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Magnonics: Selective heat production in nanocomposites with different magneticnanoparticlesYu Gu and Konstantin G. Kornev Citation: Journal of Applied Physics 119, 095106 (2016); doi: 10.1063/1.4943067 View online: http://dx.doi.org/10.1063/1.4943067 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/119/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Sensing magnetic nanoparticles using nano-confined ferromagnetic resonances in a magnonic crystal Appl. Phys. Lett. 106, 232406 (2015); 10.1063/1.4922392 Tuning of the spin pumping in yttrium iron garnet/Au bilayer system by fast thermal treatment J. Appl. Phys. 115, 17C511 (2014); 10.1063/1.4864046 A probabilistic model for the interaction of microwaves with 3-dimensional magnetic opal nanocomposites J. Appl. Phys. 113, 173901 (2013); 10.1063/1.4803127 Magnetic properties of ZnFe2O4 ferrite nanoparticles embedded in ZnO matrix Appl. Phys. Lett. 100, 122403 (2012); 10.1063/1.3696024 Magnetic properties of metallic ferromagnetic nanoparticle composites J. Appl. Phys. 96, 519 (2004); 10.1063/1.1759073
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Magnonics: Selective heat production in nanocomposites with differentmagnetic nanoparticles
Yu Gu1 and Konstantin G. Kornev2
1Institute of Optoelectronics and Nanomaterials, College of Materials Science and Engineering,Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China2Department of Materials Science and Engineering, Clemson University, Clemson,South Carolina 29634, USA
(Received 9 November 2015; accepted 18 February 2016; published online 7 March 2016)
We theoretically study Ferromagnetic Resonance (FMR) in nanocomposites focusing on the
analysis of heat production. It is demonstrated that at the FMR frequency, the temperature of
nanoparticles can be raised at the rate of a few degrees per second at the electromagnetic (EM)
irradiation power equivalent to the sunlight power. Thus, using FMR, one can initiate either
surface or bulk reaction in the vicinity of a particular magnetic inclusion by purposely delivering
heat to the nanoscale at a sufficiently fast rate. We examined the FMR features in (a) the film with
a mixture of nanoparticles made of different materials; (b) the laminated films where each layer is
filled with a particular type of magnetic nanoparticles. It is shown that different nanoparticles can
be selectively heated at the different bands of EM spectrum. This effect opens up new exciting
opportunities to control the microwave assisted chemical reactions depending on the heating rate.VC 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4943067]
I. INTRODUCTION
The current understanding of interactions of nonmag-
netic metal nanocomposites with the electromagnetic (EM)
waves has been significantly advanced and enabled new
exciting engineering applications mostly in the band of EM
spectrum corresponding to the optical frequency range.1–9
EM waves passing through a metal nanoparticle excite the
oscillations of the charge carries, plasmons. Plasmon oscilla-
tions, in turn, change the properties of EM waves. A new
field, plasmonics, has been created to study these effects.
Plasmonics has attracted significant attention in the recent
decades guiding development of new light-responsive mate-
rials and devices.1,10,11 In contrast to plasmonics, magnetic
excitations, magnons, in magnetic nanomaterials remain
poorly studied and, hence, the field of magnonics is still at
the beginning of its development.3,9,12–19
A ferromagnetic single domain nanoparticle demon-
strates an enhanced absorption of the EM irradiation within
the microwave frequency range. The absorption becomes
significant at the ferromagnetic resonance (FMR) frequency
when the magnetic moment of nanoparticle vigorously pre-
cesses about its easy axis;20,21 the nanoparticles at FMR
effectively dissipate the energy into heat.22,23 Absorption of
the microwave irradiation by ferromagnetic nanoparticles is
therefore attractive due to the broad potential applications in
different technologies.3,15,16,18,24–37
Some polymer films or nonpolar solutions are poor
microwave absorbers; therefore, embedding ferromagnetic
nanoparticles into the matrix appears attractive.38 At FMR,
the embedded fillers can produce heat while the matrix can
be transparent within a particular FMR band. At FMR, the
heat can be purposely delivered to the point of care where
the magnetic inclusions are being concentrated. This way,
the rate of chemical reactions can be controlled at the scale
associated with the size of the inclusion or it can be triggered
at the inclusion surface.39,40 Thus, the FMR heating of nano-
composites opens up new opportunities for magnonics.
In this paper, we theoretically study ferromagnetic reso-
nance in non-magnetic films loaded with the single domain
magnetic nanoparticles. Closely following the classical
works of Landau and Lifshitz41 and Kittel,22 we derived the
constitutive equation for the magnetic induction B in a nano-
particle subject to the AC field. The difference is that we set
up the FMR theory considering the corresponding boundary
value problem of electrodynamics. This approach allows one
to employ the averaging technique developed for non-
magnetic nanocomposites4,10,21 and derive the constitutive
equation relating magnetic induction with magnetization in
nanocomposites.
We then examine the heat production in nanocompo-
sites. Laminated films where each layer is filled with a par-
ticular type of magnetic nanoparticles and films with a
mixture of different nanoparticles are examined to show the
selectivity of heating of different nanoparticles at a selected
EM frequency. It is demonstrated that applying the EM irra-
diation at the FMR frequency, one can significantly enhance
the heating rate at the nanoscale level.
II. FERROMAGNETIC RESONANCE IN A SINGLEDOMAIN NANOPARTICLE
A schematic of the FMR setup is shown in Figure 1.
During the experiment, the specimen, a composite film in
our case, is subject to a bias DC magnetic field, Hex. The
film properties are probed by applying the EM irradiation
with the AC magnetic field h0 perpendicular to the bias mag-
netic field Hex, i.e., h0?Hex. The incident EM wave has the
wave vector k0 directed along the z-axis, see Fig. 1. The EM
wave is partially absorbed by the material, hence the
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amplitude of the outgoing wave decreases, h3< h0. In the
FMR experiments,42 the microwave absorption by the sam-
ple is measured by changing the strength of the bias mag-
netic field Hex and keeping the frequency f of the AC field h0
non-changed. The FMR is detected when a maximum
absorption is observed at a certain magnetic field
Hex¼Hc.22,41–43 One can also fix Hex and scan over the fre-
quency to observe an absorption peak.42
Conventional FMR experiments are conducted on the
ferromagnetic samples for which the constitutive equations
are known. In the case of composites containing ferromag-
netic single domain nanoparticles, the fields inside the sam-
ple and non-magnetic host are perturbed and hence they are
different from the external applied field Hex. Therefore, the
constitutive equation for such a material has to be derived.
To simplify the problem, we assume that the easy axes
of the nanoparticles are directed parallel to the direction of
the bias field Hex, the z-direction in Figure 1(b). This
assumption is not strong and, as shown in Refs. 29, 34, and
44–46, this type of alignment can be easy realized in experi-
ments. The film perturbs the bias magnetic field Hex.
Denoting the volume fraction of nanoparticles by u, we can
find the field inside the film as Hin¼Hex – uM, where M is
the magnetization of a single nanoparticle (Figure 1(c)). This
internal bias field Hin acts on each nanoparticle to align all
magnetic moments of the nanoparticles parallel to this field.
In order to find the permeability of a composite film, we
have to look at the reaction of a single nanoparticle on the
applied field Hin.
