Dionne G.F. Magnetic Oxides
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332 6 Electromagnetic Properties
the phase change that occurred initially will not be recovered (unless a reversal of
the M direction takes place). This feature of rf propagation is unique in a magne-
tized medium and is termed “nonreciprocal.” Because the velocity of either mode
will change to that of the opposite mode if the magnetization direction is reversed,
a “differential phase shift” .D
C
/ of varying amount can be obtained by
switching the magnetization within the full range of M
s
M M
s
,sothat[89]
' D
ˇ
p
ˇ
pC
z
p
"
2c
4 .M
M
C
/ z D
p
"
c
4 .M/z; (6.115)
since 0 M 2M
s
from (6.114), in the context of nonreciprocity.
The unique propagation properties of circular polarization form the basis of mi-
crowave devices that can produce adjustable phase shift through choice of frequency
and magnetic field by moving the magnetization to selected points on the hysteresis
loop. Other applications rely on the presence of both circular modes of a linearly
polarized wave that were introduced for the discussion of the classical model of
paramagnetic resonance in Chap. 1. Linear polarization in magnetic media has found
important applications at optical as well as radio frequencies.
6.4.5 Linear Polarization and Faraday Rotation
Mentioned initially in Chap. 1, linearly polarized waves comprise both circular
modes, and are also used in applications where nonreciprocal propagation is re-
quired. Because ˇ
p
and ˇ
pC
can have different magnitudes, a rotation of the
polarization axis can occur for the same configuration of dc field and propagation
axis, as depicted in Fig. 6.34. The resultant H
rf
of the counterrotating H
rfC
(RHCP)
and H
rf
(LHCP) vectors about is rotated through an angle that is half of the differ-
ential phase shift of (6.115), given by
F
D
'
2
D
ˇ
p
ˇ
pC
2
p
"
2c
!
M
D
p
"
2c
.4M / ; (6.116)
where
F
is the Faraday rotation, here expressed as an angle per unit length. Note
that
F
is dependent on the magnetization, but independent of the frequency (or
wavelength) in this approximation. The corresponding difference in the absorption
constant is a measure of the “ellipticity” that results from the unequal amplitudes of
H
rfC
and H
rf
. It is generally overlooked in rf situations away from resonance, but
can be readily derived from (6.113a):
F
D
˛
p
˛
pC
2
D
p
"
4c
4!
2
!
0
!
M
2!
!
2
0
!
2
2
2
p
"
c
!
0
!
!
!
M
!
: (6.117)
Appendix 6A 333
A quasioptical three-port microwave Faraday rotation isolator/circulator was
demonstrated at 35 GHz [90]. After exiting a polarizer plate that defines a lin-
early polarized beam, the axis of polarization is first rotated 45
ı
as it passes through
an axially magnetized ferrite medium designed thickness. After reflection, the re-
turn beam undergoes an additional 45
ı
rotation when it traverses the ferrite for
the second time. Because of the nonreciprocal property of the magnetized ferrite,
the polarization axis is then orthogonal to the axis of the polarizer plate and is de-
flected, thereby accomplishing its isolation function of protecting the signal source
and directing the return signal into a receiver channel when desired.
In Chap. 7, this topic is expanded to include quantum mechanical transitions.
At wavelengths in the far-infrared region, resonant permeability effects occur from
exchange field interactions. In the visible and ultraviolet bands, electrical permittiv-
ity frequency spectra arise from selected electric dipole transitions from spin–orbit
multiplet structure.
Appendix 6A Transverse Permeability Tensor
The Polder–Smit tensor that relates rf magnetization with transverse rf magnetic
field H
rf
D H
x
rf
i C H
y
rf
j in a dc field H along the z-axis can be expressed in
general form as M
rf
D ΠH
rf
,or
0
@
M
x
rf
M
y
rf
M
z
rf
1
A
D
2
4
xx
xy
0
yx
yy
0
000
3
5
0
@
H
x
rf
H
y
rf
0
1
A
: (6.118)
In terms of the corresponding rf permeability, the tensor can be written as
ΠD
2
4
i0
Ci0
001
3
5
: (6.119)
For Larmor-related effects of a plane-wave propagating along the z-axis, the com-
ponents transverse to the z-axis comprise the RHCP and LHCP circular polarization
modes .1=2/
H
x
rf
iH
y
rf
. Standard solutions of the precession equation of motion
dM
dt
D .M H
i
/ (6.120)
in a semi-infinite medium and without damping were derived as [91–93]
xx
D
yy
D !
M
!
r
!
2
r
!
2
;
xy
D
yx
Di!
M
!
!
2
r
!
2
; (6.121)
334 6 Electromagnetic Properties
where !
M
D .4M/and !
r
D H
r
, guided where applicable by the various ge-
ometric and anisotropic situations described in Sect. 6.2.2. The following solutions
are approximations for the case of a semi-infinite medium in which demagnetiz-
ing influences on the rf signal are ignored. Effects of demagnetization on M
rf
are
examined in a model outlined in Lax and Button [91].
