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An interpretation of the magnetic properties of the perovskite-type mixed crystals La1−xSrxCoO3−λ
J. Phys. Chw.
Solids
Pergamon
Press 1958. Vol. 6. pp. 287-297.
AN INTERPRETATION OF THE MAGNETIC PROPERTIES OF THE PEROVSKITE-TYPE MIXED CRYSTALS Lal_,Sr,Co03_~* JOHN Lincoln (Received
Laboratory,
Massachusetts
3 October 1957;
B. GOODENOUGH Institute
of Technology,
revised 27 November
1957;
Cambridge,
revised 2 January
Mass. 1958)
Abstract-It is pointed out that trivalent cobalt with an octahedral co-ordination of oxygen ions has a low-spin and a high-spin state of comparable energy. The anomalous magnetic properties of the perovskite-type mixed crystals Lal-$r,CoOs-l are interpreted in the light of this fact. It is assumed that the tetravalent cobalt ion is always in the low-spin state in this system. It is shown how the geometry of the perovskite-type lattice lends itself to the formation of two sublattices, each consisting of a set of (111) planes, in one of which the cobalt ions are predominantly in a high-spin state and in the other in a low-spin state. Experiments indicate that the magnetic interaction between trivalent cobalt ions in the high-spin state is antiferromagnetic, whereas the interaction between the low-spin tetravalent cobalt ions and the high-spin trivalent cobalt ions is ferromagnetic. These interactions are considered in relation to modified superexchange and the semicovalent-exchange models. The ferromagnetic CO~+-CO’~ interaction permits a sharp distinction to be made between the two, although there is always some ambiguity in qualitative models. Further, the ferromagnetic interaction can be understood without recourse to ZENER’S double-exchange mechanism. 1. INTRODUCTION THE
magnetic properties of the perovskite-type mixed crystals LaI_,3+Sr,2+Co03_A have been studied by several workers,(l-3) but there has been no satisfactory interpretation of the interesting experimental results. The perovskite-type compounds are of particular interest to the theory of magnetism because the structure is ideal for unambiguous, indirect magnetic-exchange interaction via an intervening oxygen ion. This structure is shown in Fig. 1. As illustrated, the cobalt cations of the system under discussion constitute a simplecubic array with oxygen ions at the centers of the cube edges. An indirect exchange interaction between the magnetic moments of two neighboring cobalt ions must involve the intermediary oxygen ion; this interaction is optimized by the perovskitetype configuration, since the two neighboring cations perturb the same p-orbital of their common oxygen ion. There will probably be no other ex-
FIG. 1. Ideal perovskite-type structure La,-,Sr&oOs.
for the system
change interaction than that between a cobalt ion and its six nearest magnetic neighbors. The interesting properties of this series of mixed crystals are summarized below. Any interpretation of the magnetic interaction must be consistent with all of these properties. ASKHAMet al.@) studied the crystal structure of
* The research in this document was supported by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology. 287
288
JOHN
B.
GOODENOUGH
the whole series of mixed crystals and found that it is perovskite-tame with a smalf rhombohedral deviation from the cubic form, the distortion decreasing with increasing strontium content up to about 50 per cent strontium, a number of extra lines appearing in the X-ray diagrams at high strontium content. They found only a small variation of the pseudo-cubic-cell axes, from 3.82 A for LaCoOs to 3.84 A for Sr&o&s. Further, the lattice parameter 3.82 A for LaCoOs is to be comp.u-ed with 3.838 A for LaNi03, recently measured by WOLD and BANKS.(~) ?VATANABE~~) has found that the crystal structure of the strontium-Rich compounds is cubic if the compounds are prepared in a stream of oxygen, but is indeterminate if prepared in vacua. JONKER and VAN SANEN(~) reported that it is extremely difficult to obtain the correct oxygen content for a perfect perovskite-type lattice. In LaCoOs there were always some tetravalent cobalt ions present. However, as strontium was added, they were unable to obtain more than 45 per cent of the cobalt as a tetravalent ion, this maximum being attained in a mixture with x = O-6. Strontium cobaltite tends toward the formula SrzCoeO,, WATANABE(~)reported that he could vary the tetravaient cobalt content from 6.3 to 59.3 per cent in SrCoOa_^ and from 11.3 to 61-O per cent in La,.,Sr,.jCoOa_~ by heating in zlacuo or in a stream of oxygen. JONKER and VAN SANTEN(~) also measured the paramagnetic susceptibility, the paramagnetic Curie temperature, and the saturation magnetization below the Curie temperature. In Fig. 6 are shown their measurements, expressed in numbers of Bohr magnetons per cobalt atom, for the saturation magnetization ns and the effective paramagnetic moment peR. The effective ~aramagnetic moment per cobalt ion is considerably less than that predicted from spin-only contributions of Coa+ and Co4+ ions in their high-spin state (see Fig. 6(a)). Further, the paramagnetic Curie temperatures, though negative for small tetravalentcobalt concentrations (indicating an antiferromagnetic Co3+-Co3+ interaction), are positive if more than 10 per cent of the cobalt content is tetravalent. The paramagnetic Curie temperature rises sharply from below -200°K for small tetravafent cobalt concentrations to above room temperature for over 40 per cent tetravalent cobalt.
Although this indicates a predominantly positive coupling between trivalent and tetravalent cobalt ions, the saturation magnetization at the absolute zero of temperature is less than 1.5 PB per cobalt ion. WATANABE(~)found similarly that the perovskite-type compounds which resulted from heating in an oxygen stream had lower effective paramagnetic moments than would be predicted from a spin-only calculation of the Cos+ and Co44 ions in their high-spin state. For x = 0.5, he found h = --0*055 and peff = 3.65, as against a predicted value of 5.52. For x = I, he found h==0*208 and pefP = 3.32, as against a predicted vaIue of 5.50. compounds heated & ‘UCIEUO and of undetermined crystal structure, however, had effective paramagnetic moments which were very close to those predicted from spin-only values of Coz+ and Co”+ in their high-spin states. Further, WATANABE found that the samples treated in vacua had a negative paramagnetic Curie temperature, indicating a predominantly antiferromagnetic interaction, whereas samples heated in oxygen had a spontaneous magnetization below room temperature. KOEHLERand WOLLAN~~)studied the compound LaCoOs.os with neutron diffraction and found no magnetic ordering at temperatures down to 42°K. JO~YKER and VAN S~~~~~(s)observed a discontinuity in the paramagnetic susceptibility/temperature curve in LaCoOa.sl; at about 500°K the susceptibility remains constant over an interval of about loo”, obeying a Curie-Weiss law above and below this region. Finally, electrical-resistivity measurementsus2) indicate that as the amount of tetravalent cobalt is increased, the temperature coefficient of the resistivity goes gradually from a negative value (normal semi-conductor) to a positive vatue. This result is qualitatively in agreement with wellknown ideas on semiconduction in oxidic systems.@) From these facts it is apparent that any interpretation of this system must answer the following questions: (1) Why is the pseudo-cubic-cell edge of LaCoOs_h smaller than that for LaNiOs, and why is its variation with strontium substitution SOsmall? (2) Why is it difficult to obtain more than 50 per cent tetravalent cobalt in the perovskite-type lattice?
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
(3) Why are the magnetic interactions in perovskite-type compounds containing both tetravalent and trivalent cobalt predominantly positive? (4) Why are the magnetic interactions in vacuum-prepared SrCoOe.5 (structure undetermined) definitely negative, whereas neutrondiffraction patterns in perovskite-type LaCoOs.os show no magnetic ordering down to temperatures of 4*2”K? (5) What is the origin of the curious anomaly in the magnetic susceptibility of LaCoOs.01 at 500”Kt (6) For the perovskite-type compounds, why are the measured magnitudes of peff and ?zg so low?
MIXED
CRYSTALS
289
Magnetic measurements and optical spectra of compounds and complexes containing trivalent cobalt indicate that if this ion is in an octahedral interstice of oxygeri ions, the exchange splitting and the crystal-field splitting are approximately e,(+f
_s-f
3ais
-&
_*---_--
_ egi*)
t&*1 --I__
-
%I(*) &ex
1_
The purpose of this paper is to show that a satisfactory answer to these questions can be given if one takes account of the stabilities of the lowspin states of trivalent and tetravalent cobalt in octahedral interstices of an oxygen lattice and of the particular geometrical properties of the perovskite-type structure. 2. LOGICAL
FOU~ATI~NS
If a transition-element cation is located in an octahedral interstice of a cubic anion lattice, the electrostatic fields associated with the cations will cause the atomic d level to split into a stable, triply degenerate state t,, (composed of dzyr dy d,,) and an upper, doubly degenerate state eg (composed of d,z_,e and d,2).(7) The magnitude of the splitting A depends upon the particular anion complex and the valency of the cation. Cations of higher valence will be located in a stronger electrostatic field, and the magnitude of A is greater. The two halves of the d level are, as in the atom, split by the intraatomic exchange effects, caused by the Pauli exclusion principle, which favor a parallel alignment of spins within the atom. If the magnitude of this exchange splitting is denoted by E,, then the energy-level diagram for the d orbitals of a transition-element cation in an octahedral anion interstice will be as shown in Fig. 2, where are illustrated the conditions for the occurrence of Co3+ or Co4+, high-spin states with E,, > A, and for Co”’ or Colv, low-spin states with ,?& < A. It should be realized that although the eg orbital is a stable orbital for localized d electrons, a hybrid (ezsp3) orbital may be more stable for nonlocalized electrons which are shared between a cation and an anion. a,
FIG. 2. Schematic energy-level diagram for the 3d orbitaIs of a cobalt cation in a perovskite-type lattice with (a) E,, > A and (b) E,, < 5, where the states eg and tBs are doubly and triply degenerate, respectively.
equal,(*) or Eex M A. This means that secondorder effects may determine whether trivalent cobalt is in a high-spin or a low-spin state. COSSEE(g) has recently performed an elegant series of experiments which demonstrate that trivalent cobalt in the octahedral interstices of the spine1 Co304 are diamagnetic, whereas JONKJZR and VAN SANTE&@) experiments suggest that in the ideally stoichiometric perovskite LaCoOa, the trivalent cobalt ions are in their high-spin state, carrying a magnetic moment nearly that of their spin-only value of 4 1~3 per cobalt ion. Since E, is the same
290
JOHN
B.
