piezoelectric

• 圧電気の、圧電の

関

piezoelectricity

　

WordNet

1. relating to or involving piezoelectricity; "piezoelectric plates"
2. electricity produced by mechanical pressure on certain crystals (notably quartz or Rochelle salt); alternatively, electrostatic stress produces a change in the linear dimensions of the crystal (同)piezoelectric effect, piezo effect
3. a crystal that can be used as a transducer

PrepTutorEJDIC

1. ピエゾ電気,圧電気(ある種の石への圧力で生ずる電気)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/10/12 18:15:10」(JST)

wiki en

[Wiki en表示]

Piezoelectricity /piˌeɪzoʊˌilɛkˈtrɪsɪti/ is the electric charge that accumulates in certain solid materials (such as crystals, certain ceramics, and biological matter such as bone, DNA and various proteins) in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure and heat. It is derived from the Greek πιέζειν piezein, which means to squeeze or press, and ἤλεκτρον ēlektron, which means amber, an ancient source of electric charge. Piezoelectricity was discovered in 1880 by French physicists Jacques and Pierre Curie.

The piezoelectric effect is understood as the linear electromechanical interaction between the mechanical and the electrical state in crystalline materials with no inversion symmetry. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field). For example, lead zirconate titanate crystals will generate measurable piezoelectricity when their static structure is 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 to the material. The inverse piezoelectric effect is used in the production of ultrasonic sound waves.

Piezoelectricity is found in useful applications, such as the production and detection of sound, generation of high voltages, electronic frequency generation, microbalances, to drive an ultrasonic nozzle, and ultrafine focusing of optical assemblies. It is also the basis of a number of scientific instrumental techniques with atomic resolution, the scanning probe microscopies, such as STM, AFM, MTA, SNOM, etc., and everyday uses, such as acting as the ignition source for cigarette lighters, and push-start propane barbecues, as well as the time reference source in quartz watches.

History

Discovery and early research

The 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.

The first demonstration of the direct piezoelectric effect was in 1880 by the brothers Pierre Curie and Jacques Curie. 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, cane sugar, and Rochelle salt (sodium potassium tartrate tetrahydrate). Quartz and Rochelle salt exhibited the most piezoelectricity.

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. The Curies immediately confirmed the existence of the converse effect, 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. 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), which described the 20 natural crystal classes capable of piezoelectricity, and rigorously defined the piezoelectric constants using tensor analysis.

World War I and post-war

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 ultrasonic submarine detector. 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. 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 reflectometers (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, Russia, and Japan discovered a new class of synthetic materials, called 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 didn't 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 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 car models as an echolocation device, helping the driver determine the distance from the car to any objects that may be in its path.

Mechanism

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 moments in solids. The latter may either be induced for ions on crystal lattice sites with asymmetric charge surroundings (as in BaTiO₃ and PZTs) or may directly be carried by molecular groups (as in cane sugar). The dipole density or polarization (dimensionality [C·m/m³] ) may easily be calculated for crystals by summing up the dipole moments per volume of the crystallographic unit cell. 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 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.

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 cm³ cube of quartz with 2 kN (500 lbf) of correctly applied force can produce a voltage of 12500 V.

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.

Mathematical description

Linear piezoelectricity is the combined effect of

• The linear electrical behavior of the material:

$\mathbf {D} ={\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies \quad D_{i}=\varepsilon _{ij}\,E_{j}\;$

where D is the electric charge density displacement (electric displacement), ε is permittivity (free-body dielectric constant), E is electric field strength, and $\nabla \cdot \mathbf {D} =0\,\,,\,\nabla \times \mathbf {E} =\mathbf {0}$.

• Hooke's Law for linear elastic materials:

${\boldsymbol {S}}={\mathsf {s}}\,{\boldsymbol {T}}\quad \implies \quad S_{ij}=s_{ijkl}\,T_{kl}\;$

where S is strain, s is compliance under short-circuit conditions, T is stress, and

$\nabla \cdot {\boldsymbol {T}}=\mathbf {0} \,\,,\,{\boldsymbol {S}}={\frac {\nabla \mathbf {u} +\mathbf {u} \nabla }{2}}$.

These may be combined into so-called coupled equations, of which the strain-charge form is:

${\begin{aligned}{\boldsymbol {S}}&={\mathsf {s}}\,{\boldsymbol {T}}+{\mathfrak {d}}^{t}\,\mathbf {E} \quad \implies \quad S_{ij}=s_{ijkl}\,T_{kl}+d_{kij}\,E_{k}\\\mathbf {D} &={\mathfrak {d}}\,{\boldsymbol {T}}+{\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies \quad D_{i}=d_{ijk}\,T_{jk}+\varepsilon _{ij}\,E_{j}\,.\end{aligned}}$

In matrix form,

${\begin{aligned}\{S\}&=\left[s^{E}\right]\{T\}+[d^{\mathrm {t} }]\{E\}\\\{D\}&=[d]\{T\}+\left[\varepsilon ^{T}\right]\{E\}\,,\end{aligned}}$

where [d] is the matrix for the direct piezoelectric effect and [d^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 transposition of a matrix.

