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Ordering in spinels and perovskites
Reactivity of Solids, 5 (1988) 279-291 Elsevier Science Publishers B.V., Amsterdam
279 - Printed
in The Netherlands
ORDERING IN SPINELS AND PEROVSKITES
E. POLLERT Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 180 40 Praha 8 (Czechoslovakia) (Received
March 3rd, 1987; accepted
January
14th, 1988)
ABSTRACT Effects of Coulomb energy, crystal field stabilization energy and the contribution of volume changes on the formation of an ordered state in spinels and perovskites are discussed. It can be created by the following possible mechanisms: ordering of cations, anions, valencies and electronic states. Attention is particularly paid to the ordering of the electronic states leading to the cooperative Jahn-Teller effect.
INTRODUCTION
The stability of an ion in a particular site in the crystal lattice and a crystal, respectively, is given by the general condition of the Gibbs energy minimum G=U+PV-TS=E,+E,+E,,+PV-TS
where the Coulomb energy E,, Born repulsion energy E, (it will be neglected in the following discussion), the contribution of the energy changes due to the crystal field stabilization and covalency E,, and the contribution of volume changes PV reflect the properties of cations entering the oxygen lattice, namely their valency, electron configuration and size. Mutual effects of the contributions can be various in the individual cases and an effort to minimalize them has the following basic consequences: - Distribution of cations in the oxygen sublattice in order to achieve the maximum distances among the cations for the ratio of cations and anions given by the structural type - Preference of cations for a specific kind of site - Formation of an ordered state which generally leads to the lowering of the original symmetry. On the other hand, the entropy contribution TS acts against these tendencies because it is connected with an effort to increase the number of equivalent sites. 0168-7336/88/$03.50
0 1988 Elsevier Science Publishers
B.V.
280
Let us now deal in more detail with the ordering tendency and its influence on the spine1 and perovskite structures, respectively. There are the following possible mechanisms of the formation of an ordered state: - Ordering of cations connected with the migration of cations. Consequently, it can be realized only at relatively high temperatures; Coulomb energy and PV contribution, respectively, are minimized. _ Ordering of anions by anion displacements from their ideal positions in order to optimize cation-anion-cation distances. Usually, it is realized only at a short distance and mainly E, and PV contributions are decisive. The ordered state is stable up to relatively high temperatures. - Ordering of valencies, effected by the redistribution of electrons. It is a diffusionless process with a much lower activation energy in comparison with the previous cases. Formation of the ordered state can be effected at low temperatures. - Ordering of the electronic states, called the cooperative Jahn-Teller effect. This diffusionless process is conditioned by the presence of ions with orbitally degenerated ground states. This degeneracy can be removed below the critical temperature Tctit by lowering the symmetry of their surroundings. When the concentration of active ions is sufficient, then below the critical temperature the total symmetry of the crystal is lowered. The resulting equilibrium represents a compromise between the increase in elastic and decrease in electronic energies.
SPINELS
Ordering of cations
Formation of an ordered state due to an effort to minimize the Coulomb energy contribution is conditioned by sufficient differences in the valencies of cations present in the relevant sublattice. Thus, Zn[LiNb]O,, i.e. I-V spinel, is known only in the ordered state, 1 : 1 in the octahedral sites, in II-IV inverse spinels both ordered and disordered phases exist and II-III inverse spinels are not ordered. An example of the 1 : 3 cation ordering in octahedral sites are the lithium spinels of the general formula M3+[Li&M:.f]04 (M = metal) one of them is lithium ferrite. Its structure is disordered above 1020 K but it remains inverse and it is not completely random. Below 1020 K sequences of the lithium and three iron ions in (110) and (170) directions are generated in the octahedral sublattice [l] (Fig. 1). A correlation between the extent of ordering and the size of ions present in octahedral sites can be seen in the series of M2+[M2+M4+]04 spinels. If the ratio of the octahedral cationic radii r(M2’)/r(M4’) increases, the
281
Fig. 1. Ordering
of cations
in the octahedral
sublattice
of lithium
ferrite. 0 Li+, o Fe3+.
tetragonal distortion of the ordered spinels also increases (Table 1). The influence of the cation radii applies likewise in the LiMe2+M5+0, (M5+ = Sbsf, Nb’+) spinels [6,7]. While the ordering of Sb5+ ions is affectuated by alternating the (001) planes of the B-sublattice occupied by Li+ and Sb5+ ions, respectively, the arrangement of the Nb5+ cations gives rise to a more complicated structure due to the influence of both the Coulomb contribution and the steric term PV. The result is the formation of rows of octahedral sites in the (110) and (110) directions sequentially occupied by Li-Nb-Li-Nb and rows in the (lOl), (lOl), (011) and (011) directions with occupation Li-Li-Nb-Nb. Displacement
of oxygen anions
The presence of cations of various sizes and valencies in the tetrahedral and octahedral sites, respectively, causes a shift of oxygen anions from their TABLE
1
Steric distortion
in M2+ [M2+ Ti4+ 10, spinels
Spine1
rV’(M2+)/rV’(M4+)
T,* (K)
c/a
Mg 2Ti04 Zn ,TiO, Mn ,TiO,
1.19 1.22 1.37
770 770-820 1040
0.995 0.991 0.982
*
temperature.
