Abstract
Skyrmions are particlelike topological entities in a continuous field that have an important role in various condensed matter systems, including twodimensional electron gases exhibiting the quantum Hall effect, chiral ferromagnets and Bose–Einstein condensates. Here we show theoretically, with the aid of numerical methods, that a highly chiral nematic liquid crystal can accommodate a quasitwodimensional Skyrmion lattice as a thermodynamically stable state, when it is confined to a thin film between two parallel surfaces imposing normal alignment. A chiral nematic liquid crystal film can thus serve as a model Skyrmion system, allowing direct investigation of their structural properties by a variety of optical techniques at room temperatures that are less demanding than Skyrmion systems discussed previously.
Introduction
Skyrme showed that particlelike topological excitations can be stable in certain types of continuous nonlinear fields, and can account for the existence of protons and neutrons in the field of pions^{1}. When one considers a continuous vector field m in three dimensions of fixed magnitude (that is, m ∈ S^{2}), a Skyrmion is roughly understood to be a localized variation in the field, in which m rotates continuously through an angle π or π/2 (in the case of a halfSkyrmion) from its center to its boundary. Figure 1a illustrates the vector field profile of a halfSkyrmion.
Although the original idea of Skyrme was applied to a continuous field of pions to explain the existence of protons and neutrons^{1} as mentioned above, Skyrmions have found a place in a wide variety of fields of physics, in particular in condensed matter systems. In the mid1990 s Skyrmions were predicted and observed in quantum Hall devices^{2,3,4}, with the realization that quantum effects are best understood in terms of topological features. Later, spinor Bose–Einstein condensates were shown to accommodate Skyrmions due to the internal spin degrees of freedom^{5,6,7}. More recently, great attention has been paid to Skyrmions in helical ferromagnets^{8}, such as MnSi^{4,9,10} and Fe_{1x}Co_{x}Si^{11,12}, characterized by a lack of inversion symmetry (chirality) and the presence of DzyalonshinskiiMoriya spin–orbit interaction. Skyrmions in chiral ferromagnets could stimulate potential applications, because local twisted magnetic structures coupled to electric or spin currents could be used to manipulate electrons and their spins.
Among the intriguing features of helical ferromagnets is their similarity^{9,13,14} to cholesteric blue phases (BPs) in chiral liquid crystals^{15}. BPs have attracted much interest as one of the fascinating nontrivial structures in condensed matter systems, where the unit vector n, known as the director that denotes the average orientation of liquid crystal molecules, exhibits threedimensional ordering. The delicate balance between the local preference for a doubletwist structure over a singletwist in a helical phase and the global topological constraint that prevents a doubletwist structure from filliing the whole space without introducing discontinuities yields threedimensional regular stacks of socalled doubletwist cylinders^{15} (Fig. 1b) and topological defect lines^{15}. In Figure 1c and d, we show the structures of bulk BPs with cubic symmetry. BP I in Figure 1c is characterized by a space group O^{8} (I4_{1}32), and contains straight defect lines that do not intersect each other. Defect lines in BP II (Fig. 1d, space group O^{2} or P4_{2}32) form a doublediamond structure with an array of fourarm junctions. Doubletwist cylinders can be regarded as a type of Skyrmion excitation with π/4 rotation. A π/4 rotation enables the n field to remain continuous at the point of contact between two neighbouring orthogonal doubletwist cylinders. Possibly due to the delicate balance mentioned above, most liquid crystals exhibiting a BP have a limited stability range in temperature as small as a few Kelvin or less. However, recently discovered polymerstabilized BPs allow a temperature range wider than 60 K^{16}, which has stimulated numerous experimental studies aimed at practical applications of BPs such as fastswitching displays. Close similarity is apparent in the free energy densities, which take the form of a(∇×a) that is allowed in the absence of inversion symmetry^{8,15}, where a=n (director) in the case of liquid crystals and a=m (magnetization) in the case of ferromagnets. Possible ferromagnetic structures consisting of stacks of doubletwist cylinders have also been discussed^{9,14}.
Here we show theoretically, with the aid of numerical calculations, that a highly chiral liquid crystal can be host to a stable quasitwodimensional (2D) Skyrmion lattice when it is confined to a thin film between two parallel plates imposing normal alignment (anchoring) at the surfaces. It should be stressed that liquid crystals possess head–tail symmetry, thus n is equivalent to −n; therefore, in comparison with magnets, they allow for a wider variety of ordered structures. Our present study is motivated by our discovery of a wide variety of ordered structures in a confined chiral liquid crystal^{17,18}, and a very recent realspace observation of regular Skyrmion lattices in a thin film of Fe_{1x}Co_{x}Si^{12}. With the aid of extensive numerical calculations, we construct a phase diagram in terms of temperature and film thickness, and demonstrate that regular quasi2D Skyrmion lattices with hexagonal or rectangular symmetry are indeed thermodynamically stable over a certain range of temperatures and film thicknesses.
Results
Skyrmion structures
Calculations are carried out in the temperature range −1.05 K<T−T*<0.32 K, and the film thickness range 50 nm<d<200 nm. Here T* is the temperature below which an isotropic phase becomes unstable to any small perturbations. Details of our calculations are described in Methods. We present several thermodynamically stable profiles in our calculations in Figure 2a–f. Here, the director (n) profile at z=d/2 (midplane of the system) and topological defects are depicted. Confining surfaces with strong normal anchoring no longer allow straight defect lines to exist, in contrast to bulk BPs, thus yielding various exotic ordered structures not found in bulk liquid crystals. In our previous study at a fixed temperature^{17,18}, we reported the observation of an array of doublehelix defect lines (Fig. 2e) or inchwormlike fragmented defects (Fig. 2f). By varying the temperature, we find a much wider variety of ordered structures as shown in Figure 2a–d. Particularly interesting are those with hexagonal symmetry (Fig. 2a,b). In the case of small d (Fig. 2a), a 2D lattice of halfSkyrmion excitations (Fig. 1a) embedded in an array of straight topological defect lines is clearly observed, resembling those found in a recent realspace observation of thin chiral ferromagnets^{12}. With increasing d, variation of the profile in the zdirection is allowed, which results in further exotic structures (Fig. 2b–d). The profile in Figure 2c can also be regarded as a Skyrmion lattice because halfSkyrmion excitations are still clearly visible as a centeredrectangular lattice in an array of topological defects. In Figure 2b, a labyrinth of topological defects with hexagonal symmetry prevents the formation of Skyrmion excitations. Although possible similar hexagonal structures in chiral liquid crystals have been discussed before^{19,20,21,22}, they are bulk structures (without confinement) under an electric field; the difference between the local nature of the surface anchoring and the longrange nature of the electric field should be stressed. Moreover, the confinement does not allow screw axes to arise^{17}, which are required for some observed and proposed bulk hexagonal structures^{19,20,21,22}. Indeed, structures seen in Figure 2b and f cannot exist as bulk ones. It is interesting to note that in Figure 2d, fourarm junctions of topological defects can be seen clearly, reminiscent of those in BP II (Fig. 1d). Here, and also in Figure 2e and f, halfSkyrmion excitations forming a rectangular lattice can be observed. Typical lattice constants are 113 nm for the hexagonal structure of Figure 2a, and 91×101 nm for the centeredrectangular structure of Figure 2d. The dependence of the lattice constant on the temperature and film thickness is not strong; in the case of Figure 2a, the lattice constant ranges from 106 to 120 nm. Smaller film thicknesses and lower temperatures yield a larger lattice constant.
Phase diagram
In Figure 2g, we present the phase diagram with respect to temperature T and film thickness d. For higher temperatures, that is, T−T*>0.25 K, an alignment normal to the confining surfaces with reduced ordering in the middle of the system is the most stable state (we will refer to this state as NA or 'normal alignment'). Note that the orientational order in the NA state is induced by confining surfaces^{23}; the temperature is too high to allow orientational order in the bulk. In the case of a lower temperature or small d, a helical orientational order corresponding to a cholesteric phase, whose pitch is parallel to the confining surfaces, attains thermodynamic stability (often referred to as 'uniform lying helix'). In between the normal alignment state and uniform lying helix, structures presented in Figure 2a–f can be thermodynamically stable. For smaller d, a 2D hexagonal Skyrmion lattice (Fig. 2a) is energetically preferable. For larger d, thermodynamic stability of hexagonal structures (Fig. 2b) and rectangular structures (Fig. 2d,e) depends on temperature; the former is more stable for higher temperatures. The stable regions for centeredrectangular structures (Fig. 2c) and for fragmented defects (Fig. 2f) are limited in extent.
Discussion
Here we discuss the similarities and differences between the 2D hexagonal Skyrmion lattice in Figure 2a and the square one in ref. 8. In both systems, halfSkyrmion excitations are embedded in a regular lattice of topological defects. The vectorial nature of the magnetization in the latter allows only Skyrmions with +z magnetization to neighbour ones with −z magnetization; otherwise, the boundary between two Skyrmions must be discontinuous. Therefore, a hexagonal lattice is prohibited because it would lead to frustrations. For a hexagonal Skyrmion lattice to exist, Skyrmions with +z magnetization must adopt a rotation of m in the sea of −z magnetization^{10,12}. On the other hand, in liquid crystalline systems with n and –n symmetry, such frustrations do not exist, which allows for the formation of a hexagonal lattice of halfSkyrmions.
From the phase diagram presented in Figure 2g, it is clear that if one wishes to observe 2D hexagonal Skyrmion lattices, the temperature must be carefully controlled within a range of 0.5 K at most, just as BPs in the bulk are in most cases stable over a 1 K temperature range^{15}. The stable temperature range of rectangular Skyrmion lattices is relatively large, compared with hexagonal lattices.
We find from a separate calculation that the thermodynamic stability of 2D hexagonal Skyrmion lattices is significantly reduced when the chirality of the liquid crystal is lower (larger cholesteric pitch). This observation corresponds to the wellknown fact that cholesteric BPs can be found only in highly chiral liquid crystals^{15}. Notice also that with weak surface anchoring, a structure similar to a bulk BP is the most stable^{17}, because the surfaces disturb only slightly the bulk structure. Together with the phase diagram in Figure 2g, we conclude that for the observation of 2D Skyrmion lattices in a chiral nematic liquid crystalline system, the following conditions must be fulfilled: high chirality; accurate temperature control (close to the transition towards the isotropic phase); and very small film thickness with strong surface anchoring.
Finally, we comment on the possibility of observing the structures discussed here experimentally. The transition from a rectangular to a hexagonal structure induced by a temperature change would be clearly detected by scattering experiments. Transmission electron microscopy with a freeze fracture technique^{24} could help identify the orientational profile of a confined liquid crystal together with its spatial symmetry. A more recent and promising technique that allows realspace observation is confocal microscopy, which has been applied to observe a bulk BP^{25}, and defect structures in simple nematics and cholesterics^{26}. We note that liquid crystals have been exploited for modelling physical systems that are difficult to access experimentally and in which topology has a substantial role^{27,28}. Experimental observation of liquid crystalline Skyrmion systems could also shed light on the properties of other experimentally demanding Skyrmion systems, for example, 2D electron gases exhibiting quantum Hall effect, spinor Bose–Einstein condensates and chiral ferromagnets, and clarify the similarities and differences between these systems. We thus encourage experimental studies for the observation of Skyrmion lattices in a liquid crystal. Further theoretical studies may be devoted to several interesting but challenging problems on, say, the dynamics of phase transitions induced by temperature change, and possible effects of thermal fluctuations on the stability of Skyrmion structures^{10}. We further note that photopolymerization, which was applied to the stabilization of BPs^{16}, could open a route to various Skyrmion structures stable over a wide temperature range. The Skyrmion structures could be templates for assembling complex structures of colloidal particles that can be further stabilized by photopolymerization for possible photonic applications.
