  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.
