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PiezoD Tutorial

This tutorial walks through using PiezoD to calculate the properties of a single-crystal silicon piezoresistive cantilever. It covers an epitaxial design; diffusion, ion-implantation, polycrystalline, and piezoelectric variants follow the same pattern with a different Cantilever* class.

Install

pip install piezod

PiezoD requires Python 3.12 or later. The examples/ directory in the repository has runnable scripts that mirror this tutorial.

Import

from piezod import CantileverEpitaxy

Define the Cantilever

All dimensions are in MKS units (meters, Hz, V, cm^-3 for concentrations).

c = CantileverEpitaxy(
    freq_min=1,                      # measurement bandwidth (Hz)
    freq_max=1000,
    l=100e-6,                        # length: 100 um
    w=4e-6,                          # width: 4 um
    t=1e-6,                          # thickness: 1 um
    l_pr_ratio=0.8,                  # piezoresistor extends 80% of length
    t_pr_ratio=0.5,                  # piezoresistor is 50% of thickness
    v_bridge=2.0,                    # 2V Wheatstone bridge bias
    doping_type="phosphorus",        # n-type, <100> orientation
    dopant_concentration=1e18,       # cm^-3
)

Choosing "phosphorus" (n-type) implicitly uses the <100> orientation where the piezoresistive coefficient peaks. Choosing "boron" (p-type) uses the <110> direction (E = 169 GPa) with the corresponding coefficients.

Attributes can also be set after construction:

c.fluid = "vacuum"
c.number_of_piezoresistors = 2

Calculate Properties

Mechanical

c.stiffness()              # spring constant (N/m)
c.omega_vacuum_hz()        # vacuum resonant frequency (Hz)
f_d, Q = c.omega_damped_hz_and_Q()  # damped frequency (Hz) and Q

Electrical

c.sheet_resistance()       # ohms/square
c.resistance()             # ohms
c.number_of_carriers()     # carriers in the piezoresistor

Noise

c.integrated_noise()       # total integrated voltage noise (V)
c.johnson_integrated()     # Johnson-Nyquist contribution (V)
c.hooge_integrated()       # 1/f (Hooge) contribution (V)
c.amplifier_integrated()   # amplifier contribution (V)
c.plot_noise_spectrum()    # voltage-noise PSD vs frequency

Sensitivity and Resolution

c.force_sensitivity()        # V/N
c.beta()                     # piezoresistor efficiency factor
c.force_resolution()         # minimum detectable force (N)
c.displacement_resolution()  # minimum detectable displacement (m)

Summary

c.print_performance()

prints a formatted block covering geometry, frequencies, sensitivity, resolution, resistances, noise breakdown, and thermal estimates.

Change the Operating Environment

c.fluid = "water"   # "vacuum", "air", or "water"

This changes the hydrodynamic damping and lowers the damped resonant frequency.

Inspect the Doping Profile

Every cantilever exposes a doping_profile() method that returns the depth coordinate, net active carrier concentration, and total dopant concentration:

import matplotlib.pyplot as plt

z, active, total = c.doping_profile()

fig, ax = plt.subplots()
ax.semilogy(z * 1e9, total, label="Total")
ax.semilogy(z * 1e9, active, "--", label="Net active")
ax.set_xlabel("Depth (nm)")
ax.set_ylabel("Concentration (cm$^{-3}$)")
ax.legend()
plt.show()

Other Cantilever Types

The same pattern applies to the other implementations:

Class Doping / sensing approach
CantileverEpitaxy Step-function epitaxial doping
CantileverDiffusion POCl3 / boron diffusion (kink profile)
CantileverImplantation Ion-implant profile from lookup tables
CantileverPoly Polycrystalline thin-film (poly-Si, Ti, Al)
CantileverPiezoelectric AlN or PZT piezoelectric sensing

See python/examples/ for a runnable script for each.

Optimization

Design optimization is built around three pieces:

  • A goal callable that maps a cantilever to the scalar to minimize. PiezoD ships factories that match the MATLAB tutorial's units (pN, nm, pN/sqrt(Hz), etc.):
from piezod import (
    force_resolution_goal,
    displacement_resolution_goal,
    force_noise_density_goal,
    surface_stress_resolution_goal,
)

  • Optional parameter_constraints that override the default state-variable bounds (e.g. clamp thickness, cap bias voltage). Keys are min_<name> / max_<name> for any state variable returned by c.optimization_state_vars().

  • Optional metric_constraints -- inequality constraints on derived quantities like power dissipation, resonant frequency, or stiffness. Each constraint is a CantileverMetricConstraint referencing a CantileverMetric enum value.

from piezod import (
    CantileverMetric,
    CantileverMetricConstraint,
    optimize_performance,
)

constraints = [
    CantileverMetricConstraint(CantileverMetric.POWER_DISSIPATION, maximum=2e-3),
    CantileverMetricConstraint(CantileverMetric.OMEGA_VACUUM_HZ, minimum=5 * c.freq_max),
    CantileverMetricConstraint(CantileverMetric.STIFFNESS, minimum=1e-3, maximum=1e1),
]

result = optimize_performance(
    c,
    force_resolution_goal(),
    parameter_constraints={"max_v_bridge": 10.0},
    metric_constraints=constraints,
    n_starts=5,
    max_iterations=10,
    random_seed=0,
)

optimize_performance runs SciPy's SLSQP (or L-BFGS-B when there are no nonlinear constraints) from random initial conditions and returns the best result once two starts agree within convergence_tolerance (1% by default), capped at max_iterations total runs. Use optimize_performance_from_current for a single-shot refinement of the existing design.

print(f"force_resolution: {c.force_resolution() * 1e12:.1f} pN -> "
      f"{result.optimized.force_resolution() * 1e12:.2f} pN")

Default geometric sanity constraints (l/w >= 2, w/t >= 2, l_pr/w_pr >= 2, l_pr >= 2 um) are added automatically; pass default_aspect_constraints=False to opt out.

For a runnable end-to-end example with full output, see python/examples/optimization.py. The example mirrors the MATLAB tutorial flow and typically improves Harley-1999-style force resolution by several hundred times.

For implant-process-only optimization (geometry fixed, only annealing_time/temp and implantation_energy/dose varied), CantileverImplantation.optimize_doping_for_hooge_noise is also available with a Hooge-noise-limited default objective.

Next Steps

  • Browse python/examples/ for runnable end-to-end scripts.
  • Read the docstrings on Cantilever, CantileverEpitaxy, etc. -- every public method is documented inline.
  • For low-level customisation (custom doping profiles, manual sheet-resistance calculations) see PiezoresistorFromProfile.