A Domain Wall Theory for Ferroelectric Hysteresis

A Domain Wall Theory for Ferro electric Hysteresis

Ralph C. Smith

Center for Research in Scienti c Computation

Department of Mathematics

North Carolina State University

Raleigh, NC 27695-8205

[email protected]

Craig L. Hom

Advanced Technology Center

Lo ckheed Martin Missiles & Space

Palo Alto, CA 94304

[email protected]

Abstract

This pap er addresses the mo deling of hysteresis in ferro electric materials through consideration

of domain wall b ending and translation. The development is considered in two steps. In the rst

step, dielectric constitutive relations are obtained through consideration of Langevin, Ising spin and

preferred orientation theory with domain interactions incorp orated through mean eld relations. This

yields a mo del for the anhysteretic p olarization that o ccurs in the absence of domain wall pinning.

In the second step, hysteresis is incorp orated through the consideration of domain wall dynamics and

the quanti cation of energy losses due to inherent inclusions or pinning sites within the material. This

yields a mo del analogous to that develop ed by Jiles and Atherton for ferromagnetic materials. The

viability of the mo del is illustrated through comparison with exp erimental data from a PMN-PT-BT

actuator op erating at a temp erature within the ferro electric regime. i

1 Intro duction

An inherent prop erty of ferro electric materials is the presence of hysteresis in the relation b etween

an applied eld and the resulting p olarization. This phenomenon is asso ciated with the domain

structure of the materials and must be accommo dated in applications which employ ferro electric

comp onents. In certain materials, such as relaxor ferro electrics, hysteresis can be minimized in a

di use transition region through the choice of stoichiometry and op erating conditions. For example,

a signi cant advantage of PMN-PT actuators lies in the fact that they pro duce large strains with

minimal hysteresis when employed near their Curie p oint. Similarly the hysteresis exhibited by PZT

actuators at low to mo derate drive levels is suciently small to p ermit the use of linear design and

control metho dologies. However, at the high drive levels and general temp eratures encountered in

many current applications, the level of hysteresis in b oth PMN-PT and PZT elements is signi cant

and must be accommo dated to attain the full p otential of these materials in high p erformance

actuator design. At a fundamental level, the presence and nature of hysteresis lo ops represents a

basic prop ertycharacterizing ferro electric materials. At the system level, hysteresis generates internal

heat in the actuator which can cause thermal runaway in insulated systems. Highly hysteretic devices

also require closed lo op control for precision displacement actuation. To b etter understand and utilize

these actuator materials, the fundamental mechanisms underlying the hysteresis must b e ascertained

and eventually mo deled.

In an e ort to provide a mo del which incorp orates asp ects of the underlying physical mecha-

nisms and is appropriate for high p erformance actuator and control design, we consider a theory for

ferro electric hysteresis which is based on domain wall dynamics and energy losses due to inherent in-

clusions in the material. Although various typ es of hysteresis curves can b e generated in ferro electric

materials, sigmoid curves such as that shown in Figure 1 are common in applications, and the form

of hysteresis that we shall address here. This theory is the ferro electric analogue of that develop ed

by Jiles and Atherton for ferromagnetic materials [27 ].

Ferro electric Domain Theory

It has long b een recognized that domain and domain wall mechanisms within ferro electric materi-

als result in phenomena suchashysteresis, aging and fatigue. Initial investigations fo cussed primarily

on mechanisms leading to domain formation, nucleation and growth in single crystal BaTiO . Cas-

pari and Merz provided an early analysis of the mechanisms leading to linear and quadratic strains

in single crystal ferro electric crystals [5 ]. They concluded that the sp ontaneous p olarization and

domain e ects on the electromechanical resp onse of the material are signi cant and demonstrated

that a linear piezo e ect results when domain e ects were minimized through the application of an

external eld which aligned domains. Caspari and Merz p ostulated that hysteresis is due to the

presence of parallel and antiparallel domains which switchatvarious eld strengths.

Fundamental investigations regarding the nucleation and growth of domains in BaTiO were

provided by Merz [34 ] and Little [32 ], and were extended by Drougard [11 ] and Miller and Weinreich

[35 , 37 ]. In concert, these investigations established that b oth nucleation and domain wall motion

contributed to p olarization reversal in BaTiO . The dep endence of p olarization reversal on domain

wall motion illustrated that in certain asp ects, this e ect is analogous to magnetization reversal in

ferromagnetic materials. Subsequent exp eriments have elucidated further details regarding domain

nucleation and growth e.g., see [40 , 41] and numerous mo dels have b een prop osed to quantify

domain phenomena and their e ect on hysteresis and fatigue [20 , 23 , 24 , 43 , 45 ].

An imp ortant comp onent in domain and domain wall theories is the e ect due to impurities or

inclusions in the materials. Miller and Savage observed that domain wall motion is dep endentupon 1

the impurity levels of the crystal and considered a mo del based on a surface layer adjacent to the

domain wall [36 ]. Laikhtman extended this idea to develop a mo del for the b ending of domain walls

pinned by inclusions or defects in the material [30 ]. This mo del was develop ed under the premise

that the application of an external eld pro duces a force which acts as a pressure on the wall and

causes it to b end in a manner analogous to a pinned membrane. The resulting mo del was derived

using elasticity theory and accounts for the inertia of the domain wall as well as certain anisotropy

and temp erature e ects. Finally, Laikhtman observed from exp erimental evidence in [14 , 19] that

domain walls exhibit an \e ective viscosity" which dep ends on the concentration of defects. This

provides an irreversible comp onent to the p olarization which Laikhtman incorp orated through the

inclusion of factors mo deling the dissipation of heat. Finally, it is noted that this theory breaks down

if the mo deled wall comes in contact with remote defects. We note that an overview of the \friction"

e ects and dielectric losses due to domain wall motion can b e found in [6] and [50 , pp. 402-427].

The ideas underlying the in uence of pinning sites on domain wall dynamics were extended by

Fedosov and Sidorkin [12 ]. They concluded that for low displacement levels on the order of the

lattice constant, domain walls can be considered as moving as a whole. For higher displacement

levels, however, pinning e ects b ecome signi cant and lo cal b ending b etween pinning sites should b e

considered. A comprehensive analysis of eld, frequency and thermal e ects on domain wall dynamics

in relaxor ferro electrics has b een provided by Chen and Wang [7 ]. These authors note that at low

eld levels with high frequencies, the dielectric b ehavior of the material can b e primarily attributed

to the b ending of domain walls pinned at inclusions. For large applied or internal elds, the lo cal

energy barriers asso ciated with pinning sites are overcome and the domain walls move for extended

distances. We note that these lo calized energy barriers provide a thermo dynamic mechanism for

describing the \e ective viscosity" and \frictional" domain wall e ects rep orted in [30 , 50 ].

The combination of the b ending mechanisms describ ed in [7 , 12 , 30 ], and the energy barriers

detailed in [7 ] indicates that b oth reversible and irreversible mechanisms contribute to ferro electric

hysteresis. Furthermore, the analysis and exp eriments rep orted in these investigations indicate that

these mechanisms are due to the pinning of domain walls at defects in the material and the energy

input required to move domain walls across pinning sites. The discussion in the subsequent section

summarizes certain previous approaches used to quantify these e ects and provides a background

for the mo del presented here.

0.25

0.2

0.15

0.1 ) 2 0.05

−0.05 Polarization (C/m

−0.1

−0.15

−0.2

−0.25

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Electric Field (MV/m)

Figure 1. Hysteresis lo op measured at a temp erature of T = 20 C in a PMN-PT-BT sample

whose Curie temp erature is T =45 C.

c 2

Hysteresis Mo dels

The nature of currenthysteresis mo dels for ferro electric materials can b e roughly categorized as

follows: 1 microscopic theories based on quantum mechanics, classical elasticity and electromagnetic

relations, or thermo dynamic laws, 2 macroscopic theories based up on phenomenological principles,

or 3 semi-macroscopic theories which are derived using a combination of approaches.

