Ti-3d0 configu- where E0 is the energy of the HS phase. The u vari-
m
ration of the n=4 compound (that is our La Ti O , the
2 2 7 ables represent the structural distortions of the refer-
family member with smallest n reported in the litera-
ence configuration; for the study of a FE transition, one
ture) to the Ti-3d1 configurationof the n=∞compound
can typically restrict oneself to the distortions compati-
(which has the prototype perovskite structure). The va-
ble with the reference unit cell, i.e., those associated to
riety of electronic phenomena observed for intermediate
the Γ-point of the Brillouin zone of the PE phase. The
valuesofn–includingphasesofsemi-conducting,normal so-calledforce-constantmatrixK is the centralquantity
metallic,andlow-dimensionalmetalliccharacter19–con- oneneedstocompute,asitsnegativeeigenvalues(ifany)
stitutes an additional motivation to study in detail the correspond to unstable structural distortions – i.e., soft
structural behavior of the relatively simple end member modes – that may result in a phase transition. Let us
La2Ti2O7. use the term mode stiffness to refer to the eigenvalues
The paperis organizedasfollows. InSectionII we de- of K, which we denote by κ with s running from 1 to
s
scribethetheoreticalapproachandfirst-principlesmeth- the dimension of the force-constant matrix. Note that
ods used in this work. We present and discuss our re- the magnitude of a negative κ determines the strength
s
sults in Section III, which is split in the following way. of the structural instability, and thus the likelihood of
In Section III.A we describe our results for the struc- observing it experimentally. (In general, one may find
ture of the high-temperature paraelectric (PE) and FE several, possibly competing, instabilities of a HS phase;
phases of LTO. In Section III.B we show that the PE it is by no means guaranteed that all of them will lead
phase presents a strong FE instability whose topologi- to experimentally observable phase transitions.) Once a
cal nature is discussed in detail. In Section III.C we soft mode is identified, one can readily study the corre-
describe the phase transition between the PE and FE sponding low-symmetry (LS) phase – by distorting the
phases, and discuss the energetics of the transformation. crystal according to the soft mode eigenvector and re-
In Section III.D we present our results for the sponta- laxing the resulting structure – and the energetics of the
neous polarization, dielectric, and piezoelectric proper- instability – by computing the usual double-well poten-
ties. HavingestablishedLTO’speculiarity,whichismade tial connecting the HS and LS phases.
manifest by means of a detailed comparison with proto- In order to apply this program to LTO, we had to
typecompoundBTO,inSectionIII.EweshowthatLTO tackle one fundamental difficulty: We couldnot find any
also presents some features that are clearly reminiscent experimental information on the structure of the high-
of the usual FE oxides with the ideal perovskite struc- temperature PE phase of this compound.22 Note that
ture. InSectionIII.Fweoutlinetheexcitingimplications thisisneveraproblemwhenoneworkswiththeusualFE
3
a22-atomprimitiveunit cell,ismostlikelyassociatedto
O(7)
aPEphasewiththesameprimitivecellandspacegroup
Ti(2)
O(6) Cmcm (see sketch in Fig. 2). We thus performed our
first-principles study using this Cmcm phase as our HS
reference structure.
Ti(1) A few additional points are in order: (1) The pseudo-
O(6)
symmetryanalysisresultedinrelativelylargeatomicdis-
O(7) placementsconnectingthe HSandLSphases,withmax-
O(5) LLaa((22)) O(5) imum values of about 0.4 ˚A corresponding to the La
atoms. Note that the magnitude of such distortions
is expected to reflect the associated HS-LS transition
O(2) temperature,26 whichisindeedveryhighinthiscase. (2)
La(1)
Experimental studies of the compounds Sr Nb O and
2 2 7
O(1) O(1) Ca Nb O ,27,28 which share with LTO the same layered
2 2 7
O(3) y O(3) structure and a similar (nd0) electronic configuration of
the transition metal atoms, show that they present a
z O(4) high-temperature phase with Cmcm space group. (3)
x ThechoiceofCmcmasourPEspacegroupresultsinspe-
Cmcm Cmc2 cific predictions for the symmetry of the softmodes that
1
would be associated with a FE transition to the Cmc2
1
FIG. 2: (Color on-line.) Structures of the Cmc21 (FE) and phase. Indeed, the leading instability should transform
Cmcm (PE) phases of La2Ti2O7 studied in this work. The with the B1u irreducible representation of mmm, the
small (red), medium (blue), and big (violet) balls represent point group of the Cmcm phase. As we will see, our
O,Ti,andLaatoms,respectively. Theshadedpolyhedraare first-principlesresultsconfirmedtheseexpectations,thus
top-viewedO6groups. Thetwophaseshavethesame44-atom supporting our choice of space group for the PE phase.
conventionalcelldepictedinthefigure;the22-atomunitcell,
which is also common to both, is indicted with dashed lines
in the Cmcm case. The defined Cartesian axes x, y, and
z (which follow the convention adopted in Ref. 32) are used
throughoutthepaper. Thesymmetry-independentatomsare B. Details of the calculations
labeled as in Table I; note that oxygens O(1) and O(3) of
the Cmcm structure split, respectively, into O(1)/O(2) and
O(3)/O(4) in the Cmc21 structure. We used the local density approximation (LDA) to
density functional theory (DFT) as implemented in the
first-principles package VASP.29 To represent the ionic
perovskites,wherethe idealcubic perovskitestructure is cores we used the projector-augmented wave scheme,30
the reference phase of choice. Such a choice is obviously solvingexplicitly for the followingelectrons: La’s 5p,5d,
correctin cases like those of BTO or PbTiO , where the and6s;Ti’s 3p,3d,and4s;andO’s2sand2p. The elec-
3
cubic PE phase is experimentally accessible for T >T ; tronic wave functions were represented in a plane-wave
C
more remarkably, this choice is also the most physically basistruncatedat400eV.Wealwaysworkedwiththe44-
sound one for materials (e.g., BiFeO ) whose cubic PE atomcellofLTOsketchedinFig.2(whichistheconven-
3
phaseisnoteasytoaccessexperimentally,asthesamples tionalcellfor bothCmcm andCmc21 phases),andused
tendtomeltbeforereachingthecorrespondingtransition a6×1×5k-pointgridforBrillouinzoneintegrations. We
temperature. In the general case, the problem of choos- checkedthese calculationconditions were well-converged
ing an appropriate HS reference phase for the theoret- by monitoring the computed equilibrium structure, bulk
ical study of displacive phase transitions has long been modulus, and phonon frequencies of the Cmcm phase.
