Physica B 306 (2001) 216–220

Configurational anisotropy and thermally activatedswitching in magnetic nanosquares

L. Torres*, L. Lopez-Diaz, E. Mart!ınez, J. I *niguez

Departamento de F!ısica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain

Abstract

Switching processes in magnetic nanosquares of size 10lexoao64lex and thickness t ¼ 1:5lex have been systematicallyanalyzed for different applied field angles. The effect of configurational anisotropy in all the cases is discussed.Thermally activated switching is studied in nanosquares of a ¼ 20lex and 40lex with t ¼ 1:5lex by solving the stochasticLandau–Lifschitz–Gilbert equation. The dependence of the relaxation time on applied field and temperature isinvestigated. Superparamagnetic effects are found to be significant for a ¼ 20lex:r 2001 Elsevier Science B.V. All rightsreserved.

Keywords: Micromagnetism; Nanostructures; Magnetic switching; Thermal activation

1. Introduction

Patterned magnetic nanostructures (PMN) arebeing extensively investigated for applications inultra-high density storage, MRAM, field sensorsand logic gates [1–4]. Computational micromag-netic investigations and experimental measure-ments in PMN have shown that even for verysmall sizes in the order of 10lexðlex ¼ ðA=2pM2

s Þ1=2Þ

slight deviations from uniform magnetization arefound [1–6]. The magnetization configurationdepends on the PMN size and thickness as wellas on the applied field history. These smalldeviations are responsible for the so-called config-urational anisotropy [4–6].Magnetic nanosquares are adequate to study the

effect of configurational anisotropy because, ifmagnetocrystalline anisotropy is neglected, the

configurational anisotropy is the only one to takeinto account [5]. In a previous work the micro-magnetic switching with the field applied along theedge of the squares was analyzed in detail [6]. Twoswitching modes were found (quasi-coherent andincoherent) depending on the size and thickness ofthe nanosquare. The agreement with experimentalmeasurements was good although the smallernanosquares, probably due to not taking intoaccount the effect of temperature, showed compu-tational coercivities larger than the experimentalones [5,6].In this paper the effect of applied field direction

and temperature on the micromagnetic switchingof the nanosquares is studied.

2. Model and computational details

A standard micromagnetic finite differencemodel has been used. The nanosquare is discretizedin a 2D regular square mesh, the magnetization is

*Corresponding author. Tel.: +34-923-294-436; fax: +34-

923-294-584.

E-mail address: luis@usal.es (L. Torres).

0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 0 0 7 - 9

assumed uniform in each cell and it is allowed torotate in 3D. Different mesh sizes Dx ¼ Dy ¼0:25lex; 0:5lex; 1:0lex have been applied leading tothe same results so that no further extrapolationwas needed. The exchange energy is computedusing four-neighbor dot product representation [7]and the demagnetizing field is calculated using theexpressions by Newell [8]. Jacobi and conjugategradient micromagnetic solvers have been usedobtaining the same equilibrium states [9,10]. Somesimulations using a 3D mesh were also performed,up to eight cells were considered in the ‘z’ directionobtaining similar results than in 2D.The effect of the thermal activation is taken into

account by means of a stochastic three-dimen-sional zero-mean Gaussian field (xðtÞ) introducedin the Landau–Lifschitz–Gilbert equation [11,12]:

d %M

dt¼ �gð %M�½ %Heff þ %xðtÞ�Þ �

lgMs

%M�ð %M� %Heff Þ;

ð1Þ

where %Heff is the effective field, g is the electrongyromagnetic ratio, and l is the damping para-meter (l ¼ 0:1). The correlation function for thenoise is, according to the fluctuation–dissipationtheorem [13]:

/xaðtÞ xbðt0ÞS ¼

2lkBT

gMsVdabdðt � t0Þ; ð2Þ

where a; b ¼ x; y; z and V is the volume of thecomputational cell.The corresponding Langevin equation is solved

numerically using a first order Euler scheme andthe Ito interpretation [13]. For each temperature,200 simulations are performed and a statisticalanalysis is carried out. In some cases 2000realizations were calculated to test convergenceof the average value. A more complete explanationabout computational details can be found inprevious articles [1,6,10–12].

