Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).
Kelton, K. F. Quasicrystals: structure and stability. Int. Mater. Rev. 38, 105–137 (1993).
Steurer, W. Quasicrystals: what do we know? What do we want to know? What can we know? Acta Crystallogr. Sect. A: Found. Adv. 74, 1–11 (2018).
Shimasaki, T. et al. Anomalous localization in a kicked quasicrystal. Nat. Phys. 20, 409–414 (2024).
Jeon, S. Y., Kwon, H. & Hur, K. Intrinsic photonic wave localization in a three-dimensional icosahedral quasicrystal. Nat. Phys. 13, 363–368 (2017).
Nagai, Y. et al. High-Temperature atomic diffusion and specific heat in quasicrystals. Phys. Rev. Lett. 132, 196301 (2024).
Ghadimi, R., Sugimoto, T., Tanaka, K. & Tohyama, T. Topological superconductivity in quasicrystals. Phys. Rev. B: Condens. Matter Mater. Phys. 104, 144511 (2021).
Uri, A. et al. Superconductivity and strong interactions in a tunable moiré quasicrystal. Nature 620, 762–767 (2023).
Han, I. et al. Formation of a single quasicrystal upon collision of multiple grains. Nat. Commun. 12, 5790 (2021).
Dyson, F. Birds and frogs. Not. AMS 56, 212–223 (2009).
Torquato, S., Zhang, G. & De Courcy-Ireland, M. Hidden multiscale order in the primes. J. Phys. A: Math. Theor. 52, 135002 (2019).
Steurer, W. Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal, and dodecagonal quasicrystals. Z. Krist. Cryst. Mater. 219, 391–446 (2004).
Mukhopadhyay, N. K. & Yadav, T. P. Quasicrystals: a new class of structurally complex intermetallics. J. Indian Inst. Sci. 102, 59–90 (2022).
Sun, W. et al. The thermodynamic scale of inorganic crystalline metastability. Sci. Adv. 2, e1600225 (2016).
Kalugin, P. Growth of entropy stabilized quasicrystals. Eur. Phys. J. B 18, 77–84 (2000).
Je, K., Lee, S., Teich, E. G., Engel, M. & Glotzer, S. C. Entropic formation of a thermodynamically stable colloidal quasicrystal with negligible phason strain. Proc. Natl Acad. Sci. USA 118, e2011799118 (2021).
Widom, M. & Mihalkovič, M. Quasicrystal structure prediction: a review. Isr. J. Chem. https://doi.org/10.1002/ijch.202300122 (2023).
Fayen, E., Filion, L., Foffi, G. & Smallenburg, F. Quasicrystal of binary hard spheres on a plane stabilized by configurational entropy. Phys. Rev. Lett. 132, 48202 (2024).
Tsai, A. P., Guo, J. Q., Abe, E., Takakura, H. & Sato, T. J. A stable binary quasicrystal. Nature 408, 537–538 (2000).
Canfield, P. C. et al. Solution growth of a binary icosahedral quasicrystal of Sc12Zn88. Phys. Rev. B 81, 020201 (2010).
Henley, C. L., De Boissieu, M. & Steurer, W. Discussion on clusters, phasons and quasicrystal stabilisation. Philos. Mag. 86, 1131–1151 (2006).
Tsai, A. P. Icosahedral clusters, icosaheral order and stability of quasicrystals—a view of metallurgy. Sci. Technol. Adv. Mater. 9, 013008 (2008).
Tsai, A. P. A test of Hume-Rothery rules for stable quasicrystals. J. Non-Cryst. Solids 334, 317–322 (2004).
Tsai, A. P. Discovery of stable icosahedral quasicrystals: progress in understanding structure and properties. Chem. Soc. Rev. 42, 5352–5365 (2013).
Tsai, A. P., Inoue, A., Bizen, Y. & Masumoto, T. Kinetics of the amorphous to icosahedral structure transition in Al-Cu-V and Al-Mn-Si alloys. Acta Metall. 37, 1443–1449 (1989).
Okumura, H., Tsai, A. P., Inoue, A. & Masumoto, T. The observation of mechanical relaxation in a quasicrystalline Al75Cu15V10 alloy. Mater. Sci. Eng.: A 181, 781–784 (1994).
Weisbecker, P., Bonhomme, G., Bott, G. & Dubois, J. M. The oxidation at 500 °C of AlCuFe quasicrystalline powders: a X-ray diffraction study. J. Non-Cryst. Solids 351, 1630–1638 (2005).
Hennig, R. G., Carlsson, A. E., Kelton, K. F. & Henley, C. L. Ab initio Ti-Zr-Ni phase diagram predicts stability of icosahedral TiZrNi quasicrystals. Phys. Rev. B: Condens. Matter Mater. Phys. 71, 144103 (2005).
Berns, V. M. & Fredrickson, D. C. Problem solving with pentagons: Tsai-type quasicrystal as a structural response to chemical pressure. Inorg. Chem. 52, 12875–12877 (2013).
