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Frequency-induced Negative Magnetic Susceptibility in
Epoxy/Magnetite Nanocomposites
Che-Hao Chang1, Shih-Chieh Su2, Tsun-Hsu Chang1, 2*, & Ching-Ray Chang3**
1Interdisciplinary Program of Sciences, National Tsing Hua University, Hsinchu, Taiwan
2Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
3Department of Physics, National Taiwan University, Taipei, Taiwan
*To whom the correspondence should be address: thschang@phys.nthu.edu.tw
**To whom the correspondence should be address: crchang@phys.ntu.edu.tw
ABSTRACT
The epoxy/magnetite nanocomposites express superparamagnetism under a static or a low-frequency
electromagnetic field. At the microwave frequency, said the X-band, the nanocomposites reveal an unexpected
diamagnetism. To explain the intriguing phenomenon, we revisit the Debye relaxation law with the memory effect.
The magnetization vector of the magnetite is unable to synchronize with the rapidly changing magnetic field, and
it contributes to superdiamagnetism, a negative magnetic susceptibility for nanoparticles. The model just developed
and the fitting result can not only be used to explain the experimental data in the X-band, but also can be used to
estimate the transition frequency between superparamagnetism and superdiamagnetism.
Introduction
The paramagnetic materials will be aligned with the direction of the externally applied magnetic field, and hence the real
part of the susceptibility will be positive. The diamagnetic materials will be arrayed in the opposite direction, and hence result
in the negative real susceptibility. Most metals, like copper and silver, will express a weak diamagnetism since the averaged
orbital angular momentum of electrons will be in the opposite direction of the magnetic field, called the Langevin
diamagnetism1-2. The generated magnetic susceptibility is usually very small, around 10-4-10-5. For magnetic materials with
small particle sizes, the superparamagnetism will be observed from the thermal agitation3-6. The epoxy/magnetite (Fe3O4)
nanocomposites show superparamagnetism as expected at low frequencies3-6, but our experiment revealed that the Fe3O4 nano-
powder also exhibits peculiar diamagnetism in the X-band. A similar phenomenon was observed on La0.7Sr0.3MnO3
nonocomposites7 and FeNi3/C nanocapsules8. As the order of the negative susceptibility of those materials is much higher than
10-4 from the Langevin diamagnetism, there must be a different mechanism accounting for the transition from paramagnetism
to diamagnetism in our epoxy/magnetite (Fe3O4) nanocomposites.
The electromagnetic properties of nano-materials are different from their bulk counterpart. A well-known example is that
the color of gold particles depends on the sizes9. Besides, the ferromagnetic particles, like magnetite, display
superparamagnetism when the size is in the nanoscale. The superparamagnetic effect is similar to the paramagnetic effect but
with much higher magnetic susceptibilities10. The existence of superparamagnetic property arises since the relaxation time
scale of the nanoparticles becomes much smaller than usual due to the reduction of the particle volume11-13. However, this
work shows that the superparamagnetic property become diamagnetism with high magnetic susceptibilities which is at least 4
orders of magnitude higher than the Langevin diamagnetism at high frequencies as the reason we call this phenomenon
superdiamagnetism. Under usual circumstances, the permeability of superparamagnetic particles can be explained by the
Debye relaxation model14, such as the FeAl@(Al, Fe)2O3 nanoparticles15, the Fe/Ag/citrate nano-composites in 2-18 GHz16,
and magnetite in kHz frequency range17. Nevertheless, the Debye relaxation model fails to explain the negative susceptibility,
and hence an amended physics mechanism is needed.
Depending on the structure of the materials, the spins will naturally be in stable states, that is, in the directions that make
the free energy reach a minimum. For a nanoscale material, the thermal fluctuation allows the spins to transit between stable
states with the volume reduction of energy barriers. The transition of states contributes to the magnetic susceptibility of
superparamagnetic particles. The work of Klik et al.18 shows that the transition of states may contribute to the negative
magnetic susceptibility by considering the memory effect of nanoparticles. They considered the correction of the master
equation of the spins by an exponential memory kernel. Reference [18] predicts that the material with the uniaxial anisotropy
will express diamagnetism when the frequency is greater than a threshold frequency. However, there is no detailed analysis of
the physics origins of the high-frequency diamagnetism and also no experimental evidence to support the prediction of the
memory induced diamagnetism.
