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Spin- and valley-polarized one-way Klein tunneling in photonic topological insulators

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Science Advances  11 May 2018:
Vol. 4, no. 5, eaap8802
DOI: 10.1126/sciadv.aap8802
  • Fig. 1 Schematic geometries and corresponding band structures.

    (A) Nonreciprocal two-dimensional (2D) photonic crystal composed of ferromagnetic rods arranged in a honeycomb lattice with magnetic bias applied along the z direction. (B) Topological 2D photonic crystal made of bianisotropic triangulated rods arranged in a triangular array in air. Dimensionless (normalized to lattice constants a and a′) geometric parameters of the rods are rA = 0.191, δ = rArB = 0.01, μ = 2, κ = 0.6, ϵA = ϵB = 14 (A) and r0 = 0.34, rf = 0.27, ϵ|| = μ|| = 14, ϵz = μz = 1 (B). A bianisotropic response ξxy = 0.2 is introduced in the background. In (B), red bands correspond to (pseudo)spin-up states, whereas blue bands correspond to (pseudo)spin-down states.

  • Fig. 2 One-way Klein tunneling in nonreciprocal photonic topological insulator.

    (A) Dispersion bands for a symmetric (nondimerized) structure of nonmagnetized rods (μ = 2, κ = 0) (top), a nonreciprocal PT-preserving crystal where cylinders of equal radii are magnetized in opposite directions (μ = 2, κ = 0.6) (middle), and a nonreciprocal PT-violating crystal where cylinders of slightly detuned radii are magnetized in the same direction (parameters are as in Fig. 1) (bottom). (B) Photonic bands near K and K′ points (left) and transmission coefficients (right). Top and bottom panels correspond to the backward and forward wave propagation, respectively. Numerically calculated transmission is plotted with a blue line. The analytically retrieved dependence with second-order correction in k·p method is shown with a green dashed line. The fitting parameters of the spectra extracted from the numerically calculated band diagrams are as follows: At the top (K′ valley), the frequency of the Dirac crossing at the K′ point ω0a/2πc = 0.233, and Fermi velocity vD/2πc = 0.020. In domain (1), u1 = 0, m1 = 0, α1 = −0.12vD, β1 = 0.54vD; in domain (2), u2a/2πc = −0.007, m2a/2πc = 0.005, α2 = −0.005vD, β2 = 0.06vD;. At the bottom (K valley), parameters are the same as those in domain (1) at the K′ valley, except that the frequency of the Dirac crossing at the K point becomes ω0a/2πc = 0.228, and in domain (2), m2 = 0, u2a/2πc = −0.004. αi, βi are the coefficients of Embedded Image in the effective Hamiltonian with their numerical values expressed in term of vD. (C) Simulated electric field intensity |E|2 distributions in the strip for backward (top and middle) and forward (bottom) wave propagation. The strip consists of three domains: Domain (1) is the nonreciprocal PT-preserving honeycomb lattices and separated by domain (2), which is composed of the inequivalent-sites lattice with magnetic field applied perpendicular to the lattice, and domains (1) and (2) contain 2 × 42 and 2 × 12 unit cells, respectively. The boundaries of three crystal regions are marked by black vertical lines. The modes are excited by current sheets at cuts indicated by the arrows and white lines.

  • Fig. 3 Spin-valley–coupled Klein tunneling in bianisotropic photonic topological insulator.

    (A) Photonic band diagrams for dual triangular lattice without any symmetry reduction and with ϵ|| = μ|| = 14 (top) and triangulated triangular lattice with ϵ|| = μ|| = 14 with bianisotropic response ξxy = 0.2 introduced along with fillet of radius rf = 0.27 at each vertex (other parameters are the same as before) (bottom). Red bands correspond to the states of (pseudo)spin-up, and blue lines correspond to the states of (pseudo)spin-down. (B) Photonic band diagrams of the symmetric triangular lattice (black lines) overlapped with the triangulated rods and bianisotropy introduced. Numerically calculated transmittance for the spin-polarized source generating selectively (pseudo)spin-up or (pseudo)spin-down state for the transport at K (K′) valleys, as shown in the upper (lower) right panel. The analytically retrieved transmittance with second-order correction in k·p method at the K valley for pseudospin-down (spin up) is shown with a green (black) dashed line. The fitting parameters of the spectra extracted from the numerically calculated band diagrams are as follows: In the case of pseudospin-down, the frequency of the Dirac crossing at K point ω0a/2πc = 0.358, and Fermi velocity vD/2πc = 0.039. In domain (1), u1 = 0, m1 = 0, α1 = −0.05vD, β1 = −0.4vD; in domain (2), u2a/2πc = −0.003, m2a/2πc = 0.018, α2 = −0.001vD, β2 = −0.07vD;. In the case of pseudospin-up, parameters are the same as those in pseudospin-down, except m2 = 0, α2 = −0.04vD, β2 = −0.25vD. αi and βi are the coefficients of Embedded Image in effective Hamiltonian with their numerical values expressed in term of vD. (C) Simulated electric field intensity |E|2 distributions along the strip excited by the sources containing only one pseudospin component. The upper (lower) three panels correspond to pseudospin-down (spin-up) excitation. The strip consists of two domains of the first type on the left and right (1), which represent triangular lattices of circular rods with ϵ = μ, separated by domain type (2), in the middle, which is composed of the triangulated rods with bianisotropy. The domains (1) and (2) contain 2 × 25 and 2 × 20 unit cells, respectively. Black dashed lines denote the boundaries between domains. The source is placed in the right domain (1) in the upper panel and in the left domain (1) in the lower panel, as indicated by black arrows and white lines.

