Skyrmion lattice in a two-dimensional chiral magnet

We develop a theory of the magnetic field-induced formation of Skyrmion crystal state in chiral magnets in two spatial dimensions, motivated by the recent discovery of the Skyrmionic phase of magnetization in thin film of ${\text{Fe}}_{0.5}{\text{Co}}_{0.5}\text{Si}$ and in the A phase of MnSi. Ginzburg-Landau functional of the chiral magnet rewritten in the ${\text{CP}}^1$ representation is shown to be a convenient framework for the analysis of the Skyrmion states. Phase diagram of the model at zero temperature gives a sequence of ground states, helical $\text{spin}\to \text{Skyrme}$ $\text{crystal}\to \text{ferromagnet}$, as the external field $B$ increases, in good accord with the thin-film experiment. In close analogy with Abrikosov’s derivation of the vortex lattice solution in type-II superconductor, the ${\text{CP}}^1$ mean-field equation is solved and shown to reproduce the Skyrmion crystal state.

Figure 1

(a) A typical (anti-)Skyrmion configuration given by Eq. (8) with $R=2$. (b) $z_1$ component of the ${\text{CP}}^1$ Skyrmion ${\mathbf{z}}_{\text{Sk}}$. When expressed as a planar spin $\left(\text{Re}\left[z_1\right],\text{Im}\left[z_1\right]\right)$, it is an antivortex.

Figure 2

(Color online) Dependence of the free energy density $F_{\text{SkX}}$ on the radical cutoff $R$ at magnetic field $\beta =1$ (red curve) and $\beta =0$ (black curve), respectively. We choose $\kappa =1$. The angle $\theta$ measured from the $z$ axis as a function of the distance $r$ from the center of the Skyrmion for two different $R$’s are shown in the insets. Optimal $R_0$ for $\beta =1$ leads to an almost linear dependence of $\theta \left(r\right)$ on $r$ while larger $R=\rho$ leads to a long ferromagnetic tail outside the Skyrmion core region with $\text{radius}\cong R_0$.

Figure 3

(Color online) (color online) The energies of the three states, i.e., (i) helical state, (ii) Skyrmion crystal, and (iii) ferromagnetic state. The free energy of ferromagnetic order is set to be zero, which is labeled in green. The helical state is energetically favored at low magnetic field while the Skyrmion phase emerges at intermediate magnetic field. Ferromagnetism is favored at larger fields.

Figure 4

(Color online) The inter-Skyrmion distance in SkX (in red), and the first-order derivative of SkXs free energy with respect to the magnetic field $\beta$ (in blue). The distance increases rapidly near the transition from SkX to FM phase. As the derivative is nonvanishing at the critical field ${\beta }_{c2}=1.6$, this transition is first order.

Figure 5

(a) A typical Skyrme-crystal spin configuration given by Eq. (19) with $B=D^2/J$, with the optimized lattice spacing $\kappa l_H=\sqrt{3/2}$. (b) Skyrme-crystal spin configuration obtained by Monte Carlo method from the lattice spin model (Ref. 10).

Figure 6

(Color online) (a) Self-consistently induced field $H$ (in the dimensionless unit ${\kappa }^{2}H$) and the ferromagnetic polarization $⟨n^z⟩$, against external magnetic field $b=B/\left(D^2/J\right)$. Taking $\kappa l_H=\sqrt{3/2}$, two different values of $u$ [see Eq. (22) for definition], $u=0$ and $u=\left(1/8\right)\left(D^2/J\right)$ were used with little differences in the results. (b) Energies of helical, Skyrme crystal, and ferromagnetic spin states against $b$, both measured in units of $D^2/J$. SkX energy ${\mathcal{E}}_{\text{SkX}}$ goes up with $u$ while the other energies remain insensitive to $u$. $b_{c1}$ and $b_{c2}$ define the two first-order transitions.

Figure 7

(Color online) (a) Induced field distribution $H_{\text{SkX}}\left(x,y\right)$ from the Skyrmion lattice solution ${\mathbf{z}}_{\text{SkX}}$. Bright area is the Skyrmion core where the intensity is the maximum. (b) Distribution of $n^z\left(x,y\right)={\mathbf{z}}_{\text{SkX}}^{†}{\sigma }^z{\mathbf{z}}_{\text{SkX}}$. Dark area means reversed spins at the Skyrmion core.