CERAMICS, PIEZOELECTRIC AND ELECTROSTRICTIVE

ANDREIKHOLKIN

BAHRAMJADIDIAN

AHMADSAFARI Rutgers University Piscataway, NJ

INTRODUCTION

In a rapidly developing world, the use of smart materials becomes increasingly important when executing sophis- ticated functions within a designed device. In a common definition (1), smart materials differ from ordinary materials because they can perform two or several functions, sometimes with a useful correlation or feedback mecha- nism between them. For piezoelectric or electrostrictive materials, this means that the same component may be used for both sensor and actuator functions. Piezoelec- tric/electrostrictive sensors convert a mechanical variable (displacement or force) into a measurable electrical quan- tity by the piezoelectric/electrostrictive effect. Alternately, the actuator converts an electrical signal into a useful displacement or force. Typically, the term transducer is used to describe a component that serves actuator (trans- mitting) and sensor (receiving) functions. Because piezo- electrics and electrostrictors inherently possess both direct (sensor) and converse (actuator) effects, they can be considered smart materials. The degree of smartness can vary in piezoelectric/electrostrictive materials. A merely smart material (only sensor and actuator functions) can often be engineered into a “very smart” tunable device or further, into an “intelligent structure” whose sensor and actuator functions are intercorrelated with an integrated process- ing chip.

Recent growth in the transducer market has been rapid and, it is predicted will continue on its current pace through the turn of the century. The sensor market alone rose to $5 billion in 1990, and projections are

$13 billion worldwide by the year 2000 and an 8% annual growth rate during the following decade (2). Piezoelectric/

electrostrictive sensors and actuators comprise a signifi- cant portion of the transducer market. There is a growing trend due especially to automobile production, active vibration damping, and medical imaging. In this article, the principles of piezoelectric/electrostrictive sensors and actuators are considered along with the properties of the most useful materials and examples of successful devices.

PIEZOELECTRIC AND ELECTROSTRICTIVE EFFECTS IN CERAMIC MATERIALS

Piezoelectricity, first discovered in Rochelle salt by Jacques and Pierre Curie, is the term used to describe the ability of certain crystals to develop an electric charge that is directly

proportional to an applied mechanical stress (Fig. 1a) (3).

Piezoelectric crystals also show the converse effect: they deform (strain) proportionally to an applied electric field (Fig. 1b). To exhibit piezoelectricity, a crystal should belong to one of the twenty noncentrosymmetric crystallographic classes. An important subgroup of piezoelectric crystals is ferroelectrics, which possess a mean dipole moment per unit cell (spontaneous polarization) that can be reversed by an external electric field. Above a certain temperature (Curie point), most ferroelectrics lose their ferroelectric and piezoelectric properties and become paraelectrics, that is, crystals that have centrosymmetric crystallographic structures do not spontaneously polarize. Electrostriction is a second-order effect that refers to the ability of all materials to deform under an applied electrical field. The phe- nomenological master equation (in tensor notation) that describes the deformations of an insulating crystal sub- jected to both an elastic stress and an electrical field is

xi j=si jklXkl+dmi jEm+Mmni jEmEn,

i,j,k,l,m,n=1,2,3, (1) where xi j are the components of the elastic strain, si jkl

is the elastic compliance tensor, Xkl are the stress components, dmi j are the piezoelectric tensor components,

Mmni j are the electrostrictive moduli, and Emand En are

the components of the external electrical field. Here, the Einstein summation rule is used for repeating indexes.

Typically, the electrostriction term (∝ EmEn) is more than an order of magnitude smaller than the piezoelectric term in Eq. (1), that is, the electrostrictive deformations are much smaller than the piezoelectric strains. In this case, under zero stress, Eq. (1) simply transforms to

xi j≈dmi jEm, i,j,m=1,2,3. (2a) Eliminating symmetrical components simplifies the relationship in matrix notation (4) expressed as

xi≈dmiEm, m=1,2,3, i =1,2,3,4,5,6,

(2b)

where i=4,5, and 6 describe the shear strains perpen- dicular to the crystal axis resulting from application of the electrical field. Equations (2a) and (2b) describe the converse piezoelectric effect where the electrical field induces a change in the dimensions of the sample (Fig. 1b).

