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J. Phys. Chem. Solids Pergamon Press 1959. Vol. 10. pp. 87-98. SUPEREXCHANGE INTERACTION AND SYMMETRY PROPERTIES OF ELECTRON ORBITALS JUNJIRO KANAMORI* Department of Physics, Osaka University, Osaka, Japan (Received 31 May 1958; revised 6 November 1958) Abstract-The relation between the symmetry of electron orbitals and superexchange interaction is discussed. It is shown that the sign of the superexchange interaction is closely connected with the cation orbital state, when the cation is subject to the crystalline field arising from octahedrally or tetrahedrally surrounding anions. In some cases, the sign of the superexchange interaction is de- finitely determined from the symmetry relations. The cases in which each cation is subject to an octahedra1 cubic field and the lines connecting the interacting cations to the intervening anion make an angle of either 180” or 90” are discussed in particular. Our discussion of the 180” case is applicable to crystals of the perovskite type and NaCl type and that of the 90” case to anhydrous chlorides. In the case where each cation is subject to a tetrahedral cubic field, there is a definite relation between the symmetry of the cation electron orbitals and superexchange interaction, if only the s-orbital of anion participates in the superexchange interaction. TiHs is an example of this case. The interaction between nearest-neighbor cations in the crystal of the NaCl type is also discussed. 1. INTRODUCTION IN 1950 ANDERSON(~) fo rmulated KRAMERS’ id ea(s) of the superexchange interaction in magnetic com- pounds and showed that KRAMERS’ m echanism gives rise to an antiferromagnetic interaction be- tween cations with a more-than-half-filled d-shell and a ferromagnetic interaction between cations with a less-than-half-filled d-shell. His conclusion for the less-than-half-filled d-shell is, however, not always supported experimentally, though it proved to be valid in a few compounds such as CrTe, CrOs, etc., which are ferromagnetic. In the mean- while, various other mechanisms which give rise to antiferromagnetic interaction even in the less-than- half case have been proposed by SLATERR), GOODENOUGH a nd LOMB, GOODENOUGH( and ANDERSON a nd HASEGAWA(@. T he present paper will show that more attention has to be paid to the symmetry relation between the occupied d- orbitals of the cation and thep- or s-orbitals of the anion before it is possible to conclude whether the superexchange mechanism gives rise to ferro- magnetic or antiferromagnetic interaction. In - * Now at the Institute for the Study of Metals, University of Chicago, Chicago 37, Ill., on leave from Osaka University. 87 certain cases it can be shown that the sign of the superexchange interaction is so closely connected with the orbital state of the cation that it does not depend on any specific mechanism conceivable. In the crystal the orbital of the cation and that of the anions overlap each other considerably, so that we cannot generally consider them to be ortho- gonal to each other, except when they have certain different symmetry properties. In the case in which each cation is surrounded octahedrally or tetra- hedrally by anions, there exist definite orthogonal- ity relations between the occupied d-orbitals of the cation and the occupied orbitals of the anion through their different angular dependences. These orthogonalities determine, in the first place, the pair of the d-orbital and anion orbital forming a partial covalent bond and, in the second place, the sign of the exchange integral between the occupied d-orbital and anion orbital. As an illustrative ex- ample of the discussions in the following sections, we shall here present the case of CaMnOs. CaMnOs is an antiferromagnet having the crystal structure of the perovskite type.(v) Each Mn4+ is surrounded octahedrally by Os-, and each Os- is situated at the center of each line connecting two nearest-neighbor Mn4+. Each Mn4f is subject
88 JUNJIRO KANAMORI to a crystalline field of cubic symmetry, in which the lowest state of the free Mn*+, *F, is split into three sublevels. The lowest of these sublevels is non- degenerate, with such an electronic configuration that three 3d+orbitals, d(q), d(y,z), and d(zx), are singly occupied, where yz, etc., represent the angular dependences of the d-orbitals and the co- ordinates refer to the cubic principal axes. It is usually understood that in the octahedral cubic field the &-level, whose wave functions are &cY), 4~4 and 441 ies lower than the &-level whose wave functions are d(xz-~2) and d(2z2- --x2-9). Now, let the anion orbital which par- takes in the superexchange mechanism be the p-orbital whose axis points to one of the Mn*+ (pa-orbital). This pa-orbital does not change its sign when the coordinate axes are rotated about the line connecting these two ions, whereas the oc- cupied de-orbitals of Mn*+ do change their sign (see Fig. 1). Therefore the po-orbital is orthogonal FIG. 1. The symmetry relation between the dc- and PO- orbitals. to the de-orbitals. As for the dy-orbitals, which are unoccupied, we can construct an appropriate linear combination of them in such a way that it does not change their sign under the same rotation; for instance, 42.x2--y2 -2s) does not change sign by rotating the coordinate axes about the x-axis. Therefore, a partial covalent bond involving the pa-orbital can be formed with the &-orbital, i.e. electron transfer from pu can take place to the &orbital, but not to the dc-orbitals. Furthermore, the exchange integrals betweenpu and the occupied da’s are positive (ferromagnetic), as they are mutually orthogonal. ANDERSON’S m echanism is described in the scheme of perturbation theory by virtual processes in which an electron is first transferred from the pa-orbital to one of the neighboring Mn*+ and then the unpaired spin left on Os- couples with the spin of the other neighbor- ing Mn*+ through exchange interaction. It follows from the above consideration that the transfer occurs to the unoccupied dy and, since the ex- change interaction within Mn*+ favors parallel spins, the unpaired spin left on Oz- is antiparallel to the spin of that Mn*+, and this unpaired spin couples ferromagnetically with the spin of the other Mn*+. The resultant superexchange interaction is antiferromagnetic. Other mechanisms, such as those proposed by SLATER@), G OODENOUGH(5),* a nd AHDER~ON a nd HASEGAWA(@ a lso give rise to antiferromagnetic interaction. SLATER’S m echanism takes into ac- count the polarization of the electron cloud of the anion through the exchange interaction with the cations. If the spins of the interacting cations are antiparallel to each other, the electrons with + spin and - spin polarize in opposite directions ito lower the exchange energy with the cations. If the spins of the interacting cations are parallel, such polarizations cannot occur. In the present case, the polarization occurs through a mixing of the 3s- orbital with the 2pa-orbita1.t Since these orbitals are orthogonal to the occupied de, the electron with + spin is pulled toward the cation with + spin and the electron with - spin toward the cation with - spin. GOODENOUGH’S m echanism takes into account the simultaneous partial bond formation of the anion with the cations on both side. If the spins of the cations are antiparallel to each other, the electron in pu with + spin forms the partial bond with the cation with + spin and the electron with - spin with the cation with - spin, while if the spins of the cations are parallel, such a bond formation cannot occur. ANDERSON a nd HASEGAWA have considered such virtual processes in which an electron is first transferred from the po-orbital to * GOODENOUGH discussed in detail the mechanism which was proposed by himself and LoEB(~). I n the following, we call it GOODENOUGH’S m echanism. 7 The mixing of an unoccupied dr- into the po-orbital is also conceivable. However, this mechanism is essenti- ally the same as GOODENOUGH’S m echanism.
