1-s2.0-S0304885303001318-main

Journal of Magnetism and Magnetic Materials 264 (2003) 1429

Memory effects and slow dynamics in ultra thin magnetic films

S.P. Bromleya, J.P. Whiteheada, K. DeBellb,*, A.B. MacIsaacc

aDepartment of Physics and Physical Oceanography, Memorial University of Newfoundland, St. Johns, Newfoundland, Canada A1B 3X7bDepartment of Mathematical Sciences, University of New Brunswick at Saint John, Saint John, New Brunswick, Canada E2L 4L5

cDepartment of Applied Mathematics, University of Western Ontario, London, Ont., Canada N6A 5B9

Received 17 September 2002; received in revised form 17 January 2003

Abstract

Monte Carlo studies of a uniaxial spin system on a square lattice with both ferromagnetic exchange interactions and

dipolar interactions reveal three distinct dynamical regimes at low temperatures. In the first regime the magnetization

decays through nucleation and growth of spin islands, in the second regime, despite the magnetization being effectively

zero, there is a persistent memory effect most clearly revealed by the asymmetry between the numbers of up and down

islands. The third regime is characterized by slow crystallization of the equilibrium smectic phase from the quenched

polycrystalline state.

r 2003 Elsevier Science B.V. All rights reserved.

PACS: 75.70; 75.60.L

Keywords: Memory effects; Magnetic; Ultra-thin film; Time dependence; Dynamics

1. Introduction

Ultrathin magnetic films are of considerableinterest because of their potential technologicalapplications in data storage media and magneto-electronics [1]. While the rapid technologicaldevelopment of these materials is almost concur-rent with the investigation of the underlyingphysics [2], many aspects of these system remainto be elucidated. In particular, a more completeunderstanding of the stability and dynamics of thedomain structures, that determine many of the

important properties of these materials, is highlydesirable.

A key feature of these materials is the inherentmagnetic frustration which results from thecompetition between the (ferromagnetic) exchangeinteraction, the dipolar interaction and the surfaceanisotropy [3]. This frustration can result indomain phases and, as a result, these materialscan exhibit a variety of phase transitions. At lowtemperatures the system is typically mono-domainwith spins oriented perpendicular to the plane ofthe film, as the temperature is raised a smecticmulti-domain regime occurs and this is usuallyevidenced by a loss of the net magnetic moment.The static, equilibrium properties of these filmshave been studied extensively both experimentallyand theoretically (for a review see Ref. [4]).

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*Corresponding author. Department of Mathematics,

University of New Brunswick at Saint John, Saint John,

New Brunswick, Canada E2L 4L5. Tel.: +506-648-5615; fax:

+506-648-5650.

0304-8853/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0304-8853(03)00131-8

Dynamical properties, such as relaxation phe-nomena, have been much less studied, howeverseveral significant experimental studies have beenperformed. Berger and Hopster have examinedthe relaxation of the magnetization fromsaturation in Fe/Ag(1 0 0) [5]. Their experimentswere performed in the multi-domain regime.Berger and Hopster observed that relativelysmall external fields E90 Oe were required tosaturate the magnetization of the system;however, after the field was switched off, severalseconds were required for the magnetization tovanish at lower temperatures. It was arguedthat the dynamics of these systems could becharacterized in terms of a single activation energyat low temperatures. However, Berger and Hop-ster found that the activation energy variedsignificantly between samples. Cowburn and cow-orkers have studied dynamical effects in anultrathin wedge of Fe between thin layers of Ag[6] and found evidence for a thermally nucleateddomain structure. Venus et al. [7] have alsodiscussed the importance of energy barrier domi-nated dynamics in interpreting the results of lowfrequency AC susceptibility measurements on Fe/2ML Ni/W(1 1 0).

Here we report the results of a Monte Carlostudy of the dynamics of a system of Ising spins ona square lattice in which the spins are alignedperpendicular to the plane of the lattice. Specifi-cally we examine the case in which the system isprepared as a single saturated (ferromagnetic)domain and is then allowed to relax to theequilibrium phase.

At low temperatures, our results demonstratethat there are three distinct dynamicalregimes during the relaxation of this system. Atearly times the relaxation process is characterizedby nucleation and growth of islands of spinswith the opposite orientation to the initialmagnetization and the corresponding decay ofthe magnetization.

