In numerous functional materials, such as steels, ferroelectrics and magnets, new functionalities can be achieved through the engineering of the domain structures, which are associated with the ordering of certain parameters within the material. The recent progress in technologies that enable imaging at atomic-scale spatial resolution has transformed our understanding of domain topology, revealing that, along with simple stripe-like or irregularly shaped domains, intriguing vortex-type topological domain configurations also exist. In this Review, we present a new classification scheme of ‘Zm Zn domains with Zl vortices’ for 2D macroscopic domain structures with m directional variants and n translational antiphases. This classification, together with the concepts of topological protection and topological charge conservation, can be applied to a wide range of materials, such as multiferroics, improper ferroelectrics, layered transition metal dichalcogenides and magnetic superconductors, as we discuss using selected examples. The resulting topological considerations provide a new basis for the understanding of the formation, kinetics, manipulation and property optimization of domains and domain boundaries in functional materials.
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Catalan, G., Seidel, J., Ramesh, R. & Scott, J. F. Domain wall nanoelectronics. Rev. Mod. Phys. 84, 119–156 (2012). A comprehensive review of emergent properties at domain boundaries in ferroelectrics and multiferroics.
Ma, E. Y. et al. Mobile metallic domain walls in an all-in-all-out magnetic insulator. Science 350, 538–541 (2015).
Sluka, T., Tagantsev, A. K., Bednyakov, P. & Setter, N. Free-electron gas at charged domain walls in insulating BaTiO3 . Nat. Commun. 4, 1808 (2013).
Farokhipoor, S. et al. Artificial chemical and magnetic structure at the domain walls of an epitaxial oxide. Nature 515, 379–383 (2014).
Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y. Observation of skyrmions in a multiferroic material. Science 336, 198–201 (2012).
Maksymovych, P. et al. Dynamic conductivity of ferroelectric domain walls in BiFeO3 . Nano Lett. 11, 1906–1912 (2011).
Van Aert, S. et al. Direct observation of ferrielectricity at ferroelastic domain boundaries in CaTiO3 by electron microscopy. Adv. Mater. 4, 523–527 (2012).
Seidel, J. et al. Conduction at domain walls in oxide multiferroics. Nat. Mater. 8, 229–234 (2009).
Rubio-Marcos, F., Del Campo, A., Marchet, P. & Jose, F. F. Ferroelectric domain wall motion induced by polarized light. Nat. Commun. 6, 6594 (2015).
Emori, S., Bauer, U., Ahn, S.-M., Martinez, E. & Beach, G. S. D. Current-driven dynamics of chiral ferromagnetic domain walls. Nat. Mater. 12, 611–616 (2013).
Cheong, S.-W. & Mostovoy, M. Multiferroics: a magnetic twist for ferroelectricity. Nat. Mater. 6, 13–20 (2007).
Usui, T. et al. Observation of quadrupole helix chirality and its domain structure in DyFe3(BO3)4 . Nat. Mater. 13, 611–618 (2014).
Khomskii, D. I. Multiferroics: different ways to combine magnetism and ferroelectricity. J. Magn. Magn. Mater. 306, 1–8 (2006).
Choi, T. et al. Insulating interlocked ferroelectric and structural antiphase domain walls in multiferroic YMnO3 . Nat. Mater. 9, 253–258 (2010). This paper presents the discovery of Z2 × Z3 domains and Z6 vortices in hexagonal manganites and demonstrates that the interlocking nature of DVBs and APBs results in Z6 vortices.
Dong, S., Liu, J.-M., Cheong, S.-W. & Ren, Z. Multiferroic materials and magnetoelectric physics: symmetry, entanglement, excitation, and topology. Adv. Phys. 64, 519–626 (2015).
Aizu, K. Possible species of ferromagnetic, ferroelectric, and ferroelastic crystals. Phys. Rev. B 2, 754–772 (1970).