INTERACTION AND SYMMETRY PROPERTIES OF ELECTRON ORBITALS 89 one of the neighboring Mn4+ and then two elec- trons are transferred simultaneously, one from the cation back to the po and another from the pu to another Mn4+ and finally the electron is back to the pa. This can be regarded as a kind of simultaneous formation of two partial bonds similar to that con- sidered by GOODENOUGH, a nd therefore gives rise to the same sign of the superexchange interaction as in GOODENOUGH’S m echanism. Thus, it can be concluded that all the mechan- isms of the superexchange interaction hitherto pro- posed give rise to antiferromagnetic interaction in CaMnOs. In the present paper various cases are investi- gated from a point of view similar to that pre- sented above. In some cases we shall be able to determine the sign of the superexchange interaction merely by investigating the symmetry relations among the electron orbitals. Also, we shall be able to obtain some insight into the nature of the partial covalent bond by comparing the results of our qualitative discussion with the experimental results, Recently YAMASHITA and RONDO(~) have for- mulated a theory of the superexchange interaction which takes rigorously into account the effect of the nonorthogonality among the electron orbitals. They show that the nonorthogonality itself can produce superexchange interaction without any configuration mixing, in other words, without any electron transfer. They also show that there are several combined effects of nonorthogonality and electron transfer. These arise because, if a d- orbital and a p-orbital are nonorthogonal to each other, the Pauli principle will produce effective deformations of these orbitals. In the conven- tional Heitler-London model, on which our follow- ing discussion is based, these effects are nothing other than that the exchange integral between nonorthogonal orbitals is negative or, in other words, that parallel spins in nonorthogonal orbitals repel each other.” Since the gist of the present theory is the consideration of the symmetry re- lations among the electron orbitals, the result will be valid even when nonorthogonality is taken into account. In order to simplify the following discussion, we * The effect of nonorthogonality can be interpreted also as an effective electron transfer from anion to cation in some cases.(s) shall define here the sign of the partial covalent bond in ANDERSON’S m echanism. If the transfer of an electron with spin parallel to the spin of the bonding cation is favored, that is, if the unpaired spin left on the anion is favored when it is anti- parallel, we define the sign of the bond to be -. If, in SLATER’S mechanism, the electron cloud with spin parallel to the spin of the cation is pulled towards the latter, the sign of the polarization is -. Conversely, if the partial covalent bond or the polarization produces unpaired spin on the anion parallel to the spin of the cation, the bond or the polarization is +. By so defining the sign, the sign of the superexchange interaction is determined in the following way. In ANDERSON’S m echanism, the sign of the superexchange interaction is the product of the sign of the bond and the sign of the exchange integral between anion and cation; in CaMnOs, the bond is - and the exchange integral is +, so that the resultant is -. In GOODENOUGH’S or SLATER’S mechanism the sign of the superexchange interaction is the product of the signs of the two simultaneous bonds or polarizations multi- plied further by - sign which arises from the Pauli principle within the intervening anion orbital. In Section 2 we shall discuss the case in which each cation is surrounded octahedrally by anions and the lines connecting the interacting cations to the intervening anion make an angle of 180” (180’ case). This case includes the crystals of the perovskite type, NaCl type and ReOs type. In Section 3 will be discussed the case in which each cation is again surrounded octahedrally by anions, but the lines connecting the interacting cations to the intervening anion make an angle of 90” (90 case); it is applicable to anhydrous chlorides. The interaction between the nearest-neighbor cations in the NaCl-type crystal, which also belongs to this case, will be discussed in Section 5. Section 4 will treat the case where each cation is surrounded tetrahedrally by anions, the typical example of such a case being the interaction in TiHs. 2. SUPEREXCHANGE INTERACTION BETWEEN CATIONS IN OCTAHEDRAL SITES: THE 180 CASE The electronic configuration of the lowest orbital state of the cation which is subject to an octahedral cubic field is the following:(s) drl for Tis+; d2 for
90 JUNJIRO KANAMORI Vaf, Cr3+ and Mn4f; dyldes for Crs+ and Mns+; d$dys for Mnsf and Fes+ ; (dG)dcsdy2 for Fes+ ; (dG)dy2 for Nis+; (d&)(dyz)dyl for Cu2+; those in parentheses indicate paired electrons. The con- figuration of the ions of d2 and d7, such as Tis+ and Cos+, cannot be represented by a single one- electron configuration, because the Coulomb inter- action between the electrons causes a configuration mixing. However, in Tisf the configuration da2 and in Cos+ the configuration (d8)dcldy2 amount to 80 N 90 per cent in their respective total wave functions. Besides the po-orbital of the anion, the prr- orbitals and s-orbital can take part in the super- exchange interaction. The psr-orbitals, whose axis is perpendicular to the line connecting the anion and cation, are orthogonal to the dy-orbitals (see Fig. 2), but nonorthogonal to one of the de- orbitals, and therefore can form a partial covalent Y c ,d(X2-Y’i (--/I$ (j ---- y + / + -X 9% - - FIG. 2. The symmetry relation between the dr a nd pr orbit&. bond with the latter. (For instance, p(x) of the anion