Microscopic theories for ferro electric materials generally address the previously describ ed p olar-

ization pro cesses at the lattice or domain level. Mo del development at the lattice level is illustrated

by the theory of Omura et al [42 ]. In their approach, the free energy for a one-dimensional ferro-

electric lattice system is represented in terms of the summed moments and coupling co ecients at

the atoms in the lattice. Hysteresis is incorp orated through the inclusion of viscosity co ecients at

each lattice site. Simulation results demonstrate that the mo del yields reasonable families of hys-

teresis curves but requires a large number of degrees of freedom to accommo date the atoms in the

one-dimensional lattice the examples were generated with N = 250.

For simple stoichiometries, microscopic mo dels of this typ e are theoretically plausible. For the

crystalline conditions encountered in present actuator designs, however, it is dicult to construct

tractable microscopic mo dels which accommo date attributes such as grain b oundaries and intergran-

ular stresses as well as the p olycrystalline nature of the materials. Moreover, microscopic mo dels at

the atomic or lattice level are often dicult to employ in control design due to the large number of

required states and parameters.

The domain wall mo dels of Fedosov and Sidorkin [12 ] and Laikhtman [30 ] are also lo cal in

nature and hence would require either macroscopic averaging or a large numb er of parameters when

employed in actuator characterization or design. However, the concepts underlying these mo dels are

in accordance with the approach employed in this pap er for mo deling the reversible comp onent of the

p olarization with the primary di erence in philosophy residing in our use of macroscopic parameters

which quantify \average" prop erties of the material.

Numerous macroscopic mo dels for hysteresis pro cesses in ferro electric materials have b een de-

rived using empirical or phenomenological principles. In some cases, these mo dels are motivated by

physical domain pro cesses while other mo dels circumvent p o orly understo o d physics through purely

mathematical characterizations. Mo dels in the latter category are advantageous when the underlying

physics is dicult to quantify. Due to their nature, however, it is dicult to employ known physics

or physical measurements to directly construct such purely phenomenological mo dels or up date them

to accommo date changing op erating conditions.

A large class of hysteresis mo dels have b een derived through ts obtained with hyp erb olic tangent

and cotangent functions. As detailed in Section 2 of this pap er, Langevin functions involving the

hyp erb olic cotangent and Ising spin mo dels yielding the hyp erb olic tangent arise when Boltzmann

statistics are used to derive the anhysteretic hysteresis-free resp onse in ferro electric and ferromag-

netic materials. These representations provide natural envelops for phenomenological quanti cation

of hysteresis families.

Hom and Shankar have demonstrated that certain hysteresis curves can be generated through

the appropriate choice of parameters in the Ising spin mo del [22 ]. As illustrated by their results,

however, this metho d pro duces hysteresis curves in which the transition from one saturation region

to the other is much steep er than that observed in typical ferro electric applications.

More gradual transitions can b e obtained by employing Langevin or hyp erb olic cotangent func-

tions which are shifted to coincide with the remanence p olarizations. This technique was employed

by Zhang and Rogers when mo deling hysteresis in piezo ceramic materials [54 ] and Miller et al for

characterization hysteresis in ferro electric capacitors [38 ]. While this technique often provides ac-

curate mo del ts with a small number of required parameters in some cases two, its capabilities 3

for general applications involving symmetric and asymmetric minor lo ops app ears limited. In the

context of the underlying physics, the shifting of the anhysteretic curves provides a phenomenological

mechanism for mo deling the \viscosity" or energy losses which o ccur during domain growth through

domain wall movement.

Another class of phenomenological hysteresis mo dels are based up on the determination of empir-

ical laws which quantify the rate of dip ole or p olarization switching. Based on observations by Merz

[34 ], Landauer et al [31 ] prop osed a mo del which quanti es the rate of p olarization switching in terms

of the current and saturation p olarization and applied eld. Applying a similar philosophy, Chen

and Montgomery [6] develop ed a mo del in which the rate at which dip oles align with the applied

eld is sp eci ed as a nonlinear function of the eld. Both approaches provide phenomenological

characterizations of the rate at which domain walls move.

Preisach and generalized Preisach mo dels have also b een widely used to characterize hysteresis in

ferro electric materials. In their classical form, Preisach op erators are constructed from linear com-

binations of multivalued kernels [33 ]. The co ecients for each kernel represent the input level and

magnitude when switching o ccurs b etween two saturation states which de ne the kernel. When used

to quantify magnetic hysteresis, the kernel has b een interpreted as representing diametrically opp o-

site dip ole con gurations with the co ecients interpreted as quantifying the eld inputs required for

dip ole switching and degree to which they switch at the p oints. While a similar interpretation can

be made for ferro electric pro cesses, the techniques are more commonly considered as phenomeno-

logical and hence provide a purely mathematical mo del for the pro cess. Generalized Preisach or

Krasnoselskii-Pokrovskii mo dels di er through the choice of kernel. As detailed in [2 ], the classical

Preisach op erators are discontinuous with resp ect to b oth the parameters and time. This is allevi-

ated in Krasnoselskii-Pokrovskii op erators through the use of smo oth ridge functions which provide

an envelop of admissible families. We note that the construction of Krasnoselskii-Pokrovskii kernels

through translations of ridge functions is quite similar in philosophy to the use of shifted anhysteretic

curves employed in the mo dels of Miller et al [38 ] and Zhang and Rogers [54 ].

The mo di cations necessary for employing the classical Preisach techniques for ferro electric hys-

teresis are illustrated by the mo dels develop ed by Ge and Jouaneh [17 , 18 ]. As illustrated by b oth nu-

merical and exp erimental examples, this technique has b een used accurately characterize b oth ma jor

and minor hysteresis lo ops in piezo ceramic actuators. The development of Krasnoselskii-Pokrovskii

hysteresis mo dels for piezo ceramic actuators has b een considered by Galinaitis and Rogers [15 , 16].

References [16 ] also provides an overview concerning the construction of inverse op erators for control

metho ds based on output linearization.

Finally, semi-macroscopic mo dels are typically derived using a combination of thermo dynamic

and phenomenological principles. The goal in these approaches is to employ energy-based relations to

the extent p ossible so that known physics can b e used to facilitate mo del construction and up dating.

Phenomenological principles contribute to asp ects of the constitutive equations and the development

of macroscopic averages of microscopic prop erties such as the intergranular stresses and domain wall

pinning site energies.

An illustration of this typ e of approach can be found in the hysteresis mo del of Yoo and Desu

for ferro electric thin lms [53 ]. The hysteresis is characterized through consideration of three stages

in the p olarization pro cess: domain nucleation and growth, domain merging, and domain shrink-

age. The dynamics in these stages are develop ed through the combination of physical mechanisms

underlying p olarization reversal and macroscopic averages of certain domain prop erties. The do-

main wall mo del of Chen and Wang [7 ] for relaxor ferro electric materials can also b e considered as

semi-macroscopic. In their development, Chen and Wang employ energy relations in combination

with phenomenological relations to obtain microscopic p olarization rate equations and corresp ond-

ing macroscopic constitutive equations. Asp ects of the mo del are formulated in the framework of 4

dislo cation theory for plasticity and anelasticity which leads to a fairly general mo del for domain

wall dynamics. While low temp erature hysteresis lo ops are used to provide evidence of domain wall

motion, this investigation do es not address p er se the mo deling of hysteresis in relaxor ferro electric

materials.

