The corresponding gyrotropic frequencies for nm and also for nm are shown in Figure 7, versus . These were obtained from simulations the same as those described for circular nanodisks. Note that the shapes of these curves are very similar to the curves of in Figure 6, which is to be expected if the Thiele theory (43) is valid. The additional results for nm are included to demonstrate the dependence on disk thickness. With thicker disks having a greater restoring force and , due to the extra area on the disk edges, the dependence of results is gyrotropic frequencies increasing roughly linearly in . The results can be presented in another view in Figure 8, showing versus ellipticity for different . One again gets a clear and quantitative verification of the Thiele theory result (43), seeing that with the correct constant of proportionality.
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.
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.  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  of the magnetic Langevin equation (21), the dimensionless temperature is required. For the simulation units being used, determines the energy scale and depends on the disk thickness. As an example, we consider a disk with nm, nm, and thickness nm, at temperature K. For Py parameters (pJ/m), the energy unit is zJ, while the thermal energy scale is zJ, which gives the dimensionless temperature,
This was used to determine the variance of the random magnetic fields, Eq. (37), together with a damping parameter . A dimensionless time step for the second-order Heun method was used. The resulting vortex core coordinates are displayed in Figure 9, out to a time of . From Figure 9, a clockwise orbital motion takes place, together with a noisy background, and there are about 15 complete orbits for (period ). The period is somewhat longer than that found at zero temperature, . This softening of the mode with temperature is to be expected. In addition, the amplitudes of and motions are not equal, as expected from the elliptical disk shape. The gyrotropic orbital motion continues indefinitely; it was followed out to to get vortex statistics.
For comparison, an identical simulation but with the disk thickness increased by a factor of 2 to nm in shown in Figure 10, again starting the vortex from the disk center. The greater thickness approximately quadruples , but also doubles the gyrovector, thereby resulting in the frequency being double that for nm. It is also clearly apparent that the amplitude of the thermally induced motion is reduced in the thicker nanodisk (the graphs have different vertical scales). These differences then are primarily driven by the modifications to the force constants and to .
Next, we consider the statistical mechanics of the vortex core position , based on an effective Lagrangian and Hamiltonian that give back the Thiele equation. The analysis  makes use of the general elliptic potential in Eq. (4). It is straightforward to check that a Lagrangian whose Euler-Lagrange variations gives back the Thiele equation is 
This is a particular choice of gauge and this Lagrangian is not unique (see Ref.  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 . As P is determined by and , without any time derivatives, one can interpret this as a pair of constraint relations between components of and . It means that of four original coordinates plus momenta, only two are independent.
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 , which is clearly wrong. This singular situation comes about because of the constraint (49) between momentum and position components. In order to get a true dynamics, one needs to rewrite the Hamiltonian half as a potential part and half as a kinetic part, that is,
This is exactly equal to in Eq. (50), but now it does give back the Thiele equation when its time dynamics is found from Eq. (51). Because of the constraint, the vortex motion depends on only two independent coordinates, or degrees of freedom, rather than four. For the purposes of statistical mechanics, then, the thermalized vortex motion contains an average energy, .
One can assume that any of the coordinates, , as well as effective circular coordinate , obey a Boltzmann distribution, whose parameters are determined by the average energy,
This directly gives the mean squared effective circular radius for an elliptic disk,
This becomes the usual mean squared radius, , in the limit of a circular disk. Using expression (50) for , with the energy shared equally between and motions (equipartition theorem for quadratic degrees of freedom) implies that each coordinate has a mean square value,
For the systems we study, with b < a and , this implies a wider range of motion for the coordinate, as could obviously be expected. These relations for the mean square values indicate the importance of the force constants for describing the statistical distribution of vortex position.
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 . We can suppose that each possible location has a probability determined from a Boltzmann factor, , where . Employing the circular symmetry for this coordinate, the probability of finding the vortex core within some range centered at radius is proportional to the area in a ring of radius , and the Boltzmann factor :
By including a normalization constant, the unit normalized probability distribution function is easily found to be
The root-mean-square radius implied from relation (54) can be verified with this probability function. One can also get the mean radius and the most probable radius:
For the simulations shown in Figures 9 and 10, with nm, nm, K, position data out to was used to find histograms of vortex core position, and thereby get the radial probability distribution to compare with Eq. (57). The results are shown in 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 . nm, those data correspond to many more orbits of the vortex, equivalent to a more complete averaging. Even so, the distributions for both thicknesses follow very closely to the expected form that depends on the validity of the Thiele equation, with no adjustable parameters (see Figure 12).
