Dionne G.F. Magnetic Oxides
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5.3 Magnetocrystalline Anisotropy and Magnetostriction 231
and
@.
Kc
C
H
/
@
D2K
1
2 sin
2
3 sin
2
2
K
2
1
2
sin
2
6 sin
4
11 sin
2
cos
2
C sin
4
(5.27)
H
Kc
M cos .ı / D 0:
Examples of the magnetization as a function of field where the fields required to
magnetize the specimens vary according to material and crystal orientation with
respect to the applied field H can be found in [54]. Calculations of these measured
curves can be carried out by finding the roots of angle for a fixed value of ı by
solving the equation or by numerical computation to determine M cos for each
value of H , given values of M , angle ı, and the appropriate K constants. The field
required to fully magnetize by rotating M between symmetry axes in a particular
crystallographic plane is used to characterize the difficulty in rotating M and is
treated as an intrinsic anisotropy field. This effective internal “bias” field equates
to the rotational “stiffness” of the anisotropy torque in the limit where ! ı,
according to
H
Ku
2K
u
M
cos 2ı
H
Kc
2K
1
M
1
1
2
sin
2
ı
3
2
sin
2
2ı
(5.28)
C
K
1
2M
sin
2
ı
6 sin
4
ı 11 sin
2
ı cos
2
ı C sin
4
ı
:
The above example for H
Kc
was originally derived by Bickford [55], and applies
to locations of the vector in a plane of the
f
110
g
family, with angle ı referred to a
h100i axis. The values of E
K
and H
K
expressed in terms of anisotropy constants
along principal axes of symmetry are summarized in Table 5.7 for rotation in the
three most important families of planes in a cubic lattice:
f
100
g
,
f
110
g
,and
f
111
g
.
The axes of minimum and maximum E
K
, i.e., easy and hard directions, are (100)
and (110) in the
f
100
g
planes, (100), (110), and (111) in the
f
110
g
planes, and (111)
and (112) in the
f
111
g
planes. In Appendix 5D, the complete relations for H
K
as
functions of angle are listed for torques both in-plane and out-of-plane. The H
K
out-of-plane is necessary in ferromagnetic resonance, which is reviewed in Chap.6.
5.3.2 Phenomenological Magnetostriction Theory
The second part of the magnetoelastic phenomena is the reaction of the lattice to
the internal stress imparted by the torque when the magnetization is rotated from its
minimum energy or easy axis position. The result is called magnetostriction and is
manifested by a measurable deformation strain l=l of the lattice, usually in the
232 5 Anisotropy and Magnetoelastic Properties
Table 5.7 Magnetocrystalline anisotropy energies and anisotropy fields in cubic lattices
fPlaneghAxisi
.
•
/
E
K
H
K
(in plane) H
K
(out of plane)
f100gh100i (0
ı
, 90
ı
)0 2K
1
=M
s
2K
1
=M
s
h110i (45
ı
) K
1
=4 2K
1
=M
s
K
1
=M
s
C K
1
=2M
s
f110gh100i (0
ı
)0 2K
1
=M
s
2K
1
=M
s
h112i (35
ı
) K
1
=4 C K
2
=54 K
1
=M
s
C
K
1
=18M
s
K
2
=3M
s
h111i (55
ı
) K
1
=3 C K
1
=27 4K
1
=3M
s
4K
1
=3M
s
4K
1
=9M
s
4K
1
=9M
s
h110i (90
ı
) K
1
=4 K
1
=M
s
C
K
1
=2M
s
2K
2
=M
s
f111gh112i (0
ı
, 60
ı
) K
1
=4 C K
2
=54 K
2
=3M
s
K
1
=M
s
C K
1
=18M
s
h110i (30
ı
, 90
ı
) K
1
=4 K
2
=3M
s
K
1
=M
s
C K
1
=2M
s
f112gh111i (0
ı
) K
1
=3 C K
1
=27 4K
1
=3M
s
4K
1
=3M
s
4K
1
=9M
s
4K
1
=9M
s
h110i (90
ı
) K
1
=4 K
2
=3M
s
K
1
=M
s
C K
2
=6M
s
range of 10
5
–10
6
, depending on the amount of anisotropy energy involved and
the elasticity of the lattice. The formal treatment of the origins of magnetostriction
will not be included here because their mathematical details would unnecessarily
burden the text. This information can be found in most standard texts on magnetism,
and in particular, the review article by Lee [56] and the book by du Tr´emolet de
Lacheisserie [57]. For our purposes, the discussion will focus on the results of these
analyses from two standpoints: (a) the phenomenology of magnetostriction and its
relation to magnetocrystalline anisotropy and (b) the effect on the anisotropy from
applied stress, known as inverse magnetostriction.