This pap er describ es a hysteresis mo del which is semi-macroscopic in nature and is based up on

the quanti cation of the domain wall energy in the material. For an ideal material which is free

from inclusions or second-phase materials, domain wall movement is unimp eded and the relation

between applied stresses, electric elds and temp eratures, and the resulting hysteresis-free or anhys-

teretic p olarization is mo deled by nonlinear constitutive equations. As detailed in [22 ], appropriate

constitutive relations are derived through thermo dynamic theory combined with phenomenological

relations such as Ho oke's law. In actual materials, however, domain wall movement is imp eded by

inherent inclusions within the material. The mo del presented here quanti es the resulting hysteresis

through consideration of the energy required to translate domain walls and break pinning sites. An

`average' measure of this energy is employed to obtain a mo del which requires the identi cation of

only ve parameters. Due to the small numb er and physical nature of certain parameters e.g., the

saturation p olarization P , the mo del is easily constructed and up dated. We note that this mo del

is the ferro electric analogue of that develop ed by Jiles and Atherton for ferromagnetic materials

[27 ]. While asp ects of the Jiles/Atherton theory were employed in [51 ] to characterize hysteresis in

stress/strain relations for LiCsSO crystals, this is the rst application of this approach to quantify

hysteresis in the E -P relation for ferro electric materials. We also p oint out that while certain ferro-

electric actuator materials provide the motivation, the mo del is general in nature and can b e applied

toavariety of ferro electric materials including capacitor and electro-optic materials.

Certain asp ects of the constitutive theory of Hom and Shankar are summarized in Section 2.

This discussion includes the development of the classical Langevin and Ising spin mo dels as well

as a third mo del which combines attributes of the two. This section also provides a discussion

concerning the incorp oration of interdomain coupling to provide the e ective eld observed at the

domain level and outlines the extension of the mo dels from the microscopic to macroscopic levels.

The domain wall theory contributing to b oth reversible and irreversible changes in p olarization is

presented in Section 3. To simplify the development, the nal hysteresis mo del is formulated through

scalar-valued relations between the applied eld E and resulting p olarization P . This formulation

is suitable for a variety of current applications and materials. Atvarious p oints in the development,

however, intermediate vector-valued relations are required and the notation E and P are used to

denote the resp ective eld and p olarization in lR . This also provides a framework for developing the

corresp onding vector-valued hysteresis mo del if required by the application. In the nal section, the

viability of the mo del is illustrated through comparison with exp erimental data from a PMN-PT-BT

actuator.

2 Constitutive Theory for Ferro electric Materials

In this section we consider the development of constitutive equations which describ e the anhysteretic

hysteresis-free mechanical and dielectric b ehavior of certain ferro electric materials. Hysteresis

e ects are included in the next section through quanti cation of energy loss due to pinning sites in

the material.

Following classical developments for electrostrictive materials, the constitutive relations are as-

sumed to b e of the form

= Y e Q P

E = 2Q P + FT; P :

33 5

Here denotes the axial stress, e denotes the longitudinal strain, Y is the closed-circuit elastic mo du-

lus, T is the temp erature, P denotes the p olarization and Q denotes the longitudinal electrostrictive

coupling co ecient. We note that these constitutive relations can b e mo di ed for piezo electric ma-

terials by replacing the quadratic p olarization term in the mechanical relation by a linear term.

In the remainder of the section, we summarize three techniques for mo deling the function F

which quanti es the dep endence of the electric eld E on the temp erature and p olarization. All three

mo dels are based up on the application of Boltzmann's lawtovarious dip ole orientations within the

ferro electric crystal. This follows the approach employed by Hom and Shankar [21 , 22 ] and in one case

provides a Langevin mo del analogous to that typically employed in ferromagnetic applications. We

note that other techniques can b e employed for mo deling the anhysteretic constitutive relations for

certain ferro electric materials and refer the reader to [8 , 44 ] for discussion concerning two alternative

mo dels.

2.1 Noninteracting Cells

We view a ferro electric crystal as a lattice of cells, where each cell p ossesses a p ermanent dip ole

moment with an asso ciated direction. Normally, the cell has a nite numb er of p ossible orientations

eight for a tetragonal system. When free of electric eld and ab ove the Curie temp erature, thermal

agitation creates a random distribution of orientations and the ceramic has an average p olarization of

zero. The application of an electric eld forces some dip ole moments to switch to an orientation closer

to the direction of the eld. The resulting distribution is then no longer random and a macroscopic

p olarization develops. We p oint out that collections of neighb oring cells having the same p olarization

form the domain structure of the crystal. The degree of switching increases with increasing eld until

the global p olarization eventually saturates when all the cells have optimally oriented their dip ole

moments relative to the eld. For the development in this section, we will assume that the cells do

not interact and use statistical mechanics to derive three nonlinear mo dels for the dielectric resp onse

of this system. The mo dels di er only in the assumptions made ab out the cells' p ossible orientation.

The e ects of cell interaction are then considered in Section 2.2.

For a dip ole moment p in an electric eld E, the p otential energy is given by

E = p E = p E cos 2

0 0

where p = jp j;E = jEj. For noninteracting cells and moments, classical Boltzmann statistics can

0 0

be used to express the probability of dip oles o ccupying certain energy states. Letting k denote

Boltzmann's constant, the thermal energy is k T and the probability that a dip ole o ccupies the

energy state E is

E =k T

E =Ce : 3

The parameter C is later sp eci ed to ensure that integration over all p ossible con gurations yields

the total numb er of moments p er unit volume N . Wenow consider three mo dels based on di ering

assumptions concerning the moment orientations.

Langevin Mo del

The rst mo del is obtained under the assumption that the material is isotropic and a cell's

orientation can be in any direction. This situation is equivalent to a dielectric uid [13 ] where the

dip ole moments are free to moveinany direction up on application of the electric eld and yields the

Langevin mo del commonly employed in ferromagnetic applications. 6

Under the assumption of an isotropic material, the a priori number of moments between and

+ d is prop ortional to the surface area 2 sin d on a unit sphere. With the probability of

o ccupying that state given by 3, the numb er of moments in this con guration can b e expressed as

p E cos =k T

dN =2sin d C e :

To evaluate C , we note that the integration of dN over all p ossible con gurations must equal the

total numb er of moments p er unit volume N ; hence

C =

p E cos =k T

2 e sin d

= :

p E

4k T

sinh

p E kT

Because each cell contributes p cos to the p olarization, the total p olarization is

p E cos =k T

P = 2 p Ce cos sin d

k T p E k T p E

B 0 B 0

cosh sinh : = rp C

p E k T p E k T

0 B 0 B

Using 4 to eliminate C ,we obtain the Langevin relation

p E k T

0 B

P = p N coth 5

k T p E

B 0

between the eld and p olarization.

Ising Spin Mo del

The second mo del examined in this section is the Ising spin mo del which is derived under the

assumption that only two orientations are p ossible in one cell: one in the direction of the electric

eld and the other in the direction directly opp osite the eld. Letting N denote the numb er of cells

with the same orientation as the eld and N denote the numb er of cells with opp osing orientation,

the total numb er of cells is

N = N + N : 6

The application of Boltzmann's equation then yields the expressions

p E=k T p E=k T

0 0

B B

N = Ce ; N = Ce 7

for the distribution of cells in each orientation. Combining 6 and 7, we obtain

p E

; 8 N =2Ccosh

k T

which relates N and C for the Ising spin mo del. The p olarization is the sum of each cell's contribution

p E

P = p N + p N =2p Csinh :

0 + 0 0

k T

B 7

Using 8 to eliminate the parameter C yields the Ising spin mo del

p E

9 P = p N tanh

k T

relating P and E . This form of the anhysteretic p olarization was used by Hom and Shankar in the

development of a constitutive mo del for electrostrictive relaxor ferro electrics [21 , 22 ]. The reader is

referred to [25 , pages 214-215] for a discussion of the quantum mechanical rami cations of the Ising

spin mo del for ferromagnetic materials.

