The influence of intergranular interaction on the magnetization of the ensemble of oriented Stoner-Wohlfarth nanoparticles
A.A. Timopheev, S.M. Ryabchenko, V.M. Kalita, A.F. Lozenko, P.A. Trotsenko, V.A. Stephanovich, A.M. Grishin, M. Munakata
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec The influence of intergranular interaction on the magnetization ofthe ensemble of oriented Stoner-Wohlfarth nanoparticles
A.A. Timopheev, ∗ S.M. Ryabchenko, V.M. Kalita, A.F. Lozenko, P.A. Trotsenko, V.A. Stephanovich, A.M. Grishin, and M. Munakata Institute of Physics NAS of Ukraine,Prospect Nauki str. 46, Kiev, 03028, Ukraine Institute of Mathematics and Informatics,Opole University, Oleska 48, 45-052 Opole, Poland Royal Institute of Technology, Electrum 229,S-164 40 Kista, Stockholm, Sweden Energy Electronics Laboratory, Sojo University, Kumamoto 860-0082, Japan (Dated: October 27, 2018) bstract We consider the influence of interparticle interaction on the magnetization reversal in the orientedStoner-Wohlfarth nanoparticles ensemble. To do so, we solve a kinetic equation for the relaxationof the overall ensemble magnetization to its equilibrium value in some effective mean field. Latterfield consists of external magnetic field and interaction mean field proportional to the instantaneousvalue of above magnetization. We show that the interparticle interaction influences the temperaturedependence of a coercive field. This influence manifests itself in the noticeable coercivity at
T > T b ( T b is so-called blocking temperature). The above interaction can also lead to a formation of the”superferromagnetic” state with correlated directions of particle magnetic moments at T > T b .This state possesses coercivity if the overall magnetization has a component directed along theeasy axis of each particle. We have shown that the coercive field in the ”superferromagnetic”state does not depend on measuring time. This time influences both T b and the temperaturedependence of coercive field at T < T b . We corroborate our theoretical results by measurementson nanogranular films (CoFeB) x -(SiO ) − x with concentration of ferromagnetic particles close, butbelow percolation threshold. PACS numbers: 61.18.Fs, 61.46.+w, 75.50.Tt, 75.60.Ej, 75.30.Gw, 75.75+aKeywords: superparamagnetic state, interparticle interaction, nanogranular films, coercitivity ∗ e-mail: [email protected] . INTRODUCTION The consideration of interparticle interaction in an ensemble of single-domain superpara-magnetic particles is important both from theoretical [1, 2] and practical points of view[3, 4, 5, 6, 7]. The above interaction alters (as compared to the case of noninteracting par-ticles) the magnetization curves, the coercive fields and the temperature dependence of anensemble magnetic susceptibility. The intergranular interaction is always of dipole nature,although there are cases where an additional exchange interaction occurs also.To be more specific, if metallic ferromagnetic (FM) granules are embedded in conductinghost, they can have exchange interaction via common electronic system of a composite evenif the granules concentration is lower than the percolation threshold. In the latter case, theinteraction is of RKKY type being small and of alternating sign. If the host is dielectricand the granules concentration is lower than the percolation threshold the interaction willbe of entirely dipole nature. On the other hand, close to and/or above this threshold, theinteraction between contacting granules may have exchange contribution of both RKKY andusual ion-ion types.A possibility for homogeneous long-range FM order to appear due to presence of dipole-dipole interaction has been discussed by many authors beginning from Ref. [8]. It has beenshown in Ref. [8], that in simple cubic lattice of magnetic dipoles the latter interactiongenerates long-range antiferromagnetic order rather then FM one. The same conclusion hasalso been drawn in Ref. [9]. The question about stability of FM long-range order in thecases when corresponding static mean field solution predicts the appearance of FM orderhas been considered in Ref. [10]. This analysis shows that FM order is unstable with respectto 3D perturbations, but is stable with respect to 2D ones. Despite of the above discussions,the appearance of long-range FM order in granular systems with dielectric matrix has beendetected experimentally in many systems with granules concentration both below and abovepercolation threshold. The properties