Janovec, V. & Privratska, J. in International Tables for Crystallography Vol. D (ed. Authier, A. ) 449–505 (Wiley, 2003). A comprehensive introduction of a systematic approach to the symmetry analysis of domain states.
Wei, X.-K. et al. Ferroelectric translational antiphase boundaries in nonpolar materials. Nat. Commun. 5, 3031 (2013).
McKenna, K. P. et al. Atomic-scale structure and properties of highly stable antiphase boundary defects in Fe3O4 . Nat. Commun. 5, 5740 (2014).
Meier, D. et al. Translation domains in multiferroics. Phase Trans. 86, 33–52 (2013).
Wu, W. et al. Formation of pancakelike Ising domains and giant magnetic coercivity in ferrimagnetic LuFe2O4 . Phys. Rev. Lett. 101, 137203 (2008).
Eggebrecht, T. et al. Light-induced metastable magnetic texture uncovered by in situ Lorentz microscopy. Phys. Rev. Lett. (in the press).
Choi, Y. J. et al. Giant magnetic coercivity and ionic superlattice nano-domains in Fe0.25TaS2 . Europhys. Lett. 86, 37012 (2009).
Mori, S., Chen, C. H. & Cheong, S. W. Pairing of charge-ordered stripes in (La, Ca)MnO3 . Nature 392, 473–476 (1998).
Fennie, C. J. & Rabe, K. M. Ferroelectric transition in YMnO3 from first principles. Phys. Rev. B 72, 100103 (2005).
Van Aken, B. B., Palstra, T. T. M., Filippetti, A. & Spaldin, N. A. The origin of ferroelectricity in magnetoelectric YMnO3 . Nat. Mater. 3, 164–170 (2004).
Demus, D. Schlieren textures in smectic liquid crystals. Krist. Techn. 10, 933–946 (1975).
Dierking, I., Marshall, O., Wright, J. & Bulleid, N. Annihilation dynamics of umbilical defects in nematic liquid crystals under applied electric fields. Phys. Rev. E 71, 061709 (2005).
Matsumoto, T. et al. Multivariate statistical characterization of charged and uncharged domain walls in multiferroic hexagonal YMnO3 single crystal visualized by a spherical aberration-corrected STEM. Nano Lett. 13, 4594–4601 (2013).
Zhang, Q. et al. Direct observation of multiferroic vortex domains in YMnO3 . Sci. Rep. 3, 2741 (2013).
Han, M.-G. et al. Ferroelectric switching dynamics of topological vortex domains in a hexagonal manganite. Adv. Mater. 25, 2415–2421 (2013).
Meier, D. et al. Anisotropic conductance at improper ferroelectric domain walls. Nat. Mater. 11, 284–288 (2012).
Wu, W., Horibe, Y., Lee, N., Cheong, S. W. & Guest, J. R. Conduction of topologically protected charged ferroelectric domain walls. Phys. Rev. Lett. 108, 077203 (2012).
Geng, Y., Lee, N., Choi, Y. J., Cheong, S. W. & Wu, W. Collective magnetism at multiferroic vortex domain walls. Nano Lett. 12, 6055–6059 (2012).
Geng, Y. et al. Direct visualization of magnetoelectric domains. Nat. Mater. 13, 163–167 (2013).
Zhang, Q. H. et al. Direct observation of interlocked domain walls in hexagonal RMnO3 (R = Tm, Lu). Phys. Rev. B 85, 020102 (2012).
Katsufuji, T. et al. Dielectric and magnetic anomalies and spin frustration in hexagonal RMnO3 (R = Y, Yb, and Lu). Phys. Rev. B 64, 104419 (2001).
Kumagai, Y. & Spaldin, N. A. Structural domain walls in polar hexagonal manganites. Nat. Commun. 4, 1540 (2013).
Fiebig, M., Lottermoser, T., Frohlich, D., Goltsev, A. V. & Pisarev, R. V. Observation of coupled magnetic and electric domains. Nature 419, 818–820 (2002).
