A novel high-sensitivity triaxial accelerometer based on 3-D phononic crystals

Abstract

High-sensitivity triaxial acceleration sensing is widely applied from consumer electronics to inertial navigation. However, Conventional triaxial accelerometers are mainly rely on assembling multiple single-axis transducers, which increases device size and complexity. The acceleration-sensing mechanism of phononic crystals (PnCs) has demonstrated significant potential for high-sensitivity detection, owing to the high-quality factor of their resonant cavities. In this paper, based on the theory of 2-D PnCs single-axis acceleration sensing, we propose a novel triaxial accelerometer based on 3-D PnCs. Specifically, a point defect is introduced along the x and y axis, while a line defect is created along the z axis. Defective PnCs create a resonant cavity that produces sharp transmission peaks within the band gap. Acceleration causes a frequency shift in these peaks. This study employs the finite element method (FEM) to calculate the band structure of the 3-D PnCs, as well as the transmission spectra along all three axes. Both numerical and experimental results indicate that this approach is suitable for high-sensitivity triaxial acceleration sensing. The experimentally measured sensitivities along the x, y, and z axes are 1.58 kHz/g, 0.62 kHz/g, and 0.43 kHz/g, respectively, with corresponding measurement bandwidths of 37.1 Hz, 15.9 Hz, and 37.4 Hz. This innovative design based on 3-D PnCs enables triaxial high-sensitivity acceleration sensing within a single structural configuration and holds promise for future integration with ultrasonic transducers to form a compact microsystem.

Introduction

Accelerometers are key components in a wide range of applications, including consumer electronics¹, inertial navigation systems², structural health monitoring³, and aerospace engineering⁴. With the increasing demand for motion perception and inertial sensing, accurate multi-axis acceleration measurement has become essential in both civilian and aerospace systems. Most triaxial accelerometers were realized by fabricating three sensing structures oriented in different directions on a single silicon substrate, thereby forming a monolithic device^5–7. However, such designs often suffer from increased structural complexity and potential cross-axis coupling, which may degrade measurement accuracy.

According to the sensing principles, accelerometers can be broadly classified into capacitive^8,9, piezoelectric¹⁰, piezoresistive¹¹, resonant¹², and optical¹³ types. Capacitive accelerometers are widely used due to their simple structure and low power consumption, but their sensitivity is fundamentally constrained by parasitic capacitance, electrical noise, particularly in microscale devices. Optical accelerometers can achieve ultra-high sensitivity and immunity to electromagnetic interference; nevertheless, they usually require complex optical components and precise alignment, which significantly increases system complexity and hinders large-scale integration.

In recent years, phononic crystals (PnCs), characterized by periodic elastic structures and frequency band gaps¹⁴, have attracted growing interest in sensing applications^15–26. By introducing defects into PnCs, highly localized resonant modes with high quality factors can be formed within the band gap. The application of PnCs in sensing was first introduced by Lucklum et al. for liquid sensing^15–19. Since then, extensive efforts have been made to extend PnCs-based sensing to other domains, including gas sensing^20,21, multimodal physiological monitoring²², and biological sensing applications^23–26, demonstrating the broad versatility of PnCs in sensing technologies. Furthermore, in our previous work, we have demonstrated single-axis high- sensitivity acceleration sensing using two-dimensional (2-D) defect PnCs^27,28. Although PnCs-based accelerometers have demonstrated promising performance in single-axis sensing, their extension to triaxial acceleration measurement remains largely unexplored. In particular, how to simultaneously achieve multi-axis sensitivity and structural compactness using a unified PnCs architecture is still an open challenge.

In this paper, we propose a novel triaxial accelerometer based on three-dimensional (3-D) PnCs. By introducing point defects along the x- and y-directions and a line defect along the z-direction, a compact resonant sensing structure capable of triaxial acceleration detection is realized. The band structure and transmission characteristics of the proposed 3-D PnCs are investigated using the finite element method (FEM), and both numerical and experimental results demonstrate the feasibility and effectiveness of the proposed approach for high-sensitivity triaxial acceleration sensing.

Results

Principle of acceleration sensing

The proposed device comprises two primary components: an acoustic module and a mechanical module. The acoustic module includes a fixed PnCs slab and a movable PnCs slab, while the mechanical module consists of a proof mass, four folded beam and four anchors, as illustrated in Fig. 1a. In the PnCs section, the structure is composed of periodically arranged cross-shaped PnCs unit cells, as shown in Fig. 1b. The transverse characteristic dimension of the PnCs is denoted as d₁, the longitudinal characteristic dimension as d₂, and the lattice constant as a. The top view of the PnCs is illustrated in Fig. 1c, where a point defect (1 mm×1 mm) is introduced in the x-y plane. Correspondingly, the side view shown in Fig. 1d indicates that a line defect (0.4 mm) is constructed along the z direction. Through this defect-engineering strategy, acoustic resonant cavities are formed along the three orthogonal directions, namely x, y, and z.

Fig. 1. Structural design of the 3-D PnCs device.

a Schematic view of the PnCs device. b The 3-D unit cell of PnCs. c Top view of PnCs with a point defect. d Side view of PnCs with a line defect

When the PnCs anchored to the proof mass is subjected to accelerations along the x, y, and z axes, the structure undergoes corresponding displacements. The displacements lead to variations in the effective lengths of the point and line defects, which in turn induce shifts in the resonance frequencies of the acoustic cavities. Consequently, three-axis acceleration sensing can be achieved by monitoring the resonance frequency variations of the PnCs resonators. The structural parameters of the 3-D PnCs device design are outlined in Table 1, while the material parameters are detailed in Table 2.

Table 1.

