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Part III · Module 18

CMUT, PMUT and new transducer materials

Module 19 introduced micromachined transducers with a lumped parallel-plate model. Here the membrane becomes a real clamped plate. That changes the pull-in voltage, the coupling and the collapse behaviour, and it explains where CMUT bandwidth, crosstalk and charging come from. The same plate physics, driven by a thin piezoelectric film, gives the PMUT. The last part updates the bulk-material choices: single crystals and their composites, lead-free ceramics, AC poling and transparent crystals, piezoelectric films and PVDF, with a sourced property table.

You will be able to

  • Compute CMUT pull-in, snap-back and coupling from a distributed plate model, and say why the lumped piston overestimates \(V_{\mathrm{PI}}\) by about 43 %
  • Explain collapse-mode operation: contact radius, hysteresis, higher annulus resonance, and why the insulator thickness sets the snap-back voltage
  • Predict CMUT bandwidth from \(k_{\mathrm{eff}}\), membrane mass and the water load, and compare it with a matched PZT stack
  • Recognise the dispersive interface wave that carries CMUT crosstalk, and the effects of charging and the \(V^{2}\) force
  • Choose between sacrificial-release and wafer-bonded CMUTs, and between monolithic CMUT-on-CMOS and flip-chip 2D integration
  • Design a PMUT: clamped-plate resonance, neutral axis, water loading, electrode radius 0.71 a, film figure of merit
  • Rank PZT-5H, PMN-PT, PIN-PMN-PT, composites, KNN, AlN, ScAlN, PZT film and PVDF on the numbers that matter for a given probe

Assumes

  • Module 16, Piezoelectrics & KLM: piezoelectric constants, \(k_t\), KLM/Mason, the Smith–Auld composite model
  • Module 17, Matching, backing, lens, FEA: matching layers and backing; Module 21, Simulation & characterization, Section 21.6: bandwidth definitions (−6 dB pulse-echo)
  • Module 19, Arrays: 1D to 2D, RCA, CMUT: Section 19.7, lumped CMUT model (pull-in at \(g_0/3\), \(k_T^{2} = 2\xi/(1 - \xi)\)), and Section 19.4, the 2D channel-count problem
Exact formulaclosed form, or tabulated published values, plotted as isSimulationnumerical solution run in this page or in the Octave scripts (axisymmetric finite-difference plate with Newton continuation and eigen-solves)Approx. modelsimplified physics; assumptions under the badgeSchematicdrawing for intuition, labelled with typical dimensions, not to scale

18.1The CMUT cell as a clamped plate

A CMUT cell is a capacitor whose top plate is a membrane. From the bottom up: a conductive (doped) silicon substrate that serves as the bottom electrode, an insulating layer of silicon nitride or oxide, a sealed vacuum gap, and a membrane of silicon or silicon nitride carrying the top electrode. Typical numbers: membrane thickness 0.5–2 µm (1 µm in the Stanford arrays described by Ergun et al., SPIE 2005), radius 10–50 µm, and gaps from about 40 nm (Park et al., MEMS 2008, ±2 nm across the wafer) to about 1 µm. One array element is tens to hundreds of such cells wired in parallel.

Schematic Figure 18.A · CMUT cell cross-sections, conventional (left) and collapse mode (right); vertical scale exaggerated about 20×
Conventional: 0 < VDC < VPI Collapse mode: VDC > VPI (stays down to VSB) water / lens (1.5 MRayl) p doped Si substrate = bottom electrode vacuum gap g₀ ≈ 0.04–1 µm membrane 0.5–2 µm (Si, Si₃N₄) top electrode post / support insulation 0.1–0.2 µm 2a ≈ 20–100 µm water / lens substrate (bottom electrode) contact, radius b free annulus a − b vibrates higher f, higher k², larger strokehysteresis between VSB and VPI

With a bias \(V\) the membrane deflects toward the substrate by \(w(r)\). Each patch of membrane feels the local parallel-plate pressure, so the static problem is the clamped-plate equation with a nonlinear load (18.1)–(18.2):

\[ \begin{aligned} D\,\nabla^{4} w(r) &= \frac{\varepsilon_0 V^{2}}{2\left(g_{\mathrm{eff}} - w(r)\right)^{2}} \\ w(a) &= w'(a) = 0 \end{aligned} \]
(18.1)
\(D\)
flexural rigidity of the membrane, N·m
\(w(r)\)
membrane deflection toward the substrate at radius \(r\), m
\(\varepsilon_0\)
vacuum permittivity, F/m
\(V\)
bias voltage, V
\(a\)
membrane radius, m
\(g_{\mathrm{eff}}\)
effective (electrical) gap, m
\(r\)
radial coordinate from the membrane centre, m
\(\nabla^{4}\)
biharmonic operator (Laplacian applied twice), 1/m⁴
\(w\'(a)\)
radial slope at the edge (clamped: \(w\'(a) = 0\))
\[ g_{\mathrm{eff}} = g_0 + \frac{t_i}{\varepsilon_r}, \qquad D = \frac{E h^{3}}{12\left(1 - \nu^{2}\right)} \]
(18.2)
\(g_0\)
vacuum gap, m
\(t_i,\ \varepsilon_r\)
insulator thickness (m) and relative permittivity
\(E,\ \nu\)
Young's modulus (Pa) and Poisson ratio of the membrane
\(h\)
membrane thickness, m
The insulator of thickness \(t_i\) and permittivity \(\varepsilon_r\) adds \(t_i/\varepsilon_r\) to the electrical gap but not to the mechanical one: the membrane stops at \(w = g_0\). Example: 0.1 µm of Si₃N₄ (\(\varepsilon_r \approx 7.5\)) adds 13 nm. Fringing fields, residual stress and the electrode's own stiffness are neglected.
\[ \begin{aligned} k &= \frac{192\pi D}{a^{2}}, \qquad A = \pi a^{2} \\ V_{\mathrm{PI}}^{2} &= \frac{8\,k\,g_{\mathrm{eff}}^{3}}{27\,\varepsilon_0 A} = 56.9\,\frac{D\,g_{\mathrm{eff}}^{3}}{\varepsilon_0 a^{4}} \end{aligned} \]
(18.3)
\(k\)
piston-equivalent spring of the clamped plate, N/m
\(A\)
electrode (membrane) area, m²
\(V_{\mathrm{PI}}\)
pull-in voltage, V
\(D,\ g_{\mathrm{eff}},\ \varepsilon_0,\ a\)
as in (18.1)
Lumped piston.
\[ \begin{aligned} V_{\mathrm{PI}}^{2} &= 27.8\,\frac{D\,g_{\mathrm{eff}}^{3}}{\varepsilon_0 a^{4}} \\ V_{\mathrm{PI,plate}} &= 0.699\, V_{\mathrm{PI,piston}} \end{aligned} \]
(18.4)
\(V_{\mathrm{PI}}\)
pull-in voltage, V
\(D\)
flexural rigidity, N·m
\(g_{\mathrm{eff}}\)
effective gap, m
\(a\)
membrane radius, m
\(\varepsilon_0\)
vacuum permittivity, F/m
\(V_{\mathrm{PI,plate}},\ V_{\mathrm{PI,piston}}\)
pull-in voltage of the full plate model (18.4) and of the piston model (18.3)
Distributed plate (this module, FD). The piston spring \(k = 192\pi D/a^{2}\) is the force \(\pi a^{2}P\) divided by the average deflection \(Pa^{4}/(192D)\) of a clamped plate under uniform pressure \(P\). The distributed constant 27.8 comes from the finite-difference solution below (27.76; unchanged to 0.02 % between 60, 120 and 240 radial intervals). It holds for any \(a\), \(h\), \(g\) as long as the plate stays in small deflection (\(w \ll h\)) and the gap is uniform. Default cell (Si, \(a\) = 20 µm, \(h\) = 1 µm, \(g_0\) = 0.2 µm, 0.1 µm nitride): \(V_{\mathrm{PI}}\) = 53.1 V, against 76.0 V from the piston.

Why the piston is wrong by 30 %. The piston puts the whole electrode at the average gap. A plate concentrates its deflection at the centre, where the gap is smallest and the pressure, \(\propto 1/(g - w)^{2}\), is largest; the centre therefore softens first. At pull-in the centre has moved ≈ 0.46 \(g_{\mathrm{eff}}\) (0.459 converged; coarse grids read slightly higher; the insulator enters only through \(g_{\mathrm{eff}}\)) but the average only 0.15 \(g_{\mathrm{eff}}\), less than half the piston's \(g/3\) (compare (18.3) and (18.4)). The numerical method: the conventional branch is traced by fixing the centre deflection \(w(0)\) and solving for both the shape and the voltage that holds it (a Newton solve on \([w;\ V^{2}]\)). Because \(w(0)\) increases monotonically along the branch, the solver walks through the pull-in fold, where \(V\) reaches its maximum, and continues on the unstable side.

Pitfall

"Pull-in at a third of the gap" is a statement about a rigid piston. For a clamped membrane, quote the centre deflection (≈ 0.46 \(g_{\mathrm{eff}}\)) or the average deflection (≈ 0.15 \(g_{\mathrm{eff}}\)), and say which. Datasheets and FEM papers usually give the centre value, so do not compare it with \(g/3\).

Demo 18.1

Distributed CMUT plate: both branches, pull-in, coupling versus bias

SimulationAxisymmetric finite-difference clamped plate (48 radial intervals, JS), electrostatic pressure on every node, Newton continuation in centre deflection (conventional) and in contact radius (collapsed); coupling from the two lowest eigenvalues. Silicon membrane \(E\) = 169 GPa, \(\nu\) = 0.22, \(\rho\) = 2330 kg/m³, Si₃N₄ insulator \(\varepsilon_r\) = 7.5, in vacuum. Dotted: lumped piston.
conventional (stable)unstable side of the foldcollapsed branchlumped piston\(f/f_0\)
Try: at 0.9 \(V_{\mathrm{PI}}\) the distributed plate gives \(k^{2} \approx 0.26\) where the piston promises 0.43; the gap between the curves is the cost of a bending membrane. Thicken the insulator and watch the snap-back voltage (left end of the red branch) rise toward \(V_{\mathrm{PI}}\): the hysteresis window closes. Halve the gap: \(V_{\mathrm{PI}}\) falls by \(2^{1.5} \approx 2.8\) (it scales as \(g_{\mathrm{eff}}^{3/2}\)).

18.2Coupling, collapse mode and hysteresis

Coupling as a stiffness ratio. Linearize the plate about its biased shape. Held at constant voltage (short circuit), the electrostatic pressure adds a negative stiffness at every node, \(-\varepsilon_0 V^{2}/(g_{\mathrm{eff}} - w)^{3}\) per unit area. Held at constant charge (open circuit), the electrode voltage readjusts as the capacitance changes, which adds back a rank-one term. The coupling coefficient is the fractional stiffness difference of the fundamental mode (18.5)–(18.6):

\[ \begin{aligned} \mathbf{K}_V &= \mathbf{K} - V^{2}\,\mathrm{diag}\!\left(\frac{\varepsilon_0 W_i}{(g_{\mathrm{eff}} - w_i)^{3}}\right) \\ \mathbf{K}_Q &= \mathbf{K}_V + \frac{V^{2}\,\mathbf{c}\mathbf{c}^{T}}{C} \end{aligned} \]
(18.5)
\(\mathbf{K}\)
mechanical stiffness matrix of the FD plate
\(\mathbf{K}_V,\ \mathbf{K}_Q\)
stiffness at constant voltage (short circuit) and at constant charge (open circuit)
\(W_i\)
ring area of node \(i\), m²
\(w_i\)
deflection of node \(i\), m
\(C\)
capacitance, F
\(V,\ \varepsilon_0,\ g_{\mathrm{eff}}\)
as in (18.1)
\(\mathbf{c}\)
capacitance-sensitivity vector, as in (18.6)
\((\cdot)^{T}\)
transpose
\(\mathrm{diag}(\cdot)\)
diagonal matrix of the node values
\[ \begin{aligned} c_i &= \frac{\partial C}{\partial w_i} = \frac{\varepsilon_0 W_i}{(g_{\mathrm{eff}} - w_i)^{2}} \\ k^{2} &= 1 - \frac{\lambda_{\min}(\mathbf{K}_V, \mathbf{M})}{\lambda_{\min}(\mathbf{K}_Q, \mathbf{M})} \end{aligned} \]
(18.6)
\(\mathbf{c}\)
vector of capacitance sensitivities \(c_i\), F/m
\(\mathbf{M}\)
mass matrix
\(\lambda_{\min}\)
lowest generalized eigenvalue
\(k^{2}\)
electromechanical coupling coefficient
\(C\)
total capacitance of the cell, F
\(\partial\)
partial derivative
\(\varepsilon_0,\ W_i,\ g_{\mathrm{eff}},\ w_i\)
as in (18.5)
\(\mathbf{K}_V,\ \mathbf{K}_Q\)
as in (18.5)
\(C = \sum \varepsilon_0 W_i/(g_{\mathrm{eff}} - w_i)\). For a piston the rank-one term exactly cancels the softening (\(K_Q = k\)) and \(k^{2} = 1 - k_{\mathrm{eff}}/k\), the Module 19 formula. For a plate it does not cancel fully, and \(k^{2}\) is lower: 0.04, 0.16, 0.26 and 0.37 at 0.5, 0.8, 0.9 and 0.95 \(V_{\mathrm{PI}}\) for the default cell (piston: 0.08, 0.28, 0.43, 0.56). Bayram et al. (IEEE TUFFC 2003) found the same character with FEM: \(k_T^{2}\) exceeds 0.35 only above 85 % of the collapse voltage.

Collapse mode. Above \(V_{\mathrm{PI}}\) the centre lands on the insulator. The contact disc of radius \(b\) grows with bias and the free annulus between \(b\) and \(a\) carries the vibration (Figure 18.A, right). Three things change. The annulus is a much stiffer plate, so the resonance rises steeply with \(b\). The field across the thin insulator in the contact region is enormous, so the coupling of the annulus edge is high; Bayram et al. report \(k_T^{2} \approx 0.7\) at about 60 % of the collapse voltage after collapse, versus 0.3 averaged over a conventional swing. And the average displacement available for transmit is larger (0.25–0.45 µm versus 0.05–0.15 µm in their 1 µm-gap design). The collapsed branch is traced here by fixing the contact radius and requiring zero contact force at its edge, which again turns the fold into a monotonic path.

Snap-back and hysteresis. Lowering the bias, the membrane stays down until the contact force at the edge can no longer be maintained, at the snap-back voltage \(V_{\mathrm{SB}}\). Since the electrostatic pressure in contact is \(\varepsilon_0 V^{2}/\left(2(t_i/\varepsilon_r)^{2}\right)\), \(V_{\mathrm{SB}}\) is controlled by the insulator: the thinner the electrical gap at contact, the lower \(V_{\mathrm{SB}}\) and the wider the hysteresis loop. For the default cell \(V_{\mathrm{SB}}\) = 25.8 V = 0.49 \(V_{\mathrm{PI}}\); for a 1 µm gap with the same 0.1 µm nitride it is 0.26 \(V_{\mathrm{PI}}\). Bayram et al. computed 140 V and 68 V (0.49) for their 50 µm-radius, 1 µm-gap cell, which also had a 50 nm contact offset; adding that offset to this model moves the ratio from 0.26 to 0.43, so the offset matters as much as the insulator.

Demo 18.2

Bias history: C–V hysteresis, contact radius and annulus resonance

SimulationSame finite-difference plate and geometry as Demo 18.1. Resonance: lowest eigenvalue of the plate with contact nodes fixed, in vacuum. On the conventional branch it includes the electrostatic softening; on the collapsed branch it is the mechanical annulus only, because the softening concentrated at the contact edge is not resolved on a 48-interval grid (it lowers the value; refine before quoting a number).
conventionalcollapsedcurrent state
Try: sweep up: the membrane snaps down at \(V_{\mathrm{PI}}\) and \(C\) jumps. Sweep down: it stays collapsed well below \(V_{\mathrm{PI}}\) and releases only at \(V_{\mathrm{SB}}\). A probe biased between \(V_{\mathrm{SB}}\) and \(V_{\mathrm{PI}}\) can sit in either state, which is why collapse-mode systems collapse the membrane with a brief overdrive and then hold the bias. The geometry comes from the Demo 18.1 sliders.

