3D metamaterials

Walser, R. M. Electromagnetic metamaterials. Proc. SPIE, Complex Mediums II: Beyond Linear Isotropic Dielectr. (eds Lakhtakia, A., Weiglhofer, W. S. & Hodgkinson, I. J.) 4467 (2001).

Bendsøe, M. & Sigmund, O. Topology Optimization: Theory, Methods and Applications (Springer, 2004).

Browning, V. DARPATech 2002 7, 791–795 (2002).

Kittel, C. Introduction to Solid State Physics (Wiley, 2004).

Milton, G. W. The Theory of Composites (Cambridge Univ. Press, 2002). This textbook provides a comprehensive theoretical introduction to man-made composite materials in electromagnetism and optics, acoustics and mechanics, and transport.

Golden, K., Grimmett, G., James, R., Milton, G. & Sen, P. Mathematics of Multiscale Materials (Springer, New York, 2012).

Chen, H.-T., Taylor, A. J. & Yu, N. A review of metasurfaces: physics and applications. Rep. Prog. Phys. 79, 076401 (2016).

McCall, M. et al. Roadmap on transformation optics. J. Opt. 20, 063001 (2018).

Craster, R. V., Kaplunov, J. & Pichugin, A. V. High-frequency homogenization for periodic media. Proc. R. Soc. Lond. Ser. A 466, 2341–2362 (2010).

Bensoussan, A., Lions, J. & Papanicolaou, G. Asymptotic Analysis for Periodic Structures (American Mathematical Society, 2011).

Jikov, V., Yosifian, G., Kozlov, S. & Oleinik, O. Homogenization of Differential Operators and Integral Functionals (Springer, Berlin Heidelberg, 2012).

Bakhvalov, N. & Panasenko, G. Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials (Springer, Netherlands, 2012).

Pham, K., Maurel, A. & Marigo, J.-J. Two scale homogenization of a row of locally resonant inclusions — the case of anti-plane shear waves. J. Mech. Phy. Solids 106, 80–94 (2017).

Brassart, M. & Lenczner, M. A two-scale model for the periodic homogenization of the wave equation. J. Math. Pures Appl. 93, 474–517 (2010).

Harutyunyan, D., Milton, G. W. & Craster, R. V. High-frequency homogenization for travelling waves in periodic media. Proc. R. Soc. Lond. Ser. A 472, 20160066 (2016).

Krushlov, E. Y. The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain. Math. USSR Sb. 35, 266–282 (1979).

Auriault, J. L. & Boutin, C. Deformable porous media with double porosity III: acoustics. Transp. Porous Med. 14, 143–162 (1994).

Zhikov, V. V. On an extension of the method of two-scale convergence and its applications. Sb. Math. 191, 31–72 (2010).

Pendry, J. B., Holden, A. J., Stewart, W. J. & Youngs, I. Extremely low frequency plasmons in metallic mesostructures. Phys. Rev. Lett. 76, 4773–4776 (1996).

Belov, P. A. et al. Strong spatial dispersion in wire media in the very large wavelength limit. Phys. Rev. B 67, 113103 (2003).

Menzel, C. et al. Validity of effective material parameters for optical fishnet metamaterials. Phys. Rev. B 81, 035320 (2010).

Soukoulis, C. & Wegener, M. Past achievements and future challenges in the development of three-dimensional photonic metamaterials. Nat. Photonics 5, 523–530 (2011). This reference is an extensive review on 3D optical metamaterials, with emphasis on negative refractive indices and chirality; it also provides a more detailed history of the field.

Laude, V. Phononic Crystals: Artificial Crystals for Sonic, Acoustic, and Elastic Waves (De Gruyter, 2015). This is a textbook introduction to acoustic and elastic metamaterials.

Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014).

Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).

Süsstrunk, R. & Huber, S. D. Classification of topological phonons in linear mechanical metamaterials. Proc. Natl Acad. Sci. USA 113, E4767–E4775 (2016).

Landau, L. D. & Lifshitz, E. M. Electrodynamics of Continuous Media (Pergamon Press, 1960).

