Harmonic oscillator model for current- and field-driven magnetic vortices

Benjamin Krüger,1 André Drews,2 Markus Bolte,2 Ulrich Merkt,2 Daniela Pfannkuche,1 and Guido Meier2

1I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany2Institut für Angewandte Physik und Zentrum für Mikrostrukturforschung, Universität Hamburg,

Jungiusstrasse 11, 20355 Hamburg, GermanyReceived 12 April 2007; published 26 December 2007

In experiments, the distinction between spin-torque and Oersted-field-driven magnetization dynamics is stillan open problem. Here, the gyroscopic motion of current- and field-driven magnetic vortices in small thin-filmelements is investigated by analytical calculations and by numerical simulations. It is found that for smallharmonic excitations, the vortex core performs an elliptical rotation around its equilibrium position. The globalphase of the rotation and the ratio between the semiaxes are determined by the frequency and the amplitude ofthe Oersted field and the spin torque.

DOI: 10.1103/PhysRevB.76.224426 PACS numbers: 75.60.Ch, 72.25.Ba

I. INTRODUCTION

Recently, it has been found that a spin-polarized currentflowing through a magnetic sample interacts with the mag-netization and exerts a torque on the local magnetization.1,2

A promising system for the investigation of the spin-torqueeffect is a vortex in a micro- or nanostructured magneticthin-film element. Vortices are formed when the in-planemagnetization curls around a center region. In this few na-nometer large center region,3 called the vortex core, the mag-netization turns out of plane to minimize the exchangeenergy.4 It is known that these vortices precess around theirequilibrium position when excited by magnetic fieldpulses,5,6 and it was shown that spin-polarized electric cur-rents can cause the same precession.7–10 The spatial restric-tion of the vortex core as well as its periodic motion aroundits ground state yield an especially accessible system forspace- and time-resolved measurements with scanning probeand time-integrative techniques, such as soft x-ray micros-copy or x-ray photoemission electron microscopy.5,6,11–13

Magnetic vortices also occur in vortex domain walls. Themotion of such walls has recently been investigatedintensively.14,15 Understanding the dynamics of confined vor-tices can give deeper insight in the mechanism of vortex-wallmotion.16 An in-plane Oersted field accompanying the cur-rent flow also influences the motion of the vortex core. Forthe interpretation of experimental data, it is crucial to distin-guish between the influence of the spin torque and of theOersted field.17

In this paper, we investigate the current- and field-drivengyroscopic motion of magnetic vortices in square thin-filmelements of size l and thickness t, as shown in Fig. 1, andpresent a method to distinguish between spin torque andOersted-field-driven magnetization dynamics.

II. ANALYTICAL CALCULATIONS

In the presence of a spin-polarized current, the time evo-lution of the magnetization is given by the extended Landau-Lifshitz-Gilbert equation

dM

dt= − M H eff +

MsM

dM

dt

−bj

Ms2 M M j · M −

bj

MsM j · M ,

1

with the coupling constant bj = PB / eMs1+2 betweenthe current and the magnetization, where P is the spin polar-ization, MS the saturation magnetization, and the degree ofnonadiabaticity.18 If the vortex keeps its static structure, itsmotion with the velocity v can be described using the Thieleequation.19 This equation was expanded by Thiaville et al.20

to include the action of a spin-polarized current flowing inthe sample,

F + G v + bjj + Dv + bjj = 0. 2

Denoting the out-of-plane angle of the magnetization with and the angle of the in-plane magnetization with , the forcedue to the external and the stray field is

F = − 0 dV

+

H sz · M . 3

The gyrovector

(a)

l

X

(b)

FIG. 1. a Scheme of the magnetization in a square magneticthin-film element with a vortex that is deflected to the right. bMagnetization of a vortex in its static ground state. The heightdenotes the z component, while the gray scale corresponds to thedirection of the in-plane magnetization.

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G = −Ms0

dV sin = −

2Ms0tp

ez = G0ez

4

indicates the axis of precession and points out of plane. Thedissipation tensor is given by

D = −Ms0

dV + sin2 . 5

It is diagonal with

Dxx = Dyy = D0 −Ms0t lnl/a

, Dzz = 0. 6

The constant a is the lower bound of the integration. It is inthe order of magnitude of the radius of the vortexcore.3,16,21,22 A polarization p of +1 −1 denotes that themagnetization in the vortex core is parallel antiparallel tothe z axis. The velocity of the vortex core is in plane and,hence, perpendicular to the gyrovector. Thus, Eq. 2 can berewritten as

G F − G02v + bjj + D0G v + bjj = 0. 7

By calculating G v from Eq. 7 and inserting the result inEq. 2,

G02 + D0

22v = G F − D0F − G02 + D0

2bjj

+ bjD0G j − , 8

we can derive the velocity of the vortex core. As for anysquare-symmetric confining potential, the stray-field energyfor small deflections can be modeled as a parabolic potential,

Es =1

2mr

2X2 + Y2 , 9

with the coordinates X and Y of the vortex core see Fig.1a.

