Valley photonic crystals for control of spin and topology

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1 In the format provided by the authors and unedited. DOI: /NMAT4807 Valley photonic crystals for control of spin and topology Jian-Wen Dong 1,,, Xiao-Dong Chen 1,, Hanyu Zhu, Yuan Wang,3,3, 4, & Xiang Zhang 1. State Key Laboratory of Optoelectronic Materials and Technologies & School of Physics, Sun Yat-Sen University, Guangzhou 51075, China.. NSF Nanoscale Science and Engineering Center (NSEC), University of California, Berkeley, CA 9470, USA. 3. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 9470, USA. 4. Department of Physics, King Abdulaziz University, Jeddah 1589, Saudi Arabia These authors contributed equally to this work. Corresponding author. Supplement A: The low-energy Hamiltonian In this section, we will derive the low-energy Hamiltonian of valley photonic crystal (VPC) with the lowest monopolar dispersion bulk bands. Consider a two-dimensional bianisotropic medium with the reciprocal constitutive relations of D 0εrE ζh and B 0μrH ζe. The relative permittivity, the relative permeability and the bianisotropic coefficient tensors are assumed to be // εr μ, r // z xy i // / c ζ yx i // / c. (1SA) Then the corresponding full-vector Maxwell equations are written: E i( 0μrH ζe ), i 0 r H ( ε E ζh ). (SA) Here, the harmonic time dependence is assumed to be e -iωt. By expressing the in-plane electromagnetic (EM) components in terms of the z-components, we finally reach a set of reduced equations: NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

2 [ k ] H i[ ] E 0 z x x y y z x y y x z [ k ] E i[ ] H, (3SA) 0 z x x y y z x y y x z where and ( r) 1/ //(1 ). In VPC, utilizing the periodicity of the ( r) / //(1 ) honeycomb crystalline structure, we expand the fields and the constitutive parameters as follow, E(; rq) Ee z iq ( G) r, G G H(; rq) He z iq ( G) r, G G igr igr ( r) e, ( r) e. (4SA) G G G G Here, the z-components of the permittivity and permeability are assumed to be spatially constant. Then, we reach the linear equations for the Fourier components of the fields, ' ' x x y y y y x x k H q G q G q G q G H ' ' 0 z G G' G ' x G x x x y y y y G' i q G q G q G q G E G' GG' ' ' x x y y y y x x k E q G q G q G q G E ' ' 0 z G G' G ' x G x x x y y y y G' i q G q G q G q G H G' GG' G' G'. (5SA) To describe the spin split behaviours, we focus on different valleys. For K valleys, we truncate the plane wave basis to the first three plane waves. These plane waves correspond to three equal-length reciprocal vectors: K G each rotating π/3 with respect to one another, i where K [K, 0] ( K 4 /3a ) and G0 [0,0], G1 [ K/, K/ 3], G [ K/,K/ 3] components of the fields:. Then a 6 6 Hermitian equation can be obtained for the Fourier ˆ ˆ E E k0 z ˆ, (6SA) ˆ H H where E E ; E ; E G0 G1 G, and H H ; H ; H G0 G1 G. In the vicinity of the K valley q K k x x and qy ky, the matrices ˆ and ˆ are found to be NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

3 G 0 G G G G G ˆ K G 1 G 0 G K k x G 1 G 0 G G G 1 G0 G G 1 G G G 1 G 0 G G1 3 Kky G G k 0 x k y G 1 G 0 G 3 G G 1 G0 G 0 3 G 0 0 G G G G 1 G G1 ˆ 3i 3i 3i K G G Kk x G 1 G Kk yg. (7SA) G G1 G G1 G As to discuss the monopolar modes, we expand ˆ and ˆ up to and k i 1 k 1 i terms and neglect other higher order terms. Using the transformation Uˆ,0;0, ˆ ˆ Uˆ where Uˆ 1 / 3 1,1,1;1,, ;1,, and exp( i / 3), one can arrive at one copy of the monopolar singlet band by eliminating the doublet bands: ( k ) Es Es ( k ), (8SA) ( k ) Hs Hs where s( k ) s(k) ( k ) z s(k), ck 1 1 s(k) Kc G / 0 G 1 G z 1/, (K) K s 3i G G G, s(k) G G 0 1. Using the transformation 1 Uˆ 1 / 1,1;1, 1, we arrive at the dispersion for the two pseudo-spins ( pseudo-spin is abbreviated as spin for short) for K valley: ( k ) sk sk ( k ), (9SA) ( k ) sk sk NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