solved. The basic idea is to look for pseudo-symmetries For the calculation of the force-constant matrix K and
(i.e., slightly brokensymmetries)of the knownLS struc- the dielectric andpiezoelectric responses,we employeda
ture; we canthus identify possible HSphases thatwould simplefinite-displacementschemeandthecorresponding
transforminto the known LS structure upon a relatively linear-response tensor formulas, which can be found e.g.
small distortion. This is a very powerful strategy that in Ref. 31. We only computed the lattice-mediated part
canlead to the discoveryof complex phase transitionse- of the static dielectric response, which has been repeat-
quences (e.g., when more than one possible HS phases edly shownto dominate the effect in ferroelectricoxides.
are found) and, in particular, has been used to identify OurBTOsimulationswereanalogoustothe abovede-
previously unnoticed FE transitions.23,24 scribed ones. We solved explicitly for Ba’s 5s, 5p, and
Using widely available crystallographictools,25 we ap- 6s electrons (treating Ti and O as above), and used a
pliedthepseudo-symmetryanalysistoLTOandobtained 400eV cut-off for the plane-wavebasis. We workedwith
a very clear prediction: We found that the FE phase of the 5-atom unit cell of BTO, and used a 6×6×6 k-point
this compound, which presents space group Cmc2 and grid for Brillouin zone integrations.
1
4
III. RESULTS AND DISCUSSION
TABLEI:ComputedequilibriumstructuresoftheCmcmand
Cmc21 phases of La2Ti2O7 discussed in the text. We show
A. Structure of the Cmcm and Cmc21 phases inparenthesistheexperimentalvaluesreportedinRef.32for
theCmc21 phase.
Table I shows our results for the equilibrium struc-
ture ofthe Cmcm (PE)andCmc2 (FE)phases ofLTO Cmcm a = 3.891 ˚A b = 25.720 ˚A c = 5.465 ˚A
1
considered in this study. For Cmc2 we also report the α = β = γ = 90◦
1
experimental structure measured at 1173 K by Ishizawa
Atom Wyc. x y z
et al.32 using X-ray diffraction.
La(1) 4c 0 0.2924 1/4
While the overall agreement between the experimen-
tal and theoretical FE structures is acceptable, the de- La(2) 4c 0 0.4453 3/4
viations affecting some atomic positions and lattice con- Ti(1) 4c 1/2 0.3389 3/4
stantsarelargerthanwhatisusualforthistypeofcalcu- Ti(2) 4c 1/2 0.4437 1/4
lations. Forexample, the predicted positionofthe La(1)
O(1) 8f 1/2 0.2881 0.9931
atomdiffersnotablyfromtheexperimentallydetermined
O(3) 8f 1/2 0.3982 0.9869
one;moresignificantly,thecomputedaandclatticecon-
O(5) 4a 1/2 1/2 1/2
stants deviate from the experimental values by almost a
3%. We attribute these discrepancies to the fact that O(6) 4c 0 0.3458 3/4
we are comparing our first-principles results, which cor- O(7) 4c 0 0.4525 1/4
respondtothelimitof0K,withstructuraldatatakenat
very high temperatures. Given that thermal expansion Cmc21 a = 3.845 ˚A b = 25.626 ˚A c = 5.464 ˚A
and other temperature-driveneffects are not included in (a = 3.954 ˚A b = 25.952 ˚A c = 5.607 ˚A)
our simulations, our results seem compatible with the α = β = γ = 90◦
experimental information.
Atom Wyc. x y z
La(1) 4a 0 0.2978 0.1675
B. Ferroelectric instabilities of the Cmcm phase (0.2981) (0.1757)
La(2) 4a 0 0.4464 0.7515
AsdescribedinSectionII.A,westudiedthestructural (0.4461) (0.7500)
stability of the Cmcm phase against Γ-point distortions
Ti(1) 4a 1/2 0.3369 0.7069
(compatiblewiththePEunitcell)bycomputingthecor-
(0.3370) (0.7095)
respondingforce-constantmatrixK. Fromthediagonal-
ization of K we obtained two negative eigenvalues that Ti(2) 4a 1/2 0.4407 0.2422
correspondtotwostructuralinstabilities. Thecomputed (0.4404) (0.2452)
mode stiffnesses are −0.81 eV/˚A2 and −0.01 eV/˚A2, re- O(1) 4a 1/2 0.2803 0.9245
spectively. Hence, according to our calculations, LTO’s (0.2818) (0.9350)
Cmcmphasepresentsastructuralinstabilitycomparable
O(2) 4a 1/2 0.2960 0.4399
instrengthwiththeFEsoftmodeofBTO’sPEphase(for
(0.2964) (0.4580)
whichweobtained−2.74eV/˚A2),aswellasamarginally
O(3) 4a 1/2 0.3843 0.0415
unstable mode with nearly zero stiffness.
