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.

Designing perturbative metamaterials from discrete models

Abstract

Identifying material geometries that lead to metamaterials with desired functionalities presents a challenge for the field. Discrete, or reduced-order, models provide a concise description of complex phenomena, such as negative refraction, or topological surface states; therefore, the combination of geometric building blocks to replicate discrete models presenting the desired features represents a promising approach. However, there is no reliable way to solve such an inverse problem. Here, we introduce ‘perturbative metamaterials’, a class of metamaterials consisting of weakly interacting unit cells. The weak interaction allows us to associate each element of the discrete model with individual geometric features of the metamaterial, thereby enabling a systematic design process. We demonstrate our approach by designing two-dimensional elastic metamaterials that realize Veselago lenses, zero-dispersion bands and topological surface phonons. While our selected examples are within the mechanical domain, the same design principle can be applied to acoustic, thermal and photonic metamaterials composed of weakly interacting unit cells.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Design method concept.
Fig. 2: Veselago lens metamaterial example.
Fig. 3: Zero-group-velocity metamaterial.
Fig. 4: Topological insulator metamaterial.

References

  1. 1.

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

    Article  Google Scholar 

  2. 2.

    Grbic, A. & Eleftheriades, G. V. Overcoming the diffraction limit with a planar left-handed transmission-line lens. Phys. Rev. Lett. 92, 117403 (2004).

    Article  Google Scholar 

  3. 3.

    Liu, Z., Lee, H., Xiong, Y., Sun, C. & Zhang, X. Far-field optical hyperlens magnifying sub-diffraction-limited objects. Science 315, 1686 (2007).

    Article  Google Scholar 

  4. 4.

    Li, J., Fok, L., Yin, X., Bartal, G. & Zhang, X. Experimental demonstration of an acoustic magnifying hyperlens. Nat. Mater. 8, 931–934 (2009).

    Article  Google Scholar 

  5. 5.

    Kaina, N., Lemoult, F., Fink, M. & Lerosey, G. Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials. Nature 525, 77–81 (2015).

    Article  Google Scholar 

  6. 6.

    Silva, A. et al. Performing mathematical operations with metamaterials. Science 343, 160–164 (2014).

    Article  Google Scholar 

  7. 7.

    Bückmann, T., Thiel, M., Kadic, M., Schittny, R. & Wegener, M. An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun. 5, 4130 (2014).

    Article  Google Scholar 

  8. 8.

    Schurig, D. et al. Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006).

    Article  Google Scholar 

  9. 9.

    Tretyakov, S. Analytical Modeling in Applied Electromagnetics (Artech House, Boston, MA, USA, 2003).

  10. 10.

    Werner, D. H. & Kwon, D.-H. Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, London, UK, 2014).

  11. 11.

    Sihvola, A. H. Electromagnetic Mixing Formulas and Applications (The Institution of Engineering and Technology, London, UK, 1999).

  12. 12.

    Gibson, W. C. The Method of Moments in Electromagnetics (Chapman and Hall/CRC, Boca Raton, FL, USA, 2007).

  13. 13.

    Hao, Y. & Mittra, R. FDTD Modeling of Metamaterials: Theory and Applications (Artech House, Boston, MA, USA, 2008).

  14. 14.

    Monk, P. Finite Element Methods for Maxwell’s Equations (Clarendon Press, Oxford, UK, 2003).

  15. 15.

    Caloz, C. & Itoh, T. Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, Hoboken, NJ, USA, 2005).

  16. 16.

    Volakis, J. L., Sertel, K. & Usner, B. C. Frequency Domain Hybrid Finite Element Methods for Electromagnetics (Morgan & Claypool, San Rafael, CA, USA, 2006).

  17. 17.

    Reis, P. M., Jaeger, H. M. & van Hecke, M. Designer matter: a perspective. Extrem. Mech. Lett. 5, 25–29 (2015).

    Article  Google Scholar 

  18. 18.

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

    Article  Google Scholar 

  19. 19.

    Kalinin, S. V., Sumpter, B. G. & Archibald, R. K. Big–deep–smart data in imaging for guiding materials design. Nat. Mater. 14, 973–980 (2015).

    Article  Google Scholar 

  20. 20.

    Schumacher, C. et al. Microstructures to control elasticity in 3D printing. ACM Trans. Graph. 34, 136 (2015).

    Article  Google Scholar 

  21. 21.

    Sun, Y., Edwards, B., Alù, A. & Engheta, N. Experimental realization of optical lumped nanocircuits at infrared wavelengths. Nat. Mater. 11, 208–212 (2012).

    Article  Google Scholar 

  22. 22.

    Li, Y., Liberal, I., Giovampaola, C. D. & Engheta, N. Waveguide metatronics: lumped circuitry based on structural dispersion. Sci. Adv. 2, e1501790 (2016).

    Article  Google Scholar 

  23. 23.

    Zhang, S., Xia, C. & Fang, N. Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106, 24301 (2011).

    Article  Google Scholar 

  24. 24.

    Schrieffer, J. R. & Wolff, P. A. Relation between the Anderson and Kondo Hamiltonians. Phys. Rev. 149, 491–492 (1966).

    Article  Google Scholar 

  25. 25.

    Bravyi, S., DiVincenzo, D. P. & Loss, D. Schrieffer–Wolff transformation for quantum many-body systems. Ann. Phys. 326, 2793–2826 (2011).

    Article  Google Scholar 

  26. 26.

    Winkler, R. Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Appendix B). Springer Tracts in Modern Physics, Vol. 191 (Springer-Verlag, Berlin, Germany, 2003).

  27. 27.

    Wagner, M. Unitary Transformations in Solid State Physics (North Holland, Amsterdam, 1986).

    Google Scholar 

  28. 28.

    Nemat-Nasser, S. & Srivastava, A. Negative effective dynamic mass-density and stiffness: Micro-architecture and phononic transport in periodic composites. AIP Adv. 1, 1–10 (2011).

    Article  Google Scholar 

  29. 29.

    Veselago, V. G. The electrodynamics of substances with simultaneously negative values of ϵ and μ. Sov. Phys. Uspekhi 10, 509–514 (1968).

    Article  Google Scholar 

  30. 30.

    Chalker, J. T., Pickles, T. S. & Shukla, P. Anderson localization in tight-binding models with flat bands. Phys. Rev. B 82, 1–5 (2010).

    Article  Google Scholar 

  31. 31.

    Mukherjee, S. et al. Observation of a localized flat-band state in a photonic Lieb lattice. Phys. Rev. Lett. 114, 245504 (2015).

    Article  Google Scholar 

  32. 32.

    Vicencio, R. A. et al. Observation of localized states in Lieb photonic lattices. Phys. Rev. Lett. 114, 1–5 (2015).

    Article  Google Scholar 

  33. 33.

    Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  Google Scholar 

  34. 34.

    Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  Google Scholar 

  35. 35.

    Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  Google Scholar 

  36. 36.

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

    Article  Google Scholar 

  37. 37.

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

    Article  Google Scholar 

  38. 38.

    Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 1–5 (2015).

    Google Scholar 

  39. 39.

    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).

    Article  Google Scholar 

  40. 40.

    Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 19 (2015).

    Article  Google Scholar 

  41. 41.

    Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alù, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

    Article  Google Scholar 

  42. 42.

    McHugh, S. Topological insulator realized with piezoelectric resonators. Phys. Rev. Appl. 6, 14008 (2016).

    Article  Google Scholar 

  43. 43.

    Pal, R. K., Schaeffer, M. & Ruzzene, M. Helical edge states and topological phase transitions in phononic systems using bi-layered lattices. J. Appl. Phys. 119, 84305 (2016).

    Article  Google Scholar 

  44. 44.

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

    Article  Google Scholar 

  45. 45.

    Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature https://doi.org/10.1038/nature25156 (2018).

  46. 46.

    Lai, A. & Itoh, T. Composite right/left-handed transmission line metamaterials. IEEE Microw. Mag. 5, 34–50 (2004).

    Article  Google Scholar 

  47. 47.

    Semouchkina, E. A., Semouchkin, G. B., Lanagan, M. & Randall, C. A. FDTD study of resonance processes in metamaterials. IEEE Trans. Microw. Theory Tech. 53, 1477–1487 (2005).

    Article  Google Scholar 

  48. 48.

    Sihvola, A. Homogenization principles and effect of mixing on dielectric behavior. Photon. Nanostruct. 11, 364–373 (2013).

    Article  Google Scholar 

  49. 49.

    La Spada, L. et al. Surface wave cloak from graded refractive index nanocomposites. Sci. Rep. 6, 1–8 (2016).

    Article  Google Scholar 

  50. 50.

    Lin, Z., Pick, A., Lončar, M. & Rodriguez, A. W. Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals. Phys. Rev. Lett. 117, 107402 (2016).

    Article  Google Scholar 

  51. 51.

    Petyt, M. Introduction to Finite Element Vibration Analysis (Cambridge University Press, New York, NY, USA, 2015).

Download references

Acknowledgements

This work was partially supported by the ETH Postdoctoral Fellowship to K.H.M., and by the Swiss National Science Foundation. The authors would like to thank R. Süsstrunk and O. Bilal for discussions.

Author information

Affiliations

Authors

Contributions

K.H.M. and M.S.G. performed the optimization and simulation of the materials. K.H.M. wrote the manuscript. M.S.G. proposed the reduction approach and optimization algorithm. A.P. designed and performed the dynamic condensation. S.H. selected and interpreted the reduced-order models. C.D. provided guidance during all stages of the project. All authors contributed to the discussion and interpretation of the results, and to the editing of the manuscript.

Corresponding author

Correspondence to Marc Serra-Garcia.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Information.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Matlack, K.H., Serra-Garcia, M., Palermo, A. et al. Designing perturbative metamaterials from discrete models. Nature Mater 17, 323–328 (2018). https://doi.org/10.1038/s41563-017-0003-3

Download citation

Further reading

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