Preferred Orientation Mo del

Neither the Langevin mo del nor the Ising spin mo del completely represents the actual situation

in a ferro electric lattice due to the xed number of orientations attained by the cell. Since a p oly-

chrystalline ceramic has a random distribution of grains, these orientations should also b e random.

For our third mo del, whichwe call the Preferred Orientation mo del, we assume that a cell has two

p ossible orientations that have opp osite directions like the Ising spin mo del. However, the axis of

orientation is uniformly distributed to re ect the random distribution of the grains. For cells with

a given axis of orientation, the distribution is

+ p Ecos =k T

N =C e 10

and

p Ecos =k T

N =C e 11

where N and N are the numb er of cells in the two p ossible orientations. The total number

of cells with that axis is

N =2C coshp E cos =k T = 12

0 B

where N again denotes the total number of cells in the lattice. The p olarization for the lattice is

then

P =2 [N N ]p cos sin d

which simpli es to

p N

P = tanhp E cos =k T cos sin d

0 B

through the use of 10 through 12. Performing the integration yields the formula

h i

p N k T

0 B

2Ep =k T 2Ep =k T

0 0

B B

P = log 1+e + log 1+e

2 Ep

h i

1 k T

2Ep =k T 2Ep =k T

0 0

B B

+ Li e Li e

2 Ep

where Li represents the dilogarithm function given by

log 1 x

Liz = dx :

The mo del 13 provides a characterization of the E -P relation which incorp orates attributes of b oth

the Langevin and Ising spin mo dels. 8

Macroscopic Relations

The three mo dels 5, 9 and 13 are formulated in terms of microscopic parameters suchas p

and N . To obtain macroscopic versions for use as constitutivelaws, the relations must b e normalized

to re ect the bulk prop erties of the material. We rst assume that all three mo dels saturate to a

common p olarization P . Secondly,we assume that they exhibit the same initial p ermittivity

P E T

s 0

= ;

where E and T are a scaling electric eld and the Curie temp erature, resp ectively. Under these

0 c

assumptions, the dielectric functions for the Langevin, Ising spin and Preferred Orientation mo dels

are resp ectively given by

3T E E T

c 0

coth P = P

E T 3T E

0 c

T E

P = P tanh

E T

and

h i

2E T

3ET =E T 3ET =E T

c c

0 0

P = P log 1+e + log 1+e

3ET

2 E T

3ET =E T 3ET =E T

c c

0 0

: Li e + Li e

9 ET

The relative b ehavior of the three mo dels is illustrated in Figure 2. The Langevin mo del saturates

the slowest since its cells have the most freedom in selecting orientations that are partially aligned

with the electric eld. The Ising Spin mo del saturates more quickly since the cells have very little

freedom and must completely orient in the direction of the eld while the Preferred Orientation

mo del provides a compromise b etween the other two mo dels. We note that the Langevin and Ising

spin mo dels have equivalent rst-order terms but di er in the third and higher order terms.

0.25

0.2 ) 2

0.15 Polarization (C/m

0.1

0.05 Ising Spin Preferred Orientation Langevin

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Electric Field (MV/m)

Figure 2. Ising spin, Preferred Orientation and Langevin mo dels for the anhysteretic p olarization

with no cell interactions. 9

2.2 Cell Interactions

The theoretical development of the previous subsection ignores mutual interaction b etween the p olar

moments of the cells. These interactions are often signi cant and can lead to b oth second-order

phase transitions of the typ e describ ed by Devonshire [10 ] and domain nucleation and growth. To

incorp orate the coupling due to neighb oring domains, one typically considers the e ective eld acting

on the cell rather than fo cusing solely on the applied eld E .

E ective Fields

The issues concerning the computation of an e ective eld are illustrated by the Clausius-Mossotti

equation which mo dels dip ole interactions in dielectric uids and certain dielectric solids [29 ]. As

detailed in [1 , 13 ], this mo del quanti es the changes in the lo cal electric eld due to the p olarization

in the surrounding dielectric through the approximation of the eld inside a sphere with a surface

charge equivalent to the p olarization. The resulting e ective eld is

E = E + P

where denotes the p ermittivity of free space. Through the incorp oration of the p olarization comp o-

nent, the e ective eld accounts for mutual cell interactions and thus provides a b etter approximation

of the eld which actually in uences individual cells.

For materials such as BaTiO ,however, the validity of the Clausius-Mossotti is doubtful [9 ] which

has prompted the development of more general e ective eld mo dels for dielectric solids [1 , 39].

Hom and Shankar [22 ] have also extended this analysis to ferro electric materials whose anhysteretic

b ehavior is mo deled by the Ising spin mo del. Under the assumption that it is energetically favorable

for neighb oring cells to have the same spin, the e ective eld in the absence of an applied stress is

E = E + P 16

where E =P . This mo del can b e extended to materials sub jected to an external stress through

0 s

consideration of the e ective eld

E = E + P +2Q P 17

e 33

where Q again denotes the longitudinal electrostrictive co ecient.

For the development of subsequent domain wall theory, it will b e necessary to relate the e ective

electric eld E to a corresp onding eld D . For general elds, D is related to E and P through the

e e

equation

D = E + P; 18

where denotes the free space p ermittivity, or the relation

D = E 19

where is the p ermittivity of the substance. It should b e noted that for ferro electric materials, the

p ermittivity is in general b oth nonlinear and multivalued due to hysteresis. For such materials, the

relation 19 has limited value as a constitutive relation until is characterized. When quantifying

hysteresis losses in the material, we employ the eld

D = E + P +2Q P : 20

e 0 33 10

This de nition is analogous to the expression B = H + M = H employed for ferromagnetic

e 0 0 e

materials e.g., see [26 , page 1266] along with [4 , 27 , 28 , 47 ] where H and B resp ectively denote

e e

the e ective magnetic eld and induction.

Macroscopic Constitutive Relations and Phase Transitions

Because the e ective eld E incorp orates the contributions due to interdomain coupling and

applied stresses, we will employ it when describing the general mo del for the anhysteretic p olarization

and developing constitutive relations for the material. The consideration of the e ective eld E in

14 yields the Langevin mo del

3T E E T

c e 0

coth P = P 21

E T 3T E

0 c e

and the Ising spin mo del

T E

c e

P = P tanh 22

E T

for the anhysteretic p olarization, with analogous treatment in the Preferred Orientation Mo del. In

the case of constant temp erature, the anhysteretic p olarization provided by the Langevin and Ising

spin mo dels can b e expressed as

a E

P = P coth 23

a E

and

P = P tanh 24

E T E T

0 0

where a = or a = are constants whichmust b e estimated for a given material or device.

3T T

c c

To obtain a constitutive dielectric equation commensurate with the mechanical constitutive re-

lation 1, the characterization must b e p olarization-based. This means that the dielectric relations

develop ed in this section must be inverted to obtain the electric eld as a function of the p olariza-

tion. Of the three mo dels, this can be accomplished in closed form only for the Ising spin mo del.

