Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

Designing disorder into crystalline materials

Nature Reviews Chemistry volume 4, pages 657–673 (2020)Cite this article

Subjects

Abstract

Crystals are a state of matter characterized by periodic order. Yet, crystalline materials can harbour disorder in many guises, such as non-repeating variations in composition, atom displacements, bonding arrangements, molecular orientations, conformations, charge states, orbital occupancies or magnetic structure. Disorder can sometimes be random but, more usually, it is correlated. Frontier research into disordered crystals now seeks to control and exploit the unusual patterns that persist within these correlated disordered states in order to access functional responses inaccessible to conventional crystals. In this Review, we survey the core design principles that guide targeted control over correlated disorder. We show how these principles — often informed by long-studied statistical mechanical models — can be applied across an unexpectedly broad range of materials, including organics, supramolecular assemblies, oxide ceramics and metal–organic frameworks. We conclude with a forward-looking discussion of the exciting link between disorder and function in responsive media, thermoelectrics and topological phases.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Canonical disordered crystals.
Fig. 2: Common local degrees of freedom, interactions and lattice types.
Fig. 3: Models of disorder and competing interactions.
Fig. 4: Correlated disorder from electronic instabilities and compositional complexity.
Fig. 5: Correlated disorder by self-assembly.
Fig. 6: Molecular shape and low-energy dynamics.
Fig. 7: Data storage in disordered states.

Similar content being viewed by others

References

  1. Warren, B. E. The diffraction of X-rays in glass. Phys. Rev. 45, 657–661 (1934).

    CAS  Google Scholar 

  2. Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935).

    CAS  Google Scholar 

  3. Cartwright, J. H. E. & Mackay, A. L. Beyond crystals: the dialectic of materials and information. Phil. Trans. R. Soc. A 370, 2807–2822 (2012).

    PubMed  Google Scholar 

  4. Parsonage, N. G. & Staveley, L. A. K. Disorder in Crystals (Clarendon, 1978).

  5. Ziman, J. M. Models of Disorder. The Theoretical Physics of Homogeneously Disordered Systems (Cambridge Univ. Press, 1979).

  6. Weller, M. T., Weber, O. J., Henry, P. F., Di Pumpo, A. M. & Hansen, T. C. Complete structure and cation orientation in the perovskite photovoltaic methylammonium lead iodide between 100 and 352 K. Chem. Commun. 51, 418–4183 (2015).

    Google Scholar 

  7. Comès, R., Lambert, M. & Guinier, A. Désordre linéaire dans les cristaux (cas du silicium, du quartz, et des pérovskites ferroélectriques). Acta Cryst. A 26, 244–254 (1970).

    Google Scholar 

  8. Senn, M. S., Keen, D. A., Lucas, T. C. A., Hriljac, J. A. & Goodwin, A. L. Emergence of long-range order in BaTiO3 from local symmetry-breaking distortions. Phys. Rev. Lett. 116, 207602 (2016).

    CAS  PubMed  Google Scholar 

  9. Perversi, G. et al. Co-emergence of magnetic order and structural fluctuations in magnetite. Nat. Commun. 10, 2857 (2019).

    PubMed  PubMed Central  Google Scholar 

  10. Onsager, L. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944).

    CAS  Google Scholar 

  11. Welberry, T. R. Diffuse X-ray scattering and models of disorder. Rep. Prog. Phys. 48, 1543–1593 (1985).

    CAS  Google Scholar 

  12. Moessner, R. & Ramirez, A. P. Geometrical frustration. Phys. Today 59, 24–29 (2006).

    CAS  Google Scholar 

  13. Froufe-Pérez, L. S. et al. Role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials. Phys. Rev. Lett. 117, 053902 (2016).

    PubMed  Google Scholar 

  14. Damasceno, P. F., Engel, M. & Glotzer, S. C. Predictive self-assembly of polyhedra into complex structures. Science 337, 453–457 (2012).

    CAS  PubMed  Google Scholar 

  15. Ising, E. Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253–258 (1925).

    CAS  Google Scholar 

  16. Brush, S. G. History of the Lenz-Ising model. Rev. Mod. Phys. 39, 883–893 (1967).

    CAS  Google Scholar 

  17. Potts, R. B. Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc. 48, 106–109 (1952).

    Google Scholar 

  18. Šimenas, M., Balčiunas, S., Maczka, M., Banys, J. & Tornau, E. E. Structural phase transition in perovskite metal–formate frameworks: a Potts-type model with dipolar interactions. Phys. Chem. Chem. Phys. 18, 18528–18535 (2016).

    PubMed  Google Scholar 

  19. Ahmed, M. R. & Gehring, G. A. Potts model for the distortion transition in LaMnO6. Phys. Rev. B 74, 014420 (2006).

    Google Scholar 

  20. Vaks, V. G. & Larkin, A. I. On phase transitions of second order. Sov. Phys. JETP 22, 678–687 (1966).

    Google Scholar 

  21. Sirota, E. B. The rotator phases of neat and hydrated 1-alcohols. J. Chem. Phys. 105, 7763–7773 (1996).

    CAS  Google Scholar 

  22. Meier, Q. N. et al. Global formation of topological defects in the multiferroic hexagonal manganites. Phys. Rev. X 7, 041014 (2017).

    Google Scholar 

  23. Abrikosov, A. A. The magnetic properties of superconducting alloys. J. Phys. Chem. Solids 2, 199–208 (1957).

    CAS  Google Scholar 

  24. Cairns, A. B. et al. Encoding complexity within supramolecular analogues of frustrated magnets. Nat. Chem. 8, 442–447 (2016).

    CAS  PubMed  PubMed Central  Google Scholar 

  25. Joyce, G. S. Classical Heisenberg model. Phys. Rev. 155, 478–491 (1967).

    CAS  Google Scholar 

  26. Leguy, A. M. A. et al. The dynamics of methylammonium ions in hybrid organic–inorganic perovskite solar cells. Nat. Commun. 6, 7124 (2015).