Both instabilities transform with the B irreducible (0.3891) (0.0390)
1u
representationofthemmmpointgroupofthePEphase; O(4) 4a 1/2 0.4072 0.5549
more precisely, they are infra-red active (polar) modes (0.4077) (0.5420)
that involve the development of a polarization along the
O(5) 4a 1/2 0.4905 0.9724
z direction defined in Fig. 2. Such B modes break the
1u (0.4912) (0.9750)
C , C , I, and σ point symmetries of the PE phase
2y 2x z
O(6) 4a 0 0.3460 0.7426
(where I is the spatial inversion, C stands for a two-
2α
foldrotationaroundaxisα,andσ isamirrorplaneper- (0.3472) (0.7200)
α
pendicular to direction α), leading to the Cmc2 space O(7) 4a 0 0.4507 0.2604
1
group. Hence, the obtained B1u soft modes are exactly (0.4511) (0.2550)
the kind of instabilities that can drive a ferroelectric
phase transition between the Cmcm and Cmc2 phases.
1
Our first-principles results thus support the correctness
of our working hypothesis, i.e., that the studied Cmcm ξ involvesarotationoftheO oxygenoctahedraaround
1 6
structureisindeedthehigh-symmetryPEphaseofLTO. the x axis, as sketched in Fig. 3(a). Hence, the displace-
Wecangaininsightintotheoriginofferroelectricityin ments of the equatorial oxygens amount to most of the
LTObyinspectingtheeigenvectorofthestrongestinsta- eigenvector(11% of the norm of ξ is associatedto O(1)
1
bility mode, denoted by ξ in the following. In essence, displacements, 51% to O(3), and 28% to O(5), using the
1
5
p to form a cubic perovskite lattice) and chemical (as
1
in the Bi-based compounds that display the so-called
p
3 stereochemical activity)effectsareusuallyresponsiblefor
p the occurrence of AFD distortions.35 Hence, it is not a
5
surprise to find that layered perovskite structures may
p
3 present AFD-like instabilities that allow for a better ful-
p fillment of steric and/or chemical constraints at a local
1
level. Itisnotourgoalheretodiscusstheoccurrenceand
originofsuchsoftmodesinLTO-likelayeredperovskites;
FIG. 3: (Color on-line.) Panel (a): sketch of the largest such a study should probably involve consideration of a
atomic displacements associated to the strongest instability number of representative crystals (e.g., a few members
mode (ξ1) obtained for the Cmcm phase of La2Ti2O7. We of the AnBnO3n+2 family) and falls beyond the scope of
showthedisplacementscorresponding toonelayercomposed this work. Suffice it to say that the leading FE instabil-
of n = 4 perovskite-like planes. The arrows on the side rep- ity that we found in LTO is essentially analogous to the
resent the electric dipoles associated to the displacement of
AFD distortions that are ubiquitous among perovskite
oxygens in different y-planes (see text). The dipoles are la-
oxides.
beled as the oxygen atoms in Fig. 2. Panel (b): sketch of a
typical anti-ferrodistortive mode occurring in an ideal (non- However, we must note a critical difference between
layered) perovskite structure. LTO’sAFD-likemodedepictedinFig.3(a)andtheAFD
modesoccurringinidealperovskitestructures[Fig.3(b)]:
the former causes a spontaneous polarization, while the
labels defined in Fig. 2 for the oxygen atoms); there are latter do not. To better explain this, in Fig. 3(a) we in-
alsosignificantLadisplacements(9%ofthenorm),while dicate with arrows the electric dipoles that appear as a
the Ti atoms and apical oxygens O(6) and O(7) have a consequenceofthedisplacementoftheoxygenatomsfol-
negligible participation in ξ1. The character of the sec- lowingtheξ1eigenvector. Theoxygensineachy-oriented
ond soft mode (ξ ) is more complex: It is dominated by plane give raise to a dipole along the z direction. Using
2
the displacements of oxygens O(1) and O(5) (with 21% the definitions inFig.3(a),the totaldipole associatedto
and35%ofthe norm,respectively)andinvolvesa signif- onelayerwouldbeplayer =p5+2p3+2p1. Sincethereis
icant deformation of the O octahedra; it also presents no symmetry relationship between the displacements of
6
a large participation of the La atoms (about 24% of the theO(1),O(3),andO(5)oxygensintheξ1eigenvector,it
norm). followsimmediatelythatplayerwillbedifferentfromzero.
Moreover, even if oxygens of different types were to dis-
These FE instabilities are very different from the ones
that are usual among ABO3 oxides with the ideal per- pwleacweobuyldthsetilslamhaevaempolauynert,=sopth6=at0p,5a=s −thpe3n=ump1be=r pof,
ovskite structure. The inset of Fig. 4(b) shows the rep-
oxygen planes in each layer is odd. Finally, note that
resentative case of BTO: the Ti cation moves away from
LTO’s unstable mode ξ involves an identical distortion
the center of the O6 octahedron, which is only slightly 1
ofalllayersinthestructure,andthusgivesraisetoanet
distorted, giving raise to a large electric dipole. Fur-
macroscopic polarization.
ther, the FE instability in BTO and related materials is
known to originate from the strong interactions between Onthe other hand, AFD distortions ofthe ideal(non-
suchlocaldipoles.18 In contrast,the dominantFE insta- layered) perovskite structure do not result in a macro-
bility found in LTO consists of O octahedra rotations, scopic polarization. In Fig. 3(b) we show a pattern of
6
the off-centering of the Ti atoms being negligible. As O6 rotations around the indicated xc axis; the situation
quantified in Section III.D, such a pattern of atomic dis- is very similar to the one depicted in Fig. 3(a), except
placements does not lead to a large local dipole, which that in this case the network of O6 octahedra is not
suggests that the mechanisms responsible for the ferro- truncated. Again, we indicate with arrows the electric
electricity in LTO have to be of a different nature. dipoles that appear as a consequence of the oxygen dis-
Interestingly, structural instabilities involving O6 oc- placements: oxygens within a [011]c-oriented plane give
tahedra rotations are very common among ABO3 per- raise to a dipole p along [01¯1]c. It is apparent that the
ovskites, and are usually termed anti-ferrodistortive addition of all such dipoles gives no net polarization in
(AFD). Indeed, AFD modes as the one sketched in this case, a result that can be viewed as a consequence
Fig. 3(b) are the driving force for most of the struc- of the three-dimensional nature of the O6 network.
