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PAGE 1 1 ON THE CORRELATION OF DYNAMIC ELECTRICFIELDINDUCED STRUCTURAL CHANGES AND PIEZOELECTRICITY IN FERROELECTRIC CERAMICS By ABHIJIT PRAMANICK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Abhijit Pramanick PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS I would like to thank m y advisor Dr. Jacob L. Jones for providing me all the opportunities for completing this work and his guidance and as sistance throughout. I would also like to express my gratitude to Dr. Dragan Damjanovic and Dr John E. Daniels for their collaboration and helpful discussions, which contributed to sign ificant portions of this work. I sincerely acknowledge the intellectual i nputs from my committee members Dr Juan C. Nino, Dr. David P. Norton, Dr. Aravind Asthagiri and Dr. Franky So. I would like to acknowledge the following funding sources: funding from the National Science Foundation (NSF) under award number DMR0746902; funding for travel to the Swiss Federal Institute of Technology (EPFL) thr ough NSF award number OISE0755170; funding to travel to Oak Ridge National Laboratories th rough the Oak Ridge Associated Universities (ORAU) Powe Junior Faculty Enhancement Aw ard; travel funds to access the Australian Nuclear Science and Technology Organisation (ANSTO) provided by th e University of Tennessee's International Materials Institute (IMI) ANSWER program and supported by NSF under award number DMR0231320. I want to thank all the presen t and past members of Dr. Jacob Jones research group for their company and support Shelley Cottrell, An derson D. Prewitt, Wei Qiu, Martin McBriarty, Prateek Maheswari, Sandhya Nallamada, Chri s Dosch, Elena Aksel, Chris Brink, Krishna Nittala, Kyle Calhoun and Paul Draper. Special th anks to Elena for her assistance while writing this dissertation. Also, I would like to thank Lu Cai for helping me with ferroelectric measurements in Dr. Ninos laboratory. During this work, I had opportunities to spend time at other facilities as part of my research and education. I want to acknowledge members of the Ceramics La boratory at the Swiss Federal Institute of Technology for providing company and assistance during a very productive and PAGE 5 5 enjoyable stay at EPFL. I want to especia lly thank Soodkhet (Bond) fo r showing me around in Sydney during my offhours at ANSTO. I would also like to thank the organizers of the 10th National School on Neutron and Xray Scattering, in which I had a chance to participate. I want to thank the fellow attendees of the school Aiming, Vibhor and others for the good times. For all the varied and engaging discussions, notsoapparent atte mpts at comedies, rivalries at the bowling alley, dinner and lunch times and all sort of things during th e last four years, I want to thank Krishna (KP), Mahesh, Purushotta m, Kalyan, Rakesh, Shobit, Sankara, David and Peter. Finally, I want to acknowledge my family fo r all of their support during the good and bad times. I want to express my most heartfelt gratitu de to my parents for which no amount of words would be enough. Their encouragement, support and guidance have been instrumental in all my life. This work is my humble tribute to all their love and support. PAGE 6 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES.......................................................................................................................10 ABSTRACT...................................................................................................................................14 1 INTRODUCTION TO PIEZOELECTRICITY...................................................................... 16 1.1 Piezoelectric Ceramics..................................................................................................... .16 1.2 Structural Origin of Pi ezoelectricity in Ceram ics............................................................. 17 1.2.1 Intrinsic Mechanisms..............................................................................................20 1.2.2 Extrinsic Mechanisms............................................................................................22 1.2.2.1 Rayleigh law................................................................................................. 23 1.2.2.2 Preisach model.............................................................................................24 1.2.2.3 Micromechanical models............................................................................ 26 1.3 Diffraction Studies of Pi ezoelectric Mechanism s.............................................................27 2 OBJECTIVE AND OVERVIEW...........................................................................................42 3 SYNTHESIS OF LEAD ZIRCO NATE TITANATE CERAMICS .......................................45 3.1 Phase Diagram and Dopants for PZT............................................................................... 45 3.2 Synthesis of PZT Ceramics..............................................................................................46 4 NONLINEARITIES AND LOSSES IN CONVE RSE PIEZ OELECTRIC EFFECT OF PZT CERAMICS....................................................................................................................56 4.1 Experimental Procedure.................................................................................................... 56 4.1.1 Converse Piezoelectric Measurements................................................................... 56 4.1.2 Application of Rayleigh Law................................................................................. 57 4.1.3 Calculation of Piezoelectric Losses and Complex Coefficients .............................58 4.1.4 Measurement of the Harmonic Component s of Converse Piezoelectric Strain ..... 61 4.2 Results and Discussion..................................................................................................... 62 4.2.1 Piezoelectric Nonlinearity and Application of Rayleigh Law ................................ 62 4.2.1.1 Effect of crystal stru cture on nonlinear behavior ......................................... 62 4.2.1.2 Effect of dopants on nonlinear behavior......................................................64 4.2.1.3 Prediction of piezoelectric hys teresis from Rayleigh relations.................... 65 4.2.2 Piezoelectric Losses and Com plex Coefficients..................................................... 66 4.2.3 Harmonics of the Converse Piezoelectric Strain.................................................... 67 4.3 Conclusions.......................................................................................................................68 PAGE 7 7 5 CHARACTERIZATION OF ELECTRICFIELDINDUCED STRUCTURAL CHANGES USING LABORATO RY XRAY DIFFRACTI ON........................................... 81 5.1 Experimental Procedure.................................................................................................... 81 5.1.1 In situ Xray Diffraction Under Application of Electric Field ............................... 82 5.1.1.1 Diffraction under static electric fields..........................................................82 5.1.1.2 Diffraction under cyclic electric fields.........................................................82 5.1.2 Quantification of Structural Changes.....................................................................84 5.2 Results and Discussions.................................................................................................... 87 5.2.1 Structural Changes Under Static Electric Fields.................................................... 87 5.2.2 Structural Changes Unde r Cyclic Electric Fields ................................................... 89 5.3 Conclusions.......................................................................................................................92 6 CHARACTERIZATION OF ELECTRICFIELDINDUCED STRUCTURAL CHANGES USING HIGHENERGY SYNC HROTRON XRAY DIFFRACTION .......... 101 6.1 Experimental Procedure.................................................................................................. 101 6.1.1 Sample Synthesis and Preparation........................................................................ 101 6.1.2 TimeResolved Diffraction...................................................................................102 6.2 Analysis of Diffraction Data...........................................................................................103 6.2.1 Binning of Diffraction Data at Different Azim uthal Angles................................ 103 6.2.2 Fitting of ( hkl ) Diffraction Peaks .........................................................................103 6.2.3 Analysis of Ferroelectric Phases..........................................................................104 6.2.4 Binning of Diffraction Data with Respect to Tim e.............................................. 105 6.2.5 Quantification of Structural Changes...................................................................107 6.2.5.1 Non180 domain switching....................................................................... 107 6.2.5.2 Electricfieldindu ced lattice strains ........................................................... 109 6.3 Results and Discussions.................................................................................................. 110 6.3.1 Non180 Domain Switching............................................................................... 110 6.3.2 Calculation of Strain from Non180 Dom ain Switching.................................... 113 6.3.3 Field Dependence of Non180 Dom ain Switching Strains................................. 114 6.3.4 ElectricFieldInduced Lattice Strains..................................................................116 6.4 Conclusions.....................................................................................................................118 7 NONLINEAR CONTRIBUTIONS FROM STRUCTURAL MECHANISMS UNDER CYCLIC ELECTRIC FIELDS ............................................................................................. 135 7.1 Effect of Non180 Domain Switching Strains..............................................................135 7.2 Effect of Lattice Strains..................................................................................................138 7.3 Cumulative Nonlinear Contributions fr om Different Structural Mechanisms............... 141 7.4 Conclusions.....................................................................................................................143 8 SUMMARY AND FUTURE WORK.................................................................................. 152 8.1 Summary.........................................................................................................................152 8.2 Suggestions for Future Work in this Area...................................................................... 157 8.2.1 Nonlinearity in Direct Piezoelectric Behavior..................................................... 157 PAGE 8 8 8.2.2 Frequency Dependence of Piezoelectric Coefficients.......................................... 158 8.2.3 Temperature Dependence of Piezoel ectric Nonlinearity and Frequency Dispersion .................................................................................................................. 159 8.2.4 Piezoelectric Nonlinearity and Fr equency Dispersion in LeadFree Piezoceram ics............................................................................................................ 160 8.2.5 Piezoelectric Nonlinearity and Freque ncy Dispersion in Ferroelectric Thin Film s..........................................................................................................................161 APPENDIX A EFFECT OF SAMPLE DISPLACEMEN T ON DIFFRACT ION PEAK POSITION FOR A CPS DETECTOR..................................................................................................... 165 B STROBOSCOPIC DATA COLLECTION USING MCDWIN........................................... 168 C PENETRATION DEPTH OF XRAYS IN PZT CERAMICS ............................................ 170 D IGOR CODES FOR ANALYSIS OF 2DIMENSIONAL DIFFRACTION DATA ........... 173 REFERENCES............................................................................................................................195 BIOGRAPHICAL SKETCH.......................................................................................................200 PAGE 9 9 LIST OF TABLES Table page 31. The measured densities and the percent th eoretical densities for the different sam ples... 54 32. List of the samples used for experim ents in this work....................................................... 55 41. Harmonics of the converse pi ezoelectric strains for PLZT52 48....................................... 80 71. d and non180 for the different samples studied............................................................. 150 72. The values of Thkl and mhkl for the different hkl crystallographic poles used to calculate the total eff ective lattice strain.......................................................................... 151 PAGE 10 10 LIST OF FIGURES Figure page 11. Unit cell of a cubic pero vskite crystal structure................................................................. 28 12. Domain variants in ferroelectric ceramics shown as arrangement of polarization axes of adjacent dom ains...........................................................................................................29 13. The different structural changes in a f erroelectric ceramic fr om a paraelectric state above TC to a poled ferroelectric state w ith reference to a tetragonal crystal structure..............................................................................................................................30 14. The effect of reorientation of dom ains under applied electric field on A) the polarization, and B) the strain of a piezoelectric ceramic.................................................. 31 15. The elastic free energy profiles fo r polarization contraction/elongation and polarization rotation in tetragonal PZT with Zr/Ti ratio of 40/60. ....................................33 16. The electricfieldinduced lattice strain coefficien ts for different crystal orientations within a polycrystalline tetragonal lead zirconate titanat e piezoelectric ceramic under cyclic electric actuation......................................................................................................34 17. Illustration of energy profile for dom ain wall motion in a medium with random pinning centers...................................................................................................................35 18. Application of Rayleigh law for direct piezoelectric behavior of 4 atom ic % Nbdoped PZT ceramics with Zr/Ti ratio of 63/37.................................................................. 36 19. Energy profile (left), and square hysteresis (right) of an elem entary bistable unit, as assumed in Preisach model for hysteresis. m represents generalized susceptibility.......... 37 110. Energy profile and hysteresis of bistable units as a function of their position on the Preisach plane at zero applied field. ..................................................................................38 111. Preisach distribution function for differe nt characteristic piezo electric behaviors. .......... 39 112. Simulated and measured xE hystere sis loops for Ladoped PZT ceram ics using micromechanical models based on energy criterion for switching, as described in Eq. 17................................................................................................................................40 113. Non180 domain switching fraction (002) in tetragonal PZT ceramics under applied unipolar and bipolar electric fields, as a function of field amplitude.................... 41 31. Phase diagram and piezoelectric coefficients of PZT ceramics. PbZrO3 PbTiO3 phase diagram over compositional range cl ose to the morphotropic phase boundary (MPB) is shown in the bottom half of the plot.................................................................. 50 PAGE 11 11 32. A) Scanning electron micrograph of fracture surface of PLZT5248; the g rain size is of the order of 12 m. B) Xray diffrac tion patterns of the sintered ceramics................51 33. A) Polarizationelectric field, and B) strainelectric field hysteresis loops of the various samples. .................................................................................................................52 41. Experimental setup for meas uring converse piezoelectric strain. ...................................... 70 42. The piezoelectric coefficient d33 was calculated from the measured peaktopeak longitudinal strain, indicated as xmax, using Eq. 41..........................................................71 43. Converse piezoelectric coefficients of PLZT6040, PLZT5248 and PLZT4060 as a function of applied sinusoidal electric field am plitudes.................................................... 72 44. Estimate of the irreversible extrinsic contributions for Ladoped PZT ceram ics as a function of applied sinusoidal electric field amplitudes.................................................... 73 45. Converse piezoelectric coefficients and fractional extrinsi c contributions f or PLZT5248, PUZT5248 and PFZT4753............................................................................. 74 46. The strainelectric field hysteresis loop s for the PZT ceram ic samples at electric field amplitudes 400 V/mm and 600 V/mm: A) PLZT5248, B) PLZT6040, C) PLZT4060, D) PUZT5248....................................................................................................................75 47. Hysteresis area and imagin ary piezoelectric coef ficient.................................................... 79 51. Experimental setup a nd synchronization settings for in situ dif fraction experiments. .... 94 52. The fit for the (002)type diffraction peaks for an unpoled sample of the type PLZT5248.. ........................................................................................................................95 53. Structural changes measur ed under the application of step wise static electric fields: A) 002 and B) 111, as a function of the field amplitude................................................... 96 54. Schematic of the proposed 90 domain sw itching process under static electric fields E0........................................................................................................................................97 55. Structural changes as a function of tim e during the appli cation of cyclic square wave electric field of amplitude 650 V/mm and of frequency 0.3 Hz......................................98 56. Measured and fit patt erns of the A) (002)type, a nd B) (111) diffraction peaks, during th e positive and the negative parts of the applied electric field waveform............ 99 57. Dependence of non180 domain switchi ng and 1 11 lattice strains on amplitude of applied cyclic electric fields............................................................................................. 100 61. Details about experimental setup and tim ing electronics for in situ diffraction experiments......................................................................................................................120 PAGE 12 12 62. Specific diffraction peaks for the different sam ples used in the current experiments..... 121 63. Timeresolved structural cha nges under cyclic electric fields. ........................................ 122 64. 002 as a function of field amplitude as we ll as orientation with respect to the direction of applied field, for an unpoled Ladoped tetragonal PZT ceramic under the application of static electric fields................................................................................... 125 65. Change in volume fraction of non180 dom ains in Ladoped PZT ceramics under application of cyclic electric fields.................................................................................. 126 66. 002 for different amplitudes of applied cyclic square wave electr ic fields, for A) sample PUZT5248, and B) sample PFZT4753................................................................ 129 67. Illustration of strain due to non180 switching of ferroelastic dom ains in a single grain.................................................................................................................................130 68. The coefficient of strain due to non180 dom ain switching dnon180 for PZT ceramics....................................................................................................................... ....131 69. Electricfieldinduced lattice st rain coefficients in PZT ceram ics................................... 133 71. Nonlinear contributions to macroscopi c electricfield induced strains and nonlinear contributions from non180 domain switc hing strains for samples A) PLZT6040, B) PLZT5248, C) PUZT5248, D) PFZT4753...................................................................... 144 72. A simplified schematic illustration of cum ulative addition of strains with different hkl orientations towards macroscopi c electricfieldinduced strain................................ 146 73. The total effective lattice strain coefficient as a function of the amplitude of the applied cyclic electric field. ............................................................................................. 147 74. Nonlinear contributions to macroscopi c electricfield induced strain in sample PLZT5248 as a function of amplitude of applied electric field....................................... 148 75. Illustration of intergranular interactions lead ing to elastic strains for grains with different orientations........................................................................................................ 149 81. Variation of converse piezoelectric coefficients of PLZT5248, as a function of frequency of applied electric fields. ................................................................................. 162 82. The timedependent changes in the 111 lattice strains in PLZ T5248 ceramic, parallel to the direction of the applied electr ic fields of different frequencies............................. 163 83. The electricfieldinduced 111 lattice strains 111, as a function of time during application of a bipolar elec tric waveform of amplitude .4 kV/mm and frequencies of A) 1 Hz, B) 10 Hz, C) 100 Hz, and D) 500 Hz, for EC65 ceramics........................... 164 PAGE 13 13 A1. Diffraction geometry and peak shift due to sam ple displacement, for a CPS detector... 167 B1. Sequential steps for enabli ng stroboscopic data collection.............................................. 169 C1. Fraction of diffracted intensities as a function of thickness of diffracting volum e for A) Xrays from a CuK source, B) high energy Xrays (87.80 keV) from a synchrotron source........................................................................................................... 172 PAGE 14 14 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON THE CORRELATION OF DYNAMIC ELECTRICFIELDINDUCED STRUCTURAL CHANGES AND PIEZOELECTRICITY IN FERROELECTRIC CERAMICS By Abhijit Pramanick May 2009 Chair: Jacob L. Jones Major: Materials Science and Engineering Piezoelectric ceramics are used in various de vices such as sensors and actuators for conversion of electrical signals into mechanical signals and vice versa. In order to gain more insight into the fundamental behavior of these materials, the present work reconciles macroscopic piezoelectric nonlinearities with respect to the electricfieldinduced microstructural and crystallographic changes in ferroelectric lead zirconate titanate (PZT) ceramics. Nonlinearity and hysteresis in piezoelectric response under cyclic electric fields are investigated as a function of crystallographic phase (rhombohedral and te tragonal) and different dopants (undoped, La, and Fe). For longitudinal piezoelectric response, the amount of nonlinear contributions from irreversible extrinsic mechan isms in the rhombohedral phase is observed to be greater than in tetragonal phases for all am plitudes of applied electric fields. Likewise, Ladoping increases the amount of nonlinear contributions from irreversible extrinsic mechanisms as compared to both undoped and Fe doped ceram ics. The largest irreversible extrinsic contribution is observed to be a bout 55% of the total electricfiel dinduced strain for 2 atomic % Ladoped rhombohedral PZT ceramics, for an applied electric field of V/mm. Structural changes under the ap plication of cyclic electric fields are measured by timeresolved diffraction using both laboratory and sy nchrotron Xray sources. From timeresolved X PAGE 15 15 ray diffraction experiments, the largest non180 domain switching strains are observed for 2 atomic % Ladoped tetragonal PZT ceramics of compositions close to the morphotropic phase boundary. It is further observed that both non180 domain switching and la ttice strains can be nonlinear with respect to the applied electric fields. The nonlinear contributions from different structural changes to macroscopic electricfieldinduced strains are subsequently anal yzed. Quantitative analyses show only partial contribution from non180 domain switching towa rds macroscopic piezoe lectric nonlinearity for Ladoped and undoped compositions. This is c onsistent with certain additional nonlinear contributions determined from different hkl lattice strains. It is propos ed that the nonlinear lattice strains originate through inte rgranular constraints and are therefore influenced by non180 domain switching in the material. It is therefore concluded that the displ acement of non180 domain walls is the primary driving mechanism but not necessarily contribute exclusively towards macroscopic piezoelectric nonlinearity in ferroelectric PZT ceramics. PAGE 16 16 CHAPTER 1 INTRODUCTION TO PIEZOELECTRICITY A brief description of the funda m entals of piezoelectricity is provided in this chapter along with an overview of the present understandi ng of the different structural mechanisms contributing to piezoelectric effect in ferroelec tric ceramics. This chapter is intended as a background for the concepts developed later in the disserta tion. Readers with advanced understanding of piezoelectri city in solids are referred to Chap ter 2 for objective and overview of the present work. 1.1 Piezoelectric Ceramics Piezoelectric ceram ics are an important part of many modern technologies. Actuators based on piezoelectric ceramics transform electrical energy to an induced linear or rotary motion and are valuable in applications which require fast response, high precision, high structural endurance, and high powertoweight ratio [1]. Some devices th at use this technology include precise positioning systems, fuel in jectors, sonar and ultrasonics. Piezoelectricity is the coupling between the dielectric and the elastic displacements in a material. The linear relations for direct and conve rse piezoelectric effects are shown in Eqs. 11 and 12 respectively, jkijkiXdD =, (11) iijkjkEdx =, (12) where dijk is a constant called the piezoelectric coefficient, Di is the induced charge density, Xjk is the applied mechanical stress, xjk is the induced strain and Ei is the applied electric field. The piezoelectric coefficient dijk is a third rank tensor that relates second rank tensors Xjk and xjk with PAGE 17 17 first rank tensors Di and Ei, respectively. The tensor dijk is often represented in a simplified matrix form of dmn [2]. The phenomenon of converse piezoelect ric effect is utilized to achieve controlled actuation under applied electric fields. An accurate descripti on of electricfieldinduced strains is necessary in highprecision actuator applications. In ma ny instances, the apparent converse piezoelectric coefficient dmn of a material is found to be a function of the applied electric field amplitude Ei [35]. Under such circumstances, the linear relationship describe d in Eq. 12 no longer remains valid. Insight into the physical origins of nonlin ear converse piezoelectric responses can be obtained from an analysis of induced structur al changes such as domain switching and lattice strains under the application of electric fields. More specifically, nonlin ear contributions from different possible structural mechanisms can be distinguished by direct measurements of such structural changes under the appl ication of electric fields. 