GOODENOUGH
in both compounds, it is apparent that the difference must lie in the variation in A in the two instances; the larger A should be associated with the larger crystalline field, or for a given lattice structure and set of ionic charges, with the shorter distance between the positively charged cation and the center of charge density on the negatively charged anion. Paramagnetic-susceptibility data by LOTGERING(~~) on MnCozOa and ZnCozOa (Co-02distances N 2.08 A and N 2.10 A, respectively) indicate that only 5.5 and 15.5 per cent of the cobalt is in a high-spin state. In Co304 the octahedral C0~~‘-02- distance isclo) 2.02 a; in LaCoOs the Cos+-O*distance appears to be I.91 A. The contribution from nearest neighbors to the crystalline field about the trivalent cobalt ion in LaCoOs exceeds that in CosO4, MnCozOa, and ZnCozOd; hence it is reasonable to expect that although trivalent cobalt in LaCoOs is apparently in a high-spin state, in this lattice A e _Ee’,,,so that whether a particular trivalent cobalt remains in the high-spin state or is converted to the low-spin state, may depend upon small changes in the crystalline field. These changes could be induced by thermal expansion or contraction of the lattice or by the introduction of ions of different charge. The introduction of a tetravalent cobalt ion into an octahedral interstice will increase the electrostatic-field splitting probably by as much as 10,000 crn-l;(ll) this should be more than enough to offset anv increase in exchange energy. Therefore, for tetiavalent cobalt in an octahedral site of a perovskite-type lattice, it is reasonable to assume that A > E,, and that Co’v is in a low-spin state with a spin-only magnetic moment of 1 pi. If a CoIv ion is introduced into an ideal LaCoOs lattice, the crystal fields about the neighboring trivalent cobalt ions will be altered in two ways: first, the presence of the higher charge on CoIv will increase the crystalline fields at the neighboring cobalt sites; second, the presence of CO’~ with empty eB orbitals overlapping the filled 2p orbitals of the neighboring 02~ ions will induce some electron sharing, or covalence, in the CoTv-02- bonds. This, in turn, will shift the center of negativecharge density about the 02- ions towards the CoIv ion, so that the crystalline field at the trivalent cobalt ions on the opposite side of the 02- ion is reduced. Therefore it is postulated that, because of the large positive charge, the introduction of a
Co’” ion into the LaCoOs lattice should increase the crystalline field in its environment sufficiently to induce the neighboring trivalent cobalt ions to go into their low-spin state Co”‘. However, the resulting covalent bonding decreases the crystalline fields about the nearest neighbors sufficiently to stabilize these in the high-spin state, so that only next-nearest neighbors are changed to the lowspin state Co III. Again, covalent bonding about the Co”’ will stabilize further the near neighbors of the Colv in a high-spin state, as well as the next-next-near neighbors of the Co”’ ion. It is therefore postulated that the introduction of a Colv ion into LaCoO3 induces the transformations Co3+ -+ Co”’ which are illustrated in Fig. 3. Ill
1
3+
3+
In
1m FIG. 3. Postulated cobalt-ion distribution about a Co’” ion introduced into an ideal La3+Co3+0, lattice. Two features of this diagram are to be noted: first, there is a tendency for the cobalt ions in alternate (111) planes to be in a low-spin state and for those in the other set of (111) planes to be in a high-spin state; second, the introduction of a CoIv ion may be reasonably expected to induce approximately 18 neighboring, trivalent cobalt ions to become Co”‘. Since the Co”’ ions are diamagnetic, the introduction of a Colv ion into a matrix of Co3+ ions causes a cluster consisting of the CoIv ion and its six near-neighbor Cos+ ions to become magnetically isolated from the rest of the lattice. If a sufficient number of CoIv ions are present so that the CoIv ions have near-neighbor or next-near-neighbor Co’v ions, then the additional
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
MIXED
CRYSTALS
291
excited states of the cation-anion-cation configuration. Magnetic interactions cannot be accounted for on the basis of a purely ionic model of the lattice. In the excited configurations, one or both of the electrons in a 2p oxygen orbital are excited into the empty, or partially filled, cation orbitals which overlap it on either side. This situation is shown schematically in Figs. 4 and 5. PRATTu4) has drawn attention to the fact that configurations in which two electrons are simultaneously excited, one to each of the neighboring cations, may contribute as strongly, or more strongly, to the magnetic interaction as those configurations in which only one oxygen electron is excited from the purely ionic ground state. Others(15s1s) have called atten3. MAGNETIC INTERACTIONS tion to the importance of the configuration of the In order to interpret the magnitude of the empty or partially filled cation orbitals in deterspontaneous magnetization which is developed in mining which of the excited configurations are the some of the compounds containing Corv ions, as more stable. In Fig. 4 the cation eBorbitals, which well as to understand the signs of the paramagnetic point toward the six near-neighbor oxygen ions, are Curie temperatures, it is necessary to review briefly indicated, these being the critical d orbitals in the the theory of indirect magnetic-exchange intersuperexchange modelu’) for magnetic interactions, actions. The double-exchange mechanism of since they overlap the filled oxygen 2p orbital. The ZENER(~~) is not considered relevant, even though hybrid (e,%p3) is considered to have too high an ferromagnetic coupling is developed between two energy to play any role (see Fig. 4(a)). In the semicobalt ions of different ionization, which is just the covalent-exchange modelos) for magnetic intercondition required for the double-exchange mechactions, on the other hand, the hybrid (e,%p3) anism. This mechanism is discounted on the orbitals are assumed to be relatively stable, as following grounds. First, as pointed out by JONKER shown in Fig. 5. The difference between the superand VAN SANTEN,(~) there is no anomaly in the exchange model of KRAMRRS~~)and ANDERSON~~: electrical resistance at the Curie temperature. and the semicovalent-exchange modelus) lies in Second, of the several rocksalt-type and perovskitethe fact that the hybridized orbitals, (e,%p3) in type structures that have been investigated which octahedral sites or (tv83s) and (sp3) in tetrahedral have a transition element in two valence states, only sites, are less than half filled, whereas the eB or the perovskites containing Mr?+-Mn4+ and teg orbitals may be half filled. This difference is Co3'co4+ interactions contain ferromagnetic significant for Co3+, the trivalent cobalt ion of interactions. ANDERSON and HASEGAWA~~) have high-spin state. shown that even in the perovskites containing In Fig. 4 are shown the various magnetic interferromagnetic Mn3+-Mn4+ interactions, the be- actions which are predicted if the hybrid (e,%p3) havior of the paramagnetic susceptibility is in- orbitals are of too high an energy to be relevant to compatible with a double-exchange mechanism. the discussion. From this model the Co3+-Co3+ Finally, it will be shown that a ferromagnetic interaction is unambiguously antiferromagnetic. Co3+-Corv interaction follows from modified The oxygen electrons are perturbed by the elecsuperexchange considerations, so that there is no trons in the near-neighbor cation eB orbitals, the need to have recourse to ZENER’Sdouble-exchange oxygen electron with spin parallel to the net cation mechanism to account for the ferromagnetic spin being excluded from the neighborhood of the coupling. cation e, orbital, provided that there is a low-energy In indirect magnetic-coupling interactions be- excited state available to admix with the ground tween two cations via an intervening oxygen ion, state. If the Co3+ ion on the opposite side of the the origin of the interaction lies in the existence of oxygen ion has a magnetic moment antiparallel to Corv ions tend to order in the (111) planes of lowspin state, an electron interchange being all that is necessary for ordering. It is only as a result of regions of relatively large Sr2+ or La3+ concentrations that mismatch occurs. If z is the fraction of the total cobalt ions which are present as Co3+ ions in the (Ill) planes of lowspin state in a region of perfect ordering of lowspin and high-spin states, all the Corv and Co”’ ions order in the alternate set of (111) planes and 2x N e-r*P. The parameter 5 represents the fraction of total cobalt ions which are Corv, and the expression approximates the boundary conditions 22 = l-l@ for small 5 and z = 0 at 5 = 0.5.
292
JOHN
B. GOODENOUGH -9
(e2gspX-_
_.-Ae2gsp3)
8 -+e2,sp3) _Li!J!_ Q t29-e -u.!-Q9
-etl
@g----
(e2gsp3)
+B
t
(b)
e2gspX...--
, ~ezqsp3)_-..k.-t eg_.&...
(e* sp3)
-9
eg
k!” SP3)
-
eg.
