See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/224174962

Low-temperature onset of the spin glass correlations in the ensemble of

oriented Stoner–Wohlfarth nanoparticles

Article in Journal of Applied Physics · August 2010

DOI: 10.1063/1.3466991 · Source: IEEE Xplore

CITATIONS READS

13 218

7 authors, including:

A. A. Timopheev S. M. Ryabchenko
Crocus Technology, Genoble, France Institute of Physics of the National Academy of Science of Ukraine
51 PUBLICATIONS 618 CITATIONS 161 PUBLICATIONS 1,296 CITATIONS

SEE PROFILE SEE PROFILE

A. F. Lozenko V. A. Stephanovich
Institute of Physics of the National Academy of Science of Ukraine Opole University
50 PUBLICATIONS 376 CITATIONS 245 PUBLICATIONS 2,292 CITATIONS

SEE PROFILE SEE PROFILE

All content following this page was uploaded by V. A. Stephanovich on 27 May 2014.

The user has requested enhancement of the downloaded file.

 JOURNAL OF APPLIED PHYSICS 108, 033919 共2010兲

Low-temperature onset of the spin glass correlations in the ensemble of

oriented Stoner–Wohlfarth nanoparticles
A. A. Timopheev,1,a兲 S. M. Ryabchenko,1 V. M. Kalita,1 A. F. Lozenko,1 P. A. Trotsenko,1
V. A. Stephanovich,2 and M. Munakata3
1
Institute of Physics NAS Ukraine, Prospect Nauki str. 46, 03028 Kyiv, Ukraine
2
Institute of Physics, Opole University, Oleska 48, 45-052 Opole, Poland
3
Energy Electronics Laboratory, Sojo University, Kumamoto 860-0082, Japan
共Received 9 March 2010; accepted 23 June 2010; published online 13 August 2010兲
We reveal the low-temperature anomaly in the temperature and angular dependencies of the
coercivity in granular 共CoFeB兲x – 共SiO2兲1−x films with oriented in-plane anisotropy. Namely, at T
⬍ 100 K the in-plane angular dependence of coercive field acquires two maxima corresponding to
easy and hard 共in a film plane兲 directions. This signifies the emergence of coercivity for hard
direction in a film plane. The experimental results are explained in terms of a random field model,
which describes the onset of spin glass-like correlations in the ensemble of oriented weakly
interacting Stoner–Wohlfarth particles. © 2010 American Institute of Physics.
关doi:10.1063/1.3466991兴

I. INTRODUCTION sitions to spin glass-like state in SPM media really exist, they
should be revealed not only in dynamical magnetic proper-
The influence of the interparticle interactions on the ties, but in static 共or quasistatic兲 properties also. Moreover,
magnetic properties of the superparamagnetic 共SPM兲 media during the onset of spin glass correlations between neighbor-
is a very significant question both from theoretical and prac- ing particles, one can expect the joint influence 共or competi-
tical points of view. It is well known, that as interaction tion兲 of the collective ultrametric potential profile and indi-
strength grows, an ensemble of SPM particles undergoes a vidual one due to single-particle anisotropy. This competition
transition from the state, where everything is determined by should change the temperature and angular dependences of
the identical contributions of individual particles, to the state the coercive field in blocked state of the SPM ensemble. At
with collective properties, where correlated orientations of least, it should be noticeable in the ensembles with oriented
particles 共granules兲 magnetic moments play a decisive role. granules anisotropy.
One of such collective states is so-called superspin glass To the best of our knowledge, any manifestation of onset
共SSG兲 one 共in other words it is a spin glass-like state of of above spin glass-like temperature-driven correlations in
“macrospins”—magnetic moments of single-domain gran- the coercivity of the blocked state of the SPM particle en-
ules兲, where the frustrated collective behavior of granules sembles has not been reported in the literature. So, the ques-
with randomly distributed magnetic moments is due to either tions about the onset of such metastable configurations and
pure magnetic dipole interaction or its combination with ex- the manifestations of the SSG correlations in such ensembles
change one. If the exchange interaction of ferromagnetic are still remaining unanswered. The clarification of this point
type dominates or dipole interaction undergoes certain is a subject of the present paper.
modifications,1 the so-called superferromagnetic 共SFM兲 state
共the ferromagnetic ordering of the granules magnetic mo-
ments兲 appears. The comprehensive experimental studies of II. EXPERIMENT AND DISCUSSION
above ensembles were performed in a number of papers 共see To be specific, in this paper we report the low-
Refs. 2–6 for example兲. They had shown that as ferromag- temperature studies of physically nonpercolated
netic granules concentration grows 共and consequently the 共Co0.25Fe0.66B0.09兲x共SiO2兲1−x nanogranular film with uni-
strength of interparticle interaction兲 an ensemble of interact- formly oriented in-plane uniaxial anisotropy 共the anisotropy
ing SPM particles could reveal a crossover from SPM to field Ha␸ can be from 50 to 300 Oe depending on x value兲.
SSG and further to SFM state. It has also been found, that at This anisotropy reveals itself in the measurements of magne-
temperature lowering the SPM-SFM-SSG transitions can tization in the film plane and in angular dependence of the
take place. The main source of these findings is experimental ferromagnetic resonance 共FMR兲 line positions for the mag-
studies of the dynamical properties of the above ensembles. netic field directed in this plane. The films of about 500 nm
In general case the formation of a spin glass state is thickness with nominal atomic content of the magnetic frac-
related to the onset of ultrametric potential profile in the tion x = 0.55, 0.60, and 0.70 were grown by magnetron
system, i.e., the profile with multiple quenched metastable sputtering7,8 on the substrates fixed on the rotating drum. The
configurations of the mutual orientations of magnetic mo- x-ray diffraction studies7 have shown that granules are amor-
ments of the constituents. So, if the temperature-driven tran- phous. One could suppose that discussed in- plane anisotropy
may be related to small deviation of granules shape from
a兲
Electronic mail: timopheev@iop.kiev.ua. axially symmetric in the direction normal to film plane. The

