View source for Piezoelectricity

{{Short description|Electric charge generated in certain solids due to mechanical stress}}
[[File:Piezoelectric balance presented by Pierre Curie to Lord Kelvin, Hunterian Museum, Glasgow.jpg|thumb|upright=1.1|Piezoelectric balance presented by [[Pierre Curie]] to [[Lord Kelvin]], [[Hunterian Museum and Art Gallery|Hunterian Museum]], [[Glasgow]]]]

'''Piezoelectricity''' ({{IPAc-en|ˌ|p|iː|z|oʊ|-|,_|ˌ|p|iː|t|s|oʊ|-|,_|p|aɪ|ˌ|iː|z|oʊ|-}}, {{IPAc-en|us|p|i|ˌ|eɪ|z|oʊ|-|,_|p|i|ˌ|eɪ|t|s|oʊ|-}})<ref>{{cite LPD|3}}</ref> is the [[electric charge]] that accumulates in certain [[solid]] materials—such as [[crystal]]s, certain [[ceramic]]s, and biological matter such as [[bone]], [[DNA]], and various [[protein]]s—in response to applied [[stress (mechanics)|mechanical stress]].<ref name='InstrumentAnalysis'>{{cite book |title=Principles of Instrumental Analysis |edition=6th |date=2007 |first1=F. James|last1=Holler |last2=Skoog|first2=Douglas A. |last3=Crouch|first3=Stanley R. |name-list-style=amp |page=9 |publisher=[[Cengage]] Learning |isbn=978-0-495-01201-6}}</ref> The word ''piezoelectricity'' means [[electricity]] resulting from [[pressure]] and [[latent heat]]. It is derived {{etymology|grc|''{{wikt-lang|grc|πιέζω}}'' ({{grc-transl|πιέζω}})|to squeeze or press||''{{wikt-lang|grc|ἤλεκτρον}}'' ({{grc-transl|ἤλεκτρον}})|[[amber]]}} (an ancient source of static electricity).<ref>{{OEtymD|piezoelectric}}</ref><ref>{{LSJ|pie/zw|πιέζειν}}, {{LSJ|h)/lektron|ἤλεκτρον|ref}}.</ref>  The German form of the word (''Piezoelektricität'') was coined in 1881 by the German physicist [[Wilhelm Gottlieb Hankel]]; the English word was coined in 1883.<ref>{{OEtymD|piezoelectric}}</ref><ref>{{cite journal |last1=Hankel |first1=W. G. |title=Elektrische Untersuchungen. Fünfzehnte Abhandlung. Über die aktino- und piezoelektrischen Eigenschaften des Bergkrystalles und ihre Beziehung zu den thermoelektrischen. |journal=Abhandlungen der Mathematisch-Physischen Klasse der Königlichen-Säschsischen Gesellschaft der Wissenschaften |date=1881 |volume=12 |pages=459–547 |trans-title=Electrical researches.  Fifteenth treatise.  On the radiative- and piezoelectric properties of rock crystal [i.e., quartz] and their relation to the thermoelectric [ones]. |language=German}}  From p. 462:  ''"Da die durch Druck erzeugte Elektricität sonach auch besonderen Gesetzen unterliegt, so wird es angemessen sein, derselben gleichfalls einen besonderen Namen beizulegen, und es dürfte sich dazu die Bezeichnung Piezoelektricität eignen."'' (Since the electricity [which is] generated by pressure is therefore also subject to special laws, then it will likewise be appropriate to give it a special name, and for this, the designation "piezoelectricity" might be suitable.)
* The above article was also published separately as a pamphlet: {{cite book |last1=Hankel |first1=W. G. |title=Elektrische Untersuchungen. Fünfzehnte Abhandlung. Über die aktino- und piezoelektrischen Eigenschaften des Bergkrystalles und ihre Beziehung zu den thermoelektrischen. |date=1881 |publisher=S. Hirzel |location=Leipzig, Germany |url=https://books.google.com/books?id=kXXWkyYs8cEC&pg=PA455 |language=German}}  See p. 462.</ref> 

The piezoelectric effect results from the linear [[electromechanical]] interaction between the mechanical and electrical states in crystalline materials with no [[centrosymmetry|inversion symmetry]].<ref name='Piezoelectric Sensorics'>{{cite book |title=Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and Acoustic Emission Sensors, Materials and Amplifiers |isbn=978-3-662-04732-3 |doi=10.1007/978-3-662-04732-3 |date=2002 |first=G.|last=Gautschi |publisher=[[Springer Science+Business Media|Springer]]}}</ref> The piezoelectric effect is a [[reversible process (thermodynamics)|reversible process]]:  [[List of piezoelectric materials|materials exhibiting the piezoelectric effect]]  also exhibit the reverse piezoelectric effect, the internal generation of a mechanical strain resulting from an applied [[electric field]]. For example, [[lead zirconate titanate]] crystals will generate measurable piezoelectricity when their static structure is [[Deformation (physics)|deformed]] by about 0.1% of the original dimension. Conversely, those same crystals will change about 0.1% of their static dimension when an external electric field is applied. The inverse piezoelectric effect is used in the production of [[Ultrasound|ultrasound wave]]s.<ref name='UT of Materials'>{{cite book
|title=Ultrasonic Testing of Materials |date=1990
|last1=Krautkrämer |first1=J. |last2=Krautkrämer |first2=H. |name-list-style=amp
|publisher=Springer |isbn=978-3-662-10680-8 |pages=119–49}}</ref>

French physicists [[Jacques Curie|Jacques]] and [[Pierre Curie]] discovered piezoelectricity in 1880.<ref name="Manbachi, A. and Cobbold R.S.C. 2011 187–96">{{cite journal | last1=Manbachi |first1=A. |last2=Cobbold |first2=R.S.C. |name-list-style=amp |title= Development and Application of Piezoelectric Materials for Ultrasound Generation and Detection |journal= Ultrasound |volume=19 |issue=4 |pages=187–96 |date=2011|doi= 10.1258/ult.2011.011027|s2cid=56655834 }}</ref> The piezoelectric effect has been exploited in many useful applications, including the production and detection of sound, piezoelectric [[inkjet printing]], generation of high voltage electricity, as a [[clock generator]] in electronic devices, in [[microbalance]]s, to drive an [[ultrasonic nozzle]], and in ultrafine focusing of optical assemblies. It forms the basis for [[scanning probe microscopy|scanning probe microscopes]] that resolve images at the scale of [[atom]]s. It is used in the [[Pickup (music technology)|pickups]] of some [[Guitar amplifier|electronically amplified guitars]] and as [[Trigger (drums)|triggers]] in most modern [[electronic drum]]s.<ref>{{Cite web |url=https://www.dawsons.co.uk/blog/how-do-electronic-drums-work |title=How Do Electronic Drums Work? |first=Joe |last=Dempsey |date=June 24, 2020 |website=Studio D |access-date=July 27, 2021 |archive-date=October 1, 2019 |archive-url=https://web.archive.org/web/20191001070929/https://www.dawsons.co.uk/blog/how-do-electronic-drums-work |url-status=dead }}</ref><ref>{{Cite web |url=https://www.sparkfun.com/tutorials/330 |title=Piezo Drum Kit Quickstart Guide |first = C. |last=Taylor |date=October 31, 2011 |website=SparkFun Electronics |access-date=July 27, 2021}}</ref> The piezoelectric effect also finds everyday uses, such as generating sparks to ignite gas cooking and heating devices, torches, and [[lighters|cigarette lighters]].

==History==
===Discovery and early research===
The [[pyroelectricity|pyroelectric effect]], by which a material generates an [[electric potential]] in response to a temperature change, was studied by [[Carl Linnaeus]] and [[Franz Aepinus]] in the mid-18th century. Drawing on this knowledge, both [[René Just Haüy]] and [[Antoine César Becquerel]] posited a relationship between mechanical stress and electric charge; however, experiments by both proved inconclusive.<ref>{{cite web|last=Erhart |first=Jiří |url=https://moodle.fp.tul.cz/nano/pluginfile.php/2476/mod_resource/content/3/FPM_Piezo_lecture1.pdf |title=Piezoelectricity and ferroelectricity: Phenomena and properties |publisher=Department of Physics, Technical University of Liberec |url-status=unfit |archive-url=https://web.archive.org/web/20140508030117/https://moodle.fp.tul.cz/nano/pluginfile.php/2476/mod_resource/content/3/FPM_Piezo_lecture1.pdf |archive-date=May 8, 2014 }}</ref>

[[File:Top view of Curie piezo electric compensator.jpg|thumb|View of piezo crystal in the top of a Curie compensator in the Museum of Scotland.]]
The first demonstration of the direct piezoelectric effect was in 1880 by the brothers [[Pierre Curie]] and [[Jacques Curie]].<ref>{{cite journal|first1=Jacques|last1=Curie|author1-link=Jacques Curie |first2=Pierre |last2=Curie|author2-link=Pierre Curie |date=1880 |title=Développement par compression de l'électricité polaire dans les cristaux hémièdres à faces inclinées |trans-title=Development, via compression, of electric polarization in hemihedral crystals with inclined faces |journal=Bulletin de la Société Minérologique de France |volume=3 |issue=4|pages=90–93|doi=10.3406/bulmi.1880.1564}}<br />
Reprinted in: {{cite journal |first1=Jacques |last1=Curie |author1-link=Jacques Curie |first2=Pierre |last2=Curie |author2-link=Pierre Curie |date=1880 |url=http://gallica.bnf.fr/ark:/12148/bpt6k30485/f296.image |title=Développement, par pression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées |journal=Comptes Rendus |volume=91 |pages=294–295 |language=fr |url-status=live |archive-url=https://web.archive.org/web/20121205083302/http://gallica.bnf.fr/ark:/12148/bpt6k30485/f296.image |archive-date=2012-12-05 }}<br />
See also: {{cite journal|first1=Jacques|last1=Curie|author1-link=Jacques Curie|first2=Pierre|last2=Curie|author2-link=Pierre Curie|date=1880|url=http://gallica.bnf.fr/ark:/12148/bpt6k30485/f385.image|title=Sur l'électricité polaire dans les cristaux hémièdres à faces inclinées|trans-title=On electric polarization in hemihedral crystals with inclined faces|journal=Comptes Rendus|volume=91|pages=383–386|language=fr|url-status=live|archive-url=https://web.archive.org/web/20121205090430/http://gallica.bnf.fr/ark:/12148/bpt6k30485/f385.image|archive-date=2012-12-05}}</ref> They combined their knowledge of pyroelectricity with their understanding of the underlying crystal structures that gave rise to pyroelectricity to predict crystal behavior, and demonstrated the effect using crystals of [[tourmaline]], [[quartz]], [[topaz]], [[sugar cane|cane]] [[sugar]], and [[Rochelle salt]] (sodium potassium tartrate tetrahydrate). Quartz and Rochelle salt exhibited the most piezoelectricity.

[[Image:SchemaPiezo.gif|thumb|A piezoelectric disk generates a voltage when deformed (change in shape is greatly exaggerated).]]

The Curies, however, did not predict the converse piezoelectric effect. The converse effect was mathematically deduced from fundamental thermodynamic principles by [[Gabriel Lippmann]] in 1881.<ref>{{Cite journal|first=G.|last=Lippmann|title=Principe de la conservation de l'électricité|trans-title=Principle of the conservation of electricity|journal=[[Annales de chimie et de physique]]|volume=24|page=145|url=http://gallica.bnf.fr/ark:/12148/bpt6k348640|year=1881|language=fr|url-status=live|archive-url=https://web.archive.org/web/20160208081244/http://gallica.bnf.fr/ark:/12148/bpt6k348640|archive-date=2016-02-08}}</ref> The Curies immediately confirmed the existence of the converse effect,<ref>{{cite journal|first1=Jacques|last1=Curie|author1-link=Jacques Curie|first2=Pierre|last2=Curie|author2-link=Pierre Curie|date=1881|url=http://gallica.bnf.fr/ark:/12148/bpt6k3049g/f1131.image|title=Contractions et dilatations produites par des tensions dans les cristaux hémièdres à faces inclinées|trans-title=Contractions and expansions produced by voltages in hemihedral crystals with inclined faces|journal=Comptes Rendus|volume=93|pages=1137–1140|language=fr|url-status=live|archive-url=https://web.archive.org/web/20121205084840/http://gallica.bnf.fr/ark:/12148/bpt6k3049g/f1131.image|archive-date=2012-12-05}}</ref> and went on to obtain quantitative proof of the complete reversibility of electro-elasto-mechanical deformations in piezoelectric crystals.

For the next few decades, piezoelectricity remained something of a laboratory curiosity, though it was a vital tool in the discovery of polonium and radium by Pierre and [[Marie Curie]] in 1898. More work was done to explore and define the crystal structures that exhibited piezoelectricity. This culminated in 1910 with the publication of [[Woldemar Voigt]]'s ''Lehrbuch der Kristallphysik'' (''Textbook on Crystal Physics''),<ref>{{cite book |first=Woldemar |last=Voigt |author-link=Woldemar Voigt |url=https://books.google.com/books?id=SvPPAAAAMAAJ&pg=PR1 |title=Lehrbuch der Kristallphysik |location=Berlin |publisher=B. G. Teubner |date=1910 |url-status=live |archive-url=https://web.archive.org/web/20140421051401/http://books.google.com/books?id=SvPPAAAAMAAJ&pg=PR1 |archive-date=2014-04-21 }}</ref> which described the 20 natural crystal classes capable of piezoelectricity, and rigorously defined the piezoelectric constants using [[tensor analysis]].

===World War I and inter-war years===
The first practical application for piezoelectric devices was [[sonar]], first developed during [[World War I]]. In [[France]] in 1917, [[Paul Langevin]] and his coworkers developed an [[ultrasound|ultrasonic]] [[submarine]] detector.<ref>{{cite journal |title= Who knew piezoelectricity? Rutherford and Langevin on submarine detection and the invention of sonar |last= Katzir |first= S. |journal= Notes Rec. R. Soc. |date= 2012 |volume= 66 |issue= 2 |pages= 141–157 |doi= 10.1098/rsnr.2011.0049 |doi-access= free }}</ref> The detector consisted of a [[transducer]], made of thin quartz crystals carefully glued between two steel plates, and a [[hydrophone]] to detect the returned [[Echo (phenomenon)|echo]]. By emitting a high-frequency pulse from the transducer, and measuring the amount of time it takes to hear an echo from the sound waves bouncing off an object, one can calculate the distance to that object.

The use of piezoelectricity in sonar, and the success of that project, created intense development interest in piezoelectric devices. Over the next few decades, new piezoelectric materials and new applications for those materials were explored and developed.

Piezoelectric devices found homes in many fields. Ceramic [[phonograph]] cartridges simplified player design, were cheap and accurate, and made record players cheaper to maintain and easier to build. The development of the ultrasonic transducer allowed for easy measurement of viscosity and elasticity in fluids and solids, resulting in huge advances in materials research. Ultrasonic [[time-domain reflectometer]]s (which send an ultrasonic pulse through a material and measure reflections from discontinuities) could find flaws inside cast metal and stone objects, improving structural safety.

===World War II and post-war===
During [[World War II]], independent research groups in the [[United States]], [[USSR]], and [[Japan]] discovered a new class of synthetic materials, called [[ferroelectricity|ferroelectrics]], which exhibited piezoelectric constants many times higher than natural materials. This led to intense research to develop [[barium titanate]] and later lead zirconate titanate materials with specific properties for particular applications.

One significant example of the use of piezoelectric crystals was developed by [[Bell Telephone Laboratories]]. Following World War I, Frederick R. Lack, working in radio telephony in the engineering department, developed the "AT cut" crystal, a crystal that operated through a wide range of temperatures. Lack's crystal did not need the heavy accessories previous crystal used, facilitating its use on aircraft. This development allowed Allied air forces to engage in coordinated mass attacks through the use of aviation radio.

Development of piezoelectric devices and materials in the United States was kept within the companies doing the development, mostly due to the wartime beginnings of the field, and in the interests of securing profitable patents. New materials were the first to be developed—quartz crystals were the first commercially exploited piezoelectric material, but scientists searched for higher-performance materials. Despite the advances in materials and the maturation of manufacturing processes, the United States market did not grow as quickly as Japan's did. Without many new applications, the growth of the United States' piezoelectric industry suffered.

In contrast, Japanese manufacturers shared their information, quickly overcoming technical and manufacturing challenges and creating new markets. In Japan, a temperature stable crystal cut was developed by [[Issac Koga]]. Japanese efforts in materials research created piezoceramic materials competitive to the United States materials but free of expensive patent restrictions. Major Japanese piezoelectric developments included new designs of piezoceramic filters for radios and televisions, piezo buzzers and audio transducers that can connect directly to electronic circuits, and the [[piezo ignition|piezoelectric igniter]], which generates sparks for small engine ignition systems and gas-grill lighters, by compressing a ceramic disc. Ultrasonic transducers that transmit sound waves through air had existed for quite some time but first saw major commercial use in early television remote controls. These transducers now are mounted on several [[automobile|car]] models as an [[Acoustic location|echolocation]] device, helping the driver determine the distance from the car to any objects that may be in its path.

==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|doi = 10.1007/BF01313055|bibcode = 1995ZPhyB..96..333B|url = https://www.researchgate.net/publication/226272268|issue = 3|s2cid = 122393358|url-status = live|archive-url = https://web.archive.org/web/20161030075927/https://www.researchgate.net/publication/226272268_Crystal-field_induced_dipoles_in_heteropolar_crystals_II_physical_significance|archive-date = 2016-10-30}}</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 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.

-->

===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 |s2cid=250873563 }}</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 |s2cid=95995717 }}</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"/>

==Crystal classes==

[[File:Capacitor schematic with dielectric.svg|thumb|Any spatially separated charge will result in an [[electric field]], and therefore an [[electric potential]]. Shown here is a standard dielectric in a [[capacitor]]. In a piezoelectric device, mechanical stress, instead of an externally applied voltage, causes the charge separation in the individual atoms of the material.]]

Of the 32 [[crystal class]]es, 21 are non-[[centrosymmetric]] (not having a centre of symmetry), and of these, 20 exhibit direct piezoelectricity<ref>{{cite web|title=Piezoelectric Crystal Classes|url=https://www.staff.ncl.ac.uk/j.p.goss/symmetry/PP_Piezo.html|publisher=Newcastle University, UK|access-date=8 March 2015|url-status=live|archive-url=https://web.archive.org/web/20150402160441/https://www.staff.ncl.ac.uk/j.p.goss/symmetry/PP_Piezo.html|archive-date=2 April 2015}}</ref> (the 21st is the cubic class 432). Ten of these represent the polar crystal classes,<ref>{{cite web|title=Pyroelectric Crystal Classes|url=https://www.staff.ncl.ac.uk/j.p.goss/symmetry/PP_Pyro.html|publisher=Newcastle University, UK|access-date=8 March 2015|url-status=live|archive-url=https://web.archive.org/web/20150402183459/https://www.staff.ncl.ac.uk/j.p.goss/symmetry/PP_Pyro.html|archive-date=2 April 2015}}</ref> which show a spontaneous polarization without mechanical stress due to a non-vanishing electric dipole moment associated with their unit cell, and which exhibit [[pyroelectricity]]. If the dipole moment can be reversed by applying an external electric field, the material is said to be [[ferroelectric]].
* The 10 polar (pyroelectric) crystal classes: 1, 2, m, mm2, 4, {{Not a typo|4mm}}, 3, 3m, 6, {{Not a typo|6mm}}.
* The other 10 piezoelectric crystal classes: 222, {{overline|4}}, 422, {{overline|4}}2m, 32, {{overline|6}}, 622, {{overline|6}}2m, 23, {{overline|4}}3m.

For polar crystals, for which '''''P'''''&nbsp;≠&nbsp;0 holds without applying a mechanical load, the piezoelectric effect manifests itself by changing the magnitude or the direction of '''''P''''' or both.

For the nonpolar but piezoelectric crystals, on the other hand, a polarization '''''P''''' different from zero is only elicited by applying a mechanical load. For them the stress can be imagined to transform the material from a nonpolar crystal class ('''''P'''''&nbsp;=&nbsp;0) to a polar one,<ref name="ZPB1995a"/> having '''''P'''''&nbsp;≠&nbsp;0.

==Materials==<!-- This section is linked from [[Crystal Oscillator]] -->
{{See also|List of piezoelectric materials}}
Many materials exhibit piezoelectricity.
<!--===Bone===
Dry [[bone]] exhibits some [[Piezoelectric material properties|piezoelectric properties]]. Studies of Fukada ''et al.'' showed that these are not due to the [[apatite]] crystals, which are centrosymmetric, thus non-piezoelectric, but due to [[collagen]]. Collagen exhibits the polar uniaxial orientation of molecular dipoles in its structure and can be considered as bioelectret, a sort of dielectric material exhibiting quasipermanent space charge and dipolar charge. Potentials are thought to occur when a number of collagen molecules are stressed in the same way displacing significant numbers of the charge carriers from the inside to the surface of the specimen. Piezoelectricity of single individual collagen fibrils was measured using piezoresponse force microscopy, and it was shown that collagen fibrils behave predominantly as shear piezoelectric materials.<ref>{{cite journal |volume=20 |year=2009|page=085706|last1=Minary-Jolandan |first1=M. |last2=Yu |first2=Min-Feng |name-list-style=amp |journal=Nanotechnology |doi=10.1088/0957-4484/20/8/085706 |pmid=19417467 |title=Nanoscale characterization of isolated individual type I collagen fibrils: Polarization and piezoelectricity |issue=8|bibcode = 2009Nanot..20h5706M }}</ref>

The piezoelectric effect is generally thought to act as a biological force sensor.<ref>{{cite web |url=http://silver.neep.wisc.edu/~lakes/BoneElectr.html |title=Electrical Properties of Bone: A Review |first=Roderic |last=Lakes |publisher=[[University of Wisconsin–Madison]] |url-status=live |archive-url=https://web.archive.org/web/20131105105946/http://silver.neep.wisc.edu/~lakes/BoneElectr.html |archive-date=2013-11-05 }}</ref><ref>{{cite book |first=Robert O. |last=Becker |last2=Marino |first2=Andrew A. |title=Electromagnetism & Life |chapter=Chapter 4: Electrical Properties of Biological Tissue (Piezoelectricity) |publisher=State University of New York Press |location=Albany, NY |isbn=978-0-87395-560-7 |chapter-url=http://www.ortho.lsuhsc.edu/Faculty/Marino/EL/EL4/Piezo.html |year=1982 |archive-url=https://web.archive.org/web/20090802194455/http://www.ortho.lsuhsc.edu/Faculty/Marino/EL/EL4/Piezo.html |archive-date=2009-08-02 }}</ref> This effect was exploited by research conducted at the [[University of Pennsylvania]] in the late 1970s and early 1980s, which established that sustained application of electrical potential could stimulate both resorption and growth (depending on the polarity) of bone ''in vivo''.<ref>{{cite book|last1=Pollack |first1=S. R. |last2=Korostoff |first2=E. |last3=Starkebaum |first3=W. |last4=Lannicone |first4=W. |year=1979 |chapter=Micro-Electrical Studies of Stress-Generated Potentials in Bone|isbn=978-0-8089-1228-6 |title=Electrical Properties of Bone and Cartilage |editor1-last=Brighton |editor1-first=C. T. |editor2-last=Black |editor2-first=J. |editor3-last=Pollack |editor3-first=S. R. |publisher=Grune & Stratton|location=New York}}</ref> Further studies in the 1990s provided the mathematical equation to confirm long bone wave propagation as to that of hexagonal (Class 6) crystals.<ref>{{cite journal |last=Fotiadis |first=D. I. |last2=Foutsitzi |first2=G. |last3=Massalas |first3=C. V. |year=1999 |title=Wave propagation modeling in human long bones |journal=Acta Mechanica |volume=137 |issue=1–2 |pages=65–81 |doi=10.1007/BF01313145}}</ref>-->

===Crystalline materials===
* [[Langasite]] (La<sub>3</sub>Ga<sub>5</sub>SiO<sub>14</sub>)&nbsp;– a quartz-analogous crystal
* [[Gallium orthophosphate]] (GaPO<sub>4</sub>)&nbsp;– a quartz-analogous crystal
* [[Lithium niobate]] (LiNbO<sub>3</sub>)
* [[Lithium tantalate]] (LiTaO<sub>3</sub>)
* [[Quartz]]
* [[Berlinite]] (AlPO<sub>4</sub>)&nbsp;– a rare [[phosphate]] [[mineral]] that is structurally identical to quartz
* [[Potassium sodium tartrate|Rochelle salt]]
* [[Topaz|Topaz&nbsp;–]] Piezoelectricity in Topaz can probably be attributed to ordering of the (F,OH) in its lattice, which is otherwise centrosymmetric: orthorhombic bipyramidal (mmm). Topaz has anomalous optical properties which are attributed to such ordering.<ref>{{cite journal|last1=Akizuki |first1=Mizuhiko |first2=Martin S. |last2=Hampar |first3=Jack |last3=Zussman |year=1979|title=An explanation of anomalous optical properties of topaz|journal= Mineralogical Magazine |volume=43|pages= 237–241|url=http://rruff.geo.arizona.edu/doclib/MinMag/Volume_43/43-326-237.pdf|doi=10.1180/minmag.1979.043.326.05|issue=326|bibcode=1979MinM...43..237A|citeseerx=10.1.1.604.6025 |s2cid=4945694 }}</ref>
* [[Tourmaline|Tourmaline-group minerals]]
* [[Lead titanate]] (PbTiO<sub>3</sub>)&nbsp;– Although it occurs in nature as mineral macedonite,<ref>{{cite journal |last1=Radusinović |first1=Dušan |last2=Markov |first2=Cvetko |name-list-style=amp |year=1971 |title=Macedonite – lead titanate: a new mineral |journal=American Mineralogist |volume=56 |pages=387–394 |url=http://www.minsocam.org/ammin/AM56/AM56_387.pdf |url-status=live |archive-url=https://web.archive.org/web/20160305091225/http://www.minsocam.org/ammin/AM56/AM56_387.pdf |archive-date=2016-03-05 }}</ref><ref>{{cite journal|last1=Burke |first1=E. A. J. |last2=Kieft |first2=C. |name-list-style=amp |year=1971|title=Second occurrence of makedonite, PbTiO<sub>3</sub>, Långban, Sweden|journal=Lithos|doi= 10.1016/0024-4937(71)90102-2 |volume=4|issue=2 |pages=101–104|bibcode = 1971Litho...4..101B }}</ref> it is synthesized for research and applications.

===Ceramics===
[[Image:Perovskite.svg|thumb|Tetragonal unit cell of lead titanate]]
Ceramics with randomly oriented grains must be ferroelectric to exhibit piezoelectricity.<ref>{{cite book|first1=B. |last1=Jaffe |first2=W. R. |last2=Cook |first3=H. |last3=Jaffe |title=Piezoelectric Ceramics |location=New York |publisher=Academic |date=1971}}{{ISBN missing}}</ref> The occurrence of [[abnormal grain growth]] (AGG) in sintered polycrystalline piezoelectric ceramics has detrimental effects on the piezoelectric performance in such systems and should be avoided, as the microstructure in piezoceramics exhibiting AGG tends to consist of few abnormally large elongated grains in a matrix of randomly oriented finer grains. Macroscopic piezoelectricity is possible in textured polycrystalline non-ferroelectric piezoelectric materials, such as AlN and ZnO.
The families of ceramics with [[Perovskite (structure)|perovskite]], [[tungsten]]-[[bronze]], and related structures exhibit piezoelectricity:
* [[Lead zirconate titanate]] ({{chem2|auto=1|Pb&#91;Zr_{''x''}Ti_{1−''x''}]O3}} with 0&nbsp;≤&nbsp;''x''&nbsp;≤&nbsp;1)&nbsp;– more commonly known as PZT, the most common piezoelectric ceramic in use today.
* [[Potassium niobate]] (KNbO<sub>3</sub>)<ref>{{Cite journal|last1=Ganeshkumar|first1=Rajasekaran|last2=Somnath|first2=Suhas|last3=Cheah|first3=Chin Wei|last4=Jesse|first4=Stephen|last5=Kalinin|first5=Sergei V.|last6=Zhao|first6=Rong|date=2017-12-06|title=Decoding Apparent Ferroelectricity in Perovskite Nanofibers|journal=ACS Applied Materials & Interfaces|volume=9|issue=48|pages=42131–42138|doi=10.1021/acsami.7b14257|pmid=29130311|issn=1944-8244}}</ref>
* [[Sodium tungstate]] (Na<sub>2</sub>WO<sub>3</sub>)
* Ba<sub>2</sub>NaNb<sub>5</sub>O<sub>5</sub>
* Pb<sub>2</sub>KNb<sub>5</sub>O<sub>15</sub>
* [[Zinc oxide]] (ZnO)&nbsp;– [[Wurtzite crystal structure|Wurtzite structure]]. While single crystals of ZnO are piezoelectric and pyroelectric, polycrystalline (ceramic) ZnO with randomly oriented grains exhibits neither piezoelectric nor pyroelectric effect. Not being ferroelectric, polycrystalline ZnO cannot be poled like barium titanate or PZT. Ceramics and polycrystalline thin films of ZnO may exhibit macroscopic piezoelectricity and pyroelectricity only if they are [[Texture (crystalline)|textured]] (grains are preferentially oriented), such that the piezoelectric and pyroelectric responses of all individual grains do not cancel. This is readily accomplished in polycrystalline thin films.<ref name="DD1998" />

===Lead-free piezoceramics===
* [[Sodium potassium niobate]] ((K,Na)NbO<sub>3</sub>). This material is also known as NKN or KNN. In 2004, a group of Japanese researchers led by Yasuyoshi Saito discovered a sodium potassium niobate composition with properties close to those of PZT, including a high ''T''<sub>C</sub>.<ref>{{cite journal |title=Lead-free piezoceramics |first1=Yasuyoshi |last1=Saito |last2=Takao |first2=Hisaaki |last3=Tanil |first3=Toshihiko |last4=Nonoyama |first4=Tatsuhiko |last5=Takatori |first5=Kazumasa |last6=Homma |first6=Takahiko |last7=Nagaya |first7=Toshiatsu |last8=Nakamura |first8=Masaya |journal=[[Nature (journal)|Nature]] |volume=432 |issue=7013 |pages=81–87 |doi=10.1038/nature03028 |pmid=15516921 |date=2004-11-04 |bibcode=2004Natur.432...84S |s2cid=4352954 }}</ref> Certain compositions of this material have been shown to retain a high mechanical quality factor (''Q''<sub>m</sub>&nbsp;≈&nbsp;900) with increasing vibration levels, whereas the mechanical quality factor of hard PZT degrades in such conditions. This fact makes NKN a promising replacement for high power resonance applications, such as piezoelectric transformers.<ref name="GurdalUral2011">{{cite journal|last1=Gurdal|first1=Erkan A.|last2=Ural|first2=Seyit O.|last3=Park|first3=Hwi-Yeol|last4=Nahm|first4=Sahn|last5=Uchino|first5=Kenji|title=High Power (Na<sub>0.5</sub>K<sub>0.5</sub>)NbO<sub>3</sub>-Based Lead-Free Piezoelectric Transformer|journal=Japanese Journal of Applied Physics|volume=50|issue=2|year=2011|page=027101|issn=0021-4922|doi=10.1143/JJAP.50.027101|bibcode = 2011JaJAP..50b7101G |s2cid=123625588 }}</ref>
* [[Bismuth ferrite]] (BiFeO<sub>3</sub>) &nbsp;– a promising candidate for the replacement of lead-based ceramics.
* Sodium niobate (NaNbO<sub>3</sub>)
* [[Barium titanate]] (BaTiO<sub>3</sub>)&nbsp;– Barium titanate was the first piezoelectric ceramic discovered.
* [[Bismuth titanate]] (Bi<sub>4</sub>Ti<sub>3</sub>O<sub>12</sub>)
* [[Sodium bismuth titanate]] (NaBi(TiO<sub>3</sub>)<sub>2</sub>)
The fabrication of lead-free piezoceramics pose multiple challenges, from an environmental standpoint and their ability to replicate the properties of their lead-based counterparts. By removing the lead component of the piezoceramic, the risk of toxicity to humans decreases, but the mining and extraction of the materials can be harmful to the environment.<ref>Ibn-Mohammed, T., Koh, S., Reaney, I., Sinclair, D., Mustapha, K., Acquaye, A., & Wang, D. (2017). "Are lead-free piezoelectrics more environmentally friendly?" ''MRS Communications'', '''7'''(1), 1-7. doi: 10.1557/mrc.2017.10</ref> Analysis of the environmental profile of PZT versus sodium potassium niobate (NKN or KNN) shows that across the four indicators considered (primary energy consumption, toxicological footprint, eco-indicator 99, and input-output upstream greenhouse gas emissions), KNN is actually more harmful to the environment. Most of the concerns with KNN, specifically its Nb<sub>2</sub>O<sub>5</sub> component, are in the early phase of its life cycle before it reaches manufacturers. Since the harmful impacts are focused on these early phases, some actions can be taken to minimize the effects. Returning the land as close to its original form after Nb<sub>2</sub>O<sub>5</sub> mining via dam deconstruction or replacing a stockpile of utilizable soil are known aids for any extraction event. For minimizing air quality effects, modeling and simulation still needs to occur to fully understand what mitigation methods are required. The extraction of lead-free piezoceramic components has not grown to a significant scale at this time, but from early analysis, experts encourage caution when it comes to environmental effects.

Fabricating lead-free piezoceramics faces the challenge of maintaining the performance and stability of their lead-based counterparts. In general, the main fabrication challenge is creating the "morphotropic phase boundaries (MPBs)" that provide the materials with their stable piezoelectric properties without introducing the "polymorphic phase boundaries (PPBs)" that decrease the temperature stability of the material.<ref>Wu, Jiagang. (2020). "Perovskite lead-free piezoelectric ceramics." ''Journal of Applied Physics'', '''127''' (19). doi: 10.1063/5.0006261</ref> New phase boundaries are created by varying additive concentrations so that the phase transition temperatures converge at room temperature. The introduction of the MPB improves piezoelectric properties, but if a PPB is introduced, the material becomes negatively affected by temperature. Research is ongoing to control the type of phase boundaries that are introduced through phase engineering, diffusing phase transitions, domain engineering, and chemical modification.

===III–V and II–VI semiconductors===
A piezoelectric potential can be created in any bulk or nanostructured semiconductor crystal having non central symmetry, such as the [[List of semiconductor materials#Types of semiconductor materials|Group]] [[Boron group|III]]–[[Nitrogen group|V]] and [[Group 12 element|II]]–[[Chalcogen|VI]] materials, due to polarization of ions under applied stress and strain. This property is common to both the [[zincblende]] and [[wurtzite]] crystal structures. To first order, there is only one independent piezoelectric coefficient in [[zincblende]], called e<sub>14</sub>, coupled to shear components of the strain. In [[wurtzite]], there are instead three independent piezoelectric coefficients: ''e''<sub>31</sub>, ''e''<sub>33</sub> and ''e''<sub>15</sub>.
The semiconductors where the strongest piezoelectricity is observed are those commonly found in the [[wurtzite]] structure, i.e. [[GaN]], [[InN]], [[AlN]] and [[ZnO]] (see [[piezotronics]]).

Since 2006, there have also been a number of reports of strong [[non linear piezoelectric effects in polar semiconductors]].<ref>{{cite conference |title=A Review of Non Linear Piezoelectricity in Semiconductors |first=Max |last=Migliorato |conference=AIP Conference Proceedings|volume=1590 |issue=N/A |pages=32–41 |doi=10.1063/1.4870192|display-authors=etal|series=AIP Conference Proceedings |year=2014 |bibcode=2014AIPC.1590...32M |doi-access=free }}</ref>
Such effects are generally recognized to be at least important if not of the same order of magnitude as the first order approximation.

===Polymers===

The piezo-response of [[polymer]]s is not as high as the response for ceramics; however, polymers hold properties that ceramics do not. Over the last few decades, non-toxic, piezoelectric polymers have been studied and applied due to their flexibility and smaller [[Acoustic impedance|acoustical impedance]].<ref name=":0">{{Cite book |title=Piezoelectricity : evolution and future of a technology |date=2008 |publisher=Springer |editor-last1=Heywang |editor-first1=Walter |editor-last2=Lubitz |editor-first2=Karl |editor-last3=Wersing |editor-first3=Wolfram |isbn=978-3540686835|location=Berlin|oclc=304563111}}</ref> Other properties that make these materials significant include their [[biocompatibility]], [[Biodegradation|biodegradability]], low cost, and low power consumption compared to other piezo-materials (ceramics, etc.).<ref name=":1">{{Cite journal |last1=Sappati |first1=Kiran |last2=Bhadra |first2=Sharmistha |last3=Sappati |first3=Kiran Kumar |last4=Bhadra |first4=Sharmistha |date=2018|title=Piezoelectric Polymer and Paper Substrates: A Review|journal=Sensors |volume=18 |issue=11 |pages=3605|doi=10.3390/s18113605|pmid=30355961|pmc=6263872|bibcode=2018Senso..18.3605S |doi-access=free }}</ref> Piezoelectric polymers and non-toxic polymer composites can be used given their different physical properties.

Piezoelectric polymers can be classified by bulk polymers, voided charged polymers ("piezoelectrets"), and polymer composites. A piezo-response observed by bulk polymers is mostly due to its molecular structure. There are two types of bulk polymers: [[Amorphous solid|amorphous]] and [[Semi-crystalline polymer|semi-crystalline]]. Examples of semi-crystalline polymers are [[Polyvinylidene fluoride|Polyvinylidene Fluoride]] (PVDF) and its [[copolymer]]s, [[Polyamide]]s, and [[Parylene|Parylene-C]]. Non-crystalline polymers, such as [[Polyimide]] and [[Polyvinylidene chloride|Polyvinylidene Chloride]] (PVDC), fall under amorphous bulk polymers. Voided charged polymers exhibit the piezoelectric effect due to charge induced by poling of a porous polymeric film. Under an electric field, charges form on the surface of the voids forming dipoles. Electric responses can be caused by any deformation of these voids. The piezoelectric effect can also be observed in polymer composites by integrating piezoelectric ceramic particles into a polymer film. A polymer does not have to be piezo-active to be an effective material for a polymer composite.<ref name=":1" /> In this case, a material could be made up of an inert matrix with a separate piezo-active component.

PVDF exhibits piezoelectricity several times greater than quartz. The piezo-response observed from PVDF is about 20–30 pC/N. That is an order of 5–50 times less than that of piezoelectric ceramic lead zirconate titanate (PZT).<ref name=":0" /><ref name=":1" /> The thermal stability of the piezoelectric effect of polymers in the PVDF family (i.e. vinylidene fluoride co-poly trifluoroethylene) goes up to 125&nbsp;°C. Some applications of PVDF are pressure sensors, hydrophones, and shock wave sensors.<ref name=":0" />

Due to their flexibility, piezoelectric composites have been proposed as energy harvesters and nanogenerators. In 2018, it was reported by Zhu et al. that a piezoelectric response of about 17 pC/N could be obtained from PDMS/PZT nanocomposite at 60% porosity.<ref>{{Cite journal|last1=Ma|first1=Si Wei|last2=Fan|first2=You Jun|last3=Li|first3=Hua Yang|last4=Su|first4=Li|last5=Wang|first5=Zhong Lin|last6=Zhu|first6=Guang|date=2018-09-07|title=Flexible Porous Polydimethylsiloxane/Lead Zirconate Titanate-Based Nanogenerator Enabled by the Dual Effect of Ferroelectricity and Piezoelectricity|journal=ACS Applied Materials & Interfaces|volume=10|issue=39|pages=33105–33111|doi=10.1021/acsami.8b06696|pmid=30191707|s2cid=52171041|issn=1944-8244}}</ref> Another PDMS nanocomposite was reported in 2017, in which BaTiO<sub>3</sub> was integrated into PDMS to make a stretchable, transparent nanogenerator for self-powered physiological monitoring.<ref>{{Cite journal|last1=Chen|first1=Xiaoliang|last2=Parida|first2=Kaushik|last3=Wang|first3=Jiangxin|last4=Xiong|first4=Jiaqing|last5=Lin|first5=Meng-Fang|last6=Shao|first6=Jinyou|last7=Lee|first7=Pooi See|date=2017-11-20|title=A Stretchable and Transparent Nanocomposite Nanogenerator for Self-Powered Physiological Monitoring|journal=ACS Applied Materials & Interfaces|volume=9|issue=48|pages=42200–42209|doi=10.1021/acsami.7b13767|pmid=29111642|issn=1944-8244}}</ref> In 2016, polar molecules were introduced into a polyurethane foam in which high responses of up to 244 pC/N were reported.<ref>{{Cite journal|last1=Moody|first1=M. J.|last2=Marvin|first2=C. W.|last3=Hutchison|first3=G. R.|date=2016|title=Molecularly-doped polyurethane foams with massive piezoelectric response|journal=Journal of Materials Chemistry C|volume=4|issue=20|pages=4387–4392|doi=10.1039/c6tc00613b|issn=2050-7526}}</ref>

===Other materials===
Most materials exhibit at least weak piezoelectric responses. Trivial examples include [[sucrose]] (table sugar), [[DNA]], viral proteins, including those from [[bacteriophage]].<ref>{{cite journal|last1=Lee|first1=B. Y.|last2=Zhang |first2=J. |last3=Zueger |first3=C. |last4=Chung |first4=W. J. |last5=Yoo |first5=S. Y. |last6=Wang |first6=E. |last7=Meyer |first7=J. |last8=Ramesh |first8=R. |last9= Lee |first9=S. W. |title=Virus-based piezoelectric energy generation|journal=Nature Nanotechnology|date=2012-05-13|pmid=22581406|doi=10.1038/nnano.2012.69|volume=7|issue=6|pages=351–356|bibcode = 2012NatNa...7..351L }}</ref><ref>{{cite journal |title=Stable and Optoelectronic Dipeptide Assemblies for Power Harvesting |first1=Kai |last1=Tao |last2=et |first2=al |journal=Materials Today |volume=30 |pages=10–16 |doi=10.1016/j.mattod.2019.04.002 |pmid=31719792 |pmc=6850901 |year=2019 }}</ref> An actuator based on wood fibers, called [[cellulose fiber]]s, has been reported.<ref name=":1" /> D33 responses for cellular polypropylene are around 200 pC/N. Some applications of cellular polypropylene are musical key pads, microphones, and ultrasound-based echolocation systems.<ref name=":0" /> Recently, single amino acid such as β-glycine also displayed high piezoelectric (178 pmV<sup>−1</sup>) as compared to other biological materials.<ref>{{Cite journal|last1=Guerin|first1=Sarah|last2=Stapleton|first2=Aimee|last3=Chovan|first3=Drahomir|last4=Mouras|first4=Rabah|last5=Gleeson|first5=Matthew|last6=McKeown|first6=Cian|last7=Noor|first7=Mohamed Radzi|last8=Silien|first8=Christophe|last9=Rhen|first9=Fernando M. F.|last10=Kholkin|first10=Andrei L.|last11=Liu|first11=Ning|date=February 2018|title=Control of piezoelectricity in amino acids by supramolecular packing|url=http://www.nature.com/articles/nmat5045|journal=Nature Materials|language=en|volume=17|issue=2|pages=180–186|doi=10.1038/nmat5045|pmid=29200197|issn=1476-1122}}</ref>

[[Ionic liquid]]s were recently identified as the first piezoelectric liquid.<ref>{{cite news |last1=Choi |first1=Charles Q. |title=Liquid Salts Bring Push-button Lenses Into Focus – IEEE Spectrum |url=https://spectrum.ieee.org/piezoelectric-liquid |access-date=13 April 2023 |work=[[IEEE Spectrum]] |date=25 March 2023 |language=en}}</ref>

==Application==
<!-- "Piezoelectric network for smart roads" links here -->
===High voltage and power sources===

Direct piezoelectricity of some substances, like quartz, can generate [[potential difference]]s of thousands of volts.
* The best-known application is the electric [[Lighter|cigarette lighter]]: pressing the button causes a spring-loaded hammer to hit a piezoelectric crystal, producing a sufficiently high-voltage [[electric current]] that flows across a small [[spark gap]], thus heating and igniting the gas. The portable sparkers used to ignite [[gas stove]]s work the same way, and many types of gas burners now have built-in piezo-based ignition systems.
* A similar idea is being researched by [[DARPA]] in the United States in a project called [[energy harvesting]], which includes an attempt to power battlefield equipment by piezoelectric generators embedded in [[soldier]]s' boots. However, these energy harvesting sources by association affect the body. DARPA's effort to harness 1–2&nbsp;watts from continuous shoe impact while walking were abandoned due to the impracticality and the discomfort from the additional energy expended by a person wearing the shoes. Other energy harvesting ideas include [[Crowd Farm]], harvesting the energy from human movements in train stations or other public places<ref>{{cite web | url = http://www.treehugger.com/files/2006/08/japan_ticket_gates.php | title = Japan: Producing Electricity from Train Station Ticket Gates | last = Richard | first = Michael Graham | website = TreeHugger | publisher = Discovery Communications, LLC | date = 2006-08-04 | url-status = live | archive-url = https://web.archive.org/web/20070709200844/http://www.treehugger.com/files/2006/08/japan_ticket_gates.php | archive-date = 2007-07-09 }}</ref><ref>{{cite web | url = http://web.mit.edu/newsoffice/2007/crowdfarm-0725.html | title = MIT duo sees people-powered "Crowd Farm" | last = Wright | first = Sarah H. | date = 2007-07-25 | website = MIT news | publisher = [[Massachusetts Institute of Technology]] | url-status = live | archive-url = https://web.archive.org/web/20070912201419/http://web.mit.edu/newsoffice/2007/crowdfarm-0725.html | archive-date = 2007-09-12 }}</ref> and converting a dance floor to generate electricity.<ref>{{cite web | url = https://abcnews.go.com/International/story?id=5358214&page=1 | title = How to Save the World One Dance at a Time | last = Kannampilly | first = Ammu | work = ABC News | date = 2008-07-11 | url-status = live | archive-url = https://web.archive.org/web/20101031010929/https://abcnews.go.com/International/story?id=5358214&page=1 | archive-date = 2010-10-31 }}</ref> Vibrations from industrial machinery can also be harvested by piezoelectric materials to charge batteries for backup supplies or to power low-power microprocessors and wireless radios.<ref>{{cite journal|url=http://www.linear.com/docs/29984 |title=True Grid Independence: Robust Energy Harvesting System for Wireless Sensors Uses Piezoelectric Energy Harvesting Power Supply and Li-Poly Batteries with Shunt Charger |last=Barbehenn |first=George H. |page=36 |date=October 2010 |journal=Journal of Analog Innovation}}</ref><ref>{{Cite journal |last1=Bahl |first1=Shashi |last2=Nagar |first2=Himanshu |last3=Singh |first3=Inderpreet |last4=Sehgal |first4=Shankar |date=2020-01-01 |title=Smart materials types, properties and applications: A review |url=https://www.sciencedirect.com/science/article/pii/S2214785320331278 |journal=Materials Today: Proceedings |series=International Conference on Aspects of Materials Science and Engineering |volume=28 |pages=1302–1306 |doi=10.1016/j.matpr.2020.04.505 |s2cid=219435304 |issn=2214-7853}}</ref>
* A piezoelectric [[transformer]] is a type of AC voltage multiplier. Unlike a conventional transformer, which uses magnetic coupling between input and output, the piezoelectric transformer uses [[acoustic coupling]]. An input voltage is applied across a short length of a bar of piezoceramic material such as [[Lead zirconate titanate|PZT]], creating an alternating stress in the bar by the inverse piezoelectric effect and causing the whole bar to vibrate. The vibration frequency is chosen to be the [[resonance|resonant]] frequency of the block, typically in the 100&nbsp;[[kilohertz]] to 1&nbsp;megahertz range. A higher output voltage is then generated across another section of the bar by the piezoelectric effect. Step-up ratios of more than 1,000:1 have been demonstrated.{{Citation needed|date=May 2013}} An extra feature of this transformer is that, by operating it above its resonant frequency, it can be made to appear as an [[inductor|inductive]] load, which is useful in circuits that require a controlled soft start.<ref>{{cite web | url = http://www.eetimes.com/electronics-news/4197064/Piezoelectric-Technology-A-Primer | archive-url = https://web.archive.org/web/20101006002651/http://www.eetimes.com/electronics-news/4197064/Piezoelectric-Technology-A-Primer | archive-date = 2010-10-06 | title = Piezoelectric Technology: A Primer | last = Phillips | first = James R. | date = 2000-08-10 | website = eeProductCenter | publisher = TechInsights }}</ref> These devices can be used in DC–AC inverters to drive [[cold cathode fluorescent lamp]]s. Piezo transformers are some of the most compact high voltage sources.

===Sensors===

[[File:Piezoelectric pickup1.jpg|thumb|Piezoelectric disk used as a [[guitar pickup]]]]
[[File:RPG-7 detached.jpg|thumb|Many rocket-propelled grenades used a piezoelectric [[wikt:fuse|fuse]]. Pictured, a Russian [[RPG-7]]<ref>{{cite web |last=Speck |first=Shane |date=2004-03-11 |url=http://science.howstuffworks.com/rpg3.htm |title=How Rocket-Propelled Grenades Work by Shane Speck |website=HowStuffWorks.com |access-date=2012-05-04 |url-status=live |archive-url=https://web.archive.org/web/20120429064528/http://science.howstuffworks.com/rpg3.htm |archive-date=2012-04-29 }}</ref>]]
{{main|Piezoelectric sensor}}
The principle of operation of a piezoelectric [[sensor]] is that a physical dimension, transformed into a force, acts on two opposing faces of the sensing element. Depending on the design of a sensor, different "modes" to load the piezoelectric element can be used: longitudinal, transversal and shear.

Detection of pressure variations in the form of sound is the most common sensor application, e.g. piezoelectric [[microphone]]s (sound waves bend the piezoelectric material, creating a changing voltage) and piezoelectric [[Pick up (music technology)|pickups]] for [[acoustic-electric guitar]]s. A piezo sensor attached to the body of an instrument is known as a [[contact microphone]].

Piezoelectric sensors especially are used with high frequency sound in ultrasonic transducers for medical imaging and also industrial [[nondestructive testing]] (NDT).

For many sensing techniques, the sensor can act as both a sensor and an actuator—often the term ''transducer'' is preferred when the device acts in this dual capacity, but most piezo devices have this property of reversibility whether it is used or not. Ultrasonic transducers, for example, can inject ultrasound waves into the body, receive the returned wave, and convert it to an electrical signal (a voltage). Most medical ultrasound transducers are piezoelectric.

In addition to those mentioned above, various sensor and transducer applications include:
* Piezoelectric elements are also used in the detection and generation of sonar waves.
* Piezoelectric materials are used in single-axis and dual-axis tilt sensing.<ref>{{cite journal | last1 = Moubarak | first1 = P. | display-authors = etal | year = 2012 | title = A Self-Calibrating Mathematical Model for the Direct Piezoelectric Effect of a New MEMS Tilt Sensor | journal = IEEE Sensors Journal | volume = 12 | issue = 5| pages = 1033–1042 | doi=10.1109/jsen.2011.2173188| bibcode = 2012ISenJ..12.1033M| s2cid = 44030488 }}</ref>
* Power monitoring in high power applications (e.g. medical treatment, [[sonochemistry]] and industrial processing).
* [[Piezoelectric microbalance]]s are used as very sensitive chemical and biological sensors.
* Piezoelectrics are sometimes used in [[strain gauge]]s. More commonly however, a [[Piezoresistive effect]] element is used. 
* A piezoelectric transducer was used in the penetrometer instrument on the [[Huygens Probe]].
* Piezoelectric [[transducer]]s are used in [[electronic drum|electronic drum pads]] to detect the impact of the drummer's sticks, and to detect muscle movements in medical [[acceleromyography]].
* Automotive [[Engine Control Unit|engine management systems]] use piezoelectric transducers to detect Engine knock (Knock Sensor, KS), also known as detonation, at certain hertz frequencies. A piezoelectric transducer is also used in fuel injection systems to measure manifold absolute pressure (MAP sensor) to determine engine load, and ultimately the fuel injectors milliseconds of on time.
* Ultrasonic piezo sensors are used in the detection of acoustic emissions in [[acoustic emission testing]].
* Piezoelectric transducers can be used in transit-time [[ultrasonic flow meters]].

===Actuators===

[[File:2007-07-24 Piezoelectric buzzer.jpg|thumb|Metal disk with piezoelectric disk attached, used in a [[buzzer]]]]

As very high electric fields correspond to only tiny changes in the width of the crystal, this width can be changed with better-than-[[micrometre|µm]] precision, making piezo crystals the most important tool for positioning objects with extreme accuracy—thus their use in [[actuators]].<ref name=shabestari>{{cite journal
|last1=Shabestari |first1=N. P.
|title=Fabrication of a simple and easy-to-make piezoelectric actuator and its use as phase shifter in digital speckle pattern interferometry
|journal=Journal of Optics
|date=2019 |volume=48 |issue=2 |pages=272–282
|doi=10.1007/s12596-019-00522-4|s2cid=155531221
}}</ref>
Multilayer ceramics, using layers thinner than {{nowrap|100 µm}}, allow reaching high electric fields with voltage lower than {{nowrap|150 V}}. These ceramics are used within two kinds of actuators: direct piezo actuators and [[amplified piezoelectric actuator]]s. While direct actuator's stroke is generally lower than {{nowrap|100 µm}}, amplified piezo actuators can reach millimeter strokes.
* [[Piezoelectric loudspeaker|Loudspeakers]]: Voltage is converted to mechanical movement of a metallic diaphragm.
* [[Ultrasonic cleaning]] usually uses piezoelectric elements to produce intense sound waves in liquid.
* [[Piezoelectric motor]]s: Piezoelectric elements apply a directional force to an [[axle]], causing it to rotate. Due to the extremely small distances involved, the piezo motor is viewed as a high-precision replacement for the [[stepper motor]].
* Piezoelectric elements can be used in [[laser]] mirror alignment, where their ability to move a large mass (the mirror mount) over microscopic distances is exploited to electronically align some laser mirrors. By precisely controlling the distance between mirrors, the laser electronics can accurately maintain optical conditions inside the laser cavity to optimize the beam output.
* A related application is the [[acousto-optic modulator]], a device that scatters light off soundwaves in a crystal, generated by piezoelectric elements. This is useful for fine-tuning a laser's frequency.
* [[Atomic force microscope]]s and [[scanning tunneling microscope]]s employ converse piezoelectricity to keep the sensing needle close to the specimen.<ref>{{cite book|chapter=The scanning mechanism for ROSETTA/MIDAS from an engineering model to the flight model |last1=Le Letty |first1=R. |last2=Barillot |first2=F. |last3=Lhermet |first3=N. |last4=Claeyssen |first4=F. |last5=Yorck |first5=M. |last6=Gavira Izquierdo |first6=J. |last7=Arends |first7=H. |bibcode=2001ESASP.480...75L |pages=75–81 |isbn=978-92-9092-761-7 |year=2001 |title=Proceedings of the 9th European Space Mechanisms and Tribology Symposium, 19–21 September 2001, Liège, Belgium |journal=9th European Space Mechanisms and Tribology Symposium |editor-first=R. A. |editor-last=Harris |series=ESA SP-480 |volume=480}}</ref>
* [[Inkjet printer]]s: On many inkjet printers, piezoelectric crystals are used to drive the ejection of ink from the inkjet print head towards the paper.
* [[Diesel engine]]s: High-performance [[common rail]] diesel engines use piezoelectric [[fuel injector]]s, first developed by [[Robert Bosch GmbH]], instead of the more common [[solenoid valve]] devices.
* Active vibration control using amplified actuators.
* [[X-ray]] shutters.
* XY stages for micro scanning used in infrared cameras.
* Moving the patient precisely inside active [[X-ray computed tomography|CT]] and [[Magnetic resonance imaging|MRI]] scanners where the strong radiation or magnetism precludes electric motors.<ref name="piezact">{{cite web |last=Simonsen |first=Torben R. |url=http://elektronikbranchen.dk/nyhed/dansk-elektronikvirksomhed-goer-klar-til-rumeventyr-efter-koeb-af-piezoteknologi |title=Piezo in space |website=Electronics Business |language=da |date=27 September 2010 |access-date=28 September 2010 |url-status=dead |archive-url=https://web.archive.org/web/20100929113331/http://elektronikbranchen.dk/nyhed/dansk-elektronikvirksomhed-goer-klar-til-rumeventyr-efter-koeb-af-piezoteknologi |archive-date=29 September 2010 }}</ref>
* [[Crystal earpiece]]s are sometimes used in old or low power radios.
* [[High-intensity focused ultrasound]] for localized heating or creating a localized [[cavitation]] can be achieved, for example, in patient's body or in an industrial chemical process.
* [[Refreshable braille display]]. A small crystal is expanded by applying a current that moves a lever to raise individual braille cells.
* Piezoelectric actuator. A single crystal or a number of crystals are expanded by applying a voltage for moving and controlling a mechanism or system.<ref name=shabestari/>
* Piezoelectric actuators are used for fine servo positioning in hard disc drives.<ref>{{cite web|url=https://documents.westerndigital.com/content/dam/doc-library/en_us/assets/public/western-digital/collateral/tech-brief/tech-brief-western-digital-micro-actuator.pdf|title=Second-generation Micro Actuator for Better Head-positioning Accuracy|website=Documents.westerndigital.com|access-date=March 10, 2022}}</ref><ref>{{cite web |url=https://www.seagate.com/files/staticfiles/docs/pdf/whitepaper/seagate-acutrac-TP624.1-1110US.pdf|title=Exceeding Capacity, Speed and Performance Expectations|website=Seagate.com|access-date=March 10, 2022}}</ref>

===Frequency standard===
The piezoelectrical properties of quartz are useful as a [[Frequency standard|standard of frequency]].
* [[Quartz clock]]s employ a [[crystal oscillator]] made from a quartz crystal that uses a combination of both direct and converse piezoelectricity to generate a regularly timed series of electrical pulses that is used to mark time. The quartz crystal (like any [[elasticity (physics)|elastic]] material) has a precisely defined natural frequency (caused by its shape and size) at which it prefers to [[oscillator|oscillate]], and this is used to stabilize the frequency of a periodic voltage applied to the crystal.
* The same principle is used in some [[radio]] [[transmitter]]s and [[receiver (radio)|receivers]], and in [[computer]]s where it creates a [[clock pulse]]. Both of these usually use a [[frequency multiplier]] to reach gigahertz ranges.

===Piezoelectric motors===
[[File:Slip-stick actuator operation.svg|thumb|A stick-slip actuator]]
{{Main|Piezoelectric motor}}
Types of piezoelectric motor include:
* The [[ultrasonic motor]] used for [[auto-focus]] in [[single lens reflex|reflex cameras]]
* [[Inchworm motor]]s for linear motion
* Rectangular four-quadrant motors with high power density (2.5&nbsp;[[watt|W]]/cm<sup>3</sup>) and speed ranging from 10&nbsp;nm/s to 800&nbsp;mm/s.
* Stepping piezo motor, using [[stick-slip]] effect.

Aside from the stepping stick-slip motor, all these motors work on the same principle. Driven by dual orthogonal vibration modes with a [[phase (waves)|phase]] difference of 90°, the contact point between two surfaces vibrates in an [[ellipse|elliptical]] path, producing a [[friction]]al force between the surfaces. Usually, one surface is fixed, causing the other to move. In most piezoelectric motors, the piezoelectric crystal is excited by a [[sine wave]] signal at the resonant frequency of the motor. Using the resonance effect, a much lower voltage can be used to produce a high vibration amplitude.

A stick-slip motor works using the inertia of a mass and the friction of a clamp. Such motors can be very small. Some are used for camera sensor displacement, thus allowing an anti-shake function.

===Reduction of vibrations and noise===

Different teams of researchers have been investigating ways to reduce vibrations in materials by attaching piezo elements to the material. When the material is bent by a vibration in one direction, the vibration-reduction system responds to the bend and sends electric power to the piezo element to bend in the other direction. Future applications of this technology are expected in cars and houses to reduce noise. Further applications to flexible structures, such as shells and plates, have also been studied for nearly three decades.

In a demonstration at the Material Vision Fair in [[Frankfurt]] in November 2005, a team from [[Darmstadt University of Technology|TU Darmstadt]] in [[Germany]] showed several panels that were hit with a rubber mallet, and the panel with the piezo element immediately stopped swinging.

Piezoelectric ceramic fiber technology is being used as an electronic damping system on some [[Head (company)|HEAD]] [[tennis racket]]s.<ref>{{cite web|url=http://www.head.com/tennis/technology.php?region=eu&tag=intelligence |title=Isn't it amazing how one smart idea, one chip and an intelligent material has changed the world of tennis? |access-date=2008-02-27 |publisher=Head.com |url-status=dead |archive-url=https://web.archive.org/web/20070222224436/http://www.head.com/tennis/technology.php?region=eu&tag=intelligence |archive-date=February 22, 2007 }}</ref>

All piezo transducers have a fundamental resonant frequency and many harmonic frequencies. Piezo driven Drop-On-Demand fluid systems are sensitive to extra vibrations in the piezo structure that must be reduced or eliminated. One inkjet company, Howtek, Inc solved this problem by replacing glass(rigid) inkjet nozzles with Tefzel (soft) inkjet nozzles. This novel idea popularized single nozzle inkjets and they are now used in 3D Inkjet printers that run for years if kept clean inside and not overheated (Tefzel creeps under pressure at very high temperatures)

===Infertility treatment===
In people with previous [[total fertilization failure]], piezoelectric activation of [[oocyte]]s together with [[intracytoplasmic sperm injection]] (ICSI) seems to improve fertilization outcomes.<ref>{{cite journal|title=The effectiveness of intracytoplasmic sperm injection combined with piezoelectric stimulation in infertile couples with total fertilization failure |journal=Fertil. Steril. |volume= 94|issue= 3|pages= 900–904|year=2009|pmid=19464000 |doi=10.1016/j.fertnstert.2009.03.107|last1=Baltaci|first1=Volkan|last2=Ayvaz|first2=Özge Üner|last3=Ünsal|first3=Evrim|last4=Aktaş|first4=Yasemin|last5=Baltacı|first5=Aysun|last6=Turhan|first6=Feriba|last7=Özcan|first7=Sarp|last8=Sönmezer|first8=Murat |doi-access=free}}</ref>

===Surgery===
[[Piezosurgery]]<ref name="Manbachi, A. and Cobbold R.S.C. 2011 187–196">{{cite journal | last1=Manbachi |first1=A. |last2=Cobbold |first2=R.S.C. |name-list-style=amp|title= Development and Application of Piezoelectric Materials for Ultrasound Generation and Detection|journal= Ultrasound |volume=19 |issue=4 |pages=187–96 |year=2011|doi= 10.1258/ult.2011.011027|s2cid=56655834 }}</ref> is a minimally invasive technique that aims to cut a target tissue with little damage to neighboring tissues. For example, Hoigne ''et al.''<ref>{{cite journal |last1=Hoigne |first1=D.J. |last2=Stubinger |first2=S. |last3=von Kaenel |first3=O. |last4=Shamdasani |first4=S. |last5=Hasenboehler |first5=P. |title=Piezoelectric osteotomy in hand surgery: first experiences with a new technique |journal=BMC Musculoskelet. Disord. |volume= 7|page= 36|year=2006|pmid= 16611362|pmc=1459157 |doi=10.1186/1471-2474-7-36 |doi-access=free }}</ref> uses frequencies in the range 25–29&nbsp;kHz, causing microvibrations of 60–210&nbsp;μm. It has the ability to cut mineralized tissue without cutting neurovascular tissue and other soft tissue, thereby maintaining a blood-free operating area, better visibility and greater precision.<ref>{{cite journal |last1=Labanca |first1=M. |last2=Azzola |first2=F. |last3=Vinci |first3=R. |last4=Rodella |first4=L. F. |title=Piezoelectric surgery: twenty years of use |journal=Br. J. Oral Maxillofac. Surg. |volume= 46 |issue = 4 | pages=265–269 |year=2008 |pmid=18342999|doi=10.1016/j.bjoms.2007.12.007}}</ref>

===Potential applications===
In 2015, Cambridge University researchers working in conjunction with researchers from the National Physical Laboratory and Cambridge-based dielectric antenna company Antenova Ltd, using thin films of piezoelectric materials found that at a certain frequency, these materials become not only efficient resonators, but efficient radiators as well, meaning that they can potentially be used as antennas. The researchers found that by subjecting the piezoelectric thin films to an asymmetric excitation, the symmetry of the system is similarly broken, resulting in a corresponding symmetry breaking of the electric field, and the generation of electromagnetic radiation.<ref>{{Cite journal|first1=Dhiraj|last1=Sinha|first2=Gehan|last2=Amaratunga|title=Electromagnetic Radiation Under Explicit symmetry Breaking|journal=Physical Review Letters|volume=114|issue=14|page=147701|year=2015|doi=10.1103/physrevlett.114.147701|pmid=25910163|bibcode = 2015PhRvL.114n7701S |url=https://zenodo.org/record/1038659}}</ref><ref>{{cite web|title=New understanding of electromagnetism could enable 'antennas on a chip'|url=http://www.cam.ac.uk/research/news/new-understanding-of-electromagnetism-could-enable-antennas-on-a-chip|website=cam.ac.uk|url-status=live|archive-url=https://web.archive.org/web/20160304113152/http://www.cam.ac.uk/research/news/new-understanding-of-electromagnetism-could-enable-antennas-on-a-chip|archive-date=2016-03-04|date=2015-04-09}}</ref>

Several attempts at the macro-scale application of the piezoelectric technology have emerged<ref>{{cite journal|last1=Takefuji|first1=Y.|title=And if public transport does not consume more of energy?|journal=Le Rail|date=April 2008|pages=31–33|url=http://neuro.sfc.keio.ac.jp/publications/pdf/rail.pdf|access-date=2018-12-15|archive-date=2021-01-15|archive-url=https://web.archive.org/web/20210115165450/http://neuro.sfc.keio.ac.jp/publications/pdf/rail.pdf|url-status=dead}}</ref><ref>{{Cite conference|last1=Takefuji|first1=Y.|title=Known and unknown phenomena of nonlinear behaviors in the power harvesting mat and the transverse wave speaker|conference=international symposium on nonlinear theory and its applications|date=September 2008|url=http://neuro.sfc.keio.ac.jp/publications/pdf/nolta.pdf|access-date=2018-12-15|archive-date=2020-10-21|archive-url=https://web.archive.org/web/20201021073953/http://neuro.sfc.keio.ac.jp/publications/pdf/nolta.pdf|url-status=dead}}</ref> to harvest kinetic energy from walking pedestrians.

In this case, locating high traffic areas is critical for optimization of the energy harvesting efficiency, as well as the orientation of the tile pavement significantly affects the total amount of the harvested energy.<ref>{{cite journal |last1=Deutz |first1=D.B. |last2=Pascoe |first2=J.-A. |last3=van der Zwaag |first3=S. |last4=de Leeuw |first4=D.M. |last5=Groen |first5=P.|title=Analysis and experimental validation of the figure of merit for piezoelectric energy harvesters |journal=Materials Horizons |volume=5 |date=2018 |issue=3 |pages=444–453 |doi=10.1039/c8mh00097b|s2cid=117687945 |url=http://resolver.tudelft.nl/uuid:3ebce2f3-e2d8-433c-bbcd-684e0835c14b |hdl=10044/1/60608 |hdl-access=free }}</ref> A density flow evaluation is recommended to qualitatively evaluate the piezoelectric power harvesting potential of the considered area based on the number of pedestrian crossings per unit time.<ref name="ReferenceA">{{cite journal|last1=Li|first1=Xiaofeng|last2=Strezov|first2=Vladimir|year=2014|title=Modelling piezoelectric energy harvesting potential in an educational building|url=http://www.researchonline.mq.edu.au/vital/access/services/Download/mq:33352/DS01|journal=Energy Conversion and Management|volume=85|pages=435–442|doi=10.1016/j.enconman.2014.05.096}}</ref> In X. Li's study, the potential application of a commercial piezoelectric energy harvester in a central hub building at Macquarie University in Sydney, Australia is examined and discussed. Optimization of the piezoelectric tile deployment is presented according to the frequency of pedestrian mobility and a model is developed where 3.1% of the total floor area with the highest pedestrian mobility is paved with piezoelectric tiles. The modelling results indicate that the total annual energy harvesting potential for the proposed optimized tile pavement model is estimated at 1.1&nbsp;MWh/year, which would be sufficient to meet close to 0.5% of the annual energy needs of the building.<ref name="ReferenceA"/> In Israel, there is a company which has installed piezoelectric materials under a busy highway. The energy generated is enough to power street lights, billboards, and signs.{{citation needed|date=June 2016}}

Tire company [[Goodyear Tire and Rubber Company|Goodyear]] has plans to develop an electricity generating tire which has piezoelectric material lined inside it. As the tire moves, it deforms and thus electricity is generated.<ref>{{cite magazine|title=Goodyear Is Trying to Make an Electricity-Generating Tire|url=https://www.wired.com/2015/03/goodyear-trying-make-electricity-generating-tire/|magazine=WIRED|access-date=14 June 2016|url-status=live|archive-url=https://web.archive.org/web/20160511070323/http://www.wired.com/2015/03/goodyear-trying-make-electricity-generating-tire|archive-date=11 May 2016|date=2015-03-12}}</ref>

The efficiency of a hybrid [[photovoltaic cell]] that contains piezoelectric materials can be increased simply by placing it near a source of ambient noise or vibration. The effect was demonstrated with organic cells using [[zinc oxide]] nanotubes. The electricity generated by the piezoelectric effect itself is a negligible percentage of the overall output. Sound levels as low as 75 decibels improved efficiency by up to 50%. Efficiency peaked at 10&nbsp;kHz, the resonant frequency of the nanotubes. The electrical field set up by the vibrating nanotubes interacts with electrons migrating from the organic polymer layer. This process decreases the likelihood of recombination, in which electrons are energized but settle back into a hole instead of migrating to the electron-accepting ZnO layer.<ref>{{cite web |url=http://www.gizmag.com/vibration-sound-efficient-hybrid-solar-cell-arrays/29679/ |title=Good vibrations lead to efficient excitations in hybrid solar cells |publisher=Gizmag.com |access-date=2013-11-11 |author=Heidi Hoopes |date=November 8, 2013 |url-status=live |archive-url=https://web.archive.org/web/20131111193548/http://www.gizmag.com/vibration-sound-efficient-hybrid-solar-cell-arrays/29679/ |archive-date=November 11, 2013 }}</ref><ref>{{Cite journal | last1 = Shoaee | first1 = S. | last2 = Briscoe | first2 = J. | last3 = Durrant | first3 = J. R. | last4 = Dunn | first4 = S. | title = Acoustic Enhancement of Polymer/ZnO Nanorod Photovoltaic Device Performance | doi = 10.1002/adma.201303304 | journal = Advanced Materials | volume = 26 | issue = 2 | pages = 263–268 | year = 2013 | pmid = 24194369| s2cid = 40624518 | url = http://qmro.qmul.ac.uk/xmlui/handle/123456789/12456 }}</ref>

==See also==
{{div col|colwidth=20em}}
* [[Charge amplifier]]
* [[Electret]]
* [[Electronic component]]
* [[Electrostriction]]
* [[Flexoelectricity]]
* [[Magnetostriction]]
* [[Photoelectric effect]]
* [[Piezoelectric speaker]]
* [[Piezoluminescence]]
* [[Piezomagnetism]]
* [[Piezoresistive effect]]
* [[Piezosurgical]]
* [[Quartz crystal microbalance]]
* [[Sonomicrometry]]
* [[Surface acoustic wave]]
* [[Thermoelectric generator]]
* [[Triboluminescence]]
{{div col end}}

==References==
{{Reflist}}

== Further reading ==
* EN 50324 (2002) Piezoelectric properties of ceramic materials and components (3 parts)
* ANSI-IEEE 176 (1987) Standard on Piezoelectricity
* IEEE 177 (1976) Standard Definitions & Methods of Measurement for Piezoelectric Vibrators
* IEC 444 (1973) Basic method for the measurement of resonance freq & equiv series resistance of quartz crystal units by zero-phase technique in a pi-network
* IEC 302 (1969) Standard Definitions & Methods of Measurement for Piezoelectric Vibrators Operating over the Freq Range up to 30&nbsp;MHz

==External links==
{{Commons category|Piezoelectricity}}
* {{cite book|last=Gautschi|first=Gustav H.|date=2002|title=Piezoelectric Sensorics|publisher=Springer|isbn=978-3-540-42259-4}}
* [https://web.archive.org/web/20180905175615/http://www.aimspress.com/fileOther/PDF/Materials/matersci-05-05-845.pdf Piezoelectric cellular polymer films: Fabrication, properties and applications]
* [https://www.ncbi.nlm.nih.gov/pubmed/19163430 Piezo motor based microdrive for neural signal recording]
* [https://web.archive.org/web/20080925074311/http://www.scientificblogging.com/news_account/research_new_piezoelectric_materials Research on new Piezoelectric materials]
* [https://archive.today/20110809024157/http://www.techonline.com/showArticle.jhtml?articleID=192201162&queryText=22 Piezo Equations]
* [https://web.archive.org/web/20120229184729/http://medicaldesign.com/motors-motion-control/precision-via-piezo-20100501/ Piezo in Medical Design]
* [https://web.archive.org/web/20120106021343/http://vega.org.uk/video/programme/195 Video demonstration of Piezoelectricity]
* [http://www.doitpoms.ac.uk/tlplib/piezoelectrics/index.php DoITPoMS Teaching and Learning Package&nbsp;– Piezoelectric Materials]
* [https://web.archive.org/web/20140221180044/http://piezomat.org/ PiezoMat.org] – Online database for piezoelectric materials, their properties, and applications
* [[commons:File:Piezomotor type inchworm.gif|Piezo Motor Types]]
* [https://web.archive.org/web/20140319163719/https://www.piezosystem.com/piezo_actuator_nanopositioning/downloads_publications/technical_information/piezo_theory/ Piezo-Theory & Applications]

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[[Category:Condensed matter physics]]
[[Category:Electrical phenomena]]
[[Category:Energy conversion]]
[[Category:Transducers]]
[[Category:Energy harvesting]]