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Velocity of Transverse Domain Wall Motion AlongThin, Narrow Strips

D. G. Porter, M. J. DonahueNational Institute of Standards and Technology,

Gaithersburg, MD 20899

Abstract Micromagnetic simulation of domain wall mo-

tion in thin, narrow strips leads to a simplied analyticalmodel. The model accurately predicts the same domainwall velocity as full micromagnetic calculations, includingdependence on strip width, thickness, and magnitude of ap-plied eld pulse. Domain wall momentum and retrogradedomain wall motion are both observed and explained by theanalytical model.

I. Introduction

The effects of the shape and small dimensions on magne-todynamics are important so that devices can be producedto meet magnetization reversal design requirements. Inthis study we rst use micromagnetic simulation to exam-ine domain wall motion in thin, narrow strips of magneticmaterial. Inspired by the simulation results, we then pro-duce a simpler analytical model that agrees with the fullmicromagnetic simulation remarkably well. Both modelspredict a few unexpected behaviors.

II. Simulation

Using the OOMMF micromagnetic software package [1],we examined domain wall motion in a strip T = 5 nm thickand L = 1250 nm long. Our simulations included strips of width W ranging from 5 nm to 35 nm. Material param-eters approximating Permalloy were chosen, with satura-tion magnetization M S = 800 kA/m and exchange energycoefficient A = 13 pJ/m. Crystalline anisotropy was notincluded in the simulation of this soft material. Landau-Lifshitz magnetization dynamics are computed:

dmdt

=

1 + 2m H eff

1 + 2

m H eff m, (1)

where = 221 kHz/(A/m) is the gyromagnetic constant, is a dimensionless phenomenological damping parameter,m = M/M S is normalized magnetization, and H eff is theeffective eld representing the effect of all energies includedin the simulation.

From a prior simulation study [2] of static domain wallsin thin, narrow strips, we expect head-to-head domains tobe separated by a transverse domain wall as illustrated inFig. 1. We have used the same technique as in that priorstudy to suppress edge effects, to focus on the behavior of a domain wall down the length of a strip, far removed fromthe ends.

The initial transverse domain wall is established in theelement. A eld pulse is applied along the strip axis,

0 H x (t) = 0 H app (1 cos2f t ), 0 < t < 1 ns (2)

Hx

y

z

Fig. 1. Transverse domain wall in thin, narrow strip.

where f = 1 GHz, so that the 1 ns pulse includes one fullcosine period. Pulse magnitudes 0 H app from 1 to 10 mTwere applied. In response to each applied eld pulse, thetransverse domain wall moves in the positive x direction.After the pulse ends, the domain wall continues to move

with a momentum of its own. Simulations with = 0demonstrate that the domain wall motion is primarily aprecessional effect.

Because the transverse domain wall holds its shape andthe domains remain uniformly magnetized along the stripaxis, the wall velocity can be derived from the average mag-netization of the whole element,

v(t) =L2

d < m x (t) >dt

. (3)

When = 0, the domain wall momentum moves the wallat constant velocity. When > 0, the domain wall stopssome time after the pulse ends, although for large enoughapplied eld pulses, the wall velocity increases after thepulse ends before slowing to a stop.

Figure 2 graphs constant wall velocities observed when = 0 for several values of strip width W and pulse magni-tude 0 H app . For each W , there is a pulse magnitude thatmaximizes wall velocity. For wider strips, a lesser pulsemagnitude produces the maximum velocity and that max-imum velocity is greater.

Another set of simulations applied a constant eld ratherthan a pulse. The remarkable observation was that for largeenough applied eld, the domain wall velocity becomes neg-ative part of the time, leading to a retrograde motion of the

domain wall.III. Domain Wall Structure

As a rst step toward deriving an analytical model toexplain these simulation results, we examine the structureof the domain wall itself. First we note that within thescales under study, the magnetization can be considered tovary along only the x axis, M (x,y,z ) = M (x).

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domain wall velocity. The time rate of change of the tiltangle is

(1 + 2 )ddt

= | |(H app H D ). (14)

For the applied eld pulses, the total tilt angle achievedby the end of the pulse is proportional to the area underthe applied eld pulse. It is also clear that larger veloci-ties are expected as ( N z N y ) grows larger; that is, as thewidth-to-thickness ratio of the strip increases. These rela-tionships explain the features of the micromagnetic simula-tion results in Fig. 2. When the applied eld pulse createsa tilt angle greater than that which maximizes velocity,the model predicts that after the pulse, as damping de-creases , the wall velocity will actually increase before itdecreases and the wall comes to a stop, just as observed inmicromagnetic simulation.

