Degeneracies and fluctuations of Néel skyrmions in confined geometries

The recent discovery of tunable Dzyaloshinskii-Moriya interactions in layered magnetic materials with perpendicular magnetic anisotropy makes them promising candidates for stabilization and manipulation of skyrmions at elevated temperatures. In this article, we use Monte Carlo simulations to investigate the robustness of skyrmions in these materials against thermal fluctuations and finite-size effects. We find that in confined geometries and at finite temperatures skyrmions are present in a large part of the phase diagram. Moreover, we find that the confined geometry favors the skyrmion over the spiral phase when compared to infinitely large systems. Upon tuning the magnetic field through the skyrmion phase, the system undergoes a cascade of transitions in the magnetic structure through states of different number of skyrmions, elongated and half-skyrmions, and spiral states. We consider how quantum and thermal fluctuations lift the degeneracies that occur at these transitions, and find that states with more skyrmions are typically favored by fluctuations over states with less skyrmions. Finally, we comment on electrical detection of the various phases through the topological and anomalous Hall effects.

Figure 1

Anisotropy-external field phase diagram for a wire geometry with $L=16$ and $p=8$ at $k_{B}T/J=0.5$ (solid line) and zero temperature (dashed line). The dotted line corresponds to the case of an infinite system at zero temperature [40]. These lines encircle the region where the winding number is larger than one-half. The susceptibility of the winding number at finite temperature ${\chi }_w$ is also indicated.

Figure 2

Simulations snapshots of spin systems of size $L\times L=32\times 32$ for different parameter values and temperatures with the parameter $D/J$ chosen such that the pitch length $p=8$. The periodic boundary is drawn as a red solid line and spin vectors are represented by colored arrows where the color scales linearly with the $\widehat{\mathbf{z}}$ component of the vector to highlight spin textures. From top to bottom $k_{B}T/J$ takes on values $5.00, 0.50$, and $0.01$, respectively, and the system undergoes a transition from an unpolarized state to an ordered state in this temperature range. In all cases $KJ/D^2$ is zero and from left to right $BJ/D^2$ has values $0.0, 0.5$, and $1.5$, resulting in a spiral, skyrmion, and polarized state, respectively, for low enough temperatures.

Figure 3

Magnetization (top row), energy density ${\rho }_E$ (middle row), and topological charge density ${\rho }_w$ (bottom row) for different values of the field for a square system with linear size $L=16$ with periodic boundaries in the horizontal direction. The pitch length is $p=7$ and the temperature is zero. The color coding indicates out of plane magnetization (top row) and energy and topological charge densities (middle and bottom row).

Figure 4

(a) Difference in energy (upper panel: classical energy; lower panel: energy including quantum correction) between configurations with skyrmions and the state without skyrmions at zero temperature vs pitch length, for various numbers of skyrmions and system size $L\times L=16\times 16$ with one periodic and one open boundary and parameters $KJ/D^2=0.0$ and $BJ/D^2=0.5$. The blue circular, yellow rectangular, green diamond-shaped, and red triangular data points correspond to systems with 0, 1, 2, and 4 skyrmions, respectively. (b),(c) Magnetic configuration of two classically degenerate ground states with energy $E=-596.17J$ of a system at pitch length $p=9.336$ deep in the skyrmion regime with one periodic (horizontal direction) and one open (vertical direction) boundary.

Figure 5

Energy of various spin configurations, with pitch length $p=9.336$ in (a) and $p=7.284$ in (b), as a function of temperature $k_{B}T/J$ and where $E_{\mathrm{cl}}$ is put equal to zero. In both cases the parameters are $KJ/D^2=0.0$ and $BJ/D^2=0.5$. The data points represent data from Monte Carlo simulations that are in agreement with results from the equipartition theorem at low temperatures (shown by the black dashed line). The solid black lines represent the energies resulting from the spin-wave analysis.

Figure 6

Ratio of probabilities of having a certain amount of skyrmions for two different situations with parameters $KJ/D^2=0.0$ and $BJ/D^2=0.5$ and pitch length $p=9.336$ for (a) and $p=7.284$ for (b) as a function of temperature. The dots correspond to results from the Monte Carlo simulations, the solid line from the spin-wave analysis not including translation entropy, and the dashed line to the zero-temperature classical limit.

Figure 7

Winding number (left axis) and average magnetization in the $z$ direction (right axis) as a function of magnetic field for the same parameters as Fig. 3. Labels a–f refer to states in Fig. 3. All parameters are taken the same as for the results in Fig. 3.