Theory of skyrmion states in liquid crystals

Within the Oseen-Frank theory we derive numerically exact solutions for axisymmetric localized states in chiral liquid crystal layers with homeotropic anchoring. These solutions describe recently observed two-dimensional skyrmions in confinement-frustrated chiral nematics [P. J. Ackerman et al., Phys. Rev. E 90, 012505 (2014)]. We stress that these solitonic states arise due to a fundamental stabilization mechanism responsible for the formation of skyrmions in cubic helimagnets and other noncentrosymmetric condensed-matter systems.

Figure 1

Axisymmetric skyrmion in a chiral liquid crystal layer of thickness $L$ confined by surfaces $z=\pm L/2$ with homeotropic anchoring. The director $\mathbf{n}$ is designated by arrows to demonstrate a fixed rotation sense. (a) Due to homeotropic anchoring the cores of the surface layers ($s$) are smaller than in the central ($c$) layer. (b) Cut in the $\rho z$ half-plane shows rotation of $\mathbf{n}$ along the $\rho$ axes for different values of $z$. Isoclines $\theta =\pi i/6$ $\left(i=0...6\right)$ calculated for solutions $\theta (\rho ,z)$ in Fig. 2 are indicated with solid (blue) lines. Characteristic widths of the skyrmion core $R\left(z\right)$ are defined by Eq. (9).

Figure 2

Solutions of Eqs. (6) and (7) in a layer with confinement ratio $\nu =L/p=1.8$, surface anchoring $k_s=7.0$, and the applied field $E=1.02E_0$. Functions $\theta (\rho ,z)$ are plotted as a set of profiles $\theta \left(\rho \right)$ for different values of $z$ $\left(0<z<L/2\right)$. Thick (red) lines show solutions in the center (c), $\theta (\rho ,0)$ and on the surfaces (s) of the layer, $\theta (\rho ,\pm L/2)$. The corresponding core sizes $R_{\left(c\right)},R_{\left(s\right)}$ are derived from Eq. (9). Inset shows corresponding profiles $d\theta /d\rho \left(\rho \right)$ indicating the inflection points of $\theta \left(\rho \right)$ profiles.

Figure 3

Equilibrium shapes $R\left(z\right)$ of spherulites in a layer with thickness $L=1.8p$ and for different values of homeotropic anchoring: (a) $E=1.02E_0$. Near the unwinding transition spherulites have extended sizes; (b) $E=1.5E_0$. For higher fields their cores become strongly localized. Inset in panel (a) introduces the effective sizes of profiles $\theta \left(\rho \right)$ [Eq. (9)]; inset in panel (b) shows spherulites shapes in layers with thickness $L/p=0.01\left(a\right),0.005\left(b\right),0.0025\left(c\right)$ $\left(E=1.02E_0,K_s=7.0K_{s0}\right)$.

Figure 4

Typical solutions for confined helicoids (a). Top and right axes show profiles $\theta \left(z\right)$ in equidistant planes $xz$. Lower and left axes show energies densities: dashed (blue) lines correspond to surface layers $\left(z=\pm L/2\right)$ and the solid (red) line is for the center of the layer $\left(z=0\right)$. Corresponding energy densities for an isolated sherulite and a skyrmion lattice are plotted in panels (b) and (c).

Figure 5

The phase diagram of the equilibrium states in reduced values of applied field $\left(E/E_0\right)$ and homeotropic anchoring $\left(k_s\right)$, Eq. (8), indicate the existence areas of the helicoid and the skyrmion lattice in a layer with the confinement ratio $\nu =1.8$.