# Scaling Theory of Magnetic Order and Microwave Absorption in Amorphous and
Granular Ferromagnets

Eugene M. Chudnovsky and Dmitry A. Garanin  Physics Department, Herbert H.
Lehman College and Graduate School, The City University of New York, 250
Bedford Park Boulevard West, Bronx, New York 10468-1589, USA

(August 28, 2024)

###### Abstract

Magnetic order and microwave absorption in amorphous ferromagnets and
materials sintered from nanoscale ferromagnetic grains are investigated
analytically and numerically within the random-anisotropy model. We show that
a scaling argument specific to static randomness allows one to make
conclusions about the behavior of a large system with a weak disorder by
studying a smaller system with a strong disorder. The breakdown of the scaling
on increasing the strength of the magnetic anisotropy and/or the size of the
grain separates two distinct regimes in magnetic ordering and frequency
dependence of the absorbed microwave power. Analytical results are confirmed
by numerical experiments on spin lattices containing up to ten million spins.
Our findings should help design materials with desired magnetic and microwave
properties. The method can be extended to other systems with quenched
randomness.

##  I Introduction

Static randomness is a feature of many systems with a continuous order
parameter. They include amorphous and sintered magnets (see, e.g., Refs. RA-
book, ; CT-book, ; Marin-MMM2020, ; GC-JPhys2022,  and references therein),
arrays of magnetic bubbles bubbles , pinned charge-density waves Efetov-77 ;
Okamoto-2015  and flux lattices in superconductors Blatter-RMP1994 ; EC-
PRB1989 , liquid crystals and polymers LC , thin films on imperfect substrates
EC-PRB1986 , superfluid 3He-A in aerogel Volovik-JLTP2008 ; Li-Nature2013 ;
Volovik-JETPlett2018 , and others. The effect of static disorder on the order
parameter is stronger than the effect of thermal fluctuations IM . It usually
destroys the long-range order even at zero temperature on a scale that is
inversely proportional to the disorder’s strength. Consequently, observation
of the effect of weak static randomness on the long-range order requires a
large system. This often inhibits numerical studies of the effect because they
demand a prohibitively large computing time even with the use of the most
powerful computers. In this paper, using an example of amorphous and granular
ferromagnets described by the random-anisotropy (RA) model Harris-PRL1973 , we
demonstrate how this problem can be solved by applying scaling that is
specific to static randomness. Both statics and dynamics of random magnets
will be studied and practical results based on that scaling method will be
derived that must assist experimentalists in designing materials with desired
magnetic and microwave properties.

Theoretical research on amorphous and sintered magnets has been largely based
on the Imry-Ma concept IM . Amorphous ferromagnets are obtained by rapid
quenching from the melt of a substance that would become a crystalline
ferromagnet on slow cooling. This does not allow the neighboring atoms to
develop a crystalline lattice. Nevertheless, the neighboring spins still
develop parallel orientation due to the strong overall ferromagnetic exchange.
Directions of local magnetic anisotropy axes are random, however, or are
correlated at short distances determined by the amorphous structure factor.
The RA strength $D_{R}$ is typically small compared to the exchange $J$
because magnetic anisotropy is produced by the relativistic spin-orbit
interaction while the exchange is due to the Coulomb interaction between
electrons CT-book . Nevertheless, as suggested by Imry and Ma, weak random
pushes from the magnetic anisotropy make the direction of the magnetization
slightly wander from one spin to the other, disordering it at large distances
as in the random walk problem. The corresponding ferromagnetic correlation
length is $R_{f}\sim(J/D_{R})^{2/(4-d)}a$, with $a$ being the average
interatomic distance and $d$ being the dimensionality of the system, implying
the destruction of the long-range order in less than four dimensions. This
magnetic state received the name of the correlated spin glass (CSG) CSS-1986 ;
CT-book ; PCG-2015 .

The question of the ground state of the RA model has never been settled to
everyone’s satisfaction. Numerous studies of systems with static disorder
Cardy-PRB1982 ; Villain-ZPB1984 ; Nattermann ; Kierfield ; Korshunov-PRB1993 ;
Giamarchi-95 ; LeDoussal-PRL07 ; Bogner ; Orland-EPL ; Garel-PRB  that used
renormalization-group, scaling, replica-symmetry-breaking, and variational
methods often suggested types of the ordering that deviated from the random-
walk picture. So did numerical studies Gingras-Huse-PRB1996 ; Zeng ; Rieger ;
Itakura-05 ; Perret-PRL2012 ; Fisch-1998 ; Fisch-2000  that assumed full
thermal equilibrium. This effort subsided after it was realized that from the
practical perspective, the problem of the ground state (or the low-temperature
state) of a metastable system with quenched randomness and an exponentially
large number of local energy minima was largely irrelevant. Besides, even if
the full thermal equilibrium was achieved, the long-range correlations would
be strongly affected by the presence of topological defects GCP-PRB2013 ; PGC-
PRL ; CG-PRL . The low-temperature state of any many-body system that exhibits
hysteresis depends on history and initial condition.