0021-8979/2010/108共3兲/033919/7/$30.00 108, 033919-1 © 2010 American Institute of Physics

033919-2 Timopheev et al. J. Appl. Phys. 108, 033919 共2010兲

value of 4␲ M fm-bulk 共M fm-bulk is saturation magnetization of

the ferromagnetic granule material, which we suppose to be
equal to the magnetization of a single-domain granule m p兲 is
about 22000 G for the bulk CoFeB.9 Thus, one can obtain
that the deviation of granules shape from the above axially
symmetric one may be not more than 1%. However, it is
difficult to expect that the deviation would be of similar
magnitude and direction for all granules. Possible explana-
tion can be sought in variation in intergranular distances in
different directions and in their 共granules兲 small elongation
emerging in the fabrication process similar to those described
in Ref. 10. Namely, the rotation of a substrate holder can
probably cause the above elongation and different inter-
granular distances along tangential and radial directions due FIG. 1. Temperature dependence of coercive field for film with x = 0.60.
to different related velocities of the precipitable atoms on the Curve 1 corresponds to easy magnetization direction. Dotted line shows the
high temperature asymptote Hceasy共T兲 ⬵ ␣共1 − 冑T兲. Curve 2 corresponds to
substrate. Their joint effect yields not only the anisotropy of hard 共in-plane兲 magnetization direction. The magnetization curves for easy
isolated granules but the anisotropy of intergranular interac- 共1兲 and hard 共in-plane兲 共2兲 directions at T = 380 K are shown in the inset.
tion also.
Additionally to the above in-plane anisotropy, the films
have strong easy-plane anisotropy related to their demagne- b, and c in the x, y, and z directions, respectively.13 Unfor-
tization factor 4␲ M eff. Due to the latter, the granules magne- tunately, our attempts to check the a:b:c ratio by the scanning
tization does not leave a film plane at magnetization reversal electron microscope measurements were not succesful. So,
by the external magnetic field lying in this plane. The films the uncertainty of our knowledge of the real f v values for
with nominal values 共determined by the technologists during x = 0.55, 0.60, and 0.70 is within the differences of the above
samples fabrication兲 of x = 0.55, 0.60, and 0.70 should have estimations. Namely, if granules are nonspherical, then we
the volume fractions, f v, of the ferromagnetic granules about have above nominal x values and conversely, if granules are
0.26, 0.30, and 0.40, respectively. In the case of spherical spherical, we have noticeable larger x. Of course, the esti-
granules, the easy-plane anisotropy field should be equal to mated ratio of three granule axes 1:1:0.55 may indicate the
4␲ M eff = f v4␲m p. The different experimental techniques averaged diameter/thickness ratio for oblate 共probably non-
共measurements of FMR and the film magnetization at mag- ellipsoidal兲 granules’ clusters, although an assumption about
netic field directed normally to the film plane兲 show that the a noticeably larger x seems to us more plausible. Neverthe-
values of 4␲ M eff equal to 9700 G, 9500 G, and 12 000 G, less, even the latter f v estimation for the samples with nomi-
respectively. Therefore, if the granules shape is spherical, the nal x = 0.55 and 0.60 共f v = 0.44 and 0.43兲 is lower than the
volume fractions f v of ferromagnetic material in our films percolation threshold for the granular films 共f vp ⬇ 0.55兲
should be about 0.44, 0.43, and 0.55, respectively. These which is pointed out in review article of Chien.14 Latter
values are substantially different from those following from value of f vp is higher than theoretical one f vp = 0.3 obtained
the nominal atomic content x. The origin of such difference for the case of three-dimensional uniform random distribu-
may be either incorrectness in determination of the actual tion of granules positions. The reasons for such discrepancy
value of x 共and, consequently, f v兲 or strong enough deviation may be some correlations in the granules positions 共as com-
of granules shape from sperical 共their oblateness in the nor- pared to uncorrelated random ones兲 or/and possible presence
mal to film plane direction兲. Really, the demagnetization en- of the oxide insulating shell on the granules surfaces.
ergy density of a granular film in saturating magnetic field is The films with nominal contents x = 0.55 and 0.60 have
equal to Ud = 共1 / 2兲f vm high electric resistance 共about 100– 150 m⍀ cm versus
ជ p关共1 − f v兲N̂g + f vN̂ f 兴m
ជ p 共see Refs. 11
60 ␮⍀ cm for the bulk alloy兲 and a giant tunnel magnetore-
and 12兲. Here, N̂g and N̂ f are demagnetization tensors of a
sistance without anisotropic part.15 Latter fact indicates ad-
single-granule 共we assume that they are the same for all
ditionally that these films are physically nonpercolated. The
granules兲 and of the film as a whole respectively. The main
film with x = 0.70 reveals the simultaneous presence of tunnel
components of the N̂ f are 兵Nfxx , Nfyy , Nfzz其 = 兵0 , 0 , 4␲其, where and anisotropic magnetoresistance and, is propably partly
z direction coinsides with normal to a film plane and x, y lie percolated.
in it. Let us assume that main axes of N̂g tensor obey the The hysteresis loops at different temperatures T and di-
relation 2Ngxx = 2Ngyy = Ngzz − 4␲. This means that we neglect rections of external magnetic field He, have been measured
a small difference of Ngxx and Ngyy taking into account that in the above films by vibration sample magnetometer LDJ-
in-plane anisotropy is much less than easy-plane one. In this 9500. These magnetization curves are almost rectangular in
case the effective field of easy-plane anisotropy is 4␲ f vm p the easy direction and linear at −Ha␸ ⬍ H ⬍ Ha␸ in the hard
+ 共3 / 2兲共1 − f v兲m p共Ngzz − 4␲ / 3兲. This equation implies, that 共in-plane兲 direction for the films with x = 0.55 and 0.60 at
the coincidence with the observed values 4␲ M eff for the T ⬎ 110 K. The inset to Fig. 1 shows the example of such
above films with nominal x values is achieved if Ngzz ⬇ 2␲ curves for the film with x = 0.60 at the T = 380 K. The shapes
and Ngxx ⬇ Ngyy ⬇ ␲. Such Ng components correspond to the of these curves do not show appreciable signs of dispersion
ratio a : b : c = 1 : 1 : 0.55 for the ellipsoidal granule with axes a, of particles sizes and/or shape, which would lead to disper-
033919-3 Timopheev et al. J. Appl. Phys. 108, 033919 共2010兲