1.2 Structural Origin of Piezoelectricity in Ceramics An understanding of the phenomenon of pi ezoelectricity in solids requires an understanding of the internal struct ure of the material or more pr ecisely the change of structure under electrical or mechanical fields In principle, the existence of piezoelectricity in a material is decided by the absence of a center of symmetry in its crystal structure. In addition, most piezoelectric ceramic materials ar e ferroelectric, which means that each crystallographic unit cell has a spontaneous and reversible polarization or dipole moment under certain conditions. The spontaneously polarized crystall ites are further arranged in a co mplex threedimensional pattern within a polycrystalline ceramic material. The electricfieldinduced macroscopic strain in a piezoelectric ceramic is therefore a result of different structural changes, as described below. PAGE 18 18 The crystal structures of mo st widely used piezoelectric ceramics, such as BaTiO3 and Pb(Zr,Ti)O3, belong to the perovskite family. In simple terms, the structure can be described as a network of cornerlinked oxygen octahedra, with the smaller cation filling the octahedral holes and the larger cation filling the dodecahedral holes. The un it cell of a cubic perovskite structure can be represented as shown in Figure 11; the large cations (A) occupy the corners, the smaller cation (B) occupies the body center while the oxygen ions (O) occupy the face centers of the unit cell. When a ferroelectric ceramic is cooled th rough a certain temperature, the crystallographic unit cell typically undergoes distortion from a c ubic structure to a lower symmetry structure with a spontaneous polarization. The te mperature at which this transf ormation occurs coincides with the Curie temperature. For example, BaTiO3 undergoes a second order phase transformation from a cubic paraelectric phase to a tetragonal ferroelectric phase at 130 C. In addition, a spontaneous polarization develops within each unit cell due to a shift in th e relative positions of the ions. As the strains arising from crystallogra phic distortions give rise to internal stresses within the ceramic, twinlike patterns called domains are formed. A domain is a homogeneous area in the material having a uniform orientation of the pol arization axis. The boundaries between the domains are defined by the angle be tween the polarization axes of the adjacent domains. Mainly they are classified as 180 dom ain walls (adjacent domains are antiparallel with each other) and non180 domain walls (polarization directions of adjacent domains are arranged in headtotail configurations at angles different than 180), as shown in Figure 12. In the assintered state, a ceramic is nonpiezoelectric though the individual domains can exhibit piezoelectricity due to further lattice distortions under external fields. This happens because the polarization and strain components of all the randomly oriented domains cancel each other and no significant net macr oscopic polarization or strain develops in the material. PAGE 19 19 However, the volume fractions of the differently oriented domains in a material can be changed under the application of electric fields of high enough amplitudes. The process of creating a preferred orientation of domains within a ferro electric/piezoelectric ceramic is called poling. When poled, the polarization and strain compon ents of the different domains in the ceramic contribute towards a net dielectric and piezoelectric response. The different structural changes in a ferroelectric ceramic from a paraelectric state above TC to a poled ferroelectric state are shown in Figure 13 with reference to a tetra gonal crystal structure. Some ceramics with orthorhombic, rhombohedral and monoclinic crystal st ructures are also know n to be ferroelectric and exhibit similar structural changes as a func tion of temperature and applied electric field [6,7]. The above discussion provides only a partial description of the structural changes in a piezoelectric ceramic under the application of elec tric fields. The domai n configuration in the material achieved after poling is not rigid but can be subjected to furt her rearrangements under subsequent applications of electric fields. Such rearra ngements can be brought about by vibration or displacement of the existing domain walls. A change in the volume fraction of the differently oriented domains wi ll disrupt the existing continuity of the uniformly polarized regions in the material and consequently will contribute to its macroscopic polarization and strain. When the applied electric field exceeds a certain critical limit called the coercive field EC, a largescale switching of domains towards an energetically preferred direction takes place leading to a nonlinear change in macroscopic pola rization and strain. Figure 14 illustrates the effect of reorientation of doma ins under applied electric field on the polarization and strain of a piezoelectric ceramic [4]. PAGE 20 20 In some actuator applications, such as fuel injectors in diesel e ngines, piezoelectric ceramics are subjected to cyclic electric fields of subcoercive amplit udes [7,8]. Under such loading conditions, both lattice distortions a nd domain wall motions can contribute to the macroscopic electricfieldinduced st rain in the material [9]. The former is generally referred to as an intrinsic mechanism while the latter is referr ed to as an extrinsic m echanism [4]. In order to better predict the material behavior for such appl ications, it is helpful to understand the various structural mechanisms operating in piezoelectric cer amics under the applicati on of cyclic electric fields. A brief overview of the present understa nding of intrinsic and extrinsic piezoelectric mechanisms is provided in Sections 1.2.1 and 1.2.2. 1.2.1 Intrinsic Mechanisms The intrinsic piezoelectric response can be defi ned as that arising due to distortion of crystallographic unit cells within a single domain, single crystal material [5,10]. The intrinsic piezoelectric properties of a material can be meas ured at low temperatures at which the various extrinsic contributions can be pr esumably excluded [11]. Alternately, they can be determined based on firstprinciple calculations [1216] or from macroscopic phys ical properties using thermodynamic phenomenological models. Th e LandauGinsburgDe vonshire (LGD) thermodynamic theory has been used in the past fo r interpretation of piezoel ectric properties of many perovskite ferroelectric materials. An exam ple of the application of the LGD theory is provided here for lead zirconate tit anate (PZT) solid solutions [1724]. The tetragonal and hightemperature rhom bohedral phases of the PZT solid solutions undergo proper ferroelectric transitio ns from the paraelectric cubic ( m3m ) state, where the spontaneous polarization is the order parameter fo r the phase transition. Therefore, the elastic Gibbs free energy of a PZT crystal can be expres sed as a function of powers of the spontaneous PAGE 21 21 ferroelectric polarizations, the coupling between the various polarizations and the coupling between the polarizations and stress, as [5,19] ( ) ( ) ( ) ()()()()[]() () () ()() ()()()[]()21613532444 2 2 2 13 2 1 2 32 2 3 2 2112 2 33 2 22 2 1111 2 6 2 5 2 444 13322112 2 3 2 2 2 111 2 3 2 2 2 1123 2 2 2 1 4 3 2 1 2 3 4 2 2 3 2 2 4 1112 6 3 6 2 6 1111 2 1 2 3 2 3 2 2 2 2 2 112 4 3 4 2 4 111 2 3 2 2 2 11++ +++++++++ 2 1 ++++ 2 1 + +++++ ++++ ++ ++++++= PP PP PP Q PP PP PP Q P P P Q s s sPPP PPPPPPPPP P PP PPPPPP PPP PPP G D D D (13) where s are the ferroelectric dielectric stiffnesses at constant stress, Pi are the spontaneous polarizations, i are the stress component in the Voigt notation, sij D are the elastic compliances at constant polarization, and Qij are the electrostrictive coup ling coefficients between the ferroelectric polarization and stress. Eq. 13 is described in the c oordinate system of the parent cubic phase ( m3m ). The various parameters in Eq. 13 can be obtained through measurement of relevant physical properties su ch as spontaneous strains a nd subsequently relating these properties to the various parameters. Once the vari ous parameters could be evaluated, the elastic Gibbs free energy along the various crystallographic directions can be calculated following Eq. 13. In effect, such a profile s hows the variation of the elastic energy due to atomic shifts in a particular crystallographic direct ion. A flatter profile in a cert ain direction indicates a higher susceptibility of the system to atomic displacements in that pa rticular direction, leading to anisotropic dielectric and piezoelectric properties. The calculation of the longitudinal piezoelectric coefficients d33 along the various crystallographic directions for pi ezoelectric materials, using LGD theory, has been demonstrated [5,2527]. For example, in tetragonal PZT with Zr/Ti ratio of 40/60, the elastic free energy PAGE 22 22 profiles for the following two situations: (a) for elongati on or contraction of polarization parallel to the polar axis of [001] (P2=0; P30), and (b) polarization rotation away from the polar axis of [001] ( P20; P3=P3, P3 is the equilibrium polarization at 298 K), are shown in Figures 15A and 15B, respectively. The longitudinal piezoel ectric coefficients, calculated from the G profiles, are shown in Figure 15C as a pol ar plot. It can be seen from Figure 15C, that for tetragonal PZT the largest piezoelectric cons tants can be expected parallel to the polar axis of [001] and decreases as one moves away from th e direction of the polar axis [26]. In polycrystalline bulk ceramics, the observe d piezoelectric response along the various crystallographic directions can di ffer from the predictions of the LGD theory. The electricfieldinduced lattice strain coefficien ts for different crystal orientations within a polycrystalline tetragonal lead zirconate titanate piezoelectric ceramic under cyclic electric actuation are shown in Figure 16 [28]. It can be seen that the maximum electricfieldinduced lattice strains are observed for crystallographic orientat ions with polar axis oriented away from the direction of the applied electric field. This is in contrast with LGD theory wh ich predicts larg est longitudinal piezoelectric coefficients along the polar axis of [001] in tetragonal PZT. The possible reasons for disagreements between thermodynamic pred ictions and measured lattice strains in piezoelectric ceramics can be speculated to be due to the following: (1) deviations of crystallographic symmetry from a tetragonal stat e [29], (2) nonzero stress state for the grains within a polycrystalline matrix [26], (3) inter/intragranul ar strains under el ectrical cycling [30,31]. 1.2.2 Extrinsic Mechanisms Contributions to piezoelectric responses in ce ramics which are not attributed to intrinsic mechanisms can be termed as extrinsic. Extr insic contribution to di electric properties of PAGE 23 23 ferroelectrics was recognized very early from th eir polarizationelectric field hysteresis loops [32]. The principal mechanisms proposed for diel ectric extrinsic contribu tions included domain switching and domain wall vibra tions [33]. Later studies con centrated on describing the relationship between the dielectric constant ( ) and the amplitude of the applied electric field (E0) for ferroelectrics [3441]. Linear [3741] or quadratic [36] dependence of on E0 had been reported earlier for PZT ceramics. Extrinsic contributions to piezo electric properties of ferroelectrics were undertaken in subsequent studies. A brief desc ription of different approaches for interpretation of extrinsic piezoelectric contributions in fe rroelectrics is given below. 1.2.2.1 Rayleigh law Systematic studies of extrinsic contributions to piezoelectric properties in ferroelectric ceramics were pioneered by Damjanovic and coworkers during the late 1990s. The direct longitudinal piezoelectric coefficients (d33) of different ferroelectric ceramics were studied under subswitching mechanical stresses [4244]. It was shown for the studied materials that the Rayleigh law for ferromagnetic materials [45] can be successfully applied to describe the behavior of the piezoelectric charge and piezo electric coefficients under low amplitudes of applied mechanical stresses. Within a certain ra nge of applied stress amplitudes the piezoelectric coefficient d33 was observed to be a linear function of the amplitude of the applied mechanical stress (X0), as described below ( ) 0 033+= X dXdinit, (14) where dinit is the piezoelectric coefficient at the lim it of zero amplitude of applied mechanical stress, and describes the linear dependence of the piezoelectric coefficient d33 on the amplitude of the applied stress. In effect, dinit is the contribution to the piezoelectric coefficient from intrinsic lattice strains and reversib le domain wall displacements, and X0 is the contribution PAGE 24 24 originating from the irreversible displacemen t of domain walls. Th e distinction between reversible and irreversible displacements of domain walls can be described from the energy profile for domain wall motion in a medium with random pinning centers, as is illustrated in Figure 17. Reversible domain wall displacements involve motion around an equilibrium position at the bottom of a potenti al well, while irreversible do main wall displacements occur across potential energy barriers. Since a variable d33 will result in departure from the linear relationship described in Eq. 11, the piezoelectric response of the material becomes nonlinear with respect to the applied mechanical stress X. It was further shown that relationship between non linearity and hysteresis in ferroelectric ceramics can be related following, ()22 0 02 += XX X)X d(Dinit, (15) for an applied cyclic mechanical stress of amplitude X0, where at a particul ar instant D is the piezoelectric charge density and X is th e applied mechanical stress. [4244] Studies on nonlinearity and hysteresis of converse piezoel ectric effect were also undertaken in subsequent years [41,46,47]. It wa s shown that, for certain piezoelectric materials and within certain range of applied electric field amplitudes, relations simi lar to Eqs. 14 and 15 can be applied to describe converse piezoelectri c behavior. Figure 18A shows the longitudinal piezoelectric d33 coefficient of 4 atomic % Nbdoped PZT ceramics with Zr/Ti ratio of 63/37, as a function of amplitude of applied a.c. pre ssure [43]. Figure 18B shows the corresponding hysteresis loop for this material calculated for 0 2.6 MPa, using Eq. 15 [43]. 1.2.2.2 Preisach model The physical interpretation of Rayleigh law for magnetism in terms of magnetic regions or units was attempted by several investigators [4850]. These type of models are most well PAGE 25 25 known as Preisachtype models [ 51]. Application of Preisach m odel for ferroelectric materials was done by Turik in 1960s [5254]. In recent years, Preisach model has been successfully applied to describe piezoelectric nonlineari ty and hysteresis in ferroelectric materials by Damjanovic and coworkers [4,5557]. A brief descri ption of the Preisach model is provided here. It is assumed in this model that a hysteretic piezoelectric system is composed of simple bistable units (such as domain walls in a media with pinning centers) where each unit can be characterized by following two parameters: (a) a bias field Fi, and (b) a coercive field Fc. Each state of the unit is assumed to contribute to the macroscopic piezoelectric response R by an amount R0, as illustrated in Figure 19. Further, it is defined that < Fi < + and 0 < Fi. The possible values of Fi and Fc can be represented on a plane called Preisach plane as shown in Figure 110. The density of the bistable units in the Preisach plane can be defined by a distribution function f(Fi,Fc), which obeys the following normalization condition, () 0 + 1=cicidFdFF,Ff. (16) The Preisach description is only applicable for hysteretic systems which follow the necessary conditions of returnpoint memory and congruency [51,58]. This condition is usually satisfied for ferroelectric materials [59,60]. In the absence of any external field, the Prei sach plane can be divided into three regions, as shown in Figure 110: (a) Fi > Fc with a negative zero field state (R0); (b) Fc > Fi with a indefinite zero field state; and (c) Fi < Fc with a positive zero field state (+R0). When an external field is applied, it acts similar to a homogene ous bias field and accordingly moves the region (b) upwards or downwards. For increasi ng fields, the region (b) is shif ted up and all bistable units with the condition Fi + Fc < F are switched to a positive state thereby contributing to the total response by +2R0. For decreasing fields, the region (b) is shifted down and all bistable units with PAGE 26 26 the condition Fi Fc < F are switched to a positive state thereby contributing to the total response by +2R0. Integration of the distribution function f(Fi,Fc), over a portion of the Preisach plane, as defined by the applied field profile, can describe the shape of the hysteresis loop. It can be shown that the Rayleigh law for piezoel ectrics corresponds to a uniform distribution function, as shown in Figure 111A [4]. For cases where the hysteresi s cannot be described by Rayleigh law, such as in hard PZT ceramics, the shape of the hysteresis loop can be described by a different distribution function as shown in Figure 111B [4,57]. 1.2.2.3 Micromechanical models Rather than an extensive review of micr omechanical modeling for piezoelectrics, a general approach for such modeling is given in th is section. In this approach, a piezoelectric ceramic is considered to be made up from an array of single domain grains. Ferroelectric or ferroelastic switching in these gr ains can occur when the energy provided by the applied electric or stress field exceeds a certain critical value. The energy criterion can be described as follows, crits jkjkiiEPxXP E 2+ (17) where Ps is the spontaneous polarization, Ei is the applied electric field, Xjk and xjk are the stress and strain states, and Ecrit is the critical field for polarizat ion switching [61]. Figure 112 shows a comparison between the simulated and measured xE relationships for Ladoped PZT ceramics. Further development of such models could incorpor ate the effect of static stress and grain to grain interaction on polari zation switching [62]. The main prin cipal shortcoming of these models is the various simplifying assumptions required fo r their implementation such as single domain grains. PAGE 27 27 1.3 Diffraction Studies of Piezoelectric Mechanisms In situ and ex situ diffraction of piezoelectrics have proved to be a highly useful experimental technique in order to gain insight into the relative electromechanical contributions from intrinsic and extrinsic mechanisms under the application of st atic electric fields. [9,30,31,6371] Most of these studies in piezoelec tric ceramics have been conducted for the lead zirconate titanate (PZT) solid solutions, which have wide technological application as well as being studied as a model system for fundament al understanding of piezo electric phenomenon in ceramics. These experiments have demonstrated that changes in the volume fractions of the non180 domains as a result of domain wall motion can be characterized from the changes in the relative intensities of the diffraction peaks which are characteristic of a pa rticular ferroelectric distortion. For example in tetragonal PZT with a ferroelectric distorti on along [001], relative intensities of (00h) and (h00) diffraction peaks, where h is an integer can be used to quantify changes in the volume fractions of the 90 domains At the same time, by noting the shift in the angular position of the (hkl) diffraction peaks it is possible to calculate the amount of hkl lattice strains in response to an applie d electric field. While structural measurements of piezoelectric materials under static electric fields provide valuable insight in to the various electromechanical mechanisms, such measurements do not provide information about ma terial behavior under cyclic electric fields, as is us ed in various actuation devices. For the measurement of structural changes in piezoelectric ceramics under cyclic electric fields, timeresolved diffraction techniques have been demonstrated to be useful [9,7274]. It has been shown that under increasing amplitudes of applied cyclic elect ric fields, amount of non180 domain switching increases linearly, as shown in Figure 113 [72,73]. This type of linear relationship bear semblance to the linear variati on of piezoelectric coefficients as described by Rayleigh law. PAGE 28 28 Figure 11. Unit cell of a cubic perovskite crystal st ructure. The A atoms occupy the cell corners (shown in brown), the B atom is in the body center (shown in green), and the O atoms are in the face centers (shown in blue). The BO3 octahedron is shown in blue. PAGE 29 29 A B Figure 12. Domain variants in ferroelectric ceramics shown as arrangement of polarization axes of adjacent domains. A) 180 domains with antiparallel arrangeme nt of polarization axes, B) non180 domains with headtotail arrangement of polarization axes at an angle different than 180 (90 in this case). PAGE 30 30 Figure 13. The different struct ural changes in a ferroelectric ceramic from a paraelectric state above TC to a poled ferroelectric state w ith reference to a tetragonal crystal structure. The ceramic is piezoelectric af ter being poled under high electric field. Cooled through Tc Poled under high electric field Above Tc Nonpiezoelectric Piezoelectric Nonpiezoelectric PAGE 31 31 A Figure 14. The effect of re orientation of domains under app lied electric field on A) the polarization, and B) the strain of a piezoelectric ceramic. 42024 60 40 20 0 20 40 60 Polarization (C/cm2)E (kV/mm) PR PS EC+EC PAGE 32 32 B Figure 14 Continued. 42024 0.0 0.1 0.2 0.3 0.4 0.5 Strain (%)E (kV/mm) EC+EC PAGE 33 33 50100150 50 100 150 d33((pC/N)d33((pC/N) (deg) [001] [111] 54.70.0 0.5 25 0 25 50 G (MJ)P2 (C/m2) 0.00.5 25 0 25 50 G (MJ)P3 (C/m2)(a) (b) (c) Figure 15. The elastic free energy profile s for polarization contraction/elongation and polarization rotation in tetr agonal PZT with Zr/Ti ratio of 40/60. A) Free energy profile for elongation or cont raction of polarization along the polar axis of [001] (P2=0; P3 0), and B) Free energy profile for polar ization rotation away from the polar axis of [001] (P2 0; P3=P3, P3 is the equilibrium polarization at 298 K) [from ref. 26]. A B C PAGE 34 34 Figure 16. The electricfieldinduced lattice strain coefficients for different crystal orientations within a polycrystalline tetr agonal lead zirconate titanat e piezoelectric ceramic under cyclic electric actuation. The tetragon al unit cells in the figure represent the crystallographic orientation of grains and domains in real space, with the (hkl) plane perpendicular to the electric field [from ref.28]. PAGE 35 35 Figure 17. Illustration of energy profile for domain wall motion in a medium with random pinning centers Energy Position, x Reversible displacement Irreversible displacement PAGE 36 36 0123 130 140 150 160 170 d33 (pC/N)Amplitude of ac pressure, 0 (MPa) d33 = 118.5 + 17.70 A 3210123 600 400 200 0 200 400 600 exp. data calculated Charge density, Q (pC/mm2)Pressure, (MPa) B Figure 18. Application of Rayl eigh law for direct piezoelectric behavior of 4 atomic % Nbdoped PZT ceramics with Zr/Ti ratio of 63/37. A) Linear variation of the longitudinal piezoelectric d33 coefficient with respect to the amp litude of applied a.c. pressure. The d.c. bias pressure is 15 MPa. B) The corresponding hysteresis loop calculated for 0 2.6 MPa, using Eq. 15 [from ref.43]. PAGE 37 37 G R R0R0 mFCR0mFiR0 mFiR0 R0 R0R F FiFC Figure 19. Energy profile (left), and square hysteresis (right) of an elementary bistable unit, as assumed in Preisach model for hysteresis. m represents generalized susceptibility [from ref. 4]. PAGE 38 38 a b c F > 0 F < 0 FCFiFi = F + FCFi = F FC A Figure 110. Energy profile and hysteresis of bist able units as a functi on of their position on the Preisach plane at zero applied field [from ref. 4]. PAGE 39 39 A B Figure 111. Preisach distribution function for different characteristi c piezoelectric behaviors. A) Flat (uniform) Preisach distribution function corres ponding to Rayleigh relations [from ref. 4]. B) Proposed polynomial Prei sach distribution function for pinched hysteresis loops as is observed for ferroele ctrics containing def ect dipoles with two preferred directions. [from ref. 4]. PAGE 40 40 1.00.50.00.51.0 0.1 0.0 0.1 0.2 0.3 Measured Calculated Longitudinal Strain x / %E / kV mm1 Figure 112. Simulated and measured xE hys teresis loops for Ladoped PZT ceramics using micromechanical models based on energy crit erion for switching, as described in Eq. 17 [from ref. 61]. PAGE 41 41 Figure 113. Non180 domain switching fraction ( 002) in tetragonal PZT ceramics under applied unipolar and bipolar electric fields, as a function of field amplitude [from ref. 73]. PAGE 42 42 CHAPTER 2 OBJECTIVE AND OVERVIEW As described in Chapter 1, both intrinsic structural m echanisms (such as piezoelectric lattice strains) and extrinsic structural mechan isms (such as displacement of domain walls) can contribute to the overall piezoelect ric response in ferroelectric cer amics. Extrinsic structural mechanisms are believed to cause piezoelectric n onlinearity. However, the relative contributions of different extrinsic mechanisms towards macros copic piezoelectric non linearity are currently not well understood. The objective of this disserta tion is to reconcile converse piezoelectric nonlinearity and hysteresis in ferroelectric cer amics with respect to the induced crystallographic and microstructural changes that occur during application of cy clic electric fields. In order to perform such an analysis, measurements of macroscopic piezoelectric properties as well as timeresolved diffraction under the application of cyclic electric fields ar e performed for ferroelectric PZT ceramics of different compositions (i.e., Zr/Ti ratio) and dopant variants (i.e., undoped, La, and Fe). These various compositions were chosen because they yield different crystal structures and different amounts of domain wall displacements. In order to measure crystallographic and microstructural changes under the ap plication of cyclic electric fields, timeresolved diffraction techniques are developed and utilized at bot h laboratory and synchrotron Xray sources. The layout of the rest of the dissertation is provided below. Chapter 3 describes the phase diagram of the PZT solid solution and the procedure followed for synthesis of PZT ceramics. The cr ystallographic phases of the PZT ceramics of different compositions are analyzed from their respective Xray diffractio n patterns. A general description of ferroelectric and piezoelectric properties of the synthesized ceramics is also included. PAGE 43 43 The nonlinearities and losses in the converse piezoelectric effect of PZT ceramics are presented in Chapter 4. For describing piezoelect ric nonlinearity and hyste resis, Rayleigh law is mostly adopted. In addition, complex piezoelectric coefficients are calculated from a Fourier expansion of relations describing piezoelectric hysteresis. It is shown that the separate components of piezoelectric coefficients, from Rayleigh relations and possible viscoelastic mechanisms, can be computed using such an approach. Measurements of electricfieldinduced structur al changes under the ap plication of cyclic electric fields are presented in Chapters 5 a nd 6. Chapter 5 describes the development of a stroboscopic timeresolved diffraction technique for a laboratory diffractometer. This technique is used to measure the structural changes in PZT ceramics under the application of cy clic electric fields. In Chapter 6, a similar structural ch aracterization of PZT ceramics using highenergy Xray diffraction is presented. Hi ghenergy Xray diffraction in the transmission mode is used to obtain structural information from the bulk of a ceramic in different sample directions. In both Chapters 5 and 6, particular relationships be tween the measured structural changes and the applied electric field amplitudes are evaluated. Chapter 7 provides a reconciliation of the m acroscopic piezoelectric properties presented in Chapter 4 and the electricfieldinduced struct ural changes presented in Chapters 5 and 6. The relative contributions from non180 domain switching and lattice distortions towards macroscopic piezoelectric nonlinearity are subseq uently evaluated using similar formulations developed in Chapter 4. A summary of all the results is provided in Chapter 8. The significance and implication of the findings, in the context of piezoelectric proper ties of ferroelectric ceramics, are discussed. In addition, some potential areas of future inves tigations are suggested where an approach of PAGE 44 44 reconciling macroscopic propertie s with timeresolved structur al measurements can provide valuable insights into the material physics of ferroelectrics. PAGE 45 45 CHAPTER 3 SYNTHESIS OF LEAD ZIRCO NATE TITANATE CERAMICS Lead zircon ate titanate (PZT) ceramics are chos en as the model system in this work for investigation of structureprope rty correlations in piezoelectric ceramics. The solid solution of lead zirconate (PbZrO3) and lead titanate (PbTiO3) is a well known ferro electric system with widespread technological applications [7]. A brief description of the phase diagram of the system is provided here, along with th e synthesis procedure followed for producing PZT ceramics with different crystallographic phase s and with different dopants. 