The thermally activated magnetic relaxation in two-dimensional ensembles of
dipolar interacting nanoparticles with large uniaxial perpendicular anisotropy
is studied by a numerical method and within the mean-field approximation. The
role that the correlation effects in the presence of a bias magnetic field and
taking into account the lattice structure play in magnetic relaxation is
revealedComment: 10 pages, 3 figure

Denysov, Stanislav Ivanovych

Liutyi, Taras Volodymyrovych

Trohidou, K.N.

Денисов, Станислав Иванович

Денисов, Станіслав Іванович

Лютий, Тарас Володимирович

Лютый, Тарас Владимирович

English

arXiv

The thermally activated magnetic relaxation in two-dimensional lattices of dipolar interacting nanoparticles with large uniaxial perpendicular anisotropy is studied by a numerical method and
within the mean-field approximation for comparison. The role that the correlation effects play in
magnetic relaxation and the influence of lattice structure and bias magnetic field on the relaxation
process are revealed. The correlations of the nanoparticle magnetic moments enhance relaxation on
small times, delay it on large times, and reduce the steady-state absolute magnetization at nonzero
bias fields. In a hexagonal lattice, magnetic relaxation on small times occurs faster and the
steady-state absolute magnetization has the larger magnitude than in a square lattice with the same
lattice spacing.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/262