on the z-axis can form a bond with d(m).) However, this d+m bond should be weaker than the dy-pa bond owing to a smaller overlap. The dy-s bond has the same symmetry property as the dy-po bond, and therefore we shall confine our- selves in the following to the dy-po bond. In fact, we have no mean of clearly distinguishing be- tween them, because an electron transfer can gener- ally occur from an s-p hybridized orbital. Further- more, the 4s and 4~0 cation orbitals can mix to dy and 4~77 to dc. We can disregard all these mixings in the following discussion without affecting any qualitative conclusion. We shall consider the super- exchange interaction between cations of different kinds as well as cations of the same kind. (a) Interaction between the cations of de3 CaMnOs and LaCrOa(10) belong to this case; they are known to exhibit antiferromagnetism. The superexchange interaction via the 2pu-orbital was discussed in Section 1. The sign of the de-prr bond is + according to the definition of Section 1, since all the de’s are occupied. The average exchange integral between the pn and de’s may be negative, because a pr has the largest overlap with a de which is nonorthogonal to it, and moreover the ex- change integral between them is considered to be negative. Thus ANDERSON’S mechanism via the prr-orbital gives rise to antiferromagnetic interac- tion. It can readily be seen that all the other mechanisms also give rise to antiferromagnetic interaction. So, together with the consideration given in Section 1, we can conclude that all the conceivable mechanisms give rise to antiferro- magnetic interaction between ions Mn4+ or Crsf. (b) Interaction between the cations of (des)dy2 This is a typical more-than-half-filled case to which NiO(lr) belongs, Only the dy-pa bond can take part in the superexchange interaction. The sign of this bond is +, while the average exchange integral between pa and dy’s is negative. Thus ANDERSON’S mechanism gives rise to antiferro- magnetic interaction. Other mechanisms also give rise to antiferromagnetic interaction. (c) Interaction between the cations of drsdy2 LaFeOs(10) and MnO(ll) belong to this case; they are both antiferromagnetic. In this case, the superexchange interaction via the pa-orbital is antiferromagnetic for all conceivable mechanisms. The dE-p+ bonds can also take part in the super- exchange interaction, but there is an ambiguity in the sign of the average exchange integral between per and 3d’s. The exchange integral between non- orthogonal pr and de will be negative, but other pn-dc exchange integrals, as well as pr-dy ex- change integrals, are positive. Thus the sign of the superexchange interaction arising from ANDER- SON’S mechanism via prr is uncertain. Other mech- anisms give rise to antiferromagnetic interaction. However, since the dy-pu bond is stronger than
INTERACTION AND SYMMETRY PROPERTIES OF ELECTRON ORBITALS 91 the de-pn bond, the total superexchange inter- action will nevertheless be antiferromagnetic. (d) In~action between Niz+((d@)dy2) and Vs+(des) Though there is no observation for this case, we expect here a ferromagnetic interaction. We shall attach a prime to the 3d-orbitals of Vs+ and take Os- as the intervening anion. In Table 1 are listed Table 1. 2%~ 180” int~~~i~ better Nia+ and V2+ / dy-pa bond , J(P@-v2+) / A -I- + + dy’-pa bond J(pa-Niz+) A -____-~-- - - + ----~~2~ ~-- G and A-H ~-___ + i L.__j + -/ de’-prr bond J(pn-Ni2+) / ! A + / -/ I + I+ A =ANDERSON’S m echanism, G = GOODENOUGH’S mech- anism, A-H =ANDERSON a nd HASEGAWA’S me chanism, S=SLATER’S m echanism, J(pu-V2+) is the average ex- change integral between po and V2+. the sign of the bond, the sign of the average ex- change integral between cation and anion, the sign of the polarizations in SLATER’S m echanism, and finally the sign of the resultant superexchange interaction in each mechanism. The sign of the resultant superexchange interaction can be ob- tained by multiplying the signs in the same row for ANDERSON’S m echanism, by multiplying the signs in the upper part of the same column and then attaching - for GOODENOUGH’S, ANDERSON and HASEGAWA’S an d SLATER’S m echanisms. In SLATER’S mechanism, the signs of the exchange polarizations are just the reverse of those of J’s in the upper part of the same column. As shown in Table 1, all the mechanisms give rise to ferromagnetic interaction. (e) Interaction between Fes+(d~s~~s) and Cr3+(dcr3) This case occurs in mixed crystals, such as LaCr,Fel_zOa. It is known(rs) that the para- magnetic Curie temperature, 0, defined by x = C/( T+O), is smaller in this mixed crystal than in either LaFeOa or LaCrOa. This suggests that the interaction between Fes+ and Crs+ is ferromagnetic. As for the superexchange interaction via pu, we can draw the same conclusion as that which we reached in case (d); it is ferromagnetic. The sign of the d~-prr bond is positive for both the cations, while the average exchange integral between pi and Crsf is negative and that between prr and Fea+ is uncertain. Thus GOODENOUGH’S an d ANDERSON and HASEGAWA’S m echanisms via prr give rise to antiferromagnetic interaction, while the sign in ANDERSON’S m echanism is uncertain. The ferro- magnetic interaction between Fes+ and Crs+ may be explained by considering that the po-dy bonds are more effective than the dE-pw bonds. (f) Interaction between the cations of dyldc3 This case, as well as that in which Cu2+((d@) x x (dy2)dyl) is involved, is complicated because the lowest level in the cubic field is doubly degenerate. GOODENOUGH@) d iscussed in detail the case of LaMnOs, in which he considered that each Mns~t forms a square covalent bond with the four surrounding Os- and the plane of this square bond alternates between being