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PAPIERS INVITÉSSTRUCTURE AND TRANSITIONS IN PEROVSKITES
H. Megaw
To cite this version:
H. Megaw. PAPIERS INVITÉSSTRUCTURE AND TRANSITIONS IN PEROVSKITES. Journal de
Physique Colloques, 1972, 33 (C2), pp.C2-1-C2-5. �10.1051/jphyscol:1972201�. �jpa-00214937�
PA PIERS INVITES
STRUCTURE AND TRANSITIONS IN PEROVSKITES
H, D.
MEGAWCrystallographic Laboratory, Cavendish Laboratory, Cambridge, England
RBsumB.
-I1 est utile de distinguer les unites de structure
t(dures
1)et
((douces
:les
((douces
))sont celles qui varient facilement avec la temperature et la composition, alors que les
t(dures
1)restent constantes. Dans les perovskites les octakdres
0 6sont eux-mbmes
t(durs
)),leurs bascule- ments les uns par rapport aux autres et le deplacement du cation B par rapport
ason centre peuvent btre doux. Les basculements sont souvent determines par I'entourage du cation A
;les deplacements sont en premier lieu determines par le cation B mais peuvent &re modifies par d'autres facteurs.
Basculements et deplacements bien qu'originalement independants, interagissent . On peut rapporter chacun
atrois composantes ass oci6es a I'axe d'ordre quatre de l'octa&dre
;ils peuvent Stre double- ment dkgeneres (equivalents
au n axe d'ordre deux ou trois), et ils peuvent Stre couplBs par symetrie.
Les systkmes de basculements laissent supposer l'existence de mode de reseau. Uniquement quand ils sont couples avec les deplacements des cations B les basculements affectent les proprietes ferro- electriques d'une manikre non negligeable. Le renversement des basculements entre dew couches adjacentes d'octabdres tend
abtre difficile. Ceci est illustre par les orthoferrites de terres rares et les diverses phases de NaNbO3 qui comprennent la phase basse temperature Net la phase la plus haute non cubique
Tz.Abstract.
-It is useful to distinguish hard and soft structure-building units
:the soft are those which vary easily with temperature or composition, while the hard stay constant.
Inperovskites, the
0 6octahedra themselves are hard, their tilts relative to one another, and the displacement of the B-cation from its centre, may be soft. Tilts are often determined by the A-cation environment
;displacements are primarily determined by the B-cation, but may be modified by other factors. Tilts and displacements, though independent in origin, interact. Each may be resolved into three compo- nents associated with the tetrad axes of the octahedron
;they may be doubly or triply degenerate (equivalent to diad or triad axes), and they may be coupled by symmetry. The tilt systems throw light on possible lattice modes. Only when they are coupled with B-cation displacements do the tilts affect the ferroelectric properties significantly. Reversal of tilts between adjacent layers of octahedra tends to be difficult. Illustrations include the rare-earth orthoferrites and various phases of NaNbOs including the low-temperature phase Nand the highest non-cubic phase
T2.Introduction. -
Phase transitions in perovskites are not basically ferroelectric in character. The ferro- electricity is, as it were, accidental. They can be understood using rather simple ideas of atomic radii and interatomic forces.
Hard and soft structural features.
- In a perovskite ABO, the fundamental feature is the B octahedron.
We have to imagine the B(OX), group cut out from the structure, x representing the fraction of the electronic charge of a point atom 0 associated with one B neigh- bour. This group has a permanent physical reality, in that it keeps its identity when built into different structures. In the structures which concern us this is a hard unit, in the sense that the 0, group is little distorted from a regular octahedron, even when other distortions of the structure are quite considerable. To a first approximation, we can treat it as a rigid unit.
I t is true that, at a second approximation, the small dis- tortions of the octahedron itself are often systematic and can be very interesting, but they are only intelli- gible after the rigid-body behaviour of the group has been studied. The remainder of the structure, the
do not inquire about the value or the constancy of x, and in future we shall write the hard group as BO,.