In the intermediate regime the magnetizationhas dropped to a few percent of its saturationvalue, however the system is not in itsequilibrium state, since the number of islands ofup spins is observed to differ significantly fromthe number of islands of down spins. In addition

to, and closely correlated with, the asymmetrybetween the spin up and spin down islands,a small remnant of the magnetizationpersists.

This asymmetry and the associated magnetiza-tion decay on a much longer time scale thanthe initial decay of the magnetization associatedwith the island nucleation and growth. In thissense, the system retains a memory of its initialstate which gradually decays in this intermediateregime.

At late times the spin up and spin down islandsarrange to form regions consisting of the stripes ofalternating spins that characterize the equilibriumsmectic phase. However the regions are notoriented along a single axis but instead areanalogous to crystallites in a polycrystallinesample. The dynamical processes which result inthe equilibrium single smectic crystal phase areextremely slow.

The possibility of a relatively long lived memoryeffect in ultrathin magnetic films is intriguing.First, in experimental studies to probe thedynamics it is important to be aware thatrelaxation to a system with close to zero magne-tization does not necessarily imply that the systemhas reached a state that is close to the equilibriumstate. Moreover the existence of more than onetime scale in the non-equilibrium regime hasimportant consequences for the interpretation ofthe dynamical results. As we show below, mea-surements of the relaxation time even for themagnetization may be a useful diagnostic toolwhen probing the nature of the multi-domainstate. Second, analogous memory effects inalloys have potential technological applications(however the nature of the long rangeinteraction is different in those systems). While itis not clear if the memory effect reported here hasdirect technological uses, understanding such aneffect is likely to be of importance in theapplication of ultrathin magnetic films in devices.The very slow dynamics observed in the thirdregime, in addition to their intrinsic interest in thecontext of magnetic films, may provide insight intothe more general problem of slow dynamics in, forexample, solids with a crystalline equilibriumphase.

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S.P. Bromley et al. / Journal of Magnetism and Magnetic Materials 264 (2003) 1429 15

2. The simulation model

The interaction Hamiltonian for the model isgiven by

H JX/ijS

sisj gXiaj

sisjR3ij

; 1

where the variable s 71 denotes the direction ofthe magnetic moment perpendicular to the planeof the film. The first term represents the effect ofthe exchange interaction, with exchange constantJ: The exchange interaction is assumed to beferromagnetic and J is therefore positive. Thesecond term arises as a consequence of the dipolarinteraction.

Earlier simulation studies of the static propertiesof this system revealed that it exhibits a multi-domain smectic phase at low temperatures. At acritical temperature TO the smectic phase isreplaced by a tetragonal phase and the orienta-tional order associated with the smectic phase islost [8]. In subsequent experimental studies thistransition was directly observed by direct imagingof the magnetic domains [9]. A transition from anorientationally ordered phase to a phase withoutorientational order is also observed in simulationsof the corresponding triangular lattice system[10,11]. However the exact nature of the phasesand transitions is dependent on the lattice.

A value of J 8:9g is used throughout the workreported here corresponding to a ground statestripe width of h 8 and a transition from thesmectic striped phase to the tetragonal phase atTOE4:8g [8]. A square L L lattice is used withperiodic boundary conditions. L is chosen to be amultiple of h in order that the stripes will becommensurate with the periodicity imposed by theboundary conditions. The dipolar interaction istreated using the Ewald summation technique [4].The simulations are performed using standardGlauber dynamics and the initial spin configura-tion is fully polarized fsigt0 f1g: As discussedbelow, the initial decay of the magnetization isstrongly dependent on the size of the system and asystem size of 256 256 appears to be theminimum system size for reliable dynamical results[12]. All of the calculations described in this workare performed on lattices of size N 256 256:

3. Relaxation kinetics

In these simulations the system is allowed torelax from its initial saturated ferromagnetic statestarting at time t 0 toward its equilibrium phase.Fig. 1 shows the spin configurations of a 256 256lattice at various times for a typical simulation runat T 4:25g: This is below the critical temperatureTOE4:80g: For this value of T the equilibriumphase therefore corresponds to the smectic phaseconsisting of stripes of alternating spin orientedalong a common crystallographic axis, with no netmagnetization.

The magnetization for several simulation runs atT 4:25g is plotted as a function of time t inFig. 2, together with the average (over the simula-tion runs) of the magnetization. The time t is givenin units of Monte Carlo steps (mcs).

