Deterministic Scheme for Two-Dimensional Type-II Dirac Points and Experimental Realization in Acoustics

Low-energy electrons near Dirac/Weyl nodal points mimic massless relativistic fermions. However, as they are not constrained by Lorentz invariance, they can exhibit tipped-over type-II Dirac/Weyl cones that provide highly anisotropic physical properties and responses, creating unique possibilities. Recently, they have been observed in several quantum and classical systems. Yet, there is still no simple and deterministic strategy to realize them since their nodal points are accidental degeneracies, unlike symmetry-guaranteed type-I counterparts. Here, we propose a band-folding scheme for constructing type-II Dirac points, and we use a tight-binding analysis to unveil its generality and deterministic nature. Through realizations in acoustics, type-II Dirac points are experimentally visualized and investigated using near-field mappings. As a direct effect of tipped-over Dirac cones, strongly tilted kink states originating from their valley-Hall properties are also observed. This deterministic scheme could serve as a platform for further investigations of intriguing physics associated with various strongly Lorentz-violating nodal points.

Figure 1

Band-folding creation of type-II DPs. (a) Initial sonic crystal, an acoustic-hard plate with bow-tie shaped blind holes arranged in a rectangular lattice. Green (orange) shaded region denotes a primitive (enlarged) unit cell. Perspective view of primitive cell in bottom-right inset, sectional side view in top-right inset. (b) Calculated band structure of (a) using enlarged unit cells depicted in inset, comprising sublattices A and B. Dashed lines: sound cone. (c) Folding mechanism of the first Brillouin zone when considering enlarged unit cells. Outer green region ($X{X}_P{M}_{P}M$) of the original FBZ are folded back into the inner orange region ($\mathrm{\Gamma }XMY$) along the fold line ($XM$). (d) Rotation (90°) of left holes (sublattice A) in enlarged unit cells transform them into real unit cells (orange shaded region). Inset: schematic of transformed unit cell. (e) Band structure of sonic crystal in (d). The 1st (red) and 2nd (blue) bands linearly touch each other in midpoint $D$ of the $XM$ direction, highlighted by the green dashed rectangle. Inset: distribution of type-II DPs in the Brillouin zone. Because of time-reversal symmetry, another inequivalent type-II DP exists at $D^{\prime }$. (f) 3D band structure around the $D$ point. At Dirac frequency 6.39 kHz (green, semi-transparent plane) of the type-II DP, its iso-frequency contour features a pair of crossing lines contacting at $D$. (g),(h) Pressure fields of degenerate doublets at $D$, corresponding to eigenvalues $\pm 1$ of mirror symmetry ${\mathrm{\Sigma }}_A$ indicated by orange dashed lines. Green scale bar: 10 mm.

Figure 2

Experimental imaging of type-II DPs. (a) Photograph of experimental setup for near-field mappings (see note S15 in the Supplemental Material for details [40]). Inset: close top view of the 3D printed sample. (b) Fourier spectra along high-symmetry directions of FBZ. Green dots indicate numerical dispersion. Only regions outside the sound cone are shown. (c) Fourier spectra at selected frequencies. Bright strips represent excited SSAW modes, indicating their IFCs in reciprocal space. (d) Numerically calculated IFCs at corresponding frequencies. Type-II DPs manifest themselves as the contacts of “electronlike” (red IFCs) and “holelike” (blue IFCs) Fermi pockets. Note that a Brillouin zone is topologically equivalent to a torus [43].

Figure 3

Tight-binding analysis of band-folding mechanism. (a) Tight-binding model of sonic crystal in Fig. 1. The $s$-orbital SSAW modes in sublattice A (B) are represented by red (blue) lattice points. Only nearest-neighbor hoppings are considered. (b),(c) Band structure near the $D$ point along the $x$ direction (b) or $y$ direction (c), calculated from simulations (sold lines) or the tight-binding model (scatters) in (a). (d) The tight-binding model when sublattice B is shifted with displacement $\delta \mathbf{r}=(\delta x,0)$, which breaks mirror symmetry ${\mathrm{\Sigma }}_A$ and creates hopping difference $\delta t_x$ along the $x$ direction. Inset: detailed schematic of displacement. (e) Band structure near the $D$ point along the $y$ direction for type-$P$ ($\delta x=+1\mathrm{mm}$) and type-$N$ ($\delta x=-1\mathrm{mm}$) unit cells, calculated from simulations (sold lines) or tight-binding model (scatters) in (d). (f) Berry curvature distributions for type-$P$ and type-$N$ unit cells, calculated from the tight-binding model in (d) (see Fig. S6 and Note S8 in the Supplemental Material [40]).

Figure 4

Observation of type-II valley-Hall kink states (VHKSs). (a),(b) Schematics of supercells periodic along the $y$ direction, formed by type-$P$ and type-$N$ unit cells, referred to as the PN-configuration (a) and NP-configuration (b), respectively. Dashed lines indicate kinks between two domains. (c) Numerically calculated projected band structure. Gray shaded regions correspond to bulk bands. Type-II VHKS of the PN-configuration (NP-configuration) is denoted by red (blue) solid line. (d),(e) Simulated pressure fields of type-II VHKSs at the projected $D$ point ($k_y=\pi /(2a_y)$), for the PN-configuration (d) and NP-configuration (e), respectively. Black arrows indicate where pressure is the strongest. (f),(g) Fourier spectra obtained from measured pressure field maps for the PN-configuration (f) and NP-configuration (g), excited with matched sublattice polarizations. Dark red (blue) dots indicate numerical dispersion of type-II VHKSs in the PN-configuration (NP-configuration). (h), (i) Experimentally imaged pressure fields at 6.40 kHz for the PN-configuration (h) and NP-configuration (i). Yellow stars indicate where source sounds are injected. (j),(k) Enlarged views of selected areas enclosed by black dashed rectangles in (h),(i), revealing sublattice polarizations of type-II VHKSs in the PN-configuration (j) and NP-configuration (k). Green scale bar: 16 mm.