Two circular-rotational eigenmodes and their giant resonance asymmetry in vortex gyrotropic motions in soft magnetic nanodots

(1)

Two circular-rotational eigenmodes and their giant resonance asymmetry in vortex gyrotropic motions in soft magnetic nanodots

Ki-Suk Lee and Sang-Koog Kim

*

Research Center for Spin Dynamics and Spin-Wave Devices, Seoul National University, Seoul 151-744, Republic of Korea and Nanospinics Laboratory, Department of Materials Science and Engineering, College of Engineering, Seoul National

University, Seoul 151-744, Republic of Korea 共Received 9 April 2008; published 3 July 2008兲

We found, by micromagnetic numerical and analytical calculations, that the clockwise共CW兲and counter- clockwise共CCW兲circular-rotational motions of a magnetic vortex core in a soft magnetic circular nanodot are the elementary eigenmodes existing in the gyrotropic motion with respect to the corresponding CW and CCW circular-rotational-field eigenbasis. The oppositely rotating eigenmodes show a giant asymmetric resonance behavior, i.e., for the up-core orientation the CCW eigenmode shows a strong resonance at the field frequency equal to the vortex eigenfrequency, but the other CW eigenmode shows nonresonance. This asymmetric resonace effect is reversed by changing the vortex polarization. The orbital radius amplitudes and phases of the two circular eigenmodes vary with the polarization and chirality of the given vortex state as well as the field frequency. The overall linear-regime steady-state vortex gyrotropic motions driven by arbitrary polarized oscillating in-plane magnetic field in the linear regime can be perfectly understood according to the superposition of the two circular eigenmodes.

DOI:10.1103/PhysRevB.78.014405 PACS number共s兲: 75.40.Gb, 75.40.Mg, 75.75.⫹a INTRODUCTION

The magnetic vortex consists of the in-plane curling mag- netizations共M’s兲and the out-of-planeM’s at the center re- gion, the so-called vortex core共VC兲,^1–3which is known to be a ground state in soft magnetic elements of micron size or smaller. When, in such a confined system, magnetic fields共or currents兲with harmonic oscillations or pulses are applied to the vortex, its VC rotates around its equilbrium position at a characteristic eigenfrequency, ␻D/2␲, typically, of several hundred megahertz.^4–10 The responsible force is the gyroforce exerting on the VC, which is in balance with the re- storing force due mainly to the long-range dipole-dopole in- teraction dominating in a confined magnetic element.⁵Such vortex excitation is known to be the translation mode or gyrotropic motion of a VC in the dot plane, and the rotation sense of such gyrotropic motion is determined by the polar- izationp of a given vortex, which is represented by the VC Morientation关p= + 1共−1兲for up共down兲-core orientation兴. If the angular frequency ␻Hof an oscillating field共or current兲 is close to the␻D,⁵the VC motion is resonantly excited.^8–12 Recently, this resonantly excited VC gyrotropic motions under harmonic oscillating field,^8,9 ac current,¹⁰ or both of them¹¹have been intensively studied. Moreover, the resonant VC motion has attracted much attention on account of its related ultrafast VC switching applicable to information storage.^13–18In addition, the variation of the circular and elliptical shapes of the orbital trajectories of the on- and off- resonance VC motions driven by linearly polarized oscillating magnetic fields was observed.^7,8However, the underlying physics has not been clearly understood since the true eigenmodes of the vortex gyrotropic motions remain incompletely understood. In this paper, having considered the results of the present theoretical and numerical simulation studies, we posit that these VC motions can be clearly understood by considering them to be the superposition of counterclockwise 共CCW兲 and clockwise 共CW兲 circular-rotational eigenmodes

and also by considering their aysmmetric resonance effect.

The orbital-trajectory radius ampltidues and phases of the oppositely rotating eigenmodes’ VC motions are presented as a fucntion of the frequecy of oscillating fields according to the different vortex polarizations and chiralities.

MICROMAGNETIC SIMULATIONS

In the present study, we employed micromagnetic numerical simulations of vortex M dynamics by using theOOMMF

