Ghodssi R., Lin P., MEMS Materials and Processes Handbook

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274 R.G. Polcawich and J.S. Pulskamp

Fig. 5.1 The relationship of

piezoelectric materials, polar

materials, and ferroelectric

materials relative to the 32

crystal point groups

of strain proportional to an applied electric ﬁeld. This unique material property is

limited to a small subset of materials. Examination of the 32 point groups shows that

only 20 point groups allow piezoelectricity; s ee Fig. 5.1 [4]. Of these 20, ten of the

point groups can be classiﬁed as polar materials in which an electric dipole moment

can be present in the absence of an electric ﬁeld (i.e., spontaneous polarization).

The spontaneous polarization also changes with temperature enabling pyroelec-

tricity. In addition, there is a special class of materials in which the spontaneous

polarization can be permanently reoriented between crystallographically deﬁned

states by applying an electric ﬁeld. This is the key parameter that distinguishes

ferroelectric materials such as (PZT) from strictly polar materials (i.e., quartz,

AlN, and ZnO).

The most commonly utilized piezoelectric material is α-quartz because of its

stable piezoelectric characteristics and its extremely low acoustic losses [5]. In

addition, zero-temperature coefﬁcient properties can be achieved in certain orien-

tations or cuts of quartz, which are extremely important for military and aerospace

applications. Quartz-based devices can be found in military and aerospace, research

and metrology, industrial, consumer, and automotive applications. Of the variety of

applications in which quartz is of use, resonators using both bulk and surface waves

are in high demand for everything from low-cost timing devices to high-precision

oscillators for global positioning systems. The global market for quartz timekeep-

ing devices is in excess of a billion US dollars (USD) [5]. In contrast to quartz,

lead zirconate titanate (PZT) ceramics are used in a variety of actuator applications

including positioning actuators, ultrasonic transducers, and inkjet printers to name

a few. The extremely large piezoelectric coefﬁcients of PZT make it nearly ideal for

these applications. Continued use of piezoelectric materials is expected to increase

as the need for increased miniaturization and integration of sensor and actuators

continues to be required.

5 Additive Processes for Piezoelectric Materials 275

5.1.1 Direct and Converse Piezoelectricity

Piezoelectricity is comprised of two electromechanical responses, a direct effect

and a converse effect. The direct effect refers to the generation of charge under the

application of an applied stress and is governed by Equation (5.1) where the induced

electric displacement (D

) is proportional to the piezoelectric coefﬁcient (d

ijk

) and

the applied stress (T

). The converse effect refers to the generation of strain under

an applied electric ﬁeld and is governed by Equation (5.2), where the induced strain

) is proportional to the piezoelectric coefﬁcient and electric ﬁeld (E

= d

ijk

(5.1)

= d

ijk

(5.2)

In applications, the direct effect can be used for sensing and energy harvesting in

which applied stresses are used to generate surface charges on the piezoelectric (see

Fig. 5.2). For actuators, an applied electric ﬁeld generates a strain in the piezoelectric

layer producing ﬂexure in unimorph cantilevers (see Fig. 5.3). The tensor notation

for the aforementioned relationships can be simpliﬁed using matrix notation where

the subscripts i, j, and k having values 1–3 are replaced with the subscripts p and

q with values of 1–6 [6]. As a result, d

ijk

is transformed to d

.Thed

(strain) and

(stress) piezoelectric coefﬁcients are the most useful for thin ﬁlm sensor and

actuator applications and are related by the elastic constant, c

, Equation (5.3).

= d

(5.3)

Fig. 5.2 The direct piezoelectric effect applied to a piezoelectric unimorph membrane

276 R.G. Polcawich and J.S. Pulskamp

Fig. 5.3 The converse piezoelectric effect applied to a unimorph cantilever

For thin-ﬁlm composites where the piezoelectric layer is rigidly attached to a

supporting elastic layer (i.e., thin ﬁlm or substrate), the out-of-plane stress X

generally set to zero and the longitudinal strain x

varies from a combination of

the piezoelectric effect and elastic coupling from the Poisson effect. As a result,

an effective thin-ﬁlm piezoelectric coefﬁcient, e

31,f

, can be related to the common

piezoelectric coefﬁcients, d

and e

(see Equation (5.4)) by compensating for

the clamping effect experienced by piezoelectric thin ﬁlms in composite structures

[7–9].