Ie$w3f
_.Lco3’ay-cg)yJj -;rcp_llL2g- eg
2g
J
t*,M_
f 4+3
*2-,
--
\
‘c--w
1
ut2,
‘“8 (cf
te2gSp3L-
-..-.fe2,sp31
-eg it -t2g
Iesp3) P
_!I!_b FE. 4. Superexchange model for indirect magnetic interactions. The dashed arrow in brackets represents a n bonding alternate excited state. (c) and (d) represent, respectively, the competitive Co3+-CoI’ ferromagnetic and a&ferromagnetic interactions.
that of the first Co3+ ion, then two low-energy excited states are avai’lable for admixing, viz. an eg of higher energy but larger over@ CCbond) and a tsB of lower energy but smalfer overlap (W bonds). Since the importance of the excited state varies with the amount of overlap and inversely with the energy difference between it and the ground state,
the relative impurtance of the D and v bonding is not completely unambiguous. However, in the Cos+ case the two excited configurations are cooperative, and no ambiguity arises. Similarly the other oxygen 2~ electron is excluded from the neighborhood of the second Cos+ ion, and its orbital is altered by the admixture of excited states
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
involving the first Co3+ ion. An antiferromagnetic interaction results between the Co3+ ions on either side of the oxygen ion. This is pictured schematically in Fig. 4(a), the dashed arrows enclosed in brackets depicting T bonding and the other dashed arrows indicating IS bonding. A general conclusion which follows from this discussion is that whenever octahedral-site cations are located on opposite sides of a common anion, they interact antiferromagnetically if they have a halffilled eg orbital (e.g. Mn3+, Fez+, Fe3+, Co3+, C03+, Co4+, Nis+). Further, the strength of this interaction increases with decreasing E,,, since the energy difference between the ground and excited configurations decreases with E,,. This trend is experimentally observed in the series of compounds MnO(T, = 122”K), FeO(T, = 198”K), CoO(T, = 291”K), and (NiO(T, = 523°K). The CoIv cation, on the other hand, has a completely empty set of eg orbitals. This means that the oxygen 2p electrons are not excluded from the neighborhood of the cation e0 orbitals; in fact the overlap of the empty e0 orbitals with the full oxygen 2p orbitals means that considerable e bonding can take place via excited configurations in which an oxygen electron is excited into the cation esl orbitals. Because of the energy difference E,, between eg states with spin parallel or antiparallel to the net cation moment, the oxygen electron with parallel spin will predominate in the o bond. However, the tz, orbitals are not completely occupied, so that the oxygen electron with antiparallel spin predominates in the rr bonding, especially as the oxygen electron with parallel spin is excluded from the neighborhood of the tz, orbitals. Since a magnetic interaction across the oxygen ion depends upon the net difference in the stabilities of the excited states with spin parallel and antiparallel to the net cation spin, the unto-operative character of the o and n bonding weakens the magnetic interactions involving cations with a completely empty eg orbital. Although there is a possible ambiguity as to which type of bonding predominates, any interaction between identical cations via an intervening, symmetrically bonded oxygen ion must, by symmetry, be antiferromagnetic, the identical configuration on the opposite side of the oxygen ion being possible only if the electron of opposite spin predominates in the bonding on this side. This is illustrated in Fig. 4(b). The relative magnitudes of the cr and rr
MIXED
CRYSTALS
293
bonding change oppositely with increasing crystalfield splitting A. Therefore Cr3+-Cr3+, Mn4+-Mn4+ and Corv-Corv interactions in a perovskite-type lattice should be antiferromagnetic, and the Cr3+-Cr3+ interaction, which has the smaller crystal field for a given anion sublattice, should be stronger than the Mn4+-Mn4+ interaction if e bonding predominates. This conclusion agrees with the experimental observations forus) CaMnO (T, = 105°K) and t2s) LaCrO3(T, = 300”K), o bonding apparently predominating. Because of the ambiguity, from qualitative arguments alone, as to the relative strengths of the cr and 7r bonding, there is an ambiguity in the sign of the Co3+-Corv interaction. If rr bonding dominates the Corv-Osinteraction, then the Co3+-Corv interaction is antiferromagnetic, as indicated in Fig. 4(d). However, the large overlap of the cation eB and oxygen 2p orbitals argues strongly for the predominance of the c bonding, or a ferromagnetic Co3+-Corv interaction as depicted in Fig. 4(c). This is also the conclusion reached from the experimentally observed magnitudes of the Cr3+-Cr3+ and Mn4+-Mn4f interactions. This conclusion is significant, as the model provides a ferromagnetic C03f-Co~~ interaction without the need to invoke ZENER’S double-exchange mechanism. From this discussion and the observation of ferromagnetic coupling between the trivalent and tetravalent cobalt ions in the lanthanum-strontium cobaltites, it is possible to infer a third general conclusion : if octahedral-site magnetic cations are located on opposite sides of a common anion, they interact ferromagnetically if one cation has completely empty eg orbitals and the other has half-filled es orbitals. In Fig. 5 are shown the various magnetic interactions which are predicted if the hybrid (e,3sp3) orbitals are of sufficiently low energy that c bonding predominates in the magnetic interactions. Since these orbitals are not completely filled, interactions are of the type described above for Corv-Corv interactions, the e bonding predominating. Although this energy-level scheme is compatible with the antiferromagnetic coupling of the Co3+-Co3+ and CO’~‘-CO’~ interactions, it is not compatible with a ferromagnetic Co3+-Corv interaction. Thus the experimental observation of a ferromagnetic CO~+-CO’~ interaction rules out the energy-level-scheme requirement for semicovalent
-e
B
(b)
Fro. 5. ~em~c~~ale~t-e~c~n~e model for indirect magnetic interactions. brackets represents
exchange.*
a ?T bonding afternate excited state. (c) represents ferromagnetic Co”*_CoTv interaction.
It is concluded that t.he modified super-
--_* The arguments in reference (16) remain valid. Howevc~r, empty eg orbitals may replace empty (d2spS) orbitals about Mn4+ ions, empty cL?-.,~ orbitals may replace empty (dsp2) orbitals in the discussion on magnetic coupiing, and the lattice distortians can be accounted for by the fahn-Teller effect, this effect being due to the same symmetry arguments for the d wave functions. The two descriptions are supplementary.
The dashed arrow in a highly improbabte,
exchange model depicted in Fig. 4 is essentially the correct model, provided that due attention is paid to the spatial distributions and occupation of the cation TV and eg orbitals. Finally, it should be noted that the cation Co”’ has the outer-electron configuration tz,e and is diamagnetic. Therefore there can be no magnetic interaction between a Co”I ion and a Cos+ or Co’” ion.
MAGNETIC 4. ~P~I~A~O~
PROPERTl
5.S OF
PEROVSKITE-TYPE
OF THEORY TO THE SYSTEM
The ground work for an interpretation of the systemLa~_,Sr,CoOs_hisnowestablished. Ideally, the cobalt lattice may be separated into two sublattices, each representing the cobalt ions on a set of alternate (111) planes. Thus the chemical formula for the system might be more appropriately written as:
CRYSTALS
29.5
given at the end of Section 2 is therefore amended to: 2x’ N e-r*~+lz(&y)[l-2(y+x’)] 22”N
e-nJ(E-@+
12y[l-2(5-y +s”)].
It is now possible to calculate pee and na as a number of Bohr magnetons per cobalt ion. As a first approximation, it is assumed that the orbital angular momentum is quenched, so that the spectroscopic splitting factor is g = 2. In the case of cobalt this approximation is undoubtedly incorrect, but the errors introduced by taking the spin-only values should be relatively small. From expression
where &(x--2X) represents the fraction of total cobalt ions which are Corv. Since all magnetic interactions are between sublattices A and B, the spins associated with the various cations are denoted symbolically. These orientations follow immediately from the fact that the Cos+-Coa+ and CO~~-C~‘~ interactions are antiferromagnetic, while the CO~+-CO’~ interaction is ferromagnetic. The fraction y, where 0 < y < 05.$, is a measure of the ordering of the Corv ions on the sets of alternate (111) planes depicted as sublattice A or sublattice B. Therefore an ordering parameter v is defined as 77z 1-(2yl,$ such that 0 < 77 < 1. For low values of f there can be no long-range order, as there are not sufficient low-spin-state cations available; in this range of compositions y = 0.55 and 7 = 0. At higher f values, the longrange order is never complete even though only electron transfers among the cobalt ions are involved, since there will always be local regions in the crystal of high Srst- density in which some Corv ions are forced into sublattice A. It is assumed that the crystal field associated with tetravalent cobalt is always sufficiently strong so that this ion is in its low-spin state, Coxv, regardless of its location in the lattice. Finally, in the estimate for Z’ and z”, it is assumed that high-spin-state Coaf is practically always correlated with Cotv; i.e. that the Corv cations have Cosf near neighbors. This is consistent with the concept that a low-spin-state cation stabilizes its near neighbors in a high-spin state; electrostatic considerations minimize the number of Corv-Corv pairs. The expression for z
MIXED
(l),
it follows
,u&spin
only)
=
[24(z’+x”)+3@
(2)
?zg = 4(2’-ax”)+&+
(3)
o-n, 0
that
a-2
0-t
0'3
EFIG. 6. Paramegnetic magnetization ne for
moment
pen and
saturation
as a function of 6, the percentage of cobalt ions that are Corv. Theoretical curvea are calculated from spin-only values of the atomic moments. The parameter 7 3 l-(2yjf) is a measure of the degree of order of Coxv on a given set of alternate (111) planes. Experimental pomts (0 for pee, n for no) are taken from JONKER
and VAN SANTEN.(*)
296
JOHN
B.
