Type-II Dirac point and extreme dispersion in one-dimensional plasmonic-dielectric crystals with off-axis propagation

Dirac cones exhibit linear crossing dispersion, which results in extraordinary topological properties in the energy bands. When the dispersion around the Dirac points is tilted, type-II or type-III Dirac points are present, which lead to intriguing transport phenomena. With the off-axis wave vector, a two-dimensional (2D) momentum space for one-dimensional (1D) crystals can be constructed. In this work, we revealed that the band structure in the extended 2D momentum space of carefully designed 1D plasmonic-dielectric crystals can be tuned to be highly tilted, forming type-II Dirac cones. In particular, an approximately upright dispersion can be achieved, which is referred to as a class-II Dirac point. Gapless interface states occur across the interface between two connected lattices formed by two distinct inversion-symmetry-broken crystals (i.e., type A and type B). The tilt of the band structure has a crucial influence on the formation of interface states and their sensitivity to the period, which results from the extraordinary properties of the surface impedance in the bulk gap. More importantly, class-II Dirac points exhibit peculiar topological properties. The wavelength area is divided into two regions, each of which can only support interface states for one configuration, type-AB or type-BA connections. Our results show that type-II Dirac points, especially those that are class II, have an underlying impact on the surface impedance and determine the existence of interface states.

Figure 1

(a) Schematic diagram of the multilayered lattices composed of four alternating stacks. (b) The dispersion of an inversion-symmetric crystal with $d_1=d_3=d_0=52$ nm. (c) The dispersion of the symmetric and antisymmetric states on plane $q=0$ of (b). (d) The trajectory of the DPs over the $(\lambda ,d_0)$ plane, where $d_2=8.5$ nm and $\mathrm{\Lambda }=d_1+d_2+d_3+d_4=308.5$ nm.

Figure 2

(a) Schematic plots of the inversion-broken lattices: type A and type B, connected in BA- and AB-type configurations. (b) The band structure of the inversion-broken crystal with $d_a=\pm 10$ nm for $d_0=52$ nm. (c), (d) $\mathrm{Im}\left(Z_l\right)$ for the left semi-infinite crystal (marked in blue) and $\mathrm{Im}\left(Z_r\right)$ for the right semi-infinite crystal (marked in red) as a function of $k_y$ for wavelength $\lambda =1450$ nm. Here $Z_0=376.73\mathrm{\Omega }$ is the vacuum impedance. The crossing points of the red and blue lines in (c), (d) indicate that $\mathrm{Im}\left(Z_l\right)+\mathrm{Im}\left(Z_r\right)=0$.

Figure 3

(a) Band structure of the inversion-symmetry-broken crystal for $d_0=116.37$ nm. (b) Band structure of symmetric and antisymmetric states for $q=0$ in (a). (c), (d) $\mathrm{Im}\left(Z_l\right)$ for the left semi-infinite crystal (blue curves) and $\mathrm{Im}\left(Z_r\right)$ for the right semi-infinite crystal (red curve) as a function of $k_y$ for $\lambda =1450$ nm.

Figure 4

(a) Band structure of the inversion-symmetry-broken crystal for $d_0=140$ nm. (b) Band structure of symmetric and antisymmetric states for $q=0$ in (a). (c), (d) $\mathrm{Im}\left(Z_r\right)$ for the right semi-infinite crystal (red curves) as a function of $k_y$ for four selected wavelengths on both sides of the DP in (b).

Figure 5

(a) Finite lattice configuration of type BA and type AB surrounded by air. The trajectories of the interface states in the gap regions for (b) $d_0=52$ nm and (c) $d_0=116.37$ nm crystal for type A ($d_a=10$ nm) and type B ($d_a=-10$ nm), respectively. The gray region indicates the projection of the bulk bands.

Figure 6

Schematic diagram of the multilayered lattice composed of four alternating stacks. The origin of the supercell is fixed at the center of metal layer B.

Figure 7

Dispersions near the DPs. The lines are the rigorous theoretical results obtained by the TMM, and the black squares are the results from the effective Hamiltonian.

Figure 8

The values of $\mathrm{Im}\left(Z_l\right)$ for a left semi-infinite crystal (marked in blue) and $\mathrm{Im}\left(Z_r\right)$ for a right semi-infinite crystal (marked in red) as a function of $k_y$ for a fixed wavelength $\lambda =1650$ nm with (a), (b) $d_0=52$ nm and (c), (d) $d_0=116.37$ nm.

Figure 9

The values of $\mathrm{Im}\left(Z_l\right)$ for a left semi-infinite crystal (marked in blue) and $\mathrm{Im}\left(Z_r\right)$ for a right semi-infinite crystal (marked in red) as a function of $k_y$ for a fixed $d_0=140$ nm with (a), (b) $\lambda =1200$ nm and (c), (d) $\lambda =1450$ nm.

Figure 10

$H_z$ distributions of the topological interface states for the BA-type connection with ten periods at $\lambda =1450$ nm from full-wave simulation. The crystals have $d_0=52$ nm and $d_a=10$ nm for the right type-A lattices. The left type-B lattices have (a) $d_a=-10$ nm, (b) $d_a=-20$ nm, (c) $d_a=-30$ nm, and (d) $d_a=-40$ nm.

Figure 11

The band structure for this 1D crystal with $q=0$ obtained by Eq. (1). The geometrical parameters of the lattice for (a), (b) are same as that in Fig. 2, i.e., $d_0=52$ nm and $|d_a|=10$ nm, and the same for (c), (d) is with $d_0=52$ nm and $|d_a|=20$ nm.

Figure 12

The field distribution of magnetic component $H_z$ for the BA-type connection. The geometrical structure for the left panel is the same as that in Fig. 10, and the right panel is the same as that in Fig. 10.