Abstract
Graphene, a two-dimensional honeycomb lattice of carbon atoms, has been attracting much interest in recent years. Electrons therein behave as massless relativistic particles, giving rise to strikingly unconventional phenomena. Graphene edge states are essential for understanding the electronic properties of this material. However, the coarse or impure nature of the graphene edges hampers the ability to directly probe the edge states. Perhaps the best example is given by the edge states on the bearded edge that have never been observed—because such an edge is unstable in graphene. Here, we use the optical equivalent of graphene—a photonic honeycomb lattice—to study the edge states and their properties. We directly image the edge states on both the zigzag and bearded edges of this photonic graphene, measure their dispersion properties, and most importantly, find a new type of edge state: one residing on the bearded edge that has never been predicted or observed. This edge state lies near the Van Hove singularity in the edge band structure and can be classified as a Tamm-like state lacking any surface defect. The mechanism underlying its formation may counterintuitively appear in other crystalline systems.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).
Beenakker, C. W. J. Colloquium: Andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys. 80, 1337–1354 (2008).
Zhang, Y., Tan, Y-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
Guinea, F., Katsnelson, M. I. & Geim, A. K. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nature Phys. 6, 30–33 (2010).
Montambaux, G., Piéchon, F., Fuchs, J-N. & Goerbig, M. O. Merging of Dirac points in a two-dimensional crystal. Phys. Rev. B 80, 153412 (2009).
Fefferman, C. & Weinstein, M. Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25, 1169–1220 (2012).
Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G. & Esslinger, T. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012).
Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C. Designer Dirac fermions and topological phases in molecular graphene. Nature 483, 306–310 (2012).
Wertz, E. et al. Spontaneous formation and optical manipulation of extended polariton condensates. Nature Phys. 6, 860–864 (2010).
Goldman, N., Beugnon, J. & Gerbier, F. Detecting chiral edge states in the hofstadter optical lattice. Phys. Rev. Lett. 108, 255303 (2012).
Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).
Kuhl, U. et al. Dirac point and edge states in a microwave realization of tight-binding graphene-like structures. Phys. Rev. B 82, 094308 (2010).
Bellec, M., Kuhl, U., Montambaux, G. & Mortessagne, F. Topological transition of Dirac points in a microwave experiment. Phys. Rev. Lett. 110, 033902 (2013).
Peleg, O. et al. Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett. 98, 103901 (2007).
Sepkhanov, R. A., Bazaliy, Y. B. & Beenakker, C. W. J. Extremal transmission at the Dirac point of a photonic band structure. Phys. Rev. A 75, 063813 (2007).
Bahat-Treidel, O., Peleg, O. & Segev, M. Symmetry breaking in honeycomb photonic lattices. Opt. Lett. 33, 2251–2253 (2008).
Polini, M., Guinea, F., Lewenstein, M., Manoharan, H. C. & Pellegrini, V. Artificial honeycomb lattices for electrons, atoms and photons. Nature Nanotech. 8, 625–633 (2013).
Bahat-Treidel, O. et al. Klein tunneling in deformed honeycomb lattices. Phys. Rev. Lett. 104, 063901 (2010).
Ablowitz, M. J., Nixon, S. D. & Zhu, Y. Conical diffraction in honeycomb lattices. Phys. Rev. A 79, 053830 (2009).
Bartal, G. et al. Brillouin zone spectroscopy of nonlinear photonic lattices. Phys. Rev. Lett. 94, 163902 (2005).
Eisenberg, H. S., Silberberg, Y., Morandotti, R., Boyd, A. R. & Aitchison, J. S. Discrete spatial optical solitons in waveguide arrays. Phys. Rev. Lett. 81, 3383–3386 (1998).
Fujita, M., Wakabayashi, K., Nakada, K. & Kusakabe, K. Peculiar localized state at zigzag graphite edge. J. Phys. Soc. Jpn 65, 1920–1923 (1996).
Özyilmaz, B. et al. Electronic transport and quantum Hall effect in bipolar graphene p–n–p junctions. Phys. Rev. Lett. 99, 166804 (2007).
Ritter, K. A. & Lyding, J. W. The influence of edge structure on the electronic properties of graphene quantum dots and nanoribbons. Nature Mater. 8, 235–242 (2009).
Yao, W., Yang, S. A. & Niu, Q. Edge states in graphene: From gapped flat-band to gapless chiral modes. Phys. Rev. Lett. 102, 096801 (2009).
Kohmoto, M. & Hasegawa, Y. Zero modes and edge states of the honeycomb lattice. Phys. Rev. B 76, 205402 (2007).
Kobayashi, Y., Fukui, K., Enoki, T., Kusakabe, K. & Kaburagi, Y. Observation of zigzag and armchair edges of graphite using scanning tunneling microscopy and spectroscopy. Phys. Rev. B 71, 193406 (2005).
Tao, C. et al. Spatially resolving edge states of chiral graphene nanoribbons. Nature Phys. 7, 616–620 (2011).
Klein, D. J. Graphitic polymer strips with edge states. Chem. Phys. Lett. 217, 261–265 (1994).
Efremidis, N. K., Sears, S., Christodoulides, D. N., Fleischer, J. W. & Segev, M. Discrete solitons in photorefractive optically induced photonic lattices. Phys. Rev. E 66, 046602 (2002).
Szameit, A. et al. Discrete nonlinear localization in femtosecond laser written waveguides in fused silica. Opt. Express 13, 10552–10557 (2005).
Tamm, I. E. On the possible bound states of electrons on a crystal surface. Phys. Z. Sowjetunion 1, 733–735 (1932).
Shockley, W. On the surface states associated with a periodic potential. Phys. Rev. 56, 317–323 (1939).
Lederer, F. et al. Discrete solitons in optics. Phys. Rep. 463, 1–126 (2008).
Fleischer, J. W., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422, 147–150 (2003).
Malkova, N., Hromada, I., Wang, X., Bryant, G. & Chen, Z. Observation of optical Shockley-like surface states in photonic superlattices. Opt. Lett. 34, 1633–1635 (2009).
Garanovich, I. L., Sukhorukov, A. A. & Kivshar, Y. S. Defect free surface state in modulated photonic lattices. Phys. Rev. Lett. 100, 203904 (2008).
Szameit, A. et al. Observation of defect-free surface modes in optical waveguide arrays. Phys. Rev. Lett. 101, 203902 (2008).
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).
Acknowledgements
The Technion team is part of the Israeli Center of Research Excellence ‘Circle of Light’ supported by the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation. M.C.R. is grateful to the Azrieli foundation for the Azrieli fellowship. This research was financially supported by an Advanced Grant from the European Research Council; the Israel Science Foundation; the USA-Israel Binational Science Foundation; the German Ministry of Education and Science (ZIK 03Z1HN31); the 973 Programs (2013CB328702, 2013CB632703) and the Program for Changjiang Scholars and Innovative Research Teams in China; and by the NSF and AFOSR in the USA.
Author information
Authors and Affiliations
Contributions
All authors contributed significantly to this work.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Information (PDF 628 kb)
Rights and permissions
About this article
Cite this article
Plotnik, Y., Rechtsman, M., Song, D. et al. Observation of unconventional edge states in ‘photonic graphene’. Nature Mater 13, 57–62 (2014). https://doi.org/10.1038/nmat3783
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nmat3783
This article is cited by
-
The higher-order topological pumping explored in the 2D acoustic crystal
Science China Physics, Mechanics & Astronomy (2024)
-
Steady-state Peierls transition in nanotube quantum simulator
npj Quantum Information (2023)
-
Two-dimensional Shiba lattices as a possible platform for crystalline topological superconductivity
Nature Physics (2023)
-
Spin-dependent properties of optical modes guided by adiabatic trapping potentials in photonic Dirac metasurfaces
Nature Nanotechnology (2023)
-
Photonic graphene with reconfigurable geometric structures in coherent atomic ensembles
Frontiers of Physics (2023)