Notice that the third order tensor ${\mathfrak {d}}$ 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 4mm (C_4v) crystal class (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO₃) as well as the 6mm crystal class may also be written as (ANSI IEEE 176):

${\displaystyle {\begin{bmatrix}S_{1}\\S_{2}\\S_{3}\\S_{4}\\S_{5}\\S_{6}\end{bmatrix}}={\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\s_{21}^{E}&s_{22}^{E}&s_{23}^{E}&0&0&0\\s_{31}^{E}&s_{32}^{E}&s_{33}^{E}&0&0&0\ ==Wikipedia preview== ==Wikipedia preview==$出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/12/05 12:10:06」(JST) ====[http://en.wikipedia.org/wiki/piezoelectric wiki en]====

Piezoelectricity /piˌeɪzoʊˌilɛkˈtrɪsɪti/ is the electric charge that accumulates in certain solid materials (such as crystals, certain ceramics, and biological matter such as bone, DNA and various proteins) in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure and latent heat. It is derived from the Greek πιέζειν piezein, which means to squeeze or press, and ἤλεκτρον ēlektron, which means amber, an ancient source of electric charge. Piezoelectricity was discovered in 1880 by French physicists Jacques and Pierre Curie.

The piezoelectric effect is understood as the linear electromechanical interaction between the mechanical and the electrical state in crystalline materials with no inversion symmetry. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field). For example, lead zirconate titanate crystals will generate measurable piezoelectricity when their static structure is 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 to the material. The inverse piezoelectric effect is used in the production of ultrasonic sound waves.

Piezoelectricity is found in useful applications, such as the production and detection of sound, generation of high voltages, electronic frequency generation, microbalances, to drive an ultrasonic nozzle, and ultrafine focusing of optical assemblies. It is also the basis of a number of scientific instrumental techniques with atomic resolution, the scanning probe microscopies, such as STM, AFM, MTA, SNOM, etc., and everyday uses, such as acting as the ignition source for cigarette lighters, and push-start propane barbecues, as well as the time reference source in quartz watches.

History

Discovery and early research

The 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.

The first demonstration of the direct piezoelectric effect was in 1880 by the brothers Pierre Curie and Jacques Curie. 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, cane sugar, and Rochelle salt (sodium potassium tartrate tetrahydrate). Quartz and Rochelle salt exhibited the most piezoelectricity.

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. The Curies immediately confirmed the existence of the converse effect, 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. 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), which described the 20 natural crystal classes capable of piezoelectricity, and rigorously defined the piezoelectric constants using tensor analysis.

World War I and post-war

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 ultrasonic submarine detector. 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. 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 reflectometers (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, Russia, and Japan discovered a new class of synthetic materials, called 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 didn't 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 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 car models as an echolocation device, helping the driver determine the distance from the car to any objects that may be in its path.

Mechanism

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 moments in solids. The latter may either be induced for ions on crystal lattice sites with asymmetric charge surroundings (as in BaTiO₃ and PZTs) or may directly be carried by molecular groups (as in cane sugar). The dipole density or polarization (dimensionality [C·m/m³] ) may easily be calculated for crystals by summing up the dipole moments per volume of the crystallographic unit cell. 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 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.

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 cm³ cube of quartz with 2 kN (500 lbf) of correctly applied force can produce a voltage of 12500 V.

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.

Mathematical description

Linear piezoelectricity is the combined effect of

• The linear electrical behavior of the material:

$\mathbf {D} ={\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies \quad D_{i}=\varepsilon _{ij}\,E_{j}\;$

where D is the electric charge density displacement (electric displacement), ε is permittivity (free-body dielectric constant), E is electric field strength, and $\nabla \cdot \mathbf {D} =0\,\,,\,\nabla \times \mathbf {E} =\mathbf {0}$.

• Hooke's Law for linear elastic materials:

${\boldsymbol {S}}={\mathsf {s}}\,{\boldsymbol {T}}\quad \implies \quad S_{ij}=s_{ijkl}\,T_{kl}\;$

where S is strain, s is compliance under short-circuit conditions, T is stress, and

$\nabla \cdot {\boldsymbol {T}}=\mathbf {0} \,\,,\,{\boldsymbol {S}}={\frac {\nabla \mathbf {u} +\mathbf {u} \nabla }{2}}$.

These may be combined into so-called coupled equations, of which the strain-charge form is:

${\begin{aligned}{\boldsymbol {S}}&={\mathsf {s}}\,{\boldsymbol {T}}+{\mathfrak {d}}^{t}\,\mathbf {E} \quad \implies \quad S_{ij}=s_{ijkl}\,T_{kl}+d_{kij}\,E_{k}\\\mathbf {D} &={\mathfrak {d}}\,{\boldsymbol {T}}+{\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies \quad D_{i}=d_{ijk}\,T_{jk}+\varepsilon _{ij}\,E_{j}\,.\end{aligned}}$

In matrix form,

${\begin{aligned}\{S\}&=\left[s^{E}\right]\{T\}+[d^{\mathrm {t} }]\{E\}\\\{D\}&=[d]\{T\}+\left[\varepsilon ^{T}\right]\{E\}\,,\end{aligned}}$

where [d] is the matrix for the direct piezoelectric effect and [d^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 transposition of a matrix.

