Topological phononics arising from fluid-solid interactions

Nontrivial band topologies have been discovered in classical systems and hold great potential for device applications. Unlike photons, sound has fundamentally different dynamics and symmetries in fluids and solids, represented as scalar and vector fields, respectively. So far, searches for topological phononic materials have only concerned sound in either fluids or solids alone, overlooking their intricate interactions in “mixtures”. Here, we report an approach for topological phononics employing such unique interplay, and demonstrate the realization of type-II nodal rings, elusive in phononics, in a simple three-dimensional phononic crystal. Type-II nodal rings, as line degeneracies in momentum space with exotic properties from strong tilting, are directly observed through ultrasonic near-field scanning. Strongly tilted drumhead surface states, the hallmark phenomena, are also experimentally demonstrated. This phononic approach opens a door to explore topological physics in classical systems, which is easy to implement that can be used for designing high-performance acoustic devices.


Supplementary Note 1 Band structures slightly away from high-symmetry planes
To confirm the touching points between the 1st and 2nd bands are nodal rings, we calculate band structures along directions that are slightly away from high-symmetry planes. For comparison, the band structures on high-symmetry planes k z = 0 and k z = π/a z are plotted in Figs. S1(a) and S1(b). The calculated band structure with k z = 0.1π/a z is shown in Fig. S1(c), which confirms that the red touching points in Fig.   S1(a) are now gapped. We then consider the band structure with k z = 0.9π/a z , and the result in Fig. S1(d) confirms that the blue and green touching points in Fig. S1(b) are also gapped when slightly away from the high-symmetry planes.

Supplementary Note 2 Calculation of bands without fluid-solid interaction
To demonstrate that the fluid-solid interaction cannot be ignored in the "mixture" phononic crystal, we calculate the band structure without the interaction. In this case, the phononic crystal are modeled only with the Pressure Acoustics module in COMSOL Multiphysics. The longitudinal sound speed of aluminum c l = 6100 m/s is employed in the numerical calculations, and the calculated band structure is shown in Resultantly, the SH modes and Lamb modes (both FL and EL modes) that involve transverse components disappear.

Mode profiles of the first four bands around Γ point
We plot the mode profiles at k = (0.1π/a 0 , 0, 0) for the modes of the first four bands in Fig. S3. It can be seen that when k z = 0, the FL modes and WG modes have opposite parities with respect to the mirror symmetry M z (z → −z).

Emergence of blue and green nodal rings
To illuminate the origin of blue and green nodal rings, we start from metallic plates without holes as considered in Fig. 2 The green and blue nodal rings exist for these through holes, and the diameter of holes d 0 only slightly affect them, which suggest their emergence are highly dependent on the acoustic resonance of the through holes. For more discussion, please see Supplementary Note 17.

Supplementary Note 7
Band structure with no mirror symmetries We consider a unit cell with no mirror symmetries as shown in Fig. S7(a). It consists of two holes with different diameters on the two sides of the aluminum plate, which breaks the mirror symmetry M z (z → −z). To break the other mirror symmetries M x (x → −x) and M y (y → −y), we also shift the holes on the back side of the aluminum plate with the displacement on the x (y) direction being δ x (δ y ), as indicated in the inset of Fig. S7(a). The detailed geometric parameters we choose are d 1 = 2.4 mm, d 2 = 0.8 mm, t 1 = 1.0 mm, t 2 = 1.0 mm, δ x = 0.5 mm, and δ y = 0.5 mm. The calculated band structure along high-symmetry directions is shown in Fig. S7

Supplementary Note 8 Comparison with the phononic crystal immersed in air
We calculate the band structure of the phononic crystal immersed in air, and it can be seen that the deterministic type-II nodal ring on k z = 0 plane still exists, as shown in Fig. S8(a). However, it is obvious that the size of the nodal ring is significantly shrunken due to the smaller sound speed of air compared with water. Further, the sound is also much more difficult to be transmitted due to the extreme impedance mismatch between air and aluminum which leads to very weak fluid-solid interaction.
In fact, we have simulated excitation of the phononic crystal in water and air at the corresponding frequency of the red nodal ring on k z = 0, respectively. In simulations, the phononic crystal is comprised of 10 layers of the drilled aluminum plates, and the average out-of-plane displacement for each layer is shown in Fig. S8(b). It is seen that the sound can hardly be transmitted when the phononic crystal is immersed in air, since the specific acoustic impedance of aluminum is four orders of magnitude larger than that of air.