parallel to the zx-plane and parallel to the yz-plane when one goes along any one of the cubic axes. The square bond in the XX- ( oryx-) plane can be formed by constructing the hybridized orbitals from d(z2-S)(or d($--x2)) and 4s, 4p(z>, and 4~(~)(or 4~(~)). The four 3d-electrons occupy the remaining four 3d-orbitals. Along the x-direction, the intervening Os- forms simultaneous bonds with the cations on both its side, and therefore GOODENOUGH’S m echanism gives rise to antiferro- magnetic interaction. In the X- or y-direction, the intervening Os- forms a covalent bond with only one cation; therefore GOODENOUGH a ssumed that in these directions ANDERSON’S m echanism would give rise to ferromagnetic interaction. This assumption does not conflict with our own picture: the sign of the covalent bond, say in the x-direc- tion, is - and the exchange integral between PO and the non-bonding cation is negative, because the singly occupied dy of this cation has the largest density in the x-direction.
92 JUNJIRO KANAMORI (g) Interaction in Fe0 and CoO(l3) In the above cases (a)-(f), interactions involving Fea+(ds) and Co2+(d7) have not been included. In the lowest orbital state of Fez+, only one of the three dc is doubly occupied and in that of Coa+ approximately two of the three de are doubly occupied. These lowest states are triply degenerate. If the po-dy bond only is considered in these cases, one will come to the same conclusion as that of the case of Nia+. However, the existence of a pr-dc bond makes the situation somewhat complicated. Also, compared with LaMnOs, in which the lowest state of Mns+ is doubly degenerate and hence the Jahn-Teller effect determines its orbital state as implied by GOODENOUGH’S theory, the present case is different in that the dr-pr bond is weaker than the dy-pu bond and that there is a strong spin- orbit coupling which was absent in LaMnOs. In FeO, the spins orient to the [l 111 direction in the antiferromagnetic state, and the spin-orbit coup- ling determines the orbital state in such a way that the orbital electronic current produces a magnetic field which points in the direction of the spin, i.e. the wave function of an electron outside the half- filled d-shell is a linear combination of the three de-orbitals: d(zx)+exp(4+3) d(xy)+exp(2&/3)+ +d(yz). Thus the extra one electron occupies the three de’s equally. Consequently, there is no differ- ence among the superexchange interactions along the x-, y- and z-directions. In COO, the spins orient approximately to [OOl], and the spin-orbit coupling energy deter- mines the orbital of the only unpaired dc-electron approximately as l/dL!(d(zx)+id(yx)). Therefore the electrons of the two pn-orbitals of the Os- can be transferred into the cation which is next to this Oa- in the z-direction, while in the x- or y-direction the transfer into the cation can occur only from one of the two pm orbitals. Thus one expects a differ- ence between the superexchange interaction in the z-direction and that in the x- or y-direction. In the paramagnetic state of Fe0 and COO, the three dt are occupied with equal probability, and there can be no anisotropy of the superexchange interaction. The same situation will occur in the case in which Tis+, V3f, and Tia+ are involved. To conclude the above discussion for the 180” case, we may say that we expect generally antiferro- magnetic interaction between cations of the same kind, except in the case of cations of d4, in which case the sign of the superexchange interaction depends on the direction of the line of superex- change, and ferromagnetic interaction between a cation with more-than-half-filled d-shell and a cation with less-than-half-filled d-shell. Table 2 summarizes the discussion of this section. 3. SUPEREXCHANGE INTERACTION BETWEEN CATIONS IN OCTAHEDRAL SITES: THE 90” CASE In this section we treat the case in which the two lines connecting the interacting cations to the intervening anion are at right angles and the cations are in an octahedral cubic field. Interactions within the same layer of a crystal of layer structure, such as the anhydrous chlorides of the iron-group elements, belong to the present type. Interactions between nearest-neighbor cations in crystals of the NaCl type are also of the same type, but we defer consideration of them to Section 5. In the anhydrous chlorides of the iron-group elements, the anions form a cubic close-packed structure whose [ill] axis coincides with the trigonal axis of the whole crystal, and the cations occupy the alternate (1 lljinterstitial planes, form- ing a set of two-dimensional triangular lattices in the case of dichlorides and of honeycomb lattices in the case of the trichlorides. Strictly speaking, the cations are subject to a crystalline field of trigonal symmetry, but the six anions adjoining each cation form an almost regular octahedron, and the octahedral cubic field arising from them determines the main orbital configuration of the cation. It is known experimentally that FeCla, CoCl2, and NiCIa are antiferromagnets whose spin struc- ture is such that the spins within each layer are all parallel and the spins of the neighboring layers antiparallel.(l4) This structure itself suggests that cations in the same layer interact ferromagnetically with each ether and cations in the adjacent layers interact antiferromagnetically. The magnetic prop- erties of these compounds are well explained by assuming that the intra-layer ferromagnetic inter- action is much stronger than the inter-layer anti- ferromagnetic interaction. We list their NPel temperature T,v and paramagnetic Curie tempera- ture 0 in Table 3. It can be considered that both TN and 0 are determined mainly by the