GOODENOUGH
In Fig. 6 are pIotted peff and ns as calculated from equations (2) and (3) after the ordering parameter 17has been adjusted to give the best fit with the experimental data. At low Srs+ concentrations, 71= 0 because there is no long-range interaction between ferromagnetic clusters about the Corv ions. As the Co’v concentration increases (t > O.l), long-range order sets in, but the ferromagnetically ordered regions are not everywhere matched. Also, for values of S: just greater than that required for the onset of long-range order, there are clusters which remain magnetically isolated, The parameter v, then, essentially provides a measure of the degree of long-range matching of the ferromagnetic regions, or the degree of ordering of the individual ferromagnetic clusters. The variation of 7) with composition given in Fig. 6 is certainly reasonable. 5. DISCUSSION It is now
possible to discuss the several questions listed in Section 1 which were presented by the experimental data, The small pseudo-cubic-cell edge of LaCoOa_x can be understood as due to the presence of considerahle Co”“. Since it is extremely difficult to prepare stoichiometric LaCoOa and the value of z falls off rapidly with departures from stoichiometry, it is reasonable to assume that in those crystals which have been studied by X-ray diffraction, E N 0.1 and nearly 0.4-0.5 of the cobalt ions were in their low-spin state at room temperature. Further, the ionic size of octahedrally located Corr’ and Corv ions should be simiIar, since in each case the eg orbital which points toward the six nearneighbor anions is completely empty. Since the ratio of Coa* to low-spin-state cobalt ions remains approximately the same all the way across the system for the range of compositions studied by X-ray diffraction, the variation in lattice parameter should be largely determined by the difference in size of the Las+ and Sr*+ ions, which is small. Further, the compound LaNiOa is peculiar because it gave(a) no detectable coherent neutron-diffraction peaks at 4.2°K. Recent measurements in the author’s laboratory indicate that below room temperature LaNiOa has a temperature-independent paramagnetic susceptibility of only 5 i: 10-G c.g.s. units. These data, coupled with definite evidence”“) that Nis+ is in a low-spin state in the system La(Mn, Ni)Oa, suggest that Nis+ is in a
low-spin state in LaNiOs. But even Iow-spin-state Nis+ has one electron in the eg orbital and should therefore have a larger ionic radius than either Coul or CoIv. The curious anomaly in the magnetic susceptibility of LaCoOs.sl at 500°K can be attributed to a variation through this temperature range in the number of trivalent cobalt ions in the low-spin state. Such a change could be induced by the thermal expansion of the lattice. If the energy relations in trivalent cobalt are, as postulated, such that B cx m il, an expansion of the lattice, which would reduce h, could induce transitions of the type Co’rr + Coa+. Such transitions would be in the correct direction to account for the observed anomaly. If this interpretation is correct, however, the Curie-Weiss plot of the magnetic susceptibilities above 500°K should give a peE corresponding to spin-only values for all the cobalt ions in their high-spin state, whereas the Curie-Weiss plot below 500°K should give a lower value for I-L&. Further, if there is a tendency for the Corv ions to be correiated with Co”+ near neighbors, Co’vCorv pairs being minimized, this would be manifest as a greater difficulty to attain more than 50 per cent of the cobalt ions as Corv. However, it is probably possible to force a considerably larger percentage of cobalt ions into the Cor‘l’ state if stronger oxidizing conditions are used in the preparation. The introduction of a Corv into a Cos-I+ matrix produces a ferromagnetic “ciuster” which is isolated magnetically from the rest of the matrix by the diamagnetic Co”’ ions which are next-near neighbors to the Corv ion. In LaCoOa.es, t -_=;O*I and nearly 30 per cent of the cobalt ions are diamagnetic Co”‘. This means that the original Cos-, matrix is completely broken up into paramagnetic clusters, or multiple clusters, so that all long-range magnetic ordering is eliminated. This can account for the lack of magnetic order in the compound studied by neutron diffraction(sJ at 4.2”K. The large paramagnetic background scattering can be attributed to the parnmagnetic clusters. Further, the paramagnetic-susceptibility data(l) on vacuum-prepared SrCoOs.5 give a strongly negative paramagnetic Curie temperature, as is predicted for this lattice since the cobalt matrix is almost entirely Co”. However, as Co’v ions are
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
added to the antiferromagnetic Cos+ matrix, as in the case of LaCoO,_h or Lal_zSr,CoO,_h, the negative paramagnetic Curie temperature must decrease in absolute magnitude, approaching zero at that composition in which the antiferromagnetic Co3+-Co3+ interactions are destroyed and the matrix consists of paramagnetically coupled clusters. The predominantly paramagnetic state should correspond to 4 w 0.1, as was observed.@) In addition, as [ increases (< > 0.1) and the ferromagnetic clusters become coupled into ferromagnetic regions through the replacement of intercluster Co1” ions by CoIv ions, a net moment begins to appear, and the paramagnetic Curie temperature becomes positive since the long-range order is ferromagnetic. It is only at the transition regions of mismatch between ferromagnetic regions that antiferromagnetic coupling between clusters or regions appears. These mismatchings cause a significant decrease in the net magnetic moment. However, the paramagnetic Curie temperature remains positive and the 11~ curve has a predominantly ferromagnetic character because the number of antiferromagnetic interactions within the transition regions represents a very small fraction of the total number of magnetic interactions. Also, the measured magnitudes for peff and nB can be quite adequately accounted for by this analysis, as is indicated in Fig. 6. It should be noted that the calculation for pefl is only weakly dependent upon the value of 77 which is chosen; at 5 = O-4 with71 = 0 a value of pefi = 3.4 is calculated from equation (2). The fit of p&calc.) with experimental data will be as good as that shown in Fig. 6 regardless of the value of 7 in the range 0 < 17 < 1, so that the agreement here does not depend upon any arbitrary parameter. As has already been pointed out, the values of 7 which were chosen to give a best fit for nB are reasonable. Finally, the metallic-like conductivity in ssmples with substantial [ values is reasonable even though the usual conduction mechanism assumes a random distribution of Co3+ and CoIv ions, whereas the model of this paper assumes ordering between low-spin-state and high-spin-state ions; the electrons of the high-spin-state Cos+ are, in the pres-
MIXED
CRYSTALS
297
ence of an externally applied electric field, readily transferred through neighboring 02- p orbitals, across the cation near-neighbors of low-spin state via the empty eg orbitals, through a next-nearneighbor 02-p orbital, to another Cosf + Co4+ --f Co3+ ion. That is, a transfer of an electron from a Cos+ through a Co Iv ion without a change in the arrangement of low-spin-state and high-spin-state cations is envisaged. REFERENCES 1. WATANABEH. j. Whys. Sot. &xzn 12, 515 (1957). 2. JONKERG. H. and VANSANTENJ. H. Physica 19,120 (19.53). 3. KOEHLER W. C. and WOLLAN E. 0. r. Phys. Chem. Solids 2, 100 (1957). 4. ASKHAMF., FANKUCHENI., and WARD R. J. Amer. Chem. Sot. 72, 3799 (1950). 5. WOLD A., POST B., and BANKS E. r. Amer. Chem. sot. 79, 4911 (1957). 6. VERWEY E. J. W. and DEBOERJ. H. Rec. Trav. chim. Pays-Bus 55, 531 (1936). 7. VAN VLECK J. H. Theory of Electric and Magnetic Susceptibilities. Oxford University Press, London (1932). SCHLAPP R. and PENNEY W. G. Phys. Rev. 42, 666 (1932). BETHE H. A. Ann. Phys. 3, 133 (1929). 8. TANABEY. and SUGA~OS. 3. Phis. So;. rapan 9,753, 766 (1954). JORGE&N 6. K. Acta Chem. Stand. 8, 1502 (1954). GRIFFITH J. S. and ORCEL L. E. Trans. Faraday Sot. 53, 601 (1957). 9. COSSEEP. Rec. Trav. chim. Pays-Bus X5,1089 (1956). 10. LOTGERINGF. K. Philips Res. Rep. 11, 337 (1956). 11. JORGENSENK. Solvay Internat. Chem. Inst. 10, 355 (1956). 12. ZENER C. Phys. Rev. 82, 403 (1951). 13. ANDEMONP. W. and HASEGAWAH. Phys. Rev. 100, 675 (1955). 14. PRATT G. W. Jr. Phys. Rev. 97, 926 (1955). 15. GOODENOUGH J. B. and LOEB A. L. Phys. Rev. 98, 391 (1955). 16. GOODENOUGH J. B. Phys. Rev. 100, 564 (1955). 17. ANDERSONP. W. Phys. Rev. 79, 350 (1950). 18. KRAMERSH. A. Physica 1, 182 (1934). 19. WOLLAN E. 0. and KOEHLER W. C. Phys. Rev. 100, 545 (1955). 20. JONKER G. H. Physica 22, 707 (1956). 21. FONER S. Unpublished measurements on a sample prepared by A. WOLD. 22. WOLD A., ARNOTT R., and GOODENOUGHJ. B. Conference on Magnetism and Magnetic Materials, Washington, D.C., Nov. 18-21 (1957); to be published in 7. Appl. Phys.