    PubMed  PubMed Central  Google Scholar 

  27. Zhong, W., Vanderbilt, D. & Rabe, K. M. First-principles theory of ferroelectric phase transitions for perovskites: the case of BaTiO3. Phys. Rev. B 52, 6301–6312 (1995).

    CAS  Google Scholar 

  28. James, H. M. & Keenan, T. A. Theory of phase transitions in solid heavy methane. J. Chem. Phys. 31, 12–41 (1959).

    CAS  Google Scholar 

  29. Frenkel, D. in Liquides, Cristallisation et Transition Vitreuse: Les Houches, Session LI, 1989 Ch. 9 (eds Hansen, J. P., Levesque, D. & Zinn-Justin, J.) 689–762 (Elsevier, 1997).

  30. Karney, C. F. F. Quaternions in molecular modeling. J. Mol. Graph. Model. 25, 595–604 (2007).

    CAS  PubMed  Google Scholar 

  31. Pato-Doldán, B. et al. Magnetic transitions and isotropic versus anisotropic magnetic behaviour of [CH3NH3][M(HCOO)3] M = Mn2+, Co2+, Ni2+, Cu2+ metal–organic perovskites. J. Mater. Chem. C 4, 11164–11172 (2016).

    Google Scholar 

  32. Evans, N. L. et al. Control of multipolar and orbital order in perovskite-like [C(NH2)3]CuxCd1−x(HCOO)3 metal–organic frameworks. J. Am. Chem. Soc. 138, 9393–9396 (2016).

    CAS  PubMed  Google Scholar 

  33. den Hertog, B. C. & Gingras, M. J. P. Dipolar interactions and origin of spin ice in Ising pyrochlore magnets. Phys. Rev. Lett. 84, 3430–3433 (2000).

    Google Scholar 

  34. Läuchli, A., Mila, F. & Penc, K. Quadrupolar phases of the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice. Phys. Rev. Lett. 97, 087205 (2006).

    PubMed  Google Scholar 

  35. Lipscomb, W. N. Comment on crystal structures of α-CO and α-N2. J. Chem. Phys. 60, 5138 (1974).

    CAS  Google Scholar 

  36. Dzyaloshinsky, I. A thermodynamic theory of “weak” ferromagnetism of antiferromagnets. J. Phys. Chem. Solids 4, 241–255 (1958).

    CAS  Google Scholar 

  37. Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91 (1960).

    CAS  Google Scholar 

  38. Sauvage, M. & Parthé, E. Vacancy short-range order in substoichiometric transition metal carbides and nitrides with the NaCl structure. II. Numerical calculation of vacancy arrangement. Acta Cryst. 28, 607–616 (1972).

    CAS  Google Scholar 

  39. Urones-Garrote, E., Gómez-Herrero, A., Landa-Cánovas, Á. R., Withers, R. L. & Otero-Díaz, L. C. Order and disorder in rocksalt and spinel structures in the MgS–Yb2S3 system. Chem. Mater. 17, 3524–3531 (2005).

    CAS  Google Scholar 

  40. Nussinov, Z. & van den Brink, J. Compass models: theory and physical motivations. Rev. Mod. Phys. 87, 1–59 (2015).

    CAS  Google Scholar 

  41. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

    CAS  Google Scholar 

  42. Lieb, E. & Mattis, D. C. Mathematical Physics in One Dimension (Academic, 1966).

  43. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C Solid State Phys. 6, 1181–1203 (1973).

    CAS  Google Scholar 

  44. Wannier, G. H. Antiferromagnetism. The triangular Ising net. Phys. Rev. 79, 357–364 (1950).

    Google Scholar 

  45. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

    CAS  PubMed  Google Scholar 

  46. Bak, P. Commensurate phases, incommensurate phases and the devil’s staircase. Rep. Prog. Phys. 45, 587–629 (1982).

    Google Scholar 

  47. Yeomans, J. The theory and application of axial Ising models. Solid State Phys. 41, 151–200 (1988).

    CAS  Google Scholar 

  48. Cheng, C., Needs, R. J. & Heine, V. Inter-layer interactions and the origin of SiC polytypes. J. Phys. C Solid State Phys. 21, 1049–1063 (1988).

    CAS  Google Scholar 

  49. Bramwell, S. T. & Harris, M. J. Frustration in Ising-type spin models on the pyrochlore lattice. J. Phys. Condens. Matter 10, L215–L220 (1998).

    CAS  Google Scholar 

  50. Jaubert, L. D. C., Haque, M. & Moessner, R. Analysis of a fully packed loop model arising in a magnetic Coulomb phase. Phys. Rev. Lett. 107, 177202 (2011).

    CAS  PubMed  Google Scholar 

  51. Baise, M. et al. Negative hydration expansion in ZrW2O8: microscopic mechanism, spaghetti dynamics, and negative thermal expansion. Phys. Rev. Lett. 120, 265501 (2018).

    CAS  PubMed  Google Scholar 

  52. Anderson, P. W. Resonating valence bonds: a new kind of insulator? Mater. Res. Bull. 8, 153–160 (1973).

    CAS  Google Scholar 

  53. Overy, A. R. et al. Design of crystal-like aperiodic solids with selective disorder–phonon couplings. Nat. Commun. 7, 10445 (2016).

    CAS  PubMed  PubMed Central  Google Scholar 

  54. Keen, D. A. & Goodwin, A. L. The crystallography of correlated disorder. Nature 521, 303–309 (2015).

    CAS  PubMed  Google Scholar 

  55. Mydosh, J. A. & Oppeneer, P. M. Hidden order, superconductivity, and magnetism: the unsolved case of URu2Si2. Rev. Mod. Phys. 83, 1301–1322 (2011).

