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\begin{document}
\makeatletter
\renewcommand{\@oddhead}{Mater.Phys.Mech. {\bf 1} (2000) 00-00\hfil\thepage}
\renewcommand{\@oddfoot}{\copyright 2000 Advanced Study Center Co. Ltd.\hfil}
\makeatother

\vspace*{2cm}
\begin{center}
{\Huge\sf\bf
AC Magnetic Properties of Compacted FeCo Nanocomposites}
\end{center}
\vspace{0.3cm}


\centerline{\bf Anit K. Giri, Krishna M. Chowdary and Sara A. Majetich}
\vspace*{2ex}

\centerline{\small
Department of Physics, Carnegie Mellon University
Pittsburgh, PA 15213-3890, USA}

\centerline{\small{\it E-mail}: sm70+@andrew.cmu.edu}
\vspace*{2ex}

\centerline{\small Received November 17, 1999}
\begin{quote}
{\small{\bf Abstract.}
Here we report the AC magnetic properties of soft magnetic 
nanocomposites made from compaction of Fe$_{10}$Co$_{90}$ nanoparticles. Following a 
discussion of previous work on soft magnetic nanocomposites, the sample 
preparation and experimental characterization by AC permeametry are 
described. The permeability is constant and equal to the DC value for low 
frequencies, but drops off sharply above a characteristic frequency $f_c$. A model 
is developed to explore the relation between $f_c$ and material parameters 
including the effective anisotropy, exchange coupled volume, temperature, and 
saturation magnetization.}
\end{quote}

\section{Introduction}

Many recent developments in soft magnets, which are used in power applications, have 
been based on amorphous and nanocrystalline materials [1-5].  In amorphous solids the 
magnetocrystalline anisotropy $K$ is by definition equal to zero, eliminating the main material-
dependent contribution to the coercivity. The effective anisotropy $K_{eff}$ can also be low in 
nanocrystalline materials if conditions specified by the Random Anisotropy Model are met [1-
3, 6]. The grain size must be smaller than the magnetic exchange length $L_{ex}$, and the grains 
must be exchange coupled with their easy axes for magnetization randomly oriented. The 
combination of random orientation and averaging over multiple grains makes the preference 
for magnetization in a particular direction, and therefore $K_{eff}$, very small. Theoretically, low $K$ 
materials become even softer if the grain size $D$ is small enough. The DC magnetic properties 
of nanocomposites have been modeled in terms of the Random Anisotropy Model [1-3]. The 
magnetization reversal field or coercivity $H_c$ is predicted to be proportional to $D^6$ , while the 
permeability $\mu=B/H$ is predicted to be proportional to $D^{-6}$. Here we extend this approach to 
understand the AC magnetic properties of nanocomposites.


Previous work on the AC magnetic properties of bulk magnetic materials has shown 
that the permeability can be split into in-phase ($\mu'$) and out-of-phase ($\mu''$) components, just like 
the permittivity of dielectric materials:
\begin{equation}
\mu=\mu'-i\mu''.
\end{equation}
These components obey a Kramers-Kronig relationship [7], so that by knowing the 
value of one for all frequencies, the other is uniquely determined. The real part $\mu'$ is typically 
flat up to a cutoff frequency, and then drops off, while the imaginary part $\mu''$ shows a peak at 
the same frequency. In the best high frequency ferrite materials, the cutoff frequency is in the 
GHz range, and is associated with the precession frequency of spins about the applied 
magnetic field. While high frequency magnetic materials are highly insulating to reduce eddy 
current losses, their properties have not been analyzed in terms of their microstructure.
A benchmark for soft magnetic materials is the total power loss, $P_{tot}$, as the material is 
magnetized and demagnetized. It depends on the hysteresis loss, losses due to the eddy 
currents created by the changing magnetic field, and anomalous losses associated with domain 
wall motion [8]. These contributions have different frequency dependencies:
\begin{equation}
P_{tot}  = P_{hys} + P_{eddy\,\,\, current} + P_{anom}
  = W_{hys} f + A f^2/\rho+ P_{anom}(f).   
\end{equation}
$P_{hys}$ varies linearly with the frequency $f$ and the area within the hysteresis curve $W_{hys}$,
$P_{eddy\,\,\, current}$ is quadratic in frequency and inversely proportional to the resistivity $\rho$. A is a 
constant. The anomalous loss $P_{anom}$ has no fixed power law dependence on the frequency.
The AC magnetic properties of ferrofluids, which are small ($< 10$~nm) magnetic 
particles dispersed in a liquid, show distinctly different behavior from that of bulk materials. 
There are two characteristic relaxation times, one due to Brownian relaxation caused by 
rotation of the particles in the liquid, and the other due to Neel relaxation caused by coherent 
rotation of the atomic spins within the particle [9,10]. Both processes occur at frequencies well 
below the Larmor frequency $f_o$.