The concept of hard and soft structural features is useful for a wide variety of structures, and not merely for the perovskites which are the subject of the present paper.
Tilts.
- In the ideal perovskite structure, all octa- hedra are parallel. The bond angles at their corners are however soft, and tilting is easy - except in so far as it is prevented by the A cation, acting as a spacer.
The simplest tilt systems are those in which all octa- hedra tilt about the same axis, alternate octahedra tilting in opposite directions. There are three such systems, about the three kinds of symmetry axis of the octahedron
:(i) the tetrad-axis tilt, (ii) the diad-axis tilt, (iii) the triad-axis tilt, shown in figures la, b, c. In the triad-axis and diad-axis systems, the sense of tilt of one octahedron fixes that of all the others, three- dimensionally. In the tetrad-axis system, it only k e s that of a sheet of octahedra
:perpendicular to the axis, in the next sheet above, the tilt may be in the same or the opposite sense.
A(01/4-x~2)12 group, is soft. For present purposes we If the tilts are small, it is possible to think of them as
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972201
C2-2 H. D. MEGAW
FIG. 1. - Tilts of octahedra : (a) about tetrad axis normal to paper (projection of one layer) ; (b) about diad axis in plane of paper (projection of one layer, or two related by mirror plane in paper) ; (c) about triad axis normal to paper (projection of one octahedron in each of three layers, at heights 46, 2 46, 3 46).
Outline of octahedra and edges of upper faces are shown.
made up of components which are tilts about the three tetrad axes [I]. Then a triad-axis tilt requires three equal non-zero components, a diad-axis tilt two equal non-zero components.
We may however have unequal tilt components, and then interesting possibilities arise. A systematic classifi- cation has been derived by A. M. Glazer [unpublished], using three orthogonal axes, and indicating by the superscript +, -, or 0, whether successive octahedra along an axis have the same, opposite, or zero tilts about the axis. In directions perpendicular to the axis, successive octahedra are constrained to have opposite tilts aboht it. Then if no repeat period consists of more than two octahedra, there are 10 possible tilt systems, as follows
:With 3 non-zero tilts
:(i) a + b+ c + , (ii) a + b + c-, (iii) a + b- c-, (iv) a- b- c-
;With 2 non-zero tilts
: (v)a0 b + c + , (vi) a0 bf c-, (vii) a0 b- c-
;With 1 non-zero tilt
:(viii) a0 a 0 c f , (ix) a0 a0 c-
;With no tilts
:(x) a0 a0 a'.
Most known perovskites have rather simplified tilt systems, as follows
:Tetrad-axis tilt, face-centred lattice
:a0 a0 c- (ix), SrTiO, below 110 OK [2]
;Tetrad-axis tilt, base-centred lattice
:a0 a0 c + (x), NaNbO, (T,) at about 600 OC [3].
Diad-axis tilt
:a- b0 a- (vii), PrAlO, (ortho- rhombic) at 172 OK [4].
Triad-axis tilt
:a - a - a- (iv), LaAlO, at about 5000C 151, LiNbO, [6], NaNbO,(N) at about 130 OK 171.
Combined diad-axis and tetrad-axis
:a- b + a- (iii), CaTiO,, GdFeO,, NaNbO,(Q). This system has a (010) mirror plane through the 0 atoms common to the two layers.
Combined diad-axis and tetrad-axis
: a- b- a- (iv),alternate pairs of layers in NaNbO,(P) [8]. This arrangement has a [loll diad-axis, perpendicular to the [loll tilt axis, through the 0 atoms common to the two layers.
The last example indicates how systems may be combined when there are more than two octahedra in the repeat unit
;in NaNbO,(P), a - b + a- (iii) alter- nates with a - b- a - (iv). (For the nomenclature of NaNbO, phases, see reference [9]).
The a0 a0 c + system (viii) has not previously been reported in oxide perovskites. The structure of NaNbO,(T,) has been determined by A. M. Glazer [3], using X-ray diffraction methods
;it is in agreement with lattice-dynamics results [lo].