Moreover, Ref. [18] focused their discussion on uniaxial particles having just two stable states. The material of interest
(i.e., the magnetite) carries the cubic anisotropy which has six or eight states, depending on the sign of anisotropy constant19.
As a result, the nano Fe3O4 has three distinct transition rates with multiple relations20. Nevertheless, when it comes to the
permeability, the lowest transition rate dominates the magnetization. This work proposes a model considering the cubic
anisotropy, and the results agree well with our experimental findings.
Results
Permittivity
The transmission/reflection methods can characterize materials’ electromagnetic (EM) properties over a broad frequency
range. The measured transmission/reflection coefficients using a network analyzer uniquely determine the complex
permittivity
ε
and the complex permeability
µ
, when the sample thickness d is smaller than a quarter of the guide wavelength
(
g
λ
)21. Here, we adapt the relative permittivity
0
/
εε
(
i
εε
′ ′′
= +
) and permeability
0
/
µµ
(
i
µµ
′ ′′
= +
).
0
ε
and
0
µ
denote
the permittivity and the permeability of vacuum. Four different volume fractions of epoxy/magnetite composites (0%, 6%,
12%, and 18%) are measured. Then, the effective medium theory is introduced to extract the EM properties of the nano Fe3O4
powder.
While there are several different effective medium theories, Chang et al. discussed three different models and concluded
that the Looyenga model22-23 is a suitable model to fit the permittivity of the epoxy/Fe3O4 nanocomposites3. It reads,
13 13 13
f
eff h f
(1 ) ,
f
vv
ε εε
=−+
(1)
where
eff
ε
,
h
ε
and
f
ε
are the permittivity of the composites, the host medium (epoxy), and the filled material (Fe3O4),
respectively.
f
v
is the volume fraction of the filled materials.
The measured effective complex permittivity of nanocomposites with different volume fractions are presented in Fig.
1(a). In our experiment, epoxy was chosen to be the host medium. It carries relative permittivity
3.1 0.1i+
and relative
permeability near 1.0 with high stability within a broad frequency range. The EM properties of epoxy are confirmed by our
transmission/reflection method. We calculate the corresponding
f
ε
based on Eq. (1). The extracted
f
ε
is shown in Fig. 1(b).
The extracted complex permittivities
f
ε
differ slightly for different volume fractions
f
ν
. The difference might be attributed
to the error of the measured volume fraction
f
v
.
Permeability
The measured permeability of nanocomposites with different volume fractions are presented in Fig. 2(a). As mentioned,
the permeability of pure epoxy is near 1.0, so the magnetic susceptibility comes strictly from the existence of Fe3O4. To explain
the data, we reasonably assume that the magnetic susceptibility is linearly proportional to the volume fraction of the nano
Fe3O4 powder. That is,
eff eff f f f
( ),ivi
χ χ χχ
′ ′′ ′ ′′
+= +
(2)
where
eff eff
i
χχ
′ ′′
+
and
ff
i
χχ
′ ′′
+
are the magnetic susceptibility of the composites and the filled material (Fe3O4), respectively.
The extracted
f
µ
using Eq. (2) is shown in Fig. 2(b) with
ff f f
+ (1 )ii
µµ χ χ
′ ′′ ′ ′′
=++
.
For the Fe3O4 structure, the free energy of the particles will be23
2 2 22 2 2 0
( ) cos ,
s
E KV HM V
αβ βγ γα µ ϕ
= ++ −
(3)
where
K
is the anisotropy constant,
V
is the volume,
,,
αβγ
are the direction cosines of the magnetization vector along the
x, y, and z axes,
s
M
is the saturated magnetization,
H
is the amplitude of the applied field, and
cos
ϕ
is the direction cosine
of the magnetization vector along the H-field.
