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Aperiodic topological order in the domain configurations of functional materials

Abstract

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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Figure 1: Zm × Zn domains and Zl vortices.
Figure 2: Z2 × Z3 domains and Z6 vortices in h-RMnO3.
Figure 3: Z2 × Z3 domains and Z6 vortices in 2H-Fe1/3TaS2.
Figure 4: Z4 × Z2 domains and Z3 vortices in bilayered perovskite RP327.
Figure 5: Z1 × Z4 domains and Z3 vortices.
Figure 6: Other examples of domain configurations.

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References

  1. 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.

    CAS  Google Scholar 

  2. Ma, E. Y. et al. Mobile metallic domain walls in an all-in-all-out magnetic insulator. Science 350, 538–541 (2015).

    CAS  Google Scholar 

  3. Sluka, T., Tagantsev, A. K., Bednyakov, P. & Setter, N. Free-electron gas at charged domain walls in insulating BaTiO3 . Nat. Commun. 4, 1808 (2013).

    Google Scholar 

  4. Farokhipoor, S. et al. Artificial chemical and magnetic structure at the domain walls of an epitaxial oxide. Nature 515, 379–383 (2014).

    CAS  Google Scholar 

  5. Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y. Observation of skyrmions in a multiferroic material. Science 336, 198–201 (2012).

    CAS  Google Scholar 

  6. Maksymovych, P. et al. Dynamic conductivity of ferroelectric domain walls in BiFeO3 . Nano Lett. 11, 1906–1912 (2011).

    CAS  Google Scholar 

  7. Van Aert, S. et al. Direct observation of ferrielectricity at ferroelastic domain boundaries in CaTiO3 by electron microscopy. Adv. Mater. 4, 523–527 (2012).

    Google Scholar 

  8. Seidel, J. et al. Conduction at domain walls in oxide multiferroics. Nat. Mater. 8, 229–234 (2009).

    CAS  Google Scholar 

  9. Rubio-Marcos, F., Del Campo, A., Marchet, P. & Jose, F. F. Ferroelectric domain wall motion induced by polarized light. Nat. Commun. 6, 6594 (2015).

    CAS  Google Scholar 

  10. 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).

    CAS  Google Scholar 

  11. Cheong, S.-W. & Mostovoy, M. Multiferroics: a magnetic twist for ferroelectricity. Nat. Mater. 6, 13–20 (2007).

    CAS  Google Scholar 

  12. Usui, T. et al. Observation of quadrupole helix chirality and its domain structure in DyFe3(BO3)4 . Nat. Mater. 13, 611–618 (2014).

    CAS  Google Scholar 

  13. Khomskii, D. I. Multiferroics: different ways to combine magnetism and ferroelectricity. J. Magn. Magn. Mater. 306, 1–8 (2006).

    CAS  Google Scholar 

  14. 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.

    CAS  Google Scholar 

  15. 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).

    CAS  Google Scholar 

  16. Aizu, K. Possible species of ferromagnetic, ferroelectric, and ferroelastic crystals. Phys. Rev. B 2, 754–772 (1970).

    Google Scholar 

  17. 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.

    Google Scholar 

  18. Wei, X.-K. et al. Ferroelectric translational antiphase boundaries in nonpolar materials. Nat. Commun. 5, 3031 (2013).

    Google Scholar 

  19. McKenna, K. P. et al. Atomic-scale structure and properties of highly stable antiphase boundary defects in Fe3O4 . Nat. Commun. 5, 5740 (2014).

    CAS  Google Scholar 

  20. Meier, D. et al. Translation domains in multiferroics. Phase Trans. 86, 33–52 (2013).

    CAS  Google Scholar 

  21. Wu, W. et al. Formation of pancakelike Ising domains and giant magnetic coercivity in ferrimagnetic LuFe2O4 . Phys. Rev. Lett. 101, 137203 (2008).

    Google Scholar 

  22. Eggebrecht, T. et al. Light-induced metastable magnetic texture uncovered by in situ Lorentz microscopy. Phys. Rev. Lett. (in the press).

