1601.0Atomic Design of Three-Dimensional Photonic Z2 Dirac and Weyl Points2276v1

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Atomic Design of Three-Dimensional Photonic Z2 Dirac and Weyl Points

HaiXiao Wang,1 Xu Lin,1 HuanYang Chen,1 and Jian-Hua Jiang1,

1School of Physical Science and Technology, Soochow University, 1 Shizi Street, Suzhou 215006, China(Dated: January 12, 2016)

Topological nodal points such as Dirac and Weyl points in photonic spectrum offer unique abilitiesin manipulating light propagation. However, designing topological nodal points in photonic crystalsis much more difficult than in electronic systems due to lack of the atomic picture. We proposean atomic approach for the design of three-dimensional Dirac and Weyl points via Mie resonanceswhich can be viewed as photonic local orbits. Using connected-hollow-cylinder structure as anexample we demonstrate how to design topological degeneracy points in photonic energy bands bytuning the geometric shape. We discover a new type of topological degeneracy three-dimensionalphotonic spectrum, the Z2 Dirac points, which are monopoles of the SU(2) Berry-flux protectedby the parity-time symmetry. Upon breaking the inversion symmetry each Dirac point splits intoa pair of Weyl points with opposite chirality. Our study provides new methodology and examplefor future topological photonics where monopoles and surface states offer unprecedented control oflight flow.

Introduction. Stimulated by the discovery of topo-logical insulators[14] and topological semimetals[57],the study of topological nodal points in electronic en-

ergy spectrum has attracted a lot of attention in thepast decade. Recently, research interest is spread toatomic[810], photonic[1115], and acoustic[1618] sys-tems. Pseudo-spin S= 1 Dirac cone was found in two-dimensional (2D) photonic crystals[19], offering uniqueproperties for manipulation of light via, e.g., zero re-fractive index[19]and Klein tunneling[20]. Lately, Weylpoints are predicted[11, 15]and observed[14, 15]in three-dimensional (3D) photonic crystals. It is found that 3DWeyl points can provide effective angle and frequency se-lective transmission[21]. These progresses, alongside withthe progresses in 2D[2225]and 3D[26] photonic topolog-ical insulators, are unfolding a revolutionary platform of

topological photonics where propagation of light can becontrolled via topology and Berry phases to realize un-precedented applications.

The manifestation of topology and Berry phases inelectronic and photonic systems usually comes alongwith symmetry protection. Electronic topological insu-lators, for instance, have nontrivial topology protectedby the time-reversal symmetry. In electronic and otherfermionic systems, this protection is because T2 = 1where Tis the time-reversal operator. Thus time-reversaloperation enables and protects double degeneracy andthe appearance of chiral edge states. For photons andother bosons, T2 = 1 forbids emergence of topologicalinsulators due to lack of such double degeneracy. Thus

in photonic systems topological insulators emerge only insystems with additional lattice symmetries, such as rota-tion symmetry[25]and nonsymmorphic symmetry[26].

However, due to the complexity of photonic energybands, up till now the design of topological propertiesin photonic crystals remains as arbitrary and acciden-tal. This is due to the key difference between electronic

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bands and photonic energy bands: The photonic energybands are consequences of multiple Bragg scattering ofthe vectorial electromagnetic waves as no dielectric ma-

terial can trap light[2729]. In contrast, electronic bandstructure can mostly be understood as hybridization oflocal atomic orbits. For materials with inversion sym-metry, theZ2 topological index can be calculated simplyby counting band (parity) inversion at high symmetrypoints in the Brillioun zone. Such a simplified pictureis not available in photonic crystals, creating lots of ob-stacles in designing and understanding the topologicalproperties of photonic bands. Nevertheless, it was notedthat Mie resonances play similar roles in understandingphotonic energy bands as that of atomic orbits for elec-tronic band structures[30].