Ferromagnets, and RA ferromagnets in particular, are a good example of that
Serota-1986 ; Dieny-PRB1990 ; DC-PRB1991 ; GC-JPhys2022 . Since the uniaxial
anisotropy, unlike the magnetic field, has two preferred directions, one can
argue that the global ordering of all spins into a hemisphere should be
preferred energetically in the RA magnet due to the ferromagnetic exchange. In
real experiments, as well as in the numerical work on systems below four
dimensions, however, the magnetic state of an RA ferromagnet depends on the
initial condition. Starting with a collinear initial condition (CIC), that is
with all spins looking in the same direction, the minimization of the energy
in zero field leads to the state with a finite magnetization. On the contrary,
energy minimization from a random initial condition (RIC), with spins oriented
randomly (which mimics cooling down from a paramagnetic state), leads to a
fully disordered state with the magnetization wandering on a sphere on the
spatial scale $R_{f}$ that decreases on increasing $D_{R}/J$. For a magnetized
state, that evolved from the CIC in the absence of the field, $R_{f}$
describes the scale of wandering of the transversal component of the
magnetization.

Until now, computer simulations of the RA problem, limited by the system size,
addressed values of $D_{R}/J$ that were large compared to $D_{R}/J\sim
10^{-6}-10^{-3}$ relevant to disorder at the atomic scale in amorphous
magnets. In this paper, we will show that due to the scaling, the physical
case of a very small $D_{R}$ is mathematically equivalent to $D_{R}\sim J$.
This allows one to greatly reduce the computing time when studying the RA
model numerically in application to amorphous ferromagnets. A similar
advantage of the proposed scaling must exist for many other systems with the
static disorder. For a granular ferromagnet, the effective $D_{R}$ can be
large. We shall demonstrate that there exists a critical value of $D_{R}/J$,
that is, a critical grain size, at which the scaling breaks down and the
behavior of the RA system changes qualitatively.

The above picture relates to the static properties of random magnets that have
been intensively investigated in the past. By comparison, studies of their
dynamic properties were scarce. They have been mainly focused on the
ferromagnetic resonance Saslow2018  (FMR) for which there are experimental
data Monod ; Prejean ; Alloul1980 ; Schultz ; Gullikson , and on the the
localization of spin modes Fert ; Levy ; Henley1982 ; HS-1977 ; Saslow1982 ;
Bruinsma1986 ; Serota1988 ; Ma-PRB1986 ; Zhang-PRB1993 ; Alvarez-PRL2013 ; Yu-
AnnPhys2013 ; Nowak2015  that has been reported in various magnetic systems
with static disorder Amaral-1993 ; Suran1-1997 ; Suran2-1997 ; Suran-1998 ;
McMichael-PRL2003 ; Loubens-PRL2007 ; Du-PRB2014 .

More recently, it was shown GC-PRB2021 ; GC-PRB2022 ; GC-PRB2023L ; CG-
PRB2023IP  that nonconducting amorphous ferromagnets, or granular materials
comprised of coated magnetic particles of size below the skin depth, can be
promising systems for strong broadband microwave absorption. In this paper, we
apply the scaling method to make predictions about the dependence of the
absorbed microwave power on frequency, RA, and the size of the grain. We test
these predictions in numerical experiments using the model of atomic spins on
the lattice with RA and obtain good agreement with the scaling theory The
numerical method we used is described in detail in Refs. GC-PRB2021, ; GC-
PRB2022, .

The paper is organized as follows. Scaling of the static properties of the RA
magnets is discussed in Section II. It is worked out for an amorphous
ferromagnet with the atomic-scale disorder in Subsection II.1), and for a
magnet sintered of nanoscale ferromagnetic grains in Subsection II.2. Evidence
of the transition on the strength of the RA and/or the size of the grain is
presented. Microwave absorption by granular ferromagnets is studied in Section
III, with the scaling argument for the peak frequency presented in Subsection
III.1 and the scaling of the power absorption worked out in Subsection III.2.
Our findings are summarized in Section IV.

