# Piezoelectricity **Piezoelectricity** is the appearance of electric charge on the faces of certain crystals when they are stressed, and the strain those crystals show when an electric field is applied. The two are one effect from two sides: the direct effect turns force into charge, the converse turns [[Voltage|voltage]] into displacement, and the constant connecting them is the same number both ways.[^mitofsky-ch2] Only crystals without a centre of symmetry show it, so the property is a matter of [[Crystallography|crystal symmetry]] before one of chemistry. In the microsim below a crystal is squeezed along one axis and the reader turns the applied force. Two equations answer. The direct effect is `D = d·T`, displacement proportional to stress, which for a disc under load reads `Q = d_33·F`: charge is the coefficient times the force, and the area cancels.[^spec-m41] The converse effect is `S = d·E`. The presets are [[Quartz|quartz]] at d₁₁ = 2.3 pC/N and [[Lead_zirconate_titanate|lead zirconate titanate]] (PZT) at d₃₃ ≈ 400 pC/N, with [[Barium_titanate|barium titanate]] between; the home position is a 1 kN press on a 1 cm² PZT disc, reading 0.4 μC.[^spec-m41][^derived-pz] A second readout takes that charge to a voltage through `V = Q/C`, giving the page's most instructive number: because PZT's permittivity is hundreds of times quartz's, the far smaller charge from a quartz disc sits on a far smaller [[Capacitor|capacitance]] and yields a comparable voltage.[^derived-pz] On the [[Materials_science]] flagship this article serves the *Piezoelectric materials* section of Part VII — Research. Its spine neighbour is the [[Thermoelectric_effect|thermoelectric effect]], whose sim sits on the Energy flagship; the Portal Book is explicit that the Carnot ceiling limiting a thermoelectric device does not apply to a piezoelectric one.[^mitofsky-carnot][^manual05-94] ## Etymology The name was built from the Greek *piezein*, to press, and means "pressure electricity". It was coined for the direct effect, observed first; the converse was predicted afterwards and kept the name with a qualifier. The related *pyroelectricity*, from the Greek for fire, is charge appearing on heating; every pyroelectric crystal is piezoelectric, needing the same absent symmetry centre, but not the reverse. ## History The subject has three phases. A nineteenth-century discovery established the effect but left it a curiosity: the crystals were weak or fragile, and the electronics to use millivolt signals did not exist. A wartime application, underwater sound, made it a technology. A post-war materials development, the poled ferroelectric ceramic, made it cheap. ### Discovery and early research Jacques and Pierre Curie reported in 1880 that compressing certain hemihedral crystals with inclined faces produced charge proportional to the pressure, and that reversing the compression reversed its sign.[^curie1880] They worked with [[Quartz|quartz]], tourmaline, topaz, cane sugar and Rochelle salt, the last giving much the largest effect and so dominating early practice despite being soluble. The converse effect was not found by experiment but deduced: the thermodynamics of a crystal that converts stress to charge requires that it also convert field to strain, with the same coefficient, and the prediction was then confirmed. That the two coefficients are equal follows from energy conservation, and survives in modern units as an exact identity — one picocoulomb per newton is one picometre per volt, C/N and m/V being the same combination written two ways.[^derived-pz] For forty years the effect's main value was as evidence about [[Crystal_structure|crystal structure]], reporting a symmetry optical methods could not. ### World War I and inter-war years The application that changed the subject was underwater sound. A transducer made by bonding quartz between steel plates can be driven at its [[Resonance|resonance]] to radiate an [[Acoustic_wave|acoustic pulse]] into water and then used in reverse to detect the echo, giving the first practical [[Sonar|echo-ranging]] system; the work is generally credited to Paul Langevin and collaborators in France around 1917, but no Portal Book on this page's shelf documents it and the claim is left to source.