    CAS  Google Scholar 

  56. Henley, C. L. The “Coulomb phase” in frustrated systems. Annu. Rev. Condens. Matter Phys. 1, 179–210 (2010).

    CAS  Google Scholar 

  57. Castelnovo, C., Moessner, R. & Sondhi, S. L. Spin ice, fractionalization, and topological order. Annu. Rev. Condens. Matter Phys. 3, 35–55 (2012).

    CAS  Google Scholar 

  58. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).

    PubMed  Google Scholar 

  59. Oakes, T., Garrahan, J. P. & Powell, S. Emergence of cooperative dynamics in fully packed classical dimers. Phys. Rev. E 93, 032129 (2106).

    Google Scholar 

  60. Pearson, R. G. Concerning Jahn–Teller effects. Proc. Natl Acad. Sci., USA 72, 2104–2106 (1975).

    CAS  PubMed  Google Scholar 

  61. Goodenough, J. B. Jahn-Teller phenomena in solids. Annu. Rev. Mater. Sci. 28, 1–27 (1998).

    CAS  Google Scholar 

  62. Goodenough, J. B. Theory of the role of covalence in the perovskite-type manganites [La, M(II)]MnO3. Phys. Rev. 100, 564–573 (1955).

    CAS  Google Scholar 

  63. Qiu, X., Proffen, T., Mitchell, J. F. & Billinge, S. J. L. Orbital correlations in the pseudocubic O and rhombohedral R phases of LaMnO3. Phys. Rev. Lett. 94, 177203 (2005).

    PubMed  Google Scholar 

  64. Sartbaeva, A., Wells, S. A., Thorpe, M. F., Božin, E. S. & Billinge, S. J. L. Quadrupolar ordering in LaMnO3 revealed from scattering data and geometric modelling. Phys. Rev. Lett. 99, 155503 (2007).

    CAS  PubMed  Google Scholar 

  65. Pissas, M., Margiolaki, I., Papavassiliou, G., Stamopoulos, D. & Argyriou, D. N. Crystal and magnetic structure of the La1−xCaxMnO3 compound (0.11 ≤ x ≤ 0.175). Phys. Rev. B 72, 064425 (2005).

    Google Scholar 

  66. Rao, C. N. R. & Cheetham, A. K. Giant magnetoresistance in transition metal oxides. Science 272, 369–370 (1996).

    CAS  Google Scholar 

  67. Wolpert, E. H. et al. Hybrid local-order mechanism for inversion symmetry breaking. Phys. Rev. B 97, 134106 (2018).

    CAS  Google Scholar 

  68. Melot, B. C. et al. Large low-temperature specific heat in pyrochlore Bi2Ti2O7. Phys. Rev. B 79, 224111 (2009).

    Google Scholar 

  69. Du, H., Yao, X. & Wang, H. Relaxor-like behavior of bismuth-based pyrochlores containing Sn. J. Electroceram. 21, 222–225 (2007).

    Google Scholar 

  70. Liu, Y. et al. Displacive disorder and dielectric relaxation in the stoichiometric bismuth-containing pyrochlores Bi2MIIINbO7 (M=In and Sc). J. Solid State Chem. 182, 2748–2755 (2009).

    CAS  Google Scholar 

  71. Morin, F. J. Oxides which show a metal-to-insulator transition at the Neel temperature. Phys. Rev. Lett. 3, 34–36 (1959).

    CAS  Google Scholar 

  72. Corr, S. A., Shoemaker, D. P., Melot, B. C. & Seshadri, R. Real-space investigation of the structural changes at the metal-insulator transition in VO2. Phys. Rev. Lett. 105, 056404 (2010).

    PubMed  Google Scholar 

  73. Attfield, J. P. Orbital molecules in electronic materials. APL Mater. 3, 041510 (2015).

    Google Scholar 

  74. Blaha, P., Singh, D. J. & Schwarz, K. Geometric frustration, electronic instabilities, and charge singlets in Y2Nb2O7. Phys. Rev. Lett. 93, 216403 (2004).

    CAS  PubMed  Google Scholar 

  75. McQueen, T. M. et al. Frustrated ferroelectricity in niobate pyrochlores. J. Phys. Condens. Matter 20, 235210 (2008).

    CAS  PubMed  Google Scholar 

  76. Shinaoka, H., Motome, Y., Miyake, T. & Ishibashi, S. Spin-orbital frustration in molybdenum pyrochlores A2Mo2O7 (A = rare earth). Phys. Rev. B 88, 174422 (2013).

    Google Scholar 

  77. Thygesen, P. M. M. et al. Orbital dimer model for the spin-glass state in Y2Mo2O7. Phys. Rev. Lett. 118, 067201 (2017).

    PubMed  Google Scholar 

  78. Senn, M. S., Wright, J. P. & Attfield, J. P. Charge order and three-site distortions in the Verwey structure of magnetite. Nature 481, 173–176 (2012).

    CAS  Google Scholar 

  79. Browne, A. J., Kimber, S. A. J. & Attfield, J. P. Persistent three- and four-atom orbital molecules in the spinel AlV2O4. Phys. Rev. Mater. 1, 052003(R) (2017).

    Google Scholar 

  80. Bozin, E. S. et al. Local orbital degeneracy lifting as a precursor to an orbital-selective Peierls transition. Nat. Commun. 10, 3638 (2019).

    CAS  PubMed  PubMed Central  Google Scholar 

  81. Folkers, L. C., Simonov, A., Wang, F. & Lidin, S. The mystery of the AuIn 1:1 phase and its incommensurate structural variations. Inorg. Chem. 57, 2791–2796 (2018).

    CAS  PubMed  Google Scholar 

  82. Vegard, L. Die konstitution der mischkristalle und die raumfüllung der atome. Z. Phys. 5, 17–26 (1921).

    CAS  Google Scholar 

  83. Pauling, L. The sizes of ions and the structure of ionic crystals. J. Am. Chem. Soc. 49, 765–790 (1927).

    CAS  Google Scholar 

  84. Castellanos, M. & West, A. R. Deviations from Vegard’s law in oxide solid solutions. J. Chem. Soc. Faraday Trans. 1 76, 2159–2169 (1980).