18.3Bandwidth, the water load and crosstalk

Per unit area, a CMUT membrane is a mass \(m_a\) on a spring \(k_a\) radiating into water with resistance \(Z_w = \rho_w c_w\) = 1.5 MRayl. Using the piston-equivalent constants of the clamped plate (spring \(192D/a^{4}\), modal mass \(1.8\rho h\) for the average displacement), a 1 µm silicon membrane of 20 µm radius has \(m_a\) = 4.2 g/m² and \(k_a = 1.8\times 10^{13}\) Pa/m. Its vacuum resonance is 10 MHz, and its quality factor in water is \(Q = \sqrt{k_a m_a}/Z_w \approx 0.18\): deeply over-damped. The velocity response per unit force is then flat between two corner frequencies (18.7):

\[ \begin{aligned} f_{\mathrm{low}} &\approx \frac{\eta\, k_{a,\mathrm{eff}}}{2\pi Z_w}, \qquad f_{\mathrm{high}} \approx \frac{Z_w}{2\pi\eta\, m_a} \\ m_a &= 1.8\,\rho h + \frac{8}{3\pi}\,\rho_w a \end{aligned} \]
(18.7)
\(\eta\)
fraction of the element area covered by cells
\(k_{a,\mathrm{eff}}\)
spring per unit area including bias softening, Pa/m
\(m_a\)
moving mass per unit area, kg/m²
\(Z_w\)
water impedance \(\rho_w c_w\), Rayl
\(\rho,\ \rho_w\)
membrane and water density, kg/m³
\(f_{\mathrm{low}},\ f_{\mathrm{high}}\)
lower and upper edges of the usable band, Hz
\(h,\ a\)
membrane thickness and radius, m
\(\eta\) is 0.8 here. \(k_{a,\mathrm{eff}}\) includes spring softening, so \(f_{\mathrm{low}}\) falls as the bias rises. The second mass term is the water added mass of an isolated baffled cell, \((8/3)\rho_w a^{3}\) per cell; in an array moving in phase it is smaller, so the true \(f_{\mathrm{high}}\) lies between the two estimates. For the 20 µm cell at 0.9 \(V_{\mathrm{PI}}\): \(f_{\mathrm{low}} \approx\) 0.8 MHz, \(f_{\mathrm{high}} \approx\) 14 MHz with added mass. A PZT element, in contrast, is a half-wave resonator with mechanical impedance ≈ 34 MRayl and \(Q \approx (\pi/2)Z_p/(Z_B + Z_L)\) that needs matching layers to broaden (Modules 16–17).
Demo 18.3

CMUT versus PZT stack: pulse-echo spectrum and impulse response

Approx. model1D, plane-wave radiation resistance ρcA for both transducers (no diffraction, no lens), 50 Ω source and receiver, element 0.27 × 5 mm, plane reflector, pulse-echo \(H = S_{21}^{2}\). PZT: Mason three-port, PZT-5H (\(v\) 4580 m/s, \(k_t\) 0.51, \(\varepsilon^{S}\) \(1470\varepsilon_0\); Ma et al. 2015), \(f_a\) = 5 MHz, lossless Desilets quarter-wave layers. CMUT: one piston membrane mode with lumped bias physics (Module 19), no higher membrane modes, no crosstalk, no parasitic capacitance. A bandwidth upper bound for the CMUT, not a prediction.
PZT-5H stackCMUT−6 dBmeasured CMUT band, 96 % at 4.6 MHz (Oralkan 2002, Fig. 10)
Try: the PZT stack with two layers reaches about 63 % around 4.6 MHz; remove the layers and it collapses to 17 %. The CMUT band spans 0.8–14 MHz in this model (about 180 %), its peak sensitivity within a few dB of the PZT stack at 0.9 \(V_{\mathrm{PI}}\), falling 8 dB at 0.7 \(V_{\mathrm{PI}}\). Untick the added mass and the upper edge moves past 25 MHz: the moving mass, not a resonance, sets the top of the band. Measured immersion arrays reach 91–105 % (Oralkan et al. 2002, 2003); higher membrane modes, parasitic capacitance, diffraction and crosstalk eat the rest.

Crosstalk rides an interface wave. In a CMUT array, the neighbouring membranes and the water above them form a compliant surface. Bayram et al. (IEEE TUFFC 2007) measured and simulated 1D arrays and identified the main crosstalk path as a dispersive guided mode along the fluid–membrane interface: its phase velocity approaches the water sound speed at low frequency, falls with frequency and cuts off (near 4 MHz in conventional mode); first-neighbour crosstalk was about −17 dB, strongest near 2.1 MHz. In collapse mode the average crosstalk fell to about −39 dB and the cutoff rose to about 10 MHz. A homogenized model reproduces that behaviour: treat the array as a locally reacting surface of membranes (mass \(m_a\), spring \(k_a\) per area) under a fluid half-space and look for a wave \(e^{j(\omega t - kx)}\) that decays as \(e^{-\alpha z}\) into the water (18.8):

\[ \begin{aligned} j\left(\omega m_a - \frac{k_a}{\omega}\right)\frac{\alpha\, p}{j\omega\rho_w} &= -p \\ \Rightarrow\quad \alpha &= \frac{\omega^{2}\rho_w}{k_a - \omega^{2} m_a} \\ k_x &= \sqrt{k_w^{2} + \alpha^{2}} \\ c_{\mathrm{ph}} &= \frac{\omega}{k_x} < c_w \end{aligned} \]
(18.8)
\(\alpha\)
decay constant of the wave into the water, 1/m
\(k_a,\ m_a\)
membrane spring and mass per unit area
\(k_x,\ k_w\)
interface-wave and water wavenumbers, 1/m
\(c_{\mathrm{ph}},\ c_w\)
phase velocity of the interface wave and sound speed in water, m/s
\(\omega,\ j\)
angular frequency (rad/s) and imaginary unit
\(p\)
pressure amplitude at the surface, Pa
\(\rho_w\)
water density, kg/m³
Membrane impedance times the fluid's normal velocity equals minus the pressure. A real decay constant \(\alpha > 0\) exists only below the membrane resonance \(\omega_c = \sqrt{k_a/m_a}\), where the surface is stiffness-controlled (a compliant wall slows sound, as in a fluid-filled elastic tube). As \(\omega \to \omega_c\), \(\alpha \to \infty\) and \(c_{\mathrm{ph}} \to 0\): the cutoff. The fluid here enters through the evanescent field, so \(m_a\) is the membrane mass alone. Spring softening lowers the cutoff; collapse, which stiffens the annulus, raises it, as Bayram et al. observed.
Demo 18.4

The crosstalk mode: phase velocity and decay depth of the interface wave

Approx. modelHomogenized locally reacting membrane surface (cells much smaller than the interface wavelength), water half-space, lossless; piston-equivalent \(k_a\) with lumped spring softening; collapsed state from the finite-difference annulus resonance at the chosen contact radius (no softening).
\(c_{\mathrm{ph}}/c_w\)decay depth \(1/\alpha\) (µm, right axis)cutoff
Try: raise the bias: the cutoff drops and the slow, strongly confined part of the curve moves into the imaging band, where it couples elements. Switch to collapsed: the cutoff jumps several-fold. The decay depth shows why a lossy layer (Bayram et al. used PDMS) attenuates this wave: it lives within a fraction of a wavelength of the surface.

18.4Making CMUTs: fabrication, charging, nonlinearity and CMOS integration

Two process families. In sacrificial release (surface micromachining) the gap is a sacrificial layer (polysilicon or oxide) deposited on the insulated substrate, covered by a deposited silicon nitride membrane, removed through etch holes and sealed under vacuum; the gap is the sacrificial thickness minus the nitride deposited during sealing. In wafer bonding the cavities are etched into a thermal oxide on one wafer and a silicon-on-insulator wafer is bonded on top; removing the handle and buried oxide leaves the single-crystal device layer (under 1.5 µm, uniform to better than 10 nm on 100 mm wafers) as the membrane (Ergun et al., SPIE 2005, Figs. 3 and 5). Bonding gives better membrane uniformity and material properties; because the silicon membrane is conductive, it needs a thicker insulator than a nitride membrane. Park et al. (MEMS 2008; J. Microelectromech. Syst. 2011) combined local oxidation (LOCOS) with direct bonding to set the gap and the oxide post height independently, reaching 40 nm gaps with ±2 nm uniformity, fewer breakdown paths and lower parasitic capacitance because most of the bonded area is oxide.

Schematic Figure 18.B · Sacrificial release (top) versus wafer bonding (bottom), four steps each, after the flows in Ergun et al. 2005
Sacrificial release 1 · sacrificial layer 2 · nitride membrane 3 · etch holes, release 4 · seal, metallize Wafer bonding (SOI device layer = membrane) 1 · oxidize, etch cavity 2 · bond SOI in vacuum 3 · remove handle, BOX 4 · isolate, metallize gap: sacrificial thickness (top) or oxidation (bottom); membrane: Si₃N₄ (top) or single-crystal Si (bottom)

Charging. The vacuum gap sees fields of order \(V_{\mathrm{DC}}/g_{\mathrm{eff}}\), about 50 V over 0.21 µm, i.e. 240 V/µm; by continuity of \(D\) the nitride carries \(1/\varepsilon_r\) of that, about 30 V/µm (0.3 MV/cm), and in contact the full bias across \(t_i\), about 500 V/µm. Charge injected into it adds a bias offset \(V_q\): the membrane responds to \(V - V_q\), so the C–V curve, the pull-in voltage and the sensitivity drift with use. Remedies are structural (insulation only on posts or patterned, as in the LOCOS process), electrical (bipolar or alternating bias) and operational (stay below collapse).

The force is quadratic. With \(V = V_{\mathrm{DC}} + v\cos\omega t\) the electrostatic force contains \(V_{\mathrm{DC}}^{2} + v^{2}/2\) (static), \(2V_{\mathrm{DC}}v\cos\omega t\) (linear) and \((v^{2}/2)\cos 2\omega t\). The second-harmonic force is \(v/(4V_{\mathrm{DC}})\) of the fundamental: −32 dB at \(v = 0.1\,V_{\mathrm{DC}}\), −20 dB at \(v = 0.4\,V_{\mathrm{DC}}\). On top of that the \(1/(g - w)^{2}\) gap nonlinearity grows with stroke. For tissue-harmonic imaging the transmit harmonic must stay well below what tissue generates, which pushes CMUT drive schemes toward small AC on a large bias, predistortion, or pulse inversion that accounts for the \(V^{2}\) term.

The gap is the stroke. A membrane cannot move further than its gap. Its plane-wave surface pressure is \(Z_w\omega\bar x\) for an average displacement amplitude \(\bar x\): 50 nm at 5 MHz gives 2.4 MPa. Large gaps cost bias voltage (\(V_{\mathrm{PI}} \propto g^{3/2}\)), small gaps cost output pressure; collapse mode and tall-gap designs with pre-charged or high-voltage bias are the standard answers to this trade.

Integration for 2D arrays. A \(\lambda/2\)-pitch 2D aperture at 3 MHz has 257 µm elements; 32 × 32 such elements cannot each be cabled (Module 19, Section 19.4). CMUTs offer two ways out. Flip-chip: through-wafer vias bring each element's contact to the back of the CMUT chip, which is bump-bonded to a fan-out or an ASIC; Oralkan et al. (IEEE TUFFC 2003) built 128 × 128 arrays at 420 µm pitch this way and imaged in 3D with a 32 × 64 sub-array flip-chip bonded to a glass fan-out (their Figs. 1–2 show the via and the assembly). Monolithic CMUT-on-CMOS: the cells are built directly on the passivation of a finished CMOS wafer with a back-end-compatible, low-temperature process, so that every element has its pulser, preamplifier and often part of the beamformer underneath it; a commercial whole-body handheld probe built this way is described by Rothberg et al. (PNAS 2021). The same logic applies to PMUTs bonded to CMOS. In both cases micro-beamforming in the probe (Module 19, Section 19.5) is what makes the channel count tractable.

Schematic Figure 18.C · Two routes to a 2D CMUT array with per-element electronics; not to scale
Monolithic CMUT-on-CMOS Flip-chip: through-wafer vias + bumps CMOS: pulser, LNA, T/R switch per element interconnect metal stack (BEOL) CMUT cells built on the passivation, low-temperature process via from each element to the circuit underneath it CMUT array chip (cells on top) ASIC or fan-out (glass, PCB) through-wafer via per element, bump-bonded Oralkan et al. 2003: 128 × 128 elements, 420 µm pitch pitch ≈ λ/2 (257 µm at 3 MHz)

18.5PMUT: a flexural plate driven by a piezoelectric film

A piezoelectric micromachined ultrasonic transducer is a clamped plate made of a passive layer (Si, SiO₂ or SiN, 1–5 µm) and a piezoelectric film (AlN or ScAlN 0.5–2 µm, sol-gel or sputtered PZT up to a few µm) between electrodes. A field across the film makes it contract laterally through \(e_{31,f}\); because the film sits off the neutral axis of the stack, the lateral stress becomes a bending moment and the plate flexes (Qiu et al., Sensors 2015, Fig. 1(c)). No DC bias is needed, drive voltages suit CMOS, and AlN processing is CMOS-compatible.