Using expressed in terms of both and , the probability to find the vortex core within some range and of the location is , where the normalized probability function is found to be
This is a product of Gaussian distributions in each coordinate, , with zero mean values, but variances given by
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 . This is simplest if we use the effective circular coordinate , and consider that fact that its velocity in Eq. (36) implies a speed given by
As is proportional to , so are their probability distributions. If is the desired speed probability distribution, then conservation of probability states that
Thus, the speed distribution is derived from the effective circular coordinate distribution by
With , one obtains
This depends on the root-mean-square speed, determined from ,
The function is a Maxwellian speed distribution similar to that for an ideal gas. One could consider the factor in the exponent as depending on a kinetic energy term for a particle, where is some mass associated with that particle in gyrotropic motion. From Eq. (64), one can read off the value needed for this mass,
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, is linearly proportional to thickness [see expression (25)], whereas tends to increase approximately with [see expression (41) and also Section 4.2], making this mass nearly independent of L. Probably, is more strongly determined by the disk area, . In the case of circular disks, using the approximate expression (41) for and the definition (25) of gives a quantitative result,
Thus, the mass is determined primarily by the disk radius , and it does not depend on the material parameters such as the exchange stiffness or saturation magnetization. For a radius nm, the mass is kg, an extremely small value. Even so, the mass can be taken to represent how a vortex responds dynamically to the potential. With the gyrotropic frequency given by , the mass is written equivalently as
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 magnetization points primarily within the plane of the disk, except within the vortex core, and it has only weak dependence on the coordinate perpendicular to the plane. This allowed for the transformation to an equivalent 2D problem, which has been studied here using a form of micromagnetics, converting the continuum problem to one on a square grid.
The Lagrange-constrained spin alignment scheme was used to find static vortex energies while securing the vortex in a desired location , thereby allowing for the calculation of vortex potential within the disk. For a vortex near the center of an elliptic disk, the force constants and for displacements along the principal axes a, b are found, with when . However, the disk ellipticity must be above a lower limiting value for a vortex to stable; a very narrow disk will prefer the formation of a quasi-single-domain state, or even a multi-domain state, but not a vortex. A vortex energetically prefers a displacement along the longer axis of the disk; that is consistent with the shape of its elliptic orbits, which have the same ellipticity as the disk itself [see Eq. (33)].
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 by the dynamics of only two Cartesian components in the vortex core location, . This works best for a vortex near the disk center, where it is unlikely to be destabilized by deformations caused by the boundaries. For zero temperature, the dynamics from RK4 integration of the LLG equations is completely consistent with that from the Thiele equation. The Thiele equation predicts the vortex gyrotropic frequencies to be , which is confirmed in the dynamics simulations while using force constants from the Lagrange-constrained static vortex structures. Generally, the zero-temperature gyrotropic frequencies are roughly proportional to with only a weak dependence on disk ellipticity, as can be concluded from the results in Figure 7. The frequencies are determined by the geometric mean force constant, , which shows why knowledge of the vortex potential is important for this problem.
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 , because the presence of spin waves weakens the exchange stiffness of the system. The resulting distribution of vortex position can be predicted using an effective Lagrangian and Hamiltonian that result from the Thiele equation. That Hamiltonian can be expressed in a form in Eq. (50) containing only a potential energy. This then shows that the distributions (and variances) of effective radial coordinate and Cartesian coordinates and depend on , where is either or or , respectively [see Eqs. (57) and (60)]. Surprisingly, large vortex rms displacements on the order of 1–10 nm can result, with the larger values taking place in the weaker potentials of thinner disks (Figure 11) and in disks with larger radii . However, these noisy elliptical motions simply reflect the equipartition of energy into the two collective degrees of freedom available to the vortex (), with each receiving an average thermal energy of . The radial coordinate, in contrast, receives a full of energy on average. The theoretical probability distributions are confirmed in simulations provided the time evolution averages over a large number of gyrotropic orbits.
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 can be characterized by a mass proportional to the disk radius , but independent of material properties. The mass has the sense that as the vortex position fluctuates, it has some Maxwellian speed distribution, with a kinetic energy that enters in the Boltzmann factor. This is in contrast to the Thiele equation, which has been used here with no intrinsic mass term. Indeed, the vortex gyrotropic frequency is the same as that for a corresponding 2D harmonic oscillator of mass and spring constant , that is,
Portions of this work benefited substantially from discussions with Afranio Pereira and Winder Moura-Melo at the Universidade Federal de Viçosa, Minas Gerais, Brazil, and Wagner Figueiredo at the Universidade Federal de Santa Catarina, Florianópolis, Brazil, and from use of computation facilities at both universities.
© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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