##  II Magnetic order in a random-anisotropy ferromagnet

The RA model reflects the fact that the exchange interaction and the magnetic
anisotropy are determined by the local atomic environment. If the exchange is
predominantly ferromagnetic (favoring parallel orientation of neighboring
spins), in the first approximation one can choose a single constant $J$ for
all pairs of neighboring spins. Similarly, one can choose a single constant
$D_{R}$ for the strength of the magnetic anisotropy for all spins. However,
its direction ${\bf n}$ must be random due to the absence of the global
crystallographic order. The profound effect on the magnetic ordering comes
from that random orientation of the anisotropy axes. In what follows we shall
distinguish between an RA magnet with anisotropy axes disordered at the scale
of individual spins, see Fig. 1, and the magnet sintered from randomly
oriented nanocrystallites. We begin with the first case.

Figure 1: Equilibrium spin structure obtained numerically in a 2D amorphous
ferromagnet with random anisotropy axes of individual spins and RIC. In-plane
spin components are shown by white arrows. The out-of-plane component is shown
by orange/green corresponding to positive/negative.

###  II.1 Amorphous ferromagnet disordered at the atomic scale

Here we consider an amorphous ferromagnet described by the RA model in $d$
dimensions with the Hamiltonian

${\cal{H}}=\frac{1}{2}\int\frac{d^{d}r}{a^{d}}\left[Ja^{2}(\nabla{\bf
s})^{2}-D_{R}({\bf n}\cdot{\bf s})^{2}\right],$ (1)

where ${\bf s}({\bf r})$ is the unit vector of the dimensionless spin-field
density and ${\bf n}({\bf r})$ is the unit vector of the magnetic anisotropy
that is assumed to be random on a spatial scale $a$ representing the average
interatomic distance.

The Imry-Ma argument describing the effect of this type of disorder on
magnetic ordering goes like this. Let $R_{f}$ be the characteristic distance
where the directions of spins begin to deviate strongly from the ferromagnetic
alignment. From the Hamiltonian (1) the density of the exchange energy per
spin would be of order $J\left({a}/{R_{f}}\right)^{2}$, and the total exchange
energy in a system of size $L$ in $d$ dimensions would be

$E_{ex}\sim J\left(\frac{a}{R_{f}}\right)^{2}\left(\frac{L}{a}\right)^{d}.$
(2)

If directions of the anisotropy axes were uniformly distributed on a
$d$-dimencional sphere, the anisotropy energy per spin would be a constant
$D_{R}/d$ due to $\langle n_{i}n_{j}\rangle=\delta_{ij}/d$. The dependence of
the anisotropy energy on $R_{f}$ comes from statistical fluctuations in the
distribution of anisotropy axes which define the direction of average
magnetization in the volume of size $R_{f}$. They add the term of order
$E_{an}\sim-D_{R}\left({a}/{R_{f}}\right)^{d/2}$ to the anisotropy energy per
spin, which, in turn, adds

$E_{an}\sim-
D_{R}\left(\frac{a}{R_{f}}\right)^{d/2}\left(\frac{L}{a}\right)^{d}$ (3)

to the total anisotropy energy. The energy minimum of $E=E_{ex}+E_{an}$ on
$R_{f}$ is achieved at

$R_{f}=k_{d}\left(\frac{J}{D_{R}}\right)^{{2}/{(4-d)}}a,$ (4)

where $k_{d}$ is a $d$-dependent numerical factor. The ferromagnetic
correlation length $R_{f}$ is finite for all $d<4$, no matter how weak the RA
is. Various approximations show that $k_{d}$ is large compared to one so that
the condition of validity of Eq. (4), $R_{f}\gtrsim a$, is satisfied even for
$D_{R}>J$. A rough estimate of the limiting value of $D_{R}$ can be obtained
from the condition that the anisotropy energy of an individual spin, $D_{R}$,
is less than its total exchange energy $2dJ$ with the nearest neighbors. Above
that value of the RA, the problem simplifies: directions of spins are
determined by the directions of the local anisotropy axis.

The dependence of $R_{f}$ on $D_{R}/J$ in two dimensions (thin film), computed
on lattices of size $200\times 200$ and $500\times 500$ is shown in Fig. 2. It
has been obtained from the general formula for the fluctuations of the
magnetization GC-JPhys2022 , ${\bf m}=N^{-1}\sum_{i}{\bf s}_{i}$, in a finite-
size system of $N$ spins,

$\langle m^{2}\rangle=\frac{1}{N}\sum_{j}\langle{\bf s}_{i}\cdot{\bf
s}_{i+j}\rangle=\frac{1}{N}\int_{0}^{\infty}\frac{d^{d}r}{a^{d}}C(r).$ (5)