Notice that the third order tensor ${\mathfrak {d}}$ 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 4mm (C_4v) crystal class (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO₃) as well as the 6mm crystal class may also be written as (ANSI IEEE 176):

${\begin{bmatrix}S_{1}\\S_{2}\\S_{3}\\S_{4}\\S_{5}\\S_{6}\end{bmatrix}}={\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\s_{21}^{E}&s_{22}^{E}&s_{23}^{E}&0&0&0\\s_{31}^{E}&s_{32}^{E}&s_{33}^{E}&0&0&0\&0&0&s_{44}^{E}&0&0\&0&0&0&s_{55}^{E}&0\&0&0&0&0&s_{66}^{E}=2\left(s_{11}^{E}-s_{12}^{E}\right)\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}0&0&d_{31}\&0&d_{32}\&0&d_{33}\&d_{24}&0\\d_{15}&0&0\&0&0\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}&0&0&s_{44}^{E}&0&0\&0&0&0&s_{55}^{E}&0\&0&0&0&0&s_{66}^{E}=2\left(s_{11}^{E}-s_{12}^{E}\right)\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}0&0&d_{31}\&0&d_{32}\&0&d_{33}\&d_{24}&0\\d_{15}&0&0\&0&0\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}}$

${\begin{bmatrix}D_{1}\\D_{2}\\D_{3}\end{bmatrix}}={\begin{bmatrix}0&0&0&0&d_{15}&0\&0&0&d_{24}&0&0\\d_{31}&d_{32}&d_{33}&0&0&0\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}{\varepsilon }_{11}&0&0\&{\varepsilon }_{22}&0\&0&{\varepsilon }_{33}\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}&0&0&d_{24}&0&0\\d_{31}&d_{32}&d_{33}&0&0&0\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}{\varepsilon }_{11}&0&0\&{\varepsilon }_{22}&0\&0&{\varepsilon }_{33}\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}}$

where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.

Although the above equations are the most used form in literature, some comments about the notation are necessary. Generally, D and E are vectors, that is, Cartesian tensors 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 → 1; 22 → 2; 33 → 3; 23 → 4; 13 → 5; 12 → 6. (Different conventions may be used by different authors in literature. For example, some use 12 → 4; 23 → 5; 31 → 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-4 tensor. Such a relabeled notation is often called Voigt notation. Whether the shear strain components S₄, S₅, S₆ 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(sE

11 − sE

12). Engineering shear strains are double the value of the corresponding tensor shear, such as S₆ = 2S₁₂ and so on. This also means that s₆₆ = 1/G₁₂, where G₁₂ is the shear modulus.

In total, there are four piezoelectric coefficients, d_ij, e_ij, g_ij, and h_ij defined as follows:

${\begin{aligned}d_{ij}&=\left({\frac {\partial D_{i}}{\partial T_{j}}}\right)^{E}&&=\left({\frac {\partial S_{j}}{\partial E_{i}}}\right)^{T}\\e_{ij}&=\left({\frac {\partial D_{i}}{\partial S_{j}}}\right)^{E}&&=-\left({\frac {\partial T_{j}}{\partial E_{i}}}\right)^{S}\\g_{ij}&=-\left({\frac {\partial E_{i}}{\partial T_{j}}}\right)^{D}&&=\left({\frac {\partial S_{j}}{\partial D_{i}}}\right)^{T}\\h_{ij}&=-\left({\frac {\partial E_{i}}{\partial S_{j}}}\right)^{D}&&=-\left({\frac {\partial T_{j}}{\partial D_{i}}}\right)^{S}\\\end{aligned}}$

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. 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_ij from electrostatic lattice constants or higher-order Madelung constants.

Crystal classes

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 classes, 21 are non-centrosymmetric (not having a centre of symmetry), and of these, 20 exhibit direct piezoelectricity (the 21st is the cubic class 432). Ten of these represent the polar crystal classes, 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 the application of an electric field, the material is said to be ferroelectric.

• Polar crystal classes: 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm.
• Piezoelectric crystal classes: 1, 2, m, 222, mm2, 4, 4, 422, 4mm, 42m, 3, 32, 3m, 6, 6, 622, 6mm, 62m, 23, 43m.

For polar crystals, for which P ≠ 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 = 0) to a polar one, having P ≠ 0.

Materials

Many materials, both natural and synthetic, exhibit piezoelectricity:

Naturally occurring crystals

• Quartz
• Berlinite (AlPO₄), a rare phosphate mineral that is structurally identical to quartz
• Sucrose (table sugar)
• Rochelle salt
• Topaz
• Tourmaline-group minerals
• Lead titanate (PbTiO₃). Although it occurs in nature as mineral macedonite, it is synthesized for research and applications.

The action of 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.

Bone

Dry bone exhibits some 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.

The piezoelectric effect is generally thought to act as a biological force sensor. 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. Further studies in the 1990s provided the mathematical equation to confirm long bone wave propagation as to that of hexagonal (Class 6) crystals.

Other natural materials

Biological materials exhibiting piezoelectric properties include:

• Tendon
• Silk
• Wood due to piezoelectric texture
• Enamel
• Dentin
• DNA
• Viral proteins, including those from bacteriophage. One study has found that thin films of M13 bacteriophage can be used to construct a piezoelectric generator sufficient to operate a liquid crystal display.

