Ground state in applied magnetic field
We apply a magnetic field perpendicular to the xy plane. As we know, in systems where some spin orientations are incompatible with the field such as in antiferromagnets, the down spins cannot be turned into the field direction without losing its interaction energy with the up spins. To preserve this interaction, the spins turn into the direction almost perpendicular to the field while staying almost parallel with each other. This phenomenon is called "spin flop” . In more complicated systems such as helimagnets in a field, more complicated reaction of spins to the field was observed,
Figure 15.4 GS spin configurations for Jmf = — 0.45 (a), —1.2 (b), with H = 0. Angles between NN are schematically zoomed (c). See text for comments.
leading to striking phenomena such as partial phase transition in thin helimagnetic films . In the present system, there is a competition between the applied field which wants to align the spins along the z direction, and the DM interaction which wants the spins to be perpendicular which each other in the x plane. As a consequence, spins find a compromise which is the structure of skyrmions as shown below.
Figure 15.5a shows the ground state configuration for Jmf = —1.1 for first (surface) magnetic layer, with external magnetic layer H =0.1. Figure 15.5b shows the 3D view. We can observe the beginning of the birth of skyrmions at the interface and in the interior magnetic layer.
Figure 15.6a shows the ground state configuration for Jmf = -1.1 for first (surface) magnetic layer, with external magnetic layer H = 0.2. Figure 15.6b shows the 3D view. We can observe the skyrmions for the surface and interior magnetic layer.
Figure 15.5 GS configuration of the surface magnetic layer for (a) ] mf = — 1.1 and H =0.1, (b) 3D view of the surface GS configuration.
Note that the skyrmions are found here in a range of sufficiently strong interface coupling and the applied field. The skyrmions are distributed in 3D space (not on a plane) in the magnetic layer. Figure 15.6 shows a cut in xy plane so that the projected sizes are not uniform. We have made a single magnetic layer: In that case, skyrmions are uniform on a 2D sheet (not shown). We note that the skyrmion and anti-skyrmion textures are not degenerate due to the DM asymmetry [see Eq. (15.10)]: Choosing the direction of P will fix the skyrmion turning direction, i.e., [S, x Sy], Changing P will change skyrmions into antiskyrmions or vice versa.
Figure 15.6 (a) GS configuration for the surface magnetic layer for Jmt = — 1.1 and H = 0.2, (b) 3D view.
Figure 15.7 shows the GS configuration of the interface magnetic layer (top) for ] mf = —1.1, with external magnetic layer H = 0.33. The bottom figure shows the configurations of the second (interior) magnetic layer. We can observe skyrmions on both the interface and the interior magnetic layers.
Figure 15.8 shows the 3D view of the GS configuration for J mf = — 1.1, with H =0.33 for the first (interface) magnetic layer and the second (interior) magnetic layer. We can observe skyrmions very pronounced for the surface layer but less contrast for the interior magnetic layer. For fields stronger than H = 0.33, skyrmions disappear in interior layers. At strong fields, all spins are parallel to the field, thus no skyrmions anywhere.
Figure 15.7 (a) GS configuration for the interface magnetic layer for J mf =
— 1.1 and H = 0.33, (b) GS configurations forthe second and third magnetic layers (they are identical). See text for comments.
Figure 15.8 (a) 3D view of the GS configuration of the interface, (b) 3D view
of the GS configuration of the second and third magnetic layers, for J and
H = 0.33.