The lattice strain resulting from the magnetization of a single crystal of cubic
symmetry is expressed as
l
l
D
3
2
100
˛
x
2
ˇ
x
2
C ˛
y
2
ˇ
y
2
C ˛
z
2
ˇ
z
2
1
3
C3
111
˛
x
˛
y
ˇ
x
ˇ
y
C ˛
y
˛
z
ˇ
y
ˇ
z
C ˛
z
˛
x
ˇ
z
ˇ
x
; (5.29)
where
100
and
111
are the magnetostriction constants, and ˛
1
, ˛
2
,and˛
3
are the
direction cosines of the M vector, while ˇ
1
, ˇ
2
,andˇ
3
are the direction cosines of
astrainl. Inspection of (5.29) reveals that l=l D
100
or
111
when M and l
are collinear and directed along the corresponding h100i and h111i axes. Further
inquiry into the meaning of
100
and
111
shows that they depend on the elastic
constants of the crystal with standard designations of c
11
, c
12
,andc
44
, and on the
magnetoelastic constants B
1
and B
2
which are dependent on the configuration of
nearest neighbors, according to
100
D
2
3
B
1
c
11
c
12
111
D
1
3
B
2
c
44
: (5.30)
5.3 Magnetocrystalline Anisotropy and Magnetostriction 233
An additional relation that is useful for polycrystalline materials defines an average
magnetostriction constant
s
D
2
5
100
C
3
5
111
: (5.31)
The merging of the anisotropy and magnetostriction concepts into a single theory
was first formalized by Kittel [58]. The relation that combines the interactive effects
between strain and anisotropy in the free energy is given by
E
K
D
K
0
1
C K
1
˛
x
2
˛
y
2
C ˛
y
2
˛
z
2
C ˛
z
2
˛
x
2
; (5.32)
where K
1
D
9
4
h
.c
11
c
12
/
100
2
2c
44
111
2
i
. Note that K
1
can enhance
or reduce the total K
1
depending on the values of the magnetostriction and elastic
constants.
The influence of magnetostriction on anisotropy is particularly important where
external stress is involved. With a uniaxial compressive stress applied to a mag-
netized body, the combined energy of anisotropy and inverse magnetostriction
E
K
D K
1
˛
x
2
˛
y
2
C˛
y
2
˛
z
2
C˛
z
2
˛
x
2
C
3
2
100
˛
x
2
x
2
C ˛
y
2
y
2
C ˛
z
2
z
2
C3
111
˛
x
˛
y
x
y
C ˛
y
˛
z
y
z
C ˛
z
˛
x
z
x
; (5.33)
where
x
,
y
,and
z
are the direction cosines of the uniaxial stress vector. If the M
and are coincident, (5.33) can be simplified to
E
K
D ŒK
1
C 3 .
111
100
/
˛
x
2
˛
y
2
C ˛
y
2
˛
z
2
C ˛
z
2
˛
x
2
C
3
2
100
:
(5.34)
Special cases have been analyzed for situations where the magnetostriction is com-
parable to the anisotropy. In particular, the variation of the easy axis direction when
H and are applied collinearly is of interest in forecasting the effects of stress on
the magnetization process, more specifically, the hysteresis loop. An example of this
is shown in Fig. 5.16 for a single crystal of YIG with stress applied along a <100>
hard axis .K
1
<0/[59]. Note how the anisotropy field has become smaller and
the remanence ratio R.D M
r
=M
s
/ has increased through the action of the uniaxial
anisotropy component caused by the stress. In this case,
H
K
D
2K
1
M
s
C
3
100
2M
s
;
R D
M
r
M
s
D
s
1
3
C
100
K
1
; (5.35)
for which the room-temperature saturation magnetization 4M
s
D 1; 780 G, K
1
6; 000 erg=cm
3
,and
100
D1:3 10
6
.