A. Landau-Lifshitz-Gilbert dispersion of magneticpermeability of a nanoparticle
The EM waves of interest have the wavelengths which
are much greater than the nanoparticle size. For example, a
10 GHz frequency wave in vacuum has the wavelength k of
about k¼ 3� 10�2 m, which is much greater than the size of
the single domain nanoparticles ranging between 10�9 m and
10�8 m.42 Therefore, each wave period covers thousands and
thousands of nanoparticles in the nanocomposites. EM
waves cannot recognize nanoparticles seeing the material as
it would be a continuum. In turn, each nanoparticle does not
recognize the profile of the EM wave. The nanoparticle is
able to recognize only magnitude of the EM wave at the
particle location. Thus, for a nanoparticle, the EM wave is
merely an external uniform AC field.
When an EM wave penetrates the nanoparticle, the mag-
netization vector M is forced to deflect from the easy axis.
As a result, the magnetization vector rotates around the easy
axis. This precession is schematically depicted in Figure
1(b). In continuum electrodynamics, the precession of mag-
netization vector is described by the Landau-Lifshitz-Gilbert
(LLG) equation21,42
dM
dt¼ �cl0 M � Hs þ hmð Þ½ � þ a
jMj M � dM
dt
� �; (1)
where c> 0 is the gyromagnetic ratio, l0 is the permeability
of vacuum, a is the phenomenological damping coefficient,
and hm is the AC component of magnetic field inside the
nanoparticle. We assume that the particle is spherical and
has an uniaxial magnetocrystalline anisotropy. In case of a
single domain nanoparticle, jMj ¼M¼Ms, where Ms is the
saturation magnetization of the material. Then, the bias DC
component of magnetic field inside the nanoparticle, Hs (par-
allel to Hin) is calculated as42
Hs ¼ jHsj ¼ Hin þ2K1
l0Ms
�Ms
3; (2)
where K1 is the constant of magnetocrystalline anisotropy.
For interpretation of the FMR experiments, the magnet-
ization vector M is represented as M¼M0þm, where M0 is
the time independent component of the magnetization vector
which is set by the bias field (M0 jj Hin) and m is the
dynamic time-dependent component of the magnetization
vector. The amplitudes of vectors m and hm are typically
much smaller than the corresponding bias components,
hm� Hs, m�M. Therefore, Eq. (1) can be linearized as42
M0�Hs¼0;
dm
dt¼�cl0 m�HsþðM0�hm½ Þ�þ a
jM0jM0�
dm
dt
� �
M2s ¼jMj
2¼jM0j2þ2M0 �mþjmj2�M02:
8>>>><>>>>:
(3)
FIG. 1. (a) Schematic of the ferromagnetic resonance experiment. A bias
DC field, Hex, is applied to a composite film. An EM wave with the wave
vector k0 and magnetic field vector h0 propagates through the material. The
AC component of the applied field, h0, is perpendicular to the bias field Hex.
Detector receives and analyzes the wave h3. (b) Schematic of a circular pre-
cession of the magnetization vector M about the easy axis of a spherical
nanoparticle: m is the 2D vector rotating in the xy-plane, it corresponds to
the xy-projection of the magnetization vector M moving over the circular
orbit. The easy axis of a nanoparticle is directed parallel to the local bias DC
field, Hin. The local AC component of magnetic field, �h, is perpendicular to
the local bias field Hin. (c) A schematic of the distribution of magnetic field
in a thin nanocomposite film. (d) A schematic of the field and wave vector
directions in a system of reflected and transmitted EM waves outside and
inside the probed film.
095106-2 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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The first equation implies that the time independent compo-
nent of magnetization vector, M0, is always parallel to the
bias magnetic field, Hs. The second equation is satisfied only
by a 2D vector m rotating perpendicularly to the Hs and M0
vectors which are parallel to each other. This also implies
that the magnitude of magnetization M0 is equal to the satu-
ration magnetization, the third equation (3).
In a wave, the field inside the nanoparticle oscillates
periodically, hm / eixt, hence the magnetization vector
follows the same time dependence m / eixt. Solving Eq. (3)
for m with this particular form of the time dependence, one
can represent the dynamic magnetic induction inside the
nanoparticle bm¼l0(mþ hm) as22,23
bm ¼ l0lðxÞhm þ l0igðxÞðhm � zÞ; hm ¼ heixt (4)
l xð Þ ¼ 1þ cl0Mxr þ iax
xr þ iaxð Þ2 � x2; (5a)
g xð Þ ¼ cl0Mx
xr þ iaxð Þ2 � x2; (5b)
xr ¼ cl0Hs; (6)
where z is the unit vector directed parallel to the z-axis, h is
a 2D time-independent vector specifying the field inside the
nanoparticle, and l(x) and g(x) are the two characteristic
functions describing the dependence of relative permeability
on the wave frequency. As follows from the derivation of
Eqs. (4) and (5), these equations are applicable for any ferro-
magnetic materials.15,18,43 The nanomaterial features come
from the definition of xr by Eq. (6), where the field Hs
carries all the information about the nanoparticle nature.
Therefore, it is instructive to analyze first the behavior of
functions l(x) and g(x) on x considering xr as a phenome-
nological parameter, and then discuss their behavior in a
nanoparticle explicitly specifying xr through Eq. (2).
B. Functions l(x) and g(x)
It is convenient to introduce the circularly polarized
waves as hm6¼ (ex 6 iey) h exp(ixt),21 where the plus wave
with subscript “þ” and the minus wave with subscript “�”
are defined as the left- and right-handed circularly polarized
waves, respectively. As shown in Appendix A, Equations (4)
and (5) can be rewritten as
bm6 ¼ l0 l xð Þ7g xð Þ½ �hm
6 l xð Þ7g xð Þ ¼ 1þ cl0M
xr6xþ iax;
(7)
where bm6 are the right- and left-handed magnetic inductions
defined as bm6¼ (ex 6 iey) b exp(ixt). Figure 2 illustrates the
characteristic features of the relative permeabilities of a
ferromagnetic material probed by the two distinctly polar-
ized waves.
As follows from Eq. (7), the resonance peak appears only
for the minus-wave for which the m-vector spins in the anti-
clockwise direction (Figure 11 in Appendix A). When the micro-
wave frequency x approaches the natural precession frequency
xr, the denominator in Eq. (7) goes to zero and the effective per-
meability (lþ g) significantly increases. Close to this natural fre-
quency, the permeability can become negative (Figure 2). In the
limit x � xr, the permeability approaches 1 and the ferromag-
netic material behaves as a nonmagnetic material.
For a bulk material, the negative permeability implies
that the EM wave cannot penetrate the material. When the
EM penetrates the distance d below its surface, the amplitude
of the EM wave exponentially decays, h / exp(�2pd/k).21
Therefore, when an EM wave hits a ferromagnetic film, the
absorption significantly increases at the resonance frequency
xr and finally the material becomes almost impermeable for
the waves when the frequency is further increased. This is a
signature of the ferromagnetic resonance in the bulk materials.
However, if the diameter of a ferromagnetic nanopar-
ticle is much smaller than the wavelength k, the EM wave
will be able to penetrate the nanoparticle. This effect is spe-
cific for the nanoparticles and we discuss it below.