    CAS  Google Scholar 

  85. Page, K. et al. Local atomic ordering in BaTaO2N studied by neutron pair distribution function analysis and density functional theory. Chem. Mater. 19, 4037–4042 (2007).

    CAS  Google Scholar 

  86. Yang, M. et al. Anion order in perovskite oxynitrides. Nat. Chem. 3, 47–52 (2011).

    CAS  PubMed  Google Scholar 

  87. Camp, P. J., Fuertes, A. & Attfield, J. P. Sub-extensive entropies and open order in perovskite oxynitrides. J. Am. Chem. Soc. 134, 6762–6766 (2012).

    CAS  PubMed  Google Scholar 

  88. Johnston, H. et al. Dimensional crossover of correlated anion disorder in oxynitride perovskites. Chem. Commun. 54, 5245–5247 (2018).

    CAS  Google Scholar 

  89. Yang, M., Oró-Solé, J., Kusmartseva, A., Fuertes, A. & Attfield, J. P. Electronic tuning of two metals and colossal magnetoresistances in EuWO1+xN2−x perovskites. J. Am. Chem. Soc. 132, 4822–4829 (2010).

    CAS  PubMed  Google Scholar 

  90. Goto, Y. et al. Pressure-stabilized cubic perovskite oxyhydride BaScO2H. Inorg. Chem. 56, 4840–4845 (2017).

    CAS  PubMed  Google Scholar 

  91. Kageyama, H. et al. Expanding frontiers in materials chemistry and physics with multiple anions. Nat. Commun. 9, 772 (2018).

    PubMed  PubMed Central  Google Scholar 

  92. Whitaker, M. J. & Greaves, C. Magnetic ordering in the pyrochlore Ho2CrSbO7 determined from neutron diffraction, and the magnetic properties of other RE2CrSbO7 phases (RE = Y, Tb, Dy, Er). J. Solid State Chem. 215, 171–175 (2014).

    CAS  Google Scholar 

  93. Fennell, T. et al. Multiple Coulomb phase in the fluoride pyrochlore CsNiCrF6. Nat. Phys. 15, 60–66 (2019).

    CAS  Google Scholar 

  94. Kim, S. W. et al. RbFe2+Fe3+F6: Synthesis, structure, and characterization of a new charge-ordered magnetically frustrated pyrochlore-related mixed-metal fluoride. Chem. Sci. 3, 741–751 (2012).

    CAS  Google Scholar 

  95. Zhang, X. et al. Hierarchy of exchange interactions in the triangular-lattice spin liquid YbMgGaO4. Phys. Rev. X 8, 031001 (2018).

    CAS  Google Scholar 

  96. Wen, J.-J. et al. Disordered route to the Coulomb quantum spin liquid: random transverse fields on spin ice in Pr2Zr2O7. Phys. Rev. Lett. 118, 107206 (2017).

    PubMed  Google Scholar 

  97. Menéndez, E. et al. Direct magnetic patterning due to the generation of ferromagnetism by selective ion irradiation of paramagnetic FeAl alloys. Small 5, 229–234 (2009).

    PubMed  Google Scholar 

  98. Lee, J. et al. Unlocking the potential of cation-disordered oxides for rechargeable lithium batteries. Science 343, 519–522 (2014).

    CAS  PubMed  Google Scholar 

  99. Ji, H. et al. Hidden structural and chemical order controls lithium transport in cation-disordered oxides for rechargeable batteries. Nat. Commun. 10, 592 (2019).

    CAS  PubMed  PubMed Central  Google Scholar 

  100. Jones, M. A. et al. Short-range ordering in a battery electrode, the ‘cation-disordered’ rocksalt Li1.25Nb0.25Mn0.5O2. Chem. Commun. 55, 9027–9030 (2019).

    CAS  Google Scholar 

  101. Clément, R. J., Lun, Z. & Ceder, G. Cation-disordered rocksalt transition metal oxides and oxyfluorides for high energy lithium-ion cathodes. Energy Environ. Sci. 13, 345–373 (2020).

    Google Scholar 

  102. Yabuuchi, N. et al. Origin of stabilization and destabilization in solid-state redox reaction of oxide ions for lithium-ion batteries. Nat. Commun. 7, 13814 (2016).

    CAS  PubMed  PubMed Central  Google Scholar 

  103. Paściak, M., Welberry, T. R., Kulda, J., Kempa, M. & Hlinka, J. Polar nanoregions and diffuse scattering in the relaxor ferroelectric PbMg1/3Nb1/3O3. Phys. Rev. B 85, 224109 (2012).

    Google Scholar 

  104. Battle, P. D., Evers, S. I., Hunter, E. C. & Westwood, M. La3Ni2SbO9: a relaxor ferromagnet. Inorg. Chem. 52, 6648–6653 (2013).

    CAS  PubMed  Google Scholar 

  105. Boström, H. L. B., Bruckmoser, J. & Goodwin, A. L. Ordered B-site vacancies in an ABX3 formate perovskite. J. Am. Chem. Soc. 141, 17978–17982 (2019).

    PubMed  Google Scholar 

  106. Simonov, A. et al. Hidden diversity of vacancy networks in Prussian blue analogues. Nature 578, 256–260 (2020).

    CAS  PubMed  PubMed Central  Google Scholar 

  107. Veal, B. W. et al. Time-dependent superconducting behavior of oxygen-deficient YBa2Cu3Ox: Possible annealing of oxygen vacancies at 300 K. Phys. Rev. B 42, 4770–4773 (1990).

    CAS  Google Scholar 

  108. Fang, Z., Bueken, B., De Vos, D. E. & Fischer, R. A. Defect-engineered metal–organic frameworks. Angew. Chem. Int. Ed. 54, 7234–7254 (2015).

