Impact of octahedral rotations on Ruddlesden-Popper phases of antiferrodistortive perovskites

0.46 MB·English

by  Daniel A. Freedman

Impact of octahedral rotations on
Ruddlesden–Popper phases of antiferrodistortive perovskites
Daniel A. Freedman1,2 and T.A. Arias1,2
1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
2Cornell Center for Materials Research, Cornell University, Ithaca, NY 14853
(Dated: December 31, 2008)
This work presents the most detailed and extensive theoretical study to date of the structural
configurations of Ruddlesden–Popper (RP) phases in antiferrodistortive (AFD) perovskites and
formulatesaprogramofstudywhichcanbepursuedforRPphasesofanyAFDperovskitesystem.
WesystematicallyinvestigatetheeffectsofoxygenoctahedralrotationsontheenergiesofRPphases
in AFD perovskites (A B O ) for n=1...30, providing asymptotic results for n→∞ that
9 n+1 n 3n+1
give both the form of the interaction between stacking faults and the behavior of such stacking
0
faults in isolation. We find an inverse-distance interaction between faults with a strength which
0
variesbyasmuchasafactoroftwodependingontheconfigurationoftheoctahedra. Wefindthat
2
thestrengthofthiseffectcanbesufficientto(a)stabilizeordestabilizetheRPphasewithrespectto
n dissociationintothebulkperovskiteandthebulkA-oxideand(b)affecttheenergyscalesoftheRP
a phasesufficientlytoconstraintherotationalstatesoftheoctahedraneighboringthestackingfaults,
J
even at temperatures where the octahedra in the bulk regions librate freely. Finally, we present

1 evidence that the importance of the octahedral rotations can be understood in terms of changes in
the distances between oxygen ions on opposing sides of the RP stacking faults.
]
i
c PACSnumbers: 61.50.Ah,61.50.Lt,61.50.Nw,61.66.Fn,61.72.Nn,68.35.Ct,68.35.Dv,68.65.Cd,81.05.Je
s
-
l
r I. INTRODUCTION cess (001) planes of AO to create stacking faults of the
t
m form .../AO/BO /AO/AO/BO /AO/... in the normal
2 2
.../AO/BO /AO/BO /AO/... stackingsequenceofthe
. Perovskites possess a vast range of scientifically inter- 2 2
t
a esting and technologically important properties. These material. Across each such stacking fault, the bulk per-
m ovskiteslabsarealternatelydisplacedbyin-planevectors
materials are highly valued for their dielectricity,1–4
- ferroelectricity,5–7 semiconductivity,8–11 superconductiv- oftheform a20[±1,±1,0]. Figure1depictsfourmembers
d of the homologous series A B O , with n = 1, 2,
ity,12–14 catalytic activity,15–17 and colossal magnetore- n+1 n 3n+1
n 3, and ∞.
sistance.18 Such physical properties allow for the use
o
c of perovskites in diverse technological applications, in- RP phases, specifically, manifest additional properties
[ cluding tunneling semiconductor valves and magnetic beyond those of bulk perovskites; these include high di-

1 tunnel junctions in spintronics,19–21 dielectric insulators electricity,35,36 thermoelectricity,37 photocatalysis,38 un-
v in dynamic random access memory,22,23 thin films in conventional superconductivity,39 quantum criticality,40
7 graded ferroelectric devices (GFDs),24,25 and alterna- metamagnetism,41 ferromagnetism,42 colossal magneto-
5 tive gates in metal-oxide-semiconductor field transistors resistance,43 and x-ray optic adaptability.44 Despite the
1 (MOSFETs).26,27 interest in such compounds, experimentalists have faced
0
Cationstoichiometryhasalargeimpactonthephysics difficulties growing RP phases for “intermediate” val-
.
1 of these materials, but is difficult28,29 to control dur- ues of n (n (cid:38) 4).45–48 For example, strontium titanate
0 ing film growth. Cation non-stoichiometries in the per- (SrTiO ) compounds with such stoichiometries tend not
3
9
ovskites can potentially take the form of point or pla- toformgrowthsofasingleuniformRPphase,butrather
0
nar defects, with perovskites experimentally found to re- materials with multiple regions, each with RP phases of
:
v sist point defect formation upon sufficient cation non- different values of n, so-called “intergrowths.”49–51 In-
Xi stoichiometry.30–32 In A2+B4+O23− materials (such as deed, conventional ceramic powder sintering has only
SrTiO ) with an excess of species A (or, more pre- produced single-phase RP samples for n ≤ 3: Sr TiO ,
r 3 2 4
a cisely, additional AO to maintain charge neutrality), Sr Ti O , and Sr Ti O .49,52 More recently, successful
3 2 7 4 3 10
the resulting planar defects produce a series of homol- creation of intermediate members of the Sr Ti O
n+1 n 3n+1
ogouscompoundsoftheformA B O . (Notethat series has been accomplished with advanced techniques
n+1 n 3n+1
bulk ABO corresponds to one end member of this se- forepitaxialgrowthofthin-filmsunderthermodynamical
3
ries, n = ∞, and bulk AO corresponds to the other non-equilibrium conditions, such as sputtering,29 pulsed
end member, n = 0.) The A B O compounds, laserdeposition(PLD),53–55 andmolecularbeamepitaxy
n+1 n 3n+1
known as the Ruddlesden–Popper (RP) phases,33,34 re- (MBE),36,56–58 with characterization by high-resolution
flect the modification of the cation stoichiometry by the transmission electron microscopy (HRTEM)57 and x-ray
addition of an extra unit of AO per n units of bulk diffraction (XRD).59 Such recent observations renew the
ABO material. Structurally, they take the form of bulk urgencyforacarefultheoreticalstudyofRPphaseswith
3
ABO , separated every n layers by the insertion of ex- intermediate and large values of n.
3
2
(a)A2BO4 (b)A3B2O7 (c)A4B3O10 (d)ABO3
FIG.1: FourmembersofseriesofhomologouscompoundsofformA B O ,showingnlayersofbulkperovskitebetween
n+1 n 3n+1
stacking faults: (a) A BO with n=1, (b) A B O with n=2, (c) A B O with n=3, and (d) bulk ABO with n=∞.
