Statistics of gyrotropic vortex dynamics in sub-micron magnetic disks

G. M. Wysin∗Department of Physics, Kansas State University, Manhattan, KS 66506-2601

(Dated: July 3, 2016)

Topological vortex excitations in thin magnetic nano-disks have attracted a lot of attention be-cause of their dynamic stability and various charge-like properties, which makes them suitableobjects for data storage. They also have a natural gyrotropic orbital motion that can be describedrather well by an approximate Thiele gyrotropic equation for the magnetization dynamics. Thegyrotropic oscillation makes them available as a basis for natural oscillators at close to gigahertzfrequencies. This gyrotropic motion is excited naturally even by thermal fluctuations. In addition,the gyrotropic oscillation frequency can be affected by external perturbations, which allows possi-bilities for the design of nano-scale detectors. The vortex moves in an effective potential, stronglydetermined by the shape anisotropy of the magnetic disk, which then determines the force appearingin the Thiele equation of motion. The motion of an individual vortex within a disk of circular orelliptical shape is considered theoretically, including stochastic thermal effects together with thedeterministic gyrotropic effects. From simulations of the motion at different parameter values, apicture of the typical vortex position and velocity distribution within the disk is developed, andcompared with what is expected from the Thiele equation.

Keywords: magnetic vortex, topological charge, vortex potential, magnetic resonance, magnetic dots

1. INTRODUCTION: VORTICES IN THINSUB-MICRON MAGNETIC DISKS

A cylindrically shaped thin disk of soft ferromagneticmaterial with a radius a on the order of 100 nm to a fewmicrons and a thickness L � a on the order of 10 nmto 50 nm provides an interesting system for the studyof vortices [1, 2]. A magnetic configuration is describedby its local magnetization �M(r), which is the magneticdipole moment per unit volume, at position r. A ma-terial is considered with saturation magnetization Ms,which is the magnitude when the medium is completelymagnetized along some axis. Due to the demagnetiza-tion effect, which is responsible for shape anisotropy, anysuch magnetic system tends to avoid the formation ofmagnetic poles on the surfaces, if possible, which wouldraise the total energy. For a thin circular disk, the localmagnetization �M(r) as a function of position r may tendto do two things: (1) �M(r) will have a strong tendencyto point within the plane of the disk [3], if possible; (2)�M(r) may then follow the curved circular boundary atthe disk edge, thereby completely avoiding the genera-tion of any poles on the edge. This prevents any strongmagnetic field lines from passing through the space sur-rounding the disk edge.

Within the disk the forces of ferromagnetic exchangecause �M(r) also to have a circular structure and remainclose to the disk plane. At the disk center, which is asingular point, remaining in the disk plane is impossible,and �M(r) then points perpendicular to the plane of thedisk, forming tiny north/south poles on opposite faces

∗Electronic address: wysin@phys.ksu.edu;URL: http://www.phys.ksu.edu/personal/wysin

X=0, Y=0, it=1250, E=14.752 J

FIG. 1: The magnetization field �M(r) = Msm(r) of a vortexcentered in a nanodisk with principal axes a = 60 nm, b = 30nm, thickness L = 10 nm, from the spin alignment relaxationscheme for the micromagnetics model, Sec. 2.4. The cell sizeis acell = 2 nm. Arrows show only the in-plane projection,(mx

i , myi ). Blue line (red open) arrows indicate positive (neg-

ative) values of out-of-plane component mzi . The core, where

mzi is larger, appears as a hole in this projection. Even though

the system is elliptical, note that the core region remains closeto circular.

at the disk center. The resulting circular form of �M(r),together with its central poles in a core region, is a mag-netic vortex. It is a type of magnetic excitation that istopologically stable and acts in many ways like a particle,when exposed to forces.

One can also consider deviations from circular symme-try, such as in elliptic nanodisks, where magnetic vortexdynamics has been studied by measuring their radio fre-quency oscillations [4] and even by direct electrical con-tact [5] to a nanodisk. An example of a magnetic vortexcentered in a thin elliptical nanodisk is shown in Fig. 1.It has been obtained from a numerical relaxation algo-

2

rithm [6], see Sec. 2.4 below. Although there is a ten-dency for �M(r) to follow the boundary, one sees that theexchange forces are more dominating, and expecially thecore structure of the vortex retains a circular shape. Thelocus where the perpendicular component Mz changessign is obvious as a circle in Fig. 1.

1.1. Vortex charges

The magnetization profile �M(r) of a vortex may pointeither in a counterclockwise (CCW) or clockwise (CW)direction around the disk. This two-fold degeneracy isassociated with its circulation charge c = ±1, which isalso refered to as chirality. It provides one topologicallystable geometric property, that could be used for datastorage in a vortex, if it can be reliably controlled anddetected.

In principle, a vortex profile �M(r) also has a vortic-ity charge q = ±1, which corresponds to the direction ofrotation of �M(r) as one moves along a closed path encir-cling the core. The value q = +1 holds for the vorticesdescribed here, which are controlled by demagnetizationeffects ( �M being forced to follow the boundary). Thevalue q = −1, known as an antivortex, would only beenergetically stable if demagnetization effects were notpresent. The limit of zero thickness would eliminate therelevant demagnetization and make antivortices energet-ically possible.

The magnetization at the vortex core can take one oftwo values, �M(0) = ±Msz = pMsz, where p = ±1 is thepolarization charge. Because there is an energy barrierto flip the core polarization from p = +1 to p = −1, itoffers yet another charge that could be useful for datastorage and manipulation.

1.2. Vortex potential and forces

The above-described magnetic vortex will have its min-imum energy when it is centered in the disk. The loca-tion of the poles (where �M points perpendicular to thedisk) defines the location of the vortex core, which wedenote by position vector R = (X, Y ), measured alongthe x, y Cartesian axes within the disk plane. Becausethe system is assumed to be thin, only two coordinatesX, Y are needed to locate the core. Further, the mag-netization itself has little dependence on the coordinate(z) perpendicular to the disk. We take R = (0, 0) for thevortex at the center of the disk. It is possible to imag-ine that the vortex core becomes slightly displaced fromthe disk center. In that case, a slight deformation of thevortex structure �M(r) takes place, while the demagneti-zation effects at the disk edge still try to maintain �M(r)parallel to the edge. The net result of the displacement isa slight increase in total energy. The vortex, as a quasi-particle, lives in some effective potential U(R), which has

something approximating a parabolic form [6], with theminimum at the disk center,

U(R) ≈ 12kF R2, (1)

where kF is a force constant. This further implies aneffective force F back towards the disk center, accordingto the gradient of the potential,

F = −�∇U(R) ≈ −kF R. (2)

For a circular disk, the potential is circularly symmetric,and then small displacements lead to an circularly sym-metric Hooke’s Law type of force. It is also possible toconsider magnetic vortices in a cylindrical disk of ellip-tical shape [7], defined by principal axes a and b < a:

x2

a2+

y2

b2= 1. (3)

This situation leads correspondingly to a modification ofthe potential also to an approximately elliptic form [8],

U(R) ≈ 12

(kxX2 + kyY 2

). (4)

The parabolic functional form now has separate forceconstants kx, ky along the two principal axes. It givesa force,

F = (−kxX,−kyY ). (5)

While a vortex in a nanodisk experiences a force di-rected roughly towards the disk center, its motion tendsto be in an orbital sense, which is the gyrotropic oscilla-tion mode [9, 10]. This is discussed further in Sec. 3 ondynamics. Before coming to that we begin by a quanti-tative description of the calculation of vortex structures.

