Merging mechanical bound states in the continuum in high-aspect-ratio phononic crystal gratings

Mechanical bound states in the continuum (BICs) present an alternative avenue for developing high-frequency, high-Q mechanical resonators, distinct from the conventional band structure engineering method. While symmetry-protected mechanical BICs have been realized in phononic crystals, the observation of accidental mechanical BICs -- whose existence is independent of mode symmetry and tunable by structural parameters -- has remained elusive. This challenge is primarily attributed to the additional radiation channel introduced by the longitudinal component of elastic waves. Here, we employ a coupled wave theory to predict and experimentally demonstrate mechanical accidental BICs within a high-aspect-ratio gallium arsenide phononic crystal grating. We observe the merging process of accidental BICs with symmetry-protected BICs, resulting in reduced acoustic radiation losses compared to isolated BICs. This finding opens up new possibilities for phonon trapping using BIC-based systems, with potential applications in sensing, transduction, and quantum measurements.

Bound states in the continuum (BICs) constitute a unique category of states existing within the continuum spectrum of open systems, yet effectively decoupled from radiating waves that could dissipate energy.BICs have been extensively studied across various physical domains, including optics [1][2][3], acoustics [4][5][6], and mechanics [7,8].The distinctive features of BIC modes, including high quality factors and substantial mode sizes, have found applications in low-threshold lasing [9][10][11], ultrasensitive sensing [12][13][14], and efficient harmonic generation [15][16][17].The introduction of the BIC concept to mechanical systems has opened a new avenue for the creation of high-quality factor, high-frequency mechanical resonators, departing from the conventional approach of band engineering with suspended structures.Mechanical BICs can be realized due to symmetry mismatch between the mechanical mode and the outgoing radiation field [7,18]-termed symmetry-protected BICs-and destructive interference between two coexisting modes [8], known as Friedrich-Wintgen BICs.Notably, recent developments have enabled coupled mechanical BICs with colocalized optical modes in optomechanical systems [8,18].Compared to released mechanical micro-resonators, BICbased mechanical systems offer advantages such as enhanced thermal capacity and macroscopic mode sizes.These characteristics hold potential for mitigating thermal noises in quantum measurements [19][20][21][22] and enabling high-throughput sensing [23][24][25].
However, a crucial breed of BICs, referred to as accidental BICs, has yet to be realized in mechanical systems.Unlike their counterparts, accidental BICs do not necessarily require radiation-forbidden symmetry or depend on the interference of two coupled modes.Instead, they emerge "accidentally" under specific system parameters that permit zero-radiation solutions [1].Achieving accidental BICs in mechanical systems poses a substantial challenge, primarily due to an additional loss channel introduced by the longitudinal component of elastic waves [7,26], in contrast to electromagnetic waves.For a mechanical resonance to manifest as an accidental BIC, it must simultaneously decouple from both transverse and longitudinal radiative waves.Because of the distinct mechanism, accidental BICs can exist in proximity to symmetry-protected BICs by tuning system parameters, resulting in suppressed radiation loss for all surrounding modes.This merging-BIC mechanism has been shown to effectively mitigate radiation losses in structures affected by disorder-induced intermodal scattering [3,11,27,28].
Here we employ a coupled wave theory to predict and experimentally realize mechanical accidental BICs within a high-aspect-ratio gallium arsenide (GaAs) phononic crystal gratings.Additionally, the Love-wave accidental BIC can merge with the symmetry-protected BIC in the same acoustic band through single-parameter tuning.We observe the BIC merging process, resulting in enhanced mechanical quality factor of the merged BIC compared with isolated BICs.The demonstrated Lovewave BICs hold great promise for various applications.Particularly, Love waves offer advantages over Rayleigh waves in sensing as they effectively decouple from compressional waves in liquids [29][30][31].The BIC structure eliminates the need for multi-layered materials and complicated fabrication processes, potentially offering an improved method for utilizing Love waves in sensing applications.Moreover, the BIC phononic crystal gratings, featuring a large mode volume and a high f Q−product, provide a promising platform for exploring macroscopic quantum mechanical oscillators [32][33][34].