Solids
Pergamon
Press 1958. Vol. 6. pp. 287-297.
AN INTERPRETATION OF THE MAGNETIC PROPERTIES OF THE PEROVSKITE-TYPE MIXED CRYSTALS Lal_,Sr,Co03_~* JOHN Lincoln (Received
Laboratory,
Massachusetts
3 October 1957;
B. GOODENOUGH Institute
of Technology,
revised 27 November
1957;
Cambridge,
revised 2 January
Mass. 1958)
Abstract-It is pointed out that trivalent cobalt with an octahedral co-ordination of oxygen ions has a low-spin and a high-spin state of comparable energy. The anomalous magnetic properties of the perovskite-type mixed crystals Lal-$r,CoOs-l are interpreted in the light of this fact. It is assumed that the tetravalent cobalt ion is always in the low-spin state in this system. It is shown how the geometry of the perovskite-type lattice lends itself to the formation of two sublattices, each consisting of a set of (111) planes, in one of which the cobalt ions are predominantly in a high-spin state and in the other in a low-spin state. Experiments indicate that the magnetic interaction between trivalent cobalt ions in the high-spin state is antiferromagnetic, whereas the interaction between the low-spin tetravalent cobalt ions and the high-spin trivalent cobalt ions is ferromagnetic. These interactions are considered in relation to modified superexchange and the semicovalent-exchange models. The ferromagnetic CO~+-CO’~ interaction permits a sharp distinction to be made between the two, although there is always some ambiguity in qualitative models. Further, the ferromagnetic interaction can be understood without recourse to ZENER’S double-exchange mechanism. 1. INTRODUCTION THE
magnetic properties of the perovskite-type mixed crystals LaI_,3+Sr,2+Co03_A have been studied by several workers,(l-3) but there has been no satisfactory interpretation of the interesting experimental results. The perovskite-type compounds are of particular interest to the theory of magnetism because the structure is ideal for unambiguous, indirect magnetic-exchange interaction via an intervening oxygen ion. This structure is shown in Fig. 1. As illustrated, the cobalt cations of the system under discussion constitute a simplecubic array with oxygen ions at the centers of the cube edges. An indirect exchange interaction between the magnetic moments of two neighboring cobalt ions must involve the intermediary oxygen ion; this interaction is optimized by the perovskitetype configuration, since the two neighboring cations perturb the same p-orbital of their common oxygen ion. There will probably be no other ex-
FIG. 1. Ideal perovskite-type structure La,-,Sr&oOs.
for the system
change interaction than that between a cobalt ion and its six nearest magnetic neighbors. The interesting properties of this series of mixed crystals are summarized below. Any interpretation of the magnetic interaction must be consistent with all of these properties. ASKHAMet al.@) studied the crystal structure of
* The research in this document was supported by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology. 287
288
JOHN
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GOODENOUGH
the whole series of mixed crystals and found that it is perovskite-tame with a smalf rhombohedral deviation from the cubic form, the distortion decreasing with increasing strontium content up to about 50 per cent strontium, a number of extra lines appearing in the X-ray diagrams at high strontium content. They found only a small variation of the pseudo-cubic-cell axes, from 3.82 A for LaCoOs to 3.84 A for Sr&o&s. Further, the lattice parameter 3.82 A for LaCoOs is to be comp.u-ed with 3.838 A for LaNi03, recently measured by WOLD and BANKS.(~) ?VATANABE~~) has found that the crystal structure of the strontium-Rich compounds is cubic if the compounds are prepared in a stream of oxygen, but is indeterminate if prepared in vacua. JONKER and VAN SANEN(~) reported that it is extremely difficult to obtain the correct oxygen content for a perfect perovskite-type lattice. In LaCoOs there were always some tetravalent cobalt ions present. However, as strontium was added, they were unable to obtain more than 45 per cent of the cobalt as a tetravalent ion, this maximum being attained in a mixture with x = O-6. Strontium cobaltite tends toward the formula SrzCoeO,, WATANABE(~)reported that he could vary the tetravaient cobalt content from 6.3 to 59.3 per cent in SrCoOa_^ and from 11.3 to 61-O per cent in La,.,Sr,.jCoOa_~ by heating in zlacuo or in a stream of oxygen. JONKER and VAN SANTEN(~) also measured the paramagnetic susceptibility, the paramagnetic Curie temperature, and the saturation magnetization below the Curie temperature. In Fig. 6 are shown their measurements, expressed in numbers of Bohr magnetons per cobalt atom, for the saturation magnetization ns and the effective paramagnetic moment peR. The effective ~aramagnetic moment per cobalt ion is considerably less than that predicted from spin-only contributions of Coa+ and Co4+ ions in their high-spin state (see Fig. 6(a)). Further, the paramagnetic Curie temperatures, though negative for small tetravalentcobalt concentrations (indicating an antiferromagnetic Co3+-Co3+ interaction), are positive if more than 10 per cent of the cobalt content is tetravalent. The paramagnetic Curie temperature rises sharply from below -200°K for small tetravafent cobalt concentrations to above room temperature for over 40 per cent tetravalent cobalt.
Although this indicates a predominantly positive coupling between trivalent and tetravalent cobalt ions, the saturation magnetization at the absolute zero of temperature is less than 1.5 PB per cobalt ion. WATANABE(~)found similarly that the perovskite-type compounds which resulted from heating in an oxygen stream had lower effective paramagnetic moments than would be predicted from a spin-only calculation of the Cos+ and Co44 ions in their high-spin state. For x = 0.5, he found h = --0*055 and peff = 3.65, as against a predicted value of 5.52. For x = I, he found h==0*208 and pefP = 3.32, as against a predicted vaIue of 5.50. compounds heated & ‘UCIEUO and of undetermined crystal structure, however, had effective paramagnetic moments which were very close to those predicted from spin-only values of Coz+ and Co”+ in their high-spin states. Further, WATANABE found that the samples treated in vacua had a negative paramagnetic Curie temperature, indicating a predominantly antiferromagnetic interaction, whereas samples heated in oxygen had a spontaneous magnetization below room temperature. KOEHLERand WOLLAN~~)studied the compound LaCoOs.os with neutron diffraction and found no magnetic ordering at temperatures down to 42°K. JO~YKER and VAN S~~~~~(s)observed a discontinuity in the paramagnetic susceptibility/temperature curve in LaCoOa.sl; at about 500°K the susceptibility remains constant over an interval of about loo”, obeying a Curie-Weiss law above and below this region. Finally, electrical-resistivity measurementsus2) indicate that as the amount of tetravalent cobalt is increased, the temperature coefficient of the resistivity goes gradually from a negative value (normal semi-conductor) to a positive vatue. This result is qualitatively in agreement with wellknown ideas on semiconduction in oxidic systems.@) From these facts it is apparent that any interpretation of this system must answer the following questions: (1) Why is the pseudo-cubic-cell edge of LaCoOs_h smaller than that for LaNiOs, and why is its variation with strontium substitution SOsmall? (2) Why is it difficult to obtain more than 50 per cent tetravalent cobalt in the perovskite-type lattice?
MAGNETIC
PROPERTIES
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PEROVSKITE-TYPE
(3) Why are the magnetic interactions in perovskite-type compounds containing both tetravalent and trivalent cobalt predominantly positive? (4) Why are the magnetic interactions in vacuum-prepared SrCoOe.5 (structure undetermined) definitely negative, whereas neutrondiffraction patterns in perovskite-type LaCoOs.os show no magnetic ordering down to temperatures of 4*2”K? (5) What is the origin of the curious anomaly in the magnetic susceptibility of LaCoOs.01 at 500”Kt (6) For the perovskite-type compounds, why are the measured magnitudes of peff and ?zg so low?
MIXED
CRYSTALS
289
Magnetic measurements and optical spectra of compounds and complexes containing trivalent cobalt indicate that if this ion is in an octahedral interstice of oxygeri ions, the exchange splitting and the crystal-field splitting are approximately e,(+f
_s-f
3ais
-&
_*---_--
_ egi*)
t&*1 --I__
-
%I(*) &ex
1_
The purpose of this paper is to show that a satisfactory answer to these questions can be given if one takes account of the stabilities of the lowspin states of trivalent and tetravalent cobalt in octahedral interstices of an oxygen lattice and of the particular geometrical properties of the perovskite-type structure. 2. LOGICAL
FOU~ATI~NS
If a transition-element cation is located in an octahedral interstice of a cubic anion lattice, the electrostatic fields associated with the cations will cause the atomic d level to split into a stable, triply degenerate state t,, (composed of dzyr dy d,,) and an upper, doubly degenerate state eg (composed of d,z_,e and d,2).(7) The magnitude of the splitting A depends upon the particular anion complex and the valency of the cation. Cations of higher valence will be located in a stronger electrostatic field, and the magnitude of A is greater. The two halves of the d level are, as in the atom, split by the intraatomic exchange effects, caused by the Pauli exclusion principle, which favor a parallel alignment of spins within the atom. If the magnitude of this exchange splitting is denoted by E,, then the energy-level diagram for the d orbitals of a transition-element cation in an octahedral anion interstice will be as shown in Fig. 2, where are illustrated the conditions for the occurrence of Co3+ or Co4+, high-spin states with E,, > A, and for Co”’ or Colv, low-spin states with ,?& < A. It should be realized that although the eg orbital is a stable orbital for localized d electrons, a hybrid (ezsp3) orbital may be more stable for nonlocalized electrons which are shared between a cation and an anion. a,
FIG. 2. Schematic energy-level diagram for the 3d orbitaIs of a cobalt cation in a perovskite-type lattice with (a) E,, > A and (b) E,, < 5, where the states eg and tBs are doubly and triply degenerate, respectively.
equal,(*) or Eex M A. This means that secondorder effects may determine whether trivalent cobalt is in a high-spin or a low-spin state. COSSEE(g) has recently performed an elegant series of experiments which demonstrate that trivalent cobalt in the octahedral interstices of the spine1 Co304 are diamagnetic, whereas JONKJZR and VAN SANTE&@) experiments suggest that in the ideally stoichiometric perovskite LaCoOa, the trivalent cobalt ions are in their high-spin state, carrying a magnetic moment nearly that of their spin-only value of 4 1~3 per cobalt ion. Since E, is the same
290
JOHN
B.