3.1 Phase Diagram and Dopants for PZT A portion of the phase diagram for the solid solu tion of lead zirconate and lead titanate is shown in Figure 31. The ratio of mole percentages of PbTiO3 and PbZrO3 can be equivalently described in terms of Zr/Ti molar ratio. Depending on the Zr/Ti ratio, a Pb(Zr,Ti)O3 solid solution undergoes a specific phase transformation when cooled from its cubic phase to below its Curie temperature (TC). At room temperature, for compositions Zr/Ti (0.53/0.47), a tetragonal phase is observed whereas for compositions Zr/Ti > (0.53/0.47), a rhombohedral phase is observed [6]. A monoclinic phase separating te tragonal and rhombohedral regions has been reported in PZT [75]. However, the presen ce of a monoclinic phase in PZT remains controversial, since the appearance of a monoclinic phase has also been explained by tetragonal/ rhombohedral nanodomains [76]. The crystallographic phase of PZT highly infl uences its piezoelectric properties. For example, Figure 31 shows the portion of phase diagram of PZT around MPB and the piezoelectric coefficient d33 of PZT bulk ceramics as a func tion of Zr/Ti ratio at room temperature [6]. PAGE 46 46 For a particular crystallographic phase, the overall piezoelectric response is also influenced by domain wall motion under an applied external field. The contribution from domain wall motion and therefore the total piezoelectric resp onse of a PZT ceramic can be modified by addition of small amounts of impurities or dopant s. The displacement of domain walls can be enhanced by substitution of Pb2+ or Zr4+/Ti4+ with dopants of higher va lence (donor dopants) like La3+ (for Pb2+) and Nb5+ (for Zr4+/Ti4+). Substitution of Pb2+ or Zr4+/Ti4+ with dopants of lower valence (accep tor dopants) like Na+ (for Pb2+) and Fe3+ (for Zr4+/Ti4+) can restrict the displacement of the domain walls. The effect of the dopants on domain wall displacements is not well understood, however this maybe related to the concentration of oxygen vacancies in the material. While addition of donor dopants redu ces the concentration of the existing oxygen vacancies in the system, addition of acceptor dopants creates new oxygen vacancies [6,7]. 3.2 Synthesis of PZT Ceramics Dense ceramics of the following nominal compositions were produced by solid state synthesis route: Pb(ZrxTi1x)O3 (PUZT, undoped) with x = 0.52, Pb11.5zLaz(ZrxTi1x)O3 (PLZT, Ladoped) with x = 0.60, 0.52, 0.40 and z = 0.02, and Pb1+0.5z[(ZrxTi1x)1zFez)]O3 (PFZT, Fedoped) with x = 0.47 and z = 0.02. The powde rs of these compositions were produced by reactive calcination method [24,25] from reagent grade oxides: TiO2 (Fisher Scientific), ZrO2 (AlfaAesar), PbO (AlfaAesar), La2O3 (AlfaAesar) and Fe2O3 (AlfaAesar). The stoichiometric amounts of the different oxide reagents for synt hesis of the desired final compositions were obtained from their respective chemical formulas. In addition, loss of igniti on (LoI) at particular calcination temperatures and purity of the r eagent oxides were taken into account while preparing each batch of powder mixtures for further processing. Precursors of compositions (ZrxTi1x)O2 and [(ZrxTi1x)1zFez]O2 were first obtained by ball milling with ZrO2 milling media in ethanol for at least 12 hours a nd calcining the stoichiometric mixtures of the oxides at 1300C PAGE 47 47 for 4 hours in closed alumina crucibles of vol ume 10 ml and 40 ml. Stoichiometric amounts of PbO (and La2O3 for Ladoped compositions) were subsequently mixed with the precursors using ball milling with ZrO2 milling media in ethanol for at least 12 hours and the mixtures were calcined at 950C for 2 hours in closed alumin a crucibles of volume 10 ml and 40 ml. During the final calcination step, the heating rate was ma intained at 10C/minute. The calcined powders were cooled at a rate of 10C/minute above 600C and were subsequently furnace cooled to room temperature. The final PZT powders obtai ned were ball milled an additional time with ZrO2 milling media in ethanol for at least 12 hours. For all ball milling procedures, the powders in batches of about 1530 grams were added inside a Nalgene jar (500 ml) containing equal volume fractions of spherical ZrO2 milling media of diameters 5 mm and 10 mm and subsequently about 7585 ml of etha nol was poured inside the jar. The milling media covere d about onethird volume of the Nalgene jar. The milling media was cleaned in deionized water subsequent to ball milling powder mixtures each time. Separate Nalgene jars were used for powders of different compositions. Subsequent to ball milling, the slurry was poured into a Polytetrafluoroethylene (PTFE) sheet placed over a glass pan. The glass pan along with the slurry on the PTFE sheet was covered with pe rforated aluminum sheet and put inside an oven at 100C for at least 3 hour s to vaporize ethanol. Th e powder obtained from the dried slurry was ground using a porcelai n mortar and pestle and the ground powder was sieved through a 212 m test sieve (Fisher Scientific). Green pellets were obtained by mixing the powde rs with organic bind er polyvinyl acetate (PVA) using a mortar and pestle (typically 45 drops of PVA for 5 grams of PZT) and pressing first uniaxially in 10 mm dies using a force of 3 metric tones (stress of 400 MPa) and then isostatically using 200 MPa. A CARVER 3850 pr ess was used for uniaxial pressing while a PAGE 48 48 Fluitron press was used for isos tatic pressing of pellets; in ea ch batch up to 4 pellets were pressed inside the chamber. For each pellet, ap proximately 0.5 grams total of the powder and binder blend was used. The green pe llets were sintered at 1200C for 2 hours in a bed of PbZrO3 + 8 weight % ZrO2 in a closed alumina circular dish, with a heating/cooling rate of 4C/minute. During sintering the pellets were covered sufficiently within the bed. The particular composition of the bed is chosen to prevent loss of PbO from the material and to maintain Zr/Ti ratio during sintering. The size of the circular dish used for sintering was 10 ml in volume. The sintered pellets were disc shaped with ~1 mm thickness and were ~8 mm in diameter. A scanning electron micrograph (JEOL JSM 6400, operating voltage of 15kV) of fracture surface of PLZT5248 is shown in Figure 32A. The estimated grain size, based on visual examination, is of the order of 12m. The crystallographic phases of the processed ceramics were determined from their respective Xray diffraction pattern s, shown in Figure 32B. The crystallographic phase of the sintered ceramics is tetra gonal for compositions with x 0.52, and rhombohedral for compositions with x = 0.6. The densities of the sintered ceramics were measured using the Archimedean immersion method. The theoretical densities for the different compositions were calculated from the lattice parameters determined from (002)type Xray diffr action peaks. For tetragonal crystal structures, the volume of the unit cell was calculated as ( a2c ) where a and c are the interplanar spacings for the (100) and (001) planes, respec tively. For rhombohedral crystal structure, the volume of the unit cell was calculated as a3, while assuming a pseudocubic struct ure with a lattice parameter of a. Table 31 lists the actual measured densities and the percent theoretical densities for the different samples. PAGE 49 49 The sintered pellets were po lished to a final thickness of 0.75 1 mm using SiC grinding papers (Buehler P1200 and P4000). The polished cer amics were subsequently annealed, at 600 C for 2 hours, to relieve surface residual stresse s. Gold was deposited, in an Argon atmosphere using a sputter coater, on the top and bottom surfaces of the disc samples in order to get 300400 nm thick electrodes. The polarizationelectric fiel d and strainelectric field hysteresis loops for the ceramics were measured using a SawyerTower circuit a nd a linear variable disp lacement transducer (LVDT), respectively. The applied electric fields were of sinusoidal waveforms with a frequency of 100 mHz. The polarizationelectr ic field and strainelectric field hysteresis loops are shown in Figures 33A and 33B, respectively. The coercive field (EC) of ceramics of a particular composition was determined from the macrosc opic strainelectric field hysteresis loops. EC was determined to be equal to the nonzero amplitude of the applied electric field when the fieldinduced strain is minimum. For samples with a broad minimum for the fieldinduced strain, EC was not defined. The samples were then poled under high electric fields at elevated te mperatures using a hot plate and a high voltage power supply. The electric fields us ed for poling were chosen for the different samples based on data in prior literatu re regarding poling char acteristics of donorand acceptordoped ferroelectrics [68], experimental trial and error and electrical breakdown strengths. Table 32 lists all the materials synt hesized for this work, the coercive fields EC of the samples, the conditions under which they were pole d and their direct piezoelectric coefficients measured using an APC International YE2730A Be rlincourt meter. The measurements for the piezoelectric measurements were made 24 hours after poling. PAGE 50 50 Figure 31. Phase diagram and piezoelectr ic coefficients of PZT ceramics. PbZrO3 PbTiO3 phase diagram over compositional range cl ose to the morphotropic phase boundary (MPB) is shown in the bottom half of the plot. The MPB for the system is ~47 mole % PbTiO3 (x=0.53) at room temperature. Th e region for the different phases are indicated by the following letters: T (t etragonal), R (rhombohed ral) and C (cubic). The plot on the top shows the piezoelectric coefficient d33 of undoped PZT ceramics as a function of composition, at room temperature, adapted from Ref. [6]. PAGE 51 51 1.41.61.82.02.22.4 Intensity (A.U)Lattice spacings, {200} {111}PLZT5248PLZT6040 PLZT4060 PUZT5248 PFZT4753 {102} {112} {220} B Figure 32. A) Scanning elec tron micrograph of fracture surface of PLZT5248; the grain size is of the order of 12 m. B) Xray diffracti on patterns of the sintered ceramics. The different {hkl} peaks are consistent with a tetragonal phase for compositions with Zr/Ti 52/48, and a rhombohedral phase for co mpositions with Zr/Ti ratio equal to 60/40. The legends for the different ceramic samples are provided in Table 31. A PAGE 52 52 42024 Electric Field (kV/mm) 42024 Electric Field (kV/mm) 42024 60 40 20 0 20 40 60 Polarization (C/cm2)Electric Field (kV/mm) PLZT6040 PLZT5248 PLZT4060 42024 10 5 0 5 10 Electric Field (kV/mm) 42024 20 0 20 Polarization (C/cm2)Electric Field (kV/mm) PUZT5248 PFZT4753 A Figure 33. A) Polariza tionelectric field, and B) strainele ctric field hysteresis loops of the various samples. The dotted lines represent polarization/srain measurements for the first applied electric field cycle on an ass intered ceramic and the solid line represent measurements for subsequent cycles. PAGE 53 53 42024 0.0 0.1 0.2 0.3 0.4 0.5 Strain (%) Electric Field (kV/mm) 42024 Electric Field (kV/mm) 42024 Electric Field (kV/mm) PLZT6040 PLZT5248 PLZT4060 42024 0.000 0.002 0.004 0.006 0.008 0.010 Electric Field (kV/mm) 42024 0.00 0.02 0.04 0.06 0.08 0.10 Strain (%)Electric Field (kV/mm) B Figure 33. Continued PAGE 54 54 Table 31. The measured densitie s and the percent theoretical densities for the different samples Sample notation Dopants Zr/Ti ratio Measured density (g/cm3)* Percent theoretical density* PUZT5248 Undoped 52/48 7.64 (.05) 95.7 (.66) PLZT6040 2 at. % Ladoped 60/40 7.72 (.03) 97.1 (.43) PLZT5248 2 at. % Ladoped 52/48 7.77 (.04) 97.1 (.50) PLZT4060 2 at. % Ladoped 40/60 7.62 (.08) 95.3 (.00) PFZT4753 2 at. % Fedoped 47/53 7.81 (.02) 97.8 (.20) The numbers in the braces indicate the standard deviation for the reporte d values. The standard deviation for each composition was calculated fro m the densities measured on three different samples of the same composition. PAGE 55 55 Table 32. List of the samples used for experiments in this work. The d33 values were measured using a Berlincourttype d33 meter. Sample notation Coercive field, EC Poling conditions d33 (pC/N) PUZT5248 not defined 4 kV/mm at 100 C for 60minutes 180 PLZT6040 ~ 0.85 kV/mm 3.5 kV/mm at 50 C for 15minutes 208 PLZT5248 ~ 1.35 kV/mm 3.5 kV/mm at 50 C for 15minutes 480 PLZT4060 ~ 2.6 kV/mm 3.5 kV/mm at 50 C for 15minutes 130 PFZT4753 not defined 4 kV/mm at 100 C for 60minutes 204 PAGE 56 56 CHAPTER 4 NONLINEARITIES AND LOSSES IN CONVERS E PIEZ OELECTRIC EFFECT OF PZT CERAMICS Converse piezoelectric response of Pb(ZrxTi1x)O3 ceramics is investigated as a function of material composition. The effects of crys tallographic phase a nd different dopants on piezoelectric nonlinearity are separately examined. For a linear dependence of d33 on E0, Rayleigh law is applied to describe the material behavi or. The observed piezoelectric nonlinearities are described in term s of contributions from extrin sic mechanisms. The effect of piezoelectric nonlinearities on strainelectric field hysteresis is subsequently examined. In order to calculate the complex piezoelectric coefficien ts, a method based on Four ier expansion of the Rayleigh relations is adopted. Fina lly, a description of first and hi gher order harmonics in used to show that Rayleigh component is dominant in th e overall piezoelectric strain of the material. 4.1 Experimental Procedure 4.1.1 Converse Piezoelectric Measurements The experimental setup for converse piezoel ectric measurements unde r cyclic electric fields is shown in Figure 41. A Trek 609C6 highvoltage amplifier driven by a Stanford Research System DS360 function generator was used to apply si nusoidal electric fields of frequency 1 Hz across the thickn ess of a ceramic sample. The e xpansion and contraction of the sample along the direction of the applied electric field was measured by detecting the vertical displacement of the sample surface using a MTI 2000 photonic sensor. The signals from the function generator and the photonic sensor were measured with a Tektronix TDS410 oscilloscope connected to a computer. Acquisi tion of the measured signals on the oscilloscope was performed using a National Instruments Labview program. Figure 42 shows a typical strainelectric field hysteresis loop obtained using this experimental setup. The longitudinal PAGE 57 57 piezoelectric coefficient d33 was calculated from the strainelect ric field hysteresis loop using the following equation: 0 332 = E x dmax, (41) where xmax is the peaktopeak longit udinal measured strain and E0 is the amplitude of the applied bipolar electric field E0 parallel to the direction of the measured strain. This is illustrated in Figure 42. Since only this coe fficient was measured in the present work, d33 and d are used interchangeably th roughout this dissertation. In order to determine the electric field de pendence of the piezoelectric coefficient d33, the amplitude of the applied electric field E0 was steadily increase d after each successive measurement. The maximum value of E0 was limited to below the macroscopic coercive field of the sample, EC. 4.1.2 Application of Rayleigh Law The following Rayleigh relations have been prev iously observed to satisfactorily describe the linear dependence of converse longitudinal piez oelectric coefficients in certain ferroelectric ceramics: 0 0+= E d)E(ddinit, (42) )EE( E)E d(xd dinit 22 0 02 +=, (43) where dinit is the piezoelectric coefficient at th e limit of zero applied field amplitude E0, x is the measured macroscopic strain res ponse under applied field E, and d describes the linear dependence of the piezoelectric coefficient d on the amplitude of the applied field E0 [46,47]. In effect, dinit is the contribution to the piezoelectric co efficient from intrinsic lattice strains and reversible domain wall displacements, and dE0 is the contribution originating from the PAGE 58 58 irreversible displacement of domain walls. Relations similar to Eqs. 42 and 43 have also been used to describe the direct piezoelectric effect [42,43]. The applicability of the Rayleigh relations, Eq s. 42 and 43, was investigated for the longitudinal converse piezoelectric properties of all the measured data. The fractional contribution to piezoelectric nonlinearity from irreversible extrinsic mechanisms was calculated as d33/d33(E0), where d33(E0) is the measured piezoelectric coe fficient at electric field amplitude E0. For materials which were observed to follow Rayleigh law, d33 was calculated as [d33(E0)dinit]; otherwise d33 was calculated as [d33(E0)d33,min], where d33,min is the piezoelectric coefficient measured for applied el ectric field of lowest amplitude. d33 is expected to be equal to dE0 when Rayleigh behavior is followed. 4.1.3 Calculation of Piezoelectric Losses and Complex Coefficients The operation of piezoelectric ceramics under cy clic electric fields of amplitude 0 < E0 < EC can involve significant energy dissipation. Th e total energy dissipatio n or piezoelectric loss per unit volume of the material can be correlated to the total area of the piezoelectric strainfield hysteresis loop [4]. The total area of the straine lectric field hysteresis loop and the various contributing factors are calculated following th e method described in reference [3]. For a material that follows Rayleigh law, the hysteresis area AR can be calculated from the integral of the hysteresis loop as predicted by the Rayleigh relation Eq. 43, 3 4 ==3 0E xdEAd cycle R. (44) However, in reality there can be additional nonR ayleigh contributions to the total area of the strainelectric field hysteresis l oop from the generalized susceptibili ty of the material [80]. For the converse piezoelectric effect, this additional contribution can be pres ent due to the linear viscoelastic component of domain wall motion wh ich is not represented in the Rayleigh law. PAGE 59 59 The response of a linear viscoelastic system w ith a certain phase lag under the application of a sinusoidal electric field E = E0 sin( t), can be described by ) t sin(xx =0, (45) where x0 is the amplitude of the piezoelectric strain response of the system. The piezoelectric coefficient dlin of this system can be represented as a complex variable lin lin liniddd =, (46) where d and d are respectively the real a nd imaginary components and d* is the modulus of the complex piezoelectric coefficient. The hysteresis area for such a system Alin is given by the integral of Eq. 45, 2 0 00= ==Ed sinEx xdEA" lin cycle lin. (47) The total area of strainelectric field hysteresis loop of a nonlinear viscoelastic piezoelectric material can therefore be represented as a sum of the areas contribute d due to Rayleightype nonlinearity and due to linear viscoelastic effects, linR totalAAA + =. (48) Eq. 48 is valid if there is no coupling between the different loss contributions. This is a reasonable first approximation a nd has been previously made in magnetic materials [51]. The application of this equation in present work is an assumption, but does not significantly affect the conclusions drawn in this dissertation. The different contributions to the overall elec tricfieldinduced respons e of a material can also be identified from th e real and imaginary components of its complex piezoelectric coefficient. The real and imagin ary components of piezoelectric co efficient of a material that follows Rayleigh law, can be identified from a Fourier expansion of Eq. 43 for E = E0 sin ( t) PAGE 60 60 ()()() () () ()]..t cost cos[ E t cos E t sinEE dtxd d dinit+5 35 1 3 5 1 3 4 3 4 +=2 0 2 0 00. (49) By combining Eqs. 45 and 49, the total strain of a nonlinear piezoelect ric under the application of electric field E = E0 sin ( t) can be expressed as () ( ) () ( ) () ( ) ()...t cosE]Ed[t cosE]Ed[t sinE]Ed[tx" lin R R+ =00 00 00, (410) where d R and d R are the real and the imaginary component s of the piezoelectric coefficient of an ideal Rayleigh system, respectively, and d" lin is the imaginary component of the viscoelastic susceptibility of the material. The terms in Eq. 410 associated with an id eal piezoelectric materi al following Rayleigh law can be compared with the corresponding terms in Eq. 49. If the first harmonic component in Eq. 49 is over an order of magnitude greater than the higher order harmonics, we may further assume that the higher order harmonics are rela tively insignificant in the description of the complex piezoelectric coefficients and therefore can be neglected. We can then arrive at the following relations ( ) ( ) 0 0+= E dEddinit R, () E Edd R3 4 =0 0. (411) A summary of the data analysis routine is described below. The piezoelectric coefficient d R(E0) is calculated from the peaktopeak amp litude of the measured strainelectric field hysteresis loop at electric field amplitude E0, as shown in Figure 42. The Rayleigh coefficient d is obtained from a linear fit of multiple measurements of d R(E0) at several the electric field amplitudes E0. The area of the strainele ctric field hysteresis loop fo r an ideal Rayleigh type response is then calculated using Eq. 44 and th e complex coefficient a ssociated with Rayleigh PAGE 61 61 behavior is given by Eq. 411. The area associated with viscoelastic nature of the material deformation, Alin is obtained from the difference between the area of the experimental hysteresis loop and the area predicted by Eq. 44. The imaginary component of the viscoelastic susceptibility of the material at electric field amplitude E0 is obtained using Eq. 47 as 2 0= E A dlin lin. (412) Note that since AR is proportional to E0 3 and Alin is proportional to E0 2, a distinction between the contributions of each to the overall strainele ctric field hysteresis ar ea can be obtained. 4.1.4 Measurement of the Harmonic Components of Converse Piezoelectric Strain The other essential characteristics of a ma terial system followi ng Rayleigh law can be identified from Eq. 49 [4]. First, it can be observed that the Fourier expansion of the Rayleigh law for a sinusoidal electric fi eld contains only odd harmonics. Further, the amplitude of the third and higher harmonics should be proportional to the square of the amplitude of the applied electric field. Moreover, since the third and all other higher harmonics have no component in phase with the applied electric fi eld, their phase angle should be 90 for ideal Rayleigh behavior. For such an ideal Ra yleigh behavior, every domain wall displacement needs to be hysteretic as well as also contribute to a linear in crease in piezoelectric coefficient with applied electric field amplitude. All of the above featur es of Rayleightype beha vior can therefore be experimentally verified from the measured piezoelectric strains. In order to measure the harmonics, the electr onic signals from the photonic sensor were measured using a Stanford Research System SR 830 lockin amplifier. The amplitude and phase of the first, second and third harmonics of th e piezoelectric strain were measured under sinusoidal electric fields and are presented in the section 4.2.3. PAGE 62 62 4.2 Results and Discussion 4.2.1 Piezoelectric Nonlinearity a nd Application of Rayleigh Law The piezoelectric coefficients measured at increasing amplitudes of subcoercive cyclic electric fields for the ceramic materials listed in Table 31 ar e presented in this section. 4.2.1.1 Effect of crystal structure on nonlinear behavior Figure 43 shows the measured conv erse piezoelectric coefficients d33 of 2 at% Ladoped PZT ceramic samples identified as PLZT6040, PLZT5248 and PLZT4060. The piezoelectric coefficients of all the three co mpositions show linear dependence on the amplitude of the applied electric field over a wide range, as indicated in Figure 43. Eq. 42 is therefore applied to describe the converse piezoelect ric coefficients under subcoerc ive electric fields for these materials. The value of the parameter d is ~0.43, ~0.36, and ~0.07 (pm/V)(V/mm)1 for PLZT5248, PLZT6040 and PLZT4060 respectiv ely. [The unit of (pm/V)(V/mm)1 is equivalent to 1015 m2/V2]. The linear increase in the piezoelect ric coefficients can be explained by irreversible motion of non180 domain walls in a medium with random ly distributed pinning centers [44]. In a polyc rystalline ceramic, these pinning centers can be point defects such as vacancies or line defects such as dislocations Ladoping is generally understood to promote the displacement of the non180 domain walls [6,7]. With increasing amplitude of applied electric fields, motion of the domain wall s is increasingly promoted and therefore higher contributions from domain wall motion to the macroscopic fi eldinduced strains are made. While non180 domain wall displacement may be the dominan t influencing factor for the increase in piezoelectric coefficient with increasing electric field amplitude, other extrinsic contributing factors cannot be ruled out. This includes 180 domain wall motion through dynamic poling [81] and nonlinear misfit lattice strains arising due to non180 domain wall displacement in neighboring grains [30]. PAGE 63 63 In order to estimate the absolute and the fractional amounts of irreversible extrinsic contributions to the total electric fieldinduced stra in, the factor d33 and the ratio d33/d33(E0) are calculated for all samples. The values ar e plotted in Figure 44A and Figure 44B. The amount of irreversible extrinsic contribution can be seen to steadily increa se with the amplitude of the electric field. The largest fractional extr insic contribution is found to be about 55% for Ladoped rhombohedral PZT ceramics, for an applied electric field amplitude of V/mm. For all amplitudes of applied electric fields, the largest and smallest fractions of irreversible extrinsic contributions are observed for the rhom bohedral PLZT6040 and the tetragonal PLZT4060 samples, respectively. The difference in irrevers ible extrinsic contributions for the different samples can be interpreted in terms of strain contributions due to non180 domain wall displacement in the material. For the rhombohedral phase the number of possi ble spontaneous ferroelastic distortions is 4 whereas in tetragonal phase this number is 3. In other words, more alte rnate ferroelastic states are available for domains with non180 domain wa lls in the rhombohedral phase as compared to the tetragonal phase. This may imply compar atively greater possible non180 domain wall motion in the rhombohedral phase. However, th e amount of strain cr eated due to non180 domain wall motion will also depend on the spontaneous strain associated with a rhombohedral or tetragonal ferroelastic distortion, which can be determined from crystallographic lattice parameters. The specific lattice sp acings of the different ceramics were determined from their respective Xray diffraction pattern s. In the rhombohedral phase, the spontaneous strain is defined by the ratio of the lattice spacings 111 111d/d which is equal to 1.0068(1) for PLZT6040. In the tetragonal phase, the spont aneous strain is defined by th e ratio of the lattice spacings 200 002d/d which are equal to 1.0209(1) and 1.0290(1) for PLZT5248 and PLZT4060, PAGE 64 64 respectively. The values for 111 111d/d and 200 002d/d were calculated from the position of the (111)type and the (002)type diffr action peaks of these ceramics, as presented in Figure 32B. Therefore the same amount of non180 domain wa ll displacement in the tetragonal phase would create a greater strain as compared to a rhombohedral phase. The observed trends in Figure 44A and Figure 44B are therefore contributed by bot h factors discussed here, namely amount of non180 domain wall motion and spontaneous strains. Further elaboration on this point will be provided in Chapter 6. Since the rhombohedral phase w ith the smallest spontaneous distortion shows relatively higher values of extrinsic contributions, a hi gh degree of non180 domain wall motion in this material is implied. Among the samples with a tetragonal phase, PLZT5248 shows high degree of extrinsic contributions as compared to PLZT 4060. This trend when in terpreted in conjunction with the fact that PLZT4060 has a higher spont aneous strain than PLZT5248 would indicate comparatively less amount of non180 domain wall displacement in PLZT4060. This can be explained due to greater clamping of 90 domain walls due to internal stresses caused by high spontaneous distor tion in PLZT4060 (200 002d/d = 1.0290) as compared to PLZT5248 (200 002d/d = 1.0209). Another possible interpreta tion for this effect could be lower domain wall energies in materials with compositions near the MPB [82] which makes domain wall displacement easier in PLZT5248 as compared to PLZT4060. Additiona l improvements in micromechanical models [61,62] can help to understand the predominance of either of the above listed mechanisms. 