We discuss thermally activated switching between micromagnetic states in nanodisks with perpendicular anisotropy and a large strength of the interfacial Dzyaloshinskii-Moriya interaction, D. For large D, the magnetization on infinite planes acquires a cycloidal configuration in the ground state. However, Isolated singularities that are analogous to disclinations of cholerestic phases [1] are topologically protected and prevent the system from reaching the cycloidal phase (for examples see Fig. 1). In confined geometries, a rich variety of possible configurations with complex magnetization textures that are also energy minima of the energy landscape [2] has been observed. We argue that characterizing these different states as arrangements of disclinations is an effective way for understanding them. We support this statement on results of String Method [3] calculations done using OOMMF to examine the energy landscape of nanodisks (Fig. 2). We found that the transition states correspond to motion of individual merons across the device's edge (first maximum in Fig. 2), and to merging of disclinations inside the structure (2nd, 3rd and 4th maxima in Fig. 2). An additional advantage of emphasizing the importance of these magnetic singularities is the possibility of adopting them for collective coordinate descriptions which can be treated using low-dimensional stochastic methods to study their dynamics in the presence of thermal noise. 

**References** 
[1] Phys. Rev. B 99, 214408 (2019)
[2] The Philosophical Magazine. 22:178, 739-749
[3] J. Chem. Phys. 126, 164103 (2007)
This research was supported in part by the National Science Centre Poland under OPUS funding Grant No. 2019/33/B/ST5/02013 (GDC, PK) and U.S. National Science Foundation Grant
DMR 1610416 (DLS). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/bc16da61cb6378eeaa848cb0834b87f6.png) 

Examples of disclinations on the cycloidal phase of a ferromagnet with large DMI. (a) cycloidal phase without singularities. Disclinations occur when the helical axis rotates by a multiple
of π. (b,c,e,f) Examples of disclinations, respectively their values are: (π,2π,-π,-2π). (d) a skyrmion shown as a 2π disclination embedded in a cycloidal background, (g) a meron shown as a
π disclination embedded in a cycloidad background. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/55824035ff74275789c3c3d8097028e3.png) 
Results of String Method calculations on nanodisks with high DMi for thermally activated switching between two different micromagnetic configurations. The highest maxima occurs
when a meron reaches the in-plane edge. The other maxima are the result of -π disclinations merging briefly into a +2π disclination resulting on re-acommodation of adjacent skyrmionic
domains.

The Role of Disclinations in the Thermal Stability of Magnetization Textures in Confined Helimagnetic Nanostructures

The topological number of micromagnetic textures (Q) is a useful quantity for information encoding and transport [1,2], and determines the dynamics of magnetic objects such as skyrmions, merons and domain walls. Recent developments allow us to tune the strength of the interfacial Dzyaloshinskii-Moriya interaction, thereby inducing the formation of different states and generating a rich energy landscape for a given physical device, leading to new and potentially useful applications [3]. This energy landscape determines the thermal stability of a given state which is strongly dependent on the barrier that separates neighboring minima [4-6]. In this work we use the String Method for the Study of Rare Events to determine transition states and thermal barriers of a circular nanodisk at different magnitudes of the Dzialoshinskii-Moriya constant (D). Low values of D are associated with high energies of domain wall formation; in this regime mostly uniform magnetizations with isolated skyrmions correspond to the lowest energy states. The transition between these states occurs by the motion of entire skyrmions across the material's edge. At intermediate values of D, domain walls require less energy, skyrmions have large radii and leave the nanodisk via an intermediate minima in which a meron is expelled from the film (Figures 1 and 2). Very large D values cause complex textures to form but transtions are still mediated by meron motion across the device's edge. This work emphasizes the importance of considering meron-mediated-transitions to appropriately create the initial path used in methods that use chain-of-states to estimate transition barriers.

![](https://assets.underline.io/uploads/markdown_image/1/image/1eaa5816c8a78055e4675a6f428397e1.jpg)

String Method results showing energy and skyrmion number along the transition path. The images above correspond to points indicated in the images. The disk diameter is 80nm and its thickness is 0.4nm.

![](https://assets.underline.io/uploads/markdown_image/1/image/43f1086b40436f45582a732b611bd679.jpg)

Skyrmion Number vs Energy for the same transition path of Fig.1. The skyrmion numbers of energy minima differ by roughly 1/2, highlighting the importance of meron motion across the edges as a mechanism for switching between energy minima. The values are not half-integer due to magnetization tilting at the sample's edge.

Thermally Activated Transitions in Confined Terromagnets with Perpendicular Magnetic Anisotropy as Function of the Interfacial Dzialoshinskii-Moriya Interaction Strength.

Droplet solitons are magnetization fluctuations that preserve their shape as they precess with uniform frequency ω=0. They satisfy a delicate balance between anisotropy and exchange interactions, and decay in the presence of dissipation. To prevent this, a spin polarized current σ can be applied via a nanocontact of radius ρ*. The magnitude of the current can be increased to induce switching between uniform precession at the ferromagnetic resonance frequency (ω=1), and a stable precession at a frequency larger than the Zeeman frequency (ω=0, in zero applied field). In the absence of dissipation, conservative solitons of frequency ω0 are described by a function Θ(ρ;ω0), where Θ is the angle of the magnetization with the easy axis and ρ is the distance to the center of the nanocontact1. We introduce an effective energy ξ that quantifies the work done (against damping and spin torque) to create a fluctuation of arbitrary shape Θ(ρ). We show that, for specific values of σ, some conservative soliton solutions are saddles of ξ. This allows us to calculate activation barriers Δξ between uniform precession at the ferromagnetic resonance and stable solitons. We present results of Δξ as a function of σ for a variety of nanocontact radii ρ* and spin-torque anisotropy parameters ν. We present micromagnetic simulations with a modified Spin-Transfer-Torque extension in OOMMF which uses the effective energy ξ introduced here. In this way, the difference of energies between transition states and metastable configurations can be used to estimate annihilation rates of droplet solitons.References: [1] Gabriel D. Chaves-O’Flynn and D. L. Stein, Phys. Rev. B 101, 184421 (2020) [2] M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010).

Thermal activation barriers for creation and annihilation of magnetic droplet solitons in the presence of spin transfer torque

HOD: Skyrmions: Characterization, Dynamics and Control

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Gabriel Chaves-O'Flynn

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Presentations

The Role of Disclinations in the Thermal Stability of Magnetization Textures in Confined Helimagnetic Nanostructures

Thermally Activated Transitions in Confined Terromagnets with Perpendicular Magnetic Anisotropy as Function of the Interfacial Dzialoshinskii-Moriya Interaction Strength.

Thermal activation barriers for creation and annihilation of magnetic droplet solitons in the presence of spin transfer torque

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