Results
Phononic crystal gratings with accidental me-  chanical BICs The GaAs phononic crystal grating with a double-slot unit cell is illustrated in Fig. 1a.For a one-dimensional phononic crystal grating aligned with the crystal axis of GaAs, the mechanical modes propagating along the x direction can be divided into two categories: Love-wave modes and Rayleigh-wave modes.The Love-wave mode is xz−plane-odd and thus only has y−displacement, while the Rayleigh-wave mode is xz−plane-even and thus only has coupled x− and z−displacements.For the Rayleigh-wave mode, regardless of its symmetry with respect to the yz−plane, it couples with radiation fields, because the x−polarized plane wave is odd and the z−polarized plane wave is even with respect to the yz−plane.For the Love-wave mode, which only has the y−displacement, it decouples from the radiation field if it is odd with respect to the yz−plane, because the y−polarized plane wave is even.Based on the symmetry analysis, the 1D mechanical grating only supports symmetry-protected BICs that are odd Love-wave modes.Table I lists common piezoelectric materials and their compatibility with symmetry-protected BICs in 1D phononic crystal gratings.We choose GaAs because it also has a shear piezoelectric component for excitation of the Love-wave modes.
We develop a coupled wave theory [3,37,38] for elastic waves to identify mechanical grating structures with accidental BICs (SI).The displacement field of the Bloch mode of the phononic crystal grating can be written as where r ∥ is the in-plane spatial vector, G = 2πme x /a (m ∈ Z) is the reciprocal vector and Q G (z) is the corresponding Fourier component.For Bloch modes in the vicinity of the 2nd Γ point, i.e., (± 2π a , 0), or equivalently on the 1st folded band near the Γ point (0, 0), Q ±2π/a (z) are the dominant Fourier components and Q 0 (z) is the only radiating component [3,[37][38][39].According to the acoustic coupled wave theory, the radiation amplitude Q 0 (z) of the Lovewave mode at the 2nd Γ point can be calculated by, to the leading order, where ρ G and C G 44 are the Fourier components of the density and elastic tensor component C 44 of the GaAs grating layer, ω is the frequency of the Bloch mode, is the Green's function, and the integral is performed in the grating layer (SI).Symmetryprotected BIC is realized for Q G (z) = −Q −G (z), which leads to cancellation of the two terms corresponding to G = (±2π/a, 0) in Eq. 1 and thus a vanishing radiation amplitude.On the other hand, when each individual term of Eq. 1 is zero, an accidental BIC is realized.Using Eq. 1, we can predict grating structures with accidental BICs, which are close to the ab initio simulation (SI).
Guided by the coupled wave theory and using finiteelement simulations (COMSOL), we designed GaAs phononic crystal gratings with both symmetry-protected BICs and accidental BICs on the same Love-wave band.Fig. 1b shows the mechanical band structure of the grating with a lattice constant a = 1198 nm, slot width g = 100 nm, slot depth t = 1000 nm and center pillar width w = 120 nm.The lowest Love-wave band (L 0 ) supports a symmetry-protected BIC at the Γ point and accidental BICs along the Γ − X line at k x = ±0.18π/a.The mode profiles of the BICs are shown in Fig. 1c.

