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- 3 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 3 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 3 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 −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) o Figure 1. Hysteresis lo op measured at a temp erature of T = 20 C in a PMN-PT-BT sample o 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.