Calculation of dispersions and derivation of effective Hamiltonian
Here, we use a transfer matrix method to model the phononic crystal that is comprised of periodic solid plates (aluminum) without holes immersed in the background fluid (water). To obtain an analytical model, we first calculate the dispersions of the first two modes of the phononic crystal in the long-wavelength limit, which allows us to model the solid plates based on the thin plate theory. We assume the time-harmonic condition and follow the e -iωt sign convention. As shown in Fig.   S9(a), we assume k y = 0 for simplicity as the system is rotation-invariant, and the acoustic pressure in the n-th water region can be written as The out-of-plane displacement in the n-th solid plate can be written as (S3) The thin plate theory then gives the dynamic equation for the n-th solid plate is the bending stiffness of the thin plate, and ( , ) the gradient on the plate surface. From Eq. (S4), we can solve C n as the function of A n , B n , A n+1 , and B n+1 Then, we use the boundary conditions of continuous velocity on two surfaces of the n-th solid plate, which give and it can be simplified under the time-harmonic condition as Combining Eqs. (S5) and (S7), we can eliminate C n and solve A n+1 and B n+1 as the function of A n and B n , and we obtain the transfer matrix M with the matrix element where * denotes the complex conjugate. Since the system is periodic and Hermitian, we can utilize the criteria on the trace of the transfer matrix which states that [1] tr( ) 2 cos( ) where k z is the Bloch wave number in z direction. Then, it gives the equation which determines the dispersions of the modes (S11) When k z = 0, one solution gives the WG mode, which coincides with the sound cone projected over k z . The other solution in the long-wavelength limit when k z = 0 gives the FL mode, (S13) 14 As can be seen, this dispersion is similar to that of a thin plate in free space , except its quadratic coefficient is reduced due to the fluid-solid interaction. We then plot the first two bands around Γ point from full-wave simulations and from Eqs. (S12) and (S13) in Fig. S9(b), and they agree quite well.
We then derive the effective Hamiltonian of the red nodal ring. Equating the two , it is found that the two modes will cross each other at (S14) Based on Eq. (S14), we plot the evolution of k x0 with respect to t w and compare that with values extracted from full-wave simulations, and the results agree quite well.
Next, we expand Eq. (S11) around k 0 = (k x0 , 0, 0) and ω 0 = ck x0 to the second order of δk x , k z , and δω, and we obtain the equation where δk x = k x −k x0 and δω = ω−ω 0 . Comparing Eq. (S15) with the characteristic equation of the effective Hamiltonian which is we can conclude that the parameters are and obviously we have |v 0 /v r |>1, confirming the touching points form a type-II nodal ring. We need to point out that, since k x0 is not small enough for the thin plate assumption regarding our geometric parameters, the derivation above based on the thin plate assumption overestimate the slope of the FL mode v 0 +v x , and also underestimate the value of k x0 , as shown in Fig. S9(c). In contrast, the expression for v z can quantitatively predict the dispersions along k z direction since we start from k z = 0, as shown in Fig. S9(d). The effective Hamiltonian in the main text can be obtained after we replace the x direction with a general direction in x-y plane, that is, k x is replaced by k r , k x0 is replaced by k r0 , etc.