4.2.1.2 Effect of dopants on nonlinear behavior As indicated in Chapter 3, the domain wall displacements are known to be highly influenced by the addition of dopants. While addition of La as a donor dopant is known to enhance the displacement of the domain walls, addition of Fe as an acceptor dopant is known to PAGE 65 65 have an opposite effect [6,7]. The influence of dopants on the amount of irreversible extrinsic contributions to the converse piezoelectric strain for tetragonal PZT ceramics with compositions close to the MPB is discussed in this section. Figure 45A shows the field dependent c onverse piezoelectric coefficients of PZT ceramics with compositions slightly on the tetragonal side of the MPB. It is observed that for MPBadjacent tetragonal phases th e piezoelectric coefficients s how drastic increase with the addition of La. Though both undoped and Ladoped co mpositions can be seen to follow Rayleigh type behavior, there is more than a thre efold increase in the Rayleigh coefficient d with Ladoping and consequently an increase in the amount of extrinsic contributions. For MPBadjacent tetragonal PZT ceramics, irreversible extrinsic co ntributions for the highest applied electric field amplitudes increases from ~25% in undoped composition to ~45% in Ladoped compositions (Figure 45B). For Fedoped compositions, dependence of the piezoelectric coefficient on the amplitude of the electric field is not describe d by Rayleigh law. The fraction of irreversible extrinsic contributions, calculated from [d33(E0)d33,min], is a maximum of ~8% at the highest applied electric field amplitude for Fedoped MP Badjacent tetragonal ceramics. The effect of dopants on the irreversible extrin sic contributions is therefore clearly noted for piezoelectric ceramics with similar crystallographic phases. 4.2.1.3 Prediction of piezoelectric hysteresis from Rayleigh relations As indicated in section 4.1.2, the straine lectric field hysteresis loops at subcoercive electric fields can be calcul ated using the second Rayleigh re lation, Eq. 43. The measured hysteresis loops were compared with the hysteresis loops calc ulated using Eq. 43 for undoped and Ladoped PZT ceramics. Measured and calc ulated hysteresis l oops for Ladoped and undoped PZT ceramics are presented in Figure 46 as examples. For Fedoped PZT ceramics, no significant hysteresis was observed. PAGE 66 66 4.2.2 Piezoelectric Losses and Complex Coefficients The method for calculating the different contribu tions to the overall piezoelectric losses, as described in section 4.1.3, is applied here for the samp le PLZT5248. The areas of the measured strainelectric field hysteresis loops at different amplitude s were obtained using numerical integration and are shown in Figure 47A. The linear slope of the loglog plot of the measured areas with respect to the electric field amplitude should be 3 for a nonlinear piezoelectric material fo llowing Rayleigh law, according to Eq. 44. However, this slope is approximately 2.84 for the experimentally determ ined hysteresis areas. The differences between the measured hysteresis areas and the calculated areas in this case can arise from additional nonRayleigh mechanisms for domain wall motion. On e such possible nonRayleigh mechanism can be viscoelastic motion of domain wa lls, as described in section 4.1.3. The differences between the measured hystere sis areas and the calculated areas from Eq. 44 are shown by the square data points in Figur e 47A. The best fit for the square data points with respect to field ampitudes is obtained with a linear slope of 1.7.3 on a loglog plot. If we assume that the nonRayleigh co ntribution to the hysteresis ar ea comes solely from a linear viscoelastic process, the slope of the difference plot should be e qual to 2 according to Eq. 47. The observed nonRayleigh contri butions to the hysteresis ar ea could deviate from the predictions of Eq. 47 due to additional nonRayleigh mechanisms that are not described by a linear viscoelastic process. Nevertheless, it can be observed th at additional mechanisms, not described by Rayleigh law, contribute to the total piezoelectri c hysteresis in the material. However, at field amplitudes > 650 V/mm the measured hysteresis areas closely correspond to the predictions of Eq. 44 which implies that Ra yleigh type nonlinearity becomes more dominant at higher amplitudes of applied electric field. PAGE 67 67 In order to calculate the comp lex piezoelectric coefficients of this material, Eq. 411 is applied. This is justified by the general observation that the component of the first order harmonic is two orders of magnitude higher th at the second and third order harmonics, as presented in the following section 4.2.3. The im aginary piezoelectric coeffi cient associated with Rayleightype nonlinearity, dR is plotted as a function of el ectric field amplit ude in Figure 47B. In order to make an estimate of the imagin ary piezoelectric coefficien ts associated with nonRayleigh processes, Eq. 412 is applied while assuming that the nonRayleigh losses can be described mostly by a linear vi scoelastic process. The average value for the coefficient dlin calculated from the difference plot of Figure 47A and using Eq. 412, is observed to be 6.5 pm/V. The calculated average value for dlin is one order of magnitude less than the estimated values of dR particularly for E0 > 500 V/mm. Since dlin << dR for moderate to high amplitudes of applied electric field, it is therefore concluded that appr oximate values of the complex piezoelectric coefficients can be calculated us ing Eq. 411 under such co nditions. It should be noted from the above discussion that dlin can be disregarded in th e description of complex piezoelectric coefficients, only if a Rayleightype nonlinear proce ss is the dominating mechanism for the overall elect ricfieldinduced strain. 4.2.3 Harmonics of the Converse Piezoelectric Strain In this work, the harmonics are measured only for the sample PLZT5248 which has the highest piezoelectric coefficien ts. General agreement between nonlinearity and hysteresis in other samples are noted and furt her investigation into the harm onic components of their strain responses was not undertaken. The amplitudes and the phase angles of th e first three harmonics of the converse piezoelectric strains for sample PLZT5248 at three different electric field amplitudes are listed in Table 41. The most dominant is the first harm onic with phase angle close to 0. The third PAGE 68 68 harmonic is proportional to the square of the amp litude of the applied electric field and has a phase angle ~75. This is different from 90 as pr edicted for ideal Rayleigh behavior described in Eq. 49. A significant presence of the second harmoni c is also observed like ly as an effect of 180 domain wall displacement in the material [4 ]. Deviation of the pha se angle of the third harmonic from 90 and the presence of the seco nd harmonic indicate deviation of the material behavior from an ideal Rayleigh response. However, since the pha se angle of the third harmonic is closer to 90 than 0, it can be concluded that the Rayleigh component is dominant in the description of the piezoelectric strain. Further, according to Eq. 49, the value fo r the amplitude of the third harmonic divided by the square of the amplitude of the applied electric field should be a constant. As shown in Table 2, the value of this factor is ~317 m2/V2, which is consistent with the value of 3.617 m2/V2 for the factor (4 d/15 ) in Eq. 11 calculated using d = 0.431015 m2/V2 (the value of d used here is the Rayleigh coe fficient for the sample PLZT5248). 4.3 Conclusions The converse piezoelectric response of PZT ceramics of different compositions was examined under the applica tion of cyclic electric fields. Th e effects of crystal structure and addition of different dopants on d33 were studied. The fractional nonlinear contributions from irreversible extrinsic mechanisms for all the investigated materi als were calculated from the variation of d33 with the driving electric field amplitude, E0. For Ladoped PZT ceramics, the longitudinal piezoelectric coefficient is observed to follow Rayleigh law. The largest irreversible extrinsic contribution is found to be about 55% for Ladoped rhombohedral PZT ceramics. In genera l, the fraction of i rreversible extrinsic contributions is found to be greater for the rhom bohedral phase than the tetragonal phase. This PAGE 69 69 has been interpreted due to the greater possibili ty of non180 domain wall displacement in the rhombohedral phase. Extrinsic contributions are also observed to be greatly enhanced with La doping in comparison to undoped and Fedoped ceramics. For compositions near the MPB, a maximum of 45% contribution from irreversible extrinsi c mechanisms can be estimated for Ladoped ceramics as compared to a maximum of 25% and 8% for undoped and Fedoped ceramics respectively. This is consistent with the prio r understanding that Ladoping promotes while Fedoping restricts domain wall motion in PZT ceramics. The piezoelectric losses for Ladoped tetragona l PZT ceramics of compositions adjacent to the MPB were described from the area of th e strainelectric field hysteresis loops. Although Rayleightype nonlinearity is found to be the do minating mechanism for piezoelectric hysteresis, it is observed that the total hysteresis area is inadequately described from the Rayleigh relations. It is proposed that additional mechanisms, possibl y in the form of the vi scoelastic nature of domain wall motion in the material, contribute to the total hysteresis area. In order to calculate the complex piezoelectric coefficients, the compone nts from Rayleigh relati ons and viscoelastic mechanisms were considered. An approximate description of the complex coefficients is provided from the Rayleigh relations. Finally, in order to further investigate th e applicability of the Rayleigh law, the harmonics of the converse piezoelectric strain were analyzed for Ladoped tetragonal ceramics with compositions adjacent to the MPB. The Rayl eigh component is found to be dominant in the prediction of the amplitude and the phase of the piezoelectric st rain components. PAGE 70 70 Figure 41. Experimental setup for m easuring converse piezoelectric strain. Sample stage HV MTI 2000 Photonic Sensor TREK 609C6 HV Amplifier SRS DS 360 Function generator Tektronix TDS 410 Oscilloscope Computer Probe for MTI2000 PAGE 71 71 4000400 0.04 0.02 0.00 0.02 0.04 Strain (%)Electric Field (V/mm) xmax E0 Figure 42. The piezoelectric coefficient d33 was calculated from the measured peaktopeak longitudinal strain, indicated as xmax, using Eq. 41. PAGE 72 72 02004006008001000 200 400 600 800 PLZT5248 PLZT6040 PLZT4060 d33 (pm/V)Electric Field (V/mm) d(E0) = dinit + dE0 PLZT5248: dinit = 461.5 4.3d = 0.43 0.01 PLZT6040: dinit = 207.3 4.5d = 0.36 0.01 PLZT4060: dinit = 103.3 3.9d = 0.07 0.01 Figure 43. Converse piezoelec tric coefficients of PLZT6040, PLZT5248 and PLZT4060 as a function of applied sinusoidal electric field amplitudes. The fits to Eq. 42 are shown with bold lines. The maximum Raylei gh coefficient of ~0.43 (pm/V)(V/mm)1 is observed for the sample PLZT5248. PAGE 73 73 02004006008001000 0 50 100 150 200 250 300 350 PLZT5248 PLZT6040 PLZT4060 d33 (pm/V)Electric Field (V/mm) A 02004006008001000 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 PLZT6040 PLZT5248 PLZT4060 d33/d33(E0)Electric Field (V/mm) B Figure 44. Estimate of the irreversible extrin sic contributions for La doped PZT ceramics as a function of applied sinusoidal electric fiel d amplitudes. A) Absolute irreversible extrinsic contributions d33 B) Fractional irreversible extrinsic contributions d33/d33(E0); the solid lines show the predictions from the Rayleigh relation described in Eq. 42. PAGE 74 74 02004006008001000 200 400 600 800 PLZT5248 PUZT5248 PFZT4753 d33 (pm/V)Electric Field (V/mm) d(E0) = dinit + dE0 PLZT5248: dinit = 461.5 4.3d = 0.43 0.01 PUZT5248: dinit = 242.4 1.0d = 0.13 0.002 A 2004006008001000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 PLZT5248 PUZT5248 PFZT4753 d33/d33(E0)Electric Field (V/mm) B Figure 45. Converse piezoelectr ic coefficients and fractiona l extrinsic contributions for PLZT5248, PUZT5248 and PFZT4753. A) Converse piezoelectric coefficients as a function of applied sinusoidal electric field amplitudes. B) fractional extrinsic contributions d33/d33(E0). The solid lines show the predictions from the Rayleigh relation described in Eq. 42. PAGE 75 75 4002000200400 0.024 0.012 0.000 0.012 0.024 measured calculated Strain (%)Electric Field (V/mm) 6004002000200400600 0.048 0.032 0.016 0.000 0.016 0.032 0.048 measured calculated Strain (%)Electric Field (V/mm) A Figure 46. The strainelectric fi eld hysteresis loops for the PZT ceramic samples at electric field amplitudes 400 V/mm and 600 V/mm: A) PLZT5248, B) PLZT6040, C) PLZT4060, D) PUZT5248. The measured loops and the loops calculated using Eq. 43 are in general agreement with each other. PAGE 76 76 4002000200400 0.018 0.012 0.006 0.000 0.006 0.012 0.018 measured calculated Strain (%)Electric Field (V/mm) 6004002000200400600 0.03 0.02 0.01 0.00 0.01 0.02 0.03 measured calculated Strain (%)Electric Field (V/mm) B Figure 46. Continued PAGE 77 77 4002000200400 0.006 0.004 0.002 0.000 0.002 0.004 0.006 measured calculated Strain (%)Electric Field (V/mm) 6004002000200400600 0.009 0.006 0.003 0.000 0.003 0.006 0.009 measured calculated Strain (%)Electric Field (V/mm) C Figure 46. Continued PAGE 78 78 4002000200400 0.012 0.006 0.000 0.006 0.012 measured calculated Strain (%)Electric Field (V/mm) 6004002000200400600 0.024 0.012 0.000 0.012 0.024 measured calculated Strain (%)Electric Field (V/mm) D Figure 46. Continued PAGE 79 79 100 400700 103104105 Measured area From Eqn. (6) Difference between measured and Eqn.(6) Area (V/mm)Electric Field (V/mm)slope = 2.84 0.02 slope = 1.7 0.3 slope 3 A 250500750 0 30 60 90 120 150 180 From Rayleigh law Viscoelastic component d'' (pm/V)Electric Field (V/mm) B Figure 47. Hysteresis area and imaginary piezoelectric coefficient. A) Contribution to strainfield hysteresis area from different mechanis ms B) imaginary piezoelectric coefficient due to Rayleightype nonlinearity and visc oelastic losses, for sample PLZT5248, at different amplitudes of applied sinusoidal electric fields. PAGE 80 80 Table 41. Harmonics of the c onverse piezoelectric strains for PLZT5248. Note that the phase angle of third harmonic is close to 90 fo r the three different amplitudes of applied electric field. Also, the value for the am plitude of the third harmonic divided by the square of the applied el ectric field is ~317 m2/V2 which is consistent with the Rayleigh coefficient of the material. Electric Field Amplitudes (E0) 500 V/mm 600 V/mm 700 V/mm Harmonics Strain amplitude (105) Phase Strain amplitude (105) Phase Strain amplitude (105) Phase 1st 34.04 8 42.9 8.8 53.9 9.6 2nd 0.34 105 0.354 128 0.41 160 3rd 0.73 73 1.11 72.5 1.4 74.7 3rd harmonic/E0 2 (1017 m2/V2 ) 2.94 3.08 2.86 PAGE 81 81 CHAPTER 5 CHARACTERIZATION OF ELECTRICFIELDINDUCED STRUCTURAL CHANGES USI NG LABORATORY XRAY DIFFRACTION In order to gain insight into the relative electromechanical contributions from intrinsic and extrinsic mechanisms, both in situ and ex situ diffraction experiments in PZT ceramics have been carried out in the past [30,6371]. Recently, in situ neutron an d diffraction experiments under cyclic electric fields using timeresolved stroboscopic techniques have been reported [7274]. In the next chapter, use of highenergy Xra y diffraction for timeresolved structural characterization of piezoelectrics is described. However, over subscrip tion of various neutron and highenergy Xray sources can be a limiting factor towards a wider adoption of timeresolved techniques for struct ural characterization of piezo electrics. Development of timeresolved diffraction capabilities at smallerscale laborator y Xray systems is therefore necessary. The objective of this chapter is to present the development of timeresolved diffraction techniques for a laboratory X ray source and its utilization for in situ characterization of piezoelectrics under static and dynamic electric fields. An inhouse experimental setup is devel oped for collection of timeresolved Xray diffraction patterns in synchroniza tion with an applied periodic stimulus. Using this technique, structural changes in tetragona l Ladoped PZT ceramics are measur ed under different electrical loading conditions. In particular, non180 domai n switching and lattice strains are measured parallel to the direction of the applied electric field of the following types: (1) static electric fields of amplitudes up to 2 kV/mm and (2) subcoercive (< EC) cyclic electric fields. Correlations between the structural changes a nd the amplitude of the applied el ectric field are also studied. 5.1 Experimental Procedure Sintered pellets of nominal composition Pb0.97La0.02(Zr0.52Ti0.48)O3 were polished down to thickness in the range of 0.65 1 mm and we re subsequently annealed above the Curie PAGE 82 82 temperature to relieve surface resi dual stresses. Gold electrodes were sputtered on the top and bottom surfaces of the disc samples. Some of the samples were then poled using an electric field of 3.5 kV/mm at 50C for 15 minutes. Unpoled and poled ceramics were used for all the experiments described here. 5.1.1 In situ Xray Diffraction Under Application of Electric Field In situ Xray diffraction of the ceramic sample s under the application of electric fields was performed using a INEL CPS120 diffractometer in reflection geometry. The diffractometer is equipped with a curved position sensitive (CPS) detector for simultaneous measurement of multiple diffraction peaks. The incident radiatio n used for the experiments was a monochromatic CuK 1 Xray beam. The samples were positioned with re spect to the incident beam such that the 002 and 111 diffraction vectors were or iented to within 5 of the applied electric field, as is illustrated in Figure 51A. The diffe rent techniques adopted for in situ diffraction under static and cyclic electric fields are described below. 5.1.1.1 Diffraction under stat ic electric fields Discshaped ceramic samples in their unpoled stat e were loaded under sta tic electric fields of incremental amplitudes using a Stanford Research System PS350 high voltage supply. Diffraction patterns were collected in situ as a function of the amplitude of the applied electric field. The amplitude of the applied static field was stepwise and was vari ed within the range 2 kV/mm. The collection time for the diffraction pa tterns at each field amplitude was 720 seconds after an initial delay time of 60 seconds at each electric field amplitude. 5.1.1.2 Diffraction under cycl ic electric fields In situ diffraction of poled ceramics under cyclic electric fields were performed using the following experimental technique. A sample was driven with cycl ic squarewave electric fields and simultaneous diffraction patterns were collecte d at predefined time intervals. In order to PAGE 83 83 achieve synchronization between the data acquisiti on and the applied electr ic fields the following scheme was adopted. The CPS detector of the diffractometer was conn ected to a computer through a data acquisition board (P7889 PCI board, FAST Co mTec GmbH). The acquisition electronics of the instrument outputs a logi cal signal each time the acquisition of a new diffraction pattern commences. This signal was used to trigger th e start of a square waveform from an Agilent 33220A function generator at sp ecified intervals. The waveform from the function generator was used to drive a DEI PVX4140 pulse generator which was further used to obtain a high voltage square waveform across th e sample. The highvoltag e input to the pulse generator was switched between two Stanford Research Syst em (SRS PS350) highvoltage supplies in accordance with the signals received from the function generator. A cyclic square wave electric waveform was therefore obtained across the sample at intervals defined by the acquisition sequence of the diffr action patterns. The layout of the various components of the experimental setup is shown in Figure 51A. An acquisition cycle was programmed so that ten diffraction patterns at equal intervals were obtained within each cycle of an applied el ectric field waveform of frequency ~0.3 Hz. The acquisition sequence of the diffrac tion patterns with respect to the applied electric field is illustrated in Figure 51B. For the specified settings, the total length of a single cycle of the applied electric field waveform is equal to the total time for ten acquisition windows as well as the delay time between each acquisition window. In order to improve the statisti cs of the collected diffraction patterns, the process of data collection was repeated over multiple cycles and the diffraction patterns collected during similar time intervals with respect to the applied electric field waveform were summed. This method of data collection resulted in ten highstatistics timeresolved diffraction patterns which are PAGE 84 84 representative of the material structure at various instants within a single cycle of a continuous square wave electric field. Summation of di ffraction patterns over multiple cycles of applied electric fields can be adopted if the measured structural changes are repeatable on average over multiple cycles of applied electric fields. This is a reasonable assumption for structural changes such as domain wall displacement and lattice stra ins under subcoercive bipolar cyclic electric fields, and has been adopted in earlier works [69,7274]. Further details about the settings used in the software (MCDWIN) for stroboscopic data collection are provided in APPENDIX B. 5.1.2 Quantification of Structural Changes The intensities of the different (hkl) diffraction peaks of a mate rial are representative of the structure factor, multiplicities, and the volume frac tions of the corresponding (hkl) lattice planes along their respective diffraction vectors. In te tragonal PZT, the 90 domains can be defined by the orientation of their 002 and 200 crystallographic poles. Therefore, in the absence of changing volume fractions of any coexisting phases and/or any large structural distortions, the volume fractions of the existing 90 domains along a partic ular specimen direction can be correlated to the intensities of the (002) and (200) diffraction p eaks. In an earlier work [69], the particular correlation between the (002)type diffraction peak intensities and the volume fraction of the 002 domains (002v ) along a specimen direction has been shown to be given by 2+ =200 200 002 002 002 002 002' 'I I I I I I v (51) where hklI is the integrated area of the (hkl) diffractio n peak for a given sample with a preferred orientation of 90 domains and hklI is the integrated area of the same peak for a sample with no PAGE 85 85 preferred orientation of 90 domains. Eq. 51 takes into account the different structure factors and multiplicities of the (002) and (200) planes while calculating the domain volume fractions. As 002v is equal to 1/3 for a random distribution of 90 domains, the volume fraction of the domains switched by 90 under a give n condition is given by [69] 3 1 2+ =200 200 002 002 002 002 002' 'I I I I I I (52) According to Eq. 52, 002 can have a maximum value of 2/3 (0.67) for a complete reorientation of all the 002 domain s parallel to the measured direct ion. In order to estimate the volume fraction of the 90 switched domains parallel to an applied electric field, the value of 002 was calculated from the measur ed intensities of the (002)typ e diffraction peaks with their diffraction vectors oriented paralle l to the electric field. The inte grated intensities of the 002 and 200 peaks were obtained from the best fit of the (002)type diffraction peaks using two asymmetric Pearson VII functions [83]. Using asymmetric Pears on VII functions, the fit for the (002)type diffraction peaks for an unpoled sample of the type PLZT5248 is shown in Figure 52. Under the application of cyclic electric fields, the amount of 90 domain switching between the positive and negative electric field states (002 ) was further obtained by [72,73] negative positive002 002 002= (53) The term domain switching is not intende d here to necessarily imply crystallographic reorientation of entire domains. Instead, in the present context, 90 domai n switching refers to a change in the crystallographic orientation of certain material vol ume fraction as a result of both reversible and irreversible 90 domain wall displacements. PAGE 86 86 In addition to the changing volume fractions of the non180 domains, expansion or contraction of the lattice planes also contribute to the macroscopic electricfieldinduced strain in a piezoelectric material. The component of lat tice strain for the grains with their hkl poles oriented parallel to the direction of the applied electric field can be obtained from the change in the ( hkl ) lattice plane spacings measured parallel to the same direction. Following Braggs law for diffraction, the change in the ( hkl ) lattice plane spacing is obtai ned from the shift in the angular position of the ( hkl ) diffraction peak. For tetragonal P ZT ceramics, the largest lattice strains are observed away from the polarization ax is of [001] [66,30,74]. Th erefore, in order to perform a comparative analysis of lattice strains in PZT ceram ics under different electrical loading conditions, the component of lattice strain was measured for the grains with their 111 poles oriented parallel to the dir ection of the applied electric fiel d. This was ensured by orienting the sample such that the 111 diffr action vector was within 5 of the direction of the applied electric field. The 111 lattice plane spacings were obt ained from the best fit of the (111) diffraction peaks using a pseudoVoigt functi on. A symmetric pse udoVoigt function was adopted due to the symmetric nature of the (111) diffraction peaks. For static electric fields, the 111 lattice strain, 111, is defined as () ( ) () () 0 0=111 1110111 0111d dEd E (54) where ( ) 0111Ed is the (111) lattice spacing under an static electric field of amplitude E0 and ()0111d is the (111) lattice spacing for an unpoled sample with no ap plied electric field. Under the application of cyclic bipolar electric fields, the 111 lattice strain is calculated from the measured (111) lattice spacings under the positive a nd negative parts of th e applied electric field waveform and is given by PAGE 87 87 () ( ) ( ) () 0111 01110111 0111+ = Ed EdEd E (55) where ( ) 0111+ Ed and ( ) 0111Ed are the 111 lattice spacings under positive and negative electric fields respectively. It should be mentioned here th at, for the diffraction geometry us ed in this work, a vertical displacement of the diffraction volume results in a change of the 2 position of the diffraction peaks. APPENDIX A presents typi cal expected shifts in the 2 positions caused by a vertical displacement of diffraction volume due to electri cfieldinduced macroscopic strain in PZT ceramics. As explained in APPENDIX A, peak shifts due to this effect are at least two orders of magnitudes less than the peak shift due to electri cfieldinduced lattice strains and are therefore neglected in the current calculation of 111 lattice strains. 