C. FMR in nanoparticles: Analogy withnanoplasmonics
Knowing magnetic permeability of the ferromagnetic
nanoparticle, we can analyze interactions between the micro-
wave and the nanoparticle. For a gigahertz wave, the wave-
length k is measured in centimeters. The characteristic time
scale for the wave penetration is s¼D/c, where D is the par-
ticle diameter and c is the speed of light. Taking D¼ 10�8
m, we have s 10�9/108¼ 10�17 s. The characteristic time
of rotation of spins in the gigahertz wave is inversely propor-
tional to the frequency of applied field, hence it is roughly
estimated as sx 10�9 s. Since s is much smaller than sx,
one can safely assume that the nanoparticle is subject to a
quasi-static magnetic field, i.e., all time-dependent terms in
the Maxwell equations can be safely dropped and the field
distribution outside and inside the nanoparticle is obtained
from the magnetostatic field equations:21 Mathematically,
the magnetostatic problem is analogous to an electrostatic
problem solved in Ref. 4. The details can be found in
Appendix B. The solution suggests that inside the nanopar-
ticle, the magnetic fields of both polarizations hm6 are uni-
form. These fields are related to the average fields �h6 inside
the film through the following equation (see Appendix B):
FIG. 2. Typical behavior of the permeability l 6 g. (a) In the minus-wave,
the relative permeability is defined as lþ g, (b) In the plus wave, the relative
permeability is defined as l - g. Calculations are given for a ferromagnetic
nanoparticle with the following parameters: l0Hs¼ 0.31 T, M¼ 4.3� 105A/m,
and a¼ 0.05.
095106-3 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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hm6 ¼
3
2þ l xð Þ7g xð Þ½ ��h6 ¼
xr6xþ iaxxr6xþ cl0M=3þ iax
�h6:
(8)
It becomes clear that the magnetic field inside the nanopar-
ticle tends to infinity when the denominator of Eq. (8) goes
to zero. In the limit of no damping, this condition is satisfied
at x ¼ xc where xc is the root of the equation
2þ Re½lðxcÞ þ gðxcÞ� ¼ xr � xc þ cl0M=3 ¼ 0: (9)
A similar condition is used in nanoplasmonics dealing
with the light-induced excitations of charges in metal
particles suspended in the air10,11 (one has to replace the per-
meability with the dielectric function, provided that the
dielectric function of the air is equal to 1). In metals, these
excitations are called “plasmons,” thus emphasizing the na-
ture of the electron oscillations described in terms of the
solid state plasma.4,47–52 In ferromagnetic nanoparticles, it is
natural to talk about magnons as the carriers of excitations.9
Condition (9) is the resonance condition for magnonics.
Such a resonance is caused by the interactions of magnons
with the surrounding media when the real part of the effec-
tive permeability of the minus-wave changes the sign from
positive to negative. Accordingly, the field strength changes
resulting in a transition of materials properties from the field
permeable—to impermeable—to the field-enhanced states,
Figure 3 (b! c! d).
Solving Eq. (9) for xc, we obtain
xc ¼ xr þ cl0M=3: (10)
The distribution of dimensionless fields hi�=
�h� outside
(i¼ l) and inside (i¼m) a single domain ferromagnetic
nanoparticle for the right-handed circularly polarized
wave (minus – wave) is shown in Figure 3. The distinct
behavior of magnetic fields is illustrated with three different
frequencies. In calculations, we assumed that the snapshots
are taken at a certain time moment t when the average
magnetic field in the nanocomposite, �h�, is pointing in the
x-direction. All physical parameters are chosen the same as
those used to graph in Figure 2: l0Hs¼ 0.31 T, M¼ 4.3
� 105 A/m, and a¼ 0.05. Under these conditions, the two
resonance frequencies xr and xc appear very different: xr/
2p¼ 10.1 GHz, xc/2p¼ 16 GHz.
The frequency of the 1 GHz wave is much lower than the
natural precession frequency xr/2p resulting in l(x)þ g(x)
¼ 2.7. The nanoparticle behaves as a normal permeable
material. Accordingly, the field inside the nanoparticle is co-
directed with the average field �h� and is weaker than �h� ,
Figure 3(b).
The frequency of the 9.7 GHz wave is close to the natu-
ral precession frequency xr/2p. Therefore, the dynamic per-
meability of the nanoparticle l(x)þ g(x) is positive and
reaches its maximum (Figure 2(a)) thus effectively shielding
the nanoparticles from the external field. This effect can be
also explained by considering two field components.
Calculations show that the demagnetization field, �m/3, is
counter directed to the average field �h�; the strengths of the
demagnetization and average fields are about the same.
Summing up these two vectors, we observe a significant
drop of the field inside the nanoparticle, hm� ¼ 0.1 �h�, Figure
3(c). Therefore, the nanoparticle can be considered almost
impermeable to the field.
The frequency of the 16 GHz wave is close to the reso-
nance frequency xc/2p. In this case, the demagnetization
field is almost perpendicular to the applied field �h� and its
magnitude is much greater than that of �h�. As a result, the
magnetic field inside the nanoparticle is significantly
enhanced, Figure 3(d). Appendix B contains a detailed
analysis of the spectral behavior of the dynamic magnet-
ization and internal field for frequencies from 1 GHz to
25 GHz.
FIG. 3. Distribution of dimensionless
fields hi�=
�h� outside (i¼ l) and inside
(i¼m) a single domain ferromagnetic
nanoparticle with magnetization
M¼ 4.3� 105 A/m. The static field is
taken as l0Hs¼ 0.31 T. (b) x/2p¼ 1 GHz, (c) x/2p¼ 9.7 GHz, (d) x/2p¼ 16 GHz. The black circle represents
the surface of the nanoparticle; the field
strength is measured according to the
color bar. The vector diagrams under
the pictures illustrate the vector relation
between the applied field �h� and inter-
nal field hm� corrected by the demagnet-
ization field, hm� ¼ �h� �m=3.
095106-4 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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D. The difference between two resonance frequenciesof a nanoparticle
According to Sec. II C, for a single domain nanoparticle
embedded in a nonmagnetic matrix, the natural precession
frequency xr is different from the resonance frequency xc.
For a spherical nanoparticle, these two frequencies are
related through Eq. (10).
Equation (10) can interpreted as follows. Based on Eqs.
(2) and (6), the natural precession frequency is written as
xr ¼ cl0Hs ¼ cl0ðHin þ Ha �M=3Þ , where Ha ¼ 2K1=ðl0MÞ is the field of crystalline anisotropy.42 Using Eq. (9),
the resonance frequency xc for the embedded nanoparticle is
rewritten as xc ¼ xr þ cl0M=3 ¼ cl0ðHin þ HaÞ: Thus, the
resonance frequency xc is independent of the demagnetiza-
tion field of an individual nanoparticle Hd ¼ �M=3!
The resonance frequency for a composite film is imme-
diately obtained by specifying the magnetic field inside the
film Hin ¼ Hex � uM. Therefore, the resonance frequency
for an embedded nanoparticle takes the form
xc ¼ cl0 Hex � uM þ 2K1
l0M
� �: (11)
This formula can be used to tune the FMR frequency by
varying the external magnetic field Hex, the volume concen-
tration of nanoparticles u, or by choosing materials with dif-
ferent magnetization M and the anisotropy constant K1.
