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5. Strain and Pressure Sensors

Piezoresistivity

Applied stress gives the change in resistance = F/A = x/x R/R (stress) (strain)

In the case of elastic deformations the Hooke’s law obeys.For a sample with the shape of a rod of length x and cross secion A one can write

E – Young’s modulus of the material

AF

Exx 1

Metallic cylidrical conductor (a wire) changes its resistance under the influence of applied stress

The resistance x - length of a conductorA – cross sectional area

After differentiating

or

Because

then

Introducing the Poisson’s number one obtains

R xA

dR

Rdxx

dAA

d

A r 2 dA rdr2

dR

Rdxx

drr

d 2

d

AxdA

A

xdxA

dRdAARdx

xRdR

2

drrdxx

drr

dR

Rd 2

d121R

dR

eS

Using one can write

In practice one uses the gauge factor Se (relative changein resistance for unit deformation):

material constant

For most metals Se ~ 2 (for platinum about 6)The change in resistance is not exceeding 2%.

Metallic strain gauges should reveal:• appreciable R• high Se

• low TCR (TCR = ΔR/RΔT) • high mechanical durability

Manganin – alloy consisting of: 84%Cu + 12%Mn + 4%NiConstantan: 60%Cu + 40%Ni

Characteristics of typical alloy strain gauges manganin (solid line), Se = 2 constantan (dashed line), Se = 0.8

Examples of metallic strain gauges

Foil - type(etched metallic foilon a backing film)

Rosette - type Thin film

Piezoresistance in semiconductors

Semiconductor strain gauges have about 50 times higher gauge factor than metals (typical value of Se is 100).

Drawbacks:• Se depends on (nonlinearity)• strong temp. dependence• lower dynamic range of .

For a given semiconductor Se depends on its crystallographic orientation and doping. In this case the variations of / are important

ddRdR

eS1121

Stresses cause change in a band structure of the silicon crystal what influences the mobility and concentration of current carriers. In effect the resistivity changes but the current density vector j and electric field vector E are no longer parallel (effect of anisotropy – tensor description).

)(j)(jE

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П - tensor of piezoresistane coefficientsσ - stress

Piezoresistance in silicon

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Only one stress comp., longitudinal effect

LL

In general the piezoresistive coeff. depend on crystal orientation, the type of doping and change significantly from one direction to the other.

Piezoresistance in silicon

.compstressorthogonalandparallel, TL

.coefftivepiezoresislarperpendicu

.coefftsivepiezoresisparallel

T

L

TTLL

L TDiffusive piezoresistor under parallel and orthogonal stress

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Examples of semiconductor strain gauges

Semiconductor strain gauges printed on a thick cantilever for measurements of force P.The stress above neutral axis is positive, below – negative.The resistors are connected in a Wheatstone brigde configuration.

Strain gauges in a bridge connection

Wheatstone bridge with two active arms and identical strain gauges.

εt - strechingεc - compression

Wheatstone bridge with four active arms (increase in sensitivity, temperature offset compensation).Identical sensors undergo the influence of compressive and tensile stresses.

thermmech R

R

R

R

R

R

Strain gauges in a bridge connection, cont.

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Changing doping one can change sign of the effect

Compensation of nonlinearity in semiconductor piezoresistors

Fully compensated bridge based on n-Si and p-Si piezoresistors

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Membrane pressure sensors

Two resistors have their primary axes parallel to the membrane edge,resulting in a decrease in resistance with membrane bending. The other two resistorshave their axes perpendicular to the edge, which causes the resistance to increasewith the pressure load.

Distribution of stresses in a circular membrane under the influence of applied pressure.

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Pressure sensor with diffusedpiezoresistive sense elements in a Wheatstone bridge configuration.

Silicon micromachined pressure sensors

National Semiconductor Corp. of Santa Clara, California was the first company which began the high-volume production of this kind of pressure sensor in 1974. Recently this market has grown to tens of million sensors p.a.The vast majority use piezoresistive elements to detect stress in a thin silicon diaphragm in response to a pressure load.

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Technology of micromachined pressure sensors

The fabrication process of a typical pressure sensor.Technological steps are characteristic to the integrated circuit industry, with the exception of the precise formingof the thin membrane using electrochemical etching.

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High temperature pressure sensors

Most of commercially available silicon micromachined pressure sensors are working in a temperature range –40° to +125ºC, which covers the automotive and military specifications. Above 125ºC the increased leakage current across the p-n junction between the diffused piezoresistor and the substrate significantly degrades performance. At elevated temperatures the silicon-on-insulator (SOI) technology can be used.

High-temperature pressure sensor in SOI technology (GE NovaSensor ).

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An example of pressure sensor used in vaccum measurements, working as a differential capacitor.

M

pr px

10-4 < p < 103 TrΔCmin = 10-5 pF (Δd~ nm)

Vacuum measurements