T. Magn. Soc. Jpn. (Special Issues)., 3, 12-18 (2019)<Paper>Local strain dependence of uniaxial magnetic anisotropyin Mtype ferrites
J. Inoue*,**, H. Nakamura***, and H. Yanagihara**Institute of Applied Physics, University of Tsukuba,

Tsukuba 305-8573, Japan**Department of Applied Physics, Tohoku University,

Sendai 980-8577, Japan*** Department of Materials Science and Engineering, Kyoto University,

Kyoto 606-8501, Japan

We study the magnetic anisotropy (MA) of magnetoplumbite (M)type ferrites by calculating the electronic
structure of clusters formed with an Fe3+ ion and O2 ions. Spinorbit interaction (LS coupling) is taken into account.
It is shown that the pd mixing between the Fe and O ions and local lattice distortion are crucial to understand theMA of Mtype ferrites. Calculations of MA for Mtype ferrites doped with Fe2+ or Co2+ ions demonstrate the importance of local lattice distortion on MA. It is shown that Fe2+ ions doped into 2asite produce metamagnetic character when a compressive deformation along caxis is introduced. It is also shown that the Co2+ ions doped onto 4f1 site may enhance the uniaxial MA when a compressive deformation is introduced.
Key words : hexagonal ferrite, uniaxial magnetic anisotropy, local lattice distortion, cluster model

1. Introduction
Development of high performance permanent magnets is one of the important technological challenges forthe energy-conserving society. Hexagonal ferrite magnets 1),2) are used most widely on a commercial basis.
Even a 10 percent increase in the performance of ferrite
magnets makes a great contribution towards energysaving. It was recently found that doping of divalent
Co ions into magnetoplumbite (M)type ferrites increases magnetic anisotropy (MA).3)5) However, relation
between the increase in MA and preferential sites forCo2+ ions has not been clarified 6)9) and hence, no systematic development of highperformance ferrite magnets has been successful.
Requirements for large MA in permanent magnets
are large value of magnetization, strong spinorbit interaction (SOI), and low crystal symmetry. It is wellknown that rareearth Fe (Co) compounds have tetragonal or hexagonal lattice symmetry. Recently, it wasreported that thin films of tetragonally distorted spinel
ferrites, Fe(Fe-Co)2O4, on suitable substrates produce
extremely high uniaxial MA.10) The mechanism of thishigh MA has been elucidated by both a phenomenological theory and an electron theory for an octahedralcluster containing a single Co ion and surrounding six O
ions.11),12)Mtype ferrites, AFe12O19 with A = Ba, Sr, or La,
have a hexagonal lattice structure and their unit cell
contains two formula units (fu's).1),2) There are fivenonequivalent sites for Fe ions called 2a, 2b, 4f1, 4f2, and12k sites. Fe ions are trivalent in Mtype ferrites withdivalent A (Ba2+ and Sr2+) ion. On the other hand, one
Fe ion in LaFe12O19 is divalent because La ions are trivalent. Magnetic moments of Fe ions at 2a, 2b, and 12k
sites are parallel to the bulk moment, while those at 4f 1

12

and 4f2 sites are antiparallel to the bulk moment. An
important point is the shape of the clusters formed by asingle Fe ion and surrounding O ions. Clusters for 2a,
4f2, and 12kFe ions are of sixcoordinate (octahedral),and the clusters for 2b and 4f1Fe ions are offivecoordinate (hexahedral) and fourcoordinate (tetrahedral), respectively, as shown in Fig. 1. It is notedthat the octahedral and tetrahedral clusters are slightly
distorted compared to the regular clusters existing in a
cubic spinel ferrite.
In this work, using the electron theory for FeO
clusters with SOI (LS coupling), we will show that thesmall lattice distortion of the clusters is crucial to themagnitude of MA in Mtype ferrites and that the MA
produced by Co2+ ions doped into Mtype ferrite is also
affected by the cluster distortion. To begin with, we first
explain how the trivalent Fe ions produce MA of Mtype
ferrites in the present cluster model. This is nontrivial
because 3d5 electron configuration of Fe3+ gives rise tonull orbital angular momentum resulting in no LS coupling. We show that the pd mixing of wave functions on
O and Fe ions is essential to understand the MA in oxide
magnets.

Fig. 1 Types of clusters formed by a TM ion (Fe or Co)
and surrounding O ions. (a) Sixcoordinate (octahedral)cluster, (b) fourcoordinate (tetrahedral) cluster, and (c)fivefold (hexahedral) cluster. Direction of caxis of theMtype ferrite is shown by an arrow.

Transaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 2019

In next section we explain a model (cluster model)
for the electron theory to calculate MA. In section III,we will provide the calculated results of MA for undopedMtype ferrites, followed by those produced for Fe2+ andCo2+ ions in Mtype ferrites. To clarify the effect ofcluster distortion on MA of doped ferrites, the MA energy is calculated as a function of cluster distortion.
Implications of the calculated results will be discussed,and conclusions are given in section IV.
2. Method of calculation
We perform numerical calculations for clusters witha single Fe3+, Fe2+, or Co2+ ion surrounded by six (octahedral cluster), five (hexahedral cluster), and four (tetrahedral cluster) O2 ions using full orbitaltightbinding (TB) model with SOI (LS coupling) on Fe
and Co ions. The 3dorbitals of the transition metal(TM) ions are spinpolarized and the ions have local
moments. The clusters are simplified ones but satisfy
the local symmetry.
Parameters of the intersite pd hopping between
five 3dorbitals on a TM ion and three 2porbitals on
oxygen ions are determined from Harrison's textbook.13),14) Intersite pp hopping between O ions is also
taken into consideration. The atomic energy levels of
2p and 3dorbitals and the magnitude of the exchange
splitting are fixed by considering the results of theelectronic states of the spinel ferrites calculated using
firstprinciples.15), 16) Number of electrons on 3d statesof Fe3+, Fe2+, and Co2+ are five, six, and seven per ion,
respectively, and that of O2 is six per ion. The groundstate energy is calculated by diagonalizing the Hamiltonian matrix as a function of magnetization direction.
It is noted that the mixing of p and dorbitals reproduces the correct symmetrydependence of energy levels as in the crystal field potentials.
The values of the SOI of Fe3+, Fe2+, and Co2+ ions are
fixed as 0.057, 0.050, and 0.063 eV per ion, respectively.17) The values are consistent with the dependence of
SOI on Z/r3, where Z and r are the atomic number andionic radius of the TM ion, respectively.
In the following, "positive uniaxial MA" of the cluster
is defined in such a way that the easydirection of thecluster magnetization coincides with the caxis shownin Fig. 1.
3. Calculated results and discussions
3.1 Magnetic anisotropy of AFe12O19
Figure 2 shows the calculated results of MA energy(K u) for each cluster in Mtype ferrites with A = Ba, Sr,and La. Positive K u indicates the uniaxial MA. The
shapes of the clusters used in the calculations are thesame as those in the lattices observed for A = Ba, Sr, andLa.18)20) Two sets of parameter values are used, among
which, one includes no pp hopping and the other includes pp hopping with a weighing factor of 0.5. Although the Laferrite includes one Fe2+ ion in fu because

Fig. 2 Calculated results of MA energy, K u for Fe3+ oneach nonequivalent site in Mtype ferrites. Closed
symbols are for those without pp hopping in the clustermodel, and open symbols are for those with pp hopping
and a weighing factor of 0.5. "av" indicates the averaged
value over the K u values calculated for five nonequivalent sites. 0.1 meV/ion corresponds to 5.6 Merg/cm3.
La ion is trivalent, we neglected the existence of Fe2+
assuming that all the Fe ions are trivalent, in order tostudy the dependence of K u on the small local lattice
distortion
As shown in Fig. 2, the 2b site contributes mostdominantly to the uniaxial MA because of its lowest
cluster symmetry. Nevertheless, the other sites, especially 12k site, may contribute to the total MA becauseof the abundance of the 12k site. The tendency has already been discussed,1) and is consistent with that obtained in the theoretical analysis using crystal fieldmodel 21) and with that in the firstprinciple calculations.22)
The calculated and experimental results of MA energy per unit volume are presented in Table 1. The calculated results are obtained by summing up the contribution of each cluster. We find that the results are ingood agreement with the experimental ones. The result
calculated by the firstprinciples is nearly half of theexperimental values.22) The tendency observed in our
calculation suggests the validity of our cluster model. It
is noted that the calculated values shown in Table 1
include the effect of pp hopping and that the inclusionof pp hopping enhances the magnitude of K u values.
Table 1 Experimental and calculated values of K u in
Merg/cm3 for A = Ba, Sr, and La. The result calculatedusing firstprinciples is also presented. The value of K uof Laferrite includes a contribution from Fe2+ and islarger than those of Ba and Srferrites by a factor 1~ 2.
A
Exp. 23)
Present cal.
Firstprinciples 22)