For the inclusion of damping with (6.120), the two commonly considered ver-
sions are the Gilbert (G) form of the Landau–Lifshitz term from (6.58), and the
Bloch–Bloembergen (B–B) function listed, respectively, as
dM
dt
Š .M H /
˛
M
M
dM
dt
ŒG (6.122)
and
dM
dt
.M H/
M
z
M
1
ŒB B (6.123)
From (6.122), the Gilbert relations with damping parameter ˛ follow if we replace
!
r
with !
r
C i!˛ (where !˛ D 1=
1
/ (6.121) to obtain for the complex
xx
D
0
xx
i
00
xx
0
xx
D !
M
(
!
r
!
2
r
!
2
1 ˛
2
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
;
00
xx
D !
M
(
!˛
!
2
r
C !
2
1 C ˛
2
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
(6.124)
and for
xy
D
00
xy
C i
0
xy
, chosen to comply with the sign convention for
xy
Di [91],
0
xy
D !
M
(
!
!
2
r
!
2
1 C ˛
2
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
;
0
xy
D !
M
(
2!
r
!
2
˛
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
: (6.125)
Conformance to the permeability elements from (6.119) can then be established as
D
0
i
00
,and D
0
i
00
,
0
D 1 C !
M
(
!
r
!
2
r
!
2
1 ˛
2
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
;
00
D !
M
(
!˛
!
2
r
C !
2
1 C ˛
2
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
(6.126)
Appendix 6A 335
and
0
D !
M
(
!
!
2
r
!
2
1 C ˛
2
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
;
00
D !
M
(
2!
r
!
2
˛
!
2
r
!
2
.1 C ˛
2
/
2
C 4!
2
r
!
2
˛
2
)
: (6.127)
For the B–B model, damping is introduced into (6.121) by the substitution of
! i .1=
1
/ for ! instead of !
r
C i .1=
1
/. The permeability relations are
00
D 1 C !
M
(
!
r
!
2
r
!
2
C 1=
2
1
!
2
r
!
2
C 1=
2
1
2
C 4!
2
=
2
1
)
;
00
D !
M
(
2!
r
!=
1
!
2
r
!
2
C 1=
2
1
2
C 4!
2
=
2
1
)
(6.128)
and
0
D !
M
(
!
!
2
r
!
2
1=
2
1
!
2
r
!
2
C 1=
2
1
2
C 4!
2
=
2
1
)
;
00
D !
M
(
!
2
r
C !
2
C 1=
2
1
=
1
!
2
r
!
2
C 1=
2
1
2
C 4!
2
=
2
1
)
: (6.129)
For the two counterrotating modes of circular polarization, we define
˙
D
0
˙
i
00
˙
D ˙ , with the Gilbert result
0
˙
D 1 C !
M
.!
r
!/
.!
r
!/
2
C 4.!˛/
2
;
00
˙
D !
M
2!˛
.!
r
!/
2
C 4.!˛/
2
: (6.130)
Solutions equivalent to those of the B–B model in terms of a relaxation time
1
can
be obtained by substitution for the dimensionless damping parameter ˛ D .!
1
/
1
or !=! in terms of half-linewidth .!/, according to
0
˙
D 1 C !
M
.!
r
!/
.!
r
!/
2
C 4.!/
2
;
00
˙
D !
M
2!
.!
r
!/
2
C 4.!/
2
: (6.131)
336 6 Electromagnetic Properties
In another investigation, a term was added to the B–B model to make it more com-
patible with the G version, provided that ˛ D .!
r
1
/
1
D !=!
r
[94]
dM
dt
Š .M H /
M
z
M
1
C
ˇ
ˇ
ˇ
ˇ
M
H
ˇ
ˇ
ˇ
ˇ
H
1
: (6.132)
In this case, the circular polarization permeability components are expressed as
0
˙
D 1 C !
M
8
ˆ
<
ˆ
:
.!
r
!/
.!
r
!/
2
C 4
!
!
r
!
2
9
>
=
>
;
;
0
˙
D !
M
8
ˆ
<
ˆ
:
2!
.!
r
!/
2
C 4
!
!
r
!
2
9
>
=
>
;
: (6.133)
Many microwave applications are operated away from resonance, usually where it
can be assumed that ! !
r
!. In these situations, the two damping schemes
can be compared directly as
0
˙
1
!
M
!
;
00
˙
2!
M
!
!
!
˛ D !=!; (6.134a)
0
˙
1
!
M
!
;
00
˙
2!
M
!
r
!
!
˛ D !=!
r
: (6.134b)
Based on inspection of (6A.134), the choice of relation for ˛ will influence mainly
the absorption component through the introduction of frequency dependence to the
linewidth term.
Appendix 6B Classical Instability Threshold
From (6.80)and(6.81) for the transverse rf susceptibility, an expression for the
z-axis magnetization component in a significant H
rf
field can be expressed as
.4M
z
/
2
D.4/
2
M
2
M
2
T
D .4M /
2
.4M /
2
.H
rf
/
2
.H N
Dz
4M
z
!/
2
C
2
2k!0
:
(6.135)
Appendix 6B 337
Since 4M
z
D .4/ .M –M
z
/,whereM
z
is the change in the z component
caused by H
rf
, we can write
.4M
z
/
2
.4M /
2
D
.4M /
2
.H
rf
/
2
ŒH N
Dz
4 .M M
z
/ !