2 4 3 2 7 4 3 10 3
Previous theoretical work on RP phases of perovskites terrelated effects: (a) the interaction between different
includes a limited number of both ab initio electronic possibleantiferrodistortivereconstructionsoneitherside
structure calculations and empirical potential studies. ofthestackingfaultsand(b)thepossibilitythatthesere-
A pioneering empirical-potential study60 in the 1980’s constructions will assume different orientations and thus
considered all values of n ≤ 12 for strontium titanate prefer lattice structures in potential conflict across the
but did not report sufficient precision to resolve differ- stackingfault. Ifthedifferentpossiblereconstructionsof
ences among any of the phases above n = 1. A decade octahedral rotations are not considered, the true ground
later, a subsequent empirical-potential study61 was able state of the RP phases may be missed. If the lattice
to resolve differences among phases but only considered vectors are not allowed to relax fully, the extraction of
phases for n≤4. Total energy electronic structure stud- the formation energy of the RP phases will suffer errors
ies have also been carried out, primarily for strontium which scale with the size of the bulk regions (i.e., lin-
titanate62–68 (n ≤ 3) but also for three additional per- early with n), as we demonstrate below in Section VA.
ovskite transition-metal oxides66 (n = 2 only). None of The aforementioned theoretical works thus have limited
these studies have explored the significant interactions applicability, even for the small values of n which they
which we find to exist between the rotational states of consider.
the oxygen octahedra on opposite sides of the AO/AO Indeed,experimentsrevealthattheseeffectsareknown
stacking faults. to be present in RP phases of perovskites; for example,
Many perovskites that form RP phases exhibit anti- both Srn+1TinO3n+1 and Can+1TinO3n+1 exhibit rota-
ferrodistortive (AFD) behavior in which the oxygen oc- tions of their oxygen octahedra, for n=2 and n=3 RP
tahedra tend to rotate in an alternating spatial pattern phases.69 Previous work has considered general group-
from their ideal orientations with a relatively low energy theoretical studies of the space groups of possible oc-
scale, so that numerous structural phases exist for the tahedral rotation patterns in RP phases,70,71 but with
correspondingbulkmaterials. Unlessafullquantumsta- restrictions either to the n = 1 RP phase or to a lim-
tistical treatment of the RP phases is considered in the ited number of possible relative rotational orientations
theoretical calculations, the bulk regions in these phases on opposite sides of the stacking faults.
will tend to relax to the zero-temperature ground state In this work, we provide a careful, comprehensive
within the model used to describe the material. One atomistic study of the Ruddlesden–Popper phases of a
must therefore carefully and explicitly consider two in- physical model of an antiferrodistortive perovskite, con-
3
sidering a wide range of n (specifically, n = 1...30), each other and thus are high in energy, whereas others
properly accounting for the full lattice relaxation of the result in movement of oxygen ions past each other and
RP phases, and exploring all combinatorial possibilities thus are low in energy.
of different orientations of oxygen octahedra on opposite Finally, in Section VD, exerting care in tracking the
sides of the AO/AO stacking faults. In Section II, we chemical potentials of the various reference systems, we
introduce the class of materials which we consider, those verify that the formation of isolated planar stacking
antiferrodistortive perovskites which form intrinsic RP faults is favored over the formation of point defects in
phases and belong to the Glazer system of the greatest our model material, regardless of the rotational state of
symmetry commonly found in nature: a−a−a−.72 Sec- the oxygen octahedra.
tion III then presents an exhaustive catalogue of the dif-
ferent possible structures for RP phases in perovskites
within this system. In generating this catalogue, we in- II. BACKGROUND
troduceaconvenientsymmetryalgebrawhichallowsone
toquicklyenumeratethestructures,andweuseittofind
In this work, we embark on a study of the generic ef-
a total of five symmetry-distinct possibilities (consistent
fects of octahedral rotations on RP phases. For this first
with distorted phases Nos. 18–22 which Hatch et al.71
such study, we shall focus on the most highly symmetric
found for the n = 1 RP phase after extensive searching
of the twenty-three possible Glazer systems.72 There are
with a computer program). Section IV then introduces
twosuchsystems,denotedinGlazernotationasa+a+a+
the shell-potential model and the numerical techniques
anda−a−a−,onlythelatterofwhichiscommonlyfound
which we use.
in nature. Examples of a−a−a− perovskites include
Section VA explores the aforementioned five distinct LaAlO ,72NdAlO ,73,74CeAlO ,74BiFeO ,75LiNbO ,75
3 3 3 3 3
configurations for all RP phases for n = 1...30. We LiTaO ,75 PbZr Ti O (PZT),76 and many others
3 0.9 0.1 3
find that the energies of RP phases are indeed quite sen- catalogued by Glazer72 and Megaw and Darlington.77
sitive to octahedral rotations, sufficiently sensitive that The Glazer notation refers to the relative state of ro-
unfavorable configurations become unstable relative to tation of neighboring oxygen octahedra in antiferrodis-
phase separation into bulk perovskite and bulk A-oxide. tortive reconstructions of the perovskite structure. In
Infact,theeffectissufficientlystrongtosuggestsomein- actual fact, the motion of the octahedra within such re-
triguing possibilities. For low densities of stacking faults constructionsisonlyapproximatelyarotationastheoxy-
(highn),therotationalstatesoftheoctahedraneighbor- genatoms areconfined tothe facesof eachcube; regard-
ingthefaultsmightbeconstrained,evenattemperatures less, we shall refer to their motion as rotational through-
wheretheoctahedrainthebulklibratefreely,sothatdif- out. Glazerexhaustivelycataloguedalltwenty-threepos-
ferent degrees of order are observed in the bulk and at siblepatternsoftheseoctahedralrotations(“tilts”inhis
the interfaces. For high densities of stacking faults (low terminology),72,76 witheachcategoryassignedanappro-
n), this effect may increase the transition temperatures priate nomenclature to denote the axis of rotation and
associated with the octahedral rotations. the relative sign of successive rotations along that axis.
