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Long-Hua Wu, [Xiao Hu](https://orcid.org/0000-0001-6880-402X)

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[Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material](https://mdr.nims.go.jp/datasets/7d00c82d-8938-494e-924c-d06143a6184e)

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untitledScheme for Achieving a Topological Photonic Crystal by Using Dielectric MaterialLong-Hua Wu and Xiao Hu*International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science,Tsukuba 305-0044, JapanGraduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan(Received 10 February 2015; published 3 June 2015)We derive in the present work topological photonic states purely based on conventional dielectricmaterial by deforming a honeycomb lattice of cylinders into a triangular lattice of cylinder hexagons.The photonic topology is associated with a pseudo-time-reversal (TR) symmetry constituted by the TRsymmetry supported in general by Maxwell equations and the C6 crystal symmetry upon design, whichrenders the Kramers doubling in the present photonic system. It is shown explicitly for the transversemagnetic mode that the role of pseudospin is played by the angular momentum of the wave function of theout-of-plane electric field. We solve Maxwell equations and demonstrate the new photonic topology byrevealing pseudospin-resolved Berry curvatures of photonic bands and helical edge states characterizedby Poynting vectors.DOI: 10.1103/PhysRevLett.114.223901 PACS numbers: 42.70.Qs, 03.65.Vf, 73.43.-fIntroduction.—The discovery of the quantum Hall effect(QHE) opened a new chapter of condensed matter physicswith topology as the central concept [1–11]. Topologicalstates are not only interesting from an academic point ofview, but also expected to yield significant impacts toapplications because robust surface (or edge) states pro-tected by bulk topology provide possibilities for spintronicsand quantum computation [12–17]. However, electronicsystems with nontrivial topology confirmed so far are stilllimited in number, and most of them exhibit topologicalproperties only at very low temperatures, which hinderstheir better understanding and manipulation indispensablefor practical applications.Photonic crystals are analogues of conventional crystalswith the atomic lattice replaced by a medium of periodicelectric permittivity and/or magnetic permeability [18].Metamaterials are designed to generate electromagnetic(EM) properties such as negative index, magnetic lens, andso on, which are not available in nature [19]. Recently, ithas been recognized that topological states characterizedby unique edge propagations of an EM wave can berealized in photonic crystals based on gyromagnetic mate-rials under external magnetic field, bi-anisoctroic meta-materials with coupled electric and magnetic fields wherebi-anisotropy acts as effective spin-orbit coupling, andcoupled resonator optical waveguides (CROWs) [20–31](for a review see Ref. [32]).In the present work, we propose a two-dimensional (2D)photonic crystal purely made of conventional dielectricmaterial. We notice that a honeycomb lattice is equivalentto a triangular lattice of hexagonal clusters composed by sixneighboring sites, and that, taking this larger hexagonal unitcell instead of the primitive rhombic unit cell of two sites(see Fig. 1), the Dirac cones at K and K0 points in the firstBrillouin zone of honeycomb lattice are folded to doublydegenerate Dirac cones at the Γ point. It is then intriguingto observe that at the Γ point there are two 2D irreduciblerepresentations in the C6 symmetry group associated withodd and even parities respective to spatial inversion oper-ation. Based on these properties, we propose opening atopologically nontrivial band gap by deforming the honey-comb lattice in a way that keeps the hexagonal clusters andpreserves the C6 symmetry (see Fig. 1). Solving Maxwellequations, we reveal explicitly that harmonic transversemagnetic (TM) modes hosted by the hexagonal cluster,working as “artificial atom” in the present scheme, exhibitelectronic orbital-like p- and d-wave shapes and formphotonic bands. We clarify that there is a pseudo-time-reversal (TR) symmetry constituted by the TR symmetryrespected by Maxwell equations and the C6 crystalFIG. 