Henley, C. L., Mihalkovič, M. & Widom, M. Total-energy-based structure prediction for d(AlNiCo). J. Alloys Compd. 342, 221–227 (2002).
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 (2015).
Mihalkovič, M. & Widom, M. Spontaneous formation of thermodynamically stable Al-Cu-Fe icosahedral quasicrystal from realistic atomistic simulations. Phys. Rev. Res. 2, 013196 (2020).
Han, I. et al. Dynamic observation of dendritic quasicrystal growth upon laser-induced solid-state transformation. Phys. Rev. Lett. 125, 195503 (2020).
Kurtuldu, G. & Bernet, M. The relation between the glass forming ability and nucleation kinetics of metastable quasicrystals in Mg–Zn–Yb liquid. J. Alloys Compd. 945, 168930 (2023).
Fiorentini, V. & Methfessel, M. Extracting convergent surface energies from slab calculations. J. Phys.: Condens. Matter 8, 6525 (1996).
Gavini, V., Knap, J., Bhattacharya, K. & Ortiz, M. Non-periodic finite-element formulation of orbital-free density functional theory. J. Mech. Phys. Solids 55, 669–696 (2007).
Dreyer, C. E., Janotti, A. & Van De Walle, C. G. Absolute surface energies of polar and nonpolar planes of GaN. Phys. Rev. B: Condens. Matter Mater. Phys. 89, 081305 (2014).
Navrotsky, A. Nanoscale effects on thermodynamics and phase equilibria in oxide systems. ChemPhysChem 12, 2207–2215 (2011).
Navrotsky, A. Energetics at the nanoscale: impacts for geochemistry, the environment, and materials. MRS Bull. 41, 139–145 (2016).
Takakura, H., Gómez, C. P., Yamamoto, A., De Boissieu, M. & Tsai, A. P. Atomic structure of the binary icosahedral Yb–Cd quasicrystal. Nat. Mater. 6, 58–63 (2007).
Yamada, T. et al. Atomic structure and phason modes of the Sc–Zn icosahedral quasicrystal. IUCrJ 3, 247–258 (2016).
Motamarri, P. et al. DFT-FE–A massively parallel adaptive finite-element code for large-scale density functional theory calculations. Comput. Phys. Commun. 246, 106853 (2020).
Das, S., Motamarri, P., Subramanian, V., Rogers, D. M. & Gavini, V. DFT-FE 1.0: a massively parallel hybrid CPU-GPU density functional theory code using finite-element discretization. Comput. Phys. Commun. 280, 108473 (2022).
Motamarri, P. & Gavini, V. Configurational forces in electronic structure calculations using Kohn-Sham density functional theory. Phys. Rev. B 97, 165132 (2018).
Motamarri, P., Nowak, M. R., Leiter, K., Knap, J. & Gavini, V. Higher-order adaptive finite-element methods for Kohn–Sham density functional theory. J. Comput. Phys. 253, 308–343 (2013).
Zhou, Y., Saad, Y., Tiago, M. L. & Chelikowsky, J. R. Self-consistent-field calculations using Chebyshev-filtered subspace iteration. J. Comput. Phys. 219, 172–184 (2006).
Das, S. et al. Fast, scalable, and accurate finite-element based ab initio calculations using mixed precision computing: 46 PFLOPS simulation of a metallic dislocation system. In Proc. International Conference for High Performance Computing, Networking, Storage and Analysis 1–11 (ACM, 2019).
Hamann, D. R. Optimized norm-conserving Vanderbilt pseudopotentials. Phys. Rev. B: Condens. Matter Mater. Phys. 88, 085117 (2013).
Prandini, G., Marrazzo, A., Castelli, I. E., Mounet, N. & Marzari, N. Precision and efficiency in solid-state pseudopotential calculations. npj Comput. Mater. 4, 72 (2018).
Das, S. et al. Large-scale materials modeling at quantum accuracy: ab initio simulations of quasicrystals and interacting extended defects in metallic alloys. In Proc. International Conference for High Performance Computing, Networking, Storage and Analysis 1–12 (ACM, 2023).
Service, R. F. Exascale computers show off emerging science. Science 382, 864–865 (2023).
Curtarolo, S., Morgan, D. & Ceder, G. Accuracy of ab initio methods in predicting the crystal structures of metals: a review of 80 binary alloys. Calphad 29, 163–211 (2005).
Brown, I. D. Chemical and steric constraints in inorganic solids. Acta Crystallogr. Sect. B: Struct. Sci. Cryst. Eng. Mater. 48, 553–572 (1992).
Lalvani, H. Non-periodic space structures. Int. J. Space Struct. 2, 93–108 (1987).
Palenzona, A. & Manfrinetti, P. The phase diagram of the Sc-Zn system. J. Alloys Compd. 247, 195–197 (1997).
Sun, W., Jayaraman, S., Chen, W., Persson, K. A. & Ceder, G. Nucleation of metastable aragonite CaCO3 in seawater. Proc. Natl Acad. Sci. USA 112, 3199–3204 (2015).