Zurek, W. H. Cosmological experiments in superfluid-Helium. Nature 317, 505–508 (1985).
Tosi, G. et al. Onset and dynamics of vortex–antivortex pairs in polariton optical parametric oscillator superfluids. Phys. Rev. Lett. 107, 036401 (2011).
Artyukhin, S., Delaney, K. T., Spaldin, N. A. & Mostovoy, M. Landau theory of topological defects in multiferroic hexagonal manganites. Nat. Mater. 13, 42–49 (2013). This paper presents the first application of the Landau theory to the formation of domain boundaries and topological vortices in hexagonal manganites.
Huang, F.-T. et al. Domain topology and domain switching kinetics in a hybrid improper ferroelectric. Nat. Commun. 7, 11602 (2016). The first report of vortex–antivortex nucleation–annihilation during polarization switching in a new ferroelectric, the domain configurations of which are not topologically protected.
Oh, Y. S., Luo, X., Huang, F.-T., Wang, Y. & Cheong, S.-W. Experimental demonstration of hybrid improper ferroelectricity and the presence of abundant charged walls in (Ca,Sr)3Ti2O7 crystals. Nat. Mater. 14, 407–413 (2015).
Schröder, M. et al. Conducting domain walls in lithium niobate single crystals. Adv. Funct. Mater. 22, 3936–3944 (2012).
Sluka, T., Tagantsev, A. K., Damjanovic, D., Gureev, M. & Setter, N. Enhanced electromechanical response of ferroelectrics due to charged domain walls. Nat. Commun. 3, 748 (2012).
Bode, M. et al. Atomic spin structure of antiferromagnetic domain walls. Nat. Mater. 5, 477–481 (2006).
Seidel, J., Vasudevan, R. K. & Valanoor, N. Topological structures in multiferroics — domain walls, skyrmions and vortices. Adv. Electron. Mater. 2, 1500292 (2015).
Das, H., Wysocki, A. L., Geng, Y. & Wu, W. Bulk magnetoelectricity in the hexagonal manganites and ferrites. Nat. Commun. 5, 2998 (2014).
Huang, F.-T. et al. Topological defects at octahedral tilting plethora in bi-layered perovskites. npj Quantum Mater. 1, 16017 (2016).
Ma, E. Y. et al. Charge-order domain walls with enhanced conductivity in a layered manganite. Nat. Commun. 6, 7595 (2015).
Huang, F.-T. et al. Delicate balance between ferroelectricity and antiferroelectricity in hexagonal InMnO3 . Phys. Rev. B 87, 184109 (2013).
Chae, S. C. et al. Direct observation of the proliferation of ferroelectric loop domains and vortex–antivortex pairs. Phys. Rev. Lett. 108, 167603 (2012).
Chae, S. C. et al. Self-organization, condensation, and annihilation of topological vortices and antivortices in a multiferroic. Proc. Natl Acad. Sci. USA 107, 21366–21370 (2010). The paper provides the first report of a graph theoretical analysis of domains and domain-boundary configurations providing mathematical guidance for the investigation of a network of topological defects.
Chae, S. C. et al. Evolution of the domain topology in a ferroelectric. Phys. Rev. Lett. 110, 167601 (2013).
Horibe, Y. et al. Color theorems, chiral domain topology, and magnetic properties of FexTaS2 . J. Am. Chem. Soc. 136, 8368–8373 (2014). This paper reports the discovery of Z2 × Z3 domains and Z6 vortices in a transition metal dichalcogenide and the application of tensorial colouring for its domain topology analysis.
Cho, D. et al. Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T-TaS2 . Nat. Commun. 7, 10453 (2016).
Ma, L. et al. A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2 . Nat. Commun. 7, 10956 (2016).
