Magnetic Vortex Dynamics

Roman ANTOS1, YoshiChika OTANI1;2�, and Junya SHIBATA3

1RIKEN FRS, Wako, Saitama 351-01982Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581

3Kanagawa Institute of Technology, Atsugi, Kanagawa 243-0292

(Received November 15, 2007; accepted November 26, 2007; published March 10, 2008)

We review the recent theoretical and experimental achievements on dynamics of spin vortices inpatterned ferromagnetic elements. We first demonstrate the theoretical background of the research topicand briefly list the analytical and experimental approaches dealing with magnetic vortices. Then wereport on the most remarkable studies devoted to steady state vortex excitations, switching processes, andcoupled-vortex dynamic phenomena including the design of artificial crystals where the micromagneticenergy transfer takes place via the magnetic dipolar interaction among excited vortices. Finally wesummarize the present state of the research with respect to novel prospects from both the fundamentaland the application viewpoints.

KEYWORDS: magnetization, Landau–Lifshitz–Gilbert equation, spin vortex, polarity, chirality, dynamic switching,spin waves, time-resolved Kerr microscopy, permalloy, spin transfer torque

DOI: 10.1143/JPSJ.77.031004

1. Introduction

One of the most remarkable manifestations of the recentprogress in magnetism is the establishment of microfabri-cation procedures employing modern magnetic materials.Electron or ion beam lithographies combined with theconventional thin film deposition techniques yield a varietyof laterally patterned nanoscale structures such as arraysof magnetic nanodots or nanowires.1,2) Among them, sub-micron ferromagnetic disks have drawn particular interestdue to their possible applications in high density magneticdata storage,3) magnetic field sensors,4) logic operationdevices,5) etc.

It has been revealed both theoretically and experimentallythat for particular ranges of dimensions of cylindrical andother magnetic elements (Fig. 1) a curling in-plane spinconfiguration (vortex) is energetically favored, with a smallspot of the out-of-plane magnetization appearing at the coreof the vortex.6–8) Such a system, which is sometimes referredto as a magnetic soliton9) and whose potentialities havealready been discussed in a few recent review papers,10,11) isthus characterized by two binary properties (‘‘topologicalcharges’’), a chirality (counter-clockwise or clockwisedirection of the in-plane rotating magnetization) and apolarity (the up or down direction of the vortex core’smagnetization), each of which suggests an independent bitof information in future high-density nonvolatile recordingmedia. For this purpose various properties have beeninvestigated such as the appearance and stability of vorticeswhen subjected to quasistatic or short-pulse magnetic fieldsand variations of those properties when the dots are denselyarranged into arrays. The properties are identified withexperimentally measured and theoretically calculated quan-tities called nucleation and annihilation fields, effectivemagnetic susceptibilities, etc.

Most recently, the time-resolved response to appliedmagnetic field pulses or spin-polarized electrical currents

with sub-nanosecond resolution has been extensively stud-ied, providing results on the time-dependence of thelocation, size, shape, and polarity deviations of the vortexcores, eigenfrequencies and damping of time-harmonictrajectories of the cores, the switching processes, and thespin waves involved. In this paper we will review the recentachievements in this research area with a particular interestin submicron cylindrical ferromagnetic disks with negligiblemagnetic anisotropy, for which permalloy (Py) has beenchosen as the most typical material. We will demonstrate thetheoretical background of the research topic according to thedescription by Hubert and Schafer6) (§2) and briefly describethe achievements in analytical approaches (§3) and exper-imental techniques (§4). Then we will review the researchof various authors devoted to steady state excitations (§5),dynamic switching of vortex states (§6), and excitations ofmagnetostatically coupled vortices (§7). Finally we willsummarize the present state of the research with respectto future prospects and possible applications (§8). We willaccompany our description by our simulations using theObject-Oriented Micromagnetic Framework (OOMMF),12)

and in some cases by demonstrative examples provided bytheir original authors.

SPECIAL TOPICS

Fig. 1. (Color online) Examples of vortices appearing in a cylindrical (a),

rectangular (b), elliptic (c), multilayered (d), and ring-shaped (e)

elements. Each vortex’s center contains an out-of-plane polarized core

except for the ring. Classical multidomain structures appear in larger

elements where the anisotropy energy is predominant (f).