tural phase transitions occurring in these compounds, In conclusion, we have found that the strongest struc-
theexamplesincludingcrystalsaswell-knownasSrTiO , turalinstability in LTO’s high-symmetry phase is a very
3
LnMnO and LnNiO (where Ln is a lanthanide), mul- common and simple one: it involves concerted rotations
3 3
tiferroics BiMO (where M is a 3d transition metal), ofthe O octahedra,muchalikethe AFD modes respon-
3 6
etc. Accordingly, there is extensive literature devoted sible for the structural phase transitions in most ABO
3
to the study and classification AFD modes in ABO crystals with the ideal perovskite structure. AFD dis-
3
perovskites,33,34 and it is known that size (e.g., the in- tortions in the usual perovskites are well-known to be
compatibility of the ionic radii of the A and B cations non-polar,afeaturethatcanbeviewedasaconsequence
6
pose the PE volume (so that the FE cell is allowed to
TABLEII:EnergydifferencebetweentheFEandPEphases
deform and acquire a c/a 6= 1 ratio), the energetics of
of La2Ti2O7 and BaTiO3. The FE phases are considered the transformation is not strongly affected. In contrast,
under several elastic constraints (see text), which we denote
the analogous elastic constraints have a relatively small
“fully-relaxed” (noconstraint), “PEvolume”, and“PE cell”.
effectinthecaseofLTO(i.e.,theenergygaindropsfrom
For both LTO and BTO, the energies are normalized to the
volume of the PE phase so that a comparison between com- −1.097meV/˚A3 to−0.938meV/˚A3 whenweimposethe
poundscan bemade. Results given in meV/˚A3. PE cell). Such a weak coupling between strain and the
FE distortion is probably reflecting the AFD-like char-
La2Ti2O7 BaTiO3 acter of the instability;36 this result also suggests that
fully-relaxed −1.097 −0.110 LTO will display relatively small piezoelectric effects as
PE volume −1.042 −0.099 compared with regular FE perovskites.
PE cell −0.938 −0.051 Let us now consider the atomic displacements that
characterize the Cmcm-to-Cmc2 transformation. In
1
most materials undergoing displacive transitions, the
of the symmetry and three-dimensional character of the atomic distortion connecting the HS and LS phases is
lattice. Incontrast,becausethestructureofLTOissplit essentially captured by the eigenvector of the instabil-
inlayerscomprisinganevennumberofperovskiteplanes, ity mode that triggers the transformation. One can eas-
the O6-rotationmode gives raise to a net polarizationin ily quantify this by constructing a distortion vector ∆X
this case. Hence, since the occurrence of a spontaneous comprisingalltheatomicdisplacementsassociatedtothe
polarizationin LTO relieson the layeredtopologyof the HS-to-LStransition,andexpressingitinthebasisformed
lattice, it seems appropriate to describe this compound bytheeigenvectorsξ oftheforce-constantmatrixofthe
s
as a topological ferroelectric. Let us stress that LTO’s HS phase. (For simplicity, we constructed our distortion
peculiar lattice topology does not seem essential for the vectors∆X usingtheatomicpositionsofaLSphasethat
structural instability to exist, but it is critical for it to isforcedtohavethesameunitcellastheHSphase.) For
have a polar character. BTO,thisanalysisledustotheexpectedresult: thesoft
modeofthePEphasecaptures98%oftheatomicdistor-
tion involved in the Pm¯3m-to-P4mm transition. How-
C. Nature of the Cmcm-to-Cmc21 transition ever, the result for LTO was qualitatively different: the
strongest instability mode (ξ , with κ =−0.81 eV/˚A2)
1 1
Inadditiontothealreadydiscussedtopologicalnature captures 81% of the total distortion, and the second in-
oftheleadingFEinstability,thetransformationbetween stability mode (ξ2, with κ2 =−0.01 eV/˚A2) contributes
LTO’s Cmcm and Cmc2 phases presents a number of with a 15%. (For both compounds, no other mode con-
1
interestingaspectsthatwediscussinthefollowing. Most tributesmorethana1%.) Hence,ourresultssuggestthat
importantly,ourresultsshowthatthetransitionisdriven the two soft modes ξ and ξ play an important role in
1 2
bythecombinedactionofthetwosoftmodesdiscussedin LTO’s FE phase transition.
thepreviousSection,andsuggestthatsuchacooperation
Suchalargecontributionofξ tothestructuraltrans-
is critical for it to occur at a very high temperature. 2
formation seems incompatible with the computed mode
To study a structural phase transition quantitatively,
stiffnesses;indeed, thevaluesofκ andκ wouldsuggest
one can start by comparing the energies of the phases 1 2
that ξ is about 80 times weaker than ξ as an instabil-
involved. Table II shows our results for LTO,along with 2 1
ity. Toresolvethisapparentcontradiction,wecomputed
theanalogousdataforthePm¯3m(PE)andP4mm(FE)
howLTO’senergychangeswhentheCmcmphaseisdis-
phases of BTO. Note that the energy change per unit
torted according to the individual ξ and ξ modes. As
volume involved in LTO’s transition is about one or- 1 2
shown in Fig. 4(a), we found a large energy reduction
der of magnitude greaterthan the correspondingone for
associated with the ξ distortion, while the ξ instabil-
BTO. Such an enormous difference is compatible with 1 2
ity is almost negligible. We also computed the energy
the experimentally measured Curie temperatures, which
variation as the total distortion ∆X is frozen in. Re-
are 1770 K and 400 K for LTO and BTO, respectively.
markably, as compared with the results for ξ , the ∆X
Hence, our first-principles results for the energetics of 1
curve in Fig. 4(a) presents a much deeper minimum cor-
LTO’s FE transition seem consistent with experiments.
