SaM - Piezoelectric Effect in Details • Direct and Inverse Piezoelectric Effect

Remeber:

Piezoelectricity (ChatGPT):
When mechanical stress is applied to a piezoelectric material, the material’s internal structure becomes distorted. This distortion causes a redistribution of electrical charges within the material, leading to the generation of an electric potential across its surfaces. This phenomenon is known as the direct piezoelectric effect:

• We have seen already two equations for the piezoelectric effect:$\{\begin{array}{lc}\vec{D}=\overline{\overline{{\epsilon }^T}}\vec{E}+\overline{\overline{d}}\overline{\overline{T}}& \text{(direct piezoelectric effect)}\\ \overline{\overline{S}}=\overline{\overline{s}}\overline{\overline{T}}+\overline{\overline{d^T}}\vec{E}& \text{(inverse piezoelectric effect)}\end{array}$Where:
  • ==The first equation describes the direct piezoelectric effect==.
  • ==While the second equation describes the inverse piezoelectric effect==.
    ==The inverse piezoelectric effect states that even if there is no stress applied to a piezoelectric material, but is present an external electrical field ⇒ then we will see a deformation, or strain of the material itself==
  • $\vec{D}$ : Electric Displacement Field, size: $[3\times 1]$.
  • $\overline{\overline{\epsilon }}$ : Electric Permettivity, size: $[1\times 1]$.
  • $\vec{E}$ : Electric Field, size: $[3\times 1]$.
  • $\overline{\overline{d}}$ : Piezoelectric Coefficient Matrix, size: $[3\times 6]$.
  • $\overline{\overline{T}}$ : Stress Tensor or Vector, size: $[3\times 3]$, however we can reduce it to a $[6\times 1]$, in collapsed notation, since by definition is a symmetrical matrix.
  • $\overline{\overline{S}}$ : Strain Tensor or Vector, size: $[3\times 3]$, however we can reduce it to a $[6\times 1]$, in collapsed notation, since by definition is a symmetrical matrix.
  • $\overline{\overline{s}}$ : Compliance Tensor, or Yieldness Tensor, size: $[6\times 6]$.
    NOTE: These are the sizes we have seen, but they are specific to isotropic materials.
    since for anisotropic materials for example the yieldness tensor has 4 dimensions, and not only 2.
    ==Also typically piezoelectric materials are anisotrpic (Source: Google), take these sizes only as a reference==.
  • We have already seen the realtion between Stress Tensor and Strain Tensor, and we defined it as the Hooke’s Law:$\{\begin{array}{l}\overline{\overline{S}}=\overline{\overline{s}}\overline{\overline{T}}\\ \overline{\overline{T}}=\overline{\overline{c}}\overline{\overline{S}}\end{array}$Where: - $\overline{\overline{s}}$ is the yieldness tensor. (“tensore di cedevolezza”) - $\overline{\overline{c}}$ is the stifness tensor, (“tensore di rigidità”) it describes the elastic constants of the material. - Also $\overline{\overline{c}}$ and $\overline{\overline{s}}$ are inverse of each other.

We have different materials which show piezoelectric effect:IMPORTANTE

• Quartz $(Si{O}_2)$ : small losses, almost a perfect dielectic.
• PZT (Lead Zirconate Titanate, polycrystal) : large piezo-coefficients $(d_{33}=500\frac{\text{pC}}{\text{N}})$
• PVDF (polymer) : low cost

Acutally:NOT_SURE_ABOUT_THIS (this is my reasoning)

• The quartz has: ${\epsilon }_r^{\prime }\approx 4.0,{\epsilon }_r^{\prime \prime }=0.001$ and $d_{12}\approx 2\frac{\text{pC}}{\text{N}}$.
• The PZT has: ${\epsilon }_r^{\prime }\in [500÷2000],{\epsilon }_r^{\prime \prime }\in [0.01÷0.05]$ and $d_{12}\approx 500\frac{\text{pC}}{\text{N}}$.
• Their complex electric permettivity ratio is almost the same $\frac{{\epsilon }_r^{\prime }}{{\epsilon }_r^{\prime \prime }}\sim 10^{-3}$ but the PZT has higher $d$ value, meaning if we take into the direct and inverse piezoelectric effect formual, it will result in a higer response given the same electric field $E$ and the same stress value $T$.
  The same is true for the capacitance value $C_E=\epsilon \frac{A}{h}$ and charge output $Q=d\cdot F$, the PZT will have higher value for both.
  ==So for “raw performance” the PZT is better==.
• However we have studied how we can vary some characteristics of the quartz, if we cut it in a different way.
  Also we have seen how the AT-cut quartz has a linear relationship between its thickness $(t)$ and the wavelength $({\lambda }_{\omega })$: $t\approx \frac{{\lambda }_{\omega }}{2}$.
  So the quartz give us "more flexibility".