For a Gaussian spin-spin correlation function $C(r)=\exp(-r^{2}/R_{f}^{2})$
that we used it yields $m^{2}=\pi R_{f}^{2}/(Na^{2})$. The power $-0.85$,
obtained numerically and shown in Fig. 2, differs from power $-1$ predicted by
theory. We attribute it to the energy barriers and topological defects seen in
Fig. 1 that are not accounted for in the Imry-Ma approximation. They prevent
the system from relaxing to the IM state PGC-PRL . The difference in the power
is not large, which allows one to use scaling formulas based on the IM concept
with fair accuracy. Deviation observed on decreasing $D_{R}/J$, which depends
on the system size, is clearly related to the violation of condition $R_{f}\ll
L$ used for deriving theoretical formulas.

Figure 2: Dependence of the ferromagnetic correlation length on the strength
of random anisotropy in two dimensions.

A remarkable property of the RA model is that it is non-perturbative on
$D_{R}/J$. As we show below, the case of $D_{R}\ll J$ is mathematically
equivalent to $D_{R}\sim J$. Thus, the RA problem is a strongly correlated
problem regardless of the RA strength. The only difference is in the
ferromagnetic correlation length $R_{f}$. On one hand, this makes it difficult
to compute the exact spin-spin correlation function analytically. On the other
hand, it greatly simplifies the numerical work. Indeed, for $D_{R}\ll J$ of
practical interest, $R_{f}$ can be very large. To distinguish such a state
from the long-range ferromagnetic order, one needs to do computations on a
system of size $L\gg R_{f}$ containing a very large number of spins. Luckily,
the mathematical equivalence of this case to $D_{R}\sim J$ allows one to have
an insight into the properties of the system by considering a much smaller
$R_{f}$. With that purpose, instead of considering unit cells of size $a$ with
the local exchange $J$ and the local anisotropy $D_{R}$, let us consider
blocks of spins of size $r$ satisfying $a<r<R_{f}$, coupled by the exchange
$J_{eff}$ and characterized by the effective anisotropy $D_{eff}$. The
corresponding rescaled exchange and anisotropy energy become

$E^{\prime}_{ex}\sim
J_{eff}\left(\frac{r}{R_{f}}\right)^{2}\left(\frac{L}{r}\right)^{d}$ (6)

and

$E^{\prime}_{an}\sim-
D_{eff}\left(\frac{r}{R_{f}}\right)^{d/2}\left(\frac{L}{r}\right)^{d}$ (7)

Equating the expressions for exchange and anisotropy energies before and after
rescaling, we obtain

$J_{eff}=J\left(\frac{r}{a}\right)^{d-2},\qquad
D_{eff}=D_{R}\left(\frac{r}{a}\right)^{d/2}.$ (8)

The last equation for $D_{eff}$ is an obvious consequence of statistical
fluctuations of anisotropy directions inside the block of spins of size $r$.
The expression for $R_{f}$ that follows from the minimization of
$E^{\prime}=E^{\prime}_{ex}+E^{\prime}_{an}$ is the same as before

$R_{f}=k_{d}\left(\frac{J_{eff}}{D_{eff}}\right)^{{2}/{(4-d)}}r=k_{d}\left(\frac{J}{D_{R}}\right)^{{2}/{(4-d)}}a,$
(9)

confirming the invariance of the problem with respect to rescaling to bigger
blocks. It must be related to the fractal structure of the random walk that is
behind the Imry-Ma argument.

Since the scaling requires $a<r<R_{f}$, it is clear that it breaks at
$R_{f}\sim a$, that is, at

$D_{R}=D_{R}^{(c)}\sim k_{d}^{(4-d)/{2}}J$ (10)

on increasing $D_{R}$. The practical consequence of this is that for any
$D<D_{c}$, the answer for any global characteristic of the RA system can be
obtained numerically by making computation at $D=D_{c}$, which should provide
an enormous computational advantage because at $D=D_{c}$ the condition
$R_{f}<L$ can be satisfied for a relatively small system as compared to a very
large system needed to satisfy that condition at $D\ll J$.

Fig. 3 shows the average magnetic moment per spin of a two-dimensional RA
system, obtained by relaxing from the initial state with collinear spins. This
can be achieved by aligning all spins in a strong magnetic field and then
switching the field off. As soon as the condition $R_{f}\ll L$ is satisfied,
the curves in a broad range of $D_{R}$ tend to the universal value of
$m=0.715$ per spin, confirming the scaling property of the RA model
demonstrated above. It breaks at the critical value of the RA that is close to
$D_{R}^{(c)}=5J$. Substituting it into the fit $R_{f}/a=12(J/D_{R})^{0.85}$
shown in Fig. 2, we get $R_{f}\approx 3a$ for the value of the ferromagnetic
correlation length below which the scaling breaks down. On increasing $D_{R}$
the magnetic moment per spin tends to $m=0.5$ as is expected for three-
component spins directed randomly in a hemisphere.