Chen, C. H., Gibson, J. M. & Fleming, R. M. Microstructure in the incommensurate and the commensurate charge-density-wave states of 2H-TaSe2: a direct observation by electron microscopy. Phys. Rev. B 26, 184–205 (1982).
Alden, J. S. et al. Strain solitons and topological defects in bilayer graphene. Proc. Natl Acad. Sci. USA 110, 11256–11260 (2013).
Choi, S. et al. First observation of plaquette antiferromagnetic order and manipulation of their domain walls in iron-based superconductors. Preprint at arXivhttps://arxiv.org/abs/1608.00884 (2016).
Gruverman, A. et al. Vortex ferroelectric domains. J. Phys. Condens. Matter 20, 342201 (2008).
Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).
Jia, C. L., Urban, K. W., Alexe, M., Hesse, D. & Vrejoiu, I. Direct observation of continuous electric dipole rotation in flux-closure domains in ferroelectric Pb(Zr,Ti)O3 . Science 331, 1420–1423 (2011).
Tang, Y. L. et al. Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348, 547–551 (2015).
Ivry, Y., Chu, D. P., Scott, J. F. & Durkan, C. Flux closure vortex-like domain structures in ferroelectric thin films. Phys. Rev. Lett. 104, 207602 (2010).
Bodnarchuk, M. I., Shevchenko, E. V. & Talapin, D. V. Structural defects in periodic and quasicrystalline binary nanocrystal superlattices. J. Am. Chem. Soc. 133, 20837–20849 (2011).
Maciá, E. The role of aperiodic order in science and technology. Rep. Prog. Phys. 69, 397–441 (2005).
Xue, F. et al. Evolution of the statistical distribution in a topological defect network. Sci. Rep. 5, 17057 (2015).
Griffin, S. M. et al. Scaling behavior and beyond equilibrium in the hexagonal manganites. Phys. Rev. X 2, 041022 (2012).
Lin, S.-Z. et al. Topological defects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics. Nat. Phys. 10, 970–977 (2014). The first report of a quantitative analysis of topological defects in hexagonal manganites, and the first application of the Higgs mechanism to the condensation of topological defects.
Weiss, P. L'hypothèse du champ moléculair et la propriété ferromagnétique. J. Phys. Radium 6, 661–690 (in French) (1907).
Stewart, K. H. in Ferromagnetic Domains 1–2; 82 (Cambridge Univ. Press, 1954).
von Hámos, L. & Thiessen, P. A. Über die sichtbarmachung von Bexirken verschiedenen ferromagnetischen zustandes fester körper. Z. Phys. 71, 442–444 (in German) (1931).
Zwicker, B. & Scherrer, P. Elektrooptische eigenschaften der seignette-elektrischen kristalle KH2PO4 und KD2PO4 . Helv. Phys. Acta 17, 346–373 (in German) (1944).
Ubbelohde, A. R. & Woodward, I. Laue photographs of sub-crystalline regions in ‘hybrid’ crystals of potassium dihydrogen phosphate. Nature 156, 20–21 (1945).
Kay, H. F. Preparation and properties of crystals of barium titanate, BaTiO3 . Acta Cryst. 1, 229–237 (1948).
Matthias, B. & von Hippel, A. Domain structure and dielectric response of barium titanate single crystals. Phys. Rev. 73, 1378–1384 (1948).
Johansson, C. H. & Linde, J. O. Röntgenographische und elektrische untersuchungen des CuAu-systems. Ann. Phys. 417, 1–48 (in German) (1936).
Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979). This is a comprehensive review of the topological theory of defects, which introduces a topological approach for condensed matter physics problems.
Huang, F.-T. et al. Duality of topological defects in hexagonal manganites. Phys. Rev. Lett. 113, 267602 (2014).
Bednorz, J. G. & Müller, K. A. Perovskite-type oxides — the new approach to high-Tc superconductivity. Rev. Mod. Phys. 60, 585–600 (1988).
