## 1. Introduction: vortices in thin submicron magnetic disks

A cylindrically shaped thin disk of soft ferromagnetic material with a radius *L* ≪ *shape anisotropy*, any such magnetic system tends to avoid the formation of magnetic poles on the surfaces, if possible, which would raise the total energy. For a thin circular disk, the local magnetization

Within the disk, the forces of ferromagnetic exchange cause

One can also consider deviations from circular symmetry, such as in elliptic nanodisks, where magnetic vortex dynamics has been studied by measuring their radio frequency oscillations [4] and even by direct electrical contact [5] to a nanodisk. An example of a magnetic vortex centered in a thin elliptical nanodisk is shown in **Figure 1**. It has been obtained from a numerical relaxation algorithm [6], see Section 2.4 below. Although there is a tendency for **Figure 1**.

### 1.1. Vortex charges

The magnetization profile *circulation charge* *chirality*. It provides one topologically stable geometric property that could be used for data storage in a vortex, if it can be reliably controlled and detected.

In principle, a vortex profile *vorticity charge* *direction of rotation* of *antivortex*, would only be energetically stable if demagnetization effects were not present. The limit of zero thickness would eliminate the relevant demagnetization and make antivortices energetically possible.

The magnetization at the vortex core can take one of two values, *polarization charge*. Because there is an energy barrier to flip the core polarization from

### 1.2. Vortex potential and forces

The above-described magnetic vortex will have its minimum energy when it is centered in the disk. The location of the poles (where *vortex core*, which we denote by position vector

where *effective force*

For a circular disk, the potential is circularly symmetric, and then small displacements lead to a circularly symmetric Hooke’s law type of force. It is also possible to consider magnetic vortices in a cylindrical disk of *elliptical* shape [7], defined by principal axes

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

The parabolic functional form now has separate force constants

While a vortex in a nanodisk experiences a force directed roughly towards the disk center, its motion tends to be in an orbital sense, which is the gyrotropic oscillation mode [9, 10]. This is discussed further in Section 3 on dynamics. Before coming to that, we begin by a quantitative description of the calculation of vortex structures.

## 2. Analysis of quasi-stationary vortices in a nanodisk

The theoretical analysis is based on the statics and dynamics of the magnetization field, which is now assumed to keep a uniform magnitude

### 2.1. Energetics of a continuum nanomagnet

The system is governed by ferromagnetic exchange energy and interactions of

where

For Py,

### 2.2. The demagnetization field H → M in a thin magnetic film

The demagnetization field is determined by the global configuration of the magnetization of the system; it is derived from considerations of magnetostatics (see Ref. [13] for the details of the approach used here). In the absence of an external applied magnetic field, one has magnetic induction

By assuming the demagnetization field comes from a scalar potential via

Therefore, the magnetization

where

### 2.3. Discretization and micromagnetics for simulations

For numerical solutions of the magnetization field _{cell} × _{cell} × *L*, with _{cell}*L* = 5 nm and L = 10 nm. At the center of each cell is a unit direction vector *L*_{cell}^{2}*M*_{s} and direction *micromagnetics* approach [16, 17] then represents the original continuum system, but with a discretized 2D micromagnetics Hamiltonian,

where the effective exchange constant and energy scale between nearest-neighbor cells is

The presence of the factor _{cell}^{2}_{cell} should then be less than the exchange length. The micromagnetics approach, with the assumption that only the direction of

The Hamiltonian can be used to define the net magnetic inductions that act on each cell’s magnetic dipole

where the dimensionless magnetic inductions are

The first term involves a sum over the nearest neighbors

In the results presented here with _{cell}

### 2.4. Static vortex configurations from a relaxation scheme

Static vortex configurations are derived as the stationary solutions of the dynamic equations of motion. At zero temperature, the undamped dynamic equation of motion is a simple torque equation for each magnetic dipole, which interacts with its local net field:

Note that this holds because

The unit of time used for simulations is

For static configurations, however, the time derivatives in Eq. (17) are zero. This implies that each dipole