Electronic Sumy State University Institutional Repository

Dipolar interaction effects on the thermally activated magnetic relaxationof two-dimensional nanoparticle ensemblesS. I. Denisova) and T. V. LyutyyDepartment of Mechanics and Mathematics, Sumy State University, 2, Rimskiy-Korsakov Street,40007 Sumy, UkraineK. N. Trohidoub)Institute of Materials Science, NCSR ‘‘Demokritos,’’ 15310 Athens, Greece~Received 21 October 2003; accepted 13 April 2004; published online 19 May 2004!The thermally activated magnetic relaxation in two-dimensional lattices of dipolar interactingnanoparticles with large uniaxial perpendicular anisotropy is studied by a numerical method andwithin the mean-field approximation for comparison. The role that the correlation effects play inmagnetic relaxation and the influence of lattice structure and bias magnetic field on the relaxationprocess are revealed. The correlations of the nanoparticle magnetic moments enhance relaxation onsmall times, delay it on large times, and reduce the steady-state absolute magnetization at nonzerobias fields. In a hexagonal lattice, magnetic relaxation on small times occurs faster and thesteady-state absolute magnetization has the larger magnitude than in a square lattice with the samelattice spacing. © 2004 American Institute of Physics. @DOI: 10.1063/1.1759782#The two-dimensional ~2D! ensembles of uniaxial ferro-magnetic nanoparticles, whose easy axes of magnetizationare perpendicular to the nanoparticles’ plane, represent animportant class of perpendicular magnetic recording media.1From a technological point of view, the main characteristicof these media is the average time of data storage, which isone of the numerical parameters that describe the thermallyactivated magnetic relaxation. Due to the dipolar interactionbetween nanoparticles, its analytical description in such en-sembles is a very complicated problem, which was solvedonly within the mean-field2,3 and fluctuating4 theories ofmagnetic relaxation. The former deals with the mean com-ponent of the dipolar field acting on the nanoparticles, andthe latter accounts of both the mean and fluctuation compo-nents. However, the correlations of the nanoparticle magneticmoments, arising from the dipolar interaction, also play animportant role, especially on the final phase of magnetic re-laxation. To study the features of magnetic relaxation in suchensembles, which are conditioned by the correlation effects,we have developed a method for its numerical simulation atzero bias field.5 Usually, magnetic relaxation in ensembles ofdipolar interacting nanoparticles is simulated either by theconventional Monte Carlo method,6,7 where the role of timeplays the Monte Carlo steps, or by the improved MonteCarlo methods,8–11 that use the short-time stochastic behav-ior of the nanoparticle magnetic moments to convert theMonte Carlo steps to real time units. The main advantage ofthe method proposed in Ref. 5 is that it does not use theMonte Carlo approach, so the last complicated problem doesnot appear. This feature of the method and its applicability tothe study of magnetic relaxation over ten decades of time5make it a very useful tool in this domain.In this letter, we extend our method to the case of non-zero bias magnetic fields, and we present results obtainedwithin its framework, which clarify the role that the correla-tion effects, bias field, and lattice structure play in the mag-netic relaxation. We consider the 2D ensembles of sphericalnanoparticles with a radius r , which occupy the sites of asquare or hexagonal lattice with lattice spacing d , and whoseeasy axes of magnetization are perpendicular to the latticeplane (xy plane!. We also assume that: ~a! the nanoparticlemagnetic moments mi(t) ~the index i labels the nanopar-ticles! perform a coherent rotation (umi(t)u5m), ~b! a biasmagnetic field H0k ~k is the unit vector along the z axis! actson each mi(t), and ~c! the initial condition mi(0)5mk holdsfor all i .If the potential barriers between the equilibrium direc-tions of all mi(t) are much larger than the thermal energykBT (kB is the Boltzmann constant, T is the absolute tem-perature!, then the vectors mi(t) fluctuate within small vi-cinities of the k and 2k directions, and they are reorientedoccasionally. In this case the average numbers N1(t) andN2(t) of positively and negatively oriented magnetic mo-ments in a lattice region that contains