4 where the spin-up is sk Es H and the spin-down is s sk Es H. The same procedure s can be repeated for K valley, and we have the dispersion for the two spins for K valley,. (10SA) ( k ) sk ' sk ' ( k ) ( k ) sk ' sk ' By introducing valley and spin Pauli matrices ˆi and s ˆi, respectively, we have the low energy Hamiltonian of the lowest monopolar dispersion bulk bands, as shown in Eq. (1) of the text: H ˆ ˆ 0 ( k) zs. (11SA) z Supplementary Figure 1a re-plots the bulk band structure of VPC whose detailed constitutive parameters have been shown in the main text. Due to EM duality symmetry, i.e. ε r μ, the r full-vector Maxwell equations can be decomposed into two subspaces by introducing the pseudo-fields, 0 0, 0 0 P P E H E H. Regarding to the spin-up (spin-down) states being with nonzero ( P, P, P ) T [ (, P P, P ) T ], the EM components E z and H z x y z x y z should be in-phase (out-of-phase). The eigen-field patterns of the second modes at K and K points are plotted in the left and right panels of Supplementary Fig. S1b, respectively. Inferring from the relation between the field distributions, the mode at K point is spin-up while that at K point is spin-down. With same procedure, all bands in Supplementary Fig. S1a can be classified as spin-up and spin-down, marked with blue and red curves, respectively. The valley dependent spin-split of the two lowest monopolar bulk bands is clearly illustrated by the eigen-frequency surfaces around K and K valleys (Supplementary Fig. S1c). In addition, the equi-frequency contour at f = (c/a) inside the spin gap is also plotted at Supplementary Fig. S1d, indicating that the spin-up (spin-down) states will propagate paraxial to ΓK (ΓK) direction. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

5 Supplementary Figure S1 Bulk band dispersions for honeycomb VPC with staggered bianisotropic response. a, Dispersions of the spin-up (blue) and spin-down (red) modes. Two reciprocal lattice vectors G and 1 G in the first Brillouin zone, which are used for the plane wave expansion, are marked by dashed-pink and dashed-purple arrows in the middle inset. b, The eigen-field patterns: E z and H z of the second modes at K (left panels) and K (right panels) points are in-phase and out-of-phase, respectively. c, Eigen-frequency surfaces around K and K valleys showing the valley dependent spin-split. d, Equi-frequency contour at the frequency of (c/a), indicating that the spin-up (spin-down) states propagate paraxial to ΓK (ΓK) direction. Supplement B: Photonic valley Hall effect In this section, we show photonic valley Hall effect for bulk state with higher frequency, e.g., f = 0.4 (c/a). Small local density states lead to low coupling efficiency and hence the amplitude of the electric filed (NormE) in Supplementary Fig. Sa is smaller than that in Fig. b. However, the phase difference is still preserved around the value of π (0) along ΓK (ΓK ) direction (Supplementary Fig. Sb). Spin flow from the two inequivalent valleys is routed to different NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