Schelkunoff, S. A. & Friis, H. T. Antennas: the Theory and Practice (Wiley, 1952).

Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).

Schneider, H. J. & Dullenkopf, P. Slotted tube resonator: a new NMR probe head at high observing frequencies. Rev. Sci. Instrum. 48, 68–73 (1977).

Ghim, B. T., Rinard, G. A., Quine, R. W., Eaton, S. S. & Eaton, G. R. Design and fabrication of copper-film loop–gap resonators. J. Magn. Reson. A 120, 72–76 (1996).

Lagarkov, A. N. & Sarychev, A. K. Electromagnetic properties of composites containing elongated conducting inclusions. Phys. Rev. B 53, 6318–6336 (1996).

Rose-Innes, A. & Rhoderick, E. Introduction to Superconductivity (Pergamon Press, 1978).

Meade, R. & Diffenderfer, R. Foundations of Electronics: Circuits and Devices (Delmar Cengage Learning, 2002).

Dolling, G., Enkrich, C., Wegener, M., Soukoulis, C. M. & Linden, S. Simultaneous negative phase and group velocity of light in a metamaterial. Science 312, 892–894 (2006).

Veselago, V. The electrodnamics of substances with simultaneously negative values of and μ. Sov. Phys. Usp. 10, 509–514 (1968). This is an inspiring ‘what if’ paper, showing some of the enhanced possibilities with negative magnetic permeability and negative refractive index in optics; it stimulates much of the early work on electromagnetic and optical metamaterials.

Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).

Shelby, R. A., Smith, D. R. & Schultz, S. Experimental verification of a negative index of refraction. Science 292, 77–79 (2001). This experimental paper on negative refractive indices at microwave frequencies is a catalyser for the metamaterial field.

Zhang, S. et al. Experimental demonstration of near-infrared negative-index metamaterials. Phys. Rev. Lett. 95, 137404 (2005).

Soukoulis, C. M., Linden, S. & Wegener, M. Negative refractive index at optical wavelengths. Science 315, 47–49 (2007).

García-Meca, C. et al. Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths. Phys. Rev. Lett. 106, 067402 (2011).

Kinsler, P. & McCall, M. W. Causality-based criteria for a negative refractive index must be used with care. Phys. Rev. Lett. 101, 167401 (2008).

Zheludev, N. I., Prosvirnin, S. L., Papasimakis, N. & Fedotov, V. A. Lasing spaser. Nat. Photonics 2, 351–354 (2008).

Noginov, M. A. et al. Demonstration of a spaser-based nanolaser. Nature 460, 1110–1112 (2009).

Fang, A., Koschny, T., Wegener, M. & Soukoulis, C. M. Self-consistent calculation of metamaterials with gain. Phys. Rev. B 79, 241104 (2009).

Wuestner, S., Pusch, A., Tsakmakidis, K. L., Hamm, J. M. & Hess, O. Overcoming losses with gain in a negative refractive index metamaterial. Phys. Rev. Lett. 105, 127401 (2010).

Valentine, J. et al. Three-dimensional optical metamaterial with a negative refractive index. Nature 455, 376–379 (2008).

Nguyen, V. C., Chen, L. & Halterman, K. Total transmission and total reflection by zero index metamaterials with defects. Phys. Rev. Lett. 105, 233908 (2010).

Moitra, P. et al. Realization of an all-dielectric zero-index optical metamaterial. Nat. Photonics 7, 791–795 (2013).

Javani, M. H. & Stockman, M. I. Real and imaginary properties of epsilon-near-zero materials. Phys. Rev. Lett. 117, 107404 (2016).

Wegener, M., Dolling, G. & Linden, S. Plasmonics: backward waves moving forward. Nat. Mater. 6, 475–476 (2007).

Yao, J. et al. Optical negative refraction in bulk metamaterials of nanowires. Science 321, 930 (2008).

Cui, T., Smith, D. & Liu, R. Metamaterials: Theory, Design, and Applications (Springer, 2009).

Krishnamoorthy, H. N., Jacob, Z., Narimanov, E., Kretzschmar, I. & Menon, V. M. Topological transitions in metamaterials. Science 336, 205–209 (2012).