  23. Choi, Y. J. et al. Giant magnetic coercivity and ionic superlattice nano-domains in Fe0.25TaS2 . Europhys. Lett. 86, 37012 (2009).

    Google Scholar 

  24. Mori, S., Chen, C. H. & Cheong, S. W. Pairing of charge-ordered stripes in (La, Ca)MnO3 . Nature 392, 473–476 (1998).

    CAS  Google Scholar 

  25. Fennie, C. J. & Rabe, K. M. Ferroelectric transition in YMnO3 from first principles. Phys. Rev. B 72, 100103 (2005).

    Google Scholar 

  26. 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).

    CAS  Google Scholar 

  27. Demus, D. Schlieren textures in smectic liquid crystals. Krist. Techn. 10, 933–946 (1975).

    CAS  Google Scholar 

  28. 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).

    CAS  Google Scholar 

  29. 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).

    CAS  Google Scholar 

  30. Zhang, Q. et al. Direct observation of multiferroic vortex domains in YMnO3 . Sci. Rep. 3, 2741 (2013).

    Google Scholar 

  31. Han, M.-G. et al. Ferroelectric switching dynamics of topological vortex domains in a hexagonal manganite. Adv. Mater. 25, 2415–2421 (2013).

    CAS  Google Scholar 

  32. Meier, D. et al. Anisotropic conductance at improper ferroelectric domain walls. Nat. Mater. 11, 284–288 (2012).

    CAS  Google Scholar 

  33. 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).

    Google Scholar 

  34. 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).

    CAS  Google Scholar 

  35. Geng, Y. et al. Direct visualization of magnetoelectric domains. Nat. Mater. 13, 163–167 (2013).

    Google Scholar 

  36. Zhang, Q. H. et al. Direct observation of interlocked domain walls in hexagonal RMnO3 (R = Tm, Lu). Phys. Rev. B 85, 020102 (2012).

    Google Scholar 

  37. 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).

    Google Scholar 

  38. Kumagai, Y. & Spaldin, N. A. Structural domain walls in polar hexagonal manganites. Nat. Commun. 4, 1540 (2013).

    Google Scholar 

  39. Fiebig, M., Lottermoser, T., Frohlich, D., Goltsev, A. V. & Pisarev, R. V. Observation of coupled magnetic and electric domains. Nature 419, 818–820 (2002).

    CAS  Google Scholar 

  40. Zurek, W. H. Cosmological experiments in superfluid-Helium. Nature 317, 505–508 (1985).

    CAS  Google Scholar 

  41. Tosi, G. et al. Onset and dynamics of vortex–antivortex pairs in polariton optical parametric oscillator superfluids. Phys. Rev. Lett. 107, 036401 (2011).

    CAS  Google Scholar 

  42. 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.

    Google Scholar 

  43. 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.

    CAS  Google Scholar 

  44. 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).

    CAS  Google Scholar 

  45. Schröder, M. et al. Conducting domain walls in lithium niobate single crystals. Adv. Funct. Mater. 22, 3936–3944 (2012).

    Google Scholar 

  46. 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).

    Google Scholar 

  47. Bode, M. et al. Atomic spin structure of antiferromagnetic domain walls. Nat. Mater. 5, 477–481 (2006).

    CAS  Google Scholar 

  48. Seidel, J., Vasudevan, R. K. & Valanoor, N. Topological structures in multiferroics — domain walls, skyrmions and vortices. Adv. Electron. Mater. 2, 1500292 (2015).

    Google Scholar 

  49. Das, H., Wysocki, A. L., Geng, Y. & Wu, W. Bulk magnetoelectricity in the hexagonal manganites and ferrites. Nat. Commun. 5, 2998 (2014).

    Google Scholar 

  50. Huang, F.-T. et al. Topological defects at octahedral tilting plethora in bi-layered perovskites. npj Quantum Mater. 1, 16017 (2016).