\makeatletter
\renewcommand{\@evenhead}{\thepage\hfil {\it Anit K. Giri et al}\hfil}
\renewcommand{\@oddhead}{\thepage\hfil {\it AC Magnetic Properties}\hfil}
\renewcommand{\@oddfoot}{}
\makeatother

Nanocomposites have potential for two niche applications of soft magnetic materials. In 
low frequency applications, amorphous materials should continue to dominate, since they have 
the smallest hysteretic losses. Without significant eddy currents, only the small percentage of 
atoms required for glass forming do not contribute to the magnetization. However, at higher 
temperatures amorphous materials crystallize. In the current nanocrystalline materials [1-5], 
grain growth is limited by precipitation of additional phases, and a larger fraction of 
nonmagnetic atoms is needed. At higher frequencies, nanocomposites have potential 
advantages because they can have higher resistivity to reduce eddy current losses. While 
ferrites are currently used for high frequency applications because of their high resistivities, 
they have much lower magnetizations than iron-based alloys.

Iron-cobalt alloys are of interest for high temperature soft magnets because they have a 
high Curie temperature, high magnetization, and low coercivity. The value of the Curie 
temperature depends on the relative Co abundance. Iron-cobalt alloys have the largest magnetic 
moment per atom, and therefore large saturation magnetizations [11]. They have a low 
coercivity due to the low magnetocrystalline anisotropy $K$ of the body-centered cubic structure. 
There is a zero crossing of the anisotropy near the equiatomic composition [12].


We investigate nanolaminates formed by compaction of FeCo nanoparticles with a very 
thin coating of a protective carbon or oxide. By choosing a particle or grain diameter well 
below the maximum monodomain size, the coercivity and hysteretic power losses can be 
minimized [1-3]. The coatings act as a barrier to eddy currents, reducing the eddy current 
power losses at high frequency. The goals are to minimize the portion of nonmagnetic atoms 
while retaining a stable nanocrystalline structure at high temperature, and to determine the 
degree of coating which retains significant exchange coupling for minimum $K_{eff}$ while creating 
significant barriers to eddy currents.

\section{Experimental}

\noindent
{\bf 2.a. Particle Synthesis}
\vspace*{2ex}

Here the alloy nanoparticles were synthesized by the polyol method [13], a chemical 
route which leads to highly monodisperse metal particles. Iron chloride tetrahydrate and 
sodium hydroxide were dissolved in ethylene glycol and heated to 110 $^\circ$C while stirring. A 
second solution of cobalt hydroxide in ethylene glycol at the same temperature was added, and 
the mixture was heated to 195 $^\circ$C, where a precipitate formed. The total metal ion 
concentration was 0.2 M, and the hydroxide concentration was 2.0 M. Water and other 
reaction products were distilled off, and the solvent was refluxed for one hour at the maximum 
temperature. The precipitate was thoroughly washed with methanol, and dried.