2. ANALYSIS OF QUASI-STATIONARYVORTICES IN A NANODISK

The theoretical analysis is based on the statics and dy-namics of the magnetization field, which is now assumedto keep a uniform magnitude Ms, but a spatially varyingdirection, by writing �M(r) = Msm(r), where m(r) is aunit vector. From the energetics expressed in terms ofm(r), equations for the vortex structure and motion canbe developed. See Ref. [11] for a general discussion of thecalculation of magnetic vortex structures and properties.

2.1. Energetics of a continuum nano-magnet

The system is governed by ferromagnetic exchange en-ergy and interactions of �M generally with the demagne-tization field �HM (self-generated by �M) and any possible

3

externally applied field �Hext. A continuum Hamiltonianfor the system is

H =∫

dV{A∇�m · ∇�m − µ0

(�HM + 1

2�Hext

)· �M

},

(6)where µ0 is the permeability of free space, and A isthe exchange stiffness. One commonly used material isPermalloy-79 (Py, 79% nickel, 21% iron), with exchangestiffness about 13 pJ/m and saturation magnetizationMs = 860 kA/m [12]. The magnetization changes its di-rection over a length scale λex called the exchange length.Exchange energy of the order A/λ2

ex competes with de-magnetization energy of the order 1

2µ0M2s . Equating

these terms gives the definition of the length scale,

λex =

√2A

µ0M2s

. (7)

For Py, λex ≈ 5.3 nm. Exchange forces dominate overlengths less than λex but demagnetization dominates overlarger lengths, allowing the �M(r) field to change its di-rection. At a boundary, the exchange effects are muchless present, and demagnetization helps �M(r) to pointparallel to the boundary, if possible.

2.2. The demagnetization field �HM in a thinmagnetic film

The demagnetization field is determined by the globalconfiguration of the magnetization of the system; it isderived from considerations of magnetostatics see Ref.[13] for the details of the approach used here. In theabsence of an external applied magnetic field one hasmagnetic induction �B = µ0( �HM + �M). Gauss’ Law,�∇�B = 0 then becomes

�∇ · �HM = −�∇ · �M. (8)

By assuming the the demagnetization field comes from ascalar potential via �HM = −�∇ΦM , a Poisson equationfor the magnetostatics is obtained:

−∇2ΦM = ρM , ρM ≡ −�∇ · �M. (9)

Therefore, the magnetization �M(r) produces an effectivemagnetic charge density ρM , that is the source in thisPoisson equation. The solution for scalar potential ΦM

can be obtained be various numerical methods. Gener-ally, we have used a scheme based on discretization ofthe system (introduction of a spatial grid), together withappropriate Green’s functions for the Poisson equation.In addition, it is extremely helpful to use the approxi-mation that the disk is very thin, L � a, where a is theradius (or the semi-major axis for an elliptical disk). Inthis case, both �M and �HM do not depend on the verticalcoordinate z along the disk axis. Then the problem can

be solved by using effective two-dimensional 2D Green’sfunctions [14]. The components α = x, y, z of the de-magnetization field can be expressed as 2D convolutionintegrals,

HMα (r) =

∫d2r′

∑β=x,y,z

Gαβ(r − r′)Mβ(r′) (10)

where Gαβ(r − r′) represents a tensor Green’s func-tion, and the integration is over 2D positions, e.g., nowr = (x, y). The evaluation of these integrals can be ac-celerated through the use of fast Fourier transforms [15].

2.3. Discretization and micromagnetics forsimulations

For numerical solutions of the magnetization field�M(r) = Msm(r), it is necessary to partition the systeminto cells labeled by index i for positions ri. We use asquare grid of cells of individual size acell×acell×L, withacell = 2.0 nm, and disk thickness L = 5 nm and L = 10nm. At the center of each cell is a unit direction vectormi, whose motion is to be followed. Each cell contains amagnetic dipole moment �µi of magnitude µ = La2

cellMs

and direction mi. This micromagnetics approach [16, 17]then represents the original continuum system, but witha discretized 2D micromagnetics Hamiltonian,

H = −J

∑

(i,j)

mi · mj +a2cell

λ2ex

∑i

(Hext

i + 12HM

i

)· mi

,

(11)where the effective exchange constant and energy scalebetween nearest-neighbor cells is J = 2AL, and magneticfields have been brought to dimensionless forms,

Hexti = �Hext

i /Ms, HMi = �HM

i /Ms. (12)

The presence of the factor a2cell/λ2

ex gives the relativestrength of demagnetization effects compared to ex-change effects. For the micromagnetics approach to bevalid, this factor should be much less than one. Thetransverse cell size acell should then be less than the ex-change length. The micromagnetics approach, with theassumption that only the direction of �M is changing,makes the implicit assumption that demagnetization ef-fects are a perturbation on exchange effects. Obviously,the Green’s functions Gαβ(r − r′) must also be broughtto a discrete form to carry out the calculation of HM

i .The Hamiltonian can be used to define the net mag-

netic inductions that act on each cell’s magnetic dipole�µi, according to

�Bi = − δHδ�µi

=J

µ�bi, (13)

where the dimensionless magnetic inductions are

�bi =∑

j∈z(i)

mj +a2cell

λ2ex

(Hext

i + HMi

). (14)

4

The first term involves a sum over the nearest neighborsz(i) of cell i; it is the exchange field. The second termrepresents the combination of external and demagnetiza-tion fields. The effective strength of magnetic inductionsis indicated by the unit we use for simulations,

B0 ≡ J/µ = µ0Msλ2ex/a2

cell. (15)

In the results presented here with acell = 2.0 nm, and Pyparameters, one has µ0M

2s ≈ 1.08 T and B0 ≈ 7.59 T.

B0 gives the order of magnitude of the exchange fields;the demagnetization fields are considerably weaker.

2.4. Static vortex configurations from a relaxationscheme

Static vortex configurations are derived as the station-ary solutions of the dynamic equations of motion. Atzero temperature, the undamped dynamic equation ofmotion is a simple torque equation for each magneticdipole, which interacts with its local net field:

d�µi

dt= γ�µi × �Bi. (16)

Note that this holds because �µi/γ = �Si is the spin angu-lar momentum of the cell, whose time derivative is thetorque, �µi × �Bi. The equation can be written in terms ofthe dimensionless quantities, also defining a dimension-less time τ ,

dmi

dτ= mi ×�bi, τ ≡ γB0t. (17)

The unit of time used for simulations is t0 ≡ (γB0)−1.For Py parameters, it takes the value t0 ≈ 0.75 ps. Itsreciprocal also defines a simulation frequency unit, f0 ≡γB0 ≈ 1.336 THz.

For static configurations, however, the time derivativesin (17) are zero. This implies that each dipole �µi or itsunit vector mi must align with the local field in that cell,�bi.

