PMUT Unit cell - Ultrasound 001

What you’ll learn

• How to import a variables CSV, build a parameterized PMUT stack with boolean geometry operations, and assign materials using mass rule create and computed regions.
• How to define symmetry and electrode surface regions with the rule selector, then set up Elastic waves, Acoustic waves, and Electrostatics with piezoelectricity and acoustic structure couplings.
• How to mesh thin layers with simple mesh extrusion, run a transient simulation with Generalized alpha timestepping, and plot drive voltage and current.

In this step-by-step tutorial, a single PMUT (piezoelectric micromachined ultrasonic transducer) unit cell is modeled and simulated.

The cell consists of a thin-film stack — silicon dioxide substrate, aluminum nitride piezo layer, molybdenum electrodes, and a polysilicon membrane — coupled to a water load above. Symmetry boundary conditions on the lateral faces represent a periodic array, while an absorbing boundary condition (ABC) truncates the water domain at the top.

Model definition

The unit cell is built entirely in the Allsolve geometry editor. Each geometry element is given a name, which is then used as a region rule attribute so materials and boundary conditions are easy to set up.

Layer stack

Layer	Geometry name	Material
Water	vol_water	Water
Polysilicon membrane	vol_poly	Polycrystalline Silicon
Top Mo electrode	vol_mo_top	Molybdenum
AlN piezo layer	vol_aln	Aluminum Nitride
Bottom Mo electrode	vol_mo_bot	Molybdenum
SiO₂ substrate (with cavity)	vol_sio2	Silicon Dioxide

Key variables

Import all 29 project variables in Step 1. The table below summarizes the most important ones:

Group	Variables	Role
Geometry	pitch, cell_d, bot_elec_d, t_sio2, t_aln, t_mo, t_poly	Stack dimensions
Derived geometry	cell_r, bot_elec_r, half_pitch, z_sio2_top, z_aln_top, z_poly_top, z_water_top, eps	Centers, radii, and region selectors
Mesh	lateral_mesh_size, elem_per_wl, lambda_water	Lateral element size and wavelength meshing in water
Simulation	freq, dt, t_end, V_amp, delay, ppc, n_cycles	Transient settings and drive waveform

At the default values, the excitation frequency is 12 MHz and the water domain height equals one acoustic wavelength (lambda_water = c_water / freq).

Physics summary

Physics	Target region	Key interactions
Elastic waves	vol_solid	Clamp on sur_base; symmetry on sur_sym_x0, sur_sym_xL, sur_sym_y0, sur_sym_yL; Piezoelectricity on vol_aln
Acoustic waves	vol_water	Acoustic structure coupling; Absorbing boundary on sur_water_top
Electrostatics	vol_aln	Constraint (0 V) on sur_top_elec_intf; Lump V/Q drive on sur_bot_elec_intf

The voltage drive uses V_amp * wavelet(freq, delay). The function wavelet creates a Ricker wavelet with a given frequency and delay, and is often used as an excitation/load in time-dependent simulations.

Output results

• Drive-port signals: voltage (drive.V), charge (drive.Q), and current (dt(drive.Q))
• Front-face averages: acoustic pressure, normal displacement, and normal velocity averaged over the radiating surface
• Field outputs: displacement magnitude (norm(u)) and pressure (p), saved every 25th time step

Step-by-step guide

Here you’ll find a detailed step-by-step tutorial on how to build and simulate a PMUT unit cell in Quanscient Allsolve.

Step 1 - Create the project and import variables

1. Create a new project and name it PMUT unit cell.

2. Open the Common sidebar, Definitions tab.

3. Import the variables from a CSV file. In the Variables section, use Import CSV and upload a file with the following contents:

   Name,Expression,Description
   freq,12e6,Centre excitation frequency (Hz)
   V_amp,1.0,Drive voltage amplitude (V)
   delay,1.2,Wavelet delay in periods (peak at delay/freq)
   pitch,60e-6,Inter-cell pitch (m)
   cell_d,50e-6,Cavity / cell diameter (m)
   bot_elec_d,25e-6,Bottom electrode diameter (m)
   t_sio2,2e-6,SiO2 substrate thickness (m)
   t_aln,1e-6,AlN piezo layer thickness (m)
   t_mo,200e-9,Molybdenum electrode thickness (m)
   t_poly,2e-6,Polysilicon elastic layer thickness (m)
   c_water,1500,Speed of sound in water (m/s)
   lambda_water,c_water / freq,Acoustic wavelength in water (m)
   elem_per_wl,6,Mesh elements per wavelength in water
   lateral_mesh_size,10e-6,Lateral mesh element size (m)
   ppc,25,Time-steps per cycle
   dt,1/(freq*ppc),Time-step size (s)
   n_cycles,10,Number of excitation cycles to simulate
   t_end,n_cycles/freq,Total simulation time (s)
   cell_r,cell_d/2,Cell radius (m)
   bot_elec_r,bot_elec_d/2,Bottom electrode radius (m)
   half_pitch,pitch/2,Half of inter-cell pitch (m)
   eps,0.05e-6,Tolerance for bounding-box region selection (m)
   z_sio2_bot,0,z of SiO2 bottom face (base of stack)
   z_sio2_top,t_sio2,z of SiO2 top face / AlN bottom face
   z_aln_bot,z_sio2_top,z of AlN bottom face
   z_aln_top,z_aln_bot + t_aln,z of AlN top face
   z_mo_top_top,z_aln_top + t_mo,z of top Mo electrode upper face
   z_poly_top,z_mo_top_top + t_poly,z of polysilicon top face (front face)
   z_water_top,z_poly_top + lambda_water,z of water domain top face

Step 2 - Build the geometry

Go to the Geometry section and build the stack from the bottom up. Give each element the Name shown below — these names are used later in region rules and boolean targets.

2.1 — SiO₂ substrate and cavity

1. Add a Box named sio2_raw:

   Setting	Value
   Size (X, Y, Z)	pitch, pitch, t_sio2
   Center (X, Y, Z)	0, 0, t_sio2/2

2. Add a Cylinder named cavity_void:

   Setting	Value
   Radius	cell_r
   Height	t_sio2
   Center (X, Y, Z)	0, 0, t_sio2/2

   Build both elements. The substrate block and cavity cutter should look like this:

   

3. Add a Difference operation named vol_sio2:

   Setting	Value
   Set 1	/sio2_raw
   Set 2	/cavity_void
   Delete	Unchecked

   This subtracts the cavity cylinder from the substrate without deleting the cutter volume from the model tree yet.

   

4. Add a Remove operation named remove_cavity with Target /cavity_void.

   

2.2 — Bottom Mo electrode

1. Add a Cylinder named vol_mo_bot:

   Setting	Value
   Radius	bot_elec_r
   Height	t_mo
   Center (X, Y, Z)	0, 0, z_aln_bot - t_mo/2

   

2.3 — Remaining layers

Add a Box for each layer:

Name	Size (X, Y, Z)	Center (X, Y, Z)
vol_aln	pitch, pitch, t_aln	0, 0, z_aln_bot + t_aln/2
vol_mo_top	pitch, pitch, t_mo	0, 0, z_aln_top + t_mo/2
vol_poly	pitch, pitch, t_poly	0, 0, z_mo_top_top + t_poly/2
vol_water	pitch, pitch, lambda_water	0, 0, z_poly_top + lambda_water/2

Build each element. Your geometry list should contain vol_sio2, vol_mo_bot, vol_aln, vol_mo_top, vol_poly, and vol_water:

1. Add Fragment all (finalize) and select Confirm model changes.

   

Step 3 - Create material regions

Return to the Common sidebar, Definitions tab.

1. Under Regions, use Mass rule create:

   • Attribute: name
   • Entity type: Volume

   This creates one volume region for each named geometry element (vol_sio2, vol_mo_bot, vol_mo_top, vol_aln, vol_poly, vol_water).

   

2. Add a Computed region named vol_mo:

   Setting	Value
   Entity type	Volume
   Operation	Union (OR)
   Regions	vol_mo_top, vol_mo_bot

Step 4 - Assign materials

Go to the Physics section and assign materials from the global library:

Material	Target region
Silicon Dioxide	vol_sio2
Aluminum Nitride	vol_aln
Molybdenum	vol_mo
Polycrystalline Silicon	vol_poly
Water	vol_water

Step 5 - Create physics regions

Back in the Common sidebar, define surface and volume regions for boundary conditions.

5.1 - Solid domain

Add a Computed region named vol_solid (entity type Volume, operation Union):

Regions to union
vol_sio2, vol_aln, vol_poly, vol_mo

5.2 - Boundary surfaces

Create Surface regions by picking on the model view:

Region	Selection hint
sur_base	Bottom face of the stack at z = z_sio2_bot (this surface has a circular cut-out due to the cylinder cavity)
sur_water_top	Top face of the water box at z = z_water_top
sur_top_elec_intf	Interface between the top Mo electrode and the polysilicon layer
sur_bot_elec_intf	Top face of the bottom Mo electrode disc
sur_front_face	Front face of the unit cell (boundary surface between water and poly-Si)

5.3 - Symmetry surfaces

For the four lateral symmetry faces, use Rule selector with Bounding box and Pick similar:

1. Create a surface region named sur_sym_x0.
2. Select top-most and bottom-most surfaces in the x0 plane, so that the bounding box includes all 4 solid region surfaces in that plane.
3. Repeat for sur_sym_xL, sur_sym_y0, sur_sym_yL.

Step 6 - Define physics and boundary conditions

In the Physics section, add three modules to your physics set: Elastic waves, Acoustic waves, and Electrostatics.

Physics 1 — Elastic waves

• Set Elastic waves target to vol_solid.

• Add a Clamp interaction named ClampBase on sur_base.

  

• Add four Symmetry interactions:

  Name	Target
  SymmetryX0	sur_sym_x0
  SymmetryXL	sur_sym_xL
  SymmetryY0	sur_sym_y0
  SymmetryYL	sur_sym_yL

  

• Add Piezoelectricity with target vol_aln.

  

Physics 2 — Acoustic waves

• Set Acoustic waves target to vol_water.

• Add Acoustic structure (AW_EW_Coupling) to couple acoustic and elastic domains.

• Add an Absorbing boundary named WaterTopABC on sur_water_top.

  

Physics 3 — Electrostatics

• Set Electrostatics target to vol_aln.

• Add a Constraint named GroundTopElectrode on sur_top_elec_intf with Constraint value 0.

  

• Add a Lump V/Q named DrivePort:

  Setting	Value
  Namespace	drive
  Target	sur_bot_elec_intf
  Actuation mode	Voltage
  Voltage	V_amp * wavelet(freq, delay)

  

Step 7 - Generate the mesh

Go to the Simulations section and create a new mesh:

1. Set Autorefine to Disabled.

2. Under Mesh element size, set Max size to lateral_mesh_size with type Absolute.

3. Under Runtime, set Node type to Fast start.

4. Under Customizations, add Mesh extrusion → Simple extrusion:

   • Target: select all six volumes.
   • Sublayers (from top down; water first): elem_per_wl, 1, 1, 1, 1, 1.

   See Simple extrusion for more on layered extruded meshing.

5. Apply settings and mesh. Check the preview:

   

Step 8 - Create the simulation and run

1. Create a new simulation named PMUT Unit Transient.

2. Set Analysis type to Transient.

3. Configure Generalized alpha timestepping:

   Setting	Value
   Start time	0
   End time	t_end
   Timestep size	dt
   Target frequency	freq

4. Select the extruded mesh.

5. Add Outputs. First, add the drive-port value outputs:

   Name	Expression
   V_drive	drive.V
   Q_drive	drive.Q
   I_drive	dt(drive.Q)

   Then add surface-averaged value outputs on the front face of the unit cell:

   Name	Expression
   p_avg_front	average(reg.sur_front_face, p, 3)
   uz_avg_front	average(reg.sur_front_face, compz(u), 3)
   vz_avg_front	average(reg.sur_front_face, dt(compz(u)), 3)

   These track the average acoustic pressure, normal displacement, and normal velocity on the radiating surface.

   Finally, add field outputs for visualization. Set the filter to Output every Nth step with an interval of 25 on both:

   Name	Expression
   u_mag	norm(u)
   pressure	p

   

6. Run the simulation.

Step 9 - Plot & visualize results

Once the simulation completes, inspect the outputs to verify that the PMUT behaves as expected.

In the Simulations section:

1. Add Plots for the drive-port Voltage, Charge, and Current custom values over time.

2. Add a Visualization for the pressure field p in the water load (for example with a Slice filter through the cell center).

3. Add a Visualization for displacement u in the solid stack.

Results

Drive-port signals

The drive port plots show the applied Ricker wavelet voltage, the charge accumulated on the bottom electrode, and the resulting current. The wavelet peaks at delay / freq ≈ 100 ns, after which the current decays rapidly — confirming that the excitation is compact in time and well-suited for broadband characterization.

Front-face averages

The surface-averaged pressure, displacement, and velocity on the radiating front face reveal the transducer’s impulse response. After the initial excitation, the membrane rings down over several cycles. The peak pressure reaches approximately 2 kPa and the normal velocity is on the order of 1 mm/s — typical values for a PMUT driven at 1 V.

Frequency-domain performance

By applying an FFT to the time-domain results, you can extract the transmit sensitivity spectrum (Pa/V) and the electrical impedance spectrum. The sensitivity peaks near 13–14 MHz, indicating the fundamental flexural resonance of the membrane. The impedance curve shows a predominantly capacitive response, as expected for a thin piezoelectric film operating below its thickness-mode resonance.