6 directions, i.e. photonic valley Hall effect, can be still maintained for higher frequency inside the spin gap. Supplementary Figure S Photonic valley Hall effect at higher frequency with f = 0.4 (c/a) inside the spin gap. a, The amplitude of the electric field (NormE) is smaller than that in Fig. b due to small local density state and low coupling efficiency. b, The phase difference is preserved around π (0) along ΓK (ΓK ) direction, showing the spin-maintainable feature in the spin gap. Supplement C: Polarization characteristics and the valley dependence In this section, we present detailed results of the polarization characteristics and the valley dependence. Here, we change the plotted unit-cell to be the purple bianisotropic rod on the top while the blue one on the bottom, for a better illustration of valley dependent polarization characteristics (Supplementary Fig. S3a). For the polarization ellipse, the ellipticity angle () and the orientation angle () are defined as follows (illustrated in Supplementary Fig. S3b), where arctan E0y / E0x sin( ) sin( ) sin( ), tan( ) tan( )cos( ), (1SC) and y correspond to the amplitude ratio and the phase x difference between E y and E x. The ellipticity angle ( /4 /4) specifies the shape and temporal evolution of the polarization ellipse. The orientation angle ( 0 ) determines the angle between the major axis of polarization ellipse and the coordinate axis Ox. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

7 Supplementary Figure S3c shows the polarization characteristics of the second modes at K point. As we can see, for the C 3 -rotation related positions, the ellipticity angles are exactly the same and the orientation angles between them are differed by /3. That is to say, the whole polarization ellipse is threefold rotational symmetry. It is in good agreement with the fact that the VPC is C 3 -rotation invariant. In addition, polarization singularities in some specific positions are found. For example, L-lines (linear polarization with undetermined polarization handedness, i.e. = 0) are marked by dashed brown lines and cut the unit-cell into six domains. C-points (circular polarization with undetermined orientation angle, i.e. /4) are outlined by yellow or green circles. There are two kinds of C-points here. The type-i C-points exist at the centres of the purple (blue) rods, with the RCP (LCP) feature marked by yellow (green) solid circles. The type-ii C-points (yellow and green dashed circles) surround concentrically about the centre of the unit-cell and locate in the air background. Supplementary Figure S3d shows the polarization characteristics of the second modes at K point. Comparing polarization characteristics of these two modes at different valleys, we find that the ellipticity angles are just different by the sign of while their orientation angles are exactly the same. The physics of such valley dependence can be understood as follows: due to * T-symmetry, the electric fields of modes at K and K valleys satisfy: E ( r) E ( r). That is to say, for a mode at K valley with an in-plane rotating polarization, the time-reversal counter-propagating mode at K valley has the orthogonal rotating polarization. As a result, the colours in ellipticity angle patterns between Supplementary Fig. S3c and Fig. S3d are interchanged while those in orientation angle keep the same in between. K K' NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

8 Supplementary Figure S3 Polarization characteristics and the valley dependence. a, Schematic of the unit-cell of VPC with EM duality symmetry but staggered bianisotropy. For a better illustration of valley dependent polarization characteristics, the unit-cell is chosen to be purple rod locating on the top while the blue one locating on the bottom. b, Definition of polarization ellipse. The ellipticity angle (-π/4 χ π/4) specifies the shape and temporal evolution of the polarization ellipse. The orientation angle (0 ψ < π) determines the angle between the major axis of polarization ellipse and the coordinate axis Ox. c, Polarization characteristics of the second mode at K point. Left panel: ellipticity angle distributions with threefold rotational symmetry. Linear polarizations (so-called L-lines) with =0 are marked by dashed brown lines. Circular polarizations (so-called C-points) are outlined by yellow or green circles. As to highlight the circular polarization singularities, we use a saturated wavelight colour scale to exaggerate values near ellipticity angle extrema. Right panel: orientation angles at the C 3 -rotation related positions are differed by /3. d, Polarization characters of the second mode at K point. Valley dependent polarization is shown by the comparison between (c) and (d). Supplement D: Experimental proposal In this section, we illustrate the experimental designs of the VPCs proposed in the main text. The bulk band structure of the VPC in Fig. 1b is redrawn in Supplementary Fig. S4a, of which the eigen-field distributions are homogeneous at valleys to show the monopolar characteristics (Supplementary Fig. S4b). On the other hand, the case in Fig. 4a is redrawn in Supplementary Fig. S4c and the dipolar feature of the bulk band is illustrated in Supplementary Fig. S4d. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