T, = transition
ti
Ref.
PI L3.41
[51
282 TABLE 2 Size of cations
and oxygen parameters
Spine1
O’),t r(M”) r(M’)+ u
6% (A) r(M”)
(p\)
in some spinels
CdV,O,
MoAg 204
ZnFe,O,
0.95
0.41
0.49
0.60
0.64
1.15
0.78
0.62
1.59 0.394
1.56 0.364
1.27 0.379
1.22 0.385
ideal positions. This effect, however, cannot be considered as the formation of an ordered state. The relationship between sizes of the respective cations and the parameters u characterizing the position of oxygen anions in the lattice (ideal value of u = 0.375) is illustrated by some examples in Table 2. Ordering of valemies
The superstructure due to the ordering of the d electrons of iron ions in the octahedral sites arises in magnetite Fe3+[Fe2+Fe3+]0, below 119 K (Verwey transition). Originally a charge ordering scheme in which Fe3+ and Fe2+ ions form chains lined up in (710) and (110) directions in alternating layers was assumed. The recent detailed studies showed, however, another arrangement given in Fig. 2. Stability of the ordering is very sensitive to small substitutions or oxygen non-stoichiometry. The reason is, probably, and the rise of imperfections in the the ‘changes in the ratio Fe2+/Fe3+ ordered phase due to the different size, valency and electron configuration of cations replacing the iron ions.
Fig. 2. Electron ordering O’-. 0 Fe3+. o Fe’+.
of C, and C,
planes,
respectively,
in Mizoguchi’s
model
[8]. 0
283 ORDERING
OF THE ELECTRONIC
STATES
The relationship between the electron configuration of the cation and its coordination has a decisive role. Electrons have a particularly important influence on the orbitals directed towards the nearest oxygen ions, i.e. eg electrons in an octahedral field and t, electrons in a tetrahedral field, see Fig. 3. Nevertheless, some additional effects on the resulting distortion must be considered. Thus a strong effect of Mn3+( d4) (medium field) and Cu2+( d9) ions in octahedral positions is a consequence of the degenerated eg levels. Lowering of E,, contribution does not favour any type of distortion and the change in the elastic energy (PV term) is decisive. Obviously the arrangement leading to c/a > 1 is more advantageous, see Fig. 4. The manganese-chromium spinels, Mn$r,_,O, can be considered as the model system for the relationship between the concentration of Mn3’ ions
I+ i\ $=l
i
: \’
1
- - - _ _ ___ I
,I
I
@,
dv &z d,. ---__d “~
-___
/ 5-l
6
___-_-
____
\_1
a
dxy dv’
Fig. 3. Tetragonal distortion of octahedron (a) and tetrahedron energy levels. (For simplicity in (b) only the d,, orbital is drawn.)
(b) and
corresponding
284
(b)
(a)
Fig. 4. Ordering respectively.
of occupied
dX2_Y2 orbitals
(c/a
< 1) (a) and
d12
orbitals
(c/a
> 1) (b),
and the macroscopic Jahn-Teller distortion [9,10]. The reason is the fixed valency of the chromium ions and the remarkable preference of Mn3+ and Cr3+ ions to occupy the octahedral sites determining unambiguously the distribution of cations in the sublattices: Mn2+ [ Mnz
ICr:f,]
0,
The critical concentrations of Mn3+ ions giving rise to the tetragonal distortion are [Mn3+], = 0.4 at T = 298 K and [Mn3+], = 0.35 at T = 4.5 K, respectively. These values are substantially lower than, e.g., those of spine1 solid solutions with Zn2+ or Cd2+ on tetrahedral sites, MgxMel_,[Mn2_,,Al,,]O,, where [Mn3+], = 0.68 and 0.72, respectively [ll]. Covalent bonds of d” ions stabilize the cubic structure and act against the distortion. Moreover, in the latter case the large size of the Cd2+ ions causes an increase in the elastic forces which have to be overcome. Similarly, high valency Sn4+ ions stabilize octahedra in the systems containing tin ions and, consequently, inhibit macroscopic distortion [12]. In contrast to the chromite series tendencies towards the formation of a mixed structure and to the electron transfer between the present cations exist, e.g., in the solid solution series Mn,Fe,_,O,, according to the equations Fe:+ + Mnv
= Fep
Fer
= Fe;+ + Mnv
+ Mny
+ MnT
Furthermore, a clustering tendency of Mn3’ ions appears which is suppressed in chromites due to the high stability of Cr3+ ions in octahedral sites and influences significantly the formation of the ordered state, see c/a vs. composition dependence of the tetragonality in Fig. 5. The mutual interactions of isolated clusters are weaker than those of randomly distributed distorting ions (1.75 < x < 2.1), while at high concentrations of Mn3’ ions,
285
25
Fig. 5. Dependence series [9].