The structural parameters of the 3-D PnCs device design

Parameters	Symbol	Value	Unit
Proof mass length	L1	36	mm
Proof mass width	W1	19	mm
Proof mass thickness	h1	7	mm
Folded beams length	L2	17	mm
Folded beams width	W2	0.5	mm
Folded beams thickness	h2	2.5	mm
Gap of suspension	L4	0.5	mm
Air gap	L5	5	mm
PnCs lattice constant	a	1	mm
PnCs diameter 1	d1	0.6	mm
PnCs diameter 2	d2	0.3	mm
Row of PnCs	R	9	–
Column of PnCs	C	9	–

Table 2.

The material parameters of the 3-D PnCs device design

Parameters	Symbol	Value	Unit
Density of air	ρair	1.29	Kg/m3
Density of resin	ρresin	1050	Kg/m3
Sound speed in air	cair	340	m/s
Sound speed in resin	cresin	1512	m/s
Elasticity modulus of resin	Eresin	2.4	GPa

Mechanical model

The mechanical component consists of a mass block and four folded beams, as shown in Fig. 1a. Accelerations parallel to the x,y and z axial cause the proof mass to move in the x-y, x-y and x-z plane. The static response of the PnCs device with a suspended mass of M (proof mass) and spring constant of k is defined by combining Newton’s second law and Hook’s law and is given by

$$a=-\frac{k}{M}\mathrm{\Delta }L=-{w_n}^2\mathrm{\Delta }L$$ 1

where a is the acceleration, w_n is the natural frequency of the system in the sensing direction, and ΔL is the relative displacement of the suspended mass (movable PnCs) with respect to the fixed PnCs. The stiffness of the folded beam can be theoretically calculated from its geometric parameters, and the equivalent stiffness can be expressed as

$$k_x=N\frac{E{h}_2{W_2}^3}{{L_2}^3}$$ 2

$$k_y=\frac{E{h}_2{W_2}^3}{N{L_2}^3}$$ 3

$$k_z=\frac{E{W}_2{h_2}^3}{N{L_2}^3}$$ 4

where N is the number of folds in the folded beam, E is the elasticity modulus of resin, L₂ is the folded beams length, W₂ is the folded beams width, and h₂ is the folded beams thickness.

To gain a deeper understanding of the impact of vibrational modes on the acceleration working bandwidth, a finite element model (FEM) was developed using COMSOL software. The first three mechanical vibrational modes of the PnCs device, with corresponding resonant frequencies of 19.05 Hz, 37.93 Hz, and 38.56 Hz, are illustrated in Fig. 2. In the modal analysis, the four anchors and the fixed PnCs are fixed, the movable PnCs slab together with proof mass are free. As the numerical results in Fig. 2, the first three mechanical modes correspond to translational motions along the y, x, and z axes, respectively. This gives a x-axial mechanical sensitivity of s_mx = 1/w_nx² = 1/ (2π×37.93 Hz) = 173μm/g (g = 9.81 m/s²), y-axial mechanical sensitivity of s_my = 1/w_n² = 685μm/g, z-axial mechanical sensitivity of s_mz = 1/w_n² = 167μm/g.

Fig. 2. First three mechanical vibration modes obtained from FEM.

a Mode 1 (19.05 Hz): Translational vibration along the Y-axis. b Mode 2 (37.93 Hz): Translational vibration along the X-axis. c Mode 3 (38.56 Hz): Translational vibration along the Z-axis

Band structure of the 3-D PnCs

The acoustic wave equation for a homogeneous isotropic fluid or gas medium can be written as follows

$$\mathrm{\nabla }\cdot \left(\frac{1}{\rho \left(\mathit{r}\right)}\mathrm{\nabla }p\left(\mathit{r}\right)\right)+\frac{w^2}{\rho \left(\mathit{r}\right)c^2\left(\mathit{r}\right)}p\left(\mathit{r}\right)=0$$ 5

where ρ is the density, c is the speed of sound and w is the angular frequency, r is a spatial vector, expressed as x₁e₁ + x₂e₂ + x₃e₃. According to Bloch’s theorem, the displacement of periodic structures can be expressed by

$$p\left(\mathit{r}+\overline{\mathit{r}}\right)=p\left(\mathit{r}\right)e^{i\mathit{k}\cdot \overline{\mathit{r}}}$$ 6

where $\stackrel{\circledR }{\mathit{r}}$ is a lattice vector, k = (k_x, k_y, k_z) is the Bloch wave vector. Due to the periodic nature of the PnCs, the governing equations can be formulated and solved within a single unit cell by applying periodic boundary conditions. Substituting Eq. (6) into Eq. (5) and performing finite-element discretization of the unit cell transforms the problem into a standard eigenvalue formulation given by

$$\left[\mathit{K}\left(\mathit{k}\right)-w\left(\mathit{k}\right)\mathit{M}\right]p\left(\mathit{r}\right)=0$$ 7

where p(r) is the eigenvector. K and M are the global stiffness matrix and mass matrix. Under periodic boundary conditions, the eigenfrequencies in Eq. (7) can be computed for an arbitrary wave vector k. it is sufficient to evaluate the wave vectors only along the edges of the irreducible Brillouin zone (BZ) to obtain the complete band structure. As shown in Fig. 3a, the irreducible BZ of the cubic lattice is indicated by the light-blue contour. Since only wave propagation along the Γ–X and Γ–R directions are considered in this work, the FEM calculations are performed for wave vectors sweeping along the boundary path Γ–X–M–Γ–R of the irreducible BZ. Along the boundary path Γ–X–M–Γ–R, k_x∈{[0, π/a], [π/a, π/a], [π/a, 0], [0, 0]}, k_y∈{[0, 0], [0, π/a], [π/a, 0], [0, 0]}, and k_z∈{[0, 0], [0, 0], [0, 0], [0, π/a]}. The calculated band structure obtained by the FEM method is shown in Fig. 3b, ranging from 0 kHz to 400 kHz. The light green shading in the figure indicates the frequency ranges of the band gaps. Along the Γ–X direction, the band gap spans from 126.15 kHz to 201.59 kHz, while along the Γ–R direction it ranges from 160.29 kHz to 177.52 kHz.