Thus, an algorithm that iteratively points each *spin alignment* relaxation scheme [18]. To carry it out, some initial state must be chosen from which to begin the iteration. Assume that the direction vectors are defined in terms of *spherical planar angles*

In this notation, *in-plane angle* and *out-of-plane* angle, which can be positive or negative. The approximate in-plane structure of a vortex located at position

where (*x _{i}, y_{i}*) is the 2D location of micromagnetics cell

For a circular or elliptic disk, if the vortex is initiated away from the center, as the spin alignment relaxation proceeds, the vortex will be found to both develop an out-of-plane component and also move to the disk center. Spin relaxation is an energy minimization algorithm; the system moves to its nearest minimum energy state, which is that configuration centered in the disk. A profile of a vortex obtained this way in an elliptic disk is shown in **Figure 2**. The projection of the dipoles onto the disk plane is shown. Note that there is a core region with a radius of the order of

### 2.5. Effective potentials of a vortex in a nanodisk

Spin alignment relaxation can also be used to estimate the effective potential

Rather than using a uniform applied field, it is possible to imagine the application of a spatially varying field, which primarily acts on the core region of the vortex. These fictitious extra fields are the undetermined Lagrange multipliers; they are determined through course of the calculation. Simultaneously, another constraint is applied that ensures unit length for the direction vectors **Figure 1**, the vortex has been relaxed by this scheme to positions 16 nm from the center, in **Figure 2**. Note that the energy is higher for a displacement along the shorter axis of the ellipse [8].

The work here considers stable vortex states. It should be kept in mind that for some parameters or disk sizes, the vortex could become unstable towards the formation of a lower energy quasi-single-domain state (nearly uniform

Typical vortex potentials obtained from Lagrange-constrained spin alignment for circular nanodisks are shown in **Figure 3**, for various radii with fixed thickness **Figure 3**, one can observe that

In elliptical disks [4, 7, 8], the force constant for displacement along the longer disk axis is found to be weaker than that along the shorter disk axis; see **Figures 1** and **2**. Thus, the potential acquires an elliptical shape that is determined by the original geometrical shape of the disk. For a disk with semi-major axes

This has the correct limit for a circular disk, *geometric ellipticity*, *b*/*energetic ellipticity*,

## 3. Magnetic dynamics and the Landau-Lifshitz-Gilbert-Langevin equations

The dynamics described by Eq. (16) or its dimensionless form in Eq. (17) is not completely realistic, because it does not include the effects of damping nor of temperature and its statistical fluctuations. Both the damping and thermal effects could be quite large on a vortex. When only damping with a dimensionless parameter

The changes in

Here, *δ _{λλ′}* is a Kronecker delta and the indices λ,λ′ refer to any of the Cartesian coordinates;

where

### 3.1. The Thiele equation for vortex core motion

Magnetic excitations such as domain walls and vortices do not obey Newtonian dynamics. Instead, it can be shown from magnetic torque considerations (i.e., analysis of the *Thiele equation* [22],

The motion is governed by the *gyrovector* ^{−1} s^{−1} of an electron (its magnetic moment divided by its angular momentum), the gyrovector of a vortex is

For the vortices in a disk,

Here, we suppose that a vortex is moving within a nanodisk of elliptical shape, at position

A vector identity is useful,

The vortex velocity points in the plane of the disk, but

With

where the gyrotropic frequency is determined by the geometric mean of the force constants,

The negative square root is used, because a vortex with

With

Dividing through by the constant,

Their ratio is then

The last approximate result in terms of the disk axes *energetic ellipticity* (not to be confused with the eccentricity),

determines the ratio of the orbital axes. Indeed, the potential can be brought to a circular form, with a new coordinate

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

where

### 3.2. Numerical methods for magnetization dynamics

The analysis of vortex motion via the Thiele equation is expected to be approximate. Numerical simulations can be used to give a more complete and reliable description of the dynamics. We require the time evolution from Eq. (21) solved either for zero temperature or finite temperature. These results are generated for Py parameters, based on the exchange length of _{cell}