Ba
3.25
2.45


Sr
3.57
2.60
1.8

Transaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 2019

La

2.81


13

Fig. 3 A schematic diagram of the energy levels of acluster formed by a TM ion at the center and O ionssurrounding the TM ion. Shaded boxes represent theunhybridized levels and open boxes indicate the bonding
and antibonding states after pd mixing. VSOI ↑↓ andVSOI↓↓denote updown spin and downdown spin components of the LS coupling.
To study the effect of the local lattice distortion in
AFe12O19 on the MA, we performed similar cluster calculations for an ideal Mtype ferrite in which the octahedral and tetrahedral clusters are regular octahedrons
and tetrahedrons, respectively. The K u value is 1.6
Merg/cm3, which is much smaller than those calculatedfor AFe12O19. The result indicates the importance of thelocal lattice distortion on MA.
The appearance of the MA in Fe3+ ions is explainedin terms of pd mixing between the spinpolarized 3d
states of an Fe ion and 2p states of O ions. Figure 3 is aschematic presentation of the energy levels for thespinpolarized 3d states and unpolarized 2p states.
Shaded boxes represent the energy levels of unhybridized states and white boxes indicate the bonding andantibonding states after pd mixing. The downspin
bonding states include 3dorbital components; consequently, the LS coupling becomes active between bothupdown and downdown spin components, resulting ina weak MA in Fe3+ ions.
Some discussions on the validity of the present calculations are ready. The bulk K u of Baferrite was calculated to be 0.6 Merg/cm3 by using the crystal fieldmodel,21) in which the local K u values were reported to
be 17.4 × and –2 ~ 6 ×105 eV/ion for 2bFe and theother Fe ions, respectively. Although the tendency of thelocal K u values agrees with the present results, the bulk
K u is smaller than that in the present model. The
result may be attributed to the negative value of local
K u for the abundant 12ksite Fe ions.
The first principles calculation of the local K u for
Srferrite in the Wien2K with GGA+U formalism22)
gives 24 × and 1 ~ 2 × 105 eV/ion for 2bFe and theother Fe ions, respectively. Although the bulk K u value
shown in Table 1 is consistent with the present result,the local K u values except for 2bFe are not necessarily
consistent with ours. We attribute the difference to thelongrange effect neglected in our cluster model. Resultant numerical error inherent to the present modelseems be of the order of 105 eV/ion, because the accu-

14

Fig. 4 MA energy, calculated in the cluster model, as afunction of angle θ of the magnetization direction
measured from caxis shown in Fig. 1 for (a) Fe2+ and(b) Co2+ ions in the nonequivalent sites of Laferrite.
racy in the numerical calculation is better than 10 7
eV/ion.
3.2 MA of divalent Fe and Co ions
To clarify the sites occupied by Co2+ in Mtype ferrites is crucial in the fabrication of high performance
ferrite magnets, however, the experimental results arecontroversial:6),7),24)27) Mössbauer and Raman measurements suggest 4f2 site as the occupied site, while
Neutron diffraction and Extended Xray Absorption
Fine Structure (EXAFS) measurements suggest 4f1site. As for the occupation of Fe2+ ions, on the other
hand, both experimental and theoretical results suggest
2a site as the most preferential site.22),28) In addition, ametamagnetic transition observed in the magnetization
process for Fe2+ doped (Ca-La)Fe12O19 might be a clue toidentify the occupation sites of Fe2+ ions.28)
To examine the effects of Fe2+ and Co2+ ions on MA,
we perform numerical calculation of MA energy forclusters which contain Fe2+ or Co2+. The shape of thecluster is assumed to be the same as that in Laferrite.
Calculated results for Fe2+ and Co2+ ions as a function ofmagnetization direction θ, the angle measured fromcaxis, are presented in Fig. 4(a) and 4(b), respectively.
As shown in figure 4(a), Fe2+ ions on 2a, 2b, and4f1sites enhance the uniaxial K u, while those on 4f2and
12ksites suppress it. However, no such tendency has
been observed in experiments. Furthermore, theθdependence of MA energy on 2aFe shows no tendency of a metamagnetic transition. As for Co2+ ions inLaferrite, they, except for those on 2a site, suppress
the uniaxial MA as shown in Fig. 4(b). The characteristic feature shown in Fig. 4(b) has been found to be unchanged for Co2+ ions doped in the other Mtype ferrites.
The MA energy thus calculated for Co2+ ion doped into
any site in Mtype ferrites would not explain the observed increase in the uniaxial K u.
As mentioned, the shape of the clusters used in thecalculations is assumed to be the same as that in thelatticestructureofLaferrite.
However,in(La-Ca)ferrite for example, random distribution of Fe3+
and Fe2+ ions may give rise to additional lattice distor-

Transaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 2019

Table 2. Estimated magnitude of cluster distortion inMtype ferrites, normalized rc/ra, and shift of Fe ion|δz|. Values for 12ksite cluster are omitted because of
the complexity of the cluster distortion.