Since the initial saturated ferromagnetic state(all spins up) is a metastable state of the system,the initial stage of the relaxation process is verysimilar to other thermally activated nucleationprocesses. Specifically our results show that at veryearly times, many islands, each consisting of asmall number of reversed spins (spin down), formthroughout the system. However, there exists anenergy barrier preventing evolution towards theequilibrium state. This energy barrier, limiting thegrowth of the islands, is overcome only oncethe size of the islands exceeds some critical value.We refer to the time during which the systemremains on the metastable side of the energybarrier as the wait time.

In practice, we use the time taken for themagnetization to decay by 1% as providing somequantitative measure of the wait time. As therelaxation process is stochastic, the wait times willvary for different simulation runs at the sametemperature. The histogram of wait times isapproximately a Poisson distribution for systemsof size 256 256; however the distribution devi-ates significantly from this if smaller system sizesare used [12]. Earlier work also demonstrated thesensitivity of results for this system to system size,however, the conclusions of this earlier work werestill based on simulations of relatively smallsystems [13]. This combined with the unusualboundary conditions employed in that work

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S.P. Bromley et al. / Journal of Magnetism and Magnetic Materials 264 (2003) 142916

makes the usefulness of conclusions drawn in thatearlier work somewhat uncertain. Subsequentwork which also concentrated on the effect ofthe ratio J=g (in our notation) used larger latticesbut truncated the dipolar interaction [14]. Tomitigate system size effects, all of the simulations

described here use a minimum system size of N 256 256:

The effect of the nucleation process on themagnetization is manifest in the initial slopeof the magnetization curve shown in Fig. 2(limt-0dM=dtE0).

As the number of islands that exceed the criticalsize increases, the islands begin to grow and themagnetization decays rapidly. This rapid decay ofthe magnetization following the initial nucleationregime is clearly seen in Fig. 2. In Fig. 3 we showthe average magnetization temperatures. The plotsshow the decay rate of the magnetization increas-ing with increasing temperature.

However, while the magnetization in Fig. 2appears to decay to effectively zero after aboutt 2000 mcs; it is quite obvious from the spinconfigurations shown in Fig. 1, that the system isstill far from its equilibrium phase. In addition acareful examination of the magnetization datashows that m has not decayed to zero, but insteada small remnant magnetization persists; thisremnant magnetization decays over a much longertime scale.

Inspecting the spin configuration in Fig. 1, fort 2000mcs we see that the spin configurationconsists of a large number of disconnected spin

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1.0

0.8

0.6

0.4

0.2

0.0

mag

netis

atio

n

2000150010005000t (mcs)

T = 4.25g

Fig. 2. A plot of the magnetization as a function of Monte

Carlo time steps (mcs) for several runs. The parameters are

J 8:9g; T 4:25g and L 256: The solid line shows theaverage magnetization.

t = 0 mcs t = 250 mcs t = 750 m

t = 2000 mcs t = 20000 mcs t = 30000

Fig. 1. Snapshots of the spin configuration at successive times for a typical Monte Carlo run, with J 8:9; T 4:25 and L 256:


down islands within a single connected island ofup spins. This asymmetry between the number ofspin up and spin down islands serves as a veryuseful quantitative measure of the memory thesystem has of its initial configuration. We there-fore find it useful to define the island asymmetry asthe difference between the number of spin up andspin down islands, which we denote by a: Theisland asymmetry may be elegantly and efficientlycalculated by generating the winding number for aparticular spin configuration. The details havebeen given previously [15].

The average island asymmetry is plotted forseveral temperatures in Fig. 4. All the curves showthe same qualitative behaviour of a rapid rise inthe asymmetry followed by a slow decay.

The time dependence of the island asymmetryprovides some useful insights into the relaxationprocess. The rapid initial growth of the islandasymmetry seen in Fig. 4, is directly related to thenucleation of islands, since to begin with the islandasymmetry is approximately equal to the numberof isolated spin reversed islands. This is readilyunderstood, since at the earliest times the spin upsites, except for short lived individual spin flips,are all connected to form a single island which,because of the periodic boundary conditions, doesnot contribute to the parameter a; the island

asymmetry is therefore given by the number ofspin down islands.

As the islands grow the process of islandnucleation abates and the spin down islands startto coalesce to form larger more elongated islands.At this point the island asymmetry reaches amaximum and begins to decrease. As the spindown islands begin to coalesce, not only do thenumber of spin down islands begin to fall, but alsothe spin up sites are beginning to becomedisconnected forming isolated pools surroundedby spin down sites, thus the island asymmetrybegins to fall. A comparison of the time scales inFigs. 2 and 4 shows that the time scale associatedwith the decay of the island asymmetry is at leastan order of magnitude greater than that associatedwith the decay of the magnetization. This slowdynamic associated with the decay of the islandasymmetry defines the intermediate regime.