234 5 Anisotropy and Magnetoelastic Properties
Fig. 5.16 Trace of oscilloscope photograph showing remanence and approach to saturation
regions from a hysteresis loop of a Y
3
Fe
5
O
12
single-crystal with magnetic field and uniaxial
compressive stress applied along [001] “hard” axis. Effects of stress on remanence ratio (R) and
anisotropy field
.
H
K
/
are indicated with their measured values (after Dionne [59]). Figure reprinted
from [59].
c
1969 by the IEEE
If the anisotropy induced by external stress is large enough to dominate the un-
stressed anisotropy, it is possible to create a uniaxial component that can produce
an easy axis normal to the plane of a magnetic thin film, i.e., K
u
K
1
,even
one in which a demagnetizing field of magnitude 4M of a single domain must be
overcome [60].
With this brief phenomenological background, we can proceed to examine
sources of the anisotropy and magnetostriction parameters in magnetic oxides.
5.3.3 Dipolar Pair Model of Magnetic Anisotropy
Among the earliest attempts to analyze the molecular origins of magnetocrystalline
anisotropy was the dipole pair model of Van Vleck [61]. In two dimensions, the spin
configurations in an ordered lattice plane are depicted in Fig. 5.17, where the angle
is the angle between the aligned spin direction and a direction set by the bonds
connecting the spins. The pair energy can be expressed by a Legendre polynomial
expansion
E
dipole
.cos '/ D E
0
C E
1
cos
2
'
1
3
C E
2
cos
4
'
6
7
cos
2
' C
3
35
C :::
(5.36)
5.3 Magnetocrystalline Anisotropy and Magnetostriction 235
Fig. 5.17 Two-dimensional model of ferromagnetic ordering of magnetic dipoles in a lattice plane
with a non-orthogonal bonding arrangement
Fig. 5.18 Three-dimensional
model of ferromagnetic
ordering of magnetic dipoles
in a cubic cell
The zero-order term would include the isotropic exchange energy; the first and
second-order terms are the phenomenological contributors to the anisotropy that
would comprise spin–orbit–lattice interactions. For N ions per unit volume in a lat-
tice of cubic symmetry shown in Fig. 5.18, direction cosines ˛
x
, ˛
y
,and˛
z
of the
three-dimensional structure are introduced and the relation of (5.36) becomes
E
dipole
D N
2
6
6
4
E
0
C E
1
˛
x
2
1
3
C E
2
˛
4
x
6
7
˛
x
2
C
3
35
CE
1
˛
y
2
1
3
C E
2
˛
4
y
6
7
˛
y
2
C
3
35
CE
1
˛
z
2
1
3
C E
2
˛
4
z
6
7
˛
z
2
C
3
35
3
7
7
5
:
(5.37)
Recalling the identity ˛
x
2
C ˛
y
2
C˛
z
2
D 1, we can reduce (5.37)to
E
dipole
D N
E
0
C E
2
˛
4
x
C ˛
4
y
C ˛
4
z
C
9
35
; (5.38)
and then write the angle-dependent anisotropy energy as
236 5 Anisotropy and Magnetoelastic Properties
E
K
NE
2
˛
4
x
C ˛
4
y
C ˛
4
z
DK
1
˛
2
x
˛
2
y
C ˛
2
y
˛
2
z
C ˛
2
z
˛
2
x
CK
2
.higher order factors/:
(
5.39)
By inspection of (5.21)and(5.22) we recognize that K
1u
and K
1
correspond to the
E
1
and E
2
energies of the Legendre series and that the former terms correspond to
an axially symmetric component that cancels out in perfect cubic symmetry.