For that mo del, inversion of the relation 22 and combination with the mechanical relation yields

the constitutive equations

= Y e Q P

E T E P

0 0

P + arctanh E = 2Q P

P T P

s c s

so that F in 1 is given by

E E T P

0 0

F T; P = P + arctanh :

P T P

s c s

Finally, the consideration of the dielectric constitutive equation in 25 illustrates the nature of

phase transitions in the material. Consider the relation

P P T

= arctanh 26

P T P

s c s

which results in the absence of an applied eld E = 0 and stress = 0. The left hand side

represents the force driving cells to have the same orientation while the right hand side represents 11

the tendency for random orientation due to thermal agitation. When the temp erature is ab ove the

transition temp erature T , the only solution to 26 is P = 0 as shown in Figure 3a. In this case,

thermal agitation overp owers the ordering force and the random orientation of the dip oles leads

to zero average p olarization in the ceramic. Below the transition temp erature, three p olarization

solutions exist for 26. The solution P = 0 is unstable while the other two solutions are stable

and represent a sp ontaneous nonzero p olarization. These nonzero solutions determine the physical

b ehavior of the ferro electric materials at temp eratures b elow the Curie p oint. For a xed temp erature

T b elow the Curie p oint T , the relation between an input eld and the resulting p olarization is

plotted in Figure 3b. The instability at P = 0 is noted in b oth the initial p olarization and the

b ehavior at the co ercive eld values.

1 1 Stable Solutions 0.8 Unstable Solution 0.8

0.6 0.6

0.4 ) 0.4 s )

0.2 s 0.2

0 0 Bifurcation Point −0.2

Polarization (P/P −0.2 Normalized Polarization (P/P −0.4 −0.4

−0.6 −0.6

−0.8 −0.8

−1 −1 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Normalized Temperature (T/T )

c Electric Field (MV/m)

a b

Figure 3. a The phase transition predicted by the Ising spin mo del with co op erativeinteraction

between cells; b Relation b etween an input eld E and the p olarization of PMN-PT for T

3 Domain Wall Theory

The relations 23 or 24, in combination with the e ective eld relations 16 or 17, quantity the

anhysteretic relation b etween the electric eld E input to the material and the resulting p olarization

P . For most values of the parameters and a, the relation is single-valued. As illustrated in

Figure 3b, however, one can obtain a multivalued relation for certain parameter choices. While this

latter con guration pro duces a curve which quanti es certain asp ects of hysteresis, the transition

from one saturation region to the other is much steep er than that observed in typical ferro electric

applications.

The anhysteretic domain pro cess is dep endent up on unimp eded domain wall movement and is

reversible and conservative. As illustrated by the exp erimental data in [22 ], this is often not the case

and the material exhibits hysteresis and nonreversible b ehavior. In this mo del, we incorp orate this

hysteresis through consideration of energy losses at domain wall pinning sites.

3.1 Pinning Energy

As indicated in the rst section, all ferro electric materials exhibit material nonhomogeneities, voids,

and stress variations which are energetically favorable for domain wall formation and pinning. Be-

cause these inclusions inhibit domain wall movement, they lead to energy loss as the material is 12

cycled through a full E-P lo op. To incorp orate the hysteresis observed under constant temp erature,

quasistatic op erating conditions, we quantify the reversible and irreversible domain wall dynam-

ics exhibited at pinning sites. This theory follows closely that derived by Jiles and Atherton for

ferromagnetic materials [27 ].

We consider rst the energy required to break the pinning site and hence move the domain wall.

As detailed in [7 ], a pinning site is broken when sucient energy is provided to overcome a lo cal

energy barrier. Consider the domain con guration depicted in Figure 4 in which two domains are

separated by a domain wall situated at a pinning site. The dip ole moments p er unit volume in the

two domains are denoted by p and p with p assumed to b e aligned with the e ective electric eld

E this assumption simpli es the discussion and pro duces the same nal mo del as that determined

with a general con guration. The angle b etween the moments is denoted by .

For ferromagnetic materials, Jiles and Atherton hyp othesized that the energy required to over-

come the pinning sites, and hence move the domain wall, is prop ortional to the change in energy

required to align the magnetic moments with the eld [27 ]. We make the same assumption for

ferro electric materials.

For a dip ole moment p, the energy due to the e ective eld is

E = p E 27

so that the change in energy required to rotate the moments p in the direction of the eld is

E = p E + p E :

e e

Under the assumption that the energy E required to break pinning sites is prop ortional to this

change, it follows that

E = pE c 1 cos

pin e 1

where p = jpj;E = jE j and c is the constant of prop ortionality. To eliminate c ,we let E denote

e e 1 1

the energy for 180 domain walls and note that E =2pE c . This implies that

e 1

E = E 1 cos : 28

The relation 28 quanti es the energy required to break a single pinning site. To obtain a

corresp onding macroscopic relation which quanti es the b ehavior of the bulk material, we employ

the techniques of Jiles and Atherton [27 ] and let n denote the average density of pinning sites and

hE i denote the average energy for 180 walls. The energy required to break pinning sites if a domain

wall of area A moves a distance x is then

nhE i

E x= 1 cos Adx:

2 0

p pθ

Domain Wall

Figure 4. Domain wall and orientation of dip ole moments p; p and electric eld E .

e 13

Because the p olarization is the total contribution due to dip ole moments in a given volume, the

change in p olarization due to the reorientation of moments is

dP = p1 cos Adx : 29

nhE i

, 28 and 29 can b e combined to yield the expression With the de nition k =

E P =k dP 30

for the energy required to break pinning sites. Note that this relation provides a macroscopic de-

scription due to the averaging of domain density and energy in the de nition of k .

Completely analogous arguments follow for 90 domain walls as well as materials which exhibit

o o

rhomb ohedral symmetries 71 =109 domain walls with the nal expressions for the pinning energy

di ering only in the values of k . Because the numb er of pinning sites and average energy are unknown,

the parameter k must b e estimated for any of these domain con gurations through a least squares

t to data.

3.2 Work in the Polarization Pro cess

To quantify the irreversible and reversible p olarization pro cesses, we must determine the work re-

quired to attain a sp eci ed p olarization level. This can b e formally obtained from classical expressions

for the work required to con gure a sp eci ed charge distribution or derived from rst principles of

electrostatic eld theory.

As detailed in [46 ], if denotes a charge density, then the work necessary to pro duce a change

is given by

W = dV

where is the asso ciated scalar p otential. The application of Maxwell's equations then yields

W = E D dV

so that the change in energy densityisED. This yields the classical expression

w = E dD 31

for the energy required to create the eld at a p oint.

For this analysis, we are interested in the work required to attain a sp eci ed p olarization level

through the application of the interaction eld D . Such a relation can be formally obtained by

employing the relation 18 in 31 to obtain

Z Z

D D

1 1

w = D dD P dD :

0 0

Over a full cycle, the rst term on the right hand side is zero so that the losses are given by

w = P dD :

Employing the interaction eld D , one is then left with the energy functional

P dD 32 w =

for sp ecifying hysteresis losses over a p ortion of the p olarization cycle. Further details regarding the

derivation of the functional 32 are provided in App endix A. We p oint out that the energy functional

32 is analogous to that employed by Jiles and Atherton [27 ] in their magnetization mo del. 14

3.3 Irreversible Polarization

As detailed by Chen and Wang [7], the dielectric prop erties of ferro electric materials are in uenced

by b oth the b ending b ehavior of domain walls pinned by inclusions and the long-range movementof

domain walls when lo cal energy barriers are overcome. We will denote the resp ective p olarization

comp onents as the reversible p olarization and irreversible p olarization and employ the strategy of

Jiles and Atherton to mo del the two phenomena.