code¹⁹ that utilizes the Landau–Lifshitz–Gilbert equation of motion⳵M/⳵^t^{= −}␥共M⫻Hef f兲+␣/兩M兩共M⫻⳵^M/⳵t兲共Ref.20兲 with the phenomenological damping constant ␣, the gyro- magnetic ratio ␥, and the effective field H_{ef f}. Also, we carried out analytical calculations of the linear-regime VC motions,⁸ which are based on a linearized Thiele’s equation of motion.²¹ As a model system, we chose a Permalloy 共Py兲 nanodot of 2R= 300 nm diameter and L= 10 nm thickness 关Fig. 1共a兲兴. For the given Py material and circular dot geometry, a single magnetic vortex is present with either p= + 1 or −1 and with either C= + 1 or −1, where C= + 1 共−1兲 is the chirality, which is represented by the CCW共CW兲in-planeM’s around the VC. The vortex eigenfrequency and static annihilation field 共Ref. 22兲 were esti- mated to be ␻D/2␲= 330 MHz and HA= 500 Oe, respectively. We considered the application of either linearly polarized oscillating magnetic fields applied along the y axis, H_lin=H₀sin共␻Ht兲yˆ , or circularly polarized oscillating fields of either CCW or CW rotation sense in the dot plane, such that H_CCW=¹₂关H0cos共␻Ht兲xˆ +H₀sin共␻Ht兲yˆ兴 and H_CW=¹₂关−H₀cos共␻Ht兲xˆ +H₀sin共␻Ht兲yˆ兴 with the oscillating field amplitude H₀. We used relatively low field amplitudes, H₀/HA= 0.1 and 0.2, in investigations of only linear-regime vortex gyrotropic motions, so as to exclude the nonlinear effect^8,9driven by high-strength fields. We also chose a frequency range from ␻H/2␲= 100 to 825 MHz, including

␻D/2␲= 330 MHz of interest, in order to investigate both the

(2)

on- and off-resonance vortex gyrotropic motions.

First, we have to note that the orbital trajectory of the H_lin-driven steady-state VC motion for the resonance case

␻H=␻D is exactly circular in shape and relatively large in radius amplitude, even for a very weak field strength, e.g., H₀/H_A= 0.01共Ref. 23兲关see Figs.1共b兲and1共c兲and Supple- mentary Movie 1 共Ref. 24兲兴. For p= + 1, the VC motion is CCW and for p= −1, CW. In the case of off-resonance共␻H

⫽␻D兲 motion, the orbits, however, become elongated along the axis perpendicular to and parallel with the H_linaxis for

␻H⬍␻Dand␻H⬎␻D, respectively, and the degree of their elongations increases with the increasing magnitude of 兩␻H

−␻D兩关Fig.1共d兲兴.^8,11Although such behaviors have been re- ported from previous numerical and analytical studies,^7,8,11 the underlying physics has not been understood yet.

For clear understanding, it is thus necessary to find out the elementary eigenmodes of the gyrotropic motions. The application of the H_linis equivalent to the application of both the pure circular fields of H_CCW and H_CW simultaneously, with the sameH₀and with equal␻H关Fig.2共a兲兴, becauseH_lin can, in principle, be decomposed into the H_CCW and H_CW components.¹⁵ The relative phase between H_CCW and H_CW determines the axis of the linearly polarized oscillating magnetic field. Thus, the observed circular or elliptical shape of the orbital trajectories 关Fig.1共d兲兴 can be interpreted according to the superposition of the CCW and CWcirculareigen- motions in circular dots with respect to the H_CCWandH_CW eigenbasis, as seen in Fig.2共b兲关see Supplementary Movie 2 共Ref. 24兲兴. To verify this by micromagnetic simulations, in Fig.2共c兲, we plotted the individual orbital trajectories of the VC motions under the individuals of theH_CCWandH_CW, as well as the H_lin for a specific case of ␻H/␻D= 2.5, H₀/HA

= 0.2, and 共p,C兲=共+1 , + 1兲. The elliptical orbital trajectory by H_lin is in excellent agreement with that obtained by the

98.5

99.2

100

(a)

_H_lin

t

L=10 nm

2R= 300nm x

z y

(b)

t= 97.0 ns

97.7

(c)

ω ω_H/ _D=1.0

-60 -30 0 30 60 60

30 0 -30 -60

x(nm)

y(nm)

0 A= 0.01 H H

/ _D 1.0 ω ω_H =

0 A= 0.01 H H

97.0

98.5 99.2 97.7 (ns)

D 0.3

ω ωH = 0.5 1.0 1.5 2.5

-60 0 60 0 60 0 60 0 60 0 60

60 0 -60

x(nm) 60

0 -60y(nm)

0 A=0.1

H H 0.1 0.1 0.1

0.2

0.005

0.01

y(nm)

0.2 0.2 0.2

(d)

FIG. 1.共Color online兲共a兲Geometry and dimension of the model Py nanodot with a vortex-state Mdistribution withp= + 1 and C

= + 1 at equilibrium under no field.共b兲Perspective view of the local M distributions at the indicated times and共c兲 the circular orbital trajectory of the VC motion in the steady state, which is driven by H_lin. The color and height display the local in-planeMorientation, as indicated by the color wheel, as well as the out-of-plane M components, respectively. The dots in共c兲indicate the VC positions at the indicated times.共d兲Orbital trajectories of the steady-state VC motions共t⬎90 ns兲in response to theH_linwith different␻Hvalues as noted forH₀/H_A= 0.1 and 0.2.