Table I lists six different materials dividing them into
two groups, I: materials with hexagonal crystal structure and
uniaxial magnetocrystalline anisotropy, II: materials with
cubic crystal structure and cubic magnetocrystalline anisot-
ropy. The gyromagnetic ratio c¼ 2� 1011 (rad/T/s) and
damping coefficient a¼ 0.05 are taken the same for all
materials. The resonance frequencies xc and xr are calcu-
lated for a single nanoparticle substituting u¼ 0 into all
formulas.
For group I, external magnetic field Hex is set to be zero
and for group II, l0Hex¼ 0.5 T. For materials with cubic
magnetocrystalline anisotropy, the effective anisotropic field
Ha is not well defined. We assume K1¼ 0 in this case.
E. Heating of a single domain nanoparticle
The heat produced by a single domain nanoparticle per
unit time and per unit volume was calculated using different
approaches.21,37,58,59 As shown in Appendix C, all these
methods lead to the same basic formula21,59
P ¼ Ph þ PE; (12)
Ph ¼l0
2Im lmð Þxjhm
6j2; PE ¼
e0
2Im emð ÞxjEj2; (13)
where x is the angular frequency of the microwave and e0
is the permittivity of vacuum; em and lm are the relative
permittivity and permeability of the magnetic nanopar-
ticle; and E and hm6 are the electric and magnetic fields of
the microwave inside the nanoparticle. The heat produc-
tion is attributed to both electric losses PE and magnetic
losses Ph. Both losses are measured in units of W/m3.
According to estimates in Appendix C, the electric losses
are negligibly small compared to the magnetic losses.
Neglecting PE, the heating rate of a magnetic nanoparticle
is written as
KT ¼xP0
2c0qCp
axMx
x� xM=3� xrð Þ2 þ a2x2; (14)
xM ¼ cl0M; (15)
where P0 ¼ c0l0h20=2 (W/m2 units) is the microwave power,
Cp (J/kg/K) is the heat capacity at constant pressure, q is the
density, c0 is the speed of light in vacuum, and h0 is the mag-
netic field of the microwave in the free space. It is evident
that the heating rate depends on the magnetic properties of
nanoparticles through the ratio xM=xr.The dependences of
the dimensionless heating rate 2KTqCp=ðxrP0Þ on this ratio
are shown in Figure 4.
The maximum heating rate is reached at the resonance
frequency x¼xrþxM/3 and the heating rate at this fre-
quency is calculated as
KTm ¼P0xM
2c0qCpa¼ P0l0cM
2c0qCpa: (16)
TABLE I. Room temperature properties of different materials.
Materials
I II
BaO 6Fe2O353 Co53 YCo5
54 a-Fe2O354 Fe3O4
54 c-Fe2O354
Crystal structure Hex Cubic
Damping a 0.05 0.05
c (�1011 rad/T/s) 2 2
l0Hex (T) 0 0.5
M(�105 A/m) 3.8 14.4 8.5 0.024 4.8 4.3
K1 (�104 J/m3)53 33 45 550 …
l0Ha¼ 2K1/M (T) 1.74 0.63 12.94 0a
xr/2p (GHz) 50.2 0.7 400.6 15.9 9.5 10.2
xc/2p (GHz) 55.3 19.9 411.9 15.9
qcp (�106 J/m3/K) 3.5055 3.7556 … 3.4256 3.2956 3.1657
KTm (K/s) 0.91 3.2 … 0.0059 1.2 1.1
aSince the effective anisotropic field Ha is not well defined for material with cubic magnetocrystalline anisotropy, we consider only the cases when the external
field is much stronger than Ha allowing to neglect this contribution.
095106-5 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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The maximum heating rate is therefore directly proportional
to the saturation magnetization M and inversely proportional
to damping coefficient a. It does not depend on the natural
precession frequency xr or on the particle size. Using the pa-
rameters listed in Table I and taking P0¼ 1 kW/m2 to be of
the order of the power of sunlight, the maximum heating rate
of a single domain cobalt nanoparticle at 298 K with
qCP¼ 3.75� 106 J/m3/K (Ref. 56) is estimated as 3.2 K/s.
This rate is extremely high. For the applications requiring
fast heating, cobalt nanoparticles appear to be the most
promising candidates among those listed in Table I.
III. HEATING OF A COMPOSITE WITH MAGNETICNANOPARTICLES
A. Constitutive equation and ferromagnetic resonancein nanocomposites
For description of nanocomposites, one typically
employs an effective medium approximation resulting in the
concentration dependent permeability and permittiv-
ity.10,11,15,21,60–64 We use the derivation developed earlier4
for composites with non-magnetic metal nanoparticles. The
average magnetic induction �b6 and magnetic field �h6 are
defined through the following equation:4,21
�b6 � l0�h6 ¼
1
V
ð1�1
ð1�1
ð1�1
½b6 ~rð Þ � l0h6 ~rð Þ�dxdydz; (17)
where V is the sample volume. The average field �h6 is con-
sidered to be equal to the field in the host material hl6
far away from the nanoparticle. Substituting Eq. (8) into
Eq. (17), one obtains4
�b6 ¼ l0lef f6
�h6; lef f6 ¼ l0 1þ 3u
l7gð Þ � 1
2þ l7gð Þ
� �: (18)
In which, u is the volume fraction of ferromagnetic
nanoparticles, lef f6 are the effective permeabilities for the
left- and right-handed circularly polarized waves.
As an illustration, Figure 5 demonstrates the behavior of
effective permeability of nanocomposites with the volume
fraction u¼ 0.01 of the c-Fe2O3 nanoparticles. This depend-
ence is very much similar to that of a single nanoparticle.
For the plus-wave, the effect of nanoparticles is insignificant
and the main response comes from the host non-magnetic
material. For the minus-wave, the picture is completely
different. This frequency defined by Eq. (11) provides the
maximum heating rate for the entire composite film.
To illustrate the distinct behaviors of the single nanopar-
ticles and composite materials made of these nanoparticles,
we first consider the associated resonance peaks, Figure 6.
Three materials, a-Fe2O3(Hematite), Fe3O4(Magnetite), and
c-Fe2O3(Maghemite), are taken for this comparison. Their
physical parameters are listed in Table I and we assume that
the external field is equal to l0Hex¼ 0.5 T and it is much
greater than l0uM and l0Ha. The real parts of permeability
are shown in Figure 6. According to Eq. (10), the difference
between xc and xr depends solely on the materials magnet-
ization M. For the a-Fe2O3 nanoparticles, magnetization M is
much smaller than the external field Hex. Therefore, the
frequency xc is approximately equal to xr. For the Fe3O4
and c-Fe2O3 nanoparticles, magnetization M is comparable
with the external field Hex. Hence, one observes significantly
different values of xc and xr.
FIG. 4. Dimensionless heating rate of three different materials with different
xM/xr ratios. All parameters are taken from Table I.
FIG. 5. Permeability dispersion of nanocomposites with c-Fe2O3 nanopar-
ticles. Parameters used for these calculations are listed in Table I. Volume
fraction u is 0.01.
FIG. 6. Permeability dispersion for the single nanoparticles and nanocompo-
sites of different magnetic materials. Only real part of the minus-wave is
presented. The volume fraction is v¼ 0.01.