To characterize the irreversible p olarization, we note that the p olarization energy for a given

e ective eld level can be expressed as that observed in the ideal anhysteretic case minus losses

necessary to overcome lo cal energy barriers and hence break pinning sites. Employing the relation

32 for the energy and 30 for the losses at pinning sites, this yields

Z Z Z

D D D

e e e

1 1 dP

ir r

dD P E dD = P E dD k

e ir r e e an e e

0 0 0

0 0 e

where P denotes the irreversible p olarization and P is the anhysteretic p olarization given by 23

ir r an

or 24 for the e ective eld E = E + P . Di erentiation then yields

e ir r

ir r

P = P k 33

ir r an 0

where the parameter = signdE ensures that the energy required to break pinning sites always

opp oses changes in the p olarization. To obtain an expression which facilitates numerical implemen-

tation, we reformulate 33 to obtain

ir r

P P = k

an ir r 0

dP 1

ir r

= k

ir r

from which it follows that

dP P P

ir r an ir r

= : 34

dE k P P

an ir r

We note that for certain materials and op erating regimes, the indiscriminate use of the di er-

ential equation 34 can lead to nonphysical solutions when the eld is reversed in the saturation

region of the hysteresis lo op near the tip in the rst and third quadrants. Sp eci cally, for very

wide hysteresis lo ops of the typ e illustrated in Figure 1, direct solution of 34 yields a negative

when E is reversed and the magnitude of P is less than that di erential electric susceptibility

of the global anhysteretic see Figure 6. This phenomenon is not observed exp erimentally and the

mo del is mo di ed slightly for these situations. As noted in [28 ] where the analogous ferromagnetic

phenomenon is analyzed, domain walls remain pinned when the eld is initially reversed and the

primary change in p olarization is due to the reversible relaxation of bulged domain walls. Hence we

ir r

enforce the condition =0 until the p olarization crosses the anhysteretic curve. This pro duces

the more physically realistic expression

dP P P

ir r an ir r

= 35

dE k P P

an ir r

where

1 ; fdE > 0 and P >P g or fdE < 0 and P

an an

= 36

0 ; otherwise :

We p oint out that this additional constrain is unnecessary for most sigmoid hysteresis curves and is

required only for materials which exhibit large hysteresis losses and are driven to full saturation. 15

3.4 Reversible Polarization

A basic tenant of domain and domain wall theory for ferro electric materials is the observation

that walls exhibit bulging and motion b efore manifesting irreversible e ects [7, 12 , 30 ]. While this

reversible comp onent of the p olarization is typically smaller than the irreversible comp onent, the

e ects are still signi cant and must b e incorp orated to obtain an accurate mo del. The development

of the reversible p olarization follows closely the theory provided by Jiles and Atherton [27 ] for

ferromagnetic materials.

To quantify the reversible p olarization P , we rst consider the energy required to change the

rev

p olarization from P E toPE , where again E = E + P , at a xed eld level E . From 27,

an e e e

it follows that the change in energy p er unit volume required to attain this change is

E = [P E P E ] :

V e an e

Under the assumption that jPj = jP j = P and employing the law of cosines, this yields

E = PE 1 cos

V e

= P P

where P P =jPP j. For xed eld values, the force on the domain walls is then

an an

[ E ] 1

P P F =

since P =[E ]E. Neglecting higher order terms and denoting the multiplicative constant by

k then yields the expression

P = k P P ;

1 an

for the pressure on the domain wall.

To obtain the reversible p olarization due to wall movement in resp onse to this pressure, we need

to approximate the displaced volume. To sp ecify the geometry, we consider a domain wall b etween

two pinning sites separated by a distance of 2y . The wall is assumed to b ow an amount x in resp onse

to the applied pressure P with a resulting radius of curvature r see Figure 5.

If we let E denote the surface energy of the domain wall, then the pressure can b e expressed as

: P =

Using techniques analogous to those employed in Jiles and Atherton [27 ], it then follows that

2 2

x = r r y

2E 2E

S S

y =

P P

P y

Binomial Theorem

y k

= P P :

S 16

Under the assumption that the wall displaces a spherical solid angle, the change in volume is given

2 2

x3y + x . For the case in which separated domains are parallel and antiparallel to the byV =

applied eld, the reversible p olarization is

P = 2Vp

rev

4 2

p y k p y k

1 1

= P P + P P :

an an

4E 3 4E

S S

Consideration of rst-order terms then yields the expression

P = c P P 37

rev 1 an

p y k

for the reversible p olarization. The constant c = must b e estimated through a least squares

t to data since the comp onents p; y ; k and E are unknown.

1 S

Pinning Site

r y

Bowed x Domain Wall

Equilibrium Domain Wall

Pinning Site Position

Figure 5. Geometry of the domain wall with reversible b owing from Jiles and Atherton [27 ].

3.5 Total Polarization

The total p olarization can now be sp eci ed through the following algorithm. For a given eld E ,

p olarization P and constant stress , the e ective eld and anhysteretic p olarization are given by

17 and 23 or 24, resp ectively. The di erential equation 35 is then integrated to compute the

irreversible p olarization P . Finally, the total p olarization P is given by the sum

ir r

P = P + P : 38

ir r rev

The reversible p olarization P can b e sp eci ed either by 37 or by the expression

rev

P = cP P 39

rev an ir r

where c = . For the remainder of this development, we will employ the expression 39 since it

1+c

p ermits the separation of the reversible and irreversible p olarizations.

The full time-dep endent mo del leading from an input eld E t to the output p olarization P t

is summarized in Algorithm 1 with prop erties of the mo del parameters summarized in Table 1. The 17

use of the Langevin mo del yields the anhysteretic p olarization iia whereas the Ising spin mo del

yields iib.

The mo del in the form summarized in Algorithm 1 is typically employed when characterizing a

material or device. For control applications, however, it can b e advantageous to express the output

p olarization directly in terms of the input eld. When the Ising spin mo del is used to characterize

the anhysteretic magnetization, the expressions 24, 35, 38 and 39 can b e consolidated to yield

dP dP dP

ir r an

= 1 c + c

dE dE dE

P P dP

an ir r an

= 1 c + c

k P P dE

an ir r

dP P P

an an

= + c

P P k dE

h i

E + P

P tanh P

cP E + P dP

h i

+ sech 1+ =

E + P

a a dE

k ~ P tanh P

where ~ = . The magnetization at a given eld level is then sp eci ed by the solution to the

di erential equation

= G E; P

P E =P

0 0

where

h i

E + P

P tanh P

i h

G E; P =

E + P cP E + P

sech P 1 k ~ P tanh

a a a

cP E + P

+ : sech

a a

If one employs the Langevin expression 23 rather than the Ising spin relation 24 for the anhys-

teretic, the function G is given by

i h

E + P

P P L

h i

G E; P =

E + P

cP E + P a

k ~ P L P

1+ csch

a a E + P

cP E + P a

csch

a a E + P

where the Langevin function is de ned by

: Lz coth z

As detailed in [49 ], where the analogous magnetization mo del is considered, the formulation 40 is

directly amenable to inversion. This provides a mechanism for obtaining an inverse comp ensator for

linear control design. 18

Remark 1:

The previous derivation has involved two asp ects of the anhysteretic p olarization, namely, the

global anhysteretic p olarization and lo cal anhysteretic p olarization curves. To facilitate implemen-

tation, we summarize p ertinent attributes of the two comp onents.

Global Anhysteretic:

The global anhysteretic represents the p olarization attained in the absence of pinning sites with

zero remanent p olarization. As detailed in Section 2, the global anhysteretic p olarization dep ends

up on the applied eld E as indicated in the Ising spin mo del P = P tanhE + P =a or

an s an

Langevin mo del P = P LE + P =a. The slop e of the curve P is determined by the ratio of

an s an an

the parameters = E =P and a = E T=T with suciently small values of a e.g., due to T

0 s 0 c c

for PMN pro ducing the multi-valued curve plotted in Figure 3b. To illustrate the dep endence of the

global anhysteretic p olarization on these parameters, the curves obtained with =1:53 10 V m=C ,

5 6 5

a = 1:8 10 T > T for PMN and = 1:53 10 V m=C , a = 0:7 10 T < T for PMN

c c

are plotted in Figure 6a. It should be noted that the multi-valued curve obtained with the latter

parameter values is the global anhysteretic obtained in Section 4 when constructing the p olarization

mo del for a PMN actuator employed b elow the Curie temp erature.