1 (color online). Schematic plot of a triangular photoniccrystal of “artificial atoms” composed by six cylinders ofdielectric material. Dark gray (Red) dashed rhombus and hexagonare primitive cells of honeycomb and triangular lattices. The solidblack hexagon labels an artificial atom, while the dashed blackone marks the interstitial region among artificial atoms. ~a1 and ~a2are unit vectors with length a0 as the lattice constant. Right panel:enlarged view of a hexagonal cluster with R the length of thehexagon edge and d the diameter of cylinders. εd and εA aredielectric constants of cylinders and surrounding environment.PRL 114, 223901 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending5 JUNE 20150031-9007=15=114(22)=223901(5) 223901-1 © 2015 American Physical Societyhttp://dx.doi.org/10.1103/PhysRevLett.114.223901http://dx.doi.org/10.1103/PhysRevLett.114.223901http://dx.doi.org/10.1103/PhysRevLett.114.223901http://dx.doi.org/10.1103/PhysRevLett.114.223901symmetry upon design, which behaves in the same way asTR symmetry in electronic systems and renders the Kramersdoubling in the present photonic system. This intimatelygives the correspondence between the positive and negativeangular momenta of the wave function of the out-of-planeelectric field and the up and down spins of the electron.Evaluating the Berry curvatures of bulk photonic bands andthe edge states for finite systems, we demonstrate theemergence of the topological phase. With the simple designfree of requirements on any external field and gyromagneticor bi-anisotropic materials, the present topological photonicstate purely based on dielectric material is expected to bepromising for future applications.Artificial atom and pseudospin.—Let us consider har-monic TM modes of the EM wave, namely, those of finiteout-of-plane Ez and in-plane Hx and Hy components withothers being zero, in a dielectric medium (for coordinatessee Fig. 1). For simplicity, the real electric permittivitiesof both cylinders (εd) and environment (εA) are takenfrequency independent in the regime under consideration.The master equation for a harmonic mode of frequency ω isthen derived from the Maxwell equations [33]�1εðrÞ∇ ×∇×�EzðrÞẑ ¼ω2c2EzðrÞẑ; ð1Þwith εðrÞ the position-dependent permittivity and c thespeed of light. The magnetic field is given by the Faradayrelation H ¼ −½i=ðμ0ωÞ�∇ ×E, where the magnetic per-meability μ0 is presumed as that of vacuum. The Blochtheorem applies for the present system when εðrÞ isperiodic as shown in Fig. 1. Note, however, that the masterequation (1) describes the EM waves instead of electronscarrying on the spin degrees of freedom, with the mostprominent difference lying at the response upon TRoperation. Equation (1) is solved in momentum spaceusing package MIT PHOTONIC BANDS (MPB) [34]. Forsimplicity, we consider first a system infinite in the zdirection which reduces the problem to two dimensions.We start from a honeycomb lattice of dielectric cylinders,and deform it in such a way as to keep the hexagonalclusters composed by six neighboring cylinders and the C6symmetry. Now the alignment of dielectric cylinders ismore convenient to be considered as a triangular lattice ofhexagonal artificial atoms. There are two 2D irreduciblerepresentations in the C6 symmetry group associated withthe triangular lattice: E0 and E00 with basis functions xðyÞand xyðx2 − y2Þ, corresponding to odd and even spatialparities, respectively [35]. As can be seen in Fig. 2(a) forthe Ez field at the Γ point, artificial atoms carry pxðpyÞ anddxyðdx2−y2Þ orbitals, with the same symmetry as those ofelectronic orbitals of conventional atoms in solids.We now examine matrix representations of π=3 rotationand its combinations for the orbitals pxðpyÞ anddxyðdx2−y2Þ. Since pxðpyÞ behave in the same way asxðyÞ, it is easy to seeDE0 ðC6Þ�pxpy�¼ 12−ffiffi3p2ffiffi3p212!