responding to a much larger distortion amplitude. This
Table II also contains information about the elastic
is a new indication that the ξ soft mode cannot explain
deformation that accompanies the FE transitions. The 1
LTO’s FE transitionby itself. This resultcontrasts with
properties of FE oxides with the ideal perovskite struc-
thesituationforBTO[Fig.4(b)],wheretheFEsoftmode
ture are known to be strongly sensitive to cell strains.
captures the energetics of the structural transformation
This is clearly reflected in the results for BTO: If the
almost exactly.
compoundisforcedtokeepthecubiccellofthePEstruc-
ture, the energy gain involved in the FE transition is The results of Fig. 4(a) suggest that there is a strong
reduced by half (i.e., it drops from −0.110 meV/˚A3 to and cooperative coupling between the two instability
−0.051 meV/˚A3). On the other hand, if we only im- modesofLTO.Tobetter describethis effect,letuswrite
7
3) 1.0 (a) ξ inatesoverγ′′andγ′′′;wethusgotγ′ =−0.35eV/˚A4. As
eV/Å 0.5 LTO 2 ξ1 acovmepryarsetdrownigthanthhearcmomonpiuctceoduαplcinogeffithcaenttfsa,vtohrisstchleearcloymis-
m 0.0 bined ξ +ξ distortion.37
( 1 2
Ω -0.5 ∆X To the best of our knowledge, such a cooperation be-
E/ tweensoftmodes israreamongperovskiteoxides. There
∆-1.0
are many examples of materials in which several strong
0.0 0.5 1.0 1.5 2.0 2.5 3.0 instabilities exist; in most cases, the strongest one leads
Distortion amplitude (Angstrom)
to a phase transition that tends to suppress, partially or
totally, the other instabilities. The competition between
0.08
3) (b) theFEandAFDsoftmodesincompoundslikeSrTiO3is
Å a representative and well studied case.38 In contrast, we
V/ 0.04 BTO
e ∆X find that the reciprocal enhancement of the two soft FE
m
( 0.00 ξ modesiscriticaltoexplainthestructuraltransformation
Ω soft in LTO. The ξ mode is clearly the leading instability,
1
/
∆E-0.04 and it would occur even in absence of ξ2. Yet, as the re-
sultsinFig.4(a)show,themagnitudeandstrengthofthe
0.00 0.05 0.10 0.15
transformationareboostedbytheinteractionbetweenξ
1
Distortion amplitude (Angstrom) andξ . Thus,ourresultsseemtosuggestthatLTOowes
2
itsveryhighT tosuchaninteraction;indeed,inviewof
C
FIG.4: (Coloron-line.) Panel(a): Variationoftheenergyof Fig. 4(a), it is hard to imagine LTO’s T would remain
C
La2Ti2O7astheCmcmphaseisdistortedaccordingtoseveral essentially the same if the coupling between ξ and ξ
1 2
atomicdisplacementpatterns,namely,thosecorrespondingto was suppressed.
the soft modes ξ and ξ , as well as the distortion ∆X that
1 2 Studying from first-principles the temperature depen-
connects the Cmcm and Cmc21 phases. The 44-atom cell of denceoftheξ andξ distortions,andthedetailsofhow
the Cmcm phase is kept fixed in all calculations. Panel (b): 1 2
Same as panel (a) for the PE (Pm¯3m) to FE (P4mm) tran- these two soft modes freeze in at LTO’s FE transition,
sition of BaTiO3. The inset shows the atomic displacements is a challenging endeavor that remains for future work.
associated to the single FE soft mode of BTO (Ba atoms at Nevertheless, a few observations can be made based on
thecorners of thecubic cell; Ti atom at thecenterof theO6 the present results. In principle, one could try to ap-
octahedron). For both LTO and BTO, the energies are nor- proach the problem by introducing a Landau potential
malized to thevolume of the PE phase so that a comparison of the form:
between compoundscan be made.
1 1
F −F = A (T −T )Q2+ B Q4+
0 2 1 C 1 4 1 1
the energy of the crystal as the following Taylor series: 1A Q2+ 1B Q4+ (3)
2 2 2 4 2 2
1 1 1 1
E =ECmcm+ 2κ1u21+ 2κ2u22+ 4α1u41+ 4α2u42 (2) C′Q31Q2+C′′Q21Q22+C′′′Q1Q32,
+γ′u3u +γ′′u2u2+γ′′′u u3+O(u6),
1 2 1 2 1 2 which is written in terms of two one-dimensional order
where ECmcm is the energy of the Cmcm phase, and u parameters,Q1andQ2,thathavethesamesymmetry. In
1
thefollowingheuristicargumentwewillconsideronlythe
andu are,respectively,theamplitudes(inAngstrom)of
2 Q3Q crossed term – in accordance with our above con-
the ξ and ξ distortions. Note that this expression for 1 2
1 2
jecture regarding the couplings in Eq. (2), and because
the energy is greatly simplified by symmetry, and that
this is the most relevant crossed term in vicinity of the
the quadratic parameters coincide with the mode stiff-
phase transition–,39 thus assuming that C′′ =C′′′ = 0.