5.2 Results and Discussions 5.2.1 Structural Changes Under Static Electric Fields Figures 53A shows the variation of 002 during the application of st atic electric fields. As discussed in the preceding section, 002 represents the volume fraction of existing domains in the material that reorient by 90 under an applied el ectric field. During the initial application of positive electric fields of increasing amplitude across an unpoled sample, a sharp increase in 002 (from ~0.06 to ~0.17) is obser ved between the field amplitudes of 0.8 kV/mm and 0.9 kV/mm. This indicates that a threshold field, Ethreshold, of ~0.9 kV/mm exists fo r initial 90 switching of domains in an unpoled sample. The maximum volum e fraction of domains that switch by 90 (002 ) under an electric field of amp litude 2 kV/mm is observed to be ~0.35. This is consistent with the values reported earlier for similar materials [69]. Upon subse quent decrease of the amplitude of the applied electric field to zero, 002 decreases from 0.35 to 0.24, which indicates PAGE 88 88 that 30% of the domains that initially switched forw ard switch to a perpen dicular direction upon release of the electric field. Further decrease in 002 is observed with a revers al in the direction of the applied electric field and the minimum value of 002 is observed to be 0.09 at a negative field amplitude of 1 kV/mm, beyond which another sharp increase in 002 is evident. This indicates that with increasing negative electric fields, an increasing number of doma ins reorient away from the direction of the applied elec tric field. At negative electric fields exceeding 1 kV/mm, the material becomes polarized in the opposite dire ction, achieving the sa me ferroelastic domain state that was observed in the previous electric fi eld direction. This is illustrated in Figure 54. The critical field at which 002 is found to be the minimum upon re versal of field direction is defined here as the coercive field EC of the material, which is obser ved to be equal to 1 kV/mm. In general, Figure 53A closely resembles the butterfly shape of the macr oscopic strainelectric field loop of sample PLZT5248 shown in Figure 33B. However, a few discrepancies between the two can be noted. Under a sinu soidal electric field of freque ncy 100 mHz, the threshold field, Ethreshold, is observed to be about 1.2 kV/mm which is higher than the value of ~0.9 kV/mm observed from in situ Xray diffraction experiments under static electric fields. Similarly, the macroscopic coercive field of 1.35 kV/mm is higher than the coerci ve field of 1 kV/mm observed with Xray diffraction. The coercive field EC is not an absolute threshold field but depends on the rate or frequency of the applied el ectric field under which it is determined [4]. Therefore the currently observed difference in EC, determined from macroscopic and Xray diffraction measurements, may be c ontributed by the fact that the Xray measurements are taken under static electric fi elds unlike a continuous cyclic field as adopted for macroscopic measurements. This point further emphasizes th e importance of character izing the structural response of these materials under cyclic electric fields. PAGE 89 89 The component of lattice strain for the grains with their 111 poles oriented parallel to the direction of the applied electric field is calculated from the (111) lattice plane spacings using Eq. 54. Figure 53B shows the 111 lattice strains, 111, as a function of the applied electric field amplitude. The effect of the electric field amplitude on 111 correlates with the fielddependent 90 domain switching shown in Figure 53A. Such behavior for 111 la ttice strains in PZT ceramics have been reported prev iously [66]. The 111 lattice strains in poled PZT ceramics were previously described in terms of misfit strains between the differently oriented grains in the material which undergoes different amounts of 90 domain switching under the application of electric fields [30]. In particular, since the grai ns with their 111 poles or iented parallel to the applied electric field do not have an energetic preference for domain switching, they experience tensile or compressive stresses du e to the extension or compression of the neighboring grains that are more favorably oriented for 90 domain switchi ng. It is therefore possi ble that the 111 lattice strains under the appli cation of static electric fields are influenced by 90 domain switching in the neighboring grains. 5.2.2 Structural Changes Under Cyclic Electric Fields Figure 55A shows the (002) and (200) diffrac tion peaks as a function of time during the application of a cyclic square wave electric field of amplitude V/mm and frequency 0.3 Hz. It is observed that the intensity of the (002) peak increases duri ng the positive part of the applied electric field waveform along with a simultaneous decrease in th e intensity of the (200) peak. During the negative part of the electric field wa veform the situation is reversed whereby the (200) peak increases in intens ity at the expense of the (002) peak. The observed intensity interchanges between the (002) and (200) diffract ion peaks clearly indicate changes in the volume fraction of the 90 domains in the materi al under the application of subcoercive cyclic electric fields. It should be st ated here that no measurable amount of strain can be observed for PAGE 90 90 the (002) and (200) lattice planes within the error of the peak fitting procedure. Instead, a significant strain for (111) planes under the same electrical loading conditi ons is observed, as is indicated by arrows in the contour plot of Figure 55B. It is evident from Figure 55B, that the (111) peak position shifts to a lower 2 angular position during the positive part of the applied electric field waveform, indi cating an increase in the (111) lattice plane spacings. In both Figures 55A and 55B, discrete changes in the diffr action patterns can be observed when the polarity of the electr ic field is reversed within the applied waveform. The timing resolution of the collected diffraction patterns was inadequate to quantify the timedependence of the measured structural changes. Therefore, in or der to quantify the measured structural changes, two representative diffraction patterns corresponding to the positive and the negative states of the applied square waveform were obtained by a su mmation of the diffraction patterns during the corresponding portions of the applied waveform. Figure 56A shows the (002)type diffraction peaks during the positive and negative states of the applied square wave electric field of amplitude V/mm. The volume fractions of the 90 domains in the material were determined from the integrated areas of the (002) and the (200) diffraction peaks. Using Eqs. 52 and 53, the change in the volume fraction of the 90 domains,002 is found to be ~0.05 or 5% parallel to the applied electric fi eld. This value is consistent with previous measurements of do main switching under cyc lic electric fields in commercial PZT ceramics [72,73]. The 111 lattice strain measured under the same electrical loading conditions is highlighted in Figure 56B. The 111 lattice stra in parallel to the applied elect ric field is calculated to be ~0.07% using Eq. 55. This value corresponds to a lattice strain coefficient of 530 pm/V, a value PAGE 91 91 of comparable magnitude to the direct piezoele ctric coefficient of 480 pC /N measured for this material using a Berlincourt d33 meter. In order to investigat e the relationship between the amplit ude of the applied square wave electric field and the different structural changes, the amount of 90 domain switching (002 ) and the 111 lattice strains (111) were measured as a function of field amplitude. The measured values for 002 and 111 are plotted in Figures 57A and 57B, resp ectively. It is observed that 002 monotonically increases with increasing electric field amplitude. This is in agreement with the generally accepted understanding of nonlinear electromechanical contributions from non180 domain wall motion in polycrystalline piezoelectric ceramics under subcoercive electric fields [35]. It is generally understood that the contributions of non180 domain wall motion to the macroscopic properties can in crease with the amplitude of the applied external field. In order to understand the fielddependence of 111, the coefficient of the electricfieldinduced 111 lattice strain, 111, is calculated from 111 and the electric field amplitude E0 using ( ) 0 0111 1112 = E E (56) It should be noted here that a value of two is in the denominator because the electric field is bipolar and the value of E0 represents only half of the total el ectric field amplitude. The values of ( ) 0111E calculated from Eq. 56 are shown in Figure 56B. It is observed that 111 increases with an increase in E0. A linear equation of the type ( ) 0 111 0111+= E E init (57) is used to fit the linear dependence of the stra in coefficient with resp ect to the electric field amplitude. In Eq. 57, ( ) 0111E is the 111 lattice strain coeffici ent for an electric field amplitude of E0, is the coefficient for the linear dependence, and the value of init111 represents the value PAGE 92 92 of 111 as E0 0. The values of init111 and are 117 pm/V and 0.7 (pm/V)(V/mm)1 respectively. Eq. 57 implies a 111 lattice strain dependence of ( ) 00 0111+2= EE d)E( The observed changes in the 111 lattice strain and 111 lattice strain coefficients under static and cyclic electric fields may indicate that, in addition to their dependence on thermodynamic variables [1727], th e lattice strain coefficients in a polycrystalline matrix may be influenced by other extrinsic factors such as misfit elastic strain s arising from non180 domain switching [30]. 5.3 Conclusions A laboratory Xray diffract ometer was used to measure the structural changes in tetragonal Ladoped PZT ceramics under the application of static and dynamic electric fields in situ. The amount of 90 domain switching pa rallel to the applied field was calculated from the intensity interchange between the 002 and 200 diffraction peak s. In addition, in order to calculate the electric field induced lattice st rains under certain app lied electric fields, the change in the 111 lattice spacings were measured. Under application of st atic electric fields, 002 is found to have a maximum value of 0.35 for an electric field of amplitude 2 kV/mm. In general, under static electric fields, the 90 domain switching and the 111 lattice strains are observed to corr elate with the macroscopic strainelectric field hysteresis loop. However, differences in Ethreshold and EC calculated from macroscopic strain measurements and Xray diffr action analysis are noted, which are attributed to the frequency dependence of the coercive field EC. A timeresolved Xray diffraction technique was adopted to measure the structural changes under applied cyclic electric fields in situ. For a square wave electric field of amplitude V/mm and frequency 0.3 Hz, 002 is observed to be 0.05, along with ~0.07% strain for the 111 PAGE 93 93 lattice planes. Both the amount of 90 domain switching and the 111 lattic e strains are observed to increase with an increase in the amplitude of the cyclic electric field. The field dependence of the 111 lattice strains are described in terms of misfit strains between th e differently oriented grains under the application of electric fields. PAGE 94 94 A 10123 650 0 650 1300 Voltage, VTime, seconds 1 cycle of applied square waveform Acquisition window for diffracted signal B Figure 51. Experimental setup and synchronization settings for in situ diffraction experiments. A) Schematic layout of the experimental setup for in situ diffraction under applied electric field. The sample is oriented su ch that the 002 and the 111 diffraction vectors are oriented to within 5 of the applied electric field. B) Th e sequential acquisition windows (shown at the bottom) for the collect ed diffraction patterns with respect to the applied electric fiel d (shown on the top). Data acquisition board, P7889 Waveform generator Oscilloscope XY Sample Incident beam Diffracted beam }{k002 E Detector HV switch HV supply +ve HV supply ve ()111k PAGE 95 95 4 3.344.144.945.746.5 Measured intensities Fit intensities 2002 200 Figure 52. The fit for the ( 002)type diffraction peaks for an unpoled sample of the type PLZT5248. Two asymmetric Pear son VII functions were us ed for peak fitting in order to account for 002 and 200 diffraction p eaks (shown by dotted lines in black). PAGE 96 96 21012 0.0 0.1 0.2 0.3 0.4 002Electric Field (kV/mm)1 2 3 4 5 6 7 EC~1kV/mm A 21012 0.0 0.1 0.2 0.3 (%)Electric Field (kV/mm)1 2 3 4 5 6 7 EC~1kV/mm B Figure 53. Structural changes measured under the application of stepwise static electric fields: A) 002 and B) 111, as a function of the field amplitude. The path followed for incrementing electric field amplitude is indi cated by the arrows sequenced 1 to 7. The coercive field EC for domain switching is observed to be ~1 kV/mm from the plot of 002 vs the electric field amplitude. PAGE 97 97 Figure 54. Schematic of the proposed 90 domain switching process under static electric fields E0. The 90 domain states are indicated by the orientation of the rectangles. The numbers refer to the specific region of the 002 electric field plot shown in Figure 53A: (1) 0 < E0 < +EC, (2) EC < E0 < +Emax, (3) EC < E0 < 0, (4) Emax < E0 < EC; Emax refers to the maximum amplitude of applied electric field. 1 2 3 4 PAGE 98 98 1 0 1 2 44.0 44.8 45.6 T i m e s e c o n d s2 Diffracted Intensity (002) (200) +E E E A 38.438.739.0 2 1 0 1 2Time, seconds E +E E (111) B Figure 55. Structural changes as a function of time during the application of cyclic square wave electric field of amplit ude V/mm and of freque ncy 0.3 Hz. A) Intensity interchange between the (002) and (200) peaks can be obs erved indicating change in volume fraction of 90 domains. B) Contour plot of the (111) diffraction peak shows a shift towards lower 2 under positive state of the applied electric field waveform. PAGE 99 99 43.344.144.945.746.5 positive negative difference Diffracted Intensity2(200) (002) A 38.038.539.039.5 positive negative difference Diffracted Intensity2(111) B Figure 56. Measured and fit patterns of the A) (002)type, and B) (111) diffraction peaks, during the positive and the negative parts of the applied electric field waveform. The difference between the positive and the negative states are plotted at the bottom of the each plot. The difference plot in A demonstr ates an intensity in terchange between the (002) and the (200) peaks, whereas the diffe rence plot in B demonstrates a shift of (111) peak with no apparent change in peak profile. PAGE 100 100 550600650700750800 0.03 0.04 0.05 0.06 0.07 0.08 002Field Amplitude +/(V/mm) A 550600650700750800 0.02 0.04 0.06 0.08 0.10 0.12 111 111 (%) Field Amplitude +/(V/mm) 400 500 600 700 800 111 111 (pm/V) B Figure 57. Dependence of non180 domain switc hing and 111 lattice strains on amplitude of applied cyclic electric fields. A) 002 as a function of the amplitude of the applied square wave electric field. A monotonic increase in the amount of non180 domain switching can be observed with increasing fi eld amplitude. B) 111 lattice strain and lattice strain coefficient as a function of the applied square wave electric field amplitude. The dashed line in black and th e bold line in red shows the corresponding fits for 111 and 111, respectively. PAGE 101 101 CHAPTER 6 CHARACTERIZATION OF ELECTRICFIELDINDUCED STRUCTURAL CHANGES USI NG HIGHENERGY SYNCHROTRON XRAY DIFFRACTION In situ diffraction experiments using a laboratory Xray source provide an understanding of the nature of different structural changes c ontributing to the overall electricfieldinduced strain in piezoelectric ceramics, as presented in Chapter 5. However, the penetration depth of CuK radiation in PZT ceramics is of the order of micrometers. In contrast, highenergy Xrays (~80 keV) have much larger penetration depths of the order of millimeters in PZT ceramics (See APPENDIX B). Therefore, diffraction in tran smission mode using highenergy Xrays is necessary for measuring electr icfieldinduced structural ch anges from the bulk of these materials. Measurement of structural changes such as non180 domain switchi ng and lattice strains in Pb(Zr,Ti)O3 ceramics, using timeresolved highenergy Xray diffraction, are presented in this chapter. The contributions to the electricfieldin duced strains from non180 domain switching and lattice distortions are cal culated as a function of mate rial composition and dopants. Correlations between the structural changes and th e amplitude of the applied electric fields are also studied. 6.1 Experimental Procedure 6.1.1 Sample Synthesis and Preparation The sintered and poled ceramics were cut in dimensions of 1 mm 1 mm 3 mm for in situ electric field induced structural studies. In or der to verify any possibl e change in the poled state of the materials subsequent to cutting, piezoelectric meas urements were taken before and after the cutting process. No changes were observed in the piezoelectric coefficient d33 of the ceramic samples subsequent to their dimensioning. Electric fields were to be applied across the 1 PAGE 102 102 mm 3 mm faces. The surfaces across which elec tric fields were to be applied during in situ studies were coated with an airdry commercial silver paint (RS components 1863600). 6.1.2 TimeResolved Diffraction Highenergy Xray diffraction experiments were carried out at beamline ID15B of the European Synchrotron Radiation Facility (ESRF), located at Gr enoble, France. A beam energy of 87.80 keV was generated from a bent Laue (5 11) silicon monochramtor. A schematic of the experimental setup and sample diffraction image are shown in Figure 61. The incident beam size was set to 300 300 m2 using a set of Tungsten slits just prior to the sample stage. An insitu electric field cell was used, whereby an electric field of up to 3 kV/mm can be applied to a sample in air [84]. Diffracted intensities were collected using a Pixium 4700 flat panel detector [85] at a distance of approximately 1130 mm. The cen ter of the Debye rings was approximately coincident with the center of the det ector panel, as s hown in Figure 61A. Time resolved diffraction images were collected during cyclic actuation of the sample. The stroboscopic data acquisition process was ac hieved by the following pr ocess. An electric field signal was applied across the sample using a function ge nerator (FLUKE PM5193) and a highvoltage amplifier (TREK 10/10B). A delay ge nerator was simultaneously triggered by this signal and applied a set delay ( t in Figure 61B). This delayed si gnal was then used to trigger a detector frame, the length of which decided th e time resolution of the experiments (Frame Length in Figure 61B). A number of detector frames were then summed together into a single image, and the delay signal ( t), adjusted. In this way, large area detector images are collected as a function of time, and can be represente d in a single electric field cycle [84]. PAGE 103 103 6.2 Analysis of Diffraction Data 6.2.1 Binning of Diffraction Data at Different Az imuthal Angles The 2D images from the area detector were first analyzed using software Fit2D Ver. 11.012 [86]. The beam center and tilt were refine d for each pattern with respect to a standard diffraction pattern of CeO2 (part of standard reference material 674b from National Institute of Standards and Technology), collected under identical settings. The DebyeScherrer diffraction rings from the sample were segmented into azimut hal sectors of 10 width (as indicated in Figure 61). The pattern within each azimuthal sector was integrated to obtai n equivalent diffracted intensities as a function of 2 [87]. The pattern of the hkl diffracted intensity within each azimuthal sector is indicative of the structural state of grains with their hkl poles approximately parallel to an angle to the applied electric field (within 5). This effect is highlighted in Figure 61C wher e the induced domain texture is clearly visible at selected azimuthal bins. 6.2.2 Fitting of ( hkl ) Diffraction Peaks In order to measure the structural changes in piezoelectric ceramics during the application of cyclic electric fields, calculations of the intensities and the 2 positions of the different ( hkl ) diffraction peaks are necessary. Tw o different approaches can be taken to obtain the relevant information from the measured di ffraction patterns. In one approach a leastsquare fitting of the entire diffraction pattern over a large 2 range can be performed such as Rietveld refinement [88]. However, it is known that stresses along non180 domain walls [8991] as well as intergranular strains [30] in pol ycrystalline ferroelectrics can lead to different asymmetric distortions of specific ( hkl ) diffraction peaks. The asymmetr ic nature of the different ( hkl ) diffraction peaks can be difficult to take into account for the Ri etveld refinement approach. Fitting of individual ( hkl) diffraction peaks can instead be applied to model the diffraction PAGE 104 104 patterns in these materials [83]. In this approach, each individual ( hkl ) diffraction peak can be fit separately using a suitable profile shape function in order to obtain information about the 2 position and intensity of the corr esponding peak. In the current wo rk, the structural information of the state of the non180 domain s and lattice strains are obtained from th e best fits of the individual ( hkl ) diffraction peaks. The different (hkl ) diffraction peaks were fit using split Pearson VII peak profile functions [83,92]. Peak fitting routines for the different ( hkl ) peaks were executed within the commercial software package Igor Pro 5.05A 6.2.3 Analysis of Ferroelectric Phases Diffraction patterns of all th e ceramic samples were found to be consistent with a pseudocubic perovskite structure; no impurities or secondary phases were identified from the measured diffraction patterns. An analysis of the crystallographic phases present in the synthesized ceramics is necessary prior to their structural charact erization under applied electric fields. This was performed by an alyzing the various diffraction peaks of the unpoled and poled ceramic samples. The (002)type and the (111)type diffraction peaks of the different ceramic samples in their unpoled state are shown in Fi gures 62A and 62B, respectivel y. Ceramics with Zr/Ti ratio less than or equal to 52/48 show a single (111) and two (002)t ype diffraction peaks, while the ceramics with Zr/Ti ratio equal to 60/40 show a single (002) and tw o (111)type diffraction peaks. These observations are consistent with the presence of a tetragonal phase for compositions with Zr/Ti ratio 52/48 and a rhombohedral phase for compositions with Zr/Ti ratio equal to 60/40. The coexistence of tetragona l and rhombohedral phases is not evident from the collected diffraction patterns in any of the ceramic samples. These results agree with the phase diagram of PAGE 105 105 Pb(Zr,Ti)O3 solid solution (Figure 31) and the di ffraction patterns re ported elsewhere for materials with similar compositions [93]. Recently, a monoclinic phase has been proposed for PZT ceramics with compositions close to the morphotropic phase boundary (MPB) a nd Zr/Ti ratio equal to 53/47 [94]. In order to verify the presence or absence of any monoclinic phase, the (022)type di ffraction peaks of the different ceramic samples were analyzed. A monoc linic phase would be evident with the splitting of the (022)type diffracted intens ities into three different peak s. Figure 62C shows the (022)type diffraction peaks of the samples P LZT5248, PUZT5248 and PFZT4753, none of which show this characteristic. The presence of a mono clinic phase in samples with compositions close to the MPB is therefore not considered here. The diffraction of the poled samples under cyclic electric fiel ds showed similar characteristics with regard to their phase anal ysis. Therefore, the coexistence of different crystallographic phases or the pr esence of a monoclinic phase wa s not considered in further analysis of electric field induced structural changes for the ceramics used in this study. 6.2.4 Binning of Diffraction Da ta with Respect to Time The time dependence of the structural changes in the material within each cycle of an applied electric field was characterized from the timeresolved diffraction patterns. During the in situ Xray diffraction experiments, the applied el ectric fields across the ceramic samples were monitored using a Tektronix TDS3012 oscillosc ope. The waveforms for the bipolar electric fields were observed to have a square profile with complete positivetonegative transition within one microsecond. Figure 63A shows timereso lved (002)type diffraction pe aks of the sample PLZT5248 during the application of a cyclic square wave electric field of amplitude V/mm and frequency 1 Hz, for = 0, an angle approximately parallel to the applied electric field. It is PAGE 106 106 observed that the (002) diffracti on peak intensity increases duri ng the positive segment of the applied electric field waveform along with a simultaneous decrease in the intensity of the (200) diffraction peak. The situation is reversed durin g the negative segment of the applied electric field waveform, i.e., the (200) peak intensity increas es at the expense of the intensity of the (002) peak. The time dependence of the intensity interchange between the (002) and the (200) peaks was obtained from a simple ratio of their integrated peak intensities I002/I200. Figure 63B shows the values for I002/I200 as a function of time and as a function of angle to the direction of the applied electric field. A tran sition period of < 0.05 s is obser ved for the measured intensity interchanges between the (002)type peaks, as illustrated by the sh aded portion in Figure 63B. I002/I200 is observed to be generally stable beyond this time period, as is illustrate d in the unshaded portions in Figure 63B. The time dependence of the (111) lattice spacings d111 parallel to the electric field is shown in Figure 63C. d111 shows a time depende nce similar to one observed for I002/I200. In general, a transition period of < 0.05 s is also observed for the 2 positions of all other (hkl ) peaks analyzed for this mate rial. The time dependence of the diffraction patterns described here for the sample PLZT5248 is also observed for other materials investigated in the work, under cyclic squarewave electric fields of frequency 1 Hz. From the above discussion, it is evident that the structural changes in the investigated materials is quite rapid and shows a transition period of less than 0.05 s under the conditions of the applied electric fields. For statistical advantag es in subsequent analyses, two representative diffraction patterns corresponding to the positive and the negative st ates of an applied square wave electric field were obtained by summing diffrac tion data within each portion of the electric field. For each state of applied electric field, the representative diffr action pattern was obtained PAGE 107 107 by a summation of the diffraction pa tterns during the last 0.045 s of the corresponding section of the electric field waveform, af ter the region of transition. 