�pxpy�: ð2ÞIt is noticed that U ¼ ½DE0 ðC6Þ þDE0 ðC26Þ�=ffiffiffi3p ¼ −iσywithDE0 ðC26Þ≡D2E0 ðC6Þ is associated with the π=2 rotationof pxðpyÞ (σy being the Pauli matrix). Therefore,U2ðpx; pyÞT ¼ −ðpx; pyÞT, which is consistent with theodd parity of pxðpyÞ with respect to spatial inversion.Similarly, one hasDE00 ðC6Þ�dx2−y2dxy�¼ − 12−ffiffi3p2ffiffi3p2− 12!�dx2−y2dxy�; ð3Þwhich is same as DE0 ðC26Þ because the basis functionsare now bilinear of xðyÞ. It is then straightforwardto check that ½DE00 ðC6Þ −DE00 ðC26Þ�=ffiffiffi3p ¼ U is asso-ciated with a π=4 rotation of dxyðdx2−y2Þ, which yieldsU2ðdx2−y2 ; dxyÞT ¼ −ðdx2−y2 ; dxyÞT.We compose the antiunitary operator T ¼ UK, where Kis the complex conjugate operator associated with the TRoperation respected by Maxwell systems in general. SinceT 2 ¼ −1 is guaranteed by U2 ¼ −1, T can be taken as apseudo-TR operator that provides Kramers doubling in thesame way as TR symmetry in electronic systems. It is clearthat the crystal symmetry plays an important role in thispseudo-TR symmetry [36].The two pseudospin states are given byp� ¼ ðpx� ipyÞ=ffiffiffi2p; d� ¼ ðdx2−y2 � idxyÞ=ffiffiffi2p; ð4ÞFIG. 2 (color online). (a) Electric fields Ez of the pxðpyÞand dxyðdx2−y2Þ photonic orbitals hosted by the artificial atom atthe Γ point. (b) Magnetic fields associated with Ez fieldswith wave functions of positive and negative angular momentap� ¼ ðpx� ipyÞ=ffiffiffi2pand d� ¼ ðdx2−y2 � idxyÞ=ffiffiffi2p. The angularmomentum of the wave function of the Ez field constitutes thepseudospin in the present photonic crystal.PRL 114, 223901 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending5 JUNE 2015223901-2which are related to the above basis functions by unitarytransformation (see Supplemental Material [37]). Namely,the up and down pseudospins correspond to positive andnegative angular momenta of the wave function of the Ezfield. The in-plane magnetic fields associated with p� andd� in Eq. (4) are shown in Fig. 2(b). The physics discussedabove applies also for K and K0 points with 2D irreduciblerepresentations.Pseudospins discussed so far in photonic systemsinclude bonding (antibonding) states of electric and mag-netic fields [24,25], left-hand (right-hand) circular polar-izations of EM waves [28], and clockwise (anticlockwise)circulations of light in CROWs [29,30].Photonic bands.—Now we calculate the photonic banddispersions described by the master equation (1) imposingperiodic boundary conditions along unit vectors ~a1 and ~a2given in Fig. 1. As shown in Fig. 3, double degeneracyin the band dispersions appears at the Γ point, which canbe identified as p� and d� states, consistent with thesymmetry consideration. For large lattice constant a0, thephotonic band below (above) the gap is occupied by p�(d�) states [see Fig. 3(a) for a0=R ¼ 3.125 with R thelength of hexagon edge].Reducing the lattice constant to a0=R ¼ 3, the p and dstates become degenerate at the Γ point, and two Diraccones appear as shown in Fig. 3(b). This is because at thislattice constant the system is equivalent to the honeycomblattice of individual cylinders, and the doubly degenerateDirac cones are nothing but those at the K and K0 point inthe Brillouin zone of honeycomb lattice based on theprimitive rhombic unit cell of two sites [31].When the lattice constant is further reduced, a globalphotonic band gap is reopened near the Dirac point asshown in Fig. 3(c) for a0=R ¼ 2.9. Now the Ez field at thelow- (high-) frequency side of the band gap exhibits d�(p�) characters around the Γ point, opposite to the orderaway from the Γ point. Namely, a band inversion takesplace upon reducing the lattice constant in the presentsystem. Quantitatively, the band gap is Δω ¼ 5.47 THz atω ¼ 138.77 THz with a0 ¼ 1 μm, with all the quantitiesscaling with the lattice constant.In order to see what happens in the system around theband inversion, we check the real-space distribution ofthe pseudospin specific Poynting vector ~S ¼ Re½~E × ~H��=2averaged over a period τ ¼ 2π=ω, which describes theenergy flow in the present EM system. It is found that thePoynting vector is circling around individual atoms asshown in Fig. 4(a) for a0=R ¼ 