---

Here’s a table with some properties for quartz, PTZ and PVDF: (NOT IMPORTANT)

Material	$d\left(\text{pC}\cdot {\text{N}}^{-1}\right)$	${\epsilon }_r$	$g\left(\frac{{\text{m}}^2}{\text{C}}\right)$	$E\left(10^9\frac{\text{N}}{{\text{m}}^2}\right)$	$\lambda \left(\frac{\text{s}\cdot \text{N}}{\text{m}}\right)$
Quartz $(Si{O}_2)$	$d_{11}\in [2÷3]$	$\in [4÷5]$	$5.8\cdot 10^{-2}$	$80$	$2\cdot 10^{-5}$
PZT (Lead Zirconate Titanate, polycrystal)	$d_{33}=500$	${\epsilon }_{r_{11}}=2400$	$2.4\cdot 10^{-2}$	$80$	$7\cdot 10^{-3}$
PVDF (polymer)	$d_{31}=23$	$12$		$2$	$5\cdot 10^{-2}$

---

• So remember that piezoelectric effect can be seen in materials which have a crystal structure, and this crystal structure is made of ions.
• For instance, in the figure you can see the structure of a PZT material where the blue spheres are positive ions, metal ions, where has the negative spheres, negative ions, oxygen ions.
• When the crystal gets deformed, you will have a shift of the two centers of mass, such that polarization appears because the positive charge and the negative charge are “separated” in space.
  therefore as a macroscopic effect, you will find a separation of charge.

The mathematical description of this effect:

• We have two different equations that describes the direct and the inverse piezoelectric effect, this is the first one, called the “direct piezoelectric effect”.
• $D$ : field ???
• $E$ : electrical field.
• ${\overline{\overline{d}}}^E\overline{\overline{T}}$: piezoelectric effect.
  • $\overline{\overline{T}}$: stress tensor (previously called $\sigma$, i changed the name since $\sigma$ is used for conductivity)
  • ${\overline{\overline{d}}}^E={\overline{\overline{d}}}^T=d$ it’s a symmetrical tensor.NOT_SURE_ABOUT_THIS
• So you can polarize the material either with an electrical field, or this is particular for piezoelectric effect and piezoelectric material with an applied stress.
• NOTE: this is a static relationship, and a linear relationship.
• NOTE: In this formula there is an error, if we consider the second term $\overline{\overline{d}}\overline{\overline{T}}$ its resulting size is $[3\times 3\times 3]$ which is not the same size as $\vec{D}$ which is $[3\times 3]$

While the inverse piezo electric effect:

• $S$ was called before $\epsilon$ sometimes and it is the strain tensor.
  Its dimension is 3 by 3 or in collapse notation can sometimes be replaced by a 6 by 1 vector.
• this small $s$ here which is the inverse of the matrix of the elastic constant, this is called the compliance tensor.
• We are describing our piezoelectric material as an elastic material, it is a crystal so we know that it is true.
• The inverse piezoelectric effect and states that even if there is no stress applied to the material, but there is an external electrical then we will see a deformation, a strain of the material itself.
• I drop this ${}^E$ and ${}^T$ for the $d$ tensor, I put here just $d$ (it is still a 3x3x3 tensor), the measurement unit can be meter over volt or Coulomb divided by Newton.
• So now we can for instance take this equation here and write it more in a clear manner, so I have the $i$-th component of the displacement field $D$ is equal to:$D_i={\sum }_{j=1}^3{\epsilon }_{ij}{E}_{ij}+{\sum }_{j=1}{\sum }_{k=1}{d}_{ijk}{T}_{ijk}$

Sometimes instead of using this $d$ coefficient to describe the piezoelectric behavior, we can write this, and now I write it as if it is a scalar, so 1D:

• $g$ : voltage piezoelectric coefficients.
• Originally the $d$ coefficients were called charge piezoelectric coefficients.
• For $g$ obviously you have a tensor.

I remember you that we have different materials which show piezoelectric effect:

---