While the slow decay of the island asymmetryserves to define the intermediate regime, a carefulexamination of the magnetization data in thisregime shows that it does not fluctuate about itsequilibrium value but instead a small remnant, ofthe order of 12% of the saturation magnetization,persists long after the bulk of the magnetizationhas disappeared.

This long lived tail is clearly seen in Fig. 5 inwhich the magnetization is plotted on an expandedscale along with the island asymmetry for several

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1.0

0.8

0.6

0.4

0.2

0.0

mag

netis

atio

n

10008006004002000t (mcs)

T=4.0g

T=4.25g

T=5.0g

T=6.0g

Fig. 3. A plot of the average magnetization as a function of

Monte Carlo time steps (mcs) for several temperatures for J 8:9g and L 256:

100

80

60

40

20

0

isla

nd a

sym

mte

ry

20103151050

t (mcs)

T = 4.0g

T = 4.25g

T = 3.75g

T = 3.5g

Fig. 4. A plot of the island asymmetry as a function of Monte

Carlo time steps (mcs) for several temperatures for J 8:9gand L 256:


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40

30

20

10

0

(i

slan

d sy

mm

etry

)

6010350403020100t (mcs)

1510-3

10

5

0

m (m

ag

ne

tisatio

n)

T = 3.75g

60

50

40

30

20

10

0

(i

slan

d sy

mm

etry

)

6010350403020100t (mcs)

2510-3

20

15

10

5

0

m (m

agnetisation)

T = 4.0g

30

25

20

15

10

5

0

(i

slan

d sy

mm

etry

)

6010350403020100t (mcs)

1210-3

10

8

6

4

2

0

m (m

agnetisation)

T = 3.50g

800

600

400

200

0M (

tota

l mag

netis

atio

n)

30252015105

(island symmetry)

T = 3.75

500

400

300

200

100

0M (

tota

l mag

netis

atio

n)

222018161412108 (island symmetry)

T = 3.50

800

600

400

200

0

M (

tota

l mag

netis

atio

n)

3020100 (island symmetry)

T = 4.00

Fig. 5. A plot of the island asymmetry against time for several temperatures with J 8:9g and L 256: Also shown is the tail of themagnetization data (dots) on an expanded scale. The insets show the island asymmetry as a function of the magnetization.


temperatures. Indeed this graph not only showsthe long lived tail but also that the magnetizationin the intermediate regime is strongly correlated tothe island asymmetry. The correlation between theisland asymmetry and the magnetization is moreclearly seen in inset in each of the graphs in Fig. 5where a is plotted as a function of the totalmagnetization M m L2; in the intermediateregime. The data shows a linear relationshipbetween the island asymmetry and the magnetiza-tion. This suggests that the long lived tail in themagnetization is a feature of the low temperaturerelaxation process and reflects the fact that thesystem has not yet quite relaxed to its equilibriumstate.

Describing the linear relationship between theisland asymmetry a and the total magnetization Min the intermediate regime by the coefficients Oand o; we write

M O oa: 2

Values for O and o are given in Table 1 for severaltemperatures. The results show that o is virtuallyindependent of temperature. (The estimates of Oshow no systematic variation. This is probably dueto a high degree of sensitivity to noise in the datafor this parameter.)

The slope o defined by Eq. (2), sheds some lighton the nature of the relaxation process in theintermediate regime. If we assume that therelaxation process in the intermediate regimeconsists of blocks of spins flipping over asislands coalesce, then the slope o provides somemeasure of the number of spins involved in theprocess, and possibly gives some insight into thecollective excitation driving the decay of the islandasymmetry.

At times greater than the relaxation time for athe system has essentially lost its memory of theinitial state; nonetheless, inspection of spin con-figurations reveals there is a substantial timeperiod before the system reaches its equilibriumstate. Fig. 6 shows typical spin configurations froma single simulation run at several hundredthousand (Monte Carlo) time steps. Inspection ofthe configurations, such as those in Fig. 6, at latetimes in the simulation shows that the smecticphase has formed locally; however the regionscontaining these sections of smectic phase domainsdo not have a common orientation.

