Introduction to Piezoresistive Sensors

1. EMT 452/3 Micro-Electro-Mechanical-Systems (MEMS)

2. CHAPTER 4 Piezoresistive Sensors

3. Introduction • The piezoresistive effect is the changing of an electrical resistance of a material due to applied mechanical stress. • Electrical resistor will change its resistance when it experiences a strain and deformation. • The piezoresistive effect only causes a change in resistance, it does not produce electrical charges.

4. Introduction • The piezoresistive effect of semiconductor materials can be several magnitudes larger than the geometrical piezoresistive effect in metals and is present in materials like germanium, polycrystalline silicon, amorphous silicon, silicon carbide, and single crystal silicon. • Resistance value of a resistor with length, l and the cross-sectional area, A :

5. Introduction • Resistance value is determined by bulk resistivity, ρ and the dimensions. • Two important ways by which the resistance value change with applied strain; • The dimensions including l and A will change with strain. • The resistivity of certain materials may change as a function of strain. The magnitude of resistance change from this principle is much greater than the dimension changes.

6. B’ A A’ B pressure pressure tension compression tension compression A-A’ cross section (stress along y-axis) B-B’ cross section (stress along y-axis) Dimension changes of a resistor under longitudinal stress Fixed boundaries

7. Piezoresistive sensors • The change in resistance is linearly related to the applied strain. • The proportional constant G is called gauge factor and defined in the below equation.

8. F F R R F F R F F Longitudinal piezoresistor Transverse piezoresistor Three cases of piezoresistive force sensors (longitudinal and transverse piezoresistor configurations)

9. Wheatstone bridge • Resistance changes often read using Wheatstone bridge. • Consists of 4 resistor connected in loop. • Vin is applied across two junctions that are separated by 2 resistors. • Voltage drop across the two junctions forms the output. • One or more resistors in the loop may be sensing the resistors, whose resistances change with the intended variables.

10. Wheatstone bridge

11. Stress Analysis on Mechanical Elements: Stress in Flexural Cantilevers • Under a transverse loading of a concentrated force at the free end, the torque distribution through the beam is; • non-uniform- zero at free end and maxim at the fixed end. • At any cross section, the signs of the longitudinal stresses change across the neutral axis. • The magnitude of stresses at any point on the cross section is linearly proportional with the respect to the distance to the neutral axis. • The magnitude of max stresses + individual cross sections changes linearly with respect to the distance to the free end, reaching a section-wide max at the top and bottom surfaces. • The piezoresistors are commonly found on the surface of cantilever and near the fixed end.

12. Stress distribution in a uniform & symmetric cantilever beam

13. Stress distribution in a uniform & symmetric cantilever beam

14. Stress in Flexural Cantilevers The total torque given by the area integral of normal force acting on given area dA, called dF(x,h) multiplied by the arm distance between the force and the neutral plane: The magnitude of stress linearly related to h and is the greatest at the surface (denoted σmax(x)) at any given cross section, the torque balance equation at any given cross section yields:

15. Stress in Flexural Cantilevers The maximum strain for the entire cantilever occurs at the fixed end, where x=L. In fact, the sole interest to find the magnitude of the maximum stress/strain at the fixed end. The maximum strain is expressed as a function of total torque M(x):

16. doped resistor l deposited resistor ineffective design #1 ineffective design #2 Design of cantilevers with piezoresistors

17. Stress Analysis on Mechanical Elements: Stress in the membrane The governing equation for membrane under a uniform pressure loading p is, w is the normal displacement of the membrane at a location (x,y). D is the rigidity of the membrane and given by

18. Stress on the membrane Normalized displacement (left) and stress in the x-axis (right)

19. Stress on the membrane The max displacement at the center (wcenter) of a rectangular diaphragm (a x b) under a uniform pressure, p is α is determined by the ratio of a to b. The max stress (at the center point of the long edge) and the stress in the center of the plate are

20. Bending of rectangular plate under uniform stress

21. Application of Piezoresistive Sensor • Inertia Sensors • Under an applied acceleration, a proof mass experiences an inertial forces, which in turn deforms mechanical support elements connected to proof mass and introduce stress and strain. By measuring the magnitude of stress, value of acceleration can be determined. • Pressure Sensors • For an example bulk micromachined pressure sensor with Wheatstone Bridge configuration. These 4 piezoresistor are located correspond to regions of maximum tensile stress when the diaphragm is bent by a uniform applied pressure difference across the diaphragm.

22. Application of Piezoresistive Sensor • Tactile Sensors • Used to measure contact forces and to characterize surface profiles and roughness. Micromachined tactile sensor has the potential of high density integration. • Flow Sensors • Microstructure is used for flow sensing. Fluid flow around microstructure can balance a lifting force, drag force or momentum transfer on a floating element.