#### 3.2.1. Zero temperature: fourth-order Runge-Kutta

At zero temperature, a stable integration scheme is the well-known fourth-order Runge-Kutta (RK4) scheme, which we have used. A time step in dimensionless simulation units of Δ*τ* = 0.04 is sufficient to insure good energy conserving dynamics (at zero damping), resulting in energy conservation to one part in

#### 3.2.2. Finite temperature: Langevin dynamics via second-order Heun method

For finite temperatures, the Eq. (21) has been solved effectively by a second-order Heun method (H2) [21, 23]. This scheme is equivalent to a two-stage predictor-corrector algorithm, where the predictor stage is an Euler step and the corrector stage is the trapezoid rule. Both stages use the same random fields

This is a result of the FD theorem Eq. (22), and it is used to replace the stochastic fields integrated over a time step, by the relation

The vectors

## 4. Vortex gyrotropic motion at zero temperature

In a circular nanodisk at zero temperature, with a radial force

The minus sign indicates that a vortex with positive gyrovector (along

where

The frequency unit

This is

With vorticity

Simulations can verify the frequency predictions from the Thiele equation. As shown below in some examples, the dimensionless periods

We use this below to convert the raw numerical data (

Of course, to get precise estimates of the frequency, the vortex must be instantaneously located to high precision. That is a two step process. The first step is to use the singularity in the in-plane magnetization angle

For the micromagnetics square grid, the vorticity center falls between the four cells that have a net

For better efficiency, the sum is restricted to cells within four exchange lengths of the vorticity center. This avoids using useless data from the core of interest (e.g., spin wave oscillations near the edge of the disk should be ignored). As the vortex moves, the resulting estimate for

### 4.1. Circular nanodisks simulations

Some typical vortex motions in circular nanodisks of radius **Figure 4**, as obtained from integration of the LLG equations by the RK4 scheme. The initial states came from Lagrange-constrained spin alignment to the initial position

For the motions displayed in **Figure 4**, the dimensionless periods for *L* = 5 nm, 10 nm, and 20 nm are *k*_{f }_{cell} _{cell} = 2.0 nm. Rescaling by a factor λ_{ex}/_{cell} = 5.3/2.0 converts them into **Figure 5**). Note that for a given radius *dynamics* simulations, is very close to linearly related to *static* simulations, with a unit slope for these units. This gives a strong verification of the Thiele equation being applicable to vortex motion in nanodisks where the vortex is stable. Note that all simulations here used a reasonably small vortex orbital radius of about 4.0 nm, avoiding having the vortex core approach the disk edge, which would tend to destabilize the vortex.

### 4.2. Elliptical nanodisk simulations

Simulations for elliptic nanodisks [7] offer an even wider range of possibilities, because the variations with geometric ellipticity **Figure 6**, for the particular case

The corresponding gyrotropic frequencies **Figure 7**, versus **Figure 6**, which is to be expected if the Thiele theory (43) is valid. The additional results for **Figure 8**, showing

## 5. Spontaneous gyrotropic motion from thermal fluctuations

Now we consider the effects of temperature on a vortex. Specifically, the temperature effectively acts as a bath of random magnetic fields that exchange torques and energy with the vortex. Even though that exchange is somewhat random, one sees that it is able to spontaneously initiate the organized gyrotropic motion of the vortex. That motion proceeds over a noisy background of spin waves. Even so, it is readily apparent and persistent. Here, we show typical time evolutions, and then later discuss statistical properties.

### 5.1. Simulation of a vortex initially at disk center

A vortex that has been relaxed to its minimum energy configuration (e.g., by the spin alignment scheme) is situated in the disk center, whether it be circular or elliptical. This assumes that a quasi-single-domain state is not lower in energy. Then, in the absence of any external forces or forces due to a thermal environment, it would statically remain centered in the disk and exhibit no dynamics. However, Machado et al. [25] noticed that finite temperature micromagnetics simulations demonstrate the spontaneous motion of the vortex, even if it starts in it minimum energy location. This is rather surprising, although it is really not much different than any spin wave mode from being excited thermally in an equilibrium system with temperature. From the point of view of statistical mechanics, any excitable modes (i.e., independent degrees of freedom) should share equally in available thermal energy, and because the energy present in the vortex gyrotropic motion is quite small, rather large orbital motions can develop solely due to the effects of temperature.