It is not always possible to say why a particular tilt system is chosen
-why, for example, should LaAlO,, SrTiO,, and NzNbO, have different systems imme- diately below the cubic transition
?Some generalisa- tions can however be made. We may reasonably expect systems with only one independent tilt to be less common than those with two independent tilts
-i. e.
two independent
soft parameters,capable of adjust- ment to provide a suitable environment for the
Acation. Moreover, the mirror-plane symmetry of a- b + a - is more adaptable for this purpose than the diad-axis symmetry of a- b- a-. The series of rare- earth ferrites [ l l ] provides an elegant example. In figure 2 the values of
oand
q ~ /JZ calculated from the
3 0 L l ~ l l l l l ~ ~ l l l l '
Np P r N d SmEuGdTb D y H o E r TmYbLu ( 1 ~ 1 6 ~ ~ 1 ~ 1 4 h ( 1 ~ 1 2 1 \ )
FIG. 2. -Tilt angles in rare-earth orthoferrites, versus atomic number, calculated from data of Marezio, Remeika and Der- nier [ll]. The point for Na in NaNbO3(P) has been added assum- ing that the dependence is actually on ionic radius for 8-coordina- tion, which is nearly linear with atomic number for the rare earths. Open circles and triangles, and full lines, give wand q /
J2
respectively calculated from atomic position parameters ; upright and diagonal crosses, and dashed lines, give values calculated from lattice parameters, JZa/c and a!b respectively.
STRUCTURE AND TRANSITIONS IN PEROVSKITES C2-3
position parameters of the oxygen atoms are plotted
against the atomic number of the A cation. There is a fairly smooth variation from moderately small tilts for the large Pr atom to larger tilts for the small Lu atom.
The near-equality of the two independent components of tilt throughout the series is an interesting point. If, instead of atomic number, we had plotted the tilts against mean A-0 distance for the eight nearest
neighbours, we should have obtained a very similar curve which could be extrapolated to go through the point for the mirror-plane Na in NaNbO, Phase (P), where the tilt components are 8.00 and 6.70 [8]. The similarity of this Na and Pr in PrFeO, is rather strikingly shown by a comparison of their A-0 distances (in A) (the note [2] indicating 2 bonds related by the mirror plane).
Pr in PrFeO, 2.37, 2.39[2], 2.48, 2.63[2], 2.73[2], 3.16, 3.19, 3.39[2]
Na(2) in NaNbO,(P) 2.39, 2.52 [2], 2.52, 2.66 [2], 2.74 [2], 3.07, 3.15, 3.17 [2]
Na in NaNbO,(N) 2.40 [3], 2.57 [3], 3.05 [3], 3.12 [3]
Again we may ask
:why should the large La and the small Li both give rise to triad-axis tilts in LaAlO, and LiNbO, -with tilt angles of about 6O and 23O respectively
-while for a whole range of intermediate- sized cations in orthoferrites we have the two-para- meter tilt system
?One possible answer is that an allowed coordination number for La is 9, for Li 6, both numbers compatible with a triad symmetry axis. On the other hand, 6-coordination is actually achieved in the PrFeO, structure for rare earths smaller than Gd.
There are, moreover, two rhombohedra1 structures, BiFeO, [I21 and NaNbO,(N) [7], with intermediate tilt angles (both about 120), where 6-coordination would not have been expected from the ionic radii
;these will be considered below. The Na-0 distances are shown in the table.
To the extent to which the octahedra remain rigid, we may calculate the tilt angles from the lattice para- meters [I], [13]. Unless one allows for distortion of octahedron shape (as in [I] for the niobates) the appro- ximation is a rough one. In figure 2, the difference between full and dotted lines shows that the octahedron is distorted and changes shape throughout the series, most conspicuously at the large-cation end. The FeO, group, with an average 0-0 edge of 2.844 A, is of course expected to be a little less hard than the NbO, group in NaNbO,, with 2.802 A. In LaAlO, the distor- tion is actually large enough to reduce the ratio c/a
belowits ideal value, whereas tilting would increase it
;calculations of the tilt angle from the lattice para- meters are here wholly misleading [13].