Fig. 3. Band structure and transmission characteristics of the PnCs device.

a Irreducible Brillouin zone for 3-D unit cells. b FEM calculated band structures of the unit cell. c Quasi-3-D defect PnCs model along the z axis. d Transmission spectra of the perfect and defect PnCs along the z axis. e Quasi-3-D defect PnCs model along the x and y axes. f Transmission spectra of the perfect and defect PnCs along the x and y axes

Transmission characteristics and resonance behavior

To evaluate the ultrasound transmission through a finite slab of the crystal lattice while reducing the computational cost, quasi-3-D models are constructed along the wave propagation direction, as shown in Fig. 3c and e. As shown in Fig. 3c, a 9×1×9 quasi-3-D defect PnCs model is constructed to simulate acoustic wave transmission along the z-axis. The light-yellow region represents the background pressure field, where a plane wave is launched from the z-direction. Periodic boundary conditions are applied on the two faces along the y-direction to emulate an infinitely periodic PnCs. Perfectly matched layers (PML) are imposed around the PnCs to suppress spurious reflections and to model an unbounded homogeneous medium after wave transmission through the crystal. The upper and lower boundaries of the PnCs along the z-axis are defined as the input integration region and the output integration region, respectively, and the transmission coefficient is calculated as

$$T_z=\frac{P_{out}}{P_{in}}=\frac{-\int I_{z}dV}{\int \frac{p^2}{2\rho c}dV}$$ 8

where p = 1 Pa is the amplitude of the incident plane wave, ρ and c are the density and sound speed of the air, respectively. I_z denotes the acoustic intensity component in the propagation z-direction. The quasi-3-D transmission model of the perfect PnCs follows the same formulation, with only the geometric configuration of the PnCs being modified. The transmission spectra of the perfect and defect PnCs are shown in Fig. 3d. FEM results show that the transmission coefficient exhibits significant attenuation in the frequency range of 157.6–183.6 kHz, corresponding to the Γ–R direction bandgap from 160.29 kHz to 177.52 kHz in Fig. 3b. This transmission behavior agrees well with the calculated band structure. A defect band appears within the bandgap, corresponding to the defect mode. FEM results reveal the z-axial defect band at a frequency of 167.6 kHz, as shown in Fig. 3d.

Then, a 9×9×1 quasi-3-D defect PnCs model is constructed to simulate acoustic wave transmission along the x-axis, as shown in Fig. 3e. Unlike the quasi-three-dimensional model with wave propagation along the z-axis, the background pressure field in this case is excited by a plane wave propagating along the x-direction. Accordingly, the transmission coefficient is calculated as

$$T_x=\frac{P_{out}}{P_{in}}=\frac{\int I_{x}dV}{\int \frac{p^2}{2\rho c}dV}$$ 9

where I_x denotes the acoustic intensity component in the propagation x-direction. And the transmission spectra of the perfect and defect PnCs are shown in Fig. 3f. FEM results show that the transmission coefficient exhibits significant attenuation in the frequency range of 124.0–204.7 kHz, corresponding to the Γ–X direction bandgap from 126.15 kHz to 201.59 kHz in Fig. 3b. This transmission behavior agrees well with the calculated band structure. Moreover, FEM results reveal the x-axial defect band at a frequency of 196.3 kHz, as shown in Fig. 3f.

Furthermore, the resonance characteristics associated with the line-defect and point-defect structures are compared in Fig. 4. As shown in Fig. 4a and c, the line-defect resonance exhibits a relatively broad transmission peak with a quality factor (Q-factor) of approximately 17, whereas the point-defect resonance presents a significantly sharper resonance peak with a much higher quality factor of 159. The Q-factor was calculated according to

$$Q=\frac{f_0}{\mathrm{\Delta }f}$$ 10

where f₀ is the resonance frequency and Δf is the full width at half maximum (FWHM) of the resonance peak. This difference mainly originates from the distinct energy confinement conditions of the two defect configurations.

Fig. 4. Resonance characteristics between the line-defect and point-defect PnCs structures.

a Transmission spectrum of the line-defect structure. b Acoustic pressure field distribution of the line-defect resonance mode. c Transmission spectrum of the point-defect structure. d Acoustic pressure field distribution of the point-defect resonance mode

For the line-defect structure, although a localized acoustic cavity resonance is still formed, the introduction of the defect channel required for x-direction displacement inevitably increases the acoustic coupling between the localized cavity mode and the surrounding propagating modes. As a result, part of the acoustic energy leaks through the defect pathway, leading to enhanced radiation loss and a reduced Q-factor. In contrast, the point defect forms a more isolated acoustic microcavity inside the PnCs lattice, which provides stronger acoustic confinement and suppresses energy leakage into the surrounding structure. Consequently, the point-defect resonance exhibits a substantially higher Q-factor and stronger resonance localization. The corresponding acoustic pressure field distributions are illustrated in Fig. 4b and d. In both cases, the acoustic energy is primarily localized around the defect regions, confirming the formation of defect-state resonances within the bandgap. However, compared with the point-defect structure, the line-defect configuration exhibits a relatively more extended energy distribution due to the existence of the displacement pathway, which further explains the increased acoustic leakage and the lower Q-factor.