The magnetization vector of Fe3O4 has eight stable states, all having the form
αβγ
= =
. We, therefore, denote the
stable states as the form
[111]
or
111
. The numbers are used to represent the ratios of direction cosines, and we put a bar
above the first, second, or third number if
,
αβ
or
γ
is negative. To describe the susceptibility, we define
1234567
,,,,,,nnnnnnn
, and
8
n
as the probability of occupation of
[111], 111 , 1 11 , 11 1 , 111 , 1 1 1 , 111 ,
and
111
. Since the antiparallel occupation will cancel each other out, we can further simplify the expression by defining
1 1 52 2 63 3 7
,,m n nm n nm n n
=−=−=−
, and
4 48
mnn= −
. The four variables give us enough information to get the
magnetization.
We consider the case that direct transitions happen only between two adjacent states16. When considering the memory
effect, the master equation of
1234
[]
T
mmmm=m
is
3111
1 311
+ with ,
1 131
1 113
qh
−
−−−
Θ+ = Γ =Γ
−−−
−−−
00
m m fm v f
(4)
where
Θ
is the memory time and
0
exp( /12)
qΓ=Γ −
. A detailed derivation of how we deduce Eq. (4) from the exponential
memory kernel can be found in the Supplementary information Part I.
0
Γ
is a constant in the unit of frequency24, representing
the transition frequency under the high-temperature limit and
/( )
B
q KV k T=
. The term
exp( /12)q−
comes from the energy
barrier, like the Arrhenius equation. In the ground states, the particle carries free energy
/3KV
. When the magnetization
vector transits from a stable state to the others, it will go through a saddle point as the form of
[ ]
110
with energy
/4KV
, and
therefore, the energy barrier is
/12KV−
. For the second term at the right-hand side,
0
/( )
s
h MH K
µ
= −
and
v
can be
expressed as
1
2
3
4
cos
3 111
cos
13 1 1
1,
cos
11 3 1
4
cos
11 1 3
ϕ
ϕ
ϕ
ϕ
−−−
−
=
−
−
v
(5)
where
123
cos ,cos ,cos ,
ϕϕ ϕ
and
4
cos
ϕ
are the direction cosines of the magnetization vector along
[111], 111 , 1 11
and
111
. When there is an externally applied H-field, the transition coefficients will change, since the energy of stable states
change from
/3
KV
to
0
/ 3 cos
s
KV HM V
µϕ
+
and therefore the exponential terms of the transition rates change. Since the
externally applied H-field is small in our experiment, we only need the influences of the first order of
h
. The second term at
the right-hand side in Eq. (4) comes from the change of the transition rates.
If
0
exp( )h h it
ω
= −
with
0
h
as the amplitude and
ω
as the oscillating frequency, we can expect that
m
will oscillate
with the same frequency. Therefore, we can do the Fourier transformation for the left side. The
0
f
has an eigenvalue
6−Γ
corresponding to the eigenvector
[1111]
T
−−−
. In addition, the
0
f
has another eigenvalue
2−Γ
associated with three
degenerate eigenvectors
[ ]
110 0
T
,
[ ]
1010
T
, and
[ ]
1001
T
. The physical interpretation of eigenvalues and
eigenvectors can be found in the Supplementary information Part II. After the matrix operation, Eq. (4) becomes
21234
21 234
212 34
2123 4
(6 )( ) 1111
(2 )( 3 ) 13 1 1.
1 13 1
(2 )( 3 ) 1113
(2 )( 3 )
i mm mm
im mmm qh
i mm m m
i mm m m
ωω
ωω
ωω
ωω
Γ−Θ − − − − −−−
Γ−Θ − + − − −−
= Γ
−−
Γ−Θ − − + −
−−
Γ−Θ − − − +
v
(6)
To get magnetization
M
, we define the reduced magnetization
/
rs
m MM=
, and then the value of
r
m
is related to
i
m
’s by
1
2
1234
3
4
[cos cos cos cos ] .
r
m
m
mm
m
ϕϕ ϕ ϕ
=
(7)
For simplicity, we define
2
1
1/ 6 i
λ ωω
= Γ−Θ −
and
2
2
1/ 2 i
λ ωω
= Γ−Θ −
. Then, we get
11
2
1234 2
2
000
1111 1111
0 00
13 1 1 13 1 1
[cos cos cos cos ] .
