Type-II Dirac phonons in a two-dimensional phononic crystal

When the effective mass of a particle approaches zero, a linear dispersion emerges from the solution of the Dirac equation.¹ Over the past decade, extensive research has been conducted on the Dirac conical dispersion in electromagnetic,^2–4 acoustic,^5–8 and mechanical wave systems.^9,10 Numerous interesting phenomena, including zero-reflective-index properties^11–18 and topological transitions,^{2–8,19–21} are directly linked to the distinctive features of Dirac cones. When certain non-Hermitian perturbations are introduced, the Dirac point can be deformed and spawn a ring of exceptional points (EPs),^22–29 which is associated with unique opportunities such as unidirectional transparency,^30,31 super-prism effect,²⁵ and single-mode lasers.^32,33

Dirac points can be categorized into two classes: type-I Dirac points exhibit a point-like Fermi surface and vanishing density of states (DOS) at the frequency of the Dirac point, while type-II Dirac points feature strongly titled dispersion, conical-like Fermi surfaces, and non-vanishing DOS at the frequency of Dirac point.^34–38 Although less prevalent than its higher-dimensional counterpart, the type-II Weyl point,^39–46 the manifestation of a type-II Dirac point, serves to enhance comprehension and facilitate research in the realm of topological semimetals with strongly tilted dispersion.

In this study, we present a straightforward realization of a type-II Dirac point in a two-dimensional phononic crystal (PC). In contrast to previous implementations of type-II linear dispersions, which are predominantly reliant on the intricate design of unit cell structures,^34–46 our approach leverages lattice symmetry and band folding. We construct a rectangular lattice of unit cells by deforming a supercell consisting of two copies of square unit cells. This configuration induces a type-II Dirac point at the Brillouin zone boundary, accompanied by a conical-like Fermi surface in the equi-frequency curve (EFC). The PC is experimentally realized using simple LEGO bricks. When PT-symmetric non-Hermitian periodicity is introduced, the type-II Dirac point is deformed into a titled oval of EPs. Meanwhile, the response of the PC to the incident waves from the left and right sides becomes disparate and the scattering matrix of the PC undergoes a phase transition. In contrast to one-dimensional PT-symmetric systems, which yield unidirectional transparency at a single frequency, our phase transition results in two curves of EPs in the eigenvalues of the scattering matrix, corresponding to broadband unidirectional transparency. Within the titled oval of EPs, the PC with PT-symmetric non-Hermitian periodicity exhibits characteristics of both a coherent perfect absorber (CPA) and a laser oscillator when there is an appropriate phase difference in the waves incident from different sides.

We start with a two-dimensional PC depicted in the inset at the upper left corner of Fig. 1(a). The PC has a supercell comprising two solid rods, which are treated as sound-hard bodies in the air. The lattice constant is 2a in the x direction and a in the y direction. Both rods have a square cross section with side length L = a/3. We calculate the band structure of this supercell along the boundaries of the rectangular irreducible Brillouin zone, using the finite-element package COMSOL Multiphysics, and plot the results in Fig. 1(a). The dimensionless frequency, 2πc/a, is adopted, where c is the speed of sound in air. The band structure in Fig. 1(a) can be interpreted as the folded band structure of the square unit cell along the k_x = π/(2a) line, resulting in a doubly degenerate line along the XM boundary. We then modify the supercell by moving the two square rods closer until they touch, as shown in the inset of Fig. 1(b). The original double degeneracy along the XM boundary is lifted, and the bands split into two linear branches with distinct slopes, intersecting at $k x , k y = π 2 a , 0.505 π a$⁠. In other words, a type-II Dirac point is induced below the sound cone, represented by the blue lines in the band structure.

Enlarged views of the dispersion surface and the EFCs at different frequencies near the type-II Dirac point are plotted in Figs. 1(e) and 1(f), respectively. A tilted conical dispersion is clearly observed. Above or below the frequency of the Dirac point $ω D =0.33 2 π c / a$⁠, the EFC shows two untouched quadratic branches at the Brillouin zone boundary. At the frequency ω_D, the EFC exhibits a cross at the Dirac point with conical-like Fermi surfaces. Such EFCs distinguish the type-II Dirac point from type-I Dirac points, which have a point-like EFC at ω_D.