[citation needed][^langevin-cn] The physics is identical both ways, so one element serves as source and receiver. Between the wars the principle took two further roles. A quartz plate cut to a chosen orientation has a resonance so sharp, and for the right cut so insensitive to [[Temperature|temperature]], that it became the reference for frequency in radio and timekeeping. And the direct effect became the basis of the first [[Sensor|sensors]] for rapidly varying pressure, where no mechanical gauge could follow the signal. ### World War II and post-war The decisive development was that certain [[Ferroelectricity|ferroelectric]] [[Ceramic|ceramics]] could be made piezoelectric without being single crystals. A sintered ceramic has randomly oriented grains and no net effect, but heated near its [[Curie_temperature|Curie temperature]] and cooled in a strong field, the grains' polar axes align as far as they can and it is left permanently poled. [[Barium_titanate|Barium titanate]] came first, PZT afterwards, with coefficients more than a hundred times quartz's in a body that can be pressed and fired into any shape.[^mitofsky-ch2][^spec-m41] Poling is reversible, so every ceramic device is bounded by temperature, field and stress. ## Mechanism The mechanism is displacement of charge inside the unit cell. Each ion sits where the balance of forces puts it; a strain moves them, and the centres of positive and negative charge move too. If the crystal has a centre of symmetry, every displacement is matched by an equal and opposite one, the centres stay coincident, and no polarization appears however hard it is squeezed. If it does not, they separate, and the separation is a dipole moment per unit volume proportional to the strain. The converse effect is that picture backwards: a field pulls positive ions one way and negative the other, and the cell changes shape. Two neighbours must be kept distinct. Electrostriction, a strain proportional to the square of the field, occurs in every dielectric and does not reverse with the field, where piezoelectricity is linear and does. And in a ferroelectric ceramic much of the response is domain walls moving rather than ions within a domain — hence PZT's large coefficients, their dependence on drive level, and the [[Hysteresis|hysteresis]] the intrinsic effect would not show. ### Mathematical description The linear constitutive relations couple the mechanical and electrical variables both ways: `D = d·T + eps·E` and `S = s·T + d·E`, with D the electric displacement, T the stress, S the strain, E the field, s the compliance and ε the permittivity.[^mitofsky-ch2] In general d is a third-rank tensor of up to eighteen components, and d₃₃ is charge on faces normal to axis 3 from stress along it. The worked case is the sim's. A load F on a disc of area A gives T = F/A, so the electrode charge is Q = D·A = d₃₃·F: the area cancels, and a charge generator answers to force, not pressure. A 1 kN load on 1 cm² is 10 MPa, and with d₃₃ = 400 pC/N it yields 0.4 μC.[^derived-pz][^spec-m41] The voltage follows from the disc's own [[Capacitor|capacitance]], C = ε₀·ε_r·A/t: for a 1 mm-thick PZT disc of relative permittivity of order 1,500 that is about 1.3 nF, so V ≈ 300 V.[^derived-pz][^openstax-v2-cap] The same press on quartz gives only 2.3 nC, but quartz's permittivity is some 300 times smaller, its capacitance about 4 pF, and its open-circuit voltage near 580 V — higher than PZT's, on a hundred and seventy times less charge.[^derived-pz] The figure of merit therefore depends on the question: d for charge into a low impedance, g = d/ε for voltage into a high one, and the two rank materials differently. The converse direction is as blunt: `S = d·E` at d₃₃ = 400 pm/V and 100 V across 1 mm gives a strain of 4×10⁻⁵ and 40 nm of motion. Stacking N layers driven at the same voltage gives `dL = d·V·N`, independent of layer thickness, so a hundred-layer stack at 100 V moves 4 μm. The energy at the home position, ½CV², is about 60 μJ — enough to fire a spark, not to run anything.