Schematic Figure 18.D · PMUT cross-section: bilayer plate over a cavity, inner top electrode, neutral axis inside the passive layer; vertical scale exaggerated
water e₃₁,f: the film contracts laterally → the stack bends SiSi zp cavity (back-side DRIE or front-side release) first axisymmetric mode 2a ≈ 40–400 µm for MHz in water top electrode 0.71 a piezo film hp:AlN/ScAlN 0.5–2 µm,PZT 1–3 µm bottom electrode passive Si hs 1–5 µm neutral axis zn
\[ f_{\mathrm{vac}} = \frac{\lambda_1^{2}}{2\pi a^{2}}\sqrt{\frac{D}{\bar\rho h}}, \qquad \lambda_1^{2} = 10.2158 \]
(18.9)
\(\lambda_1\)
first root of \(J_0(\lambda)I_1(\lambda) + I_0(\lambda)J_1(\lambda) = 0\)
\(a\)
plate radius, m
\(D\)
flexural rigidity, N·m
\(\bar\rho h\)
mass per unit area of the plate, kg/m²
\(f_{\mathrm{vac}}\)
fundamental resonance in vacuum, Hz
\(J_0,\ J_1,\ I_0,\ I_1\)
Bessel and modified Bessel functions of the first kind, orders 0 and 1
First axisymmetric mode of a clamped circular plate in vacuum (Leissa, Vibration of Plates, NASA SP-160, 1969). The constant is verified here twice: bisection on the Bessel characteristic equation gives \(\lambda_1\) = 3.1962, and the finite-difference eigen-solve used for the CMUT converges to 10.2113, 10.2147, 10.2155 with 40, 80, 160 intervals. \(f \propto h/a^{2}\): halve the radius and the frequency quadruples.
\[ \begin{aligned} z_n &= \frac{\sum E'_i h_i z_i}{\sum E'_i h_i}, \qquad E' = \frac{E}{1 - \nu^{2}} \\ D &= \sum E'_i\left[\frac{h_i^{3}}{12} + h_i\,(z_i - z_n)^{2}\right] \\ \bar\rho h &= \sum \rho_i h_i \end{aligned} \]
(18.10)
\(z_n\)
height of the neutral axis, m
\(z_i,\ h_i\)
mid-plane height and thickness of layer \(i\), m
\(E'_i\)
plate modulus of layer \(i\), Pa
\(\rho_i\)
density of layer \(i\), kg/m³
\(\sum\)
sum over the layers \(i\)
\(E,\ \nu\)
Young’s modulus (Pa) and Poisson ratio of a layer
\(D\)
flexural rigidity of the stack, N·m
\(\bar\rho h\)
mass per unit area of the stack, kg/m²
Laminated-plate stiffness about the neutral axis, \(z_i\) the mid-plane of layer \(i\). Electrodes (0.1–0.2 µm of Mo or Pt) and residual stress are neglected; tensile stress raises the frequency, and a highly stressed film behaves as a membrane rather than a plate (Qiu et al. 2015 give both limits).
\[ f_{\mathrm{water}} = \frac{f_{\mathrm{vac}}}{\sqrt{1 + \beta}}, \qquad \beta = \frac{0.67\,\rho_w a}{\bar\rho h} \]
(18.11)
\(\beta\)
added-mass ratio (water on one side)
\(\rho_w\)
water density, kg/m³
\(f_{\mathrm{water}},\ f_{\mathrm{vac}}\)
fundamental resonance with water on one side and in vacuum (18.9), Hz
\(a,\ \bar\rho h\)
plate radius (m) and mass per unit area (kg/m²), as in (18.9)
Lamb's added virtual mass for a clamped plate with water on one side (H. Lamb, Proc. R. Soc. Lond. A 98, 1920; the form used by Ayela and Nicu to validate PMUT measurements). For \(a\) = 50 µm and 3 µm of AlN-on-Si, \(\beta\) = 4.2: water lowers the resonance by a factor 2.3.
\[ \begin{aligned} M_p &= e_{31,f}\, V z_p \\ Q_{\mathrm{gen}} &= -8\pi M_p\, \beta_e^{2}\left(1 - \beta_e^{2}\right) \\ \frac{w_c}{V} &= \frac{0.375\,\lvert e_{31,f}\rvert\, z_p a^{2}\, \beta_e^{2}\left(1 - \beta_e^{2}\right)}{D} \end{aligned} \]
(18.12)
\(M_p\)
piezoelectric bending moment per unit length, N
\(e_{31,f}\)
effective transverse film coefficient, C/m²
\(z_p\)
lever arm, film mid-plane to neutral axis, m
\(\beta_e\)
electrode radius / plate radius
\(Q_{\mathrm{gen}}\)
generalized force on the mode
\(w_c/V\)
centre deflection per volt, m/V
\(V\)
drive voltage on the piezoelectric film, V
\(D,\ a\)
as in (18.1)
\(w_c\)
centre deflection, m
The film stress \(e_{31,f}V/h_p\) acting over \(h_p\) at lever arm \(z_p\) (film mid-plane to neutral axis) is a moment per unit length \(M_p\), independent of \(h_p\). Its virtual work on the mode shape \(\varphi = (1 - r^{2}/a^{2})^{2}\) over an inner electrode of radius \(\beta_e a\) is the generalized force \(Q_{\mathrm{gen}}\) (one-term Ritz, quasi-static). It vanishes for a full electrode, because the curvature changes sign near \(r = 0.71a\), and is maximal at \(\beta_e = 1/\sqrt{2} = 0.707\): the origin of the familiar 70 % electrode.
Demo 18.5

PMUT designer: resonance in vacuum and water, lever arm, electrode radius

Approx. modelExact clamped-plate constant 10.2158 with laminated-plate D; Lamb added mass for water on one side; actuation from the one-term Ritz shape, quasi-static. Film data from Wang et al. 2016 and Roy et al. 2023 (\(e_{31,f}\), \(\varepsilon_{33,f}\), \(E\)); bulk densities; silicon passive layer \(E\) = 169 GPa. No residual stress, no electrodes.
f in waterf in vacuumactuation \(w_c/V\) (relative)
Try: set \(h_s = 0\): the neutral axis sits in the film, \(z_p = 0\) and the PMUT cannot bend. The actuation peaks when the passive layer is comparable to the film (the exact ratio depends on the stiffness ratio). Switch AlN to PZT: deflection per volt rises more than tenfold with \(\lvert e_{31,f}\rvert\) = 14 versus 1.05 C/m², but the capacitance rises a hundredfold, which is why receive sensitivity (\(e_{31,f}/\varepsilon\)) favours AlN and ScAlN.

Where PMUTs fit. Their impedance is high and their coupling and bandwidth in water are usually lower than a CMUT's, but they need no bias and integrate with CMOS. They win where size, power and cost decide: they are in products for ultrasonic time-of-flight ranging and fingerprint sensing, and are being developed for wearables and catheters. Qiu et al. (2015) show a 64-element, 5 MHz linear PMUT array (Fig. 12), a 6 × 6 matrix of 200 µm elements (Fig. 13) and a 14 Fr intracardiac catheter with a 512-element array built on through-silicon interconnects (Fig. 14). Rank the films by the figures of merit in Table 18.2: \(e_{31,f}^{2}/(\varepsilon_0\varepsilon_{33,f})\) for transmitted energy and coupling, \(e_{31,f}/(\varepsilon_0\varepsilon_{33,f})\) for receive voltage.

18.6New bulk and film materials

Module 16 gave the constitutive physics and a first property table. This section adds what changed in the last decade and puts every material a probe designer meets in one sourced comparison.

Table 18.1 · Bulk transducer materials (ranges are supplier or composition spreads; – = not given by the cited source)
Material\(d_{33}\) (pC/N)\(k_{33}\)\(k_t\)\(\varepsilon_{33}^{T}/\varepsilon_0\)Z (MRayl)Temperature limit (°C)\(E_c\) (kV/cm)Source
PZT-5H ceramic593–6500.750.513800 (\(\varepsilon^{S}\) 1470)36.0\(T_c\) 230≈ 8Ma et al. 2015, Table I (\(d_{33}\) 593, \(k_t\), \(\varepsilon^{S}\), Z); APC/Piezo Systems PSI-5H4E (\(d_{33}\) 650, \(k_{33}\), \(\varepsilon^{T}\), \(T_c\), \(E_c\))
PMN-PT [001] crystal1500–24000.91–0.960.55–0.595000–700036–39\(T_{\mathrm{rt}}\) 86–100 (\(T_c\) 130–145)1.9–2.5TRS Technologies X2B data; \(Z\) from \(v_t\) 4500–4850 m/s × \(\rho\) 8084 kg/m³ (Kim et al. 2013)
PIN-PMN-PT [001] crystal1200–21000.90–0.950.52–0.584000–600035–38\(T_{\mathrm{rt}}\) 105–130 (\(T_c\) 150–200)4.5–6.0TRS Technologies X4B data; \(Z\) from \(v_t\) 4340–4650 m/s, \(\rho \approx\) 8100 kg/m³ assumed
AC-poled PMN-28PT> 2100≈ 0.94–––––Qiu et al., Nature 2020: near-perfect optical transparency
1-3 PZT-5H / soft epoxy, \(\nu\) = 0.3–0.7––0.67–0.69\(\varepsilon^{S}\) 460–106011–22epoxy-limited–Smith–Auld model, this module (Demo 18.6, m18_composite_effective_properties.m)
1-3 PMN-28PT / soft epoxy, \(\nu\) = 0.3–0.7––0.81–0.83\(\varepsilon^{S}\) 280–65011–23\(T_{\mathrm{rt}}\) of the crystal–Smith–Auld model with Kim et al. 2013 constants
KNN-based lead-free ceramic> 300; 416 textured––––––Saito et al., Nature 2004
LiNbO₃ (high-frequency single elements)6–0.49\(\varepsilon^{S}\) 3934––Ma et al. 2015, Table I
PVDF film−30 to −33–0.148–124.0use ≤ 70–80–Measurement Specialties (\(d_{33}\) −33, \(g_{33}\) \(-330\times 10^{-3}\) V·m/N, \(k_t\) 14 %); Precision Acoustics (\(\rho\) 1780, \(c\) 2250 m/s, \(\varepsilon\) 8–10, \(\tan\delta\) 0.25 at 1 MHz)
Table 18.2 · Piezoelectric thin films for PMUTs (Roy, Lee and Lee, Microsyst. Nanoeng. 2023, Table I; ScAlN \(d_{33}\): Akiyama et al. 2009 as quoted by Arab et al. 2022)
Film\(\lvert e_{31,f}\rvert\) (C/m²)\(\varepsilon_{33,f}\)\(e_{31,f}/(\varepsilon_0\varepsilon_{33})\) (GV/m)\(e_{31,f}^{2}/(\varepsilon_0\varepsilon_{33})\) (GPa)Notes
AlN1.0510.511.311.9\(d_{33} \approx\) 6 pC/N; CMOS-compatible; \(\tan\delta\) 0.003
Sc0.15Al0.85N1.6121524\(d_{33}\) rises to 27.6 pC/N at 43 % Sc
PZT (sol-gel, sputtered)8–12300–13000.7–1.85.6–54highest transmit force per volt; high capacitance; receive FOM as printed by Roy et al. (their \(e_{31,f}\) and \(\varepsilon\) ranges give 0.7–4.5)
PZT (fibered epitaxial)14200–3005.3–7.974–110texture lowers \(\varepsilon\), raises both FOMs
KNN film8.5–14.44452.2–3.618–53lead-free film option

Single crystals. [001]-poled relaxor-PT crystals reach \(k_{33}\) above 0.9 and \(d_{33}\) above 1500 pC/N (Zhang et al., Prog. Mater. Sci. 2015). For a transducer the gain is \(k_t\) (0.55–0.59 against 0.51) and, more importantly, \(k_{33}\) in diced or composite form, which buys bandwidth and sensitivity. The price is thermal and electrical fragility: PMN-PT depoles near its rhombohedral-to-tetragonal transition at 86–100 °C and has a coercive field of only about 2 kV/cm. PIN-PMN-PT, the "second generation", raises \(T_{\mathrm{rt}}\) to 105–130 °C and \(E_c\) to 4.5–6 kV/cm at slightly lower \(d_{33}\). Mn doping raises the mechanical \(Q\) further (TRS lists \(Q_m \approx\) 500 for Mn:PIN-PMN-PT against 80 for PMN-PT), which matters for therapy and high-power pushes rather than imaging.

AC poling and transparency. Poling with an alternating field instead of DC changes the domain structure of rhombohedral PMN-PT. Qiu et al. (Nature 2020) used it to remove the light-scattering 71° domain walls, obtaining near-perfect transparency together with \(d_{33}\) above 2100 pC/N and \(k_{33}\) ≈ 0.94 (their Fig. 3 compares AC- and DC-poled crystals). Two consequences matter for ultrasound: AC poling is now a route to higher \(d_{33}\) in ordinary crystal transducers, and transparent crystals let light pass through the transducer, which simplifies co-axial photoacoustic and optical-ultrasound probes.

Composites. A 1-3 composite of single-crystal pillars combines the crystal's \(k_{33}\) with the polymer's low impedance: \(k_t\) ≈ 0.83 against 0.69 for PZT-5H pillars, at the same 11–23 MRayl (Demo 18.6). The fine-scale requirement (Smith and Auld, IEEE TUFFC 1991) becomes the manufacturing limit at high frequency: at 20 MHz a half-wave PMN-PT composite is about 90 µm thick, so pillars must be under about 45 µm wide with narrow kerfs, beyond conventional dicing, which is why high-frequency composites are made by etching or other micro-machining routes. A 2-2 composite (plates instead of pillars, \(\nu = w/p\)) is easier to dice and suits 1D arrays, with coupling between the 1-3 and plate values.

Lead-free. Regulatory pressure on lead drives work on (K,Na)NbO₃ (KNN) and Bi-based (BNT, BKT) ceramics. Saito et al. (Nature 2004) reported a KNN-based ceramic with \(d_{33}\) above 300 pC/N, 416 pC/N when textured; KNN films reach \(\lvert e_{31,f}\rvert\) = 8.5–14.4 C/m² (Table 18.2). For imaging arrays they are not yet a drop-in replacement for PZT-5H, mainly on coupling, temperature stability and processing; treat property claims in this area as composition- and process-specific.

High frequency and polymers. Above about 30 MHz single elements are thin and small, so electrical impedance, not coupling, often decides: LiNbO₃ (\(\varepsilon^{S}\) 39, \(k_t\) 0.49) gives a large element a convenient impedance where PZT-5H (\(\varepsilon^{S}\) 1470) would be a near short. PVDF sits at the other extreme: \(Z \approx\) 4 MRayl, close to water, \(k_t\) 0.14 and a tiny permittivity, which together give a weak transmitter but an excellent wideband receiver with \(g_{33} \approx\) 0.33 V·m/N, about 18 times PZT-5H's. That is why membrane and needle hydrophones (Modules 21 and 22) are PVDF.

Schematic Figure 18.E · 1-3 composite: top view and cross-section, with the 2-2 variant; typical 5 MHz dimensions, not to scale
Top view (1-3)Cross-section A–A′2-2 variant A–A′ pitch p ν = (w/p)² front electrode (to matching) back electrode (to backing) t = λ/20.37 mmat 5 MHz ceramic or crystal pillars (w/t ≲ 0.5) in soft epoxy; pitch ≪ t and ≪ polymer shear wavelength plates, ν = w/p
Demo 18.6

Composite designer: PZT-5H versus PMN-PT pillars

Approx. modelSmith–Auld effective medium (as Module 16), fine-scale limit, lossless; PZT-5H constants as Module 16, PMN-28PT constants from Kim, Joh and Roh (Sensors and Materials 2013): \(c_{11}^{E}\) 14.7, \(c_{12}^{E}\) 13.1, \(c_{13}^{E}\) 11.4, \(c_{33}^{E}\) \(11.4\times 10^{10}\) N/m², \(e_{31}\) −6.88, \(e_{33}\) 21.4 C/m², \(\varepsilon_{33}^{S}\) \(885\varepsilon_0\), \(\rho\) 8084 kg/m³. Epoxy: \(v_L\) 2500, \(v_S\) 1100 m/s, \(\rho\) 1150 kg/m³.
\(k_t\) PMN-PT pillars\(k_t\) PZT-5H pillars\(Z\) (MRayl, right axis), solid PMN-PT, dashed PZT
Try: both curves have a broad \(k_t\) plateau between \(\nu \approx\) 0.4 and 0.7, so pick \(\nu\) for impedance and dicing yield, not for coupling. Raise \(f_0\) to 20 MHz: the pillar width falls below 50 µm and the kerf to a few tens of µm, where dicing saws stop and etched composites take over. The readout turns red when the kerf is under 15 µm, a practical saw-kerf limit.
Demo 18.7

Material comparison: ranges from Tables 18.1 and 18.2

Exact formulaTabulated published values plotted as is (bars span the quoted range, a tick marks a single value); composite rows are Smith–Auld model values. Film FOMs re-plotted from Roy et al. 2023, Table I. \(g_{33} = d_{33}/\varepsilon_{33}^{T}\) from the tabulated mid values (PVDF: datasheet value).
Try: \(k_t\) ranks crystal composites first; \(Z\) ranks PVDF and composites best for matching; the temperature limit reverses the crystal advantage; \(g_{33}\) shows why PVDF is a receiver. For films, switch between the two FOMs: epitaxial PZT leads on energy, ScAlN and AlN on receive voltage.

18.7Visual summary

Fifteen questions an expert answers in five seconds, each with its number and a figure computed from the formula or from this module's models.

18.8How experts think about it

A CMUT is a nonlinear capacitor with a plate for a top electrode: bias trades stiffness for coupling until the plate folds, and the water load turns its mass-spring resonance into a broadband response. A PMUT is the same plate with a built-in moment instead of a bias, so its design is about the neutral axis, the electrode radius and the film's figure of merit. Bulk materials are chosen on three numbers, coupling, impedance and robustness, and every new material buys one of them by spending another.