In the following, a spatially homogeneous current in the xdirection is investigated. Due to possible inhomogeneities inreal samples, the current flow may vary in the out-of-planedirection. This results in an in-plane Oersted field, which is

perpendicular to the direction of the current flow. In the fol-lowing, this Oersted field is accounted for by a homogeneousmagnetic field in the y direction. Both driving forces maydepend on time. To estimate the Zeeman energy due to theOersted field H, the magnetization pattern is divided intofour triangles see Fig. 1a. Assuming that the magnetiza-tion is uniform in each of these triangles, the total Zeemanenergy is given by

Ez =0MsHltc

2 l

2+ X − l

2− X , 10

with the chirality c of the vortex. A chirality of +1 −1denotes a counterclockwise clockwise curling of the mag-netization around the vortex core. We will see that thissimple approximation describes the field-induced vortex mo-tion sufficiently well. In this case, the force is given by

F = − Es + Ez = − 0MsHltcex − mr2Xex − mr

2Yey .

11

Inserting Eq. 11 into Eq. 8 yields the equation of motionfor the vortex. In the absence of current and field, the excitedvortex performs an exponentially damped spiral rotationaround its equilibrium position with its free frequency

= −pG0mr

2

G02 + D0

22 12

and damping constant

= −D0mr

2

G02 + D0

22 . 13

From Eqs. 12 and 13, one easily obtains that

D0 =pG0

. 14

For thin-film systems t / l0.1, the resonance frequencyof a vortex is proportional to the inverse lateral dimension1 / l.23 Here, we obtain from Eq. 14 that the damping con-stant also has a characteristic length dependence, lnl /a / l. Substituting D0 using Eq. 14, the equationof motion of the vortex can be written as

X

Y = − − p

p − X

Y +

p

2 + 2

0MsHltc

G0− bjj −

2

2 + 2

−

bjj

−2

2 + 2

0MsHltc

G0+

p

2 + 2

−

bjj . 15

In the following, we assume harmonic excitations; i.e., the magnetic field and the electrical current are of the form Ht=H0ei t and jt= j0ei t. The magnetic Oersted field and the electrical current are in phase. Assuming that the squared Gilbertdamping is small 21, the damping constant of the vortex is small compared to its frequency 22. Then, Eq. 15 hasthe solution

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X

Y = A i

pe−t+it + B− i

pe−t−it −

ei t

2 + i + 2H +

j +

H + j i

jp − H +

−

j i p , 16

with H=H0lc / 2 and j=bjj0. The first two terms withprefactors A and B are exponentially damped and depend onthe starting configuration. Independent of the source of exci-tation, i.e., field or current, the sense of rotation of the vortexis given by its polarization; i.e., p= +1 p=−1 denotes acounterclockwise clockwise rotation of the vortex core.Changing the sign of the chirality has the same effect asturning the magnetic field by 180°.10 Similar to the motion ofmagnetic domain walls in thin nanowires,24 the vortex isdriven by the current and the magnetic field as well as bytheir time derivatives.

At resonance, the amplitude of the vortex core displace-ment in x and y directions is the same, and the vortex per-forms a circular rotation. A vortex that is excited with anonresonant frequency has an elliptic trajectory. The ratiobetween the semiaxes is given by the ratio between the fre-quency of the excitation and the resonance frequency.25

III. NUMERICAL CALCULATIONS

To test the applicability of the approximations leading tothe analytical result in Eq. 16, we performed micromag-netic simulations for magnetic thin-film elements with differ-ent lengths, thicknesses, polarizations, and chiralities. Thematerial parameters of Permalloy are used, i.e., an exchangeconstant of A=1310−12 J /m and a saturation magnetiza-tion of Ms=8105 A /m. For the Gilbert damping, we use avalue of =0.01, which is in the regime, as found by recentexperiments.26–28 The degree of nonadiabaticity is chosento be equal to .14,29

For the micromagnetic simulations, we extended theimplementation of the Landau-Lifshitz-Gilbert equation inthe Object Oriented Micromagnetic Framework OOMMF bythe additional current-dependent terms of Eq. 1.24,30 Thesimulation cells are 2 nm in x and y directions, which is wellbelow the exchange length of Permalloy. One cell of thick-ness t was used in the z direction. As in the analytical model,we substitute the Oersted field by a homogeneous magneticfield.