The dipolar model of ferromagnetic anisotropy has produced in some cases
reasonable estimates of actual measured values, and spawned much of the early
analysis of this phenomenon. One useful and often-cited relation between the
anisotropy and magnetization as a function of temperature comes from a calcula-
tionbyZener[62] that produced the equation
K
1
.T /
K
1
.0/
D
M.T /
M.0/
10
: (5.40)
This result was found to be accurate for metallic iron over the entire range from zero
to the Curie temperature, but not so appropriate for cobalt and nickel for which the
exponent is variable over the full range due to the larger unquenched orbital angular
momentum and spin–orbit coupling.
For further details on this aspect of the subject and the derivation and implica-
tions of (5.39), the reader is directed to Chikazumi [63]. For the present discussion,
however, the origins of the anisotropy constants are of greater interest and the focus
will be placed on the microscopic theory of single-ion anisotropy as it contributes
to the macroscopic value of K
1
.
5.3.4 Single-Ion Model of Ferrimagnetic Anisotropy
In magnetic oxides, the microscopic origins of K
1
can be extrapolated from the
single-ion properties discussed above. When the individual magnetic moments be-
come collective under the stabilizing influence of magnetic exchange, the effects
of anisotropy with respect to the crystal lattice are retained, often as competing in-
fluences between opposing sublattices in ferrites or between individual sites where
different ion species with their own particular anisotropy and magnetostriction con-
tributions, i.e., spin–orbit or Jahn–Teller ties to the lattice, become significant. The
most important yet least dramatic of these cases is logically Fe
3C
3d
5
. With
L D 0, it is an S-state ion with no first-order spin–orbit coupling and there-
fore a small single-ion interaction with the lattice, as discussed previously. Among
the more active ions are Co
2C
3d
7
,Fe
2C
3d
6
,Ni
2C
3d
8
,Mn
3C
3d
4
,and
Cu
2C
3d
9
, and they will be examined in the next section. However, because of
the practical importance of low anisotropy ferrites, the analysis of the single-ion
contributions of Fe
3C
will be summarized to serve as background to the reader’s
awareness.
For the g 2 ions, we shall describe the single-ion mechanism set forth inde-
pendently by Yosida and Tachiki [64], and Wolf [65]. Wolf began with the general
spin-Hamiltonian based on (5.20) with no first-order anisotropy terms
5.3 Magnetocrystalline Anisotropy and Magnetostriction 237
H
s
D gm
B
H S C
1
6
a
S
4
z
C S
4
x
C S
4
y
C DS
2
˛
C FS
4
˛
; (5.41)
where subscript ˛ designates the direction of a local axial distortion. For this pro-
cedure, Wolf began with the assumptions that the exchange energy dominates the
lower symmetry crystal fields, i.e, a<D gm
B
H
ex
,andtheF contribution from
equivalent sites averaged over the cubic symmetry axes is absorbed into the a term.
Equation (5.41) is then applied to the S D 5=2 case to compute the energy levels of
a single 3d
5
ion:
E
˙
5
2
D˙
5
2
gm
B
H
ex
C
1
2
a.1 5'/ C
5
3
D
3 cos
2
1
˙
D
2
4gm
B
H
ex
5 C 70 cos
2
75 cos
4
E
˙
3
2
D˙
3
2
gm
B
H
ex
3
2
a.15'/ C
1
3
D
3 cos
2
1
˙
D
2
4gm
B
H
ex
9 66 cos
2
C 57 cos
4
(5.42)
E
˙
1
2
D˙
1
2
gm
B
H
ex
Ca.15'/
4
3
D
3 cos
2
1
˙
D
2
4gm
B
H
ex
4 40 cos
2
C 36 cos
4
;
where is the angle that H
ex
makes with the direction of the distortion, l, m,
and n are the direction cosines of H
ex
with respect to the cubic axes, and D
l
2
m
2
C m
2
n
2
C n
2
l
2
. The anisotropy energy is extracted from the free energy of