A clamped plate pulls in at 70 % of the piston's voltage; its centre has moved about 0.46 of the gap, its average only 0.15.
\(V_{\mathrm{PI}}^{2} = 27.8\,Dg_{\mathrm{eff}}^{3}/(\varepsilon_0 a^{4})\)piston: 56.9ratio 0.699

The electrostatic pressure grows as \(1/(g - w)^{2}\), so the deflected centre is loaded hardest and softens first; the piston, which sees only the average gap, waits too long. The finite-difference plate gives 53.1 V for a 20 µm-radius, 1 µm silicon membrane over a 0.2 µm gap with 0.1 µm nitride, against 76.0 V from the piston with \(k = 192\pi D/a^{2}\) (18.3)–(18.4). The figure plots average deflection against bias for both.

SimulationFD plate (blue, red after collapse) versus lumped piston (grey dashed), default cell of Demo 18.1.
Small deflection (\(w \ll h\)), no residual stress (tension raises \(V_{\mathrm{PI}}\)), uniform gap, no fringing, full electrode. Square or hexagonal cells and partial electrodes change the constant.
Collapse is hysteretic: down at \(V_{\mathrm{PI}}\), up only at the snap-back voltage, and the insulator decides how low that is.
\(V_{\mathrm{SB}} = 0.49\,V_{\mathrm{PI}}\) (\(g_0\) 0.2 µm)\(0.26\,V_{\mathrm{PI}}\) (\(g_0\) 1 µm)Bayram 2003: 140 / 68 V

In contact the electrical gap is only \(t_i/\varepsilon_r\) (13 nm for 0.1 µm of nitride), so a small bias holds a large contact force. The thinner that gap relative to \(g_0\), the lower \(V_{\mathrm{SB}}\). The loop in the figure is the capacitance a C–V meter would trace on the default cell: up the blue branch, jump at \(V_{\mathrm{PI}}\), back down the red one to \(V_{\mathrm{SB}}\).

SimulationFD plate, capacitance normalized to the unbiased value.
Ideal flat contact. Roughness or a contact offset raises \(V_{\mathrm{SB}}\) (a 50 nm offset moves 0.26 to about 0.43 for the 1 µm-gap cell); trapped charge shifts both thresholds.
Coupling is a stiffness ratio: \(k^{2} = 1 - K_V/K_Q\), short circuit over open circuit.
plate: 0.26 at 0.9 \(V_{\mathrm{PI}}\)piston: 0.430.37 vs 0.56 at 0.95 \(V_{\mathrm{PI}}\)

At constant voltage the field acts as a negative spring; at constant charge the voltage readjusts and adds back \(V^{2}\mathbf{c}\mathbf{c}^{T}/C\) (18.5). For a piston the two cancel exactly, giving the Module 19 result. For a plate the softening is concentrated at the centre while the rank-one restoring term is spread over the electrode, so less of the stiffness is electrical and \(k^{2}\) is lower at every bias.

SimulationFD plate eigenvalues (blue) and the exact lumped formula \(2\xi/(1 - \xi)\) (grey dashed).
Fundamental mode, small signal, vacuum. Collapse mode needs a much finer grid at the contact edge; FEM gives \(k_T^{2} \approx 0.7\) there (Bayram et al. 2003).
In collapse the annulus rings: the resonance climbs steeply with the contact radius.
\(b/a\) = 0.3: \(4.4\,f_0\)0.5: \(8.7\,f_0\)0.6: \(13.5\,f_0\)

With its centre pinned, the vibrating part is an annulus clamped at \(b\) and at \(a\); its width \(a - b\) sets the frequency much as the radius sets it for a disc. The mechanical values above (240-interval FD) match the classical clamped–clamped annulus: \(\lambda^{2} \approx 89\) at \(b/a\) = 0.5, 8.7 times the disc's 10.22. Raising the bias grows \(b\), so a collapsed CMUT is a bias-tuned transducer.

SimulationFD eigen-solve with the contact disc fixed, mechanical only (no electrostatic softening), in vacuum.
Softening at the contact edge lowers these values; water loading lowers them further. The contact radius itself depends on bias, insulator and roughness.
A water-loaded CMUT is an over-damped resonator: its band runs from \(\eta k_{\mathrm{eff}}/(2\pi Z_w)\) to \(Z_w/(2\pi\eta m)\).
\(Q = \sqrt{km}/Z_w \approx 0.18\)0.8–14 MHz at 0.9 \(V_{\mathrm{PI}}\)

A 1 µm silicon membrane has \(m_a \approx\) 4 g/m²; at 10 MHz its inertial impedance \(\omega m\) is 0.26 MRayl, a sixth of water. The velocity per unit force of a series \(m\)–\(k\)–\(R\) circuit is flat between \(k/R\) and \(R/m\) when \(Q \ll 1\) (18.7); the figure shows \(\lvert H\rvert\) for \(Q\) = 0.18 and, for contrast, \(Q\) = 3.

Exact formula\(\left\lvert 1/\left(1 + jQ(f/f_0 - f_0/f)\right)\right\rvert\), normalized frequency.
One membrane mode; higher modes, cell-level added mass and parasitic capacitance cut the upper band in practice.
Matched PZT stacks give 60–80 %; CMUT models promise over 150 % and real immersion arrays deliver about 100 %.
PZT, 2 layers: 63 % (model)Oralkan 2002: 96 %, 105 % compensatedOralkan 2003: 91–100 %

The PZT element is a resonator whose Q the matching layers lower; the CMUT is over-damped from the start. The 1D model of Demo 18.3 shows both spectra. The difference between model and measurement for the CMUT is a list of what the model ignores: second and non-axisymmetric membrane modes, crosstalk through the interface wave, diffraction of the finite element and cable capacitance.

Approx. modelDemo 18.3 defaults: PZT-5H, 2 Desilets layers, \(Z_B\) 5 MRayl; CMUT \(a\) = 20 µm at 0.9 \(V_{\mathrm{PI}}\) with added mass.
Plane reflector, plane-wave loading, lossless layers; compare measured bandwidths only after diffraction and attenuation compensation, as Oralkan et al. did.
The gap is the stroke: a CMUT's surface pressure cannot exceed \(Z_w\omega\bar x\).
50 nm at 5 MHz → 2.4 MPa\(V_{\mathrm{PI}} \propto g^{3/2}\)

A plane piston radiates \(p = Z_w v = Z_w\omega\bar x\). The average displacement amplitude \(\bar x\) is a fraction of the gap (the centre can travel at most \(g_0\), the average much less), so a large-output CMUT needs a tall gap, and a tall gap needs a high bias. Collapse mode raises \(\bar x\) several-fold for the same gap (Bayram et al. 2003: 0.25–0.45 µm versus 0.05–0.15 µm).

Exact formula\(p = Z_w\cdot 2\pi f\cdot\bar x\) for \(\bar x\) = 25, 50, 100 nm.
Plane-wave, linear radiation; focusing gain adds on top. Gap nonlinearity and harmonics rise long before contact.
The force goes as \(V^{2}\): keep \(v\) small against \(V_{\mathrm{DC}}\); the second-harmonic force is \(v/(4V_{\mathrm{DC}})\).
\(v = 0.1\,V_{\mathrm{DC}}\): −32 dB\(v = 0.4\,V_{\mathrm{DC}}\): −20 dB

\((V_{\mathrm{DC}} + v\cos\omega t)^{2} = V_{\mathrm{DC}}^{2} + v^{2}/2\) \(+\, 2V_{\mathrm{DC}}v\cos\omega t + (v^{2}/2)\cos 2\omega t\). The ratio of the \(2\omega\) to the \(\omega\) term is \(v/(4V_{\mathrm{DC}})\). Without bias (\(V_{\mathrm{DC}} = 0\)) a CMUT transmits only at \(2\omega\), the reason unbiased drive is sometimes used deliberately. Tissue-harmonic imaging needs the transmitted \(2\omega\) well below the tissue-generated one.

Exact formula\(20\log_{10}(v/4V_{\mathrm{DC}})\), force only (gap nonlinearity adds more).
Force harmonic only; the \(1/(g - w)^{2}\) dependence adds displacement harmonics that grow with stroke.
CMUT crosstalk is a slow interface wave that cuts off at the membrane resonance.
first neighbour −17 dB near 2.1 MHzcollapse: −39 dB average, cutoff ≈ 10 MHz

Below resonance the membrane surface is a spring; water over a compliant wall supports a wave slower than \(c_w\), confined within \(1/\alpha\) of the surface (18.8). In normalized form \(c_{\mathrm{ph}}/c_w = \left[1 + \left(R\Omega/(1 - \Omega^{2})\right)^{2}\right]^{-1/2}\) with \(\Omega = f/f_c\) and \(R = Z_w/(m_a\omega_c)\). The numbers are the measurements of Bayram et al. (2007); a lossy coating, collapse mode or trenches attenuate the wave.

Approx. modelHomogenized membrane surface, \(R\) = 2, 5 and 15.
Homogenization needs cells much smaller than the interface wavelength; near cutoff the wavelength becomes comparable to the cell and the periodic structure (stop bands) matters.
Trapped charge is a bias offset: it slides the C–V curve sideways.
effective bias \(V - V_q\)insulator ≈ 30 V/µm, ≈ 500 V/µm in contact

The force depends on \((V - V_q)^{2}\), so the symmetric C–V "butterfly" of a clean device shifts by \(V_q\). A fixed bias then sits at a different fraction of pull-in, changing sensitivity and, near \(V_{\mathrm{PI}}\), risking collapse. Measuring the C–V minimum before and after a stress test is the standard charging check.

Approx. modelLumped piston \(C/C_0 = 1/(1 - \xi)\) for \(V_q = 0\) and \(0.2\,V_{\mathrm{PI}}\).
Uniform charge sheet; real charge is patchy and drifts, and patterned or post-only insulators reduce it.
Clamped plate: \(f = 10.22/(2\pi a^{2})\cdot\sqrt{D/\rho h}\); halve the radius and the frequency quadruples.
\(\lambda_1 = 3.1962\)\(f \propto h/a^{2}\)Si 1 µm, \(a\) = 20 µm: 10.2 MHz

\(\lambda_1\) is the first root of \(J_0(\lambda)I_1(\lambda) + I_0(\lambda)J_1(\lambda) = 0\) (18.9); the FD solver used here converges to it (10.2155 at 160 intervals). Since \(D \propto h^{3}\) and the mass \(\propto h\), \(f \propto h/a^{2}\). CMUT and PMUT cells are therefore tuned mainly by radius.

Exact formulaSilicon, \(h\) = 1 and 2 µm, vacuum.
Uniform isotropic plate, no residual stress, vacuum; anisotropic silicon and electrodes shift it by a few percent, stress by much more.
Water lowers a PMUT's resonance by \(\sqrt{1 + 0.67\rho_w a/\bar\rho h}\), often by a factor of two or more.
\(a\) = 50 µm, 3 µm stack: \(\beta\) = 4.2÷ 2.3

The plate drags a slug of water whose mass scales as \(\rho_w a^{3}\) while the plate's own mass scales as \(\bar\rho h a^{2}\), hence the ratio \(a/h\) (Lamb 1920, (18.11)). Thin, wide plates are dominated by the water: their frequency in water depends little on their own density, and their bandwidth is set by radiation.

Approx. modelLamb added mass, one side wetted, \(\bar\rho\) = 2640 kg/m³ (AlN on Si).
Baffled plate in an infinite fluid, first mode; arrays with mutual loading and cavities with back pressure differ.
Stop the top electrode at 0.71 a: a full electrode cancels its own moment.
\(\beta_e = 1/\sqrt{2}\)actuation \(\propto \beta_e^{2}(1 - \beta_e^{2})\)

The first mode's curvature \(\nabla^{2}\varphi = (16r^{2}/a^{2} - 8)/a^{2}\) changes sign at \(r = a/\sqrt{2}\) (18.12). A uniform piezoelectric moment does positive work inside that radius and negative work outside, so covering the whole plate gives zero; the optimum electrode ends exactly at the sign change. A ring electrode on the outer part, driven in antiphase, adds the outer contribution.

Approx. modelOne-term Ritz with \(\varphi = (1 - r^{2}/a^{2})^{2}\).
Quasi-static, first mode; the exact mode shape moves the optimum by a few percent, and receive sensitivity per unit capacitance favours a slightly smaller electrode.
A film on the neutral axis does nothing: the bending moment is \(e_{31,f}\cdot V\cdot z_p\).
\(h_s = 0 \to z_p = 0\)AlN 1 µm on Si 2 µm: \(z_p\) = 0.69 µm

The film's lateral stress is \(e_{31,f}V/h_p\); integrated over \(h_p\) it is a force per length \(e_{31,f}V\), and its moment arm about the neutral axis is \(z_p\). Deflection per volt \(\propto z_p a^{2}/D\): a thicker passive layer raises \(z_p\) but \(D\) faster, so there is an optimum passive thickness of the order of the film thickness.

Approx. modelLaminated plate, AlN 1 µm on Si, centre deflection per volt (one-term Ritz), normalized.
Electrodes and residual stress ignored; with a bimorph (two active films driven in antiphase) the neutral axis can sit between them and both contribute.
Rank films by \(e_{31,f}^{2}/(\varepsilon_0\varepsilon_{33})\) for energy and by \(e_{31,f}/(\varepsilon_0\varepsilon_{33})\) for receive voltage.
AlN 11.9 GPaSc0.15AlN 24 GPasol-gel PZT 5.6–54 GPa

Transmit force per volt scales with \(e_{31,f}\), the stored electrical energy with \(\varepsilon\), so \(k^{2}\) and transmitted energy per stored energy scale with \(e^{2}/\varepsilon\). Open-circuit receive voltage per unit stress scales with \(e/\varepsilon\), where AlN's low permittivity (10.5 against hundreds) wins despite its tenfold lower \(e_{31,f}\). Values re-plotted from Roy et al. 2023, Table I.

Exact formulaTabulated values (range mid-points), Roy et al. 2023.
Film values vary strongly with deposition, texture and thickness; loss tangent (AlN 0.003 versus PZT 0.01–0.03) also enters the receive SNR.
Put single-crystal pillars in the composite: \(k_t\) climbs from 0.69 to 0.83 at the same impedance.
\(\nu\) = 0.5: 0.687 vs 0.833\(Z\) 16.4 vs 17.0 MRayl

In a fine-scale composite the pillars work near their bar coupling \(k_{33}\) (0.75 for PZT-5H, above 0.9 for PMN-PT), while the epoxy sets the density and stiffness. The Smith–Auld model gives a broad \(k_t\) plateau for \(\nu \approx\) 0.4–0.7 with either pillar; \(Z\) hardly depends on the pillar material because the densities and stiffnesses are similar.

Approx. modelSmith–Auld, soft epoxy; PMN-28PT constants from Kim et al. 2013.
Fine-scale limit, lossless, no inter-pillar lateral modes; the crystal's \(T_{\mathrm{rt}}\) and coercive field still limit bonding temperature and bipolar drive.
PVDF is a receiver: small \(d_{33}\), huge \(g_{33}\), water-matched.
\(g_{33}\) = 0.33 V·m/N≈ 18 × PZT-5H\(Z \approx\) 4 MRayl, \(k_t\) 0.14

\(g = d/\varepsilon^{T}\) is the open-circuit field per unit stress. PVDF's \(d_{33}\) is 20 times smaller than PZT-5H's but its permittivity is 300–500 times smaller, so its receive voltage per unit thickness is larger. Its impedance is close to water's, so it barely perturbs the field it measures: the basis of membrane and needle hydrophones.

Exact formula\(g_{33} = d_{33}/(\varepsilon_0\varepsilon_{33}^{T})\) from Table 18.1 mid values; PVDF datasheet value.
Low \(k_t\) makes PVDF a poor transmitter; \(\tan\delta \approx\) 0.25 at 1 MHz and a 70–80 °C limit restrict it further.