00 0
1 13 1 1 13 1
000
1113 1113
r
m qh
λ
λ
ϕϕ ϕ ϕ λ
λ
−
−−− −−−
−− −−
= Γ
−− −−
−− −−
v
(8)
Equations (6) and (8) give us
r
m
for any direction of the externally magnetic field. However, the orientation is randomly
distributed in the experiment. Therefore what we need is the averaged
r
m
for all the possible directions, denoted as
r
m
. All
cos
nn
m
ϕ
are equal by symmetry, where n = 1, 2, 3, or 4. Therefore we only need to calculate
11
4 cos m
ϕ
.
11
1
12
2
13
2
14
2
cos cos
000
1 111 6 6 6 6
cos cos
0 00
1100 2 6 2 2
[1000] .
cos cos
00 0
1010 2 2 6 2 4
cos cos
000
1001 2226
r
qh
m
ϕϕ
λ
ϕϕ
λ
ϕϕ
λ
ϕϕ
λ
−−−
− −− Γ
=
− −−
− −−
(9)
To find
1
cos cos
n
ϕϕ
for all n’s, we can express the direction of the magnetization vector as
(sin sin , sin cos , cos )
θφ θφ θ
by setting
( , )
θφ
as a coordinate of a unit sphere such that
[111]
identities to
(1,1,1) 3
. We
can find
cos
n
ϕ
by calculating the inner product between the magnetization vector and the direction of stable states. The
average of
1
cos cos n
ϕϕ
for all possible
( , )
θφ
is what we need. The values are
11
cos cos 2 / 3
ϕϕ
=
and
1
cos cos 2 / 9
n
ϕϕ
=
for the rest.
1222 2
66666
2 6 2 22 4
[] .
2 2 6 2236 3
22262
r
qh
m qh
λλλλ λ
−−−
−− Γ
= = Γ
−−
−−
(10)
Eventually, we can write the magnetization as
2
0
2
22 .
32
s
sr b
M
KV
MMm H
kT K
i
µ
ωω
Γ−
= = −
Γ−Θ −
(11)
Our calculation shows that the only decay rate contributes to the magnetization is
2Γ
, and therefore the mathematical
form looks just like that for uniaxial particles. This result is not limited to the randomly oriented case as shown in the
Supplementary information Part II.
The magnetic susceptibility of Fe3O4 will be
22
00 2
2,
3
f
ss
ff ff
M Mq
Mm
iH Kh K f if
µµ
χχ
Γ
′ ′′
+== =
−Γ −Θ −
(12)
where we use the frequency
f
to replace
ω
,
4
f
π
Γ= Γ
, and
/2
f
π
Θ=Θ
.
The three unknown variables are
2
0
2 /(3 )
s
Mq K
µ
,
f
Γ
, and
f
Θ
. We have two curves: the real permeability and the
imaginary permeability, which give us two conditions. Since we have two conditions but three variables, it will be beneficial
if we can obtain another constraint. From Eqs. (2) and (12), the real part and imaginary part of magnetic susceptibility of
composites will be
2
2
0
eff 2 22
2
0
eff 2 22
()
2
3()
.
2
3()
ff f
sv
Bff
f
sv
Bff
f
MVf
kT ff
f
MVf
kT ff
µ
χ
µ
χ
Γ Γ −Θ
′=
+ Γ −Θ
Γ
′′ =
+ Γ −Θ
(13)
The two equations merge to
2
eff
eff .
ff
ff
χ
χ
′=Γ −Θ
′′
(14)
Since the left-hand side can be directly computed from the experimental data, we can get f
Θ
by the slope of the left-
hand side term versus
2
f
, as shown in Fig. 3(a). The
f
Θ
should be a positive quantity, and we indeed found a decreased line
in Fig. 3(a). After getting the
f
Θ
, one can get
f
Γ
by putting
2
f
fΘ
to the left side of the Eq. (14) as shown in Fig. 3(b).