GOODENOUGH
in both compounds, it is apparent that the difference must lie in the variation in A in the two instances; the larger A should be associated with the larger crystalline field, or for a given lattice structure and set of ionic charges, with the shorter distance between the positively charged cation and the center of charge density on the negatively charged anion. Paramagnetic-susceptibility data by LOTGERING(~~) on MnCozOa and ZnCozOa (Co-02distances N 2.08 A and N 2.10 A, respectively) indicate that only 5.5 and 15.5 per cent of the cobalt is in a high-spin state. In Co304 the octahedral C0~~‘-02- distance isclo) 2.02 a; in LaCoOs the Cos+-O*distance appears to be I.91 A. The contribution from nearest neighbors to the crystalline field about the trivalent cobalt ion in LaCoOs exceeds that in CosO4, MnCozOa, and ZnCozOd; hence it is reasonable to expect that although trivalent cobalt in LaCoOs is apparently in a high-spin state, in this lattice A e _Ee’,,,so that whether a particular trivalent cobalt remains in the high-spin state or is converted to the low-spin state, may depend upon small changes in the crystalline field. These changes could be induced by thermal expansion or contraction of the lattice or by the introduction of ions of different charge. The introduction of a tetravalent cobalt ion into an octahedral interstice will increase the electrostatic-field splitting probably by as much as 10,000 crn-l;(ll) this should be more than enough to offset anv increase in exchange energy. Therefore, for tetiavalent cobalt in an octahedral site of a perovskite-type lattice, it is reasonable to assume that A > E,, and that Co’v is in a low-spin state with a spin-only magnetic moment of 1 pi. If a CoIv ion is introduced into an ideal LaCoOs lattice, the crystal fields about the neighboring trivalent cobalt ions will be altered in two ways: first, the presence of the higher charge on CoIv will increase the crystalline fields at the neighboring cobalt sites; second, the presence of CO’~ with empty eB orbitals overlapping the filled 2p orbitals of the neighboring 02~ ions will induce some electron sharing, or covalence, in the CoTv-02- bonds. This, in turn, will shift the center of negativecharge density about the 02- ions towards the CoIv ion, so that the crystalline field at the trivalent cobalt ions on the opposite side of the 02- ion is reduced. Therefore it is postulated that, because of the large positive charge, the introduction of a
Co’” ion into the LaCoOs lattice should increase the crystalline field in its environment sufficiently to induce the neighboring trivalent cobalt ions to go into their low-spin state Co”‘. However, the resulting covalent bonding decreases the crystalline fields about the nearest neighbors sufficiently to stabilize these in the high-spin state, so that only next-nearest neighbors are changed to the lowspin state Co III. Again, covalent bonding about the Co”’ will stabilize further the near neighbors of the Colv in a high-spin state, as well as the next-next-near neighbors of the Co”’ ion. It is therefore postulated that the introduction of a Colv ion into LaCoO3 induces the transformations Co3+ -+ Co”’ which are illustrated in Fig. 3. Ill
1
3+
3+
In
1m FIG. 3. Postulated cobalt-ion distribution about a Co’” ion introduced into an ideal La3+Co3+0, lattice. Two features of this diagram are to be noted: first, there is a tendency for the cobalt ions in alternate (111) planes to be in a low-spin state and for those in the other set of (111) planes to be in a high-spin state; second, the introduction of a CoIv ion may be reasonably expected to induce approximately 18 neighboring, trivalent cobalt ions to become Co”‘. Since the Co”’ ions are diamagnetic, the introduction of a Colv ion into a matrix of Co3+ ions causes a cluster consisting of the CoIv ion and its six near-neighbor Cos+ ions to become magnetically isolated from the rest of the lattice. If a sufficient number of CoIv ions are present so that the CoIv ions have near-neighbor or next-near-neighbor Co’v ions, then the additional
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
MIXED
CRYSTALS
291
excited states of the cation-anion-cation configuration. Magnetic interactions cannot be accounted for on the basis of a purely ionic model of the lattice. In the excited configurations, one or both of the electrons in a 2p oxygen orbital are excited into the empty, or partially filled, cation orbitals which overlap it on either side. This situation is shown schematically in Figs. 4 and 5. PRATTu4) has drawn attention to the fact that configurations in which two electrons are simultaneously excited, one to each of the neighboring cations, may contribute as strongly, or more strongly, to the magnetic interaction as those configurations in which only one oxygen electron is excited from the purely ionic ground state. Others(15s1s) have called atten3. MAGNETIC INTERACTIONS tion to the importance of the configuration of the In order to interpret the magnitude of the empty or partially filled cation orbitals in deterspontaneous magnetization which is developed in mining which of the excited configurations are the some of the compounds containing Corv ions, as more stable. In Fig. 4 the cation eBorbitals, which well as to understand the signs of the paramagnetic point toward the six near-neighbor oxygen ions, are Curie temperatures, it is necessary to review briefly indicated, these being the critical d orbitals in the the theory of indirect magnetic-exchange intersuperexchange modelu’) for magnetic interactions, actions. The double-exchange mechanism of since they overlap the filled oxygen 2p orbital. The ZENER(~~) is not considered relevant, even though hybrid (e,%p3) is considered to have too high an ferromagnetic coupling is developed between two energy to play any role (see Fig. 4(a)). In the semicobalt ions of different ionization, which is just the covalent-exchange modelos) for magnetic intercondition required for the double-exchange mechactions, on the other hand, the hybrid (e,%p3) anism. This mechanism is discounted on the orbitals are assumed to be relatively stable, as following grounds. First, as pointed out by JONKER shown in Fig. 5. The difference between the superand VAN SANTEN,(~) there is no anomaly in the exchange model of KRAMRRS~~)and ANDERSON~~: electrical resistance at the Curie temperature. and the semicovalent-exchange modelus) lies in Second, of the several rocksalt-type and perovskitethe fact that the hybridized orbitals, (e,%p3) in type structures that have been investigated which octahedral sites or (tv83s) and (sp3) in tetrahedral have a transition element in two valence states, only sites, are less than half filled, whereas the eB or the perovskites containing Mr?+-Mn4+ and teg orbitals may be half filled. This difference is Co3'co4+ interactions contain ferromagnetic significant for Co3+, the trivalent cobalt ion of interactions. ANDERSON and HASEGAWA~~) have high-spin state. shown that even in the perovskites containing In Fig. 4 are shown the various magnetic interferromagnetic Mn3+-Mn4+ interactions, the be- actions which are predicted if the hybrid (e,%p3) havior of the paramagnetic susceptibility is in- orbitals are of too high an energy to be relevant to compatible with a double-exchange mechanism. the discussion. From this model the Co3+-Co3+ Finally, it will be shown that a ferromagnetic interaction is unambiguously antiferromagnetic. Co3+-Corv interaction follows from modified The oxygen electrons are perturbed by the elecsuperexchange considerations, so that there is no trons in the near-neighbor cation eB orbitals, the need to have recourse to ZENER’Sdouble-exchange oxygen electron with spin parallel to the net cation mechanism to account for the ferromagnetic spin being excluded from the neighborhood of the coupling. cation e, orbital, provided that there is a low-energy In indirect magnetic-coupling interactions be- excited state available to admix with the ground tween two cations via an intervening oxygen ion, state. If the Co3+ ion on the opposite side of the the origin of the interaction lies in the existence of oxygen ion has a magnetic moment antiparallel to Corv ions tend to order in the (111) planes of lowspin state, an electron interchange being all that is necessary for ordering. It is only as a result of regions of relatively large Sr2+ or La3+ concentrations that mismatch occurs. If z is the fraction of the total cobalt ions which are present as Co3+ ions in the (Ill) planes of lowspin state in a region of perfect ordering of lowspin and high-spin states, all the Corv and Co”’ ions order in the alternate set of (111) planes and 2x N e-r*P. The parameter 5 represents the fraction of total cobalt ions which are Corv, and the expression approximates the boundary conditions 22 = l-l@ for small 5 and z = 0 at 5 = 0.5.
292
JOHN
B. GOODENOUGH -9
(e2gspX-_
_.-Ae2gsp3)
8 -+e2,sp3) _Li!J!_ Q t29-e -u.!-Q9
-etl
@g----
(e2gsp3)
+B
t
(b)
e2gspX...--
, ~ezqsp3)_-..k.-t eg_.&...
(e* sp3)
-9
eg
k!” SP3)
-
eg.
