New Type of Stable Particlelike States in Chiral Magnets

We present a new type of thermodynamically stable magnetic state at interfaces and surfaces of chiral magnets. The state is a soliton solution of micromagnetic equations localized in all three dimensions near a boundary, and it contains a singularity but nevertheless has finite energy. Both features combine to form a quasiparticle state for which we expect unusual transport and dynamical properties. It exhibits high thermal stability and thereby can be considered as a promising object for fundamental research and practical applications in spintronic devices. We identified the range of existence of such particlelike states in the thickness dependent magnetic phase diagram for helimagnet films and analyzed its stability in comparison with the isolated skyrmion within the conical phase. We provide arguments that such a state can be found in different B20-type alloys, e.g., ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$, ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$, ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$.

Figure 1

Phase diagram of the ground state for isotropic helimagnetic thin film of thickness $L$ at magnetic field $H$ applied normally. The dashed area within the conical ground state corresponds to the isolated skyrmion as the lowest-energy metastable state. Arrows indicate the values of the fixed parameters used in the energy calculation presented in Figs. 3. The star symbol corresponds to the parameters used for the calculation of the nonzero anisotropy case [see Figs. 3 and 3]; the circles are for the calculation of the energy barriers (see Fig. 4).

Figure 2

Schematic representation of the vector field and cross sections of corresponding isosurfaces ($n_{\mathrm{z}}=0$) for the SkT (a) and chiral bobber (ChB) (b). The small (green) sphere indicates the position of the chiral Bloch point. The spins in red-outlined box on the right illustrate the magnetization distribution in the conical phase, which surrounds these localized metastable states. Circular blue arrows indicate the sense of magnetization rotation within the conical phase and surface-induced twist. For the exact spin structure of SkT and ChB see Ref. [16] (Movies 1 and 2).

Figure 3

Energy dependence for isolated SkT (open circles) and ChB (red solid circles) on the applied magnetic field for fixed film thickness (a,b), on film thickness for the fixed applied field (c,d), and as a function of varying values of cubic (e) and uniaxial anisotropy (f) for fixed thickness and field. The energies are presented relative to the pure conical phase. Note, the cases in (a)-(d) correspond to the isotropic case, $K_{\mathrm{c}}=K_{\mathrm{u}}=0$. The top and bottom scales in (a,b) and (c,d) are identical but given in absolute and reduced units, respectively. For these calculations, we used $J=1$, $D=0.15$ ($L_{\mathit{D}}=41.89a$) and a size of the simulated domain of $256\times 256\times L_n$ spins, where $L_n=40÷280$.

Figure 4

Minimum energy path calculated for $L_1=2.07L_{\mathit{D}}$, and $L_2=2.79L_{\mathit{D}}$ at $H/H_{\mathit{D}}=0.8$, $K_{\mathrm{c}}=K_{\mathrm{u}}=0$. The reaction coordinate is an order parameter that represents the relative distance between the states in the configuration space [35]. The local minima $a$, $c$, and $e$ correspond to the energies of SkT, a pair of ChBs, and a single ChB, respectively (see, also, the schematic representation in the corresponding insets). The difference between the energy values in the local minimum and nearest maximum corresponding to the saddle points (see $b$, $d$, and $f$) defines the energy barrier responsible for the stability of the corresponding state. The reference level ($E=0$) is the energy of the global minimum $g$ corresponding to the pure conical phase [16].