9 Supplementary Figure S4 Bulk band structures and typical eigen-field patterns of valley modes for the two proposed VPCs. a, Bulk band structure and b, eigen-field patterns of the lowest states at K and K points for the VPC with the lowest monopolar band. c and d are similar except for the VPC with the lowest dipolar band. The left and right panels in panel b and d present the E z and H z. Next we show the concrete samples. Two different meta-atoms are designed to form the VPCs, denoted by disk-atom and closed-ring-atom (CR-atom) as shown in Supplementary Fig. S5. Both meta-atoms have the shape of regular hexagonal prism with a height of 0.8a 0, where a 0 is a constant. For the disk-atom, it is constructed by two hexagonal perfect electric conductor (PEC) disks linking by one slim PEC cylinder. In contrast, the CR-atom is constructed by three closed-rings PEC. Detailed structural parameters of meta-atoms are also shown in Supplementary Fig. S5. We extract the effective permittivity and permeability of each meta-atom by standard S-parameter method S1. All the retrieved parameters (dashed curves) are mildly dispersive in the frequency from 0.08 to 0.1 (c/a 0 ). In addition, we also examine the EM duality symmetry. Ratios of // / // and z / z are roughly equal to each other for both two meta-atoms (solid curves). Thus, these two meta-atoms are well-designed to form the VPCs. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

10 Supplementary Figure S5 Effective constitutive parameters and structural parameters of the two meta-atoms. The retrieved effective constitutive parameters of (a) disk-atom and (b) CR-atom. The right schematics show the structural parameters of both meta-atoms. We first demonstrate the concrete designed VPC with monopolar bulk band, similar to that of Supplementary Fig. S4a (i.e., Fig. 1b). The unit-cell consists of CR-atoms surrounded by disk-atom, as outlined by black hexagonal prism in right panels of Supplementary Figs. S6a and S6b. The lattice constant is set to be a 1a. Bulk band structure is obtained and also 0 shown in Supplementary Figs. S6a and S6b. The bulk band dispersion is degenerate when the inversion symmetry is maintained (see in Supplementary Fig. S6a). However, when changing the CR-atom to be split-ring (SR), the bianisotropy becomes nonzero. Furthermore, the staggered bianisotropy can be achieved by reversing the gap opening orientations of the SR, see e.g. the red and blue SRs in right panel of Supplementary Fig. S6b. Consequently, the inversion symmetry is broken and the valley dependent spin split band is expected, which is confirmed by the bulk band dispersions in left panel of Supplementary Fig. S6b. In addition, one can also see the homogeneous distributions of the eigen-fields of the lowest modes at K and K points (Supplementary Figs. S6c and S6d), confirming the similar monopolar characteristic in Fig. S4b. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

11 Combining the bulk band structure and the eigen-field patterns, the valley contrasting phenomena presented in Fig. and Fig. 3 are expected in such experimental design. Supplementary Figure S6 Experimental design of VPC with monopolar bulk band. a, Spin degenerate monopolar bulk band for inversion symmetric VPC. Black hexagonal prism outlines the unit-cell consisting of CR-atoms surrounded by disk-atoms. b, Valley dependent spin split band dispersion. The inversion symmetry is broken due to the staggered bianisotropy, which is fulfilled by orientating the gap opening of the spilt rings up-side-down. c/d, Eigen-fields of the lowest modes at (c) K and (d) K points. The left and right panels plot E z and H z distributions, while the top and bottom panels show the top and perspective views. Interestingly, by changing the unit-cell to disk-atoms surrounded by CR-atoms, the first and second degenerate bulk bands will form a dipolar Dirac-typed band dispersion. The results are illustrated in Supplementary Fig. S7. In addition, to realize the hetero-spin degenerate point such that the two different-spin bulk bands are structurally degenerate at valley centre, as that in Supplementary Fig. S4c (i.e., Fig. 4a), one can change one CR-atom at the corner of hexagon to NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