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As evident from Figure 6, the resonance peak is located at
different frequency, unique for each nanoparticle. The saturation
magnetization is the main controlling parameter of the reso-
nance frequency of a single nanoparticle. The value of satura-
tion magnetization of Fe3O4 is very close to that of the c-Fe2O3
nanoparticles. Therefore, the resonance frequencies of the
Fe3O4 and c-Fe2O3 nanoparticles are sitting close to each other.
Since the volume fraction of nanoparticles is small and the field
of crystalline anisotropy is much smaller than the applied field,
the resonance frequency of the entire composites xc¼ cl0
(Hex�uMþHa) is almost the same for all three materials.
B. Heat production in a nanocomposite film
Assume that a microwave propagates perpendicularly to
the film surface and the external magnetic field Hex is
co-directed with the wave vector k (Figure 1(d)). The film
thickness is d. In Figure 1(d), E0 and h0 are the electric and
magnetic components of the incident microwave, E3, h3 are
those of the transmitted wave, and E4, h4 correspond to the
reflected wave; E1, h1 and E2, h2 are two waves travelling
inside the film in the opposite directions. Two wave vectors
k0 ¼ xffiffiffiffiffiffiffiffiffie0l0
pz and k1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffieef f lef f
pk0 are the wave vectors
corresponding to the EM waves propagating in vacuum and
in the nanocomposite, respectively. The effective permeabil-
ity leff is defined by Eq. (18) and eeff was calculated in Ref.
4. The waves are considered circularly polarized. The reflec-
tion and transmission coefficients are calculated by matching
the tangential components of electric and magnetic fields at
the two boundaries z¼ 0 and z¼ d (see Appendix D for deri-
vation). The result is
T ¼ E3
E0
exp �ik0dð Þ ¼ � 4 exp �ik1dð ÞZ1Z0
Z1 � Z0ð Þ2 exp �2ik1dð Þ � Z1 þ Z0ð Þ2
R ¼ E4
E0
¼ Z21 � Z2
0
� �exp �2ik1dð Þ � Z2
1 � Z20
� �Z1 � Z0ð Þ2 exp �2ik1dð Þ � Z1 þ Z0ð Þ2
; Z0 ¼ffiffiffiffiffil0
e0
r; Z1 ¼ Z0
ffiffiffiffiffiffiffiffilef f
eef f
r:
8>>>><>>>>:
(19)
Z0 and Z1 are the wave impedances in vacuum and in
nanocomposites, respectively. We introduce a dimensionless
parameter, the absorption coefficient g ¼ 1� jTj2 � jRj2,
which is the ratio of the energy absorbed by the film to the
energy of the EM irradiation pumped into the system. As an
illustration, we examine a paraffin film with thickness
d¼ 1 mm subject to the external magnetic field Hex¼ 0.5 T.
Three volume fractions of nanoparticles in the film were
examined: v¼ 0.03, 0.01, and 0.005. In these films, the
effects of the wave interference can be safely neglected.
Indeed, as an order of magnitude estimate of the wavelength
of the microwaves of interest, we take a 10 GHz microwave
in vacuum. Its wavelength is about k 0.2 m, which is much
greater than the film thickness k� d¼ 1 mm.
At very small concentrations of the nanoparticles, the
electric losses in nanoparticles are insignificant compared to
those of the magnetic losses. Hence, we will use the dielec-
tric constant of paraffin eeff¼ 2.2 in calculating electric
losses. Within the frequency band of interest, the dielectric
function of paraffin does not change significantly and can be
considered constant.
We assume that the plus- and minus-waves have the
same amplitude. Figure 7 illustrates the behavior of absorp-
tion coefficient as a function of the microwave frequency. As
expected, the greater the amount of magnetic material in the
film, the greater the absorption. It is worth noting that the
absorption peak (marked by the purple dots) is significantly
shifted (except for a-Fe2O3) as the volute fraction changes
from 0.005 to 0.03; this shift of the FMR frequency is deter-
mined by Eq. (11).
An analysis of the FMR absorption spectrum allows one
to evaluate the heating rate of the film. Assume that the film
has the surface area A. If P0 is the power of the EM wave
propagating through the film, the heat produced per unit time
will be P0Ag and the heating rate of the composite film KT is
defined as KT¼P0Ag/(qCpAd)¼P0g/(qCpd), where d is the
film thickness. Consequently, the heating rate KT is propor-
tional to the absorption coefficient g. Therefore, the heating
rate has the same frequency dependence as that of the
absorption coefficient.
As an illustrative example of application of this theory,
we examined the maximum heating rates of different paraffin
films (Cp¼ 2.14 J/g/K, q¼ 0.9 g/cm3) loaded with different
magnetic nanoparticles of the same volume fraction v¼ 0.03.
Assuming the same damping coefficient a¼ 0.05 for all
FIG. 7. The effect of the volume fraction of nanoparticles in a paraffin film
(eeff¼ 2.2) on the behavior of absorption coefficient as a function of micro-
wave frequency. External magnetic field is Hex¼ 0.5 T and the damping
coefficient is a¼ 0.05.
095106-7 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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nanoparticles, we calculated the heating rate of the incident
microwave of power P0¼ 1 kW/m2, Figure 8. The heating
rate of the composite film follows the trend of the heating
rate of individual nanoparticles. The a-Fe2O3 nanoparticles
are less attractive as the composite fillers for the heat genera-
tion application. The c-Fe2O3 and Fe3O4 nanoparticles have
almost identical properties. The Co nanoparticles are still the
best candidates as the fillers of composite films.
Another example is a composite film made of a mixture
of magnetic nanoparticles. Each family of nanoparticles will
support a particular reaction initiated at a given frequency.
In this case, the effective permeability is defined as follows:
lef f6 ¼ l0 1þ 3
Xi
ui
li7gið Þ � 1
2þ li7gið Þ
" #; (20)
where ui is the volume fraction of the ith type of magnetic
particles, and li and gi are defined by Eqs. (2), (5), and (6).
The magnetic field inside the film is Hin ¼ Hex �P
uiMi,
where Mi is the magnetization of the ith type of nanoparticles.
Figure 9 shows the resulting heating rates for different
frequencies. Three distinguishable heating peaks show up,
each corresponds to the resonance frequency of a particular
family of nanoparticles. These results support the idea of
selective triggering of the chemical reaction provided that
the dispersion contains different magnetic inclusions. As a
natural application of FMR, one can think about the detec-
tion of different types of magnetic clusters in a composite
film by scanning over the EM frequency.
In some applications, one needs to initiate reactions with
different kinetics to supply the heat at the different
rates.39,40,65–67 Therefore, one can think of a laminated struc-
ture with different layers having the same matrix but differ-
ent fillers. Applying an EM irradiation at the different
frequency bands to the structure, one expects that the layers
with different magnetic nanoparticles will support different
reactions. As an illustration, we choose a sandwich made of
three different paraffin films filled with ferromagnetic nano-
particles: the top layer is filled with Co, the middle layer
with BaO�6Fe2O3, and the bottom layer with c-Fe2O3.
The problem of propagation of an EM wave through this
sandwich is solved by matching the E- and H-field
components at all the four interfaces (see Appendix D for
derivation). The H-component in each layer is specified and
then the heating rate is calculated for each individual layer.