Ie$w3f
_.Lco3’ay-cg)yJj -;rcp_llL2g- eg
2g
J
t*,M_
f 4+3
*2-,
--
\
‘c--w
1
ut2,
‘“8 (cf
te2gSp3L-
-..-.fe2,sp31
-eg it -t2g
Iesp3) P
_!I!_b FE. 4. Superexchange model for indirect magnetic interactions. The dashed arrow in brackets represents a n bonding alternate excited state. (c) and (d) represent, respectively, the competitive Co3+-CoI’ ferromagnetic and a&ferromagnetic interactions.
that of the first Co3+ ion, then two low-energy excited states are avai’lable for admixing, viz. an eg of higher energy but larger over@ CCbond) and a tsB of lower energy but smalfer overlap (W bonds). Since the importance of the excited state varies with the amount of overlap and inversely with the energy difference between it and the ground state,
the relative impurtance of the D and v bonding is not completely unambiguous. However, in the Cos+ case the two excited configurations are cooperative, and no ambiguity arises. Similarly the other oxygen 2~ electron is excluded from the neighborhood of the second Cos+ ion, and its orbital is altered by the admixture of excited states
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
involving the first Co3+ ion. An antiferromagnetic interaction results between the Co3+ ions on either side of the oxygen ion. This is pictured schematically in Fig. 4(a), the dashed arrows enclosed in brackets depicting T bonding and the other dashed arrows indicating IS bonding. A general conclusion which follows from this discussion is that whenever octahedral-site cations are located on opposite sides of a common anion, they interact antiferromagnetically if they have a halffilled eg orbital (e.g. Mn3+, Fez+, Fe3+, Co3+, C03+, Co4+, Nis+). Further, the strength of this interaction increases with decreasing E,,, since the energy difference between the ground and excited configurations decreases with E,,. This trend is experimentally observed in the series of compounds MnO(T, = 122”K), FeO(T, = 198”K), CoO(T, = 291”K), and (NiO(T, = 523°K). The CoIv cation, on the other hand, has a completely empty set of eg orbitals. This means that the oxygen 2p electrons are not excluded from the neighborhood of the cation e0 orbitals; in fact the overlap of the empty e0 orbitals with the full oxygen 2p orbitals means that considerable e bonding can take place via excited configurations in which an oxygen electron is excited into the cation esl orbitals. Because of the energy difference E,, between eg states with spin parallel or antiparallel to the net cation moment, the oxygen electron with parallel spin will predominate in the o bond. However, the tz, orbitals are not completely occupied, so that the oxygen electron with antiparallel spin predominates in the rr bonding, especially as the oxygen electron with parallel spin is excluded from the neighborhood of the tz, orbitals. Since a magnetic interaction across the oxygen ion depends upon the net difference in the stabilities of the excited states with spin parallel and antiparallel to the net cation spin, the unto-operative character of the o and n bonding weakens the magnetic interactions involving cations with a completely empty eg orbital. Although there is a possible ambiguity as to which type of bonding predominates, any interaction between identical cations via an intervening, symmetrically bonded oxygen ion must, by symmetry, be antiferromagnetic, the identical configuration on the opposite side of the oxygen ion being possible only if the electron of opposite spin predominates in the bonding on this side. This is illustrated in Fig. 4(b). The relative magnitudes of the cr and rr
MIXED
CRYSTALS
293
bonding change oppositely with increasing crystalfield splitting A. Therefore Cr3+-Cr3+, Mn4+-Mn4+ and Corv-Corv interactions in a perovskite-type lattice should be antiferromagnetic, and the Cr3+-Cr3+ interaction, which has the smaller crystal field for a given anion sublattice, should be stronger than the Mn4+-Mn4+ interaction if e bonding predominates. This conclusion agrees with the experimental observations forus) CaMnO (T, = 105°K) and t2s) LaCrO3(T, = 300”K), o bonding apparently predominating. Because of the ambiguity, from qualitative arguments alone, as to the relative strengths of the cr and 7r bonding, there is an ambiguity in the sign of the Co3+-Corv interaction. If rr bonding dominates the Corv-Osinteraction, then the Co3+-Corv interaction is antiferromagnetic, as indicated in Fig. 4(d). However, the large overlap of the cation eB and oxygen 2p orbitals argues strongly for the predominance of the c bonding, or a ferromagnetic Co3+-Corv interaction as depicted in Fig. 4(c). This is also the conclusion reached from the experimentally observed magnitudes of the Cr3+-Cr3+ and Mn4+-Mn4f interactions. This conclusion is significant, as the model provides a ferromagnetic C03f-Co~~ interaction without the need to invoke ZENER’S double-exchange mechanism. From this discussion and the observation of ferromagnetic coupling between the trivalent and tetravalent cobalt ions in the lanthanum-strontium cobaltites, it is possible to infer a third general conclusion : if octahedral-site magnetic cations are located on opposite sides of a common anion, they interact ferromagnetically if one cation has completely empty eg orbitals and the other has half-filled es orbitals. In Fig. 5 are shown the various magnetic interactions which are predicted if the hybrid (e,3sp3) orbitals are of sufficiently low energy that c bonding predominates in the magnetic interactions. Since these orbitals are not completely filled, interactions are of the type described above for Corv-Corv interactions, the e bonding predominating. Although this energy-level scheme is compatible with the antiferromagnetic coupling of the Co3+-Co3+ and CO’~‘-CO’~ interactions, it is not compatible with a ferromagnetic Co3+-Corv interaction. Thus the experimental observation of a ferromagnetic CO~+-CO’~ interaction rules out the energy-level-scheme requirement for semicovalent
-e
B
(b)
Fro. 5. ~em~c~~ale~t-e~c~n~e model for indirect magnetic interactions. brackets represents
exchange.*
a ?T bonding afternate excited state. (c) represents ferromagnetic Co”*_CoTv interaction.
It is concluded that t.he modified super-
--_* The arguments in reference (16) remain valid. Howevc~r, empty eg orbitals may replace empty (d2spS) orbitals about Mn4+ ions, empty cL?-.,~ orbitals may replace empty (dsp2) orbitals in the discussion on magnetic coupiing, and the lattice distortians can be accounted for by the fahn-Teller effect, this effect being due to the same symmetry arguments for the d wave functions. The two descriptions are supplementary.
The dashed arrow in a highly improbabte,
exchange model depicted in Fig. 4 is essentially the correct model, provided that due attention is paid to the spatial distributions and occupation of the cation TV and eg orbitals. Finally, it should be noted that the cation Co”’ has the outer-electron configuration tz,e and is diamagnetic. Therefore there can be no magnetic interaction between a Co”I ion and a Cos+ or Co’” ion.
MAGNETIC 4. ~P~I~A~O~
PROPERTl
5.S OF
PEROVSKITE-TYPE
OF THEORY TO THE SYSTEM
The ground work for an interpretation of the systemLa~_,Sr,CoOs_hisnowestablished. Ideally, the cobalt lattice may be separated into two sublattices, each representing the cobalt ions on a set of alternate (111) planes. Thus the chemical formula for the system might be more appropriately written as:
CRYSTALS
29.5
given at the end of Section 2 is therefore amended to: 2x’ N e-r*~+lz(&y)[l-2(y+x’)] 22”N
e-nJ(E-@+
12y[l-2(5-y +s”)].
It is now possible to calculate pee and na as a number of Bohr magnetons per cobalt ion. As a first approximation, it is assumed that the orbital angular momentum is quenched, so that the spectroscopic splitting factor is g = 2. In the case of cobalt this approximation is undoubtedly incorrect, but the errors introduced by taking the spin-only values should be relatively small. From expression
where &(x--2X) represents the fraction of total cobalt ions which are Corv. Since all magnetic interactions are between sublattices A and B, the spins associated with the various cations are denoted symbolically. These orientations follow immediately from the fact that the Cos+-Coa+ and CO~~-C~‘~ interactions are antiferromagnetic, while the CO~+-CO’~ interaction is ferromagnetic. The fraction y, where 0 < y < 05.$, is a measure of the ordering of the Corv ions on the sets of alternate (111) planes depicted as sublattice A or sublattice B. Therefore an ordering parameter v is defined as 77z 1-(2yl,$ such that 0 < 77 < 1. For low values of f there can be no long-range order, as there are not sufficient low-spin-state cations available; in this range of compositions y = 0.55 and 7 = 0. At higher f values, the longrange order is never complete even though only electron transfers among the cobalt ions are involved, since there will always be local regions in the crystal of high Srst- density in which some Corv ions are forced into sublattice A. It is assumed that the crystal field associated with tetravalent cobalt is always sufficiently strong so that this ion is in its low-spin state, Coxv, regardless of its location in the lattice. Finally, in the estimate for Z’ and z”, it is assumed that high-spin-state Coaf is practically always correlated with Cotv; i.e. that the Corv cations have Cosf near neighbors. This is consistent with the concept that a low-spin-state cation stabilizes its near neighbors in a high-spin state; electrostatic considerations minimize the number of Corv-Corv pairs. The expression for z
MIXED
(l),
it follows
,u&spin
only)
=
[24(z’+x”)+3@
(2)
?zg = 4(2’-ax”)+&+
(3)
o-n, 0
that
a-2
0-t
0'3
EFIG. 6. Paramegnetic magnetization ne for
moment
pen and
saturation
as a function of 6, the percentage of cobalt ions that are Corv. Theoretical curvea are calculated from spin-only values of the atomic moments. The parameter 7 3 l-(2yjf) is a measure of the degree of order of Coxv on a given set of alternate (111) planes. Experimental pomts (0 for pee, n for no) are taken from JONKER
and VAN SANTEN.(*)
296
JOHN
B.
