A3 Lamb-Wave Resonators

• A3 is the third-order antisymmetric Lamb-wave mode defined by its three half-wave thickness profile and impact on acoustic dispersion in thin-film piezoelectrics.
• In Z-cut LiNbO3 devices, integrating sub-wavelength through-holes preserves key metrics—about 21 GHz frequency, ~4% k², and Q ranging from 400 to 800—while enhancing mechanical rigidity and thermal conduction.
• The design standardizes etch distances across the wafer, streamlines multi-resonator filter integration, and minimizes spurious modes for robust, scalable K-band resonators.

Searching arXiv for the specified paper and closely related A3 Lamb-wave resonator work. In thin-film piezoelectric acoustics, A3 denotes the third-order antisymmetric Lamb-wave mode of a plate resonator. In the LiNbO$_3$ implementation reported in "K-band LiNbO$_3$ A3 Lamb-wave Resonators with Sub-wavelength Through-holes" (Wu et al., 2024), the mode is realized in a Z-cut single-crystal thin film and used as the basis of K-band resonators that incorporate sub-wavelength through-holes in the suspended region. In that formulation, A3 is both a modal designation—the third root of the antisymmetric Lamb-wave dispersion relation—and a device platform whose defining claims are preservation of operating frequency, electromechanical coupling coefficient, and quality factor, together with elimination of extra spurious modes and reduction of the ineffective suspension area by means of a through-hole release strategy (Wu et al., 2024).

1. Modal definition and governing equations

The A3 Lamb-wave mode in a thin piezoelectric plate is the third-order antisymmetric flexural mode. In an isotropic or weakly anisotropic plate, the coupled piezoelectric Lamb modes satisfy the characteristic equation

$$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$

with

$$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$

where $v_L$ and $v_T$ are the longitudinal and transverse bulk acoustic velocities, and $k=\omega/v_{\text{phase}}$. The antisymmetric A3 branch is the third root of $D(k,\omega)=0$ (Wu et al., 2024).

Its displacement field is written as

$$u_z=U_0\cos(kx)e^{-i\omega t}[A\sin(pz)+B\cos(qz)],$$

$$u_x=\frac{ik}{p}U_0\cos(kx)e^{-i\omega t}\left[A\cos(pz)-\frac{p}{q}B\sin(qz)\right],$$

with coefficients $_3$0 and $_3$1 fixed by the boundary conditions. The mode has three half-waves across the plate thickness, expressed as $_3$2 (Wu et al., 2024).

These relations define A3 as a thickness-structured Lamb-wave branch rather than merely a label for a particular resonator geometry. In the cited LiNbO$_3$3 implementation, that modal structure is the basis for high-frequency operation in the K band.

2. Resonator configuration in LiNbO$_3$4

The reported A3 resonator uses a Z-cut LiNbO$_3$5 single-crystal piezoelectric plate with total thickness $_3$6. The electrodes are Au, with thickness $_3$7, arranged as interdigitated transducers perpendicular to the Y-axis. The electrode geometry is parameterized by the finger width $_3$8, gap $_3$9, aperture $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$0, electrode duty cycle

$$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$1

and a design-dependent number of finger pairs $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$2 (Wu et al., 2024).

The distinctive structural addition is a set of sub-wavelength through-holes. In Designs I–VIII, the holes are circular, have diameter $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$3, are uniformly etched in the suspended region, and are placed at spacing $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$4 along the propagation direction. The holes lie between adjacent electrodes and extend through the LN film (Wu et al., 2024).

Parameter	Value or description
Piezoelectric plate	Z-cut LiNbO$$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$5, $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$6
Electrodes	Au, $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$7, interdigitated, perpendicular to Y-axis
Through-holes	Circular, $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$8, spacing $$D(k,\omega)\equiv (q^2-k^2)^2\tan(ph)+4k^2pq\tan(qh)=0,$$9

This geometry is central to the paper’s claim that the release strategy can be altered without altering the principal acoustic figures of merit. The holes are not introduced as a secondary trimming feature; they are part of the suspension and release architecture.

3. Resonance, coupling, and quality-factor description

The resonant frequency $$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$0 is obtained by solving the dispersion equation with a wavenumber matched to the interdigital transducer periodicity $$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$1. In practice,

$$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$2

For half-thickness $$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$3 and the third antisymmetric branch, the relation

$$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$4

yields $$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$5 for LiNbO$$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$6 (Wu et al., 2024).

The electromechanical coupling coefficient is extracted from the series and anti-resonance frequencies:

$$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$7

or equivalently,

$$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$8

The quality factor is defined either through the decay constant $$p^2=\omega^2/v_L^2-k^2,\qquad q^2=\omega^2/v_T^2-k^2,$$9 or through the $v_L$0 bandwidth:

$v_L$1

or

$v_L$2

These expressions are standard descriptors of the resonator’s electromechanical and dissipative performance in the A3 implementation (Wu et al., 2024).