of tetragonality
x
30
on composition
0 Mn,Fe3_,0,
series, 0
MnxCr,_xO,
where the clusters are in contact, their occurrence can support the cooper. . ative effect of distorting ions. The important feature of the transition between the ordered and disordered states, as was observed for the chromite series, is the coexistence region of both low temperature-tetragonal and high temperature-cubic phases and a large hysteresis. The course of the transition is controlled simultaneously by the contribution of E,, and of the elastic energy Eelast, which is, in fact, a part of the PV term. The latter contribution acts against the transition and cuts off the growth of emerging crystal domains. The continuation of the transformation requires a change in the equilibrium condition, i.e. an increase or decrease in temperature in dependence on the direction of the process [13,14]. Cations with incompletely occupied t,, levels in an octahedral field, i.e. d’, d*, d3 (high spin), A’ (high spin) exhibit only a weak tendency towards the formation of an ordered state. In contrast to the previous case lowering of E,, is determining and favours c/a > 1 distortion. The electron configuration determines the resulting type of ordering for cations in tetrahedral sites having partially occupied t, levels, i.e. Ni2+(3ds) and Cu2+(3d9), see Fig. 6. On the other hand, Fe2+ cations (e3t;) do not exhibit a preference for any type of ordering and the resulting behaviour depends on the type of cations present in the octahedral sites. A tendency to lower the symmetry of the Fe2+ ions surroundings is compensated in FeAl,O, by a shift of oxygen anions, facilitated by the small size of the A13+
286
fr A
t2,
-2A
c/o
a 1
-6
0
-2A
-A
-6
0
-A
-2A
8 _-______
@a c
-6 ?A
t
ZP G
c/o-=
-A
1 6 ___-____
% c
-6
:
Fig. 6. Splitting of d levels in a tetragonally stabilization energies for A-site cations.
distorted
tetrahedral
field and one-electron
ions. Evidence for such behaviour seems to follow from the comparison of ionic radii and oxygen parameters for FeAl,O, and CdAl,O,, respectively. ~rv(Fe2+) = 0.63
u(FeAl,O,)
= 0.390
yrv(Cd2+) = 0.78
u(CdAl,O,)
= 0.384
Substantially smaller Fe2+ cations cause a higher shift of oxygen anions towards the octahedral A13+ ions. Such a compensation is excluded for FeCr,O, because of the size of the Cr3+ ions. Furthermore, due to their electron configuration, no type of ordering is prefered and, as a consequence, the orthorhombic distortion is stabilized. The V3+ ions in the octahedral sites of the spine1 FeV,O, contribute to the stabilization of the c/a > 1 distortion, due to an effort to remove the degeneracy of t&, electrons [15,16]. The influence of the cation distribution on the ordering is perceptible from the example of CuFe,_,Cr,O, solid solutions, where Cu2+ ions have a tendency to be present in both sublattices simultaneously. Then, for x < 0.4 we have c/a > 1 (Cu2+ present in B-sites), while for x > 1.4, due to the presence of Cu 2+ ions in A-sites, distortion with c/a < 1 arises. Both effects are mutually compensated in the range of 0.4 < x < 1.4 [17]. PEROVSKITES
The significant simultaneous action of E,, E,, and PV contributions exists in the orthomanganites of the general formula R,_,M,MnO,, where R = Pr, Y and M = Ca, Ba [18,19]. The resulting behaviour is controlled by the interplay of two effects provoked by the substitution of trivalent rare earth for bivalent earth cations: _ Variation of the mean size of cations placed in A-sites with 12-fold coordination and B-sites with 6-fold coordination _ Variation of the Mn3+/Mn4+ ratio
287
tolerance factor
0.81 0
I 05 composition
Fig. 7. Concentration dependence Pr,_,Ca,MnO,; 3, Y,_,Ca,MnO,.
1
of
the
tolerance
factor
1, Pr,_,Ba,MnO,;
2,
There are three different mechanisms that could contribute to ordering: (1) The displacement of anions due to an effort to optimize A cation-oxygen-B cation distances, a steric effect connected with the changes in mean ionic radii. This is known to lead to the cooperative buckling of the corner-shared octahedra, if the A cation is too small compared to the octahedrally coordinated B cation. The situation is characterized by the tolerance factor becoming less than 1, see Fig. 7, and macroscopically by the so called O-type orthorhombic symmetry with b > c/ fi> a. (2) The Coulomb interactions among 3d electrons, when two valency states i .e. Mn3’ and Mn4* coexist. When the concentrations of Mn3+ and Mn4+’ are equal, the situation might be regarded as analogous to that in magnetite, already described. (3) Ordering of the electronic states as a consequence of the strong orbit-lattice interaction of the Mn3+( d4) ions. The distorted octahedra are usually arranged perpendicular to each other in (001) planes in order to minimize the increase in the elastic energy. It seems to be sensible to distinguish at least two basic types of ordering, i.e. for x = 0 and x = OS, given in Fig. 8. While in the first one only Mn3’ ions in the B-sublattice are present at x = 0.5, the effect of the arrangement of dZ2 orbitals is combined with a long range ordering of the Mn3’ and Mn4’ ions.
(a)
Fig. 8. Ordering
(b) of occupied
d,z orbitals
in (001) planes (a) x = 0, (b) x = 0.5.