    CAS  Google Scholar 

  109. Shearer, G. C. et al. Defect engineering: tuning the porosity and composition of the metal–organic framework UiO-66 via modulated synthesis. Chem. Mater. 28, 3749–3761 (2016).

    CAS  Google Scholar 

  110. Cliffe, M. J., Hill, J. A., Murray, C. A., Coudert, F.-X. & Goodwin, A. L. Defect-dependent colossal negative thermal expansion in UiO-66(Hf) metal–organic framework. Phys. Chem. Chem. Phys. 17, 11586–11592 (2015).

    CAS  PubMed  Google Scholar 

  111. Nield, V. M., Keen, D. A., Hayes, W. & McGreevy, R. L. Structure and fast-ion conduction in α-AgI. Solid State Ion. 66, 247–258 (1993).

    CAS  Google Scholar 

  112. Wachsman, E. D. & Lee, K. T. Lowering the temperature of solid oxide fuel cells. Science 334, 935–939 (2011).

    CAS  PubMed  Google Scholar 

  113. Grindlay, J. & Ter Haar, D. On the ferroelectric behaviour of potassium dihydrogen phosphate. Proc. R. Soc. A 250, 266–285 (1959).

    CAS  Google Scholar 

  114. Di Stefano, D. et al. Superionic diffusion through frustrated energy landscape. Chem 5, 2450–2460 (2019).

    Google Scholar 

  115. Bernal, J. D. & Fowler, R. H. A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J. Chem. Phys. 1, 515–548 (1933).

    CAS  Google Scholar 

  116. Salzmann, C. G., Radaelli, P. G., Slater, B. & Finney, J. L. The polymorphism of ice: five unresolved questions. Phys. Chem. Chem. Phys. 13, 18468–18480 (2011).

    CAS  PubMed  Google Scholar 

  117. Bramwell, S. T. Ferroelectric ice. Nature 397, 212–213 (1999).

    CAS  Google Scholar 

  118. Blunt, M. O. et al. Random tiling and topological defects in a two-dimensional molecular network. Science 322, 1077–1081 (2008).

    CAS  PubMed  Google Scholar 

  119. Barth, J. V., Costantini, G. & Kern, K. Engineering atomic and molecular nanostructures at surfaces. Nature 437, 671–679 (2005).

    CAS  PubMed  Google Scholar 

  120. Stannard, A. et al. Broken symmetry and the variation of critical properties in the phase behaviour of supramolecular rhombus tilings. Nat. Chem. 4, 112–117 (2012).

    CAS  Google Scholar 

  121. Gerbode, S. J. et al. Glassy dislocation dynamics in 2D colloidal dimer crystals. Phys. Rev. Lett. 105, 078301 (2010).

    PubMed  Google Scholar 

  122. Muangnapoh, K., Avendaño, C., Escobedo, F. A. & Watson, C. M. L. Degenerate crystals from colloidal dimers under confinement. Soft Matter 10, 9729–9738 (2014).

    CAS  PubMed  Google Scholar 

  123. Haji-Akbari, A., Haji-Akbari, N. & Ziff, R. M. Dimer covering and percolation frustration. Phys. Rev. E 92, 032134 (2015).

    Google Scholar 

  124. Miyajima, D. et al. Ferroelectric columnar liquid crystal featuring confined polar groups within core–shell architecture. Science 336, 209–213 (2012).

    CAS  PubMed  Google Scholar 

  125. Fitié, C. F. C., Roelofs, W. S. C., Kemerink, M. & Sijbesma, R. P. Remnant polarization in thin films from a columnar liquid crystal. J. Am. Chem. Soc. 132, 6892–6893 (2010).

    PubMed  Google Scholar 

  126. Simonov, A., Weber, T. & Steurer, W. Experimental uncertainties of three-dimensional pair distribution function investigations exemplified on the diffuse scattering from a tris-tert-butyl-1,3,5-benzene tricarboxamide single crystal. J. Appl. Cryst. 47, 2011–2018 (2014).

    CAS  Google Scholar 

  127. Zehe, C. S. et al. Mesoscale polarization by geometric frustration in columnar supramolecular crystals. Angew. Chem. Int. Ed. 56, 4432–4437 (2017).

    CAS  Google Scholar 

  128. Blomenhofer, M. et al. “Designer” nucleating agents for polypropylene. Macromolecules 38, 3688–3695 (2005).

    CAS  Google Scholar 

  129. Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966).

    CAS  Google Scholar 

  130. Žukovič, M. & Idogaki, T. Low-temperature long-range ordering of a classical XY spin system with bilinear-biquadratic exchange Hamiltonian. Physica B Condens. Matter 329-333, 1055–1056 (2003).

    Google Scholar 

  131. Schlenk, H. & Holman, R. T. The urea complexes of unsaturated fatty acids. Science 112, 19–20 (1950).

    CAS  PubMed  Google Scholar 

  132. Playford, H. Y., Whale, T. F., Murray, B., Tucker, M. G. & Salzmann, C. G. Analysis of stacking disorder in ice I using pair distribution functions. J. Appl. Cryst. 51, 1211–1220 (2018).

    CAS  Google Scholar 

  133. Goodwin, A. L. et al. Aperiodicity, structure, and dynamics in Ni(CN)2. Phys. Rev. B 80, 054101 (2009).

    Google Scholar 

  134. Jood, P. & Ohta, M. Hierarchical architecturing for layered thermoelectric sulfides and chalcogenides. Materials 8, 1124–1149 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  135. Engel, M., Damasceno, P. F., Phillips, C. L. & Glotzer, S. C. Computational self-assembly of a one-component icosahedral quasicrystal. Nat. Mater. 14, 109–116 (2016).

    Google Scholar 

  136. Kumar, S., Manickam, M., Varshney, S. K. & Shankar Rao, D. S. Novel heptasubstituted triphenylene discotic liquid crystals. J. Mater. Chem. 10, 2483–2489 (2000).