Thus, an algorithm that iteratively points each mi

along its current value of �bi will tend to move the systemtowards a static configuration. We call this approach aspin alignment relaxation scheme [18]. To carry it out,some initial state must be chosen from which to begin theiteration. Assume that the direction vectors are definedin terms of spherical planar angles (φi, θi), according to

mi = (cos θi cosφi, cos θi sin φi, sin θi). (18)

In this notation, φi is refered to as the in-plane angleand θi is the out-of-plane angle, which can be positive ornegative. The approximate in-plane structure of a vortexlocated at position R = (X, Y ) in a disk can be expressedusing the vorticity q = +1 as

φi = q tan−1

(yi − Y

xi − X

)+ φ0, (19)

where (xi, yi) is the 2D location of micromagnetics cell i,and φ0 = cπ

2 depends on the vortex circulation charge.There isn’t a corresponding analytic form for the out-of-plane component. Instead, one can start with all θi = 0,i.e., a planar vortex. However, the iteration will be suchthat all θi will remain zero, unless some small nonzerodeviation is included. Therefore, small random values ofθi can be used for the initial state. A nonzero out-of-plane component will then grow naturally as the systemrelaxes into a vortex state. The process is repeated un-til the changes in the mi become insignificant (less thanabout one part in 108).

For a circular or elliptic disk, if the vortex is initiatedaway from the center, as the spin alignment relaxationproceeds, the vortex will be found to both develop anout-of-plane component and also move to the disk center.Spin relaxation is an energy minimization algorithm; thesystem moves to its nearest minimum energy state, whichis that configuration centered in the disk. A profile of avortex obtained this way in an elliptic disk is shown inFig. 1. The projection of the dipoles onto the disk planeis shown. Note that there is a core region with a radius ofthe order of λex (region where �M has significant out-of-plane components, appearing as a hole in the diagram).Interestingly, the core tends to keep a reasonably circularform, as seen by the locus of points where the sign of Mz

reverses.

2.5. Effective potentials of a vortex in a nanodisk

Spin alignment relaxation can also be used to estimatethe effective potential U(R) for the vortex, by including aconstraint on the vortex position R = (X, Y ). The effec-tive potential is the system energy H less any constraintenergy, for a chosen vortex position. A constraint on vor-tex position can be enforced with the use of Lagrange’sundetermined multipliers [6]. Physically, a vortex can beshifted away from the disk center, by the application ofa magnetic field within the disk plane. A uniform field�Hext along ±x will displace the vortex along ±y, andvice-versa, with the sign determined by the vortex chi-rality. Buchanan et al. [8] were able to map out thevortex potential energy numerically by using the field tomove the vortex to different equilibrium positions. Thisgives one way to obtain the effective force constants kx

and ky.Rather than using a uniform applied field, it is pos-

sible to imagine the application of a spatially varyingfield, that primarily acts on the core region of the vortex.These fictitious extra fields are the undetermined La-grange multipliers; they are determined through courseof the calculation. Simultaneously, another constraint isapplied that ensures unit length for the direction vec-tors mi. The fictitious fields exert torques on the cellsin the core region, that hold the vortex in the desiredlocation, without significantly changing the overall vor-tex structure. Thus, a quasi-static vortex structure can

5

(a) X=16 nm, Y=0, it=2000, E=15.244 J

(b) X=0, Y=16 nm, it=3500, E=16.001 J

FIG. 2: Vortex structures for an elliptical nanodisk witha = 60 nm, b = 30 nm, L = 10 nm, using cell size acell =2.0 nm in the spin alignment relaxation scheme, including aLagrange constraint on position. (a) Vortex is held at X = 16nm, Y = 0, resulting in total energy E = 15.244J after 2000iterations. (b) Vortex is held at X = 0, Y = 16 nm, resultingin total energy E = 16.001J after 3500 iterations. CompareFig. 1, where X = Y = 0 and the energy is lower.

be obtained numerically, for whatever position is desired,within reason. The approach works best for a vortex nearthe disk center. For the same elliptical disk of Fig. 1, thevortex has been relaxed by this scheme to positions 16nm from the center, in Fig. 2. Note that the energy ishigher for a displacement along the shorter axis of theellipse [8].

The work here considers stable vortex states. It shouldbe kept in mind that for some parameters or disk sizes,the vortex could become unstable towards the formationof a lower energy quasi-single-domain state (nearly uni-form �M(r)), or some other multi-domain state withouta vortex. This is especially likely in the case of ellipticdisks with a high aspect ratio (b � a), where demagne-tization will strongly favor �M aligning with the longeraxis [7]. The vortex state will also become unstable ina circular disk if it is too thin, which minimizes the de-magnetization forces from the circular edge, that usuallystabilize a vortex. Also, if the disk is too thick (L � a),again, demagnetization will cause �M to approximatelyalign with the long axis and a vortex will not be stable.

Typical vortex potentials obtained from Lagrange-

-120 -80 -40 0 40 80 120X (nm)

10

15

20

25

E /

J

40 nm

60 nm

80 nm

100 nm

a=120 nm Vortex potentials, L=4.0 nm

FIG. 3: Numerically determined vortex potentials, in unitsof the effective cell exchange constant J = 2AL, for circularPy nanodisks of thickness L = 4.0 nm and indicated radii, asfound from the Lagrange constrained spin alignment relax-ation. The vortex becomes unstable towards escaping fromthe disk in the regions of downward curvature.

constrained spin aligment for circular nanodisks areshown in Fig. 3, for various radii with fixed thicknessL = 4.0 nm. The minimum energy region is close toparabolic form, however, as the vortex is placed closerto the edge, a lack of stability (downward curvature) ap-pears. Using the interior region of the potential, the ef-fective force constants kF for circular disks or even kx

and ky for elliptical disks can be estimated quite accu-rately. In the example of Fig. 3, one can observe that kF

becomes smaller for the larger radii disks. See Refs. [6, 7]for further details.

In elliptical disks [4, 7, 8], the force constant for dis-placement along the longer disk axis is found to be weakerthan that along the shorter disk axis; see Figs. 1 and 2.Thus, the potential acquires an elliptical shape that isdetermined by the original geometrical shape of the disk.For a disk with principal axes a and b with b < a, wehave found that for adequately large nanodisks and b ofsufficient size to stabilize the vortex, the ratio of forceconstants asymptotically approaches the relation,

√kx/ky ≈ b/a. (20)

This has the correct limit for a circular disk, kx = ky.The relation is summarized by saying that the geomet-ric ellipticity, b/a, directly determines an energetic el-lipticity,

√kx/ky. The energetic ellipticity can be seen

to determine the shape of the elliptical vortex orbits atconstant energy in the phase space.