[^derived-pz] ## Crystal classes Symmetry decides which crystals can show the effect, and the rule is short. Of the thirty-two crystal classes, twenty-one lack a centre of symmetry and twenty of those are piezoelectric; the exception is a cubic class whose remaining symmetry elements cancel the effect. Within the twenty, ten have a unique polar axis and are also pyroelectric, and the ferroelectrics are those of the ten whose polarization can be reversed by an applied field — exactly the property that makes a ceramic pollable. The rule is about the [[Crystal_system|crystal class]], not the substance: the same composition in a centrosymmetric structure shows nothing, and the same composition in two polymorphs can differ absolutely. It also explains the effect's use as a diagnostic, a measurable response being direct evidence that a structure lacks an inversion centre, and why a randomly oriented polycrystal is inert until it is poled: the individual grains respond, and their responses cancel to nothing. ## Materials Useful materials fall into a few families, separated by whether the polar structure is grown, poled or stretched into them. Single crystals give stability and low loss; poled ceramics large coefficients and arbitrary shapes; polymers flexibility and an acoustic match to water. No family wins on every count, and the choice usually turns on impedance and temperature range rather than on the coefficient. ### Crystalline materials [[Quartz|Quartz]] is the reference: modest coefficients but very low mechanical loss, excellent stability, and cuts whose frequency barely changes with temperature — unbeatable for frequency control, nearly useless for force. Lithium niobate and lithium tantalate combine large coefficients with optical transparency and dominate surface-acoustic-wave devices. Rochelle salt, the Curies' largest signal, survives only as a footnote, its response outweighed by its solubility. ### Ceramics Poled ferroelectric ceramics are the workhorses. PZT is a solid solution of [[Lead|lead]] [[Zirconium|zirconate]] and lead [[Titanium|titanate]], its composition set near the boundary between two ferroelectric phases, where polarization rotates almost freely and the coefficients peak. [[Ceramic_engineering|Processing]] governs the result as much as composition: grain size, porosity from [[Sintering|sintering]] and the poling schedule all move the coefficients. Doping gives hard formulations for high-power transducers and soft ones for the largest response. ### Lead-free piezoceramics PZT is more than half lead by mass, which puts it at odds with restrictions on hazardous substances. The leading replacements are alkali niobates and bismuth-based perovskites; both reach useful coefficients but give up something — a lower Curie temperature, a narrower range, more sensitivity to processing. None matches PZT across the whole envelope, which is why exemptions have persisted. ### III–V and II–VI semiconductors Compound [[Semiconductor|semiconductors]] with the zincblende or wurtzite structure lack an inversion centre and are piezoelectric, with coefficients far below a ceramic's. Their importance is as a coupling that cannot be switched off: strain in a nitride heterostructure generates internal fields large enough to shift a light-emitting layer's emission and separate the carriers meant to recombine there. Zinc oxide and aluminium nitride, deposited as thin films by the equipment that makes the device, are the standard resonator layers. ### Polymers Polyvinylidene fluoride and its copolymers become piezoelectric when stretched to align the chains into a polar crystalline phase and then poled. Their coefficients are small, but their [[Polymer_engineering|polymer]] properties change the application: flexible, available as large thin sheets, with an [[Acoustic_impedance|acoustic impedance]] close to water's and tissue's, so a film couples into either without the matching layers a ceramic needs. ### Other materials Piezoelectricity appears outside engineering materials. [[Bone|Bone]] is piezoelectric, and the charge generated by loading has long been proposed as part of the signal directing remodelling; collagen and cellulose show it for the same symmetry reason. Composites mix families deliberately: ceramic rods in a polymer matrix give much of the ceramic's coefficient with the polymer's impedance. ## Application The applications divide on two questions: which direction of the effect is used, and whether the device runs at resonance. Direct-effect devices make a signal and are limited by the mechanical energy available; converse-effect devices make motion, limited by stroke. Resonant devices trade bandwidth for a large gain at one frequency; off-resonance devices keep a flat response and get much less out of it. ### High voltage and power sources The direct effect makes voltage easily and current with difficulty. A piezoelectric igniter — a spring-loaded hammer striking a poled ceramic — produces several kilovolts across a spark gap from a thumb's work, which is how a gas lighter runs without a battery. Scaled up, the principle harvests ambient [[Vibration|vibration]]. The ceiling is the energy available: the sim's press stores about 60 μJ, so a harvester's output is microwatts and its job is to run a sensor that sleeps.