SaM - Piezoresistivity Effect in Details

Remeber:

Let’s compare the electric filed generated by an isotropic and an anisotropic material: $\begin{array}{lc}\text{Isotropic:}& \vec{E}={\rho }_0\cdot \vec{J}\\ \text{Anisotropic:}& \vec{E}=\overline{\overline{\rho }}\cdot \vec{J}\end{array}$The difference is that for isotropic material the reisistivity is a scalar (${\rho }_0$), while for anisotropic material we are talking about a tensor of resisitivity.
More specifically for anisotropic materials: $\mathrm{size}\left\{\overline{\overline{\rho }}\right\}=[3\times 3]$

If a material (==both isotropic or anisotropic==) gets deformed, the resulting energy field changes, due to the piezoresistive effect, let’s take for simplicity an isotropic material:$\begin{array}{lc}\text{Isotropic (UNdeformed):}& \vec{E}={\rho }_0\cdot \vec{J}\\ \text{Isotropic (DEFORMED):}& \vec{E}={\rho }_0\cdot \left(\overline{\overline{I}}+\frac{\Delta \overline{\overline{\rho }}}{{\rho }_0}\right)\cdot \vec{J}\end{array}$Where $\frac{\Delta \overline{\overline{\rho }}}{{\rho }_0}$ represents the piezoresistivity effect, and is found as:$\frac{\Delta \overline{\overline{\rho }}}{{\rho }_0}=\overline{\overline{\Pi }}\cdot \overline{\overline{\sigma }}$Where:

• $\overline{\overline{\Pi }}$ piezoresisitve coefficient tensor, with $\mathrm{size}\left\{\overline{\overline{\pi }}\right\}=[3\times 3\times 3\times 3]$, for isotropic materials, we will see that for isotropic materials it can be reduced to a matric $[6\times 6]$.
• $\overline{\overline{\sigma }}$ stress tensor. (it should be a $[3\times 3]$ matrix, but we can compact it to a $[1\times 6]$ vector)

Here’s an example of a piezoresisitve coefficient tensor for Silicon (an isotropic material):

• So ==for Silicon==, we can write the special $[3\times 3\times 3\times 3]$ $\overline{\overline{\Pi }}$ piezoresistive coefficients tensor with only $3$ indipendent constants: ${\Pi }_{11},{\Pi }_{12},{\Pi }_{44}$.
  These inidipendent constants depend on the type and doping level.
• This is true for all isotropic materials, and we defined it as the “collapsed stifness tensor”.
• Here we have some values for the piezoresisitve coefficients: piezoresistive coefficients values.IMPORTANTE

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The energy field generated by the piezoelectric effect depdens on the application of the force, so we need vectors, matricies and tensors:

While for isotropic materials, vector are sufficient:

Now for silicon we have the difference for undeformed and deformed silicon, when the silicon is deformed we need to take into account the piezo-resistive effect:

Where the relative variation of resisitivity given by the piezoresisitivity effect is given by:

We can write the special $6\times 6$ $\overline{\overline{\Pi }}$ piezoresistive coefficients matrix with only $3$ indipendent constants: