Surface anchoring as a control parameter for shaping skyrmion or toron properties in thin layers of chiral nematic liquid crystals and noncentrosymmetric magnets

Existence of topological localized states (skyrmions and torons) and the mechanism of their condensation into modulated states are the ruling principles of condensed matter systems, such as chiral nematic liquid crystals (CLCs) and chiral magnets (ChM). In bulk helimagnets, skyrmions are rendered into thermodynamically stable hexagonal skyrmion lattice due to the combined effect of a magnetic field and, e.g., small anisotropic contributions. In thin glass cells of CLCs, skyrmions are formed in response to the geometrical frustration and field coupling effects. By numerical modeling, I undertake a systematic study of skyrmion or toron properties in thin layers of CLCs and ChMs with competing surface-induced and bulk anisotropies. The conical phase with a variable polar angle serves as a suitable background, which shapes skyrmion internal structure, guides the nucleation processes, and substantializes the skyrmion-skyrmion interaction. I show that the hexagonal lattice of torons can be stabilized in a vast region of the constructed phase diagram for both easy-axis bulk and surface anisotropies. A topologically trivial droplet is shown to form as a domain boundary between two cone states with different rotational fashion, which underpins its stability. The findings provide a recipe for controllably creating skyrmions and torons, possessing the features on demand for potential applications.

Figure 1

Diagram of the cone solutions for model (4) in the space of control parameters $(K_u-K_s)$ and fixed value of the thickness, $T=4\pi L_D$. Filled areas designate the regions, where the conical state has distinct internal structure: in region 1 (yellow shading), the conical state is flat across the layer thickness with ${\theta }_c=\pi /2$; in region 2 (green shading), the magnetization near the surfaces starts to align along $z$ in response to the increasing homeotropic anchoring (line $b–d$). At the line $b–e$, the spins at the surface point along $z$ and the conical structure is practically frozen. In region 3 (blue shading), the surface states acquire flat structure due to the planar anchoring, whereas the spins in the middle of the layer are in the homogeneous state along $z$. The color plots in (b) exhibit $m_z$ component in all aforementioned conical states. Black arrows show projection of the magnetization onto the $xz$ plane. Average magnetization $\langle m_z\rangle$ for the conical state is plotted within the regions 1–2 (c) and 3 (e) in dependence on the surface-induced anisotropy $K_s$ for fixed values of the bulk anisotropy $K_u$. The surface anchoring leads to the saturation of $\langle m_z\rangle$ with the magnitude different from the FM state in (c) and/or planar cone in (e): once the surface states are formed according to the strong anchoring, the core structure within the layer does not change. Magnetization profiles plotted for several values of $K_s$ in (d) and (f) show that the surface states almost do not intersect and the state in the middle of the layer is fully specified by the bulk anisotropy $K_u$.

Figure 2

Internal structure of isolated skyrmions subjected to increasing homeotropic anchoring $K_s>0$: $K_{s}A/D^2=0$ (a), 5 (c), and 25 (e). The structure of isolated skyrmions is represented as color plots of $m_z$ (b), (d), and (f) in the plane $xz$ (first column) and in the $xy$ cross sections at the surface (second column) and in the middle of the layer (third column). Skyrmion profiles in the middle of the layer remain visually unchanged, whereas the profiles at the surface acquire axisymmetric shape. Projections of the magnetization onto the corresponding plane are indicated with black arrows. The three-dimensional constructs with excluded spins of the conical phase are shown in (a), (c), and (e) for three values of the surface anchoring and fixed value of the bulk anisotropy $K_{u}A/D^2=-0.2$. In (e), a ring of formed defects is shown by the white dashed line. Blue arrows indicate that the red coil associated with the domain boundary between isolated skyrmions and a conical state is squeezed into the interior of the layer. The internal structure of isolated skyrmions implies anisotropic character of their interaction (see text for details).

