Figure 2: Skyrmions in bilayer system.

(a) The profile of the n_z component of the skyrmion solution (see text) as a function of the radius r = | r − r_c,i| where the single skyrmion is on layer i and the other layer is a perfect ferromagnetic state along n_z axis. The black (purple) curve is for the cases (A1) and (A2) (the cases (B1) and (B2)). (b) The normalized potential as a function of r_d = | r_c,1 − r_c,2| for D₁ = D₂ (black) and D₁ = −D₂ (red) corresponding to the cases (A1) and (A2), i.e., the ferromagnetic interlayer coupling. The interaction energy between the two skyrmions on different layers is given by U = |J_inter| u. (c) The same as (b) but for the cases (B1) and (B2), i.e., the antiferromagnetic interlayer coupling. (See the section Methods for the detail of numerical conditions). In the figures (d–g), the magnetic textures for the cases (A1), (A2), (B1) and (B2) which minimize the total energy are shown, respectively. The length scale is shown in the lower panel of (d). The arrows indicate the winding direction of the in-plane components of the magnetic moments. The color code (h) is used to express the magnetic textures n_{i, r}, where the color represents the direction of in-plane magnetic moments and the brightness does the out-of-plane magnetic moments, e.g., white and black correspond to n_{i,z, r} = +1 and =−1, respectively. In figure (e), the red symbols × on layer 1 and 2 represent the center-of-mass of the skyrmion on layer 2 and 1, respectively.