6.2.5 Quantification of Structural Changes 6.2.5.1 Non180 domain switching In earlier works, the volume fraction of non180 domains has been shown to be a function of the intensities of characteristic di ffraction peaks [30,6369]. The relation between the intensities of the (002)type diffraction peaks and the volume fraction of the 90 domains in tetragonal ceramics were introduced in Chapter 5. The relation between the intensities of certain characteristic diffraction peaks and volume fraction of non180 domains in tetragonal and rhombohedral perovskites are reintroduced here for the benef it of the reader. In perovskite ferroelectric materials with a tetragonal crystal structure, the volume fraction of the 002 domains along a partic ular specimen dire ction is given by 2+ =200 200 002 002 002 002 002 'I I I I I I (61) where hklI is the integrated area of the ( hkl ) diffraction peak for a give n sample with a preferred orientation of 90 domains and hklI is the integrated area of the same peak for a sample with no preferred orientation of 90 domains. The volum e fraction of the 111 domains along a particular specimen direction for a rhombohedral perovskite ferroelectric is obta ined by the following modification of Eq. 61, 3+ =111 111 111 111 111 111 111 'I I I I I I (62) PAGE 108 108 where lattice spacings for ()111 planes are larger than that of ()111 planes. In Eqs. 61 and 62, the numerical factors in the denom inator account for the multiplici ties of the particular (hkl) lattice planes. For a random distribution of non180 domains, 002 is equal to 1/3 and 111 is equal to 1/4, corresponding to equal distribution of possi ble ferroelastic domain variants. Therefore, the volume fraction of the domains reoriented by non180 along a specimen direction, under a certain condition, is given by 3 1 =002 002 (63) for a tetragonal perovskite, and 4 1 =111 111 (64) for a rhombohedral perovskite. Under the application of cyclic electric fields of subcoercive amplit udes, switching of non180 domains results in dynamic changes in 002 and 111. The term non180 domain switching is not intended here to necessarily imply crystallo graphic reorientation of entire domains. Instead, in the present context, non180 domain switching refers to a change in the crystallographic orientation of certain material vol ume fraction as a result of both reversible and ir reversible non180 domain wall displacements. The change in the volume fractions of the non180 domains under bipolar squarew ave electric fields can be given by, += (65) where + and are values of for maximum positive and minimum negative electric fields, respectively, within a cyclic squarewave electric field. PAGE 109 109 The use of a 2D detector allows calculation of and at different angles with respect to the direction of the applied electric field. Th is introduces an extra dimension for characterizing electricfieldinduced non180 domain switching in the material, in addition to the amplitude of the applied electric field. Fo r example, Figure 64 shows 002 as a function of field amplitude as well as orientation with respect to the direc tion of applied field, for an unpoled Ladoped tetragonal PZT ceramic under the ap plication of static electric fields. The measurements for non180 domain switching provided in this chap ter are therefore complementary to the measurements provided in Chapter 5. The advant age of this added dimension of orientation towards complete characterization of material beha vior will be apparent in the later sections of this chapter. In all the ceramic samples, were calculated for orient ations corresponding to 0 < < 90. The sample symmetry for a pole d polycrystalline ceramic is given by mm, with the fold rotation axis parallel to the poli ng direction. In all the experiment s, the direction of the applied electric field was kept parallel to the poling direction. It is therefore expected that during cyclic electrical loading of the poled ceramics, the structural state for the following quadrants are symmetric with respect to the quadrant 0 < < 90: 90 < < 180, 180 < < 270, and 270 < < 360. Therefore, were calculated only for orient ations corresponding to 0 < < 90. 6.2.5.2 Electricfieldindu ced lattice strains Following Braggs law for diffraction, the change in the (hkl) plane spacing can be determined from the change in the 2 position of the (hkl) diffraction peak. The peaktopeak electricfieldinduced lattice strains were then calculated following +=hkl hklhkl hkld dd, (66) PAGE 110 110 where +hkld and hkld are the (hkl) lattice spacings during the a pplication of maximum positive and minimum negative electric fields, respectively, w ithin a cyclic squarewav e electric field. The (hkl) lattice plane spacings were determin ed from the best fits of the (hkl) diffraction peaks measured during the application of electric fields. Piezoelectric properties are more often represented in terms piezoelectric coefficients rather than electricfieldinduced strains. In or der to make direct comparison with macroscopic piezoelectric coefficients, the measured lattic e strains were subsequently transformed to equivalent lattice strain coefficients dhkl, as given by 02 = E dhkl hkl, (67) for an applied electric field of amplitude E0. In Eq. 67, the letter d in dhkl is chosen because it represen ts an electricfieldinduced strain coefficient. The commonly accepted variable in the field of piezoelectric materials for such a coefficient is d. Unfortunately, the same lette r is used in the field of crystallography to represent interplanar lattice spacing, e.g. d in dhkl. 6.3 Results and Discussions 6.3.1 Non180 Domain Switching Figure 65A shows the (002)type diffracted intensities of the sample PLZT5248, corresponding to the positive and the negative segments of an applied square wave electric field of amplitude V/mm and frequency 1 Hz, para llel to the direction of the applied electric field. As described in Section 6.2.5.1 the obser ved intensity interchange between the (002) and (200) diffraction peaks in Figure 65A correlate s to the amount of 90 domain switching in the material induced parallel to the applied electric field direction. For 0 < < 90, 002 were PAGE 111 111 calculated from the observed intensity intercha nges between the (002)type diffraction peaks using Eqs. 63 and 65. Figure 65B shows the results of thes e calculations as a function of applied electric field amplitude for sample PLZT 5248. It can be observed from Figure 65B that for all amplitudes of the applied electric field, 002 decreases with increasing and becomes negative for > 50. In PLZT6040, the intensity interchanges be tween the (111)type diffraction peaks are used to calculate the amount of 71/109 doma in switching, following Eqs. 64 and 65. In PLZT4060, the amount of 90 domain switching under cyclic electric fields is calculated from the intensity interchanges between the (002)type diffraction peak s following Eqs. 63 and 65. The angular and amplitude dependence of 111 for PLZT6040 and of002 for PLZT4060 are shown in Figure 65C and Figure 65D, respectively. For PLZT6040, 111 beyond > 70 could not be calculated due to signif icantly decreased intensity of the ()111 diffraction peak for these angles. Similar to the observed trends for PLZT5248, is observed to decrease with for both PLZT4060 and PLZT6040 and becomes negative for > 50 (PLZT4060) and for > 40 (PLZT6040). Figures 66A and 66B show the angular dependence of 002 at different amplitudes of applied bipolar electric fields for undoped a nd Fedoped varieties of MPBadjacent tetragonal compositions. For both the samples PUZT5248 a nd PFZT4357, it is again observed that 002 show an inverse dependence on and 002 becomes negative for > 50. For Figures 64 and 65, the erro rs in the reported values for were calculated from the estimated standard deviations in the measured intensities of the diffraction patterns. The error PAGE 112 112 bars for are of the order of the size of the data points for = 0 and about three times the size of the data points for = 90, with a gradual transition through 0 < < 90. The angular dependence of 002 indicates the energetic pr eference for non180 domain switching for different crystallographi c orientations of grains with respect to the applied electric field. Non180 switching of domains is most fa vored when the polar ax is after switching is parallel to the direction of th e applied electric field. For > 50 in tetragonal ceramics and for > 40 in rhombohedral ceramics, a reverse tr end in non180 domain switching is observed which indicates a decreasing preference for the polar axis for these orientations. This is similar with observed trends for preferred orientation of ferroelastic domains in poled piezoelectric ceramics [30,67,68,69]. In addition to angular dependence, it can al so be inferred from the above observations that the amount of non180 doma in switching in the different cer amic samples is a function of the amplitude of the applied cyclic electric field. In Ladoped PZT ceramics, a monotonic increase in can be observed with increase in the am plitude of the applied electric field. Among the compositions studied, PLZT6040 shows the highest amount of non180 domain switching, with a value of ~0.09 for 111 parallel to the direction of an applied cyclic electric field of amplitude V/mm. Under the same conditions of applied electric fields, the measured value for 002 parallel to the electric field di rection is ~0.08 for PLZT5248. For PLZT4060,002 is measured to be only ~0.015 parallel to the direction of the applied cyclic electric field of amplitude 1350 V/mm, indicating less extent of non180 domain switching as compared to PLZT6040 and PLZT5248. The observed values of 002 for the samples PUZT5248 and PFZT4753 are less dependent on the amplitude of the applied electric field as compared to the observed behavior for PAGE 113 113 PLZT5248. Further, the amount of 90 doma in switching in samples PUZT5248 and PFZT4753 are at least three times lower than what is observed for PLZT5248. 6.3.2 Calculation of Strain from Non180 Domain Switching Non180 reorientations of ferro elastic domains within differe nt grains should contribute to the macroscopic material strain as a resu lt of anisotropic dimensional changes based on spontaneous crystallographi c strains. The spontaneous crystallographic strain tS (or rS) is given by 00 0000=l ll tSd dd (68) in tetragonal crystals, and kkk kkk kkk rS d dd = (69) in rhombohedral crystals, where hkld is the spacing of the (hkl ) lattice planes. The contribution to the electricfieldinduced macroscopic strain from non180 domain switching within a grain will depend on the orientation of the grain with respect to the applied electric field direction. For example, this is i llustrated in Figure 67 for a grain with 90 domains which has some of its domains aligned with their 001 poles oriented at angle with respect to the electric field direction. Under the application of electric field, displacement of domain walls will cause an increase in the volume fraction of the domains which are favorably oriented with respect to the applied field at the expense of the less favorably oriented domains. This will cause a change in the dimension of the grain, an exaggerated illustration of wh ich is shown in Figure 67. The projection of this distorti on parallel to the direction of th e electric field can be obtained by a secondorder tensorial transformation [95]. PAGE 114 114 The macroscopic strain due to all non180 domainwall induced distortions within a polycrystalline sample can be calculated by ()()[] d sin cos mS / non2 0 2 180=, (610) where m=3 for tetragonal ceramics and m=4 for rhombohedral ceramics. The factor m is the change in the multiple of random di stribution (MRD) of the ferroelastic hkl crystallographic pole, along the sample direction with respect to the applied electric field. The geometrical factor of cos2 in the integrand arises from a tensorial transformation of the domainswitching strain along the direction of the applied electric field. The factor sin describes the transformation from an elemental volume in the orientation space to an effective volume fraction within the sample [69,95]. Eq. 611 is applied to calculate the amount of strain due to non180 domain switching in the different samples at various amplitudes of applied electric field. The coefficient of electricfieldinduced strain due to non180 domain switching dnon180 is subsequently obtained following 0 1801802 = E dnon non, (611) for an applied electric field of amplitude E0. The dependence of dnon180 on the amplitude of applied electric fields is an alyzed to understand the nonlinear field dependence of the non180 domain switching strains. 6.3.3 Field Dependence of Non180 Domain Switching Strains Figure 68A shows the non180 domai n switching strain coefficient, dnon180, for PZT ceramics with different crystallographi c phases: PLZT6040 (rhombohedral), PLZT5248 (tetragonal with composition clos e to MPB with a c/a ratio equal to 1.0209) and PLZT4060 PAGE 115 115 (tetragonal with a larger c/a ratio equal to 1.029). The larges t strain contributions from non180 domain switching are observ ed for tetragonal ceramics close to the MPB, with dnon180 ranging from ~185 pm/V to ~352 pm/V, for applied electric field amplitudes of 00 < E0 < V/mm. Significant strain contributions due to non180 domain switching are also observed for PLZT6040 ceramics with a rhombohed ral phase, with a maximum dnon180 of ~160 pm/V for an applied electric field of amp litude V/mm. Due to the ex tremely small amount of non180 domain switching in the highly te tragonal ceramics with a large c/ a ratio, significant conclusions from diffraction experiments on these ceramics could only be determined at electric field amplitudes greater than V/mm. The maximum dnon180 measured for PLZT4060 is ~66 pm/V, for applied electric field of amplitude 350 V/mm, which is significantly lower than the non180 domain switching strains measured for rhombohedral and MPBadjacent tetragonal PZT ceramics. The effect of different dopants on the am ount of non180 domain switching strains can be observed from Figure 68B. Undoped and Fedoped tetragonal PZT ceramics, with compositions close to MPB, show comparable non180 domain switching strains under cyclic electric fields. In comparison, Ladoping for MPBadjacent tetragonal compositions lead to a more than three fold increase in the amount of non180 domain sw itching strains. For example, under cyclic electric field of amplitude V/mm, the values for dnon180 are ~339 pm/V for PLZT5248, ~71 pm/V for PUZT5248 and ~57 pm/V for PFZT4753. This is not unexpected, as Ladoping is known to enhance displacement of non180 domain walls in PZT ceramics [6,7]. The coefficient of linear increase in dnon180 with respect to the amplitude of the applied electric field E0 is determined by fitting the calculated values of dnon180 following ( ) 01800 1800180+= E dEdnon non non, (612) PAGE 116 116 where 0 180nond is the intercept at zero amplitude of app lied electric field. A nonzero value of non180 implies a nonlinear dependence of the non180 domain switching strains on the amplitude of the applied electric fields. The electricfield induced non180 domain switching strains in Ladoped PZT ceramics of rhombohedral and MPBad jacent tetragonal compositions show nearly the same amount of nonlinearity with non180 ~0.2 (pm/V)(V/mm)1. Further, for MPBadjacent tetragonal PZT ceramics, Ladoping leads to a more than two fold increase in non180 as compared to undoped compositions, leading to gr eater nonlinearity in strain due to non180 domain switching. A linear relati onship is not observed between 0 180 nond and E0 for PLZT4060 and PFZT4753 in both cases the displacemen t of non180 domain walls are restricted, possibly due to internal stresses as a result of clamping due to large spontaneous strains (PLZT4060) or due to pinning of domain walls due to oxygen vacancies (PFZT4753), as mentioned in Chapter 4. 6.3.4 ElectricFieldInduced Lattice Strains The different ( hkl ) lattice strain coefficients dhkl for sample PLZT5248, parallel to the direction of applied electr ic field of amplitude E0, are shown in Figure 69A. It can be observed that the lattice strain coefficien ts increase with the amplitude of the applied electric field. The total electricfieldinduced lattice strain is gi ven by the product of dhkl(E0) and E0. Consequently, if dhkl(E0) increases with E0, hkl lattice strain is no l onger a linear function of E0. The coefficient of linear increase in dhkl with respect to the electric field amplitude E0 is obtained through the following equation ( ) 0 0 0+= E dEdhkl hkl hkl, (613) where 0 hkld is the intercept at zero amplitude of app lied electric field. The degree of nonlinearity for the electricfieldinduced lattice st rains is provided by the parameter hkl. PAGE 117 117 For PLZT5248, the largest lattice strains ar e observed for the (111) crystallographic planes. As described in Section 1.2.1, in te tragonal PZT ceramics, the largest piezoelectric constants can be expected parallel to the polar axis of [001]. In contrast to general expectation, no significant amount of strain can be measured for the (00 h ) crystallographic planes, the normal to which is parallel to the polar axis. It is also noted that the largest non linearity is observed for the (111) lattice strains, with hkl ~ 0.4 (pm/V)(V/mm)1. For samples PLZT6040 (rhombohedral) and PLZT4060 (tetragonal), some of the ( hkl ) lattice strains cannot be quan tified since changes in the ( hkl ) peak positions were within the errors of measurement as obtained from the fitting of the hkl diffraction peaks. However, it is observed that the largest (hkl) la ttice strains are observed for th e (002) crystallographic planes for PLZT6040 and the (111) planes for PLZT4060. These observations are in agreement with previously reported experimental results for directions of maximum lattice strains in rhombohedral and tetragonal PZT ceramics [ 66,30,70,73,74]. The maximum lattice strain coefficients for the different samples are s hown in Figure 69B. The maximum lattice strain coefficient for PLZT5248 is greater than that observed in PLZT6040 and PLZT4060. In addition, d111 in PLZT5248 and d002 in PLZT6040 exhibit larger nonlinearity than d002 in PLZT4060. The (111) lattice strain coefficients for undoped, Ladoped and Fedoped tetragonal PZT ceramics with compositions close to the MPB are shown in Figure 69C. d111 in Ladoped ceramics are more than three times than d111 in undoped and Fedoped ceramics. Moreover, d111 in Ladoped ceramics show a la rger nonlinearity as compared to the undoped and Fedoped ceramics. The observed nonlinear ( hkl ) lattice strain coefficients are in contrast with the conventional assumption of linear piezoelectric lattice strains for cer amics. By comparing PAGE 118 118 Figures 67 and 68, it can be noted that field dependence of th e largest lattice strains closely corresponds to the field dependence of the non180 domain switching strains for the different samples. This could imply that non180 domain switching affects the am ount of electricfieldinduced lattice strains in the material. The impli cations for this observati on are further discussed in Chapter 7. 6.4 Conclusions Using highenergy timeresolved Xray diffrac tion, structural changes in PZT ceramics were evaluated under the application of cyclic el ectric fields. Non180 domain switching within a material was measured at different angles with respect to the direction of the applied electric field. The total amount of strain in a sample due to the displacement of non180 domain walls was evaluated through integration of the amount of non180 domain switching over the complete orientation space. The amount of non180 domain switching strain in PZT ceramics is found to be dependent on the crystal structure and composition of the material. The largest non180 domain switching strains are measured for Ladoped tetragonal ceramics with a composition close to the MPB; a maximum non180 domain switching strain coefficient of ~352 pm/V was measured for an applied cyclic electric field of V/mm. In addition, for MPBadjacent tetragonal PZT ceramics, Ladopi ng is observed to increase the amount of non180 domain switching strains by more than th ree times over the corresponding strains in undoped and Fedoped compositions. This is observed from their respective values of non180 domain switching strain coefficients, such as ~339 pm/V for PLZT5248 as compared to ~71pm/V for PUZT5248 and ~57 pm/V for PFZT 4753. Evaluation of lattice strains under the application of cyclic electric fields shows that the maximum ( hkl ) lattice strain in a sample can be correlated to the amount of stra in due to non180 domain switching. PAGE 119 119 An important observation from the current i nvestigation is that both the non180 domain switching strain coefficients and the lattice strain coefficients can be functions of the amplitude of the applied electric field. PAGE 120 120 Figure 61. Details about experiment al setup and timing electronics for in situ diffraction experiments. A) Schematic of timeresolv ed diffraction experime ntal setup showing an example large area diffraction image from a poled tetragonal PZT and the timing electronics. B) timing diagram for electric field waveform and detector frames. C) Individual cakes of the detector represent diffracti on patterns with scattering vectors at various angles to the applied elec tric field. This is outlined when observing the integrated patterns of a poled ceramic. A B C PAGE 121 121 A B C Figure 62. Specific diffr action peaks for the different samples used in the current experiments. A) (002)type, B) (111)type and C) (022)type diffrac tion peaks. The diffraction patterns show a tetragonal phase for sample compositions with Zr/Ti 52/48 and a rhombohedral phase for Zr/T i ratio equal to 60/40. 3.94.04.1 Intensity (A.U.)2 (degrees) PLZT5248 PLZT6040 PLZT4060 PUZT5248 PFZT4753 3.4 3.5 Intensity (A.U.)2 (degrees) PLZT5248 PLZT6040 PLZT4060 PUZT5248 PFZT4753 5.55.65.7 Intensity (A.U.)2 (degrees) PLZT5248 PLZT6040 PLZT4060 PUZT5248 PFZT4753 PAGE 122 122 0.0 0.5 1.0 3.87 3.97 4.07 T i m e s e c o nds2 d e g r e e s ) +E E E Diffracted Intensity A Figure 63. Timeresolved structural changes under cyclic electric fields. A) Timeresolved (002) and (200) diffraction peaks for the sample PLZT5248 during the application of a cyclic square wave electric field of amplitude V/mm and frequency 1 Hz, along the direction of the applied electric field. Intensity inte rchange between the (002) and (200) diffraction peaks occur fo r positive and negative segments of the applied electric field cycle. B) The time dependence of the ratio of the integrated intensities of the (002) and (200) diffraction peaks, plotted as I002/I200. C) Timedependent (111) lattice spacings d111, parallel to the electric field direction. For both I002/I200 and d111, a transition period of <0.05 s is observed during a change in the direction of the applied electric field. PAGE 123 123 0.00.20.40.60.81.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0 10 20 30 40 50 60 70 80 90I002/I200Time, seconds +E E E Angle to Electric Field (degrees): B Figure 63. Continued PAGE 124 124 0.00.20.40.60.81.0 2.3540 2.3545 2.3550 2.3555 2.3560 2.3565 2.3570 2.3575 2.3580 d111Time, seconds +E E E C Figure 63. Continued PAGE 125 125 Figure 64. 002 as a function of field amplitude as well as orientation with respect to the direction of applied field, for an unpoled Ladoped tetragonal PZT ceramic under the application of static electric fields. Th e measured and fitted (002)type diffraction peaks corresponding to the particular values of 002 (marked by circles and indicated by arrows) are shown in the bottom section of the figure. For the fitted diffraction patterns, the deconvoluted 200 and 002 p eaks are shown in black solid lines. 50x103 40 30 20 10 0Intensity 2.10 2.05 2.00 1.95 dspacin g s, An g stroms 50x103 40 30 20 10 0Intensity 2.10 2.05 2.00 1.95 dspacin g s, An g stroms 0102030405060708090 0.08 0.04 0.00 0.04 0.08 Angle to electric field Increasing amplitude of static electric fields 002 002 200 200 PAGE 126 126 3.853.903.954.004.05 Positive NegativeDiffracted Intensity2 (degrees) A 020406080100 0.04 0.02 0.00 0.02 0.04 0.06 0.08 100 200 300 400 500 600 700 800002Angle to Electric Field Electric field (V/mm): B Figure 65. Change in volume fraction of non180 domains in Ladoped PZT ceramics under application of cyclic electric fields. A) The (002) and ( 200) diffracted intensities for the sample PLZT5248, corresponding to the pos itive and the negative segments of an applied cyclic square wave electric fi eld of amplitude V/mm and frequency 1 Hz, measured parallel to the direction of the applied electric field. The values of PAGE 127 127 for different angles with respec t to the electric field direct ion, are calculated from the measured intensity interchanges between char acteristic ferroelastic peaks at different azimuthal angles For clarity, data for amplitudes in steps of 100 V/mm are only shown, though measurements for this samp le were taken at steps of 50 V/mm. B)002 for sample PLZT5248. C)111 for sample PLZT6040. D)002 for sample PLZT4060. Error bars are not shown on the plot for clarity. Errors for are of the order of the size of the data points for = 0 and about three times the size of the data points for = 90, with a gradual transition through 0 < < 90. PAGE 128 128 020406080100 0.02 0.00 0.02 0.04 0.06 0.08 0.10 800 700 600 500111Angle to Electric Field Electric field (V/mm): C 020406080100 0.016 0.012 0.008 0.004 0.000 0.004 0.008 0.012 0.016 0.020 1350 1250 1150 1050 950002Angle to Electric Field Electric field (V/mm): D Figure 65. Continued PAGE 129 129 020406080100 0.008 0.004 0.000 0.004 0.008 0.012 0.016 850 750 650 550002Angle to Electric Field Electric field (V/mm): A 020406080100 0.01 0.00 0.01 0.02 0.03 600 700 800 900 1000 1100 1200 1300 1400 1500002Angle to Electric FieldElectric field (V/mm): B Figure 66. 002 for different amplitudes of applied cyclic square wave electric fields, for A) sample PUZT5248, and B) sample PFZT4753. Error bars are not shown on the plot for clarity. Errors for are of the order of the si ze of the data points for = 0 and about three times the size of the data points for = 90, with a gradual transition through 0 < < 90. PAGE 130 130 Figure 67. Illustration of st rain due to non180 switching of ferroelastic domains in a single grain. The domains with their long axes oriented at angle with respect to the direction of the applied el ectric field expand, causing a change in the overall dimensions of the grain. Electric Field PAGE 131 131 2506501050 0 100 200 300 400 PLZT5248 PLZT6040 PLZT4060 dnon180(E), pm/VElectric Field, V/mmdnon180(E0) = d0 non180 + non180E0 PLZT5248: d0 non180 = 170 9non180 = 0.22 0.02 PLZT6040: d0 non180 = 9 29non180 = 0.2 0.04 A 25065010501450 100 200 300 PLZT5248 PUZT5248 PFZT4753 dnon180(E), pm/VElectric Field, V/mmdnon180(E0) = d0 non180 + non180E0 PLZT5248: d0 non180 = 170 9non180 = 0.22 0.02 PUZT5248: d0 non180 = 4 12non180 = 0.09 0.02 B Figure 68. The coefficient of stra in due to non180 domain switching dnon180 for PZT ceramics. A) dnon180 for PZT ceramics with different crystallographic phases PLZT6040 (rhombohedral), PLZT5248 (tetragonal with composition close to MPB) and PLZT4060 (tetragonal with a large c/a ratio). B) dnon180 for PZT ceramics with different type of dopants for MPBadjacent tetragonal compositions PLZT5248 (Ladoped), PUZT5248 (undoped), and PFZT4753 (Fedoped). For compositions where a PAGE 132 132 linear relationship is valid, the relationship between the lattice strain coefficients and the applied electric field am plitudes are also shown. PAGE 133 133 400600800 0 250 500 750 111 022 211 112 002dhkl(E), pm/VElectric Field, V/mm dhkl(E) = d0 hkl + hklE111: d0 hkl = 457 31 hkl = 0.39 0.04 022: d0 hkl = 355 17 hkl = 0.30 0.03 211: d0 hkl = 345 20 hkl = 0.21 0.03 112: d0 hkl = 117 23 hkl = 0.16 0.04 A Figure 69. Electricfieldinduced lattice strain coefficients in PZT ceramics. A) The different (hkl) lattice strain coefficients dhkl, for sample PLZT5248, parallel to the direction of applied electric fields of amplitude E0, B) the maximum lattice strain coefficients for Ladoped PZT ceramics with different crys tal structures as a function of the amplitude of the applied cyclic electric fields PLZT5248 (tetragonal with composition near MPB), PLZT6040 (rhombohe dral), and PLZT4060 (tetragonal with a large c/a ratio). The linear relationships between the lattice strain coefficients and the applied electric field amplitudes are al so shown. C) The maximum lattice strain coefficients for Ladoped (PLZT5248) undoped (PUZT5248), and Fedoped (PFZT4753) PZT ceramics of MPBadjacent tetragonal compositions are compared for different amplitudes of app lied cyclic electric fields. PAGE 134 134 4008001200 0 200 400 600 800 d111(PLZT5248) d002(PLZT6040) d111(PLZT4060) dhkl(E), pm/VElectric Field, V/mm dhkl(E0) = d0 hkl + hklE0PLZT5248: d0 111 = 457 31111= 0.39 0.04 PLZT6040: d0 002 = 281 21002= 0.32 0.03 B 40080012001600 200 300 400 500 600 700 800 PLZT5248 PUZT5248 PFZT4753 d111(E), pm/VElectric Field, V/mm C Figure 69. Continued PAGE 135 135 CHAPTER 7 NONLINEAR CONTRIBUTIONS FROM STRUCTURAL MECHANISMS UNDER CYCLIC ELECTRIC FIELDS In this chapter, the m acroscopic piezoelect ric nonlinearities presen ted in Chapter 4 are reconciled with the structural changes presented in Chapters 5 and 6. The objective is to develop a more fundamental understanding of piezoelectr ic nonlinearity with resp ect to the different structural mechanisms under cyc lic electrical loading. The effects of non180 domain switching and lattice strains on macroscopic piezoelectric nonlinearity are separately discu ssed in Section 7.1 and Section 7. 