δCCW^H

H_CCW H_CW

ω

_H

ω

_H

(a)

δCW^H

δCCW^F

FCCW

δCW^F

FCW XCW

ω

_H

ω

_H

XCW

XCCW

(ω ωH D,H H0/ A)=(2.5,0.2) 30

15 0 -15 -30

y(nm)

-30 -15 0 15 30 x(nm)

ω

H ωH H_CCW H_CW

H_lin (^{p C}^, )⁼^(1,1)

(b) (c)

XCCW

FIG. 2. 共Color online兲共a兲 Graphical illustrations of the CCW and CW circular eigenmodes and the corresponding H_CCW, H_CW, F_CCW, andF_CW, as well as the definitions of the orbital-radius am- plitude兩X_CCW,CW兩and phase␦CCW,CW

H of the linear-regime steady- state circular VC motions.共b兲Graphical illustration of the superposition of the CCW and CW eigenmodes, yielding an overall elliptical orbit. The black-colored ellipse 共solid line兲 indicates the resultant superposition of the individual CCW 关blue-colored 共dashed兲 line兴 and CW 关red-colored 共dotted兲 line兴 eigenmodes, which is equivalent to the elliptical VC motion driven by the H_lin 共black arrow兲. The VC positions of the CCW and CW eigenmotions at a certain time are indicated by the blue-colored square and red- colored diamond, respectively, and their vector sum is indicated by the black-colored dot on the ellipse.共c兲 Micromagnetic simulation results on the VC trajectories of the CCW共blue open squares兲and CW 共red open diamonds兲 eigenmotions driven by the individual H_CCW andH_CW, respectively, as well as the VC trajectory共black open circles兲 driven by the H_lin. The elliptical orbit 共black solid circles兲results from the superposition of the individual CCW and CW eigenmotions.

(3)

superposition of the CCW and CW circular-rotational motions.

ANALYTICAL CALCULATIONS Two circular-rotational eigenmodes

In order to theoretically verify and understand such dynamic responses of a vortex to any polarized oscillating field, now we introduce a useful quantity of the dynamic susceptibility tensor ␹^ˆX defined by X=␹^ˆXH.¹² For convenience, first let us define the orbital radius 兩XCCW,CW兩 and phase

␦CCW,CW

H of the VC positionXin the dot共x-y兲plane for the CCW and CW circular motions关Figs.2共a兲 and2共b兲兴. Here, we exclude nonsteady transient-state motions that have yet to reach the steady state, as well as the nonlinear effect.^8,9 To analytically calculate兩XCCW,CW兩and␦CCW,CW

H , we employed the linearized Thiele’s equation²¹ of motion, −G⫻X˙ −DˆX˙ +⳵W共X,t兲/⳵X= 0, with the gyrovector G= −Gzˆ, and the damping tensor Dˆ=DIˆ with the identity matrix Iˆ and the damping constant D.^5,12 The potential energy function is given byW共X,t兲=W共0兲+␬兩X兩²/2 +WH. The first termW共0兲 is the potential energy for a VC at its initial positionX= 0, and the second term is dominated by the exchange and mag- netostatic energies for the VC shift from X= 0 and for the given stiffness coefficient ␬. The last one, WH= −␮共zˆ

⫻H兲·X, is the Zeeman energy term due to a driving force, where␮⁼␲^RLMs␰^{C, with}␰= 2/3 for the “side-charge-free”

model.^5,8,12 For any polarized oscillating field H

=H₀exp共−i␻Ht兲, i.e., linearly, circularly polarized oscillating field, or their mixed polarizations, the general solution is written as X=X^trans+X^steady,¹¹ where the terms X^trans and X^steady correspond to the VC motions in the transient and field-driven steady states, respectively, given by X^trans=

−X₀exp共−i␻Dt兲exp共D␻Dt/兩G兩兲 with ␻D=␬兩G兩/共G²+D²兲共Ref. 17兲 and X^steady=X₀exp共−i␻Ht兲. For tⰇ兩G/共D␻D兲兩共t Ⰷ23 ns in this case兲, the VC motions can be represented by X⯝X^steady=X₀exp共−i␻Ht兲. For the linear polarization basis, the susceptibility tensor is X_0,L=␹^ˆX,LH_0,L with X_0,L=X_0xxˆ +X_0yyˆ andH_0,L=H_0xxˆ +H_0yyˆ , where

␹^ˆX,L共␻H兲=

冋

^␹␹^xxyx ^␹␹^xyyy

册

= ␮

共i␻HD+␬兲²−共␻HG兲²

冋

ⁱ␻⁻Hⁱ^␻D^H+^G␬ ⁻ⁱ−^␻i^H␻^DHG⁻^␬

册

^.