Figure 3: Magnetic moment per spin in a two-dimensional RA model, obtained
numerically by relaxation from the initial state with collinear spins.

Similar formulas can be easily generalized for the RA film of finite thickness
$h$. Here the exchange energy per spin remains the same, $J({a}/{R_{f}})^{2}$,
but the anisotropy energy becomes modified by statistical fluctuations in the
direction normal to the film. This adds the factor $(a/h)^{1/2}$ to the
formula (8) for $D_{eff}$ with $d=2$, making the anisotropy energy per spin of
the film $-D_{R}({a}/{h})^{1/2}({a}/{R_{f}})$. The minimum of the total energy
is achieved at

$\frac{R_{f}}{a}\sim\left(\frac{h}{a}\right)^{1/2}\frac{J}{D_{R}}.$ (11)

This formula is valid for $h<R_{f}$. At $h=R_{f}$ it gives

$\frac{R_{f}}{a}\sim\left(\frac{J}{D_{R}}\right)^{2},$ (12)

as it should be in three dimensions.

###  II.2 Granular ferromagnet

Figure 4: Spin structure of a ferromagnet sintered from randomly oriented
nanograins having the same anisotropy direction for all spins inside the
grain. The RIC has been used. The color code is the same as in Fig. 1.

The scaling argument that is similar to the one presented in the previous
subsection can be developed for a granular ferromagnet sintered from
nanocrystals. In practice, this problem introduces another source of
randomness due to the random shape and size of the nanocrystals. Here we
consider a model of a granular ferromagnet made of identical densely packed
nanocrystals (see Fig. 4) with randomly oriented easy anisotropy axes. We
believe that such a simplified model correctly catches static and dynamic
properties of sintered ferromagnets. Due to the same anisotropy strength and
direction for all spins inside the individual grain, it permits rigorous
rescaling from atomic spins to the spins of the grains that can be tested
numerically.

The argument goes like this. Keeping the spin density ${\bf s}^{2}=1$ for the
grains, Hamiltonian (1) can be rescaled by writing it in terms of the rescaled
exchange $J^{\prime}$ and anisotropy $D^{\prime}_{R}$ acting on the spins of
the grains of size $R_{a}$,

${\cal{H}}=\frac{1}{2}\int\frac{d^{d}r}{R_{a}^{d}}\left[J^{\prime}R_{a}^{2}(\nabla{\bf
s})^{2}-D^{\prime}_{R}({\bf n}\cdot{\bf s})^{2}\right].$ (13)

Hamiltonians (1) and (13) can be understood as continuous limits of the
corresponding discrete problems with atomic and grain spins. Comparing Eq.
(13) with the original expression (1) for $R_{a}=a$, we obtain

$J^{\prime}=J\left(\frac{R_{a}}{a}\right)^{d-2},\qquad
D^{\prime}_{R}=D_{R}\left(\frac{R_{a}}{a}\right)^{d}$ (14)

For the atomic disorder defined by $R_{a}=a$, the ferromagnetic correlation
length is given by Eq. (4). The same argument must apply to the blocks of
spins of size $R_{a}$. This gives

$\frac{R_{f}}{R_{a}}=k_{d}\left(\frac{J^{\prime}}{D^{\prime}_{R}}\right)^{2/(4-d)}=k_{d}\left(\frac{J}{D_{R}}\right)^{2/(4-d)}\left(\frac{a}{R_{a}}\right)^{4/(4-d)},$
(15)
$\frac{R_{f}}{a}=k_{d}\left(\frac{J}{D_{R}}\right)^{2/(4-d)}\left(\frac{a}{R_{a}}\right)^{d/(4-d)}.$
(16)

The ferromagnetic correlation length goes down as $1/R_{a}$ in two dimensions,
and as $1/R_{a}^{3}$ in three dimensions.

