SaM - MEMORIZE (Piezoresisitivity)

List of things to memorize:

SaM - Piezoresistivity

• SaM - Piezoresistivity Effect
  • SaM - Recap of Basic Components • Resistance, Capacitance, Inductance • Resistivity, Permittivity, Permeability
  • SaM - Definition of Stress and Strain • Definition of Strain and Stress • Stress Vector • Strain Vector
  • SaM - Hooke’s Law • Collapsed Yieldness Tensor for Isotropic Materials
• SaM - Piezoresistivity Effect in Details
• SaM - Definition of Isotropic and Anisotropic Materials
• SaM - Crystallographic Axis
• SaM - Collapsed Piezoresistive Coefficient Tensor for Silicon
• SaM - Dependency of the Piezoresistance Factor on the Concentration of Dopants
• SaM - Piezoresistance of Doped Silicon • Doped Silicon as a Strain Gauge

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SaM - Piezoresistivity Effect

• Formula for metals:$\frac{d\rho }{\rho }=c\frac{dV}{V}$
• Formula for anisotropic materials (like doped Silicon):$\frac{d\vec{\rho }}{{\rho }_0}=\overline{\overline{\Pi }}\cdot \overline{\overline{c}}\cdot \overline{\overline{\epsilon }}$
• Terminology:
  • $\frac{d\rho }{\rho }$ : infinitesimal change in resisitivity.
  • $\frac{dV}{V}$ : infinitesimal change in volume.
  • $\overline{\overline{\Pi }}$ 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.

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SaM - Recap of Basic Components • Resistance, Capacitance, Inductance • Resistivity, Permittivity, Permeability

• Resistance : $R=\rho \frac{l}{S}$
• Condutance : $G=\frac{1}{R}$.
• Capacitance : $C=\epsilon \frac{A}{d}$
• Inductance : $L=n^2\mu \frac{A}{l}=n^2\frac{1}{\mathcal{R}}$
• Terminology:
  • $\rho$ : resisitivity.
  • $\epsilon$ : eletric permittivity.
  • $n$ : number of turns of a coil.
  • $\mu$ : magnetic permeability.
  • $l$ : length.
  • $S$ : surface.
  • $A$ : area.
  • $d$ : distance
  • $\mathcal{R}$ : reluctance.

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Transclude of SaM---MEMORIZE-(Stress-and-Strain-•-Strain-and-Stress)#sam---definition-of-stress-and-strain

SaM - Young and Poisson Modulus

• Young Modulus:$E=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}=\frac{\sigma }{{\epsilon }_{\parallel }}$
• Poisson Modulus:$\nu =-\frac{\frac{\Delta r}{r}}{\frac{\Delta l}{l}}=\frac{{\epsilon }_{\perp }}{{\epsilon }_{\parallel }}$
• Terminology:
  • $F$ : force.
  • $A$ : area.
  • $r$ : radius.
  • $l$ : lenght.
  • $\sigma$ : stress
  • $\epsilon$ : strain.
    • ${\epsilon }_{\parallel }$ : parallel strain.
    • ${\epsilon }_{\perp }$ : perpendicular strain.

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SaM - Piezoresistivity Effect in Details

• Electric field:$\begin{array}{lc}\text{Isotropic:}& \vec{E}={\rho }_0\cdot \vec{J}\\ \text{Anisotropic:}& \vec{E}=\overline{\overline{\rho }}\cdot \vec{J}\end{array}$
• Electric field in a deformed 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:$\frac{\Delta \overline{\overline{\rho }}}{{\rho }_0}=\overline{\overline{\Pi }}\cdot \overline{\overline{\sigma }}$
• Terminology:
  • $\vec{E}$ : electric field.
  • $\rho$ : resisitvity.
  • $\vec{J}$ : current density vector, $\left[J\right]=\frac{\text{A}}{{\text{m}}^2}$
  • $\overline{\overline{I}}$ : identity matrix.
  • $\frac{d\rho }{\rho }$ : infinitesimal change in resisitivity.
  • $\overline{\overline{\Pi }}$ is the piezoresistive coefficent tensor.
    TODONOT_SURE_ABOUT_THIS We probably defined it wrong, we made an example for Silicon, and it is the same as the collapsed stifness tensor, but that is defined as $\overline{\overline{c}}$ in the Hooke’s law.
    NOT_SURE_ABOUT_THIS This is probably also called the piezoresistive factor.
  • $\overline{\overline{\sigma }}=\overline{\overline{c}}\cdot \overline{\overline{\epsilon }}$ is the Hooke’s law.