Section VB considers interactions between the stack- In brief, in the a−a−a− system, all oxygen octahedra
ing faults present in the RP phases. For each config- rotate either clockwise or counterclockwise about a fixed
uration, we examine the energy of the A B O trigonalaxisinanalternatingcheckerboardpatterninall
n+1 n 3n+1
RP phases as a function of n, which directly measures threedimensions, resultinginacell-doublingreconstruc-
the separation between stacking faults. We demonstrate tion with a 2×2×2 supercell. (Greater details appear
that the interaction is quite sensitive to the octahedral in Section III.)
rotations, differing in strength by as much as a factor The antiferrodistortive phase transition, associated
of two depending on the rotations. Next, Section VC with these rotations, occurs due to the softening of
examines the issue of the asymptotic form of this inter- the Γ optical phonons at the R corner ([111] zone
25
action for n→∞. We find that the interaction between boundary) of the Brillouin zone, as famously studied
stacking faults varies as the inverse of the distance be- experimentally78–80 and theoretically81,82 and reviewed
tween them, and we extract both the binding energies extensively.83 The transition is therefore also sometimes
of stacking faults and the formation energies of isolated known as a Zone Boundary Transition (ZBT) and was
faults. Again, we find that the interaction energy be- recognizedevenearlierasameansbywhichcrystalscould
tween faults is highly sensitive to the different possible double the size of their primitive cells.84 The rotation of
rotational states of the oxygen octahedra and may even the oxygen octahedra accompany the softening of these
lead to ordering at the stacking faults at temperatures phonons, with their rotation angles serving as the order
where the bulk regions have lost their orientational or- parameters.85Thedegeneracyofthisphononmodeinthe
der. This section then concludes with a proposal for a higher-temperature cubic phase enables the rotation of
simple physical mechanism to explain the strong depen- octahedraaboutdifferentaxes—(cid:104)100(cid:105),(cid:104)110(cid:105),and(cid:104)111(cid:105)
dence of the interfacial energy on the rotational state of —tobreakthecrystal’ssymmetrybelowitscriticaltem-
the octahedra: some configurations result in movement perature. While the a−a−a− system results from rota-
of like-charged neighboring oxygen ions directly toward tionsaboutthe(cid:104)111(cid:105)axis,allthreesystemsemergefrom
4
thesameunderlyinginstability,83,86 withanharmonicin- thanum aluminate, which closely resemble strontium ti-
teractionsdeterminingtheresultinglow-temperaturelat- tanatebutwitha−a−a− octahedralrotations. Thestan-
tice symmetry.87 As the underlying physics is so similar, dard shell potential for strontium titanate can thus rea-
we thus expect that perovskites in Glazer systems other sonably be utilized as a parameterization for generic
thana−a−a−willexhibitsimilargenericbehaviorstothe a−a−a− perovskites which form RP phases. Therefore,
results expounded below. in interpreting our results, one should be mindful that
Inseekingamodelpotentialforourstudy,wesearched details, such as the values which we find for the vari-
the literature for shell-potential parameters among the ous quantities, may not apply to any specific perovskite.
a−a−a− perovskites. Unfortunately, we discovered that However, the general phenomena that we uncover, such
thegroundstatesoftheavailablemodelsgenerallydonot as the enumeration of possible octahedral configurations
correctly match their corresponding Glazer systems.88 andthegeneralformandscaleofthevariousinteractions,
We were only able to identify a single material, lan- may be taken as representative of the class of a−a−a−
thanum aluminate (La3+Al3+O2−), with a ground state perovskitesinRPphases. Moreover,sincetheunderlying
3
inthecorrectGlazersystem.88Unfortunately,lanthanum physicalmechanismisthesameforotherGlazersystems,
aluminate does not form intrinsic RP phases, since manyofthephenomenawhichweuncovershouldbecon-
its composition as A B O would violate basic sidered for the RP phases of all AFD perovskites.
n+1 n 3n+1
charge balance; intrinsic RP formation requires per-
ovskites with an A2+B4+O2− chemical formula for the
3
additional A2+O2− layer to be neutral. Lanthanum alu-
minate, however, can form extrinsic RP phases by in-
corporation of additional neutral layers of another per-
ovskite, strontium oxide: SrO·LanAlnO3n. III. CLASSIFICATION OF OCTAHEDRAL
Forsimplicityofthisinitialtheoreticalstudy,wefocus ROTATIONS IN RP PHASES
onperovskitesthatformintrinsicRPphases,examplesof
whichfromthea−a−a− systemdoindeedexistinnature
(e.g., BaTbO 72). However, we are not aware of shell Wereturnnowtoacarefulexaminationofthespecifics
3
potentialsforanyofthesematerials. Ontheotherhand, of the a−a−a− Glazer reconstruction, as visualized in
we discovered that the shell-potential model commonly Figure 2. The oxygen octahedra each rotate around
used for strontium titanate (Sr2+Ti4+O2−), which forms one of the eight possible (cid:104)111(cid:105) axes (expressed in the
3
intrinsic RP phases, does possess a ground state of the coordinates of the closely related cubic structure), with
a−a−a− type. neighboring octahedra rotating in opposite directions in
a cell-doubling, alternating three-dimensional 2×2×2
Indeed, on a microscopic level, strontium titanate is
checkerboard pattern. The crystal itself responds to the
very similar to lanthanum aluminate. In fact, early
x-ray experiments89,90 erroneously predicted a low-tem- presenceofthesetrigonalrotationsaboutacommonaxis
perature (T (cid:46) 35 K) phase transition in strontium ti- by stretching or compressing along that axis, forming a
rhombohedralBravaislattice. Selectionofaspecificoxy-
tanate to a rhombohedral lattice (generally associated
witha−a−a− microscopicordering),onlytobecorrected gen octahedron as reference then permits eight possible
by subsequent spectroscopic studies.78,91 Strontium ti- distinct bulk configurations, each characterized and enu-
merated by the particular choice of one of the eight pos-
tanate assumes its actual tetragonal ground-state struc-
sible trigonal rotation axes for that specific octahedron.