Merging mechanical BICs
The k−position of the accidental BICs can be tuned by varying the grating parameters.For example, by reducing the lattice constant from 1198 nm to 1193 nm, the two accidental BICs shift towards the Γ point and eventually merge with the symmetry-protected BIC, forming a merged BIC (Figs. 2a and b).When the lattice constant is below 1193 nm, the grating structure only has one symmetry-protected BIC at the Γ point (Fig. 2c, a = 1165 nm).Such a merging behavior of the mechanical BICs is also illustrated by their topological charges.Because BICs have vanishing radiation amplitudes, the far-field polarization of BICs cannot be defined, which leads to singularity points in the far-field polarization map of Bloch modes (see Figs. 2a-c bottom panel).These singularity points are characterized by a topological charge, i.e., the winding number of the polarization surrounding them [7,39].During the merging process, the total topological charge is conserved [39].Figs.2a-c show the x− and y−components of the far-field polarization of Bloch modes of the L 0 band and the topological charges of mechanical BICs before, at, and after BIC merging.On the other hand, due to the odd symmetry of the Love-wave mode, the z−component 100 μm  The merging of BICs enhances the quality factor (Q) of the Bloch modes in the vicinity of the BICs [3,28,39].For an isolated BIC, the scaling of Q versus k x along the Γ − X line is given by Q ∝ 1/k 2 x for k x ≪ π/a (Fig. 2d red line) [3] (SI).In the presence of two accidental BICs adjacent to the symmetry-protected BIC, the x , leading to enhanced quality factor for the Bloch modes near the merged BIC (Fig. 2d yellow line).The Q-enhancement effect due to the merging BIC is also manifested at finite k y ≪ π/a, despite the disappearance of rigorous BICs for finite k y .Fig. 2e shows the simulated Q(k x ) for k y = π/400 µm −1 and a = 1198, 1193, 1165 nm.
This result indicates the effect of merging BIC can persist in finite grating structures.The standing-wave resonance of order {m, n} in finite grating structures is characterized by quantized momentum (k x , k y ) = (mπ/N a, nπ/L), where N is the number of grating periods and L is the length of the grating along the y di-rection.To illustrate the effect of merging BIC on the Q of standing-wave resonances, we simulate the Q of Bloch modes of finite (k x , k y ) for varying lattice constant across the merging point.In Fig. 2f, solid (dashed) lines represent the Q of Bloch modes with k y corresponding to n = 1 and L = 400 µm (L = 120 µm).For each k y , three different k x are simulated, k x = 0.09 π/a, 0.15 π/a, and 0.2 π/a, corresponding to green, purple, and gray lines.We find a peaked Q exists for all these cases as a result of merging BICs, and the value of the peaked Q becomes larger for smaller k x and k y as the Bloch mode approaches the merged BIC.
The enhanced quality factor due to the merging BIC is also manifested in grating structures with disorders [3,40].This is because the merging BIC enhances the quality factor of all adjacent Bloch modes as shown in Figs.2d and e, thus suppressing the radiation loss due to the disorder-induced intermodal scattering [3,40].To show this, we simulate a super-cell structure with 4 unit cells and each unit cell has a random variation of the slot width (standard deviation ≈ 1%).As shown in Fig. 2g, for the same disorder level, the Q of the grating with the merging BIC (a = 1193 nm) is higher than that of the grating with isolated BICs (a = 1165 nm).Such protection against disorders remains effective for a wide range   of k x and lattice constants near the merging-BIC design, which reveals the robustness of the merging-BIC mechanism.Since disorders are inevitable in fabricated structures, the merging BIC mechanism is expected to mitigate scattering losses compared to structures with only isolated BICs.Experimental demonstrations To experimentally demonstrate the accidental mechanical BIC and the BIC merging process, we fabricated GaAs phononic crystal gratings (Figs.3a and b; see Methods for device fabrication).We also fabricated interdigital transducers (IDTs) for piezoelectric excitation and probe of surface acoustic waves coupled with the phononic crystal grating (Fig. 3c).A vector network analyzer is used to measure the microwave transmission of the device via RF probes that contact with the IDTs.The mechanical mode spectrum of the phononic crystal grating thus can be inferred from |S 21 | 2 .For the IDT with a pitch about 1600 nm, a Lovewave mode with a frequency around 2 GHz can be excited (see Fig. 3d inset).A typical IDT transmission spectrum through the bulk GaAs without the phononic crystal grating is shown in Fig. 3d, where the number of IDT finger pairs is 50, resulting in a bandwidth of about 60 MHz.We then used such IDTs to probe phononic crystal gratings.The higher-frequency Love-wave bands of the grating correspond to higher-order modes along the z direction, which mode-mismatch with the fundamental Love-wave mode of the IDT.Consequently, the IDT transmission is significantly attenuated if the IDT frequency is higher than the L 0 band (along k x ) of the grating (Fig. 3e).In contrast, when the IDT frequency lies within the L 0 band (along k x ) of the grating, the IDT transmission is strongly modulated by the grating and multiple standing-wave resonances of the grating can be observed in the spectrum (Fig. 3f).
We fabricated three sets of phononic crystal gratings with different number of grating periods N and grating length L: (N = 50, L = 120 µm), (N = 100, L = 120 µm), and (N = 200, L = 400 µm).In each set, the lattice constant is scanned in a range of 60 nm around a = 1193 nm, corresponding to the merging BIC design, with a step size of 2 nm.We measured the microwave transmission of each device at room temperature and inferred the Q of the standing-wave resonances of the grating by fitting the transmission spectrum (SI).The Q of the observed lowest-order standing-wave resonance in each device is summarized in Figs.4a-c for the three sets of gratings.Within each set of gratings, the measured Q varies with the lattice constant and peaked at a lattice constant around a = 1193 nm.This is a direct demonstration of the BIC merging process and is consistent with the simulation (Fig. 2f).Comparing different sizes of gratings, the (N = 200, L = 400 µm) set has the highest Q at the merging point, because its standingwave resonances have the smallest k y , which agrees well with the simulated result shown in Fig. 2f.Figs.4d-f show the transmission spectrum of the gratings with the highest Q in the three sets.The measured transmission spectrum can be fit using a coupled mode theory (SI).For the transmission coefficient at the resonance frequency, |S 12 | 2 ∝ ( κe κ ) 2 , where κ ≡ ω/Q is the total dissipation rate of the standing-wave resonance and κ e is the coupling rate with the IDT-excited surface acoustic wave.As a result, given the similar noise floor in the transmission spectrum and signal-to-noise ratio, the lowest-order resonances observed in the three sets of gratings have a comparable κ e /κ ratio.We examine three gratings, with one from each set, that are furthest away from the merging BIC point.The Q of the lowest-order resonance of these three gratings have a ratio about ), and thus we infer Q e,N =50 : Q e,N =100 : Q e,N =200 ≈ 1 : 2 : 4 (Q e ≡ ω/κ e ).Our analysis shows that this can only be satisfied when the observed lowest-order resonances have a similar k x (= mπ/N a), which is found to be in the range of (0.04π/a, 0.08π/a) (SI).This is consistent with the lower Q observed for the N = 50 set compared to the N = 100 set, because the resonance of a smaller grating has a broader momentum distribution than that of a larger grating with the same k x (SI), leading to more radiation losses.
In conclusion, we have realized mechanical accidental BICs and their merging with symmetry-protected BICs in GaAs phononic crystal gratings.The merging BIC is experimentally verified via the observation of suppressed acoustic radiation loss as compared with isolated BICs.The acoustic coupled wave theory developed here can be used to explore accidental BICs and BIC merging in other mechanical systems beyond 1D gratings.The observation of mechanical merging BIC enables an alter-native approach for creating mechanical oscillators with high frequencies and macroscopic sizes, which hold great promises in various applications from signal transduction to sensing.