While the particle size can be varied by changing the metal salt concentration and
reaction time, the average grain diameter was roughly constant and approximately 20 nm. The 
composition was determined from electron energy loss spectroscopy [14]. The average grain 
size was obtained from Scherrer analysis of the x-ray diffraction peaks, and the particle size 
was found from transmission electron microscopy. The exchange length in magnetically soft 
materials like FeCo is large, and here is comparable to the grain size. Adjacent grains can then 
be exchange coupled to each other, leading to a further reduction in the anisotropy and 
coercivity, as explained by the random anisotropy model [1-3, 6].
\vspace*{2ex}

\noindent
{\bf 2.b. Compaction}
\vspace*{2ex}

A nanocomposite has nanoparticle-sized grains, which may be dispersed in a matrix 
that mediates their coupling. The coupling strength depends on the magnetic behavior of the 
matrix and on the degree of coherence at the interface between the nanoparticles and the 
matrix. Exchange interactions may couple nanoparticles together. We worked with Dr. S. 
Sudarshan and Mr. Sang Yoo of Materials Modification, Inc., on plasma pressure compaction, 
which has the advantage of being able to reduce or remove the thin coating surrounding FeCo 
nanoparticles under ambient conditions. Powders were compacted by a combination of 
pressure and an electric arc in a two stage process. First a voltage pulse established a 
continuous current path across the sample in a graphite mold under a pressure of 10 MPa. This 
generated a plasma in the voids, and partly removed surface impurities such as oxides. In the 
second stage a continuous current path led to resistive sintering. The pressure was increased to 
70 MPa  to deform and compact the particles. The entire process required only a few minutes, 
thereby limiting grain growth. The resistive heating concentrated in the "necks" forming as the 
particles sinter, making them deform more readily. Compacts with over 90\% theoretical density 
were made from the nanoparticle powders.
\vspace*{2ex}


{\bf 2.c. Magnetic Properties}
\vspace*{2ex}

There was a substantial amount of magnetic characterization, including SQUID 
magnetometry (with a Quantum Design MPMS) to determine the coercivity, and AC 
permeametry (with a Walker Scientific AC Hysteresisgraph) on toroidal compacts to measure 
the frequency dependence of the permeability $\mu$ and the power loss. For the AC 
measurements, the toroid outer diameter was 2.54 cm, and it had a square cross section with 
dimensions 3.0$\times$4.1 mm. There were 154 primary turns and 11 turns in the pickup coil. The 
amplitude of the AC magnetic field applied was 125 Oe, and the frequency ranged from 100 to 
100,000 Hz. The temperature of the toroid was controlled by placing it in thermal contact with 
an ice bath (0øC), or with it inserted in a furnace (100 $^\circ$C - 500 $^\circ$C). At elevated temperatures 
the coils were sheathed with fiberglass sleeving (Omega Engineering, Inc.).


\section{Experimental Results}

Fig. 1 shows a comparison of the DC hysteresis loops for the powder and compacted 
pellet of a typical sample. Compaction significantly reduces the coercivity, indicating that the 
exchange coupling is increased, as predicted by the random anisotropy model.


  
\begin{figure}[t]
\centerline{\epsfig{file=fig1.eps, height=3.5in,width=3.5in}}
\vspace*{6cm}

{\small{\it Figure 1:}
a.) Comparison of Fe$_{10}$Co$_{90}$ nanoparticles made by the polyol method, and a compact 
made from them; b.) An expanded region shows the coercivity of the pellet is on the order of 5 
Oe, much less than that of the precursor particles, which is approximately 150 Oe.
}
\end{figure}


The AC hysteresis loops were measured for toroids cut from the same compressed 
pellets, for a range of frequencies. With AC excitation, the pickup loop measures the magnetic 
induction $B= 4\pi M+H$. Fig. 2 illustrates the characteristic behavior as a function of frequency. 
At low frequencies (Fig. 2a) the applied field H was sufficient to saturate the sample, reaching 
the single-valued region of $B(H)$. As the frequency was raised, the width of the hysteresis loop 
and therefore the coercivity increased, but the maximum induction remained approximately the 
same, up to a characteristic frequency (Fig. 2b). Approaching this frequency, the sample was 
no longer saturated at the maximum applied field, and the maximum magnetic induction 
dropped (Fig. 2c).