N~@1! nanoparticlesare well defined (N1(t)1N2(t)’N), and we can introducethe reduced magnetization of the nanoparticle ensemble asr(t)52N1(t)/N21. To calculate the relaxation law r(t) atH050, we have developed the procedure5 that is based onthe analytical determination of the probability reorientationof the nanoparticle magnetic moments, and on the numericaldetermination of the nanoparticle ensemble states. Its impor-tant feature consists of the fact that the real dipolar fieldsacting on mi(t) are taken into account.Here, we generalize this procedure to the H0Þ0 case andapply it to study the magnetic relaxation in ensembles of Conanoparticles. The generalization consists of the derivationof the exact formula for the calculation of the probabilityreorientation of the nanoparticle magnetic moments forH0Þ0. We use this formula, derived from the correspondingbackward Fokker–Planck equation, and the numerical proce-a!Electronic mail: denisov@sumdu.edu.uab!Author to whom correspondence should be addressed; electronic mail:trohidou@ims.demokritos.grAPPLIED PHYSICS LETTERS VOLUME 84, NUMBER 23 7 JUNE 200446720003-6951/2004/84(23)/4672/3/$22.00 © 2004 American Institute of PhysicsDownloaded 28 May 2004 to 140.105.16.2. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspdure of Ref. 5 to calculate the relaxation law r(t) in theseensembles. We choose the basic nanoparticle ensemble forour simulation in the form of square ~for square lattice! orhexagonal ~for hexagonal lattice! box and surround it byeight or six identical ensembles, respectively. To calculatethe dipolar field acting on a nanoparticle, we consider thisnanoparticle as the central one in the box of the same size asthe basic and we include the nanoparticles, which belong tothis box. Increase of the size of the basic box leads to reduc-tion of the boundary effects and, to practically eliminatethem, we chose the size of the square and hexagonal boxesequal to 100d and 60d , respectively.To gain more insight into the magnetic relaxation, wealso calculate the relaxation law rmf(t) within the mean-fieldapproximation. As this approach ignores the correlation ef-fects, the difference between r(t) and rmf(t) results from thecorrelations of the nanoparticle magnetic moments. In whatfollows, the superscripts s and h on r(t), rmf(t), and d de-note the square and hexagonal lattice, respectively.The role that the correlation effects and the bias field H0play in magnetic relaxation is illustrated in Fig. 1. Wepresent the numerical results for ensembles of Co nanopar-ticles characterized by the parameters Ha56400 Oe (Ha isthe anisotropy field!, m/V51400 G (V is the nanoparticlevolume!, l50.2 ~l is the Landau–Lifshitz damping param-eter!, r54 nm, and T5300 K. Due to the correlations of thenanoparticle magnetic moments, for H050 the actual mag-netic relaxation occurs faster on small times and more slowlyon large times, than the mean-field theory predicts @see Fig.1~a!#. In this case the curves rs(t) and rmfs (t) are intersectedat the time t5t in , and rs(‘)5rmfs (‘)50. In equilibrium,the local dipolar fields have, on average, the same directionsas the magnetic moments and the mean dipolar field equalszero ~for H050!, therefore for H0Þ0 the condition urs(‘)u,urmfs (‘)u must hold. For H0,0 we havermfs (‘),rs(‘),0, and the curves rs(t) and rmfs (t) are in-tersected only once, as in the H050 case. If H0.0, thenrmfs (‘).rs(‘).0, for small enough values of H0 thesecurves are intersected twice @see Fig. 1~b!#, with increasingH0 the time interval (t in ,t in*) is decreasing, and rmfs (t).rs(t) (t.0) for large H0 @see Fig. 2~a!, curves 1 and 2#.The features of magnetic relaxation in ensembles of Conanoparticles, which arise from the different nanoparticle ar-rangement in square and hexagonal lattices, are demon-strated in Figs. 2 and 3. If the lattice spacing d is the samefor both lattices ~Fig. 2!, then the difference between therelaxation laws results mainly from the different number ofthe nearest sites in these lattices ~4 vs 6!. At t50 the localdipolar fields in hexagonal lattices are larger than in squareones, so the initial phase of magnetic relaxation occurs fasterin hexagonal lattices. At t5‘ and H050 the conditionsrs(‘)5rh(‘)50 and rmfs (‘)5rmfh (‘)50 hold. Each mag-netic moment in a square lattice is surrounded, on average,by four opposite oriented magnetic moments, whereas in ahexagonal lattice—by four opposite and by two similarlyoriented magnetic moments. Therefore, if H0.0 @see Fig.2~a!