Cheong, S.-W. Transition metal oxides: the exciting world of orbitals. Nat. Mater. 6, 927–928 (2007).
von Hippel, A. Ferroelectricity, domain structure, and phase transitions of barium titanate. Rev. Mod. Phys. 22, 221–237 (1950).
Young, J., Stroppa, A., Picozzi, S. & Rondinelli, J. M. Anharmonic lattice interactions in improper ferroelectrics for multiferroic design. J. Phys. Condens. Matter 27, 283202 (2015).
Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
Benedek, N. A. & Fennie, C. J. Hybrid improper ferroelectricity: a mechanism for controllable polarization–magnetization coupling. Phys. Rev. Lett. 106, 107204 (2011).
Harris, A. B. Symmetry analysis for the Ruddlesden–Popper systems Ca3Mn2O7 and Ca3Ti2O7 . Phys. Rev. B 84, 064116 (2011).
Ruddlesden, S. N. & Popper, P. The compound Sr3Ti2O7 and its structure. Acta Cryst. 11, 54–55 (1958).
Lilienblum, M. et al. Ferroelectricity in the multiferroic hexagonal manganites. Nat. Phys. 11, 1070–1073 (2015).
Mostovoy, M. A whirlwind of opportunities. Nat. Mater. 9, 188–190 (2010).
Disseler, S. M. et al. Multiferroicity in doped hexagonal LuFeO3 . Phys. Rev. B 92, 054435 (2015).
Li, J. et al. Ferroelectric annular domains in hexagonal manganites. Phys. Rev. B 87, 094106 (2013).
Wang, X., Huang, F.-T., Hu, R., Fan, F. & Cheong, S.-W. Self-poling with oxygen off-stoichiometry in ferroelectric hexagonal manganites. APL Mater. 3, 041505 (2015).
Mori, S. et al. Ferroelectric and structural antiphase domain and domain wall structures in Y(Mn,Ti)O3 . Ferroelectrics 462, 50–54 (2014).
Yu, Y. et al. Atomic-scale study of topological vortex-like domain pattern in multiferroic hexagonal manganites. Appl. Phys. Lett. 103, 032901 (2013).
Li, J. et al. Scanning secondary-electron microscopy on ferroelectric domains and domain walls in YMnO3 . Appl. Phys. Lett. 100, 152903 (2012).
Lochocki, E. B., Park, S., Lee, N., Cheong, S. W. & Wu, W. Piezoresponse force microscopy of domains and walls in multiferroic HoMnO3 . Appl. Phys. Lett. 99, 232901 (2011).
Fiebig, M. et al. Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation. Phys. Rev. Lett. 84, 5620–5623 (2000).
Schaab, J. et al. Imaging and characterization of conducting ferroelectric domain walls by photoemission electron microscopy. Appl. Phys. Lett. 104, 232904 (2014).
Wu, X. et al. Low-energy structural dynamics of ferroelectric domain walls in hexagonal rare-earth manganites. Preprint at ArXivhttp://arxiv.org/abs/1702.06205 (2017).
Liang, L., Wu, H., Li, L. & Zhu, X. Characterization of multiferroic domain structures in multiferroic oxides. J. Nanomater. 2015, 1–8 (2015).
Gupta, S. & Saxena, A. A topological twist on materials science. MRS Bull. 39, 265–279 (2014).
Fiebig, M. Phase engineering in oxides by interfaces. Phil. Trans. R. Soc. A 370, 4972–4988 (2012).
Wang, X. et al. Unfolding of vortices into topological stripes in a multiferroic material. Phys. Rev. Lett. 112, 247601 (2014).
Feynman, R. P. Application of quantum mechanics to liquid helium. Prog. Low Temp. Phys. 1, 17–35 (1955).
Kibble, T. Topology of cosmic domains and strings. J. Phys. A 9, 1387–1398 (1976).
Sonin, E. B. Magnus force in superfluids and superconductors. Phys. Rev. B 55, 485–501 (1997).