共1兲 By the diagonalization of␹^ˆX,X_0,L=␹^ˆX,LH_0,Lbecomes X_0,cir

=␹^ˆX,cirH_0,cirwith respect to the circular polarization eigenbasis, where H_0,cir=H_0,CCWeˆ_CCW+H_0,CWeˆ_CW and X_0,cir

=X_0,CCWeˆ_CCW+X_0,CWeˆ_CW with the circular eigenvectors of eˆ_CCW=_冑¹₂共xˆ +iyˆ兲 andeˆ_CW=_冑¹₂共xˆ −iyˆ兲.²⁵ X_0,cir=␹^ˆX,cirH_0,cir can also be rewritten, in matrix form, as

冉

^XX^0,CCW_0,CW

冊

⁼

冉

^␹^CCW⁰ ␹CW⁰

冊冉

^H^H^0,CCW0,CW

冊

^,

where ␹CCW共␻H兲=i␮/关␻HG−共i␻HD+␬兲兴 and ␹CW共␻H兲

=i␮/关␻HG+共i␻HD+␬兲兴. The term ␹CCW,CW can be sepa-

rated into the magnitude兩␹CCW,CW兩and phase␦CCW,CW

H by the

relation of ␹CCW,CW=兩␹CCW,CW兩e⁻ⁱ^␦^CCW,CW^H , where 兩␹CCW兩=兩␮兩/

冑

共G²+D²兲共␻H−p␻D兲²+␬²D²/共G²+D²兲, 兩␹CW兩=兩␮兩/

冑

共G²+D²兲共␻H+p␻D兲²+␬²^D²/共G²+D²兲, 共2a兲 and

␦CCW

H = − tan⁻¹关共␬⁻^p␻H兩G兩兲/共␻HD兲兴+␲ 2共1 −C兲,

␦CW

H = − tan⁻¹关共␬+p␻H兩G兩兲/共␻HD兲兴+␲

2共1 +C兲. 共2b兲 Both兩␹CCW,CW兩and␦CCW,CW

H are functions of␻H, indicating their ␻H variations.兩␹CCW,CW兩 also depend on the sign of p but independently of the sign ofC, whereas␦CCW,CW

H depend

on the sign of both p andC.

Next, the numerical calculations of the analytically de- rived equations of 兩␹CCW,CW兩 and ␦CCW,CW

H were plotted versus ␻H/␻D for four different cases of 共p,C兲 in Fig. 3, which are in excellent agreements with the simulation results 共circle symbols兲 obtained from the relation of 兩␹CCW,CW兩

=兩XCCW,CW兩/兩HCCW,CW兩. There exist strong resonances for both 兩␹CCW,CW兩 and ␦CCW,CW

H at ␻H/␻D= 1 and the resonance effects are asymmetric between the CCW and CW circular motions. Only one, either the CCW or CW motion, shows a resonance behavior, the other showing nonresonance. This asymmetric resonance is caused by the gyroforce 共G⫻X˙兲, which is essential for vortex gyrotropic motion. The presence of the gyroforce leads to a broken time-reversal symmetry, in turn, yielding a splitting of the degeneracy of the CCW and CW eigenmodes. There- fore, the vortex gyrotropic motion shows such asymmetric resonance, responding differently to the orthogo- nal CCW and CW circular fields. The asymmetric resonance effect is reversed by changing from p= + 1 to −1,

(p ,C)=(1,1) 20

10 (nm/Oe) 0

(1,-1) (-1,1)

CCW CW

1.0 0.5 0 -0.5 -1.0

(-1,-1)

(radians)

1.0 0.5 0 -0.5 (radians)-1.0

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

ω ωH D

0 1 2 30 1 2 30 1 2 30 1 2 3

CCW,CWδF CCW,CWδF CCW,CWδH CCW,CWδH CCW,CWχCCW,CWχ ππππ

0 1 2 30 1 2 30 1 2 3

0 1 2 3

FIG. 3. 共Color online兲 Numerical calculations of the analytical equations共solid and dashed lines兲of兩␹CCW,CW共␻H兲兩,␦CCW,CW

F 共␻H兲, and␦CCW,CWH 共␻H兲 compared to micromagnetic simulations共shaded circles兲 for the CCW and CW eigenmodes in response to H_CCW 关blue 共solid兲兴 and H_CW 关red 共dashed兲兴, respectively, for the given polarization and chirality共p,C兲, as noted.

(4)

i.e., the mode showing the resonance is switched by p.

