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, the relative strain is proportional to stress E – Young’s modulus of the material 1 1

Metallic cylidrical conductor (a wire) changes its resistance under the influence of applied stress The resistance x - length of a conductor A – cross sectional area After differentiating or Because then Introducing the Poisson’s number  one obtains

In practice one uses the gauge factor Se (relative change Using  one can write In practice one uses the gauge factor Se (relative change in 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 Characteristics of typical alloy strain gauges manganin (solid line), Se = 2 constantan (dashed line), Se = 0.8 Manganin – alloy consisting of: 84%Cu + 12%Mn + 4%Ni Constantan: 60%Cu + 40%Ni

Examples of metallic strain gauges Foil - type (etched metallic foil on 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

Piezoresistance in silicon 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). П - tensor of piezoresistane coefficients σ - stress

Piezoresistance in silicon Only one stress comp., longitudinal effect Diffusive piezoresistor under parallel and orthogonal stress In general the piezoresistive coeff. depend on crystal orientation, the type of doping and change significantly from one direction to the other. 8 8

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. 9 9

Strain gauges in a bridge connection Wheatstone bridge with two active arms and identical strain gauges. εt - streching εc - compression

Strain gauges in a bridge connection, cont. Mechanical and thermal interactions Wheatstone bridge with four active arms (increase in sensitivity, temperature offset compensation). Identical sensors undergo the influence of compressive and tensile stresses.

Compensation of nonlinearity in semiconductor piezoresistors Fully compensated bridge based on n-Si and p-Si piezoresistors Changing doping one can change sign of the effect 12 12

Membrane pressure sensors Distribution of stresses in a circular membrane under the influence of applied pressure. Two resistors have their primary axes parallel to the membrane edge,resulting in a decrease in resistance with membrane bending. The other two resistors have their axes perpendicular to the edge, which causes the resistance to increase with the pressure load. 13 13

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. Pressure sensor with diffused piezoresistive sense elements in a Wheatstone bridge configuration. 14 14

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 forming of the thin membrane using electrochemical etching. 15 15

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 ). 16 16

Vacuum measurements An example of pressure sensor used in vaccum measurements, working as a differential capacitor. C2 C1 10-4 < p < 103 Tr ΔCmin = 10-5 pF (Δl~ nm) ΔC = C2 – C1 = ε0εr A·2Δl/(l2 – Δl2) for Δl << l ΔC = ε0εrA·2Δl/l2, hence ΔC/C = 2Δl/l In differential connection sensitivity and linearity increase This type of pressure sensor is used for absolute vaccum measurements. 17 17