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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