FIG. 3. The family of angular dependencies of FMR line position in the film
with x = 0.60 for different temperatures.
FIG. 2. The angular dependence of Hc共␸兲 in the film with x = 0.60 for the
temperatures 4.5, 20, 40, and 115 K.
peratures 关␪H and ␸H are, respectively, the angles between
sion of in-plane anisotropies and Tb values. It is a convincing the direction of the external magnetic field He and normal to
proof 共magnetic characterization兲 of the possibility to con- a film plane 共␪H兲 and between the He projection on the film
sider these films as the ensembles of Stoner–Wohlfarth 共SW兲 plane and easy magnetization direction 共␸H兲兴. The angular
共Refs. 16兲 particles with oriented anisotropy with respect to dependence Hc共␪H = ␲ / 2 , ␸H兲 at low temperatures is similar
their in-plane magnetisation reversal. We did not see any to that at high temperatures except the additional coercivity
manifestation of magnetic ions presence in other phases both peak near ␸H = ␲ / 2 共hard in-plane direction兲. It should be
in the magnetostatic properties of our films and in their noted that the ranges of angles around both easy and hard
FMR. 共in-plane兲 directions, where the coercive field grows, are nar-
It has been shown in Refs. 17 and 18 that the samples row enough at the temperatures where the considered effect
with x = 0.55 and 0.60 demonstrate transitions from SPM to occurs. These regions widen at temperature lowering and
the coercive SFM state at Tsf ⬇ 550 K. The additional tem- become quite large at the lowest temperature. This is true for
perature dependent coercivity, which arises at T ⬍ Tsf in the all ␸H directions under Hc growth. To the best of our knowl-
latter state, was masked at temperature decreasing below the edge, any examples of similar angular dependencies of the
blocking temperature Tb共⬇350 K兲 by the blocking state co- coercivity for the blocked state of oriented SW particle en-
ercivity. Due to this, the temperature and angular dependen- sembles have not been reported earlier.
cies of these films coercivity at their in-plane magnetization Generally speaking, the angular dependence of the coer-
in the temperature interval 110–250 K did not differ notice- civity with maxima at ␸ = 0, ␲ / 2, ␲, and 3␲ / 2 could arise in
ably from those 共well known兲 for the blocked SPM state, the case where an additional type of anisotropy with ␲ / 2
Refs. 16 and 19. Below we will show that at T ⬍ 100 K the in-plane periodicity appears at low temperatures. But such
samples with x = 0.55 and 0.60 demonstrate the unusual tem- supposition does not allow to explain the shapes of the
perature and angular dependencies of the coercivity of Hc共␪H = ␲ / 2 , ␸H兲 peak near ␸ = ␲ / 2 and the curves on Fig. 2.
blocked SPM 共SFM兲 state, which we interpret as manifesta- Nevertheless, to check latter possibility, we measured the
tions of the SSG correlations. angular dependences of FMR for magnetic field directions in
The observed dependencies Heasy c 共T兲 and Hc 共T兲 关coer- a film plane 共␪H = ␲ / 2 , 0 ⱕ ␸H ⱕ ␲兲. The FMR measurements
hard