The results of these calculations are given Figure 10: It is
evident that one can tune maximum heating rate of the com-
posite film by selecting the optimum frequency.
IV. CONCLUSIONS
In this paper, we study the specifics of ferromagnetic
resonance in non-magnetic films loaded with the single
domain ferromagnetic nanoparticles.
First, we review the Landau-Lifshitz-Kittel theory of
magnetic resonance in a single domain nanoparticle consid-
ering this problem as a spectral boundary value problem of
electrodynamics. For the microwaves of interest, one can
limit the analysis to a quasi-static approximation, when mag-
netic induction is assumed solenoidal and magnetic field is
assumed potential. For a spherical nanoparticle, the bound-
ary value problem of magnetostatics with the linearized
FIG. 8. Comparison of heating rates of magnetic nanoparticles and nano-
composite paraffin films. Physical parameters are taken from Table I. The
heating rate is estimated at xc for the nanocomposites and at xr for a single
nanoparticle.
FIG. 9. The heating rate of a composite paraffin film containing a uniform
mixture of three different nanoparticles. All the particles have the same
volume fraction 1%. The film thickness is 1 mm and the external field
l0Hex¼ 0.5 T.
FIG. 10. The heating rate of a composite paraffin film made of three layers
each containing only one type of ferromagnetic nanoparticles. The volume
fraction of nanoparticles in each composite film is u¼ 0.03, each layer is
1 mm thick, the damping coefficient a is 0.05, and the external magnetic
field l0Hex¼ 0.5 T.
095106-8 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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Landau-Lifshitz-Gilbert equation for magnetization is solved
analytically. It is shown that the behavior of magnetic field
inside and outside the nanoparticle significantly depends on
the frequency of applied magnetic field. It appears that the
magnetic field inside the nanoparticle is significantly
enhanced at the resonance frequency xc given by Eq. (11).
This resonance frequency is different from the natural
precession frequency xr given by Eq. (6). The effect of the
particle heating is considered in details. The heat production
is attributed to the magnetic losses. The heating rate for a
single nanoparticle is found to be directly proportional to the
particle magnetization and the maximum rate is achieved at
the resonance frequency xc. At this frequency, the heating
rate can be quite high, for example, at the power of micro-
wave irradiation equivalent to the sunlight power, the tem-
perature of the cobalt nanoparticles is predicted to increase
with the rate of 3 deg/s. Thus, using FMR, one can purposely
deliver heat to the nanoscale at a sufficiently fast rate.
We then discuss the features of the ferromagnetic reso-
nance in nanocomposites loaded with a low volume fraction
of magnetic nanoparticles. It is shown that the absorption
coefficient of a nanocomposite film depends non-
monotonously on the microwave frequency. The resonance
frequency where the absorption coefficient reaches its maxi-
mum changes proportionally to the volume fraction of
nanoparticles. The heating rate of the nanocomposite film is
proportional to the absorption coefficient and hence the
fastest heating is achieved at the resonance frequency xc.
Examination of the heating rate of paraffin films loaded with
different magnetic particles of the same volume fraction
reveals that the heating rate does not change significantly
from one magnetic material to another. In contrast, the local
heating rate of individual nanoparticles can be quite distin-
guishable. This effect opens up new opportunities to initiate
and control chemical reactions using nanoparticles from dif-
ferent magnetic materials and selectively heating the targeted
spots in the reaction chamber. Two types of composite films,
(a) the films filled with a mixture of different magnetic nano-
particles, and (b) laminated films, where each layer contains
only one type of magnetic nanoparticles, were studied. The
results show that in both cases, one can achieve a selective
heating: the heating rate demonstrates a frequency-dependent
feature. The films can be deliberately heated at different rates:
each family of nanoparticles contribute almost independently
in the peak rates. Therefore, alternating the EM frequency,
one can selectively target the given family of nanoparticles.
This effect can be used to control the chemical reactions
occurring at the particle surfaces or in the particle vicinity.
The findings can be used to design experimental proto-
cols for the microwave assisted syntheses of new materials or
for the point of care heat delivery. Since the EM irradiation at
the FMR frequency of ferromagnetic materials is not harmful
for the human body, the proposed methods of heat generation
can be used in different biomedical applications as well.
ACKNOWLEDGMENTS
The authors are grateful for fruitful discussions with J.
R. Owens and I. Luzinov.
APPENDIX A: CIRCULARLY POLARIZED WAVES
Since magnetic field h, and magnetic induction b are the
2D-vectors oscillating in the xy-plane, constitutive equation
(4) can be written in a tensor form through its x and y compo-
nents as
bx
by
� �¼ l0
l ig�ig l
� �hx
hy
� �or bi ¼ l0lijhj; (A1)
where lij is the magnetic permeability tensor and the
frequency dependent functions l and g are defined through
Eq. (5).
In a circularly polarized EM wave, the 2D h-field can be
considered as a complex-valued vector h6¼ (ex 6 iey)
h0exp(ixt).21 The unit vectors ex and ey point in the x and ydirections, respectively. We will call the wave hþ¼ (exþ iey)
h0exp(ixt) the “plus”-wave, and the wave h�¼ (ex � iey)
h0exp(ixt) the “minus”-wave.
When a circularly polarized wave propagates through
the material along the wave vector k, magnetic field h spins
around this vector perpendicularly to it, h?k. This rotation
of vector h is schematically shown in Figure 11, where the
plus wave with subscript “þ” and the minus wave with sub-
script “�” are defined as the left- and right-handed circularly
polarized waves, respectively.
In this representation of the circularly polarized waves,
Eqs. (4) and (5) can be simplified by introducing the right- and
left-handed magnetic inductions b6¼ (ex 6 iey) b0exp(ixt) as
b6 ¼ l0 l7gð Þh6; l7g ¼ 1þ cl0M
xr6xþ iax: (A2)
APPENDIX B: SOLVING THE MAGNETOSTATICPROBLEM
r � bi ¼ 0; r� hi ¼ 0 ði ¼ l;mÞ: (B1)
The superscripts l and m stand for the host material and
magnetic nanoparticle, respectively. As known from magne-
tostatics,21 the magnetization inside an ellipsoidal particles
FIG. 11. Rotation of magnetic field in the circularly polarized waves and the
induced precessions of the magnetization vector M. In the system of Cartesian
coordinates with the z-axis pointing upward, magnetization vector M in the
right-handed wave rotates in the anti-clockwise direction. Magnetization vector
M in the left-handed wave rotates in the clock-wise direction.