In the proposed triaxial sensing architecture, the point-defect localized modes mainly serve as the sensing channels for the in-plane x- and y-axis accelerations owing to their strong lateral deformation sensitivity and highly confined acoustic localization. In contrast, the line-defect localized mode is primarily associated with the out-of-plane z-axis sensing response because the corresponding defect pathway is more sensitive to vertical structural deformation. Therefore, different defect configurations provide distinct directional mode localization characteristics, enabling partial axis selectivity and reduced cross-axis coupling in the proposed 3-D PnCs accelerometer.

Accelerations are applied independently along the x, y, and z directions, and the simulations show that the defect band shifts in frequency with the applied acceleration, as illustrated in the Fig. 5. As shown in Fig. 5, the resonance frequency of the PnCs increases linearly with the applied accelerations along the x, y and z axes, with coefficients of determination R² > 0.99 in all cases. The simulated acceleration sensitivities along the three axes are 5.74 kHz/g, 3.64 kHz/g, and 0.90 kHz/g for the x, y and z directions, respectively.

Fig. 5. Simulated acceleration sensing performance of the PnCs device.

a–c Simulated transmission spectra under acceleration (top) and the corresponding acceleration sensitivities (bottom) along the x, y, and z axes, respectively

In addition to the resonance-frequency shift, the influence of acceleration loading on the resonance characteristics of the PnCs defect cavity was further investigated by analyzing the evolution of the Q-factor under directional acceleration excitation. The simulated results are summarized in Fig. 6. For x-axis acceleration loading, the in-plane localized mode maintains a relatively stable Q-factor, varying from approximately 159 to 171 within the investigated acceleration range. The variation remains moderate and does not significantly affect the resonance identification process. Similarly, under z-axis acceleration loading, the out-of-plane mode exhibits an almost constant Q-factor around 16.7, indicating that the out-of-plane localized resonance remains highly stable under vertical acceleration excitation. In contrast, under y-axis acceleration loading, the Q-factor of the in-plane localized mode decreases from approximately 159 to 101 as the acceleration increases. This behavior is mainly attributed to the intentionally enhanced mechanical compliance along the y-direction. During the structural design process, the mechanical resonance frequency of the y-axis sensing structure was deliberately reduced in order to increase the mechanical sensitivity and minimize the sensitivity mismatch between the x- and y-axis responses. As a consequence, under the same acceleration loading, the y-axis experiences comparatively larger structural displacement and deformation. The increased deformation perturbs the acoustic confinement condition of the defect-localized resonant cavity more strongly, leading to increased acoustic leakage and a more pronounced reduction in the Q-factor. Nevertheless, despite the observed Q-factor variation, the resonance peak remains clearly identifiable throughout the investigated acceleration range, and the resonance-frequency shift still exhibits an approximately monotonic and near-linear dependence on acceleration.

Fig. 6. Evolution of the resonance quality factor (Q-factor) of the PnCs defect cavity under directional acceleration loading.

a Variation of the Q-factor under x-axis acceleration excitation. b Variation of the Q-factor under y-axis acceleration excitation. c Variation of the Q-factor under z-axis acceleration excitation

Cross-axis coupling analysis

To evaluate the cross-axis coupling behavior of the proposed 3-D PnCs accelerometer, directional acceleration loading was independently applied along the x-, y-, and z-axes, and the resonance-frequency shifts of both the in-plane localized mode (XY-mode) and the out-of-plane mode (Z-mode) were extracted from the simulated transmission spectra. The corresponding results are summarized in Fig. 7.

Fig. 7. The cross-axis coupling characteristics of the proposed 3-D PnCs accelerometer.

a Resonance-frequency responses under x-axis acceleration excitation. b Resonance-frequency responses under y-axis acceleration excitation. c Resonance-frequency responses under z-axis acceleration excitation

For x-axis acceleration loading, the XY-mode exhibits a clear approximately linear frequency shift with a main-axis sensitivity of approximately 5.74 kHz/g, whereas the Z-mode only shows a very weak perturbation with an equivalent cross-axis sensitivity of approximately −0.12 kHz/g, corresponding to a cross-axis ratio of approximately 1.98%. Similarly, under y-axis acceleration loading, the XY-mode remains the dominant sensing output with a sensitivity of approximately 3.64 kHz/g, while the Z-mode exhibits only a weak secondary response with an equivalent cross-axis sensitivity of approximately 0.24 kHz/g, corresponding to a cross-axis ratio of approximately 6.71%.

Under z-axis acceleration loading, the Z-mode exhibits a monotonic resonance-frequency shift with a sensitivity of approximately 0.90 kHz/g, indicating that the out-of-plane resonant mode serves as the primary sensing channel for z-axis acceleration. In contrast, the XY-mode only exhibits a non-monotonic perturbation rather than a stable linear frequency response. The maximum cross-axis perturbation observed in the XY-mode is approximately 0.22 kHz, which remains substantially smaller than the primary-axis frequency shift of the Z-mode. This behavior indicates that z-axis loading does not produce an effective in-plane sensing output, although a certain degree of modal perturbation remains due to the global structural deformation of the coupled PnCs structure. In addition, the comparatively lower primary-axis sensitivity of the Z-mode also contributes to the more noticeable relative perturbation observed in the XY-mode under z-axis excitation. Since the z-axis sensitivity is substantially smaller than the in-plane sensitivities, the influence of structural deformation on the XY-mode becomes relatively more pronounced under Z-mode loading.