2, respectively. The cumulative effects of non180 domain switch ing and lattice strains on macr oscopic piezoelectric behavior are discussed in Section 7.3. 7.1 Effect of Non180 Domain Switching Strains Table 71 lists the values for d and non180 for the different samples studied. The coefficient of linear increase in the macroscopic piezoelectric coefficient d33 with respect to the applied electric field amplitude is denoted by d. The corresponding coe fficient for non180 domain switching strains is denoted by non180. As explained in Chapter 4, dE0 is the electricfieldinduced strain from irreversible extrinsic mechanisms. From the listed values of d for Ladoped ceramics, it can therefore be concluded that the contribution from extrinsic mechanisms is minor in tetragonal compositions with a large c/a ratio and is significantly larger in rhombohe dral and MPBadjacent tetragonal compositions, among the investigated samples. This is consistent w ith the non180 domain switching strain coefficients for the three diffe rent compositions presen ted in Figure 69A. In addition for MPBadjacent tetragonal composition, a much higher value of d for Ladoped ceramics is noted as compared to undoped ceramics. This is consistent with the non180 domain switching strain coefficients for these cer amics (PLZT5248 and PUZT5248), measured using PAGE 136 136 timeresolved Xray diffraction (Figure 69B). Th ough the irreversible extr insic contributions for different crystallographic phases and dopants are consistent on many counts with the directly measured non180 domain switching strains, a fe w disagreements could be noted as described below. Macroscopic piezoelectric measurements show that the irreversible contributions from extrinsic mechanisms are similar for rhombohedral and MPBadjacent tetragonal PZT ceramics (Figure 44A). However, from in situ diffraction experiments it ca n be observed that MPBadjacent tetragonal PZT ceramics ha ve nearly twice as much strain contribution from non180 domain switching than what is observed for rh ombohedral PZT ceramics (Figure 68A). In order to reconcile these observations, it is important to understand that in situ diffraction provides information about the total amount of non180 do main switching in the ma terial while only the irreversible extrinsic contributions are dete rmined from the fielddependent macroscopic piezoelectric coefficients. Therefore it is possi ble that though the irreve rsible component of non180 domain switching strains are similar for MPB and rhombohedral compositions, a greater amount of total (reversible a nd irreversible) non180 doma in switching occurs in MPB compositions. This is consistent with a nonze ro intercept for non180 domain switching strains at zero amplitude of applied electric field fo r Ladoped MPB composition, as can be observed from Figure 68A. Additionally, it can be observed from Table 71 that d is greater than non180 for the different compositions of PZT ceram ics. This implies that the macroscopic electricfieldinduced strains exhibit larger nonlinearity as compared to the non180 domain switching strains. The relative nonlinear contributions from non180 domain switching st rains are evaluated using a procedure as described below. PAGE 137 137 For samples PLZT6040, PLZT5248 and PUZT5248, the relative nonlinear contributions from non180 domain switching strains are calculated as non180E0/d33(E0). This is done since these samples exhibit Rayleightype behavior fo r macroscopic electricfi eldinduced strains and a linear dependence of dnon180 with respect to E0. In the above formulation, d33(E0) is the longitudinal piezoelectric coefficient measured for applied electric field of amplitude E0. For sample PLZT4060, such an analysis was not undertaken since the non180 domain switching strains could only be measured for electric fiel ds outside the regime of linear dependence of d33(E0) on E0. In Figure 71, the nonlinear contri butions from non180 domain switching are plotted along with the nonlinea r extrinsic contributions evaluated in Chapter 4. For all the samples, it is observed that non180 domain switc hing strains only partially constitute nonlinear extrinsic contributions. This implies that factors other than non180 domain switching contribute to the macroscopi c piezoelectric nonlinearity, which can include 180 domain switching [81] or nonlinear lattice strains [30]. Since 180 domain switching does not result in characteristic changes in diffracted intensitie s, the amount of 180 domain switching in the material under subcoercive cyclic electric fields cannot be dir ectly determined from diffraction experiments. However, it is possible to measure lattice strains in the ma terial under subcoercive electric fields from the shifts in the ( hkl) diffraction peaks, as presen ted in Chapters 5 and 6. The effect of nonlinear contributions from lattice strains on macroscopic piezoelectric nonlinearity will be further explored in Section 7.3. For sample PFZT4753, no linear relation was observed between th e non180 domain switching strain dnon180 and applied electric field amplitude E0. Therefore, in this case the relative nonlinear contributions from non180 domain switching st rains were calculated as dnon180/d33(E0), where d33(E0) is the longitudinal piezoelectri c coefficient measured at the PAGE 138 138 amplitude E0 of applied electric field. dnon180 is given by [dnon180(E0) dnon180,min], where dnon180(E0) is the non180 domain sw itching strain coefficient for applied electric field E0 and dnon180,min is the non180 domain switching strain for the minimum amplitude of applied electric field. dnon180/d33(E0) is compared with d33/d33(E0) as calculated for this sample in Chapter 4. For both macroscopic and non180 dom ain switching strains, a sharp increase in nonlinear contributions can be obs erved for applied electric field amplitude of ~750 V/mm. This implies a threshold field of ~750 V/mm for non180 domain wall displacement in this material. It is also observed that th e nonlinear contributions from non180 domain switching strains compare very well with overall extrinsic contributions. This is in contrast with the behavior observed for Ladoped and undoped PZT ceramics. A possible explanation for this observation will be provided in Section 7.3. 7.2 Effect of Lattice Strains The different hkl lattice strain coefficients for samp le PLZT5248 were presented earlier in Section 6.3.3. For other samples, the calculated lattice strain coefficients for most of the hkl planes were zero within the errors of measurement and therefore further analysis of lattice strains in these samples were not carried out. A rigorous estimate of macroscopic electricfieldinduced strain from the different hkl lattice strains measured using di ffraction will require formulation and execution of appropriate micromechanical models. However, good estimates of macroscopic equivalent strain can also be made from physically realistic weighting of strains obtained from multiple single peak diffraction measurements. A method for calcu lating macroscopic equivalent strain from weighted average of different hkl strains under the application of mechanical stress has been presented earlier [96]. A similar approach is adopted here for calculating the equivalent PAGE 139 139 macroscopic electricfieldinduced strain from the different hkl strains measured under the application of cyclic electric fields. The total longitudinal strain due to the lattic e strains of all the grains within a ceramic sample can be calculated from a volumeweighted average of the different ( hkl ) lattice strains measured parallel to the applied el ectric field direction, that is, =33 hklhkl hklhkl hklhklhkl hkl,mT/ mT (71) where mhkl is the planar multiplici ty for the particular ( hkl ) crystallographic planes and Thkl is the texture factor given by th e MRD for the particular hkl crystallographic pole [73,96]. The values for Thkl are determined from the intensities of the ( hkl ) diffraction peaks in the unpoled and poled state measured parallel to the a pplied electric field direction, following the method described in reference [69]. The values of Thkl and mhkl for certain hkl crystallographic poles are listed in Table 72. Eq. 71 is equivalent to a Reuss approximation for calculating total strain in a composite of multiple constituents under isostress condition. Eq. 71 can be obtaine d based on the following assumptions: (1) the distribution of the grains with different hkl orientations within the diffracting volume is representative of the entire sample and the strains are cumulative for the different grains along a ve rtical section for the sa mple (a simplified schematic is provided in Figure 72) (2) each grain can contribute to only type of diffraction peak and that all grains contribute to a diffraction peak. Th e first assumption is not physically unrealistic if we consider uniform grains size (or distinct diffracting volumes) in the sample and the vertical section in Figure 72 to be uniform on aver age across the sample. The second assumption is not necessarily true and also is not strictly followed here since only certain hkl lattice strains are taken into account for calculation of equivale nt strains. However, despite the limitations of the second PAGE 140 140 assumption, such an approach has been shown to provide realistic estimates of macroscopic equivalent strains [96]. By diving both sides of Eq. 71 by 2E0, we obtain 33 33=hkl hkl hklhkl hkl,hklhkl hkl,mT/dmT d (72) where 02= E/ d hkl,d33 is the total effective lattice st rain coefficient calculated from different ( hkl ) lattice strain coefficients presented in Figure 69A. The subscript 33 represent the lattice strain coefficients for the longitudinal direction. The calculated values of hkl,d33 are plotted in Figure 73 as a function of amplitude of applied electric field E0. It can be observed fr om Figure 73 that hkl,d33 is a linear function of the amplitude of the applied electric field E0. The coefficient of linear increase in hkl,d33 with respect to the amplitude of the applied electric field E0 is determined by fitting the calculated values of hkl,d33 following ()033 0 330 33+=E dEdhkl, hkl, hkl,, (73) where 0 33 hkl,d is the intercept at zero amplitude of applied electric field. The parameter hkl,33provides an index of the nonlinear nature of the total equivalent lattice strains with respect to the amplitude of the applied electric fields. Therefore, usin g a similar approach as adopted earlier, the relative nonlinear contributions from to tal equivalent lattice strains can be calculated as hkl,33E0/d33(E0). The effect and origin of nonlinear contributions from lattice strains are discussed in the next section PAGE 141 141 7.3 Cumulative Nonlinear Contributions fr om Different Structural Mechanisms It is apparent from the a bove discussion that both non180 domain switching strains and lattice strains can contribute to macroscopic pi ezoelectric nonlinearity und er the application of cyclic electric fields. For sample PLZT5248, the cumulative nonlinear contributions from the two different mechanisms are plotted in Figure 74, which is slightly lower than the relative extrinsic nonlinear contributions calculated from macroscopic measurements. It should be noted here that the current cal culations for nonlinear contributions from the different structural mechanisms are based on ce rtain physical assumptions, as mentioned earlier. The Reuss approximation provides an upper limit for the calculated total e ffective lattice strain coefficient, which can cause an overestimate of the calculated nonlinea r contributions from lattice strains. The difference, between the tota l extrinsic nonlinear contributions (calculated from macroscopic d33 coefficients) and the cumulative nonlinear contributions from non180 domain switching and lattice strain s, can be explained due to othe r structural mechanisms such as 180 domain switching [81]. Nevertheless, based on the curre nt analysis it is concluded that the dominant contributions to macroscopic piezoelectric nonlinearity comes from non180 domain switching as well as nonlinear lattice strains. Nonlinear contributions from electricfieldi nduced lattice strains are in contrast with a constant value for piezoelectric lattice strain s derived from thermodynamic calculations [1725]. This suggests that electricfield induced lattice strains in polycrystalline ce ramics are possibly influenced by extrinsic mechanisms like domain switching. A possible mechanism through which non180 domain switching and lattice strains can be coupled in polycrystalline piezoelectrics has been suggested previously [30], which can be applied to interpre t nonlinear lattice strains observed in the present investigation. According to the proposed mechanism, strains in neighboring grains within a polycrystalline matrix are expected to be coupled through PAGE 142 142 intergranular strain. For a certain distribution of grain orientations within a ceramic, existence of neighboring grains with di fferent crystallographic orientations and conseq uently with different energetic preference for non180 domain switching can be expected. The grains which undergo a larger amount of non180 domain switching, u nder the application of electric fields, can impose constraints on the neighboring grains which are less favorably oriented for non180 domain switching. Such intergranular interactions can lead to elastic strains for grains with different orientations, in addi tion to the intrinsic piezoelectri c lattice strains, under applied electric fields. This mechanism is illustrated in Figure 75. A likely outcome for this proposed mechanism is a direct correlation between th e nonlinearity for non180 domain switching strains and lattice strains in piezoelectric ceramics with different compositions. This is observed for ceramics with different crysta llographic phases and dopant type s from the trends exhibited by non180 domain switching strains and lattice strains, as shown in Figures 66 and 68. Further evidence of correlation non180 domain switching and lattice st rains can be observed from measurements performed under the a pplication static electric fields as presented in Figures 53A and 53B. For the proposed mechanism, it is also likely that the lattice strains are correlated to non180 domain switching in the time domain under the application of app lied electric fields. This is observed to be the case from Figures 63B and 63C. It was earlier noted that for sample PFZT4753, nearly all th e nonlinear contributions come from non180 domain switching strains. In othe r words, no additional n onlinear contributions can be observed for this composition. However, according to the proposed mechanism in this section movement of domain walls are expected to create intergranular strain mismatch in polycrystalline ceramics which in turn can lead to nonlinear lattice strains. A possible explanation for such apparent c ontradiction can be pr ovided as follows. It is possible that for PAGE 143 143 limited displacements of domain walls, the intergranular strain mismatches are accommodated within the grain boundaries rather than causing elastic la ttice strains within the grains. However, additional verification of this hypothesis is neces sary for a clearer understanding of the issue. One effective way to verify the underlying mechanism could be in situ Transmission Electron Microscopy (TEM) or Scanning Transmission Electron Microsc opy (STEM), under application of electric fields. 7.4 Conclusions The nonlinear contributions to macroscopic el ectricfieldinduced strains from non180 domain switching in PZT ceramics were determined from in situ structural measurements under the application of cyclic electr ic fields. It was found that non180 domain switching strains only partially contribute to the macroscopic piezoel ectric nonlinearity for most compositions. For Fedoped PZT ceramics, it was observed that the no nlinear contributions from non180 domain switching strains almost exclus ively contribute to macroscopi c piezoelectric nonlinearity. The nonlinear contributions from the different ( hkl ) lattice strains were determined for sample PLZT5248. The cumulative nonlinear cont ributions from non180 domain switching and lattice strains show good ag reement with the tota l nonlinear contributi ons calculated from macroscopic piezoelectric measurements. A possible mechanism for the origin of non linear lattice strains from intergranular interactions in piezoelec tric ceramics under the application of electric fields is presented. PAGE 144 144 250500750 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Macroscopic Non180 domain switching Nonlinear ContributionElectric Field ( V/mm ) A 250500750 0.1 0.2 0.3 0.4 0.5 Macroscopic Non180 domain switching Nonlinear ContributionElectric Field ( V/mm ) B Figure 71. Nonlinear contribu tions to macroscopic electricfi eldinduced strains and nonlinear contributions from non180 domain switchi ng strains for samples A) PLZT6040, B) PLZT5248, C) PUZT5248, D) PFZT4753. PAGE 145 145 250500750 0.05 0.10 0.15 0.20 0.25 0.30 Macroscopic Non180 domain switching Nonlinear ContributionElectric Field ( V/mm ) C 2505007501000 0.00 0.05 0.10 Macroscopic Non180 domain switching Nonlinear ContributionElectric Field ( V/mm ) D Figure 71. Continued PAGE 146 146 Figure 72. A simplified schematic illustration of cumulative addition of strains with different hkl orientations towards macroscopic electricfieldinduced strain. (200) (002) (112) (111) E PAGE 147 147 400600800 250 300 350 400 d33,hkl (pm/V)Electric Field (V/mm) d33,hkl(E0) = d0 33,hkl + 33,hklE0 d0 33,hkl = 225.1 9.833,hkl = 0.17 0.02 Figure 73. The total effective lattice strain co efficient as a function of the amplitude of the applied cyclic electric field. PAGE 148 148 250500750 0.1 0.2 0.3 0.4 0.5 Macroscopic Non180 domain switching Non180 domain switching + Lattice strain Nonlinear ContributionElectric Field (V/mm) Non180 domain switching Lattice strain Figure 74. Nonlinear contribu tions to macroscopic electricfi eldinduced strain in sample PLZT5248 as a function of amplitude of applied electric field. The amount of nonlinear contribution from non180 domain switching is marked in black. The amount of nonlinear contribution from total e ffective lattice strain is marked in blue. The cumulative nonlinear contribution from th e two different mechanisms is in fair agreement with the relative nonlinear cont ributions calculated from macroscopic piezoelectric measurements. PAGE 149 149 Figure 75. Illustration of intergranular interactions leading to elastic strains for grains with different orientations. In the above figur e, domain wall displacements occur only in grain B under the application of electric field. However, grain A which shares a grain boundary with grain B is stretched du e to intergranular interactions. Electric Field A A B B PAGE 150 150 Table 71. d and non180 for the different samples studied. Sample d (pm/V)(V/mm)1 (corresponding to macroscopic strains) non180 (pm/V)(V/mm)1 (corresponding to non180 domain switching strains) PLZT6040 0.36 0.01 0.20 0.04 PLZT5248 0.43 0.01 0.22 0.02 PLZT4060 0.07 0.01 No linear relation PUZT5248 0.13 0.00 0.09 0.02 PFZT4753 No linear relation No linear relation PAGE 151 151 Table 72. The values of Thkl and mhkl for the different hkl crystallographic poles used to calculate the total effective lattice strain. Th e different hkl poles lis ted in the table are the ones for which significant la ttice strains were observed. hkl crystallographic pole Thkl (MRD) mhkl (Multiplicity factor) 111 1 1 002 1.82 0.33 200 0.59 0.67 112 1.46 0.33 211 0.77 0.67 022 1.59 0.33 220 0.70 0.67 PAGE 152 152 CHAPTER 8 SUMMARY AND FUTURE WORK 8.1 Summary Converse piezoelectric nonlinea rity in ferroelectric PZT ceramics is analyzed from a unique perspective of electricfi eldinduced structural changes measured using timeresolved Xray diffraction techniques. The abil ity to analyze the effects of unde rlying structural mechanisms on variations of piezoelectric properties in ferr oelectrics under subcoercive electric fields is demonstrated using complementary macroscopi c and timeresolved diffraction measurements. PZT ceramics of different crystallographic phas es and with different dopant additions were synthesized using a solid state synthesis route. Powders of the following nominal compositions were produced by the reactive calcination method: Pb(ZrxTi1x)O3 (PUZT, undoped) with x = 0.52, Pb11.5zLaz(ZrxTi1x)O3 (PLZT, Ladoped) with x = 0.60, 0.52, 0.40 and z = 0.02, and Pb1+0.5z[(ZrxTi1x)1zFez)]O3 (PFZT, Fedoped) with x = 0.47 and z = 0.02. Ceramics with 9597% of theoretical densities were obtained by sinter ing the processed powders. The sintered ceramics exhibited good ferroelectric and piezoelectric properties, characteristic of their compositions. Converse piezoelectric nonlin earities of the ceramic samples were investigated by measuring electricfieldinduced strains as a f unction of amplitude of applied cyclic electric fields. The amplitudes of the applied electric fields were limited to below the macroscopic coercive field for the different samples. It was observed that the measured d33 coefficients for Ladoped and undoped PZT ceramics follow Rayleigh law w ithin certain ranges of applied electric fields, in contrast to Fedoped PZT ceramic s. The amount of nonlinear contributions from irreversible extrinsic mechanisms were calculated by application of Rayleigh law for instances where it was valid. Nonlinear extrinsic contributio ns in Fedoped PZT ceramics were calculated from the fielddependence of their d33 coefficients. It was shown that the fraction of irreversible PAGE 153 153 extrinsic contributions is great er in the rhombohedral phase th an in the tetragonal phase. The largest irreversible extr insic contribution was observed to be about 55% of total electricfieldinduced strain for Ladoped rhom bohedral PZT ceramics, for an applied electric field of V/mm. In other words, ~298 pm/V of the macroscopic d33 value of 496 pm/V is contributed from extrinsic mechanisms. This can be interpreted as a result of greater possible switching of non180 domains in the rhombohedral phase. The eff ect of dopants on irreversible contributions was also determined for compositions close to the MPB. For MPBadjacent tetragonal PZT ceramics, a maximum of 45% contribution fr om irreversible extrinsic mechanisms could be estimated for Ladoped ceramics as compared to a maxi mum of 25% and 8% for undoped and Fedoped ceramics respectively. Similar variations in extrin sic contributions with di fferent crystallographic phases were reported earlier for direct piezoelectric response in PZT ceramics; under such loading, a maximum of 35% extrinsic contri bution was observed for Nbdoped rhombohedral PZT ceramics [97]. However, the explicit deriva tions of extrinsic contributions for converse piezoelectric effect in PZT ceramics, as a function of phase and dopants, is unique from this current work. The application of Rayleigh law for converse piezoelectric effect in PZT ceramics were further examined by determining the piezoel ectric losses and harmonics of converse piezoelectric strain. Hystereses of strainelectric field curves u nder subcoercive electric fields were described consistently using Rayleigh relations, for Ladoped and undoped PZT ceramics. From the calculated areas of the strainelectric field hysteresis loops, it was determined that the Rayleightype nonlinearity is th e dominating mechanism for piezoel ectric hysteresis in materials with linear variation of d33 with respect to electric field amplitude. However additional minor contributions to converse piezoelec tric hysteresis were also observ ed, possibly from viscoelastic PAGE 154 154 motion of domain walls in the material. Anal ysis of harmonic components further emphasized the dominant nature of Raylei gh component towards macrosc opic piezoelectric strain. As described in Chapter 1, the Preisach model can be applied to provide a physical explanation of the Rayleigh law. In simplified te rms, the Rayleigh behavior is obtained due to displacement of domain walls in a medium with randomly distributed en ergy barriers. Greater displacement of the domain walls is achieved across these energy ba rriers with increase in the amplitude of the applied electric field, th ereby leading to greater domain switching and consequently greater macroscopic strains [4]. A more fundamental description of piezoelect ric nonlinearity can th erefore be provided from direct measurements of electricfieldinduced struct ural changes including non180 domain switching, as presented in Chapters 5 and 6. The amount of non180 domain switching was evaluated from the intensity interchanges of characteristic ferroelectric diffraction peaks. The different hkl lattice strains were calculated from the shift in the 2 positions of the ( hkl ) diffraction peaks. Measurement of structural changes in tetragonal Ladoped PZT ceramics under the application of static and dynami c electric fields using a laborat ory diffractometer in reflection mode was described in Chapter 5. Under the application of static electr ic fields, 90 domain switching and 111 lattice strains, pa rallel to the directi on of the applied electric field, were observed to correlate with the macroscopic st rainelectric field hysteresis loop. However, differences were noted for values for Ethreshold and EC calculated from macroscopic strain measurements and Xray diffraction analyses. This effect is probably obs erved due to the fact that the Xray measurements were taken under static electric fi elds unlike a continuous cyclic field as adopted for macroscopic measurem ents. A novel timeresolved Xray diffraction PAGE 155 155 technique in the stroboscopic m ode was adopted to perform meas urement of structural changes under the application of cyclic elec tric fields of frequencies sim ilar to macroscopic piezoelectric measurements. It was observed that both the amount of 90 domain switching and the 111 lattice strains increase with an increase in the amp litude of applied cyclic electric fields. Timeresolved diffraction using highenergy Xrays in transmission mode was performed using an area detector at the European Sync hrotron Radiation Facility (ESRF). Highenergy Xrays enabled measurement of el ectricfieldinduced structural changes from the bulk of the material. The DebyeScherrer diffraction rings from the sample were collected in the stroboscopic mode using an area detector. The collected diffracti on rings were segmented into separate azimuthal sectors. The pattern of the hkl diffracted intensity within each azimuthal sector was indicative of the structural state of grains with their hkl poles approximately parallel to an angle to the applied electric field. This experimental setup enabled in situ measurements of structural change s at different angles with respect to the direction of the applied electric field. The total amount of strain due to non180 domain switching in the material was evaluated by integrating the measured changes in the volume fractions of the non180 domains at different azimuthal sections. The non180 domain switching strains in PZT ceramics were observed to be characteristic of the crystal st ructure and dopant compositi on of the material. The largest non180 domain switching strains were observed for Ladoped tetragonal ceramics with compositions close to the MPB. In addition, the non180 domain switching strain coefficients and the lattice strain coefficients showed charac teristic dependencies on applied electric field amplitudes for different crystal st ructures and dopant variants. The contributions from non180 domain sw itching towards macroscopic piezoelectric nonlinearities in PZT ceramics of different crystal structures and dopant variants were PAGE 156 156 determined in Chapter 7. For most compos itions, it was observed that the non180 domain switching strains only partially contribute towa rds macroscopic piezoelectric nonlinearity. For Fedoped tetragonal PZT ceramics, the nonlinear contributions from non180 domain switching strains were observed to be comparable with the irreversible extrinsic contributions determined from macroscopic piezoelectric measurements A quantitative determination of nonlinear contributions from non180 domain switching and lattice strains towards macroscopic piezoelectric nonlinearity was determined fo r Ladoped tetragonal PZT ceramics with compositions close to MPB. It was observed th at non180 domain switching and total effective lattice strain equally contribute to macroscopic piezoelectric nonlinearity It was proposed that nonlinear lattice strains result from intergranular c onstraints which in turn are caused by different amount of non180 domain switching in grains with different orientations. The support for this proposed mechanism was noted from the observed correlations between the non180 domain switching strains and lattice strain s for different amplitudes and tim e periods of applied electric fields. In addition, it was noted that the nonlinear contributions from non180 domain switching and lattice strains do not exactly add up to the total extrinsic contributions determined from macroscopic piezoelectric measurements. It is possible that additional minor nonlinear contributions come from dynamic reorientati on of 180 domains in the material during application of cyclic electric fields. Based on the conclusions made from measurements of converse piezoelectric properties and electricfieldinduced stru ctural changes, the following mechanism for piezoelectric nonlinearity in ferroelectric PZT can be proposed The underlying physical ba sis of piezoelectric nonlinearity is the displacem ent of domain walls, mostly separating non180 domains. PAGE 157 157 However, the preference for non180 domain sw itching depends on the hkl orientation of each individual grain with respect to the direction of the applied el ectric field. Ne ighboring grains with different hkl orientations experience different amounts of non180 domain switching under the application of electric field. Since the grai ns are constrained in a polycrystalline matrix, elastic strains are resulted from mismatch in thei r electricfieldinduced stra ins. In this scenario, the lattice strains are correlated to the amount of non180 dom ain switching in the material. Consequently, both non180 domain switching and la ttice strains can be functions of amplitudes of applied electric fields. In other words, non180 domain switching is the primary driving mechanism but not necessarily the exclusiv e contributor to macroscopic piezoelectric nonlinearity in ferroelectric PZT ceramics. 