García-Chocano, V. M., Christensen, J. & Sánchez-Dehesa, J. Negative refraction and energy funneling by hyperbolic materials: an experimental demonstration in acoustics. Phys. Rev. Lett. 112, 144301 (2014).

Ferrari, L., Wu, C., Lepage, D., Zhang, X. & Liu, Z. Hyperbolic metamaterials and their applications. Prog. Quantum Electron. 40, 1–40 (2015).

Landy, N. I., Sajuyigbe, S., Mock, J. J., Smith, D. R. & Padilla, W. J. Perfect metamaterial absorber. Phys. Rev. Lett. 100, 207402 (2008).

Liu, N. et al. Plasmonic analogue of electromagnetically induced transparency at the drude damping limit. Nat. Mater. 8, 758–762 (2009).

Watts, C. M., Xianliang, L. & Padilla, W. J. Metamaterial electromagnetic wave absorbers. Adv. Mater. 24, 98–120 (2012).

Lee, Y., Rhee, J., Yoo, Y. & Kim, K. Metamaterials for Perfect Absorption (Springer, Singapore, 2016).

Pfeiffer, C. & Grbic, A. Metamaterial huygens surfaces: tailoring wave fronts with reflectionless sheets. Phys. Rev. Lett. 110, 197401 (2013).

Liu, S. et al. Optical magnetic mirrors without metals. Optica 1, 250–256 (2014).

Lindell, I. Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Tretyakov, S., Sihvola, A., Sochava, A. & Simovski, C. Magnetoelectric interactions in bi-anisotropic media. J. Electro. Wave. Appl. 12, 481–497 (1998).

Eringen, A. Elastodynamics Vol. 2 (Academic Press, 1974).

Wegener, M. & Linden, S. in Tutorials in Metamaterials Ch. 8 (eds Noginov, M. A. & Podolskiy, V. A.) (Taylor and Francis, 2012).

Gansel, J. et al. Gold helix photonic metamaterial as broadband circular polarizer. Science 325, 1513–1515 (2009).

Gansel, J. et al. Tapered gold-helix metamaterials as improved circular polarizers. Appl. Phys. Lett. 100, 101109 (2012).

Johannes, K. et al. A helical metamaterial for broadband circular polarization conversion. Adv. Opt. Mater. 3, 1411–1417 (2015).

Kaschke, J. & Wegener, M. Gold triple-helix mid-infrared metamaterial by sted-inspired laser lithography. Opt. Lett. 40, 3986–3989 (2015).

Fernandez-Corbaton, I., Fruhnert, M. & Rockstuhl, C. Objects of maximum electromagnetic chirality. Phys. Rev. X 6, 031013 (2016).

Lefier, Y., Salut, R., Suarez, M. A. & Grosjean, T. Directing nanoscale optical flows by coupling photon spin to plasmon extrinsic angular momentum. Nano. Lett. 18, 38–42 (2018).

Chin, J. Y. et al. Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation. Nat. Commun. 4, 1599 (2013).

Fedotov, V. A. et al. Asymmetric propagation of electromagnetic waves through a planar chiral structure. Phys. Rev. Lett. 97, 167401 (2006).

Kaelberer, T., Fedotov, V., Papasimakis, N., Tsai, D. & Zheludev, N. Toroidal dipolar response in a metamaterial. Science 330, 1510–1512 (2010).

Papasimakis, N., Fedotov, V. A., Savinov, V., Raybould, T. A. & Zheludev, N. I. Electromagnetic toroidal excitations in matter and free space. Nat. Mater. 15, 263–271 (2016).

Fernandez-Corbaton, I., Nanz, S. & Rockstuhl, C. On the dynamic toroidal multipoles from localized electric current distributions. Sci. Rep. 7, 7527 (2017).

Wegener, M. Extreme Nonlinear Optics (Springer-Verlag, 2005).

Kauranen, M. & Zayats, A. V. Nonlinear plasmonics. Nat. Photonics 6, 737–748 (2012).

Lee, J. et al. Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions. Nature 511, 65–69 (2014).

Samson, Z. L. et al. Metamaterial electro-optic switch of nanoscale thickness. Appl. Phys. Lett. 96, 143105 (2010).

Buchnev, O., Ou, J. Y., Kaczmarek, M., Zheludev, N. I. & Fedotov, V. A. Electro-optical control in a plasmonic metamaterial hybridised with a liquid-crystal cell. Opt. Express 21, 1633–1638 (2013).

Khurgin, J. B. & Sun, G. Plasmonic enhancement of the third order nonlinear optical phenomena: figures of merit. Opt. Express 21, 27460–27480 (2013).

Jahani, S. & Jacob, Z. All-dielectric metamaterials. Nat. Nanotechnol. 11, 23–36 (2016).

Staude, I. & Schilling, J. Metamaterial-inspired silicon nanophotonics. Nat. Photonics 11, 274–284 (2017).

Hermans, A. et al. On the determination of χ (2) in thin films: a comparison of one-beam second-harmonic generation measurement methodologies. Sci. Rep. 7, 44581 (2017).

Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013).

Ding, Y., Liu, Z., Qiu, C. & Shi, J. Metamaterial with simultaneously negative bulk modulus and mass density. Phys. Rev. Lett. 99, 093904 (2007).

Lee, S. H., Park, C. M., Seo, Y. M., Wang, Z. G. & Kim, C. K. Composite acoustic medium with simultaneously negative density and modulus. Phys. Rev. Lett. 104, 054301 (2010).

Wu, Y., Lai, Y. & Zhang, Z. Q. Elastic metamaterials with simultaneously negative effective shear modulus and mass density. Phys. Rev. Lett. 107, 105506 (2011).

Cummer, S. A., Christensen, J. & Alu, A. Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016).

Liu, Z. et al. Locally resonant sonic materials. Science 289, 1734–1736 (2000).

Schoenberg, M. & Sen, P. N. Properties of a periodically stratified acoustic half-space and its relation to a Biot fluid. J. Acoust. Soc. Am. 73, 61–67 (1983).

Milton, G. W., Birane, M. & Willis, J. R. On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006).

Willis, J. R. Effective constitutive relations for waves in composites and metamaterials. Proc. Roy. Soc. Lond. A 467, 1865–1879 (2011).

Muhlestein, M. B., Sieck, C. F., Wilson, P. S. & Haberman, M. R. Experimental evidence of Willis coupling in a one-dimensional effective material element. Nat. Commun. 8, 15625 (2017).

Bueckmann, T., Kadic, M., Schittny, R. & Wegener, M. Mechanical metamaterials with anisotropic and negative effective mass-density tensor made from one constituent material. Phys. Status Solidi B 252, 1671–1674 (2015).

Yang, M., Chen, S., Fu, C. & Sheng, P. Optimal sound-absorbing structures. Mater. Horiz. 4, 673–680 (2017).

Ma, G. & Sheng, P. Acoustic metamaterials: from local resonances to broad horizons. Sci. Adv. 2, e1501595 (2015).

Liang, Z. & Li, J. Extreme acoustic metamaterial by coiling up space. Phys. Rev. Lett. 108, 114301 (2012).

Frenzel, T. et al. Three-dimensional labyrinthine acoustic metamaterials. Appl. Phys. Lett. 103, 061907 (2013).

Xie, Y., Konneker, A., Popa, B.-I. & Cummer, S. A. Tapered labyrinthine acoustic metamaterials for broadband impedance matching. Appl. Phys. Lett. 103, 201906 (2013).

Krushynska, A. O., Bosia, F., Miniaci, M. & Pugno, N. M. Spider web-structured labyrinthine acoustic metamaterials for low-frequency sound control. New J. Phys. 19, 105001 (2017).

Maurya, S. K., Pandey, A., Shukla, S. & Saxena, S. Double negativity in 3D space coiling metamaterials. Sci. Rep. 6, 33683 (2016).

Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alu, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).

Aurégan, Y. & Pagneux, V. 𝒫𝒯-symmetric scattering in flow duct acoustics. Phys. Rev. Lett. 118, 174301 (2017).