It is useful to think of the regions of smecticphase as being analogous to crystallites in apolycrystalline sample. Boundaries between

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t = 300,000 mcs t = 600,000 mcs t = 900,000 mcs

Fig. 6. Snapshots of the spin configuration at t 300; 000; 600; 000 and 900; 000 mcs showing the boundaries separating the regions ofhorizontal and vertical stripes, with J 8:9g; T 4:25g and L 256: The spin configurations shown are constructed by tiling fourlattices together to form a 512 512 lattice.

Table 1

Estimates of the parameters O and o in Eq. (2)

T=g O o

3.50 41.477.2 20.170.63.75 24.374.5 21.970.44.00 6.973.0 22.670.34.25 19.276.8 21.170.4


crystallites are determined by the presence oftopological defects. The extremely slow kineticsof these defects moving through the system andbeing annihilated determines evolution towardsthe equilibrium smectic crystalline phase.

The general characteristics of the descriptionabove are common to all simulations performedwith TtTO: At higher temperatures, T\TO; therelaxation process does not exhibit the memoryeffect that characterizes the low temperature,TtTO; behaviour and the system decays to thetetragonal fluid phase with both the magnetizationand the island asymmetry decaying to zero atapproximately the same rate.

In summary of this section, at temperaturesTtTO the relaxation of this system has manyanalogies with the relaxation of a crystalline solidinitially prepared in the wrong crystalline phase,however at temperatures T\TO the behaviour ofthe system is analogous to a fluid able to readilyreturn to its equilibrium phase.

4. Phenomenological model

Given the complexity of the nucleation processthat govern the dynamics of the spins at theearliest times and low temperatures (ToTO it ishardly surprising that the average magnetizationcurves, such as those shown in Fig. 3, cannot bedescribed in terms of a simple exponential decay.This is in contrast to the assumptions made inearlier studies [13,14].

To construct a phenomenological description ofthe time dependence of the magnetization we notethat the initial decay of the magnetization isgoverned by the nucleation and growth of thespin down islands. If we assume that the islandgrowth is proportional to the total perimeter of thespin reversed islands, then the rate of change in themagnetization may be determined from a straight-forward scaling argument as

dm

dtE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 m

p; 3

where n denotes the number of spin reversedislands. If we further assume that in the nucleationphase the number of spin reversed islands, n; is

given by

nE1 mF; 4

then the initial rate of change through themagnetization is related to the magnetizationthrough the relation

dm

dtE r1 mY 5

with Y 1 F=2:Following the island nucleation the magnetiza-

tion then decays to about 1% of its initial valueafter which it is proportional to the islandasymmetry. Neglecting the remnant magnetizationfor the time being and simply assuming thatthe magnetization decays exponentially to zero,requires that

limt-N

dm

dtE

m

t; 6

where t denotes the relaxation time associatedwith the decay of the magnetization following thenucleation phase.

Eqs. (5) and (6) can be combined into a simpleequation that interpolates the two regimes.

dm

dtE

m

t1 mY1 rt 1m: 7

To determine the parameters in the phenomen-ological model in Eq. (7) we fit m and mdetermined from the Monte Carlo data to theabove functional form by means of a regressionanalysis. The resultant curves and the data for fourtemperatures are shown in Fig. 7. Estimates of theparameters for a range of temperatures are givenin Table 2.

With the parameters obtained from the regres-sion analysis of the m vs. m data, it is possible tosolve the differential equation, given in Eq. (7),numerically for a given initial value of m: Theresultant function mt and its associated deriva-tive mt are plotted with the corresponding MCdata in Figs. 8 and 9 for four temperatures. Thesolution to the differential equation is insensitiveto the initial value provided it is less than but closeto 1.

Assuming that the relaxation time t and the fliprate r are governed by an activation energy andhence vary exponentially with the inverse

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temperature as

tEt0expDt=T; 8

1

rE

1

r0expDr=T; 9

we construct an Arrenhuis plot for both the relaxationtime t and the flip rate r; as shown in Fig. 10.

An examination of the data in Fig. 10 suggeststhat the relaxation process exhibits crossover atTETO: Therefore we fit the data above and belowthe transition temperature, TO; separately. Thelines of best fit are shown in Fig. 10 and yield thefollowing parameter estimates in the two tempera-ture ranges.