Methods
Device fabrication.The devices are fabricated on a GaAs wafer with [100] orientation.A 140 nm thick SiO 2 is deposited as the hard mask.The phononic crystal gratings are patterned by electron beam lithography using ZEP 520A as the mask.The pattern is transferred to the hard mask by inductively coupled plasma reactive ion etch (ICP-RIE) of SiO 2 using SF 6 and CHF 3 , and subsequently transferred to GaAs with another ICP-RIE using BCl 3 , Ar and N 2 .The IDTs are defined by a second electron beam lithography with PMMA as the mask, followed by electron beam evaporation of 10 nm chromium and 70 nm gold and subsequent lift-off process.The Green's function G(z, z ′ ) corresponding to Eq. S16 satisfies We assume z ′ ∈ (0, h).For z > z ′ , the Green's function can be taken as and for z < z ′ , we take the Green's function as Substituting Eq.S25 and Eq.S26 into Eq.S24, we obtain k 0(−1),z = Since the Green's function G and its normal derivative C 44 ∂ z G are continuous at the slab-substrate interface and C 44 ∂ z G is zero at the free surface z = h, we have and At z = z ′ , the Green's function should still be continuous, leading to In addition, integrating Eq.S24 from z = z ′ + 0 − to z = z ′ + 0 + leads to Solving these equations, we obtain where We now have the full Green's function for 0 ≤ z < z ′ as following ,zh e −ik0,z(z+z ′ ) + e −ik0,zh e ik0,z(z+z ′ ) + e ik0,zh e ik0,z(z−z ′ ) .