# then rmfs (‘).rmfh (‘).0, rh(‘).rs(‘).0, and thecurves rs(t) and rh(t) have the unique intersection point t5t1 (t1’13.59 s for H05500 Oe!. If H0,0 @see Fig. 2~b!#then rmfs (‘),rmfh (‘),0 and, since rh(‘),rs(‘),0, theFIG. 1. Plots of rs(t) ~curve 1! and rmfs (t) ~curve 2! for ds53r . FIG. 2. Plots of rs(t) ~curve 1!, rmfs (t) ~curve 2!, rh(t) ~curve 3!, and rmfh (t)~curve 4! for ds5dh53r .FIG. 3. Plots of rs(t) ~curves 1!, rmfs (t) ~curves 2!, rh(t) ~curves 3!, andrmfh (t) ~curves 4! on small ~a! and large ~b! time scales for dh53r , ds5dh31/4/21/2 and H050.4673Appl. Phys. Lett., Vol. 84, No. 23, 7 June 2004 Denisov, Lyutyy, and TrohidouDownloaded 28 May 2004 to 140.105.16.2. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspcurves rs(t) and rh(t) have two intersection points t5t1 andt5t2 (t1’0.01 s, t2’0.33 s for H052500 Oe!.Figure 3 presents the character of magnetic relaxation insquare and hexagonal lattices, with unit cells that have thesame area, i.e., (ds)25(dh)2A3/2, at H050. Despite the factthat each site in the hexagonal lattice has more nearest neigh-bors than the square one, the local and mean dipolar fields inthe square lattice exceed the corresponding fields in the hex-agonal one, since ds,dh (ds’0.931dh). Therefore the con-ditions rh(t).rs(t) and rmfh (t).rmfs (t) must hold for allt.0 @at large times ~see Fig. 3~b!! the distinction betweenrmfh (t) and rmfs (t) is not visible on the chosen scale of time#.On the other hand, for the same reasons as in the case pre-sented in Fig. 1~a!, the curves rs(t) and rmfs (t), and rh(t)and rmfh (t) are intersected once at t5t in (t in’0.505 s and6.59731023 s for the square and hexagonal lattice, respec-tively!.Sometimes the coefficient of magnetic viscosityS52dr(t)/d ln t is used, to characterize the magnetic prop-erties of nanoparticle ensembles. If S’constant on a widetime interval, as in the case of log-normal distribution fornanoparticle diameters, then the parameter S is an importantcharacteristic of the nanoparticle ensemble, which defines therelaxation law on that interval. For the considered en-sembles, S depends strongly on t , therefore this parameterdoes not play a special role in our case.In conclusion, we have generalized the method devel-oped in Ref. 5 to study the magnetic relaxation in 2D en-sembles of dipolar interacting nanoparticles subjected to abias magnetic field. We have demonstrated its efficiency forthe analysis of the relaxation law features arising from themutual action of the dipolar correlations and bias field fordifferent ensembles of Co nanoparticles and we have shownthat the correlation effects play an important role in magneticrelaxation on all times. We expect that, because of the above-mentioned unique resources of the proposed method, it canbe useful for the design of more reliable perpendicularnanoparticle media.1 For a review see, for example, The Physics of Ultra-High-Density Mag-netic Recording, edited by M. L. Plumer, J. Van Ek, and D. Weller~Springer, Berlin, 2001!; A. Moser, K. Takano, D. T. Margulies, M. Albre-cht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, J. Phys. D 35, R157~2002!.2 D. K. Lottis, R. M. White, and E. D. Dahlberg, Phys. Rev. Lett. 67, 362~1991!.3 S. I. Denisov and K. N. Trohidou, Phys. Status Solidi A 189, 265 ~2002!.4 S. I. Denisov and K. N. Trohidou, Phys. Rev. B 64, 184433 ~2001!.5 S. I. Denisov, T. V. Lyutyy, and K. N. Trohidou, Phys. Rev. B 67, 014411~2003!.6 L. C. Sampaio, R. Hyndman, F. S. de Menezes, J. P. Jamet, P. Meyer, J.Gierak, C. Chappert, V. Mathet, and J. Ferre´, Phys. Rev. B 64, 184440~2001!.7 M. Ulrich, J. Garcı´a-Otero, J. Rivas, and A. Bunde, Phys. Rev. B 67,024416 ~2003!.8 Y. Kanai and S. H. Charap, IEEE Trans. Magn. 27, 4972 ~1991!.9 R. W. Chantrell, N. Walmsley, J. Gore, and M. Maylin, Phys. Rev. B 63,024410 ~2000!.10 U. Nowak, in Annual Reviews of Computational Physics IX, edited by D.Stauffer ~World Scientific, Singapore, 2001!, p. 105.11 R. J. M. van de Veerdonk, X. W. Wu, R. W. Chantrell, and J. J. Miles,IEEE Trans. Magn. 39, 590 ~2003!.4674 Appl. Phys. Lett., Vol. 84, No. 23, 7 June 2004 Denisov, Lyutyy, and TrohidouDownloaded 28 May 2004 to 140.105.16.2. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Dipolar interaction effects on the thermally activated magnetic relaxation of two-dimensional nanoparticle ensembles