INTERACTION AND SYMMETRY PROPERTIES OF ELECTRON ORBITALS 93 Number o electrons interacti cation! ----_ d3-d” d8-d8 d5-d5 d8-d3 d5-d3 ---___ d4-d4 d6-ds ____ d7-d7 f .Id- : of ng 5 -___ ____ Table 2. The 180” interaction between cations in octahedral sites Specie8 of interacting cation8 Mn4+-Mn4+- Cr3 ‘-Cr3+ Ni2f_NiZt Mn2+_Mn21~ Fe3+-Fe3+ NiZt_V2+ Fe3 kpCr3 1 Mn3+-Mn3+ Fe0 coo -- Relevant bond and mechanism ______ u-bond and T-bond A, G, A-H, S u-bond A, G, A-H, S _ -___ o-bond A, G, A-H, S n-bond G, A-H, S n-bond A ______ o-bond and n-bond A, G, A-H, S o-bond A, G, A-H, S n-bond G, A-H r-bond A, S _-__ * o-bond A, G, A-H, S T-bond u-bond A, G, A-H, S n-bond * Depends on the direction of the line of superexchange. f Weak, but dependent on the direction of the line of superexchange. Resultant superexchange interaction -__-__ -~ Antiferro. Antiferro. Antiferro. Antiferro. (weak) Uncertain (weak) --__ Ferro. -___.______ Ferro. Antiferro. (weak) Uncertain (weak) -~ --____ Antiferro. U ncertain (weak) ____ Antiferro. t Total super- exchange interaction Antiferro. Antiferro. -~ Antiferro. Ferro. Ferro. Antiferro. Antiferro. A=ANDERSON’S mechanism, G = GOODENOUGH’s m echanism, A-H = ANDERSON a nd HASEGAWA’S m echanism, S = SLATER’S m echanism intra-layer ferromagnetic interaction, except for a cekain modification in the case of FeC12. The prop- erties of FeClz will be discussed elsewhere(l5) in detail, but here it is noted that the large anisotropy energy in the case of FeClz enhances its TN twice (or more) as much as the value that one would expect if it were absent; also, it can be shown that 0 is affected considerably by the anisotropy energy. Thus, although FeClz and CoClz have approxi- mately the same value of TX, the ferromagnetic intra-layer interaction in the former is actually smaller than in the latter. In MnC1#4J@ and FeCla,(ls) the cations in the same layer do not appear to interact ferromagnetically. From these facts, we can infer that the intra-layer ferro- magnetic interaction decreases with decreasing number of S&electrons. Mn-, Fe-, Co- and Ni-TiOs have the same crystal structure, which is similar to that of an- hydrous chlorides. They become antiferromagnetic at low temperatures. In FeTiOa the ferrous ions form an antiferromagnetic structure which is the
94 JUNJIRO KANAMORI Table 3. TN and 0 of anhydrous chZorides(l@ Substance T,v(“K) vc12 CC13 CrClz MnClz F&13 FeClz CoClz NiClz 7 16.8 40 -2 23.5 24.9 49.6 -~ 5 65 27.0 -149 ~~ 3.3 ~ 11.5 48.0 or 20.4* 38.1 68.2 * WOLTIJER H. R. and WIEHSMA E . C., Proc. Roy. Acad. Amsterdam 32, 736 (1929). same as that in FeCl2, according to an observation made by neutron diffraction.(l7) From magnetic measurements ISHIKAWA and AKIMOTO(‘S) have inferred that in Co-, Ni- and Fe-TiOa the ex- change interaction within each layer may be ferro- magnetic, but antiferromagnetic in MnTiOa. Thus this titanate series resembles the anhydrous chlorides. However, since there are no available data for their TN values, except that of FeTiOa (TN = 68”K),(19) and since their anion octahedra are considerably deformed, we shall not consider them further but will rather confine ourselves to the anhydrous chlorides in the following. We shall include those anhydrous chlorides whose cations have a less-than-half-filled d-shell, such as CrC13, CrC12 and VC12. Among them, CrCla is ferro- magnetic,(l6) but the other two seem to be anti- ferromagnetic, though this is not yet certain. We define the pa-orbital as the p-orbital which points to the cation situated to the left of the anion in Fig. 3 and the po’-orbital as the p-orbital which points to the cation situated to the right. We attach correspondingly a prime to the d-orbitals of the cation to the right. The pn-orbital is the one which is perpendicular to the plane containing the anion and the interacting cations. (a) The 90” interaction between Ni”+ ions In this case pu forms a partial covalent with dy of the cation on the left hand and the po’ with the dy’ of the cation on the right hand. The sign is + for these bonds, that is, the unpaired spin produced on the anion is parallel to the spin of the cation. Since pa is orthogonal to dy’ and pu’ is orthogonal to dy, the exchange interaction between the unpaired spin on the anion and that of the cation is ferromagnetic. Thus ANDERSON’S mechanism gives rise to ferromagnetic interaction. On the other hand, ANDERSON mechanism via the 3s- orbital of Cl- give rise to antiferromagnetic inter- action. Other mechanisms, such as GOODENOUGH’S mechanism in 180” case, cannot produce super- exchange interaction unless they are modified. If we treat them in the perturbation scheme, the superexchange interaction will be obtained in a higher order than in the 180” case. The higher- order mechanism which would correspond to GOODENOUGH’S m echanism is such that two partial covalent bonds, pa-dy and pa’-dy’, are simultane- ously formed and the unpaired spins produced on pa and pu’ interact with each other. Since pa and pu’ are orthogonal to each other, the exchange interaction between them is ferromagnetic, and this mechanism gives rise to a ferromagnetic inter- action. What corresponds to SLATER’S mechanism is more complicated. The pa electrons whose spin is parallel to the spin of the cation on the left hand mixes with the 4s-orbital of the anion through ex- change repulsion, and since this 4s-orbital is non- orthogonal to the dy’-orbital of the cation on the right hand, this mixing will increase the exchange energy between the unpaired spin in 4s and the spin in dy’ if they are parallel. Thus it gives rise to antiferromagnetic interaction. Another possibility is that the simultaneous exchange polarization of PO and pu’ occurs, and the Pauli principle works in FIG. 3. The p- and d-orbit& in the 90” case.
INTERACTION A?:D SYMMETRY PROPERTIES OF ELECTRON ORBITALS 95 the 4s-orbital favoring antiparallel spins of the cations. However, if we take into account the trigonality of crystal, there is another mechanism which will give rise to ferromagnetic interaction. The p-level of the anion is split in the trigonal field into two, one with the orbital (l/~&)@(x)+ +p@)+p(a)) and the other with (l/&)@(x)-p(~j)) and (1/2/6)(2p(z) -&x)-p(r)), the coordinates re- ferring to the coordinate axes parallel to the lines connecting the anion to the three neighboring cations. In the anhydrous chlorides, there is a potential gradient at the site of the anion, which will polarize the electron cloud towards the cations. This polarization takes place through the mixing of the 4s-orbital with the singlet orbital, (l/%%)@(x)+ +p(r)+p(z)). Thus the singlet orbital will have a larger overlap with the electron clouds of the ca- tions. This makes the exchange polarization of the singlet orbital more effective than that of the doublet orbitals. If the spins of the three cations are all parallel, the energy due to the exchange polarization would be the lowest, since the singlet orbital overlaps them equally. We cannot determine without quantitative cal- culations which one of the above mechanisms is the most effective. However, as will be discussed below in (b), the decreasing ferromagnetic interac- tion with decreasing number of the 3d-electrons indicates that ANDERSON’S mechanism is the most effective. It is, howe’ver, not excluded that the ferromagnetic interaction is a result of the can- cellation of the various effects. (b) The 90” interaction between Mns+ or Fea+ In this case, the de-orbitals can take part in the superexchange interaction. In ANDERSON’S mech- anism, the exchange interaction between the un- paired spin in po and the cation on the right hand will be smaller than that in Nis+, or even reversed in sign, as pa is nonorthogonal to the dc’-orbitals which are now singly occupied. Also pu can form a partial covalent bond with dr’; ANDERSON’S mech- anism via this bond gives rise to antiferromagnetic interaction, since the exchange integral between pa and dy is definitely negative. As GOODENOUGH’S mechanism, other than that discussed in the case of Ni2+, there is such a one that simultaneous bonds are formed between pu and dy on the left, and dc’ on the right; this gives rise to antiferromagnetic interaction, Also prr can form partial covalent bonds with de and dc’ and participate in the super- exchange interaction, which is also antiferro- magnetic. Thus, if any of the pu-dy’, p+de, and prr-de’ bonds are important in ANDERSON’S and GOODE- NOUGH’S mechanisms, the antiferromagnetic inter- action due to them will partly cancel the ferro- magnetic interaction arising from the pa-dy or pu’-dy’ bond, and therefore the total super- exchange interaction will decrease with decreasing number of Sd-electrons. On the other hand, if only pa-dy and pa’-dy’ are important, ANDERSON’S mechanism should be most effective, and since the exchange integral decreases with increasing num- ber of unpaired electrons in dr’ and de, the super- exchange interaction here again decreases. In SLATER’S mechanism, the exchange polariza- tion will be smaller compared with that in Nis+, since there are unpaired spins in the de-orbitals and they pull the 4+electron with parallel spin, whereas the spin in the dy-orbital repels it. (c) The 90” interaction between Crsf First we investigate the interaction which in- volves pa-dy and/or pu’-dy’. The sign of these bonds is negative. The exchange interaction between pu and the occupied de’ is negative, since PO is nonorthogonal to de’. Thus ANDERSON’S mechanism gives rise to ferromagnetic interaction. GOODENOUGH’S mechanism, in which both pa-dy and pa‘-dy’ are simultaneously formed, also gives rise to ferromagnetic interaction. SLATER’S mech- anism also gives rise to ferromagnetic interaction, since the occupied de’s are orthogonal to the 4s- orbital of the anion. If pa-de’ or pu’-de is considered, ANDERSON’S mechanism pertaining to it gives rise to ferro- magnetic interaction, since the sign of these bonds is positive and the exchange integral between pa and de or pu’ and d& is positive. Simultaneous bond formation of pa-dy and pa-de’ gives rise also to ferromagnetic interaction, since the sign of the former is - and the sign of the latter is +. Anti- ferromagnetic interactions can be obtained only if pr-dc and 3s-dy are considered; however, since in our discussion of the anhydrous chlorides we could do without these bonds, it appears that they make little contribution to the superexchange interac- tion.