18.9Cheat sheet

ConceptFormula or numberUse
Effective gap\(g_{\mathrm{eff}} = g_0 + t_i/\varepsilon_r\)insulator in the electrical but not the mechanical gap
Pull-in, piston\(V_{\mathrm{PI}}^{2} = 8kg_{\mathrm{eff}}^{3}/(27\varepsilon_0 A)\), \(k = 192\pi D/a^{2}\)quick estimate (43 % high for a plate)
Pull-in, clamped plate\(V_{\mathrm{PI}}^{2} = 27.8\,Dg_{\mathrm{eff}}^{3}/(\varepsilon_0 a^{4})\); \(w(0) \approx 0.46\,g_{\mathrm{eff}}\), \(\bar w \approx 0.15\,g_{\mathrm{eff}}\)bias design, comparison with FEM
Coupling\(k^{2} = 1 - \lambda(K_V)/\lambda(K_Q)\); plate 0.26 at 0.9 \(V_{\mathrm{PI}}\)bias point choice
Snap-backset by \(t_i/\varepsilon_r\) (and roughness); 0.26–0.49 \(V_{\mathrm{PI}}\) herecollapse-mode bias window
Collapsed annulus\(f/f_0 \approx\) 4.4, 8.7, 13.5 at \(b/a\) = 0.3, 0.5, 0.6bias-tuned frequency
CMUT band (over-damped)\(f_{\mathrm{low}} \approx \eta k_{\mathrm{eff}}/(2\pi Z_w)\), \(f_{\mathrm{high}} \approx Z_w/(2\pi\eta m_a)\), \(m_a = 1.8\rho h + (8/3\pi)\rho_w a\)bandwidth estimate
Interface wave\(\alpha = \omega^{2}\rho_w/(k_a - \omega^{2}m_a)\), \(c_{\mathrm{ph}} = \omega/\sqrt{k_w^{2} + \alpha^{2}}\)crosstalk band and its cutoff
Second harmonic\(F_2/F_1 = v/(4V_{\mathrm{DC}})\)harmonic-imaging drive limits
Stroke limit\(p \le Z_w\cdot 2\pi f\cdot\bar x\)output pressure versus gap
Clamped plate\(f = 10.2158/(2\pi a^{2})\cdot\sqrt{D/\bar\rho h}\)CMUT and PMUT cell frequency
Laminated plate\(z_n = \sum E'hz/\sum E'h\), \(D = \sum E'[h^{3}/12 + h(z - z_n)^{2}]\)PMUT stack design
Water loading\(f_w = f_{\mathrm{vac}}/\sqrt{1 + 0.67\rho_w a/\bar\rho h}\)PMUT in water or gel
Electrode radiusactuation \(\propto \beta_e^{2}(1 - \beta_e^{2})\), optimum 0.707PMUT layout
Film FOMs\(e_{31,f}^{2}/(\varepsilon_0\varepsilon_{33})\), \(e_{31,f}/(\varepsilon_0\varepsilon_{33})\)AlN vs ScAlN vs PZT film
CompositeSmith–Auld; \(\nu = (w/p)^{2}\) (1-3), \(w/p\) (2-2); \(w/t \lesssim 0.5\)\(k_t\), \(Z\) and dicing
Receive constant\(g_{33} = d_{33}/\varepsilon_{33}^{T}\): PVDF 0.33, PZT-5H 0.018 V·m/Nhydrophone choice

18.10Interview and qualifying-exam questions

I1What sets a CMUT's pull-in voltage, and how does it scale with membrane radius, thickness and gap? · screen

Balance of the plate's bending stiffness against the electrostatic pressure: \(V_{\mathrm{PI}}^{2} = c\cdot Dg_{\mathrm{eff}}^{3}/(\varepsilon_0 a^{4})\) with \(c\) = 27.8 for a clamped plate (56.9 for the piston idealization) (18.4), so \(V_{\mathrm{PI}} \propto h^{3/2}g_{\mathrm{eff}}^{3/2}/a^{2}\), and \(g_{\mathrm{eff}}\) includes \(t_i/\varepsilon_r\) of the insulator. A 1 µm Si membrane, 20 µm radius, 0.2 µm gap and 0.1 µm nitride: 53 V. Follow-up: "how would residual tensile stress change it?" It adds a membrane stiffness that raises \(V_{\mathrm{PI}}\) and the resonance; for thin, highly stressed membranes it dominates the bending term.

I2Why does a CMUT not need a matching layer, and what limits its bandwidth? · screen

Its moving mass per area is a few g/m², so \(\omega m\) is a fraction of water's 1.5 MRayl: the water load over-damps the mass–spring (\(Q \approx 0.2\)) and couples directly. The band runs from \(\approx \eta k_{\mathrm{eff}}/(2\pi Z_w)\) (spring) to \(\approx Z_w/(2\pi\eta m)\) (mass, including water added mass) (18.7). In practice higher membrane modes, parasitic capacitance, crosstalk and diffraction bring measured immersion bandwidths to about 100 % (Oralkan et al. 2002: 96 % uncompensated, 105 % compensated). Follow-up: "and what limits the low end?" Spring softening helps (\(k_{\mathrm{eff}}\) falls with bias), but element size limits radiation at low frequency.

I3Explain collapse-mode operation. Why would you use it, and what are the risks? · onsite

Biasing past \(V_{\mathrm{PI}}\) brings the membrane centre onto the insulator; the free annulus then vibrates at a higher, bias-tunable frequency (\(f/f_0 \approx 8.7\) at \(b/a\) = 0.5), with higher coupling (FEM: \(k_T^{2} \approx 0.7\)) and larger average stroke. The state is hysteretic: it releases only below \(V_{\mathrm{SB}}\), set by the insulator's electrical thickness. Risks: dielectric charging under very high contact fields, wear and stiction at the contact, temperature and bias drift of the operating point. Follow-up: "how do you get into collapse reliably?" Overdrive briefly above \(V_{\mathrm{PI}}\), then hold a bias inside the hysteresis window.

I4How do you define and compute the coupling coefficient of a CMUT, and why is the piston value optimistic? · onsite

\(k^{2} = 1 - K_V/K_Q\) (or \(1 - C^{S}/C^{T}\) electrically): the fraction of stiffness that is electrical. For a plate, linearize about the biased shape, form \(\mathbf{K}_V = \mathbf{K}\) minus the nodal softening and \(\mathbf{K}_Q = \mathbf{K}_V + V^{2}\mathbf{c}\mathbf{c}^{T}/C\), and compare the lowest eigenvalues (18.5)–(18.6). The piston puts all softening into one degree of freedom and gets \(2\xi/(1 - \xi) = 0.43\) at 0.9 \(V_{\mathrm{PI}}\); the plate gets 0.26 because only the centre softens strongly. Follow-up: "which one do you measure?" From \(f_r/f_a\) of the impedance or from \(C\) at constant bias, as for a piezo element (Module 16).

I5What is the main crosstalk mechanism in CMUT arrays and how do you suppress it? · onsite

A dispersive guided wave at the fluid–membrane interface, slower than \(c_w\) and cut off near the membrane resonance; Bayram et al. (2007) measured −17 dB on the first neighbour near 2.1 MHz in conventional mode. It is a property of the compliant surface, not of the substrate (the substrate Lamb modes A0 and S0 were about 23 and 48 dB lower). Suppression: operate in collapse mode (cutoff rises, average −39 dB), add a lossy coating such as PDMS, shape the cell layout or trench between elements, or compensate in the beamformer. Follow-up: "why does it get worse with bias?" Softening lowers the cutoff into the band.

I6Sacrificial release or wafer bonding for a new CMUT array: how do you choose? · onsite

Wafer bonding gives single-crystal silicon membranes with excellent thickness control (device layer under 1.5 µm, < 10 nm variation), any cell shape and fill factor, and gap control by oxidation (LOCOS: 40 nm ± 2 nm), but needs SOI wafers, a thicker insulator for the conductive membrane, and is harder to do on top of CMOS. Sacrificial release uses deposited nitride membranes, works at lower temperature and is one natural route for CMUT-on-CMOS (low-temperature bonding onto the CMOS wafer is the other), at the cost of membrane stress and thickness variation and etch-hole sealing. Follow-up: "which would you use for a monolithic 2D array?" Sacrificial release (or low-temperature bonding variants), because the CMOS limits the thermal budget.

I7Sketch a 2D CMUT-on-CMOS array for 3 MHz cardiac imaging. What dominates the design? · research

\(\lambda/2\) = 257 µm pitch; a 20 × 20 mm aperture is about 80 × 80 = 6400 elements, impossible to cable, so each element gets a pulser and LNA under it and micro-beamforming reduces the channel count to the system (Module 19). Dominant constraints: power density (a few mW per channel already gives watts on a few cm2 at the patient), element capacitance versus LNA noise, bias distribution (tens of volts on a CMOS die, isolated per element or shared), acoustic crosstalk through the interface wave, and thermal limits on the probe face (Module 22). Follow-up: "how do you test it?" Per-element impulse response and C–V maps on wafer before dicing, then pulse-echo uniformity.

I8Give the PMUT resonance formula. Why is the electrode only about 70 % of the radius? · screen

\(f = 10.22/(2\pi a^{2})\cdot\sqrt{D/\bar\rho h}\) for a clamped bilayer plate with laminated stiffness \(D\), divided by \(\sqrt{1 + 0.67\rho_w a/\bar\rho h}\) in water (18.9)–(18.11). The electrode applies a uniform moment; the mode curvature changes sign at \(r = a/\sqrt{2}\), so the work done is \(\propto \beta_e^{2}(1 - \beta_e^{2})\): zero for a full electrode, maximal at 0.707. Example: AlN 1 µm on Si 2 µm, \(a\) = 50 µm: 5.6 MHz in vacuum, 2.4 MHz in water. Follow-up: "what does residual stress do?" Tension raises \(f\) and pushes toward membrane behaviour, compression lowers it and can buckle the plate.

I9AlN or PZT film for a PMUT receiver? For a transmitter? · onsite

Receiver: open-circuit voltage per stress \(\propto e_{31,f}/(\varepsilon_0\varepsilon_{33})\): AlN 11.3 GV/m and Sc0.15AlN 15 GV/m beat sol-gel PZT (0.7–1.8 as published) because their permittivity is 10–12 rather than hundreds; AlN also has a lower loss tangent. Transmitter: force per volt \(\propto e_{31,f}\), 8–14 C/m² for PZT versus 1.05 for AlN, and energy FOM \(e^{2}/\varepsilon\) up to 110 GPa for epitaxial PZT. ScAlN doubles AlN's energy FOM while staying CMOS-friendly. Follow-up: "what about one die doing both?" Use separate receive and transmit electrodes, or a ScAlN compromise.

I10Would you put single-crystal composite in a 2D matrix TEE probe? What would worry you? · research

The gain: \(k_t \approx\) 0.8 (composite) versus ≈ 0.5–0.7, which means wider bandwidth and higher sensitivity, valuable for harmonic imaging through a small aperture. The worries: PMN-PT depoles near \(T_{\mathrm{rt}}\) 86–100 °C and has \(E_c \approx\) 2 kV/cm, so bonding, sterilization and self-heating at the probe face, and bipolar drive at high voltage, all eat the margin; PIN-PMN-PT (\(T_{\mathrm{rt}}\) 105–130 °C, \(E_c\) 4.5–6 kV/cm) is the usual answer. Also cost, dicing yield at small pitch, and the higher permittivity, which helps small elements drive the in-probe electronics. Follow-up: "Mn doping?" Raises \(Q_m\) (≈ 500), useful for therapy, less for imaging bandwidth.

I11What is AC poling and why did it attract attention? · research

Poling a relaxor-PT crystal with an alternating field instead of DC. In rhombohedral PMN-PT it changes the domain configuration; Qiu et al. (Nature 2020) removed the light-scattering 71° domain walls and obtained near-perfect optical transparency with \(d_{33}\) above 2100 pC/N and \(k_{33} \approx 0.94\). For ultrasound it offers higher \(d_{33}\) from the same crystal and transparent transducers through which light can be delivered, which simplifies photoacoustic and optical–acoustic probes. Follow-up: "what limits it?" Temperature stability and the same \(T_{\mathrm{rt}}\) and \(E_c\) constraints as any PMN-PT; poling-condition reproducibility in production.

18.11MATLAB lab

Base MATLAB only (no toolboxes); each script runs as-is in MATLAB R2019b+ and GNU Octave 8. Paste into the editor and press Run.

Download the scripts: m18_cmut_parallel_plate.m · m18_pmut_plate_resonance.m · m18_composite_effective_properties.m · m18_material_comparison.m · m18_cmut_vs_pzt_bandwidth.m. Base MATLAB R2019b+ or GNU Octave 8, no toolboxes; all scripts in the repository.

m18_cmut_parallel_plate.mLumped vs distributed CMUT: both branches, pull-in, snap-back, coupling · change \(a\), \(h\), \(g_0\), \(t_i\), \(N\) · runtime ≈ 2 s 
%% m18_cmut_parallel_plate.m
% CMUT electromechanics: lumped parallel-plate model versus a distributed
% clamped-plate model (axisymmetric finite differences), conventional and
% collapse branches, snap-back, spring softening and coupling versus bias.
%
% What it shows
%   1) Lumped piston: (V/Vpi)^2 = 27/4 xi (1-xi)^2, kT^2 = 2xi/(1-xi), pull-in at xi = 1/3.
%   2) Distributed clamped plate D*del^4 w = eps0 V^2 / (2 (g_eff - w)^2):
%      conventional branch traced by fixing the centre deflection (the voltage
%      that holds it is the unknown), collapsed branch traced by fixing the
%      contact radius (zero contact reaction at its edge). Both are Newton solves.
%   3) Pull-in voltage, centre and average deflection at pull-in, snap-back voltage.
%   4) Coupling k^2 = 1 - lambda_V / lambda_Q from the two lowest eigenvalues
%      (constant voltage vs constant charge) on the conventional branch.
% What to change: a, h, g0, ti, er (geometry), N (grid), E/nu/rho (membrane).
% Expected output (defaults: Si membrane a = 20 um, h = 1 um, g0 = 0.2 um,
%   0.1 um nitride): f0 = 10.24 MHz, Vpi = 53.1 V distributed vs 76.0 V lumped
%   (ratio 0.699), centre deflection at pull-in 0.46 g_eff (0.459 printed below), average 0.15 g_eff,
%   snap-back about 26 V, k^2 = 0.26 at 0.9 Vpi (lumped model: 0.43).
% Runtime: about 5 s in Octave.
clear; close all; rng(1);

%% Parameters (SI units)
E = 169e9; nu = 0.22; rho = 2330;     % silicon membrane
a = 20e-6; h = 1.0e-6;                % membrane radius, thickness (m)
g0 = 0.20e-6; ti = 0.10e-6; er = 7.5; % vacuum gap, insulation thickness, insulation permittivity
e0 = 8.854e-12;
N = 60;                               % radial grid intervals
D = E*h^3/(12*(1-nu^2));              % flexural rigidity (N m)
ge = g0 + ti/er;                      % effective electrical gap (m)
f0 = 10.2158/(2*pi*a^2)*sqrt(D/(rho*h));
fprintf('Membrane: a = %.0f um, h = %.2f um, D = %.3g N m, f0 (vacuum) = %.2f MHz\n', a*1e6, h*1e6, D, f0/1e6);

%% Lumped piston model (spring from the average deflection under uniform pressure)
kp = 192*pi*D/a^2; Ap = pi*a^2;
Vpi_l = sqrt(8*kp*ge^3/(27*e0*Ap));
xi = linspace(0, 1/3, 400);
Vl = sqrt(27/4*xi.*(1-xi).^2);        % V/Vpi
k2l = 2*xi./(1-xi); fl = sqrt((1-3*xi)./(1-xi));
fprintf('Lumped: Vpi = %.1f V; at 0.9 Vpi: xi = %.3f, kT^2 = %.3f, f/f0 = %.3f\n', Vpi_l, ...
  interp1(Vl, xi, 0.9), interp1(Vl, k2l, 0.9), interp1(Vl, fl, 0.9));