Figure 3

Internal structure of isolated skyrmions subjected to increasing planar anchoring $K_s<0$. The structure of isolated skyrmions is represented as color plots of $m_z$ (b), (d), and (f) in the plane $xz$ (first column) and in the $xy$ cross sections in the middle of the layer (second column) and at the surface (third column). Whereas the central profiles retain their axisymmetric shape, the surface profiles become pronouncedly nonaxisymmetric. Projections of the magnetization onto the corresponding plane are indicated with black arrows. The three-dimensional constructs with excluded spins of the conical phase are shown in (a), (c), and (e) for three values of the surface anchoring and fixed value of the bulk anisotropy $K_{u}A/D^2=0.5$. Crescents formed only near the confining surfaces imply an anisotropic character of skyrmion interaction with two possible configurations within the skyrmion chains (see text for details).

Figure 4

(a) Schematic representation of torons—spatially localized three-dimensional skyrmions composed of a skyrmion filament of finite length cupped by two point defects terminating its prolongation. According to the 3D model in (b), torons are composed of a globulelike (blue) core centered around the magnetization opposite to the field and a fragment of a (red) coil with the magnetization along the field. Color plots of the magnetization component $m_z$ exhibit the structure of an isolated toron in different crosscuts (c) and (d). Whereas in the middle of the layer [second panel in (d)], the magnetization distribution is similar to that of an isolated skyrmion [see, e.g., for comparison Fig. 2, third panel], the spins at the surface are all magnetized homogeneously along $z$ [first panel in (d)]. The internal structure of constituent torons within the toron lattice is shown in (e) and (g). Above the dotted line $A–B$ (f), such a lattice is energetically more favorable as compared with the skyrmion lattice (see text for details).

Figure 5

Phase diagram of states in the space of control parameters $(K_u,K_s)$. The regions of the thermodynamical stability are colored by red (hexagonal lattice of torons), yellow (cones), green (helicoids), blue (oblique spiral state), and white (ferromagnetic state). The detailed description of the phase transitions between modulated phases is given in the text.

Figure 6

(a) Diagram of 1D solutions for model (4) in the space of control parameters $\left(K_u\text{−}{K}_s\right)$ for the fixed value of the thickness, $T=4\pi L_D$. The color plots in (c) stand for $m_z$ components of the magnetization. $m_y$ and $m_z$ components are shown with thin black arrows. The straight spiral state [also called a chiral soliton lattice (CSL)] expands into a system of isolated solitons [third panel in (c)] only along the line $A–B$. Additionally, the strong surface anchoring leads to a formation of defects near the confining surfaces. Otherwise, the CSL transforms into the conical state: such a transition may occur either as the first-order phase transition along the line $D–C$ or via an intermediate oblique spiral state [fourth panel in (c)] along the line $D–G–F$. Then, in the latter case, the tilt angle $\alpha$ (b) smoothly decreases from the value $\pi /2$ in the CSL state to the value $\alpha =0$ in the conical state. A procedure to determine a canting angle $\alpha$ for an oblique spiral state is shown as an inset in (b) (see also text for details).

Figure 7

Internal structure of a metastable conical state arising due to the homeotropic anchoring. The color plot in (a) exhibits $m_z$ component; black arrows show projection of the magnetization onto the $xz$ plane. Magnetization profiles plotted for several values of $K_s$ in (b) demonstrate the full cycle of spin rotation from the state down at the lower surface to the state up at the upper surface. Interestingly, the domain boundary between two conical states reduces to the so called droplet—a topologically trivial particle with zero topological charge $Q=0$. The 3D model of such a droplet is shown in (c) and the magnetization distribution in the $xy$ plane in (d). For completeness, I also construct 3D models for so called chiral bobbers localized near layer surfaces and culminating in two point defects (e). I notice that the metastable conical state (a), which may actually acquire the same energy as the cone in region 2 of Fig. 1 with the increasing film thickness, would accommodate chiral bobbers with opposite polarities (see text for details).