This analytical model can also explain the response of domain walls to a constant applied eld. When the appliedeld is small enough, its tendency to increase the tilt angle will eventually be exactly balanced by the tendency of thedamping to push back to zero. Specically, for H app < max H D , a constant is reached and the wall moves atthe constant velocity determined by that tilt angle.

For larger H app , will continue to grow as precession

about the applied eld continues past the z axis ( = / 2).Once exceeds / 2, both precession and damping com-bine to accelerate the magnetization back into the plane.Though precession continues clockwise around H D , thetransverse direction of magnetization is reversed, so thatprecession moves the wall in the reverse direction. That is,the domain wall velocity becomes negative. As exceeds, the magnetization passes through the plane of the strip,and the direction of H D is reversed, causing the precessiondirection to reverse, yielding another reversal of wall di-rection. The same pattern repeats as the precession aboutthe applied eld continues for < < 2. The numberof cycles of domain wall direction reversal is exactly twice

the number of precession rotations about the applied eld.For H app > 1 max H D , we know from (13) that wallvelocity will not be negative, so for such large elds retro-grade motion will cease, though the domain wall tilt angle will continue to precess around the strip axis.

Figure 3 depicts how well the simple analytical modelsucceeds in predicting the same domain wall position as afunction of time as a full micromagnetic calculation. Thesolid line is the wall trajectory predicted by numeric in-tegration of (13). Wall width a ranges from 24 to 39 nmduring each precession cycle. Figure 4 is a direct illustra-tion of the retrograde motion of the domain wall in thepresence of a constant applied eld.

Equations (12) - (14) are substantially similar to thosederived in a previous study of domain wall dynamics innanowires [5]. However, in our work, we have derived thedependence of H D on W , T , and , while the previouswork assumed a simple uniaxial anisotropy form of the de-magnetization energy. If we made the same assumptions,our threshold for observing retrograde domain wall motionwould be H app < M S (N z N y )/ 2 which corresponds to

0

50

100

150

200

250

0 2 4 6 8 10

W a l

l p o s

i t i o n

( n m

)

Time (ns)

= 0 . 0

1

= 0.001

OOMMFTheory

Fig. 3. Comparison of the predictions of the analytical model withthe results computed by a full micromagnetic simulation. Response of a transverse domain wall to an applied eld ramped up to a constantvalue. 0 H app = 25 mT, W = 15 nm, = 0 .001 and = 0 .01.

3110 ps 3290 ps 3470 ps 3650 ps 3830 ps 4010 ps 4190 ps

Happ

Fig. 4. A sequence of magnetization patterns, illustrating retrogradedomain wall motion driven by a constant applied eld. At 3470 ps

the wall is tilted up. At 4190 ps the wall is tilted down.

the Walker eld predicted by the earlier work. It shouldbe noted that although the analytical work in Ref. 5 issound, the simulation results reported are invalid becausethe demagnetization elds are computed using a samplingtechnique rather than an averaging technique, a simulationerror we have fully described elsewhere [6].

References

[1] Michael J. Donahue and Donald Gene Porter, NISTIR 6376, Na-tional Institute of Standards and Technology, 100 Bureau Drive,Gaithersburg, MD 20899, 1999.

[2] Robert D. McMichael and Michael J. Donahue, IEEE Trans.Magn. , vol. 33, pp. 41674169, 1997.

[3] Andrew J. Newell, Wyn Williams, and David J. Dunlop, Journal of Geophysical Research - Solid Earth , vol. 98, no. B6, pp. 95519555, June 1993.

[4] Sushin Chikazumi, Physics of Magnetism , p. 348, Robert E.Krieger Publishing Company, 1964.

[5] A. Thiaville, J. M. Garc ia, and J. Miltat, Journal of Magnetism and Magnetic Materials , vol. 242, pp. 10611063, 2002.

[6] M. J. Donahue, D. G. Porter, R. D. McMichael, and J. Eicke, J.Appl. Phys. , vol. 87, no. 8, pp. 55205522, April 2000.