2a
2b
4f1
4f2
12k

Fig. 5 Types of cluster distortions denoted by the shift(small red arrows) of O and Fe ions for (a) 2a, (b) 2b, (c)4f1, (d) 4f2, and (e) 12k sites. Right and left panels ofeach figure are the projections of the cluster onto thecplane and a plane along the caxis, respectively. Two
distances ra and rc are defined as the inplane OO
distance and interplane one, respectively.
tion because of different ionic radii between Fe 3+ andFe2+ ions. Distribution of La and Ca ions may also leadto a similar effect. As shown in the previous subsection,
the MA of Mtype ferrite is sensitive to the lattice distortion of clusters, and therefore, it is expected that thecalculated results shown in Fig. 4 might be altered bylattice distortion of the clusters. In the following, we
perform cluster calculations by introducing cluster distortions.
As the details of local lattice distortion in(La-Ca)ferrite are unknown, we first survey the characteristic feature of the local lattice distortion of FeO
clusters in Ba, Sr, and Laferrites.18)20) Results are
depicted in Fig. 5, where small arrows denote the shiftsof O and Fe ions from the potions of ideal clusters. The
shifts for 2asite cluster, for example, are depicted inFig. 5(a). The left figure is a projection of the cluster on
the cplane, and shows that the O ions on both upperand lower triangles shrink, that is, the distance rashown in the figure becomes smaller than that of theregular octahedron in the cubic spinel ferrite. The rightfigure is a projection on to a plane along caxis. It showsthat O ions on the upper triangle shift upward andthose on the lower triangle shift downward, resulting in
an increase in the distance rc. Figures for the otherclusters should be understood similarly, except for theshift of the Fe ions along the caxis. It is notable that
result for the 12kcluster exhibits complicated lattice
distortion as depicted in Fig. 5(e).
The Oion position is shifted from the position of
the regular octahedron or tetrahedron by a small
amount of (δx0, δy0, δz0), which may be estimated byusing the Xray diffraction data for Ba, Sr, andLaferrites.18)20) The estimated values are δx0 = 0.01 ~0.015 Å, δy0 = 0.02 ~ 0.01 Å, and δz0 ≲0.04 Å for 2a, 4f1

Normalized rc/ra1.08 ~ 1.12
0.87 ~ 0.90
1.06 ~ 1.07
1.10 ~ 1.17


Shift of Fe ion |δz| (Å)
00.006 ~ 0.007
0.002 ~ 0.003
0.009 ~ 0.011


and 4f2 site clusters. By using these values, the range ofra and rc values defined in Fig. 5 have been evaluated.
The cluster distortion may be characterized by twoquantities, a ratio rc/ra where rc and ra are the inplane
and interplane distances between two O ions, respectively, and δz is the shift of Fe ion as defined in Fig. 5.
Here, we introduce a normalized ratio (rc/ra)n that is thevalue of rc/ra normalized by that for regular octahedronor tetrahedron. The (rc/ra)n value for 2bsite cluster isdetermined using 2bsite cluster in the ideal Mtype
ferrite in which regular octahedrons and tetrahedrons
are included. Table 2 shows values of normalized ration(rc/ra)n and |δz| for clusters with the nonequivalentFesites. In the following, we use these values as ameasure of lattice distortion to calculate the dependenceof K u on the local lattice distortion in Mtype ferrite
doped with Fe2+ or Co2+ ions.
Now, we study the dependence of K u values on local
lattice distortion for Fe2+ and Co2+ ions doped inSrferrite. We focus our attention on the Fe2+ ion on
2asite, which may be responsible for the metamagnetic
behavior, and on the Co2+ ion on 4f1 and 4f2 sites, which
are suggested to be plausible for Co2+ occupation.
Figure 6(a) shows calculated values of K u of Sr
ferrite in which 20 % Fe2+ ions are doped into 2asites.
The range of the lattice distortion is 0.9 < (rc/ra)n < 1.0
and 0.01 < δz < 0.01. The value of δz is zero because of
the lattice symmetry, however, we have introduced it
virtually for comparison with results obtained for theother clusters. The K u value on Fe2+ ion and that of thebulk Srferrite calculated in the previous subsection,
have been averaged over. Red region shows positive K u
and blue and green regions indicate negative K u. The
linear dependence of K u on (rc/ra)n is the same as thetetragonality dependence of K u given by the relation
K u = B (c/a1) in cubic lattices, where B is the magnetoelastic constant. The dependence of K u on δz is symmetrical because of the symmetry of the 2asite cluster
along the caxis.
Because θdependence of K u is given as K u ~ K1
sin2θ + K2 sin4θ, several magnetic states may appear
depending on the signs and magnitudes of K1 and K2.
Therefore, we have approximately fitted the calculated
results of K u (θ) using the equation above, and haveTransaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 201915