In the reported devices, the A3 mode is therefore characterized simultaneously by a modal-dispersion condition, an IDT-matched wavelength, a resonance–antiresonance extraction of $v_L$3, and a linewidth-based or decay-based definition of $v_L$4. This makes the designation “A3” inseparable from both the branch physics and the device metrology.

4. Through-holes as a release and stability mechanism

The principal engineering idea is the replacement of a conventional large-window release geometry by a regular array of sub-wavelength through-holes. In a conventional release, two large release windows are opened with edge-to-edge distance $v_L$5. Because the BOE etch undercuts the SiO$v_L$6 hard mask isotropically, $v_L$7 must exceed the largest lateral undercut required to free the central resonator region. The result is a wide band of thin LN around the device that is mechanically fragile and thermally insulating (Wu et al., 2024).

With a regular array of holes of spacing $v_L$8, the etchant gains distributed access across the suspended area. The new required etch distance becomes

$v_L$9

For $v_T$0, this is about $v_T$1, rather than about $v_T$2–$v_T$3 in the conventional case (Wu et al., 2024).

The cited work states that this reduces the ineffective suspension area by 50–60%. Using the rough area model

$v_T$4

the effective mass scales as $v_T$5. The paper emphasizes that the main benefit is not merely the slight mass reduction, but improved lateral support stiffness and heat-spreading cross-section (Wu et al., 2024).

The reported consequences are improved mechanical rigidity and thermal conduction, together with higher power handling and lower temperature sensitivity, while preserving acoustic performance. The paper also notes that turnover temperature can be tuned more precisely. This suggests that the through-hole geometry addresses packaging- and reliability-adjacent constraints that are often external to the narrow resonance metric set, yet decisive for filter realization.

5. Fabrication workflow and etch-distance standardization

The fabrication sequence is given as a five-step flow. First, on 260 nm LN / SiO$v_T$6 hard mask / 320 nm Cr, the release windows and through-hole pattern are defined by photolithography. Second, LN is dry-etched by ICP-RIE so that both windows and hole arrays are transferred in one etch step. Third, 50 nm Au IDTs are deposited and lifted off. Fourth, the SiO$v_T$7 under the LN is removed in BOE, with the holes enabling lateral etchant penetration such that LN between any two holes is released after a lateral etch distance of about $v_T$8 rather than the full resonator aperture. Fifth, the structure is critical-point dried (Wu et al., 2024).

A central system-level consequence is etch-distance standardization. If $v_T$9 and $k=\omega/v_{\text{phase}}$0 are kept uniform across all resonators on a wafer, then $k=\omega/v_{\text{phase}}$1 becomes identical even for devices with widely differing aperture $k=\omega/v_{\text{phase}}$2, resonance frequency, or duty cycle $k=\omega/v_{\text{phase}}$3. The paper states that this greatly simplifies filter integration, because it removes the need for per-device etch-time tuning (Wu et al., 2024).

This standardization function is one of the most consequential aspects of the A3 through-hole design. It converts a device-level geometric modification into a wafer-level manufacturing regularization strategy, which is particularly relevant for multi-resonator filter layouts.

6. Measured performance and acoustic invariance

The reported experimental outcome is that the through-hole design preserves the principal acoustic metrics of the K-band A3 resonator. Across Devices I–VIII, resonators with and without through-holes exhibit virtually identical resonant frequencies, with $k=\omega/v_{\text{phase}}$4 variation and $k=\omega/v_{\text{phase}}$5 deviation. The electromechanical coupling coefficient remains at $k=\omega/v_{\text{phase}}$6–$k=\omega/v_{\text{phase}}$7, unchanged within $k=\omega/v_{\text{phase}}$8. The quality factor remains in the range $k=\omega/v_{\text{phase}}$9–$D(k,\omega)=0$0, with differences below 10% (Wu et al., 2024).

The study also reports no new lateral or bulk-wave modes introduced by the holes, and describes the admittance as an A3 clean single-peak admittance. In summary form, the technique delivers:

• consistent K-band A3 resonators at $D(k,\omega)=0$1,
• $D(k,\omega)=0$2,
• $D(k,\omega)=0$3–$D(k,\omega)=0$4,
• zero spurious modes,
• 50–60% periphery area reduction,
• unified etch distance $D(k,\omega)=0$5 for all devices,
• improved mechanical rigidity and thermal conduction,

all without any adverse impact on acoustic performance (Wu et al., 2024).

The paper further states that the approach is directly extendable to other Lamb-wave resonators in LiNbO$D(k,\omega)=0$6, LiTaO$D(k,\omega)=0$7, AlN, AlScN, and hybrid acoustic-plate-modes, and that it paves the way for robust, high-yield, large-scale monolithic filters from 4 GHz to beyond 100 GHz. Within that framing, A3 is not only a specific Lamb-wave branch but also a practical resonator platform whose manufacturability and integration are explicitly linked to the sub-wavelength through-hole architecture.