[^derived-pz] ### Sensors A piezoelectric sensor answers to change, not level. Charge leaks away through any finite input resistance, so a static load gives a decaying signal and the device measures dynamic quantities — pressure fluctuation, force transients, acceleration. The high-frequency limit is its own resonance, the low-frequency limit the charge amplifier's time constant. Within that band the advantages are enormous stiffness, linearity over several decades, and no power needed to make the signal. ### Actuators The converse effect gives extraordinary resolution and very little travel. Forty nanometres per hundred volts on a 1 mm disc is a useless stroke and an exquisite resolution, and every actuator design trades one for the other: stacks multiply displacement by the layer count, and bimorphs bend rather than extend, giving a millimetre of tip motion at the cost of most of the force.[^derived-pz] They are unmatched at positioning against stiffness — a [[Scanning_tunneling_microscope|scanning tunnelling microscope]] moves its tip this way. ### Frequency standard A quartz plate driven electrically rings at a mechanical resonance whose frequency depends on its dimensions and cut, and whose sharpness is beyond any electrical circuit. An oscillator locked to it keeps the frequency of a radio, a clock or a watch. The watch case is arithmetic: a crystal cut to 32,768 Hz, which is 2¹⁵, divides to one pulse per second in fifteen binary stages, so the number follows from the divider, not the [[Quartz|quartz]].[^derived-pz] ### Piezoelectric motors Two elements driven a quarter cycle apart make a point on a surface travel a small ellipse, and a rotor pressed against it is carried along by friction — a motor of many small pushes per second rather than a magnetic field. Such motors are slow but give high torque without gearing, hold position unpowered, make no magnetic field, and work in vacuum and at cryogenic temperatures — hence camera lenses and spacecraft instruments. ### Reduction of vibrations and noise An element bonded to a structure removes [[Vibration|vibration]] in either mode. Passively, it is wired to a resistive or resonant shunt, so strain makes charge dissipated there: mechanical energy leaves as heat by an electrical path, a tunable damper with no moving parts. Actively, a second element senses the motion and a controller drives the first out of phase, cancelling the [[Sound|sound]]. ### Surgery A piezoelectric surgical instrument vibrates a cutting tip at ultrasonic frequency with an amplitude of tens of micrometres. Because mineralized and soft tissue respond differently there, it cuts bone while leaving nerves, vessels and membranes intact — the property that made it standard in oral and craniofacial work. The same [[Ultrasound|ultrasonic]] principle drives dental scalers. ### Piezoelectric metamaterials with electro-momentum couplings A metamaterial is a composite whose structure, not its constituents, sets its effective properties. Breaking spatial symmetry within the repeating cell of a piezoelectric composite couples the electric field to the mechanical momentum — a term with no counterpart above. The interest is directional control: such a structure transmits an [[Acoustic_wave|acoustic wave]] one way differently from the other, and describing it needs those relations extended, not re-fitted. ## See also - [[Lead_zirconate_titanate]] — the ceramic behind most devices - [[Quartz]] — the frequency standard and the sim's second preset - [[Barium_titanate]] — the first of the poled ferroelectric ceramics - [[Ferroelectricity]] — what makes a ceramic pollable - [[Thermoelectric_effect]] — the other direct converter on the spine; its sim sits on the Energy flagship - [[Curie_temperature]] — the ceiling on poling - [[Ultrasound]] - [[Sensor]] ## References [^mitofsky-ch2]: Mitofsky, Andrea (2018). *Direct Energy*. Chapter "I. Survey of Energy Conversion Devices" (pp. 33–254): piezoelectric devices, the linear constitutive relations, capacitors and the poled ferroelectric ceramics (page to