of such FM ordered states of nanogranular systemshave not been adequately explored.Regardless of the intergranular interaction nature, the question about joint influence ofsuperparamagnetism and intergranular ordering on the magnetic properties of FM particlesensembles is still opened. It is naturally to expect that intergranular interaction generatesa correlation of the particles magnetic moment directions. If this interaction is of FM3ype, then below certain temperature T sf , it should generate long-range magnetic order.Otherwise, so-called superspinglass state can be realized.Most frequently, each particle has a certain crystallographic anisotropy. If its shape isnon-spherical, the anisotropy can be of magnetostatic nature, related to demagnetizationfactors tensor for that shape. The thermal fluctuations lead to reorientation of a particlemagnetic moment between several equivalent easy magnetization directions, dictated by theabove anisotropy. In that case the observable magnetic properties of the ensemble are dif-ferent depending on the relation between the reorientation time and the period of ensembleobservation. For noninteracting particles, the thermally activated reorientations of theirmagnetic moments are mutually independent. Two kinds of ensemble superparamagneticstates can be distinguished. Namely, there are equilibrium and nonequilibrium superpara-magnetic states. The equilibrium state occurs when the magnetic moment of an averageparticle (i.e. the typical ensemble particle with some average parameters) “covers” all per-missible easy magnetization directions during the time of observation. The nonequilibriumor “blocked” superparamagnetic state occurs in the opposite case, when the particles are“blocked”, i.e. they cannot alter their magnetization orientations during the observationtime. The threshold temperature between these two states is called blocking temperature, T b . The magnetic switching in the blocked state ( T < T b ) has a hysteretic character, whileat T > T b it is almost unhysteretic.Now we “turn on” the interparticle interaction. If it has FM character, then at T
Let us consider an ensemble of interacting SW-particles with their easy axes pointingalong the direction of an external magnetic field. For this case, the energy density perparticle in a mean field approximation is: U = − K cos ( θ ) − m p ( H + λm ) cos( θ ) . (1)Here K is a uniaxial magnetic anisotropy constant, m p is a single particle saturation mag-netization (it is the same for each particle), θ is the angle between a particle magnetizationvector and external magnetic field H direction, λ is a mean field interaction parameter and m is an average magnetization per each ensemble particle. The latter quantity equals to theoverall ensemble magnetization divided by the relative volume occupied by ferromagneticparticles in a sample. Below we will use the dimensionless magnetization M = m/m p . Thisvalue will be the same both for the ensemble and for each single particle.Let us pay attention that the potential energy profile (1) has the form of double-wellpotential. According to the Neel model [20] we can describe this system in the temperaturerange typical for the magnetostatic measurements (0 < T < T b ) as a system with twopossible orientations of particles magnetic moments. If the magnetic field is directed along6asy magnetization axes of the particles, the double well potential can be substituted by itstwo lowest energy levels so that our system can be described as a two-level system. In thiscase, the transitions between levels corresponding to magnetic moment reorientations occuras thermally activated hops over energy barrier.In this model, the magnetization dynamics is of purely relaxational type. This meansthat time dependence of magnetization M at fixed temperature T and magnetic field H can be described (similar to Refs. [18, 22]) by Bloch-like equation for z-component ofmagnetization only. If the interaction term λm is absent in Eq. (1), the equation formagnetization dynamics has the form: ∂M ( t ) ∂t = 1 τ [ M ∞ − M ( t )] , (2)where M ∞ ≡ M ( t → ∞ ) is the equilibrium magnetization at fixed H and T and τ is therelaxation time. For our case of two-level system τ − = W + W , where W ij ( j = 1 ,