Notably, if one checks Fig. 2(a) carefully, the real and imaginary susceptibilities (
eff
χ
′
and
eff
χ
′′
) above 10 GHz are very close
to zero. Since
eff
χ
′′
is in the dominator of the left-hand side of Eq. (14), the very small value of
eff
χ
′′
will enlarge the uncertainty.
Therefore, the regions to the left of the dashed lines in Figs. 3(a) and 3(b) are more reliable and those are the region of interest.
The consistency between Eq. (14) and the measured data suggests the memory model with cubic anisotropic materials works
well. After finding f
Θ
and
f
Γ
, we can then determine the remaining coefficient in Eq. (12), i.e.,
2
0
2 /( 3 )
s
Mq K
µ
−
. The
fitting results of the three variables are listed in Table 1.
Figure 4(a) shows the magnetic susceptibility of nano-magnetite based on Eq. (12) and Table 1. Using Eq. (12), we can
estimate the permeability outside the range of the X-band. As an example, from the form of
f
χ
′
, one can expect that the
transition between superparamagnetism and superdiamagnetism happens at
/
t ff
f=ΓΘ
. When the frequency is higher than
t
f
, the real part of the magnetic susceptibility becomes negative. This result is similar to the frequency response of an RLC
circuit as Eq. (4) can be analogous to an RLC circuit, which is explained in the Supplementary information Part III.
t
f
will
be 5.33 GHz using the parameters in Table 1. The transition frequency
t
f
agrees with the experimental observation that the
epoxy/magnetite nanocomposite expresses superparamagnetism at 2.45 GHz3 and exhibits superdiamagnetism in the X-band
(8-12 GHz, i.e., this work).
Discussion
We have proposed a memory model with thermal agitation of the ferromagnetic nanoparticle. The superparamagnetism
is from the thermal fluctuations between stable states, while the superdiamagnetism originates from the out of phase response
with the external driven field (Supplement Part III). The experimental results of the epoxy/magnetite nanocomposites showed
the memory effects clearly at microwave frequency. An unexpected negative susceptibility can be at least four orders of
magnitude higher than the Langevin diamagnetism (Fig. 4). We just show that the memory effect yields the correct frequency
response of the susceptibility for cubic anisotropic materials where the magnetic susceptibility changes from positive at low
frequency to negative at high frequency. To demonstrate the importance of the value of
Θ
, we examine the minimum of the
real susceptibility in Eq. (12):
2
0
min
22
()
322 1
s
Mq
K
µ
χ
ΓΘ
′= − ΓΘ +
, (15)
when
( )
min
22
ω
= ΓΘ + ΓΘ Θ
. Note that the value of
2ΓΘ
, i.e., the ratio of the memory time (
Θ
) and the relaxation
time
12Γ
, affects the amplitude of the minimum. Under the limit
20ΓΘ →
,
min
0
χ
′→
with the corresponding frequency
ω
→∞
, which implies that the memory effect is feeble and the susceptibility will comply with the Debye relaxation formula.
Another extreme case is
2ΓΘ → ∞
. In this case,
min
χ
′
is negative associated with extremely large
χ
′′
. However, the
corresponding frequency
0
ω
→
and once the order of frequency is larger than that of the corresponding frequency
min
ω
,
χ
′
will be quite close to 0. The reason we can observe the superdiamagnetism phenomenon in X-band is that the memory time
scale and relaxation time scale are comparable, and hence superdiamagnetism becomes observable as
/
ω
≈ ΓΘ
.
The diamagnetism expels the magnetic fields within a material. Superconductor is a perfect diamagnetic material with
1
χ
′= −
, which results in no internal magnetic field. This study explored the memory effect, which produced a strong
diamagnetism for nanocomposites at room temperature. Such a new mechanism is called superdiamagnetism. The
superdiamagnetism significantly reduces the ac magnetic field without affecting the ac electric field. It is fundamentally
different from the Meissner effect. The limit of the negative susceptibility is yet to be uncovered. Besides, the adjustable
magneto-dielectric properties of the composite materials can be used for multilayer antireflection coating or the stealth
aircraft/warship. It deserves further theoretical and experimental studies.