Synthetic crystals

• Langasite (La₃Ga₅SiO₁₄), a quartz-analogous crystal
• Gallium orthophosphate (GaPO₄), a quartz-analogous crystal
• Lithium niobate (LiNbO₃)
• Lithium tantalate (LiTaO₃)

Synthetic ceramics

Tetragonal unit cell of lead titanate

Ceramics with randomly oriented grains must be ferroelectric to exhibit piezoelectricity. The macroscopic piezoelectricity is possible in textured polycrystalline non-ferroelectric piezoelectric materials, such as AlN and ZnO. The family of ceramics with perovskite, tungsten-bronze and related structures exhibits piezoelectricity:

• Barium titanate (BaTiO₃)—Barium titanate was the first piezoelectric ceramic discovered.
• Lead zirconate titanate (Pb[Zr_xTi_1−x]O₃ with 0 ≤ x ≤ 1)—more commonly known as PZT, lead zirconate titanate is the most common piezoelectric ceramic in use today.
• Potassium niobate (KNbO₃)
• Sodium tungstate (Na₂WO₃)
• Ba₂NaNb₅O₅
• Pb₂KNb₅O₁₅
• Zinc oxide (ZnO)–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 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.

Lead-free piezoceramics

More recently, there is growing concern regarding the toxicity in lead-containing devices driven by the result of restriction of hazardous substances directive regulations. To address this concern, there has been a resurgence in the compositional development of lead-free piezoelectric materials.

• Sodium potassium niobate ((K,Na)NbO₃). This material is also known as NKN. 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_C. Certain compositions of this material have been shown to retain a high mechanical quality factor (Q_m ≈ 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.
• Bismuth ferrite (BiFeO₃) is also a promising candidate for the replacement of lead-based ceramics.
• Sodium niobate NaNbO₃
• Barium titanate (BaTiO₃)—Barium titanate was the first piezoelectric ceramic discovered.
• Bismuth titanate Bi₄Ti₃O₁₂
• Sodium bismuth titanate NaBi(TiO₃)₂

So far, neither the environmental effect nor the stability of supplying these substances have been measured.

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 Group III–V and II–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 e₁₄, coupled to shear components of the strain. In wurtzite, there are instead three independent piezoelectric coefficients: e₃₁, e₃₃ and e₁₅. The semiconductors where the strongest piezoelectricity is observed are those commonly found in the wurtzite structure, i.e. GaN, InN, AlN and ZnO. ZnO is the most used material in the recent field of piezotronics.

Since 2006, there have also been a number of reports of strong non linear piezoelectric effects in polar semiconductors. Such effects are generally recognized to be at least important if not of the same order of magnitude as the first order approximation.

Polymers

• Polyvinylidene fluoride (PVDF): PVDF exhibits piezoelectricity several times greater than quartz. Unlike ceramics, where the crystal structure of the material creates the piezoelectric effect, in polymers the intertwined long-chain molecules attract and repel each other when an electric field is applied.

Organic nanostructures

A strong shear piezoelectric activity was observed in self-assembled diphenylalanine peptide nanotubes (PNTs), indicating electric polarization directed along the tube axis. Comparison with LiNbO₃ and lateral signal calibration yields sufficiently high effective piezoelectric coefficient values of at least 60 pm/V (shear response for tubes of ≈200 nm in diameter). PNTs demonstrate linear deformation without irreversible degradation in a broad range of driving voltages.

Application

Currently, industrial and manufacturing is the largest application market for piezoelectric devices, followed by the automotive industry. Strong demand also comes from medical instruments as well as information and telecommunications. The global demand for piezoelectric devices was valued at approximately US$14.8 billion in 2010. The largest material group for piezoelectric devices is piezocrystal, and piezopolymer is experiencing the fastest growth due to its low weight and small size.

Piezoelectric crystals are now used in numerous ways:

High voltage and power sources

Direct piezoelectricity of some substances, like quartz, can generate potential differences of thousands of volts.

• The best-known application is the electric 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 stoves 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 soldiers' boots. However, these energy harvesting sources by association affect the body. DARPA's effort to harness 1–2 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 harvesting the energy from human movements in train stations or other public places and converting a dance floor to generate electricity. 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.
• 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 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 resonant frequency of the block, typically in the 100 kilohertz to 1 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. An extra feature of this transformer is that, by operating it above its resonant frequency, it can be made to appear as an inductive load, which is useful in circuits that require a controlled soft start. These devices can be used in DC–AC inverters to drive cold cathode fluorescent lamps. Piezo transformers are some of the most compact high voltage sources.

Sensors

Piezoelectric disk used as a guitar pickup

Many rocket-propelled grenades used a piezoelectric fuse. For example: RPG-7.

Main article: 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 microphones (sound waves bend the piezoelectric material, creating a changing voltage) and piezoelectric pickups for acoustic-electric guitars. 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 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.
• Power monitoring in high power applications (e.g. medical treatment, sonochemistry and industrial processing).
• Piezoelectric microbalances are used as very sensitive chemical and biological sensors.
• Piezos are sometimes used in strain gauges.
• A piezoelectric transducer was used in the penetrometer instrument on the Huygens Probe.
• Piezoelectric transducers are used in electronic drum pads to detect the impact of the drummer's sticks, and to detect muscle movements in medical acceleromyography.
• Automotive 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.

Actuators

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-µm precision, making piezo crystals the most important tool for positioning objects with extreme accuracy — thus their use in actuators. Multilayer ceramics, using layers thinner than 100 µm, allow reaching high electric fields with voltage lower than 150 V. These ceramics are used within two kinds of actuators: direct piezo actuators and Amplified piezoelectric actuators. While direct actuator's stroke is generally lower than 100 µm, amplified piezo actuators can reach millimeter strokes.