2) areprobabilities of transition between i and j levels in the double-well potential (1). Accordingto approach [20] for SW particles, the final form of τ reads: τ = 1 f (cid:0) exp( − E b − E kT ) + exp( − E b − E kT ) (cid:1) . (3)Here k is Boltzmann constant, f ∼ ÷ s − for typical magnetic particles and E , E , E b are, respectively, the energies of minima of U ( θ ) and a barrier between them. Thequantities E , E and E b depend on K , λ and M ( t ). They are the functions of time byvirtue of M ( t ) dependence. The expressions (2) and (3) correspond to the approach, wherea fictitious particle (corresponding to magnetization) is localized exactly in the minimumof U( q ) rather then “smeared” by temperature in a wide interval of angles θ within thewell of the potential (1). For magnetic field sweeping times, taking place in magnetostaticmeasurements, this approach is well satisfied in the temperature range T < (4 ÷ T b , i.e.in the entire temperature domain. Thus magnetization reversal occurs by thermoactivationoverbarrier hopping. In a mean field approximation, the equilibrium magnetization for suchtwo-level system is determined by usual equation: M ∞ = tanh m p ( H + λm ∞ ) V p kT = tanh 2( h + λ red M ∞ ) T red . (4)7ere we introduce following dimensionless parameters: the dimensionless magnetic field, h = H/H a ( H a = 2 K/m p is the anisotropy field), the dimensionless relaxation time τ r = τ f , the temperature T red = kT / ( KV p ) ( V p is SW-particle volume) and the dimensionlessparameter of interparticle interaction λ red = λm p / (2 K ). We also introduce the dimensionlessenergy minima E / ( KV p ), E / ( KV p ), barrier maximum energy E b / ( KV p ), dimensionlesstime t r = tf and time of measurements t reg = t exp f . Here t is a real dimensional time and t exp is a characteristic dimensional time of measurements, i.e.“measuring time” - the timerequired for magnetic field sweeping in the range of H a .The introduction of the interaction term λm in the Eq.(1) modifies the character ofrelaxation. Namely, under magnetization reversal this term becomes time dependent as itcomprises the magnetization m ( t ). This means that the overall magnetization relaxes not tothe above real equilibrium magnetization M ∞ = M ∞ ( H, T ), but to certain (so far unknown)self-consistent equilibrium magnetization value, dictated by the effective magnetic field H + λm ( t ) at each time point. We denote this new hypothetical equilibrium magnetization as m ∗∞ ( t ), and its normalized value as M ∗∞ ( t r ) = m ∗∞ ( t r /f ) /m p . Here we note, that M ∗∞ ( t r )= M ∗∞ [ H, T, M ( t r )] so that the kinetic equation for magnetization assumes the form: ∂M ( t r ) ∂t r = 1 τ r [ M ∗∞ − M ( t r )] , (2a)where M ∗∞ ( t r ) is determined by the equation M ∗∞ ( t r ) = tanh m p [ H + λm ( t r /f )] V p kT = tanh 2[ h + λ red M ( t r )] T red . (4a)The equation (2a) with respect to (4a) will be solved numerically. To model the hysteresisloops, we consider, similar to Ref. [22], the linear field sweep h ( t r ) = ( t r /t reg ) − ∂M ( t r ) ∂t r = (cid:18) exp (cid:20) − ( t r /t reg − λ red M ( t r )) T red (cid:21) + exp (cid:20) − ( t r /t reg + λ red M ( t r )) T red (cid:21)(cid:19) × (cid:18) tanh (cid:20) t r /t reg − λ red · M ( t r )) T red (cid:21) − M ( t r ) (cid:19) . (5)The last brackets in the right-hand side of Eq. ((5)) define the difference between self-consistent hypothetical equilibrium magnetization (4a) and its current value.8 ,0 0,1 0,2 0,3 0,4 0,5 0,60,00,51,0 redredred red h c T red1/2 red T rb* FIG. 1: (color online) The temperature dependence of coercive field h c ( √ T red , t reg , λ red ) for t reg =10 and different values of λ red (0, 0.1, 0.2, 0.3, 0.4). Points are the results of numerical solution ofEq. (5) and the full lines correspond to low- and high temperature extrapolations of the numericalcurves. III. RESULTS OF MODELING
The solution of Eq. (5) shows that the account for the intergranular interaction term λ red M ( t r ) increases the coercivity. Also, the hysteresis loops become “more rectangular”(”harder”). The temperature dependences of the coercive field h c , extracted from the hys-teresis loops calculated with the help of Eq. (5), are shown on Fig. 1 at different valuesof interaction parameter λ red . It is seen that the interaction increases the coercive field.Besides that, at 2 λ red > T ∗ b the dependence h c ( T red ) has two linear in √ T red parts.The low-temperature part is similar to Neel-Brown law: h c ( T red ) = 1 − q T red /T ∗ b for T red < T ∗ b , (6)where T ∗ b is a dimensionless effective blocking temperature, determined by extrapolationof the low-temperature part of dependence h c ( √ T red ) up to its intersection with abscissaaxis. The value of T ∗ b depends on λ red .The high-temperature part of h c ( √ T red ) lies at T red > T ∗ b . Its origin is a consequence offormation (due to interaction λ red > T ∗ b /