More complicated tilt systems than those described above can occur if different octahedra are allowed to have tilts of different magnitude. Examples are less common, and harder to analyse unambiguously.
Off-centring of B cation. -
This is an independent process that can occur in any structure built from hard octahedra, though most distinctively when, as in perovskites, all octahedron corners are shared between two octahedra. The O, group is effectively a rigid box with opposite corners tied together by 0-B-0 bonds, and the tension in B-0 varies non-linearly with its length. If one imagines B to shrink gradually in
size, without change in 0-0 length, the tension increases until, at a certain value, the central position of B becomes unstable, and two off-centre positions stable [14]. Off-centring is accompanied by small characteristic distortions of octahedron shape [I], but we shall not consider these here. In any given structure, whether or not off-centring occurs is in the first instance an intrinsic property of the B cation. We can recognise a sequence with increasing valency and decreasing size, ranging from thoee like A13+ and Fe3+
which are rarely or never off-centre, through Ta5+
and Ti4+ which are borderline, to Nb5+ which is almost always off-centre
;beyond this are W 6 + , Mo6+, V5+, and even 17+ (as in KIO,)
;but in those extreme cases the distortion of the 0, group becomes so large that it is no longer a useful concept. Where the effect is borderline or small, the off-centre displacement is a soft parameter of the structure.
The nature and the magnitude of the displacement are temperature-dependent. At high temperatures the cation is central. Successive low-temperature forms normally have displacements along a tetrad, a diad, and a triad axis of the octahedron, i. e. with 1, 2 and 3 tetrad-axis components.
The intrinsic effect, characteristic of the B cation and the temperature, may however be modified by other structural forces. For example, in BaTiO, the large Ba increases abnormally the 0-0 edge lengths, and so makes the central position in the octahedron unstable for Ti, which with the intrinsic effect only would remain central, as in CaTiO,. Again, while NaNbO, and KNbO, show the intrinsic behaviour of Nb, with the 3-component displacement ending at about - 500C and the displacements probably vanishing completely by 500 OC, in LiNbO, the triad- axis displacement persists up to 1200 OC because the NbO, group is influenced from outside by the Li-Nb electrostatic repulsion. Again, in BiFeO,, the polarising power of Bi (tendency to covalent bond formation) is the extrinsic effect bringing about the off-centring of Fe.
The fact that these groups with off-centre cations
have a dipole moment in the absence of any external
field, and retain it in such varied structural situations
that its existence cannot reasonably be attributed to an
C2-4 H. D. MEGAW
internal field emanating from outside the group,
means that they must be treated as permanent dipoles, even though they are more sensitive to temperature and applied field than are the more familiar permanent dipoles of molecular structures.
Interaction of tilting and off-centring. -
Though tilting and off-centring have independent origins, they can interact in the structure in important and interest- ing ways. There is a strong tendency for the two effects to share a common axis and a common overall sym- metry. This is well shown in the successive phases of NaNbO,
:in Phase N it is the triad axis, and in Phase P the diad axis, the latter having also an independent tetrad-axis tilt. In Phase P, moreover, the a- b- a- tilt system has a diad axis of symmetry between alternate pairs of layers, perpendicular to the diad tilt axis within a layer
;the Nb displacements in successive layers, conforming to this, are antiparallel. Though the tilts provide an environment for Na(1) which is steri- cally rather unstable, the electrostatic energy of dipole interaction between layers stabilises it. Phase
Q,which is stabilised by an applied electric field, has layers like P but with the mirror-plane configuration a- b+ a- between all pairs of layers. Phase R, above 360 OC, has a tetrad-axis displacement, but presumably no simple tetrad-axis tilt can provide a satisfactory environment for Nb at this temperature. A more complicated tilt system is observed with octahedra which are not all symmetry-related. The P
+R transition is in fact one of the uncommon examples where increasing tempera- ture gives a reduction of symmetry because of the failure to reconcile two independent requirements without increase of unit-cell size. In BiFeO, and NaNbO,(N), it is the triad-axis displacement which stabilises the triad-axis tilt, in spite of the abnormally low coordination number which results.