In the numerical solution [13] of the magnetic Langevin equation (21), the dimensionless temperature is required. For the simulation units being used,

This was used to determine the variance of the random magnetic fields, Eq. (37), together with a damping parameter **Figure 9**, out to a time of **Figure 9**, a clockwise orbital motion takes place, together with a noisy background, and there are about 15 complete orbits for

For comparison, an identical simulation but with the disk thickness increased by a factor of 2 to **Figure 10**, again starting the vortex from the disk center. The greater thickness approximately quadruples

### 5.2. Thermal vortex motion as described by the Thiele equation

Next, we consider the statistical mechanics of the vortex core position

This is a particular choice of gauge and this Lagrangian is not unique (see Ref. [26] for a different choice). To transform to the associated Hamiltonian, one finds the canonical momentum for this symmetric gauge,

This shows that the Lagrangian can be expressed as **P** is determined by

The Hamiltonian is obtained in the usual way,

Curiously, this has no momenta present. This strange situation seems to imply that there is no dynamics, because the Hamilton equations of motion are

That would give

This is exactly equal to

### 5.3. Thermalized vortex probability distributions from the Thiele equation

One can assume that any of the coordinates,

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

This becomes the usual mean squared radius,

For the systems we study, with *b* < *a* and

Now consider determining the probability distributions for the vortex core location. The Hamiltonian is circularly symmetric when expressed in terms of the square of the effective circular coordinate

By including a normalization constant, the unit normalized probability distribution function is easily found to be

The root-mean-square radius

For the simulations shown in **Figures 9** and **10**, with **Figure 11**. To compare with theory, the force constants from spin alignment relaxations were used (see the **Figure 11** caption). Note also that as the gyrotropic frequency is considerably larger for **Figure 12**).

Using

This is a product of Gaussian distributions in each coordinate,

The distributions p(X) and p(Y) found from the simulation data of **Fig. 9** are shown in **Fig. 12**, and compare closely to the theoretical expression (59).

Clearly one could also find the corresponding distributions of the momentum components by similar reasoning.

Instead of looking at the momentum components, we can equivalently calculate a theoretical speed distribution for the vortex core [13]. This is simplest if we use the effective circular coordinate

As

Thus, the speed distribution is derived from the effective circular coordinate distribution by

With

This depends on the root-mean-square speed, determined from

The function

This curious result gives a kind of effective mass that depends on the potential experienced by the vortex. Thus, it should not be consider an intrinsic vortex mass. Generally, *L*. Probably,

Thus, the mass is determined primarily by the disk radius

With **Figures 5** and **7**.

## 6. Summary and interpretation of results

This chapter has provided an overview of some methods for finding the static, dynamic, and statistical properties of vortex excitations in thin nanodisks of soft magnetic material. By assuming the thickness is much less than the principal radius,

The Lagrange-constrained spin alignment scheme was used to find static vortex energies while securing the vortex in a desired location *a*, *b* are found, with

The vortex gyrotropic orbits can be described very well through the use of the Thiele equation (24), which replaces the dynamics of the many degrees of freedom in the magnetization field *dynamics* simulations while using force constants from the Lagrange-constrained *static* vortex structures. Generally, the zero-temperature gyrotropic frequencies are roughly proportional to **Figure 7**. The frequencies are determined by the geometric mean force constant,

Thermal effects for nonzero temperature have been included by introducing a Langevin equation (21) that results from including stochastic magnetic fields into the LLG equation. This Langevin equation gives the time evolution in the presence of thermal fluctuations. Solved numerically using a second-order Heun algorithm, a vortex initially at the disk center (the minimum energy point) will *spontaneously* undergo gyrotropic orbital motion, on top of a noisy spin wave background. The orbital motion takes place at a slightly lower frequency compared with its motion for **Figure 11**) and in disks with larger radii

A vortex speed distribution can also be derived from the position distribution, essentially because the momentum and position coordinates of a vortex are not independent. The speed distribution