�E-mail: yotani@issp.u-tokyo.ac.jp

Journal of the Physical Society of Japan

Vol. 77, No. 3, March, 2008, 031004

#2008 The Physical Society of Japan

Advances in Spintronics

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2. Theoretical Background

The unique spin distributions favored in ferromagneticmaterials are governed by the exchange interaction betweennearest neighbor spins si, sj described by the HeisenbergHamiltonian13)

H ¼ �Xi; j

Ji; jsi � sj ð1Þ

or by more general formulas if particular anisotropies aretaken into account. For the sake of solving many-spinproblems, the discrete spin distribution is replaced by mag-netization Mðr; tÞ, a continuous function of space and time,or by unit magnetization m ¼ M=Ms, where Ms is a satu-ration constant. Accordingly, the total energy of a ferro-magnet is determined as the sum

Etot ¼ Eexch þ Ed þ Eext þ Ean þ � � � ; ð2Þ

which demonstrates the competition among exchange,demagnetizing, external-field, anisotropy, and other formsof energy (such as magneto-elastic interaction or magneto-striction), whose particular integrals per the volume of aferromagnet are

Eexch ¼Z

AðrmÞ2 dV ; ð3Þ

¼ �Z�0Ms

2m �Hexch dV ð4Þ

Ed ¼ �Z�0Ms

2m �Hd dV ; ð5Þ

Eext ¼ �Z�0Msm �Hext dV ; ð6Þ

etc., where A is the exchange stiffness constant, �0 thepermeability of vacuum, and Hexch, Hd, Hext are the‘‘effective exchange field’’, demagnetizing (stray) field, andexternal magnetic field, respectively, altogether forming thetotal effective field

Heff ¼ Hexch þHd þHext þ � � � : ð7Þ

While the exchange interaction forces the nearest spins toalign into a uniform distribution, the demagnetizing fieldmakes the opposite effect on the long-range scale. It can beevaluated via a potential �d as

Hd ¼ �r�d; r2�d ¼ �Ms�d; ð8Þ

whose sources are volume and surface ‘‘magnetic charges’’

�d ¼ �r �m; �d ¼ m � n; ð9Þ

where n is a unit vector normal to the surface of themagnetized element. Hence, the magnetization tends to alignparallel to the surface in order to minimize the surfacecharges, leading to the occurrence of vortex distributions asdepicted in Figs. 1(a)–1(e). Moreover, the singularity at thecenter of a vortex is replaced by an out-of-plane magnetizedcore in order to reduce the exchange energy. On theother hand, in large samples, where the anisotropy energypredominates the surface effects of the disk edges, themagnetization forms conventional domain patterns withmagnetization aligned along easy axes [Fig. 1(f)].

When we slowly apply an external magnetic field, thecompetition among all the energies breaks the symmetry of

the vortex, shifting its core so that the area of magnetizationparallel to the field enlarges, until the vortex annihilates (atthe ‘‘annihilation field’’), resulting in the saturated (uniform)state. Then, when we reduce the external field, the uniformmagnetization changes into a curved ‘‘C-state’’, until thevortex nucleates again (at the ‘‘nucleation field’’). Reducingthe field further to negative values causes the symmetricallyanalogous process, as depicted in Fig. 2.

For the vortex dynamics the main area of interest isthe range of states before the vortex annihilates, which isrepresented in Fig. 2 by the slightly curved line whosetangent � ¼ �M=�H is called the effective magnetic sus-ceptibility defined both statically and dynamically as afunction of frequency �ð!Þ. The dynamic response to fastchanges of external field is considerably different from thatdescribed by the hysteresis loop, and is in general governedby the Landau–Lifshitz–Gilbert (LLG) equation

@m

@t¼ ��m�Heff þ �m�

@m

@t; ð10Þ

where � denotes the gyromagnetic ratio, � the Gilbertdamping parameter, and t the time.