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can be constant. Hence assuming this constancy of M, the
equation for induction r � bi¼ 0 is reduced to r � hi¼ 0. The
second equation r� hi¼ 0 is always satisfied by introducing
magneto-static potential wi as hi¼�rwi. Substituting this
relation into the first equation r � hi¼ 0, we obtain the
Laplace equation for the potential
r2wi ¼ 0; (B2)
subject to the boundary conditions at the nanoparticle sur-
face. Written in the spherical system of coordinates where
r¼ (r, h) is the position vector with the origin at the nanopar-
ticle center, this condition states that at r¼R we must have
wl Rð Þ ¼ wm Rð Þ
bl Rð Þ � RR¼ bm Rð Þ � R
R:
8<: (B3)
Far away from the nanoparticle, as r tends to 1, the field
must be equal to the average field �h in the composites,
rwi¼��h The solution to Eqs. (B2) and (B3) for a spherical
nanoparticles is sought in the form21
wi ¼ �ai �h � rð Þ � bi�h � rð Þr3
: (B4)
Using the boundary condition at infinity, rwi¼��h (r!1),
we immediately obtain al¼ 1. To avoid singularity of the
magnetic potential wm at r¼ 0, the constant bm should be set
as zero, bm¼ 0. Thus, the field inside the particle is uniform
and this field constant am has to be found from the remaining
boundary conditions. Substituting constitutive equation (7)
into Eq. (B3), and assuming that the host matrix satisfies the
following constitutive equation bl¼l0hl, the two coefficients
am, bl for left- and right-handed circularly polarized waves
are obtained as4
am6 ¼
3
2þ l xð Þ7g xð Þ½ �
bl6 ¼
1� l xð Þ7g xð Þ½ �2þ l xð Þ7g xð Þ½ �R
3:
8>>><>>>:
(B5)
Taking into account the following equality which follows
from Eq. (7):
2þ l xð Þ7g xð Þ½ � ¼ 3þ cl0M
xr6xþ iax
¼ 3xr6xþ iaxð Þ þ cl0M=3
xr6xþ iax(B6)
the dynamic magnetic fields inside the nanoparticle for the
plus and minus waves hm6 are related to �h6 through the fol-
lowing equation:
hm6 ¼
3
2þ l xð Þ7g xð Þ½ ��h6 ¼
xr6xþ iaxxr6xþ cl0M=3þ iax
�h6:
(B7)
The amplitude and phase of the magnetic field hm� and
the magnetization m of the nanoparticle for the minus-waves
from 1 to 25 GHz are shown in Figure 12. The phase of mag-
netization is always negative because it is induced by the
magnetic field �h� and there always will be a phase lag.
When the frequency of EM wave is low (x�xr), the phase
of magnetization approaches 0 meaning that the magnetiza-
tion rotates in phase with the field. In the other limit
(x�xr), the phase approaches –p meaning that magnetiza-
tion m is always antiparallel to the average field �h�.
In both limits, the amplitude of the magnetic moment is
very small indicating that the precession shown in Figure 1
is very weak. This precession becomes vigorous and signifi-
cantly changes the amplitudes of the field when the
frequency approaches the solution of Eq. (9). In our case
xc/2p¼ 16 GHz. At this point, magnetization m becomes
almost perpendicular to the field �h� (the phase shift of 1.68
is approximately p/2). Simultaneously with the magnetiza-
tion, the internal magnetic field also reaches the maximum at
the same frequency as shown in Figure 3(b). It is interesting
to observe that close to the point xr/2p (9.7 GHz for this
case), the amplitude of magnetic field attains a minimum
corresponding to Figure 3(a). As discussed above, at this fre-
quency, the demagnetization field is almost antiparallel to
the average field �h� and magnetic field �h� is strongly
shielded.
APPENDIX C: HEATING RATE OF A MAGNETICNANOPARTICLE
Assume that the nanoparticle is suspended in free space,
the power of the microwave is P0 ¼ c0l0h20=2 ¼ c0e0E2
0=2
(in the W/m2 units) where c0 is the speed of light in vacuum,
h0 and E0 are the magnetic and electric fields of the micro-
wave in the free space. Normalizing PE and Ph by the wave-
length k¼ 2pc0/x, and P0 we have
PhkP0
¼ 2pIm lmð Þ hm6
h0
2
;PEkP0
¼ 2pIm emð Þ E
E0
2
: (C1)
FIG. 12. Amplitude and phase of inter-
nal field hm� and magnetization m as a
function of frequency. Both hm� and m
are normalized by �h�. As follows from
Figure 3, the orientation and magni-
tude of the magnetization m strongly
depend on the frequency of the micro-
wave. Since microwave is circularly
polarized, the orientations of internal
field hm� and magnetization m with
respect to �h� actually corresponds to
the phase shift with respect to �h�.
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We first study the magnetic losses. The relative perme-
ability lm¼ l 6 g is interpreted by Eq. (7). Since the imagi-
nary part of lm for the plus-wave is almost zero (Figure 2),
only minus-wave will induce the magnetic losses. The mag-
netic field hm6 of microwave inside the nanoparticle is deter-
mined by Eq. (8). Since the damping coefficient a is
unknown for the materials listed in Table I except of cobalt,
we consider cobalt as an example and estimate the heat pro-
duction rate Phk/P0 induced by the minus-wave. It is instruc-
tive to study the heat production rate in the vicinity of the
two frequencies xr/2p¼ 0.7 GHz and xc/2p¼ 19.9 GHz
(Table I). As illustrated in Figure 13, the heat production rate
reaches a maximum at the resonance frequency xc/2p. At
this frequency, the magnetic field hm� inside the nanoparticle
is significantly enhanced (Figure 3(b)). No peak can be
observed in the vicinity of the natural precession frequency
xr/2p because the internal magnetic field hm� is almost zero
at this frequency (Figure 3(a)).
In the vicinity of the resonance frequency xc/2p, electric
losses are negligibly small compared to the magnetic losses.
This statement can be justified using the following argument.
Electric field E inside the nanoparticle can be found by solving
an electrostatic problem. The result takes the form of Eq. (8)
E ¼ 3el
2el þ emE0; (C2)
where el is the relative permittivity for the host material and
E0 is the electric field of the microwave far from the particle.
For a cobalt nanoparticle, the electric permittivity em in the
GHz range satisfies the relation jemj� el. Hence, the electric
field inside the nanoparticle is diminished (jE/E0j� 1). As a
result, the electric losses are negligibly small.
Using the heat production rate, we can also calculate the
heating rate (K/s) of a single domain nanoparticle.
Considering only magnetic losses Ph, the heat produced by a
nanoparticle per unit time is PhV, where V is the volume of
the nanoparticle. The heating rate is calculated as KT¼PhV/
Cpm¼Ph/qCp, where Cp (J/kg/K) is the heat capacity at con-
stant pressure, m is the mass of the particle, q is the density.
