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Effect of Demagnetizaing Field on Frequency
Dispersoin of Complex Permeability
Sho Muroga, Masahiro Yamaguchi
Graduate School of Engineering
Tohoku University
Sendai, Japan
muroga@ecei.tohoku.ac.jp
Abstract— This paper describes that the ferromagnetic
resonance (FMR) frequency shift can be calculated by an
electromagnetic field simulator based on the Maxwell’s
equation although the relative permeability and the FMR
frequency is defined by the Landau-Lifshitz-Gilbert (LLG)
and the Kittel’s equation. We evaluate the magnetic circuit
model with the leakage magnetic flux path for considering
the demagnetizing field generated in the magnetic film. As
the result, we clarify that the effect of the demagnetizing
field is considered as a reluctance of the leakage magnetic
flux path in the magnetic circuit calculation, which is
considered as the increase of anisotropy field in the Kittel’s
equation. Furthermore, we show that the simulated FMR
frequency by the electromagnetic simulator agrees with
the measured values.
Keywords—on-chip
electromagnetic
noise
suppressor;
ferromagnetic films; ferromagnetic resonance frequency;
demagnetizing field; electromagnetic simulator
I.
INTRODUCTION
The demands on compact analog and digital mixed-signal
chips are increasing due to the development of mobile IT
device. In such mixed signal chips, a high-frequency noise
generated by the digital circuits is transferred to the on-chip
analog circuits. This noise affects the analog circuit
performance and causes serious desensitization [1], [2].
To solve this problem, we proposed a new integrated soft
magnetic noise suppressor that is set on the transmission line
between the noise source and the victim circuit [3]–[7]. The
soft magnetic thin film dissipates noise power only in the radio
frequency (RF) range because of ferromagnetic resonance
(FMR) and Joule losses. A representative point is that the
insertion loss was only 4 dB at 1 GHz, and increasing sharply
to !# %!85Zr3Nb12 film integrated
! !"!%" a soda glass substrate [4].
We also reported that the FMR frequency is sensitive to
shape anisotropy of magnetic material. It is an important design
parameter of the integrated electromagnetic noise suppressor
[7]. Given the shape of a magnetic material, the FMR
frequency shift could be calculated by using the demagnetizing
factor.
Copyright 2014 IEICE
Fig. 1. Calculation diagram of FMR frequency shift in the magnetic
material.
This consideration leads to an idea that this shift could be
calculated by an electromagnetic field simulator based on the
Maxwell’s equation although the FMR frequency itself is
defined by the Landau-Lifshitz-Gilbert (LLG) equation. This
paper discusses the validity of this idea using a magnetic circuit
model of the transmission line with a magnetic film.
II.
RELATIVE PERMEABILITY
A. Theory
It is known that the LLG equations give the relative
permeability r. The relative permeability and the FMR
frequency of a magnetic film with one uniaxial anisotropy are
calculated as (1) - (3).
2M s2
1
r 2
0 2f r0 2f 2 2 jf
Ms
4 0
f r0 2
(1)
(2)
MsHk
0
(3)
Here, Ms is the magnetization, is the gyromagnetic constant,
is the damping constant, Hk is the anisotropy field.
Fig. 1 shows the calculation diagram of FMR frequency
shift when the demagnetizing field generates in the magnetic
material. In this case, the FMR frequency frd shifts to higher
frequency range by the demagnetizing field energy. The frd is
calculated by well-known Kittel’s equation as shown in (4) of
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Fig. 4. The cross-sectional view of the microstrip line.
Fig. 2. Measured magnetization curve of CoZrNb film.
Fig. 5. Ploss/Pin in the microstrip line with the CoZrNb film.
Fig. 3. Frequency dependence of the relative permeability of the CoZrNb
film.
the motion equation of the magnetization, which is described
on the left side diagram in Fig. 1 .
f rd 2
M s H k N d M s 0
(4)
III.
Here, Nd is the demagnetizing factor.
Thus, the FMR loss of the magnetic material behaves as the
band elimination filter, and the center frequency of the filter is
basically equal to the FMR frequency and shift to higher
frequency range by demagnetizing field given the shape of a
magnetic material.
B. Measurements
The 250-nm-thick CoZrNb film was deposited using RF
sputtering onto a glass epoxy substrate. The length of the
magnetic film lm is 20 mm.