 

\begin{figure}
\centerline{\epsfig{file=fig2.eps, height=3.5in,width=3.5in}}
\vspace*{7.5cm}

{\small{\it Figure 2:}
AC Hysteresis loops for Fe$_{10}$Co$_{90}$, at 0 $^\circ$C.  a.) 100 Hz, b.) 800 Hz, c.) 5000 Hz.
}
\end{figure}

The permeability as a function of the maximum applied field is illustrated in Fig. 3, for 
several representative frequencies. Defining the permeability as the maximum value of $B/H$, it 
is expected to very depending on the magnitude of $H_{max}$. If $H_{max}$ is small and the sample is 
not saturated, $B$ increases approximately linearly with $H$, and $\mu$ is large. If $H_{max}$ is large and 
the sample is never saturated, $B$ increases more slowly with increasing $H$ and $\mu$  is reduced. In 
order to understand the fundamental physics of the AC excitation of the nanocomposites, a 
value of $H_{max}$ = 125 Oe was selected for further measurements so that the samples were 
saturated at low frequency.


 

\begin{figure}
\centerline{\epsfig{file=fig3.eps, height=3.5in,width=3.5in}}
\vspace*{6cm}

{\small{\it Figure 3:}
Overall permeability for Fe$_{10}$Co$_{90}$, as a function of $H_{max}$, at 0 $^\circ$C, for 500, 900, and 
1400  Hz. The peak permeability at 22 Oe reflects where the slope of the $B(H)$ starts to 
decrease.
}
\end{figure}



Several important quantities for soft magnetic materials were measured as a function of 
frequency. The real (in-phase) and imaginary (out-of-phase) parts of the permeability $\mu=B/H$ 
were also measured as a function of frequency (Fig. 4). In the low frequency range, the phase 
lag between $B$ and $H$ is small, and $\mu''$ is negligible. $\mu'$ is roughly constant up to the characteristic 
frequency, where it drops off sharply. $\mu''$ peaks as $\mu'$ drops off, and the critical frequency $f_c$ is 
determined from the position of the peak in $\mu''$. The characteristic frequency of the 
permeability varied with temperature (Fig. 5). While the low frequency magnitude of $\mu$
dropped off slightly with increasing temperature, due to a reduction in the magnetization, $f_c$ 
rose substantially, from 1400 Hz at 0 $^\circ$C to 6000 Hz at 500 $^\circ$C. As noted in the hysteresis 
loops, the coercivity rose with increasing frequency (Fig. 6). It peaked at a frequency 
approximately 2-3 times $f_c$, and decreased somewhat at high frequencies. The overall power 
loss per cycle is shown in Fig. 7. At low frequencies it rises linearly, as expected from Eq. 2 if 
hysteretic losses are dominant. At frequencies above $f_c$ a weaker dependence is observed, 
closer to $f^{0.5}$. As the temperature is increased the power loss per cycle decreases somewhat, due 
mainly to the reduction in the coercivity.


 

\begin{figure}
\centerline{\epsfig{file=fig4.eps, height=3.5in,width=3.5in}}
\vspace*{6.5cm}

{\small{\it Figure 4:}
The real ($\mu'$) and imaginary ($\mu''$ ) parts of the permeability for Fe$_{10}$Co$_{90}$, at 0 $^\circ$C, as a 
function of frequency.
}
\end{figure}

\begin{figure}[p]
\centerline{\epsfig{file=fig5.eps, height=3.5in,width=3.5in}}
\vspace*{6cm}

{\small{\it Figure 5:}
Overall permeability as a function of frequency for Fe$_{10}$Co$_{90}$, at 0 $^\circ$C and 500 $^\circ$C.
}
%\end{figure}
%\begin{figure}[p]
\centerline{\epsfig{file=fig6.eps, height=3.5in,width=3.5in}}
\vspace*{6cm}