    CAS  Google Scholar 

  137. Sherwood, J. N. (ed.) The Plastically Crystalline State: Orientationally Disordered Crystals (Wiley, 1979).

  138. David, W. I. F., Ibberson, R. M. & Matsuo, T. High resolution neutron powder diffraction: a case study of the structure of C60. Proc. R. Soc. Lond. A 442, 129–146 (1993).

    CAS  Google Scholar 

  139. Clayton, J. O. & Giauque, W. F. The heat capacity and entropy of carbon monoxide. Heat of vaporization. Vapor pressures of solid and liquid. Free energy to 5000°K. From spectroscopic data. J. Am. Chem. Soc. 54, 2610–2626 (1932).

    CAS  Google Scholar 

  140. de Gennes, P. G. & Prost, J. The Physics of Liquid Crystals (Clarendon, 1993).

  141. MacFarlane, D. R., Huang, J. & Forsyth, M. Lithium-doped plastic crystal elecrolytes exhibiting fast ion conduction for secondary batteries. Nature 402, 792–794 (1999).

    CAS  Google Scholar 

  142. MacFarlane, D. R. & Forsyth, M. Plastic crystal electrolyte materials: new perspectives on solid state ionics. Adv. Mater. 13, 957–966 (2001).

    CAS  Google Scholar 

  143. Pringle, J. M., Howlett, P. C., MacFarlane, D. R. & Forsyth, M. Organic ionic plastic crystals: recent advances. J. Mater. Chem. 20, 2056–2062 (2010).

    CAS  Google Scholar 

  144. Pringle, J. M. Recent progress in the development and use of organic ionic plastic crystal electrolytes. Phys. Chem. Chem. Phys. 15, 1339–1351 (2013).

    CAS  PubMed  Google Scholar 

  145. Horike, S. et al. Order-to-disorder structural transformation of a coordination polymer and its influence on proton conduction. Chem. Commun. 50, 10241–10243 (2014).

    CAS  Google Scholar 

  146. Nagarkar, S. S. et al. Enhanced and optically switchable proton conductivity in a melting coordination polymer crystal. Angew. Chem. Int. Ed. 56, 4976–4981 (2017).

    CAS  Google Scholar 

  147. Pauling, L. The rotational motion of molecules in crystals. Phys. Rev. 36, 430–443 (1930).

    CAS  Google Scholar 

  148. Lewis, L. J. & Klein, M. L. Random strains and the structure of (KBr)1−x(KCN)x mixed crystals. Phys. Rev. Lett. 57, 2698–2701 (1986).

    CAS  PubMed  Google Scholar 

  149. Yamada, Y., Shibuya, I. & Hoshino, S. Phase transition in NaNO2. J. Phys. Soc. Jpn. 18, 1594–1603 (1963).

    CAS  Google Scholar 

  150. Li, W. et al. Chemically diverse and multifunctional hybrid organic–inorganic perovskites. Nat. Rev. Mater. 2, 16099 (2017).

    Google Scholar 

  151. Coates, C. S. et al. Ferroic multipolar order and disorder in cyanoelpasolite molecular perovskites. Phil. Trans. R. Soc. A 377, 20180219 (2019).

    CAS  PubMed  Google Scholar 

  152. Jain, P. et al. Multiferroic behaviour associated with an order–disorder hydrogen bonding transition in metal–organic frameworks (MOFs) with the perovskite ABX3 architecture. J. Am. Chem. Soc. 131, 13625–13627 (2009).

    CAS  PubMed  Google Scholar 

  153. Besara, T. et al. Mechanism of the order–disorder phase transition, and glassy behavior in the metal-organic framework [(CH3)2NH2]Zn(HCOO)3. Proc. Natl Acad. Sci. USA 108, 6828–6832 (2011).

    Google Scholar 

  154. Stroppa, A. et al. Electric control of magnetization and interplay between orbital ordering and ferroelectricity in a multiferroic metal–organic framework. Angew. Chem. Int. Ed. 50, 5847–5850 (2011).

    CAS  Google Scholar 

  155. Stroppa, A., Barone, P., Jain, P., Perez-Mato, J. M. & Picozzi, S. Hybrid improper ferroelectricity in a multiferroic and magnetoelectric metal-organic framework. Adv. Mater. 25, 2284–2290 (2013).

    CAS  PubMed  Google Scholar 

  156. Ye, H.-Y. et al. Metal-free three-dimensional perovskite ferroelectrics. Science 361, 151–155 (2018).

    CAS  PubMed  Google Scholar 

  157. Šimėnas, M. et al. Simulation of structural phase transitions in perovskite methylhydrazinium metal–formate frameworks: coupled Ising and Potts models. J. Phys. Chem. C 123, 19912–19919 (2019).

    Google Scholar 

  158. Lloveras, P. et al. Colossal barocaloric effects near room temperature in plastic crystals of neopentylglycol. Nat. Commun. 10, 1803 (2019).

    CAS  PubMed  PubMed Central  Google Scholar 

  159. Bermúdez-García, J. M. et al. Giant barocaloric effect in the ferroic organic-inorganic hybrid [TPrA][Mn(dca)3] perovskite under easily accessible pressures. Nat. Commun. 8, 15715 (2107).

    Google Scholar 

  160. Tishin, A. M. & Spichkin, Y. I. The Magnetocaloric Effect and its Applications (Taylor & Francis, 2003).

  161. Lorusso, G. et al. A dense metal–organic framework for enhanced magnetic refrigeration. Adv. Mater. 25, 4653–4656 (2013).

    CAS  PubMed  Google Scholar 

  162. Saines, P. J., Paddison, J. A. M., Thygesen, P. M. M. & Tucker, M. G. Searching beyond Gd for magnetocaloric frameworks: magnetic properties and interactions of the Ln(HCO2)3 series. Mater. Horiz. 2, 528–535 (2015).

    CAS  Google Scholar 

  163. Frost, J. M., Butler, K. T. & Walsh, A. Molecular ferroelectric contributions to anomalous hysteresis in hybrid perovskite solar cells. APL Mater. 2, 081506 (2014).