2
C
2
2k!0
: (6.136)
After factoring the left-hand side of (6.136),
.4 M
z
/Œ4.M
z
C M/ D
.4M /
2
.H
rf
/
2
ŒH N
Dz
4 .M M
z
/ !
2
C
2
2k!0
;
(6.137)
which reduces to
4 M
z
Š
1
2
4M .H
rf
/
2
.N
Dz
4 M
z
/
2
C
2
2k!0
: (6.138)
if we assume that 4 .M
z
C M/ Š 2.4M/and fix the initial resonance condition
.H
rf
;M
z
! 0/ as H N
Dz
4M ! D 0.
Analytically, (6.138) can be viewed as cubic in M
z
, and the possibility of triple
solutions was discussed by Anderson and Suhl [67]. The physical reality, how-
ever, dictates that there will be only one stable solution, and this can be examined
graphically by plotting the left- and right-hand sides (LHS and RHS) separately as
functions of 4M
z
, sketched in Fig. 6.35. A straightforward iteration procedure
quickly determines that convergence to a single value of 4M
z
will occur only
if the slope of the right-hand side is greater than 1. Therefore, we write for the
threshold condition
N
Dz
4M
H
crit
rf
2
N
Dz
4M
crit
z
h
N
Dz
4M
crit
z
2
C
2
2k!0
i
2
1: (6.139)
If (6.138)and(6.139) are combined, a simple relation between M
z
and H
emerges:
N
Dz
4M
crit
z
D
1
2k!0
D H: (6.140)
When N
Dz
4M
z
is replaced by
1
2k!0
in (6.139), the critical rf field at the new
resonance peak (that is now shifted to lower H by an amount H D N
Dz
4M
z
)
becomes
H
crit
rf
D 2
3=2
2k!0
.N
Dz
4M /
1=2
; (6.141)
or in terms of uniform precession half-linewidth
H
crit
rf
D 2H
3=2
k!0
.N
Dz
4M /
1=2
: (6.142)
338 6 Electromagnetic Properties
Fig. 6.35 Convergence diagram of the classical iteration solution to determine the magnitude of
4M
z
caused by a strong rf drive field
Appendix 6C Domain Wall Susceptibility Equation
Below 1 GHz, the real part of the magnetic susceptibility as a function of frequency
is controlled by two mechanisms. The more universal one is magnetization rotation
that is a simple extension of the nonresonant relaxation in paramagnets described
in Sect. 6.1.1. The more important one, however, is domain-wall resonance that
is encountered in multidomain ferromagnets. Domain walls can be shown to have
mechanical-like properties of a forced damped harmonic oscillator that is character-
ized in terms of effective wall mass m
w
, restoring force ˛
w
x when disturbed from
equilibrium in the x direction, and motional damping force ˇ
w
.dx=dt/, per unit
wall area.
An equation of motion for domain walls under the action of a z-axis alternating
magnetic field H
ac
of angular frequency ! was devised as [80,95,96]
m
w
Rx C ˇ
w
Px C ˛
w
x D 2M
s
H
ac
cos .!t/ ; (6.143)
where
m
w
D
1
4
2
r
j
K
1
j
A
.wall mass area/
Appendix 6C 339
Fig. 6.36 Model of domain-wall displacement under the influence of a longitudinal ac magnetic
drive field. Figure reprinted from [80] with permission.
c
2003 by the IEEE
A D
JS
2
a
0
.spin exchange energy/lattice parameter/
ˇ
w
D
m
w
w
.
w
D unforced damping time constant/
˛
w
D
4M
2
s
3
i
&
w
.&
w
D total domain wall surface area/volume/
i
D
2M
2
s
j
K
1
j
.K
1
<0/I
4M
2
s
3
j
K
1
j
.K
1
>0/
ı
w
D
s
A
j
K
1
j
D
s
JS
2
a
0
j
K
1
j
.wall thickness/ (6.144)
From inspection of Fig. 6.36, we reason that the wall displacement of magnitude x
driven by H
ac
sweeps out a fractional change in volume equal to
w
x and produces
a change in net magnetization of
M
ac
D 2M
s
&
w
x; (6.145)
or in terms of the real part of the initial susceptibility
0
ac
D
M
ac
H
ac
D
2M
s
&
w
x
H
ac
: (6.146)
Therefore, we can now express (6.143) as (after dropping the subscript ac)
R
0
C
1
w
P
0
C !
2
w
0
D
.2M
s
/
2
&
w
m
w
cos .!t/ ; (6.147)
where !
2
w
D ˛
w
=m
w
D .8=3/
2
p
A
j
K
1
j
&
w
for the more common case of K
1
<0.
340 6 Electromagnetic Properties
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