This can also be simply confirmed by the on-resonance case equations, 兩␹CCW共␻D兲兩=共兩␮兩/␬兲

冑

_共G²+D²兲/D² and 兩␹CW共␻D兲兩=共兩␮兩/␬兲

冑

共G²+D²兲/共4G²+D²兲, which yield 兩␹CCW兩Ⰷ兩␹CW兩 for p= + 1. Alternatively, for p= −1, 兩␹CCW共␻D兲兩=共兩␮兩/␬兲

冑

共G²+D²兲/共4G²+D²兲 and 兩␹CW共␻D兲兩

=共兩␮兩/␬兲

冑

_共G²+D²兲/D², thus yielding 兩␹CCW兩Ⰶ兩␹CW兩. As a result, forp= + 1, the CCW rotational eigenmode has a large motion amplitude, but the CW rotational eigenmode has an extremely small motion amplitude, and which effect is reversed for p= −1.

On the other hand, the ␦CCW,CW

H variations with ␻H are remarkable, owing to itsCas well asp dependences. TheC dependence of␦CCW,CW

H originates from theCdependence of applied driving forces,²⁶ such that F_CCW^H =␮共zˆ⫻H_CCW兲

=兩␮兩e^−iC^␲/2H_CCWand F_CW^H =␮共zˆ⫻H_CW兲=兩␮兩e^iC^␲/2H_CW. The p dependence, meanwhile, is due to the already-mentioned asymmetric resonance effect between 兩␹CCW兩 and兩␹CW兩. As in the dynamic response of a linear oscillator to a harmonic oscillating force, for␻H⬍␻D, the phase difference␦CCW,CW

F

between the VC positionXandF_CCW,CW^H is always zero共i.e., in phase兲and independent ofCandp. However, for the other case, ␻H⬎␻D, ␦CCW

F = −␲共1 +p兲/2, and ␦CW

F = −␲共1 −p兲/2 depend only onp. Only for the case of the eigenmode showing resonance, ␦CCW,CW

F changes from 0 共in phase兲 at ␻H

⬍␻Dto −␲ 共out of phase兲 at ␻H⬎␻D 共the second row of Fig.3兲. In addition, it is worthwhile noting that such phase change with ␻H occurs only for the eigenmode showing resonance; it does not occur for the other opposite eigenmode. Thus, the complex changes of ␦CCW,CW

H with ␻H, which also depend on p and C, can be interpreted simply according to␦CCW

H =␦CCW

F +C␲/2 and␦CW H =␦CW

F −C␲/2共the third row of Fig.3兲.

Elliptical shape and orienation of orbital trajectories According to the above-mentioned relations of兩␹CCW,CW兩 and␦CCW,CW

H with␻H/␻Das well aspandC, the CCW and CW circular-rotational motions can be expressed by the simple mathematical expressions of X_CCW=␹CCWH_CCW and X_CW=␹CWH_CW, where H_CCW=H_0,CCWexp共−i␻Ht兲eˆ_CCW and H_CW=H_0,CWexp共−i␻Ht兲eˆ_CW. Consequently, the superposition of the individual circular eigenmodes thus provides a resultant VC gyrotropic motion driven by an H_lin. For this concrete verification, the individual orbital trajectories of the X_CCW and X_CW and their superposition were also obtained from the analytical calculations, for example, for ␻H/␻D

= 0.3 and 2.5 withH₀/H_A= 0.2 and for all the cases of共p,C兲. In Fig. 4, the analytical calculations共solid lines兲 are in excellent agreements with the simulation results共open circles兲.

From the relations between the orbital trajectories of the two circular-rotational eigenmodes and their superposition, as shown in Fig. 4, we have found that the degree of elongations of the elliptical trajectories 共black-colored lines兲for

␻H/␻D= 0.3 and 2.5 is determined by the relative difference between兩XCCW兩and兩XCW兩 and is independent of共p,C兲. On the contrary, the elongation axes of the elliptical orbital trajectories corresponding to the motions under anH_linare perpendicular to the H_linfor ␻H/␻D= 0.3 and parallel with the

H_lin for ␻H/␻D= 2.5. This is associated with the relative phase of␦CCW

H and␦CW

H , i.e.,兩␦CCW H −␦CW

H 兩=␲ 共out of phase兲 for ␻H/␻D= 0.3⬍1 and 兩␦CCW

H −␦CW

H 兩= 0 共in phase兲 for

␻H/␻D= 2.5⬎1共see Fig.3and TableI兲. For the case of the out of phase共in phase兲between the two eigenmodes, the VC position vectors of the opposite eigenmodes on the applied H_lin axis are always antiparallel 共parallel兲 to each other, whereas they are always parallel 共antiparallel兲 on the axis perpendicular the H_lin direction. Thereby, the resulting orbital trajectories of the superposition of the oppositely rotating circular eigenmotions, are ellipse in shape, and their elongation axes are perpendicular共parallel兲to theH_lindirec- tion for ␻H/␻D⬍1 共␻H/␻D⬎1兲, which is independent of 共p,C兲.