each sublattice individually and then combined in a manner similar to that for the so-
lution of thermomagnetization properties. Boltzmann statistics are used to account
for the population distributions over the energy levels of (5.42), with the partition
function
Z D
2
4
mDC5=2
X
mD5=2
exp .E
m
=kT/
3
5
N
; (5.43)
where N is the number of ions, and the free energy F D – .kT / log
e
.Z/.Wolf
expanded these exponential terms in .–E
m
=kT/that contain the crystal field energy
parameters which are dominated by the exchange energy at lower temperatures and
small compared with kT except at the higher temperatures. The following general
result was obtained for the i sublattice:
F
i
D N
8
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
:
F
0i
.y/ C D cos
2
p .y/ C
D
2
gm
B
H
ex
cos
4
q .y/
Ca
l
2
m
2
C m
2
n
2
C n
2
l
2
r.y/
1
2
D
2
kT
cos
4
s .y/
C
1
2
D
2
kT
cos
4
Œp.y/
2
9
>
>
>
>
>
=
>
>
>
>
>
;
; (5.44)
238 5 Anisotropy and Magnetoelastic Properties
where F
0i
.y/ is the angle independent part of the free energy. The anisotropy terms
are given by
p.y/D .1=Z
0
/
5 y 4y
2
4y
2
y
4
C 5y
5
;
q.y/D .1=4Z
0
/
75 57y 36y
2
36y
2
57y
4
C 75y
5
;
r.y/D .5=2Z
0
/
1 C 3y 2y
2
2y
2
C 3y
4
y
5
;
s.y/D .1=Z
0
/
25 C y C 16y
2
4y
2
y
4
C 5y
5
; (5.45)
Z
0
D 1 Cy C y
2
C y
3
C y
4
C y
5
;
y D exp .gm
B
H
ex
=kT/:
Equation (5.44) for the free energy can be condensed into
F
i
D N
F
0i
.y/ C D cos
2
p .y/ C a
l
2
m
2
C m
2
n
2
C n
2
l
2
r.y/
D
2
kT
cos
4
t .y/
; (5.46)
where t.y/Dq.y/=ln .y/
1
2
s.y/C
1
2
Œp .y/
2
.Thevalueofy as a function
of temperature can be found indirectly from the Brillouin function of the relative
sublattice magnetization M
i
=M
i0
, which can be obtained from the relation
M
i
M
i0
D B .gm
B
H
eff
=kT/ D
mDS
X
mDCS
my
m
=S
mDS
X
mDCS
y
m
(5.47)
either graphically or computed by a numerical iterative procedure of the kind used
by Dionne [19] of Chap. 4. For each value of M
i
=M
i0
the functions of (5.44) can
be determined in generic form for each sublattice and for different values of S .To
apply these results to a specific situation such as yttrium–iron garnet with two iron
sublattices, values of the different sublattice magnetizations must be found from the
molecular field analysis discussed in Chap. 4. Regrettably, this accurate refinement
of the thermomagnetizationcould have extended Wolf’s model to garnet compounds
with varying levels of magnetic dilution if it had been available for use in (5.47).
The first-order cubic anisotropy for N ions can be reduced from (5.46)as
K
1
D N
ar .y/ C
D
2
=kT
t.y/
: (5.48)
where D2=3 or 4/9 for axial distortions along <100> or <111> axes, respec-
tively. The single-ion theory of anisotropy has been applied to analyze magnetic
garnets. For two sublattices, e.g., the d and a of a garnet, the resultant anisotropy is
K
1
D K
1d
C K
1a
D N
d
a
d
C
7
12
F
d
r.y
d
/ C
D
2
d
ı
kT
t.y
d
/
CN
a
a
a
C
7
12
F
a
r.y
a
/ C
D
2
a
=kT
t.y
a
/
;
(5.49)
5.3 Magnetocrystalline Anisotropy and Magnetostriction 239
Fig. 5.19 Variation of K
1
with T for yttrium–iron garnet. Image was traced from data reported
by Pauthenet ([51]ofChap.4) and later fitted closely by calculations of Rodrigue et al. [66]
Higher-order F contributions have been added to the a terms. Rodrigue et al. [66]
fitted K
1
vs. T data with the Wolf model, shown in Fig. 5.19, and produced expres-
sions for the bracketed weighting factors of r.y
i
/ of the Fe
3C
contributions from
the a and d sites in yttrium–iron garnet (YIG) using values of a and F reported by
Geschwind [67]: for site d, .0:791a
d
C 0:389F
d
/; for site a, .0:335a
a
0:259F
a
/.