2) of the state with correlated directions of granules9agnetic moments realized at T red < T ord = 2 λ red . Here T ord corresponds to dimensionlesstemperature of the long range magnetic ordering in the granules ensemble. In Section I(Introduction), the real (dimensional) temperature of such ordering had been denoted as T sf .Strictly speaking, at λ red < T ∗ b / T red = 2 λ red , the dependence h c ( √ T red )acquires additional slope as compared to that in Eq. (6). This deviation, however, cannotbe seen in the scale of Fig. 1 so we do not plot corresponding curve on Fig. 1.In the temperature range above blocking temperature, the relaxation time is much lessthan measuring time so that the appearance of coercivity in this temperature range is nota consequence of slow system response to magnetic field sweep. The coercivity at T ∗ b
To corroborate the above theoretical approach, we measure the magnetostatic charac-teristics of nano-granular films (Co . Fe . B . ) x − (SiO ) − x . The aim was to check thetransition of SW particles ensemble from relaxation regime of magnetization reversal tosteady-state regime of the intergranular “ferromagnetic” ordering (arising due to the inter-action), when the coercive field ceases to depend on measuring time. In our measurements,we use the (Co . Fe . B . ) x − (SiO ) − x films grown in the Energy Electronics Laboratory,Sojo University, Japan. The ferromagnetic granules were amorphous and their shape wasclose to the spherical. A strong easy-plane anisotropy related to the demagnetization fac-tor arose for the entire film sample. In the granular films under investigation, the uniaxialanisotropy in a film plane had been formed by special technological measures [24, 25]. Thisanisotropy was supposedly related to the small deviation of the shape of granules from thespherical one. Thus the easiest (i.e. easy in a film plane) axes of all granules have been ori-ented almost parallel to each other. Therefore such granular system can be considered as anensemble of easy axis oriented Stoner-Wohlfarth particles. The ensemble can be consideredas “noninteracting” one for x substantially lower than the percolation threshold x c and as“strongly interacting” one for x ≫ x c . According to the technologist, who had fabricated12
300 -200 -100 0 100 200 300 M agne t i z a t i on , a . u . Magnetic Field, Oe x = 0.55 T = 300K T sf = 550 K x = 0.55 H c , O e T ,K T b = 325 K H cint (T=0) FIG. 3: (color online) The temperature dependence of coercive field in the sample with x = 0 . T = 300 K. our sample, the ferromagnetic component content in it is x = 0 .
55. It corresponds to volumefraction of the ferromagnetic granules about 0.26. It is strictly lower than volume fraction ofpercolation threshold. However, the film saturation magnetization, ferromagnetic resonanceand granule magnetization [27] data have shown that real volume fraction of granules in thissample is essentially higher (then 0.26), up to 0 . ÷ .
45. We assert that this real volumefraction is a little below percolation threshold. To prove this assertion, we had measuredthe magnetoresistance curves at T =300K. The sample has high enough specific resistance, ρ ( T =300 K) = 250 mOhm/cm. The measured magnetoresistance curves contained the con-tribution from only tunneling magnetoresistance and did not contain the contribution fromthe anisotropic magnetoresistance. Since latter contribution in such films appears for x > x c only [25], this result proves above assertion. For the above sample, the experimental datafor magnetization in the film plane along easy direction (curve 1) at room temperature arepresented on the inset to Fig. 3. One can see that the hysteresis loop is close to rectangu-lar and has almost 100% remanence. The field dependence of magnetization along a harddirection in a film plane (curve 2), has no hysteresis, has jogs at the intraplane anisotropyfields and is almost linear in magnetic field between jogs. The temperature dependence ofcoercivity (Fig. 3) demonstrates two linear in √ T parts, which coincide with the results of13 ,2 0,3 0,4 0,50,350,70 h c T red red = 0.34 b
10 12 14 16 18 20 22010203040
T, K H C , O e T , K t SWEEP :1 min4 min16 min64 min a FIG. 4: (color online) a) The temperature dependencies of coercive field H c for the sample with x = 0 .