Instead of applying external field, the vortex distributioncan be excited by an electrical current propagating throughthe ferromagnetic disk.14–16) It has been revealed that thisprocess, referred to as spin-transfer torque (STT), can be (inthe adiabatic approximation) evaluated as an additional termon the right-hand side of the LLG equation

Tð1ÞSTT ¼ �ðvs � rÞm; ð11Þ

Tð2ÞSTT ¼ �gh� Ie

2em2 � ðm2 �m1Þ; ð12Þ

where the first equation corresponds to the in-plane currentwith the velocity vs ¼ jePg�B=2eMs whereas the secondcorresponds to the perpendicular current propagatingthrough a multilayer depicted in Fig. 1(d) from the bottomto the top ferromagnetic layer (with magnetization dis-tributions m1, m2, respectively), both of which are separatedby a thin nonmagnetic interlayer (F/N/F). Here je, P, g, �B,e, h� , and Ie denote the current density, spin polarization,g value of an electron, Bohr magneton, electronic charge,Planck constant, and total electrical current, respectively.

Fig. 2. (Color online) Hysteresis loop representing the process of quasi-

static switching of a cylindrical Py disk.

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3. Analytical Approaches

Although most authors have adopted OOMMF for itsgenerality, simplicity, and accuracy, the development ofanalytical approaches is very useful for analyzing variousfundamental aspects of dynamic processes. Among manyattempts to reduce the number of parameters involved invortex dynamics, perhaps the most used is treating the vortexas a quasiparticle whose motion (or motion of its centera ¼ ½ax; ay�) is described by an equation derived from theLLG equation by Thiele17) for magnetic bubbles and adoptedby Huber18) to vortex systems,

G�da

dt¼

1

R2

@Etot

@a� D$�da

dt; ð13Þ

where G ¼ �2pL�0Mszz=� is the gyrovector with p ¼ �1

denoting the vortex’s polarity (the positive value stands forthe up direction, parallel to the unit vector zz) and L denotingthe disk’s thickness, and where D

$¼ �2L��0Msðxxxxþ yyyyÞ=

� is the dissipation tensor of the second order. Thiele’s equa-tion of motion thus found the use as one of the most con-venient approaches of dealing with vortex dynamics and hasfurther been generalized to include an additional term of‘‘mass times acceleration’’19,20) or to take into account STT.21)

To perform simulation with Thiele’s equation, one needsto evaluate EtotðaÞ as a function of the vortex center’sposition. For this purpose, two approximations have beenutilized, the ‘‘rigid vortex’’ model,22–25) assuming the staticsusceptibility, and the ‘‘side charges free’’ model,26) whichassumes the magnetization on the disk edges to be constantlyparallel to the surfaces. It has been revealed that the latterapproximation applied to an isolated disk gives considerablybetter agreement with rigorous numerical simulations.27)

However, when applied to a pair or arrays of magnetostati-cally coupled disks, the rigid vortex model gives a reasonabletendency while the other model fails due to the absence of theside surface charges which are particularly responsible forthe magnetostatic interaction between disks.28)

To study excitations of vortices more precisely, someauthors have analytically solved the LLG equation byassuming small deviations of the static vortex distribution.They start with the description of the unit magnetizationvector mðr; �Þ by angular parameters ðr; �Þ, �ðr; �Þ,

mx þ imy ¼ sin ei�; mz ¼ cos ; ð14Þ

where r, � are polar coordinates determining the lateralposition within the disk. The small deviations of the staticmagnetization distribution [described as stat ¼ 0ðrÞ, �stat ¼�0ð�Þ ¼ q�, where q denotes the vorticity of the systembeing þ1 for a normal vortex or �1 for an antivortex] can bewritten as

ðr; �Þ ¼ 0ðrÞ þ #ðr; �Þ; ð15Þ�ðr; �Þ ¼ q�þ ½sin 0ðrÞ��1�ðr; �Þ; ð16Þ

leading to the solution in the form29)

#ðr; �Þ ¼Xn

Xþ1m¼�1

fnmðrÞ cosðm�þ !nmt þ �mÞ; ð17Þ

�ðr; �Þ ¼Xn

Xþ1m¼�1

gnmðrÞ sinðm�þ !nmt þ �mÞ; ð18Þ

where ½n;m� is a full set of numbers labeling magnoneigenstates and �m are arbitrary phases. This approach hasbeen successfully applied to both antiferromagnets30,31) andferromagnets,32–42) and has revealed eigenfrequencies andeigenfunctions of spin wave modes propagating in cylin-drical disks and S-matrices of magnon–vortex scattering.