5.65-
I
I1
400
600
I
800
I
1000
I
I
1200
J
T[Kl
Fig. 9. The evolution of the lattice parameters of PrMnO, with temperature. In the inset the temperature dependence of the intensity ratio of (220) Bragg reflections of both 0’ and 0 orthorhombic phases in their coexistence region is given.
The actual structure is determined by the interplay of the described interactions and its character depends on their relative strength. Thus the superposition of the Jahn-Teller effect on the cooperative buckling leads to the 0’ orthorhombic structure (b > a > c/a). The coop-
Fig. 10. A model of the 3: 1 arrangement Ba*+ ions.
for Pr,,,sBa,,sMnO,
composition.
0
Pr3+ ions, l
289
erative Jahn-Teller effect is the determining interaction for x close to 0. An example is PrMnO,, where the 0’ structure exists at room temperature and the 0 structure above the critical temperature. The transition is clearly of the first order and, similarly as in the chromium spinels, it is accompanied by the coexistence of both phases and a strong hysteresis (Fig. 9). An increase in the concentration of Mn4’ ions to a sufficient value (x- 0.4) causes the destabilization of the cooperative Jahn-Teller effect and
-*-X-X
555h
300
x ~0.25
600
900
1200 [K]
Fig. 11. Evolution of the lattice parameters + c/a, o a, x b.
with temperature
in the Y,-,Ca,MnO,
system.
290
a tendency to lower the Coulomb energy contribution by creating favourable valency distribution can predominate. Then a new type of phase transition appears, still driven by the orbit-lattice interaction of Mn3’ ions (ordering of d,z orbitals) but controlled by the frequency of the electron hopping Mn3’ e Mn4’. Below the critical temperature Tcrit the hopping frequency becomes less than the vibration frequency and Mn3+ distortions are stabilized. An important feature of the transition is the weak dependence of T,,, on x corresponding to the described mechanism. Nevertheless, Tcrit as well as the resulting distortion depend on the character of the matrix into which Mn3+ ions are placed. Since in the Pr, _,Ca,MnO, and Pr, _,Ba,MnO, solid solutions the buckling effects becomes nearly negligible, the resulting distortion ( Tctit- 230-270 K and - 300 K, respectively) is tetragonal. It was found that, in agreement with the arrangement in Fig. 8, a = b > c/a for the Ca series but in contrast a 2: b < c/d? for the Ba series. The behaviour can be explained by the collinear arrangement of the distorted octahedra, probably connected with the presence of large Ba*+ ions in A positions, eventually ordered (for the possible arrangement see Fig. 10). Due to this manganese ions in B-sites could have enough room to prefer the collinear arrangement. Large cooperative buckling is the major contribution in the Yi _,Ca, MnO, series and the Jahn-Teller centres only modify the existing structure. Then the change 0’ + 0 need not possess the character of a phase transition (see Fig. 11). The strong steric orbitals lead to a deviation of the Mn3+-O-Mn4+ bonds from 180 o which may be the source of both a larger activation energy for electron hopping and a higher temperature for the structural change 0’ + 0.
CONCLUDING
REMARKS
Formation of an ordered state is a consequence of the effort to minimize the Gibbs energy of the crystal, i.e. the contributions of the Coulomb energy, volume changes and crystal field stabilization. Each of these effects is able to become cooperative and to distort the lattice as well as acting simultaneously. Then the resulting effect cannot be regarded as a mere summation of the contributions but rather as their mutual interplay. Besides the properties of the present cations, namely their valency, size and electron configuration, the ordering is strongly influenced by the type of the respective crystal matrix. Thus the spine1 lattice exhibits a certain flexibility. The oxygen anions can be easily shifted towards the tetrahedral or octahedral positions without a distortion. Due to the existence of vacant sites ferroelastic ordering is preferred.
291
The perovskite lattice, in comparison with the spine1 one is more compact. A shift of the oxygen anions from the ideal positions causes a distortion of the crystal structure. Antiferroelastic ordering is typical, with the exception of Pr,_,Ba,MnO, perovskites, where the presence of large Ba2+ cations in sufficient concentration (X = 0.2) allows parallel ordering.