Methods
Order parameter and the Landau–de Gennes theory
Our calculations are based on a Landau–de Gennes theory describing the orientational order in terms of a symmetric and traceless tensor Q_{ij}, with i and j representing the Cartesian coordinates x, y and z. When the liquid crystal is in an isotropic state, Q_{ij}=0. In the case of uniaxial ordering, Q_{ij}=Q(n_{i}n_{j}–(1/3)δ_{ij}), where Q describes the degree of orientational order, and n_{i} is the ith component of the director. The local free energy density in the bulk f_{local} and the elastic energy f_{el} are written as expansions in Q_{ij} and its gradients, up to the lowest relevant order, as^{15}
Here summations over repeated indices are implied, and a, b, c are material parameters assumed to be positive. Below the temperature T*, an isotropic state becomes unstable with respect to small perturbations. ɛ_{ikl} is the LeviCivita symbol, K_{1} and K_{0} are elastic constants, and 2π/q_{0} is the pitch of the cholesteric helix. For the free energy density of the confining surfaces, which we assume to impose normal alignment (anchoring), we employ the following quadratic form^{29}
where W is the anchoring strength, and is the order parameter preferred by the surface, with ν being the surface normal vector and Q_{0} being the degree of surface order. In this study, Q_{0} is chosen such that (Q_{s})_{ij} minimizes f_{local} for T=T*.
Calculation of the orientation profile
We consider a thin film of a liquid crystal confined by two parallel flat surfaces separated by a distance d. When we take the z axis normal to the surfaces, the total free energy F is given by . To obtain the orientational order profile minimizing F numerically, we relax not only the order parameter Q_{ij} but also the shape and size of the numerical system (imposing periodic boundary conditions in the xy plane), using the procedures described in refs 17,18 and 30. We choose typical material parameters^{18} a=8.0×10^{4} J m^{−3}, b=5.0×10^{4} J m^{−3}, c=3.0×10^{4} J m^{−3} K^{−1}, K_{1}=K_{0}=10 pN. We set the rescaled strength of chirality to 0.7, and the rescaled anchoring strength w=2q_{0}(a/b^{2})W to 2.5, corresponding to choosing 2π/q_{0}≃161 nm (cholesteric pitch), and W≃1.0×10^{−3} J m^{−2} (anchoring strength). Thus, we are considering high chirality and strong surface anchoring, but these values are experimentally achievable, and are expected to maximize frustrations that will give rise to a rich variety of exotic ordered structures.
Additional information
How to cite this article: Fukuda, J. & Žumer, S. Quasitwodimensional Skyrmion lattices in a chiral nematic liquid crystal. Nat. Commun. 2:246 doi: 10.1038/ncomms1250 (2011).
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Acknowledgements
We thank Dr Miha Ravnik for valuable discussions. Thanks are also due to Dr Richard James who critically read the manuscript and gave useful comments. J.F. appreciates financial support from the Slovenian Research Agency (ARRS research program P10099 and project J12335) and the Center of Excellence NAMASTE for his stay at the University of Ljubljana, during which most of this work was carried out. J.F. is supported in part also by KAKENHI (GrantinAid for Scientific Research) on the Priority Area 'Soft Matter Physics' from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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S.Ž. conceived the research of a thin chiral liquid crystal film. J.F. carried out numerical calculations and analysed data. Both conceived the possibility of Skyrmion lattices in this system and discussed the results. J.F. wrote the manuscript with inputs from S.Ž.
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Fukuda, Ji., Žumer, S. Quasitwodimensional Skyrmion lattices in a chiral nematic liquid crystal. Nat Commun 2, 246 (2011). https://doi.org/10.1038/ncomms1250
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DOI: https://doi.org/10.1038/ncomms1250
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