3.125, with the chirality ofthe Poynting vector corresponding to the pseudospin(Poynting vector for pseudospin-up is not shown explicitly).The EM energy flows around individual atoms, charac-terizing a conventional “insulating” state. At a0=R ¼ 2.9,namely, after the band inversion, the Poynting vectors aremuch enhanced in interstitial regimes as shown in Fig. 4(b).It is in a sharp contrast to the case in Fig. 4(a), and hints atan unconventional insulating state.Although Dirac dispersions in photonic systems werediscussed previously in both square and triangular lattices[42–44], possible nontrivial topology was not addressed.Topological edge state.—We also consider a ribbon ofphotonic crystal after band inversion by cladding its twoedges in terms of two photonic crystals with trivial bandgap (namely, before band inversion) at the same frequencywindow, which prevents possible edge states from leakinginto free space. It should be kept in mind that, since thecluster of six cylinders is the basic block of the presentdesign, we keep it intact for discussions of the mainphysics. As displayed in Fig. 5(a), there appear additionalstates as indicated by the double degenerate red curveswithin the bulk gap. Checking the real-space distributionof the Ez field at typical momenta around the Γ point[A and B in the enlarged vision of Fig. 5(a) withkx ¼ �0.04ð2π=a0Þ], we find that the in-gap states locateat the ribbon edges and decay exponentially into bulk asdisplayed in Fig. 5(b) (two other states are localized at theother ribbon edge and are not shown explicitly). As shownin the right insets of Fig. 5(b), the Poynting vectors exhibita nonzero downward (upward) EM energy flow for thepseudospin-up (pseudospin-down) state even averagedFIG. 3 (color). Dispersion relations of the TM mode for the2D photonic crystals with εd ¼ 11.7, εA ¼ 1, and d ¼ 2R=3 for(a) a0=R ¼ 3.125 (Inset: Brillouin zone of triangular lattice),(b) a0=R ¼ 3, and (c) a0=R ¼ 2.9. Blue and red are for d� andp� bands, respectively, and rainbow for hybridization betweenthem. The case of a0=R ¼ 3 corresponds exactly to the honey-comb lattice of individual cylinders.FIG. 4 (color online). Real-space distributions of the time-averaged Poynting vector associated with the pseudospin-downstate at the Γ point below the photonic gap: (a) a0=R ¼ 3.125 inthe trivial regime and (b) a0=R ¼ 2.9 in the topological regime.Other parameters are taken same as those in Fig. 3.PRL 114, 223901 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending5 JUNE 2015223901-3over time. This indicates unambiguously counterpropaga-tions of EM energy at the sample edge associated with thetwo pseudospin states, the hallmark of a quantum spin Halleffect (QSHE) state [2,3]. Distributions of the Poyntingvectors of the bulk bands in Fig. 5(b) for the ribbon systemare similar to those in Fig. 4(b) for the infinite system. QHEhas been described by the cyclic motions of electrons understrong external magnetic field in a quasiclassic picture ofelectronic wave functions [45]. Note that the Poyntingvector describes energy flows in systems governed byMaxwell equations, and therefore the distributions shownin Figs. 4 and 5(b) can be observed in experiments. Thephotonic QSHE in the present system can also be con-firmed by evaluating theZ2 topology index based on a k · pmodel around the Γ point (see Supplemental Material [37]).Since the pseudo-TR symmetry and the pseudospinrely on the C6 symmetry, deformations in the system thatbreak the crystalline order and thus the pseudo-TR sym-metry would mix the two pseudospin channels as in otherZ2 topological photonic systems [24,29]. Actually, there isa tiny gap at the Γ point in Fig. 5(a) (unnoticeablein the present scale) due to the reduction of C6 crystallinesymmetry at the ribbon edge. However, the photonictopology remains valid up to moderate deformations asfar as the dispersions of edge states are not pushed into bulkbands (see Supplemental Material [37]).For experimental implementation of the present topo-logical state, the finite height of cylinders along the zdirection has to be taken into account. We consider a squaresample of topological photonic crystal sandwiched by twohorizontal gold plates [see Fig. 5(c)] with separation hchosen to prevent photonic bands with nonzero kz fromfalling into the topological band gap. Damping of EMwaves in gold plates is taken into account by adopting acomplex reflective index for gold. The size of the topo-logical sample is 40~a1 × 20ð~a1 þ ~a2Þ with all four edgesclad by a trivial photonic crystal. A harmonic line sourceE ¼ E0eiωtẑ is placed parallel to dielectric cylinders toinject the EM wave at the interface with the frequency inthe topological band gap. We simulate the 3D system bysolving time-dependent Maxwell equations using the finitedifference time-domain method [46] implemented in theMIT electromagnetic equation propagation (MEEP) [47].Since any harmonic source preserves TR symmetry resp-ected by the Maxwell equations, the system exhibits helicaltopological edge states as shown in Fig. 5(d). When an EMwave characterized by an Ez field with wave function ofpositive (negative) angular momentum is injected by linesource Sþ (S−) [48], leftward (rightward) unidirectionalenergy propagation takes place [see Figs. 5(e) and (f)], asexpected from the bulk topology.In conclusion, we derive a two-dimensional photoniccrystal with nontrivial topology purely based on conventio-nal dielectric material, simply by deforming a honeycomblattice of cylinders. A pseudo-time-reversal symmetry isconstructed in terms of the time reversal symmetryrespected by the Maxwell equations in general and theC6 crystal symmetry upon design, which enables theKramers doubling with the role of pseudospin playedby the angular momentum of wave functions of the out-of-plane electric field of transverse magnetic modes. TheFIG. 5 (color online). (a) Dispersion relation of a ribbon-shaped2D topological photonic crystal, which is infinite in one directionand of 45 and 6 artificial atoms for the topological and trivialregions respectively in the other direction. Right panel: enlargedview of (a) around the band gap. Dark gray (Red) curves are fortopological edge states. (b) Real-space distributions of Ez fields atpoints A and B indicated in the right panel of (a). Right panels:time-averaged Poynting vectors ~S over a period. (c) 3D photoniccrystal of height h with two horizontal gold plates placed at twoends symmetrically. (d) Distribution of energy-density of Ez fielduEzðrÞ ¼ εðrÞjEzðrÞj2=2 in the 3D topological photonic crystalin (c) stimulated by a linearly polarized source. (e) Leftward and(f) rightward unidirectional energy propagation stimulated bysource Sþ and S−, which injects Ez field with wave function ofpositive and negative, respectively, angular momentum in theregion denoted by gray (green) solid frame in (d). The latticeconstant and diameter of cylinder are kept same in the wholespace a0 ¼ 1 μm and d ¼ 0.24 μm, while the edge length ofhexagon is R ¼ 0.345a0 (a0=R ¼ 2.9) and R ¼ 0.32a0(a0=R ¼ 3.125) in topological and trivial regions, and thefrequency of all sources is ω ¼ 135.6 THz within the topologicalband gap. In the 3D system the height of cylinder is h ¼ 1 μm.Other parameters are same as those in Fig. 3.PRL 114, 223901 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending5 JUNE 2015223901-4present topological photonic crystal with simple designbacked up by the symmetry consideration can be fabricatedrelatively easily as compared with other proposals, and isexpected to leave impacts on topological physics andrelated materials sciences.The authors acknowledge K. Sakoda and T. Ochiaifor useful discussions. This work was supported by theWPI Initiative on Materials Nanoarchitectonics, Ministryof Education, Culture, Sports, Science and Technology ofJapan, and partially by Grant-in-Aid for Scientific Researchunder the Innovative Area “Topological QuantumPhenomena” (No. 25103723), Ministry of Education,Culture, Sports, Science and Technology of Japan.*To whom all correspondence should be addressed.HU.Xiao@nims.go.jp[1] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,494 (1980).[2] M. 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