D. Radiation amplitude and accidental BIC
We can now calculate the zeroth-order Fourier component of the Bloch mode at the Γ point, which corresponds to the radiation amplitude.It is obained by solving Eq.S16 using the Green's function, Insert the solution of Eqs.S18 and S35, we obtain where the scaling factor in Eq.S35 has been ignored and we only calculate the contribution from G ′ = 2π a in order to identify the accidental BIC.The contribution from G ′ = − 2π a is identical but with an opposite sign.It is easy to see that Q 0 y (0 − ) = 0 is achieved when which corresponding to an accidental BIC at the Γ point.We plot the L.H.S. and R.H.S. of Eq.S38 in Fig. S1a and their ratio in Fig. S1b for a grating with g = 100 nm, t = 1000 nm, w = 120 nm, and h = 1656 nm.We find an accidental BICs can be realized for a ≈ 1200 nm.

III. TEMPORAL COUPLED-MODE THEORY FOR IDT-COUPLED PHONONIC CRYSTAL GRATINGS
The transmission spectrum of the IDT-coupled phononic crystal grating can be modeled using the temporal coupledmode theory.Consider the setup shown in the schematic of Fig. S2a, the temporal coupled-mode equations for the surface acoustic wave is given by and ... The relation between I 2 and V 1 can be obtained by solving the temporal coupled-mode equation, which is given by where G = −µ 1 g m2 .Thus the admittance Y 21 of the phononic crystal device is Comparing to the admittance of RLC series resonators in parallel, where ω n = 1/L n C n , we find the phononic crystal grating with multiple resonances can be modeled as RLC series resonators in parallel, with the equivalent circuit components: L n = 1/2Gκ n,e , R n = κ n /2Gκ n,e and C n = 2Gκ n,e /ω 2 n .
Fig. S2b shows the full equivalent circuit corresponding to the device probed by a network analyzer, where C T is the static capacitance of the IDT and the reference impedance of the port is R 0 = 50 Ω.The transmission spectrum can be derived as [2] where Z in is the total input impedence to the network analyzer as shown in Fig. S2b.We assume the system satisfies We used Eq.S45 to fit the measured transmission spectrum.

FIG. 1 .
FIG. 1. Phononic crystal grating with accidental mechanical BICs.a.An illustration of the GaAs phononic crystal grating and the unit cell.b.Mechanical band structure near the Γ point for the unit cell dimensions indicated in the text.L and R indicate Love wave and Rayleigh wave, respectively.Stars indicate the symmetry-protected BIC at the Γ point and the accidental BIC along Γ − X. c. Side and top view of the mode profile of the symmetry-protected BIC (left) and the accidental BIC (right).

FrequencyFIG. 3 .
FIG. 3. Microwave transmission spectrum of the phononic crystal grating.a and b.Scanning electron microscopy images of the GaAs phononic crystal grating.c.Optical microscopy image of a grating and IDTs.d-f.The microwave transmission spectrum of bulk GaAs (d), a grating with the L0 band mismatched with the IDT frequency (e), and a grating with the L0 band matched with the IDT frequency (f ).The IDT Love-wave mode profile is shown in the inset of d.

FIG. 4 .
FIG. 4. Observation of merging mechanical BIC.a-c.Measured Q of the observed lowest-order standing-wave resonance in the three sets of gratings with varying lattice constants.a: (N = 50, L = 120 µm), b: (N = 100, L = 120 µm), c: (N = 200, L = 400 µm).Red lines are polynomial fitting of the Q data.d-f.Microwave transmission spectrum of the gratings with the highest Q in the three sets.Right panels are zoom-in plots in narrow frequency range.Red lines are model fitting.Background-subtracted, fitted resonances are also shown with the amplitude measured by the linear scale on the right axis.
FIG. S1. a. Unit cell of the grating.b.The left-hand side (blue) and right-hand side (red) of Eq.S38.c.The ratio of left-hand side and right-hand side of Eq.S38.Accidental BICs correspond to ratio equal 1.
FIG. S2. a. Schematic of the IDT-coupled phononic crystal grating.b.Equivalent circuit for the device probed by the network analyzer.

TABLE I .
Elastic and piezoelectric properties of common piezoelectric materials.