6

3. MAGNETIC DYNAMICS AND THELANDAU-LIFSHITZ-GILBERT-LANGEVIN

EQUATIONS

The dynamics described by (16) or its dimensionlessform (17) is not completely realisitic, because it doesnot include the effects of damping nor of temperatureand its statistical fluctuations. Both the damping andthermal effects could be quite large on a vortex. Whenonly damping with a dimensionless parameter α is in-cluded, the well-known Landau-Lifshitz-Gilbert (LLG)dynamic equation [19, 20] is obtained [Eq. (21) but withall �bs,i = 0]. Here we take that one step further andalso include stochastic magnetic fields �bs,i(t) that repre-sent the effects of temperature. This leads to a Langevinequation derived from the LLG equation [21], for an in-dividual micromagnetics cell,

dmi

dτ= mi×

(�bi +�bs,i

)−αmi×

[mi ×

(�bi +�bs,i

)]. (21)

The changes in mi are a superposition of deterministiceffects (from �bi) and stochastic effects (from �bs,i). Thestochastic fields act to bring the system to thermal equi-librium. That takes place provided their correlations fol-low the fluctuation-dissipation (FD) theorem, which canbe written for this problem in the dimensionless quanti-ties as (site index i is suppressed)

〈bλs (τ) bλ′

s (τ ′)〉 = 2αT δλλ′ δ(τ − τ ′). (22)

Here δλλ′ is a Kronecker delta and the indices λ, λ′ re-fer to any of the Cartesian coordinates; δ(τ − τ ′) is aDirac delta function. The dimensionless temperature Tis thermal energy scaled by J ,

T ≡ kT

J=

kT

2AL, (23)

where k is Boltzmann’s constant and T is the absolutetemperature. The FD relation indicates how the stochas-tic magnetic fields move energy into and out of the sys-tem, in random processes that nevertheless can be quan-titatively measured. The stochastic fields are includedonly when a study of temperature effects in real time isdesired. They can be set to zero if the zero-temperaturedynamics is of interest, producing the LLG equation. Be-low we use solutions of Eq. (21) obtained appropriatelyfor the type of system under study, be it T = 0 or T > 0.

3.1. The Thiele equation for vortex core motion

Magnetic excitations such as domain walls and vor-tices do not obey Newtonian dynamics. Instead, it canbe shown from magnetic torque considerations (i.e., anal-ysis of the T = 0 LLG equations) that the steady-statedynamics of the core velocity V = R is described by aThiele equation [22],

F + G × V = 0. (24)

The motion is governed by the gyrovector G, which forvortices is a vector that points perpendicular to the diskplane, in a direction determined by the magnetizationat the vortex core. Consider a material with saturationmagnetization Ms. In terms of the magnetization perunit area, m0 ≡ MsL, and the gyromagnetic ratio γ =−1.76×1011 T−1 s−1 of an electron (its magnetic momentdivided by its angular momentum), the gyrovector of avortex is

G = Gz = 2πpqm0z/γ. (25)

For the vortices in a disk, q = +1, while there are two corepolarizations p = ±1 possible. The gyrovector pointsperpendicular to the disk in two possible directions. Asolution of the Thiele equation then gives a description ofthe motion of a vortex, provided it remains as a particle-like stable object under the dynamic environment it isfound in. A general review of vortex motion obeying aThiele equation, even including an intrinsic mass, is givenin [11].

Here we suppose that a vortex is moving within a nan-odisk of elliptical shape, at position R = (X, Y ), withthe force in Eq. (5) acting on it. One finds that it makesan elliptical orbital motion, whose gyrotropic frequencycan be estimated from the Thiele equation [7]. A solutionfor the vortex velocity is obtained quickly by taking thecross product of G = Gz with the Thiele equation,

G× F + G× (G× V) = 0. (26)

A vector identity is useful,

G× (G × V) = (G · V)G − (G ·G)V. (27)

The vortex velocity points in the plane of the disk but Gis perpendicular to that plane, so G · V = 0. This givesthe velocity as

V =G × F

G2=

1G

(kyY,−kxX) . (28)

With V = (X, Y ), this is a pair of first order differen-tial equations, which can be directly integrated, startingfrom some initial vortex position R(0) = (X0, Y0). Anelementary calculation gives elliptical motion, with in-stantaneous coordinates

X(t) = X0 cosωGt + Y0ky

GωGsin ωGt

Y (t) = Y0 cosωGt − X0kx

GωGsin ωGt (29)

where the gyrotropic frequency is determined by the ge-ometric mean of the force constants, k ≡

√kxky:

ωG = −√

kxky

G= − k

G. (30)

The negative square root is used, because a vortex withG along +z and a centrally directed force will move in the

7

clockwise (or negative) direction in the xy plane. Thisresult applies even when the vortex equilibrium positionis displaced from the disk center by an applied magneticfield, using the effective force constants at that displacedlocation [8]. Buchanan et al. [8] found that the exper-imentally measured vortex oscillation frequency can becontrolled by the application of an in-plane field, �Hext.Especially, �Hext along the short (or hard) axis of the el-lipse displaces the vortex on the long axis, where its fre-quency increases substantially due to position-dependentincreases in both force constants with X .

With �Hext = 0, one can find the shape of the ellipti-cal orbit and compare with the shape of the nanodisk.The vortex in undamped motion must move along anequipotential centered in the disk. The orbital energy Uis found to be

U = 12

(kxX2 + kyY 2

)= 1

2

(√kxX0 +

√kyY0

)2

. (31)

Dividing through by the constant, U , this is the standardequation of an ellipse, with the semi- major and minoraxes Xmax, Ymax, given by

Xmax =√

2U

kx, Ymax =

√2U

ky. (32)

Their ratio is then

Ymax

Xmax=

√kx

ky≈ b

a. (33)

The last approximate result in terms of the disk axes a, bwas obtained by using relation (20), valid only in thelimit of larger ellipses. Thus, the shape of the vortexorbit is nearly the same as the shape of the nanodisk.The energetic ellipticity (not to be confused with theeccentricity),

e ≡√

kx

ky, (34)

determines the ratio of the orbital axes. Indeed, the po-tential can be brought to a circular form, with a newcoordinate �ρ:

U(�ρ) = 12 k�ρ2, �ρ ≡

(√eX,

1√eY

). (35)

Then it is possible to show that the velocity follows atypical expression for circular motion,

�ρ = (ρx, ρy) = �ωG × �ρ. (36)

where �ωG = ωGz. This equivalent circular coordinate �ρis useful for the analysis of vortex position statistics inelliptical disks.

3.2. Numerical methods for magnetizationdynamics

The analysis of vortex motion via the Thiele equationis expected to be approximate. Numerical simulationscan be used to give a more complete and reliable descrip-tion of the dynamics. We require the time evolution fromEq. (21) solved either for zero temperature or finite tem-perature. These results are generated for Py parameters,based on the exchange length of λex = 5.3 nm, togetherwith a micromagnetics cell size of acell = 2.0 nm.

3.2.1. Zero temperature – 4th order Runge-Kutta

At zero temperature, a stable integration scheme is thewell-known 4th order Runge-Kutta (RK4) scheme, whichwe have used. A time step in dimensionless simulationunits of ∆τ = 0.04 is sufficient to insure good energyconserving dynamics (at zero damping), resulting in en-ergy conservation to one part in 1012 over as many as5.0 × 105 time steps, in systems with up to 4000 cells.To insure this high precision control of the energy, it isessential to evaluate the demagnetization field continu-ously during every sub-step of the RK4 algorithm. In thezero temperature simulations used to estimate gyrotropicfrequencies, the intial state of the dynamics is a vortexobtained by spin alignment relaxation to a desired posi-tion. It is also helpful to run the time evolution initiallywith some weak damping (α ≈ 0.02) for a limited time,followed by energy conserving dynamics (α = 0). Theinclusion of damping for a short interval helps to removeany spin wave oscillations that may be generated by aless than perfect initial vortex state. The subsequent en-ergy conserving dynamics then gives precise estimates ofthe frequencies ωG.