The piezoelectric effect is absent in centrosymmetric materials, and the elastic strain is due only to electrostriction. In ferroelectric crystals that have a centrosymmetric paraelectric phase, the piezoelectric and electrostriction coefficients can be described in terms of their polarization and relative permittivity. For example, when the electrical field and deformation are along the orthogonal axis in a tetragonal crystal system, longitudinal piezoelectric d33

and longitudinal electrostrictive M11 coefficients can be 139

140 CERAMICS, PIEZOELECTRIC AND ELECTROSTRICTIVE

+ Voltage−

+ Charge−

Force P P

P (a)

(b)

T−∆T L+∆L P

Figure 1.Schematic representations of the direct and converse piezoelectric effect: (a) an electric field applied to the material changes its shape; (b) a stress on the material yields an electric field across it.

described in matrix notation as follows (5):

d33 =2Q11ε0ε33P3, (3a) M11 = Q11(ε0ε33)², (3b) whereε33and P3are the relative permittivity and polarization in the polar direction, ε0=8.854×10⁻¹² F/m is the permittivity of vacuum, and Q11 is the polarization electrostriction coefficient, which couples longitudinal strain and polarization (in matrix notation), as described by the general electrostriction equation,

S3= Q11P₃². (4) In matrix notation, the mathematical definition of the direct piezoelectric effect, where applied elastic stress causes a charge to build on the major surfaces of the piezoelectric crystal, is given by

Pi=di jXj, =1,2,3,

j =1,2,3,4,5,6, (5) wherePiis the component of electrical polarization. In electrostriction (centrosymmetric crystals), no charge appears on the surface of the crystal upon stressing. Therefore, the converse electrostriction effect is simply a change of the inverse relative permittivity under mechanical stress:

ε0ε33

=2Q11X3. (6) The piezoelectric and electrostrictive effects were described for single crystals in which spontaneous polarization is homogeneous. A technologically important class of materials is piezoelectric and electrostrictive ceramics, that consist of randomly oriented grains, separated by grain boundaries. Ceramics are much less expensive to process than single crystals and typically offer compa- rable piezoelectric and electrostrictive properties. The piezoelectric effect of individual grains in nonferroelectric

P E

E (a)

(b)

∆L = d₃₃EL

∆T = d₃₁ET

∆T = d₁₅ET

Figure 2. Schematic of the longitudinal (a), transverse (a) and shear deformations (b) of the piezoelectric ceramic material under an applied electric field.

ceramics is canceled by averaging across the entire sample, and the whole structure has a macroscopic center of symmetry that has negligible piezoelectric properties.

Only electrostriction can be observed in such ceramics. Sin- tered ferroelectric ceramics consist of regions that have different orientations of spontaneous polarization—so-called ferroelectric domains. Domains appear when a material is cooled through the Curie point to minimize the electro- static and elastic energy of the system. Domain boundaries or domain walls are movable in an applied electric field, so the ferroelectric can be poled. For example, domains become oriented in a crystallographic direction closest to the direction of the applied electric field. Typically, poling is performed under high electric field at an elevated temperature to facilitate domain alignment. As a result, an initially centrosymmetric ceramic sample loses its inversion center and becomes piezoelectric (symmetry∞m). There are three independent piezoelectric coefficients: d33, d31, and d15, which relate longitudinal, transverse, and shear deformations, respectively, to the applied electric field (Fig. 2).

Other material coefficients that are frequently used to characterize the piezoelectric properties of ceramics are the piezoelectric voltage coefficients gi j, which are defined in matrix notation as

Ei=gi jXj, (7) whereEiare components of the electric field that arise from external stressesXj. The piezoelectric chargedi jand volt- agegi jcoefficients are related by the following equation:

gi j=di j/(ε0εii). (8)

CERAMICS, PIEZOELECTRIC AND ELECTROSTRICTIVE 141 An important property of piezoelectric and electrostric-

tive transducers is their electromechanical coupling coefficientkdefined as

k² =output mechanical energy/input electrical energy, or

k² =output electrical energy/input mechanical energy.

(9) The coupling coefficient represents the efficiency of the piezoelectric material in converting electrical energy into mechanical energy and vice versa. Energy conversion is never complete, so the coupling coefficient is always less than unity.

MEASUREMENTS OF PIEZOELECTRIC AND ELECTROSTRICTIVE EFFECTS

Different means have been developed to characterize the piezoelectric and electrostrictive properties of ceramic materials. The resonance technique involves measuring char- acteristic resonance frequencies when a suitably shaped specimen is driven by a sinusoidal electric field. To a first approximation, the behavior of a poled ceramic sample close to its fundamental resonance frequency can be rep- resented by an equivalent circuit, as shown in Fig. 3a. The schematic behavior of the reactance of the sample as a function of frequency is shown in Fig. 3b. The equations used to calculate the electromechanical properties are described in the IEEE Standard on piezoelectricity (6). The simplest example of a piezoelectric measurement by the resonance technique can be shown by using a ceramic rod (typically 6 mm in diameter and 15 mm long) poled along its length. The longitudinal coupling coefficient (k33) for this configuration is expressed as a function of the fundamental series and parallel resonance frequencies fsand fp,

Frequency y f_a f_r

Reactance

C_e

Cm R L

(a)

(b)

Figure 3. (a) Equivalent circuit of the piezoelectric sample near its fundamental electromechanical resonance (top branch represents the mechanical part and bottom branch represents the electrical part of the circuit); (b) electrical reactance of the sample as a function of frequency.

respectively:

k33=(π/2)(fs/fp) tan[(π/2)(fp− fs)/2]. (10) Then, the longitudinal piezoelectric coefficientd33 is cal- culated usingk33, the elastic compliances33, and the low- frequency relative permitivityε33:

d33=k33(ε33s33)¹^/². (11) Similarly, other coupling coefficients and piezoelectric moduli can be derived using different vibration modes of the same ceramic sample. The disadvantage of the resonance technique is that measurements are limited to specific frequencies determined by the electromechanical resonance. Resonance measurements are difficult for electrostrictive samples due to the required application of a strong dc bias field to induce a piezoelectric effect in re- laxor ferroelectrics (see next section of the article).

Subresonance techniques are often used to evaluate the piezoelectric properties of ceramic materials at frequencies much lower than their fundamental resonance frequencies.

These include the measurement of piezoelectric charge upon the application of a mechanical force (direct piezoelectric effect) and the measurement of electric-field-induced displacement (converse piezoelectric effect) when an electric field is introduced. It has been shown that piezoelectric coefficients obtained by direct and converse piezoelectric effects are thermodynamically equivalent.

The electrostrictive properties of ceramics are easily determined by measuring displacement as a function of the electric field or polarization. Thus theMandQelectrostric- tive coefficients can be evaluated according to Eqs. (1) and (4), respectively. As an alternative, Eqs. (3b) and (6) can also be used for electrostriction measurements.

A direct technique is widely used to evaluate the sensor capabilities of piezoelectric and electrostrictive materials at sufficiently low frequencies. Mechanical deformations can be applied in different directions to obtain different components of the piezoelectric and electrostrictive ten- sors. In the simplest case, metal electrodes are placed on the major surfaces of a piezoelectric sample normal to its poling direction (Fig. 1b). Thus, the charge produced on the electrodes with respect to the mechanical load is proportional to the longitudinal piezoelectric coefficient d33

and the force Fexerted on the ceramic sample:Q=d33F.

The charge can be measured by a charge amplifier using an etalon capacitor in the feedback loop. Details of direct piezoelectric measurements can be found in a number of textbooks (7).

Electric-field-induced displacements can be measured by a number of techniques, including strain gauges, linear variable differential transformers (LVDT), the capacitance method, fiber-optic sensors, and laser interferometry. Metal wire strain gauges are the most popular sensors used to measure strain at a resolution of about 10⁻⁶m. To perform the measurement, the strain gauge is glued to the ceramic sample, and the resistance of the gauge changes according to its deformation. The resistance variation is measured by a precise potentiometer up to a frequency of several MHz. However, several gauges need to be used to obtain a complete set of piezoelectric and electrostrictive coefficients of the sample.

142 CERAMICS, PIEZOELECTRIC AND ELECTROSTRICTIVE

Magnetic core

Secondary coils

Primary coil

V Piezoelectric

V_in

V_out

Figure 4. Principle of the linear variable differential transformer (LVDT) used for measuring electric-field-induced deformations in a piezolectric sample.

Figure 4 illustrates the design of an LVDT. The moving surface of the sample is attached to the magnetic core inserted into the center of the primary and secondary elec- tromagnetic coils. The change of the core position varies the mutual inductance of the coils. An ac current supplies the primary coil, and the signal in the secondary coils is proportional to the displacement of the core. The response speed depends on the frequency of the ac signal and the mechanical resonance of the coil, which typically does not exceed 100 Hz. Generally the resolution is sufficiently high and approaches∼10⁻²–10⁻¹µm, depending on the number of turns.

The capacitive technique for strain measurements is based on the change of capacitance in a parallel-plate capacitor that has an air gap between two opposite plates.

One of the plates is rigidly connected to the moving surface of the sample, and another plate is fixed by the holder.

The capacitance change due to the vibration of sample can be measured precisely by a zero-point potentiometer and a lock-in amplifier. Therefore, high resolution (in the Å range) can be achieved by this technique. The measurement frequency must be much lower than the frequency of the ac input signal, which typically does not exceed 100 Hz.

All of the aforementioned techniques require mechanical contact between the sample and the measurement unit. This, however, limits the resolution and the maxi- mum operating frequency, which prevents accurate measurement of piezoelectric loss (the phase angle between the driving voltage and the displacement). The force ex- erted on the moving surface of the sample (especially on a thin ceramic film) may damage the sample. Therefore, noncontact measurements are often preferred to determine the electric-field-induced displacement of piezoelectric and electrostrictive materials accurately. Figure 5 shows the operating principle of a Photonic fiber-optic sensor, which can be used to examine the displacement of a flat reflecting

Lamp

Target surface

Probe Photo detector

Optical fibers

Gap

Figure 5. Schematic of the fiber-optic photonic sensor used for nondestructive evaluation of electric-field-induced strains.

surface (8). The sensor head consists of a group of trans- mitting and receiving optical fibers located in the immedi- ate vicinity of the vibrating surface of sample. The intensity of the reflected light depends on the distance between the moving object and the probe tip. This dependence al- lows exact determination of displacement in both dc and ac modes. Using a lock-in amplifier to magnify the output signal, which is proportional to the light intensity, a resolution of the order of 1 Å can be achieved (8). The frequency response is determined by the frequency band of the pho- todiode and the amplifier (typically of the order of several hundreds of kHz).

Optical interferometry is another technique that al- lows noncontact accurate measurement of the electric- field-induced displacements. Interferometric methods of measuring small displacements include the homodyne (9), heterodyne (10), and Fabri–Perot (11) techniques. The most common technique is the homodyne interferometer that uses active stabilization of the working point to prevent drift from thermal expansion. When two laser beams of the same wavelength (λ) interfere, the light intensity varies periodically (λ/2 period) corresponding to the change of optical path length between the two beams. If one of the beams is reflected from the surface of a moving object, the intensity of the output light changes, which can later be translated to the amount of displacement. Using a sim- ple Michelson interferometer (12), a very high resolution of∼10⁻⁵Å is achievable. However, the measurements are limited to a narrow frequency range because the sample is attached to a rigid substrate and only the displacement of the front surface of the sample is monitored (12). As a result of this configuration, the errors arising from the bending effect of the sample can be very high, especially in ferroelectric thin films. In response to that, a double beam (Mach–Zender) interferometer is used to take into account the difference of the displacements of both major surfaces of the sample (13). The modified version of the double-beam

CERAMICS, PIEZOELECTRIC AND ELECTROSTRICTIVE 143 interferometer, specially adapted to measure thin films,

offers resolution as high as 10⁻⁴Å in the frequency range of 10–10⁵Hz and long-term stability (<1%) (14).

COMMON PIEZOELECTRIC AND ELECTROSTRICTIVE MATERIALS

Single Crystals

A number of single crystals (ferroelectric and nonferroelectric) have demonstrated piezoelectricity. However, nonferroelectric piezoelectric crystals exhibit piezoelectric coefficients much lower than those of ferroelectric crystals. The former are still extensively used in some applications in which either high temperature stability or low loss is required. The most important nonferroelectric piezoelectric crystal is quartz (SiO2) which has small but very stable piezoelectric properties [e.g., d11=2.3 pC/N, x-cut (15)].

Ferroelectric LiNbO3and LiTaO3crystals that have high Curie temperatures (1210 and 660^◦C, respectively) are used mostly in surface acoustic wave (SAW) devices. Re- cent investigations (15) have shown that rhombohedral single crystals in the Pb(Zn1/3Nb2/3)O3–PbTiO3 system have exceptionally large longitudinal piezoelectric (d33= 2500 pm/V) and coupling (k33=0.94) coefficients. In addi- tion, ultrahigh strain of 1.7% has been observed in these materials under high electric field. These single crystals are now being intensively investigated and show signifi- cant promise for future generations of smart materials.

Piezoelectric and Electrostrictive Ceramics

As indicated earlier, the randomness of the grains in as- prepared polycrystalline ferroelectric ceramics yields nonpiezoelectric centrosymmetric material. Thus “poling” the ceramic (Fig. 6) is required to induce piezoelectricity. Due to symmetry limitations, all of the domains in a ceramic can never be fully aligned along the poling axis. However, the end result is a ceramic whose net polarization along the poling axis has sufficiently high piezoelectric properties.

The largest class of piezoelectric ceramics is made up of mixed oxides that contain corner-sharing octahedra of O²⁻ions. The most technologically important materials in this class are perovskites that have the general for- mula ABO3, where A=Na, K, Rb, Ca, Sr, Ba, or Pb, and B=Ti, Sn, Zr, Nb, Ta, or W. Some piezoelectric ceramics that have this structure are barium titanate (BaTiO3), lead titanate (PbTiO3), lead zirconate titanate

Unpoled

E_p

Poled

(a) (b)

Figure 6. Schematic of the poling process in piezoelectric ceramics: (a) in the absence of an electric field, the domains have random orientation of polarization; (b) the polarization within the domains are aligned in the direction of the electric field.

Ti ⁴⁺ Ba²⁺ O²⁻

(a) (b)

Figure 7. The crystal structure of BaTiO₃: (a) above the Curie point, the cell is cubic; (b) below the Curie point, the cell is tetragonal, and Ba²⁺and Ti⁴⁺ions are displaced relative to O²⁻ions.

(PbZrxTi1−xO3, or PZT), lead lanthanum zirconate titanate {Pb1−xLax(ZryT1−y)1−x/4O3, or PLZT}, and lead magnesium niobate{PbMg1/3Nb2/3O3, or PMN}.

The piezoelectric effect in BaTiO3was first discovered in the 1940s (3), and it became the first recognizable piezoelectric ceramic. The Curie point of BaTiO3is about 120–130^◦C. Above 130^◦C, a nonpiezoelectric cubic phase is stable, and the center of positive charges (Ba²⁺ and Ti⁴⁺) coincides with the center of the negative charge (O²⁻) (Fig. 7a). When cooled below the Curie point, a tetragonal structure (Fig. 7b) develops where the center of positive charges is displaced relative to the O²⁻ions. This induces an electric dipole. The piezoelectric coefficients of BaTiO3

are relatively high:d15=270 andd33=190 pC/N (3), and the coupling coefficient of BaTiO3 is approximately 0.5.

Due to its high dielectric constant, BaTiO3is widely used as a capacitor.

Lead titanate (PbTiO3) first reported to be ferroelectric in 1950 (3), has a structure similar to BaTiO3 but has a significantly higher Curie point (Tc=490^◦C). When cooled through the Curie temperature, the grains go through a cubic to tetragonal phase change that leads to a large strain which causes the ceramic to fracture. Thus, it is difficult to fabricate pure lead titanate in bulk form. This spontaneous strain has been decreased by adding dopants such as Ca, Sr, Ba, Sn, and W. Calcium-doped PbTiO3(16) has a relative permittivity of∼200 and a longitudinal piezoelectric coefficient (d33) of 65 pC/N. Because of its high piezoelectric coefficient and low relative permittivity, the voltage piezoelectric coefficient of lead titanate ceramic is exceptionally high. Therefore, lead titanate is used in hydrophones and sonobuoys (17).

Lead zirconate titanate (PZT) is a binary solid solution of PbZrO3 and PbTiO3 (3). It is an ABO3 perovskite structure in which Zr⁴⁺and Ti⁴⁺ions randomly occupy B sites. PZT has a temperature-independent morphotropic phase boundary (MPB) between tetragonal and rhombohedral phases, when the Zr:Ti ratio is 52:48. (Fig. 8).

This composition of PZT has efficient poling and excellent piezoelectric properties because of its large number of polarization orientations. At the MPB composition, PZT is usually doped by a variety of ions to form what are known as “hard” and “soft” PZTs (3). Doping PZT with acceptor ions, such as K⁺ or Na⁺ at the A site, or Fe³⁺, Al³⁺, or Mn³⁺at the B site, creates hard PZT. This doping reduces the piezoelectric properties and makes the PZT

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