ture through a famous cell-doubling antiferrodistortive
phase transition near 105 K,92 whose “physical origin is These eight reconstructions can be characterized either
thesame”85asthatoflanthanumaluminate. Indeed,this as ±[±1,±1,1], namely a selection of an overall ± sign
transition is “strikingly analogous in all respects”93,94 to andachoiceofoneofthefourunsigned [±1,±1,1]rota-
tion axes, or, alternatively, simply as one of eight signed
that of lanthanum aluminate, although the energy scales
[±1,±1,±1] axes.
arequitedifferent. (Thestructuralphasetransitiontem-
peratureforlanthanumaluminateis800K.85,95)Further
The former perspective is visualized in Figure 3. The
connections between these two perovskites are enabled
four(unsigned)[±1,±1,1]rotationaxesdesignatethere-
by the classical concept of the Goldschmidt tolerance
construction at an arbitrary origin of the crystal, where
factor,96 t= √12rrBA++rrOO, a normalized ratio of the radii of a final choice of sign determines whether the rotation
ABO3ionshistoricallyusedtocategorizeperovskitesand at the origin is either clockwise (“positive”) or counter-
predict their ground states.97 In fact, lanthanum alumi- clockwise (“negative”), thereby fully specifying the mi-
natehasatolerancefactorwithin1%ofthatofstrontium croscopic state of the crystal. Note that a change in this
titanate (data compiled by Shannon98). signchoicecorrespondspreciselytoarigidtranslationof
In sum, the parameters for the shell potential com- the crystal by a [111] primitive translation vector (to a
monly used for strontium titanate describe a perovskite positioninthecrystalwithoppositesignofrotation,due
which intrinsically forms RP phases within the desired tothecheckerboardpatternofthereconstruction). Such
a−a−a− Glazer system. It is not surprising, then, to a translation would not be observable on a macroscopic
discover empirical evidence for perovskites, such as lan- level.
5
(a)+OB (b)−OB
FIG.4: Signpatternofoctahedralrotationsonoppositesides
of stacking fault: octahedra in lower layer A ( ) and upper
layer B ( ) and standardized displacement D(cid:35)(→). Pan-
els (a) and(cid:50)(b) depict the two distinct choices for sign of ro-
tation (+ and −, respectively) of O .
B
FIG.2: Perovskitereconstructionina−a−a−Glazernotation,
where the material on either side of the stacking fault
showing rotations of oxygen octahedra in opposite directions
possesses a specific bulk reconstruction throughout. We
in a cell-doubling, alternating three-dimensional 2 × 2 × 2
first focus on the reconstruction of the bulk material on
checkerboard pattern. (Degree of rotation exaggerated for
illustrative purposes.) the side “below” (at lower values for eˆ3) the AO/AO
stacking fault, which we shall denote as side A. As dis-
cussed above, the microscopic configuration of bulk ma-
terial on side A can be fully specified by noting the ro-
tation axis of the octahedron at an arbitrary, but from
then onward fixed, origin O . The rotation axis of this
A
particular reference oxygen octahedron can then assume
any of eight choices among the (cid:104)111(cid:105) axes, which we can
regard as a selection of an overall ± sign (“positive” or
“negative”) and a choice of one of the four (unsigned)
[±1,±1,1] rotation axes. This specification then deter-
mines the entire microscopic structure of the bulk mate-
rialinsideA,accordingtothe2×2×2three-dimensional
checkerboardpatternofalternatingsignsfortherotation
axes. For the present purpose, we choose always to se-
lectthereferenceoctahedronO fromamongthoseocta-
A
hedra immediately neighboring the stacking fault which
have a positive rotation in the above sense of the choice
FIG. 3: “Positive” senses of the four (unsigned) [±1,±1,1]
of overall ± sign.
rotation axes for oxygen octahedra (eˆ axis out of the page).
3
Next,asnotedabove,thebulkperovskiteontheother
side, B, of the stacking fault is displaced, in general,
by a vector a0[±1,±1,0] relative to side A. Temporar-
A. Isolated stacking faults 2
ily disregarding the rotational state of all octahedra, we
note that all four of these displacements are equivalent,
The RP phases observed in experiments49,52,99 consist sincetheoctahedraonBarepositionedatthecentersof
ofsuperlatticesofbulkperovskiteslabsseparatedbythe squares formed by the octahedra on A. (See Figure 4.)
insertion of excess (001) AO planes to create stacking This equivalence (apart from rotational states of the oc-
faults of the form .../AO/BO /AO/AO/BO /AO/... in tahedra)allowsustochooseastandardizeddisplacement
2 2
the normal .../AO/BO2/AO/BO2/AO/... stacking se- D ≡ a20[+1,+1,0] from OA to select the origin OB.
quence. Across each such stacking fault, the bulk per- (Morespecifically, O =O +D+ζa eˆ , whereζ ≈3/2
B A 0 3
ovskiteslabsarealternatelydisplacedbyin-planevectors represents the vertical displacement between octahedra
oftheform a20[±1,±1,0]. Toourknowledge,however,no on opposite sides of the stacking fault, and eˆ3 is a unit
onehasyetexploredtheeffectsofdifferentcombinations vector in the [001] direction.)
ofpossiblesymmetry-relatedbulkreconstructionsonop- TherotationstateofO thencompletelyspecifiesthe
B
posite sides of the repeated AO planes. entire microscopic structure of the bulk material in side
To enumerate the distinct possible configurations for B. This rotation state can be specified as a selection of
suchstackingfaults,werestrictourconsiderationtocases an overall ± sign (“positive” or “negative”) and a choice
6
of one of the four (unsigned) [±1,±1,1] rotation axes.
