Emergence of Cascading Flat Bands in Breathing Superlattices

Abstract

Flat bands have become a pillar of modern condensed matter physics and photonics owing to the vanishing group velocity and diverging density of states. Here, we present a paradigmatic scheme to construct arbitrary flat bands on demand by introducing a new type breathing superlattice, where both the number and spectral positions of isolated flat bands can be continuously tailored by simply controlling the breathing strength. Microscopically, the momentum-independent interband scatterings near the band edge protect them robust against weak intra-cell disorder. By dimensional reduction, we establish a duality between the one-dimensional (1D) breathing superlattice and the 2D Harper-Hofstadter model, where cascade flat bands naturally emerge as the different orders of Landau levels in the weak magnetic flux limit. As a proof of concept, photonic flat bands at optical frequencies are experimentally demonstrated with all-dielectric photonic crystal slabs. Finally, we generalize our scheme to 2D systems to realize partial and omnidirectional flat bands, and discuss the achievement of high-quality factors. Our findings shed new light on the manipulation of flat bands with high band flatness and large usable bandwidth, paving the way for the development of advanced optical devices.

Figures (6)

• Figure 1: Schematic of the breathing superlattice where the flat band eigenfield is localized in the shrunken sublattice.
• Figure 2: (a) Band structures of a 1D breathing superlattice with $N=14$ for TM polarization, where $K=\pm q_0/2$ and $\Gamma=0$. The fitting TB parameters are $\mu_i = 730$, $t_0 = 60$ THz, $\delta_T = 0.25$. The fitting CM parameters are $\mu=850$ , $m^*=1.7$, and $(w_1,w_2,w_3,w_4)=(18,2.88,-4.32, 0.72)$ THz. (b) Cross-sectional view of the proposed PhCS. (c, d) Electric field distributions at the $\Gamma$ point for higher and lower flat bands.
• Figure 3: (a-h) Evolution of flat bands for the of PhCSs with $N=14$ and $\delta$ varying from $0$ to $0.14$. Here, the lattice constant is $a=2.66 ~\mu m$. (i,j) Dependence of the bandwidth ($\Delta\lambda$) of conduction/valence bands on $\delta$.
• Figure 4: (a) The CM lattice model in momentum space. (b) Permittivity distribution for $\delta=0.07$ (blue) and 0.23 (purple). (c) Fitted (purple) and numerical (dashed) evolution of the zeroth flat band $E^{(0)}_q(0)$ with $\gamma$. (d) Band structure calculated with the $\hat{\Theta}$ operator for $\delta=0.23$, $N=14$ (dashed purple), numerical diagonalization for $\delta=0.17$ (solid blue), and the effective model (dashed red) (see Supplementary SI S5 for such effective model). (e) $\phi_{\text{Zak}}$ for the first $4$ bands in (h). (f) Dependence of the energy spectrum on $q_0$, where red points correspond to $q_0=2\pi/N a_0$ with even $N$. (g) Dependence of DOS on frequency and $N$ for $\delta=0.03$. (h) Band structure calculated from $\hat{\Theta}$ for $N=40$, $\delta=0.15$.(i) Energy spectrum of the finite lattice with respect to $k_{\phi}$ (see Supplementary SI S6 for calculations' detail). (j) Topological edge states at frequencies marked in (i). Inset: Field patterns of the edge states.
• Figure 5: (a-c) SEM images of fabricated $\mathrm{TiO}_{2}$ PhCS. Simulated (d) and measured (e) photonic bands for the TM-polarization with the structure parameters $N=24, \delta=0.018, \kappa=0.6, a_0=202.95,h=144~ {nm}$ and $h_s=20~ {nm}.$ (f) Evolution of DOS of flat bands on intra-cell disorder for $100$ samples. (g) FWHM extracted from the experimentally measured spectrum (see Supplementary SI S13), which indicating the leakage of the eigen-modes. (f) and (g) share the same energy window marked in (e). (h-j) Measured and calculated bands (dashed) for the TM-polarization of PhCSs with $N=14$. The structure parameters are tabled in (see Supplementary SI Tab. S2) and the dashed line are GME band structures. (k-l) RCWA calculated high Q flat bands for $0$th and 1$st$ LLs of a free standing sample.