Piezoelectric Diaphragm Underwater Acoustic Voltage Calculator

MEMS hydrophone sensitivity estimator — Si(100) diaphragm + piezoelectric thin film — e₃₁ transduction

Overview

This tool estimates the open-circuit voltage response of a MEMS acoustic sensor consisting of a circular clamped silicon diaphragm with a thin piezoelectric film deposited on its top surface. The device is intended for underwater (submerged) operation.

Given the geometry, material properties, and the acoustic pressure amplitude at a chosen frequency, the calculator outputs the open-circuit voltage across the piezoelectric film, the resonant frequency in both air and water, and the peak center deflection. Frequency response plots (voltage and deflection, both in dB on a log-frequency axis) and the in-plane strain distribution across the diaphragm radius are also provided.

Fig. 1 Structure of MEMS hydrophone. Both electrodes are ignored in this simulation.

Fig. 2 Calculator interface: parameter panel (left), frequency response and strain distribution plots (right).

Physical Model

Mechanical analysis

The diaphragm is modeled as a clamped circular plate under uniform pressure loading, following Kirchhoff–Love thin-plate theory. The silicon substrate is P-type (100)-oriented, for which an isotropic effective modulus E = 130 GPa and Poisson’s ratio ν = 0.28 are used as representative average values for in-plane bending. The piezoelectric film thickness is assumed to be less than one-tenth of the silicon thickness, so its mechanical contribution to plate stiffness is neglected.

The static center deflection and bending strain field are given analytically:

w(ρ) = [P · a&sup4; / (64D)] · (1 − ρ²)², D = E · h³ / [12(1 − ν²)]

where ρ = r/a is the normalized radius, a is the plate radius, h is the silicon thickness, and D is the flexural rigidity.

Dynamic response at frequency f is obtained from a single-mode second-order transfer function with resonant frequency f₀ and damping ratio ζ.

Fluid loading (added mass)

When the diaphragm operates submerged, the surrounding fluid increases the effective vibrating mass, shifting the resonant frequency downward. The correction follows the Lamb (1921) added-mass model for a clamped circular plate:

f₀^water = f₀^air / √(1 + α · ρ_f · a / (ρ_Si · h))

The added-mass coefficient α = 0.6691 applies to the fundamental mode of a clamped circular plate. The fluid density ρ_f is a user input (default: 1025 kg/m³ for seawater).

In-plane resonant frequency (in air)

The fundamental resonant frequency in air is:

f₀^air = (α₀₁² / 2πa²) · √(D / ρ_Sih)

where α₀₁² = 10.2158 is the squared eigenvalue for the first axisymmetric mode of a clamped circular plate (Leissa).

Piezoelectric voltage (e₃₁ transduction)

Under open-circuit conditions (no charge flow), the electric displacement D₃ across the film thickness is zero. For a film with in-plane isotropy (e.g. c-axis AlN), this gives:

V_oc = −(e₃₁ / ε₃₃) · h_p · ⟨S_r + S_θ⟩

where h_p is the film thickness, ε₃₃ is the out-of-plane permittivity, and ⟨ ⟩ denotes the area average of the in-plane strain sum over the inner electrode region (ρ < 1/√2 ≈ 0.707), where S_r + S_θ > 0.

Note on electrode coverage: For a full-area electrode on a clamped plate, the spatial average of S_r + S_θ is identically zero (tensile center and compressive edge cancel). A practical sensor uses an inner electrode covering only the tensile region (ρ < 0.707), which is what this calculator assumes.

Calculator Interface

Input panel (left side)

Parameters are grouped into three sections. All geometric inputs are in micrometers (μm).

Section	Key parameters
Geometry	Silicon diaphragm thickness h_Si, diaphragm radius a, piezoelectric film thickness h_p. Default values correspond to a typical MEMS IR pixel structure.
Piezoelectric film	Young’s modulus, Poisson’s ratio, and either a single e₃₁ (effective) value or the full 3×6 e-matrix. Default values are for sputtered AlN from literature. The out-of-plane relative permittivity ε₃₃/ε₀ is also entered here.
Acoustic input & fluid	Operating frequency, sound pressure amplitude, damping ratio ζ (default 0.05 for submerged operation), and fluid density.
Plot range	Start and end frequency for the response plots (default 100 Hz – 200 kHz). Both axes use logarithmic scaling.

Result cards (top right)

After clicking Calculate, four summary values are shown: resonant frequency in air, resonant frequency in water, open-circuit voltage at the entered frequency, and peak center deflection.

Plots

Three plot tabs are available:

Voltage |V(f)| dB — Open-circuit voltage vs. frequency in dB (re 1 V), log-log axes. Vertical markers indicate f₀ in water (orange) and in air (purple). A green marker shows the currently entered frequency.
Deflection |w(f)| dB — Center deflection vs. frequency in dB (re 1 nm), same axis convention.
In-plane strain distribution — Radial (S_r), tangential (S_θ), and their sum as a function of normalized radius ρ = r/a at the current frequency. The node point at ρ ≈ 0.707 where the sum changes sign is indicated.

Assumptions and Limitations

Linear, small-deflection plate theory (Kirchhoff–Love). Not valid if center deflection exceeds roughly h/5.
Single-mode resonance model; higher modes are not included.
Electrode stress and residual film stress are neglected.
The piezoelectric film is assumed mechanically transparent (h_p < h_Si/10).
No acoustic radiation damping; the damping ratio ζ must be set by the user based on measurements or estimates.
Fluid loading is one-sided (single fluid half-space model).

Default Values Reference

Parameter	Default	Note
Pixel length l	35 μm	Square diaphragm side
Diaphragm radius a	1000 μm	Set independently for acoustic modeling
Si thickness h_Si	15 μm
Film thickness h_p	0.5 μm	AlN sputtered
E_p (AlN)	308 GPa	Sputtered AlN, literature
ν_p (AlN)	0.287
e₃₁ (AlN)	−1.05 C/m²	Effective value
ε₃₃/ε₀	10.4	AlN, thickness direction
Fluid density	1025 kg/m³	Seawater
Damping ratio ζ	0.05	Submerged estimate
Sound pressure	1 Pa	≈ 94 dB re 20 μPa in air

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