Using Eq. (13), the heating rate can be written as
KT ¼Ph
qCp
¼ l0
2qCp
Im lmð Þxjhm6j
2; (C3)
hm6 is related to h0 through Eq. (8) and h0 is related to P0 as
P0 ¼ c0l0h20=2. As a result, considering a linear polarized
microwave which is composed of equal amount of minus-
wave and plus-wave, the heating rate can be interpreted in
terms of P0 as
KT ¼xP0
2c0qCp
Im lmð Þ 3
2þ lm
2
: (C4)
Using the definition of the relative permeability, Eq. (7), one
can relate the heating rate with magnetic properties of nano-
particles as
KT ¼xP0
2c0qCp
axMx
x� xM=3� xrð Þ2 þ a2x2; (C5)
xM ¼ cl0M: (C6)
It is instructive to double check the validity of these results
using the dissipation function as suggested in Refs. 37 and 58
� _E ¼ l0
sj½ M0 þmÞ � Hin þHa þ h0ð Þ�
j2
1
s¼ a
cl0
jM0jjM0j � M
� �
¼ l0axM
jM0j2jM0 � h0 þm� Hin þHað Þj2
ignore the m� h0 termð Þ
¼ l0axMM0
jM0j� h0 þm�Hin þHa
jM0j
2
¼ l0axM z � h0 þm� zxr þ xM=3
xM
2
xr ¼ cl0jHin þHa �M0=3jð Þ: (C7)
Using Eqs. (5), (7), and (8), m can be related to h0 as follows
(only for minus wave):
m ¼ lþ g� 1ð Þh ¼ lþ g� 1ð Þ 3
2þ lþ gh0
¼ xM
xr � xþ cl0M=3þ iaxh0: (C8)
Substituting Eq. (C8) into Eq. (C7) yields
� _E¼l0axM z�h0þxrþxM=3
xM
xM
xr�xþxM=3þiaxh0� z
2
¼l0axM 1� xrþxM=3
xr�xþxM=3þiax
2
jh0j2
¼l0axM1þa2ð Þx2
xr�xþxM=3ð Þ2þa2x2jh0j2: (C9)
It is the instantaneous energy dissipation rate. If we average
over one period of the microwave, the heat production rate
will take the form
� _E ¼ l0axM
2
1þ a2ð Þx2
xr � xþ xM=3ð Þ2 þ a2x2jh0j2
� l0axM
2
x2
xr � xþ xM=3ð Þ2jh0j2: (C10)
FIG. 13. The rate of heat production for a single domain cobalt nanoparticle
(a) in the vicinity of natural precession frequency xr/2p, (b) in the vicinity
of the resonance frequency xc/2p. Parameters used for calculations are listed
in Table I.
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It can be proved that this energy dissipation rate is equivalent
to the magnetic heat production rate defined by Eq. (13).
One can rewrite Eq. (13) as
Ph¼l0
2Im lmð Þxjhm
6j2¼l0x
2Im lmð Þ 3
2þlm
2
jh0j2
¼l0axM
2
x2
xr�xþxM=3ð Þ2þa2x2jh0j2
�l0axM
2
x2
xr�xþxM=3ð Þ2jh0j2: (C11)
APPENDIX D: WAVE REFLECTION, TRANSMISSION,AND ABSORPTION IN A COMPOSITE FILM
As shown in Figure 1(d), the tangential components of
both electric and magnetic fields of the microwave must be
continuous at the two interfaces z¼ 0 and z¼ d. Therefore,
the boundary conditions to be satisfied at the interface are as
follows:
E0 � E4 ¼ E1 � E2e�ik1d
h0 þ h4 ¼ h1 þ h2e�ik1dðz ¼ 0Þ;
((D1)
E1e�ik1d � E2 ¼ E3
h1e�ik1d þ h2 ¼ h3
ðz ¼ dÞ:(
(D2)
The electric field and magnetic field are related through the
following relations:
E0
h0
¼ E3
h4
¼ E4
h4
¼ffiffiffiffiffil0
e0
r¼ Z0;
E1
h1
¼ E2
h2
¼ffiffiffiffiffiffiffiffiffiffiffiffiffil0
e0
lef f
eef f
r¼ Z1: (D3)
Substituting Eq. (D3) into Eqs. (D1) and (D2) yields
E0 � E4 ¼ E1 � E2e�ik1d
ðE0 þ E4Þ=Z0 ¼ ðE1 þ E2e�ik1dÞ=Z1
E1e�ik1d � E2 ¼ E3
ðE1e�ik1d þ E2Þ=Z1 ¼ E3=Z0:
8>>>><>>>>:
(D4)
This system of linear equations can are solved to obtain the
transmission and reflection coefficients equation (19). For a
multilayer system shown in Figure 14, a similar formulation
of the problem is built; the wave system is illustrated in
Figure 14.
For this problem, there are four interfaces z¼ 0, z¼ d1,
z¼ d1þ d2, z¼ d1þ d2þ d3 and the system will be com-
posed of eight linear equations. Following the strategy used
for the single layer problem, this system of equations is writ-
ten as
E1 � e�ik1d1 E2 þ E8 ¼ E0
1
Z1
E1 þe�ik1d1
Z1
E2 �1
Z0
E8 ¼1
Z0
E0
e�ik1d1 E1 � E2 � E3 þ e�ik2d2 E4 ¼ 0e�ik1d1
Z1
E1 þ1
Z1
E2 �1
Z2
E3 �e�ik2d2
Z2
E4 ¼ 0
e�ik2d2 E3 � E4 � E5 þ e�ik3d3 E6 ¼ 0e�ik2d2
Z2
E3 þ1
Z2
E4 �1
Z3
E5 �e�ik3d3
Z3
E6 ¼ 0
e�ik3d3 E5 � E6 � E7 ¼ 0e�ik3d3
Z3
E5 þ1
Z3
E6 �1
Z0
E7 ¼ 0:
8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:
(D5)
Or, using the matrix form Aij Ej¼Bi, it is rewritten as
1 �e�ik1d1 0 0 0 0 0 11
Z1
e�ik1d1
Z1
0 0 0 0 0 � 1
Z0
e�ik1d1 �1 �1 e�ik2d2 0 0 0 0e�ik1d1
Z1
1
Z1
� 1
Z2
� e�ik2d2
Z2
0 0 0 0
0 0 e�ik2d2 �1 �1 e�ik3d3 0 0
0 0e�ik2d2
Z2
1
Z2
� 1
Z3
� e�ik3d3
Z3
0 0
0 0 0 0 e�ik3d3 �1 �1 0
0 0 0 0e�ik3d3
Z3
1
Z3
� 1
Z0
0
0BBBBBBBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCCCCCCA
E1
E2
E3
E4
E5
E6
E7
E8
0BBBBBBBBBBBB@
1CCCCCCCCCCCCA¼
E0
E0
Z0
0
0
0
0
0
0
0BBBBBBBBBBBBB@
1CCCCCCCCCCCCCA: (D6)
FIG. 14. A schematic of the field and wave vector directions in a multilayer
system composed of three successive layers made of different materials.
095106-12 Y. Gu and K. G. Kornev J. Appl. Phys. 119, 095106 (2016)
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All the reflecting and transmitting waves (Ej/E0 j¼ 1, 2…8)
can be specified numerically by a matrix operation. Then the
absorption coefficient of a certain layer is obtained by calcu-
lating the net energy flowing into the layer. Take layer 1 as
an example, waves E0 and E4 correspond to the inward
energy flux (the energy flow into the layer), while waves E8
and E3 correspond to the outward energy flux (the energy
flow out of the layer). We then calculated the Poynting vec-
tor of each wave; the energy flux corresponds to the time-
averaged magnitude of the vector59
hS0i ¼1
2Z0
jE0j2; hS3i ¼1
2Re Z2ð ÞjE3j2
hS4i ¼1
2Re Z2ð ÞjE4e�ik2d2 j2; hS8i ¼
1
2Z0
jE8j2: (D7)
The absorption coefficient g1, which is the ratio of the energy
absorbed by layer 1 to the energy of the EM irradiation
pumped into the system, is defined as
g1 ¼hS0i þ hS4i � hS3i � hS8i
hS0i: (D8)
The absorption coefficient of layers 2 and 3 can be defined in
a similar way.
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