Rotation-sign Specifier Algebraic Specifier
Figure 4 illustrates exactly these two possible choices of
sign, where the reference octahedron on side B is either [0+] (111=⇒111)
positive (Figure 4(a)) or negative (Figure 4(b)). Generi- [0−] (111=⇒¯1¯1¯1)
cally below, we shall refer to these two possible sign pat- [π+] (111=⇒¯111)
2
ternsinthestacking-faultconfigurationas“+”and“−”, [π−] (111=⇒1¯1¯1)
2
respectively, reflecting the overall sign of the rotation of [π+] (111=⇒¯1¯11)
O . The combination of these two sign patterns and [π−] (111=⇒11¯1)
B
the four possible unsigned rotation axes (not indicated [3π+] (111=⇒1¯11)
2
in Figure 4) for each origin, OA and OB, leads to a total [3π−] (111=⇒¯11¯1)
2
of 2×(4×4) = 32 distinct possible configurations for
this stacking fault. TABLE I: Enumeration of interfacial stacking faults remain-
Next, we consider equivalence of stacking faults under ing after application of C4z and translational symmetry, la-
applicationofC symmetry. Asdiscussedabove,thero- beled by either relative rotation and sign of the bulk regions
4z
onoppositesidesofthestackingfault(rotation-signspecifier)
tationaxisofO canalwaysbeselectedtobeamongthe
A
or rotation state of the two reference octahedra (algebraic
four (unsigned) [±1,±1,1] axes. Because C symme-
4z specifier).
triesinterconvertallfouroftheseaxes(C ◦[111]=[¯111],
4z
C2 ◦[111]=[¯1¯11],C3 ◦[111]=[1¯11]),anoverallrotation
4z 4z
of the coordinate system can be found to make the O
A
rotationaboutthe[111]axis. Wemaythusdefinethero-
tationaxisofO toalwaysbethe(unsigned)[+1,+1,1]
A
axis. This then narrows the phase space of thirty-two
configurations listed above to now only 2×(1×4) = 8
distinct configurations.
These eight distinct configurations can be enumerated
using two related nomenclatures (seen in Table I), one
which is algebraically explicit and suitable for symme-
try arguments and the other which is more compact and (a)[π+] (b)[π−]
2 2
convenient for communication. In the former case of the
algebraic specifier, we enumerate each configuration by
specification of the rotation states of each reference oc-
tahedron, O and O , expressing the rotation states in
A B
terms of (signed) [±1,±1,±1] vectors. For example, if
O is in rotation state [111] and O is in rotation state
A B
[11¯1], we write (111 =⇒ 11¯1). For a more compact no-
tation, we can take, without loss of generality as shown
above, the rotation state of O to always be [111]. The
A
unsigned rotation axis of O is then related to that of
B
OA by one of the four C4z rotations of angles {0, π2, π, (c)C2x◦[π2+]
3π}. Todenotetheconfigurationofthestackingfault,we
t2henappendtheremainingchoiceof±signforO toits FIG.5: Fullspecificationofoctahedralrotationsonopposite
B
sidesofstackingfault: octahedrainlowerlayerA( )andup-
rotationangletoproduceacompoundrotation-signspec-
ifier. For example, the (111=⇒11¯1) configuration from perlayerB( )andstandardizeddisplacementD(cid:35)(→),with
explicit(signe(cid:50)d)[±1,±1,±1]rotationaxestospecifyfullcon-
the preceding example may be more compactly written
figuration. Panels(a)and(b)depictdifferentconfigurations,
as [π−] (since −C2 ◦[111] = −[¯1¯11] = [11¯1]). Finally,
4z withPanel(c)showingtheapplicationofC2x rotationto(a)
Table I enumerates all eight configurations according to to produce a configuration equivalent to (b).
both their rotation-sign and algebraic specifiers.
The above enumerated eight stacking-fault configura-
tions can be reduced yet further by considering the ef- figurations are actually equivalent, related simply by a
fects of additional, more subtle, symmetry operations — C rotation.
2x
specifically, the C , C , and C rotations and inver- The general strategy which we shall employ is to ex-
2x 2y 4z
sion I. To demonstrate the effects of such symmetry ploit the fact that the algebraic specifier not only deter-
operations,wefirstconsidertwooftheaboveeightstack- mines the rotation state of all of the octahedra in the
ing-fault configurations, [π+] in Figure 5(a) and [π−] in crystal but also transforms in relatively simple ways un-
2 2
Figure 5(b). From the figure, these two configurations dertheapplicationofsymmetryoperationstotheoverall
clearlydifferonlyintheoverallchoiceofsignpatternfor crystal. Figure 5(c) illustrates the state of the crystal
the upper layer B ( ’s). We now use symmetry argu- in Figure 5(a) after the application of a C symmetry
2x
ments to demonstrate that these two stacking-fault con- aboutaspecific axis. Toensurethatthebasiccrystalline
(cid:50)
7
structure maps back onto itself, this axis is chosen to
Algebraic Specifier
passatequaldistancestothetwoneighboringAOlayers Symmetry
in the stacking fault and through the point immediately Original Final
abovetheOAreferenceoctahedron(baseofthestandard- C (abc=⇒αβγ) (αβ¯γ¯=⇒ a¯bc)
2x
ized displacement D in Figure 5(a)). The application of C (abc=⇒αβγ) (α¯βγ¯=⇒ a¯bc)
2y
C2x effects the following changes to the configuration in C (abc=⇒αβγ) (a¯¯bc =⇒α¯β¯γ)
2z
Figure 5(a): C (abc=⇒αβγ) (¯bac =⇒βα¯γ¯)
4z
I (abc=⇒αβγ) (αβγ =⇒ abc)
1. the position of O maps from the bottom layer A
A
to the top layer B (this changes the central in
Figure 5(a) into the central in Figure 5(c)); TABLE II: Rotation (C) and inversion (I) symmetry opera-
(cid:35)
tions applied to general stacking-fault configuration.
2. the rotation state of O map(cid:50)s from [111] to [1¯1¯1];
A
3. the position of the B-side octahedron ( ) in the of the vector connecting the two reference octahedra O
A
lower-right quadrant of Figure 5(a) maps(cid:50)to the A- and OB, requires special care: it does not change the
sideoctahedron( )intheupper-rightquadrantof direction of pseudo-vectors such as the rotation axes of
Figure 5(c); (cid:35) the different octahedra but simply interchanges the two
sides, A and B.)
