Paths to collapse for isolated skyrmions in few-monolayer ferromagnetic films

Magnetic skyrmions are topological spin configurations in materials with chiral Dzyaloshinskii-Moriya interaction (DMI), that are potentially useful for storing or processing information. To date, DMI has been found in few bulk materials, but can also be induced in atomically thin magnetic films in contact with surfaces with large spin-orbit interactions. Recent experiments have reported that isolated magnetic skyrmions can be stabilized even near room temperature in few-atom-thick magnetic layers sandwiched between materials that provide asymmetric spin-orbit coupling. Here we present the minimum-energy path analysis of three distinct mechanisms for the skyrmion collapse, based on ab initio input and the performed atomic-spin simulations. We focus on the stability of a skyrmion in three atomic layers of Co, either epitaxial on the Pt(111) surface or within a hybrid multilayer where DMI nontrivially varies per monolayer due to competition between different symmetry breaking from two sides of the Co film. In laterally finite systems, their constrained geometry causes poor thermal stability of the skyrmion toward collapse at the boundary, which we show to be resolved by designing the high-DMI structure within an extended film with lower or no DMI.

Figure 1

Oblique view of the 3ML skyrmion structure, with color-coded spin directions. The monolayer-resolved DMI (shown in the left) corresponds to 3ML Co on Pt. The remaining parameters are taken to be $J=10$ meV per bond and $K=0.1$ meV per atom. An fcc stacking sequence is used, where interlayer distances are not drawn to scale.

Figure 2

Magnetic state diagram at zero temperature and in the absence of magnetic field, as a function of anisotropy and exchange interaction, for DMI taken as in Fig. 1. Data points (circles and triangles) are retrieved from numerical simulations. The transition boundaries separate regions of metastable skyrmions (shaded in red) from the uniform (FM) ground state (white), and the skyrmion ground state (shaded in blue). Skyrmions are shown for different locations in the parameter space, denoted by crosses.

Figure 3

(a) Skyrmion energy and (b) skyrmion size as a function of magnetic anisotropy for fixed values of the exchange coupling, $J$. The dotted line corresponds to the case of constant DMI, with parameter values taken from Ref. [22].

Figure 4

Relative difference between the number of core spins in the first and the third monolayers $\left[\right(N1-N3)/N1)]$ as a function of magnetic anisotropy for weak and strong exchange interactions. Oscillations in values result from the lattice discretization, where the $S_z$ spin component switches sign for a given set of spins at different values of the anisotropy in the first and third monolayers.

Figure 5

(a) Line profile of an individual skyrmion for constant DMI and for monolayer-resolved DMI (see Fig. 1). Strong exchange coupling is assumed $(J=30$ meV per bond) and the anisotropy is chosen to be $K=0.036$ meV per atom. The labels refer to the individual Co monolayers that are closest to (Co1), in between (Co2), and farthest from (Co3) the Pt substrate. (b) Total energy difference (exchange, DM, and anisotropy) per Co atom with respect to the uniform FM state as a function of distance from the skyrmion center. (c)–(e) Energy difference due to exchange interaction, magnetic anisotropy, and DM interaction, respectively.

Figure 6

(a) Layer-resolved DMI strength for the 3ML Co in considered sandwich multilayers, compared to the case of epitaxial 3ML Co on Pt, retrieved from ab initio calculations [26, 27]. (b) Skyrmion energy and (c) skyrmion size (diameter) as a function of magnetic anisotropy for strong exchange coupling $(J=30$ meV per bond).

Figure 7

(a) Line profile of an individual skyrmion for the 3ML Co in considered sandwich multilayers and the case of epitaxial 3ML Co on Pt. (b) The energy difference per Co atom with respect to the FM ground state as a function of distance from the skyrmion core. Strong exchange coupling is assumed $(J=30$ meV per bond) and the anisotropy is taken as $K=0.036$ meV per atom. The symbols correspond to the individual Co monolayers that are closest to (squares), in between (circles), and farthest from (triangles) the Pt substrate.