Dt 17:5g70:5g for ToTO29:5g70:2g for T > TO:

(10

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Table 2

Estimates of the phenomenological model parameters obtained

at various values of temperature

T=g Y F 2Y F

3.50 0.55770.016 0.32670.004 0.78870.0324.00 0.60570.009 0.28670.005 0.92470.0194.25 0.63370.008 0.27570.003 0.99170.0174.75 0.68270.012 0.26770.002 1.09770.0245.00 0.65970.014 0.24570.005 1.07370.028

-1.610-3

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

dm/d

t

1.00.80.60.40.2m

T = 4.0g

-2.010-3

-1.5

-1.0

-0.5

0.0

dm/d

t

1.00.80.60.40.2m

T = 4.25g

-610-3

-5

-4

-3

-2

-1

0dm

/dt

1.00.80.60.40.20.0m

T=5.0g

-60010-6

-500

-400

-300

-200

-100

0

dm/d

t

1.00.80.60.40.20.0m

T=3.5g

Fig. 7. A plot of m vs. m for several temperatures.


The corresponding values for t0; the inverse of theattempt frequency, are

t0 3:6 mcs70:4 mcs for ToTO

0:46 mcs70:07 mcs for T > TO:

(11

Similarly the energies and inverse attempt fre-quencies associated with the flip rate are foundto be

Dr 32:8g71:1g for ToTO14:0g72:4g for T > TO;

(12

1

r0

0:05 mcs70:01 mcs for ToTO1:8 mcs70:6 mcs for T > TO:

(13

We note that the difference in the estimatedactivation energies cannot be ascribed to statisticaluncertainty in the data. A more detailed w2

analysis comparing the above fitting procedurewith a simple linear fitting function, indicates thatusing the separate fitting functions provides abetter description of the data.

While the above arguments provide anadequate phenomenological description of the

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1.0

0.8

0.6

0.4

0.2

0.0

m

1400120010008006004002000

t (mcs)

T = 4.25g

1.0

0.8

0.6

0.4

0.2

0.0

m

5004003002001000

t (mcs)

T = 5.0g

1.0

0.8

0.6

0.4

0.2

0.0

m

500040003000200010000

t (mcs)

T = 3.5g

1.0

0.8

0.6

0.4

0.2

0.0

m

2000150010005000

t (mcs)

T = 4.0g

Fig. 8. A plot of m as a function of time for several temperatures. The solid line is that obtained by numerically solving Eq. (7) using

parameters obtained from a regression analysis.


magnetization, a similar analysis of the islandasymmetry is more difficult. In part this is becausethe fluctuations in Monte Carlo data for the islandasymmetry is greater proportionally than thecorresponding fluctuations in the magnetization.Our studies also show that the slow decay of theisland asymmetry at long times cannot be de-scribed in terms of a simple exponential. Despitethis we can nevertheless construct a simple modelfor the asymmetry based on some reasonableassumptions about the relaxation process. Asdiscussed earlier we can see from the data thatthe island asymmetry initially rises rapidly asislands of opposite spin orientation nucleate. As

the islands continue to grow the nucleation processeffectively stops (or, at least, is negligable) and theisland asymmetry begins to decrease as the islandsbegin to connect together and the backgrounddisconnects into multiple islands. Quantitativelywe can describe this by

at it Z t0

gt t0at0 dt0; 14

where it describes the rapid nucleation of theislands to some saturation value, while the secondterm describes the decay of the island asymmetrydue to the islands coalescing and is characterizedby a relaxation function gt: This separation into

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-1.610-3

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

dm/d

t

25002000150010005000

t (mcs)

T = 4.0g

-610-3

-5

-4

-3

-2

-1

0dm

/dt

6004002000 t (mcs)

T = 5.0g

-2.010-3

-1.5

-1.0

-0.5

0.0

dm/d

t

12008004000 t (mcs)

T = 4.25g

-600x10-6

-500

-400

-300

-200

-100

0

dm/d

t

500040003000200010000 t (mcs)

T = 3.5g

Fig. 9. A plot of m as a function of time for several temperatures. The solid line shows the numerical solution of Eq. (7).


two distinct processes is only useful if the timescales describing the nucleation and decay arequite distinct.