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
P.B. Braun, Nature, 170 (1952) 1123. P. Delanoye and A. Michel, C.R. Acad. Sci., 269 (1969) 837. Y. Billiet and P. Poix, Bull. Sot. Chim. Fr., (1963) 477. Y. Billiet, P. Poix and A. Michel, CR. Acad. Sci., 256 (1963) 4217. A. Hardy, A. Lecerf, M. Ranet and G. Villers, C.R. Acad. Sci., 259 (1964) 3462. C. Haas, J. Phys. Chem. Solids, 26 (1965) 1225. G. Blasse, J. Inorg. Nucl. Chem., 25 (1963) 136. M. Mizoguchi, J. Phys. Sot. Jpn., 44 (1978) 1501. R. Holba, M. NevIiva and E. Pollert, Mat. Res. Bull., 10 (1975) 853. E. Pollert and Z. Jir& Czech. J. Phys., B26 (1976) 481. A. Bhaduri, H.V. Keer and A.B. Biswas, Ind. J. Pure Appl. Phys., 12 (1974) 745. M. Inagaki, A. Nakanishi, Y. Takeda and S. Naka, J. Solid State Chem., 20 (1977) 323. E. Pollert, J. Solid State Chem., 33 (1980) 305. E. Pollert, P. Novak, Z. Jirak and M. Nevi-iva, Proc. Symp. Reactivity of Solids, Cracow, 1980. J.B. Goodenough, J. Phys. Chem. Solids, 25 (1964) 151. R.J. Arnott, A. Wold and D.B. Rogers, ibid., 25 (1964) 161. H. Ohnishi and T. Teranishi, J. Phys. Sot. Jpn., 16 (1961) 35. E. Pollert, S. Krupizka and E. KuzmiEova, J. Phys. Chem. Solids, 43 (1982) 1137. E. Pollert, S. Krupil(ka and J.C. Grenier, Proc. Int. Symp. on Jahn-Teller Effect, Liblice, 1983.
279 - Printed
in The Netherlands
ORDERING IN SPINELS AND PEROVSKITES
E. POLLERT Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 180 40 Praha 8 (Czechoslovakia) (Received
March 3rd, 1987; accepted
January
14th, 1988)
ABSTRACT Effects of Coulomb energy, crystal field stabilization energy and the contribution of volume changes on the formation of an ordered state in spinels and perovskites are discussed. It can be created by the following possible mechanisms: ordering of cations, anions, valencies and electronic states. Attention is particularly paid to the ordering of the electronic states leading to the cooperative Jahn-Teller effect.
INTRODUCTION
The stability of an ion in a particular site in the crystal lattice and a crystal, respectively, is given by the general condition of the Gibbs energy minimum G=U+PV-TS=E,+E,+E,,+PV-TS
where the Coulomb energy E,, Born repulsion energy E, (it will be neglected in the following discussion), the contribution of the energy changes due to the crystal field stabilization and covalency E,, and the contribution of volume changes PV reflect the properties of cations entering the oxygen lattice, namely their valency, electron configuration and size. Mutual effects of the contributions can be various in the individual cases and an effort to minimalize them has the following basic consequences: - Distribution of cations in the oxygen sublattice in order to achieve the maximum distances among the cations for the ratio of cations and anions given by the structural type - Preference of cations for a specific kind of site - Formation of an ordered state which generally leads to the lowering of the original symmetry. On the other hand, the entropy contribution TS acts against these tendencies because it is connected with an effort to increase the number of equivalent sites. 0168-7336/88/$03.50
0 1988 Elsevier Science Publishers
B.V.
280
Let us now deal in more detail with the ordering tendency and its influence on the spine1 and perovskite structures, respectively. There are the following possible mechanisms of the formation of an ordered state: - Ordering of cations connected with the migration of cations. Consequently, it can be realized only at relatively high temperatures; Coulomb energy and PV contribution, respectively, are minimized. _ Ordering of anions by anion displacements from their ideal positions in order to optimize cation-anion-cation distances. Usually, it is realized only at a short distance and mainly E, and PV contributions are decisive. The ordered state is stable up to relatively high temperatures. - Ordering of valencies, effected by the redistribution of electrons. It is a diffusionless process with a much lower activation energy in comparison with the previous cases. Formation of the ordered state can be effected at low temperatures. - Ordering of the electronic states, called the cooperative Jahn-Teller effect. This diffusionless process is conditioned by the presence of ions with orbitally degenerated ground states. This degeneracy can be removed below the critical temperature Tctit by lowering the symmetry of their surroundings. When the concentration of active ions is sufficient, then below the critical temperature the total symmetry of the crystal is lowered. The resulting equilibrium represents a compromise between the increase in elastic and decrease in electronic energies.
SPINELS
Ordering of cations
Formation of an ordered state due to an effort to minimize the Coulomb energy contribution is conditioned by sufficient differences in the valencies of cations present in the relevant sublattice. Thus, Zn[LiNb]O,, i.e. I-V spinel, is known only in the ordered state, 1 : 1 in the octahedral sites, in II-IV inverse spinels both ordered and disordered phases exist and II-III inverse spinels are not ordered. An example of the 1 : 3 cation ordering in octahedral sites are the lithium spinels of the general formula M3+[Li&M:.f]04 (M = metal) one of them is lithium ferrite. Its structure is disordered above 1020 K but it remains inverse and it is not completely random. Below 1020 K sequences of the lithium and three iron ions in (110) and (170) directions are generated in the octahedral sublattice [l] (Fig. 1). A correlation between the extent of ordering and the size of ions present in octahedral sites can be seen in the series of M2+[M2+M4+]04 spinels. If the ratio of the octahedral cationic radii r(M2’)/r(M4’) increases, the
281
Fig. 1. Ordering
of cations
in the octahedral
sublattice
of lithium
ferrite. 0 Li+, o Fe3+.