cive field for easy and hard 共in-plane兲 directions of the ex- were carried out by x-band electron paramagnetic resonance
ternal magnetic field, He, respectively兴 are reported on the spectrometer. The family of angular dependences of FMR
Fig. 1. One can see that at T ⬎ 120 K Hhardc 共T兲 is almost zero line position for sample with x = 0.60 is presented on Fig. 3.
while for T ⬍ 100 K it becomes nonzero and is accompanied These dependences are well described by the usual equations
by a modification of characteristic temperature dependence for FMR in a film with strong easy-plane anisotropy and
Heasy 冑
c ⬵ ␣共1 − T兲 in easy magnetization direction. The theo- weak in-plane anisotropy with ␲ periodicity. The anisotropy
retical models invoked to interpret the temperature and an- can be described by the sum of two contributions to granule
gular dependencies of coercivity for the oriented SW par- magnetic energy density Ua = U␪ + U␸, where U␪ =
ticles ensembles do not admit it to occur for hard 共in-plane兲 −1 / 2m pHa␪ sin2␪ and U␸ = −1 / 2m pHa␸ sin2␪ cos2 ␸. Here m p
direction of magnetization. The above hard 共in-plane兲 direc- is a single-domain granule magnetization 共we suppose that
tion is an orientation of absolute instability for granules mag- m p and granule volume V p are the same for each particle兲,
netization due to the presence of in-plane anisotropy. This Ha␪ and Ha␸ are the effective fields of strong easy-plane and
result is valid both for the ensembles of noninteracting SW the weak in-plane anisotropy respectively; ␪ and ␸ are, re-
particles16,19 and for weakly interacting ensembles of such spectively, the angles between particle magnetization direc-
particles resulting in SFM state.17,18 tion and film normal 共␪兲 and easy axis in a film plane 共␸兲.
The low-temperature growth of Heasy c and onset of Hhard
c The angular dependencies of FMR from Fig. 3 demonstrate
are accompanied by an unexpected angular dependence, that Ha␪ and Ha␸ are weakly temperature dependent; the dis-
Hc共␪H = ␲ / 2 , ␸H兲, which is shown on Fig. 2 for several tem- cussion of this dependence is out of scope of present paper.
033919-4 Timopheev et al. J. Appl. Phys. 108, 033919 共2010兲

The above dependencies, however, do not give any addi-

tional contributions to the in-plane anisotropy, which could
be responsible for unusual in-plane angular dependence
Hc共␸H兲 in low-temperature region of the Fig. 2. To compare
the physics underlying the angular dependencies of FMR and
coercivity we recollect that FMR is measured near the satu-
ration magnetic field while the coercive field is much smaller
as it determines the limit of stability of metastable state in
magnetization reversal process.
In an ensemble of weakly interacting SW particles the
main part of anisotropy, determining possible metastable
states, have a single-granular nature 共analogous to single-ion
one in the ordinary magnets兲, it is responsible for Ha␸ in our
case. In such ensembles, the uniform and isotropic part of
FIG. 4. MC simulated angular dependencies of coercive field with aniso-
intergranular interaction is responsible for the formation of tropy energy density profile corresponding to Eq. 共10兲 and parameters values
SFM state. For our films, it has been considered in Refs. 17 a = 0.023, B = 10, and C = 4 for the different reduced temperatures 关Tred
and 18. At the same time the anisotropic part of the inter- = kT / 共KV p兲兴.
granular interaction can lead to the nonuniform configura-
tions of the mutual orientations of the granules magnetic One can see that with lowering temperature the model
moments. Due to spatial disorder in the above magnetic mo- curves acquire the additional contributions to coercivity near
ments ensemble such configurations create random network ␸ = 0 and ␲ / 2 while for intermediate angles their variation is
so that their energy can form the local 共much shallower then sufficiently small. This behavior is similar to the experimen-
initial global minimum兲 minima, which could stabilize the tal curves evolution for 20, 40, and 115 K shown on the Fig.
states corresponding to the maximum 共i.e. direction of abso- 2. The absence of the effect of potential 共1兲 on the FMR line
lute instability兲 of initial energy profile. In other words, the positions may be understood if we suppose that strong
anisotropic magnetic interactions between randomly posi- enough 共close to saturation one兲 magnetic field completely
tioned granules in a film can lead to quasiultrametric20,21 polarizes the system so that the above shallow energy profile
energy profile. It could stabilize the above 共initially unstable兲 disappears.
states at low temperatures, as the minima of this profile are
shallow.
To check this supposition, we add a hypothetical contri- III. MODEL AND DISCUSSION
bution of random effective fields, related to the mutual ori-
To explain the observed effect theoretically, we consider
entation of granules magnetic moments to the above single-
an ensemble of SW particles with account for the interpar-
particle anisotropy energy density Ua = U␪ + U␸. We choose
ticle interaction. The studied films have biaxial anisotropy—
the above contribution in the form
easy-plane U␪ and intraplanar U␸. For simplification of the
problem we consider the system with uniaxial anisotropy
UR共␸兲 = a共cos B␸兲C共m pHa␸/2兲, 共1兲 only supposing that it should capture the main physical fea-
tures of our films as the granules magnetic moments do not
with B Ⰷ 1, a Ⰶ 1, and arbitrary C ⱖ 1. We include the factor leave the film plane due to strong easy-plane anisotropy U␪.
m pHa␸ / 2 in this equation for coefficient a to be dimension- Choosing for definiteness the He direction along z-axis,
less. 共which coincides with similarly directed easy axes of all par-
The contribution 共1兲 determines shallow and frequent ticles兲, we can write the Hamiltonian of such system in the
ripples of the system energy angular dependence, which form
emulate the partial 共quasiultrametric兲 energy minima corre-
1
sponding to different mutual granules magnetization orienta-
tions. We perform Monte Carlo 共MC兲 simulations of magne-
H=− 兺 J␣␤共rij兲Si␣S␤j + g␮B兺
2 ij␣␤ i␣
He␣Si␣