    Google Scholar 

  164. Herz, L. How lattice dynamics moderate the electronic properties of metal-halide perovskites. J. Phys. Chem. Lett. 9, 6853–6863 (2018).

    CAS  PubMed  Google Scholar 

  165. Laurita, G., Fabini, D. H., Stoumpos, C. C., Kanatzidis, M. G. & Seshadri, R. Chemical tuning of dynamic cation off-centering in the cubic phases of hybrid tin and lead halide perovskites. Chem. Sci. 8, 5628–5635 (2017).

    CAS  PubMed  PubMed Central  Google Scholar 

  166. Zhu, T. & Ertekin, E. Mixed phononic and non-phononic transport in hybrid lead halide perovskites: glass-crystal duality, dynamical disorder, and anharmonicity. Energy Environ. Sci. 12, 216–229 (2019).

    CAS  Google Scholar 

  167. Delaire, O. et al. Giant anharmonic phonon scattering in PbTe. Nat. Mater. 10, 614–619 (2011).

    CAS  PubMed  Google Scholar 

  168. Christensen, S., Bindzus, N., Sist, M., Takata, M. & Iversen, B. B. Structural disorder, anisotropic microstrain and cation vacancies in thermoelectric lead chalcogenides. Phys. Chem. Chem. Phys. 18, 15874–15883 (2016).

    CAS  PubMed  Google Scholar 

  169. Watson, G. W. & Parker, S. C. Origin of the lone pair of α-PbO from density functional theory calculations. J. Phys. Chem. B 103, 1258–1262 (1999).

    CAS  Google Scholar 

  170. Walsh, A., Payne, D. J., Egdell, R. G. & Watson, G. W. Stereochemistry of post-transition metal oxides: revision of the classical lone pair model. Chem. Soc. Rev. 40, 4455–4463 (2011).

    CAS  PubMed  Google Scholar 

  171. Skelton, J. M., Parker, S. C., Togo, A., Tanaka, I. & Walsh, A. Thermal physics of the lead chalcogenides PbS, PbSe, and PbTe from first principles. Phys. Rev. B 89, 205203 (2014).

    Google Scholar 

  172. Li, C. W. et al. Phonon self-energy and origin of anomalous neutron scattering spectra in SnTe and PbTe thermoelectrics. Phys. Rev. Lett. 112, 175501 (2014).

    CAS  PubMed  Google Scholar 

  173. Sangiorgio, B. et al. Correlated local dipoles in PbTe. Phys. Rev. Mater. 2, 085402 (2018).

    CAS  Google Scholar 

  174. Božin, E. S. et al. Entropically stabilized local dipole formation in chalcogenides. Science 330, 1660–1663 (2010).

    PubMed  Google Scholar 

  175. Zhang, Y., Ke, X., Kent, P. R. C., Yang, J. & Chen, C. Anomalous lattice dynamics near the ferroelectric instability in PbTe. Phys. Rev. Lett. 107, 175503 (2011).

    PubMed  Google Scholar 

  176. Keiber, T., Bridges, F. & Sales, B. C. Lead is not off center in PbTe: the importance of r-space phase information in extended X-ray absorption fine structure spectroscopy. Phys. Rev. Lett. 111, 095504 (2013).

    CAS  PubMed  Google Scholar 

  177. Lee, S. et al. Resonant bonding leads to low lattice thermal conductivity. Nat. Commun. 5, 3525 (2014).

    PubMed  Google Scholar 

  178. Esquivel-Elizondo, J. R., Brooks Hinojosa, B. & Nino, J. C. Bi2Ti2O7: it is not what you have read. Chem. Mater. 23, 4965–4974 (2011).

    CAS  Google Scholar 

  179. Mukhopadhyay, A. B., Oligschleger, C. & Dolg, M. Low-frequency vibrational excitations in zeolite ZSM-5 and its partially crystalline derivatives. Phys. Rev. B 69, 012202 (2004).

    Google Scholar 

  180. Hammonds, K. D., Deng, H., Heine, V. & Dove, M. T. How floppy modes give rise to adsorption sites in zeolites. Phys. Rev. Lett. 78, 3701–3704 (1997).

    CAS  Google Scholar 

  181. Sartbaeva, A. & Wells, S. A. Framework flexibility and rational design of new zeolites for catalysis. Appl. Petrochem. Res. 2, 69–72 (2012).

    CAS  Google Scholar 

  182. Mary, T. A., Evans, J. S. O., Vogt, T. & Sleight, A. W. Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2O8. Science 272, 90–92 (1996).

    CAS  Google Scholar 

  183. Keen, D. A. et al. Structural description of pressure-induced amorphization in ZrW2O8. Phys. Rev. Lett. 98, 225501 (2007).

    PubMed  Google Scholar 

  184. Duan, N., Kameswari, U. & Sleight, A. W. Further contraction of ZrW2O8. J. Am. Chem. Soc. 121, 10432–10433 (1999).

    CAS  Google Scholar 

  185. Perottoni, C. A. & da Jornada, J. A. H. Pressure-induced amorphization and negative thermal expansion in ZrW2O8. Science 280, 886–889 (1998).

    CAS  PubMed  Google Scholar 

  186. Otto, S., Furlan, R. L. E. & Sanders, J. K. M. Selection and amplification of hosts from dynamic combinatorial libraries of macrocyclic disulfides. Science 297, 590–593 (2002).

    CAS  PubMed  Google Scholar 

  187. Reynolds, E. M. et al. Function from configurational degeneracy in disordered framework materials. Faraday Discus. https://doi.org/10.1039/D0FD00008F (2020).

    Article  Google Scholar 

  188. Serre, C. et al. Role of solvent-host interactions that lead to very large swelling of hybrid frameworks. Science 315, 1828–1831 (2007).