At first, the four ground states with c±1 and p±1 arecalculated for each l and t. The ground states are then excitedby a short current pulse. The free frequency and the damp-ing constant are obtained by fitting the subsequent freeoscillation with the first two terms in Eq. 16. Results arepresented in Fig. 2 and exhibit a good agreement between theanalytical model and the micromagnetic simulations.31

For the driven oscillation, we choose a magnetic film el-ement with length l=200 nm and thickness t=20 nm. Thissystem size allows for reasonable computing time. The mag-

netization is excited with harmonic currents with a spin-polarized current density jP=2.51010 A /m2 in the x direc-tion. The field excitation was performed with a harmonicfield of H=250 A /m in the y direction. The amplitudes andthe phases of the oscillation in x and y directions of a vortexwith positive polarization and chirality are depicted in Fig. 3.In the numerical calculations, the position of the vortex isdefined by the maximum amplitude of the out-of-plane mag-netization. To determine this maximum, the simulation cellwith maximum out-of-plane magnetization and its nextneighbors are interpolated with a polynomial of second or-der. In the current-driven oscillation, an excellent accordancebetween analytical calculations and numerical simulations isfound. In the field-driven case, the amplitudes of the analyti-cal solution are smaller than the amplitudes obtained fromthe micromagnetic simulations. These deviations are causedby the differences between the approximate magnetizationdepicted in Fig. 1 and the exact state. The phases betweenthe maximum of the exciting magnetic field and the maxi-mum deflection in x and y directions agree very well. Vorti-ces with other polarization and chirality not shown yieldthe same accordance.9

IV. DISCRIMINATION BETWEEN OERSTED FIELDAND SPIN TORQUE

From Eq. 16, one can see that the current- and field-induced forces on the vortex are of the same form. For ex-periments, it is important to separate the Oersted-field-drivenand the spin-torque-driven cases. We describe the ratio be-

0

2

4

6

8

10

200 300 400 5000

20

40

60

80

100

120

ω(G

Hz)

Γ(M

Hz)

l (nm)

ω10 nmω20 nmω30 nmΓ10 nmΓ20 nmΓ30 nm

FIG. 2. Color online Dependence of the free frequency andthe damping constant on the length l for various thicknesses t ofthe system. The symbols denote numerical results, while the linesare fits with the analytical results.

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tween the field- and current-induced forces on the vortex bytan =FOe /Fst; i.e., mixing angles of =0 and = ± /2 de-note the fully spin-torque-driven and the fully field-drivencase, respectively. There are two possibilities to determinethe ratio of both forces. On the one hand, for nonresonantexcitations the trajectory of the vortex core is elliptical, asillustrated in Fig. 4. According to Eq. 16, the direction ofthe major axis of the ellipse is determined by . The ampli-tude of the vortex motion decreases very fast when the exci-tation frequency deviates from resonance; i.e., for experi-mental observation, very high current densities withfrequencies close to resonance are needed. On the other

hand, the excitation mechanisms can be distinguished usingthe phase of the vortex deflection.17 As indicated by the dotsin Fig. 4, the position of the vortex at the maximum currentdepends on , which can be determined from Eq. 16. Thelatter method is also applicable with excitations at resonancefrequency.

V. CONCLUSION

In conclusion, we derived an analytical expression for thecurrent- and field-driven trajectories of a vortex in thin-filmelements. The analytical result is compared to micromagneticsimulations. The accordance between both approaches isvery good. The analytical expression enables us to determinethe ratio between spin-torque-driven and Oersted-field-driven motions.

ACKNOWLEDGMENTS

Financial support by the Deutsche Forschungsgemein-schaft via SFB 668 “Magnetismus vom Einzelatom zurNanostruktur” and via Graduiertenkolleg 1286 “FunctionalMetal-Semiconductor Hybrid Systems” is gratefully ac-knowledged.

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FIG. 3. Color online Amplitude of the a current-driven andb field-driven vortex oscillations in the x direction solid red line,pluses and the y direction dashed blue line, crosses for a spin-polarized current density of jP=2.51010 A /m2 and a field of H=250 A /m. The insets show the phases between the maximum ofthe applied current or field and the core displacement in the x di-rection solid red line, pluses and the y direction dashed blue line,crosses. The symbols denote numerical results, while the lines arederived from the analytical expression in Eq. 16.

180180180180

240240240240

300300300300

360360360360

10101010-2-2-2-2 101010100000 101010102222 101010104444 101010106666

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γγγγHHHH0000lc/(2lc/(2lc/(2lc/(2ππππbbbbjjjjjjjj0000))))

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0000

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Y(n

m)

Y(n

m)

Y(n

m)

Y(n

m)

X (nm)X (nm)X (nm)X (nm)

FIG. 4. Color online Analytically calculated phase between themaximum current or magnetic field and the x deflection of the vor-tex core for a 20020020 nm3 permalloy square excited with afrequency of =4.8 GHz above the resonance frequency of =4.4 GHz. The inset shows a section of the sample with the simu-lated trajectories of the vortex core excited with i solid red linea spin-polarized current density with an amplitude of jP=1.21011 A /m2 and ii dashed blue line a magnetic field with anamplitude of H=1000 A /m. Points denote the position of the vor-tex at the maximum current i and the magnetic field ii,respectively.

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