3.2.2. Finite temperature – Langevin dynamics via 2ndorder Heun method

For finite tempeatures the equations (21) have beensolved effectively by using a 2nd order Heun method(H2) [21, 23]. This scheme is equivalent to a two-stagepredictor-corrector algorithm, where the predictor stageis an Euler step and the corrector stage is the trapezoidrule. Both stages use the same random fields �bs,i, thatare produced by a random number generator. Any ofthe Cartesian components of these fields are to be ran-dom deviates with a zero mean value and a variance thatmust depend on both the dimensionless time step andtemperature according to

σs =√

2αT ∆τ . (37)

This is a result of the FD theorem (22), and it is used toreplace the stochastic fields integrated over a time step,

8

by the relation

∫ τn+∆τ

τn

dτ �bs,i(τ) −→ σs �wi. (38)

The vectors �wi are triplets of random deviates with zeromean and unit variance, for each site i. In usual pro-gramming there are standard random number gneratorswhich return a uniform deviate from 0 to 1, with a vari-ance found to be 1/

√12. These can be shifted into the

range from −0.5 to +0.5 and then rescaled by√

12σs toget stochastic field components of the correct mean andvariance (it does not need to be a Gaussian distribution)[7, 13].

4. VORTEX GYROTROPIC MOTION ATZERO-TEMPERATURE

In a circular nanodisk at zero temperature, with a ra-dial force F as in (2), the analysis from the Thiele equa-tion (24) shows that the vortex velocity always pointsalong the azimuthal direction:

V =Gz × F

G2= − γkF R

2πpqLMsφ. (39)

The minus sign indicates that a vortex with positive gy-rovector (along z) will move in the clockwise or nega-tive sense, in uniform circular motion, and oppositelyfor those with negative gyrovector. More generally, forelliptic nanodisks, the predicted gyrotropic frequency ob-tained from the Thiele equation is

ωG = − k

G= − γk

2πpqLMs, (40)

where k is the geometric mean of the force constantsalong the principal axes. For a circular system, k → kF .These results depend strongly on the force constants,which can be estimated from the static vortex configura-tions. It has been found [9, 13, 24] that for sufficientlylarge circular disks far from any stability limits of thevortex, the force constants are very roughly proportionalto L2/R, that is,

kF ≈ 14

L2

R

A

a2cell

≈ 0.878 µ0M2s

L2

R. (41)

The frequency unit f0 = t−10 = γB0 used in the sim-

ulations depends on the cell size, which is inconvenientfor comparison with experiment. Thus, it is important toconvert the results to a commonly used frequency unit,

ω0 ≡ µ0

4πγMs. (42)

This is ω0 = γMs in the centimeter-gram-second systemof units. With the help of definition (7) for the exchange

length, expression (40) for gyrotropic frequency can bewritten,

pωG = −(µ0

4πγMs

) (k

λex

A

) (λex

L

). (43)

With vorticity q = +1 assumed, the sign of ωG is deter-mined by the core polarization p. This expression sug-gests using k0 ≡ A/λex as the unit of force constant andλex as the unit of length.

Simulations can verify the frequency predictions fromthe Thiele equation. As shown below in some examples,the dimensionless periods τG of gyrotropic motion can beestimated precisely, in simulation time units (τ = t/t0).Dimensionless angular frequencies are then 2π/τG, whichare given physical values by multiplying by f0 = γB0.Using (15), these can then be converted into units of ω0

as follows:

ωG =2π

τGf0 =

2π

τGγµ0Ms

λ2ex

a2cell

=2π

τG

λ2ex

a2cell

4π ω0. (44)

We use this below to convert the raw numerical data (τG)into frequencies in ω0 units.

Of course, to get precise estimates of the frequency,the vortex must be instantaneously located to high pre-cision. That is a two step process. The first step is touse the singularity in the iin-plane magnetization angleφ, to locate the four cells nearest to the vorticity center,Rv, defined implicitely according to the relation

�∇× �∇φ(r) = 2πzδ(r − Rv). (45)

For the micromagnetics square grid, the vorticity centerfalls between the four cells that have a net 2π circulationin φ. This gives the location R ≈ Rv only to a precisionequal to the cell size. It can be greatly improved bymaking a weighted average of the cell locations ri, usingtheir squared out-of-plane components, which are largestin the vortex core, as the weighting function:

R =

∑|ri−Rv|<4λex

(mzi )

2 ri∑|ri−Rv|<4λex

(mzi )2

. (46)

For better efficiency, the sum is restricted to cells withinfour exchange lengths of the vorticity center. This avoidsusing useless data from from the core of interest (for ex-ample, spin wave oscillations near the edge of the diskshould be ignored). As the vortex moves, the resultingestimate for R changes smoothly. This algorithm evenworks very well for vortices moving in response to ther-mal fluctuations.

4.1. Circular nanodisks simulations

Some typical vortex motions in circular nanodisks ofradius a = 120 nm are presented in Fig. 4, as ob-tained from integration of the LLG equations by the

9

0 10000 20000 30000 40000τ

-5

0

5

10

15

20

25X

(nm

)

L=5 nm

L=10 nm

L=20 nm

all with X0= 4 nm

circular nanodisksa=120 nm, T=0 K

FIG. 4: Motions for one component of vortex position in cir-cular nanodisks from RK4 integration of the LLG equations(shifted for clarity). The damping α = 0.02 was turned offafter dimensionless time τ = 1000. The periods can be cal-culately accurately from the undamped motion. Graphs ofY (τ ) are of the same amplitudes and frequencies but shifteda quarter of a period.

0 0.2 0.4 0.6 0.8 1

kFλ

ex

2 / (AL)

0

0.2

0.4

0.6

0.8

1

ωG

/ ω0

slope =

1

circular nanodiskszero temperature

a=30 nma=60 nma=90 nma=120 nm

FIG. 5: Vortex gyrotropic frequency magnitudes from RK4(dynamics) simulations for circular nanodisks, with thick-nesses ranging from L = 2.0 nm to 20 nm, and indicateddisk radii, versus force constants scaled by disk thickness,obtained from Lagrange constrained spin alignment (static)simulations. The dashed line is the theoretical result (43)from the Thiele equation.

RK4 scheme. The initial states came from Lagrangeconstrained spin alignment to the initial position R =(4.0, 0) nm. A weak damping with parameter α = 0.02was included but turned off at dimensionless time τ =1000. The remaining evolution was used to estimate theperiods, τG, which are then converted using (44).

For the motions displayed in Fig. 4, the dimen-sionless periods for L = 5 nm, 10 nm, and 20 nmare τG ≈ 5800, 3270 and 1872, respectively. Fromstatics calculations of the effective potentials as de-

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1b / a

0

0.2

0.4

0.6

0.8

1

1.2

k Fλ ex

/ A

ky

kx

k=√kxk

y

_____

a=120 nm, L=10 nm

Elliptic nanodisks

FIG. 6: Effective potential force constants versus geometricellipticity b/a, for elliptic nanodisks of semi-major axis a =120 nm, and thickness L = 10 nm. Results were found byLagrange constrained spin alignment relaxation.

scribed earlier, the corresponding raw force constantsare kF acell/A ≈ 0.033863, 0.120192 and 0.419143, re-spectively, using acell = 2.0 nm. Rescaling by a factorλex/acell = 5.3/2.0 converts them into kF λex/A, that ap-pears in the Thiele theory expression (43). For these andother similar vortex motion simulations with L rangingfrom 2.0 nm to 20 nm, one can compare to the Thiele pre-diction by plotting the frequency ωG versus kF /L withunits as suggested from Eq. (43), see Fig. 5. Note thatfor a given radius a, the disk with the smallest L hasthe largest frequency. The result is that ωG, obtainedfrom dynamics simulations, is very close to linearly re-lated to kF /L, obtained from static simulations, with aunit slope for these units. This gives a strong verificationof the Thiele equation being applicable to vortex motionin nanodisks where the vortex is stable. Note that allsimulations here used a reasonably small vortex orbitalradius of about 4.0 nm, avoiding having the vortex coreapproach the disk edge, which would tend to destabilizethe vortex.

4.2. Elliptical nanodisk simulations

Simulations for elliptic nanodisks [7] offer an evenwider range of possibilities, because the variations withgeometric ellipticity b/a can be studied. For instance,the variation of the effective potential force constantshas a behavior like that in Fig. 6, for the particular casea = 120 nm and L = 10 nm. Both kx and ky were deter-mined from the potentials derived by spin alignment witha position constraint. Their geometric mean k, whichdetermines gyrotropic frequencies, is also shown. Thecurves for these force constants do not go below a mini-mum value of b/a, where the vortex becomes unstable.

The corresponding gyrotropic frequencies ωG for L =

10

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1b / a

0

0.1

0.2

0.3ω

G / ω

0

L=10 nm

L=5.0 nm

Elliptic nanodisksa=120 nm

FIG. 7: Gyrotropic frequency magnitudes versus geometricellipticity b/a, for elliptic nanodisks of semi-major axis a =120 nm, and thicknesses L = 5.0 nm and L = 10 nm. Resultswere found by simulations of the LLG equations using RK4integration. For L = 10 nm, compare the similar shape of thecurve of k in Fig. 6, as expected from ωG ∝ k in Eq. (43).

10 nm and also for L = 5.0 nm are shown in Fig. 7, ver-sus b/a. These were obtained from simulations the sameas those described for circular nanodisks. Note that theshapes of these curves are very similar to the curves ofk in Fig. 6, which is to be expected if the Thiele theory(43) is valid. The additional results for L = 5.0 nm areincluded to demonstrate the dependence on disk thick-ness. With thicker disks having a greater restoring forceand k ∝ L2, due to the extra area on the disk edges,the dependence of G ∝ L results is gyrotropic frequen-cies increasing roughly linearly in L. The results can bepresented in another view in Fig. 8, showing ωG/k ver-sus ellipticity for different L. One again gets a clear andquantitative verification of the Thiele theory result (43),seeing that ωG/k ∝ λex/L with the correct constant ofproportionality.

5. SPONTANEOUS GYROTROPIC MOTIONFROM THERMAL FLUCTUATIONS

Now we consider the effects of temperature on a vor-tex. Specifically, the temperature effectively acts as abath of random magnetic fields that exchange torquesand energy with the vortex. Even though that exchangeis somewhat random, one sees that it is able to sponta-neously initiate the organized gyrotropic motion of thevortex. That motion proceeds over a noisy backgroundof spin waves. Even so, it is readily apparent and per-sistent. Here we show typical time evolutions, and thenlater discuss statistical properties.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1b / a

0

0.2

0.4

0.6

0.8

1

1.2

ωG

/ k

(uni

ts o

f ω

0 / k 0 ) L=5.0 nm

L=10.0 nm

Elliptic nanodisks, a=120 nm- - - - from Thiele equation

FIG. 8: Gyrotropic frequency magnitudes (from dynamics)scaled by mean force constants (from statics), versus geomet-ric ellipticity b/a, for elliptic nanodisks with a = 120 nm andtwo different thicknesses. The results confirm the predictionsfrom the Thiele theory, dashed lines from Eq. (43), using ex-change length λex = 5.3 nm, with no adjustable parameters.

5.1. Simulation of a vortex initially at disk center

A vortex that has been relaxed to its minimum energyconfiguration (by the spin alignment scheme, for exam-ple) is situated in the disk center, whether it be circularor elliptical. This assumes that a quasi-single-domainstate is not lower in energy. Then, in the absence ofany external forces or forces due to a thermal environ-ment, it would statically remain centered in the disk andexhibit no dynamics. However, Machado et al. [25] no-ticed that finite temperature micromagnetics simulationsdemonstrate the sponteneous motion of the vortex, evenif it starts in it minimum energy location. This is rathersurprising, although it is really not much different thanany spin wave mode from being excited thermally in anequilibrium system with temperature. From the point ofview of statistical mechanics, any excitable modes (i.e.,independent degrees of freedom) should share equally inavailable thermal energy, and because the energy presentin the vortex gyrotropic motion is quite small, ratherlarge orbital motions can develop soley due to the effectsof temperature.

In the numerical solution [13] of the magnetic Langevinequation (21), the dimensionless temperature is required.For the simulation units being used, J = 2AL determinesthe energy scale and depends on the disk thickness. Asan example, we consider a disk with a = 60 nm, b = 30nm, and thickness L = 5.0 nm, at temperature T = 300K. For Py parameters (A = 13 pJ/m), the energy unit isJ = 130 zJ, while the thermal energy scale is kT = 4.14zJ, which gives the dimensionless temperature,

T =kT

2AL= 0.032, for T = 300 K, L = 5.0 nm. (47)

This was used to determine the variance of the random

11

0 10000 20000 30000 40000 50000τ

-10

-5

0

5

10

15X

, Y(n

m)

a=60 nm, b=30 nm, L= 5.0 nm, T=300 K

FIG. 9: Spontaneous vortex core motion caused by thermalfluctuations, as found by H2 integration of the LLG-Langevinequations (21) for a nanodisk with thickness L = 5.0 nm. Thevortex was initiated at the disk center, X = Y = 0. Thisgraph shows only 1/5 of the total data generated and usedsubsequently for analysis of vortex statistics, correspondingto hundreds of gryotropic periods.

0 10000 20000 30000 40000 50000τ

-5

0

5

X, Y

(nm

)

a=60 nm, b=30 nm, L= 10 nm, T=300 K

FIG. 10: Spontaneous vortex core motion caused by thermalfluctuations, for a nanodisk simulation identical to that inFig. 9, but with double the thickness, L = 10 nm. Note theconsiderably smaller amplitude of gyrotropic oscillations, andthe much higher frequency.

magnetic fields, Eq. (37), together with a damping pa-rameter α = 0.02 . A dimensionless time step ∆τ = 0.01for the second order Heun method was used. The result-ing vortex core coordinates (X(τ), Y (τ)) are displayed inFig. 9, out to a time of τ = 50000. From Fig. 9, a clock-wise orbital motion takes place, together with a noisybackground, and there are about 15 complete orbits forτ < 50000 (period τG ≈ 3300). The period is somewhatlonger than that found at zero temperature, τG ≈ 2970.This softening of the mode with temperature is to be ex-pected. In addition, the amplitudes of X and Y motionsare not equal, as expected from the elliptical disk shape.The gyrotropic orbital motion continues indefinitely; itwas followed out to τ = 2.5×105 to get vortex statistics.