Figure 8

Top view of magnetization $\left(S_z\right)$ distribution for (a–e) isotropic and (f–j) boundary collapse paths of a skyrmion in 3ML Co on Pt, with monolayer-resolved DMI. The skyrmion is initially 4.7 nm in diameter, for taken $J=30$ meV per bond and $K=0.036$ meV per atom. $S_z$ is shown color coded from white to red to green to blue (shown above). The initial images (a,f) are followed by saddle points (b,g), and the remaining images show configurations on the subsequent energy decline towards the FM ground state (not shown). Numbers on the top right correspond to ordering of images in minimum-energy path curves for the isotropic and boundary collapse (k). The reaction coordinate is defined as the normalized (geodesic) displacement along the minimum energy path.

Figure 9

Energy barriers for (a) isotropic and (b) boundary collapse of a skyrmion as a function of magnetic anisotropy. Each point corresponds to a unique energy barrier retrieved from the maximum (saddle point) in a minimum-energy path.

Figure 10

Minimum-energy paths for (a) isotropic and (b) boundary skyrmion collapse in the case of strong exchange coupling $(J=30$ meV per bond). Paths are shown for different magnetic anisotropies with respect to the FM ground state. Solid lines are cubic polynomial interpolations [39] of the discrete images (dots). Black lines correspond to the case of constant DMI.

Figure 11

Minimum-energy paths for (a) isotropic and (b) boundary skyrmion collapse in case of weak exchange coupling $(J=10$ meV per bond). Paths are shown for different magnetic anisotropies with respect to the FM ground state. Solid lines are cubic polynomial interpolations of the discrete images (dots).

Figure 12

(a) Side view of atomic spins along the right boundary of the system for the FM ground state, during boundary collapse, and the Sk state left to right respectively. The variation of the angle that boundary spins make with the substrate is shown in panel (d) as a function of position along the boundary and in panel (e) as a function of magnetic anisotropy, for the central spin on the boundary. Calculations are performed for low exchange coupling $(J=10$ meV per bond).

Figure 13

Magnetic state diagram from Fig. 2, but with a dashed line separating the region where stability of the skyrmion is dictated by boundary collapse (I) from the region where the isotropic collapse dominates (II). The discrimination between collapse mechanisms is obtained on energy grounds, and does not take into account the skyrmion lifetime.

Figure 14

Top view of spatial magnetization distribution $\left[S_z(x,y)\right]$ in a 3ML Co on Pt, for skyrmion collapse at the interface with the boundary region (of width 8 nm; see inset) where DMI vanishes. The parameters used are $J=30$ meV per bond for exchange and $K=0.050$ meV per atom for anisotropy. Individual images correspond to the initial Sk state (a), path towards the saddle point (b–d), and subsequent stages (e,g) towards the FM ground state (h).

Figure 15

(a) Energy barriers for the collapse of a skyrmion in the sample shown in Fig. 14 (3ML Co on Pt, with monolayer resolved DMI), at a lateral (east/west) interface where DMI vanishes, at the north/south open boundary of the film, and for the isotropic collapse, as a function of magnetic anisotropy. Each point corresponds to a unique energy barrier retrieved from the maximum (saddle point) in a minimum energy path. (b) Same as (a), but for the Pt-3ML Co-MgO sandwich, with monolayer resolved DMI, and removed MgO on lateral sides of the sample (thus lower DMI, particularly in the top Co monolayer). In either sample, stability of the skyrmion is dictated by boundary collapse at low anisotropy and isotropic collapse otherwise. The barrier for collapse at DMI interface greatly exceeds the one for boundary collapse at all $(J,K)$.

Figure 16

(a,b) Same as Figs. 15 and 15 respectively, but for periodic boundary conditions used at north/south edges of the sample, so that effectively an elongated track with laterally suppressed DMI is considered. (c) The lifespan of an isolated skyrmion [given by Eq. (2)] calculated for the most favorable mechanism of collapse in (a,b), for standard exchange coupling for Co $(J=30$ meV per bond), as a function of anisotropy, at room temperature.