96 JUNJIRO KANAMORI CrCls and VCls are suspected to be antiferro- .magnets. In the former the Jahn-Teller effect, such as occurs in LaMnOs, will play an important role in the superexchange interaction, since there is a doubly degenerate lowest orbital state. We shall not enter into the details of this case. As for VCls, we might expect ferromagnetism, since Vs+ is iso- electronic with Crs+. However, a very large nega- tive paramagnetic Curie temperature is observed, for which we have no explanation, (d) The 90” interaction between Nis+ and Vsf Contrary to the 180” case, we expect antiferro- magnetic interaction between a cation with a less- than-half-filled d-shell and a cation with a more- than-half-filled d-shell in the 90” case. Here we shall discuss only the case of Nis+-Vs+. In Table 4 the signs of the bonds, the exchange integrals and the resultant superexchange interacticn in ANDER- SON’S mechanism are listed. Attaching a prime to the Sd-orbitals of Vs+, we see that ANDERSON’S mechanism gives rise to antiferromagnetic inter- action. By a similar consideration it can be shown Table 4. The 90” interaction between Nisf and V”+ .I 1 -: pa-dy bond J(po-V2+) A 4 _ pa’-dy’ bond I J(pu’-Ni”+) A - + pa-d<’ bond J(po-Nia+) A + . // - A=AIWERSON’S mechanism. that GOODENOUGH’S mechanism gives rise to anti- ferromagnetic interaction. For SLATER’S mechan- ism, it is hard to conclude the sign definitely. We may assume that ANDERSON’S mechanism is most effective and the superexchange interaction is anti- ferromagnetic. Table 5 summarizes the discussion of this section. Number of 3d- electrons of interacting cations da-d8 Table 5. The 90” interaction between cations in octahedral sites Species of interacting cations Relevant bond and mechanism I / Total super- Resultant superexchange i exchange interaction interaction Ni2 I _Ni2+ pa-dy bond I A, G I Ferro. s 1 Uncertain Ferro. s-dy bond A, G, A-H, S Antiferro. ~ d5-d5 Mna+-Mna+ ~ Uncertain ~ * 1 Fe3+-Fe3+ d3-d3 cr3+_cr3+ pa-dy and pa-de’ A, G, A-H, S Ferro. pn-dr and s-dy A, G, A-H, S Antiferro. (weak) -___ -~ -__- da-d3 Ni2 i_V2+ pa-dy and po-dc’ A, G, A-H Antiferro. S Uncertain (weak) pn-dr’ and s-dy, s-dy’ A, G, ApEI Ferro. (weak) * Tendency towards antiferromagnetic interaction with decreasing number of 3d-electrons. Ferro. Antiferro.
INTERACTION AND SYMMETRY PROPERTIES OF ELECTRON ORBITALS 97 4. SUPEREXCHANGE INTERACTION BETWEEN CATIONS IN TETRAHEDRAL SITES The discussion in this section applies to crystals of the ZnS type, such as ,!I-MnS, and crystals of the CaFs type, such as TiHs.(s@ The orbital level splitting of the cation in a tetrahedral cubic field is such that the &-level lies lower than the &level. Therefore, the electronic configuration of Tisf is dys and that of Cos+ (dy4)d3. The cubic axes and the positions of the surrounding anions are shown in Fig. 4. FIG. 4. The cubic axes and the positions of the surround- ing anions in the tetrahedral case. The symmetry relations among the orbitals are such that the pu- and s-orbitals of the anion are orthogonal to the dy-orbital and nonorthogonal to an appropriate linear combination of the &- orbitals. For d(&‘-y2), this orthogonality is easily understood, since if we rotate the coordinate’axea about the line connecting the anion to the cation, d(x2-y2) changes its sign, while the wave func- tions of pu and s do not change sign. As for d(2$-x2-y2), we can write it as a linear com- bination of d(.S- a?-)‘and d(y2-X2), both of which are orthogonal to pa and s. Unfortunately, however, we cannot give in this case such a simple qualitative discussion as in the octahedral case if the p-orbitals of the anion take part in the superexchange interaction. This is be- cause the po-orbital referred to one cation has no definite symmetry relation with another cation. In a particular crystal of TiHs, in which only the s- orbital takes part in the superexchange interaction, the situation is simple, and we can determine the sign of the superexchange interaction. TiHs has a crystal structure of the CaFs type,(sr) in which each Tis+ is situated at the body-center site of the cube made of tight H-. Since the two 3d-electrons of Tis+ occupy the dy-orbitals, the s-orbital of H- is orthogonal to the occupied cation orbitals. Thus this case resembles that previously discussed for CaMnOa except that s replaces po. The discussion proceeds similarly, therefore, and we can conclude that all the mechanisms give rise to antiferromagnetic interaction in TiHs. In actuality ‘i’iHs is an antiferromagnet.(ss) The interaction between Tia+ and Cos+ in the mixed crystal, Tir_&oZHs, is ferromagnetic, since in Cos+ the unpaired electrons cccupy the three dc-orbitals which are nonorthogonal to the s- orbital of H-, and therefore the interaction be- tween Tis+ and Co”+ corresponds to that between Nis+ and Vs+ in octahedral sites. Furthermore, we can expect that the interaction between Tis+ and a cation with a more-than-half-filled d-shell is always ferromagnetic in hydrides (see the discus- sion of Cr3+-Fe3+ interaction in the octahedral case). 5. FURTHER DISCUSSION In Section 3, the sign of the superexchange interaction within a layer in the anhydrous chlor- ides was discussed qualitatively. Another example of the 90” case is the exchange interaction between the nearest-neighbor cations in a crystal of the NaCl type. We list in Table 6 the experimental Table 6. TN and 0 of crystals of the NaCl type(11,22,23) Substance TN( “I<) MnO 122 a-MnS 165&, 154b coo 293 NiO 540 0(x = C/(T+@))(“K) 610 528&, 300” 280 2000 a-Ref. (11). b-Ref. (23). valurs of TN and 0 of these crystals. As each cation is surrounded by an equal number of + and - spins, TN is determined by the 180” interaction between the next-nearest neighbors, while the paramagnetic Curie temperature 0 is determined by the total exchange interactions including both