In this work we propose a hexagonal photonic crystalto realize 3D Dirac and Weyl points using dielectric ma-terial with isotropic permittivity. The photonic crystalconsists of connected hollow cylinders which supports p-and d-like Mie resonances. In hexagonal lattices withC6v and inversion symmetry, both p- and d- orbits aredoubly degenerate and hence can be regarded as pseudo-spin S = 12 . They develop into four photonic energybands where the double degeneracy is kept along the -A line in the Brillouin zone. We discover that a pairof unavoidable, accidental degeneracy points emerge be-

tween the andApoints. A k ptheory reveals that suchnodal points are topological Dirac points protected by theparity-time symmetry. The Dirac points found here areZ2 monopoles of the SU(2) Berry-flux. They are quitedifferent from the Weyl points in photonic crystals[11, 15]which are monopoles of the U(1) Berry-flux and have aZ topological charge. The Z2 topological charge of theDirac point make it topologically stable and robust inthe presence of parity-time symmetry. They are totallydifferent from Dirac points with trivial Z2 topologicalcharge which can be removed by local perturbations. TheZ2 Dirac points can be moved, created or annihilated inpair only by significant tuning of the hollow cylinder ge-

arXiv:1601.02276v1

[cond-mat.mtrl-sc

i]10Jan2016
mailto:[email protected]:[email protected]


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FIG. 1. a, Structure in real-space unit cell of hexagonal pho-tonic crystals with Dirac points in photonic energy bands.The blue hollow cylinders and yellow micropillars are madeof the same material with isotropic permittivity. b (top-down

view) and c (bird view), The hexagonal photonic crystal withlattice constant a. a1 and a2 are the two lattice vectors inthe x-y plane. The diameter of each micropillar is 0.1a witha being the lattice constant in the x-y plane. The outter andinner radii of the hollow cylinder isRoutand Rin. d, Brillouinzone with a pair of Dirac points along the -Aline.

ometry.

Core-shell hexagonal photonic crystals. The hexag-onal photonic crystal consists of hollow cylinders (withouter radius Rout and inner radius Rin) connected bysix micropillars. The height of each unit cell (depictedin Fig. 1a) is h = 0.6a with a being the lattice con-stant in the x-y plane. The micropillars are of the sameheight 0.2a and diameter 0.1a. The height of each hol-low cylinder is 0.4a. The hollow cylinders support localelectromagnetic (Mie) resonances of s, p, d, f ... sym-metries (see Supplementary Information). In hexagonallattice with C6v symmetry, only the Lz = 0, 1, 2, 3 (i.e.,s, p, d, and f) orbits are distinguishable, higher angularmomentum orbits are mixed with those lower ones.

Along the z direction, the photonic wavefunctioncan have zero, one, or multiple nodes due to the latticetranslation symmetry. The wavefunctions of the photonicbands of interest (the p- and d- bands) have zero nodes,i.e., they are parity-even along the z direction. The mi-cropillars modulate the hybridization between the localresonances in adjacent hollow cylinders along the z di-rection. By tuning their radii and height, the first fewphotonic energy bands can be moved in frequency. Wefound that those energy bands mainly consist of TM po-larization (i.e., Ez polarization). This is consistent withthe observation that Mie resonances in long hollow cylin-ders for TE polarization have much higher frequenciesthan that for the TM polarization. Thus the degeneracy

FIG. 2. a, Photonic energy band structures for the core-shell hexagonal photonic crystals with inversion symmetry(the purple curve indicate the light-line). b, The band struc-ture along the -A line. Thep-bands (red) cross thed-bands

(green) at (0, 0,Kz) with Kz = 0.342c

a . The gray curvesrepresent the f bands. c, The Z2 topological number as afunction ofkz. d, Isosurface plot of the Ez field of the dxy or-bit, showing the d-orbit symmetry in a unit cell (depicted bythe yellow dashed lines). e, Isosurface of the Ez field of the pxorbit. f, The dispersion close to the Dirac point. Parameters:Rout/a= 0.5 andRin/a= 0.4, and = 12.

of thep- andd- bands come from space symmetry ratherthan spin degree of freedom. The topological property ofthose photonic bands is the same as that of the energybands of spinless bosonic systems.

We calculate the photonic energy bands for the hexag-onal photonic crystal and display the results in Figs. 2aand 2b. The p and d bands cross at certain point be-tween the and A points. This crossing is accidentalbut unavoidable since the p and d bands are reverselyordered at the and A points. The accidental degen-eracy takes place at two points related by inversion,(0, 0, Kz), which are the only p-ddegeneracy points inthe whole Brillouin zone. These two points are identifiedas the topologicalZ2 Dirac points below.

The contour surface of the electric field Ez of the panddbands at point is presented in Figs. 2d and 2e. Theirspatial symmetries can be clearly identified as p and d

orbits, which correspond to the two doubly degeneraterepresentations of the C6v symmetry group. Indeed the

p bands are doubly degenerate along the -A line (samefor the d bands). This degeneracy is lifted in k-spaceaway from the -Aline.