12 be split-ring, then the bianisotropy-nonbianisotropy configuration can be achieved and the resulting bulk band dispersion is plotted in Supplementary Fig. S7b. Once again, the valley dependent spin split behaviour is confirmed, and the hetero-spin degenerate point at valley centre is also verified. In addition, Supplementary Figs. S7c and S7d show the dipolar nature of the lowest modes at K and K points, respectively, similar to those in Supplementary Fig. S4d. Therefore, the experimental design with the lowest dipolar bulk band is achieved. Supplementary Figure S7 Experimental design of VPC with hetero-spin degenerate dispersion. a, Spin degenerate dipolar bulk band for inversion symmetric VPC. Black hexagonal prism outlines the unit-cell consisting of disk-atoms surrounded by CR-atoms. b, Valley dependent spin split band dispersion. The inversion symmetry is broken in the bianisotropy-nonbianisotropy configuration, leading to the hetero-spin degenerate point at K and K points, such that the two different-spin bulk bands are structurally degenerate at valley centre. c/d, Eigen-fields of the lowest modes at (c) K and (d) K points. The left and right panels plot E z and H z distributions, while the top and bottom panels show the top and perspective views. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

13 Supplement E: Evolution of topology in valley photonic crystals In this section, we show how the bands evolve from the trivial VPC in Fig. 1 to the nontrivial VPC in Fig. 4. As we know, the bulk band structure can be tuned by the permittivity of the two rods, e.g. ε // and ε z. Supplementary Fig. S8a illustrates the frequency spectra of eigen-modes at K/K points along either the change of ε // (left cyan panel) or the change of ε z (right green panel). Here, the permittivity of the two rods are set be identical (ε 1// = ε // and ε 1z = ε z ) to keep inversion symmetry invariant. In addition, zero bianisotropy is assumed. When the value of ε // increases from 1 to 8 (left panel in Supplementary Fig. S8a), the frequencies of the monopolar modes (dash) are always lower than those of dipolar modes (solid). Supplementary Figure S8b shows a typical bulk band structure when the constitutive parameters of the two rods are ε 1// = ε // = 8 and ε 1z = ε z = 1, indicating the monopolar fact of the lowest bulk band. These monopolar bulk bands are well known to have all-zero topological invariants. Even if introducing staggered bianisotropy with κ 1 = -κ = -0.5 (Fig. 1b), these monopolar bulk bands will not have any mode exchange or band inversion. Hence, all topological invariants can still keep to be zero to guarantee the topological triviality of the VPC. On the contrary, when we change ε z from 1 to 14 instead of ε //, see e.g. right panel in Supplementary Fig. S8a, the frequencies of the dipolar modes will become lower than those of monopolar modes. A representative example is shown in Supplementary Figure S8c when ε 1// = ε // = 1 and ε 1z = ε z = 14. The lowest bulk band is dipolar and a linear Dirac cone is present around the frequency of 0.5c/a. Consequently, when we take 1 = 0.9 and = 0 for such VPC in Fig. 4, it will be topologically nontrivial and characterized by C s = +1. NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

14 Supplementary Figure S8 Band evolution of VPC. a, Frequency spectra of the monopolar and dipolar modes at K/K points along either the change of ε // (left panel shaded in cyan) or the change of ε z of the two rods (right panel shaded in green). b-c, Bulk band structure for VPC when the parameters of the two rods are (b) ε 1// = ε // = 8 and ε 1z = ε z = 1 or (c) ε 1// = ε // = 1 and ε 1z = ε z = 14. References: S1. Smith, D. R., Schultz, S., Markoš, P. & Soukoulis, C. M. Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients. Phys. Rev. B 65, (00). NATURE MATERIALS Macmillan Publishers Limited, part of Springer Nature. All rights reserved.