tetragonal distortion of the ordered spinels also increases (Table 1). The influence of the cation radii applies likewise in the LiMe2+M5+0, (M5+ = Sbsf, Nb’+) spinels [6,7]. While the ordering of Sb5+ ions is affectuated by alternating the (001) planes of the B-sublattice occupied by Li+ and Sb5+ ions, respectively, the arrangement of the Nb5+ cations gives rise to a more complicated structure due to the influence of both the Coulomb contribution and the steric term PV. The result is the formation of rows of octahedral sites in the (110) and (110) directions sequentially occupied by Li-Nb-Li-Nb and rows in the (lOl), (lOl), (011) and (011) directions with occupation Li-Li-Nb-Nb. Displacement
of oxygen anions
The presence of cations of various sizes and valencies in the tetrahedral and octahedral sites, respectively, causes a shift of oxygen anions from their TABLE
1
Steric distortion
in M2+ [M2+ Ti4+ 10, spinels
Spine1
rV’(M2+)/rV’(M4+)
T,* (K)
c/a
Mg 2Ti04 Zn ,TiO, Mn ,TiO,
1.19 1.22 1.37
770 770-820 1040
0.995 0.991 0.982
*
temperature.
T, = transition
ti
Ref.
PI L3.41
[51
282 TABLE 2 Size of cations
and oxygen parameters
Spine1
O’),t r(M”) r(M’)+ u
6% (A) r(M”)
(p\)
in some spinels
CdV,O,
MoAg 204
ZnFe,O,
0.95
0.41
0.49
0.60
0.64
1.15
0.78
0.62
1.59 0.394
1.56 0.364
1.27 0.379
1.22 0.385
ideal positions. This effect, however, cannot be considered as the formation of an ordered state. The relationship between sizes of the respective cations and the parameters u characterizing the position of oxygen anions in the lattice (ideal value of u = 0.375) is illustrated by some examples in Table 2. Ordering of valemies
The superstructure due to the ordering of the d electrons of iron ions in the octahedral sites arises in magnetite Fe3+[Fe2+Fe3+]0, below 119 K (Verwey transition). Originally a charge ordering scheme in which Fe3+ and Fe2+ ions form chains lined up in (710) and (110) directions in alternating layers was assumed. The recent detailed studies showed, however, another arrangement given in Fig. 2. Stability of the ordering is very sensitive to small substitutions or oxygen non-stoichiometry. The reason is, probably, and the rise of imperfections in the the ‘changes in the ratio Fe2+/Fe3+ ordered phase due to the different size, valency and electron configuration of cations replacing the iron ions.
Fig. 2. Electron ordering O’-. 0 Fe3+. o Fe’+.
of C, and C,
planes,
respectively,
in Mizoguchi’s
model
[8]. 0
283 ORDERING
OF THE ELECTRONIC
STATES
The relationship between the electron configuration of the cation and its coordination has a decisive role. Electrons have a particularly important influence on the orbitals directed towards the nearest oxygen ions, i.e. eg electrons in an octahedral field and t, electrons in a tetrahedral field, see Fig. 3. Nevertheless, some additional effects on the resulting distortion must be considered. Thus a strong effect of Mn3+( d4) (medium field) and Cu2+( d9) ions in octahedral positions is a consequence of the degenerated eg levels. Lowering of E,, contribution does not favour any type of distortion and the change in the elastic energy (PV term) is decisive. Obviously the arrangement leading to c/a > 1 is more advantageous, see Fig. 4. The manganese-chromium spinels, Mn$r,_,O, can be considered as the model system for the relationship between the concentration of Mn3’ ions
I+ i\ $=l
i
: \’
1
- - - _ _ ___ I
,I
I
@,
dv &z d,. ---__d “~
-___
/ 5-l
6
___-_-
____
\_1
a
dxy dv’
Fig. 3. Tetragonal distortion of octahedron (a) and tetrahedron energy levels. (For simplicity in (b) only the d,, orbital is drawn.)
(b) and
corresponding
284
(b)
(a)
Fig. 4. Ordering respectively.
of occupied
dX2_Y2 orbitals
(c/a
< 1) (a) and
d12
orbitals
(c/a
> 1) (b),
and the macroscopic Jahn-Teller distortion [9,10]. The reason is the fixed valency of the chromium ions and the remarkable preference of Mn3+ and Cr3+ ions to occupy the octahedral sites determining unambiguously the distribution of cations in the sublattices: Mn2+ [ Mnz
ICr:f,]
0,
The critical concentrations of Mn3+ ions giving rise to the tetragonal distortion are [Mn3+], = 0.4 at T = 298 K and [Mn3+], = 0.35 at T = 4.5 K, respectively. These values are substantially lower than, e.g., those of spine1 solid solutions with Zn2+ or Cd2+ on tetrahedral sites, MgxMel_,[Mn2_,,Al,,]O,, where [Mn3+], = 0.68 and 0.72, respectively [ll]. Covalent bonds of d” ions stabilize the cubic structure and act against the distortion. Moreover, in the latter case the large size of the Cd2+ ions causes an increase in the elastic forces which have to be overcome. Similarly, high valency Sn4+ ions stabilize octahedra in the systems containing tin ions and, consequently, inhibit macroscopic distortion [12]. In contrast to the chromite series tendencies towards the formation of a mixed structure and to the electron transfer between the present cations exist, e.g., in the solid solution series Mn,Fe,_,O,, according to the equations Fe:+ + Mnv
= Fep
Fer
= Fe;+ + Mnv
+ Mny
+ MnT
Furthermore, a clustering tendency of Mn3’ ions appears which is suppressed in chromites due to the high stability of Cr3+ ions in octahedral sites and influences significantly the formation of the ordered state, see c/a vs. composition dependence of the tetragonality in Fig. 5. The mutual interactions of isolated clusters are weaker than those of randomly distributed distorting ions (1.75 < x < 2.1), while at high concentrations of Mn3’ ions,
285
25
Fig. 5. Dependence series [9].