The Z2 topological number can be calculated for eachkz. Since the photonic crystal has inversion symmetry,the Z2 topological number can be calculated by count-ing the parity inversion at the time-reversal invariantmomenta[31]. In hexagonal photonic crystals there are


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FIG. 3. Photonic dispersion in kx-ky plane around the(0, 0, kz) point for (a)kz < Kz, (b)kz =Kz, and (c)kz > Kz.There are two nearly degenerate bands above and below theDirac point. The arrows indicate the orbital angular momen-tum distribution for the pseudo-spin up bands. Note that theorbital angular momentum has substracted an universal con-stant 3

2(i.e., the average orbital angular momentum between

thep+ andd+) state.

only four such points for fixed kz: three of them areequivalent to the M point (

a, 0, kz), the other is the

point (0, 0, kz). The parity inversion can only take placeat the point. If the p- and d-bands reverse order atthe point (i.e., the p-bands are above the d-bands),then the Z2 topological number is 1, otherwise it is zero.We plot the Z2 topological number as a function ofkz in

Fig. 2c. The topological phase transistion as a function ofkzis illustrated in Fig. 3. Forkz < Kz the band structureand spin configuration resembles that of a negative massDirac electron. For kz = Kz the band gap closes and aDirac cone emerges. Forkz > Kz the cone is gaped againwhere the spin configuration resembles that of a positivemass Dirac electron.

The Dirac points (its conical dispersion is shown inFig. 2f) are identified as the kink of the Z2 topologi-cal number. The two Dirac points are then identifiedas monopoles of the SU(2) Berry-flux, which are differ-ent from trivial Dirac points in semimetals[32] that arenot robust to perturbations. The parity-time symmetry

of the system only allows nonzero SU(2) Berry-phase,whereas the U(1) Berry-phase vanishes since it is oddunder P T operation (P stands for inversion and Tdenotes the time-reversal operation).

Recently, it was found theoretically that in spinless sys-tems with parity-time symmetry the Z2 Dirac points arestable and robust[33]. Theoretical connection betweenthe topological surface states and the Riemann surfaceis established only very recently[34], which put forwardthe understanding of the Z2 topological Dirac points.

FIG. 4. Chiral hexagonal photonic crystals. a (lateral) andb (top-down) view of the structure in real-space unit cell. c,Photonic energy bands along the -A line. Two Weyl pointsin the kz >0 region are found at (0, 0, Kz1) and (0, 0, Kz2)with Kz1 = 0.2

2c

a

andKz2= 0.262c

a

. d, Depicting the fourWeyl points in the Brillouin zone. e and f, Photonic spectrumclose to the two Weyl points in the kz > 0 region.

Here we propose the first realization of the Z2topologicalDirac points in photonic crystals.

Characterizing the Dirac points usingk P theory.Near each = (0, 0, kz) point the doubly degenerate pbands can be organized as p+= px+ipyandp= pxipybands, which we define as the spin-up and spin-downstates of the p (valence) bands. Similarly, we definethe d+ = dx2y2 + idxy and d = dx2y2 idxy asthe spin-up and spin-down states of the d (conduc-tion) bands. The coupling between those bands near

the points can be obtained via a k P theory derivedfrom the Maxwell equations[35] (see Supplementary In-formation). Explicitly, the eigenvalue problem for theMaxwell equations in the photonic crystal can be writ-ten as 1

(r)

n,k(r) = c22

n,kn,k

(r) (c is the

speed of light in vacuum). The Hamiltonian is defined asH = 1

(r) and the photon wavefunction for the

nth band with wavevector k is defined as its magnetic

field, n,k

(r) = Hn,k

(r)eikr. The wavefunction is nor-

malized asu.c.

dr| Hn,k

(r)|2 = 1 (u.c. stands for integral

in a unit cell). According to the Bloch theorem Hn,k

(r)

is invariant under lattice translation when k is confinedin the first Brillouin zone. It should be a representa-tion of the C6v and the inversion symmetries. We foundthat those symmetries dictate the dominant coupling be-tween the p- andd-bands is within the same pseudo-spinpolarization (e.g., p+ couples with d+). In the basis of

(d+, p+, d, p)T the k Pphotonic Hamiltonian near the


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FIG. 5. (Color online) a-d, Spatial distribution of the Poynt-ing vectors in a volume slightly larger than the unit cell forthed,d+, p+, andp bands at the point.

points is given by

H =20

c2

2d(k)

20vke

ik 0 0

vkeik

2p(k)