nesses. We have included in Eq. (2) only the lowest-
If we were dealing with a simple transition, we would
order couplings between u and u , which are quantified
1 2
by the primed γ parameters. By fitting to the ξ and have a primary order parameter Q1 corresponding to
1
ξ energy curves of Fig. 4(a), we got κ =−0.79 eV/˚A2 the unstable eigenmode of the K matrix of the high-
2 1
and κ = 0.00 eV/˚A2, in fair agreement with the val- symmetry phase; the temperature dependence of the
2
ues obtained from the diagonalization of K; we also got Landau potential would be restricted to the Q21 term,
α = 0.65 eV/˚A4 and α = 0.66 eV/˚A4. Then, to fit as indicated in Eq. (3), and we would have positive A1
1 2
the ∆X curve we considered distortions characterized and B1 coefficients. Further, a secondary order param-
eter Q would be stable by itself, with positive A and
by u /u = 0.81/0.15, as it corresponds to the Cmc2 2 2
1 2 1
B coefficients. Hence,the correspondingLandautheory
phase. Note that, in this case, the fitted quartic coef- 2
would predict a phase transition at T =T , with
ficient is a combination of the α and γ parameters of C
Eq. (2). Given the large u /u ratio associated with the
1 2
∆X distortion, and in view of further tests discussed in A1 1/2
SectionIII.D,itseemsreasonabletoassumethatγ′dom- Q1 =(cid:20)B (TC−T)(cid:21) (4)
1
8
belowthetransitiontemperature. (Theindicatedformu-
TABLE III: Non-zero components of the spontaneous polar-
las were derived assuming that we remain close to the
ization (PS, given in C/m2), lattice-mediated dielectric ten-
transition temperature.) Then, Q would present the z
2 sor (ǫ ), and piezoelectric tensor (d , where l labels strain
αβ αl
following temperature dependence below T :
C components in Voig notation, given in pC/N) of the Cmc21
phase of La2Ti2O7. For the dielectric tensor, the clamped-
Q =−C′Q3 =−C′ A1(T −T) 3/2 . (5) cell response is given in parenthesis.31 For the piezoelectric
2 A 1 A (cid:20)B C (cid:21) tensor, the clamped-ion response is given in parenthesis. We
2 2 1 alsoshowexperimental14 andtheoretical16 valuesfortheP21
Inasimplecase,thecoefficientsinEq.(2)maybeagood phase of LTO that is stable at room temperature (Troom).
Note that the results in Refs. 14 and 16 have been adapted
approximationtothecoefficientsinEq.(3)inthelimitof
to ourchoice of Cartesian axes.
low temperatures. Thus, by supplementing the Eq. (2)
deduced from first-principles with a piece experimental Cmc21 phase P21 phase
information – i.e., the value of the transition tempera- this work exp. (Troom)14 theory16
ture T –, we could construct the corresponding Lan-
C PS 0.29 0.05 0.08
dau potential and obtain a quantitative description of z
the temperature dependence of Q and Q . ǫxx 62 (61) 52 -
1 2
The doubts quickly appear when one tries to apply ǫyy 44 (44) 42 -
this model to LTO.It seems naturalto associateQ1 and ǫzz 65 (54) 62 -
Q2 with our computed FE soft modes ξ1 and ξ2, respec- dz1 12 (0) 3 -
tively. However,thatidentificationimpliesthatthetran-
dz2 4 (1) 6 -
sition temperature for Q1 is independent of Q2; such an dz3 −22 (0) 16 -
approximationwouldbeanawkwardone,giventhelarge
influence that ξ has in the energetics of the Cmcm-to- dx5 −2 (0) - -
2
Cmc2 transformation. Additionally,wewouldbeforced dy4 1 (0) - -
1
to speculate regarding the T-dependence of the A coef-
2
ficient, as our guess for this parameter – i.e., the κ of
2
thatthecellstrainsplayindeterminingthe energeticsof
Eq.(2)whichwasfoundtobe essentiallyzero–doesnot
the Cmcm-to-Cmc2 phase transition, as mentioned in
comply with the usual requirements for the quadratic 1
the discussion of Table II.
coefficient of a secondary mode. Note that, for exam-
Thelargespontaneouspolarizationobtainedmayseem
ple, if we had a small value of A and thus a dominant
2
incompatible with the AFD-like character of the struc-
B term, the behavior with temperature of both Q and
2 1
tural instability (ξ ) that dominates the Cmcm-to-
Q2 would vary: Q1 would have the same functional T- 1
Cmc2 transition. Toclarifythispoint,weperformedal-
dependence butwithadifferentprefactor,andQ would 1
2
go as (T −T)1/2 instead of (T −T)3/2. The results ternativecalculationsusinglinear-responseexpressions31
C C ∗
basedontheBorneffective-chargetensorsZ thatquan-
of this study do not allow us to resolve such details of i
tify the polarization change associated to the displace-
LTO’s transition.
ment of an individual atom i.
The polarization reported in Table III was obtained
in the standardway: We used the Berry-phasetheory of
D. Polarization and response properties
King-SmithandVanderbilt40 tocomputethevariationof
P as the Cmc2 structure is deformed into a symmetry-
1
Table III shows the computed spontaneous polariza- equivalent one with opposite polar distortion; then, PS
tion, lattice-mediated dielectric tensor, and piezoelectric ishalfofthe computedpolarizationchange. PS canalso
tensor for the Cmc2 phase of LTO. For comparison,we
1 be estimated using the approximate formula
alsoshowresultsfromtheliteraturecorrespondingtothe
P21 phase that is stable at room temperature. PS ≈ Z∗ ∆X , (6)
We obtained a value of 0.29 C/m2 for the sponta- α i,αβ iβ
Xiβ
neous polarization, which is comparable with the result
of0.38C/m2thatweobtainedforthetetragonalphaseof where ∆X is the vector capturing the distortion that
BTO. The dielectric response is also very significant, as connects the Cmcm and Cmc2 phases, i labels the
1
the obtainedvalues are comparablewith those typicalof atoms in the unit cell, and α and β label spatial direc-
FE perovskites (e.g., we got ǫ = 23 for the z-polarized tions. Usingdifferentchoicesfortheeffective-chargeten-
zz
tetragonalphaseofBTO).Asregardspiezoelectricity,the sors (i.e., those computed for the Cmcm phase, as well
computedresponsesareconsiderablebutnotparticularly asthecorrespondingresultsfortheCmc2 phasesubject
1
large; for example, for the low-temperature rhombohe- to different cell constraints) we obtained values of PS in
z
dral phase of BTO, the piezoelectric coefficients reach the0.25-0.32C/m2range,whichareperfectlycompatible
values of 200 pC/N,31 while for LTO we got maximum with the result in Table III.