Fig. 2 shows the in-plain M-H curve of CoZrNb film
measured with a vibrating sample magnetometer (VSM). The
CoZrNb film has in-plane uniaxial anisotropy and exhibit high
permeability only in the direction of the hard axis of
magnetization. The easy axis is an energetically favorable
direction of the magnetization and perpendicular to the hard
axis. The saturation magnetization Ms is about 1 T, and the
magnetic anisotropy Hk is 1.2 kA/m.
Copyright 2014 IEICE
Fig. 3 shows the frequency dependence of the relative
permeability of the CoZrNb film measured by a high
frequency permeameter with a side-open TEM cell to generate
RF field and a planar shielded-loop type one turn coil to detect
magnetic flux as the induced voltage (Model PMM-9G1,
Ryowa Electronics Co.) [9]. The complex permeability, r =
r'' - r'", is obtained through the S-parameters in terms of the
flux voltage. The solid line shows the real part and the dashed
line shows the imaginary part of the relative permeability,
respectively. The real part of the permeability at low
frequencies is 630 and the imaginary part of the permeability
is 1520 at an FMR frequency of 1.1 GHz.
TRANSMISSION LINE
The MSL is fabricated on a glass epoxy substrate having a
permittivity of about 3.8. Fig. 4 shows the cross-sectional view
of the MSL with a magnetic film. The signal and ground lines
are about 35 % thick. The width of the signal line and ground
gap is 160 % and 30 %, respectively. The design considers
impedance matching for standard lossless transmission lines
without the magnetic film. The length of the MSL is 20 mm.
Fig. 5 shows the ratio of the measured power loss to the
incident power Ploss/Pin in the MSL covered with and without
the CoZrNb film. The loss increases with increasing frequency
in MSL covered with the CoZrNb film. These losses are
mainly caused by the eddy currents, and smaller FMR losses
are generated around 3 GHz. The peak frequency of the FMR
loss is 3.1 GHzand is higher than the intrinsic FMR frequency
of 1.1 GHz because of demagnetizing field [7].
IV.
MAGNETIC CIRCUIT MODEL
Fig. 6 shows the magnetic flux around the signal line. There
are large magnetic dipoles because of the local magnetic flux
caused by the current in signal line [8]. The leakage magnetic
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Fig. 6. Leakage magnetic flux above signal line.
Fig. 8 Comparison with the measured and calculated complex
permeabilities of a CoZrNb with Ms = 1 T, Hk = 1.2 kA/m
considering demagnetizing field.
Thus, the reluctance of the leakahe magnetic flux Rl is
calculated as
Fig. 7. Magnetic circuit network model considering leakage magnetic flux
from the magnetic film.
flux above and below the magnetic film is generated by these
dipoles. Fig. 7 shows the magnetic circuit model. R'm and R"m
are real and imaginary parts of the magnetic reluctance in
magnetic film and calculated as
Rm Rm' jR""m
Rm ' ws
'r
2
2
' r "r 0 t m l m
Rm " ws
"r
2
' "r 0 t m l m
2
r
(5)
R total R' m jR" m (8)
are the average magnetic flux density in the magnetic film and
the around the magnetic film. Therefore, the demagnetizing
factor Nd is calculated by equation (8).
Copyright 2014 IEICE
(9)
(10)
This equation shows that the reluctance of the leakage
magnetic flux path is described using the demagnetizing factor.
This is because that the distribution of the demagnetizing field
is determined by the distance of the magnetic poles.
(7)
The reluctance of the leakage magnetic flux Rl is evaluated
as follows. The flux generated by the demagnetizing field in
the magnetic film is distributed as shown in Fig. 4. Here, the
distance where the leakage magnetic flux density becomes 1/e
compared with that on the surface of the magnetic film and is
calculated to be 80 %. In this case, the magnetic flux $ is
satisfied the eq. (8).
B
t
Nd l m
Bm
2z l
ws
ws
Nd
0 alm
0tmlm
Thus, the combined reluctance Rtotal of the magnetic film
including the leakage flux path was calculated in order to
obtain the relative permeability considering the demagnetizing
field. Rtotal is defined as
(6)
Where the total magnetic flux 0 is constant supposing the
reluctance of the air gap is large enough because of the shape
and the size of the magnetic film and the MSL.
B m t m l m Bl 2 z l l m
Rl ws
1
1
Rl 0 reff t m l m
2
(11)
Here, r_cir is the relative permeability considering the
demagnetizing field. Therefore, r_cir is calculated as
r_cir ws
t l
0 R' m jR"m Rl m m
(12)
The FMR frequency considering the demagnetizing field is
calculated by analyzing the combined reluctance of the
magnetic film and the leakage magnetic flux path.
In this case, the equation (14) needs to be satisfied in order
to obtain the FMR frequency [9].
ReR total R' m 1
Rl 0
2
(13)
Thus, the FMR frequency fr_cir is calculated with (1)-(3), (6),
(9) and (13).
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f rc 2
M s H k N d M s 0
16P2-B4
VI.
(14)
This equation is equal to the Kittel’s equation. This agreement
between (4) and (14) shows that the FMR frequency shift by
the demagnetizing field can be calculated by the magnetic
circuit.
V.
MEASUREMENT AND CALCULATION RESULTS
Fig. 8 compares the permeabilities of a CoZrNb film. Solid
line shows the measured r for a film large enough to neglect
demagnetization and the measured r_w/8kA/m of a large film
with a static magnetic field of 8 kA/m applied along the easy
axis.
The intensity of 8 kA/m is identical to the demagnetization
field Hd calculated as equation (15) which is defined as the
increase of the magnetic anisotropy field in equation (4).
Hd Nd M s
(15)
In this measurement, the Nd is supposed 0.001 because the
dimensions of the magnetic film are; width wm = %
length lm = %! tm = 100 nm.
It is well known that the effective anisotropy field is linear
sum of the intrinsic anisotropy field Hk and the demagnetizing
field Hd.
H keff H k H d
(16)
Therefore, the effective FMR frequency in the case that the
magnetic field as same as the Hd is applied to parallel direction
to the hard axis of the magnetic film large enough to neglect
demagnetization is equal to the FMR frequency in the case that
the demagnetizing field generated in the magnetic film. This is
the known-way method in the magnetic engineering.
The dashed line is shows the relative permeability r_cir
given by the magnetic circuit and calculated by equation (12).
The dotted line shows the simulated permeability r_sim using
the electromagnetic simulator.
Fig. 7 shows the good agreement between the solid, dashed
and dotted lines. The result shows the good agreement with the
FMR frequency fr_w/8kA/m measurement result with applied field
of 8 kA/m, the fr_cir calculated by the magnetic circuit and the
fr_sim simulated by the electromagnetic simulation. Furthermore,
the entire waveform of the real and imaginary part of the
relative permeability is matched with each other. Additionally,
the FMR frequency considering the demagnetizing field is
successfully obtained at the 3.1GHz as same frequency of the
measured FMR loss peak in Fig. 5.
From these results, frequency dependence of the complex
permeability when the demagnetizing field generates in the
magnetic material is calculated by the magnetic circuit and the
electromagnetic field simulator. In another word, the
electromagnetic field simulator can calculate the relative
permeability in the case that the demagnetizing field generates
in the magnetic material only with the intrinsic permeability
and shape of the magnetic material.
Copyright 2014 IEICE
CONCLUSIONS
In this paper, we discussed that the FMR frequency shift
could be calculated by an electromagnetic field simulator
based on the Maxwell’s equation although the relative
permeability is defined by LLG equation.
We clarified that the effect of the demagnetizing field is
considered as a reluctance of the leakage magnetic flux path in
the magnetic circuit calculation, which is considered as the
increase of anisotropy field in the Kittel’s equation.
Furthermore, we showed that the calculated relative
permeability by magnetic circuit model and simulated value
agree with the measured value.
From these results, the electromagnetic field simulator can
calculate the relative permeability in the case that the
demagnetizing field generates in the magnetic material only
with the intrinsic permeability and shape of the magnetic
material.
This result not limited to the fundamental and technological
view for the on-chip electromagnetic noise suppressor using
the magnetic material, and also important knowledge for all
device design using the ferromagnetic resonance.
ACKNOWLEDGMENTS
This work was supported in part by Development of
Technical Examination Services Concerning Frequency
Crowding from the Ministry of Internal Affairs and
Communications of Japan. This work was also supported in
part by KAKENHI 25820131.
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