• Loudspeakers: Voltage is converted to mechanical movement of a metallic diaphragm.
• Piezoelectric motors: 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 microscopes and scanning tunneling microscopes employ converse piezoelectricity to keep the sensing needle close to the specimen.
• Inkjet printers: On many inkjet printers, piezoelectric crystals are used to drive the ejection of ink from the inkjet print head towards the paper.
• Diesel engines: High-performance common rail diesel engines use piezoelectric fuel injectors, 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 CT and MRI scanners where the strong radiation or magnetism precludes electric motors.
• Crystal earpieces 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.

Frequency standard

The piezoelectrical properties of quartz are useful as a standard of frequency.

• Quartz clocks 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 elastic material) has a precisely defined natural frequency (caused by its shape and size) at which it prefers to oscillate, and this is used to stabilize the frequency of a periodic voltage applied to the crystal.
• The same principle is critical in all radio transmitters and receivers, and in computers where it creates a clock pulse. Both of these usually use a frequency multiplier to reach gigahertz ranges.

Piezoelectric motors

A slip-stick actuator.

Main article: Piezoelectric motor

Types of piezoelectric motor include:

• The traveling-wave motor used for auto-focus in reflex cameras
• Inchworm motors for linear motion
• Rectangular four-quadrant motors with high power density (2.5 W/cm³) and speed ranging from 10 nm/s to 800 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 difference of 90°, the contact point between two surfaces vibrates in an elliptical path, producing a frictional 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 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 tennis rackets.

Infertility treatment

In people with previous total fertilization failure, piezoelectric activation of oocytes together with intracytoplasmic sperm injection (ICSI) seems to improve fertilization outcomes.

Surgery

A recent application of piezoelectric ultrasound sources is piezoelectric surgery, also known as piezosurgery. Piezosurgery is a minimally invasive technique that aims to cut a target tissue with little damage to neighboring tissues. For example, Hoigne et al. reported its use in hand surgery for the cutting of bone, using frequencies in the range 25–29 kHz, causing microvibrations of 60–210 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.

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.

In recent years, several attempts at the macro-scale application of the piezoelectric technology have emerged to harvest kinetic energy from walking pedestrians. The piezoelectric floors have been trialed since the beginning of 2007 in two Japanese train stations, Tokyo and Shibuya stations. The electricity generated from the foot traffic is used to provide all the electricity needed to run the automatic ticket gates and electronic display systems. In London, a famous nightclub exploited the piezoelectric technology in its dance floor. Parts of the lighting and sound systems in the club can be powered by the energy harvesting tiles. However, the piezoelectric tile deployed on the ground usually harvests energy from low frequency strikes provided by the foot traffic. This working condition may eventually lead to low power generation efficiency.

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. 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. 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 MW h/year, which would be sufficient to meet close to 0.5% of the annual energy needs of the building. In Israel, there is a company which has installed piezoelectric materials under a busy highway. The energy generated is adequate and powers street lights, billboards and signs.

Tire 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.

Photovoltaics

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 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.

See also

• Charge amplifier
• Electronic component
• Electret
• Electrostriction
• Energy harvesting, methods of converting other forms of energy to electricity.
• Ferroelectricity
• Flexoelectricity
• Magnetostriction
• Photoelectric effect
• Piezoluminescence
• Piezomagnetism
• Piezosurgical
• Piezoresistive effect
• Sonomicrometry
• Surface acoustic wave
• Triboluminescence
• Piezotronics
• Piezoelectric speaker

References

International standards

• 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 MHz

External links

• Gautschi, Gustav H. (2002). Piezoelectric Sensorics. Springer. ISBN 3-540-42259-5. 
• Piezo motor based microdrive for neural signal recording
• Research on new Piezoelectric materials
• Piezo Equations
• Piezo in Medical Design
• Video demonstration of Piezoelectricity
• DoITPoMS Teaching and Learning Package – Piezoelectric Materials
• PiezoMat.org – Online database for piezoelectric materials, their properties, and applications
• Piezo Motor Types
• Piezo-Theory & Applications

&0&0&s_{44}^{E}&0&0\<add_contents_exp><m=6 date=20130112>&0&0&0&s_{55}^{E}&0\<add_contents_exp><m=6 date=20130112>&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}\<add_contents_exp><m=6 date=20130112>&0&d_{32}\<add_contents_exp><m=6 date=20130112>&0&d_{33}\<add_contents_exp><m=6 date=20130112>&d_{24}&0\\d_{15}&0&0\<add_contents_exp><m=6 date=20130112>&0&0\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}}</annotation>

 </semantics>

</math>

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}D_{1}\\D_{2}\\D_{3}\end{bmatrix}}={\begin{bmatrix}0&0&0&0&d_{15}&0\<add_contents_exp><m=6 date=20130112>&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\<add_contents_exp><m=6 date=20130112>&{\varepsilon }_{22}&0\<add_contents_exp><m=6 date=20130112>&0&{\varepsilon }_{33}\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}}</annotation> </semantics> </math>

where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.

Although the above equations are the most used form in literature, some comments about the notation are necessary. Generally, D and E are vectors, that is, Cartesian tensors 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 → 1; 22 → 2; 33 → 3; 23 → 4; 13 → 5; 12 → 6. (Different conventions may be used by different authors in literature. For example, some use 12 → 4; 23 → 5; 31 → 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-4 tensor. Such a relabeled notation is often called Voigt notation. Whether the shear strain components S₄, S₅, S₆ 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(sE
11 − sE
12). Engineering shear strains are double the value of the corresponding tensor shear, such as S₆ = 2S₁₂ and so on. This also means that s₆₆ = 1/G₁₂, where G₁₂ is the shear modulus.