The rotation sense of the elliptical VC motions is CCW direction because of 兩XCCW兩⬎兩XCW兩 for all p= + 1 cases,

-80 -40 0 40 80 -40 -20 0 20 40

x(nm) x(nm)

80 40 0 -40 -80

40 20 0 -20 -40

y(nm) (1,-1)

HCCW

HCW

HCCW

HCW

/ 2 π

− π/ 2

+ +π/ 2 +π/ 2

CW 2

δ^H = + π 80

40 0 -40 -80

40 20 0 -20 -40

y(nm)

(-1,-1)

HCCW

HCW

HCCW

HCW

π/ 2

−

CCW / 2 δ^H = −π

π/ 2

− δCW^H

(

, 0/

)

(0.3,0.2)

D H HA

ω ω

=

H

80 40 0 -40 -80

40 20 0 -20 -40

y(nm)

80 40 0 -40 -80

40 20 0 -20 -40

y(nm)

(p ,C)

= (1,1)

(-1,1)

H_CCW H_CW

HCCW

H_CW

H_CCW HCW

π/ 2

−

CW

π/ 2

= − −π/ 2

/ 2 π +

π/ 2

−

π/ 2

+ +π/ 2

VC position (t= 98.2 ns) VC position (t= 90.0 ns)

CCW lin. (CCW+CW) Sim. Analy.

(2.5,0.2)

CCW / 2 δ^H = +π

FIG. 4. 共Color online兲Analytical共solid lines兲 calculations and simulations共open circles兲of the orbital trajectories of the VC motions driven byH_CCW关blue共dark gray兲兴andH_CW关red共light gray兲兴, as well asH_lin共black兲for each case of共p,C兲. The phase relation between the VC position共closed dots兲and the circular field direc- tion共arrows兲 is illustrated. The arrows on the circular or elliptical orbits indicate their rotation senses.

(5)

whereas thus the rotation sense is CW direction for all the cases of p= −1 because of 兩XCCW兩⬍兩XCW兩共see Fig. 3 and TableI兲. As a result, it is evident that the degree of elongation and its major axis of the elliptical orbital trajectories of those linearly oscillating-field-driven vortex motions shown in Fig.1共d兲are determined by the relations of兩␹CCW,CW兩and

␦CCW,CW

H with␻H.

For more quantitative understanding, it is convenient to define the ellipticity ␩G as the ratio of the length of the major 共a兲to that of the minor 共b兲 axis, and the rotation␪G

as the angle of the ellipse’s major axis from the H_lin axis 关the y axis in our case, see Fig. 5共a兲兴, as in the Kerr or Faraday ellipticity and rotation in magneto-optics.²⁷ The numerical values of ␩G and ␪G, which are ␩G=共兩X_CCW兩

−兩XCW兩兲/共兩XCCW兩+兩XCW兩兲=共兩␹CCW兩−兩␹CW兩兲/共兩␹CCW兩+兩␹CW兩兲 and ␪G=共␦CCW

H −␦CW

H 兲/2 by definition, were plotted versus

␻H/␻Din Fig.5共b兲for the all cases of共p,C兲. Note that the analytical calculations 共solid lines兲of ␩G and␪G are in excellent agreements with the simulation results 共open sym- bols兲. Owing to the resonance characteristics of either the CCW or CW eigenmode for a given p,␩G and␪G dramati- cally change across ␻H/␻D= 1 such that ␩G= + 1 or −1共in- dicating that the orbital trajectory shape is circular兲 and␪G

= +␲/4 or −␲/4 共Fig. 5兲.²⁸ ␩G⬎0 共␩G⬍0兲 represents the CCW 共CW兲 rotation of the resultant VC gyrotropic motion driven by a linearly polarized oscillating field. It is worthwhile noting again that since the relative magnitude of 兩␹CCW兩 and 兩␹CW兩 is determined by p, not by C, 共i.e., 兩␹CCW兩⬎兩␹CW兩 for p= + 1 and 兩␹CCW兩⬍兩␹CW兩 for p= −1兲,

␩G⬍0 for p= −1 and ␩G⬎0 for p= + 1, and by contrast␪G

depends on both p and C because of their dependences of

␦CCW,CW

H . The sharp variation of␪G from␲/2 to 0 with increasing ␻H across ␻Dindicates that the major axis of the ellipses changes from the x axis to the y axis. From the calculations of␦CCW,CW

H for each case of共p,C兲, we can esti- mate that␪G= +␲/2 or −␲/2 共the major axis is perpendicular to theH_lin axis兲for ␻H⬍␻D, and that␪G= 0共the major axis is parallel to theH_linaxis兲for␻H⬎␻D, regardless of p andC. For all the cases of 共p,C兲, those results are summa- rized in Table I for the two different cases of ␻H/␻D= 0.3 and 2.5.