tization reversal loops for different He directions in a film

plane. The simplified uniaxial model with U = − D 兺 Siz2 ; ␣, ␤ = x,y,z. 共2兲
i
−共1 / 2兲m pHa␸ cos2 ␸ + UR共␸兲 − m pHe cos共␸ − ␸H兲 has been
used. This model should be appropriate for the description of Here Si is a classical spin 共dimensionless granule magnetic
in-plane magnetization reversal in studied films as in this moment兲 vector of ith granule, J␣␤共rij兲 is a potential of 共gen-
case the granules magnetization do not leave film plane due erally speaking anisotropic, where indices ␣ and ␤ enumer-
to Ha␪ Ⰷ Ha␸. Our MC simulations show that at suitable ate all nine components of the interaction tensor兲 the inter-
choice of a, B and C, it is possible to obtain the dependence granular interaction, D is a uniaxial magnetic anisotropy
hc共␸H兲 qualitatively similar to that experimentally observed parameter, g and ␮B is g-factor and Bohr magneton, respec-
in our samples at low temperatures. The example of such tively. The interaction J␣␤共rij兲 can be both of exchange 关in
simulations is reported on the Fig. 4. For MC stimulation we which case it is isotropic, i.e., J␣␤共rij兲 ⬅ J共rij兲␦␣␤, ␦␣␤ is Cro-
use the dimensionless variables: h = H / Ha␸, Tred = kT / 共KV p兲, necker delta兴 and magnetic dipole nature. In the latter case
h c共 ␸ H兲 = H c共 ␸ H兲 / H a␸. the amplitude of J␣␤ is comparable with D so that the inter-
033919-5 Timopheev et al. J. Appl. Phys. 108, 033919 共2010兲

action term can be regarded as multi-ion 共multigranular in Hamiltonian H1 = 共H + g␮BHe兲S, where H is the argument of
our case兲 anisotropy. distribution function 共6兲. Note, that ⌿共␳兲 is indeed a Fourier-
The adequate description of the experimental situation in image 共calculated self-consistently兲 of f共H兲. The Eq. 共6兲 can
the above SW particles ensemble can be delivered if we take be solved approximately 共e.g. iteratively or numerically兲.
into account the randomness in the directions of elementary Having solved this equation, any spin-dependent macro-
granules moments Si. This situation can be considered utiliz- scopic quantity 具A典 共like ensemble magnetization, average
ing so-called random field 共RF, instead of mean field兲 ap- energy per particle, etc.兲 can be obtained by averaging of its
proach, developed earlier, see, e.g., Refs. 22 and 23 for de- single-particle version 具A典1 with the distribution function, de-
tails. Formally, in RF approach, the Hamiltonian 共2兲 contains rived by the solution of Eq. 共6兲. Namely, this quantity has the
two “kinds of randomness.” The first is the above spatial form
disorder. It means that spin Si can be randomly present or
absent in the specific ith place in an ensemble. The second is