In total, there are four piezoelectric coefficients, d_ij, e_ij, g_ij, and h_ij defined as follows:

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d_{ij}&=\left({\frac {\partial D_{i}}{\partial T_{j}}}\right)^{E}&&=\left({\frac {\partial S_{j}}{\partial E_{i}}}\right)^{T}\\e_{ij}&=\left({\frac {\partial D_{i}}{\partial S_{j}}}\right)^{E}&&=-\left({\frac {\partial T_{j}}{\partial E_{i}}}\right)^{S}\\g_{ij}&=-\left({\frac {\partial E_{i}}{\partial T_{j}}}\right)^{D}&&=\left({\frac {\partial S_{j}}{\partial D_{i}}}\right)^{T}\\h_{ij}&=-\left({\frac {\partial E_{i}}{\partial S_{j}}}\right)^{D}&&=-\left({\frac {\partial T_{j}}{\partial D_{i}}}\right)^{S}\\\end{aligned}}}</annotation> </semantics> </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. 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_ij from electrostatic lattice constants or higher-order Madelung constants.

Crystal classes

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 classes, 21 are non-centrosymmetric (not having a centre of symmetry), and of these, 20 exhibit direct piezoelectricity (the 21st is the cubic class 432). Ten of these represent the polar crystal classes, 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 the application of an electric field, the material is said to be ferroelectric.

• Polar crystal classes: 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm.
• Piezoelectric crystal classes: 1, 2, m, 222, mm2, 4, 4, 422, 4mm, 42m, 3, 32, 3m, 6, 6, 622, 6mm, 62m, 23, 43m.

For polar crystals, for which P ≠ 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 = 0) to a polar one, having P ≠ 0.

Materials

Many materials, both natural and synthetic, exhibit piezoelectricity:

Naturally occurring crystals

• Quartz
• Berlinite (AlPO₄), a rare phosphate mineral that is structurally identical to quartz
• Sucrose (table sugar)
• Rochelle salt
• Topaz
• Tourmaline-group minerals
• Lead titanate (PbTiO₃). Although it occurs in nature as mineral macedonite, it is synthesized for research and applications.

The action of 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.

Bone

Dry bone exhibits some 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.

The piezoelectric effect is generally thought to act as a biological force sensor. 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. Further studies in the 1990s provided the mathematical equation to confirm long bone wave propagation as to that of hexagonal (Class 6) crystals.

Other natural materials

Biological materials exhibiting piezoelectric properties include:

• Tendon
• Silk
• Wood due to piezoelectric texture
• Enamel
• Dentin
• DNA
• Viral proteins, including those from bacteriophage. One study has found that thin films of M13 bacteriophage can be used to construct a piezoelectric generator sufficient to operate a liquid crystal display.

Synthetic crystals

• Langasite (La₃Ga₅SiO₁₄), a quartz-analogous crystal
• Gallium orthophosphate (GaPO₄), a quartz-analogous crystal
• Lithium niobate (LiNbO₃)
• Lithium tantalate (LiTaO₃)

Synthetic ceramics

Tetragonal unit cell of lead titanate

Ceramics with randomly oriented grains must be ferroelectric to exhibit piezoelectricity. The macroscopic piezoelectricity is possible in textured polycrystalline non-ferroelectric piezoelectric materials, such as AlN and ZnO. The family of ceramics with perovskite, tungsten-bronze and related structures exhibits piezoelectricity:

• Barium titanate (BaTiO₃)—Barium titanate was the first piezoelectric ceramic discovered.
• Lead zirconate titanate (Pb[Zr_xTi_1−x]O₃ with 0 ≤ x ≤ 1)—more commonly known as PZT, lead zirconate titanate is the most common piezoelectric ceramic in use today.
• Potassium niobate (KNbO₃)
• Sodium tungstate (Na₂WO₃)
• Ba₂NaNb₅O₅
• Pb₂KNb₅O₁₅
• Zinc oxide (ZnO)–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 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.

Lead-free piezoceramics

More recently, there is growing concern regarding the toxicity in lead-containing devices driven by the result of restriction of hazardous substances directive regulations. To address this concern, there has been a resurgence in the compositional development of lead-free piezoelectric materials.

• Sodium potassium niobate ((K,Na)NbO₃). This material is also known as NKN. 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_C. Certain compositions of this material have been shown to retain a high mechanical quality factor (Q_m ≈ 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.
• Bismuth ferrite (BiFeO₃) is also a promising candidate for the replacement of lead-based ceramics.
• Sodium niobate NaNbO₃
• Barium titanate (BaTiO₃)—Barium titanate was the first piezoelectric ceramic discovered.
• Bismuth titanate Bi₄Ti₃O₁₂
• Sodium bismuth titanate NaBi(TiO₃)₂

So far, neither the environmental effect nor the stability of supplying these substances have been measured.

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 Group III–V and II–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 e₁₄, coupled to shear components of the strain. In wurtzite, there are instead three independent piezoelectric coefficients: e₃₁, e₃₃ and e₁₅. The semiconductors where the strongest piezoelectricity is observed are those commonly found in the wurtzite structure, i.e. GaN, InN, AlN and ZnO. ZnO is the most used material in the recent field of piezotronics.

Since 2006, there have also been a number of reports of strong non linear piezoelectric effects in polar semiconductors. Such effects are generally recognized to be at least important if not of the same order of magnitude as the first order approximation.