Kumagai, Y. et al. Observation of persistent centrosymmetricity in the hexagonal manganite family. Phys. Rev. B 85, 174422 (2012).
Li, J. et al. Homotopy-theoretic study and atomic-scale observation of vortex domains in hexagonal manganites. Sci. Rep. 6, 28047 (2016).
Cano, A. Hidden order in hexagonal RMnO3 multiferroics (R = Dy–Lu, In, Y, and Sc). Phys. Rev. B 89, 214107 (2014).
Wilson, J. A., Di Salvo, F. J. & Mahajan, S. Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides. Adv. Phys. 50, 1171–1248 (2001).
Castro Neto, A. H. Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides. Phys. Rev. Lett. 86, 4382–4385 (2001).
Moncton, D. E., Axe, J. D. & Disalvo, F. J. Neutron scattering study of the charge-density wave transitions in 2H–TaSe2 and 2H–NbSe2 . Phys. Rev. B 16, 801–819 (1977).
Morris, R. C., Coleman, R. V. & Bhandari, R. Superconductivity and magnetoresistance in NbSe2 . Phys. Rev. B 5, 895 (1972).
Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol. 7, 699–712 (2012).
Li, H. et al. Fabrication of single- and multilayer MoS2 film-based field-effect transistors for sensing NO at room temperature. Small 8, 63–67 (2011).
Yu, Y. et al. Gate-tunable phase transitions in thin flakes of 1T-TaS2 . Nat. Nanotechnol. 10, 270–276 (2015).
Qi, Y. et al. Superconductivity in Weyl semimetal candidate MoTe2 . Nat. Commun. 7, 11038 (2016).
Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015).
Klemm, R. A. Pristine and intercalated transition metal dichalcogenide superconductors. Phys. Rev. B 514, 86–94 (2015).
Parkin, S. S. P. & Friend, R. H. 3d transition-metal intercalates of the niobium and tantalum dichalcogenides. I. Magnetic properties. Philos. Mag. B 41, 65–93 (2006).
Van Laar, B., Rietveld, H. M. & Ijdo, D. J. W. Magnetic and crystallographic structures of MexNbS2 and MexTaS2 . J. Solid State Chem. 3, 154–160 (1971).
Gevers, R., Blank, H. & Amelinckx, S. Extension of the Howie–Whelan equations for electron diffraction to non-centro symmetrical crystals. Phys. Stat. Sol. 13, 449–465 (1966).
Friend, R. H., Beal, A. R. & Yoffe, A. D. Electrical and magnetic properties of some first row transition metal intercalates of niobium disulphide. Philos. Mag. 35, 1269–1287 (2006).
Parkin, S. S. P. & Friend, R. H. 3d transition-metal intercalates of the niobium and tantalum dichalcogenides. II. Transport properties. Philos. Mag. B 41, 95–112 (2006).
Mulder, A. T., Benedek, N. A., Rondinelli, J. M. & Fennie, C. J. Turning ABO3 antiferroelectrics into ferroelectrics: design rules for practical rotation-driven ferroelectricity in double perovskites and A3B2O7 Ruddlesden-Popper compounds. Adv. Funct. Mater. 23, 4810–4820 (2013).
Xu, B. et al. Hybrid improper ferroelectricity in multiferroic superlattices: finite-temperature properties and electric-field-driven switching of polarization and magnetization. Adv. Funct. Mater. 25, 3626–3633 (2015).
Ghosez, P. & Triscone, J.-M. Coupling of three lattice instabilities. Nat. Mater. 10, 269–270 (2011).
Benedek, N. A., Mulder, A. T. & Fennie, C. J. Polar octahedral rotations: a path to new multifunctional materials. J. Solid State Chem. 15, 11–20 (2012).
Gureev, M. Y., Tagantsev, A. K. & Setter, N. Head-to-head and tail-to-tail 180° domain walls in an isolated ferroelectric. Phys. Rev. B 83, 184104 (2011).