Thermal effects. -
Soft structural parameters (a better name than
(<order parameters >>) are tempera- ture-dependent. Tilt angles and off-centre displace- ments both tend to zero with increasing temperature, the rate of change increasing as the magnitude of the parameter decreases. Figures 3 and 4 show effects in NaNbO,. Figure 3 shows lattice parameters in the TI-T,-cubic range, and the tilt angles deduced from them for Phase T,, after correction for the effect of abnormally large thermal amplitudes on interatomic distances [3]. Figure 4 shows lattice parameters, pseudo- cubic unit-cell sides and angles, in the N-P range, where there is very large thermal hysteresis
;the calculated tilt angles are much larger (about 120) and their variation much slower than in the T, region.
If a structure is rigidly determined except for one tilt parameter cp, we expect most of the thermal energy to be in a mode where only
cpvaries. The structure vibrates between extreme configuration a, + 8, cp - 8,
6 being smaller than cp. As the temperature rises,cp decreases and 8 increases
;when they become equal there is a near-second-order transition. This is the
Orthor hombic
(7)
J
FIG. 3. - (i) Lattice parameters, (ii) tetragonal tilt angle, i n NaNb03 (Glazer [3]).
( i v )
:::I, , , , , , , , , , J ' ,
0 - 2 0 0 -100 0 100 200 300
400
"C FIG. 4. - Lattice parameters in NaNbOs : (i) Phase P, a = c ; (ii) Phase P, b ; (iii) Phase N, a = b = c ; (iv) Phase P, B - 90° ; (v) Phase N, 90°-
y. Points marked by dots and crosses areobservations by Darlington [7].
situation described by Axe [15], from the high- temperature side, as the condensing out of a phonon.
It is, however, more informative to approach from the low-temperature side, where we may have some hope, from the statics rather than the dynamics of structure- building, of predicting what distortions - tilts or off- centre displacements
-are likely, and therefore which modes will be soft.
If there is more than one independent soft parameter
- whether tilts or displacements - their interaction
may at some temperature allow a discontinuous change
to a new combination of lower energy.
A great vitrietySTRUCTURE AND TRANSITIONS IN PEROVSKITES C2-5
of changes are possible. A parameter which is hard at
low temperatures may gradually become soft with increasing temperature, and influence the approach to a transition primarily associated with another soft para- meter. In NaNbO,(N) the triad-axis tilt is a consequence of the triad-axis displacement
;when, with increasing temperature, the three-fold degeneracy of the latter is lost, the tilt system also changes - one of the three components of tilt is effectively uncoupled and allowed t o vary independently in the next phase, P. When the second component of the displacement disappears, near 360
OC,all three tilts are uncoupled, but the only combination they can give to fit the Na requirements is a complicated one with a larger unit cell. When the dis- placements disappear finally near 450 OC, it is hard to explain the very small differences of spacings in the next phase, S. Above this, phase TI (Fig. 3) has certainly two independent variables
;but T,, we now know, has only one, and the T2-cubic transition is probably a soft-phonon transition.