    Google Scholar 

  51. Ma, E. Y. et al. Charge-order domain walls with enhanced conductivity in a layered manganite. Nat. Commun. 6, 7595 (2015).

    Google Scholar 

  52. Huang, F.-T. et al. Delicate balance between ferroelectricity and antiferroelectricity in hexagonal InMnO3 . Phys. Rev. B 87, 184109 (2013).

    Google Scholar 

  53. Chae, S. C. et al. Direct observation of the proliferation of ferroelectric loop domains and vortex–antivortex pairs. Phys. Rev. Lett. 108, 167603 (2012).

    CAS  Google Scholar 

  54. 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.

    CAS  Google Scholar 

  55. Chae, S. C. et al. Evolution of the domain topology in a ferroelectric. Phys. Rev. Lett. 110, 167601 (2013).

    CAS  Google Scholar 

  56. 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.

    CAS  Google Scholar 

  57. Cho, D. et al. Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T-TaS2 . Nat. Commun. 7, 10453 (2016).

    CAS  Google Scholar 

  58. Ma, L. et al. A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2 . Nat. Commun. 7, 10956 (2016).

    CAS  Google Scholar 

  59. 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).

    CAS  Google Scholar 

  60. Alden, J. S. et al. Strain solitons and topological defects in bilayer graphene. Proc. Natl Acad. Sci. USA 110, 11256–11260 (2013).

    CAS  Google Scholar 

  61. 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).

  62. Gruverman, A. et al. Vortex ferroelectric domains. J. Phys. Condens. Matter 20, 342201 (2008).

    Google Scholar 

  63. Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).

    CAS  Google Scholar 

  64. 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).

    CAS  Google Scholar 

  65. Tang, Y. L. et al. Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348, 547–551 (2015).

    CAS  Google Scholar 

  66. 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).

    CAS  Google Scholar 

  67. 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).

    CAS  Google Scholar 

  68. Maciá, E. The role of aperiodic order in science and technology. Rep. Prog. Phys. 69, 397–441 (2005).

    Google Scholar 

  69. Xue, F. et al. Evolution of the statistical distribution in a topological defect network. Sci. Rep. 5, 17057 (2015).

    CAS  Google Scholar 

  70. Griffin, S. M. et al. Scaling behavior and beyond equilibrium in the hexagonal manganites. Phys. Rev. X 2, 041022 (2012).

    Google Scholar 

  71. 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.

    CAS  Google Scholar 

  72. Weiss, P. L'hypothèse du champ moléculair et la propriété ferromagnétique. J. Phys. Radium 6, 661–690 (in French) (1907).

    Google Scholar 

  73. Stewart, K. H. in Ferromagnetic Domains 1–2; 82 (Cambridge Univ. Press, 1954).

    Google Scholar 

  74. von Hámos, L. & Thiessen, P. A. Über die sichtbarmachung von Bexirken verschiedenen ferromagnetischen zustandes fester körper. Z. Phys. 71, 442–444 (in German) (1931).

    Google Scholar 

  75. Zwicker, B. & Scherrer, P. Elektrooptische eigenschaften der seignette-elektrischen kristalle KH2PO4 und KD2PO4 . Helv. Phys. Acta 17, 346–373 (in German) (1944).

    CAS  Google Scholar 

  76. Ubbelohde, A. R. & Woodward, I. Laue photographs of sub-crystalline regions in ‘hybrid’ crystals of potassium dihydrogen phosphate. Nature 156, 20–21 (1945).

    CAS  Google Scholar 

  77. Kay, H. F. Preparation and properties of crystals of barium titanate, BaTiO3 . Acta Cryst. 1, 229–237 (1948).

    CAS  Google Scholar 

  78. Matthias, B. & von Hippel, A. Domain structure and dielectric response of barium titanate single crystals. Phys. Rev. 73, 1378–1384 (1948).

    CAS  Google Scholar 

  79. Johansson, C. H. & Linde, J. O. Röntgenographische und elektrische untersuchungen des CuAu-systems. Ann. Phys. 417, 1–48 (in German) (1936).