4. Experimental Techniques

Experimental measurements of quasistatic properties ofmagnetic elements giving clear evidence of vortex struc-tures, including the core’s shapes and quasistatic switchingprocesses, have been carried out by magnetic force mi-croscopy (MFM),7) spin-polarized scanning tunneling mi-croscopy,8) magnetoresistance and Hall effect measure-ments,43–48) Lorentz transmission electron microscopy,49,50)

magneto-optical Kerr effect (MOKE) measurements,51–59)

photoelectron emission microscopy,60–62) scanning electronmicroscopy with spin-polarization analysis (SEMPA),63,64)

and others.On the other hand, different techniques have to be

employed for time-resolved dynamic measurement suchas time-resolved Kerr microscopy (TRKM) in the scan-ning65–72) or wide-field mode,73–75) photoemission electronmicroscopy combined with pulsed x-ray lasers,76) Brillouinlight scattering (BLS),77) time-resolved MFM,78) ferro-magnetic resonance (FMR) technique,79,80) vector networkanalyzers,81) superconducting quantum interference devicemagnetometry,82) and others.

The most typical measurement technique, TRKM, oftenreferred to as a ‘‘pump–probe’’ technique, combines a Kerrmicroscope of high space–time resolution achieved byultrashort-pulse laser light source and high-quality micro-scopic imaging (the ‘‘probe’’), and a system for operatingultrafast pulse excitations achieved practically via varioustransmission line configurations as depicted in Fig. 3 (the‘‘pump’’). The source for the excitation current can begenerated either by a pulse generator (triggered by thelaser control device) or by a photoconductive switch (whenlaser pulses are split between probe and pump pulses). Thewavelength of light is often halved by a second harmonicgeneration device to increase the spatial resolution ofmeasurement. The time dependence of magnetization evo-lution after excitation is determined by changing the delaytime between the pump and the probe. To obtain anappropriate signal-to-noise ratio, the pump–probe measure-

Fig. 3. (Color online) Various types of ultrafast excitations: in-plane

(a) or out-of-plane (b,c) magnetic field pulses are generated by electrical

current pulses propagating through transmission lines with appropriate

geometries; out-of-plane (d) and in-plane (e) currents induce excitations

based on spin transfer torque.

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ment must be repeated many times with exactly the sameinitial condition, referred to as a stroboscopic method. Anexample of TRKM measurement,63,64) showing radial modespropagating from the edges of a Co cylindrical dot excitedby out-of-plane field, is displayed in Fig. 4.

5. Steady State Motion Phenomena

Various authors have studied dynamic excitations ofvortices in cylindrical disks to observe a rich spectrum ofmodes.83) Besides the existence of the radial modes excitedby out-of-plane field (Fig. 4),63,64) it has been also revealedthat low-energy modes (those near the ground state) excitedby in-plane field can be classified into two elementary types.

The first type, referred to as the gyrotropic mode, is anoscillatory motion of the vortex core around its positionin equilibrium, whose numerical simulation is displayedin Fig. 5 for a Py disk with the diameter of 100 nm andthickness of 20 nm. This type of motion has been predictedas the solution of Thiele’s equation [eq. (13)], as theanalytical solution of the LLG equation in the angularvariables [eqs. (17) and (18)], and has been extensivelystudied by micromagnetic simulations and various experi-ments. It has been revealed that the core’s initial motionis parallel or antiparallel to the applied magnetic fieldpulse, depending on the vortex’s ‘‘handedness’’ (the polarityrelative to the chirality).76) However, the clockwise orcounter-clockwise sense of the core’s spiral motion onlydepends on the vortex’s polarity and is independent of thechirality. Owing to this rule, the vortex polarity can bemagneto-optically measured via this dynamic motion, eventhough the small size of the vortex core makes the staticmagneto-optical measurement very difficult. In our example(Fig. 5) the motion’s frequency is slightly above 1 GHz, andis decreased with reducing the disk’s aspect ratio (thicknessover diameter).27)

The second type, referred to as magnetostatic modes, arehigh-frequency spin-wave excitations. It has been theoret-ically predicted29) and experimentally evidenced84) that thereare azimuthal modes with degenerated frequency (frequencydoublets), corresponding to the two values of the azimuthal

magnon number m ¼ �jmj in eqs. (17) and (18). In smalldisks (where the size of the out-of-plane polarized corebecomes comparable to the entire size of the disk) thisdegeneracy is lifted (i.e., the frequency doublet becomessplit), which has been explained via spin wave–vortex (ormagnon–soliton) interactions.85) However, it has also beenshown that removing the core (replacing the disk by a widering or introducing strong easy-plane anisotropy) retains thedegeneracy of the doublet, so that no splitting occurs.84,86)

The whole process of the gyrotropic motion and a higher-frequency doublet is displayed in Fig. 6.

The dynamic manipulation of vortex states by meansof STT has become one of the most attractive subjectsfrom both the fundamental and the application viewpoints.Therefore, the current-induced motion of vortices has alsobeen investigated to reveal phenomena analogous to thosemanaged by the field excitation.87,88)

6. Dynamic Switching

During the last few months, immensely intensive workhas been carried out to study the process of dynamicswitching of vortex polarities and chiralities, which is parti-cularly important for the data storage application. Tradi-tionally, to switch the vortex core’s polarity, an extremelylarge quasistatic out-of-plane magnetic field was required.Moreover, to control chirality, the disk had to be fabricated

Fig. 4. Example of TRKM measurement performed by Acremann

et al.63,64) on a Co disk with an optical micrograph (a), SEMPA meas-

urement of the static domain configuration (b), multilayer specification of

the sample (c), time-resolved evolution of the Mz, My differences (d), and

snapshots of Mz at particular times revealing the propagation of radial

modes from the edges (which are excited by out-of-plane field) towards

the center.

(a)

(b)

Fig. 5. (Color online) Trajectory of the vortex core during and after an

externally applied in-plane magnetic field pulse of the strength

Bx ¼ 60 mT and duration of 200 ps (a). After turning the field off, the

vortex core exhibits a typical spiral motion around its equilibrium

position at the disk’s center (½0; 0� nm). The asterisks denote the time

points of 0.2, 0.5, 1, and 1.5 ns. The time evolution of the normalized

total magnetic moment is displayed in (b). For the simulation the typical

values for Py were used, Ms ¼ 860� 103 A/m, A ¼ 1:3� 10�11 J/m,

� ¼ 2:2� 105 m/(A�s), and � ¼ 0:01.

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with a geometric asymmetry, e.g., with a ‘‘D-shape’’49,89)

or other shapes.61) Unlike that, dynamic processes haverevealed considerable advantages.

Several authors have recently demonstrated90–92) that ashort pulse of in-plane magnetic field of a certain amplitudeand duration excites the vortex so that a pair of a new vortexand an antivortex is created, the new vortex possessingthe opposite polarity, and that the antivortex annihilatestogether with the old vortex core, as depicted in Fig. 7 for acylindrical disk with the diameter of 200 nm and thicknessof 20 nm. This process is fully controllable by applyingan appropriate filed pulse whose amplitude is considerablysmaller than those which are necessary for quasistaticswitching. The process has also been successfully observedby experimental measurements,93) and its variations andfurther details are presently researched.94,95)

For applications in spintronics, to control the switchingprocess via an electrical current is of particular interest.In this respect various authors have recently carried outtheoretical96–99) end experimental78) studies to reveal that thesimilar dynamic switching processes are possible by apply-ing STT excitations in the both configurations as describedby eqs. (11) and (12).

Similarly to the quasistatic case, to change the vortexchirality requires introducing some geometric asymmetryinto the process. For this purpose, Choi et al.100) haveapplied a perpendicular current pulse to an F/N/F nano-pillar, where the asymmetry is due to the magnetostaticinteraction between the vortices in the two ferromagneticlayers. However, the full and reproducible control of theboth binary parameters of a vortex is still a demandingtask.

7. Magnetostatically Coupled Vortices

Many studies of quasistatic processes have been per-formed on pairs, chains, and two-dimensional arrays ofmagnetostatically coupled vortices, but little attention waspaid to their dynamic properties, although putting magneticdisks near each other is of high importance for improvingthe density of data storage and studying the propagationof micromagnetic excitations through such arrays. Amongdynamic studies, the pioneering experiments have beenperformed by means of BLS.101,102)

Recently the dynamics of magnetostatically coupledvortices was studied in pairs of disks placed near each otherlaterally,28) vertically (as an F/N/F nanopillar),103) and as apair of two vortices located inside a single elliptic dot.104,105)

It has been revealed, e.g., that the eigenfrequency of thesynchronized steady-state motion of two vortices in thelateral arrangement is split into four distinct levels whosevalues depend on the lateral uniformity of the vortices’excitation and on the combination of their polarities (but areindependent on chiralities).28)

Large arrays of coupled vortices have also been inves-tigated to reveal a close analogy with crystal vibrations (orphonon modes) in two-dimensional atomic lattices.106,107)

The dispersion relations and the corresponding densitiesof states of propagating waves of vortex excitations werefound to vary with different ordering of vortex polaritiesregularly arranged within nanodisk arrays. Moreover, forarrays of disks with two different alternating diameters aforbidden band gap has been observed (Fig. 8), pointing atthe analogy with the band gap between the acoustic andoptical phonon branches in atomic lattices.108,109) Collectiveexcitation modes have also been studied in small (3� 3)arrays of nanodisks110) and as analytical calculations usingthe Bloch theorem which enables dealing with infinitearrays.41)

Fig. 6. Trajectory of the vortex core obtained by Ivanov et al.29) using

the analytical approach as described by eqs. (14)–(18) with damping

neglected, showing that the spiral motion is affected by high-frequency

oscillations. The inset in the middle shows the eigenfrequency spectrum

of the doublet jmj ¼ 1 (denoting the ‘‘quantum number of the angular

momentum’’) as functions of intrinsic easy-plane anisotropy � . For

� < �c (high easy-plane anisotropy) the vortex only possess the in-plane

components of magnetization (no out-of-plane core appears); for � > �c

(low anisotropy) the vortex with an out-of-plane polarized core becomes

responsible for significant frequency splitting.

Fig. 7. (Color online) Process of dynamic switching of the vortex polar-

ity. An external field pulse of the strength Bx ¼ 100 mT and duration

of 30 ps is applied to the vortex at t ¼ 0 (a). Shortly after turning the

field off (t ¼ 33 ps) the vortex distribution becomes slightly deviated (b),

at t ¼ 57 ps a pair of a new vortex and an antivortex is created (c), at

t ¼ 67 ps the antivortex annihilates together with the old vortex (d) which

creates a point source of spin waves which are scattered by the new vortex

at t ¼ 78 ps (e). Finally, (f) shows a later distribution of the new vortex

with the opposite polarity at t ¼ 1 ns. Each arrow in (b–e) represents the

in-plane component of the magnetization vector in a grid of 2� 2 nm,

whereas the shades of gray (red–blue in the online version) correspond to

the out-of-plane component.

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8. Conclusions and Perspectives

We have reviewed the most fundamental achievements onthe dynamic properties of magnetic vortices with a particularinterest in soft cylindrical ferromagnetic disks. We havedemonstrated the basic theoretical background widely usedin analytical and numerical simulations, and briefly listed theutilized experimental approaches (including a description ofTRKM as the most typical method of measurement). Thenwe have demonstrated the most significant results achievedby various authors. First, we have shown that the elemen-tary excitations near the ground vortex state (steady statemotions) are the important starting point to understand thewhole principle of spin vortex dynamics. Then we havereported the recent results of ultrafast magnetic-field andSTT based switching of the vortex binary properties (polar-ity and chirality) which are of high importance for thepossible future application in nonvolatile magnetic recordingmedia. Finally we have briefly presented a few potentialitiesof vortices densely arranged into arrays or multilayers,which were found important, e.g., for designing novelartificial metamaterials which possess propagation modesbased on magnetostatic interactions between nearest neigh-bor elements.

The contemporary research continues in the tendency ofpushing the limits of the up-to-date theoretical and measure-ment capabilities and exploring new directions of studyingfundamental physical phenomena and utilizing them innew or higher-level applications. As regards the theoreticalcapabilities, reducing the element sizes to the true nanoscalerequires the generalization of models to allow for the effectof surfaces and interfaces on atomic scale,111) quantum andnonlinear effects (such as nonlinear optical excitations ofspins usable for entirely optical switching112,113)), etc., forwhich the first-principle calculation will probably beemployed. As regards experiment, the tendency of reducingsizes will require not only the improvement of the spatialresolution114) (for which novel techniques are interestingsuch as magnetic exchange force microscopy with atomicresolution115)), but also further increase of the time reso-lution of dynamic measurements.11) Moreover, since thestroboscopic measurement of the ultrafast dynamics requiresrepetition with always equal initial conditions, this methodcannot be used to study possible stochastic processes, which

determines a challenge to develop new experimental con-ceptions. Another challenge from both the theoretical and theexperimental viewpoints is the modulation of the dynamicproperties of vortices by virtual fab10) or artificial defectsdesigned by tricky methods of deposition116) or etching,117)

leading to considerable increase of vortices’ sensitivity tofields with particular strengths or frequencies. We can thusconclude that the results of numerous dynamics studieshave pointed out various advantages and new potentialitiesin data storage, nanoscale probing of magnetic thin-filmstructures, and other types of sensing or controlling.

Acknowledgment

The authors thank Yves Acremann and Boris Ivanov forfruitful discussions and for providing some of the data usedin figures. One of the authors (R.A.) thanks the StanfordLinear Acceleration Center for the hospitality.

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Roman Antos was born in Czechoslovakia in

1975. He obtained his M. Sc. (2000) and Ph. D.

(2007) degrees from Charles University in Prague,

Czech Republic, and D. Eng. (2006) degree from

Shizuoka University. Since 2006 he has been a

postdoctoral researcher in the Quantum Nano-Scale

Magnetics Laboratory at the Frontier Research

System (FRS) of the Institute of Physical and

Chemical Research (RIKEN). He has worked on the

optical and magneto-optical spectroscopic analyses

of periodically patterned nanostructures (nanogratings) and spin-dynamics

related analyses of magnetic nanostructures.

YoshiChika Otani was born in Tokyo, Japan, in

1960. He obtained his B. Sc. (1984), M. Sc. (1986),

and Ph. D. (1989) degrees from Keio University. He

was a research fellow (1989–1991) at the Physics

Department of the Trinity College, University of

Dublin, a researcher (1991–1992) at the Laboratoire

Louis Neel, CNRS. Then he was appointed to a

research instructor (1992–1995) at the Department

of Physics, Keio University, an associate professor

at the Department of Materials Science, Graduate

School of Engineering, Tohoku University, and a head (since 2002) of the

Quantum Nano-Scale Magnetics Laboratory at FRS-RIKEN. Since 2002

he has also been a professor at ISSP, University of Tokyo. He has been

primarily working on experimental studies on spin electronics such as

magnetic and transport properties of nanostructured magnetic/nonmagnetic

(superconductive) hybrid systems including vortex dynamics confined in

magnetic nanodisks.

Junya Shibata was born in Osaka Prefecture,

Japan, in 1974. He obtained his B. Sc. (1996)

degree from Ritsumeikan University, M. Sc. (1998),

and D. Sc. (2001) degrees from Tohoku University.

He was a postdoctoral researcher (2001–2002) at

Osaka University and (2002– 2007) at the Institute

of Physical and Chemical Research (RIKEN). Since

2007 he has been an associate professor at Kana-

gawa Institute of Technology. He has worked on the

theory of condensed matter physics in nanoscale

magnetism. His research is now focused on current-induced magnetization

dynamics and current generation induced by magnetization dynamics.

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