Question

A fixed-free cantilever is made of single crystal silicon. The resistor is made by diffusion dop- ing, with a longitudinal gauge factor of 50, The depth of the doped region is less than 0.5 μm. The length, width, and thickness of the cantilever are 200 μm, 20pm, and 5 μm, respectively. If a force F 100 uN is applied in the middle of the cantilever beam, what would be the percent- age change of resistance? metal wires silicon frame cantilever doped piezoresistor silicon trane -T2#equivalent 1/2 1/2

Answer 1

If the force is applied in the middle, the portion of the cantilever from the point of force to the free end does not bear load or undergo deformation. Therefor, for the purpose of analyzing the maximum stress at the fixed end of the cantilever, an equivalent system is used. In the equivalent system, the length of the cantilever is half as long, and the force is applied at the distal end.

The resistor span a certain depth and length. The depth appears small compared with the thickness of the cantilever. The distribution of the stress in the cantilever is therefore not uniform. If we assume that a uniform stress, equating that at the top surface of the fixed end, then the maximum strain is expressed as-

Mt Ft Flt 2EI2EIEI

The Young's modulus of silicon along the longitudinal direction of the resistor is 130 GPa.

Substituting the values,

  $\varepsilon _{_{max}}=$ 0.019 %