The requirement that $R_{f}$ cannot go below $a$ establishes the range of
validity of the above formulas on the size of the grain:

$\frac{R_{a}}{a}<k_{2}\left(\frac{J}{D_{R}}\right)=\left(\frac{R_{f}}{a}\right)_{R_{a}=a}$
(17)

in two dimensions, and

$\frac{R_{a}}{a}<k_{3}^{1/3}\left(\frac{J}{D_{R}}\right)^{2/3}=\left(\frac{R_{f}}{a}\right)^{1/3}_{R_{a}=a}$
(18)

in three dimensions. As $R_{f}$ decreases on increasing $R_{a}$, this
condition breaks at $R_{f}(R_{a})\sim R_{a}$. At that point, the system
becomes equivalent to an array of densely packed, weakly interacting, single-
domain ferromagnetic particles of size $R_{a}$ with $R_{f}=R_{a}$. The
crossover of the granular ferromagnet from the collective behavior extending
over many grains to the independent magnetic ordering within individual grains
is illustrated in Fig. 5.

Figure 5: Dependence of the ferromagnetic correlation length on the size of
the grain illustrating the crossover from the collective magnetic behavior to
the independent magnetic ordering within individual grains in a two-
dimensional granular ferromagnet.

##  III Microwave absorption

In this section, we will show how the scaling arguments explored in the
previous section help understand microwave absorption by RA magnets. In
numerical work, we solve the discrete model with atomic spins. All
computations use the fluctuation-dissipation theorem as explained in Ref. GC-
PRB2022, .

###  III.1 Peak frequency

In Ref. GC-PRB2021,  we argued that the absorption of microwaves by the RA
magnet is dominated by excitation modes that can be interpreted as the FMR of
regions of size $R_{f}$. In a crystalline ferromagnet with the anisotropy
constant $D$ this would correspond to the FMR frequency $D/\hbar$ in a zero
field if dipole-dipole interactions were neglected. Along this line, the
absorption peak in an amorphous ferromagnet must be determined by statistical
fluctuations of the anisotropy inside ferromagnetically correlated regions.
The corresponding frequency is given by $(D_{R}/\hbar)(a/R_{f})^{d/2}$, which
was confirmed by numerical work for $d=1,2,3$.

Generalization for the peak frequency in a granular ferromagnet is
straightforward:

$\omega_{max}=D_{R}\left(\frac{R_{a}}{R_{f}}\right)^{d/2}.$ (19)

Substituting here Eq. (15), we obtain

$\omega_{max}=\frac{D_{R}}{k_{d}^{d/2}}\left(\frac{D_{R}}{J}\right)^{d/(4-d)}\left(\frac{R_{a}}{a}\right)^{2d/(4-d)},$
(20)

which gives

$\omega_{max}=\frac{D_{R}}{k_{2}}\left(\frac{D_{R}}{J}\right)\left(\frac{R_{a}}{a}\right)^{2}$
(21)

in two dimensions and

$\omega_{max}=\frac{D_{R}}{k_{3}^{3/2}}\left(\frac{D_{R}}{J}\right)^{3}\left(\frac{R_{a}}{a}\right)^{6}$
(22)

in three dimensions.

Notice a very strong dependence on $D_{R}/J$ and $R_{a}$ in three dimensions.
For physical values $D_{R}$ that are smaller than $J$ by orders of magnitude,
one should choose sufficiently large grains if the absorption in the GHz range
is desired. As the grain size $R_{a}$ approaches the domain wall thickness
$a\sqrt{J/D_{R}}$, the FM correlation length $R_{f}$ for all $d$ approaches
$R_{a}$, and, as follows from the above equations, the peak frequency in both
two and three dimensions tends to the FMR frequency $D_{R}/\hbar$ of the
grain.

Figure 6: Frequency dependence of the absorbed microwave power for a 2D system
composed of $3\times 3$ grains with $D_{R}/J=0.1$. Bézier functions have been
used to draw a smooth curve.

Frequency dependence of the absorbed power for a 2D system composed of
$3\times 3$ grains with $D_{R}/J=0.1$ is shown in Fig. 6. The numerical method
has been described in detail in Ref. GC-PRB2022, . For small $D_{R}$ local
modes GC-PRB2023L  are sparse and the statistics is not great even for system
sizes as large as $300\times 300$. We used Bézier functions Bezier  to draw a
smooth curve.

Figure 7: Frequency dependence of the absorbed microwave power for different
grain sizes computed on a $300\times 300$ spin lattice at $D_{R}/J=0.1$.
Bézier functions for raw data have been used.

Fig. 7 shows the frequency dependence of the absorbed power by a granular
ferromagnet with different grain sizes. For small grains, the dependence of
the peak frequency on the size of the grain, shown in Fig. 8, follows the
theoretical formula (21). In agreement with the theory, it switches to
$\omega_{max}=D_{R}/\hbar$ for large grains. The peak in Fig. 7 becomes
narrow, tending to the delta-function at $R_{a}\rightarrow\infty$ in the
absence of dissipation.

Figure 8: Dependence of the peak frequency in the absorption of microwaves by
a granular ferromagnet on the size of the grain.