Chen, C. et al. Ferroelectricity in Dion–Jacobson ABiNb2O7 (A = Rb, Cs) compounds. J. Mater. Chem. C 3, 19–22 (2014).
Fennie, C. J. & Rabe, K. M. Ferroelectricity in the Dion–Jacobson CsBiNb2O7 from first principles. Appl. Phys. Lett. 88, 262902 (2006).
Benedek, N. A. Origin of ferroelectricity in a family of polar oxides: the Dion—Jacobson phases. Inorg. Chem. 53, 3769–3777 (2014).
Perez-Mato, J. M. et al. Competing structural instabilities in the ferroelectric Aurivillius compound SrBi2Ta2O9 . Phys. Rev. B 70, 214111 (2004).
Wu, F. Y. The Potts model. Rev. Mod. Phys. 54, 235–268 (1982).
Si, Q., Yu, R. & Abrahams, E. High-temperature superconductivity in iron pnictides and chalcogenides. Nat. Rev. Mater. 1, 16017 (2016).
Dai, P. Antiferromagnetic order and spin dynamics in iron-based superconductors. Rev. Mod. Phys. 87, 855–896 (2015).
Tang, S.-C., Ding, M.-C. & Zhang, Y.-Z. Magnetic properties controlled by interstitial or interlayer cations in iron chalcogenides. Sci. Rep. 6, 19031 (2015).
Chandra, P., Coleman, P., Mydosh, J. A. & Tripathi, V. Hidden orbital order in the heavy fermion metal URu2Si2 . Nature 417, 831–834 (2002).
Ritschel, T. et al. Orbital textures and charge density waves in transition metal dichalcogenides. Nat. Phys. 11, 328–331 (2015).
Vander Griend, D. A., Malo, S., Barry, S. J. & Dabbousch, N. M. La3Cu2VO9: a surprising variation on the YAlO3 structure-type with 2D copper clusters of embedded triangles. Solid State Sci. 3, 569–579 (2001).
Chou, F. C. et al. Sodium ion ordering and vacancy cluster formation in Nax CoO2 (x = 0.71 and 0.84) single crystals by synchrotron X-ray diffraction. Phys. Rev. Lett. 101, 127404 (2008).
Van Landuyt, J., Wiegers, G. A. & Amelinckx, S. A new type of deformation modulated superstructure in 1T-VSe2 and its relation with other superstructures in transition metal dichalcogenides. Phys. Stat. Sol. (a) 46, 479–492 (1978).
Tokunaga, Y. et al. Rotation of orbital stripes and the consequent charge-polarized state in bilayer manganites. Nat. Mater. 5, 937–941 (2006).
He, Z. B., Deng, G., Tian, H. & Xu, Q. 90° Rotation of orbital stripes in bilayer manganite PrCa2Mn2O7 studied by in situ transmission electron microscopy. J. Solid State Chem. 200, 287–293 (2013).
Srolovitz, D. J. & Scott, J. F. Clock-model description of incommensurate ferroelectric-films and of nematic-liquid-crystal films. Phys. Rev. B 34, 1815–1819 (1986).
Scholten, P. D. & Irakliotis, L. J. Critical behavior of the q-state clock model in three dimensions. Phys. Rev. B 48, 1291–1294 (1993).
Baek, S. K., Minnhagen, P., Shima, H. & Kim, B. J. Phase transition of q-state clock models on hexagonal lattices. Phys. Rev. E 80, 011133 (2009).
Bazavov, A., Berg, B. A. & Dubey, S. Phase transition properties of 3D Potts models. Nucl. Phys. B 802, 421–434 (2008).
Wu, F. Y. Percolation and the Potts Model. J. Stat. Phys. 18, 115–123 (1978).
Yamaguchi, C. & Okabe, Y. Three-dimensional antiferromagnetic q-state Potts models: application of the Wang–Landau algorithm. J. Phys. A 34, 8781–8794 (2001).