Fig. 6 Calculated results of Ku (in Merg/cm3) ofSrferrite with 20 % doping of Fe2+ or Co2+ ions. Results
are plotted as a function of normalized rc/ra ratio, (rc/ra)n(see text), and δz (in Å), which characterize the cluster
distortion and the shift of Fe or Co ions. (a) Fe2+ doping
on 2a site, (b) Co2+ doping on 4f1 site, and (c) Co²⁺ doping on 4f2 site.
obtained values of K1 and K2. These values are useful tosemi-quantitatively discuss the dependence of themagnetic state on lattice distortion. Combination of thesigns of K1, K2, and K u is plotted in Fig. 7(a) as a function of (rc/ra)n and δz. For (rc/ra)n > 0.925 with δz = 0, allvalues are positive, while states with K1 > 0, K2 < 0, andK u > 0 appear with decreasing (rc/ra)n . In these states
the metamagnetic transition may occur when aninplane external magnetic field is applied.
The most

16

Fig. 7 Signs of K1, K2, and K u calculated as a function of
normalized rc/ra ratio and δz (in Å) to characterize themagnetic states of Srferrites with 20 % doping of Fe2+or Co2+ ions. (a) Fe2+ ion on 2a site, (b) Co2+ ion on 4f1
site, and (c) Co2+ ion on 4f2 site.
notable result shown in Fig. 6(a) and Fig. 7(a) is that thesign of MA is altered by a very small change in the interatomic distance.
Figure 6(b) shows the results for K u of Srferrite
with 20 % doping of Co2+ on 4f1 site. The red and blue
regions indicate positive and negative K u, respectively.
The magnitude of the local distortion of the cluster
containing 4f1 site in the undoped Srferrite corresponds to that in the blue region. With decreasing(rc/ra)n, which implies increasing the compression along
the caxis, K u values increase and become positive asshown in the figure. Similarly, K u value becomes positive as δz turns to be negative.
Combination of the signs of K1, K2, and K u is plottedin Fig. 7(b) as a function of (rc/ra)n and δz. We find theTransaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 2019

blue region with negative K u changes to be positive withdecreasing (rc/ra)n. Similar change occurs in case of signchange in δz from positive to negative. We see that the
combination of signs of K1, K2, and K u is altered by a
few % change in (rc/ra)n, as well as by a small shift δz of
Co ion.
Similar plots for Srferrite with 20 % doping of Co2+
on 4f2 site are presented in Fig. 6(c) and Fig. 7(c). It isnoted that color presentation in Fig. 6(c) is different
from that in figure 6(b): zero of K u resides in the blue
region in Fig. 6(b), while it is in the red region in Fig.
6(c). As shown in Table 2, values of (rc/ra)n and δz for 4f2
site cluster in the undoped ferrite are close to 1.1 and0.01 Å, respectively. Therefore, 20 % doping of Co2+ on
4f2 site makes K u strongly negative. The plot in Fig. 7(c)shows that the negative sign of K u would not be alteredby weak distortion of the cluster.
We have thus investigated the overall feature of theeffects of the local lattice distortion, that is, the cluster
deformation, on K u values as well as on the signs of K1,
K2, and K u. The possible magnitude of the local lattice
distortion after Fe2+ or Co2+ doping was estimated bycomparing the distortions in Ba, Sr, and Laferrites
with those in the ideal structure. The results may besummarized in the following manner.
Because the MA induced by Fe2+ and Co2+ ions ismuch larger than that induced by Fe3+ ion, dopant ofsmall amount of Fe2+ and Co2+ ions may change thecharacter of MA of Mtype ferrites. When the lattice
structure of doped ferrites is assumed to be the same asthat of undoped ferrite, the change in K u values for Fe2+
doping depends on the doped site of Fe2+, while K u forCo2+ ion is negative and independent of the doped site.
However, such change in MA of doped ferrites strongly
depends on the local lattice distortion, a very smallchange in (rc/ra)n of the cluster and/or a small shift in
the position of Fe2+ (Co2+) ions. Furthermore, such MA
change depends on the occupation sites. For Fe 2+doped
2asite, the metamagnetic feature appears under clustercompression along the caxis. For Co2+doped 4f1 site,
similar compression makes K u positive.
The results are consistent with the observed ones,
however, several discussions are possible. First, so far
no explicit observation for the relation of MA change
and local lattice distortion has been reported. Because
the lattice distortion introduced in our calculation isvery small, highprecision local probe is required tomeasure the lattice distortion.30)
Secondly, in our calculations, the distortion of eachcluster was treated independently. This treatment may
not always be correct. For example, Fe ions on 4f 2 sites
form dumbbell structure in Mtype ferrites, that is, twoneighboring octahedrons share a triangle face, resulting
in a doublecluster structure. It should be noted that,for this type of doublecluster structure, K u value induced by two Co2+ ions depends on the distance between
the two Co2+ ions. Our preliminary results show thatwith decreasing distance, direct overlap of 3delectron