S. I. Denisov

T. V. Lyutyy

K. N. Trohidou

Crossref

SocioEconomic Challenges Journal (Sumy State University)

Dipolar interaction effects on the thermally activated magnetic relaxation
of two-dimensional nanoparticle ensembles
S. I. Denisova) and T. V. Lyutyy
Department of Mechanics and Mathematics, Sumy State University, 2, Rimskiy-Korsakov Street,
40007 Sumy, Ukraine
K. N. Trohidoub)
Institute of Materials Science, NCSR ‘‘Demokritos,’’ 15310 Athens, Greece
~Received 21 October 2003; accepted 13 April 2004; published online 19 May 2004!
The thermally activated magnetic relaxation in two-dimensional lattices of dipolar interacting
nanoparticles with large uniaxial perpendicular anisotropy is studied by a numerical method and
within the mean-field approximation for comparison. The role that the correlation effects play in
magnetic relaxation and the influence of lattice structure and bias magnetic field on the relaxation
process are revealed. The correlations of the nanoparticle magnetic moments enhance relaxation on
small times, delay it on large times, and reduce the steady-state absolute magnetization at nonzero
bias fields. In a hexagonal lattice, magnetic relaxation on small times occurs faster and the
steady-state absolute magnetization has the larger magnitude than in a square lattice with the same
lattice spacing. © 2004 American Institute of Physics. @DOI: 10.1063/1.1759782#
The two-dimensional ~2D! ensembles of uniaxial ferro-
magnetic nanoparticles, whose easy axes of magnetization
are perpendicular to the nanoparticles’ plane, represent an
important class of perpendicular magnetic recording media.1
From a technological point of view, the main characteristic
of these media is the average time of data storage, which is
one of the numerical parameters that describe the thermally
activated magnetic relaxation. Due to the dipolar interaction
between nanoparticles, its analytical description in such en-
sembles is a very complicated problem, which was solved
only within the mean-field2,3 and fluctuating4 theories of
magnetic relaxation. The former deals with the mean com-
ponent of the dipolar field acting on the nanoparticles, and
the latter accounts of both the mean and fluctuation compo-
nents. However, the correlations of the nanoparticle magnetic
moments, arising from the dipolar interaction, also play an
important role, especially on the final phase of magnetic re-
laxation. To study the features of magnetic relaxation in such
ensembles, which are conditioned by the correlation effects,
we have developed a method for its numerical simulation at
zero bias field.5 Usually, magnetic relaxation in ensembles of
dipolar interacting nanoparticles is simulated either by the
conventional Monte Carlo method,6,7 where the role of time
plays the Monte Carlo steps, or by the improved Monte
Carlo methods,8–11 that use the short-time stochastic behav-
ior of the nanoparticle magnetic moments to convert the
Monte Carlo steps to real time units. The main advantage of
the method proposed in Ref. 5 is that it does not use the
Monte Carlo approach, so the last complicated problem does
not appear. This feature of the method and its applicability to
the study of magnetic relaxation over ten decades of time5
make it a very useful tool in this domain.
In this letter, we extend our method to the case of non-
zero bias magnetic fields, and we present results obtained
within its framework, which clarify the role that the correla-
tion effects, bias field, and lattice structure play in the mag-
netic relaxation. We consider the 2D ensembles of spherical
nanoparticles with a radius r , which occupy the sites of a
square or hexagonal lattice with lattice spacing d , and whose
easy axes of magnetization are perpendicular to the lattice
plane (xy plane!. We also assume that: ~a! the nanoparticle
magnetic moments mi(t) ~the index i labels the nanopar-
ticles! perform a coherent rotation (umi(t)u5m), ~b! a bias
magnetic field H0k ~k is the unit vector along the z axis! acts
on each mi(t), and ~c! the initial condition mi(0)5mk holds
for all i .
If the potential barriers between the equilibrium direc-
tions of all mi(t) are much larger than the thermal energy
kBT (kB is the Boltzmann constant, T is the absolute tem-
perature!, then the vectors mi(t) fluctuate within small vi-
cinities of the k and 2k directions, and they are reoriented
occasionally. In this case the average numbers N1(t) and
N2(t) of positively and negatively oriented magnetic mo-
ments in a lattice region that contains N~@1! nanoparticles