At low temperatures (TtTO) we can assumethat this separation of the time scales does occur asthe island nucleation appears to occur on the sametime scale as the decay of the magnetization. Ourearlier discussion implies that the two processes,the island nucleation and decay of the magnetiza-tion are strongly correlated at early times; indeedEq. (4) implies that at early times itE1 mtF:Plotting a as a function of m for severaltemperatures (Fig. 11) we see that at early timesmt1 the Monte Carlo data does suggest a powerlaw relationship. We therefore fit the data shownin Fig. 11 using

it a1 mtF1 bmt; 15

where the 1 bm term allows for the fact that aappears to saturate and even decrease somewhatbefore the magnetization has decayed to zero. Thisreflects the continued island growth even thoughthe nucleation process has ceased. The best fit linesobtained by regression for several values oftemperature, T=g; are shown in Fig. 11 togetherwith the corresponding Monte Carlo data. Thevalues for F obtained from the fit are given inTable 2, for several temperatures, together withthe quantity 2Y F: We note that, except at the

lowest temperature considered, the calculatedvalue of 2Y F is close to unity, giving goodagreement with the value predicted by the simplescaling theory that underlies the current analysis.

To determine the time dependence of at fromthe nucleation function requires the explicit formfor the relaxation function gt t0 defined byEq. (14). Thus far we have been unable to obtain asuitable function for gt t0 that describes thedecay of a over the temperature range of interest.With the absence of suitable form for gt t0we therefore make the simplest possible choiceapproximating it by a constant.

gtE1

ta: 16

While more complicated functional forms ofgt t0 yield a better fit to the data, they addlittle to the analysis and do not reveal any usefulsystematic behaviour with respect to the tempera-ture. The above form for the relaxation functionalso has the advantage that it yields an exactsolution for at in terms of the nucleation functionit

at it 1

ta

Z t0

exp t t0

ta

it0 dt0: 17

Using the value of Y in Table 2, and selecting asuitable value for ta and substituting the expres-sion for it into Eq. (17) we can determine theremaining coefficients a and b by linear regression.The results from the model are shown in Fig. 12together with the data.

The slow decay of the residual magnetization inthe intermediate regime together with its lineardependence on a suggests a simple generalizationof the phenomenological equation for m by theaddition of a term linear in a: The full system ofequations for the phenomenological model is then

dm

dt

m

t1 mY1 rt 1m ca a0; 18

at it 1

ta

Z t0

exp t t0

ta

it0 dt0; 19

it a1 mtF1 bmt; 20

where the coefficient c o=L2t is determined byEq. (2). The results for the full set of equations

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6

5

4

3

0.280.260.240.220.200.180.160.140.12

log log r

1/T

log

Fig. 10. Arrenhuis plot for the relaxation time t and flip rate r:


together with the corresponding data are shown inFig. 13 for T 4:25: (For the purposes ofillustration we use o 20; L 256 and t 216for calculating the fitting line).

While alpha decays to zero, or at least fluctuatesabout zero in our finite size system, at the longesttimes simulated in this study, Fig. 6 clearly showsthat the system has still not reached its equili-brium. Instead we see regions, which we refer to ascrystallites, in which stripes with a well definedorientation have formed, but these regions do notshare a common orientation. Thus while a 0 is anecessary condition for equilibrium it is not a

sufficient one. It would be interesting to extend thepresent phenomenological analysis into the crys-tallite regime by coupling the orientational orderparameter Ohv [8] into the analysis however, at thepresent time, it would be impractical given the timescales involved.

5. Experimental and technological implications

The results of the simulations reported heredemonstrate that the relaxation process above TOis qualitatively different from that below TO: In

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5

0

1.00.80.60.40.20.0

m

T =3.5g

200

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0

1.00.80.60.40.20.0

m

T =5.0g

70

60

50

40

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20

10

0

1.00.80.60.40.2

m

T =4.0g

80

60

40

20

0

1.00.80.60.40.2

m

T =4.25g

Fig. 11. A plot of a vs. m for several temperatures.


addition to the difference in the activation energyfor relaxation of the magnetization noted pre-viously, the relaxation process at low temperature(ToTO) exhibits three distinct regimes: the decayof the magnetization, the decay of the islandasymmetry, with the associated loss of memory ofthe initial state, and finally the growth of orienta-tional order with the gradual crystallization of theequilibrium striped phase from the quenched state.By contrast the relaxation process at high tem-peratures (T > TO) can be described in terms of asingle regime in which the magnetization and theisland asymmetry rapidly decay to zero, beyond

which the system appears to be in the equilibriumtetragonal phase.