###  III.2 Power absorption

In Ref. GC-PRB2021,  we have shown that

$P\sim
h_{0}^{2}N\frac{D_{R}^{2}}{J^{2}}\left(\frac{J}{\omega}\right)^{(4-d)/2},$
(23)

(with $h_{0}$ being the amplitude of the microwave field and $N$ being the
total number of spins) provides a good fit for the microwave absorption at
$\omega>\omega_{max}$ by a $d$-dimensional RA ferromagnet with $N$ spins and
atomic-scale disorder. Cases of $d=1,2,3$ were studied numerically. Eq. (23)
also explains the numerical finding that the peak power at
$\omega=\omega_{max}$ is weakly dependent on $D_{R}$ in all dimensions.
Indeed, for the atomic disorder with $D_{R}\ll J$, substituting Eq. (20) at
$R_{a}=a$ into Eq. (23), gives $P_{max}\sim h_{0}^{2}N$, which is independent
of $D_{R}$. The proportionality of the power to $D_{R}^{2}/\omega$ in two
dimensions and weak dependence of the peak power on $D_{R}$ are illustrated by
Fig. 9.

Figure 9: Scaled dependence of the absorbed power on the combination of
frequency and RA strength in two dimensions. The unscaled data are taken from
Fig. 4 of Ref. GC-PRB2021, .

We shall now try to come up with the dependence of the power at
$\omega>\omega_{max}$ on the size of the grain in a granular ferromagnet by
writing it in the form

$P\sim\frac{h_{0}^{2}}{J}\left(N^{\prime}\frac{{D^{\prime}}_{R}^{2}}{J^{\prime}}\right)\left(\frac{J}{\omega}\right)^{(4-d)/2},$
(24)

with the rescaled exchange and anisotropy constants given by Eq. (14) and

$N^{\prime}=N\left(\frac{a}{R_{a}}\right)^{d}$ (25)

for the number of grains. The reason we are rescaling that particular
combination $N^{\prime}{{D^{\prime}}_{R}^{2}}/{J^{\prime}}$ is that it
determines the power in the second order of the perturbation theory on RA. The
rescaling yields

$\frac{{D^{\prime}_{R}}^{2}}{J^{\prime}}N^{\prime}=\frac{{D_{R}}^{2}}{J}N\left(\frac{R_{a}}{a}\right)^{2},$
(26)

making the power absorbed by the grains given by

$P_{g}\sim
h_{0}^{2}N\frac{D_{R}^{2}}{J^{2}}\left(\frac{R_{a}}{a}\right)^{2}\left(\frac{J}{\omega}\right)^{(4-d)/2}.$
(27)

Substituting here Eq. (20) for $\omega$, we obtain for the peak power

$(P_{g})_{max}\sim h_{0}^{2}N\left(\frac{a}{R_{a}}\right)^{d-2}.$ (28)

In two dimensions it does not depend on $R_{a}$. In three dimensions it scales
as $a/R_{a}$. We speculate that the weak dependence of the peak on the grain
size at $R_{a}<R_{f}$, seen in two dimensions in Fig. 7, may be due to energy
barriers and topological defects unaccounted for in the theoretical model.

We can double-check these formulas by computing the integral power (IP), $\int
d\omega P(\omega)$, for which we have independent theory confirmed by
numerics, see Ref. CG-PRB2023IP, . The bulk of the IP comes from
$\omega>\omega_{max}$. For $d=2$ we obtain from Eq. (27)

$\displaystyle({\rm IP})_{2}$ $\displaystyle\sim$ $\displaystyle
h_{0}^{2}N\frac{D_{R}^{2}}{J^{2}}\left(\frac{R_{a}}{a}\right)^{2}J\int^{J}_{\omega_{max}}\frac{d\omega}{\omega}$
(29) $\displaystyle=$ $\displaystyle
h_{0}^{2}N\frac{D_{R}^{2}}{J}\left(\frac{R_{a}}{a}\right)^{2}\ln(J/\omega_{max})$
$\displaystyle=$ $\displaystyle
h_{0}^{2}N\frac{D_{R}^{2}}{J}\left(\frac{R_{a}}{a}\right)^{2}\ln\left[C\left(\frac{J}{D_{R}}\right)\left(\frac{a}{R_{a}}\right)\right]$

with $C$ being the integration constant of order unity. This is exactly what
we had for the IP in two dimensions CG-PRB2023IP . For $d=3$ we get

$\displaystyle({\rm IP})_{3}$ $\displaystyle\sim$ $\displaystyle
h_{0}^{2}N\frac{D_{R}^{2}}{J^{2}}\left(\frac{R_{a}}{a}\right)^{2}\sqrt{J}\int^{J}_{\omega_{max}}\frac{d\omega}{\sqrt{\omega}}$
(30) $\displaystyle\sim$ $\displaystyle
h_{0}^{2}N\frac{D_{R}^{2}}{J}\left(\frac{R_{a}}{a}\right)^{2},$

which is again exactly what we had for the IP in three dimensions CG-PRB2023IP
.