    Google Scholar 

  80. 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.

    CAS  Google Scholar 

  81. Huang, F.-T. et al. Duality of topological defects in hexagonal manganites. Phys. Rev. Lett. 113, 267602 (2014).

    Google Scholar 

  82. Bednorz, J. G. & Müller, K. A. Perovskite-type oxides — the new approach to high-Tc superconductivity. Rev. Mod. Phys. 60, 585–600 (1988).

    CAS  Google Scholar 

  83. Cheong, S.-W. Transition metal oxides: the exciting world of orbitals. Nat. Mater. 6, 927–928 (2007).

    CAS  Google Scholar 

  84. von Hippel, A. Ferroelectricity, domain structure, and phase transitions of barium titanate. Rev. Mod. Phys. 22, 221–237 (1950).

    CAS  Google Scholar 

  85. 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).

    Google Scholar 

  86. Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).

    CAS  Google Scholar 

  87. Benedek, N. A. & Fennie, C. J. Hybrid improper ferroelectricity: a mechanism for controllable polarization–magnetization coupling. Phys. Rev. Lett. 106, 107204 (2011).

    Google Scholar 

  88. Harris, A. B. Symmetry analysis for the Ruddlesden–Popper systems Ca3Mn2O7 and Ca3Ti2O7 . Phys. Rev. B 84, 064116 (2011).

    Google Scholar 

  89. Ruddlesden, S. N. & Popper, P. The compound Sr3Ti2O7 and its structure. Acta Cryst. 11, 54–55 (1958).

    CAS  Google Scholar 

  90. Lilienblum, M. et al. Ferroelectricity in the multiferroic hexagonal manganites. Nat. Phys. 11, 1070–1073 (2015).

    CAS  Google Scholar 

  91. Mostovoy, M. A whirlwind of opportunities. Nat. Mater. 9, 188–190 (2010).

    CAS  Google Scholar 

  92. Disseler, S. M. et al. Multiferroicity in doped hexagonal LuFeO3 . Phys. Rev. B 92, 054435 (2015).

    Google Scholar 

  93. Li, J. et al. Ferroelectric annular domains in hexagonal manganites. Phys. Rev. B 87, 094106 (2013).

    Google Scholar 

  94. 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).

    Google Scholar 

  95. Mori, S. et al. Ferroelectric and structural antiphase domain and domain wall structures in Y(Mn,Ti)O3 . Ferroelectrics 462, 50–54 (2014).

    CAS  Google Scholar 

  96. Yu, Y. et al. Atomic-scale study of topological vortex-like domain pattern in multiferroic hexagonal manganites. Appl. Phys. Lett. 103, 032901 (2013).

    Google Scholar 

  97. Li, J. et al. Scanning secondary-electron microscopy on ferroelectric domains and domain walls in YMnO3 . Appl. Phys. Lett. 100, 152903 (2012).

    Google Scholar 

  98. 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).

    Google Scholar 

  99. Fiebig, M. et al. Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation. Phys. Rev. Lett. 84, 5620–5623 (2000).

    CAS  Google Scholar 

  100. Schaab, J. et al. Imaging and characterization of conducting ferroelectric domain walls by photoemission electron microscopy. Appl. Phys. Lett. 104, 232904 (2014).

    Google Scholar 

  101. 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).

  102. Liang, L., Wu, H., Li, L. & Zhu, X. Characterization of multiferroic domain structures in multiferroic oxides. J. Nanomater. 2015, 1–8 (2015).

    Google Scholar 

  103. Gupta, S. & Saxena, A. A topological twist on materials science. MRS Bull. 39, 265–279 (2014).

    CAS  Google Scholar 

  104. Fiebig, M. Phase engineering in oxides by interfaces. Phil. Trans. R. Soc. A 370, 4972–4988 (2012).

    CAS  Google Scholar 

  105. Wang, X. et al. Unfolding of vortices into topological stripes in a multiferroic material. Phys. Rev. Lett. 112, 247601 (2014).