As noted earlier, direct imaging of the domainshas recently been used to observe the smectic totetragonal transition [9]. However, indirect probesof the magnetic properties which would also allowthe smectic and tetragonal phases to be distin-guished will greatly facilitate the characterizationof ultra thin magnetic films and our present studiesindicate that dynamical phenomena such asrelaxation are candidates for such indirect probes.Early experiments used direct domain imaging tofollow spin reversal and nucleation following the

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0

6010350403020100 t (mcs)

T = 3.5g

200

150

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2000150010005000

t (mcs)

T =5.0g

80

60

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0

500040003000200010000 t (mcs)

T = 4.25g

70

60

50

40

30

20

10

0

20x103151050

t (mcs)

T = 4.0g

Fig. 12. A plot of a as a function of time t for several temperatures. The solid line is the function determined from thephenomenological model described in the text.


reversal of an external magnetic field [16].Although somewhat different to the process ofrelaxation to a zero external field state simulated inour study the results are qualitatively consistent.In particular the system does not return to itsvirgin domain phase (in this case a singledomain the size of the sample imaged by Faradayrotation) but forms a more complex domainpattern which locks in if the external field isswitched off. The experiments performed by

Berger and Hopster [5] deal with relaxation froma saturated state to a zero external field state in atemperature range where (we believe) the equili-brium state would be multi-domain. The variationof the magnetization with time observed by Bergerand Hopster is not a simple exponential curve butrather appears to have horizontal regions whichlast for macroscopic periods of time. This may bean indication that different regions of the sampletake different lengths of time to relax and is

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1.00.80.60.40.2 m

T = 4.25g

1.0

0.8

0.6

0.4

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m

2000150010005000

t (mcs)

T = 4.25g

Fig. 13. m as a function of t and a as a function of m using the full phenomenological model of Eq. (20).


consistent with the idea that several differentregimes of relaxation behaviour occur. Howevera comparison of the simulations reported herewith the work of Berger and Hopster is compli-cated by the fact that the system they studied alsoexhibits a reorientation transition. In order tosimulate this it would be necessary to consider aHeisenberg model. A very interesting, thoughlargely qualitative, simulation study of this kindhas recently been reported by Iglesias and Labarta[17]. A quantitative analysis of such a study todetermine to what extent a phenomenologicalmodel of the type used here can be applied wouldaid in making comparisons with experimentalsystems many of which exhibit a reorientationtransition.

Technology based on either the magneticproperties alone or in combination with electronicproperties offer the potential for devices withhigher speed and capacity, and lower powerconsumption than conventional microelectronicdevices [1]. The development of such devices hasbeen greatly advanced by recent innovations inmaterials engineering including using compositesof layers with different magnetic properties. Thepresent work indicates that some care must beapplied in interpreting experimental results and intechnological uses of these materials as (at lowtemperatures) the abrupt drop of the magnetiza-tion following the switching on and subsequentswitching off of an external magnetic field does notof itself indicate a return to equilibrium but ratherthe system may retain a memory of the initial statefor a time considerably longer than the initialdecay of the magnetization would indicate. Whilethis memory effect may be a difficulty in someapplications the possibility that there may beapplications of the effect in its own right (perhapsthe ability to recover data even after very longtimes) is an intriguing one.

Acknowledgements

This work is supported in part by the NaturalSciences and Engineering Research Council ofCanada. One of us (KDB) thanks the Centre forInterdisplinary Studies in Chemical Physics, Uni-versity of Western Ontario, for a Senior VisitingFellowship during the early stages of this work.

References

[1] S.A. Wolf, D.D. Awshalom, R.A. Buhrman, J.M. Daugh-

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D.M. Treger, Science 294 (2001) 1488.

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[5] A. Berger, H. Hopster, Phys. Rev. Lett. 76 (1996) 519.

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[13] L.C. Sampaio, M.P. de Alburquerque, F.S. de Menezes,

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[14] T.G. Rappoport, F.S. de Menezes, L.C. Sampaio, M.P.

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[15] I.N. Booth, Honours Thesis, Memorial University of

Newfoundland, 1995.

[16] J. Pommier, P. Meyer, G. P!enissard, J. Ferr!e, P. Bruno,

D. Renard, Phys. Rev. Lett. 65 (1990) 2054.

[17] O. Iglesias, A. Labarta, J. Magn. Magn. Mater. 221 (2000)

149.

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Memory effects and slow dynamics in ultra thin magnetic filmsIntroductionThe simulation modelRelaxation kineticsPhenomenological modelExperimental and technological implicationsAcknowledgementsReferences

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