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SaM - Definition of Isotropic and Anisotropic Materials

• Isotropic materials exhibit the same mechanical properties in all directions.
• Anisotropic materials, on the other hand, have different mechanical properties in different directions.

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SaM - Crystallographic Axis

• $\left[hkl\right]$ defines a vector.
• $\left(hkl\right)$ defines a plane $\perp \left[hkl\right]$.
• $⟨hkl⟩$ defines a direction $\parallel \left[hkl\right]$.
• Typical cell for Silicon:
  
• Real World Measure:
  • Typical surfaces: $\left(100\right)$, $\left(111\right)$, $\left(110\right)$, used in micromachining technology
  • For Silicon: $a=543\text{pm}$ (Remember : $p=10^{-12}$)
• Terminology
  • $a$ : length of the unit cell.

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SaM - Collapsed Stiffness Tensor for Isotropic Materials

Collapsed stiffness tensor:

• $c_{11}=c_{12}+2c_{44}$
• Or we can write:
  • $c_{11}=\lambda +2\mu$
  • $c_{12}=\lambda$ (lame modulus)
  • $c_{44}=\mu$ (shear modulus)

While the collapsed yieldness tensor:

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SaM - Dependency of the Piezoresistance Factor on the Concentration of Dopants

• The following fenomena happen for weakely doped silicon, meaning: $N_D\text{or}{N}_A<10^{19}{\text{cm}}^3$
• Change in the piezoresisitivity factor ($c$ or $\Pi$) due to the doping concentration:
  
• Change in the TCR (Temperature Coefficient of Resisitance) due to the doping concentration:
  
• Change in the temperature coefficient of sensitivity due to the doping concentration:
  
• Terminology:
  • $N_A$ or $N_D$ : density of dopants, acceptors and donors.
  • Boron $(\text{III group})$, is an acceptor.

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SaM - Piezoresistance of Doped Silicon • Doped Silicon as a Strain Gauge

• How does a MOSFET work.
• Setup:
  1. Take a thin film of $n$-type doped silicon and place metal lines at both ends:
     
  2. We feed it a current:$\vec{J}=⟨J_1,0,0⟩$
  3. We apply an extensive stress:$\vec{\sigma }=⟨{\sigma }_1,{\sigma }_2,0,0,0,0⟩$
  4. Eletric field: current + piezoresistive effect:$\vec{E}=\rho \cdot \left(\overline{\overline{I}}+\overline{\overline{\pi }}\cdot \overline{\overline{\sigma }}\right)\cdot \vec{J}$
• If necessary you can rotate the $\overline{\overline{\pi }}$ tensor:${\pi }^{\prime }=N^{-1}\pi N$
• For an uniaxial stress $\left(\vec{\sigma }=⟨{\sigma }_1,0,0,0,0,0⟩\right)$:$E_1={\rho }_e(1+G{\epsilon }_1)J_1$Where: $G={\pi }_{11}^{\prime }\cdot E$
• Voltage:$V=R_0(1+G{\epsilon }_l)I$
• We can change the crystallographic axis and the voltage/current reading:
  This will change the values of the $\overline{\overline{\pi }}$ matrix:
  
• Terminology:
  • $\vec{J}$ : current density vector, $\left[J\right]=\frac{\text{A}}{{\text{m}}^2}$
  • $\vec{\sigma }$ : stress.
  • $\vec{E}$ : electric field.
  • $\overline{\overline{I}}$ : identity matrix.
  • ${\epsilon }_1$ or ${\epsilon }_l$ : longitudinal strain.
  • $G$ : gauge factor.
  • $V$, $R$, $I$ : voltage, resistance, current.

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