{\small{\it Figure 6:}
Coercivity as a function of frequency for Fe$_{10}$Co$_{90}$, at 0 $^\circ$C and 500 $^\circ$C.
}
%\end{figure}
%\begin{figure}[p]
\centerline{\epsfig{file=fig7.eps, height=3.5in,width=3.5in}}
\vspace*{6cm}

{\small{\it Figure 7:}
Power Loss per unit volume as a function of frequency for  Fe$_{10}$Co$_{90}$, at 0 $^\circ$C.
}
\end{figure}
 



\section{Model of AC Magnetic Properties}

Our model of the AC properties of nanocomposites is based on a monodomain 
magnetic particle in an oscillating magnetic field. For a uniaxial particle with the applied field 
parallel to the easy axis, the energy $E$ is given by
\begin{equation}
E=K\,V\,\sin^2\theta-M_s\,H\,\cos\theta,
\end{equation}
where $K$ is the magnetocrystalline anisotropy, $V$ is the particle volume, $\theta$ is the angle between 
the magnetization direction and the applied field $H$, and $M_s$ is the saturation magnetization. 
Stoner-Wohlfarth theory [15-17] has shown that energy minima occur at $\theta = 0^\circ$ or 180$^\circ$, and a 
maximum exists where $\cos\theta=-HM_s/2K$. In order to reverse the magnetization direction, the 
energy barrier must be overcome. If this does not occur within the measurement time, then 
hysteresis is observed. The rate of going over the barrier $\tau_o^{-1}$ is given by
\begin{equation}
\tau^{-1}=\tau_o^{-1}\exp\left(\frac{\Delta E}{kT}\right),
\end{equation}
where $\tau_o^{-1} = f_0$ is the Larmor frequency, $\Delta E$ is the magnitude of the energy barrier, k is the 
Boltzmann constant, and $T$ is the temperature.
If the magnetic field oscillates in time,
\begin{equation}
H(t)=H_0\cos(2\pi f t),
\end{equation}
then the positions of the energy minima and maxima will also change with time, as shown in 
Fig. 8. Early in the cycle state 1 is the global minimum and state 2 is a local minimum, but later 
the roles are reversed. The height of the energy barrier which must be overcome for the 
particle to switch from the local to the global minimum also oscillates in time and is greatest at 
zero applied field.  
	
 
\begin{figure}
\centerline{\epsfig{file=fig8.eps, height=3.5in,width=3.5in}}
\vspace*{7cm}

{\small{\it Figure 8:}
Energy levels as a function of time, for different parts of the cycle of $H(t)$. Here $E_1$ is 
the energy of a particle in state 1, $E_2$ is the energy of a particle in state 2, and $\Delta E$ is the 
magnitude of the barrier between the local minimum and the maximum energy states
}
\end{figure}


Given an ensemble of particles and monitoring the population $N_1$ in state 1 and $N_2$ in 
state 2 as a function of time, the magnetization can be determined:
\begin{equation}
M(t)=M_s(N_1(t)-N_2(t))=M_s(2N_1(t)-1).
\end{equation}
Here we assume that population is conserved, and that at all times
\begin{equation}
N_1+N_2=1.
\end{equation}
The population in state 1 as a function of time depends on the rate $\tau_{bkwd}^{-1}$ at which 
particles get over the barrier from state 2, and on the rate $\tau_{fwd}^{-1}$ that they leave state 1 in the 
forward direction over the barrier:
\begin{equation}
\frac{dN_1}{dt}=-\frac{dN_2}{dt}=\tau_{bkwd}^{-1}N_2-\tau_{fwd}^{-1}N_1.
\end{equation}
Substituting for the rates using Eq. 4,
\begin{equation}
\frac{dN_1}{dt}=\tau_0^{-1}[(1-N_1)\exp(\frac{-\Delta E_{bkwd}}{kT})
-N_1\exp(\frac{-\Delta E_{fwd}}{kT})].
\end{equation}
Substituting for the forward $\Delta E_{fwd}$ and reverse $\Delta E_{bkwd}$ energy barriers,
\begin{eqnarray}
\Delta E_{fwd}=E_{max}-E_1=KV\left[1+\frac{HM_s}{2K}\right]^2\nonumber\\
\Delta E_{bkwd}=E_{max}-E_2=KV\left[1-\frac{HM_s}{2K}\right]^2,
\end{eqnarray}
the differential equation becomes
\begin{equation}
\frac{dN_1}{dt}=\tau_0^{-1}\left[(1-N_1)\exp\left(\frac{-KV}{kT}\left[1+\frac{HM_s}{2K}\right]^2\right)
-N_1\exp\left(\frac{-KV}{kT}\left[1-\frac{HM_s}{2K}\right]^2\right)\right].
\end{equation}
In an oscillating magnetic field $H(t)$, Eq. 11 must in general be solved numerically. 