4. the rotation state of this latter octahedron maps
We are now able to demonstrate further symmetry re-
from [1¯1¯1] to [111].
duction of the stacking-fault configurations. Table I lists
To determine the algebraic specifier for the final config- all eight possible stacking fault configurations, provid-
uration in Figure 5(c), we now identify an octahedron ing both rotation-sign and algebraic specifiers for each.
on the lower side of the stacking fault which has a posi- Application of the symmetry operations from Table II
tiverotationfromamongthe(unsigned)[±1,±1,1]axes to these configurations, and subsequent translation T by
(baseofthestandardizeddisplacementDinFigure5(c)). a0[100] if necessary to make OA of rotation state [111],
The rotation state of this octahedron, combined with generates the following relationships,
that of the corresponding reference octahedron on the
T ◦C ◦[3π+]=[3π−] (1a)
upper side, defines the algebraic specifier for the result- 2y 2 2
ing stacking fault. For this particular case, it is evident T ◦C ◦[π+]=[π−] (1b)
from the figure that C ◦[π+] = (111 =⇒ 1¯1¯1). Ref- 2x 2 2
2x 2 C ◦I◦[3π+]=[π−]. (1c)
erence to Table I then identifies (111 =⇒ 1¯1¯1) = [π−], 4z 2 2
2
so that C2x ◦[π2+] = [π2−], as direct comparison of Fig- The above symmetry operations, which each involve in-
ures 5(b) and (c) confirms. Thus, we conclude that [π+] terchange of the A and B sides of the interface, demon-
2
and [π−] are identical and related by the C rotation. strate equivalence among the set of four stacking faults,
To2developanalgebrafortheactionofsym2xmetryoper- {[π+], [π−], [3π+], [3π−]}. Thus, onlyfiveuniquestack-
2 2 2 2
ationsonarbitrarystacking-faultconfigurations,wecon- ing-fault configurations remain, [0+], [0−], [π+], [π+],
2
sider the effects of such operations on general stacking- and [π−]. Comparison of the algebraic specifiers of these
fault configurations, (abc=⇒αβγ). For the case of C , five configurations, from Table I above, with the com-
2x
we note in preparation that the action of C on any puter-generated distorted phases listed as Nos. 18–22, in
2x
vector (xyz) is C ◦(xyz)=(xy¯z¯). Then, as above, the TableIIIofHatchetal.,71confirmsthatourresultiscon-
2x
operation of C transforms the generic algebraic speci- sistent with theirs, though theirs is limited to the n=1
2x
fier (abc=⇒αβγ) as follows: RP phase. We confine ourselves to discussion of these
five unique configurations for remainder of this work.
1. it swaps abc and αβγ;
2. it maps abc to a¯b¯c;
B. RP phases
3. it changes the sign of αβγ;
The RP phases, which we study in this work, consist
4. it maps αβγ to αβ¯γ¯
of periodic arrays of the above configurations of stack-
ing faults. While constructing such arrays from individ-
In short,
ual stacking faults, the array periodicity may be main-
C ◦(abc=⇒αβγ)=(α¯βγ =⇒a¯b¯c). tainedeitherthroughsimplealternatingpatternsofbulk
2x
regions (A/B/A/B sequencing) or through more com-
TableIIsummarizestheeffects,similarlydetermined,on plex, and possibly lower-energy, patterns (for example,
stacking-fault configurations of a number of useful sym- A/B/C/A/B/C sequencing).
metry operations: rotations C , C , C , and C and Immediately below, we demonstrate that, in fact, an
2x 2y 2z 4z
inversion I. (Note that inversion, through the midpoint A/B/A/B sequence of bulk regions always corresponds
8
to repetition of the same type of stacking-fault config- stituents. The shell model separates each ion into two
uration. Three very compelling reasons then follow for parts,acoreandanoutershell,whichpossessindividual
studying RP phases with this type of periodicity. First, charges that sum to the nominal charge of the ion. The
one should expect the preferred RP phase to consist of a total model potential U consists of three terms,
sequenceofstackingfaults,whichareall ofthelowest-en-
ergy configuration; this would then naturally correspond U ≡UP +UC +UB, (2)
to an A/B/A/B sequence. Second, actual experiments
representing, respectively, the polarizability of the ions,
find superlattice sizes consistent with small, simple re-
peatunits.49,52,99Finally,isolatedstackingfaultsarealso andtheCoulombandshort-rangeinteractionsamongthe
observedundercertainconditions,49 andthestudyofRP ions. The polarizability is captured by harmonic springs
connecting the core and shell of each ion, so that U has
phases of A/B/A/B periodicity, containing two identi- P
the form,
cal faults in each primitive unit cell, then allows for the
extraction of the behavior of individual isolated faults.