The comparison of their results given in Table 5.8 raises questions. From the in-
terpretation of Rodrigue et al., the more numerous (by a factor 3/2) tetrahedral d
sites appear to produce larger contributions to K
1
than the octahedral a sites. The
results of Geschwind are even more surprising in that the a-site contributions are of
opposite sign to Rodrigue et al., contrary to experiment. Rodrigue et al.’s result was
supported by studies of d-sublattice dilution by Ga
3C
substitutions in YIG [68].
Measurements of Ga
3C
dilution of YIG by Hansen that included a complete range
of temperature dependence also indicated a monotonic reduction of K
1
with con-
centration [69]. Adding to the contradictions, a corresponding study of a-sublattice
dilution by large radii .0:81
˚
A/ Sc
3C
ions revealed an even more dramatic reduc-
tion in K
1
[70]. In both cases of Ga
3C
and Sc
3C
, the anisotropy field 2K
1
=M should
decrease sharply.
240 5 Anisotropy and Magnetoelastic Properties
Table 5.8 Parameter values for single-ion anisotropy in yttrium–iron
garnet
Parameter Geschwind [67] Rodrigue, Meyer, and Jones [66]
cm
1
10
3
cm
1
10
3
a
a
19.4 —
F
a
2.7 —
a
d
6.5 —
F
d
4:2 —
K
1a
4.1 0:4
K
1d
3.4 4.2
Other investigations involving hysteresis properties that showed a decrease in
anisotropy field 2K
1
=M suggested that small radii .0:51
˚
A/ Al
3C
ions in d sites
are more effective at reducing K
1
than more average radii .0:62 A/ Ga
3C
,which
unexpectedly produced a rise in 2K
1
=M [71]. However, this story becomes more
confusing when it is recognized that a lower initial fraction of Al
3C
enters d sites
and therefore is less effective than Ga
3C
at lowering the net 4M
d
, but more effec-
tive in lowering K
1a
according to the site distribution studies of [47,48]ofChap4.
If jK
1a
jjK
1d
j in YIG as concluded from experiment, M can decline less than
K
1
with Al
3C
, but the reverse is true for than Ga
3C
dilution, based on the simple
relation
2K
1
M
D
2.K
1a
K
1d
/
M
a
M
d
: (5.50)
Studies of single-ion contributions to anisotropy in spinels have confirmed the
conclusion that octahedral sites produce a K
1
contribution that is not only nega-
tive, but five times greater than a small positive contribution from tetrahedral sites
[46,64,72].
It is clear that more is to be learned about the nature of the source of anisotropy,
including the role of ionic dilutant sizes on the local crystal fields. In all of this
speculation, the possible influence of the spin canting caused by the dilution has
been ignored. Recalling our earlier discussion of spin canting, we recognize that
an In
3C
or Sc
3C
ion that replaces a smaller Fe
3C
in an octahedral site may reduce
K
1
by lowering the anisotropy contributions of the neighboring d -site Fe
3C
ions in
addition to removing the a-site contribution of the Fe
3C
ion that it replaces. A sim-
ilar argument could be made for the effects of the small Al
3C
ions entering d sites,
or even more radically, the dilution of ions of different valence altogether, such as
aV
5C
or Zr
4C
ioninad or a site, respectively, whereby the actual crystal field
charges are altered. Furthermore, the ratio 2K
1
=M
s
which influences the magne-
tization process and ferrimagnetic resonance, is sharply reduced by a-site dilution
because of the increase in M
s
. As noted above, the result of d -site dilution by Ga
3C
in particular appears to have the opposite effect. Some of these issues are reflected
in the qualitative dilution guidelines provided in Tables 4.4 and 4.5.
In the foregoing analysis, emphasis has been placed on crystal fields of cubic
symmetry. In hexagonal ferrites such as M-type BaFe
12
O
19
, a minority of trigonal