冕
the thermal disorder, taking place at finite temperatures. It
means that spin Si can have random projection 共orientation兲 具A典 = 具A典1共H兲f共H兲d3H. 共7兲
with thermally activated jumps between them. Note that if
spatial disorder is inherent in disordered systems only, the For many cases 共e.g., for classical spin or spin 1/2, see
thermal one is ubiquitous also for conventional ordered sol- Refs 22 and 23 for details兲, the Eq. 共7兲 permits to reduce the
ids. Here we also note that our RF approach works for any integral Eq. 共6兲 to the set of transcendental equations for
form of J␣␤共r兲 in Eq. 共2兲, its specific form is responsible for macroscopic quantities like magnetization m̄ = 具S典, glassy or-
realization of different 共SFM, SSG, or their mixture兲 phases der parameter 具S2典, and higher order 具Sn典 spin averages of
in the substances under consideration. the system.
The first two terms of Hamiltonian 共2兲 can be rewritten Let us note that the distribution function, determined
identically through the local field Hi 共in energy units兲 acting from Eq. 共6兲, is by no means Gaussian. Our analysis shows
on each spin 共granule magnetic moment兲 from the remaining that this asymptotics occurs in SFM case when exchange
ones interaction with characteristic radius rc prevails so that the
Hi␣ = 兺 J␣␤共rij兲S␤j + g␮BHe␣ . 共3兲 quantity nr3c Ⰷ 1 共n is above granules concentration兲 and the
j␤ fluctuations due to spatial disorder 共spatial fluctuations兲 are
small. Mean field approximation corresponds to the complete
The above two sources of randomness make this field to be a absence of spatial fluctuations so that the distribution func-
random quantity. The distribution function of this quantity tion 共4兲 is simply delta function. In the present problem,
reads however, the spatial fluctuations are not small, which defines
f共H兲 = 具␦共H − Hi兲典, 共4兲 the physical picture of the problem under consideration. To
be more specific, in a system where spin glass and ferromag-
where bar denotes the averaging over spatial positions of netic long-range order coexist, the maximum of distribution
dipoles and angular brackets is thermal averaging over pos- function 共corresponding to mean value of an RF兲 determines
sible spins orientations. To perform the prescribed averag- the parallel spin configurations, promoting long-range order
ings, we use the integral representation of ␦-function while its tails correspond to nonparallel 共generally speaking

冕冕冕
nonuniform兲 spin configurations. For Gaussian distribution
f共H兲 =
1
共2␲兲3 冉 i␣
冊
d3␳ exp关i␳H兴具exp − i 兺 ␳␣Hi␣ 典, function the tails are very short 共the function decays rapidly兲
so that the probability of nonuniform configurations realiza-
tion is very small. On the contrary, for non-Gaussian distri-
␣ = x,y,z; d 3␳ ⬅ d ␳ xd ␳ y d ␳ z 共5兲 bution functions 共so-called heavy-tail distributions兲, the de-
cays are slow so that nonuniform configurations appear with
where components of Hi are determined by Eq. 共3兲. As it is
nonsmall probability. It is well known 共see, e.g., Ref. 24兲 that
not possible to perform the averagings exactly, we use an
long-range interactions of alternating sign 共like dipole-dipole
approach, borrowed from statistical theory of magnetic reso-
one兲 generate distribution functions with long tails, which, in
nance line shape.23 Namely, following Ref. 24, we assume
turn are responsible for appearance of the multiple 共meta-
the additivity of local molecular field contributions as well as
stable兲 configurations in the random ensemble of magnetic
a noncorrelative spatial distributions of spins. Under these
SW granules. In the pure spin glass case20,21 these metastable
assumptions, we can derive following self-consistent integral
configurations lead to so-called ultrametricity of phase space.
equation for f共H兲:22,23
To perform the actual calculations, we parametrize the

f共H兲 =
1
共2␲兲3
冕冕冕 exp关i␳H − ⌿共␳兲兴d3␳ , 共6兲
single-granule classical spin vector S by angular variables
Sx + iSy = S sin ␰ei␩, Sz = S cos ␰, where ␰ is the angle between
granule magnetic moment and z direction and ␩ is the azi-
where ⌿共␳兲 = n兰兰V兰␺共r , ␳兲d3r and ␺共r , ␳兲 = 兰兰兰具1 muth angle, S is a vector S modulus, which we consider to be
− exp关−iJ共r兲S␳兴典1 f共H兲d3H. constant. In this case 共we recollect here, that magnetic mo-
Here n = N / V = f vV−1
p is a ferromagnetic particles concen- ments do not leave the film plane, see the beginning of
tration 共number of the particles in a sample per unit volume兲 present section兲 only vector ␰ defines the random directions
and 具 . . . 典1 means averaging with the effective single-particle of H so that Eq. 共7兲 assumes the form 具A典
033919-6 Timopheev et al. J. Appl. Phys. 108, 033919 共2010兲

= 兰具A典1共␰1兲f共␰1兲d␰1. Since the quantity of our interest is the

mean energy density per particle of the system, the latter
equation renders it to the form

URF = − K cos2 ␰ − m p 冕 ␰

−␲/2
关He + ␭m̄共␰1兲兴cos ␰1 f共␰1兲d␰1 .