Sun, W. & Ceder, G. Efficient creation and convergence of surface slabs. Surf. Sci. 617, 53–59 (2013).
Sun, W., Kitchaev, D. A., Kramer, D. & Ceder, G. Non-equilibrium crystallization pathways of manganese oxides in aqueous solution. Nat. Commun. 10, 573 (2019).
Balluffi, R. W., Allen, S. M. & Carter, W. C. Kinetics of Materials (Wiley, 2005).
Bocklund, B. et al. ESPEI for efficient thermodynamic database development, modification, and uncertainty quantification: Application to Cu-Mg. MRS Commun. 9, 618–627 (2019).
Tang, C. et al. Thermodynamic modeling of the Sc-Zn system coupled with first-principles calculation. J. Min. Metall. Sect. B: Metall. 48, 123–130 (2012).
Palenzona, A. The ytterbium-cadmium system. J. Less Common Met. 25, 367–372 (1971).
Togo, A., Chaput, L., Tadano, T. & Tanaka, I. Implementation strategies in phonopy and phono3py. J. Phys.: Condens. Matter 35, 353001 (2023).
Carreras, A., Togo, A. & Tanaka, I. DynaPhoPy: a code for extracting phonon quasiparticles from molecular dynamics simulations. Comput. Phys. Commun. 221, 221–234 (2017).
Puchala, B. et al. CASM — a software package for first-principles based study of multicomponent crystalline solids. Comput. Mater. Sci. 217, 111897 (2023).
Edagawa, K., Kajiyama, K., Tamura, R. & Takeuchi, S. High-temperature specific heat of quasicrystals and a crystal approximant. Mater. Sci. Eng.: A 312, 293–298 (2001).
Maintz, S., Deringer, V. L., Tchougréeff, A. L. & Dronskowski, R. LOBSTER: a tool to extract chemical bonding from plane-wave based DFT. J. Comput. Chem. 37, 1030–1035 (2016).
Fredrickson, D. C. DFT-chemical pressure analysis: visualizing the role of atomic size in shaping the structures of inorganic materials. J. Am. Chem. Soc. 134, 5991–5999 (2012).
Fredrickson, R. T. & Fredrickson, D. C. Chemical pressure-derived assembly principles for dodecagonal quasicrystal approximants and other complex Frank-Kasper phases. Inorg. Chem. 61, 17682–17691 (2022).
Jiang, K. & Zhang, P. Numerical methods for quasicrystals. J. Comput. Phys. 256, 428–440 (2014).
Yin, J., Jiang, K., Shi, A. C., Zhang, P. & Zhang, L. Transition pathways connecting crystals and quasicrystals. Proc. Natl Acad. Sci. USA 118, e2106230118 (2021).
Miao, J., Ercius, P. & Billinge, S. J. L. Atomic electron tomography: 3D structures without crystals. Science 353, aaf2157 (2016).
Fredrickson, D. C., Lee, S. & Hoffmann, R. Interpenetrating polar and nonpolar sublattices in intermetallics: the NaCd2 structure. Angew. Chem. Int. Ed. 46, 1958–1976 (2007).
Dijkstra, M. & Luijten, E. From predictive modelling to machine learning and reverse engineering of colloidal self-assembly. Nat. Mater. 20, 762–773 (2021).
Zhou, W. et al. Colloidal quasicrystals engineered with DNA. Nat. Mater. 23, 424–428 (2024).
Liu, X.-Y. et al. Self-assembled soft alloy with Frank–Kasper phases beyond metals. Nat. Mater. 23, 570–576 (2024).
Zuo, Y. et al. Performance and cost assessment of machine learning interatomic potentials. J. Phys. Chem. A 124, 731–745 (2020).
Giannozzi, P. et al. QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials. J. Phys.: Condens. Matter 21, 395502 (2009).
Giannozzi, P. et al. Advanced capabilities for materials modelling with Quantum ESPRESSO. J. Phys.: Condens. Matter 29, 465901 (2017).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Schlipf, M. & Gygi, F. Optimization algorithm for the generation of ONCV pseudopotentials. Comput. Phys. Commun. 196, 36–44 (2015).
Jain, A. et al. A high-throughput infrastructure for density functional theory calculations. Comput. Mater. Sci. 50, 2295–2310 (2011).
Tran, R. et al. Surface energies of elemental crystals. Sci. Data 3, 160080 (2016).
Dinsdale, A. T. SGTE data for pure elements. Calphad 15, 317–425 (1991).
Redlich, O. & Kister, A. T. Algebraic representation of thermodynamic properties and the classification of solutions. Ind. Eng. Chem. 40, 345–348 (1948).
Eslami, H., Muggianu, Y. M., Gambino, M. & Bros, J. P. Enthalpies de formation des alliages liquides aluminium-germanium, gallium-germanium et aluminium-gallium-germanium entre 713 et 1230 K. J. Less Common Met. 64, 31–44 (1979).
Gelman, A. et al. Bayesian Data Analysis (CRC, 2013).