These results indicate that the proposed 3-D PnCs architecture provides dominant-axis sensitivity together with reduced, but non-negligible, cross-axis coupling through defect-mode localization. In particular, the strongly localized in-plane defect modes exhibit dominant sensitivity to in-plane acceleration while exhibiting reduced sensitivity to out-of-plane loading. Conversely, the out-of-plane mode is more susceptible to structural perturbation owing to its comparatively weaker confinement and lower Q-factor. Therefore, although complete mechanical/acoustic decoupling is not achieved, the dominant-axis sensitivities remain substantially larger than the corresponding cross-axis responses, enabling distinguishable multi-axis frequency-output sensing. Since the present implementation employs two resonance outputs, namely the in-plane localized mode and the out-of-plane mode, the directional sensitivity behavior can be approximately represented by the following sensitivity matrix:

$$\mathbf{S}=\left[\begin{array}{ccc}5.74& 3.64& -0.19\\ -0.12& 0.24& 0.90\end{array}\right]\mathrm{kHz}/\mathrm{g}$$ 11

where the first row corresponds to the XY-mode response and the second row corresponds to the Z-mode response under x-, y-, and z-axis acceleration loading, respectively. The non-diagonal terms remain substantially smaller than the dominant-axis sensitivities, confirming dominant-axis selectivity despite incomplete decoupling.

Structurally, the partial decoupling originates from the spatial localization characteristics of the defect-state resonances and the anisotropic deformation behavior of the coupled cavity structure. In particular, the in-plane localized defect modes are primarily sensitive to lateral structural deformation induced by x- and y-axis acceleration, whereas the out-of-plane localized mode mainly responds to deformation of the line-defect region under z-axis excitation. This directional difference in modal confinement and deformation sensitivity suppresses cross-axis coupling and enables dominant-axis frequency responses.

Static calibration

The emitted ultrasonic signal is shown in Fig. 8a, which consists of the superposition of three Gaussian pulses with a repetition period of 1 ms. Its frequency-domain characteristics are presented in Fig. 8b, indicating that the Gaussian pulse has a center frequency of 200 kHz, corresponding to the ultrasonic transducer with a nominal center frequency of 200 kHz. The static z-axis test conducted at 160 kHz follows the same principle and is therefore not described in further detail. The measured reference signal and transmitted signal through the PnCs is shown in Fig. 8c. The ultrasonic pulses transmitted through the PnCs undergo pronounced dispersion and temporal broadening as a result of multiple scattering within the periodic structure. The received signals are subsequently processed using a fast Fourier transform (FFT) to obtain the corresponding transmission spectra. These spectra are then employed for normalization of the measured transmission response, as shown in Fig. 8d-i, and can be expressed as follows

$$T\left(w\right)=A_{transmitted}\left(w\right)/A_{reference}\left(w\right)$$ 12

where T(w) is the transmission coefficient, A_transmitted(w) is the amplitude of the transmitted spectrum, and A_reference(w) is the amplitude of the spectrum obtained without a sample. As shown in Fig. 8d-i, For the x, y, and z axes, the linear scale factors for the measurements were determined to be 1.58 kHz/g, 0.62 kHz/g, and 0.43 kHz/g, respectively. All demonstrated a high degree of linearity, with R² = 0.987, 0.990 and 0.981, respectively. The nonlinearity was evaluated using the best-fit nonlinearity definition, calculated as the maximum deviation between the measured response and the least-squares fitted linear response, normalized by the full-scale output

$$Nonlinearity(\%)=\frac{\max (y_{measured}-y_{fit})}{y_{FS}}\times 100\%$$ 13

where y_measured is the experimentally measured output, y_fit is the corresponding best-fit linear response, and y_FS is the full-scale output range. The maximum deviations from the ideal linear behavior were explicitly calculated. The best-fit nonlinearities for the x-, y-, and z-axes were determined to be 6.75%, 5.41%, and 6.06%, respectively. The observed nonlinearity is mainly attributed to several practical factors, including slight fixture misalignment during manual calibration, gravitational projection errors caused by angle uncertainty, variations in the acoustic transducer response, frequency tracking uncertainty during resonance peak extraction, and minor mechanical hysteresis of the 3D-printed supporting structure. Further improvements in linearity are expected through optimization of the alignment fixture, and improved frequency-tracking techniques in future work.

Fig. 8. Experimental characterization and acceleration sensing performance of the PnCs device.

a Gaussian pulse time domain signal. b Gaussian pulse frequency domain signal. c Reference signal and transmitted signal through the PnCs. d–f Measured transmission spectra under acceleration along the x, y, and z axes, respectively. g–i Corresponding acceleration sensitivities of the PnCs device along the x-, y-, and z-axes

The discrepancy between the experimentally measured sensitivity and the simulated results can be attributed to several factors. First, a quasi-3-D numerical model was adopted in the simulations, which cannot fully capture the complexity of the actual three-dimensional experimental configuration. Second, the sensitivity measurements were performed using a tilt-table method, where gravitational components acting on non-sensitive axes introduce additional forces proportional to g·sinθ, thereby affecting the measured response. Third, plastic deformation of the supporting beams during fabrication or testing may lead to variations in structural stiffness compared with the idealized model. Finally, nonlinear structural deformation under acceleration can induce changes in the effective defect length of the PnCs, which directly influences the defect mode frequency and, consequently, the device sensitivity. Nevertheless, the overall trend of the experimental results shows good agreement with the simulations, confirming the feasibility of the proposed sensing mechanism.