values of about 20 pC/N. LTO’s relatively small piezo- Then, we used the effective-charge tensors of the
electric response seems compatible with the minor role Cmcm phase (given in Table IV) to compute the polar-
9
TABLEIV:Computedeffective-chargetensors(giveninunits TABLEV:Computedeffective-chargetensors(giveninunits
of elementary charge) for the Cmcm (PE) and Cmc21 (FE) of elementary charge) for the Pm¯3m (PE) and P4mm (FE)
phases of La2Ti2O7. To facilitate the comparison between phases of BaTiO3. The tensors are given in the conven-
phases,welistthetensorscorrespondingtoallthesymmetry- tional Cartesian axes for a rectangular lattice. They corre-
inequivalent atoms of theFE phase. Atoms are labeled as in spond to the following atoms (PE phase positions given in
Table I. relative units): Ba located at (0,0,0), Ti at (1/2,1/2,1/2),
O(1) at (1/2,1/2,0), and O(2) at (0,1/2,1/2). O(1) and O(2)
Cmcm (PE) Cmc21 (FE) are symmetry-equivalent in the PE phase, but not in the z-
4.63 0 0 4.64 0 0 polarized FE phase.
La(1) 0 4.17 0 0 4.09 -0.21
Pm¯3m (PE) P4mm (FE)
0 0 4.72 0 -0.69 4.71
2.72 0 0 2.71 0 0
4.37 0 0 4.63 0 0
Ba 0 2.72 0 0 2.71 0
La(2) 0 3.76 0 0 4.13 0.41
0 0 2.72 0 0 2.81
0 0 4.28 0 0.24 4.24
7.31 0 0 7.05 0 0
6.91 0 0 6.16 0 0
Ti 0 7.31 0 0 7.05 0
Ti(1) 0 6.70 0 0 5.60 -0.36
0 0 7.31 0 0 5.83
0 0 5.72 0 0.24 5.31
-2.13 0 0 -1.98 0 0
6.33 0 0 6.11 0 0
O(1) 0 -2.13 0 0 -1.98 0
Ti(2) 0 5.32 0 0 5.29 0.09
0 0 -5.77 0 0 -4.79
0 0 7.45 0 -0.15 6.32
-5.77 0 0 -5.62 0 0
-2.52 0 0 -2.52 0 0
O(2) 0 -2.13 0 0 -2.14 0
O(1) 0 -3.20 0.73 0 -3.14 0.53
0 0 -2.13 0 0 -1.96
0 0.86 -3.26 0 0.38 -2.92
-2.52 0 0 -2.15 0 0
O(2) 0 -3.20 -0.73 0 -2.35 -0.89
contrast,the secondsoftmodeξ isfoundtohaveacon-
0 -0.86 -3.26 0 -0.90 -3.44 2
siderably polar character. It can then be trivially shown
-2.08 0 0 -1.76 0 0 that the two soft modes have a very similar contribution
O(3) 0 -3.27 -1.72 0 -3.33 -1.36 tothetotalspontaneouspolarizationoftheCmc2 phase
1
0 -1.56 -3.98 0 -1.38 -2.95 of LTO, in spite of the fact that ξ embodies 81% of the
1
-2.08 0 0 -2.49 0 0 PE-to-FE distortion. Hence, the two-mode character of
thetransitioniscriticaltoexplaintherelativelylargePS
O(4) 0 -3.27 1.72 0 -3.19 1.28
0 1.56 -3.98 0 1.19 -3.64 of the Cmc21 phase.
Unfortunately,wewereunabletofindexperimentalre-
-2.49 0 0 -2.63 0 0
sults for the ferroelectric and response properties of the
O(5) 0 -3.15 1.19 0 -3.11 1.04
Cmc2 phase of LTO. Table III shows some results for
1
0 0.98 -3.64 0 1.00 -3.36 the P2 phase of the compound that is stable at room
1
-5.50 0 0 -5.04 0 0 temperature. Interestingly, the dielectric and piezoelec-
O(6) 0 -2.15 0 0 -2.08 0.24 tric responses measured for this phase seem compatible
(at least in magnitude) with our values for Cmc2 . As
0 0 -1.66 0 0.14 -1.75 1
regards the spontaneous polarization, the value for the
-5.10 0 0 -4.99 0 0
P2 phase is smaller than ours by a factor of 6; this ex-
O(7) 0 -1.73 0 0 -1.91 -0.03 1
perimental result was essentially confirmed by the first-
0 0 -2.41 0 -0.02 -2.20 principles study of Ref. 16, which employed a DFT the-
orysimilartoours.41 Hence,wecantentativelyconclude
that the Cmc2 -to-P2 transformation, which is experi-
1 1
izationchangeassociatedwiththecondensationoftheξ1 mentally determined to occur at1053K,32 involvesa re-
and ξ soft modes that dominate the Cmcm-to-Cmc2
2 1 duction of LTO’s spontaneous polarization. While such
transformation. The polar character of the modes is a reduction in the magnitude of PS as the temperature
quantified in terms of the mode effective charges:
decreases is not typical in FE perovskites, in principle
Z¯ = Z∗ ξ . (7) there is no reason to question this possibility.
s,α i,αβ s,iβ
Finally, let us discuss an intriguing possibility sug-
Xiβ
gestedby the very differentmagnitudes of the computed
WeobtainedZ¯ =1.8eandZ¯ =12.0e,whereeisthe modechargesZ¯ andZ¯ . SinceZ¯ quantifiesthecou-
1,z 2,z 1,z 2,z s
elemental charge. These results confirm that the ξ is pling betweena polarmode andanappliedelectric field,
1
weaklypolar,inaccordancewithitsAFD-likenature. In wecanexpectξ tobemuchmorereactivetoanexternal
2
10
bias than ξ . One can thus imagine the following pos- FE distortion freezes in: according to the results in Ta-
1
sibility: to apply an electric field to LTO in its Cmc2 ble V, Ti’s charge of 7.31e gets reduced to 5.83e, while
1
phase and switch only the part of the spontaneous polar- O’s charge of −5.77e falls to −4.79e.
