3.23. MAGNETOSTATIC SURFACE WAVES .1 C X / > 0 139
Concerning the azimuthal dependence, only the e
j n'
is allowed for reasons explained
below.
In view of the above, a general solution, which is also in agreement with that of
Sparks [21], can be obtained. us,
Ferrite disk:
‰
i
D ‰
0i
J
n
.k
/e
j n'
cos
k
zi
z
n
2
for
d
2
z
d
2
(3.183a)
Air region: ‰
e
D
(
‰
0e
J
n
.k
/e
in'
e
˛.zd=2/
for z d=2
‰
0
0e
J
n
.k
/e
in'
e
C˛.zCd=2/
for z d=2:
(3.183b)
Both continuity conditions of the tangential magnetic field .h
D @‰=@ and h D
.1=/@‰=@'/ are reduced to the same expressions:
At z D Cd=2 W ‰
0i
cos.k
zi
d=2 n=2/ D ‰
0e
(3.184a)
At z D d=2 W ‰
0i
cos.k
zi
d=2 C n=2/ D ‰
0
0e
: (3.184b)
On the other hand, the continuity of the normal flux density b
n
D b
z
D
0
@‰=@z yields
At z D Cd=2 W k
zi
‰
0i
sin.k
zi
d=2 n=2/ D ˛‰
0e
(3.185a)
At z D d=2 W Ck
zi
‰
0i
sin.k
zi
d=2 C n=2/ D C˛‰
0
0e
: (3.185b)
By expanding either the cosinus in (3.184) or the sinus in (3.185), one may easily show
that ‰
0
0e
D ‰
0e
for even modes n D 0; 2; 4; : : :; namely, even modes are symmetric with respect
to the slab midplane z D 0. Likewise, ‰
0
0e
D ‰
0e
for odd modes n D 1; 3; 5; : : :, which are
anti-symmetric with respect to z D 0. In turn, dividing (3.185) by (3.184) yields the expected
characteristic equation already given in (3.107a):
MSFVW waves: tan
k
zi
d
2
n
2
D
˛
k
zi
D
k
k
zi
: (3.186)
e remaining analysis given in this section is essentially the same and leads to the same
mode characteristics. What would be interesting here is to consider a ferrite disk of finite radius.
However, in this case, the general solution must account for the additional boundary conditions
at the ferrite disk perimeter D R. e key to this analysis is zero magnetization in the assumed
air region > R.
3.23.7 FINITE FERRITE DISKS
Consider a transversely magnetized circular ferrite disk with radius D R. e general solution
for the scalar magnetic potential ‰ inside the disk is given again by (3.183); that is, it remains
140 3. FINITE FERRITE SAMPLES
the same as that for an infinitely extending disk. However, magnetization outside the disk in the
region > R should be zero. is requirement provides the key boundary conditions at D R
which accounts for the disk’s finite extent. Recall that magnetization Nm is related to the magnetic
field
N
h through the susceptibility tensor ŒX by:
Nm D ŒX
N
h: (3.187)
e ŒX elements are given in (3.70). In turn,
N
h is related to the magnetic potential ‰,
which must be expanded now in cylindrical coordinates as follows:
N
h D r‰
i
D
O
@‰
i
@
C O'
1
@‰
i
@'
C Oz
@‰
i
@z
: (3.188)
It has already been explained in the previous sections that Nm represents the microwave
magnetization, which under the small signal approximation has a negligible component in the
DC biasing direction, that is, m
z
0. Substituting (3.188) into (3.187) yields
Nm D ŒX
r‰
i
for R and d=2 z d=2: (3.189)
By exploiting the general solution for ‰
i
given in (3.183a) and using the susceptibility
definition, the m
and m
'
components read as follows:
m
D X h
j k
1
h
'
D X
@‰
i
@
j k
1
1
@‰
i
@'
(3.190a)
m
'
D j k
1
h
'
X h
'
D j k
1
@‰
i
@
X
1
@‰
i
@'
: (3.190b)
Carrying out the differentiations, the above expressions yield
m
D
0i
e
j n'
cos
k
zi
z n
2
X k
J
0
n
k
C k
1
n
J
n
k
(3.191a)
m
'
D j
0i
e
j n'
cos
k
zi
z n
2
k
1
k
J
0
n
k
X
n
J
n
k
: (3.191b)
From (3.191) it can be observed that asking for both m
and m
'
to vanish at the disk
edge D R yields two inconsistent equations, at least at first glance. is requirement is also
known as “spin pinning” at the ferrite surface. According to Maksymowitz [43], for example, the
inconsistencies may arise due to localized non-uniformities around the disk edges. Concerning
the demagnetization factors of the infinite disk, it was assumed that N
z
D 1 and N
x
D N
y
D 0,
but this is violated around the edge, where D R. However, this non-uniformity is localized
and disappears at a small distance (inward) from the edge. Nevertheless, pinning the spins at
3.23. MAGNETOSTATIC SURFACE WAVES .1 C X / > 0 141
the disk edge requires that:
m
. D R/ D 0 $ k
X J
0
n
k
R
k
1
n
k
R
J
n
k
R
D 0 (3.192a)
m
'
. D R/ D 0 $ k
k
1
J
0
n
k
R
X
n
k
R
J
n
k
R
D 0: (3.192b)
For these equations to have a non-trivial solution, X and k
1
must be approximately equal,
X k
1
, or equivalently, the circular frequency (!) must be around !
i
, that is, ! !
i
. ese
may also be true in the limit k
R 1, which was first stated in [21]. In view of this and the
formula for the Bessel functions derivatives, e.g., Abramouvitz and Stegun [39],
J
0
n
.z/
D
J
nC1
.z/ C
n
z
J
n
.z/ D J
n1
.z/
n
z
J
n
.z/: (3.193)
Both (3.192) conditions are reduced to:
J
n1
k
R
D 0: (3.194)
Likewise, the magnetization components can be written as:
m
D
0i
X k
J
nC1
k
e
j n'
cos
k
zi
z n
2
(3.195a)
m
'
D
0i
j X k
J
nC1
k
e
j n'
cos
k
zi
z n
2
: (3.195b)
Besides the above, the boundary conditions at
z
D ˙
d=2
still require the validity
of (3.186), but now k
should satisfy (3.194). e consequence of these requirements is that
the radial wavenumber k
takes only discrete values for a finite disk instead of being a continu-
ous variable in the infinite film case. By symbolizing the mth root of the Bessel function J
nC1
with P
nC1;m
; k
is obtained as
J
nC1
.
P
nC1;m
/
D 0 $ k
D P
nC1;m
=R: (3.196)
ese roots are given in Table 3.1 below, e.g., Pozar [3].
A variational calculation applied on (3.192) yields a resonance frequency for the dominant
(n C 1 D 0) mode [42], as:
!
d
1
2
!
m
(
1
1 e
k
=d
k
d
)
: (3.197)
is expression is extracted without any pinning at the interfaces z D ˙d=2.
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