GOODENOUGH
In Fig. 6 are pIotted peff and ns as calculated from equations (2) and (3) after the ordering parameter 17has been adjusted to give the best fit with the experimental data. At low Srs+ concentrations, 71= 0 because there is no long-range interaction between ferromagnetic clusters about the Corv ions. As the Co’v concentration increases (t > O.l), long-range order sets in, but the ferromagnetically ordered regions are not everywhere matched. Also, for values of S: just greater than that required for the onset of long-range order, there are clusters which remain magnetically isolated, The parameter v, then, essentially provides a measure of the degree of long-range matching of the ferromagnetic regions, or the degree of ordering of the individual ferromagnetic clusters. The variation of 7) with composition given in Fig. 6 is certainly reasonable. 5. DISCUSSION It is now
possible to discuss the several questions listed in Section 1 which were presented by the experimental data, The small pseudo-cubic-cell edge of LaCoOa_x can be understood as due to the presence of considerahle Co”“. Since it is extremely difficult to prepare stoichiometric LaCoOa and the value of z falls off rapidly with departures from stoichiometry, it is reasonable to assume that in those crystals which have been studied by X-ray diffraction, E N 0.1 and nearly 0.4-0.5 of the cobalt ions were in their low-spin state at room temperature. Further, the ionic size of octahedrally located Corr’ and Corv ions should be simiIar, since in each case the eg orbital which points toward the six nearneighbor anions is completely empty. Since the ratio of Coa* to low-spin-state cobalt ions remains approximately the same all the way across the system for the range of compositions studied by X-ray diffraction, the variation in lattice parameter should be largely determined by the difference in size of the Las+ and Sr*+ ions, which is small. Further, the compound LaNiOa is peculiar because it gave(a) no detectable coherent neutron-diffraction peaks at 4.2°K. Recent measurements in the author’s laboratory indicate that below room temperature LaNiOa has a temperature-independent paramagnetic susceptibility of only 5 i: 10-G c.g.s. units. These data, coupled with definite evidence”“) that Nis+ is in a low-spin state in the system La(Mn, Ni)Oa, suggest that Nis+ is in a
low-spin state in LaNiOs. But even Iow-spin-state Nis+ has one electron in the eg orbital and should therefore have a larger ionic radius than either Coul or CoIv. The curious anomaly in the magnetic susceptibility of LaCoOs.sl at 500°K can be attributed to a variation through this temperature range in the number of trivalent cobalt ions in the low-spin state. Such a change could be induced by the thermal expansion of the lattice. If the energy relations in trivalent cobalt are, as postulated, such that B cx m il, an expansion of the lattice, which would reduce h, could induce transitions of the type Co’rr + Coa+. Such transitions would be in the correct direction to account for the observed anomaly. If this interpretation is correct, however, the Curie-Weiss plot of the magnetic susceptibilities above 500°K should give a peE corresponding to spin-only values for all the cobalt ions in their high-spin state, whereas the Curie-Weiss plot below 500°K should give a lower value for I-L&. Further, if there is a tendency for the Corv ions to be correiated with Co”+ near neighbors, Co’vCorv pairs being minimized, this would be manifest as a greater difficulty to attain more than 50 per cent of the cobalt ions as Corv. However, it is probably possible to force a considerably larger percentage of cobalt ions into the Cor‘l’ state if stronger oxidizing conditions are used in the preparation. The introduction of a Corv into a Cos-I+ matrix produces a ferromagnetic “ciuster” which is isolated magnetically from the rest of the matrix by the diamagnetic Co”’ ions which are next-near neighbors to the Corv ion. In LaCoOa.es, t -_=;O*I and nearly 30 per cent of the cobalt ions are diamagnetic Co”‘. This means that the original Cos-, matrix is completely broken up into paramagnetic clusters, or multiple clusters, so that all long-range magnetic ordering is eliminated. This can account for the lack of magnetic order in the compound studied by neutron diffraction(sJ at 4.2”K. The large paramagnetic background scattering can be attributed to the parnmagnetic clusters. Further, the paramagnetic-susceptibility data(l) on vacuum-prepared SrCoOs.5 give a strongly negative paramagnetic Curie temperature, as is predicted for this lattice since the cobalt matrix is almost entirely Co”. However, as Co’v ions are
MAGNETIC
PROPERTIES
OF
PEROVSKITE-TYPE
added to the antiferromagnetic Cos+ matrix, as in the case of LaCoO,_h or Lal_zSr,CoO,_h, the negative paramagnetic Curie temperature must decrease in absolute magnitude, approaching zero at that composition in which the antiferromagnetic Co3+-Co3+ interactions are destroyed and the matrix consists of paramagnetically coupled clusters. The predominantly paramagnetic state should correspond to 4 w 0.1, as was observed.@) In addition, as [ increases (< > 0.1) and the ferromagnetic clusters become coupled into ferromagnetic regions through the replacement of intercluster Co1” ions by CoIv ions, a net moment begins to appear, and the paramagnetic Curie temperature becomes positive since the long-range order is ferromagnetic. It is only at the transition regions of mismatch between ferromagnetic regions that antiferromagnetic coupling between clusters or regions appears. These mismatchings cause a significant decrease in the net magnetic moment. However, the paramagnetic Curie temperature remains positive and the 11~ curve has a predominantly ferromagnetic character because the number of antiferromagnetic interactions within the transition regions represents a very small fraction of the total number of magnetic interactions. Also, the measured magnitudes for peff and nB can be quite adequately accounted for by this analysis, as is indicated in Fig. 6. It should be noted that the calculation for pefl is only weakly dependent upon the value of 77 which is chosen; at 5 = O-4 with71 = 0 a value of pefi = 3.4 is calculated from equation (2). The fit of p&calc.) with experimental data will be as good as that shown in Fig. 6 regardless of the value of 7 in the range 0 < 17 < 1, so that the agreement here does not depend upon any arbitrary parameter. As has already been pointed out, the values of 7 which were chosen to give a best fit for nB are reasonable. Finally, the metallic-like conductivity in ssmples with substantial [ values is reasonable even though the usual conduction mechanism assumes a random distribution of Co3+ and CoIv ions, whereas the model of this paper assumes ordering between low-spin-state and high-spin-state ions; the electrons of the high-spin-state Cos+ are, in the pres-
MIXED
CRYSTALS
297
ence of an externally applied electric field, readily transferred through neighboring 02- p orbitals, across the cation near-neighbors of low-spin state via the empty eg orbitals, through a next-nearneighbor 02-p orbital, to another Cosf + Co4+ --f Co3+ ion. That is, a transfer of an electron from a Cos+ through a Co Iv ion without a change in the arrangement of low-spin-state and high-spin-state cations is envisaged. REFERENCES 1. WATANABEH. j. Whys. Sot. &xzn 12, 515 (1957). 2. JONKERG. H. and VANSANTENJ. H. Physica 19,120 (19.53). 3. KOEHLER W. C. and WOLLAN E. 0. r. Phys. Chem. Solids 2, 100 (1957). 4. ASKHAMF., FANKUCHENI., and WARD R. J. Amer. Chem. Sot. 72, 3799 (1950). 5. WOLD A., POST B., and BANKS E. r. Amer. Chem. sot. 79, 4911 (1957). 6. VERWEY E. J. W. and DEBOERJ. H. Rec. Trav. chim. Pays-Bus 55, 531 (1936). 7. VAN VLECK J. H. Theory of Electric and Magnetic Susceptibilities. Oxford University Press, London (1932). SCHLAPP R. and PENNEY W. G. Phys. Rev. 42, 666 (1932). BETHE H. A. Ann. Phys. 3, 133 (1929). 8. TANABEY. and SUGA~OS. 3. Phis. So;. rapan 9,753, 766 (1954). JORGE&N 6. K. Acta Chem. Stand. 8, 1502 (1954). GRIFFITH J. S. and ORCEL L. E. Trans. Faraday Sot. 53, 601 (1957). 9. COSSEEP. Rec. Trav. chim. Pays-Bus X5,1089 (1956). 10. LOTGERINGF. K. Philips Res. Rep. 11, 337 (1956). 11. JORGENSENK. Solvay Internat. Chem. Inst. 10, 355 (1956). 12. ZENER C. Phys. Rev. 82, 403 (1951). 13. ANDEMONP. W. and HASEGAWAH. Phys. Rev. 100, 675 (1955). 14. PRATT G. W. Jr. Phys. Rev. 97, 926 (1955). 15. GOODENOUGH J. B. and LOEB A. L. Phys. Rev. 98, 391 (1955). 16. GOODENOUGH J. B. Phys. Rev. 100, 564 (1955). 17. ANDERSONP. W. Phys. Rev. 79, 350 (1950). 18. KRAMERSH. A. Physica 1, 182 (1934). 19. WOLLAN E. 0. and KOEHLER W. C. Phys. Rev. 100, 545 (1955). 20. JONKER G. H. Physica 22, 707 (1956). 21. FONER S. Unpublished measurements on a sample prepared by A. WOLD. 22. WOLD A., ARNOTT R., and GOODENOUGHJ. B. Conference on Magnetism and Magnetic Materials, Washington, D.C., Nov. 18-21 (1957); to be published in 7. Appl. Phys.