of tetragonality
x
30
on composition
0 Mn,Fe3_,0,
series, 0
MnxCr,_xO,
where the clusters are in contact, their occurrence can support the cooper. . ative effect of distorting ions. The important feature of the transition between the ordered and disordered states, as was observed for the chromite series, is the coexistence region of both low temperature-tetragonal and high temperature-cubic phases and a large hysteresis. The course of the transition is controlled simultaneously by the contribution of E,, and of the elastic energy Eelast, which is, in fact, a part of the PV term. The latter contribution acts against the transition and cuts off the growth of emerging crystal domains. The continuation of the transformation requires a change in the equilibrium condition, i.e. an increase or decrease in temperature in dependence on the direction of the process [13,14]. Cations with incompletely occupied t,, levels in an octahedral field, i.e. d’, d*, d3 (high spin), A’ (high spin) exhibit only a weak tendency towards the formation of an ordered state. In contrast to the previous case lowering of E,, is determining and favours c/a > 1 distortion. The electron configuration determines the resulting type of ordering for cations in tetrahedral sites having partially occupied t, levels, i.e. Ni2+(3ds) and Cu2+(3d9), see Fig. 6. On the other hand, Fe2+ cations (e3t;) do not exhibit a preference for any type of ordering and the resulting behaviour depends on the type of cations present in the octahedral sites. A tendency to lower the symmetry of the Fe2+ ions surroundings is compensated in FeAl,O, by a shift of oxygen anions, facilitated by the small size of the A13+
286
fr A
t2,
-2A
c/o
a 1
-6
0
-2A
-A
-6
0
-A
-2A
8 _-______
@a c
-6 ?A
t
ZP G
c/o-=
-A
1 6 ___-____
% c
-6
:
Fig. 6. Splitting of d levels in a tetragonally stabilization energies for A-site cations.
distorted
tetrahedral
field and one-electron
ions. Evidence for such behaviour seems to follow from the comparison of ionic radii and oxygen parameters for FeAl,O, and CdAl,O,, respectively. ~rv(Fe2+) = 0.63
u(FeAl,O,)
= 0.390
yrv(Cd2+) = 0.78
u(CdAl,O,)
= 0.384
Substantially smaller Fe2+ cations cause a higher shift of oxygen anions towards the octahedral A13+ ions. Such a compensation is excluded for FeCr,O, because of the size of the Cr3+ ions. Furthermore, due to their electron configuration, no type of ordering is prefered and, as a consequence, the orthorhombic distortion is stabilized. The V3+ ions in the octahedral sites of the spine1 FeV,O, contribute to the stabilization of the c/a > 1 distortion, due to an effort to remove the degeneracy of t&, electrons [15,16]. The influence of the cation distribution on the ordering is perceptible from the example of CuFe,_,Cr,O, solid solutions, where Cu2+ ions have a tendency to be present in both sublattices simultaneously. Then, for x < 0.4 we have c/a > 1 (Cu2+ present in B-sites), while for x > 1.4, due to the presence of Cu 2+ ions in A-sites, distortion with c/a < 1 arises. Both effects are mutually compensated in the range of 0.4 < x < 1.4 [17]. PEROVSKITES
The significant simultaneous action of E,, E,, and PV contributions exists in the orthomanganites of the general formula R,_,M,MnO,, where R = Pr, Y and M = Ca, Ba [18,19]. The resulting behaviour is controlled by the interplay of two effects provoked by the substitution of trivalent rare earth for bivalent earth cations: _ Variation of the mean size of cations placed in A-sites with 12-fold coordination and B-sites with 6-fold coordination _ Variation of the Mn3+/Mn4+ ratio
287
tolerance factor
0.81 0
I 05 composition
Fig. 7. Concentration dependence Pr,_,Ca,MnO,; 3, Y,_,Ca,MnO,.