    CAS  Google Scholar 

  106. Feynman, R. P. Application of quantum mechanics to liquid helium. Prog. Low Temp. Phys. 1, 17–35 (1955).

    CAS  Google Scholar 

  107. Kibble, T. Topology of cosmic domains and strings. J. Phys. A 9, 1387–1398 (1976).

    Google Scholar 

  108. Sonin, E. B. Magnus force in superfluids and superconductors. Phys. Rev. B 55, 485–501 (1997).

    CAS  Google Scholar 

  109. Kumagai, Y. et al. Observation of persistent centrosymmetricity in the hexagonal manganite family. Phys. Rev. B 85, 174422 (2012).

    Google Scholar 

  110. Li, J. et al. Homotopy-theoretic study and atomic-scale observation of vortex domains in hexagonal manganites. Sci. Rep. 6, 28047 (2016).

    CAS  Google Scholar 

  111. Cano, A. Hidden order in hexagonal RMnO3 multiferroics (R = Dy–Lu, In, Y, and Sc). Phys. Rev. B 89, 214107 (2014).

    Google Scholar 

  112. 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).

    Google Scholar 

  113. Castro Neto, A. H. Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides. Phys. Rev. Lett. 86, 4382–4385 (2001).

    CAS  Google Scholar 

  114. 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).

    CAS  Google Scholar 

  115. Morris, R. C., Coleman, R. V. & Bhandari, R. Superconductivity and magnetoresistance in NbSe2 . Phys. Rev. B 5, 895 (1972).

    Google Scholar 

  116. 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).

    CAS  Google Scholar 

  117. 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).

    Google Scholar 

  118. Yu, Y. et al. Gate-tunable phase transitions in thin flakes of 1T-TaS2 . Nat. Nanotechnol. 10, 270–276 (2015).

    CAS  Google Scholar 

  119. Qi, Y. et al. Superconductivity in Weyl semimetal candidate MoTe2 . Nat. Commun. 7, 11038 (2016).

    CAS  Google Scholar 

  120. Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015).

    CAS  Google Scholar 

  121. Klemm, R. A. Pristine and intercalated transition metal dichalcogenide superconductors. Phys. Rev. B 514, 86–94 (2015).

    CAS  Google Scholar 

  122. 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).

    Google Scholar 

  123. 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).

    CAS  Google Scholar 

  124. 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).

    CAS  Google Scholar 

  125. 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).

    Google Scholar 

  126. 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).

    Google Scholar 

  127. 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).

    CAS  Google Scholar 

  128. 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).

    CAS  Google Scholar 

  129. Ghosez, P. & Triscone, J.-M. Coupling of three lattice instabilities. Nat. Mater. 10, 269–270 (2011).

    CAS  Google Scholar 

  130. 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).

    Google Scholar 

  131. 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).

    Google Scholar 

  132. Chen, C. et al. Ferroelectricity in Dion–Jacobson ABiNb2O7 (A = Rb, Cs) compounds. J. Mater. Chem. C 3, 19–22 (2014).

    Google Scholar 

  133. Fennie, C. J. & Rabe, K. M. Ferroelectricity in the Dion–Jacobson CsBiNb2O7 from first principles. Appl. Phys. Lett. 88, 262902 (2006).

    Google Scholar 

  134. Benedek, N. A. Origin of ferroelectricity in a family of polar oxides: the Dion—Jacobson phases. Inorg. Chem. 53, 3769–3777 (2014).

    CAS  Google Scholar 

  135. Perez-Mato, J. M. et al. Competing structural instabilities in the ferroelectric Aurivillius compound SrBi2Ta2O9 . Phys. Rev. B 70, 214111 (2004).

    Google Scholar 

  136. Wu, F. Y. The Potts model. Rev. Mod. Phys. 54, 235–268 (1982).

    Google Scholar 

  137. Si, Q., Yu, R. & Abrahams, E. High-temperature superconductivity in iron pnictides and chalcogenides. Nat. Rev. Mater. 1, 16017 (2016).