Thus, with good confidence, the scaling of the absorbed power with the
frequency and the grain size for a granular ferromagnet in $d$ dimensions at
$\omega>\omega_{max}$ is given by Eq. (27).

##  IV Discussion

The RA model has been used in the past to describe amorphous and sintered
ferromagnets. Regardless of the strength $D_{R}$ of the RA that produces the
static random potential for the spins locally ordered by the ferromagnetic
exchange $J$, the model is non-perturbative on $D_{R}/J$. In amorphous
ferromagnets, this parameter is very small due to the relativistic smallness
of the magnetic anisotropy. This always hindered numerical work on real
systems because it required a linear size of the system that is large compared
to the ferromagnetic correlation length which scales as $J/D_{R}$ in two
dimensions and $(J/D_{R})^{2}$ in three dimensions. Consequently, even a very
large (by physical standards) value of $D_{R}/J$, such as, e.g., $10^{-3}$,
requires modeling of systems that consist of the prohibitively large number of
spins.

Here we have demonstrated that due to the existence of the scaling specific to
static randomness, the RA model of an amorphous ferromagnet with the atomic-
scale disorder and $D_{R}\ll J$ can be mapped onto the model with $D_{R}\sim
J$. This makes static magnetic properties of the magnet, such as
magnetization, invariant with respect to $D_{R}/J$ up to a critical large
value of that ratio. Consequently, one can make conclusions about the behavior
of a large system with realistic small values of the magnetic anisotropy by
modeling much smaller systems with $D_{R}\sim J$. It must somehow be related
to the fractal structure of the random walk which is behind the Imry-Ma
argument, that is the invariance of the random walk with respect to the
rescaling of its trajectory to a map with a different scale.

The scaling concept is also extremely helpful in application to granular
ferromagnets for which the effective anisotropy constant $D^{\prime}_{R}$
scales as the size of the grain squared in two dimensions (thin film) and as
the size of the grain cubed in three dimensions, while the exchange
$J^{\prime}$ equals the original exchange constant in two dimensions and
scales linearly on the grain size in three dimensions. This, in principle,
permits any values of the ratio $J^{\prime}/D^{\prime}_{R}$ in experiments
with granular ferromagnets depending on the grain size. The scaling breaks
down at the critical grain size of the order of the domain wall width in a
conventional ferromagnet, $\delta=a\sqrt{J/D_{R}}$. Above such a grain size
the behavior of the granular magnet changes qualitatively. The ferromagnetic
correlation length, which was going down on increasing the grain size below
the critical size, begins to grow with the grain size above the critical size.
As a result, the static properties of the granular magnet switch from soft to
hard magnetic behavior.

Using the scaling arguments, we also studied the absorption of microwave power
by the granular ferromagnet. The peak frequency has been expressed in terms of
the magnetic anisotropy and the size of the grain and was tested in a
numerical experiment. The dependence of the power on frequency, magnetic
anisotropy, and the size of the grain have been obtained analytically and
numerically, with good agreement between the two methods. Our results apply to
insulating amorphous materials or materials made of coated metallic
ferromagnetic grains of size which is small compared to the skin depth, which
would typically be in the submicron range for the microwaves.

The breakdown of the scaling at the critical value of the grain size, besides
its effect on the static properties, has a dramatic effect on microwave
absorption. On increasing the grain size above the critical value, the
absorption peak narrows and increases in height. It makes the grain size of
the order of the domain wall width (in the ferromagnetic material of the
grain) optimal for achieving high absorption and the broad band at the same
time. These predictions can be tested on recently synthesized numerous
insulating amorphous/nanocrystalline magnetic materials AlAzzavi-2016 ; Lu-
JNCS2023 .

Scaling concepts developed in this paper can be used for other systems with
quenched randomness. They should facilitate numerical studies of disordered
materials by extending conclusions made from the modeling of small systems
with a strong static disorder to large systems with a weak disorder. In
application to random magnets, they should assist material scientists in
manufacturing materials with desired magnetic and microwave properties.

## ACKNOWLEDGEMENT

This work has been supported by Grant No. FA9550-20-1-0299 funded by the Air
Force Office of Scientific Research.

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