55 versus √ T . The legend shows the field sweeping time from -500 Oe to +500 Oe and viceversa ( t sweep ). b) The results of calculations of coercive field h c ( √ T red , t reg , λ red ) for λ red = 0 . t reg are the same as those on Fig. 4a. above theoretical modeling.At the same time, such dependence can be interpreted as a consequence of a bimodal sizedistribution of ensemble particles so that each particle group has its own T b value. To proveor disprove such possibility, we perform the number of magnetostatic measurements withdifferent rates of magnetic field scanning. Fig. 4a reports the temperature dependences ofcoercive field H c at temperatures from 100 to 470 K at magnetization along easy direction.At temperatures lower than 100 K the temperature dependence of a coercive field in this filmdemonstrates an anomaly. We will not discuss that in the present paper, having restrictedourselves by temperature region above 100 K only.The dependences H c ( T ) (in the form H c ( √ T )) on Fig.4a are obtained for different timesof a magnetic field sweeping ( t sweep =1 min, 4 min, 16 min, 64 min) from -500 Oe to +500Oe and vice versa. It is seen, that increasing of measuring time ( t exp is proportional to t sweep ; actually for the anisotropy field value of this sample H a = 80 Oe, t exp = t sweep / H c at fixed temperature. It also yields the loweringof the blocking temperature T b determined as an intersection point of an asymptote tothe linear low-temperature part of dependence H c ( √ T ) and abscissa axis. It follows fromFig. 4a that the dependences H c ( √ T ) behave like theoretical dependences from Fig. 2.In both experimental and simulation curves we observe a noticeable coercivity practically14ndependent of measuring time above blocking temperature. The main properties of thepresented curves can be formulated as follows. At low temperatures, T ≪ T b ( T b is takenfor the longest possible measuring time) the dependence of H c on measuring time becomesstronger with temperature increase. At a certain temperature, slightly lower then T b ( t exp ),a sensitivity of H c to measuring time variations reaches a maximum. At last, at T > T b (now T b is taken for the shortest possible measuring time) the H c value ceases to dependon measuring time. At all temperatures a hysteresis loops remain ”hard”, conserving arectangular form.To compare the experimental and theoretical dependences of H c on measuring time it isnecessary to account for the fact that measuring times, corresponding to the curves on Fig.2, have 10 orders of magnitude variation, while experimental data from Fig. 4a have only 64times difference. To illustrate the similarity between experimental and theoretical data, onFig. 4b we present a number of theoretical curves with the relation of measurement times,identical to that in experiment. The curves are plotted for λ red = 0 . λ = 0 . m p = 1590 Gauss [27] and in-plane ahisotropy field H a = 2 K/m p = 80 Oe. Latter value follows from the magnetization curve of our film inthe “hard–in-plane” direction (curve 2 on inset to Fig. 3). The value λ = 0 .
017 is obtainedfrom the equation H intc ( T →
0) = λm p (with respect to the value H intc ( T →
0) = 30 Oefollowing from Fig. 3), where H intc is dimensional value of h intc . The values of t reg (shown onthe legend to Fig. 4b) have been chosen from the condition of best fit between model andexperimental T b ( t reg ) values.Although the reported experimental results are not quite identical to the results of ourcalculation, there is obvious qualitative coincidence. Namely, in our opinion, they demon-strate uniquely the existence of superferromagnetic state with coercivity. This fact is alsocorroborated by Fig. 5, where the temperature dependence of relative coercivity incrementsis reported for different measuring times. We define the above increments as the differenceof coercive fields for two substantially different measuring times divided by the coercive fieldvalue at larger time. This dependence is reported both for theoretical (Fig.5a) and exper-imental (Fig. 5b) results. It is seen, that the theoretical dependence for noninteractingensemble has a sharp increase near the blocking temperature. At the same time, for theinteracting ensemble, this dependence has a peak near T b with subsequent decrease. Theexperimental dependences (see Fig. 5b) also demonstrate the maximum with decrease.15 ,00 0,05 0,10 0,150246810 ( h C - h C ) / h C T red a T b T red ( h C - h C ) / h C red red red red
70 140 210 280 350 4200,120,240,36 ( H C - H C ) / H C T, K b T b FIG. 5: (color online) The temperature dependence of relative coercive field increments at regis-tration time variations: a) - theoretical results; h c ( T red ) is taken for t reg = 10 and h c ( T red ) for t reg = 10 . Full line corresponds to noninteracting case, circles - to interacting. Inset shows thesame for wider temperature range. b) - the symbols represent experimental points, the curve isguide for eye; H c ( T ) is taken for t sweep = 1 min and H c ( T ) for t sweep = 64 min. Thus it turns out that the behaviour of experimentally observed H c ( T, t sweep ) dependenceis qualitatively similar to results of our modeling for the case when intergranular interactiongenerates the state with correlated directions of particles magnetic moments and with acoercive field independent from measuring time. The above results illustrate the case whentemperature T sf (“dimensional T ord ”) of transition to such ”superferromagnetic” state ex-ceeds