SaM - Piezoresistivity Effect

Remeber:

Piezoresisitivity : by applying a preassure on a material we will change its lattice structure, so the average free path will change, then the conductivity and resisitvity of the material will change as well.

For metals the changes in its resisitivity, due to piezoresisitivity, is calculated as:$\frac{d\rho }{\rho }=c\frac{dV}{V}$So if we also account for the variation of resisitivity given by the basic formula of resistivity, we get:NOT_SURE_ABOUT_THIS $\frac{d\rho }{\rho }=\frac{dl}{l}\left(c+1\right)+\frac{dS}{S}\left(c-1\right)$We have considered metals as isotropic materials.NOT_SURE_ABOUT_THIS

While for Semiconductor, specifically for doped Silicon (an anisotropic material), the piezoelectricity effect has to consider also the direction of preassure, to do this we use tensors:$\frac{d\vec{\rho }}{{\rho }_0}=\overline{\overline{\Pi }}\cdot \overline{\overline{c}}\cdot \overline{\overline{\epsilon }}$Where:

• $\overline{\overline{\Pi }}$ is the piezoresistive coefficent tensor.
• $\overline{\overline{c}}\cdot \overline{\overline{\epsilon }}=\overline{\overline{\sigma }}$ is the Hooke’s law.
  • $\overline{\overline{c}}$ : stifness tensor (we have seen the stiffness tensor for isotropic materials).
  • $\overline{\overline{\epsilon }}$ : strain matrix (or displament matrix).
  • $\overline{\overline{\sigma }}$ : stress matrix.
• We represent everthing with tensors, since the piezoresistive effect can change depending on the direction of the strain applied to the material, this way we also account for anisotropic materials.

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• This formula can be furhter improved, since we can write $V=l\cdot S$, then: NOT_SURE_ABOUT_THIS $\frac{d\rho }{\rho }=c\left(\frac{dl}{l}+\frac{dS}{S}\right)$Where:
  • $l$ is the lenght of the metal.
  • $S$ is the surface area of the metal.
• NOTE: This formula functions only for isotropic metals (most of them are).
• NOTE: An increase of surface area increases the resistivity ($\rho$), while also in metals an increase of surface area decreases the resistive value ($R$), as we know from the basic formula of resistance ($R=\rho \frac{l}{S}$).
  So from the same deformation we also get:NOT_SURE_ABOUT_THIS$\frac{d\rho }{\rho }=\frac{dl}{l}-\frac{dS}{S}$Then the complete variation is given byNOT_SURE_ABOUT_THIS :$\frac{d\rho }{\rho }=\frac{dl}{l}\left(c+1\right)+\frac{dS}{S}\left(c-1\right)$
• NOTE: (NOT IMPORTANT)
  Some metals display piezoresistivity that is much larger than the resistance change due to geometry.
  In platinum alloys, for instance, piezoresistivity is more than a factor of two larger, combining with the geometry effects to give a strain gauge sensitivity of up to more than three times as large than due to geometry effects alone.
  Pure nickel’s piezoresistivity is -13 times larger, completely dwarfing and even reversing the sign of the geometry-induced resistance change.
  Source: Wikipedia

For semiconductor the variation of volume is not so significative, so we can reduce the formula to:

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