1
of
the
tolerance
factor
1, Pr,_,Ba,MnO,;
2,
There are three different mechanisms that could contribute to ordering: (1) The displacement of anions due to an effort to optimize A cation-oxygen-B cation distances, a steric effect connected with the changes in mean ionic radii. This is known to lead to the cooperative buckling of the corner-shared octahedra, if the A cation is too small compared to the octahedrally coordinated B cation. The situation is characterized by the tolerance factor becoming less than 1, see Fig. 7, and macroscopically by the so called O-type orthorhombic symmetry with b > c/ fi> a. (2) The Coulomb interactions among 3d electrons, when two valency states i .e. Mn3’ and Mn4* coexist. When the concentrations of Mn3+ and Mn4+’ are equal, the situation might be regarded as analogous to that in magnetite, already described. (3) Ordering of the electronic states as a consequence of the strong orbit-lattice interaction of the Mn3+( d4) ions. The distorted octahedra are usually arranged perpendicular to each other in (001) planes in order to minimize the increase in the elastic energy. It seems to be sensible to distinguish at least two basic types of ordering, i.e. for x = 0 and x = OS, given in Fig. 8. While in the first one only Mn3’ ions in the B-sublattice are present at x = 0.5, the effect of the arrangement of dZ2 orbitals is combined with a long range ordering of the Mn3’ and Mn4’ ions.
(a)
Fig. 8. Ordering
(b) of occupied
d,z orbitals
in (001) planes (a) x = 0, (b) x = 0.5.
5.65-
I
I1
400
600
I
800
I
1000
I
I
1200
J
T[Kl
Fig. 9. The evolution of the lattice parameters of PrMnO, with temperature. In the inset the temperature dependence of the intensity ratio of (220) Bragg reflections of both 0’ and 0 orthorhombic phases in their coexistence region is given.
The actual structure is determined by the interplay of the described interactions and its character depends on their relative strength. Thus the superposition of the Jahn-Teller effect on the cooperative buckling leads to the 0’ orthorhombic structure (b > a > c/a). The coop-
Fig. 10. A model of the 3: 1 arrangement Ba*+ ions.
for Pr,,,sBa,,sMnO,
composition.
0
Pr3+ ions, l
289
erative Jahn-Teller effect is the determining interaction for x close to 0. An example is PrMnO,, where the 0’ structure exists at room temperature and the 0 structure above the critical temperature. The transition is clearly of the first order and, similarly as in the chromium spinels, it is accompanied by the coexistence of both phases and a strong hysteresis (Fig. 9). An increase in the concentration of Mn4’ ions to a sufficient value (x- 0.4) causes the destabilization of the cooperative Jahn-Teller effect and
-*-X-X
555h
300
x ~0.25
600
900
1200 [K]
Fig. 11. Evolution of the lattice parameters + c/a, o a, x b.
with temperature
in the Y,-,Ca,MnO,
system.
290
a tendency to lower the Coulomb energy contribution by creating favourable valency distribution can predominate. Then a new type of phase transition appears, still driven by the orbit-lattice interaction of Mn3’ ions (ordering of d,z orbitals) but controlled by the frequency of the electron hopping Mn3’ e Mn4’. Below the critical temperature Tcrit the hopping frequency becomes less than the vibration frequency and Mn3+ distortions are stabilized. An important feature of the transition is the weak dependence of T,,, on x corresponding to the described mechanism. Nevertheless, Tcrit as well as the resulting distortion depend on the character of the matrix into which Mn3+ ions are placed. Since in the Pr, _,Ca,MnO, and Pr, _,Ba,MnO, solid solutions the buckling effects becomes nearly negligible, the resulting distortion ( Tctit- 230-270 K and - 300 K, respectively) is tetragonal. It was found that, in agreement with the arrangement in Fig. 8, a = b > c/a for the Ca series but in contrast a 2: b < c/d? for the Ba series. The behaviour can be explained by the collinear arrangement of the distorted octahedra, probably connected with the presence of large Ba*+ ions in A positions, eventually ordered (for the possible arrangement see Fig. 10). Due to this manganese ions in B-sites could have enough room to prefer the collinear arrangement. Large cooperative buckling is the major contribution in the Yi _,Ca, MnO, series and the Jahn-Teller centres only modify the existing structure. Then the change 0’ + 0 need not possess the character of a phase transition (see Fig. 11). The strong steric orbitals lead to a deviation of the Mn3+-O-Mn4+ bonds from 180 o which may be the source of both a larger activation energy for electron hopping and a higher temperature for the structural change 0’ + 0.
CONCLUDING
REMARKS
Formation of an ordered state is a consequence of the effort to minimize the Gibbs energy of the crystal, i.e. the contributions of the Coulomb energy, volume changes and crystal field stabilization. Each of these effects is able to become cooperative and to distort the lattice as well as acting simultaneously. Then the resulting effect cannot be regarded as a mere summation of the contributions but rather as their mutual interplay. Besides the properties of the present cations, namely their valency, size and electron configuration, the ordering is strongly influenced by the type of the respective crystal matrix. Thus the spine1 lattice exhibits a certain flexibility. The oxygen anions can be easily shifted towards the tetrahedral or octahedral positions without a distortion. Due to the existence of vacant sites ferroelastic ordering is preferred.
291
The perovskite lattice, in comparison with the spine1 one is more compact. A shift of the oxygen anions from the ideal positions causes a distortion of the crystal structure. Antiferroelastic ordering is typical, with the exception of Pr,_,Ba,MnO, perovskites, where the presence of large Ba2+ cations in sufficient concentration (X = 0.2) allows parallel ordering.
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