  • Fig. 4 Robustness of transmission through bulk Dirac bands.

    (A) Transmittance as a function of the width of the photonic potential barrier for the case of TR-invariant spin-valley–polarized bulk transport at the K-valley. Numerical and fitted analytical results are shown by dotted and dashed lines, respectively, for the case outside the bandgap region for the pseudospin-down and for the same frequencies for pseudospin-up polarizations (ν = 0.115 GHz). Solid lines correspond to numerical results within the bandgap (for pseudospin-down polarization) and at the same frequency for pseudospin-up polarization, which lies near the Dirac point (ν = 0.107 GHz). Fitting parameters are the same as ones in Fig. 3. (B) Band structure (left) for the case of 10% imbalance of mass terms (mI = 1.1mT) and transmission (right) through the structure for K and K′ valleys. Narrowband and limited in magnitude effect of imbalance is evidenced by nearly unity transmission over the most frequencies. The frequency of the Dirac crossing at the K′ point ω0a/2πc = 0.233, and Fermi velocity vD/2πc = 0.0227. In domain (1), u1 = 0, m1 = 0.0052; in domain (2), u2a/2πc = −0.021, m2a/2πc = m1/10, and the potential barrier length is 13a0. (C) Effect of disorder (δ variation in ϵ = μ) introduced at sites indicated by yellow color. Red dotted line shows the persistence of Klein tunneling of pseudospin-up state even for strong disorder for frequencies close to Dirac point. Blue dotted line is the complete reflection of pseudospin-down due to bandgap. The cases of frequencies remote from Dirac frequency are shown by solid lines. Note that in this case, the parabolicity of the pseudospin-up band leads to the destruction of Klein tunneling for weaker disorders.

  • Fig. 5 Edge states in the continuum.

    (A and B) Band diagram of a supercell consisting of two domains, which have opposite signs of mass terms for TRS and SIS reduction (A) and bianisotropy and SIS reduction (B). (A) The continuum of bulk modes is shown by red-shaded area, and a dispersion branch of the edge mode supported by the domain wall is depicted by a black line. (B) Gray- and color-shaded regions show dispersion for the continuum of bulk modes. Color shading indicates (pseudo)spin states of the continuum: blue for spin-down and red for spin-up. Red and blue solid lines are the bands corresponding to the edge states supported by domain wall for (pseudo)spin-up and (pseudo)spin-down states, respectively. (C to E) Numerically calculated field distributions in the crystal illustrating the excitation of the one-way edge mode at the domain wall. (C) Schematic of spin- and valley-controlled selection of peer-to-peer or broadcast communication between receivers/transmitters performed over edge and bulk states, respectively. Photonic states excited by K-valley–polarized (D) and K′ valley–polarized (E) pseudospin-up source. The source represents spatially modulated [as exp(ikxx)] spin-polarized current line placed at the right half of the domain wall and is shown by red dotted lines. The excitation frequency is Embedded Image. The quadratic growth of energy density of the edge state occurs because of the one-way buildup of the field along the edge. To ensure reflectionless propagation of the waves at outer boundaries, we add progressively increasing material losses in the background in the region, enclosing the entire simulated structure. Accordingly, darker blue color closer to the boundaries in (D) and (E) implies exponential decay of the fields due to these custom-built crystalline perfectly matched layers.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/5/eaap8802/DC1

    section S1. Low-energy effective Hamiltonians

    section S2. Nonreciprocal tunneling in photonic graphene with a single potential barrier

    section S3. Specific designs of spin-valley–coupled metacrystals with one-way Dirac cones

    fig. S1. Photonic bands and transmission spectra.

    fig. S2. Schematics of the meta-waveguide design and band structures.

  • Supplementary Materials

    This PDF file includes:

    • section S1. Low-energy effective Hamiltonians
    • section S2. Nonreciprocal tunneling in photonic graphene with a single potential barrier
    • section S3. Specific designs of spin-valley–coupled metacrystals with one-way Dirac cones
    • fig. S1. Photonic bands and transmission spectra.
    • fig. S2. Schematics of the meta-waveguide design and band structures.

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