---
title: A3 Lamb-Wave Resonators
url: https://www.emergentmind.com/topics/a3
type: topic
---

# A3 Lamb-Wave 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" [2409.00783], 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 [2409.00783].

## 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 $D(k,\omega)=0$ [2409.00783].

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 $A$ and $B$ fixed by the boundary conditions. The mode has **three half-waves across the plate thickness**, expressed as $p\cdot h\approx 3\pi/2$ [2409.00783].

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\(_3\) implementation, that modal structure is the basis for high-frequency operation in the K band.

## 2. Resonator configuration in LiNbO\(_3\)

The reported A3 resonator uses a **Z-cut LiNbO\(_3\)** single-crystal piezoelectric plate with **total thickness $t=260\,\text{nm}$**. The electrodes are **Au**, with **thickness $50\,\text{nm}$**, arranged as interdigitated transducers perpendicular to the Y-axis. The electrode geometry is parameterized by the finger width $W_e$, gap $G$, aperture $L_e$, electrode duty cycle

$$
C=\frac{W_e}{W_e+G},
$$

and a design-dependent number of finger pairs $N$ [2409.00783].

The distinctive structural addition is a set of **sub-wavelength through-holes**. In Designs I–VIII, the holes are **circular**, have **diameter $D=1.0\,\mu\text{m}$**, are **uniformly etched in the suspended region**, and are placed at **spacing $L=10\,\mu\text{m}$** along the propagation direction. The holes lie **between adjacent electrodes** and extend **through the LN film** [2409.00783].

| Parameter | Value or description |
|---|---|
| Piezoelectric plate | Z-cut LiNbO\(_3\), $t=260\,\text{nm}$ |
| Electrodes | Au, $50\,\text{nm}$, interdigitated, perpendicular to Y-axis |
| Through-holes | Circular, $D=1.0\,\mu\text{m}$, spacing $L=10\,\mu\text{m}$ |

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 $f_{A3}$ is obtained by solving the dispersion equation with a wavenumber matched to the interdigital transducer periodicity $\Lambda=2\pi/k$. In practice,

$$
f_{A3}\simeq \frac{v_{\text{phase}}}{\Lambda}.
$$

For **half-thickness $h=130\,\text{nm}$** and the **third antisymmetric branch**, the relation

$$
\Lambda\approx \frac{2h\pi}{3}
$$

yields **$f_{A3}\approx 21\,\text{GHz}$** for LiNbO\(_3\) [2409.00783].

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

$$
k^2=2(\omega_a-\omega_r)/\omega_a,
$$

or equivalently,

$$
k^2\simeq 2(f_a-f_r)/f_a.
$$

The quality factor is defined either through the decay constant $\alpha$ or through the $3\,\text{dB}$ bandwidth:

$$
Q=\omega_r/(2\alpha),
$$

or

$$
Q\simeq f_r/\Delta f_{3\text{dB}}.
$$

These expressions are standard descriptors of the resonator’s electromechanical and dissipative performance in the A3 implementation [2409.00783].

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 $k^2$, and a linewidth-based or decay-based definition of $Q$. 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 $H_{\text{need}}$. Because the BOE etch undercuts the SiO\(_2\) hard mask isotropically, $H_{\text{need}}$ 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 [2409.00783].

With a regular array of holes of spacing $L\ll H_{\text{need}}$, the etchant gains distributed access across the suspended area. The new required etch distance becomes

$$
H'_{\text{need}}\approx \sqrt{2}\,L.
$$

For **$L=10\,\mu\text{m}$**, this is **about $14\,\mu\text{m}$**, rather than **about $50$–$100\,\mu\text{m}$** in the conventional case [2409.00783].

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

$$
A_{\text{solid}}\simeq A_{\text{tot}}-N_{\text{holes}}\pi(D/2)^2,
$$

the effective mass scales as $m_{\text{eff}}\sim \rho A_{\text{solid}}$. The paper emphasizes that the main benefit is not merely the slight mass reduction, but improved **lateral support stiffness** and **heat-spreading cross-section** [2409.00783].

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\(_2\) 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\(_2\) 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 **$L/\sqrt{2}$** rather than the full resonator aperture. Fifth, the structure is **critical-point dried** [2409.00783].

A central system-level consequence is **etch-distance standardization**. If $L$ and $D$ are kept uniform across all resonators on a wafer, then $H'_{\text{need}}$ becomes identical even for devices with widely differing **aperture $L_e$**, **resonance frequency**, or **duty cycle $C$**. The paper states that this greatly simplifies **filter integration**, because it removes the need for **per-device etch-time tuning** [2409.00783].

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 **$\pm 0.1\,\text{GHz}$ variation** and **$\lesssim 0.5\%$** deviation. The electromechanical coupling coefficient remains at **$k^2\approx 4.0$–$4.2\%$**, unchanged within **$\pm 5\%$**. The quality factor remains in the range **$Q_{3\text{dB}}=400$–$800$**, with differences **below 10%** [2409.00783].

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 $f\approx 21\,\text{GHz}$**,
- **$k^2\approx 4\%$**,
- **$Q\approx 500$–$800$**,
- **zero spurious modes**,
- **50–60% periphery area reduction**,
- **unified etch distance $H'_{\text{need}}\simeq 14\,\mu\text{m}$ for all devices**,
- **improved mechanical rigidity and thermal conduction**,

all **without any adverse impact on acoustic performance** [2409.00783].

The paper further states that the approach is directly extendable to other Lamb-wave resonators in **LiNbO\(_3\)**, **LiTaO\(_3\)**, **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.

Source: https://www.emergentmind.com/topics/a3