To correct the values of k x0 , we have also considered a rigorous model of full elasticity instead of the thin plate theory. With full elasticity, the in-plane displacement u n and out-of-plane displacement w n are expressed using the scalar potential φ n (x, z) and the vector potential (z component) ψ n (x, z) The potentials are expressed as and the velocity potential is expressed as  (S24) The normal stress and shear stress are calculated as , , After elimination of C 1n , C 2n , D 1n , and D 2n by combining Eqs.
from which we can numerically retrieve the k x0 . In this semi-analytical way, the retrieved k x0 from the full-elasticity theory are also plotted in Fig. S9(c), and they agree excellently with those from full-wave simulations. Therefore, it confirms that the error in our model is indeed due to the thin plate assumption. Nevertheless, the analytical model validates that the deterministic type-II nodal ring originates from the interplay of ultrasound in the aluminum plates and water, which results in different asymptotic behaviors of the FL and WG modes in the long-wavelength limit.
where V w and V m denoting the fluid and solid domains of the unit cell, respectively. In the integrals, p 1 (r) and p 2 (r) are periodic part of pressure field for 1 n and 2 n , e 1 (r) and e 2 (r) are periodic part of strain field for 1 n and 2 n . The elastic strain tensor e is defined as where u i (i=1,2,3) are components of displacement and x i (i=1,2,3) are coordinates, respectively. C is the stiffness tensor connecting stress and strain of the solid, which is expressed as For simplicity, the symmetric 3×3 strain tensor is rearranged into a 6-component vector e = [e 11 , e 22 , e 33 , 2e 12 , 2e 13 , 2e 23 ] T . The particle velocity of ultrasound in the fluid is w w p iwr In fact, since the phononic crystal has mirror symmetry with respect to z direction, the Zak phase Zak 1 θ can also be inferred from parities of the eigenmodes at high symmetry points along the k z path (k z = 0 and π/a z ). Namely, we have [4] Zak 1 ,1 where M z,1 are parities (±1) for mirror symmetry M z (z → −z) of the 1st band at k z = 0 and π/a z , respectively. The field maps of the eigenmodes at k z = 0 and k z = π/a z for k r = (k x , k y ) = (0.6π/a 0 , 0) and (0.8π/a 0 , 0) are shown in Figs. S10(c) and S10(d), respectively. It shows that, for mirror symmetry M z , the eigenmodes at k z = 0 and π/a z have the same parity when k r = (0.6π/a 0 , 0), but opposite parities when k r = (0.8π/a 0 , 0). This contrast confirms that the Zak phase takes the value 0 for the former case and π for the latter case, as we have numerically demonstrated. The three-dimensional band structure of the DSSs is retrieved and plotted in Fig.   S11.
FIG. S11. Three-dimensional band structure of DSSs. The cyan surface represents the three-dimensional band structure of the DSSs for the supercell terminated by water.