U = 1(cid:88)k |∆r |2, (3)
To establish that an A/B/A/B sequence of bulk re- P 2 i i
gionscorrespondstorepetitionofthesameconfiguration i
of stacking fault, we must prove the equivalence of the
where |∆r | is the core-shell separation for ion i and the
two stacking faults, (A/B) and (B/A)(cid:48), in such an RP i
k areasetofion-specificspringconstants. TheCoulomb
phase. As in Section IIIA, we take the algebraic speci- i
contributions take the form,
fier for this first stacking fault (A/B) to be the generic
(satabtces=⇒of tαhβeγt)wwohreerfeeraebncceanodctαahβeγdrraefeforrtothtishefaruoltta,tOioAn UC = 21(cid:88)(cid:48) kcrqiqj, (4)
ij
and O respectively. We must then choose two corre- i,j
B
sponding reference octahedra O(cid:48) and O(cid:48) for the second
A B where i and j range over all cores and shells (excluding
stacking fault (B/A)(cid:48). Since D≡−D, we can set O(cid:48) =
A terms where i and j refer to the same ion), q and q are
O +(n−1)ζa eˆ andO(cid:48) =O +(n−1)ζa eˆ ,wheren i j
A 0 3 B B 0 3 thecorrespondingcharges,r isthedistancebetweenthe
is the number of the RP phase (A B O ) so that ij
n+1 n 3n+1 charge centers, and k is Coulomb’s constant. Finally,
n−1layersofmaterialseparateO fromO(cid:48) andsimilarly c
A A the short-range interactions are included through a sum
separate O from O(cid:48) . (Consideration of the n=1 case
B B of Buckingham102 pair potentials (which can be viewed
should make this apparent.) The alternating octahedral
ascombinationsofBorn-Mayer103 andLennard-Jones104
rotations of this a−a−a− Glazer system thereby define
potentials) of the form,
the algebraic specifier for the (B/A)(cid:48) stacking fault as
((−1)n−1(αβγ) =⇒ (−1)n−1(abc)). However, displace- (cid:88)(cid:16) (cid:17)
ment of the arbitrary origin (reference octahedra) by UB = 21 Aije−rij/ρij −Cijri−j6 , (5)
a [100],forcaseswhenniseven,generatesanequivalent i,j
0
algebraic specifier for this (B/A)(cid:48) interface which is now
where i and j range over all shells and A , ρ , and C
insensitive to n, (αβγ =⇒abc). Finally, we apply inver- ij ij ij
are pair-specific adjustable parameters. Here, the first
sion symmetry, as introduced in Table II, to this (B/A)(cid:48)
term (Born-Mayer) serves as a repulsive short-range in-
stacking fault, I ◦(αβγ =⇒ abc) = (abc =⇒ αβγ), and
teraction to respect the Pauli exclusion principle, and
thusprovethatthe(A/B)stackingfaultisequivalentto
the second term (Lennard-Jones) models the dispersion
the (B/A)(cid:48) fault.
or van der Waals interactions.105 The specific electro-
Lastly, since all stacking faults in an A/B/A/B RP
static and short-range shell-model parameters used in
phase always possess equivalent configurations, we can
this study were fit to strontium titanate by Akhtar et
uniquely label these RP phases using the same rotation-
al.,101 with values as listed in Tables III and IV. Finally,
sign specifiers established above for the five symmetry-
we wish to emphasize again, as it is rarely mentioned
distinct stacking faults: [0+], [0−], [π+], [π+], and [π−].
2 explicitly in the shell-potential literature, that the pair-
potential terms in U apply to the shells only, and not
B
to the cores.
IV. MODEL AND COMPUTATIONAL
Shellmodelshavebeenextensivelyusedfordecadesas
METHODS
the primary empirical potential for modeling perovskites
and other oxides.106,107 We tested the correctness of our
A. Shell potential coded implementation of this potential through compar-
isons of lattice constants and elastic moduli without the
As discussed in Section II above, to study the generic AFD reconstruction and find excellent agreement. For
behavior of a−a−a− perovskites which form RP phases, instance, using the same shell potential and the same
we employ a shell-potential model100 parameterized for non-reconstructedgroundstate, wepredictavolumeper
strontium titanate.101 Shell-potential models are formu- SrTiO chemical unit of 59.18 ˚A3, which is within 0.4%
3
lated as an extension to ionic pair potentials and em- ofthevaluecalculatedbyAkhtaretal.101 Fortheelastic
ployed to capture the polarizability of the atomic con- moduli, we find C = 306.9 GPa, C = 138.7 GPa,
11 12
9
constant of the corresponding ion. This process ensures
Shell Core Spring Constant
Ion Charge [e] Charge [e] [eV·˚A−2] thatallsupercellenergiesarerelaxedwithrespecttoboth
ionic and lattice coordinates to a precision of ∼0.3 µeV
Sr2+ 1.526 0.474 11.406 per supercell.
Ti4+ −35.863 39.863 65974.0
O2− −2.389 0.389 18.41
C. Ground state
TABLE III: Electrostatic shell-model potential parameters
used in this study (from Akhtar et al.101).
To establish the ground state of the model potential
used in our calculations, we have carried out what we
Interaction A [eV] ρ [˚A] C [eV·˚A6] regard as a thorough, but not exhaustive, search for a
probable ground-state structure. Indeed, we have found
Sr2+ ⇔ O2− 776.84 0.35867 0.0
no alternative structure which relaxes to an energy less
Ti4+ ⇔ O2− 877.20 0.38096 9.0
O2− ⇔ O2− 22764.3 0.1490 43.0 thanourcandidateground-statestructurewithinourpo-
tential. We performed quenches on hundreds of random
displacementsfromtheidealizedpositionsofthe1×1×1
TABLE IV: Short-range shell-model potential parameters
used in this study (from Akhtar et al.101). primitiveunitcelltoexplorevariouspotentialreconstruc-
tions for supercells up to 6×6×6. We also considered a
number of highly ordered configurations commensurate
and C = 138.8 GPa, which are within 1.8%, 1.0%, with antiferrodistortive disordering.
44
and 0.7%, respectively, of the values from Akhtar et Among those minima which we explored, we selected
al.101 From this, we conclude that our implementation thelowest-energyconfigurationtoserveasthebulkcrys-
of the potential is correct. We also would like to note talline state throughout this study. This configuration
that, when the a−a−a− reconstruction is considered, possesses a fairly regular pattern as depicted in Fig-
significant changes occur, and we find, instead, a vol- ure 2, namely each oxygen octahedron rotates slightly
ume per SrTiO chemical unit of 58.49 ˚A3 and elastic along trigonal directions in a cell-doubling, alternating
3
moduli of C = 275.6 GPa, C = 144.3 GPa, and three-dimensional 2×2×2 checkerboard pattern, which
11 12
C = 133.5 GPa. This underscores the importance of is precisely the desired a−a−a− Glazer system from Sec-
44
consideringAFDreconstructionswhenconstructingsuch tion II. We also find that the lattice vectors assume the
potentials. rhombohedral symmetry commonly found in a−a−a−
perovskites, with the cubic lattice stretching along the
same trigonal axes about which the octahedra rotate.