CONCLUSION

We found that the CCW and CW circular-rotational eigenmodes are the elementary eigemmodes existing in the vortex

TABLE I. Numerical estimates of␦CCW,CWF 共␻H兲,␦CCW,CWH 共␻H兲,兩␹CCW,CW共␻H兲兩,␪G, and␩Gfor the results shown in Fig.4.

共p,C兲

␻H/␻D

共1, 1兲共−1, 1兲共1, −1兲共−1, −1兲

0.3 2.5 0.3 2.5 0.3 2.5 0.3 2.5

␦CCW

F 0 −␲ 0 0 0 −␲ 0 0

␦CWF 0 0 0 −␲ 0 0 0 −␲

␦CCW

H ␲/2 −␲/2 ␲/2 ␲/2 −␲/2 ␲/2 −␲/2 −␲/2

␦CWH −␲/2 −␲/2 −␲/2 ␲/2 ␲/2 ␲/2 ␲/2 −␲/2

兩X_CCW兩/兩X_CW兩 1.86 2.33 0.54 0.43 1.86 2.33 0.54 0.43

␩G 0.3 0.4 −0.3 −0.4 0.3 0.4 −0.3 −0.4

␪G ␲/2 0 ␲/2 0 −␲/2 0 −␲/2 0

1.0 0.8 0.6 0.4 0.2 0.0 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0.75

0.50 0.25 0 -0.25 0.25 0 -0.25 -0.50 -0.75 0.25 0 -0.25 -0.50 -0.75

0 1 2 3

ω ω_H D

(radians)

ππ

θGθG =b/a

η

G

η

G

0 1 2 3

ω ωH D

(-1,-1) (1,-1) (-1,1)

(a)

θ

G

b y

a

= b/a η

G

(b)

(radians)

ππ

θGθG =b/a

η

G

η

G

(p ,C)

=(1,1)

1.0 0.8 0.6 0.4 0.2 0.0 0.75

0.50 0.25 0 -0.25

(radians)

ππ

θGθG(radians)

ππ

θGθG =b/a

η

G

η

G=b/a

η

G

η

G

x

FIG. 5. 共Color online兲共a兲 Illustration of the definitions of ␪G

and␩Gdescribed in the text.共b兲Numerical estimates of␪Gand␩G

from the micromagnetic simulations共symbols兲and numerical calculations of the analytical equations 共solid lines兲 for all cases of 共p,C兲, as noted. The simulation results correspond to the cases shown in the first column in Fig.1共d兲.

(6)

gyrotropic motions in circular nanodots and that the two eigenmodes’ resonant excitations are largely asymmetric, according to the vortex polarization. The relative magnitudes in the orbital-radius amplitude and phase between the two circular eigenmodes determine the elongation and orientation, respectively, of the orbital trajectories of the vortex core motions driven by a linearly oscillating field. These results pro- vide information on how the orbital-radius amplitude and phase of a vortex core motion vary with the polarization and chirality of the given vortex state as well as the field frequency, resulting in overall linear-regime steady-state vortex gyrotropic motions driven by any polarized oscillating fields.

Due to the distinctly different asymmetric resonance effect between the CCW and CW circular motions on resonance, the CCW and CW circular fields with the resonance frequency can be used to selectively switch upward and downward oriented vortex cores, respectively, as well as to selec-

tively excite the large-amplitude CCW and CW gyrotropic motions of the upward and downward vortex cores, respectively. From a technological point of view, these allow for selective, reliable switching of the magnetization of vortex cores with low power consumption and the indirect detection of the VC orientation by monitoring the large shift of the vortex core position or the largely asymmetric in-plane M orientations through directly measuring induced voltage or tunneling magnetoresistance and giant magnetoresistance contrast with pinned reference spin configurations in the other ferromagnetic layers.

ACKNOWLEDGMENT

This work was supported by Creative Research Initiatives 共Research Center for Spin Dynamics and Spin-Wave De- vices兲of MOST/KOSEF.

*Corresponding author; [email protected]

1T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Sci- ence 289, 930共2000兲.

2A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, and R. Wiesendanger, Science 298, 577共2002兲.

3S.-K. Kim, J. B. Kortright, and S.-C. Shin, Appl. Phys. Lett. 78, 2742共2001兲; S.-K. Kim, K.-S. Lee, B.-W. Kang, K.-J. Lee, and J. B. Kortright,ibid. 86, 052504共2005兲.

