Aug. 18, 2025
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This tutorial provides an introduction to the basics of piezoelectricity. This includes an introduction to the nature of piezoelectricity, and a description of the two main families of piezoceramic materials (hard doped and soft doped). In this tutorial, you will also be introduced to the constitutive equations as well as the properties of piezoceramic material at high field. You will also find a description of the thermal properties of piezoceramic material, as well as an overview helping you select a ceramic material.
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The piezoelectric effect was discovered by Jacques and Pierre Curie in . The initial observation was the appearance of dielectric charge on a crystal proportional to an applied mechanical stress. Soon thereafter, the converse effect i.e. the geometrical strain of a crystal proportional to an applied electrical field, was discovered.
Basics on piezoelectric material
Piezoelectricity is the property of some materials to develop electric charge on their surface when mechanical stress is exerted on them. An applied electrical field produces a linearly proportional strain in these materials. The electrical response to mechanical stimulation is called the direct piezoelectric effect, and the mechanical response to electrical simulation is called the converse piezoelectric effect.
Different piezoelectric materials
Piezoelectric effect is exhibited by most of the materials that possess a non-centrosymmetric crystal structure. Some naturally occurring crystalline materials possessing these properties are quartz and tourmaline. Some artificially produced piezoelectric crystals are Rochelle salt, ammonium dihydrogen phosphate and lithium sulphate. Another class of materials possessing these properties is piezoelectric ceramics.
In contrast to the naturally occurring piezoelectric crystals, piezoelectric ceramics are of a “polycrystalline” structure. The most commonly produced piezoelectric ceramics are lead zirconate titanate (PZT), barium titanate and lead titanate. Polycrystalline ceramic materials have several advantages over single crystal piezoelectric materials, including the ease of fabrication and forming of various shapes and sizes. In contrast, single crystals must be cut along certain crystallographic directions, limiting the possible geometric shapes, but offer superior piezoelectric properties, except Curie and phase transition temperatures.
CTS Piezoelectric Products
PZT has crystal structures belonging to the perovskite family with the general formula AB03. In the following figure, the ideal, cubic perovskite structure is shown. PZT crystallites are centro-symmetric cubic (isotropic) above the Curie temperature and exhibit tetragonal symmetry (anisotropic structure) below the Curie temperature.
Poling process
Piezoelectric ceramics consist of grains (crystallites), each of these grains contains domains that are randomly oriented before poling, as shown in the left figure below. As a result, the net polarization of the material is zero and therefore ceramic does not exhibit piezoelectric properties. During poling process, adequate DC electrical field is applied and this applied electric field orients the domains in the electric field direction (as seen in the middle figure below) and lead to a remanent polarization of the material (as seen in the right figure below).
Although there are several types of piezoelectric ceramic materials available today, most can be placed into one of two general categories: “Hard” or “Soft” PZT materials. The perovskite structure is very tolerant to element substitution (doping) – therefore the terms “hard” and “soft” are used. Even small amounts of a dopant (~1%) may cause substantial changes in the properties of a material.
Characteristics of hard piezoceramic material
Hard piezoelectric ceramics have higher mechanical quality factor and are suitable for dynamic/on-resonance applications. Since higher mechanical quality factor provides more efficient energy conversion (from electrical to work), hard materials can withstand high level of electrical excitation and mechanical stress, generate less heat during this process and are not easy poled or depoled except at elevated temperature. Compared to soft piezoelectric materials, hard piezoelectric materials lack the strain because of the lower d coefficients.
Characteristics of soft piezoceramic material
Soft piezoelectric ceramics have higher piezoelectric coefficients compared to hard piezoelectric ceramics, at the expense of quality factor. Soft piezoelectric ceramics also provide higher sensitivity and permittivity and are well suited for static or semi static applications, where large strain is required. Soft piezoelectric ceramics, when operated in dynamic mode at high field suffer from high dielectric losses and low quality factors, which may lead to overheating over an extended period of operation.
Below you can see a comparison of the characteristics of the hard and soft doped piezoceramic material.
Type of ceramic Soft piezoceramic material Hard piezoceramic material Piezo constantsCTS Ceramic Material Properties
Because of the anisotropic nature of piezoelectric ceramics, properties vary depending on direction. To identify directions in a piezoelectric ceramic element, a specific coordinate system is used. Three axes are defined, termed 1, 2, and 3, analogous to X, Y, and Z of the classical three-dimensional orthogonal set of axes.
Piezoelectric coefficients and directions
The polar, or 3 axis, is determined by the direction of the poling. Unless the component needs to be utilized in shear mode, electric field is applied in direction 3. Directions 1 and 2 are physically equivalent so they can be defined arbitrarily, perpendicular to direction 3 and to each other. The directions termed 4, 5 and 6 correspond to tilting (shear) motions around axes 1, 2 and 3 respectively.