B. Numerical methods Thespecificlatticevectorswhichwefindforthe2×2×
2 reconstructed supercell of our model are the columns
Inthiswork,wecomputetheCoulombicinteraction108 of the following matrix,
from (4) using a Particle Mesh Ewald algorithm109–111
 
7.764 0.028 0.028
with all real-space pair-potential terms computed out to
a fixed cutoff distance using neighbor tables. Analytic R= 0.028 7.764 0.028 ˚A.
0.028 0.028 7.764
derivatives are used to determine the forces on the cores
and shells within the supercell, and finite differences are
used to compute the generalized forces on the superlat-
tice vectors. While such finite-difference methods occa- D. Construction of RP supercells
sionally introduce issues with numerical precision into
the resulting calculations, we are able to mitigate such Our construction of initial ionic configurations for the
effects with care in selection of the finite-difference step, A B O RP phases proceeds through a detailed
n+1 n 3n+1
scalingitproportionallytothesizesofthelatticevectors process defined in Section III. As discussed there, we
involved. carefully consider a large number of combinatorial pos-
With the gradients determined as above, we relax sibilities resulting from the different geometries due to
each system using the technique of preconditioned con- theaforementioned2×2×2cell-doubling, a−a−a− bulk
jugate gradient minimization112 (specifically, the Polak- reconstruction. We construct each configuration of the
Ribi`ere113 method), fully optimizing the ionic coordi- RP phase by stacking 2n layers of bulk perovskite along
natesinaninnerloopoftheroutineandthenoptimizing the [001] direction, with the two stacking faults (excess
the lattice vectors in an outer loop. The preconditioner AO (001) planes) so situated as to equally divide the
appliesonlytotheionicrelaxationofthisprocedureand supercell into two bulk slabs, each with n layers and po-
scales the generalized force on the geometric center of tentially different a−a−a− reconstructions. Relative to
each ion (mean position of the core and shell) separately each other, these bulk slabs are also laterally displaced
fromthegeneralizedforceonthecore-shelldisplacement, by one of the four (equivalent) a0[±1,±1,0] vectors at
2
with the latter scaled in inverse proportion to the spring each stacking fault. We perform this procedure to create
10
alleight(111=⇒αβγ)configurationsfromSectionIIIA, 5000
through combinations of reconstructions in the two bulk V] 4500
e
slabs,forRPphasesfromn=1...30,containingfrom56 nit [ 4000
tisot1h2e1n6fiuolnlys,rreelsapxeecdt,ivaeslyd.esEcarcibheRdPabsouvpee,rblaottthiceinsytestremms mical u 3500
ofinternalioniccoordinatesandlatticevectors,untilthe he 3000
c
energy is minimized to within a precision of ∼0.3 µeV. AO 2500
In all cases, we find that the material on either side of s
es 2000
the stacking fault maintains a uniform bulk reconstruc- xc
tionthroughout,sothattheenumerationofSectionIIIA er e 1500
p
is suitably complete. Finally, we test for meta-stability y 1000
g
by creating ensembles of 20–250 samples for each config- er
n 500
E
uration of octahedra rotations for all RP phases up to
0
n = 10, introducing root-mean-square displacements of 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
0.001 ˚A for each coordinate axis of every core and shell, n
and quenching each of these randomized systems. (a)
0.20
V. RESULTS eV] 0.15
nit [
As described in Section II above, the results obtained al u 0.10
c
below have been computed with the standard shell-po- emi 0.05
tential model for strontium titanate,101,106,107 but are ch
O
interpreted to represent the generic behavior of an anti- A 0.00
s
ferrodistortive perovskite (ABO3) from among the class ces -0.05
of perovskites within the a−a−a− Glazer system which ex
form RP phases. per -0.10
y
g
er -0.15
n
E
A. Formation of RP phases -0.20
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
n
After constructing the various possible RP phases ac-
(b)
cordingtotheprescriptioninSectionIVD,theidealpro-
cedure for determining the energetics of the RP phases FIG.6: FormationenergiesofRPphasesperexcessAOchem-
is to fully relax each system, in terms of both internal ical unit as a function of n for various members of homolo-
atomiccoordinatesand superlatticevectors,withoutany gous series An+1BnO3n+1: methodological error of employ-
ing (a) tetragonal supercells for RP phases or (b) fully re-
externally imposed symmetries. The energies of the ref-
laxed supercells for RP phases but unrelaxed lattice vectors
erence bulk materials, perovskite (ABO ) and A-oxide
3 for bulk reference cells. Each panel shows results for the dis-
(AO), should also be fully relaxed in the same sense.
tinct stacking-fault configurations corresponding to different
Without full relaxation of lattice vectors for both the
arrangements of octahedral rotations, as enumerated in the
RP phases and bulk reference cells, any attempts to ex- text (Section IIIA).
tract energies will introduce significant errors that scale
in proportion to the amount of bulk material between
stacking faults. (Such failures to fully relax the lattice
vectors might arise due to expediency in ab initio stud- of bulk material between stacking faults, due to an in-
ies or through attempts to simulate higher-temperature flated RP-phase energy associated with artificial strain
phases by imposing a higher-temperature lattice struc- in the bulk regions. (Note that the results in the figure
ture while relaxing ionic coordinates.) are visually indistinguishable whether or not one allows
Figure 6(a) shows actual results obtained from such the perovskite or A-oxide bulk reference cells to fully re-
a methodologically flawed procedure, where the energy lax.) Figure6(b)showsresultsobtainedwherethelattice
needed to form the RP phase from the bulk perovskite vectors of the RP phases are allowed to relax fully but
ABO andbulkA-oxideAO,E −n·E − the lattice vectors of the bulk reference cells are not. In
3 An+1BnO3n+1 ABO3
E , is computed without full relaxation of the lat- this case, excess bulk energy is not included in the RP
AO
tice vectors of the RP phase, but instead with the en- cell but instead in the bulk reference cells, and the en-
forcement of tetragonal lattice vectors associated with ergy of formation now decreases linearly with n. Either
a higher-temperature symmetry not possessed by the type of error will confound the extraction of meaningful
ground state of our model. Under these constraints, the information on the RP phases. The calculation of such
energyexhibitsaclearlinearincrease withn,theamount phases is herein seen to be quite delicate, a fact which