Methods
Transmission/reflection method
To conduct the transmission/reflection measurement, we sandwich the sample between two adaptors connecting to a
performance network analyzer (PNA). The experiment is performed using the standard WR90 waveguide with inner
dimensions of 0.9 in by 0.4 in. The two adaptors serve as the two ports, which convert the WR90 waveguide to the 2.4 SMA
(SubMiniature version A) coaxial cables. The two ports are calibrated before measurement. The sample is filled in a uniform
and hollowed WR90 waveguide. The measured transmission/reflection coefficients are then used to extract the complex
permittivity/permeability of samples.
Preparation of the nanocomposite samples
Figure 5 illustrates the sample preparing procedure:
1. Cleaning the waveguide
The standard WR90 waveguide made of copper should be oxidation-free and clean inside with low surface roughness.
The purpose of cleaning the waveguide is to avoid the conductor loss and to ensure that the loss comes strictly from the
nanocomposite.
We first soaked the waveguide into the copper polishing solution to remove the oxide. Then, the waveguides were
immersed in the distilled water to remove the remnant acid. After that, we bathed the waveguide into the acetone to remove
the organic dust and the water. Finally, we soaked it into isopropyl alcohol (IPA) to remove acetone buffer. An ultrasonic
cleaning machine is used during the process. Eventually, we obtain a clean waveguide, as shown in Fig. 5.
2. Preparation of filler:
Epoxy resin (epoxy A) mixing with hardener (epoxy B) will become very sticky and soon form a thermosetting polymer.
To achieve the nanocomposite with a uniform distribution, we first poured epoxy A into a small jar, and then placed the nano
Fe3O4 powder into it. Then, we stirred the liquid in the jar to make them mix. We would heat the jar to make Fe3O4 dissolve
more easily. Finally, we put the epoxy B into it and kept stirring until the mixture looks evenly distributed.
3. Filling the waveguide
After cleaning the waveguide and preparing the filler, we needed to fill the waveguide with the filler. During the
solidifying process, the volume of the epoxy will shrink and may result in an air gap between the sample and the waveguide.
To solve this problem, we piled the waveguide and Teflon and then filled the whole pile. After filling the pile, we heated it so
that epoxy can solidify. When it solidified several hours later, we removed Teflon, and one would find that the thickness of
the solidified filler exceeds the waveguide length. We used a grinding machine to remove the remnants of filler and polish the
surface of the waveguide.
After completing the sample preparation procedures, we then went through a scanning electron microscope (SEM)
measurement to check the uniformity of the nanocomposite. The results are shown in Fig. 6, which assures the uniformity of
samples. The SEM image of the sample is shown in Fig. 6(a). The Fe3O4 particles are rich in Fe. Using the Fe Ka X-ray
microanalysis, we can analyze the distribution of iron, as in Fig. 6(b). Figure 6(c) shows the ingredient analysis using the X-
ray emission spectrum.
Author contributions statement
T. H. conceived the experiment, C.H. and S.C. experimented, C.H., C.R., and T.H. provided theoretical model, C.H.,
C.R., and T.H. analyzed the results. All authors reviewed the manuscript.
Figure 1. (a) The measured (effective) complex permittivity versus frequency and (b) the extracted complex permittivity of
the Fe3O4 nanoparticles using the Looyenga model (Eq. (1)).
8910 11 12 13
Frequency (GHz)
2
3
4
5
6
ε'
eff
0%
6%
12%
18%
8910 11 12 13
Frequency (GHz)
0.0
0.2
0.4
0.6
0.8
1.0
ε''
eff
0%
6%
12%
18%
89 10 11 12 13
Frequency (GHz)
20
24
28
32
36
40
ε'
f
Extracted, 6%
Extracted, 12%
Extracted, 18%
8910 11 12 13
Frequency (GHz)
4
8
12
16
20
ε''
f
Extracted, 6%
Extracted, 12%
Extracted, 18%
(b)
(a)
Figure 2. (a) The measured (effective) complex permeability versus frequency and (b) the extracted complex permeability of
the Fe3O4 nanoparticles using the linearly proportional model (Eq. (2)).