    CAS  PubMed  Google Scholar 

  189. Yue, L., Wang, S., Wulf, V. & Willner, I. Stiffness-switchable DNA-based constitutional dynamic network hydrogels for self-healing and matrix-guided controlled chemical processes. Nat. Commun. 10, 4774 (2019).

    PubMed  PubMed Central  Google Scholar 

  190. Fennell, T. et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009).

    CAS  PubMed  Google Scholar 

  191. Hlinka, J. et al. Origin of the “waterfall” effect in phonon dispersion of relaxor perovskites. Phys. Rev. Lett. 91, 107602 (2003).

    CAS  PubMed  Google Scholar 

  192. Wiersma, D. S. Disordered photonics. Nat. Photon. 7, 188–196 (2013).

    CAS  Google Scholar 

  193. Mittal, S. et al. Topologically robust transport of photons in a synthetic gauge field. Phys. Rev. Lett. 113, 087403 (2014).

    CAS  PubMed  Google Scholar 

  194. Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).

    CAS  PubMed  Google Scholar 

  195. Cliffe, M. J. et al. Metal–organic nanosheets formed via defect-mediated transformation of a hafnium metal–organic framework. J. Am. Chem. Soc. 139, 5397–5404 (2017).

    CAS  PubMed  PubMed Central  Google Scholar 

  196. Morris, R. E. & Čejka, J. Exploiting chemically selective weakness in solids as a route to new porous materials. Nat. Chem. 7, 381–388 (2015).

    CAS  PubMed  Google Scholar 

  197. Roth, W. J. et al. A family of zeolites with controlled pore size prepared using a top-down method. Nat. Chem. 5, 628–633 (2013).

    CAS  PubMed  Google Scholar 

  198. Schrödinger, E. What is Life? (Cambridge Univ. Press, 1944).

  199. Billinge, S. J. L. & Levin, I. The problem with determining atomic structure at the nanoscale. Science 316, 561–565 (2007).

    CAS  PubMed  Google Scholar 

  200. Shull, C. G., Strauser, W. A. & Wollan, E. O. Neutron diffraction by paramagnetic and antiferromagnetic substances. Phys. Rev. 83, 333–345 (1951).

    CAS  Google Scholar 

  201. Anderson, P. W. Ordering and antiferromagnetism in ferrites. Phys. Rev. 102, 1008–1013 (1956).

    CAS  Google Scholar 

  202. Fairbank, V. E., Thompson, A. L., Cooper, R. I. & Goodwin, A. L. Charge-ice dynamics in the negative thermal expansion material Cd(CN)2. Phys. Rev. B 86, 104113 (2012).

    Google Scholar 

  203. Cruddas, J. & Powell, B. J. Spin-state ice in elastically frustrated spin-crossover materials. J. Am. Chem. Soc. 141, 19790–19799 (2019).

    CAS  PubMed  Google Scholar 

  204. Wells, A. F. Structural Inorganic Chemistry 5th edn (Clarendon, 1984).

  205. Delgado-Friedrichs, O., O’Keeffe, M. & Yaghi, O. M. Taxonomy of periodic nets and the design of materials. Phys. Chem. Chem. Phys. 9, 1035–1043 (2007).

    CAS  PubMed  Google Scholar 

  206. Fisher, M. E. & Selke, W. Infinitely many commensurate phases in a simple Ising model. Phys. Rev. Lett. 44, 1502–1505 (1980).

    CAS  Google Scholar 

  207. Powell, S. & Chalker, J. T. SU(2)-invariant continuum theory for an unconventional phase transition in a three-dimensional classical dimer model. Phys. Rev. Lett. 101, 155702 (2008).

    PubMed  Google Scholar 

  208. Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zero-point entropy in ‘spin ice’. Nature 399, 333–335 (1999).

    CAS  Google Scholar 

  209. Thygesen, P. M. M. et al. Local structure study of the orbital order/disorder transition in LaMnO3. Phys. Rev. B 95, 174107 (2017).

    Google Scholar 

  210. Lieb, E. H. Residual entropy of square ice. Phys. Rev. 162, 162–172 (1967).

    CAS  Google Scholar 

Download references

Acknowledgements

A.S. and A.L.G. gratefully acknowledge financial support from the E.R.C. (grant no. 788144) and the Leverhulme Trust (UK) (grant no. RPG-2015-292). A.S. thanks the Swiss National Science Foundation for Ambizione and Postdoc Mobility Fellowships (PZ00P2_180035 and P2EZP2_155608). A.L.G. acknowledges many useful discussions with collaborators past and present.

Author information

Authors and Affiliations

  1. Department of Materials, ETH Zurich, Zürich, Switzerland

    Arkadiy Simonov

  2. Department of Chemistry, University of Oxford, Oxford, UK

    Arkadiy Simonov & Andrew L. Goodwin

Authors
  1. Arkadiy Simonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew L. Goodwin
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors contributed equally to the preparation of this manuscript.

Corresponding author

Correspondence to Andrew L. Goodwin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Magnetocaloric

Material that exhibits a strong, reversible entropy change under applied magnetic field.

Relaxors

Materials with strong dielectric response characterized by broad frequency-dependent maximum of dielectric permittivity as a function of temperature and the absence of macroscopic symmetry breaking.

Frustrated magnetism

The suppression of conventional order in magnetic materials by an incompatibility between the geometric arrangement of interacting spins and the nature of the interactions themselves.

Nematic

Characterized by axial order.

Skyrmions

Topologically stable non-collinear (‘knot’) spin textures.

Trimerons

Small polaron quasiparticles characterized by the delocalization of electrons across a linear triplet of atoms.

Barocalorics

Materials that exhibit a strong, reversible entropy change under applied pressure.

Rigid-unit mode

Collective distortion of a network of connected polyhedra that propagates without geometric distortion of the polyhedra themselves.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simonov, A., Goodwin, A.L. Designing disorder into crystalline materials. Nat Rev Chem 4, 657–673 (2020). https://doi.org/10.1038/s41570-020-00228-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41570-020-00228-3

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing