Piezoresistive pressure sensor sensitivity

A piezoresistive pressure sensor turns diaphragm stress into a resistance change, and a Wheatstone bridge turns that into a voltage. The chain has three links, each with its own assumptions.

1. Stress to ΔR/R

For p-type silicon resistors aligned with ⟨110⟩ on a (100) wafer, the longitudinal and transverse piezoresistive coefficients are approximately π_l ≈ π₄₄/2 and π_t ≈ −π₄₄/2, with π₄₄ = 138.1 × 10⁻¹¹ Pa⁻¹ for lightly doped silicon (Smith, 1954). Then:

ΔR/R = π_l σ_l + π_t σ_t

Near the clamped edge, a resistor pointing across the edge sees σ_l = σ_edge and σ_t = ν σ_edge, so ΔR/R = (π₄₄/2)(1 − ν) σ_edge. A resistor parallel to the edge sees the opposite sign.

2. Doping and resistor size

π₄₄ falls with doping concentration and temperature. Kanda's factor P(N, T) is about 0.85 at 10¹⁸ cm⁻³ and 0.6 at 10¹⁹ cm⁻³ at room temperature. A real resistor also averages stress over its length, so its response is below the peak value. DiaphragmCalc exposes both as multipliers.

3. The bridge

With two resistors increasing by x and two decreasing by y in a full bridge:

V_out / V_in = (x − y) / (2 + x + y)

Sensitivity is usually quoted as mV/V per unit pressure, and full-scale output (FSO) as mV at the rated supply.

Worked example

1000 µm square, 20 µm thick silicon, 1 bar, doping factor 0.85, stress-averaging 0.8, 5 V supply:

• Edge stress 77 MPa
• ΔR/R = 3.38 % and -3.38 %
• Sensitivity 33.8 mV/V/bar, FSO 169 mV at 5 V

Commercial silicon sensors often land between a few and a few tens of mV/V at full scale. If your estimate is far above that, check the stress-averaging factor and whether the resistors really sit at the edge.

Results are analytical first-order estimates and should be independently validated before use in safety-critical or production designs.