    CAS  Google Scholar 

  138. Dai, P. Antiferromagnetic order and spin dynamics in iron-based superconductors. Rev. Mod. Phys. 87, 855–896 (2015).

    CAS  Google Scholar 

  139. 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).

    Google Scholar 

  140. Chandra, P., Coleman, P., Mydosh, J. A. & Tripathi, V. Hidden orbital order in the heavy fermion metal URu2Si2 . Nature 417, 831–834 (2002).

    CAS  Google Scholar 

  141. Ritschel, T. et al. Orbital textures and charge density waves in transition metal dichalcogenides. Nat. Phys. 11, 328–331 (2015).

    CAS  Google Scholar 

  142. 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).

    CAS  Google Scholar 

  143. 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).

    CAS  Google Scholar 

  144. 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).

    CAS  Google Scholar 

  145. Tokunaga, Y. et al. Rotation of orbital stripes and the consequent charge-polarized state in bilayer manganites. Nat. Mater. 5, 937–941 (2006).

    CAS  Google Scholar 

  146. 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).

    CAS  Google Scholar 

  147. 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).

    CAS  Google Scholar 

  148. Scholten, P. D. & Irakliotis, L. J. Critical behavior of the q-state clock model in three dimensions. Phys. Rev. B 48, 1291–1294 (1993).

    CAS  Google Scholar 

  149. 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).

    Google Scholar 

  150. Bazavov, A., Berg, B. A. & Dubey, S. Phase transition properties of 3D Potts models. Nucl. Phys. B 802, 421–434 (2008).

    Google Scholar 

  151. Wu, F. Y. Percolation and the Potts Model. J. Stat. Phys. 18, 115–123 (1978).

    Google Scholar 

  152. Yamaguchi, C. & Okabe, Y. Three-dimensional antiferromagnetic q-state Potts models: application of the Wang–Landau algorithm. J. Phys. A 34, 8781–8794 (2001).

    Google Scholar 

  153. 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).

    CAS  Google Scholar 

  154. 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).

    CAS  Google Scholar 

  155. Tanaka, S., Tamura, R. & Kawashima, N. Phase transition of generalized ferromagnetic Potts model-effect of invisible states. J. Phys. Conf. Ser. 297, 012022 (2011).

    Google Scholar 

  156. 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).

    CAS  Google Scholar 

  157. 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).

    CAS  Google Scholar 

  158. 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).

    Google Scholar 

  159. Koyama, Y., Yoshida, J., Hoshiya, H. & Nakamura, Y. Striped-type superstructure in γ-brass alloys. Phys. Rev. B 40, 5378–5386 (1989).

    CAS  Google Scholar 

  160. 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).

    CAS  Google Scholar 

  161. 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).

    CAS  Google Scholar 

  162. 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).

    CAS  Google Scholar 

  163. Yamamoto, N., Katoh, M. & Mori, S. Domain pattern formation during incommensurate–commensurate phase transition in Rb2ZnCl4 . Ferroelectrics 191, 247–252 (1997).

    CAS  Google Scholar 

  164. 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).

    CAS  Google Scholar 

  165. McMillan, W. L. Theory of discommensurations and the commensurate–incommensurate charge-density-wave phase transition. Phys. Rev. B 14, 1496–1502 (1976).

    CAS  Google Scholar 

  166. Walker, M. B. Theory of domains and dislocations in the charge-density-wave states of 2H-TaSe2 . Phys. Rev. B 26, 6208–6214 (1982).

    CAS  Google Scholar 

  167. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).

    CAS  Google Scholar 

  168. Koshibae, W. & Nagaosa, N. Theory of antiskyrmions in magnets. Nat. Commun. 7, 10542 (2016).

    CAS  Google Scholar 

  169. Gong, X. & Mele, E. J. Stacking textures and singularities in bilayer graphene. Phys. Rev. B 89, 121415 (2014).

    Google Scholar 

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Acknowledgements

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.

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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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