8 9 10 11 12 13
Frequency (GHz)
0.94
0.96
0.98
1.00
1.02
μ'eff
0%
6%
12%
18%
8 9 10 11 12 13
Frequency (GHz)
0.00
0.04
0.08
μ''eff
0%
6%
12%
18%
8 9 10 11 12 13
Frequency (GHz)
0.0
0.1
0.2
0.3
0.4
0.5
μf''
Extracted, 6%
Extracted, 12%
Extracted, 18%
8 9 10 11 12 13
Frequency (GHz)
0.6
0.7
0.8
0.9
1.0
μ'f
Extracted, 6%
Extracted, 12%
Extracted, 18%
(b)
(a)
Figure 3. (a) From Eq. (14),
f
Θ
is associated with the slope of the lines, i.e.,
eff eff
/f
χχ
′ ′′
vs.
2
f
. (b) Rewriting Eq. (14),
2
eff eff
/
ff
ff
χχ
′ ′′
Γ = +Θ
. The value of
f
Γ
is expected to be constant.
60 80 100 120 140 160
f 2 (GHz 2)
-30
-25
-20
-15
-10
-5
f χ'eff /χ''eff (GHz)
6%
12%
18%
8 9 10 11 12 13
f (GHz)
-4
0
4
8
12
Γ
f (GHz)
6%
12%
18%
Figure 4. The complex magnetic susceptibility of the nano magnetite versus frequency, extracted by fitting Eq. (12).
0 4 8 12 16
Frequency (GHz)
-0.4
0.0
0.4
0.8
1.2
χ'
f
0 4 8 12 16
Frequency (GHz)
0.0
0.4
0.8
1.2
χ''
f
5.33 GHz
5.33 GHz
Figure 5. The sample preparing procedures: The waveguide is cleaned through four steps. The filler is prepared by mixing
epoxy A, nano-magnetite powder, and epoxy B. Then, the mixture is poured into the mold consisting of the waveguide and
other fixtures. The mold is dried for days, and the sample surface is ground and polished.
Figure 6. (a) The SEM image, (b) the corresponding distribution of iron through the Fe Ka X-ray microanalysis, and (c) the
ingredient analysis using the X-ray emission spectrum.
(c)
(a) (b)
Table 1. The corresponding parameters in Eq. (12) deduced by the fitting.
2
0
2
3s
B
MV
kT
µ
f
Γ
(GHz) f
Θ
(ns)
1.167
4.55
0.16
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Supplementary Information
Che-Hao Chang1, Shih-Chieh Su2, Tsun-Hsu Chang1, 2*, & Ching-Ray Chang3**
1Interdisciplinary Program of Sciences, National Tsing Hua University, Hsinchu, Taiwan
2Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
3Department of Physics, National Taiwan University, Taipei, Taiwan
Part I. Differential form of the exponential memory kernel
In general, the correction of the master equation induced by the memory effect can be expressed as
( ) ( )( + )( ) ,
t
t K t qh d
τ ττ
−∞
=−Γ
∫
0
m fm v
(S1)
where
K
is the memory kernel, representing how the history of the system influence the current system. Here, we set
exp( / ) /Kt= −ΘΘ
in the sense that the system has a finite memory time
Θ
. Given this memory kernel, we can take the
derivatives with respect to time on both sides of Eq. (S1). Then, we will get
( ) (0)( + )( ) ( )( + )( ) .
t
t K qh t K t qh d
τ ττ
−∞
= Γ+ − Γ
∫
00
m fm v fm v
(S2)
Since
(0) 1/K= Θ
and
() ()/Kt Kt
ττ
−=− − Θ
, Combining with Eq. (S1), we can furthermore rewrite Eq. (S2) as
1
( ) (( + )( ) ( )).
t qh t t= Γ−
Θ0
m fm v m
(S3)
and one can easily rewrite it as the form in Eq. (4).
Part II. Interpretation of eigenvalues of
0
f
Note the diagonalization of
0
f
tells us that
11 234
()mm mmm
′=− ++
is corresponding to the eigenvalue
6−Γ
, while
21 234
3m m mmm
′=+ −−
,
312 34
3m mm mm
′=−+ −
, and
4123 4
3
m mmm m
′=−−+
are corresponding to the eigenvalue
2−Γ
. The master equation of
n
m′
’s when
0h=
and
0Θ=
is
11
22
33
44
6000
0200 .