Dynamic response

The dynamic response characteristics of the proposed PnCs device were experimentally investigated using a vibration calibration platform. The fabricated PnCs device was mounted on a mechanical shaker to measure the acceleration-induced resonance-frequency response under harmonic excitation, following a configuration similar to that used in the static calibration measurements. During the experiments, the excitation frequency for the x-axis and z-axis measurements was swept from 15 Hz to 60 Hz with a frequency step of 1 Hz, while the excitation frequency for the y-axis measurement was swept from 4 Hz to 40 Hz with the same step size of 1 Hz. The acceleration amplitude was maintained constant throughout the sweep process. A sweep-frequency excitation experiment was subsequently conducted to evaluate the frequency-response characteristics of the device, and the corresponding results are presented in Fig. 9. The sensor bandwidth was evaluated according to the standard +3 dB criterion based on the normalized frequency-response curves. Specifically, the response amplitude at the low-frequency reference state was first normalized to 0 dB. The bandwidth was then defined as the excitation-frequency range from the initial frequency point to the frequency at which the normalized response amplitude first reached +3 dB during the frequency sweep. To improve the stability of the extracted response curves and suppress measurement fluctuations, adjacent-point smoothing and fitted resonance curves were employed before determining the 3 dB crossing point. The resonance-frequency evolution was subsequently extracted from the fitted response curves to ensure reliable bandwidth estimation under experimental noise conditions.

Fig. 9. Measured dynamic response characteristics of the proposed PnCs device under directional vibration excitation.

a Frequency response of the x-axis mode. b Frequency response of the y-axis mode. c Frequency response of the z-axis mode

The normalized response amplitudes were calculated from the extracted resonance-frequency-shift responses and subsequently converted into logarithmic dB scale for bandwidth analysis. The raw dynamic-response data, including the excitation frequency, resonance-frequency shift, and normalized response amplitudes for all three sensing directions, are provided in the Supporting Information (Tables S1–S3).

As shown in Fig. 9a, the x-axis response exhibits a resonance peak centered at approximately 39.0 Hz, with a corresponding 3 dB bandwidth of approximately 37.1 Hz. For the y-axis response shown in Fig. 9b, a sharper resonance peak is observed at approximately 18.1 Hz, accompanied by a narrower 3 dB bandwidth of approximately 15.9 Hz, indicating a relatively higher dynamic selectivity and stronger resonance localization along the y-direction. In contrast, the z-axis response in Fig. 9c exhibits a resonance peak at approximately 38.0 Hz with a 3 dB bandwidth of approximately 37.4 Hz. The experimentally observed resonance-frequency responses exhibit good agreement with these simulated modal frequencies.

Noise performance and stability analysis

The output frequency signal of the resonant accelerometer was continuously recorded overnight under static conditions. The resonance-frequency fluctuations of the three-axis PnCs accelerometer were characterized. The measured frequency sequences were subsequently converted into equivalent acceleration fluctuations according to the calibrated frequency-to-acceleration sensitivities of each axis. The corresponding acceleration noise densities were then obtained through FFT-based spectral analysis. As shown in Fig. 10a–c, the measured acceleration noise floors of the three sensing axes are approximately 1.0 μg/√Hz, 2.1 μg/√Hz, and 4.4 μg/√Hz, respectively. The reported values represent the equivalent acceleration noise estimated from experimentally extracted resonance-frequency fluctuations under the present transmission-spectrum tracking configuration. The in-plane localized modes exhibit lower noise floors compared with the out-of-plane mode, which is attributed to their significantly higher Q-factors and stronger mode confinement within the PnCs defect cavity. In contrast, the out-of-plane mode suffers from stronger acoustic leakage and reduced confinement, resulting in increased frequency fluctuations and higher equivalent acceleration noise.

Fig. 10. Noise performance and long-term stability characterization of the proposed PnCs device.

a–c Measured acceleration noise spectral densities (PSD) of the three sensing axes. d–f Allan deviation curves of the three sensing axes

To further evaluate the long-term stability, Allan deviation analysis was performed using the same frequency-tracking data. As shown in Fig. 10d–f, the minimum Allan deviations of the three axes are approximately 0.52 μg, 1.51 μg, and 1.84 μg, respectively. The Allan deviation curves exhibit the typical decrease at short integration times due to white-frequency noise averaging, followed by an increase at longer integration times, which is likely associated with environmental perturbations, including temperature drift, ambient vibration, and material-related fluctuations in the 3-D printed structure. The obtained results demonstrate that the proposed PnCs resonant accelerometer can achieve stable low-noise frequency readout using defect-localized resonant modes. The lower noise floor and improved stability of the in-plane modes further confirm the importance of high-Q localized resonance for enhancing the frequency stability and acceleration resolution of PnCs-based resonant sensors.

Among the environmental perturbations affecting long-term stability, temperature-induced resonance drift is expected to be one of the dominant contributors in the current uncompensated prototype. A first-order estimation of the temperature-induced frequency drift was performed by considering the dependence of the defect resonance frequency on the acoustic wave velocity and the defect length. According to our previously published work^27,28, the defect-state resonance frequency approximately satisfies

$$f_0\propto \frac{c}{L}$$ 14

where c denotes the acoustic wave velocity and L represents the characteristic structural length of the defect cavity. For the 3-D printed structure used in this work, the dominant temperature-dependent contribution is expected to originate from variations in the elastic modulus, which modify the acoustic wave velocity and consequently shift the resonance frequency. Thermal expansion of the structure may also contribute to the resonance-frequency variation. Accordingly, the equivalent acceleration drift induced by temperature variation can be estimated as

$${\alpha }_T=\frac{1}{S_f}\frac{df}{dT}$$ 15

where S_f is the experimentally calibrated frequency sensitivity of the accelerometer. This analysis indicates that temperature variation is expected to be a significant contributor to long-term bias drift in the current unsealed and uncompensated prototype. Future work will incorporate temperature monitoring, material calibration, and compensation algorithms to further improve the thermal stability of the proposed PnCs accelerometer.