ization associated to ξ . Moreprecisely,if (uI,uI)repre-
2 1 2
sentstheFEphasediscussedsofar,anelectricfieldmight
As apparent from Table IV, some Ti and O atoms
allow us to take the material into a qualitatively differ-
in LTO also present anomalously large effective charges
ent polarization state (uII,uII) ≈ (uI,−uI). LTO would
1 2 1 2 that reach values similar to those obtained for BTO. In
thus be a four-state ferroelectric, as the (−uI1,−uI2) and the case of the Ti atoms, the computed Z∗ tensors are
(−uI1I,−uI2I) variants would be accessible as well. A nec- ratherisotropic,withthemaximumZ∗ values[i.e.,6.91e
essary condition for such a partial switching to occur is
for Ti(1) and 7.45e for Ti(2)] corresponding to displace-
thatthe(uII,uII)statebeaminimumoftheenergy. More
1 2 ments along the in-layer directions x and z. In the case
precisely, we need the coupling between ξ and ξ to be
dominated by the γ′′u2u2 term of Eq. (21): Not2e that, oftheOatoms,O(6)andO(7)clearlypresentthelargest
′′ 1 2 ′ ′′′ values,whichexceed−5eas inBTO.Suchgiantdynam-
for |γ | much greater than |γ | and |γ |, the four states
icalchargescorrespondstodisplacementsalongthexdi-
(±uI,±uI) would have essentially the same energy; in
cont1rast,2if γ′ or γ′′′ dominate, the “ξ -switched” state rection, for which LTO presents infinite chains of TiO6
2 octahedra (see Figs. 1 and 2) as those in the ideal per-
would have a much higher energy and might not be a
ovskite structure. Hence, as regards their polarizability
minimum of E(u ,u ). In order to confirm or disprove
1 2 properties,theresultsfortheTiandOatomsinLTOare
such a partial switching, we considered the equilibrium
strongly reminiscent of the behavior that is well-known
structure of the Cmc2 phase (i.e., the one reported in
1 for BTO.
Table I) and generated configurations in which the ξ
2
distortionwasinvertedandgivenvariousmagnitudes;we
relaxed such transformed structures and invariantly ob- Do such anomalous effective charges lead to FE insta-
tained the original Cmc21 phase as final result. Hence, bilities in LTO? In accordance with the above discus-
while we did not explore the E(u1,u2) energy landscape sion, the answer to this question is a negative one: the
in detail, our results clearly indicate there is no stable CmcmphasepresentsonlytwoΓ-pointinstabilities(i.e.,
“ξ -switched” state. Equivalently, this implies that the the already discussed ξ and ξ ) and the Cmc2 is sta-
2 1 2 1
γ′ andγ′′′ termsofEq.(2)dominateoverγ′′ (whichsup- ble against Γ-point perturbations.43 Nevertheless, by in-
ports the assumption made in Section III.C). spectingtheK-eigenmodesoftheCmcmphase,itiseasy
to find several low-energy FE modes that are BTO-like,
i.e., they involve the displacement of the Ti atoms away
E. BaTiO3-like ferroelectric modes in La2Ti2O7 from the center of nearly undistorted O6 octahedra.44,45
For example, we found a marginally stable and strongly
polarmodeforwhichwecomputedκ =0.26eV/˚A2 and
ThusfarwehavediscussedthemainfeaturesofLTO’s s
Z¯ = 21.7e (this distortion is x-polarized and has B
high-temperature FE transition, showing that this com- s,x 3u
symmetry);thismodegivesanenormouscontributionto
poundisverydifferentfromthe FEoxideswiththe ideal
the ǫ dielectric response:46 the obtained value exceeds
perovskite structure. In particular, our results seem to xx
600. Interestingly, these BTO-like modes become stiffer
suggest that LTO and BTO have little in common, in
in the Cmc2 phase; thus, they do not give raise to any
spite ofthe factthatthey sharethe samebuilding block: 1
anomalouslylargecontributiontothe dielectricresponse
TiO octahedra with Ti4+ in the 3d0 electronic configu-
6 reported in Table III.
ration. In the following we show that such a conclusion
would be a deceptive one.
Ferroelectricity in the usual FE perovskite oxides re- In conclusion, our analysis shows that LTO presents
liesonstrongdipole-dipoleinteractionswhosefingerprint obvious traces of the FE instabilities of BTO. Such a
is the anomalously large magnitude of the Born effec- transferabilityofinstabilitieswasdemonstratedbyoneof
tive chargesofsomeatomic species.42 Aprototypicalex- usinanhexagonalpolymorphofBTO.45Inthatcase,the
ample of this behavior is BTO, for which we computed FEphasetransitionwasshowntobedrivenbysoftmodes
the effective-charge tensors shown in Table V. Most no- that are an almost perfect match of the usual BTO-like
tably, in the PE phase the Ti4+ cations display Z∗ val- FE instability. In LTO, such modes are very low in en-
ues above 7e, almost doubling their nominal ionization ergy, but still stable; further, they clearly compete with
charge. Analogously, the displacement of the O atoms the dominating AFD-like instability, and become stiffer
towards the Ti has a dynamical charge of −5.77e asso- once the Cmcm-to-Cmc2 transition occurs. Neverthe-
1
ciated to it, almost tripling the nominal value of −2e. less,notingthatBTO-likeFEdistortionstendtobevery
It is well-known that this effect is related with a partial sensitive to cell strains, our results suggest the possibil-
covalent character of the Ti–O bond, and an increased ity that such instabilities might be induced in LTO by
hybridization of the O-2p and Ti-3d orbitals as the Ti suitable strain engineering or chemical substitution, or
and O atoms approach.17 Thus, given this chemical ori- that they might occur spontaneously in similar layered
gin,theeffectivechargesbecomelessanomalousoncethe perovskites.