4S. B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J.

Stohr, and H. A. Padmore, Science 304, 420共2004兲.

5K. Y. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037共2002兲; K. Y. Gus- lienko, X. F. Han, D. J. Keavney, R. Divan, and S. D. Bader, Phys. Rev. Lett. 96, 067205共2006兲.

6J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A.

Crowell, Phys. Rev. B 67, 020403共R兲共2003兲.

7K.-S. Lee, K. Y. Guslienko, J.-Y. Lee, and S.-K. Kim, Phys. Rev.

B 76, 174410共2007兲.

8K.-S. Lee and S.-K. Kim, Appl. Phys. Lett. 91, 132511共2007兲.

9K. S. Buchanan, M. Grimsditch, F. Y. Fradin, S. D. Bader, and V.

Novosad, Phys. Rev. Lett. 99, 267201共2007兲.

10S. Kasai, Y. Nakatani, K. Kobayshi, H. Kohno, and T. Ono, Phys. Rev. Lett. 97, 107204共2006兲.

11B. Kruger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche, and G.

Meier, Phys. Rev. B 76, 224426共2007兲.

12K. Y. Guslienko, Appl. Phys. Lett. 89, 022510共2006兲.

13S. Choi, K.-S. Lee, K. Y. Guslienko, and S.-K. Kim, Phys. Rev.

Lett. 98, 087205共2007兲.

14B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszc- zak, R. Hertel, M. Fähnle, H. Brückl, K. Rott, G. Reiss, I.

Neudecker, D. Weiss, C. H. Back, and G. Schütz, Nature共Lon- don兲 444, 461共2006兲; Q. F. Xiao, J. Rudge, B. C. Choi, Y. K.

Hong, and G. Donohoe, Appl. Phys. Lett. 89, 262507共2006兲; R.

Hertel, S. Gliga, M. Fähnle, and C. M. Schneider, Phys. Rev.

Lett. 98, 117201共2007兲; K. Y. Guslienko, K.-S. Lee, and S.-K.

Kim,ibid. 100, 027203共2008兲.

15S.-K. Kim, K.-S. Lee, Y.-S. Yu, and Y.-S. Choi, Appl. Phys. Lett.

92, 022509共2008兲.

16K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A.

Thiaville, and T. Ono, Nat. Mater. 6, 269 共2007兲; R. P. Cow- burn,ibid. 6, 255共2007兲; J. Thomas, Nat. Nanotechnol. 2, 206 共2007兲.

17S.-K. Kim, Y.-S. Choi, K.-S. Lee, K. Y. Guslienko, and D.-E.

Jeong, Appl. Phys. Lett. 91, 082506共2007兲.

18Denis D. Sheka, Yuri Gaididei, and Franz G. Mertens, Appl.

Phys. Lett. 91, 082509共2007兲.

19See http://math.nist.gov/oommf

20L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 共1935兲; T. L. Gilbert, Phys. Rev. 100, 1243共1955兲.

21A. A. Thiele, Phys. Rev. Lett. 30, 230 共1973兲; D. L. Huber, Phys. Rev. B 26, 3758共1982兲.

22K. Y. Guslienko, V. Novosad, Y. Otani, H. Shima, and K. Fuka- michi, Phys. Rev. B 65, 024414共2002兲.

23Note that we usedH₀/H_A= 0.01 to avoid VC escape from the dot 共Ref.7兲.

24See EPAPS Document No. E-PRBMDO-77-084817 for two movie files. Supplementary Movie 1: animation on the temporal evolution of the spatial configuration of the local magnetizations and of the vortex-core motion for the case shown in Figs.1共b兲 and 1共c兲. Supplementary Movie 2: animation on the temporal evolution of the individual CCW and CW circular rotational eigenmodes and of the corresponding circular eigenbasis of the pure circularly rotating fields, H_CCW and H_CW, and with their superposition. For more information on EPAPS, see http://

www.aip.org/pubservs/epaps.html.

25Note that such an eigenbasis can be chosen according to the symmetry of W共X兲. For an elliptical dot, an elliptical-rotating- field basis can be used as the eigenbasis.

26For the case of oscillating spin-polarized currents, there is noC dependence of applied driving forces. Consequently, the phase

␦CCW,CWI , does not depend onC.

27K.-S. Lee, D.-E. Jeong, S.-K. Kim, and J. B. Kortright, J. Appl.

Phys. 97, 083519 共2005兲; D.-E. Jeong, K.-S. Lee, and S.-K.

Kim, J. Korean Phys. Soc. 46, 1180共2005兲.

28Here, we emphasize only the linear regime for relatively low values ofH₀/H_A. As far as the linear regime is considered, the values of␩Gand␪Gdo not change withH₀/H_A.