This problem has many similarities to that of an electric dipole in an AC electric field,
which was originally solved by Peter Debye [18,19], with two main differences. In the electric 
dipole model, there was no equivalent to magnetocrystalline anisotropy, so the solution 
corresponds to the superparamagnetic limit. Also, in the electric dipole model, the factor 
equivalent to $x = M_sVH/kT$ is much less than one, and when $e^x$ is approximated by $1+x$, the 
equation corresponding to Eq. 11 can be solved analytically.
Equation 11 can be solved analytically in the limit of large $K$, where $\Delta E_{fwd}\approx\Delta E_{bkwd}$. 
With the substitution
\begin{equation}
y=\frac{KV}{kT},
\end{equation}
Eq. 11 then simplifies to
\begin{equation}
\frac{dN_1}{dt}\approx\tau_0^{-1}e^{-y}(1-2N_1).
\end{equation}
The solution to Eq. 13 is given by
\begin{equation}
N_1(t)=\frac{1}{2}\left[1+\exp\left(-2\tau_0^{-1}yt\right)\right].
\end{equation}
There will be a characteristic or resonant frequency $f_c$, where
\begin{equation}
f_c=\frac{\tau_0^{-1}}{\pi}\exp\left(\frac{-KV}{kT}\right)
\end{equation}
where the population oscillates in resonance with the driving field $H(t)$. Above this frequency, 
the population shifts cannot keep up with the rapidly changing driving field.

The model breaks down for cases where $H M_s/2K > 1$, since this requires that $|cos\theta=-
H M_s/2 K|$ be greater than one. To investigate the small $K$ limit, which is of critical importance 
for soft magnetic materials, we reduced the amplitude of the magnetic field $H_o$ where 
necessary. In experiments it is possible to have large amplitudes of the AC magnetic field when 
studying materials with very low anisotropy. Here the sample is being driven past the high 
slope portion of the hysteresis curve during each half-cycle. In applications of soft magnetic 
materials, this is undesirable since the permeability $\mu=B/H$ is reduced. Therefore the model can 
be used to understand the AC magnetic behavior in situations of greatest technological 
importance.


\section{Modeling Results and Comparison with Experiment}

The simulation results show the same characteristics as the experimental data, and 
provide insight about the mechanism responsible for the frequency dependent behavior. 
Fig. 9 illustrates the connection between the time-dependence of $M$ and the $B(H)$ curves for 
driving fields of different frequencies $H=H_o\cos(2\pi ft)$. If the frequency is low enough (Fig. 9a) 
the sample has time to equilibrate with the slowly varying field and superparamagnetism is 
observed. There is a negligible phase lag $\theta$ between $M(t)$ and $H(T)$. Because of this $M(t)$ 
reaches its maximum value well below the maximum amplitude of the driving field Ho. At 
somewhat higher frequencies $M(t)$ starts to lag behind $H(t)$ significantly, leading to hysteresis 
(Fig. 9b) . However $M(t)$ still reaches its maximum value before $H(t)$ starts to decrease. At the 
critical frequency $f_c$ this is no longer true (Fig. 9c), and the shape of $M(t)$ and the hysteresis 
loop change noticeably. $B(H)$ no longer contains single valued regions, and the maximum 
value of $B$ starts to decrease relative to its value at lower frequencies. At the highest 
frequencies $M(t)$ and $H(t)$ are almost completely out of phase and the maximum values of $M(t)$ 
and $B(H)$ are very small (Fig. 9d). The model simulates characteristics of the experimental 
results shown in Fig. 2, and shows that the shape and magnitude of $B(H)$ depend on whether 
the magnetization of the exchange coupled volume within the sample can equilibrate with the 
changing external magnetic field.