3. Results and discussion

3.1. Configurational anisotropy

In Fig. 1 the hysteresis loops computed fornanosquares of sizes a ¼ 20lex and 40lex with

thickness t ¼ 1:5lex in both cases are shown. Thetwo different switching modes can be observed.Large nanosquares present an intermediate stateleading to an increase in coercivity as previouslyanalyzed in Ref. [6]. Also an increase of coercivitywhen the field is applied closer to the diagonaldirection of the square is found (see Fig. 1).However, a closer inspection of the angulardependence of coercivity reveals a more compli-cated behavior for the large squares as can be seenin Fig. 2. For the squares of a ¼ 20lex the easy axisis along the diagonal of the square (f ¼ p=4) whilefor a ¼ 40lex a sudden increase in coercivity isfound at f ¼ 0:5 (see Fig. 2 and also hysteresisloops in Fig. 1).The underlying cause of such increase is the

modification of the micromagnetic states duringthe switching, i.e. the switching mode. In Fig. 2 theconfigurations found at H ¼ 0:95Hc (just beforeswitching) are shown for f ¼ 0:4 and 0:5: In both

H/Ms

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

M/M

s-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

M/M

s

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

φ = π /4

= 0

φ = 0.5

φ

Fig. 1. Hysteresis loops for different directions of the applied

field with respect to the square edge. From inner to outer loops,

f ¼ 0; p=8; p=4: Top: Squares of size a ¼ 20lex; thickness t ¼1:5lex: Bottom: a ¼ 40lex; t ¼ 1:5lex; for the most external loopf ¼ 0:5:

L. Torres et al. / Physica B 306 (2001) 216–220 217

cases the switching is incoherent but the inter-mediate state is different. For f ¼ 0:4 this statepresents an inner zone with Mxo0; My > 0 whilefor f ¼ 0:5 the inner zone has Mx > 0; Myo0: Forf ¼ 0:5 the intermediate state is more stable, dueto the 1801 domain walls more clearly developed,leading to an increase in the coercivity.A compilation of our results is shown in the

polar plot of Fig. 3. In this figure, the computedcoercivities Hc for squares with 10lexoao64lex;t ¼ 1:5lex and applied field angle f between0:0ofop=4 are presented (the rest of the figureis plotted by symmetry). These results are in goodagreement with experimental measurements [5]. Itcan be observed how, for small nanosquares(ao25lex), Fig. 3 presents a 4-fold symmetry. Forlarger squares, the complicated micromagneticbehavior with different intermediate configura-tions depending on size, thickness and fielddirection produces a more intricate coercivitydependence Hcða; t;fÞ:

3.2. Thermally activated switching

The thermally activated switching has beenstudied in two cases: a ¼ 20lex and 40lex: In bothcases t ¼ 1:5lex and the external field was along theedge of the square. The zero temperature equili-brium micromagnetic state for a field lower thanthe coercivity (Hc) is the initial point of thesimulation. Then the thermal noise term corre-sponding to the analyzed temperature is intro-duced and the stochastic LLG equation isresolved. If the applied field is close enough tothe zero temperature coercivity, thermal activatedswitching is observed in the computed timeinterval (10�7 s). The average value of theswitching time is stable after the 200 computa-tions [12].Magnetization configuration of nanosquares is

not always simple, or close to the uniformmagnetization, as demonstrated in the previoussection and on [6]. In fact, if the magnetization iscompletely uniform, there would not be config-urational anisotropy and then no hysteresis at allwould be observed (against experimental measure-ments [5]). However, although our system ischaracterized by many degrees of freedom and acomplicated free energy landscape, we will

Field Angle ( )

0.0 0.2 0.4 0.6 0.8

Hc/

Ms

(Oe

)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

φ

Fig. 2. Top: Micromagnetic states for a ¼ 40lex and H ¼0:95Hc: Bottom: Angular dependence of coercivity for a ¼20lex (K) and a ¼ 40lex (X). In all cases t ¼ 1:5lex:

Fig. 3. Polar density plot of Hc for squares with

10lexoao64lex; t ¼ 1:5lex: The radius gives the nanomagnet

size and the direction gives the applied field angle. White color

corresponds to the maximum value of HcD0:4Ms while black is

for Hc ¼ 0:

L. Torres et al. / Physica B 306 (2001) 216–220218

consider, as a working hypothesis, that it is acollection of bistable units separated by energybarriers with EbkT : This phenomenologicalhypothesis has proved to have far-reaching con-sequences in magnetic relaxation [14]. In thismodel the relaxation time is given by [14]

tðEÞ ¼ t0 expE

kT

� �; ð3Þ

tðEÞ is obtained in our computations from theprobability of not switching (PNS) distribution.PNS is calculated by integrating the distribution ofswitching times and is found to be well describedby a delayed stretched exponential [15]:

PNSðtÞ ¼ exp �t � t0t

� �b� �

; ð4Þ

where t0 is the delay time and b is the stretchedexponential parameter. An expression for theenergy barrier E for the different values of appliedfield is needed in order to apply Eq. (3). The simplecase of the energy barrier against magnetizationreversal (assuming coherent rotation) in a particlewith easy axis aligned to the field will be used [14]:

EðHÞ ¼ KV 1�H

HAN

� �2

; ð5Þ

where K and HAN are the anisotropy constant andfield and V is the volume. Since configurationalanisotropy is originated from magnetostatic inter-actions [5] we will assume that K is proportional toM2

s and we will identify HAN ¼ Hc:Figs. 4 and 5 show the dependence of the

relaxation time on energy barrier for all the casesstudied. Computations have been carried out byvarying the applied field at room temperature (’),and also changing T for H ¼ 0:98HcðnÞ and H ¼0:89HcðJÞ in both nanosquares with a ¼ 20lex and40lex:The first order fittings shown by the straight line

in Figs. 4 and 5 provide pre-exponential times t0 ¼9:4� 10�10ða ¼ 20lexÞ and 3:8� 10�10ða ¼ 40lexÞwhich agree with Refs. [12–15]. The relaxationtime for H ¼ 0 can also be estimated by extra-polating the fit. The results are tðH ¼ 0Þ ¼9� 104 s (around 1 day) for a ¼ 20lex whiletðH ¼ 0Þ � 1023 s for a ¼ 40lex: Consequently,for the small nanosquares superparamagnetic

effects are significant but for the large nano-squares the situation is clearly ferromagneticwhich is in agreement with experimental measure-ments [5].The results attained by changing the field and

by varying the temperature in the energyphase space do not coincide. According to theArrhenius model (3) they should coincide. Severalfactors are the cause of this discrepancy: (i)In some of the points of Figs. 4 and 5 EðHÞis in the order of kT (see x-axis scale) so thatexpression (3) where EðHÞbkT is assumed,

E/kT

0 50 100 150 200

Ln τ

(s)

e-24

e-23

e-22

e-21

e-20

e-19

e-18

e-17

e-16

e-15

Fig. 4. tðEÞ for a ¼ 20lex and t ¼ 1:5lex:Dependence on appliedfield at room temperature (’). Dependence on T for H ¼0:98HcðnÞ and for H ¼ 0:89HcðJÞ:

E/kT

0 100 200 300 400 500 600 700

Ln τ

(s)

e-25

e-24

e-23

e-22

e-21

e-20

e-19

e-18

e-17

e-16

e-15

e-14

Fig. 5. tðEÞ for a ¼ 40lex and t ¼ 1:5lex:Dependence on appliedfield at room temperature (’). Dependence on T for H ¼0:98HcðnÞ and for H ¼ 0:89HcðJÞ:

L. Torres et al. / Physica B 306 (2001) 216–220 219

is no longer valid. (ii) In the computations whereEðHÞbkT the simple modeling of the energylandscape by Eq. (5) cannot describe perfectly oursystem. It is not a trivial question how to compareboth effects since the calculation of tðEÞ impliesthe assumption of a known expression for theenergy landscape.

Acknowledgements

This work was partially supported by ProjectNo. PB98-0264 from Spanish government.

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