Lo cal Anhysteretic:

For the formulation of the irreversible p olarization develop ed in Section 3.3, the lo cal anhys-

teretic p olarization is computed using the e ective eld E = E + P rather than the equilibrium

e ir r

value E = E + P employed for the global anhysteretic. We interpret this as the hysteresis-free

e an

p olarization which would result if one started at a p oint E; P on the irreversible p olarization

ir r

curves and mo di ed the eld. Because the lo cal anhysteretic incorp orates the prevailing irreversible

p olarization, we employ it in Algorithm 1 when computing the anhysteretic resp onse to an applied

electric eld.

6 5

The lo cal anhysteretic p olarization with the parameter values =1:53 10 V m=C , a =1:810

is plotted along with the global anhysteretic and total p olarization in Figure 6b. It is noted that

the lo cal anhysteretic more closely resembles the total p olarization than do es the low temp erature,

multi-valued global anhysteretic curve depicted in Figure 6a. Finally, it should b e noted that in the

absence of coupling e ects = 0, the lo cal and global anhysteretic curves coincide.

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1 ) ) 2 2 0.05 0.05

0 0

−0.05 −0.05 Polarization (C/m Polarization (C/m

−0.1 −0.1

−0.15 −0.15

−0.2 −0.2 Polarization TT −0.25 c Global Anhysteretic

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Electric Field (MV/m) Electric Field (MV/m)

a b

Figure 6. a The global anhysteretic P = P LE + P =a, a = E T=T , of PMN for temp era-

an s an 0 c

tures T T . b The lo cal anhysteretic P = P LE + P =a , global anhysteretic,

c c an s ir r

and total p olarization of PMN for T >T .

c 19

We note that in certain applications of the Jiles-Atherton mo del, the e ective eld used to de ne

the irreversible magnetization is formulated in terms of the total magnetization rather than the

irreversible magnetization. This provides an adequate approximation when reversible e ects are

small. A similar approximation can be considered for ferro electric materials, in which case, the

expressions i and iii in Algorithm 1 are replaced by

i' E t=Et+ P t

dP dE [P t P t]

ir r an ir r

iii' t= .

ir r

dt dt

k [P t P t]

an ir r

ir r

The derivative results from expanding 33 with the total p olarization employed in the e ective

ir r

eld. For materials in which the reversible p olarization is small, 1, and the expression iii'

is adequately approximated by iii. This is the case for the magnetism in certain applications where

the Jiles-Atherton mo del is employed. For ferro electric materials in which the reversible p olarization

is signi cant, this derivativemust b e retained if the expressions i' and iii' are substituted for i

and iii.

i E t=Et+ P t

e ir r

E t a

iia P t=P coth Langevin Mo del

an s

a E t

E t

Ising Spin Mo del iib P t=P tanh

an s

dP dE [P t P t]

ir r an ir r

iii t=

dt dt k [P t P t]

an ir r

iv P t=c[P t P t]

rev an ir r

v P t=P t+P t

rev ir r

Algorithm 1. Time-dep endent mo del quantifying the output p olarization P t for =0. 20

Parameter Physical Prop erty E ects on Mo del

Quanti es domain Increased values lead to steep er slop es for

interactions anhysteretic and p olarization curves.

a Shap e parameter Increased value decreases slop e of P .

for P

k Average energy Increased value pro duces wider hysteresis

required to break curve and narrower minor lo op.

pinning sites

c Reversibility co e- Decrease in value leads to wider hysteresis

cient curve.

P Saturation Increase leads to large saturation value for

p olarization p olarization.

Table 1. Physical prop erties and e ects of the mo del parameters ; a; k ; c; P . The parameters

= signdE and given by 36 are computed directly from the measured eld and p olarization.

4 Mo del Validation

The mo del derived in Section 3 and summarized in Algorithm 1 provides a relation between elec-

tric elds input to a ferro electric actuator and the resulting p olarization. This mo del incorp orates

the nonlinear constitutive b ehavior, through either the Langevin or Ising spin relations, as well as

hysteresis due to imp eded domain wall movement. When coupled with a mechanical constitutive

relation, such as 1, this provides a characterization of the strains pro duced by the material.

To illustrate the capabilities of the mo del, we consider the characterization of a PMN-PT-BT

actuator op erating in its ferro electric range. The material was comp osed of 12 PT and 2 BT in

solid solution and had a Curie temp erature of T =45 C. The data was collected from a stress-free

sample = 0 in 25 at T = 20 C. To maintain quasistatic conditions, the frequency for the

input electric eld was taken to b e ! = 1 Hz for a complete steady-state cycle.

To employ the mo del, the anhysteretic relations must be sp eci ed and appropriate parameters

P ;a; ;k and c must b e ascertained. As noted in Section 2, a variety of mo dels for the anhysteretic

p olarization have b een develop ed and employed in ferro electric applications. For applications in

which anhysteretic p olarization data can be obtained e.g., through the choice of op erating tem-

p erature or the application of a large oscillating eld to \break through" pinning sites, the choice

of anhysteretic mo del can be validated through a t to the data. For other cases, the choice of

anhysteretic mo del can b e motivated by mo deling and theoretical e.g., thermo dynamic considera-

tions. To mo del the PMN-PT-BT actuator, we consider b oth the Langevin and Ising spin mo dels.

These mo dels di er in their third and higher order terms which pro duces the di erence in saturation

b ehavior illustrated in Figure 2.

The parameters P ;a; ;k and c must then be determined for the given material. For many

materials, the saturation p olarization P is well known and values can be sp eci ed directly. The

parameters a; ; k ; c are less easily sp eci ed, however, since they are de ned in terms of material

prop erties which are either unknown or not easily measured. Furthermore, the values of a; ; k ; c

dep end to some extent on the choice of anhysteretic mo del since they dep end up on the shap e of

the mo deled anhysteretic and energy di erences between the anhysteretic and measured data. In

applications, a; ; k ; c and in many cases P , are estimated through a least squares t to data.

s 21

The mo deled p olarization obtained with the Langevin relation and the parameter values a =

5 2 6 5 2 2

0:70 10 C=m ; = 1:53 10 V m=C ; k = 4:5 10 C=m ; c = :80;P = :296 C=m is compared

with the exp erimental data in Figure 7. We reiterate that the data was collected under quasistatic

conditions with one complete steady-state cycle plotted in the gure. The measured electric eld data

was used as input to the mo del for three complete cycles. While the initial p olarization curve for the

mo del is depicted here, one would typically bring the mo del to a p olarized state b efore employing it

for physical transducer characterization or control design. In this regime, the mo del very accurately

quanti es the nonlinear and hysteretic b ehavior of the material.

The corresp onding mo del employing the Ising spin relation is compared with the data in Figure 8.

5 2 6

In this case, the mo del was computed with the parameters a =4:210 C=m ; =210 V m=C ;

5 2 2

k =3:710 C=m ;c=:65;P = :28 C=m . As exp ected, less than 5:5 change is observed in the

saturation p olarization P . The other four parameters di er from those obtained with the Langevin

relation to accommo date the di erent qualitative b ehavior of the curves. A comparison of Figures 7

and 8 indicates that the use of the Langevin relation for the anhysteretic p olarization provides a

somewhat b etter t than the Ising spin-based mo del. This indicates that for this material, the Ising

spin mo del may saturate to o quickly due to the imp osed requirement that cells are restricted to only

two p ossible orientations.