共8兲

It is seen that in this case, the single-particle average 具A典1

⬅ m p关He + ␭m̄共␰1兲兴cos ␰1, where m p is a single-granule mag-
netization 共see above兲, ␭ is a mean field parameter of inter-
action of a given particle with the rest of their ensemble 关the
solution of Eq. 共6兲 shows that it is proportional to n and
consequently to f v, see Ref. 22 for details兴 and m̄共␰兲 is an FIG. 5. Energy density profile for random ensemble of SW particles 关Eq.
average magnetization of each granule in the ensemble. The 共10兲 with He = 0, a = 0.023, B = 10, and C = 4兴.
latter quantity equals to the overall ensemble magnetization
divided by the relative volume f v occupied by ferromagnetic
parameters on the value and direction of external magnetic
particles in a sample. We note here that as single-ion aniso-
field. We will not try to obtain this dependency explicitly in
tropy does not enter the self-consistent averagings in Eq. 共5兲,
the present paper. Rather, for the qualitative analysis of the
being actually the same for each granule, this term remains
Hc共␸兲, here we suppose that these parameters do not depend
the same after averaging with f共␰1兲, giving rise to the first
on He in the fields’ range, where it is much less then the
term in Eq. 共8兲, where K is the uniaxial anisotropy constant.
Also, the lower integration limit −␲ / 2 in Eq. 共8兲 corresponds coercive field. Note, that the parameters a, B, and C depend
to negative z direction. also on parameter ␭. Latter dependence is indeed not impor-
Substitution of magnetic dipole interaction tant as we choose a, B, and C values from the condition of
best fit to experiment. We note also, that a correction propor-
␦␣␤ − 3n␣n␤ tional to parameter ␭ should also enter the last term of Eq.
J␣␤共r兲 = , 共9兲 共10兲 due to uniform contribution to m̄共␰兲 generated by mag-
r3
netic field He. However, for the problems, considered in the
共where r ⬅ 兩r兩 and n␣ are unit vectors in ␣ direction, ␣ present paper, this correction is not important and we do not
= x , y , z兲 into Eq. 共6兲 requires the direct solution of the inte- cite it here.
gral equation for f共␰1兲, which in general case can be solved For detailed comparison with experimental situation we
only numerically. For classical spins the approximate solu- should consider the ␸-angle in the experiment as an equiva-
tion gives the log-normal function for f共␰1兲. Further substi- lent of the ␰-angle in the above theoretical part. We have
tution of this function into Eq. 共8兲 may be approximately already noticed that despite the difference between the actual
described by the following expression for the energy density biaxial anisotropy and uniaxial one considered theoretically,
per particle URF in the RF approximation: the above formalism captures the qualitative accordance of
observed and modeled processes of SSG onset in the en-
URF共␰,He兲K−1 = − cos2 ␰ + a共cos B␸兲C − m pHeK−1 , 共10兲
semble of weakly interacting SW particles.
where a, B Ⰷ 1, and C are fitting parameters similar to those The MC simulations of the Hc共␸H , ␪H = ␲ / 2兲, discussed
in Eq. 共1兲. The second term of this equation is approximately shortly after Eq. 共1兲, show that the influence of the “teeth” of
written in the form of a periodic function cos B␰ with B UR共␸兲 关Eq. 共1兲兴 共with the magnitude proportional to a兲, on
Ⰷ 1, which approximates the actual random 共ultrametric the coercivity depends on the magnetization direction. At
rather then periodic兲 rapidly oscillating function, which can- small a or high temperatures, the angular dependence
not be described precisely in the framework of above simple Hc共␸H , ␪H = ␲ / 2兲 has only one maximum in easy magnetiza-
model. Here we would like to emphasize that the form 共10兲 tion direction. As parameter a grows, the coercivity acquires
of URF reflects the essence of mean field like approaches 共our additional contributions in the regions where the derivative
RF method also belongs to that family兲, which reduce the of initial profile U0 = −cos2 ␰ 共which is equivalent to normal-
overall system energy 共ultrametric one in the case of pure ized U␸ = −1 / 2m pHa␸ sin2␪ cos2 ␸ for ␪ = ␲ / 2兲 has zeroes. In
spin glass兲 to some effective single-particle energy func- such case, the coercivity in the hard direction appears and
tional. In other words, the ripples in Eq. 共10兲 can be regarded above angular dependence acquires additional extremum in
as some approximate representation of a collective ultramet- this direction. As a result, the dependence Hc共␸H兲 acquires
ric potential of the system under consideration. two maxima in the regions of the above derivative zeroes. At
The plot of URF共␰兲 丢 in the form 共10兲 is reported on Fig. sufficiently large a ⬃ 1, the coercivity increases strongly for
5 for He = 0, showing the peculiar “saw-tooth” energy density all ␸ and angular dependence of Hc共␸H , ␪ = ␲ / 2兲 disappears.
profile. It resembles the typical ultrametric picture 共see Refs. The B and C parameters determine the sharpness of teeth and
20 and 21兲. The second term in Eq. 共10兲 coincides with Eq. are important for the coercivity angular dependence to be
共1兲 with respect to Ha␸ = 2K / m p. Of course, the above con- similar to the experimentally observed one. We note that de-
sideration can lead to additional dependence of a, B, and C spite the good qualitative description of the main character-
033919-7 Timopheev et al. J. Appl. Phys. 108, 033919 共2010兲

istics of the experimental Hc共␸H兲 dependencies, our calcula-

1
Y. G. Pogorelov, G. N. Kakazei, M. D. Costa, and J. B. Sousa, J. Appl.
Phys. 103, 07B723 共2008兲.
tions do not permit to find the set of a, B, and C parameters 2
S. Bedanta and W. Kleemann, J. Phys. D: Appl. Phys. 42, 013001 共2009兲.
which is capable to achieve the quantitative coincidence be- 3
V. V. Kokorin and I. A. Osipenko, JETP Lett. 29, 610 共1979兲.
tween the calculated and measured Hc共␸H兲 curves. We sup- 4
X. Chen, O. Sichelschmidt, W. Kleemann, O. Petracic, C. Binek, J. B.
pose that this fact is due to simplified character of our model, Sousa, S. Cardoso, and P. P. Freitas, Phys. Rev. Lett. 89, 137203 共2002兲.
5
S. Bedanta, T. Eimüller, W. Kleemann, J. Rhensius, F. Stromberg, E.
which captures the main peculiarities of the problem only. Amaladass, S. Cardoso, and P. P. Freitas, Phys. Rev. Lett. 98, 176601
The FMR turns out to be weakly sensitive to the onset of 共2007兲.
6
the teeth in the energy density profile. As it was noticed W. Kleemann, O. Petracic, C. Binek, G. N. Kakazei, Y. G. Pogorelov, J. B.
above, the saturating magnetic field eliminates quasiultra- Sousa, S. Cardoso, and P. P. Freitas, Phys. Rev. B 63, 134423 共2001兲.
7
P. Johnsson, S.-I. Aoqui, A. M. Grishin, and M. Munakata, J. Appl. Phys.
metricity, i.e., the teeth in the energy profile. Formally this is 93, 8101 共2003兲.
accomplished by the dependence of parameters a, B, and C 8
M. Munakata, M. Yagi, and Y. Shimada, IEEE Trans. Magn. 35, 3430
on He. 共1999兲.
9
M. Munakata, S. Aoqui, and M. Yagi, IEEE Trans. Magn. 41, 3262 共2005兲.
10
G. N. Kakazei, A. F. Kravetz, N. A. Lesnik, M. M. Pereira de Azevedo, Y.
IV. CONCLUSIONS G. Pogorelov, G. V. Bondorkova, V. L. Silantiev, and J. B. Sousa, J. Magn.
Magn. Mater. 196–197, 29 共1999兲.
11
To conclude, here we observe the low-temperature J. Dubowik, Phys. Rev. B 54, 1088 共1996兲.
12
G. N. Kakazei, A. F. Kravets, N. A. Lesnik, M. M. Pereira de Azevedo, Y.
anomaly of the coercivity in CoFeB: SiO2 granular film with
G. Pogorelov, and J. S. Sousa, J. Appl. Phys. 85, 5654 共1999兲.
oriented in-plane anisotropy. It was shown that this anomaly 13
A. G. Gurevich and G. A. Melkov, Magnetization Oscillation and Waves
is not related to any additional type of anisotropy at low 共CRC, New York, 1996兲.
14
temperatures. Rather, it signifies the onset of the SSG corre- C. L. Chien, Annu. Rev. Mater. Sci. 25, 129 共1995兲.
15
P. Johnsson, S. I. Aoqui, K. Nötzold, J. Allebert, M. Munakata, and A. M.
lations owing to the quasiultrametric saw-tooth system en- Grishin, Supercooled Liquids, Glass Transition and Bulk Metallic Glasses,
ergy profile. Latter profile has been also theoretically ob- MRS Symposia Proceedings No. 754 共Materials Research Society, Pitts-
tained in the framework of the RF method for the ensemble burgh, 2003兲, p. 459.
16
of weakly interacting SW particles. With this method, we E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A
240, 599 共1948兲.
were able to show that nonuniform and anisotropic parts of 17
A. A. Timopheev, S. M. Ryabchenko, V. M. Kalita, A. F. Lozenko, P. A.
intergranular interaction lead to the quasiultrametricity due Trotsenko, V. A. Stephanovich, A. M. Grishin, and M. Munakata, J. Appl.
to multiplicity of metastable configurations of the mutual Phys. 105, 083905 共2009兲.
18
orientations of granules magnetic moments. A. A. Timopheev, S. M. Ryabchenko, V. M. Kalita, A. F. Lozenko, P. A.
Trotsenko, A. M. Grishin, and M. Munakata, Solid State Phenom. 152–
153, 213 共2009兲.
19
L. Neel, Ann. Geophys. 共C.N.R.S.兲 5, 99 共1949兲.
ACKNOWLEDGMENTS 20
M. Mezard and M. A. Virasoro J. Phys. 共France兲 46, 1293 共1985兲.
21
K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 共1986兲.
This work was partly supported by the NAS of Ukraine 22
Y. G. Semenov and V. A. Stephanovich, Phys. Rev. B 66, 075202 共2002兲.
Program “Nanostructural systems, nanomaterials and nano- 23
Y. G. Semenov and V. A. Stephanovich, Phys. Rev. B 67, 195203 共2003兲.
24
technologies.” A. M. Stoneham, Rev. Mod. Phys. 41, 82 共1969兲.

View publication stats