are well defined (N1(t)1N2(t)’N), and we can introduce
the reduced magnetization of the nanoparticle ensemble as
r(t)52N1(t)/N21. To calculate the relaxation law r(t) at
H050, we have developed the procedure5 that is based on
the analytical determination of the probability reorientation
of the nanoparticle magnetic moments, and on the numerical
determination of the nanoparticle ensemble states. Its impor-
tant feature consists of the fact that the real dipolar fields
acting on mi(t) are taken into account.
Here, we generalize this procedure to the H0Þ0 case and
apply it to study the magnetic relaxation in ensembles of Co
nanoparticles. The generalization consists of the derivation
of the exact formula for the calculation of the probability
reorientation of the nanoparticle magnetic moments for
H0Þ0. We use this formula, derived from the corresponding
backward Fokker–Planck equation, and the numerical proce-
a!Electronic mail: denisov@sumdu.edu.ua
b!Author to whom correspondence should be addressed; electronic mail:
trohidou@ims.demokritos.gr
APPLIED PHYSICS LETTERS VOLUME 84, NUMBER 23 7 JUNE 2004
46720003-6951/2004/84(23)/4672/3/$22.00 © 2004 American Institute of Physics
Downloaded 28 May 2004 to 140.105.16.2. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
dure of Ref. 5 to calculate the relaxation law r(t) in these
ensembles. We choose the basic nanoparticle ensemble for
our simulation in the form of square ~for square lattice! or
hexagonal ~for hexagonal lattice! box and surround it by
eight or six identical ensembles, respectively. To calculate
the dipolar field acting on a nanoparticle, we consider this
nanoparticle as the central one in the box of the same size as
the basic and we include the nanoparticles, which belong to
this box. Increase of the size of the basic box leads to reduc-
tion of the boundary effects and, to practically eliminate
them, we chose the size of the square and hexagonal boxes
equal to 100d and 60d , respectively.
To gain more insight into the magnetic relaxation, we
also calculate the relaxation law rmf(t) within the mean-field
approximation. As this approach ignores the correlation ef-
fects, the difference between r(t) and rmf(t) results from the
correlations of the nanoparticle magnetic moments. In what
follows, the superscripts s and h on r(t), rmf(t), and d de-
note the square and hexagonal lattice, respectively.
The role that the correlation effects and the bias field H0
play in magnetic relaxation is illustrated in Fig. 1. We
present the numerical results for ensembles of Co nanopar-
ticles characterized by the parameters Ha56400 Oe (Ha is
the anisotropy field!, m/V51400 G (V is the nanoparticle
volume!, l50.2 ~l is the Landau–Lifshitz damping param-
eter!, r54 nm, and T5300 K. Due to the correlations of the
nanoparticle magnetic moments, for H050 the actual mag-
netic relaxation occurs faster on small times and more slowly
on large times, than the mean-field theory predicts @see Fig.
1~a!#. In this case the curves rs(t) and rmfs (t) are intersected
at the time t5t in , and rs(‘)5rmfs (‘)50. In equilibrium,
the local dipolar fields have, on average, the same directions
as the magnetic moments and the mean dipolar field equals
zero ~for H050!, therefore for H0Þ0 the condition urs(‘)u
,urmf
s (‘)u must hold. For H0,0 we have
rmf
s (‘),rs(‘),0, and the curves rs(t) and rmfs (t) are in-
tersected only once, as in the H050 case. If H0.0, then
rmf
s (‘).rs(‘).0, for small enough values of H0 these
curves are intersected twice @see Fig. 1~b!#, with increasing
H0 the time interval (t in ,t in*) is decreasing, and rmfs (t)
.rs(t) (t.0) for large H0 @see Fig. 2~a!, curves 1 and 2#.
The features of magnetic relaxation in ensembles of Co
nanoparticles, which arise from the different nanoparticle ar-
rangement in square and hexagonal lattices, are demon-
strated in Figs. 2 and 3. If the lattice spacing d is the same
for both lattices ~Fig. 2!, then the difference between the
relaxation laws results mainly from the different number of
the nearest sites in these lattices ~4 vs 6!. At t50 the local
dipolar fields in hexagonal lattices are larger than in square
ones, so the initial phase of magnetic relaxation occurs faster
in hexagonal lattices. At t5‘ and H050 the conditions
rs(‘)5rh(‘)50 and rmfs (‘)5rmfh (‘)50 hold. Each mag-
netic moment in a square lattice is surrounded, on average,
by four opposite oriented magnetic moments, whereas in a
hexagonal lattice—by four opposite and by two similarly
oriented magnetic moments. Therefore, if H0.0 @see Fig.