Distribution of energy in supercell for DSSs
We plot the distribution of energy for the DSSs along inward direction of the supercell at k r = (0.85π/a 0 , 0) and k r = (0.65π/a 0 , 0.65π/a 0 ) in Fig. S12. The energies are calculated by integrals of energy densities in the water and aluminum plates, respectively, and the maximum is normalized to unity. It can be seen that the energies are both distributed in aluminum plates and water for the DSSs.

Type-II Weyl points from lowering symmetries
We lower the symmetry of the unit cell by adding two orthogonal through holes on the aluminum plates, as shown in Figs. S14(a) and Fig. S14(b). The geometric parameters are t m = 3.0 mm, t w = 2.0 mm, d 0 = 1.6 mm, d 1 = 1.0 mm, and δ z = 0.9 mm.
The calculated band structure shown in Fig. S14(c) confirm that the red nodal ring is now broken, giving rise to two pairs of type-II Weyl points at the diagonals of k z = 0 plane.

Fluid-solid interaction and topological phase
We consider the case of stacked metallic plates with blind holes immersed in water. We start by modeling the aluminum plates with pressure acoustics only, the same way as water. The fluid-solid interaction won't come into play, and we simply employ the density ρ = 2700 kg/m 3 and sound of speed c = 6300 m/s for aluminum.
The calculated band structure is shown in Fig. S16(a), and it can be seen that only one mode emerges from Γ point, and there is no nodal ring for the first band and no sign of topological acoustic effects.
In comparison, we include the fluid-solid interaction properly in our work. The calculated band structure in our way, as demonstrated in Fig. S16(b), shows that the red nodal ring is formed between the 1st and 2nd bands due to the FL mode and the with the metallic plate assumed to be acoustically rigid. The calculated band structure is shown in Fig. S17(a), and it can be seen that only the blue nodal ring on k z = π/a z exists, as demonstrated in Fig. S17(b). On the other hand, the red nodal ring arising from the fluid-solid interaction is missing because the plates are now acoustically rigid and cannot support flexural Lamb modes. The results suggest that the blue nodal ring on k z = π/a z is due to the degeneracy between the waveguide (WG) mode and the acoustic resonance mode of the through holes. Both modes still exist when the plates are acoustically rigid. To demonstrate this point, we plot the field maps of the two modes around the blue nodal ring at k = (0.83π/a 0 , 0.75π/a 0 , π/a z ). As shown in Fig.   S17(c), the two modes indeed have opposite parities with respect to the mirror symmetry M z , as expected. On the other hand, if the holes are blind holes on the rigid plates, for example, with a separation t h = 1.0 mm between holes on two sides of the plates, we can see that the blue nodal rings will disappear, as shown in Fig. S17(d).
This fact is because if the holes are blind holes, the frequency of their first-order acoustic resonance will be significantly increased because the effective length of the holes are greatly reduced.
In other words, we can effectively tune the blue nodal rings by tuning the acoustic resonance mode, through changing the thickness of the plates t m . For example, we consider the case that the holes are through holes, and the thickness of rigid plates is t m = 1.0 mm, while other geometric parameters remain unchanged. The calculated band structure is shown in Fig. S17(e). The blue nodal rings expand and reconnect after touching each other, now centered around R point in the reciprocal space, as demonstrated in Fig. S17(f). The green nodal rings now also appear on k x = π/a 0 and k y = π/a 0 planes, connected with the blue nodal rings. In fact, the existence of green nodal rings, when the blue nodal rings are centered around R point, is guaranteed by the mirror symmetries with respect to the k x = π/a 0 and k y = π/a 0 planes in the reciprocal space. The mirror symmetries lead to the opposite orientations of the blue nodal rings on opposite sides of the k x = π/a 0 and k y = π/a 0 planes [5].
In summary, the through holes introduce an acoustic resonance mode, which has opposite parities with the WG mode when k z = π/a z , and they can form nodal rings on the high-symmetry planes in the reciprocal space. In contrast, if the holes are blind holes, because the resonance modes of the holes are significantly shifted towards higher frequencies, there will be no nodal rings when k z = π/a z .

Supplementary Note 19
Bulk state at low frequency range when probing surface states The states that emerge at low frequency range in the Fourier spectra when probing surface states are bulk states rather than surface states. It is because in experiments, we cannot prevent the excitation of bulk states. To confirm this point, we have performed simulations with a source to excite the phononic crystal structure, and the retrieved Fourier spectra are shown in Fig. S19(a). It can be seen that the bright stripes also emerge at low frequencies, just as they do in the experimental results ( Fig.   4(d) in the main text), which is also shown here as Fig. S19(b). We would like to mention that, because of the limited RAM of our workstation (512 GB), we can only

Control experiment with lattice constant 4 mm
We change the in-plane lattice constant of the perforated holes to a 0 = 4 mm and carried out additional experiments on the new control sample. Other geometric parameters are kept unchanged. The calculated band structure along high-symmetry directions is shown in Fig. S20(a). The photographs of the new control sample is shown in Fig. S20(b). The new Fourier spectra experimentally retrieved along high-symmetry directions for bulk bands is shown in Fig. S20(c). For the new sample, good agreement is observed between the bright stripes in the experimental Fourier spectra and the calculated bulk bands projected along k z direction. For comparison, the retrieved Fourier spectra when a 0 = 3 mm in the main text is also demonstrated here as Fig. S20(d). From the shift of the bright stripes which generally overlap with the projected bulk modes calculated from full-wave simulations, we can conclude that the observed signals are indeed owing to the bulk bands of the phononic crystals.
We can consider about the DSSs. The projected band structure and the retrieved