Ferroelectricity and antiferroelectricity. -
Ferro- electricity and antiferroelectricity only occur in struc- tures where there is an off-centre cation
;they have nothing directly to do with tilts. The tilts, so to speak, determine the framework within which ferroelectricity can occur and to which it may be coupled. The sense of B-displacement in one octahedron influences that in the next, through polarisation of the common corner oxygen (irrespective of the tilt). 1-component displace- ments give chain dipoles, 2-component sheet dipoles, 3-component a single three-dimensional dipole. The sense of adjacent chains or sheets is determined prima- rily by the tilt system, with the B-0-B bond-angle and electrostatic dipole interaction as minor influences, which are sometimes significant. Reversal of small off-centre displacements is easy if they are not coupled t o changes of tilt. When the tilt is zero, or when the only tilt axis coincides with the displacement axis, we have the 1-dimensional ferroelectrics of Abrahams and Keve
[16].When the sense of displacement is coupled by symmetry to the sense of tilt, their classifi- cation scheme is less clear. Here reversal is generally much harder.
In NaNbO,(Q), for example, the a- b+ a- tilt
MEGAW (H.
D.), Acta Cryst., 1968, A24,
589.THOMAS (H.) and M ~ ~ L L E R (K.
A,), Phys. Rev. Letters, 1968. 21. 1256.GLAZER (A.'M.) and MEGAW (H. D.),
Phil. Mag., 1972(in the press).
BURBANK
(R. D.).
J. Auul. Crvst.. 1970. 3. 11 2.M ~ ~ L L E R
(K.A.);BERL;NGER (w.) and WALDNER
(F.), Phys. Rev. Letters, 1968,21,814.ABRAHAMS (S. C.), LEVINSTEIN
(H.J.) and RED-
DY
(J. M.),
J. Phys. Chem. Solids, 1966, 27, 1019.DARLINGTON (C.
N. W.),Thesis, Cambridge,
1971.SAKOWSKI-COWLEY (A. C.),
EUKASZEWICZ(K.) and MEGAW (H. D.),
Acta Cryst., 1968, B25, 851.LEFKOWITZ (I.),
LUKASZEWICZ (K.)and MEGAW
(H.D.),
Acta Cryst., 1966,20,
670.system provides simultaneously a good environment for Na and the diad symmetry appropriate to the 2-component displacement
;it requires the dipoles to be parallel. In NaNbO,(P), pairs of layers have this tilt system
;the alternate pairs, with a- b- a-, put up with a less good Na environment for the sake of the electrically-favourable antiparallel arrangement of dipoles. The P
PQ transition involves switching the sense of tetrad-axis tilts of alternate pairs of layers. The N
+P transition involves, besides the loss of one component of displacement, a similar switching of tilts. The large thermal hysteresis seen in figure 4 becomes understandable.
When a displacement or a tilt is fairly large, it may be too hard to allow reversal. It is possible to envisage more complicated systems where the larger component may be irreversible and the smaller reversible, and each may be coupled with a tilt. In structures with less simple linkage patterns than perovskite, very wide possibilities exist.
Because of the connection between tilt angles and lattice parameters, soft tilt angles imply abnormal elasticities, whether or not dipole moments are present.
These changes are too common, but too varied in their structural origins, for special words like cr ferroelastic
)),to be very helpful. We may then get easy change of orientation (twinning) due to pressure, or even change of phase. The inherent stresses in a specimen due to defects, clamping by the support, or thermal gradients during cooling, may be large enough to produce
cr hybrid ))
crystals with coexisting domains of phases normally widely separated by temperature.
Remarks.
- The concepts of hard and soft structure- building features are very generally applicable in the study of thermal changes and transitions. The concepts of octahedron tilt, B off-centring, and A environment, and their interaction, can be used in many other struc- ture families besides the perovskites, e. g. for a compa- rison of the behaviour of Ca,Nb20, and BaMnF, [17].
Our use of simple
(cengineering
))ideas of structures is complementary to more sophisticated theories because it helps us to pick out the entities which any such theories must take into account if they are to be physically realistic.
ISHIDA
(Y.)and HONJO (G.),
J. Phys. Soc. Japan, 1971,30.
899.M A R E ~ I O
(M.), REMEIKA (J. P.) and DERNIER (P. D.),
Acta Cryst., 1970, B26,
2008.MICHEL
((2.). MOREAU (J. M.). ACHENBACH (G.).
GERSON (R.)
