7.4 Piezoelectricity

Piezoelectricity is a fascinating property where materials generate electricity when stressed and deform when electrified. This unique behavior arises from specific crystal structures and enables various applications in sensors, actuators, and energy harvesting devices.

Understanding piezoelectricity involves exploring its mechanisms, materials, and mathematical descriptions. We'll examine direct and converse effects, key constants, material types, and applications, as well as limitations and recent advancements in the field.

Piezoelectric effect

• Phenomenon in which certain materials generate an electric charge in response to applied mechanical stress or strain
• Conversely, these materials exhibit mechanical deformation when subjected to an electric field
• Fundamental principle behind various sensors, actuators, and energy harvesting devices

Direct piezoelectric effect

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• Occurs when mechanical stress or strain is applied to a piezoelectric material
• Causes a separation of positive and negative charges within the material's crystal structure
• Generates a measurable electric potential across the material
  • Magnitude of the generated voltage is proportional to the applied stress
  • Polarity of the voltage depends on the direction of the applied stress (compressive or tensile)

Converse piezoelectric effect

• Occurs when an electric field is applied to a piezoelectric material
• Induces mechanical strain or deformation in the material
• Magnitude and direction of the strain depend on the strength and polarity of the applied electric field
  • Positive electric field causes the material to expand along the polar axis
  • Negative electric field causes the material to contract along the polar axis

Piezoelectric constants

• Quantitative measures of the piezoelectric properties of a material
• Piezoelectric charge constant ($d$) relates the generated electric charge to the applied mechanical stress
  • Expressed in units of coulombs per newton (C/N) or meters per volt (m/V)
• Piezoelectric voltage constant ($g$) relates the generated electric field to the applied mechanical stress
  • Expressed in units of volts per newton (V/N) or meters per newton (m/N)
• Electromechanical coupling coefficient ($k$) represents the efficiency of energy conversion between mechanical and electrical domains
  • Dimensionless quantity ranging from 0 to 1

Piezoelectric materials

• Exhibit piezoelectric properties due to their unique crystal structures and asymmetric charge distribution
• Can be naturally occurring, synthetic, or polymeric

Naturally occurring crystals

• Certain single crystals exhibit piezoelectric properties due to their non-centrosymmetric crystal structure
• Examples include quartz (SiO2), Rochelle salt (NaKC4H4O6·4H2O), and tourmaline
• Quartz is widely used in frequency control applications (quartz oscillators) due to its excellent temperature stability and high quality factor

Synthetic piezoelectric ceramics

• Manufactured by sintering powders of specific chemical compositions
• Most common synthetic piezoelectric ceramics are based on lead zirconate titanate (PZT)
  • PZT exhibits strong piezoelectric properties and high electromechanical coupling coefficients
  • Composition and doping of PZT can be tailored to optimize specific properties (sensitivity, temperature stability, etc.)
• Other synthetic piezoelectric ceramics include barium titanate (BaTiO3) and potassium sodium niobate (KNN)

Polymers with piezoelectric properties

• Certain polymers, such as polyvinylidene fluoride (PVDF) and its copolymers, exhibit piezoelectric properties
• Piezoelectric polymers are flexible, lightweight, and can be easily fabricated into various shapes and sizes
• Exhibit lower piezoelectric constants compared to ceramics but have higher voltage sensitivity and better acoustic impedance matching
• Suitable for applications requiring conformable or wearable sensors and energy harvesters

Crystallographic requirements

• Piezoelectric effect arises from the crystal structure and symmetry of the material
• Specific crystallographic conditions must be met for a material to exhibit piezoelectricity

Non-centrosymmetric crystal structures

• Piezoelectric materials must have a non-centrosymmetric crystal structure
• Lack of a center of symmetry allows for the separation of positive and negative charges under mechanical stress
• Of the 32 crystallographic point groups, 20 are non-centrosymmetric, and 18 of these exhibit piezoelectricity