clouds increases and contributes to make K u positive.
The tendency is the same with that in the usual hexagonal Co lattices.
Thirdly, we would like to discuss the longrange effects of crystal field potential on K u values.
The bulk
K u values of Mtype ferrites doped with 20% Co2+ orFe2+ ions are of the order of 1 meV/fu as shown in Fig. 6,
which corresponds 0.53 eV per Co or Fe ion. The
longrange effects on the local K u of Co2+ have been
investigated in a simple model for tetragonally distorted
spinel ferrites in which a local trigonal symmetryaround TM ions on the so-called Bsite is produced due
to characteristic lattice structure in the second nearest
neighbor sites.31) It has been shown that the local trigonal symmetry reduces the K u value of the tetragonally
distorted spinel ferrites.12) Although Mtype ferrites
have no such local symmetry, existence of Ba or Sr ionsand distribution of Co2+ and/or Fe2+ ions may reduce thebulk K u values. Nevertheless, the local K u values of
Co2+ and Fe2+ remain sufficiently large as comparedwith those of Fe3+, and therefore the semi-quantitative
results shown in Figs. 6 and 7 would be unchanged. It isnoted that such effect is not large for Fe3+ ions because
the up and down spin states are already split by thelarge exchange field.
Finally, the clusters in ferrites are connected with
one another, and therefore, the distortion of each cluster
is affected by other clusters. Furthermore, such local
lattice distortion is influenced by the distribution ofdivalent and trivalent ions in the lattice. One possible
way to clarify such local lattice distortion in a doped
lattice could be the method of lattice relaxation in thefirstprinciples. In this case, however, highly precise
calculation is also desirable.
4. Conclusions
We explained the MA of Mtype ferrites by calculating the electronic structure of clusters with a single
Fe3+ ion and surrounding O2 ions. The SOI (LS coupling) was introduced in the magnetic ion. We foundthat the calculated results agree with the experimental
ones fairly well, and attributed the mechanism of MA of
Fe3+ ions to the pd mixing between 2p and 3dorbitalsof O2 and Fe3+ ions, respectively. It was pointed out thata small amount of lattice deformation of the clusters is
crucial for quantitative understanding of the MA.
Our cluster calculations showed that MA of Mtype
ferrites doped with Fe2+ or Co2+ ions is also strongly
affected by a small amount of lattice distortion. It was
shown that Fe2+ ions doped onto 2asite produce metamagnetic character when compressive deformation
along the caxis is introduced. It was also shown that
the Co2+ ions doped onto 4f1site may increase the uniaxial MA when the compressive deformation is introduced.
For material design of ferrite magnets, however,precise measurements of local lattice distortion andTransaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 201917

detailed firstprinciples calculations of MA with local
lattice relaxation are required.
Acknowledgements This study was partially supportedby the project "High Performance Magnets" of Japan
Science and Technology Agency, Japan.
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Received Oct. 09, 2018; Accepted Dec. 19, 2018

Transaction of the Magnetics Society of Japan (Special Issues) Vol.3, No.1, 2019