Discussion

Table 3 presents a comparison of the acceleration sensitivity between the proposed device and previously reported representative frequency-output accelerometers employing different resonance mechanisms. It should be noted that most of the reported devices are based on mechanical resonator frequency modulation, whereas the proposed sensor utilizes acoustic defect-state resonance in a 3-D PnCs cavity. Therefore, the comparison is intended to provide a qualitative benchmark of acceleration-frequency transduction capability rather than a strict comparison between identical physical resonance mechanisms.

Table 3.

Comparison of acceleration sensitivity reported for representative resonant and acoustic frequency-output accelerometers

Number	Reference	X axis (kHz/g)	Y axis (kHz/g)	Z axis (kHz/g)	Mechanism
1	Han et al.29	0.021	0.012	–	Mechanical
2	Yang et al.30	0.053	0.052	0.032	Mechanical
3	Caspani et al.31	0.260	0.23	–	Mechanical
4	Yadav et al.32	0.033	0.033	0.019	Mechanical
5	Ding et al.33	0.298	–	–	Mechanical
6	Pinto et al.34	0.022	–	–	Mechanical
7	Park et al.35	0.189	–	–	Mechanical
8	Yin et al.36	0.244	–	–	Mechanical
9	Mansoorzare et al.37	0.300	–	–	Mechanical
10	Fang et al.38	0.876	–	–	Mechanical
11	Li et al.39	0.442	–	–	Mechanical
12	Shin et al.40	0.427	–	–	Mechanical
13	This work	1.58	0.62	0.43	Acoustic

As shown in Table 3, the proposed 3-D PnCs triaxial accelerometer achieves competitive sensitivity performance among representative frequency-output accelerometers, particularly while simultaneously enabling three-axis sensing within a single compact structure. Compared with conventional triaxial accelerometers that typically rely on the integration or assembly of multiple single-axis resonant sensors, the proposed design realizes multidirectional sensing through defect-state engineering within a unified phononic crystal architecture, thereby reducing structural complexity and device size.

In addition, the proposed acoustic defect-state resonance mechanism provides stable and distinguishable resonance-frequency shifts under acceleration loading, enabling reliable frequency-output sensing. Although the sensing mechanism differs fundamentally from conventional MEMS mechanical resonant accelerometers, the obtained acceleration-frequency transduction sensitivities demonstrate that the proposed PnCs-based approach possesses strong potential for high-sensitivity triaxial inertial sensing applications.

In summary, by introducing point defects along the x and y axes and a line defect along the z axis, a compact resonant sensing architecture capable of triaxial acceleration measurement is realized within a single PnCs structure. We have explored the design, modeling, and performance of a PnCs-based acceleration sensor, highlighting its potential for high sensitivity. Compared with conventional triaxial accelerometers that rely on the assembly of multiple single-axis sensors, the proposed approach offers a more compact and structurally integrated solution with reduced system complexity. Owing to its high sensitivity and scalability, this 3-D PnCs-based accelerometer shows strong potential for future integration with ultrasonic transducers (CMUTs) or Piezoelectric Micromachined Ultrasonic Transducers (PMUTs), enabling the development of compact, high-performance microsystems for inertial sensing and related applications.

Materials and methods

Experimental setup

The experimental setup used for static calibration of the PnCs-based acceleration sensor is illustrated in Fig. 11. The PnCs device was fabricated using a nanoArch S140 3D printing system (BMF Precision Technology Co. Ltd., China). To characterize the transmission response and acceleration sensitivity along different directions, an ultrasonic immersion transmission measurement system was established, as shown in Fig. 11a, while the corresponding experimental photograph is presented in Fig. 11b.

Fig. 11. Experimental setup used for the static calibration of the proposed PnCs-based acceleration sensor.

a Schematic diagram of the ultrasonic transmission measurement system. b Photograph of the experimental measurement platform

In the measurement system, a signal generator (SIGLENT SDG1010) was employed to generate Gaussian pulse excitations, which were subsequently amplified by a high-voltage amplifier (ATA-2041, Aigtek) and applied to the transmitting ultrasonic transducer. Ultrasonic transducers with center frequencies of 200 kHz and 160 kHz were utilized for the characterization of the x-axis and the y/z-axis responses, respectively. The ultrasonic wave propagated through the PnCs device immersed in water and was detected by the receiving transducer positioned coaxially beneath the sample. The received signals were acquired and digitized using an oscilloscope (RIGOL DS1104), followed by data processing and transmission spectrum analysis on a PC.

To perform static acceleration calibration, the entire measurement system was mounted on a tiltable platform, as illustrated in Fig. 11a. By varying the inclination angle θ, a controllable gravitational acceleration component was introduced along the sensitive direction of the device, which can be expressed as

$$a=g\cdot \sin \theta$$ 16

The sensitive axis was switched among the x-, y-, and z-directions by adjusting the orientation of the PnCs device, as shown in the inset photographs of Fig. 11b. During the experiments, the variation of the acoustic resonance frequency under different inclination angles was monitored to evaluate the acceleration sensitivity of the proposed sensor.

Prior to sample measurements, a reference calibration was conducted without the PnCs device in the acoustic path in order to compensate for the intrinsic frequency response of the ultrasonic transducers and the measurement system. The reference transmission signal was subsequently used for normalization of the measured spectra.

The experimental setup used for evaluating the dynamic response characteristics of the proposed PnCs-based acceleration sensor is illustrated in Fig. 12. The setup consists of an ultrasonic transmission measurement system integrated with an electrodynamic mechanical shaker for dynamic excitation. As shown in Fig. 12a, a signal generator (SIGLENT SDG1010) was employed to generate Gaussian pulse excitation signals, which were amplified using a high-voltage amplifier (ATA-2041, Aigtek) and applied to the transmitting ultrasonic transducer. Ultrasonic transducers with center frequencies of 200 kHz and 160 kHz were used according to the selected sensing direction. The transmitted ultrasonic signals passing through the PnCs device were received by the aligned receiving transducer and subsequently acquired using an oscilloscope (RIGOL DS1104). The transmission spectra and resonance frequency variations were further processed on a PC.