For comparison, an identical simulation but with thedisk thickness increased by a factor of two to L = 10nm in shown in Fig. 10, again starting the vortex fromthe disk center. The greater thickness approximatelyquadruples k, but also doubles the gyrovector, therebyresulting in the frequency being double that for L = 5.0nm. It is also clearly apparent that the amplitude of thethermally indiced motion is reduced in the thicker nan-odisk (the graphs have different vertical scales). Thesedifferences then are primarily driven by the modifications

to the force constants and to G.

5.2. Thermal vortex motion as described by theThiele equation

Next we consider the statistical mechanics of the vor-tex core position R = (X, Y ), based on an effectiveLagrangian and Hamiltonian that give back the Thieleequation. The analysis [7] makes use of the general el-liptic potential U(R) in Eq. (4). It is straightforward tocheck that a Lagrangian whose Euler-Lagrange variationsgives back the Thiele equation is [13]

L = −12G(XY − Y X) − 1

2(kxX2 + kyY 2

)(48)

This is a particular choice of gauge and this Lagrangianis not unique, see Ref. [26] for a different choice. Totransform to the associated Hamiltonian, one finds thecanonical momentum for this symmetric gauge,

P =∂L

∂V=

12(GY,−GX). (49)

This shows that the Lagrangian can be expressed asL = P · V − U . As P is determined by X and Y , with-out any time derivatives, one can interpret this as a pairof constraint relations between components of P and R.It means that out of four original coordinates plus mo-menta, only two are independent.

The Hamiltonian is obtained in the usual way,

H = P · V − L = U =12

(kxX2 + kyY 2

)= 1

2 kρ2. (50)

Curiously, this has no momenta present. This strange sit-uation seems to imply that there is no dynamics, becausethe Hamilton equations of motion are

P = −∂H

∂R, R =

∂H

∂P. (51)

That would give V = R = 0, which is clearly wrong.This singular situation comes about because of the con-straint (49) between momentum and position compo-nents. In order to get a true dynamics, one needs torewrite the Hamiltonian half as a potential part and halfas a kinetic part, that is,

H =14

(kxX2 + kyY 2

)+

14

(2G

)2 (kxP 2

y + kyP 2x

). (52)

This is exactly equal to H in (50), but now it does giveback the Thiele equation when its time dynamics is foundfrom (51). Because of the constraint, the vortex motiondepends on only two independent coordinates, or degreesof freedom, rather than four. For the purposes of sta-tistical mechanics, then, the thermalized vortex motioncontains an average energy, 〈H〉 = 2 × 1

2kT .

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13ρ (nm)

0

0.1

0.2

0.3

0.4p(ρ)

Elliptic nanodisksa=60 nm , b=30 nm,

L=10 nm

L=5.0 nm

T=300 K

FIG. 11: The radial distribution of vortex core positions forthe simulations in Figs. 9 and 10, with a = 60 nm, b = 30,and thicknesses L = 5.0, 10 nm. Data out to final time τ =2.5×105 was used. The solid curves are the theory expression(57), using force constants k = 0.1753 k0 for L = 5.0 nm andk = 0.6832 k0 for L = 10 nm, as obtained from spin alignmentcalculations, with force constant unit k0 = A/λex.

5.3. Thermalized vortex probability distributionsfrom the Thiele equation

One can assume that any of the coordinates,X, Y, PX , PY , as well as effective circular coordinate �ρ =(ρx, ρy), obey a Boltzmann distribution, whose parame-ters are determined by the average energy,

〈H〉 = kT. (53)

This directly gives the mean squared effective circularradius for an elliptic disk,

〈ρ2〉 = 〈2H/k〉 = 2kT/k. (54)

This becomes the usual mean squared radius, 〈ρ2〉 →〈R2〉, in the limit of a circular disk. Using expression (50)for H , with the energy shared equally between X and Ymotions (equipartition theorem for quadratic degrees offreedom) implies that each coordinate has a mean squarevalue,

〈X2〉 = kT/kx, 〈Y 2〉 = kT/ky. (55)

For the systems we study, with b < a and kx < ky,this implies a wider range of motion for the X coordi-nate, as could obviously be expected. These relations forthe mean square values indicate the importance of theforce constants for describing the statistical distributionof vortex position.

Now consider determining the probability distributionsfor the vortex core location. The Hamiltonian is circu-larly symmetric when expressed in terms of the squareof the effective circular coordinate �ρ. We can supposethat each possible location has a probability determined

from a Boltzmann factor, e−βH , where β = (kT )−1. Em-ploying the circular symmetry for this coordinate, theprobability p(ρ)dρ of finding the vortex core within somerange dρ centered at radius ρ is proportional to the area2πρ dρ in a ring of radius ρ, and the Boltzmann factore−βH :

p(ρ)dρ ∝ 2πρ dρ e−βH = 2πρ dρ e−12 βkρ2

. (56)

By including a normalization constant, the unit normal-ized probability distribution function is easily found tobe

p(ρ) = βkρ e−12 βkρ2

. (57)

The root-mean-square radius ρrms =√

2kT/k impliedfrom relation (54) can be verified with this probabilityfunction. One can also get the the mean radius and themost probable radius:

〈ρ〉 =√

12πkT/k, ρmax =

√kT/k. (58)

For the simulations shown in Figs. 9 and 10, with a = 60nm, b = 30 nm, T = 300 K, position data out toτ = 2.5 × 105 was used to find histograms of vortex coreposition, and thereby get the radial probability distri-bution to compare with (57). The results are shown inFig. 11. To compare with theory, the force constantsfrom spin alignment relaxations were used, see the Fig.11 caption. Note also that as the gyrotropic frequency isconsiderably larger for L = 10 nm, those data correspondto many more orbits of the vortex, equivalent to a morecomplete averaging. Even so, the distributions for boththicknesses follow very closely to the expected form thatdepends on the validity of the Thiele equation, with noadjustable parameters.

Using H expressed in terms of both X and Y , theprobability to find the vortex core within some rangedX and dY of the location (X, Y ) is p(X, Y ) dX dY ∝e−βH dX dY , where the normalized probability functionis found to be

p(X, Y ) =

√βkx

2πe−

12βkxX2

√βky

2πe−

12βkyY 2

. (59)

This is a product of Gaussian distributions in each coor-dinate, p(X, Y ) = p(X)p(Y ), with zero mean values, butvariances given by

σx =√

kT/kx, σy =√

kT/ky. (60)

The distributions p(X) and p(Y ) from the simulationdata of Fig. 9 are shown in Fig. 12, and compare closelyto the theoretical expression (59). Clearly one could alsofind the corresponding distributions of the momentumcomponents by similar reasoning.