98 JUNJIRO KANAMORI the 180” and 90” interactions. Thus in MnO and a-MnS, for which 0 > TN, the 90” interaction is antiferromagnetic. For COO, the present author has shown previously(ls) that the value of 0 is strongly influenced by spin-orbit coupling which acts through the residual orbital angular momen- tum and that the 90” interaction is still antiferro- magnetic. The magnitude of the 90” interaction in Co0 was found to be about the same as in MnO For Fe0 there are no reliable experimental data In NiO, SINGER@~) o btained a very high 0, but the measurements were not extended to temperatures high enough to eliminate the short-range effect. Thus, at present we cannot derive from experi- mental data any definite correlation between the magnitude of the 90” interaction with the number of 3d-electrons. The absolute magnitude of this interaction is very large as compared with the corresponding interaction in anhydrous chlorides. Comparing a-MnS with MnO, we find that 0 of MnO is larger than that of a-MnS, while the values of TN are in the reversed relation.* This is also hard to understand, since the superexchange intcr- action is expected to increase with the decrease of the electronegativity of the intervening anion. problem might be obtained, if the electron transfer process in magnetitie should be theoretically in- vestigated. Difference in the contribution from the s-orbitals may also be conceivable. The trigonality of the crystal of anhydrous chlorides discussed in Section 3 would be another cause of the difference. Possible mechanisms which might give rise to the large difference of the 90” interaction in monoxides and anhydrous chlorides would be the following. One is the direct interaction between the cations, which will depend on the distance between the cations; the distance is larger in anhydrous chlorides than in oxides. Electron transfer be- tween the cations is also conceivable. This is the mixing of the excited configuration M+-MS+- into the ground configuration M2+-M2+. If the spins of the cations are antiparallel to each other, the un- paired electrons can move from a cation to another, while if they are parallel, this cannot occur. Thus this will give rise to antiferromagnetic interaction. The excitation energy required for this process is about 10 eV in monoxides. With a transfer integral of the order of 1000 cm-l, this mechanism gives the interaction of the right order. A clue to this *According to LINDSAY and BANEWIC~‘S measure- ment,(z3) the susceptibility of a-MnS does not follow the Curie-Weiss law even in the temperature range above room temperature. They estimated the value of 0 at the high temperature limit from the Ising-Bethe approxima- tion (see Table 6). Ackmxvledgentent-The author would like to express his sincere thanks to Professor NAGAMIYA f or valuable dis- cussions and to Professor STOUT for helpful comments on the manuscript. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. REFERENCES ANDERSON P . W., Phys. Rev. 79, 350 (1950). KRAMEKS H ., Physica 1, 182 (1934). SLATBH J . C., Quart. Progr. Rep. M.Z. T. July 15,l; Oct. 15,l (1953). GOODENOUGH J. B. and LOEB A. L., Phys. Rev. 98, 391 (1955). GOODENOUGH J. B., Phys. Rev. 100,564 (1955). See NACAMIYA T.. YOSIDA K. and KUBO R.. Advames in Phys.‘4, 1 (1955). WOLLAN E. Q. and KOEHLER W . C., Phys. Rev. 100. 545 (1955). YAMASHITA J . and KONDO J., Phys. Rev. 109, 730 (1958). See, for example, BLEANEY B . and STEVENS K . W. H., Rep. Progr. Phys. 16, 108 (1953). KOEHLER W. C. and WOLLAN E. Q., J. Phys. Chem. Solids 2, 100 (1957). See ref. 6, part I. TSUBOI~AWA I. Private communication. KANAMORI J., Progr. Theor. Phys. 17, 177, 197 (1957). KOEHLER W. C., WILKINSON M. K., CABLE J. W. and WOLLAN E. Q., Collogue Znternat. de Mag- netisme de Grenoble (1958).- KANAMORI 1. To be nublished in Prwr. Theor. Phys. See also NAC~MIYA T., Cello& Znternat. de Magnetisme de Grenoble (1958). STARR C., BITTER F. and KAUFMANN A . R., Phys. Rev. 58, 977 (1940). SHIRANE G. Unpublished work. ISMIKAWA I . and AKIMOTO, S. Private communica- tion. NAGAMIYA T.‘, Colloque Internat. Magnetisme de Grenoble (1958). BIZETTE H. and TSAI B., C.R. Acad. Sci. (Paris) 242, 2124 (1956). WOHLFARTH E . P., Acta Met. 4, 225 (1956). SIDHU S. S., HEATON I,. and ZAUBERIS D . D., Acta Cryst. 9, 607 (1956). SINGER J . R., Phys. Rev. 104, 929 (1956). I,INDSAY R. and BANEWICZ J . J., Phys. Rev. 110,634 (1958).
