1 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

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

Using one can write In practice one uses the gauge factor S e (relative change in resistance for unit deformation): material constant For most metals S e ~ 2 (for platinum about 6) The change in resistance is not exceeding 2%.

Metallic strain gauges should reveal: appreciable R high S e low TCR (TCR = ΔR/RΔT) high mechanical durability Manganin – alloy consisting of: 84%Cu + 12%Mn + 4%Ni Constantan: 60%Cu + 40%Ni Characteristics of typical alloy strain gauges manganin (solid line), S e = 2 constantan (dashed line), S e = 0.8

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 S e is 100). Drawbacks: S e depends on  (nonlinearity) strong temp. dependence lower dynamic range of . For a given semiconductor S e depends on its crystallographic orientation and doping. In this case the variations of  /  are important

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

8 Only one stress comp., longitudinal effect 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 Diffusive piezoresistor under parallel and orthogonal stress

9 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. ε t - streching ε c - compression

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

12 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

13 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 resistors have their axes perpendicular to the edge, which causes the resistance to increase with the pressure load. Distribution of stresses in a circular membrane under the influence of applied pressure.

14 Pressure sensor with diffused piezoresistive 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 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.

15 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.

16 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 ).

17 An example of pressure sensor used in vaccum measurements, working as a differential capacitor < p < 10 3 Tr ΔC min = pF (Δd~ nm) Vacuum measurements