In shear mode, after poling, electrodes are stripped and redeposited perpendicular to axis 1. In this case, once electric field is applied, the component shears in one dimension without any change in other dimensions.
Piezoelectric materials can be characterized by several coefficients. Piezoelectric coefficients with double subscripts link electrical and mechanical quantities. The first subscript provides the direction of the electric field, or the dielectric charge produced. The second subscript provides the direction of the mechanical stress or strain.
The piezoelectric constants relating the mechanical strain produced by an applied electric field are termed the piezoelectric deformation constants, or the “d” coefficients. They are expressed in meters per volt [m/V]. Conversely, these coefficients which are also called piezoelectric charge constants may be viewed as relating the charge collected on the electrodes to the applied mechanical stress. The units can therefore also be expressed in Coulombs per Newton [C/N].
In addition, several piezoelectric material constants may be written with a “superscript” which specifies either a mechanical or an electrical boundary condition. The superscripts are T, E, D, and S, signifying:
Here are three examples of parameters used in the piezoelectric equations together with an explanation of their notation:
Fundamental piezoelectric equations
There are different ways of writing the fundamental equations of the piezoelectric materials, depending on which variables are of interest. The two most common forms are (the superscript t stands for matrix-transpose):
These matrix relationships are widely used for finite element modelling. For analytical approaches, in general only some of the relationships are useful so the problem can be further simplified. For example this relationship, extracted from line 3 of the first matrix equation, describes strain in direction 3 as a function of stress and field.
Just like any other elastic material, strain is proportional to the applied stress. But in addition for piezoelectric materials, an additional piezoelectric term is present, relating strain to electric field also.
Limitations of the linear constitutive equations
There are a number of limitations of the linear constitutive equations. The piezoelectric effect is actually non-linear in nature due to hysteresis and creep.
Furthermore, the dynamics of the material are not described by the linear constitutive equations. Piezoelectric coefficients are temperature dependent. Piezoelectric coefficients show a strong electric field dependency.
Piezoelectric materials exhibit non-linearity, hysteresis and creep. This section provides typical material data to understand and compensate these effects.
Linearity: Actuators (individual and stacked multilayer) and benders
The stroke versus applied voltage relationship for piezo electric actuators is not perfectly linear as predicted by the piezoelectric equations. Typical performances are shown in the following figures. As it can be seen, the extension vs voltage curve is actually slightly S-shaped. At low voltage, the curve for increasing voltage is concave upward and the shape is close to quadratic.
The example below shows the displacement during charging of an actuator using the piezoelectric material NCE57. Higher resolution curves can be found in the “hysteresis” section. Non-linearity implies that stroke at 1kV/mm is less than expected from the linear extrapolation using stroke at the maximum recommended field (which corresponds to 3kV/mm).
Very high electric field material data: Actuators (individual and and stacked multilayer) and benders
In some applications, it is desirable to archive maximum strain from the piezo electric element only by applying a very high electric field. In some cases the maximum recommended field strength of 3kV/mm may be exceeded e.g. for short-term use applications or static applications. Operating field of 4kV/mm is normally acceptable, however testing is recommended.
The figure below shows how strain evolves with electric field for our different materials up to a maximum electrical field strength of 9kV/mm. The drawback of applying a very high electric field is that the actuator lifetime is reduced drastically.
The data in the figure are only of informative character and we recommend to contact our R&D before designing actuators based on very high electric field.
Linearity: Shear plates
The peak to peak stroke versus peak applied voltage relationship for shear plates is not linear. Typical measurements are shown in the following figure. As it can be seen, the displacement increases when the actuator is used close to the maximum recommended voltage.
The polynomial trend follows the experimental relationship. With d being the displacement, t the height of the actuator and E the applied electric field (Voltage/height):
Hysteresis: Actuators (individual and stacked multilayer) and benders
All piezoelectric materials exhibit a mechanical hysteresis as the strain does not follow the same track upon charging and discharging. The hysteresis is expressed as the maximum difference between the two tracks divided by the maximum strain, as can be seen in the figure below. Hysteresis tends to decrease with ageing. If hysteresis is a problem for a specific application, it is common to use a model-based compensation or a feedback loop to compensate it. Feedback signal can be position, force or dielectric charge.
The hysteresis depends on the type of ceramics and the amplitude of the input signal and can vary from 13% to 20%.
Material Hysteresis (%) NCE46 20 NCE51/51F 19 NCE57 19 NCE59 13On bending actuators the same hysteresis is present. However, because of the push-pull configuration, it has a symmetrical shape.
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Hysteresis: Shear plates
Due to their high non-linearity, shear plates exhibit much higher hysteresis than other actuator types. Hysteresis at full voltage amplitude is in the order of 35%. Reducing the amplitude of the voltage will reduce hysteresis.
Operation under reverse bias: Actuators (individual and stacked multilayer) and benders
In addition to the normal hysteresis curve AB when the applied voltage is positive, the butterfly diagram CDEFG defines the behavior of the material through a complete cycle of positive and negative operating electric fields. Negative electric fields produce negative strain along curve C until the depoling field (coercive field) where the extension suddenly turns positive following the curve D. The process is repeated along curves EFG when the electric field is made positive again. The “butterfly” diagram provides a complete characterization of the depoling and repoling process.
Most hard piezoelectric materials can only be fully poled or depoled at elevated temperatures so once poled, they can tolerate high reverse fields without difficulty.
We do not recommend operation under reverse field for quasi-static actuators. However, in some applications this can bring some additional strain. The drawbacks are the lower linearity, increased hysteresis and losses. In addition, temperature must be monitored as the coercive field varies with temperature (refer to “thermal properties”).
Soft piezoelectric materials are easily depoled when subjected to an electrical field opposite to the poling direction. The effect of cycling between positive and negative voltages for various piezoelectric materials is shown in the following figures:
Creep
Piezoelectric materials exhibit a creep effect i.e. the material continues to expand for some time upon application of voltage. Correspondingly the material does not immediately return to the initial strain level after return to 0V. While creeping, the material continues to draw charge at very low levels. The creep effect for different actuator materials is compared in the following figure, where the maximum electric field is established after 1s, corresponding to the baseline for displacement (relative displacement = 1).
Creep always occurs in the same direction as the dimensional change produced by the voltage step. The effect is logarithmic so the additional expansion between 10s and 100s will be similar to the expansion obtained between 1s and 10s. For linear/stacked actuators, typical values are 4% per decade when using the piezoelectric material NCE51/51F and 9% per decade when using NCE46. Values are 2-3 times higher for bending actuators. Creep is related to the long-time average that the actuator has experienced in its life.
The electrical and piezoelectric properties are affected by temperature variations. Each piezoelectric material is affected differently by temperature changes, according to the method of manufacture and chemical composition of the material.
Maximum temperature
Piezoelectric materials should be used below the Curie temperature to avoid depoling. Rule of thumb is half Curie temperature. If the temperature were to that raise close to the Curie temperature or above, it will cause the piezoelectric material to become partially or completely depoled and severely degrade the performance. For applications that require operation at elevated temperature a material with a sufficiently high Curie temperature must be chosen. Maximum recommended operating temperatures are specified for each product. It is important to monitor temperature, in particular for dynamic applications, where the component can heat-up during operation due to internal dissipation.
Minimum temperature
Our multilayer products can be used at cryogenic temperatures and have been demonstrated down to 4mK. For these applications a specific preparation (wires, adhesive etc.) is required.
The mechanical and electrical properties of piezoelectric ceramic are greatly reduced at cryogenic temperatures. When piezoelectric actuators are cooled down to cryogenic temperatures, the piezoelectric ceramic behaves like a very hard piezoelectric material featuring:
Improvement of the coercive field at low temperature enables a piezoelectric actuator to become extremely stable against electrical depoling. Therefore, a much wider bipolar operation compared to room temperature is possible. Therefore, drop in strain coefficient at low temperatures can be partially compensated for.
Below is an example of cryogenic measurements at two different temperatures showing the relationship between stroke (displacement) and voltage. As it can be seen, the stroke at 77 K is approximately reduced to half the value at room temperature. Due to the strong increase of the coercive field, it can also be observed that the actuator exhibits a fairly linear voltage-displacement characteristic at negative voltage. The piezoelectric actuator becomes extremely stable against electrical depoling and the loss in stroke at low temperature can be partially compensated by using a wide bipolar operation.
A more problematic parameter is the thermal expansion coefficient for ceramics, important to consider when designing devices where piezoelectric actuators will be part of a composite structure and where the other elements of constructions are e.g metals. The thermal expansion coefficient for ceramics is similar to many ceramics and glasses and is typically in the range of 10-5 meter/meter °C to 10-6 meter/meter °C. A major difference with common materials is that the thermal expansion coefficient is anisotropic with respect to the poling direction.
How to Choose a Piezoceramic Material
The table below gives an overview of the characteristics of two different piezoceramic materials.
The range goes from — to ++, where — is low, and ++ is high.
When mechanical strain is applied to piezoelectric material, the dipoles present in its crystalline structure get distorted, leading to the generation of electric charges.
Based on its structure, piezoelectric materials are classified into four types: ceramics, single crystals, polymers, and composites.
Piezoelectric characteristics that need to be evaluated before selecting piezoelectric energy harvester material include the piezoelectric charge constant, piezoelectric voltage constant, electromechanical coupling factor, mechanical quality factor, permittivity constant or dielectric constant, and Young's Modulus.
Heel-strike system used for LED lighting in shoes
As I was searching for a heel-strike system, I came across piezoelectric energy harvesters, which generate electrical power from heel strikes. Piezoelectric materials are used in heel strike circuits, which convert kinetic energy into electrical energy. The devices that utilize piezoelectric materials to harvest electrical energy are called piezoelectric harvesters. Piezoelectric materials are capable of converting oscillating or vibrational mechanical energy into electrical energy. The piezoelectric energy harvester is able to meet electrical energy needs from motion. Before selecting piezoelectric energy harvester material, it is important to understand its charge and voltage constant, electromagnetic coupling factor, mechanical quality factor, permittivity constant, and Young’s Modulus.
Piezoelectric materials are employed in electronic devices that convert human motion, low-frequency seismic vibrations, and acoustic noises into electrical energy. Piezoelectric energy harvesters have become popular since they eliminate the need for a battery replacement process in electronic devices. Compared to other mechanical or electrical energy conversion principles, such as electromagnetic induction and electrostatic induction, piezoelectric energy harvesting possesses high power density and high integration capability.
Piezoelectric material can be considered the best source to generate power in the mW or µW range in electronic devices. Piezoelectric materials are crystalline structures with non-overlapping positive and negative charges. As the centers of the charges are not overlapping, they form dipole moments in the materials. When mechanical strain is applied to the piezoelectric material, the dipoles are distorted, leading to the generation of electric charges. If the electric charges are allowed to flow, they form the electric current and power the circuit. Otherwise, the charge can be stored in capacitors or batteries.
Based on the structure, piezoelectric materials are classified into the following types:
Ceramics-Ceramics are polycrystalline materials with grains that have the same chemical composition. Lead zirconate titanate (PZT), barium titanate (BT), and strontium titanate (ST) are examples of piezoelectric ceramics. They are widely used in motion sensors, ultrasonic power transducers, high-frequency loudspeakers, and watches.
Single crystals-In single crystal materials, positive and negative ions follow a periodic arrangement. They find applications in sensors, transducers, and actuators. The solid solution of lead magnesium niobate and lead titanate, known as PMN-PT and Lithium Niobate ( LiNbO3), are examples of piezoelectric crystals.
Polymers-Carbon-based materials forming chains with repeated structures called monomers are good piezoelectric polymers. The flexibility of polymers makes them suitable to withstand high strain compared to ceramics and single crystals. The piezoelectric polymers are suitable for applications where the host device undergoes frequency bends. Polyvinylidene fluoride (PVDF) is an example of a piezoelectric polymer.
Composite materials-By combining ceramics, polymers, and single crystals, improved piezoelectric characteristics can be achieved. This idea has led to the formation of piezoelectric composite materials. PZT-polymer composites are one of the most commonly used piezoelectric composites, formed by combining PZT ceramics with polymers.
The piezoelectric property of materials differs between ceramics, single crystals, polymers, and composites. The selection of piezoelectric materials used in piezoelectric energy harvesters is not only based on the piezoelectric property, but also based on the application and design of the host device or harvesting system. The input frequency, occupiable volume, and type of mechanical input are some factors of the application system that need to be considered in piezoelectric material selection. Some of the piezoelectric characteristics that need to be evaluated for the selection of the piezoelectric energy harvester are below:
Piezoelectric Charge Constant, d -The piezoelectric charge constant indicates the suitability of the material for actuator applications. It is a constant that represents the polarization induced per stress applied.
Piezoelectric voltage constant, g-The piezoelectric charge constant assesses whether the piezoelectric material is suitable for sensor applications. It is the constant indicating the electric field generated per unit of mechanical stress applied.
Electromechanical coupling factor, k-The electromechanical coupling factor, k, is an index that is used to show the effectiveness of piezoelectric material in converting mechanical strain to electrical energy. It can be calculated by finding the square root of mechanical-electrical energy conversion efficiency.
Mechanical quality factor, Q-The mechanical quality factor, Q, is the representation of the sharpness of the resonance frequency in piezoelectric materials. It is the ratio of the reactance to the resistance, which is obtained from the series equivalent circuit of a piezoelectric material. This constant indicates the degree of damping in the piezoelectric material.
Permittivity constant or Dielectric constant,
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