2~a!# then rmf
s (‘).rmfh (‘).0, rh(‘).rs(‘).0, and the
curves rs(t) and rh(t) have the unique intersection point t
5t1 (t1’13.59 s for H05500 Oe!. If H0,0 @see Fig. 2~b!#
then rmf
s (‘),rmfh (‘),0 and, since rh(‘),rs(‘),0, the
FIG. 1. Plots of rs(t) ~curve 1! and rmfs (t) ~curve 2! for ds53r . FIG. 2. Plots of r
s(t) ~curve 1!, rmfs (t) ~curve 2!, rh(t) ~curve 3!, and rmfh (t)
~curve 4! for ds5dh53r .
FIG. 3. Plots of rs(t) ~curves 1!, rmfs (t) ~curves 2!, rh(t) ~curves 3!, and
rmf
h (t) ~curves 4! on small ~a! and large ~b! time scales for dh53r , ds
5dh31/4/21/2 and H050.
4673Appl. Phys. Lett., Vol. 84, No. 23, 7 June 2004 Denisov, Lyutyy, and Trohidou
Downloaded 28 May 2004 to 140.105.16.2. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
curves rs(t) and rh(t) have two intersection points t5t1 and
t5t2 (t1’0.01 s, t2’0.33 s for H052500 Oe!.
Figure 3 presents the character of magnetic relaxation in
square and hexagonal lattices, with unit cells that have the
same area, i.e., (ds)25(dh)2A3/2, at H050. Despite the fact
that each site in the hexagonal lattice has more nearest neigh-
bors than the square one, the local and mean dipolar fields in
the square lattice exceed the corresponding fields in the hex-
agonal one, since ds,dh (ds’0.931dh). Therefore the con-
ditions rh(t).rs(t) and rmfh (t).rmfs (t) must hold for all
t.0 @at large times ~see Fig. 3~b!! the distinction between
rmf
h (t) and rmfs (t) is not visible on the chosen scale of time#.
On the other hand, for the same reasons as in the case pre-
sented in Fig. 1~a!, the curves rs(t) and rmfs (t), and rh(t)
and rmf
h (t) are intersected once at t5t in (t in’0.505 s and
6.59731023 s for the square and hexagonal lattice, respec-
tively!.
Sometimes the coefficient of magnetic viscosity
S52dr(t)/d ln t is used, to characterize the magnetic prop-
erties of nanoparticle ensembles. If S’constant on a wide
time interval, as in the case of log-normal distribution for
nanoparticle diameters, then the parameter S is an important
characteristic of the nanoparticle ensemble, which defines the
relaxation law on that interval. For the considered en-
sembles, S depends strongly on t , therefore this parameter
does not play a special role in our case.
In conclusion, we have generalized the method devel-
oped in Ref. 5 to study the magnetic relaxation in 2D en-
sembles of dipolar interacting nanoparticles subjected to a
bias magnetic field. We have demonstrated its efficiency for
the analysis of the relaxation law features arising from the
mutual action of the dipolar correlations and bias field for
different ensembles of Co nanoparticles and we have shown
that the correlation effects play an important role in magnetic
relaxation on all times. We expect that, because of the above-
mentioned unique resources of the proposed method, it can
be useful for the design of more reliable perpendicular
nanoparticle media.
1 For a review see, for example, The Physics of Ultra-High-Density Mag-
netic Recording, edited by M. L. Plumer, J. Van Ek, and D. Weller
~Springer, Berlin, 2001!; A. Moser, K. Takano, D. T. Margulies, M. Albre-
cht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, J. Phys. D 35, R157
~2002!.
2 D. K. Lottis, R. M. White, and E. D. Dahlberg, Phys. Rev. Lett. 67, 362
~1991!.
3 S. I. Denisov and K. N. Trohidou, Phys. Status Solidi A 189, 265 ~2002!.
4 S. I. Denisov and K. N. Trohidou, Phys. Rev. B 64, 184433 ~2001!.
5 S. I. Denisov, T. V. Lyutyy, and K. N. Trohidou, Phys. Rev. B 67, 014411
~2003!.
6 L. C. Sampaio, R. Hyndman, F. S. de Menezes, J. P. Jamet, P. Meyer, J.
Gierak, C. Chappert, V. Mathet, and J. Ferre´, Phys. Rev. B 64, 184440
~2001!.
7 M. Ulrich, J. Garcı´a-Otero, J. Rivas, and A. Bunde, Phys. Rev. B 67,
024416 ~2003!.
8 Y. Kanai and S. H. Charap, IEEE Trans. Magn. 27, 4972 ~1991!.
9 R. W. Chantrell, N. Walmsley, J. Gore, and M. Maylin, Phys. Rev. B 63,
024410 ~2000!.
10 U. Nowak, in Annual Reviews of Computational Physics IX, edited by D.
Stauffer ~World Scientific, Singapore, 2001!, p. 105.
11 R. J. M. van de Veerdonk, X. W. Wu, R. W. Chantrell, and J. J. Miles,
IEEE Trans. Magn. 39, 590 ~2003!.
4674 Appl. Phys. Lett., Vol. 84, No. 23, 7 June 2004 Denisov, Lyutyy, and Trohidou
Downloaded 28 May 2004 to 140.105.16.2. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp


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Dipolar interaction effects in the thermally activated magnetic relaxation of two-dimensional nanoparticle ensembles

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