Fig. 12. Experimental setup used for the dynamic characterization of the proposed PnCs-based acceleration sensor.

a Schematic diagram of the dynamic measurement system. b Photograph of the experimental dynamic calibration platform

For dynamic characterization, the PnCs device was rigidly mounted on an electrodynamic mechanical shaker, as shown in Fig. 12b. A reference accelerometer was attached directly to the shaker platform adjacent to the sensor in order to monitor the actual excitation acceleration during the frequency sweep. The accelerometer signal was fed back to the vibration controller to realize closed-loop control, thereby maintaining a nearly constant acceleration amplitude throughout the measurements. This procedure ensures that the measured frequency response reflects the intrinsic dynamic behavior of the sensor rather than variations in the excitation level. The sensitive axis of the device was changed by adjusting the orientation of the PnCs structure, as shown in the inset photographs in Fig. 12b, enabling characterization along the x-, y-, and z-directions.

The dynamic response was characterized by monitoring the resonance-frequency variation of the transmission spectrum under harmonic vibration excitation. Specifically, the output quantity used to construct the amplitude–frequency response was the amplitude of the resonance frequency shift extracted from the transmission spectra at each excitation frequency. During the measurements, the excitation frequency was swept over the target frequency range while maintaining a constant acceleration amplitude using the closed-loop feedback system. At each frequency point, the transmission spectrum was recorded and the resonance frequency was determined through peak tracking of the transmission minimum/maximum. The dynamic response curve was subsequently obtained from the extracted resonance-frequency-shift amplitude as a function of excitation frequency.

Geometric tolerance sensitivity analysis

To evaluate the robustness of the proposed 3-D PnCs accelerometer against fabrication uncertainty associated with the 3-D printing process, a geometric tolerance sensitivity analysis was performed by introducing ±5% perturbations to several representative structural parameters, including the lattice constant a and the PnCs diameters d1 and d2, which is consistent with the typical dimensional uncertainty of the employed fabrication process. The corresponding resonance frequencies, Q-factors, and directional acceleration sensitivities are summarized in Table 4.

Table 4.

Geometric tolerance analysis of the proposed 3-D PnCs accelerometer

Case	f0,xy (kHz)	f0,z (kHz)	Qxy	Qz	Sx (kHz/g)	Sy(kHz/g)	Sz (kHz/g)
Nominal	196.26	167.32	159	17	5.74	3.64	0.90
a = 0.95 mm	207.10	175.76	220	18	6.64	4.19	1.01
a = 1.05 mm	187.02	159.63	179	16	5.24	3.55	0.82
d1 = 0.57 mm	196.04	167.75	141	17	5.63	4.56	0.87
d1 = 0.63 mm	196.71	166.82	224	17	5.99	3.52	0.94
d2 = 0.285 mm	196.71	167.15	240	17	5.99	3.62	0.93
d2 = 0.315 mm	196.74	167.45	226	17	6.00	3.64	0.87

The results indicate that the defect-localized resonance modes remain preserved under all investigated perturbation cases. For the in-plane localized mode, the resonance frequency f_0,xy varies within approximately 5% under lattice-constant perturbation, while the variations induced by d₁ and d₂ perturbations remain comparatively small. Similarly, the out-of-plane resonance frequency f_0,z also exhibits stable behavior under moderate geometric perturbation. These results demonstrate that the proposed defect-cavity resonance mechanism is structurally robust against moderate dimensional variation. The Q-factor exhibits comparatively larger variation than the resonance frequency and sensitivity. This behavior is physically expected for defect-localized resonant systems, since small geometric perturbations can significantly modify the acoustic confinement condition and the associated energy leakage of the localized cavity modes. Nevertheless, despite the observed Q-factor variation, the resonance peaks remain clearly identifiable in all investigated cases, and the resonance-frequency shifts maintain approximately linear relationships with acceleration loading, as shown in Supporting Information.

The extracted directional sensitivities also remain comparatively stable under geometric perturbation. In particular, the x-axis and z-axis sensitivities vary only moderately across all investigated cases, while the y-axis sensitivity exhibits relatively larger variation. This behavior mainly originates from the intentionally enhanced mechanical compliance of the y-axis sensing structure. During the structural design process, the effective mechanical resonance frequency along the y-direction was deliberately reduced in order to improve the y-axis mechanical sensitivity and reduce the sensitivity mismatch between the x- and y-axis responses. Consequently, the y-axis experiences comparatively larger structural deformation under the same acceleration loading, making its resonance characteristics more sensitive to geometric perturbation. Overall, the proposed 3-D PnCs accelerometer maintains stable resonance characteristics and acceleration-sensing functionality under moderate fabrication tolerances. These results indicate that the proposed sensing mechanism possesses acceptable robustness for practical fabrication using 3-D printing techniques.

The detailed transmission spectra together with the corresponding resonance-frequency fitting results for all perturbation configurations and sensing directions are provided in the Supporting Information (Figs. S1–S9), further confirming the stability of the defect-localized resonance modes and the robustness of the acceleration-frequency response under geometric perturbations.

Author contributions

X.G. developed the research framework, implemented the modeling approach, and prepared the manuscript draft. J.W. assisted in model validation. J.N. and Y.J. contributed to methodology refinement and interpretation of the results. B.M. and W.L. provided strategic direction for the research and oversaw the overall project development. Y.Y. supported the preparation of figures and supplementary materials. Z.Y. provided overall supervision of the project and critically reviewed the manuscript.

Data availability

Data will be made available on request.

Conflict of interest

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41378-026-01390-2.