%% Distributed model: axisymmetric FD operators
dr = a/N; r = (0:N)'*dr;
L = zeros(N+1, N);                    % Laplacian at nodes 0..N acting on unknowns w(0..N-1); w(N) = 0, w'(a) = 0
L(1,1) = -4/dr^2; L(1,2) = 4/dr^2;
for i = 2:N
  rp = r(i) + dr/2; rm = r(i) - dr/2;
  L(i,i) = -(rp+rm)/(r(i)*dr^2); L(i,i-1) = rm/(r(i)*dr^2);
  if i < N, L(i,i+1) = rp/(r(i)*dr^2); end
end
L(N+1, N) = 2/dr^2;                   % clamped edge via ghost node
Wt = 2*pi*r*dr; Wt(1) = pi*(dr/2)^2; Wt(end) = pi*r(end)*dr;
K = D*(L'*diag(Wt)*L); Wf = Wt(1:N); M = rho*h*diag(Wf);
lam0 = min(real(eig(K, M)));
fprintf('FD check: f0 = %.3f MHz (exact 10.2158 constant gives %.3f MHz)\n', sqrt(lam0)/2/pi/1e6, f0/1e6);

modal = @(KV, KQ, F) [min(real(eig(KV(F,F), M(F,F)))), min(real(eig(KQ(F,F), M(F,F))))];

%% Conventional branch: control the centre deflection wc, unknowns w and lambda = V^2
wcs = linspace(0.01, 0.99, 99)*g0;
w = zeros(N,1); lam = 0;
Vc = zeros(size(wcs)); k2c = Vc; fc = Vc; wav = Vc;
for s = 1:numel(wcs)
  for it = 1:50
    fe = Wf*e0./(2*(ge-w).^2); fp = Wf*e0./(ge-w).^3;
    R = [K*w - lam*fe; w(1) - wcs(s)];
    J = [K - lam*diag(fp), -fe; [1, zeros(1,N-1)], 0];
    dx = -J\R; w = w + dx(1:N); lam = lam + dx(end);
    if max(abs(dx(1:N))) < 1e-15, break; end
  end
  Vc(s) = sqrt(lam); wav(s) = sum(Wf.*w)/sum(Wt);
  fp = lam*Wf*e0./(ge-w).^3; KV = K - diag(fp);
  gC = Wf*e0./(ge-w).^2; C = sum(Wf*e0./(ge-w));
  KQ = KV + lam*(gC*gC')/C;
  ev = modal(KV, KQ, 1:N);
  k2c(s) = 1 - ev(1)/ev(2); fc(s) = sqrt(max(ev(1),0))/2/pi;
  if s == 1, wsave = zeros(N,0); end
  if any(abs(wcs(s)/g0 - [0.2 0.4 0.49]) < 1e-9), wsave(:,end+1) = w; end %#ok<AGROW>
end
[Vpi, ip] = max(Vc);
fprintf('Distributed: Vpi = %.1f V (lumped %.1f V, ratio %.3f)\n', Vpi, Vpi_l, Vpi/Vpi_l);
fprintf('  at pull-in: centre deflection %.3f g_eff, average deflection %.3f g_eff\n', wcs(ip)/ge, wav(ip)/ge);
fprintf('  closed form fit: Vpi^2 = %.2f D g_eff^3 / (eps0 a^4)\n', Vpi^2*e0*a^4/(D*ge^3));
st = 1:ip;                             % stable part of the conventional branch
for fr = [0.5 0.8 0.9 0.95]
  fprintf('  V/Vpi = %.2f: k^2 = %.3f, f/f0 = %.3f, wc/g0 = %.3f\n', fr, interp1(Vc(st)/Vpi, k2c(st), fr), ...
    interp1(Vc(st)/Vpi, fc(st)/f0, fr), interp1(Vc(st)/Vpi, wcs(st)/g0, fr));
end

%% Collapsed branch: control the contact radius (nodes 1..nc at w = g0, zero reaction at node nc)
w(1) = g0; Vcol = []; bcol = []; wavc = [];
for nc = 1:N-4
  F = (nc+1:N)'; e = nc;
  for it = 1:60
    w(1:nc) = g0;
    fe = Wf*e0./(2*(ge-w).^2); fp = Wf*e0./(ge-w).^3;
    Kf = K - lam*diag(fp); Rv = K*w - lam*fe;
    R = [Rv(F); Rv(e)];
    J = [Kf(F,F), -fe(F); Kf(e,F), -fe(e)];
    dx = -J\R; sstep = 1;
    while any(w(F) + sstep*dx(1:end-1) >= ge - 1e-12) && sstep > 1e-4, sstep = sstep/2; end
    w(F) = w(F) + sstep*dx(1:end-1); lam = lam + sstep*dx(end);
    if max(abs(sstep*dx(1:end-1))) < 1e-15, break; end
  end
  Vcol(end+1) = sqrt(lam); bcol(end+1) = r(nc); wavc(end+1) = sum(Wf.*w)/sum(Wt); %#ok<AGROW>
end
[Vsb, isb] = min(Vcol);
fprintf('Collapsed branch: snap-back at %.1f V (%.3f Vpi); contact radius b/a = 0.5 at %.2f Vpi\n', ...
  Vsb, Vsb/Vpi, interp1(bcol(isb:end)/a, Vcol(isb:end), 0.5)/Vpi);
fprintf('Hysteresis window: %.1f V to %.1f V\n', Vsb, Vpi);

%% Figures
figure('Position', [80 80 1100 380]);
subplot(1,3,1);
plot(Vc(st), wav(st)/ge, 'b-', 'LineWidth', 2); hold on;
plot(Vc(ip:end), wav(ip:end)/ge, 'b--');
plot(Vcol, wavc/ge, 'r-', 'LineWidth', 2);
plot(Vl*Vpi_l, xi, 'k:', 'LineWidth', 1.5);
xlim([0 2*Vpi]); ylim([0 1]);
xlabel('bias V (V)'); ylabel('average deflection / g_{eff}');
title('Both branches and hysteresis'); legend('conventional, stable', 'conventional, unstable', 'collapsed', 'lumped piston', 'Location', 'northwest');
subplot(1,3,2);
plot(Vl, k2l, 'k:', 'LineWidth', 1.5); hold on; plot(Vc(st)/Vpi, k2c(st), 'b-', 'LineWidth', 2);
xlabel('V / V_{PI}'); ylabel('k^2'); ylim([0 1]); title('Coupling vs bias'); legend('lumped piston', 'distributed', 'Location', 'northwest');
subplot(1,3,3);
plot(r(1:N)/a*1e0, wsave/g0, 'LineWidth', 1.5); xlabel('r / a'); ylabel('w / g_0'); title('Profiles at w_c = 0.2, 0.4, 0.49 g_0');

What to look for: \(f_0\) = 10.24 MHz (FD 10.242), \(V_{\mathrm{PI}}\) = 53.1 V against 76.0 V lumped (ratio 0.699), \(V_{\mathrm{PI}}^{2} = 27.76\,Dg_{\mathrm{eff}}^{3}/(\varepsilon_0 a^{4})\), \(k^{2}\) = 0.042, 0.157, 0.262, 0.374 at 0.5, 0.8, 0.9, 0.95 \(V_{\mathrm{PI}}\), snap-back 25.8 V (0.486 \(V_{\mathrm{PI}}\)).

m18_pmut_plate_resonance.mBessel root 10.2158, FD check, bilayer PMUTs in vacuum and water, electrode optimum · change film, thicknesses, radius · runtime ≈ 1 s 
%% m18_pmut_plate_resonance.m
% PMUT: clamped circular plate resonance, bilayer (piezo film on passive layer),
% water loading (Lamb added mass) and the optimum inner-electrode radius.
%
% What it shows
%   1) The clamped-plate eigenvalue: J0(L) I1(L) + I0(L) J1(L) = 0 gives L = 3.1962,
%      L^2 = 10.2158; an FD eigen-solve of the same plate converges to it.
%   2) f = (L^2 / (2 pi a^2)) sqrt(D / (rho h)) for a bilayer with the neutral-axis D.
%   3) Water on one side: f_w = f_vac / sqrt(1 + 0.67 rho_w a / (rho h))  (Lamb 1920).
%   4) Actuation from a uniform piezoelectric moment over r < beta*a is proportional to
%      beta^2 (1 - beta^2) (one-term Ritz): optimum beta = 0.707, zero for a full electrode.
% What to change: film material, film and passive thickness, radius list.
% Expected output: lambda^2 = 10.2158 (FD 10.2113, 10.2147, 10.2155 for N = 40, 80, 160);
%   1 um film on 2 um Si, a = 50 um: f_vac = 5.58 / 4.81 / 2.98 MHz (AlN / ScAlN / PZT),
%   in water 2.44 / 2.11 / 1.54 MHz (Lamb factor 4.2, 4.2, 2.7); optimum beta = 0.707.
% Runtime: about 2 s.
clear; close all; rng(1);

%% 1) Clamped plate eigenvalue from the Bessel characteristic equation
g = @(L) besselj(0,L).*besseli(1,L) + besseli(0,L).*besselj(1,L);
lo = 3.0; hi = 3.4;                     % bracket around the first root
for it = 1:60, mid = (lo+hi)/2; if sign(g(mid)) == sign(g(lo)), lo = mid; else hi = mid; end, end
L1 = (lo+hi)/2;
fprintf('Clamped circular plate: lambda = %.4f, lambda^2 = %.4f\n', L1, L1^2);

%% 2) FD check of the constant (unit plate, D = rho h = 1, a = 1)
for N = [40 80 160]
  dr = 1/N; r = (0:N)'*dr; Lp = zeros(N+1,N);
  Lp(1,1) = -4/dr^2; Lp(1,2) = 4/dr^2;
  for i = 2:N
    rp = r(i)+dr/2; rm = r(i)-dr/2;
    Lp(i,i) = -(rp+rm)/(r(i)*dr^2); Lp(i,i-1) = rm/(r(i)*dr^2);
    if i < N, Lp(i,i+1) = rp/(r(i)*dr^2); end
  end
  Lp(N+1,N) = 2/dr^2;
  Wt = 2*pi*r*dr; Wt(1) = pi*(dr/2)^2; Wt(end) = pi*dr;
  K = Lp'*diag(Wt)*Lp; M = diag(Wt(1:N));
  fprintf('  FD N = %3d: lambda^2 = %.4f\n', N, sqrt(min(real(eig(K,M)))));
end

%% 3) Bilayer plates: film on silicon
% Film data: e31,f and eps_r from Wang et al. (Hilton Head 2016) / Roy et al. (2023);
% E as listed by Wang et al.; densities are typical bulk values.
films = {'AlN', 390e9, 0.24, 3260, -1.05, 10.5;  'ScAlN', 200e9, 0.24, 3300, -1.6, 12; ...
         'PZT film', 65e9, 0.30, 7600, -14.0, 1200};
Es = 169e9; nus = 0.22; rhos = 2330;    % silicon passive layer
rhow = 1000;
hs = 2e-6; hp = 1e-6; a = 50e-6;
fprintf('\nBilayer: %.1f um film on %.1f um Si, a = %.0f um\n', hp*1e6, hs*1e6, a*1e6);
fprintf('%-9s %8s %8s %9s %9s %10s %12s\n', 'film', 'zn (um)', 'zp (um)', 'fvac MHz', 'fwat MHz', 'beta_Lamb', 'wc/V (nm/V)');
res = zeros(size(films,1), 3);
for k = 1:size(films,1)
  Ep = films{k,2}; nup = films{k,3}; rhop = films{k,4}; e31 = films{k,5};
  E1 = Es/(1-nus^2); E2 = Ep/(1-nup^2);          % plate moduli
  z1 = hs/2; z2 = hs + hp/2;                      % layer mid-planes from the bottom
  zn = (E1*hs*z1 + E2*hp*z2)/(E1*hs + E2*hp);     % neutral axis
  Db = E1*(hs^3/12 + hs*(z1-zn)^2) + E2*(hp^3/12 + hp*(z2-zn)^2);
  mh = rhos*hs + rhop*hp;                         % mass per area
  fv = L1^2/(2*pi*a^2)*sqrt(Db/mh);
  bL = 0.67*rhow*a/mh; fw = fv/sqrt(1 + bL);
  zp = z2 - zn;                                   % moment arm of the film
  wcV = 0.375*abs(e31)*zp*a^2*0.25/Db;            % centre deflection per volt at beta^2 = 1/2 (one-term Ritz)
  fprintf('%-9s %8.2f %8.2f %9.2f %9.2f %10.2f %12.2f\n', films{k,1}, zn*1e6, zp*1e6, fv/1e6, fw/1e6, bL, wcV*1e9);
  res(k,:) = [fv fw wcV];
end

%% 4) Electrode coverage factor
beta = linspace(0,1,2001); q = beta.^2.*(1-beta.^2);
[~, ib] = max(q); fprintf('\nOptimum inner-electrode radius: beta = %.3f a (factor %.3f); full electrode gives %.3f\n', beta(ib), q(ib), q(end));

%% 5) Frequency versus radius for the three films, in water
aa = linspace(20e-6, 200e-6, 100); F = zeros(size(films,1), numel(aa));
for k = 1:size(films,1)
  Ep = films{k,2}; nup = films{k,3}; rhop = films{k,4};
  E1 = Es/(1-nus^2); E2 = Ep/(1-nup^2); z1 = hs/2; z2 = hs+hp/2;
  zn = (E1*hs*z1 + E2*hp*z2)/(E1*hs + E2*hp);
  Db = E1*(hs^3/12 + hs*(z1-zn)^2) + E2*(hp^3/12 + hp*(z2-zn)^2); mh = rhos*hs + rhop*hp;
  F(k,:) = L1^2./(2*pi*aa.^2).*sqrt(Db/mh)./sqrt(1 + 0.67*rhow*aa/mh);
end
fprintf('Water-loaded f at a = 100 um: AlN %.2f, ScAlN %.2f, PZT %.2f MHz\n', interp1(aa,F(1,:),100e-6)/1e6, ...
  interp1(aa,F(2,:),100e-6)/1e6, interp1(aa,F(3,:),100e-6)/1e6);

figure('Position', [80 80 1000 380]);
subplot(1,2,1); loglog(aa*1e6, F/1e6, 'LineWidth', 1.8); grid on;
xlabel('radius a (\mum)'); ylabel('f in water (MHz)'); legend(films(:,1)); title('PMUT resonance, 2 \mum Si + 1 \mum film');
subplot(1,2,2); plot(beta, q/max(q), 'LineWidth', 2); grid on; xlabel('electrode radius / a'); ylabel('relative actuation');
title('Inner electrode: optimum at 0.707 a');

What to look for: \(\lambda^{2}\) = 10.2158; FD 10.2113 / 10.2147 / 10.2155; 1 µm film on 2 µm Si, \(a\) = 50 µm: 5.58 / 4.81 / 2.98 MHz in vacuum and 2.44 / 2.11 / 1.54 MHz in water for AlN / ScAlN / PZT; optimum electrode 0.707 a.

m18_composite_effective_properties.mSmith–Auld for PZT-5H and PMN-28PT pillars · change the polymer, the pillar constants, \(f_0\) · runtime < 1 s 
%% m18_composite_effective_properties.m
% 1-3 piezocomposites: Smith-Auld effective thickness-mode properties for
% PZT-5H pillars and PMN-28PT single-crystal pillars in the same soft epoxy.
%
% What it shows: kt, Z, v and clamped permittivity versus ceramic volume fraction,
%   the plateau of kt near nu = 0.4-0.7, and how much a crystal pillar adds.
% What to change: polymer (vL, vS, rho), pillar constants, f0 for the thickness table.
% Constants: PZT-5H as in Module 16 (c11E 12.6, c12E 7.95, c13E 8.41, c33E 11.7 x1e10 N/m^2,
%   e31 -6.5, e33 23.3 C/m^2, eps33S 1470, rho 7500); PMN-28PT from Kim, Joh & Roh,
%   Sensors and Materials 25 (2013) 539 (c11E 14.7, c12E 13.1, c13E 11.4, c33E 11.4,
%   e31 -6.88, e33 21.4, eps33S 885, rho 8084). Soft epoxy vL 2500, vS 1100 m/s, rho 1150.
% Expected output: nu = 1: kt 0.513 / 0.582, Z 34.5 / 37.3 MRayl;
%   nu = 0.5: kt 0.687 / 0.833, Z 16.4 / 17.0 MRayl.
% Runtime: < 1 s.
clear; close all; rng(1);
e0 = 8.854e-12;
pz = [12.6e10 7.95e10 8.41e10 11.7e10 -6.5 23.3 1470 7500];   % c11 c12 c13 c33 e31 e33 epsS rho
pm = [14.7e10 13.1e10 11.4e10 11.4e10 -6.88 21.4 885 8084];
rp = 1150; vL = 2500; vS = 1100; er_p = 4;
p11 = rp*vL^2; p44 = rp*vS^2; p12 = p11 - 2*p44;

nus = linspace(0.05, 1, 96);
out = zeros(2, numel(nus), 4);
mats = {pz, pm};
for m = 1:2
  c = mats{m};
  for i = 1:numel(nus)
    nu = nus(i);
    Dn  = nu*(p11 + p12) + (1-nu)*(c(1) + c(2));
    c33E = nu*(c(4) - 2*(1-nu)*(c(3) - p12)^2/Dn) + (1-nu)*p11;
    e33 = nu*(c(6) - 2*(1-nu)*c(5)*(c(3) - p12)/Dn);
    eS  = nu*(c(7)*e0 + 2*(1-nu)*c(5)^2/Dn) + (1-nu)*er_p*e0;
    c33D = c33E + e33^2/eS; rho = nu*c(8) + (1-nu)*rp;
    v = sqrt(c33D/rho);
    out(m,i,:) = [sqrt(e33^2/(c33D*eS)), rho*v/1e6, v, eS/e0];
  end
end
fprintf('%5s | %18s | %18s\n', 'nu', 'PZT-5H: kt  Z  epsS', 'PMN-28PT: kt  Z  epsS');
for nu = [1 0.7 0.6 0.5 0.4 0.3 0.2]
  [~, i] = min(abs(nus - nu));
  fprintf('%5.2f | %5.3f %5.1f %6.0f | %5.3f %5.1f %6.0f\n', nus(i), out(1,i,1), out(1,i,2), out(1,i,4), out(2,i,1), out(2,i,2), out(2,i,4));
end
f0 = 5e6; [~, i] = min(abs(nus - 0.5));
fprintf('At nu = 0.5, f0 = 5 MHz: half-wave thickness %.0f um (PZT) and %.0f um (PMN-PT); pillar width <= 0.5 t: %.0f um\n', ...
  out(1,i,3)/(2*f0)*1e6, out(2,i,3)/(2*f0)*1e6, 0.5*out(2,i,3)/(2*f0)*1e6);

figure('Position', [80 80 1000 380]);
subplot(1,2,1); plot(nus, squeeze(out(1,:,1)), 'b-', nus, squeeze(out(2,:,1)), 'r-', 'LineWidth', 2); grid on;
xlabel('ceramic volume fraction \nu'); ylabel('k_t'); legend('PZT-5H / epoxy', 'PMN-28PT / epoxy', 'Location', 'south'); ylim([0.3 0.9]);
title('Effective thickness coupling (Smith-Auld)');
subplot(1,2,2); plot(nus, squeeze(out(1,:,2)), 'b-', nus, squeeze(out(2,:,2)), 'r-', 'LineWidth', 2); grid on;
xlabel('ceramic volume fraction \nu'); ylabel('Z (MRayl)'); title('Acoustic impedance');

What to look for: \(\nu = 1\): \(k_t\) 0.513 / 0.582, \(Z\) 34.5 / 37.3 MRayl; \(\nu = 0.5\): 0.687 / 0.833 and 16.4 / 17.0 MRayl; half-wave thickness at 5 MHz 379 / 367 µm.

m18_material_comparison.mTables 18.1–18.2 with \(g_{33}\), transmission into water and film FOMs · add your supplier data · runtime < 1 s 
%% m18_material_comparison.m
% Transducer materials side by side: bulk (d33, kt, eps, Z, temperature limit)
% and thin films (e31,f and the PMUT figure of merit e31,f^2 / (eps0 eps33)).
%
% What it shows: why single crystals win on coupling, composites on impedance,
%   PVDF on receive voltage (g33) and water match, and ScAlN/PZT films on FOM.
% What to change: add your own supplier values to the tables.
% Sources (see the Module 18 table): PZT-5H: Ma et al. IEEE TUFFC 2015 Table I
%   (d33 593, kt 0.51, epsS 1470, Z 36.0) and APC/Piezo Systems PSI-5H4E (k33 0.75,
%   epsT 3800, Tc 230 C); PMN-PT and PIN-PMN-PT: TRS Technologies X2B/X4B datasheet
%   ranges; composites: Smith-Auld model (m18_composite_effective_properties.m);
%   KNN: Saito et al. Nature 2004; PVDF: Measurement Specialties data (d33 -33,
%   g33 -330e-3, kt 0.14) and Precision Acoustics (rho 1780, c 2250, eps 8-10);
%   films: Roy, Lee & Lee, Microsyst. Nanoeng. 2023, Table I.
% Expected output: g33 (1e-3 V m/N) PZT-5H 18.5, PMN-PT 36.7, PIN-PMN-PT 37.3,
%   PVDF 356 (datasheet 330); film FOM (GPa) AlN 11.9, Sc0.15AlN 24.1,
%   sol-gel PZT 5.6-54.2, epitaxial PZT 74-111, KNN film 18-53.
% Runtime: < 1 s.
clear; close all; rng(1);
e0 = 8.854e-12;

%% Bulk materials: [low high] ranges; NaN = not given by the cited sources
names = {'PZT-5H', 'PMN-PT', 'PIN-PMN-PT', '1-3 PZT/epoxy', '1-3 crystal/epoxy', 'KNN (textured)', 'PVDF'};
d33  = [593 650; 1500 2400; 1200 2100; NaN NaN; NaN NaN; 300 416; 30 33];        % pC/N (PVDF magnitude)
kt   = [0.51 0.51; 0.55 0.59; 0.52 0.58; 0.67 0.69; 0.81 0.83; NaN NaN; 0.14 0.14];
epsT = [3800 3800; 5000 7000; 4000 6000; NaN NaN; NaN NaN; NaN NaN; 8 12];
Z    = [36.0 36.0; 36.4 39.2; 35.2 37.8; 11.0 21.9; 11.3 22.8; NaN NaN; 4.0 4.0];  % MRayl
Tmax = [230 230; 86 100; 105 130; NaN NaN; NaN NaN; NaN NaN; 70 80];             % C: Tc (PZT), Trt (crystals), max use (PVDF)
mid = @(x) mean(x, 2);
fprintf('%-18s %10s %6s %8s %8s %8s %12s\n', 'material', 'd33 pC/N', 'kt', 'epsT', 'Z MRayl', 'Tmax C', 'g33 mVm/N');
g33 = mid(d33)*1e-12./(mid(epsT)*e0)*1e3;
for i = 1:numel(names)
  fprintf('%-18s %10.0f %6.2f %8.0f %8.1f %8.0f %12.1f\n', names{i}, mid(d33(i,:)), mid(kt(i,:)), mid(epsT(i,:)), mid(Z(i,:)), mid(Tmax(i,:)), g33(i));
end
fprintf('PVDF g33 from the datasheet: 330 mV m/N (computed above from d33 and eps: %.0f)\n', g33(end));
fprintf('Intensity transmission into 1.5 MRayl at the mid impedance: ');
T = 4*mid(Z)*1.5./(mid(Z)+1.5).^2; fprintf('%s ', sprintf('%.2f', T(1))); fprintf('(PZT), %.2f (composite), %.2f (PVDF)\n', T(4), T(7));

%% Thin films for PMUTs (Roy et al. 2023, Table I)
fn  = {'AlN', 'Sc0.15AlN', 'PZT (sol-gel)', 'PZT (epitaxial)', 'KNN film'};
e31 = [1.05 1.05; 1.6 1.6; 8 12; 14 14; 8.5 14.4];      % |e31,f| C/m^2
epf = [10.5 10.5; 12 12; 300 1300; 200 300; 445 445];   % eps33,f
fprintf('\n%-16s %12s %10s %14s %14s\n', 'film', '|e31,f|', 'eps33', 'e31/eps0eps GV/m', 'e31^2/eps0eps GPa');
fomT = zeros(numel(fn),2);
for i = 1:numel(fn)
  rx = e31(i,:)./(e0*epf(i,[2 1]))/1e9;                 % range: low e31 with high eps, high e31 with low eps
  fom = e31(i,:).^2./(e0*epf(i,[2 1]))/1e9; fomT(i,:) = fom;
  fprintf('%-16s %5.2f-%5.2f %4.0f-%4.0f %6.1f-%6.1f %6.1f-%6.1f\n', fn{i}, e31(i,1), e31(i,2), epf(i,1), epf(i,2), rx(1), rx(2), fom(1), fom(2));
end

%% Figure
figure('Position', [80 80 1100 380]);
subplot(1,3,1); barh(mid(kt)); set(gca, 'YTick', 1:numel(names), 'YTickLabel', names); xlabel('k_t'); title('Thickness coupling'); grid on;
subplot(1,3,2); barh(mid(Z)); set(gca, 'YTick', 1:numel(names), 'YTickLabel', names); xlabel('Z (MRayl)'); title('Acoustic impedance'); grid on;
subplot(1,3,3); barh(mean(fomT,2)); set(gca, 'YTick', 1:numel(fn), 'YTickLabel', fn); xlabel('e_{31,f}^2/(\epsilon_0\epsilon_{33}) (GPa)'); title('Film FOM (range mid)'); grid on;

What to look for: \(g_{33}\) (mV·m/N) 18.5 PZT-5H, 36.7 PMN-PT, 37.3 PIN-PMN-PT, 356 PVDF (datasheet 330); intensity transmission into 1.5 MRayl 0.15 (PZT), 0.31 (composite), 0.79 (PVDF); film FOM 11.9 (AlN), 24.1 (ScAlN), 5.6–54 (sol-gel PZT).

m18_cmut_vs_pzt_bandwidth.mApproximate 1D model: Mason PZT stack vs lumped CMUT, spectra and impulse responses · change \(Z_B\), layers, bias, \(a\) · runtime ≈ 2 s 
%% m18_cmut_vs_pzt_bandwidth.m
% Pulse-echo bandwidth of a PZT-5H stack (Mason model, backing, 0-2 quarter-wave
% matching layers) versus a CMUT element (lumped piston membrane, water load).
% APPROXIMATE MODEL: 1D, plane-wave radiation resistance rho*c*A for both,
% no diffraction, no lens, lossless layers; the CMUT membrane is one piston mode
% (no higher membrane modes, no crosstalk), with an optional cell-level water
% added mass (8/3) rho a^3 per cell.
%
% What it shows: the PZT stack is a resonator that matching layers broaden;
%   the CMUT is an over-damped mass-spring whose band runs from k_eff/(rho c)
%   to rho c / m, so its upper edge is set by the moving mass (membrane + water).
% What to change: ZB (backing), nL (layers), bias fraction Vf, a, h.
% Expected output (defaults): PZT 2 layers, ZB = 5 MRayl: FBW 63 %;
%   CMUT a = 20 um, h = 1 um, 0.9 Vpi with added mass: FBW 178 % (model upper
%   bound; measured immersion CMUT arrays report about 90-105 %).
% Runtime: about 2 s.
clear; close all; rng(1);
e0 = 8.854e-12; rw = 1000; cw = 1500;
A = 0.27e-3*5e-3; Rs = 50; ZL = rw*cw*A;     % element 0.27 mm x 5 mm, 50 ohm source and receiver
f = (1:2000)'*12.5e3; w = 2*pi*f;              % 12.5 kHz to 25 MHz

%% PZT-5H stack, Mason model (Ma et al. 2015: v 4580 m/s, kt 0.51, epsS 1470; rho 7500)
v = 4580; rho = 7500; kt = 0.51; er = 1470; f0 = 5e6;
d = v/(2*f0); Z0 = rho*v*A; C0 = er*e0*A/d; hh = kt*sqrt(rho*v^2/(er*e0));
Zp = rho*v/1e6; cm = 2500;
SP = zeros(numel(f), 3);
for nL = 0:2
  if nL == 2, Zm = [Zp^(4/7)*1.5^(3/7), Zp^(1/7)*1.5^(6/7)];   % Desilets (Module 17)
  elseif nL == 1, Zm = (Zp*1.5^2)^(1/3); else, Zm = []; end
  ZB = 5e6*A;
  for k = 1:numel(f)
    th = w(k)*d/v; aa = Z0/(1i*tan(th)); bb = Z0/(1i*sin(th)); cc = hh/(1i*w(k));
    Zin = ZL; T = eye(2);
    for m = numel(Zm):-1:1                      % from the water side towards the plate
      Zc = Zm(m)*1e6*A; t = w(k)/(4*f0);         % quarter wave at f0 (t = omega d / c, d = c/(4 f0))
      Zin = Zc*(Zin + 1i*Zc*tan(t))/(Zc + 1i*Zin*tan(t));
      T = [cos(t), 1i*Zc*sin(t); 1i*sin(t)/Zc, cos(t)] * T;   % plate-side layer first after the loop
    end
    Mx = [aa+ZB, bb, cc; bb, aa+Zin, cc; cc, cc, 1/(1i*w(k)*C0)+Rs];
    x = Mx\[0;0;1];
    Fv = [-Zin*x(2); -x(2)];                     % force and outward velocity at the plate front
    out = T\Fv;                                  % at the water
    SP(k, nL+1) = 2*sqrt(Rs/ZL)*out(1);
  end
end

%% CMUT element: lumped piston (average-displacement spring, 1.8 rho h modal mass)
E = 169e9; nus = 0.22; rs = 2330; a = 20e-6; hm = 1e-6; g0 = 0.15e-6; ti = 0.1e-6; eri = 7.5; eta = 0.8;
Dm = E*hm^3/(12*(1-nus^2)); ge = g0 + ti/eri;
k = 192*Dm/a^4*A*eta; mm = 1.8*rs*hm*A*eta; madd = (8/3)*rw*a^3/(pi*a^2)*A*eta;
Vpi = sqrt(8*k*ge^3/(27*e0*eta*A));
xi = linspace(0, 1/3, 20001); Vr = sqrt(27/4*xi.*(1-xi).^2);
SC = zeros(numel(f), 3); Vfs = [0.7 0.9 0.9]; adds = [1 1 0];
for j = 1:3
  Vf = Vfs(j); x = interp1(Vr, xi, Vf)*ge; Vdc = Vf*Vpi;
  C0c = e0*eta*A/(ge - x); n = C0c*Vdc/(ge - x); keff = k*(1 - 2*(x/ge)/(1 - x/ge));
  mt = mm + adds(j)*madd;
  Zmech = 1i*w*mt + keff./(1i*w);
  Aa = 1/n; B = Zmech/n; Cc = 1i*w*C0c/n; Dd = 1i*w.*C0c.*Zmech/n + n;
  SC(:, j) = 2*sqrt(Rs*ZL)./(Aa*ZL + B + Rs*(Cc*ZL + Dd));
end
fprintf('CMUT: f0(vac) = %.2f MHz, Vpi = %.1f V, membrane mass %.2g, cell added mass %.2g kg/m^2\n', ...
  10.2158/(2*pi*a^2)*sqrt(Dm/(rs*hm))/1e6, Vpi, 1.8*rs*hm, (8/3)*rw*a/pi);

%% Bandwidth metrics on the pulse-echo response H = S21^2
labels = {'PZT, no matching', 'PZT, 1 layer', 'PZT, 2 layers', 'CMUT 0.7 Vpi', 'CMUT 0.9 Vpi', 'CMUT 0.9 Vpi, no added mass'};
H = [SP.^2, SC.^2];
for j = 1:size(H,2)
  m = abs(H(:,j)); [pk, ip] = max(m);
  il = find(m(1:ip) < pk/2, 1, 'last'); ih = ip - 1 + find(m(ip:end) < pk/2, 1, 'first');
  if isempty(il), fl = f(1); else, fl = f(il) + (pk/2 - m(il))/(m(il+1) - m(il))*(f(il+1) - f(il)); end     % linear interpolation of the -6 dB edges
  if isempty(ih), fh = f(end); else, fh = f(ih-1) + (m(ih-1) - pk/2)/(m(ih-1) - m(ih))*(f(ih) - f(ih-1)); end
  fprintf('%-28s fc = %5.2f MHz, -6 dB band %5.2f-%5.2f MHz, FBW = %4.0f %%, peak %6.1f dB\n', labels{j}, ...
    (fl+fh)/2/1e6, fl/1e6, fh/1e6, 100*(fh-fl)/((fh+fl)/2), 20*log10(pk));
end

%% Impulse responses: inverse FFT of H, receive low-pass (cos^2 roll-off 9-12 MHz, as in Module 17), 0.5 us delay
Nf = 4096; fs = Nf*12.5e3; Hfull = zeros(Nf, 2);
Hfull(2:numel(f)+1, 1) = H(:,3); Hfull(2:numel(f)+1, 2) = H(:,5);
win = (f < 9e6) + (f >= 9e6 & f < 12e6).*cos(pi/2*(f - 9e6)/3e6).^2;
win = win.*exp(-1i*w*0.5e-6);
Hfull(2:numel(f)+1, :) = Hfull(2:numel(f)+1, :).*[win win];
Hfull(Nf:-1:Nf/2+2, :) = conj(Hfull(2:Nf/2, :));
hpe = real(ifft(Hfull));
t = (0:Nf-1)/fs*1e6;

figure('Position', [80 80 1000 380]);
subplot(1,2,1);
plot(f/1e6, 20*log10(abs(H(:,3))/max(abs(H(:,3)))), 'b-', f/1e6, 20*log10(abs(H(:,5))/max(abs(H(:,5)))), 'r-', 'LineWidth', 2);
grid on; ylim([-30 2]); xlabel('frequency (MHz)'); ylabel('pulse-echo |H| (dB, normalized)');
legend('PZT-5H, 2 layers, Z_B 5 MRayl', 'CMUT 0.9 V_{PI}', 'Location', 'south'); title('Approximate 1D model');
subplot(1,2,2);
plot(t, hpe(:,1)/max(abs(hpe(:,1))), 'b-', t, hpe(:,2)/max(abs(hpe(:,2))) - 2.2, 'r-');
xlim([0.3 2.0]); xlabel('time (\mus)'); ylabel('normalized'); title('Impulse responses, 9-12 MHz receive low-pass'); grid on;

What to look for: PZT: 17 %, 39 %, 63 % FBW with 0, 1, 2 layers (\(Z_B\) = 5 MRayl); CMUT: 171 % at 0.7 \(V_{\mathrm{PI}}\), 178 % at 0.9 \(V_{\mathrm{PI}}\) with added mass (band 0.80–13.99 MHz), upper edge beyond 25 MHz without it. These are model upper bounds; the page explains why measured values are about 100 %.

18.12Self-check

Q1You double the membrane thickness and halve the radius of a CMUT cell. By what factor does \(V_{\mathrm{PI}}\) change, and the vacuum resonance?

\(V_{\mathrm{PI}} \propto h^{3/2}/a^{2} \to 2^{1.5}\times 4 = 11.3\) times higher. \(f \propto h/a^{2} \to 2\times 4 = 8\) times higher. Both at fixed gap and insulator.

Q2A colleague's piston model gives \(V_{\mathrm{PI}}\) = 100 V for a circular clamped cell. What do you expect from FEM, and where will the centre be at pull-in?

About 0.699 × 100 ≈ 70 V. The centre will have moved ≈ 0.46 \(g_{\mathrm{eff}}\), the average ≈ 0.15 \(g_{\mathrm{eff}}\) (not \(g/3\)). Residual tension or partial electrodes move this.

Q3A CMUT biased at 100 V is driven with 20 V amplitude. What is the second-harmonic force relative to the fundamental?

\(v/(4V_{\mathrm{DC}}) = 20/400 = 0.05\), i.e. −26 dB, before any gap nonlinearity.

Q4An AlN-on-Si PMUT (1 µm on 2 µm) has \(a\) = 40 µm. Estimate its resonance in vacuum and in water.

\(f\) scales as \(1/a^{2}\) from the \(a\) = 50 µm value 5.58 MHz: \(5.58\times(50/40)^{2} = 8.72\) MHz. \(\bar\rho h = 2330\times 2 + 3260\times 1\ \mathrm{\mu m}\) = 7.92 g/m², \(\beta = 0.67\times 1000\times 40\times 10^{-6}\) \(/\,7.92\times 10^{-3} = 3.38\), \(f_w = 8.72/\sqrt{4.38} = 4.17\) MHz (18.11).

Q5Design a 10 MHz PMN-PT/epoxy 1-3 composite at \(\nu\) = 0.5 with pillar width/thickness 0.5. Thickness, pillar width, pitch, kerf?

\(v \approx\) 3673 m/s (Smith–Auld) → \(t = v/(2f_0)\) = 184 µm. \(w = 0.5t\) = 92 µm, \(p = w/\sqrt{\nu}\) = 130 µm, kerf = \(p - w\) = 38 µm: still dicable. At 30 MHz everything shrinks threefold to a 13 µm kerf, below a practical saw.

Q6What average displacement amplitude does a CMUT need to produce 1 MPa at its surface at 3 MHz?

\(\bar x = p/(Z_w\cdot 2\pi f)\) \(= 10^{6}/(1.5\times 10^{6}\times 2\pi\times 3\times 10^{6})\) = 35 nm. With an average travel of about \(0.15\,g\) before pull-in, that asks for a gap of a few hundred nm, or collapse mode.

Q7Why does a PVDF membrane hydrophone give a larger open-circuit voltage than a PZT-5H disc of the same thickness, although PZT's \(d_{33}\) is 20 times larger?

Open-circuit voltage \(= g_{33}\cdot t\cdot p\) with \(g = d/\varepsilon^{T}\). PVDF: 0.33 V·m/N; PZT-5H: 622 pC/N / (\(3800\varepsilon_0\)) = 0.0185 V·m/N. The ratio is about 18 in PVDF's favour; PVDF also matches water (4 MRayl), so it disturbs the field less.

18.13Figures from the literature

Published figures with real data that anchor this module's models. Links only; no images are reproduced. Figure numbers were checked against the full texts.

Measured CMUT pulse-echo bandwidth and wire images

Ö. Oralkan, A. S. Ergun, J. A. Johnson, M. Karaman, U. Demirci, K. Kaviani, T. H. Lee, B. T. Khuri-Yakub, "Capacitive micromachined ultrasonic transducers: next-generation arrays for acoustic imaging?" IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49(11), 1596–1610, 2002. doi:10.1109/TUFFC.2002.1049742

Figs. 9 and 10: pulse-echo frequency responses of a 128-element immersion array: 80 % at about 3 MHz from a wire, 96 % at 4.59 MHz from a plane reflector, 105 % at 4.62 MHz after diffraction and attenuation compensation. Fig. 12: B-scans of a wire phantom at 40 and 60 dB. Compare with Demo 18.3: the measured band is the model band minus everything a one-mode model omits. PDF (author site)

Collapse regime: membrane shapes, coupling and snap-back

B. Bayram, E. Hæggström, G. G. Yaralioglu, B. T. Khuri-Yakub, "A new regime for operating capacitive micromachined ultrasonic transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50(9), 1184–1190, 2003.

Fig. 2: FEM membrane shapes from conventional to collapsed. Fig. 4: \(k_T^{2}\) versus bias, rising to about 0.7 after collapse when the bias is lowered. Fig. 5: average and peak displacement with the hysteresis between 140 V (collapse) and 68 V (snap-back) for a 50 µm-radius, 1 µm-gap cell. Compare the shape of the loop with Demo 18.2. PDF (author site)

Crosstalk: the dispersive interface mode, measured

B. Bayram, M. Kupnik, G. G. Yaralioglu, Ö. Oralkan, A. S. Ergun, D.-S. Lin, S. H. Wong, B. T. Khuri-Yakub, "Finite element modeling and experimental characterization of crosstalk in 1-D CMUT arrays," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(2), 418–430, 2007. doi:10.1109/TUFFC.2007.256

Fig. 4: conventional-mode crosstalk in time–space and frequency–wavenumber, FEM against laser-vibrometer measurement, with and without PDMS. Fig. 7: crosstalk versus element number (−17 dB first neighbour). Fig. 8: phase velocity of the guided mode versus frequency, approaching water speed at low frequency and cutting off. This is the measured counterpart of Demo 18.4. PDF (author site)

Wafer-bonded CMUT with LOCOS: the cross-section and the bias-dependent impedance

K. K. Park, H. J. Lee, M. Kupnik, Ö. Oralkan, B. T. Khuri-Yakub, "Fabricating capacitive micromachined ultrasonic transducers with direct wafer-bonding and LOCOS technology," Proc. IEEE MEMS 2008, 339–342; journal version: K. K. Park, H. Lee, M. Kupnik, B. T. Khuri-Yakub, J. Microelectromech. Syst. 20(1), 95–103, 2011, doi:10.1109/JMEMS.2010.2093567.

Fig. 4 (MEMS 2008): SEM cross-section of a single cell (130 nm gap, 0.5 µm silicon membrane, oxide posts), the real version of Figure 18.A. Fig. 6: measured electrical input impedance of a 40 nm-gap device at several bias voltages, the bias-dependent resonance that the coupling formulas of Section 18.2 describe. PDF (author site)

2D CMUT arrays with through-wafer vias, flip-chip bonded, and volumetric images

Ö. Oralkan, A. S. Ergun, C.-H. Cheng, J. A. Johnson, M. Karaman, T. H. Lee, B. T. Khuri-Yakub, "Volumetric ultrasound imaging using 2-D CMUT arrays," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50(11), 1581–1594, 2003. doi:10.1109/TUFFC.2003.1251142

Fig. 1: cell cross-section, 3D model and SEM of the through-wafer via. Fig. 2: the 32 × 64 sub-array of a 128 × 128 array (420 µm pitch) flip-chip bonded to a glass fan-out. Figs. 8 and 10: reconstructed 3D images of plate and sphere phantoms; the echoes have 91–100 % bandwidth. The flip-chip route of Figure 18.C. PDF (author site)

PMUT structures, a measured element response and a 512-element ICE catheter

Y. Qiu, J. V. Gigliotti, M. Wallace, F. Griggio, C. E. M. Demore, S. Cochran, S. Trolier-McKinstry, "Piezoelectric micromachined ultrasound transducer (PMUT) arrays for integrated sensing, actuation and imaging," Sensors 15, 8020–8041, 2015. doi:10.3390/s150408020 (open access, CC BY 4.0)

Fig. 1: cross-sections of a bulk piezo transducer, a CMUT and a \(d_{31}\)-mode PMUT side by side. Fig. 11: receive and transmit responses of one PMUT array element measured in water. Fig. 14: through-silicon-interconnect PMUTs and a 14 Fr ICE catheter with a 512-element array. doi.org/10.3390/s150408020

Thin-film figures of merit (re-plotted in Demo 18.7)

K. Roy, J. E.-Y. Lee, C. Lee, "Thin-film PMUTs: a review of over 40 years of research," Microsystems & Nanoengineering 9, 2023. doi:10.1038/s41378-023-00555-7

Table I: \(\lvert e_{31,f}\rvert\), \(\varepsilon_{33,f}\), \(e_{31,f}/(\varepsilon_0\varepsilon_{33})\) and \(e_{31,f}^{2}/(\varepsilon_0\varepsilon_{33})\) for PZT variants, AlN, Sc0.15AlN, ZnO and KNN films. Table 18.2 and the film modes of Demo 18.7 re-plot these values; badge Exact formula (tabulated values as published). doi.org/10.1038/s41378-023-00555-7

AC-poled transparent PMN-PT

C. Qiu, B. Wang, N. Zhang, S. Zhang, J. Liu, D. Walker, Y. Wang, H. Tian, T. R. Shrout, Z. Xu, L.-Q. Chen, F. Li, "Transparent ferroelectric crystals with ultrahigh piezoelectricity," Nature 577, 350–354, 2020. doi:10.1038/s41586-019-1891-y

Fig. 3: AC- versus DC-poled PMN-28PT crystals: appearance, optical transmittance, birefringence and \(d_{33}\) (above 2100 pC/N for AC poling). Fig. 1: phase-field simulation of domain evolution under DC and AC poling, the mechanism. doi.org/10.1038/s41586-019-1891-y

18.14Go deeper

  • M. I. Haller, B. T. Khuri-Yakub, "Electrostatic ultrasonic transducer," US Patent 5,619,476 (filed 1994, granted 1997): the Stanford micromachined capacitive transducer.
  • I. Ladabaum, X. Jin, H. T. Soh, A. Atalar, B. T. Khuri-Yakub, "Surface micromachined capacitive ultrasonic transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45(3), 678–690, 1998. doi:10.1109/58.677612
  • A. S. Ergun, X. Zhuang, Y. Huang, Ö. Oralkan, G. G. Yaralioglu, B. T. Khuri-Yakub, "Capacitive micromachined ultrasonic transducer technology for medical ultrasound imaging," Proc. SPIE 5750, 58–68, 2005. doi:10.1117/12.595692 (sacrificial release and wafer-bonding flows, Figs. 3 and 5)
  • J. M. Rothberg, T. S. Ralston, A. G. Rothberg et al., "Ultrasound-on-chip platform for medical imaging, analysis, and collective intelligence," PNAS 118(27), e2019339118, 2021. doi:10.1073/pnas.2019339118
  • A. W. Leissa, Vibration of Plates, NASA SP-160, 1969 (clamped circular and annular plate eigenvalues).
  • H. Lamb, "On the vibrations of an elastic plate in contact with water," Proc. R. Soc. Lond. A 98, 1920.
  • Q. Wang, Y. Lu, S. Fung, X. Jiang, S. Mishin, Y. Oshmyansky, D. A. Horsley, "Scandium doped aluminum nitride based piezoelectric micromachined ultrasound transducers," Solid-State Sensors, Actuators and Microsystems Workshop, Hilton Head, 2016, 436–439 (film \(e_{31,f}\), \(\varepsilon\), \(E\)).
  • S. Zhang, F. Li, X. Jiang, J. Kim, J. Luo, X. Geng, "Advantages and challenges of relaxor-PbTiO₃ ferroelectric crystals for electroacoustic transducers – a review," Prog. Mater. Sci. 68, 1–66, 2015. doi:10.1016/j.pmatsci.2014.10.002
  • W. A. Smith, B. A. Auld, "Modeling 1-3 composite piezoelectrics: thickness-mode oscillations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38(1), 40–47, 1991. doi:10.1109/58.67833
  • J. Kim, C. Joh, Y. Roh, "Evaluation of all the material constants of PMN-28%PT piezoelectric single crystals for acoustic transducers," Sensors and Materials 25(8), 539–552, 2013.
  • T. Ma, M. Yu, Z. Chen, C. Fei, K. K. Shung, Q. Zhou, "Multi-frequency intravascular ultrasound (IVUS) imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 62(1), 97–107, 2015. doi:10.1109/TUFFC.2014.006679 (Table I: PZT-5H, PMN-PT, LiNbO₃)
  • Y. Saito, H. Takao, T. Tani, T. Nonoyama, K. Takatori, T. Homma, T. Nagaya, M. Nakamura, "Lead-free piezoceramics," Nature 432, 84–87, 2004. doi:10.1038/nature03028
  • TRS Technologies, single-crystal PMN-PT and PIN-PMN-PT property ranges (X2B, X4B, Mn:PIN-PMN-PT), trstechnologies.com; Measurement Specialties and Precision Acoustics PVDF data sheets.