Banerjee, B. An Introduction to Metamaterials and Waves in Composites (Taylor and Francis, 2011).

Walpole, L. On bounds for the overall elastic moduli of inhomogeneous systems — I. J. Mech. Phy. Solids 14, 151–162 (1966).

Milton, G. W. Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots. J. Mech. Phy. Solids 61, 1543–1560 (2013).

Milton, G. W. & Cherkaev, A. V. Which elasticity tensors are realizable? J. Eng. Mater. Technol. 117, 483–493 (1995).

Kadic, M., Bückmann, T., Stenger, N., Thiel, M. & Wegener, M. On the practicability of pentamode mechanical metamaterials. Appl. Phys. Lett. 100, 191901 (2012).

Kadic, M., Schittny, R., Bückmann, T. & Wegener, M. On anisotropic versions of three-dimensional pentamode metamaterials. New J. Phys. 15, 023029 (2013).

Bueckmann, T. et al. On three-dimensional dilational elastic metamaterials. New J. Phys. 16, 033032 (2014).

Biot, M. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28, 168–178 (1956).

Biot, M. & Willis, D. The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957).

Gatt, R. & Grima, J. N. Negative compressibility. Phys. Status Solidi RRL 2, 236–238 (2008).

Qu, J., Kadic, M. & Wegener, M. Poroelastic metamaterials with negative effective static compressibility. Appl. Phys. Lett. 110, 171901 (2017).

Qu, J., Gerber, A., Mayer, F., Kadic, M. & Wegener, M. Experiments on metamaterials with negative effective static compressibility. Phys. Rev. X 7, 041060 (2017).

Sommerfeld, A. Mechanics of Deformable Bodies (Academic Press, 1950).

Eringen, A. Microcontinuum Field Theories I: Foundations and Solids. (Springer, New York, 1999). This is a textbook on theoretical paths towards generalizing linear Cauchy elasticity.

Frenzel, T., Kadic, M. & Wegener, M. Three-dimensional mechanical metamaterials with a twist. Science 358, 1072–1074 (2017).

Rueger, Z. & Lakes, R. S. Strong cosserat elasticity in a transversely isotropic polymer lattice. Phys. Rev. Lett. 120, 065501 (2018).

Zhu, H. et al. Observation of chiral phonons. Science 359, 579–582 (2018).

Coulais, C., Kettenis, C. & van Hecke, M. A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials. Nat. Phys. 14, 40–44 (2017).

Kadic, M., Frenzel, T. & Wegener, M. Mechanical metamaterials: when size matters. Nat. Phys. 14, 8–9 (2018).

Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015).

Hassanpour, S. & Heppler, G. R. Theory of micropolar gyroelastic continua. Acta Mech. 227, 1469–1491 (2016).

Carta, G., Jones, I. S., Movchan, N. V., Movchan, A. B. & Nieves, M. J. Deflecting elastic prism and unidirectional localisation for waves in chiral elastic systems. Sci. Rep. 7, 26 (2017).

Abdoul-Anziz, H. & Seppecher, P. Strain gradient and generalized continua obtained by homogenizing frame lattices. Math. Mech. Complex Syst. 6, 213–250 (2018).

Gudmundson, P. A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004).

Olive, M. & Auffray, N. Symmetry classes for odd-order tensors. J. Appl. Math. Mech. 94, 421–447 (2014).

Cordero, N. M., Forest, S. & Busso, E. P. Second strain gradient elasticity of nano-objects. J. Mech. Phys. Solids 97, 92–124 (2016).

Liebold, C. & Mueller, W. H. Comparison of gradient elasticity models for the bending of micromaterials. Comput. Mater. Sci. 116, 52–61 (2016).

Lecoutre, G., Daher, N., Devel, M. & Hirsinger, L. Principle of virtual power applied to deformable semiconductors with strain, polarization, and magnetization gradients. Acta Mech. 228, 1681–1710 (2017).

Bertram, A. Compendium on gradient materials. Redaktion http://www.redaktion.tu-berlin.de/fileadmin/fg49/publikationen/bertram/Compendium_on_Gradient_Materials_Dec_2017.pdf (2017).

Wang, P., Casadei, F., Shan, S., Weaver, J. C. & Bertoldi, K. Harnessing buckling to design tunable locally resonant acoustic metamaterials. Phys. Rev. Lett. 113, 014301 (2014).

Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014).

Kang, S. H. et al. Complex ordered patterns in mechanical instability induced geometrically frustrated triangular cellular structures. Phys. Rev. Lett. 112, 098701 (2014).

Bauer, J. et al. Nanolattices: an emerging class of mechanical metamaterials. Adv. Mater. 29, 1701850 (2017).

Bertoldi, K., Vitelli, V., Christensen, J. & van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017). This is a further reading on recent progress in 2D and 3D elastic metamaterials.

Tobias, F., Claudio, F., Muamer, K., Peter, G. & Martin, W. Tailored buckling microlattices as reusable light-weight shock absorbers. Adv. Mater. 28, 5865–5870 (2016).

Coulais, C., Teomy, E., de Reus, K., Shokef, Y. & van Hecke, M. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016).

Schenk, M. & Guest, S. D. Geometry of miura-folded metamaterials. Proc. Natl Acad. Sci. USA 110, 3276–3281 (2013).

Wei, Z. Y., Guo, Z. V., Dudte, L., Liang, H. Y. & Mahadevan, L. Geometric mechanics of periodic pleated origami. Phys. Rev. Lett. 110, 215501 (2013).

Waitukaitis, S., Menaut, R., Chen, B. G.-G. & van Hecke, M. Origami multistability: from single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015).

Silverberg, J. L. et al. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nat. Mater. 14, 389–393 (2015).

Coulais, C., Sounas, D. & Alu, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017).

Gao, H., Ji, B., Jaeger, I. L., Arzt, E. & Fratzl, P. Materials become insensitive to flaws at nanoscale: lessons from nature. Proc. Natl Acad. Sci. USA 100, 5597–5600 (2003).

Schaedler, T. A. et al. Ultralight metallic microlattices. Science 334, 962–965 (2011).

Zheng, X. et al. Ultralight, ultrastiff, mechanical metamaterials. Science 344, 1373–1377 (2014).

Meza, L. R., Das, S. & Greer, J. R. Strong, lightweight, and recoverable three-dimensional ceramic nanolattices. Science 345, 1322–1326 (2014).

Bauer, J., Hengsbach, S., Tesari, I., Schwaiger, R. & Kraft, O. High-strength cellular ceramic composites with 3D microarchitecture. Proc. Natl Acad. Sci. USA 111, 2453–2458 (2014).

Bauer, J. et al. Nanolattices: an emerging class of mechanical metamaterials. Adv. Mater. 29, 1701850 (2002).

Popovic, R. S. Hall Effect Devices (Institute of Physics Publishing, Philadelphia, 2004).

Briane, M., Milton, G. W. & Nesi, V. Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal. 173, 133–150 (2004).

Briane, M. & Milton, G. W. An antisymmetric effective Hall matrix. SIAM J. Appl. Math. 70, 1810–1820 (2010).

Tornow, M. et al. Anisotropic magnetoresistance of a classical antidot array. Phys. Rev. Lett. 77, 147–150 (1996).

Kadic, M., Schittny, R., Bückmann, T., Kern, C. & Wegener, M. Hall-effect sign inversion in a realizable 3D metamaterial. Phys. Rev. X 5, 021030 (2015).

Kern, C., Kadic, M. & Wegener, M. Experimental evidence for sign reversal of the Hall coefficient in three-dimensional metamaterials. Phys. Rev. Lett. 118, 016601 (2017).

Kern, C., Graeme, W. M., Kadic, M. & Wegener, M. Theory of the Hall effect in three-dimensional metamaterials. New J. Phys. 20, 083034 (2018).

Bergman, D. J. & Strelniker, Y. M. Calculation of strong-field magnetoresistance in some periodic composites. Phys. Rev. B 49, 16256–16268 (1994).

Strelniker, Y. M. & Bergman, D. J. Thermoelectric response of a periodic composite medium in the presence of a magnetic field: angular anisotropy. Phys. Rev. B 96, 235308 (2017).

Liu, L. Feasibility of large-scale power plants based on thermoelectric effects. New J. Phys. 16, 123019 (2014).

Schittny, R., Kadic, M., Guenneau, S. & Wegener, M. Experiments on transformation thermodynamics: molding the flow of heat. Phys. Rev. Lett. 110, 195901 (2013).

Ros, A. et al. Brownian motion: absolute negative particle mobility. Nature 436, 928–928 (2005).

Kern, C., Schuster, V., Kadic, M. & Wegener, M. Experiments on the parallel Hall effect in three-dimensional metamaterials. Phys. Rev. Appl. 7, 044001 (2017).

Onsager, L. Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931).

Debord, J. D. & Lyon, L. A. Thermoresponsive photonic crystals. J. Phys. Chem. B 104, 6327–6331 (2000).

Stuart, M. A. C. et al. Emerging applications of stimuli-responsive polymer materials. Nat. Mater. 9, 101–113 (2010).

Schroden, R. C., Al-Daous, M., Blanford, C. F. & Stein, A. Optical properties of inverse opal photonic crystals. Chem. Mater. 14, 3305–3315 (2002).

Theato, P., Sumerlin, B. S., O’Reilly, R. K. & Epps, T. H. III Stimuli responsive materials. Chem. Soc. Rev. 42, 7055–7056 (2013).

Skylar, T. 4D printing: multi-material shape change. Archit. Des. 84, 116–121 (2014).

Hao, Z. et al. Light-fueled microscopic walkers. Adv. Mater. 27, 3883–3887 (2015).

Martella, D. et al. Light activated non-reciprocal motion in liquid crystalline networks by designed microactuator architecture. RSC Adv. 7, 19940–19947 (2017).

Momeni, F., Hassani, S. M. M., Liu, X. & Ni, J. A review of 4D printing. Mater. Des. 122, 42–79 (2017).

Akihiro, N., Ahmed, M., Hang, Z. & Martin, M. In-gel direct laser writing for 3D-designed hydrogel composites that undergo complex self-shaping. Adv. Sci. 5, 1700038 (2017).

Park, H. et al. Mechanical metamaterials with thermoresponsive switching between positive and negative poisson’s ratios. Phys. Status Solidi RRL 12, 1800040 (2018).

Hess, O. et al. Active nanoplasmonic metamaterials. Nat. Mater. 11, 573–584 (2012).

Shadrivov, I., Lapine, M. & Kivshar, Y. Nonlinear, Tunable and Active Metamaterials (Springer, 2014).

Fan, K. & Padilla, W. J. Dynamic electromagnetic metamaterials. Mater. Today 18, 39–50 (2015).

Tong, X. Functional Metamaterials and Metadevices (Springer, 2017).

Rout, S. & Sonkusale, S. Active Metamaterials: Terahertz Modulators and Detectors (Springer, 2017).

Yu, K., Fang, N.-X., Huang, G. & Wang, Q. Magnetoactive acoustic metamaterials. Adv. Mater. 30, 1706348 (2018).

Estep, N. A., S, J., Sounas, D. L. & Alu, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nat. Phys. 10, 923–927 (2014).

Bacot, V., Labousse, M., Eddi, A., Fink, M. & Fort, E. Time reversal and holography with spacetime transformations. Nat. Phys. 12, 972–977 (2016).

Deck-Léger, Z.-L., Akbarzadeh, A. & Caloz, C. Wave deflection and shifted refocusing in a medium modulated by a superluminal rectangular pulse. Phys. Rev. B 97, 104305 (2018).

Halimeh, J. C., Thompson, R. T. & Wegener, M. Invisibility cloaks in relativistic motion. Phys. Rev. A. 93, 013850 (2016).

Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photonics 6, 782–787 (2012).

Nassar, H., Chen, H., Norris, A. N. & Huang, G. L. Quantization of band tilting in modulated phononic crystals. Phys. Rev. B 97, 014305 (2018).

Milton, G. W. & Mattei, O. Field patterns: a new mathematical object. Proc. Roy. Soc. Lond. A 473, 20160819 (2017).

Xu, S. & Wu, C. Space-time crystal and space-time group. Phys. Rev. Lett. 120, 096401 (2018).

Burckel, D. B. et al. Micrometer-scale cubic unit cell 3D metamaterial layers. Adv. Mater. 22, 5053–5057 (2010).

Sreekanth, K. V., Luca, A. D. & Strangi, G. Experimental demonstration of surface and bulk plasmon polaritons in hypergratings. Sci. Rep. 3, 3291 (2013).

Correa, D. et al. Negative stiffness honeycombs for recoverable shock isolation. Rapid Prototyp. J. 21, 193–200 (2015).

Wang, Q. et al. Lightweight mechanical metamaterials with tunable negative thermal expansion. Phys. Rev. Lett. 117, 175901 (2016).

Ozaki, M., Shimoda, Y., Kasano, M. & Yoshino, K. Electric field tuning of the stop band in a liquid-crystal-infiltrated polymer inverse opal. Adv. Mater. 14, 514–518 (2002).

Kamenjicki, M., & Lednev, I. & Asher, S. A. Photoswitchable spirobenzopyran-based photochemically controlled photonic crystals. Adv. Funct. Mater. 15, 1401–1406 (2005).

Cui, T. J., Qi, M. Q., Wan, X., Zhao, J. & Cheng, Q. Coding metamaterials, digital metamaterials and programmable metamaterials. Light Sci. Appl. 3, e218 (2014).

Zhao, J. et al. Controlling spectral energies of all harmonics in programmable way using time-domain digital coding metasurface. Preprint at arXiv https://arxiv.org/abs/1806.04414 (2018).

Gracias, D. H. Stimuli responsive self-folding using thin polymer films. Curr. Opin. Chem. Eng. 2, 112–119 (2013).

Laude, V. et al. Extraordinary nonlinear transmission modulation in a doubly resonant acousto-optical structure. Optica 4, 1245–1250 (2017).

Roy, D., Cambre, J. N. & Sumerlin, B. S. Future perspectives and recent advances in stimuli-responsive materials. Prog. Polym. Sci. 35, 278–301 (2010).

Courjal, N. et al. Acousto-optically tunable lithium niobate photonic crystal. Appl. Phys. Lett. 96, 131103 (2010).

Shin, D. et al. Scalable variable-index elasto-optic metamaterials for macroscopic optical components and devices. Nat. Commun. 8, 16090 (2017).

Babaee, S., Viard, N., Wang, P., Fang, N. X. & Bertoldi, K. Harnessing deformation to switch on and off the propagation of sound. Adv. Mater. 28, 1631–1635 (2016).

Zhang, X., Liu, J., Chu, M. & Chu, B. Flexoelectric piezoelectric metamaterials based on the bending of ferroelectric ceramic wafers. Appl. Phys. Lett. 109, 072903 (2016).

Weissman, J. M., Sunkara, H. B., Tse, A. S. & Asher, S. A. Thermally switchable periodicities and diffraction from mesoscopically ordered materials. Science 274, 959–963 (1996).

Kubo, S. et al. Tunable photonic band gap crystals based on a liquid crystal-infiltrated inverse opal structure. J. Am. Chem. Soc. 126, 8314–8319 (2004).

Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–13 (2012).

Qu, J., Kadic, M., Naber, A. & Wegener, M. Micro-structured two-component 3D metamaterials with negative thermal-expansion coefficient from positive constituents. Sci. Rep. 7, 40643 (2017).

Zhang, H., Guo, X., Wu, J., Fang, D. & Zhang, Y. Soft mechanical metamaterials with unusual swelling behavior and tunable stress-strain curves. Sci. Adv. 4, eaar8535 (2018).

Kamenjicki, M., Ladnev, I. K., Mikhonin, A., Kesavamoorthy, R. & Asher, S.-A. Photochemically controlled photonic crystals. Adv. Funct. Mater. 13, 774–780 (2003).