200 0

0 0 2d(

k)20

vkeik

0 0 vkeik

2p(k)

20

,

(1)where 0 is the frequency of the Dirac point. To the

lowest orders in k and kz, d(k) = d0(kz) + Ak2,

p(k) = p0(kz) + Bk

2

. The interband coupling isnonzero only between the p+ and d+ bands as well asbetween the p andd bands (see Supplementary Infor-mation). k = Arg(kx+ iky) and |v| is the velocity inthe x-y plane. The parameters A, B, and v are deter-mined by the Bloch functions at the points (hence theyare kz dependent). Ifp0 > d0, the above Hamiltonianis similar to the Hamiltonian of the quantum spin Hallinsulator[2]and henceZ2 = 1. Otherwise the topology istrivial, Z2 = 0. Therefore,p0 = d0 determines a pairof kinks of the Z2 topology.

The conical dispersion (see Fig. 2f) near the Diracpoints is described by the following Hamiltonian that re-sembles the Dirac equation for electrons,

HDirac = 20

c21

+20c2

vdkz vkeik 0 0

vkeik vpkz 0 0

0 0 vdkz vkeik

0 0 vkeik vpkz

,(2)

where kz = kz Kz for kz > 0 (or kz = kz + Kz ifkz 0 and vp < 0 are the group velocity of the

FIG. 6. (Color online) a, Phase diagram of the hexagonalphotonic crystal with inversion symmetry. b and c, Depictingthe Dirac points and topology induced surface states in thekz-kx plane for systems with Z2 and trivial Dirac points. d

and e, Illustrating the Weyl points and the topological surfacestates (photon arcs) for systems derived from Z2 and triv-ial 3D Dirac points by introducing chiral symmetry brokenperturbations.

pand d bands along thez direction. The Dirac points at(0, 0, Kz) consists of a pair of Weyl points with oppositechirality. For kz >0 the p+ and d+ (p and d) bandsform a Weyl point with chirality +1 (-1).

We then introduce a chiral symmetry breaking mecha-nism by twisting the micropillars (see Fig. 4a-b). The re-sulting photonic bands along the -Aline (Fig. 4c) breakthe degeneracy between the the p+ and p bands as wellas that between the d+ and d. We find that, in consis-

tent with the k P theory, the crossing between the p+andd+ (p and d) bands form a Weyl point with con-ical dispersion (Fig. 4d-e). In contrast, the crossing be-tweenp+ and d (p and d+) bands is quadratic. Hencethere are four Weyl points lying between the and the Apoints (Fig. 4f ). The chirality of the four photonic bandsare identified from the spatial distributions of Poyntingvectors in a unit cell (see Fig. 5).

Phase diagrams. To show the robustness of the Diracpoints, we study the dependence of the Dirac points onthe inner and outer radii of the hollow cylinders [seeFig. 6a]. We find that the Dirac points emerge in a largeregion of various cylinder geometry, mainly in the regimewith large outer radius Rout. The position of the Diracpoints, characterized by Kz, can be tuned by Rin andRout. The arrows in Fig. 6a indicate the tendency thatthe Dirac points are moved toward the point wherethey are created or annihilated. The robustness of theDirac points are garanteed by the global feature of the

p-d band inversion along the -A line, which is difficultto be removed by local perturbations. The surface states


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comprises of kz dependent helical edge states (Calcula-tion of edge states is given in Supplementary Informa-tion). Hence they connect the two Dirac points. Theparity-time symmetry and the Z2 topology ensure thatthe surface states always form a closed iso-frequency cir-cle (as depicted in Fig. 6b)[34]. This is totally differentfrom Dirac points with trivial Z2 topology (there is notopologically protected surface states, see Fig. 6c), such

as in electronic Dirac semimetals[32]. The trivial Diracpoints are not robust and can be removed by local per-turbations.

The Weyl points emerging in chiracl symmetry brokenphotonic crystals are also found to be stable. In Fig. 6dwe give the phase diagram for the Weyl points in theRout-Rin parameter space which is similar to that of theDirac points. Their robustness is inhered from the topo-logical Dirac points. Fig. 6e illustrates that the topo-logical surface states connect a pair of Weyl points withopposite kz. This is quite different from the four Weylpoints induced by breaking chiral symmetry in systemswith trivial Dirac points (Fig. 6f): The surface states con-nect the two Weyl points in the half-plane with kz > 0(or kz


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1601.0Atomic Design of Three-Dimensional Photonic Z2 Dirac and Weyl Points2276v1