Polymers

• Polyvinylidene fluoride (PVDF): PVDF exhibits piezoelectricity several times greater than quartz. Unlike ceramics, where the crystal structure of the material creates the piezoelectric effect, in polymers the intertwined long-chain molecules attract and repel each other when an electric field is applied.

Organic nanostructures

A strong shear piezoelectric activity was observed in self-assembled diphenylalanine peptide nanotubes (PNTs), indicating electric polarization directed along the tube axis. Comparison with LiNbO₃ and lateral signal calibration yields sufficiently high effective piezoelectric coefficient values of at least 60 pm/V (shear response for tubes of ≈200 nm in diameter). PNTs demonstrate linear deformation without irreversible degradation in a broad range of driving voltages.

Application

Currently, industrial and manufacturing is the largest application market for piezoelectric devices, followed by the automotive industry. Strong demand also comes from medical instruments as well as information and telecommunications. The global demand for piezoelectric devices was valued at approximately US$14.8 billion in 2010. The largest material group for piezoelectric devices is piezocrystal, and piezopolymer is experiencing the fastest growth due to its low weight and small size.

Piezoelectric crystals are now used in numerous ways:

High voltage and power sources

Direct piezoelectricity of some substances, like quartz, can generate potential differences of thousands of volts.

• The best-known application is the electric 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 stoves 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 soldiers' boots. However, these energy harvesting sources by association affect the body. DARPA's effort to harness 1–2 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 harvesting the energy from human movements in train stations or other public places and converting a dance floor to generate electricity. 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.
• 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 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 resonant frequency of the block, typically in the 100 kilohertz to 1 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. An extra feature of this transformer is that, by operating it above its resonant frequency, it can be made to appear as an inductive load, which is useful in circuits that require a controlled soft start. These devices can be used in DC–AC inverters to drive cold cathode fluorescent lamps. Piezo transformers are some of the most compact high voltage sources.

Sensors

Piezoelectric disk used as a guitar pickup

Many rocket-propelled grenades used a piezoelectric fuse. For example: RPG-7.

Main article: 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 microphones (sound waves bend the piezoelectric material, creating a changing voltage) and piezoelectric pickups for acoustic-electric guitars. 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 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.
• Power monitoring in high power applications (e.g. medical treatment, sonochemistry and industrial processing).
• Piezoelectric microbalances are used as very sensitive chemical and biological sensors.
• Piezos are sometimes used in strain gauges.
• A piezoelectric transducer was used in the penetrometer instrument on the Huygens Probe.
• Piezoelectric transducers are used in electronic drum pads to detect the impact of the drummer's sticks, and to detect muscle movements in medical acceleromyography.
• Automotive 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.

Actuators

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-µm precision, making piezo crystals the most important tool for positioning objects with extreme accuracy — thus their use in actuators. Multilayer ceramics, using layers thinner than 100 µm, allow reaching high electric fields with voltage lower than 150 V. These ceramics are used within two kinds of actuators: direct piezo actuators and Amplified piezoelectric actuators. While direct actuator's stroke is generally lower than 100 µm, amplified piezo actuators can reach millimeter strokes.

• Loudspeakers: Voltage is converted to mechanical movement of a metallic diaphragm.
• Piezoelectric motors: 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 microscopes and scanning tunneling microscopes employ converse piezoelectricity to keep the sensing needle close to the specimen.
• Inkjet printers: On many inkjet printers, piezoelectric crystals are used to drive the ejection of ink from the inkjet print head towards the paper.
• Diesel engines: High-performance common rail diesel engines use piezoelectric fuel injectors, 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 CT and MRI scanners where the strong radiation or magnetism precludes electric motors.
• Crystal earpieces 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.

Frequency standard

The piezoelectrical properties of quartz are useful as a standard of frequency.

• Quartz clocks 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 elastic material) has a precisely defined natural frequency (caused by its shape and size) at which it prefers to oscillate, and this is used to stabilize the frequency of a periodic voltage applied to the crystal.
• The same principle is critical in all radio transmitters and receivers, and in computers where it creates a clock pulse. Both of these usually use a frequency multiplier to reach gigahertz ranges.

Piezoelectric motors

A slip-stick actuator.

Main article: Piezoelectric motor

Types of piezoelectric motor include:

• The traveling-wave motor used for auto-focus in reflex cameras
• Inchworm motors for linear motion
• Rectangular four-quadrant motors with high power density (2.5 W/cm³) and speed ranging from 10 nm/s to 800 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 difference of 90°, the contact point between two surfaces vibrates in an elliptical path, producing a frictional 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 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 tennis rackets.

Infertility treatment

In people with previous total fertilization failure, piezoelectric activation of oocytes together with intracytoplasmic sperm injection (ICSI) seems to improve fertilization outcomes.

Surgery

A recent application of piezoelectric ultrasound sources is piezoelectric surgery, also known as piezosurgery. Piezosurgery is a minimally invasive technique that aims to cut a target tissue with little damage to neighboring tissues. For example, Hoigne et al. reported its use in hand surgery for the cutting of bone, using frequencies in the range 25–29 kHz, causing microvibrations of 60–210 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.

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.