\begin{figure}[p]
\centerline{\epsfig{file=fig9.eps, height=3.5in,width=3.5in}}
\vspace*{19cm}

{\small{\it Figure 9:}
$M(t)$ and $H(t)$, and the corresponding $B(H)$ curves for a sphere of exchange coupled
volume $V$ and effective anisotropy $K$. a.) for a superparamagnetic particle, b.) for a 
monodomain but not superparamagnetic particle at low frequency  $f$, c.) at med. $f$, and d.) at 
high $f$. Note that in a.) and b.) $B(H)$ appears to saturate, but this is because $4\pi M$ is so much 
larger than $H$. These simulations used the following parameters: $M_s = 1000$ emu/cc, $H_0 = 125$ 
Oe, $K = 1.95 \cdot10^5$ erg/cc, $T=273$ K, and $V = 1.47\cdot10^{-18}$ cc. The critical frequency $f_c$ was 
1400 Hz.
}
\end{figure}

Figure 10 shows the corresponding trend in the permeability $\mu$ as a function of 
frequency. At low frequencies it is constant and equal to the DC value, but as soon as $M(t)$ is 
unable to reach its maximum value, $B_{max}$ and therefore $\mu$ begin to drop, approaching zero in the 
high frequency limit. This is in contrast to the experimental results shown in Fig. 5, where $\mu$
decays more slowly. Such a difference would arise if the sample contained a distribution of 
characteristic frequencies $f_c$, perhaps related to a distribution of grain sizes.
\looseness=-1


 

\begin{figure}[t]
\centerline{\epsfig{file=fig10.eps, height=3.5in,width=3.5in}}
\vspace*{6.5cm}

{\small{\it Figure 10:}
The permeability $\mu=B_{max}/H_{max}$ as a function of frequency, for the same parameters as 
in Fig. 9.
}
\end{figure}

Fig. 11 reveals the behavior of the simulated coercivity as a function of frequency. At 
low frequencies it follows Sharrock's Law [20]:
\begin{eqnarray}
H_c(f)=\left(\frac{2K}{M_s}\right)\left[1-\left(\frac{kT\ln(f_0/f)}{KV}\right)^{1/2}\right], \mbox{ or}\nonumber\\
H_c(f)=\left(\frac{2K}{M_s}\right)\left[1-\left(\frac{kT}{KV}\right)^{1/2}\left(\frac{1}{2}\right)\ln f_0\right]+
\left(\frac{2K}{M_s}\right)\left[\left(\frac{kT}{KV}\right)^{1/2}\left(\frac{1}{2}\right)\ln f\right],
\end{eqnarray}
rising with the log of the frequency. The peak in $H_c$ occurs between two and three times $f_c$, 
and then $H_c$ begins to decrease. This occurs when the magnetization due to the driving field is 
low enough, and the value of $H(t)$ needed to make $M(t)$ equal to zero is reduced. The 
experimental coercivity shown in Fig. 6 also rises and peaks at roughly the same frequency, but 
the rise is not proportional to $\ln f$. While there may be a region where $H_c$ obeys this functional 
dependence, there appears to be a low frequency contribution to the coercivity from a factor 
not included in the model. This could arise from particles not exchange coupled to their 
neighbors, possibly from a thicker coating.
\looseness=-1

Fig. 12 shows the hysteretic power loss $P_{hys}$ found from the are of the hysteresis loop 
$B(H)$, as a function of frequency. This rises linearly at low frequencies, as expected from Eq. 
2, but then it levels off at $f_c$. At very high frequency it should decrease when both $M_{max}$ and $H_c$ 
are decreasing.
In comparison, Fig. 7 shows the total experimental power loss. At frequencies below $f_c$,
hysteretic power losses dominate. This is expected for frequencies below 10,000 Hz.


 

\begin{figure}
\centerline{\epsfig{file=fig11.eps, height=3.5in,width=3.5in}}
\vspace*{6.5cm}

{\small{\it Figure 11:}
The coercivity  $H_c$ as a function of frequency, for the same parameters as in Fig. 9.
}
\end{figure}

 

\begin{figure}
\centerline{\epsfig{file=fig12.eps, height=3.5in,width=3.5in}}
\vspace*{6.5cm}

{\small{\it Figure 12:}
The hysteretic power loss as a function of frequency, for the same parameters as in 
Fig. 9.
}
\end{figure}

The qualitative behavior illustrated in Fig. 9-12 is similar for a wide range of input 
parameters; only the magnitudes of $\mu$, $H_c$ and $P_{hys}$ vary, along with the critical frequency $f_c$. 
The dependence of $f_c$ on the anisotropy, volume, and temperature is summarized in Fig. 13. 
From the simulated data, an empirical functional form for $f_c(K,V,T)$ was obtained:
\begin{equation}
f_c=10^{-9}\tau_0^{-1}\exp[-(5.88\cdot10^{15}\mbox{ K/erg})(\frac{KV}{T})
-(4.33\cdot10^{-6}\mbox{ cc/erg})K+(4.11\cdot10^{20}\mbox{ K/cc})(\frac{V}{T})+18.98].
\end{equation}
Note that this formula is considerably more complex than Eq. 15, which was valid only in the 
limit of large $K$. For soft magnetic materials $K$ is small, and numerical simulations are the only 
was to predict the cutoff frequency.

\begin{figure}
\centerline{\epsfig{file=fig13.eps, height=3.5in,width=3.5in}}
\vspace*{20.5cm}

{\small{\it Figure 13:}
a.) $f_c$ as a function of the anisotropy for spheres of different radii. Here $V=(4\pi/3) r^3$; 
b.) $f_c$ as a function of the volume; c.) $f_c$ as a function of the inverse temperature.
}
\end{figure}


For the experimental grain size, $r=10$ nm, and an effective anisotropy equal to $1.5\cdot10^5$~erg/cc
would be needed to fit the observed room temperature value of $f_c$. The is a much
larger value of the anisotropy estimated from the experimental coercivity and Sharrock's Law. 
Presumably the physically relevant volume $V$ is not that of a single grain, but of a small 
number of coupled grains.

\section{Conclusions}

The AC magnetic properties of soft magnetic nanocomposites made from compaction
of Fe$_{10}$Co$_{90}$ nanoparticles can be understood in terms of an effective anisotropy and a related 
exchange coupled volume. The permeability is constant and equal to the DC value for low 
frequencies, but drops off above a characteristic frequency $f_c$. For bulk materials such as 
ferrites the cutoff frequency can be extremely high, and the behavior is understood in terms of 
the Landau-Lifshitz-Gilbert equation [21-23]. However for compacted nanocomposites the 
cutoff frequency is much lower, and depends on the sample microstructure. The value of this 
frequency can be increased by raising the temperature, or by reducing the anisotropy and grain 
size. Experiments are in progress to prepare samples to further test this model.

\vspace*{2ex}

{\bf Acknowledgements.}
This work was supported in part by the National Science Foundation under Grant 
numbers DMR-9900550 and ECD-8907068, and by the Air Force Office of Scientific 
Research under grant number F49620-96-1-0454. The assistance of K. Humfeld and N. Ide in
the simulations is greatly appreciated.
\vspace*{4ex}

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\end{document}
