topological matter experimental observation of weyl points · topological matter experimental...

TOPOLOGICAL MATTER

Experimental observationof Weyl pointsLing Lu,1* Zhiyu Wang,2 Dexin Ye,2 Lixin Ran,2 Liang Fu,1

John D. Joannopoulos,1 Marin Soljačić1

The massless solutions to the Dirac equation are described by the so-called WeylHamiltonian. The Weyl equation requires a particle to have linear dispersion in all threedimensions while being doubly degenerate at a single momentum point. These Weylpoints are topological monopoles of quantized Berry flux exhibiting numerous unusualproperties. We performed angle-resolved microwave transmission measurements througha double-gyroid photonic crystal with inversion-breaking where Weyl points have beentheoretically predicted to occur. The excited bulk states show two linear dispersion bandstouching at four isolated points in the three-dimensional Brillouin zone, indicating theobservation of Weyl points. This work paves the way to a variety of photonic topologicalphenomena in three dimensions.

In 1929, Hermann Weyl derived (1) the mass-less solutions to the Dirac equation—the rel-ativistic wave equation for electrons. Neutrinoswere thought, for decades, to be Weyl fer-mions until the discovery of the neutrino

mass. Moreover, it has been suggested that low-energy excitations in condensed matter (2–12)can be the solutions to the Weyl HamiltonianHðkÞ ¼ vxkxsx þ vykysy þ vzkzsz , wherevi andki are the group velocities and momenta and siare the Pauli matrices. Recently, photons havealso been proposed to emerge as Weyl particlesinside photonic crystals (13). In all cases, twolinear dispersion bands in three-dimensional (3D)momentum space intersect at a single degen-erate point—the Weyl point. Weyl points are 3Dextensions of the 2D Dirac cones, as in graphene,possessing unique density of states and transportproperties (14).Notably,Weyl points aremonopolesof Berry flux with topological charges defined bythe Chern numbers (4, 5). These topological in-variants enablematerials containingWeyl pointsto exhibit a variety of unusual phenomena, in-cluding topological surface states (15), chiralanomaly (16), quantum anomalous Hall effect(17), and others (18, 19). Furthermore, Weyl pointsare the intermediate topological gapless phasesbetween topologically distinct bandgapmaterials(20), facilitating the search for other topologicalphases in 3D. In this work, we report on the ob-servation of Weyl points in an inversion-breaking3D double-gyroid (DG) photonic crystal withoutbreaking time-reversal symmetry.Weyl points are sources of quantized Berry

flux in the momentum space (see the supple-mentary text). Their charges can be defined bythe corresponding Chern numbers of ±1 (Fig.1A). So, Weyl points robustly appear in pairs

and can only be removed through pair annihi-lation. Because the Berry curvature is strictlyzero under PT symmetry—the product of parity(P, inversion) and time-reversal symmetry (T )—isolated Weyl points only exist when at least oneof P or T is broken. In (13), frequency-isolatedWeyl points were predicted in PT -breaking DGphotonic crystals. In our experiment, we choseto break P instead of T to avoid using lossymagnetic materials and external magneticfields. This choice also allows our approach to bedirectly extended to photonic crystals at opticalwavelengths.In Fig. 1B, we plotted the body-centered-cubic

(bcc) Brillouin zone (BZ) of the P-breaking DGshown in Fig. 1C. When T is preserved, theremust exist even pairs ofWeyl points (13). The twopairs of Weyl points illustrated in the (101) planeof the BZ, in Fig. 1B, are thus the minimum num-ber of Weyl points possible. The band structureplotted in Fig. 1D shows two linear band crossingsalong G –N and G –H; the other twoWeyl pointshave identical dispersions due to T . The fourWeylpoints are isolated in frequency andwell separatedin the momentum space, making their charac-terization easier.Wework at themicrowave frequencies around

10 GHz, where fabrication of 3D photonic crys-tals ismore accessible. Additive processes such as3D printing cannot yet fulfill the material re-quirement of having low-loss dielectrics withhigh-dielectric constants. To fabricate two inter-penetrating gyroids with a subtractive fabrica-tion process, we open up each gyroid network bylayers along the [101] direction with equal thick-ness of a=

ffiffiffi

2p

. Here, a is the cubic lattice con-stant. The cylindrical defects are introduced ineach layer of the red gyroid in Fig. 1C.We approximate each gyroid network by three

sets of hole drillings, as illustrated in Fig. 2A.Similar methods of drilling and angled etchinghave been used in the fabrication of 3D photoniccrystals in the microwave (21) and near-infraredwavelengths (22). The three cylindrical air holes

of the blue gyroid, along x, y, and z, go through(0,1/4,0)a, (0,0,1/4)a, and (1/4,0,0)a, respectively.All air holes have a diameter of 0.54a. Gyroidsapproximated by this drilling approach havealmost identical band structures as those definedby the level-set isosurfaces in (13).The second (red) gyroid is the inversion count-

erpart of the blue gyroid. To break P, we shrinkthe red gyroid at (1/4,–1/8,1/2)a to be a cylinderoriented along [101]. Shown in Fig. 2B, the defectcylinder has a diameter of 0.1a and height of0.2a. We separate the red gyroid by cutting (101)planes at the centers of the defect cylinders. Theblue gyroid, without introduced defects, is sepa-rated at the corresponding position of (–1/4,1/8,–1/2)a. Each layer is a=

ffiffiffi

2p

thick. The separatedlayers of each gyroid are identical up to trans-lations of a2 (the bcc lattice vector in Fig. 1C). Oneunit vertical period of DG consists of two layersfrom each gyroid (Fig. 2B).The materials of choice are slabs of ceramic-

filled plastics (C-STOCK AK, Cuming Micro-wave Corp., MA) of dielectric constant 16 andloss tangent 0.01. Each slab has a thickness of9.5mm (¼ a=

ffiffiffi

2p

). Both thewidth and the lengthof the slabs are 304 mm (~12 inches). The mate-rial hardness is adjusted between 80 and 90 ona Shore D gauge to be machined by carbidetools without cracking. Each layer experiencedaround 700 drillings along ±45° away from itssurface normal (in x and z) and about 40 ball-end millings in-plane (in y) on both top andbottom surfaces. These operations were per-formed using a computer-numerical-controlledthree-axis vertical milling machine (fig. S1A). Al-though all slabs of the same gyroid can in prin-ciple be machined together, we processed nomore than two layers at the same time to leavea solid frame for handling. About 20 mm wereleft undrilled from each of the four edges of theslabs. We subsequently cut two sides of theframes of each gyroid for their assembly (fig.S1B). Figure 2B illustrates that the layers of thetwo gyroids were offset vertically by half of theslab thickness (4.25 mm). A picture of the as-sembled sample is shown in Fig. 2C (more pic-tures are in fig. S1, C and D). The final samplein measurements consists of a total of 20 layers,with 10 layers from each gyroid stacked in alter-nating order (23).We performed angle-resolved transmission

measurements on the photonic crystal sampleto probe the dispersions of the 3D bulk states(24–26). The schematic of the experimentalsetup is shown in Fig. 3A (also see fig. S1E). Apair of lens antennas were placed on the twosides of the sample. Transmission amplitudes(S-parameter S21) were recorded by the networkanalyzer. The half-power beamwidths (divergentangles) of the lens antennas were 9°. The colli-mated beam impinged on the sample (101) sur-face, in which the incident angle is varied byrotating the sample around the vertical axis. Asillustrated in Fig. 3B, the component of the inci-dent wave vector parallel to the sample surface[k0sin(q)] is conserved up to a reciprocal latticevector of the sample surface, due to the discrete

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1Department of Physics, Massachusetts Institute ofTechnology (MIT), Cambridge, MA 02139, USA. 2Laboratoryof Applied Research on Electromagnetics (ARE), ZhejiangUniversity, Hangzhou, Zhejiang 310027, China.*Corresponding author. E-mail: linglu@mit.edu

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translational symmetry. All the bulk states ofthe same wave-vector projection (dashed linein Fig. 3B) and frequency could be excited inthe bulk and exit the sample in the samedirection as the incident beam. The exitingsignal was collected by the antenna at the op-posite side. A pair of prisms was used to in-crease the magnitudes of the incident wavevectors for sampling a larger volume in the BZ.We placed the angled prisms (12.4°) in contactwith the opposite surfaces of the sample (Fig.3A); the prisms are made of the same materialas the sample (e = 16).

Both the source and receiving antennas werelinearly polarized. To account for the complexpolarization response in the 3D P-breaking pho-tonic crystal, we summed up the transmittedpower of different polarizations to resolve morebulk states in the sample. Illustrated in Fig. 3A,the electric-field directions of the s and p polar-izations are vertical and parallel to the incidentplane, respectively. We first set the source an-tenna to be in one of the polarization states (e.g.,s) and measured the transmitted power twiceby setting the receiving antenna to be s and ppolarized, respectively. The sum of the two trans-

mitted signals was normalized to the transmis-sion signal without the sample. Second, we setthe source antenna to be the other polarization(e.g., p) and repeated the above procedure. Last,we summed the transmission power for bothpolarizations and plotted it in Fig. 3C. For details,see figs. S2 and S3.We mapped out all 3D bulk states, projected

along [101], by varying q and a as shown in Fig. 3,A and B. Six sets of transmission data of rep-resentative directions are plotted in Fig. 3C.The transmission data stop on the right slantedboundary, which corresponds to the maximumrotation angle in q. Close to this boundary, thetransmission intensity is low due to the smallereffective cross section of the samples at large an-gles. For comparison, the corresponding bandstructures (projected along the transmission di-rection [101]) are plotted as figure insets. In prin-ciple, all states in the projected band structurecan be observed in the transmission data, wherethe transmission amplitudes are proportional tothe bulk density of states. However, the couplingand transmission efficiencies depend on the de-tails of the Bloch mode polarizations, modal sym-metries, radiation lifetimes, finite size effects,group velocities, and Berry-curvature-inducedanomalous velocities. Although the signal strengthvaries in some areas in the experimental data, allthe transmission responses (Fig. 3C) comparevery well and are consistent with the theoreticalband structures in the insets.When a = 90° (in Fig. 3), the beam scans

through the upper Weyl point along G – H, rep-resented by a magenta sphere in Fig. 3B. Theoutline of the transmission intensity in Fig. 3Cclearly shows a linear point crossing around11.3 GHz in frequency and close to 0.45(2p/a) inwave vector. As a increases from 90° to 105° and120°,we aremoving off theWeyl point andobservethe opening of a small gap. This gap opening isexpected for a point degeneracy and excludes thepossibility of the state being associated with a linedegeneracy [line node (13)]. Because photons donot have spin or polarization degeneracies in ourcrystal, every band dispersion represents a singlestate, excluding the possibility of being a four-fold3DDirac degeneracy (27). Due to the aspect ratio ofour sample,wewere unable to scan the dispersions

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Fig. 1. Weyl pointsin the BZ of aP-breaking gyroidphotonic crystal.(A) Weyl points aremonopoles of Berryflux in the momentumspace. (B) The BZ ofthe DG photoniccrystal in (C). The(101) surfaces arehighlighted in green.Four Weyl points areillustrated on the green (101) plane along G – H and G – N, where H = (0,1,0)2p/a and N = (–1/2,0,1/2)2p/a. (C) The bcc cell of the DG with a P-breakingcylindrical defect on the red gyroid, where a1 = (–1,1,1)a/2, a2 = (1,–1,1)a/2, and a3 = (1,1,–1)a/2. (D) The photon dispersions are plotted along N – G – H.The Weyl points are the linear band touchings between the fourth and fifth bands.

ˆˆ

ˆ

Fig. 2. Fabrication of gyroids by drilling and stacking layers. (A) Illustration in a bcc unit cell that asingle gyroid structure can be approximated by drilling periodic air holes along the x, y, and z directions.(B) The DG structure can be made by stacking layers along the [101] direction.The red and blue gyroids,being inversion counterparts, interpenetrate each other. We shrink the vertical connections of the redgyroid to thin cylinders in order to break P. (C) (Left) A total of 20 layers were stacked. (Right) A zoom-in view from the top, with a ruler (in centimeters) in the background.

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along the [101] transmissiondirection.However, ourtransmission data do contain all the contributionsfrom those dispersions projected together. For ex-ample, the resonance lines below and above theWeyl point, in Fig. 3C (a = 90°), indicate the gapopening along the [101] direction as well. Similarly,the otherWeyl point, in cyan, on the right of theG –N axis, is studied by orienting a to be 0°, 15°, and30°.Although the transmission intensity of theupperbulk bands is not as prominent, the Weyl pointdispersions can still be inferred. The remainingtwoWeyl points, at the opposite k locations, relateto the two measured Weyl points by T . They havethe same projected band dispersions and the sametransmission pattern as the data shown in Fig. 3C.The photonic Weyl points observed in our ex-

periment pave the way to topological photonics(28–31) in 3D, where 3DDirac points (27, 32) andvarious gapped topological phases (33, 34) canbe accessed. Similar approaches can be readilyadopted to observe Weyl points at optical fre-quencies using 3D nanofabrication (35–37). Thesurface states, not seen in our transmission ex-periments, could potentially be characterizedthrough reflection spectra or evanescent-wavemeasurements. Close to the Weyl-point frequen-cies, photonic Weyl materials provide angularselectivity (38) for filtering light from any 3D in-cident angle. The unique density of states at theWeyl point can potentially enable devices such ashigh-power single-mode lasers (39).

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(2013).19. P. Hosur, X. Qi, C. R. Phys. 14, 857–870 (2013).20. S. Murakami, New J. Phys. 9, 356 (2007).21. E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67,

2295–2298 (1991).22. S. Takahashi et al., Nat. Mater. 8, 721–725 (2009).23. L. Lu et al., Opt. Lett. 37, 4726–4728 (2012).24. W. M. Robertson et al., Phys. Rev. Lett. 68, 2023–2026 (1992).25. E. Özbay et al., Phys. Rev. B 50, 1945–1948 (1994).26. C. Pouya, P. Vukusic, Interface Focus 2, 645–650 (2012).27. S. M. Young et al., Phys. Rev. Lett. 108, 140405 (2012).28. L. Lu, J. D. Joannopoulos, M. Soljačić, Nat. Photonics 8,

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ACKNOWLEDGMENTS

We thank A. Gallant and E. Johnson at the MIT central machineshop for machining the gyroid layers; Y. Shen, B. Zhang, J. Liu,B. Zhen, B. Wang, A. Vishwanath, H. Chen, Q. Yan, and C. Fangfor discussions; and P. Rebusco for critical reading and editingof the manuscript. J.D.J. was supported in part by the U.S. ArmyResearch Office through the Institute for Soldier Nanotechnologiesunder contract W911NF-13-D-0001. L.F. was supported by the U.S.Department of Energy (DOE) Office of Basic Energy Sciences,Division of Materials Sciences and Engineering, under awardDE-SC0010526. L.L. was supported in part by the MaterialsResearch Science and Engineering Center Program of the NSFunder award DMR-1419807. M.S. and L.L. (analysis and reading ofthe manuscript) were supported in part by the MIT Solid-StateSolar-Thermal Energy Conversion Center and Energy FrontierResearch Center of DOE under grantDE-SC0001299. Z.W, D.Y., and L.R. were supported by the ChineseNational Science Foundation (CNSF) under grants 61401395,61401393, and 61131002, respectively.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/349/6248/622/suppl/DC1Supplementary TextFigs. S1 to S3

15 February 2015; accepted 6 July 2015Published online 16 July 201510.1126/science.aaa9273

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Fig. 3. Angle-resolved transmission measurements. (A) Schematic of the microwave transmission setup. (B) The bulk states of the k vectors on thedashed line can be excited by the incident wave at incidence angle q. (C) Transmission data, summed over both s and p polarizations, as the sample isrotated along the [101] axis by the angle a. The insets show the calculated band structures projected along [101]; they are scaled to the same range andratio as the measured data.

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Experimental observation of Weyl pointsLing Lu, Zhiyu Wang, Dexin Ye, Lixin Ran, Liang Fu, John D. Joannopoulos and Marin Soljacic

originally published online July 16, 2015DOI: 10.1126/science.aaa9273 (6248), 622-624.349Science 

, this issue p. 613 and 622Sciencein a photonic crystal. The observation of Weyl physics may enable the discovery of exotic fundamental phenomena.

detected the characteristic Weyl pointset al.semimetal capable of hosting Weyl fermions. In a complementary study, Lu used photoemission spectroscopy to identify TaAs as a Weylet al.conduction bands meet at discrete ''Weyl points.'' Xu

Weyl semimetals. These materials have an unusual band structure in which the linearly dispersing valence andAlthough not yet observed among elementary particles, Weyl fermions may exist as collective excitations in so-called

were once mistakenly thought to describe neutrinos.−−massless particles with half-integer spin−−Weyl fermionsWeyl physics emerges in the laboratory

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