Instead of looking at the momentum components, wecan equivalently calculate a theoretical speed distributionfor the vortex core [13]. This is simplest if we use the

13

-15 -10 -5 0 5 10 15X, Y (nm)

0

0.05

0.1

0.15

p(X

), p

(Y)

p(X)p(Y)

Elliptic nanodiska=60 nm, b=30 nm

T=300 KL=5.0 nm

FIG. 12: Distributions of vortex core coordinates for theLLG-Langevin simulation in Fig. 9 with a = 60 nm, b = 30,and thickness L = 5.0 nm. The solid curves are the the-ory expressions from Eq. (59) based on the Thiele theoryfor vortex motion, using force constants kx = 0.1156 k0

and ky = 0.2657 k0 from spin alignment relaxation, wherek0 = A/λex.

effective circular coordinate �ρ, and consider that fact thatits velocity in Eq. (36) implies a speed u ≡ |ρ| given by

u = |ωG|ρ. (61)

As u is proportional to ρ, so are their probability distri-butions. If g(u) is the desired speed probability distribu-tion, then conservation of probability states that

g(u)du = p(ρ)dρ = p(u/|ωG|)du/|ωG|. (62)

Thus the speed distribution is derived from the effectivecircular coordinate distribution by

g(u) = |ωG|−1 p(u/|ωG|). (63)

With |ωG| = k/G, one obtains

g(u) =βG2

ku e−

12βG2u2/k =

2u

u2rms

e−u2/u2rms . (64)

This depends on the root-mean-square speed, determinedfrom ρrms,

urms = |ωG|ρrms =√

2kT k/G. (65)

The function g(u) is a Maxwellian speed distribution sim-ilar to that for an ideal gas. One could consider the factorin the exponent as depending on a kinetic energy term12mGu2 for a particle, where mG is some mass associatedwith that particle in gyrotropic motion. From (64) onecan read off the value needed for this mass,

mG = G2/k. (66)

This curious result gives a kind of effective mass that de-pends on the potential experienced by the vortex. Thus

it should not be consider an intrinsic vortex mass. Gen-erally, G is linearly proportional to thickness L [see ex-pression (25)], whereas k tends to increase approximatelywith L2 [see expression (41) and also Sec. 4.2], makingthis mass nearly independent of L. Probably mG is morestrongly determined by the disk area, πab. In the case ofcircular disks, using the approximate expression (41) fork = kF and the definition (25) of G gives a quantitativeresult,

mG ≈ (2π)2a0.878µ0γ2

. (67)

Thus, the mass is determined primarily by the disk ra-dius a, and it does not depend on the material parameterssuch as the exchange stiffness or saturation magnetiza-tion. For a radius a = 100 nm the mass is 1.2 × 10−22

kg, an extremely small value. Even so, the mass can betaken to represent how a vortex responds dynamically tothe potential. With the gyrotropic frequency given by|ωG| = k/G, the mas is written equivalently as

mG = G/|ωG| ∝ L/|ωG|. (68)

With mG depending only on disk radius or possibly areain the xy plane, and the gyrovector proportional to L,this re-expresses that |ωG| is also proportional to L, asshown implicitly in Figs. 5 and 7.

6. SUMMARY AND INTERPRETATION OFRESULTS

This chapter has provided an overview of some meth-ods for finding the static, dynamic and statistical prop-erties of vortex excitations in thin nanodisks of soft mag-netic material. By assuming the thickness is much lessthan the principal radius, L � a, the magnetizationpoints primarily within the plane of the disk, exceptwithin the vortex core, and it has only weak dependenceon the coordinate z perpendicular to the plane. Thisallowed for the transformation to an equivalent 2D prob-lem, which has been studied here using a form of micro-magnetics, converting the continuum problem to one ona square grid.

The Lagrange-constrained spin alignment scheme wasused to find static vortex energies while securing the vor-tex in a desired location R, thereby allowing for the cal-culation of vortex potential U(R) within the disk. For avortex near the center of an elliptic disk, the force con-stants kx and ky for displacements along the principalaxes a, b are found, with kx ≤ ky when b ≤ a. However,the disk ellipticity b/a must be above a lower limitingvalue for a vortex to stable; a very narrow disk will pre-fer the formation of a quasi-single-domain state, or evena multi-domain state, but not a vortex. A vortex ener-getically prefers a displacement along the longer axis ofthe disk; that is consistent with the shape of its ellipticorbits, which have the same ellipticity as the disk itself[see Eq. (33)].

14

The vortex gyrotropic orbits can be described very wellthrough the use of the Thiele equation (24), which re-places the dynamics of the many degrees of freedom inthe magnetization field M(r, t) by the dynamics of onlytwo Cartesian components in the vortex core location,R = (X(t), Y (t)). This works best for a vortex near thedisk center, where it is unlikely to be destabilized by de-formations caused by the boundaries. For zero temper-ature, the dynamics from RK4 integration of the LLGequations is completely consistent with that from theThiele equation. The Thiele equation predicts the vor-tex gyrotropic frequencies to be ωG = −k/G, which isconfirmed in the dynamics simulations while using forceconstants from the Lagrange constrained static vortexstructures. Generally the zero-temperature gyrotropicfrequencies are roughly proportional to L/a with only aweak dependence on disk ellipticity, as can be concludedfrom the results in Fig. 7. The frequencies are determinedby the geometric mean force constant, k =

√kxky, which

shows why knowledge of the vortex potential is importantfor this problem.

Thermal effects for nonzero temperature have been in-cluded by introducing a Langevin equation (21) that re-sults from including stochastic magnetic fields into theLLG equation. This Langevin equation gives the timeevolution in the presence of thermal fluctuations. Solvednumerically using a 2nd order Heun algorithm, a vortexinitially at the disk center (the minimum energy point)will spontaneously undergo gyrotropic orbital motion, ontop of a noisy spin wave background. The orbital mo-tion takes place at a slightly lower frequency comparedto its motion for T = 0, because the presence of spinwaves weakens the exchange stiffness of the system. Theresulting distribution of vortex position can be predictedby using an effective Lagrangian and Hamiltonian thatresult from the Thiele equation. That Hamiltonian canbe expressed in a form (50) containing only a potentialenergy. This then shows that the distributions (and vari-ances) of effective radial coordinate ρ and Cartesian coor-

dinates X and Y depend on√

kT/kF , where kF is eitherk or kx or ky, respectively [see Eqs. (57) and (60)]. Sur-prisingly large vortex rms displacements on the order of1 – 10 nm can result, with the larger values taking placein the weaker potentials of thinner disks (Fig. 11) and indisks with larger radii a. However, these noisy ellipticalmotions simpy reflect the equipartition of energy into thetwo collective degrees of freedom available to the vortex(X, Y ), with each receiving an average thermal energy of12kT . The radial coordinate, in contrast, receives a fullkT of energy on average. The theoretical probability dis-tributions are confirmed in simulations provided the timeevolution averages over a large number of gyrotropic or-bits.

A vortex speed distribution can also be derived fromthe position distribution, essentially because the momen-tum and position coordinates of a vortex are not inde-pendent. The speed distribution g(u) can be character-ized by a mass mG proportional to the disk radius a,but independent of material properties. The mass hasthe sense that as the vortex position fluctuates, it hassome Maxwellian speed distribution, with a kinetic en-ergy 1

2mGu2 that enters in the Boltzmann factor. This isin contrast to the Thiele equation, which has been usedhere with no intrinsic mass term. Indeed, the vortex gy-rotropic frequency is the same as that for a corresponding2D harmonic oscillator of mass mG and spring constantk, that is, ωG =

√k/mG.

Acknowledgments

Portions of this work benefited substantially from dis-cussions with Afranio Pereira and Winder Moura-Melo atthe Universidade Federal de Vicosa, Minas Gerais, Brazil,and Wagner Figueiredo at the Universidade Federal deSanta Catarina, Florianopolis, Brazil, and from use ofcomputation facilities at both universities.

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