Polar axis and polarization

• Non-centrosymmetric crystals possess one or more polar axes
• Polar axis is a unique direction along which the positive and negative charges are separated
• Spontaneous polarization occurs along the polar axis due to the asymmetric distribution of charges
• Ferroelectric materials, a subclass of piezoelectrics, exhibit switchable spontaneous polarization under an applied electric field

Electromechanical coupling

• Describes the interaction between the mechanical and electrical properties of a piezoelectric material
• Quantifies the efficiency of energy conversion between mechanical and electrical domains

Strain-charge coupling

• Relates the mechanical strain experienced by the material to the generated electric charge
• Direct piezoelectric effect: mechanical strain induces a change in the electric charge density
• Converse piezoelectric effect: applied electric field induces mechanical strain in the material

Stress-voltage coupling

• Relates the mechanical stress applied to the material to the generated electric voltage
• Direct piezoelectric effect: mechanical stress induces a voltage across the material
• Converse piezoelectric effect: applied voltage induces mechanical stress in the material
• Stress-voltage coupling is often used in sensor applications, where the generated voltage is measured to determine the applied stress

Mathematical description

• Piezoelectric behavior can be described mathematically using constitutive equations
• Constitutive equations relate the mechanical and electrical properties of the material

Piezoelectric constitutive equations

• Direct piezoelectric effect: $D = dT + \epsilon^T E$
  • $D$: electric displacement (C/m^2)
  • $d$: piezoelectric charge constant (C/N or m/V)
  • $T$: mechanical stress (N/m^2)
  • $\epsilon^T$: permittivity under constant stress (F/m)
  • $E$: electric field (V/m)
• Converse piezoelectric effect: $S = sT + dE$
  • $S$: mechanical strain (dimensionless)
  • $s$: elastic compliance (m^2/N)
  • $T$: mechanical stress (N/m^2)
  • $d$: piezoelectric charge constant (C/N or m/V)
  • $E$: electric field (V/m)

Tensor notation for piezoelectric coefficients

• Piezoelectric properties are anisotropic and depend on the direction of the applied stress or electric field
• Tensor notation is used to describe the directional dependence of piezoelectric coefficients
• Piezoelectric charge constant ($d$) is a third-rank tensor with 27 components ($d_{ijk}$)
  • $i$: direction of the generated electric displacement
  • $j$: direction of the applied mechanical stress
  • $k$: direction of the applied electric field
• Reduced matrix notation (Voigt notation) is often used to simplify the tensor representation
  • Reduces the number of components to 18 ($d_{ij}$, where $i = 1, 2, 3$ and $j = 1, 2, ..., 6$)

Applications of piezoelectricity

• Piezoelectric materials find numerous applications in sensors, actuators, and energy harvesting devices

Sensors and transducers

• Piezoelectric sensors convert mechanical stimuli (pressure, acceleration, strain) into electrical signals
• Examples include pressure sensors, accelerometers, and ultrasonic transducers
  • Quartz crystal microbalances (QCM) used for mass sensing in chemical and biological applications
  • Piezoelectric accelerometers used in automotive and aerospace industries for vibration and shock monitoring

Actuators and motors

• Piezoelectric actuators convert electrical signals into precise mechanical displacements
• Used in micro- and nanopositioning systems, such as atomic force microscopes (AFM) and scanning tunneling microscopes (STM)
• Piezoelectric motors utilize the converse piezoelectric effect to generate rotary or linear motion
  • Ultrasonic motors based on traveling wave or standing wave principles
  • Used in camera lens focusing systems, robotics, and medical devices

Energy harvesting devices

• Piezoelectric materials can convert ambient mechanical energy (vibrations, pressure fluctuations) into electrical energy
• Suitable for powering wireless sensor nodes, wearable electronics, and self-powered systems
• Examples include piezoelectric shoe insoles, floor tiles, and vibration energy harvesters
  • Piezoelectric nanogenerators based on ZnO nanowires or PVDF nanofibers for harvesting low-frequency mechanical energy

Limitations and challenges

• Despite their numerous applications, piezoelectric materials face certain limitations and challenges

Temperature dependence

• Piezoelectric properties are temperature-dependent and can degrade at elevated temperatures
• Curie temperature ($T_C$) is the temperature above which a piezoelectric material loses its spontaneous polarization and piezoelectric properties
• Operating temperature range is limited by the Curie temperature of the material
  • PZT has a Curie temperature around 300-400°C, limiting its high-temperature applications
  • Bismuth ferrite (BiFeO3) and other high-temperature piezoelectrics are being developed for extended temperature range

Aging and degradation

• Piezoelectric materials can experience aging and degradation over time, leading to a decrease in their performance
• Aging mechanisms include domain wall pinning, charge trapping, and structural changes
• Degradation can be accelerated by factors such as high electric fields, mechanical stress, and environmental conditions (humidity, temperature)
• Proper material selection, processing, and packaging techniques can help mitigate aging and degradation effects

Hysteresis and nonlinearity

• Piezoelectric materials exhibit hysteresis in their strain-electric field and polarization-electric field relationships
• Hysteresis leads to energy losses and can limit the precision and repeatability of piezoelectric devices
• Nonlinear behavior can arise at high electric fields or mechanical stresses, leading to deviations from the linear constitutive equations
• Compensation techniques, such as feedback control and inverse modeling, can be employed to mitigate hysteresis and nonlinearity effects

Measurement techniques

• Various measurement techniques are used to characterize the piezoelectric properties of materials and devices

Piezoelectric force microscopy

• Scanning probe microscopy technique that measures the local piezoelectric response of a material
• Uses a conductive AFM tip to apply an electric field to the sample and measures the resulting mechanical displacement
• Provides nanoscale mapping of piezoelectric domains, polarization switching, and local piezoelectric coefficients
• Valuable tool for studying the structure-property relationships in piezoelectric materials

Impedance spectroscopy

• Measures the frequency-dependent electrical impedance of a piezoelectric material or device
• Provides information on the dielectric, piezoelectric, and electromechanical properties of the material
• Equivalent circuit models (Butterworth-Van Dyke model) can be used to extract piezoelectric parameters from the impedance spectra
• Useful for characterizing the resonance behavior and quality factor of piezoelectric resonators and transducers

Laser interferometry

• Optical technique for measuring the mechanical displacement and vibration of piezoelectric devices
• Uses the interference of laser beams to detect sub-nanometer displacements
• Suitable for characterizing the converse piezoelectric effect and the performance of piezoelectric actuators
• Enables non-contact, high-resolution measurements of piezoelectric strain and displacement

Recent advancements

• Piezoelectric materials and devices continue to evolve with ongoing research and development efforts

Piezoelectric nanostructures

• Nanostructured piezoelectric materials, such as nanowires, nanotubes, and nanosheets, exhibit enhanced piezoelectric properties
• High surface-to-volume ratio and quantum confinement effects contribute to improved piezoelectric performance
• Examples include ZnO nanowires, BaTiO3 nanotubes, and 2D MoS2 nanosheets
• Potential applications in nanoscale sensors, actuators, and energy harvesters

Piezoelectric composites

• Composite materials combining piezoelectric ceramics with polymers or other materials
• Designed to exploit the advantages of both constituents (high piezoelectric constants of ceramics and flexibility of polymers)
• Examples include 1-3 composites (PZT rods embedded in a polymer matrix) and 0-3 composites (piezoelectric particles dispersed in a polymer matrix)
• Offer improved acoustic impedance matching, reduced brittleness, and tailored piezoelectric properties

Flexible and stretchable piezoelectrics

• Development of piezoelectric materials that can withstand large strains and deformations
• Enables the integration of piezoelectric functionality into flexible and wearable electronics
• Approaches include using intrinsically flexible piezoelectric polymers (PVDF) or embedding piezoelectric particles in a stretchable matrix
• Potential applications in wearable energy harvesters, soft robotics, and conformable sensors for human-machine interfaces