0020
0002
mm
mm
mm
mm
′′
−Γ
′′
−Γ
=
′ ′
−Γ
′′
−Γ
(S4)
Therefore, while
1
m′
has a relaxation time
1/ (6 )Γ
,
2
m′
,
3
m′
and
4
m′
has a relaxation time
1/ (2 )Γ
.
The difference can be understood by the following interpretation. If we understand
n
m
as the reduced magnetization
along the corresponding direction, then what
1
m′
represents is the difference between the reduced magnetization along
[ ]
111
and the sum of the reduced magnetizations along three adjacent directions. On the other hand,
2
m′
can be rewritten as
2 2 1 23 24
( ( ))( )( )m m m mm mm
′=−−+−+−
, where
1
m−
represents the reduced magnetizations along
111
. Therefore
what
2
m′
represents is the sum of the difference between the reduced magnetization along
111
and three directions adjacent
to the opposite direction.
3
m′
and
4
m′
just replace
111
by
111
and
111
. The
0
f
matrix allows only direct transitions
between adjacent states. For non-adjacent states, the transitions are indirect, and therefore it takes more time for the non-
adjacent states’ transitions than for the adjacent transitions. That why
1
m′
has a shorter relaxation time than the others, since
the difference between adjacent states disappears faster than that between non-adjacent states.
With the corresponding relaxation time, it is natural that the frequency response of
1
m′
will be the form
2
11
( ) / (6 )m C qh i
ωω
′= Γ Γ−Θ +H
and the frequency response of other
n
m′
will be the form
2
( ) / (2 )
nn
m C qh i
ωω
′= Γ Γ−Θ +H
. To see why we only see
2Γ
dependence in the total
r
m
, note Eq. (8) can also be written
as
1
2
1234 3
4
1 111
1100
1[cos cos cos cos ] .
1010
4
1001
r
m
m
mm
m
ϕϕ ϕ ϕ
′
′
−
=
′
−
′
−
(S4)
Hence,
1
m′
contributes to
r
m
as the form
12341
(cos cos cos cos ) 4m
ϕϕ ϕ ϕ
′
−−−
. However,
cos
n
ϕ
’s are related to each
other by
1234
cos cos cos cos
ϕϕϕϕ
=++
. As a result,
1
m′
can not contribute to
r
m
in the sense that the value of
r
m
won’t
change before and after the corresponding transition. For example, when the applied field is along
[]
111
,
1
cos 1
ϕ
=
and
cos 1/ 3
n
ϕ
=
for the rest. Hence, if
1,0 1, 0
mmm→ −∆
by the corresponding transition,
,0 ,0nn
mmm→ +∆
for the rest and
1,0 2,0 3,0 4,0
( )/3
r
mm m m m=+ ++
before and after the transition.
Part III. Understanding the memory effect using the RLC current
An RLC current satisfies
1,
LV
qq q
R RC R
++ =
(S5)
where
L
is the inductance;
R
is the resistor;
C
is the capacitance;
V
is the applied voltage, and
q
is the charge on the
capacitor. As the form looks like Eq. (4), we may explain Eq. (4) by each term of the RLC current.
For
−
0
fm
, when the applied voltage is DC, then
m
will converge to
qh−=Γ
0
fm v
just as
q
will converge to
q VC=
.
Therefore
−0
f
acts as the character of the capacitance. In addition, as the supplementary information shows, the eigenvalues
of
−0
f
represent relaxation time, just like
RC
.
For
Θm
, it acts like
/LR
. To see why the memory effect has such a meaning, note an inductor acts as the role that it keeps
current from changing. One can check the form of the memory effect in Eq. (S1) again:
( ) ( )( + )( ) .
t
t K t qh d
τ ττ
−∞
=−Γ
∫
0
m fm v
(S6)
This equation tells us what the memory kernel does is to memorize the previous value of
m
and try to keep
m
to be the
same, just as the inductor tries to keep
q
to be the same.