31, f

= e

−



+ s



(5.4)

5.1.2 Materials – Ferroelectrics and Nonferroelectrics

Pure polar materials such as AlN and ZnO do not permit the polarization vector

to change orientations relative to the crystal lattice. The polar direction is crystallo-

graphically deﬁned along the c-axis of the Wurtzite crystal structure (see Fig. 5.4). In

addition, these purely polar compounds do not undergo structural phase transitions

that lead to the large dielectric and piezoelectric properties near the Curie temper-

ature T

that is observed in ferroelectric materials. Key advantages of the purely

polar materials that are discussed in subsequent sections include the nearly con-

stant ratio of the dielectric constant and piezoelectric coefﬁcient which is extremely

important for sensing applications, process compatibility with conventional com-

plex metal-oxide semiconductor (CMOS) processing, absence of depoling or aging

of the piezoelectric coefﬁcient, and high acoustic velocities.

5 Additive Processes for Piezoelectric Materials 277

Fig. 5.4 Wurtzite crystal

structure of AlN

The identiﬁcation of ferroelectric behavior in Rochelle salt (potassium sodium

tartrate) in 1920 led to numerous research initiatives on this subject. Over the past

80-plus years, considerable progress has been made in topics such as phenomenol-

ogy, the basis of ferroelectricity, and the development of useful devices. This section

reviews some of the fundamental aspects of ferroelectricity.

As stated above, a ferroelectric material is one in which a spontaneous elec-

tric dipole moment can be reoriented between crystallographically deﬁned stable

states by a realizable electric ﬁeld (i.e., before dielectric breakdown occurs) [10]. In

the perovskite structure, ABO

(see Fig. 5.5), where, in general, A is a divalent

ion and B is a tetravalent ion, the cubic lattice can undergo distortions at tem-

peratures below the Curie temperature. These distortions result in a shift of the

octahedrally coordinated cation from the center of the unit cell. Consequently, a

dipole moment is created between the center of negative charge created by the oxy-

gen octahedra and the center of positive charge resulting from the cation sublattice.

For lead zirconate titanate (Pb(Zr

1−x

), the shift in position of the A-site, Pb

ions, as well as the B-site Zr

and Ti

ions contribute to the dipole properties

(see Fig. 5.6)[11].

In perovskites, the development of the dipole commonly occurs at a phase

transition from a nonpolar paraelectric state to a polar ferroelectric state. In

many instances the transition at the Curie temperature (T

) is associated with a

structural change from a centrosymmetric cubic phase to a noncentrosymmetric

distorted phase. Additional transition temperatures can exist between two noncen-

trosymmetric distorted phases (e.g., tetragonal to orthorhombic or orthorhombic to

rhombohedral). Figure 5.7 illustrates the lattice distortions that yield the observed

structural changes in barium titanate (BaTiO

). Expansion of the cubic unit cell

along the [001] direction results in a tetragonal structure whereas an expansion

along the [011] results in an orthorhombic structure and an expansion along the

[111] direction results in a rhombohedral structure [13].

The polarization generated at T

can be permanently reoriented into different

equilibrium states with an applied electric ﬁeld, leading to the most recognizable

278 R.G. Polcawich and J.S. Pulskamp

Fig. 5.5 Representation of

the cubic (Pm3m) prototype

structure of perovskite ABO

structure

Fig. 5.6 Tetragonal

distortion resulting in a

positive shift of the A

(0.10 Å) and B (+0.12 Å)

ions and a slight negative

shift of t he oxygen ion

(–0.03 Å) for BaTiO

(values

taken from [12])

(a) (b) (c)

Fig. 5.7 Lattice distortions that occur upon cooling below T

in perovskites: (a) in the tetragonal

phase P

develops along [001] of the original cubic structure, (b) in the orthorhombic phase P

occurs along [011], and (c) in the rhombohedral phase P

develops along [111]. The dotted cubes

are for the cubic prototype; the solid lines show the distorted unit cells of the ferroelectric phases

5 Additive Processes for Piezoelectric Materials 279

Fig. 5.8 Schematic of the

polarization electric ﬁeld

hysteresis loop illustrating the

saturation polarization (P

sat

remanent polarization (P

and coercive ﬁeld (E

)

aspect of ferroelectricity: the polarization versus electric ﬁeld hysteresis loop (see

Fig. 5.8). This loop results from the change in net polarization of the material as the

amplitude and direction of the applied electric ﬁeld are changed. The polarization

electric ﬁeld hysteresis loop is a fundamental aspect of ferroelectricity and illustrates

the reorientability of the dipole.

The saturation polarization P

sat

is deﬁned as the linear extrapolation of the high

ﬁeld polarization to zero ﬁeld and approximates the maximal amount of dipole

alignment achieved at zero electric ﬁeld. The remnant polarization P

is the retained

polarization at zero electric ﬁeld. For an ideal material, the two remnant polariza-

tion values are equal but with opposite signs. The coercive ﬁeld E

is the electric

ﬁeld that results in a net zero polarization and has both a positive and negative value

corresponding to each location the loop crosses the ﬁeld axis.

Upon cooling below T

, a spontaneous polarization develops in each unit cell of

the material. All adjacent cells polarized in the same direction (or at least nearly

so) comprise a domain. Domains form both to minimize depolarization energy and

stresses [13]. These domains have polarization orientations that are dictated by the

prototype unit cell and the symmetry elements lost at the phase transformation.

For PZT, which can possess either a tetragonal, rhombohedral, or monoclinic fer-

roelectric phase depending on the Zr:Ti ratio, the polarization direction is along

the original <001>, <110>, or <111> axes of the cubic prototype, respectively (see

Fig. 5.9)[13, 14]. Tetragonally distorted PZT possesses six equivalent polariza-

tion directions, whereas rhombohedral distortions have eight equivalent directions.

Domain boundaries (walls) form between adjacent domains of differing polarization

directions. These domain walls lie on speciﬁc crystallographic planes that minimize

the strain across the wall. Tetragonally distorted perovskites possess 90

◦

and 180

◦

domain walls, and rhombohedral perovskites have 71

◦

, 109

◦

, and 180

◦

domain walls

[12].

Most ferroelectrics possess a very high relative permittivity (ε

) or dielectric con-

stant (K). The dielectric constant is deﬁned as the ratio of the material permittivity

(ε) to the permittivity of free space (ε

= K

(5.5)

280 R.G. Polcawich and J.S. Pulskamp

(a) (b) (c)

Fig. 5.9 Possible polarization directions relative to the cubic prototype for (a) tetragonal per-

ovskite with 6 equivalent <001> directions, (b) orthorhombic distortion with 12 equivalent <110>

directions, or (c) rhombohedral with 8 equivalent <111> directions

Under AC electrical ﬁelds, the dielectric constant can be described as a complex

number with in-phase and out-of-phase components with respect to the applied volt-

age. The out-of-phase component is primarily due to dielectric absorption, resistive

leakage, and domain wall motion. The dielectric loss, represented as D or tan δ,is

the ratio of the out-of-phase and in-phase components [12]. These two parameters

are important for determining a material’s usefulness as a capacitor (i.e., its ability

to store electric charge).

Most solid solutions between PbZrO

and PbTiO

possess ferroelectricity and

show excellent piezoelectric properties. By varying the Zr

/Ti

ratio, both the

ferroelectric and piezoelectric properties can be altered signiﬁcantly. The phase

diagram in Fig. 5.10 illustrates the structural changes that occur as a result of com-

position variations. Substitutions of Zr

for Ti

in PbTiO

reduce the tetragonal

distortion and at sufﬁciently high concentrations cause the structure to transform

into a rhombohedral conﬁguration [13]. As seen in the phase diagram, the shift from

a tetragonal to a rhombohedral structure occurs at a morphotropic phase boundary

(MPB) near the Zr

/Ti

ratio of 52/48. A MPB denotes an abrupt structure change

with composition at a constant temperature [12]. From phase equilibrium, the MPB

is required to be a two-phase mixture of the tetragonal and rhombohedral struc-

tures. The presence of both of these structures in this region gives the material an

Fig. 5.10 Phase diagram for

the lead zirconate–lead

titanate solid solution [14]

(Reprinted with permission.

Physical Society)

5 Additive Processes for Piezoelectric Materials 281

Fig. 5.11 Variation in relative permittivity (K

) and the effective piezoelectric coefﬁcient (d

)

as a function of the Zr/Ti ratio and orientation for bulk PZT [15] (Reprinted with permission.

increased number of polarization directions. For MPB compositions, the high polar-

izability results in extremely large dielectric and piezoelectric properties, as seen in

Fig. 5.11 [15]. The presence of such large piezoelectric properties near the MPB

enables PZT to be an extremely attractive candidate for piezoelectric devices.

5.1.3 Fundamental Design Equations and Models

5.1.3.1 Linear Constitutive Equations of Piezoelectricity

As described in Section 5.1.1, piezoelectricity exhibits the dual electromechani-

cal responses of electric displacement and strain deﬁned by the direct and indirect

effects, respectively. The linear piezoelectric constitutive equations for isothermal

conditions, utilizing the contracted matrix notation, are deﬁned by

= d

+ ε

(5.6)

= s

+ d

(5.7).

The constitutive equation for the direct effect, Equation (5.6), deﬁnes the electric

displacement in terms of the piezoelectric strain coefﬁcient (d

), the stress (T

), the

permittivity as measured at constant stress (ε

), and the electric ﬁeld (E

). For non-

piezoelectric materials (i.e., d

= 0), Equation (5.6) reduces to Maxwell’s equation

relating electric displacement and ﬁeld. The constitutive equation for the indirect

effect, Equation (5.7), deﬁnes the generated strain in terms of the piezoelectric strain

coefﬁcient (d

), the electric ﬁeld (E

), the stress (T

), and the elastic compliance as

measured at constant electric ﬁeld (s

). For non-piezoelectric materials (i.e., d

0), Equation (5.7) reduces to the generalized Hooke’s law for linear elastic materials.

282 R.G. Polcawich and J.S. Pulskamp

The second term on the right-hand side of Equation (5.7), is the piezoelectrically

induced strain in a mechanically free structure and is commonly referred to as the

free or actuation strain. For in-plane isotropic piezoelectric materials (PZT, AlN,

ZnO), with the compressed matrix notation described above, these may be expressed

⎛

⎝

⎞

⎠

⎛

⎝

0000d

000d

000

⎞

⎠

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎝

0 ε

00ε

⎞

⎠

⎛

⎝

⎞

⎠

(5.8)

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎜

⎝

000

000s

0000s

00000s

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎜

⎝

00d

0 d

000

⎞

⎟

⎠

⎛

⎝

⎞

⎠

(5.9)

5.1.3.2 Electromechanical Coupling Factors

The electromechanical coupling inherent to piezoelectricity leads to a dependence of

the permittivity and elastic constants on both the electrical and mechanical bound-

ary conditions. The permittivity and elastic compliance that appear in Equations

(5.8) and (5.9) are therefore deﬁned for particular boundary conditions. The differ-

ence between the mechanically clamped and free permittivities and the electrically

short- and open-circuited elastic constants is a function of the piezoelectric material

property known as the electromechanical coupling factor. It is a measure of energy

transduction or more s peciﬁcally, it is the square root of the ratio between stored

converted energy to input energy. The coupling factor is useful for comparing var-

ious materials, under speciﬁc operating conditions, in terms of energy conversion.

Equation (5.10) refers to the indirect effect and Equation (5.11) refers to the direct

effect and generally these terms are identical.

Mechanical Stored Energy

Electrical Input Energy

(5.10)

Electrical Stored Energy

Mechanical Input Energy

(5.11)

(5.12)

5 Additive Processes for Piezoelectric Materials 283

Fig. 5.12 Rectangular

piezoelectric bar with parallel

plate electrodes on lw

surfaces

Equation (5.12) provides a general deﬁnition of the electromechanical coupling

factor for piezoelectrics. This term depends upon those piezoelectric coefﬁcients,

permittivities, and elastic constants associated with the transduced energy terms in

Equations (5.10) and (5.11) and the nature of the stress state in the material. For

instance, consider the bar in Fig. 5.12 of length (l), width (w), and thickness (t).

Parallel plate electrodes are placed on the two lw surfaces with the polarization

directed along the 3-axis. For such a structure with only nonvanishing stresses along

the 1-axis (T

), an electromechanical coupling factor may be easily deﬁned. The rel-

evant electromechanical coupling factor in this case is k

, referring to the use of the

piezoelectric mode. Table 5.1 deﬁnes additional coupling factors. To illustrate

the relationship between Equations (5.10) and (5.12), consider the same free bar of

Fig. 5.12 where a constant voltage is applied to the electrodes and the electrodes

have a negligible contribution to the stiffness of the bar. The stored mechanical

energy is the piezoelectric strain energy density multiplied by the volume of the

material (V

). The piezoelectric strain in this case is the free strain d

. The input

electrical energy is the electrostatic energy density associated with the capacitance

of the piezoelectric multiplied by the volume of the material (V

). As expressed in

Equation (5.13), this energy ratio simpliﬁes to Equation (5.12).



avg



(

)



(5.13)

A distinction is necessary between the “quasi-static material” coupling factor

and the “effective” coupling factor [6]. The quasi-static material coupling factor,

discussed earlier, is an intensive material property that only depends upon the

anisotropic material properties. The effective coupling factor is a design-speciﬁc

term usually referred to in the context of resonators or resonant material property

measurements. The strain ﬁeld, under static or quasi-static excitation, is generally

uniform. Hence for the previous example, the free strain is equal to the average strain