Due to the simple structure of the PC, we can construct it using LEGO bricks. As shown in Fig. 2(a), we assemble the LEGO bricks on a thin LEGO baseplate, featuring a periodic square array of fixing points for the bricks. Each unit cell measures 48 × 24 mm², with two 1 × 1 LEGO bricks affixed at the center to form a scatterer with a rectangular cross section of 16 × 8 mm² and a height of 9.6 mm. The small cylindrical studs on the baseplate and the top of the bricks do not impact our experimental results due to their small size. Since the type-II Dirac point is located outside the sound cone, the associated bands can manifest as acoustic surface waves near the PC–air interface (supplementary material). This proves advantageous as the real-space field distribution can be conveniently measured by raster-scanning the top of the PC with a microphone, as shown in Fig. 2(b). Subsequently, a two-dimensional Fourier transform is applied to convert real-space fields to k-space to obtain the EFC. In our experiment, a loudspeaker with a small opening port is placed at the edge of the PC as the sound source. In Fig. 2(c), two arcs are observed in the EFC at the frequency of Dirac point 4800 Hz. The brighter arc corresponds to the forward propagation wave, and the dimmer arc arises from the backscattering wave components in the PC. These arcs represent the different branches of phononic bands in the EFC. The numerically calculated EFC at the frequency of the Dirac point is plotted in Fig. 2(d), showing excellent agreement with the experimental results. If we position the source on the other edge of the PC, the resulting EFC will be the mirror image of the original EFC. Together, they constitute the complete EFC at the Dirac point’s frequency, with the arc crossings representing the type-II Dirac points.

The difference between S₁ and S₂ is the way the vector of the output field is written.^30,52 For the same PC slab discussed earlier, the eigenvalues $λ 1 ′$ and $λ 2 ′$ of S₂ are plotted in Figs. 5(a) and 5(b), exhibiting a unimodular $( λ 1 ′ = λ 2 ′ = 1 )$ behavior in the PT-exact phase and inverse $( λ 1 ′ = 1 / λ 2 ′ )$ in the PT-broken phase.^30,52 At the CPA-lasing point, $λ 2 ′$ goes to the maximum, while $λ 1 ′$ goes to zero. Figure 5(c) exhibits the simulated results of such a typical CPA-laser. The incident wave comes from both sides, with frequency $ω=0.332 2 π c / a$ and incident angle θ = 50.42°. When the phase difference of the incoming waves is Δφ = φ_L − φ_G = 0.5π, the coherent interference of incident waves confines the acoustic pressure field in the loss regions of the PC, resulting in perfect absorption. On the contrary, when the phase difference of the incoming waves is Δφ = 1.5π, the acoustic pressure field is confined in the gain regions of the PC, and the outgoing waves are strongly amplified. The frequency and incident angle of CPA-lasing point correspond to the position of the original Dirac point.

In summary, we have theoretically designed and experimentally demonstrated the existence of an acoustic type-II Dirac point in a two-dimensional PC. Through the implementation of a rectangular supercell, a band-folding mechanism was employed to generate doubly degenerate bands along the Brillouin zone boundary. By deforming the lattice, the band degeneracy is lifted, resulting in the emergence of a strongly tilted Dirac point. Notably, when introducing a PT-symmetric non-Hermitian perturbation, the band dispersion exhibits a titled oval of EPs, indicating the transition from the PT-symmetric phase to the PT-broken phase. The band dispersion of the PC can be accurately characterized by the $k ⃗ ⋅ p ⃗$ method. Interestingly, there exists broadband unidirectional transparency along two lines on the dispersion surfaces where the eigenstates can perfectly couple with externally incident plane waves in the air. In the PT-broken phase, the PC can function as both a laser oscillator and a CPA under specific conditions of phase difference between the left and right sides. Our proposed approach establishes a versatile platform for future investigations on the characteristics and applications of type-II Dirac points in photonic and phononic crystals.

See the supplementary material for the band structure near the type-II Dirac point in the phononic crystal, analysis of parity-time symmetric non-Hermitian perturbations in our phononic crystals, acoustic pressure field distributions of eigenstates in the PT-exact phase and the PT-broken phase, surface impedance of eigenstates associated with EPs of the scattering matrix, reverse direction of unidirectional transparency, and amplification near the CPA-lasing point.

This work was partially supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. OSR-2020-CRG9-4374 and KAUST Baseline Research Fund No. BAS/1/1626-01-01. C.X. also acknowledges the support from the Jiangsu Specially Appointed Professor Program (Grant No. 164080H00244). G.M. was supported by the Hong Kong Research Grants Council (Grant Nos. RFS2223-2S01, 12302420, and 12301822). J.M. was supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515010322).

The authors have no conflicts to disclose.

Changqing Xu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Jun Mei: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Writing – review & editing (supporting). Guancong Ma: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (supporting). Ying Wu: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.