Surungan, T., Komura, Y. & Okabe, Y. Probing phase transition order of q-state Potts models using Wang–Landau Algorithm. AIP Conf. Proc. 1617, 79–82 (2014).
Ono, I. & Ito, K. Monte Carlo simulations and pair approximations on the phase transition of the restricted orientational lattice model for liquid crystals. J. Phys. C 15, 4417–4430 (1982).
Tanaka, S., Tamura, R. & Kawashima, N. Phase transition of generalized ferromagnetic Potts model-effect of invisible states. J. Phys. Conf. Ser. 297, 012022 (2011).
Domany, E., Shnidman, Y. & Mukamel, D. Type I FCC antiferromagnets in a magnetic field: a realisation of the q = 3 and q = 4-state Potts models. J. Phys. C 15, L495–L500 (1982).
Van Landuyt, J., Van Tendeloo, G., Amelinckx, S. & Walker, M. B. Interpretation of Dauphiné-twin-domain configurations resulting from the α-β phase transition in quartz and aluminum phosphate. Phys. Rev. B 31, 2986–2992 (1985).
Amelinckx, S. The study of phase transitions and the resulting domain structures by means of electron microscopy and electron diffraction. J. Phys. Colloques 10, 83–99 (2010).
Koyama, Y., Yoshida, J., Hoshiya, H. & Nakamura, Y. Striped-type superstructure in γ-brass alloys. Phys. Rev. B 40, 5378–5386 (1989).
Koyama, Y., Hatano, M. & Tanimura, M. Antiphase boundaries, inversion, and ferroelastic domains in the striped-type superstructure of γ-brass Cu-Al alloys. Phys. Rev. B 53, 11462–11468 (1996).
Nakamura, Y., Koike, H. & Nittono, O. Structure of long period superstructure in Cu-rich γ-brass investigated by electron-diffraction. Phys. Stat. Sol. (a) 118, 389–400 (1990).
Pan, X. Q., Hu, M. S., Yao, M. H. & Feng, D. A. TEM study of the incommensurate phase and related phase-transitions in barium sodium niobate. Phys. Stat. Sol. (a) 92, 57–68 (1985).
Yamamoto, N., Katoh, M. & Mori, S. Domain pattern formation during incommensurate–commensurate phase transition in Rb2ZnCl4 . Ferroelectrics 191, 247–252 (1997).
Tsuda, K., Yamamoto, N. & Yagi, K. Electron microscope study on commensurate–incommensurate phase transition of Rb2ZnCl4 crystals. J. Phys. Soc. Jpn 57, 2057–2068 (1988).
McMillan, W. L. Theory of discommensurations and the commensurate–incommensurate charge-density-wave phase transition. Phys. Rev. B 14, 1496–1502 (1976).
Walker, M. B. Theory of domains and dislocations in the charge-density-wave states of 2H-TaSe2 . Phys. Rev. B 26, 6208–6214 (1982).
Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).
Koshibae, W. & Nagaosa, N. Theory of antiskyrmions in magnets. Nat. Commun. 7, 10542 (2016).
Gong, X. & Mele, E. J. Stacking textures and singularities in bilayer graphene. Phys. Rev. B 89, 121415 (2014).
The authors thank C. D. Batista (University of Tennessee, USA), D. Y. Jeong (Soongsil University, South Korea) and W. Wu (Rutgers University, USA) for helpful discussions, D. Cho and S. J. Lim for critical reading of the manuscript and L. L. Cheong for helping with the preparation of schematics. This work is supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4413 to the Rutgers Center for Emergent Materials.
The authors declare no competing interests.
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Huang, FT., Cheong, SW. Aperiodic topological order in the domain configurations of functional materials. Nat Rev Mater 2, 17004 (2017). https://doi.org/10.1038/natrevmats.2017.4
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