Editing Piezoelectricity (section)

==Mechanism==
[[File:Piezo bending principle.svg|thumb|Piezoelectric plate used to convert [[audio signal]] to sound waves]]
The nature of the piezoelectric effect is closely related to the occurrence of [[electric dipole moment]]s in solids. The latter may either be induced for [[ions]] on [[crystal lattice]] sites with asymmetric charge surroundings (as in [[BaTiO3|BaTiO<sub>3</sub>]] and [[PZT]]s) or may directly be carried by molecular groups (as in [[cane sugar]]). The dipole density or [[Polarization density|polarization]] (dimensionality [C·m/m<sup>3</sup>] ) may easily be calculated for [[crystals]] by summing up the dipole moments per volume of the crystallographic [[unit cell]].<ref name=ZPB1995a>{{cite journal|author = M. Birkholz|title = Crystal-field induced dipoles in heteropolar crystals&nbsp;– II. physical significance|journal = Z. Phys. B|volume = 96|pages = 333–340|year = 1995| issue=3 |doi = 10.1007/BF01313055 |bibcode = 1995ZPhyB..96..333B }}</ref> As every dipole is a vector, the dipole density '''''P''''' is a [[vector field]]. Dipoles near each other tend to be aligned in regions called [[magnetic domain|Weiss domains]]. The domains are usually randomly oriented, but can be aligned using the process of ''poling'' (not the same as [[magnet#Magnetizing ferromagnets|magnetic poling]]), a process by which a strong electric field is applied across the material, usually at elevated temperatures. Not all piezoelectric materials can be poled.<ref name="PAMTA">{{Cite book|author=S. Trolier-McKinstry|title=Piezoelectric and Acoustic Materials for Transducer Applications|publisher=Springer|year=2008|isbn=978-0-387-76538-9|editor1=A. Safari|location=New York|chapter=Chapter 3: Crystal Chemistry of Piezoelectric Materials|author-link=Susan Trolier-McKinstry|editor2=E.K. Akdo˘gan}}</ref>

Of decisive importance for the piezoelectric effect is the change of polarization '''''P''''' when applying a [[mechanical stress]]. This might either be caused by a reconfiguration of the dipole-inducing surrounding or by re-orientation of molecular dipole moments under the influence of the external stress. Piezoelectricity may then manifest in a variation of the polarization strength, its direction or both, with the details depending on: 1. the orientation of '''''P''''' within the crystal; 2. [[crystal symmetry]]; and 3. the applied mechanical stress. The change in '''''P''''' appears as a variation of surface [[charge density]] upon the crystal faces, i.e. as a variation of the [[electric field]] extending between the faces caused by a change in dipole density in the bulk. For example, a 1&nbsp;cm<sup>3</sup> cube of quartz with 2&nbsp;kN (500&nbsp;lbf) of correctly applied force can produce a voltage of 12500 [[Volt|V]].<ref>{{cite web |url=http://machinedesign.com/article/sensor-sense-piezoelectric-force-sensors-0207 |title=Sensor Sense: Piezoelectric Force Sensors |author=Robert Repas |url-status=dead |archive-url=https://web.archive.org/web/20100413205918/http://machinedesign.com/article/sensor-sense-piezoelectric-force-sensors-0207 |archive-date=2010-04-13 |website=Machinedesign.com |date=2008-02-07 |access-date=2012-05-04}}</ref>

Piezoelectric materials also show the opposite effect, called the '''converse piezoelectric effect''', where the application of an electrical field creates mechanical deformation in the crystal.<!--

The following content was added by User:James K McMahon in November 2020. It is uncited and not relevant for a basic description of the mechanism of piezoelectricity. It should be cited if possible and moved to a different section.

Applying a poling voltage will orient the polar molecules and change the physical dimensions of piezo unit cell for the entire piezo structure but will be different on each axis. A piezo structure dimension change can be seen by using a surface finish probe instrument that detects um surface finish changes. Apply an electric filed by attaching a DC voltage wire across a piezo dimension. Applying an AC voltage over a large frequency range and using a simple resistive bridge circuit to observe the impedance change with frequency will show maximum positive displacement at (Fr) Frequency of Resonance for the piezo structure. Slightly above this frequency will show the (Fa) Anti-resonant frequency of the structure and the maximum negative displacement. The minimum frequency gap between Fr and Fa will indicate the optimum piezo displacement strength and can be used to improve the formulation of piezo materials or identify differences in individual piezo samples with identical dimensions. 3D Printer's using DOD single nozzles and cylindrical PZT piezo's can be driven with lower voltage with a lower Fr-Fa spacing. In general, physical dimensions work to your advantage when applications call for the strongest displacement. There are six dimensional starting modes for a unit cell depending on poling: Unpoled dimensions with forward and reverse holding voltage, Positive poled dimensions with forward or reverse holding voltage and negatively poled dimensions with forward and reversed holding voltage. For example:  Squeeze style single nozzle inkjets work best with positive poled PZT material and a positive hold voltage prior to firing a drop. The fluid is at maximum compression with a positive hold voltage, then upon releasing the hold voltage to a lower level the fluid chamber fills with new material and finally returning the positive hold voltage squeezes the single drop out of the nozzle. This is called Fill/Fire fluid mode. The drop size and shape is best in this mode and differs when the other piezo modes are used.

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===Mathematical description===
Linear piezoelectricity is the combined effect of
* The linear electrical behavior of the material:
:: <math>\mathbf{D} = \boldsymbol{\varepsilon}\,\mathbf{E} \quad \implies</math> [[Summation|<math>\quad D_i = \sum_j \varepsilon_{ij}\,E_j \;</math>]]
: where '''D''' is the electric flux density<ref>IEC 80000-6, item 6-12</ref><ref>{{Cite web|url=http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-40|title=IEC 60050 – International Electrotechnical Vocabulary – Details for IEV number 121-11-40: "electric flux density"|website=www.electropedia.org}}</ref> ([[electric displacement]]), '''ε''' is the [[permittivity]] (free-body dielectric constant), '''E''' is the [[electric field strength]], and [[divergence|<math> \nabla\cdot\mathbf{D} = 0 </math>]] , [[Curl (mathematics)|<math>\nabla \times \mathbf{E} = \mathbf{0} </math>]].
* [[Hooke's law]] for linear elastic materials:
::<math>\boldsymbol{S}=\mathsf{s}\,\boldsymbol{T} \quad \implies \quad S_{ij} = \sum_{k,\ell} s_{ijk\ell} \,T_{k\ell} \;</math>
: where '''S''' is the linearized [[Strain (materials science)|strain]], '''s''' is [[Compliance (mechanics)|compliance]] under short-circuit conditions, '''T''' is [[Stress (physics)|stress]], and
::<math> \nabla \cdot \boldsymbol{T} = \mathbf{0} \,\,,\, \boldsymbol{S} = \frac{\nabla \mathbf{u} + \mathbf{u} \nabla}{2}, </math>
: where '''u''' is the ''displacement vector''.

These may be combined into so-called ''coupled equations'', of which the '''strain-charge form''' is:<ref name=ikeda>{{cite book|title=Fundamentals of piezoelectricity|last=Ikeda|first=T.|year=1996|publisher=Oxford University Press}}{{ISBN missing}}</ref>
:<math>
 \begin{align}
 \boldsymbol{S} &= \mathsf{s}\,\boldsymbol{T} + \mathfrak{d}^t\,\mathbf{E}\ && \implies \quad
 S_{ij} = \sum_{k,\ell} s_{ijk\ell} \,T_{k\ell} + \sum_k d^t_{ijk} \,E_k, \\[6pt]
 \mathbf{D} &= \mathfrak{d}\,\boldsymbol{T} + \boldsymbol{\varepsilon}\,\mathbf{E} && \implies \quad
 D_i = \sum_{j, k} d_{ijk} \,T_{jk} + \sum_j \varepsilon_{ij}\,E_j,
 \end{align}
</math>
where <math>\mathfrak{d}</math> is the piezoelectric tensor and the superscript t stands for its transpose. Due to the symmetry of <math>\mathfrak{d}</math>, <math>d^t_{ijk}=d_{kji}=d_{kij}</math>.

In matrix form,
:<math>
 \begin{align}
 \{S\} &= \left [s^E \right ]\{T\}+[d^\mathrm{t}]\{E\}, \\[6pt]
 \{D\} &= [d]\{T\}+\left [ \varepsilon^T \right ] \{E\},
 \end{align}
</math>
where [''d''] is the matrix for the direct piezoelectric effect and [''d''{{i sup|t}}] is the matrix for the converse piezoelectric effect. The superscript ''E'' indicates a zero, or constant, electric field; the superscript ''T'' indicates a zero, or constant, stress field; and the superscript t stands for [[Transpose|transposition]] of a [[Matrix (mathematics)|matrix]].

Notice that the third order tensor <math>\mathfrak{d}</math> maps vectors into symmetric matrices. There are no non-trivial rotation-invariant tensors that have this property, which is why there are no isotropic piezoelectric materials.

The strain-charge for a material of the [[Tetragonal crystal system|4mm]] (C<sub>4v</sub>) [[crystal system|crystal class]] (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO<sub>3</sub>) as well as the [[Hexagonal crystal system#Hexagonal crystal system|6mm]] crystal class may also be written as (ANSI IEEE 176):
:<math>
\begin{align}
& \begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix}
= \begin{bmatrix} s_{11}^E & s_{12}^E & s_{13}^E & 0 & 0 & 0 \\
s_{21}^E & s_{22}^E & s_{23}^E & 0 & 0 & 0 \\
s_{31}^E & s_{32}^E & s_{33}^E & 0 & 0 & 0 \\
0 & 0 & 0 & s_{44}^E & 0 & 0 \\
0 & 0 & 0 & 0 & s_{55}^E & 0 \\
0 & 0 & 0 & 0 & 0 & s_{66}^E=2\left(s_{11}^E-s_{12}^E\right) \end{bmatrix}
\begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix}
+
\begin{bmatrix} 0 & 0 & d_{31} \\
0 & 0 & d_{32} \\
0 & 0 & d_{33} \\
0 & d_{24} & 0 \\
d_{15} & 0 & 0 \\
0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix} \\[8pt]
& \begin{bmatrix} D_1 \\ D_2 \\ D_3 \end{bmatrix}
=
\begin{bmatrix} 0 & 0 & 0 & 0 & d_{15} & 0 \\
0 & 0 & 0 & d_{24} & 0 & 0 \\
d_{31} & d_{32} & d_{33} & 0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix}
+
\begin{bmatrix} {\varepsilon}_{11} & 0 & 0 \\
0 & {\varepsilon}_{22} & 0 \\
0 & 0 & {\varepsilon}_{33} \end{bmatrix}
\begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix}
\end{align}
</math>
where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.<ref name="DD1998">{{cite journal |last=Damjanovic |first=Dragan |year=1998 |title=Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics |journal=Reports on Progress in Physics |volume=61 |pages=1267–1324 |doi=10.1088/0034-4885/61/9/002|bibcode = 1998RPPh...61.1267D |issue=9 }}</ref>

Although the above equations are the most used form in literature, some comments about the notation are necessary. Generally, ''D'' and ''E'' are [[Vector (geometric)|vectors]], that is, [[Cartesian tensor]]s of rank 1; and permittivity ''ε'' is a Cartesian tensor of rank 2. Strain and stress are, in principle, also rank-2 [[tensors]]. But conventionally, because strain and stress are all symmetric tensors, the subscript of strain and stress can be relabeled in the following fashion: 11&nbsp;→&nbsp;1; 22&nbsp;→&nbsp;2; 33&nbsp;→&nbsp;3; 23&nbsp;→&nbsp;4; 13&nbsp;→&nbsp;5; 12&nbsp;→&nbsp;6. (Different conventions may be used by different authors in literature. For example, some use 12&nbsp;→&nbsp;4; 23&nbsp;→&nbsp;5; 31&nbsp;→&nbsp;6 instead.) That is why ''S'' and ''T'' appear to have the "vector form" of six components. Consequently, ''s'' appears to be a 6-by-6 matrix instead of a rank-3 tensor. Such a relabeled notation is often called [[Voigt notation]]. Whether the shear strain components ''S''<sub>4</sub>, ''S''<sub>5</sub>, ''S''<sub>6</sub> are tensor components or engineering strains is another question. In the equation above, they must be engineering strains for the 6,6 coefficient of the compliance matrix to be written as shown, i.e., 2(''s''{{su|b=11|p=''E''}}&nbsp;−&nbsp;''s''{{su|b=12|p=''E''}}). Engineering shear strains are double the value of the corresponding tensor shear, such as ''S''<sub>6</sub>&nbsp;=&nbsp;2''S''<sub>12</sub> and so on. This also means that ''s''<sub>66</sub>&nbsp;=&nbsp;{{sfrac|1|''G''<sub>12</sub>}}, where ''G''<sub>12</sub> is the [[shear modulus]].

In total, there are four piezoelectric coefficients, ''d<sub>ij</sub>'', ''e<sub>ij</sub>'', ''g<sub>ij</sub>'', and ''h<sub>ij</sub>'' defined as follows:

:<math>\begin{align}
d_{ij} &= \phantom{+} \left ( \frac{\partial D_i}{\partial T_j} \right )^E
&&= \phantom{+} \left ( \frac{\partial S_j}{\partial E_i} \right )^T \\[6pt]
e_{ij} &= \phantom{+} \left ( \frac{\partial D_i}{\partial S_j} \right )^E
&&= -\left ( \frac{\partial T_j}{\partial E_i} \right )^S \\[6pt]
g_{ij} &= -\left ( \frac{\partial E_i}{\partial T_j} \right )^D
&&= \phantom{+} \left ( \frac{\partial S_j}{\partial D_i} \right )^T \\[6pt]
h_{ij} &= -\left ( \frac{\partial E_i}{\partial S_j} \right )^D
&&= -\left ( \frac{\partial T_j}{\partial D_i} \right )^S
\end{align}</math>
where the first set of four terms corresponds to the direct piezoelectric effect and the second set of four terms corresponds to the converse piezoelectric effect. The equality between the direct piezoelectric tensor and the transpose of the converse piezoelectric tensor originates from the [[Maxwell relations]] of thermodynamics.<ref>{{cite journal |last=Kochervinskii |first=V. |year=2003 |title=Piezoelectricity in Crystallizing Ferroelectric Polymers |journal=[[Crystallography Reports]] |volume=48 |issue= 4 |pages=649–675|doi=10.1134/1.1595194|bibcode = 2003CryRp..48..649K }}</ref> For those piezoelectric crystals for which the polarization is of the crystal-field induced type, a formalism has been worked out that allows for the calculation of piezoelectrical coefficients ''d<sub>ij</sub>'' from electrostatic lattice constants or higher-order [[Madelung constant]]s.<ref name="ZPB1995a"/>