The nature of the Ising spin mo del is further illustrated by the global anhysteretic p olarization

plotted in Figure 6a. As discussed in Remark 1 and noted in the gure, this curve is multivalued

at temp eratures T

required to break pinning sites and hence provides an overly steep transition between saturation

states. The inclusion of domain wall e ects through the irreversible and reversible p olarizations

provides a mo del whichcharacterizes the more gradual transitions observed in the data. The global

anhysteretic for the Langevin mo del is similar to the Ising spin curve depicted in Figure 6a.

0.25

0.2

0.15

0.1 ) 2 0.05

−0.05 Polarization (C/m

−0.1

−0.15

−0.2

−0.25 Model Data

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Electric Field (MV/m)

Figure 7. Exp erimental PMN-PT-BT data and mo del t with the Langevin anhysteresis mo del. 22 0.25

0.2

0.15

0.1 ) 2 0.05

−0.05 Polarization (C/m

−0.1

−0.15

−0.2

−0.25 Model Data

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Electric Field (MV/m)

Figure 8. Exp erimental PMN-PT-BT data and mo del t with the Ising spin anhysteresis mo del.

5 Concluding Remarks

This pap er addresses the mo deling of hysteresis in ferro electric materials as well as certain relaxor

ferro electrics employed at temp eratures well b elow the Curie p oint. The mo del is derived through

the characterization of the energy required to b end and translate domain walls that are pinned at

inclusions in the material. The rst step in the development of the mo del is the determination

of appropriate constitutive relations which characterize the anhysteretic E -P curve. The relations

quantify the global equilibrium state of the material in the idealized case that domain walls are

not signi cantly inhibited by pinning sites in the material. Several techniques have b een previously

used to mo del anhysteretic p olarization or magnetization curves including the Langevin and Ising

spin relations describ ed here. An imp ortant comp onent of the anhysteretic mo del is the inclusion of

interactions due to the neighb oring domains through a mean eld approximation. Hysteresis e ects

are then quanti ed through the computation of the average energy required to move domain walls

across inclusions or pinning sites inherent to the material. The resulting quasistatic ferro electric

mo del is analogous to that develop ed by Jiles and Atherton for ferromagnetic materials.

The resulting hysteresis mo del is comprised of a simple di erential equation whose solution pro-

vides the theoretical p olarization at a given eld level. The use of the mo del for a given application

requires the determination of ve parameters ; k ; c; a and P which are macroscopic averages of

microscopic domain prop erties. Hence the mo del incorp orates the underlying domain physics but is

amenable to applications requiring real-time implementation. The determination of parameters is

facilitated by the physical nature of the parameters e.g., increased values of the average pinning en-

ergy k lead to increased hysteresis in combination with the fact that the saturation p olarization can

often be directly measured. Hence the mo del can be easily up dated to accommo date the changing

op erating conditions often encountered in control applications.

In the form presented here, the mo del strictly applies to isotropic materials. Anisotropy ef-

fects are manifested primarily in the construction of the anhysteretic curve since domains will align

with crystallographic easy axes rather than align in bip olar or uniform orientations. The choice of 23

mo del for the anhysteretic p olarization can be mo di ed to accommo date sp eci c crystallographic

anisotropies if desired. In its current form, the mo del do es incorp orate symmetric minor lo ops. We

note, however, that the present mo del do es not accommo date certain forms of asymmetric minor

lo ops, full electromechanical coupling, or frequency-dep endenthysteresis losses. While these exten-

sions are imp ortant for general design and control applications, they lie beyond the scop e of this

pap er and are under investigation.

To illustrate the capabilities of the mo del, the characterization of data collected from a PMN-

PT-BT actuator at a temp erature 65 b elow the Curie temp erature was considered. Both the

Langevin and Ising spin relations were considered for the construction of the anhysteretic curve. The

comparison of the mo del and exp erimental results illustrates that for the sp eci ed op erating regime,

the mo del provided a highly accurate characterization of the hysteresis exhibited by the material, with

the Langevin-based mo del providing a slightly b etter characterization in the saturation region of the

curve. While illustrated in the context of relaxor ferro electrics at low temp eratures, the electrostatic

basis for the mo del is suciently general to p ermit its application to a variety of ferro electric and

relaxor ferro electric materials.

Acknowledgements

The authors express sincere appreciation to David Jiles for his input regarding the development

of these mo deling techniques for ferromagnetic materials and Pamela Sadler-Hom for collecting the

exp erimental data on the PMN-PT-BT actuator. The research of R.C.S. was supp orted in part by

the Air Force Oce of Scienti c Research under the grantAFOSR-F49620-98-1-018 0.

App endix A: Work in the Polarization Pro cess

The discussion in Section 3.2 provides a somewhat formal motivation for the energy functional

32. We provide here a more rigorous analysis to motivate this choice. A similar analysis for

ferromagnetic materials can be found in Brown [3 ] with additional details regarding the magnetic

mo del presented byVenkataraman and Krishnaprasad [52 ]. Note that throughout this discussion, it

is assumed that eld and p olarization changes are suciently slow to p ermit the use of electrostatic

eld relations. In many applications, involving the characterization of materials, this is a reasonable

assumption since exp eriments are conducted under quasistatic conditions.

To quantify the e ective eld contributions, it is advantageous to consider the work required to

p olarize the b o dy rather than the work necessary to achieve a sp eci ed eld level. Let = denote

a charge density in the region V as depicted in Figure 9. As detailed in [48 , page 72], this charge

con guration then pro duces the electric eld

E = dV

4 r

and r jrj. Furthermore, the electric at the p oint s in the dielectric material . Note that r

jrj

p otential due to the dielectric is

1 P r

= d

4 r

see [48 , page 260]. Finally, the time rate of work due to the changing p olarization is

dw d

= dV :

dt dt

V 24

It then follows that

Z Z

P r dw 1 d

= V d dV

dt 4 dt r

Z Z

V r 1 dP

d dV = V

V @t 4 r

Z Z

dP V r

= dV d

dt 4 r

= E d

so that

w = E P d :

The change in p olarization p er unit volume is then E P so the energy required to attain a given

p olarization level is

w = E dP : 43

So far in this discussion, the eld E has b een considered to be generated by the charge con g-

uration . We now consider the e ective eld E = E + E where E = P quanti es the eld

e 1 1

contributions due to the p olarized dielectric b o dy. Expansion of 43 yields

w = w + w

e 1

where

w = E dP

1 1

denotes the electrostatic self-energy. This quantitycharacterizes the shap e-dep endent comp onentto

the energy which do es not contribute to the irreversible hysteresis losses since

I I

w = P dP =0: 44

Hence we have some exibility when sp ecifying an appropriate energy functional which quanti es

energy losses due to hysteresis.

P φ dτ

E dQ ρ = s r dV τ

Figure 9. Orientation of the body V with charge . This generates a eld E in the dielectric

which in turn pro duces a p otential in the charged b o dy. 25

Noting 44, the energy due to the e ective eld over a full p olarization cycle can b e expressed

w = E dP

= P dE

I I

= P dE P dP

P dD =

for arbitrary . These arguments can be extended to partial cycles by employing the fact that

P dP = 0 for any closed curve c . This motivates the use of the functional

w = P dD 45

when quantifying irreversible comp onents to the p olarization. We p oint out that 45 is identical to

the functional 32, obtained through formal arguments, and is analogous to that employed by Jiles

and Atherton [27 ] in their magnetization mo del.

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