In recent years, several attempts at the macro-scale application of the piezoelectric technology have emerged to harvest kinetic energy from walking pedestrians. The piezoelectric floors have been trialed since the beginning of 2007 in two Japanese train stations, Tokyo and Shibuya stations. The electricity generated from the foot traffic is used to provide all the electricity needed to run the automatic ticket gates and electronic display systems. In London, a famous nightclub exploited the piezoelectric technology in its dance floor. Parts of the lighting and sound systems in the club can be powered by the energy harvesting tiles. However, the piezoelectric tile deployed on the ground usually harvests energy from low frequency strikes provided by the foot traffic. This working condition may eventually lead to low power generation efficiency.

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. 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. 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 MW h/year, which would be sufficient to meet close to 0.5% of the annual energy needs of the building. In Israel, there is a company which has installed piezoelectric materials under a busy highway. The energy generated is adequate and powers street lights, billboards and signs.

Tire 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.

Photovoltaics

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 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.

See also

• Charge amplifier
• Electronic component
• Electret
• Electrostriction
• Energy harvesting, methods of converting other forms of energy to electricity.
• Ferroelectricity
• Flexoelectricity
• Magnetostriction
• Photoelectric effect
• Piezoluminescence
• Piezomagnetism
• Piezosurgical
• Piezoresistive effect
• Sonomicrometry
• Surface acoustic wave
• Triboluminescence
• Piezotronics
• Piezoelectric speaker

References

International standards

• 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 MHz

External links

• Gautschi, Gustav H. (2002). Piezoelectric Sensorics. Springer. ISBN 3-540-42259-5. 
• Piezo motor based microdrive for neural signal recording
• Research on new Piezoelectric materials
• Piezo Equations
• Piezo in Medical Design
• Video demonstration of Piezoelectricity
• DoITPoMS Teaching and Learning Package – Piezoelectric Materials
• PiezoMat.org – Online database for piezoelectric materials, their properties, and applications
• Piezo Motor Types
• Piezo-Theory & Applications

</raw> </toggledisplay>

English Journal

• Manipulator Design and Operation for a Six-Degree-of-Freedom Handheld Tremor-Canceling Microsurgical Instrument.

• Yang S1, MacLachlan RA1, Riviere CN1.
• IEEE/ASME transactions on mechatronics : a joint publication of the IEEE Industrial Electronics Society and the ASME Dynamic Systems and Control Division.IEEE ASME Trans Mechatron.2015 Apr;20(2):761-772.
• This paper presents the design and actuation of a six-degree-of-freedom (6-DOF) manipulator for a handheld instrument, known as "Micron," which performs active tremor compensation during microsurgery. The design incorporates a Gough-Stewart platform based on piezoelectric linear motor, with a specif
• PMID 25419103

• Advantages and Challenges of Relaxor-PbTiO3 Ferroelectric Crystals for Electroacoustic Transducers- A Review.

• Zhang S1, Li F2, Jiang X3, Kim J3, Luo J4, Geng X5.
• Progress in materials science.Prog Mater Sci.2015 Mar 1;68:1-66.
• Relaxor-PbTiO3 (PT) based ferroelectric crystals with the perovskite structure have been investigated over the last few decades due to their ultrahigh piezoelectric coefficients (d 33 > 1500 pC/N) and electromechanical coupling factors (k 33 > 90%), far outperforming state-of-the-art ferroelec
• PMID 25530641

• Nicotine molecularly imprinted polymer: Synergy of coordination and hydrogen bonding.

• Huynh TP1, B K C C2, Sosnowska M1, Sobczak JW3, Nesterov VN2, D'Souza F4, Kutner W5.
• Biosensors & bioelectronics.Biosens Bioelectron.2015 Feb 15;64:657-63. doi: 10.1016/j.bios.2014.09.014. Epub 2014 Sep 15.
• Two new bis(2,2'-bithienyl)methane derivatives, one with the zinc phthalocyanine substituent (ZnPc-S16) and the other with the 2-hydroxyethyl substituent (EtOH-S4), were synthesized to serve as functional monomers for biomimetic recognition of nicotine (Nic) by molecular imprinting. Formation of a p
• PMID 25441415

Japanese Journal

• Determination of temperature dependences of material constants for lead-free (Na

• Jpn. J. Appl. Phys. 55(10S), 10TD02, 2016-09-01
• NAID 150000113128

• InGaAs/GaAsP系量子井戸太陽電池におけるミニバンド形成過程とその光学的評価

• 宮崎大學工學部紀要 45, 119-123, 2016-07-31
• NAID 120005830796

• Quantification of body movement measurements in low-birthweight infants

• 金沢大学つるま保健学会誌 40(1), 11-19, 2016-07-29
• NAID 120005830556

Related Links

• piezoelectricの意味・用例｜英辞郎 on the WEB：アルク

piezoelectric 【名】《物理》圧電（性）物質 【形】圧電（性）の - アルクがお届けする進化するオンライン英和・和英辞書データベース。一般的な単語や連語から、イディオム、専門用語、スラングまで幅広く収録。

• Piezoelectric Transducers

Piezoelectric Transducers The conversion of electrical pulses to mechanical vibrations and the conversion of returned mechanical vibrations back into electrical energy is the basis for ultrasonic testing. The active element is the heart ...

Related Pictures

---

★リンクテーブル★

リンク元	「圧電気」「圧電」
拡張検索	「piezoelectricity」

「圧電気」

　　[★]

英

piezoelectricity、piezoelectric

関

圧電、ピエゾ電気

「圧電」

　　[★]

英

piezoelectric

関

圧電気

「piezoelectricity」

　　[★]

• 圧電気、ピエゾ電気

関

piezoelectric