pin). https://open.umn.edu/opentextbooks/textbooks/direct-energy [^mitofsky-carnot]: Mitofsky, Andrea (2018). *Direct Energy*. Chapter "II. Theoretical Ideas", the Carnot efficiency `eta = 1 - Tc/Th`, which the book states covers heat-to-other-energy converters but not photovoltaic or piezoelectric devices (pp. 199–200). https://open.umn.edu/opentextbooks/textbooks/direct-energy [^curie1880]: Curie, Jacques; Curie, Pierre (1880). "Développement, par compression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées." *Bulletin de la Société minéralogique de France* 3. Pages and DOI to pin. [^langevin-cn]: Citation needed. The attribution of the first practical quartz–steel ultrasonic echo-ranging transducer to Paul Langevin and collaborators in France around 1917 is not documented in any Portal Book or open text on this page's shelf. The records that would settle it are Langevin's wartime patents and the French naval reports of 1915–1918; none is cited here rather than assert a date without a source. [^openstax-v2-cap]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 8, "Capacitance": the parallel-plate result C = ε₀·ε_r·A/d and the stored energy ½CV² used for the disc calculations (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^manual05-94]: Wikitube MICROSIM_GUIDE sub-manual 05, *Chemistry and Electrochemistry*, §9.4 "A thermoelectric generator under the Carnot ceiling", which carries the page cites to [^mitofsky-carnot] and records that the Portal Book's Carnot discussion explicitly excludes piezoelectric devices. This is the cross-link to the [[Thermoelectric_effect|thermoelectric]] sim, which the contract places on the Energy flagship. [^spec-m41]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M41: new root sim in three.js, the crystal squeezed along one axis; direct effect `D = d·T` with `Q = d_33·F` and converse effect `S = d·E`; the control is the applied force (or field); presets quartz (d₁₁ 2.3 pC/N), PZT (d₃₃ ≈ 400 pC/N) and BaTiO₃; a 1 kN press on a 1 cm² PZT disc reads 0.4 μC, with the voltage across the disc from `V = Q/C`. [^derived-pz]: Computed for this article from the equations on the page and the coefficients of [^spec-m41], with ε₀ = 8.8542×10⁻¹² F/m. A 1 kN load on 1 cm² is a stress of 10 MPa; Q = d₃₃·F gives 0.4 μC for PZT and 2.3 nC for quartz. For a 1 mm-thick disc, C = ε₀·ε_r·A/t is 1.33 nF at ε_r = 1,500 and 4.0 pF at ε_r = 4.5, giving V = Q/C of 301 V and 577 V — a ratio of 1.92 in quartz's favour — and a stored energy ½CV² of 60 μJ for the PZT case. ILLUSTRATIVE: the two relative permittivities are round order-of-magnitude values chosen for this article, not read from a Portal Book; only their orders, the ~300-fold ratio between them and the conclusion that the two open-circuit voltages agree within a factor of two are claimed. Converse effect: `S = d·E` with d₃₃ = 400 pm/V and 100 V across 1 mm gives E = 10⁵ V/m, S = 4×10⁻⁵ and a displacement of 40 nm; a stack of N layers each driven at V gives `dL = d·V·N` = 4 μm for N = 100 at 100 V, independent of layer thickness. The unit identity 1 pC/N = 1 pm/V is exact, since C/N and m/V are both C·m/J. 2¹⁵ = 32,768. ## Further reading - Mitofsky, Andrea (2018). *Direct Energy*, Chapter I for piezoelectric and other direct energy-conversion devices side by side, and Chapter II for the thermodynamic limits that do and do not apply to them. https://open.umn.edu/opentextbooks/textbooks/direct-energy - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*, Chapter 8, for the capacitance and stored-energy relations used in the worked disc. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 ## External links - The Wikipedia pair's *External links* section lists coefficient tables, measurement standards and manufacturer data; the Portal Book chapters above are the sources of this page. <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Piezoelectricity.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Piezoelectricity* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Piezoelectricity.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Piezoelectricity.html" data-title="Piezoelectricity"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Piezoelectricity) : [Wikitube](https://en.wikitube.io/wiki/Piezoelectricity) · pinned revision [1372971187](https://en.wikipedia.org/w/index.php?oldid=1372971187) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M41 · sim pending (matter/Piezoelectricity).*