blocking temperature T b . The decreasing of intergranular interaction parameter λ canlead to opposite situation, when T sf < T b . In this case the variations of dependence H c ( √ T )also occur near T sf . However, they are too faint to be observed experimentally.It is possible to predict, that relaxational magnetization of the SW particles ensembleweakens if interaction energy exceeds the anisotropy energy. In this case the ensemblebehaves as a uniform ferromagnetic medium with possible occurrence of ”superdomains”consisting of many adjacent particles.Thus, the experimental dependence H c ( T ) in nanogranular magnetic film with granulesconcentration close but a bit lower than percolation threshold is in qualitative agreementwith model predictions for the system of superparamagnetic particles with intergranularinteraction described in a mean field approximation. Additionally, the similar results havebeen obtained in our studies of the (Co . Fe . B . ) x − (SiO ) − x film with nominal value x = 0 .
60. 16ote that in spite of aforementioned qualitative resemblance of the experimental andmodel data the quantitative correspondence is not so good. The obtained t reg values on Fig.4b are too small. It is the consequence of big granule size, which is needed for coincidenceof the observed T sf and T b values with those expected from the model. Particularly, tocoincide the V p value from the expression kT sf = λm p V p with λ obtained from the condition H intc ( T →
0) = λm p ≈
30 Oe, we need the mean diameter of granule 14 nm. To coincidethe observed values of T b and H a under usual assumption log( t exp f ) = 20 ÷
25, we need thisdiameter to be about 24 nm. These mean diameter values are essentially more than thoseexpected in film fabrication process. One of possible explanations of such discrepancy is therelative simplicity of used theoretical models.The last (but not the least) question is about the nature of intergranular interaction inthe studied samples. Is it of dipole-dipole or exchange nature? We do not have a convincinganswer to this question. One of possible suppositions is that exchange part of the interactioncoexists with a dipole-dipole one due to closeness of granules content in our film to thepercolation threshold.
V. CONCLUSIONS
To conclude, here we present a mean-field consideration of the magnetization of ensembleof interacting Stoner-Wohlfarth particles. We do that on the base of the kinetic equationsolution. The equation has been written for the relaxation of overall ensemble magnetizationto its self-consistent equilibrium state in the effective field consisting of external and theinteraction fields. The latter field, in turn, is proportional to instantaneous value of overallmagnetization. Numerical solution of the above kinetic equation shows that the presence ofmean-field interparticle interaction leads to the following effects:- At certain temperature, T sf , proportional to interaction parameter λ , the system ofFM granules undergoes the intergranular magnetic ordering - ”superferromagnetism”, yield-ing the additional coercivity at T < T sf . For T sf > T b , the essential coercivity arises attemperatures above blocking temperature;- At T < T sf a coercive field increases and magnetization reversal becomes ”harder”, i.e.the hysteresis loops become almost rectangular with increased remanence;- Temperature dependence of coercive field in the low temperature region resembles very17uch Neel-Broun law; for T sf > T b at increasing temperature the dependence H c ( √ T ) (or h c ( √ T red ) in the dimensionless units) has an inflexion point at blocking temperature andthen continues to go linearly in √ T up to T sf with much smaller slope;- In the system of weakly interacting particles the values of T b and H c ( T ) at T < T b depend on the measuring time as it is usual for SW particles. At the same time, in thecase of T sf > T b the dependence H c ( √ T ) in the range T b < T < T sf ceases to depend onmeasuring time.- The temperature dependence of a coercive field (Eq. (8)) is described with good accu-racy by two additive contributions. The first one (below blocking temperature) is stronglydependent on measuring time and reflects coercivity related to a metastability of the systemat finite measuring times. The second one, which is almost independent from measuringtime, has a temperature dependence described by Eq. (7). This contribution reflects achange in a mean field of intergranular interaction in the process of magnetization reversal.All above manifestations of interparticle interaction in SW particles ensemble are ob-served experimentally in magnetostatic (with 64 times difference in measuring times)measurements of a magnetic field and temperature dependencies of magnetization of(Co . Fe . B . ) . − (SiO ) . nanogranular films with FM granules content close, butbelow a percolation threshold. Thus it is firmly established that within the described ap-proach the results of our numerical modeling, are in good coincidence with the experimentaldata. Acknowledgments
This work was partly supported by the grant of NAS of Ukraine Target Program ”Nanos-tructural systems, nanomaterials and nanotechnologies”. [1] Z. Mao, D. Chen, and Z. He, J. Magn. Magn. Mater. , 642 (2008).[2] A. D. Liu and H. N. Bertram, J. Appl. Phys. , 2861 (2001).[3] C. Binns, M. J. Maher, D. Kechrakos, and K. N. Trohidou, Phys. Rev. B , 184413 (2002).[4] P. Allia, M. Coisson, P. Tiberto, F. Vinai, M. Knobel, M. A. Novak, and W. C. Nunes, Phys.Rev. B , 144420 (2001).
5] J. Escrig, S. Allende, D. Altbir, and M. Bahiana, Appl. Phys. Lett. , 023101 (2008).[6] M. Hillenkamp, G. Domenicantonio, and C. Felix, Phys. Rev. B , 014422 (2008).[7] D. Yao, S. Ge, X. Zhou, and H. Zuo, J. Appl. Phys. , 013902 (2008).[8] J. M. Luttinger and L. Tisza, Phys. Rev. , 954 (1946).[9] E. Z. Meilikhov and R. M. Farzetdinova, JETP , 751 (2002).[10] Y. G. Pogorelov, G. N. Kakazei, M. D. Costa, and J. B. Sousa, J. Appl. Phys. , 07B723(2008).[11] S.V.Vonsovsky, Magnetizm (John Wiley, New York, 1974).[12] R. S. Iskhakov, G. I. Frolov, V. S. Zhigalov, and D. J. Procof‘ev, Lett. J. Tech. Phys. , 51(2004).[13] B. J. Jonsson, T. Turkki, V. Strom, M. S. El-Shall, and K. V. Rao, J. Appl. Phys. , 5063(1996).[14] J. P. Perez, V. Dupuis, J. Tuaillon, A. Perez, V. Paillard, P. Melinon, M. Treilleux, B. Barbara,L. Thomas, and B. Boushet-Fabrer, J. Magn. Magn. Mater. , 74 (1995).[15] W. Kleemann, O. Petracic, C. Binek, G. N. Kakazei, Y. G. Pogorelov, J. B. Sousa, S. Cardoso,and P. P. Freitas, Phys. Rev. B , 134423 (2001).[16] X. Chen, S. Sahoo, W. Kleemann, S. Cardoso, and P. P. Freitas, Phys. Rev. B , 172411(2004).[17] D. L. Atherton, IEEE Trans. Magn. , 3059 (1990).[18] M. A. Chuev and J. Hesse, J. Phys.: Cond. Matt. , 506201 (2007).[19] J. J. Zhong, J. G. Zhu, Y. G. Guo, and Z. W. Lin, IEEE Trans. Magn. , 1496 (2005).[20] L. Neel, Ann. Geophys. , 99 (1949).[21] M. A. Chuev, JETP Lett. , 744 (2007).[22] A. A. Timofeev, V. M. Kalita, and S. M. Ryabchenko, Low Temp. Phys. , 446 (2008).[23] L. Wang, J. Ding, H. Z. Kong, Y. Li, and Y. P. Feng, Phys. Rev. B , 214410 (2001).[24] M. Munakata, M. Yagi, and Y. Shimada, IEEE Trans. Magn. , 3430 (1999).[25] P. Johnsson, S. I. Aoqui, A. M. Grishin, and M. Munakata, J. Appl. Phys. , 8101 (2003).[26] A. A. Timopheev and S. M. Ryabchenko, Ukr. J. Phys. , 261 (2008).[27] M. Munakata, M. Namikawa, M. Motoyama, M. Yagi, Y. Shimada, M. Yamaguchi, andK. Arai, J. Magn. Soc. Japan , 388 (2002)., 388 (2002).