8.2 Suggestions for Future Work in this Area A unique approach for analysis of fielddepe ndent piezoelectric re sponse in ferroelectric ceramics was provided in this work. It was show n that timeresolved diffraction can be employed to measure the nonlinear contributions from differ ent structural changes to macroscopic strains under different conditions of applied cyclic electri c fields. Similar approaches can be taken to understand variation of piezoelectri c properties as a function of other variables as proposed below. 8.2.1 Nonlinearity in Direct Piezoelectric Behavior Direct piezoelectric effect of ferroelectric ceramics is utilized in various sensor applications. As mentioned in Chapter 1, direct piezoelectric coefficients of PZT ceramics can exhibit Rayleigh behavior within certain range of applied ac mechanical stress. Though it is widely held that non180 domain wall motion is responsible for non linearity in direct piezoelectric behavior, dir ect experimental evidence in this re gard is lacking. It is therefore worthwhile to perform in situ structural measurements in these materials under the application of PAGE 158 158 ac mechanical stress. Experimental techniques sim ilar to ones described in Chapters 5 and 6, can be applied to perform these measurements. Re lative nonlinear contributions from different structural mechanisms can be calculated using an approach as describe in Chapter 7. Since 180 domain wall motion is absent under mechanical st ress unlike application of electric fields, it would be interesting to observe whether th at has any influence on the relative nonlinear contributions from different structural mechanisms. 8.2.2 Frequency Dependence of Piezoelectric Coefficients Characteristic dependencies of di rect piezoelectric effect on fr equencies of applied cyclic mechanical stress were observed earlier [5,80] Similar experiment on variation of converse piezoelectric coefficients as a function of frequency of applied electric field was performed for PLZT5248 samples, the result for which are s hown in Figure 81. A linear variation of piezoelectric coefficients over frequencies of two orders of magnitude coul d be observed. It is generally held that such frequency dependence of piezoelectric coefficients originate from the timedependence of domain wall motion in ferroelect ric materials. However, as it was observed in this work, domain wall motion can influence ot her induced structural ch anges such as lattice strains. A systematic investigation of different structural changes under the application of electric fields at different frequencies is therefore necessary. In situ timeresolved diffraction can be an ideal technique for making these measurements. Figures 82 and 83 show some results from preliminary investigations in this area. Figure 82 shows the timedependent chan ges in the 111 lattice strains in PLZT5248 ceramic, parallel to the direction of the applied electr ic fields of different frequencie s [98]. The data in Figure 82 was collected from measurements taken at beam line ID15B of ESRF. Similar experiments were performed on a commercial soft PZT (EC65) at the WOMBAT instrument at the Australian Nuclear Science and Technology (ANSTO). The 111 la ttice strains for EC65 ceramics parallel to PAGE 159 159 the direction of applied cyclic electrical fields at different frequencies, ar e shown in Figure 83 [99]. No significant differences in the amplitude or timedepende nce of the 111 lattice strains could be observed over a frequency range of 1 500 Hz for EC65 ceramics. This is in contrast with dynamic structural behavior observed in ot her PZT ceramics under cyclic electrical loading, where the 111 lattice strains a nd 90 domain switching were obser ved to be correlated in time [28]. This can imply the following two possibilities fo r the origin of this behavior in the current materials under applied cyclic elec tric fields: (1) the 111 lattice stra in in this material is mainly intrinsic for these frequencies, or (2) the 111 lattice strains are correlated with non180 domain switching [30,68], although doma in switching occurs at tim e scales shorter than 30 s and thus were not measurable using the experimental setup at ANSTO. 8.2.3 Temperature Dependence of Piezoelectric Nonlinearity and Frequency Dispersion It is necessary to understand piezoelectric nonlinearity and frequency dispersion in ferroelectric ceramics at different operating te mperatures. However, study of piezoelectric nonlinearity and frequency dispersion as a func tion of temperature do no t figure largely in the literature. An earlier study on the effect of temperatur e on frequency dispersion of modified lead titanate ceramics showed behavi or typical for thermally activated mechanisms following Arrhenius law, ()kTA /exp=0 (81) where is the relaxation time at temperature T, an d A is the activation energy for the underlying physical mechanism [4]. From the observed behavior, the activation energy for mechanism responsible for frequency dispersi on in modified lead titanate cer amics was estimated to be about 0.4 eV. However, the energy barrier for 90 domain wall motion in defectfree PbTiO3 ceramics PAGE 160 160 has been determined from firstprinciple calculat ions to be ~0.02 eV at room temperature [100]. It was proposed that the presence of defects in modified lead titanate cer amics leads to the large increase in the activation ener gy of 90 domain walls which ar e associated with dopants and grain boundaries [4]. However, underlying this hypothesis is the assumption that domain wall displacements contribute exclusiv ely to variable piezoelectric prope rties, which is not necessarily the case as is evident from the current work. Direct measurements of changes in th e volume fractions of non180 domains in piezoelectric ceramics at different temperatures can be obtained using timeresolved diffraction. Such measurements can also be used to dir ectly determine the activation energy for non180 domain wall motions in the material unde r application of external fields. 8.2.4 Piezoelectric Nonlinearity and Frequency Dispersion in LeadFree Piezoceramics Due to concerns about detrimental effects of l ead on environment, alternatives to PZT have been pursued intensely over the past decade [ 101]. However, for their effective adoption in sensor and actuator devices, it is necessary to study the stability of their piezoelectric properties under different applied conditions such as amplitudes and frequencies of applied electric fields or mechanical stress. Such studies are generally lack ing in the literature. In a recent article, it has been shown that Rayleigh law can be applied to describe nonlinear diel ectric and piezoelectric properties of leadfree sodiumpotassium ni obate (NKN) modified piezoceramics [102]. Interestingly, it is known that modified NKN cer amics show polymorphic behavior around room temperature [103]. It is therefore debatable whet her the nonlinear contributi ons in such materials can comprise strains due to orthorhombict otetragonal phase switching. Comprehensive analyses of nonlinear piezo electric behavior and in situ diffraction experiments under application of electric fields or mechanical stress can pot entially provide answer to such questions. PAGE 161 161 8.2.5 Piezoelectric Nonlinearity and Frequenc y Dispersion in Ferroelectric Thin Films Extrinsic contributions to electricfieldinduced strains are greatly reduced in ferroelectric thin films, particularly for film thickness of less than 1 m [104106]. This has been explained due to restricted non180 domain wall displacement s as a result of small grain sizes, residual stresses and high concentration of point and line defects in th in films as compared to bulk ceramics. Since the microstructu re and hence the energy profile for domain wall displacements can be significantly different in thin films, investigations on structural origin of nonlinear piezoelectricity in these materi als under subcoercive cyclic el ectric fields s hould provide new physical insights. Experimental techniques based on timeresolved Xray diffraction as described in this work can be adopted for such an investigation. PAGE 162 162 0.1110 620 640 660 680 700 d33 (pm/V)Fre q uenc y Hz Figure 81. Variation of conve rse piezoelectric coefficients of PLZT5248, as a function of frequency of applied electric fields The region of linear variation of d33 with respect to applied field frequency is shown by the dotted lines. PAGE 163 163 0.0 0.5 1.0 2.354 2.355 2.356 2.357 2.358 111 lattice spacingsFraction of electric field waveform Hz 3 Hz 7.5 Hz 1PLZT5248, +/0.75kV/mm Figure 82. The timedependent changes in the 111 lattice strains in PLZT5248 ceramic, parallel to the direction of the applied electric fields of different frequencies [98]. PAGE 164 164 Figure 83. The electricfiel dinduced 111 lattice strains 111, as a function of time during application of a bipolar electric waveform of amplitude .4 kV/mm and frequencies of A) 1 Hz, B) 10 Hz, C) 100 Hz, a nd D) 500 Hz, for EC65 ceramics [99]. A B C D PAGE 165 165 APPENDIX A EFFECT OF SAMPLE DISPLACEMENT ON DIFFRACTION PEAK POSITION FOR A CPS DETECTOR A curved position sensitive (CPS) detector was us ed to collect in situ diffraction patterns under the application of el ectric fields, presented in Chapter 5. The longitudinal electricfieldinduced strain causes a vertical displacement of the diffraction volume and therefore leads to a shift in the 2 position of the diffraction peaks. The amount of shift in the 2 position due to vertical sample displacement has been derived earlier [107]. The geometry for the diffraction setup is illust rated in Figure A1A. The detector is in form of a circular arc of radius R with its center coincident with the center of the diffraction volume. The angle of the incident Xray beam is with respect to the sample surface. Let us consider that the diffracted beam corresponding to a particular hkl diffraction peak is recorded at a distance S along the circumference of the de tector and it makes an angle with the surface of the sample and an angle 2 with the incident Xray beam. For this geometry, we have the following relations S S =, (A1) and 2 2 = (A2) where S is the deviation in the measured position of the peak along the circumference of the detector and 2 is the error in the measured Bragg angle 2 For a vertical displacement of the sample surface by y, S is given by PAGE 166 166 () sin siny S 2 = (A3) Eq. A3 becomes apparent from the illustration in Figure A1B. Since, for smallangle geometry ( = S/R ) and ( = ), the following relation can be arrived using Eqs. A1A3, () sinR siny 2 =2 (A4) For a sample of 1 mm thickness (t) and a longitudinal piezoelectric coefficient (d33) of 500 pm/V, the vertical displacement ( y) of the sample surface is 1 m for an applied electric field (E) of 2000 V/mm ( y = d33 E t). The values for the ot her parameters in Eq. A4 are: R ~ 1 m; ~ 10; and 2 ~ 38.7 for 111 diffraction peak. Applying Eq. A4, the value of 2 for these values is 0.0002. For the applied cyclic electric fields of amplitude V/mm, the shift in 2 position of the 111 diffraction peak was measured to be 0.02, for sample PLZT5248. Therefore, the peak shift due to electricfieldinduced lattice strains (0 .02) are at least two orders of magnitude higher th an the peak shifts due to macroscopic sample displacement (0.0002) at highest applied field amplitudes. PAGE 167 167 A B Figure A1. Diffraction geometry a nd peak shift due to sample di splacement, for a CPS detector. A) Diffraction geometry in reflectionmode with parallel incident Xray beam, used for CPS detector B) Estimation of S for a vertical sample displacement of y. B A S y 2 Incident beam Diffracted beam Incident beam Diffracted beam Sample Detector S S R B A y PAGE 168 168 APPENDIX B STROBOSCOPIC DATA COLLECTION USING MCDWIN The sequential s teps followed for enabling stroboscopic data collection using MCDWIN software are described below: 1. The windows for MCDWIN and P7889 is opened by doubleclicking the P7889 icon. 2. In the P7889 window, under settings tab, the option of system is selected. A new window, called System definition, opens. In this window, Status Dig0 check box needs is selected. On right, the following check boxes: Watch, R elease a Start and Low at sweep preset reached, are selected. The settings are saved on clicking OK. 3. A new window, called P7889 Settings, opens when the Range, Preset icon is selected in the MCDWIN window. For both Sync out TTL: and Sync out NIM:, WINDOW is selected from the dropdown menus. 4. In the P7889 Settings, the option for sequential cycles is checked and the number of cycles for which data acquisition are enabled are specified in the field of sequential cycles. The number of patterns within each cycle is spec ified in the field for yrange. For the Timepreset, the input is approximately equal to time period for electric field cycle/ number of patterns in each cycle. The se ttings are saved on clicking OK. 5. A new cycle for data collection is starte d by clicking on the Sta rt icon in MCDWIN window. 6. The TTL signal coming out of the P7889 data ac quisition board is used to trigger a function generator. The length of the acquisition windows is monitored from the NIM signal. The TTL and the NIM signals are monitore d together on an oscilloscope for synchronization of applied electric fields and collect ion of diffraction patterns. The sequential steps for enabling stroboscopi c data collection are shown in Figure B1. PAGE 169 169 Figure B1. Sequential steps for enabling stroboscopic da ta collection. PAGE 170 170 APPENDIX C PENETRATION DEPTH OF XRAYS IN PZT CERAMICS The absorption of Xrays with in ceram ic samples of Pb(Zr0.52Ti0.48)O3 composition is calculated in order to estimate the typical penetrat ion depths for Xrays in PZT ceramics used in this study. The fraction of diffracted intensity G that comes from a given layer of material of thickness t in reflection geometry, is given by [108] ) e(Gt) sin sin ( 1 + 1 1=, (C1) where is the absorption coefficient of the material for Xrays; and are the angles made by the incident and diffracted rays with respect to the sample surf ace, respectively. In case of diffraction in transmission mode, Eq. (C1) can be modified as )e(Gt 1 =. (C2) The absorption coefficient for Pb(Zr0.52Ti0.48)O3 PZT is given by ( ) PZT OOTiTiZrZrPbPb PZT w w w w +++ =, (C3) where w represents the weight fractions and represent the mass absorp tion coefficients of the respective elements; PZT is the density of Pb(Zr0.52Ti0.48)O3 (theoretical density of 8 g/cm3 is used in the present calculations). A reflection geometry is adopted for in situ experiments using a la boratory diffractometer, as presented in Chapter 5. For CuK radiation, the values are given below [108]: Pb = 207.2 cm2/g Zr = 91.2 cm2/g Ti = 47.9 cm2/g O = 16 cm2/g. PAGE 171 171 For representative hkl re flections measured at 2 ~ 40 (111 and 002 reflections), ~ 16 and ~ 24. For this geometry, G is calculated following Eqs. (C1) and (C3), and is shown in Figure C1(a) as a function material thickne ss. It is observed that most of diffracted Xray intensities come from a surface layer of less than 1 m. In other words, structural changes only within a surface layer of 1 m can be probed using a CuK source. In Chapter 6, highenergy Xrays fr om a synchrotron source are used for in situ diffraction experiments in transmission mode. For incident Xrays of energy 87.80 keV (from a synchrotron source as described in Chapter 6), the values are given below [109]: Pb ~ 1 cm2/g Zr ~ 0.7 cm2/g Ti ~ 0.3 cm2/g O ~ 0.2 cm2/g. G is calculated following Eqs. (C2) and (C3), and is shown in Figure C1(b) as a function of material thickness. It is observed that most of diffracted Xray intensity comes from material of thickness < 1 mm, which is the thickness of the samples used for highenergy diffraction experiments. Therefore, structur al changes presented in Chapter 6 are representative of the bulk of the ceramic samples. PAGE 172 172 0.000000.000250.000500.000750.00100 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of diffracted intensityThickness, mm A 0.000.250.500.751.00 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of diffracted intensityThickness, mm B Figure C1. Fraction of di ffracted intensities as a function of thickness of diffracting volume for A) Xrays from a CuK source, B) high energy Xra ys (87.80 keV) from a synchrotron source. PAGE 173 173 APPENDIX D IGOR CODES FOR ANALYSIS OF 2DIMENSIONAL DIFFRACTION DATA Som e of the codes listed here were originally written by Dr. John Daniels of the ESRF and were modified for this work by the author of this document. Other codes were written by the author. Igor code for loading text files containing intens ity values of a 2D diffraction image into waves of a Igor project #pragma rtGlobals=1 // Use modern global access method. Function loadfile1(wave0) // This function load data from a text file to waves in a Igor project string wave0 variable n,i,j,k,l variable file string line wave numwave string filename make /T sepwave make /N=(2001) numwave // N is the number of points in entire range of two theta for (n=0;n<17;n+=1) // n is the number of different amplit udes of applied electric fields make /N=(2000,36,20) $("wave"+num2str(50+50*n)) // 36 is the number of azimuthal sectors for the Debye rings for (k=0;k<20;k+=1) filename = "SamPLZT5248_0"+num2str(k+48+21*n)+".spr" // filename is the name of the text files containing intensity values open /R/P=pathname file as filename wave cd = $("wave"+num2str(50+50*n)) for(j=0;j<37;j+=1) FReadline file,line if (strlen(line)>100) LoadSpaceDellLine1 (line,sepwave,numwave) for (i=0;i<2001;i+=1) cd[i][j1][k]=numwave[i] endfor endif endfor endfor PAGE 174 174 endfor end Function LoadSpaceDellLine1(line, sepwave,numwave) // This function reads each line in a text file string line wave /T sepwave wave numwave variable j variable numtext = 0 sepwave[0] = "" For (j=0;j PAGE 175 175 Igor code for loading text files containing 2 th eta values for a 2D diffraction image into waves of a Igor project #pragma rtGlobals=1 // Use modern global access method. Function loadfileTT(wave0) // This function load data from a text fi le to waves in a Igor project string wave0 variable n,i,j,k,l variable file string line wave numwave string filename make /T sepwave make /N=(2001) numwave // N is the number of points in entire range of two theta for (n=0;n<17;n+=1) // n is the number of diffe rent amplitudes of applied electric fields make /N=2000 $("TT_") + num2str(50+n*50) wave cd = $("TT_") + num2str(50+n*50) filename = "SamPLZT5248_0"+num2str(48+21*n)+".chi // filename is the name of the text files containing 2 theta values open /R/P=pathname file as filename for (i=0;i<2000;i+=1) FReadline file,line LoadSpaceDellLineTT (line,sepwave,numwave) cd[i]=numwave[0] endfor endfor end Function LoadSpaceDellLineTT(line, sepwave,numwav e) // This function reads each line in a text file string line wave /T sepwave wave numwave variable j variable numtext = 0 sepwave[0] = "" PAGE 176 176 For (j=0;j PAGE 177 177 Igor code for extracting intensity for specific diffraction peaks for different azimuthal sectors #pragma rtGlobals=1 // Use modern global access method. function loadsegmentsang(wave0) // This functi on create waves containing diffracted intensities of specific peaks such as 002 and 200, for different azimuthal sectors string wave0 variable n,i,j,k for (k=1;k<36;k+=1) // k is the azimuthal sector make /N=(59,16) $("allfieldbi002neg"+num2str(k*10)) // 59 is the number of data points for 002 and 200 diffraction peaks wave cd =$("allfieldbi002neg"+num2str(k*10)) for (n=0;n<16;n+=1) wave tf = $("wavebineg"+num2str(100+50*n)) for (i=0;i<59;i+=1) cd[i][n] = tf[730+i][k] endfor endfor make /N=(59,16) $("allf ieldbi002pos"+num2str(k*10)) wave ce =$("allfieldbi002pos"+num2str(k*10)) for (n=0;n<16;n+=1) wave te = $("wavebipos"+num2str(100+50*n)) for (i=0;i<59;i+=1) ce[i][n] = te[730+i][k] endfor endfor endfor end PAGE 178 178 Igor code for rebinni ng diffraction intensities with respect to time #pragma rtGlobals=1 // Use modern global access method. Function addtimepos(wave0) //This function rebins diffraction intensities with respect to time string wave0 variable i,j,k,n for (n=0;n<16;n+=1) // n is the number of diffe rent amplitudes of applied electric fields make /N=(2000,36) $("wavebipos"+num2str(100+n*50)) wave cd = $("wavebipos"+num2str(100+n*50)) wave tf = $("waveb i"+num2str(100+n*50)) for (k=10;k<18;k+=1) // this c ode sums diffraction intensities fr om time period 10 to time period 18 for (j=0;j<37;j+=1) for (i=0;i<2000;i+=1) cd[i][j] = cd[i][j] + tf[i][j][k] endfor endfor endfor endfor end PAGE 179 179 Igor code for extracting peak pa rameters including ratio of inte grated intensities of diffraction peaks constituting a doublet, su ch as 002 and 200 for a tetragonal sample, for different time periods during application of cyclic electric fields of different am plitudes. The function used here is double Asymmetric Pearson VII wi th equal strain approximation. #pragma rtGlobals=1 // Use modern global access method. function fit002200f(matname) // This function fits doublet peaks such as 002 and 200 and calculate the ratio of integrated intensities of the doublet peaks wave matname variable i,j,k,field string foldername string componentpath field =Dimsize(matname,1) make /D/N=13 /O W_coef make /D/N=13 /O W_sigma FolderName = getdatafolder (0) //Make new Data Folder for the processed data NewDataFolder /O root:$(FolderName):StandardDevs NewDataFolder /O root:$(FolderName):CalibratedData NewDataFolder /O root:$(FolderName):FitData NewDataFolder /O root:$(FolderName):Coeficients NewDataFolder /O root:$ (FolderName):PeakComponents NewDataFolder /O root:$(FolderName):RunSpecs //Make new waves to hold fitted data Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Baselines") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("BackStep") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Center1") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Center2") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Y0Peak2") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Area Left Peak 1") Make/D/N=(field) /O root:$(FolderNam e):Coeficients:$("Area Right Peak 1") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Area Left Peak 2") Make/D/N=(field) /O root:$(FolderNam e):Coeficients:$("Area Right Peak 2") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("m variable left peak 1") Make/D/N=(field) /O root:$(FolderName):C oeficients:$("m variab le Right peak 1") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("m variable left peak 2") Make/D/N=(field) /O root:$(FolderName):C oeficients:$("m variab le Right peak 2") Make/D/N=(field) /O root:$(FolderN ame):Coeficients:$("ErrBaselines") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrBackStep") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrCenter1") PAGE 180 180 Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrCenter2") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrY0 Peak 2") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrArea Left Peak 1") Make/D/N=(field) /O root:$(FolderName) :Coeficients:$("ErrArea Right Peak 1") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrArea Left Peak 2") Make/D/N=(field) /O root:$(FolderName) :Coeficients:$("ErrArea Right Peak 2") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Errm variable left peak 1") Make/D/N=(field) /O root:$(FolderName):Coe ficients:$("Errm variable Right peak 1") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Errm variable left peak 2") Make/D/N=(field) /O root:$(FolderName):Coe ficients:$("Errm variable Right peak 2") //New waves to hold calculated values Make/D/N=(field) /O root:$(FolderName) :Coeficients:$("Numerical Area Peak 1") Make/D/N=(field) /O root:$(FolderName) :Coeficients:$("Numerical Area Peak 2") Make/D/N=(field) /O root:$(Fol derName):Coeficien ts:$("I002/I200") //New waves for the run specs Make /N=59 /O root:$ (FolderName):RunSpecs:tt002 //Make short reference for each of the new waves Wave Ba = root:$(FolderNam e):Coeficients:$("Baselines") Wave BS = root:$(FolderNam e):Coeficients:$("BackStep") Wave C1 = root:$(FolderNam e):Coeficients:$("Center1") Wave C2 = root:$(FolderNam e):Coeficients:$("Center2") Wave amp2 = root:$(FolderNam e):Coeficients:$("Y0Peak2") Wave AL1 = root:$(FolderName):C oeficients:$("Area Left Peak 1") Wave AR1 = root:$(FolderName):C oeficients:$("Area Right Peak 1") Wave AL2 = root:$(FolderName):C oeficients:$("Area Left Peak 2") Wave AR2 = root:$(FolderName):C oeficients:$("Area Right Peak 2") Wave mL1 = root:$(FolderName):Coefi cients:$("m variable left peak 1") Wave mR1 = root:$(FolderName):Coefi cients:$("m variable Right peak 1") Wave mL2 = root:$(FolderName):Coefi cients:$("m variable left peak 2") Wave mR2 = root:$(FolderName):Coefi cients:$("m variable Right peak 2") Wave ErrBa = root:$(FolderName) :Coeficients:$("ErrBaselines") Wave ErrBS = root:$(FolderNam e):Coeficients:$("ErrBackStep") Wave ErrC1 = root:$(FolderNam e):Coeficients:$("ErrCenter1") Wave ErrC2 = root:$(FolderNam e):Coeficients:$("ErrCenter2") Wave Erramp2 = root:$(FolderNam e):Coeficients:$("ErrY0 Peak 2") Wave ErrAL1 = root:$(FolderName):Coeficients:$("ErrArea Left Peak 1") Wave ErrAR1 = root:$(FolderName):C oeficients:$("ErrArea Right Peak 1") Wave ErrAL2 = root:$(FolderName):Coeficients:$("ErrArea Left Peak 2") Wave ErrAR2 = root:$(FolderName):C oeficients:$("ErrArea Right Peak 2") PAGE 181 181 Wave ErrmL1 = root:$(FolderName):Coeficients:$("Errm variable left peak 1") Wave ErrmR1 = root:$(FolderName):Coefi cients:$("Errm variable Right peak 1") Wave ErrmL2 = root:$(FolderName):Coeficients:$("Errm variable left peak 2") Wave ErrmR2 = root:$(FolderName):Coefi cients:$("Errm variable Right peak 2") Wave NA1 = root:$(FolderName):Coe ficients:$("Numerical Area Peak 1") Wave NA2 = root:$(FolderName):Coe ficients:$("Numerical Area Peak 2") Wave RI = root:$(FolderNam e):Coeficients :$("I002/I200") Wave d = root:$(FolderName):tt002 k=0 //First must put into new waves For(k=0; k PAGE 182 182 STDV = 1/Sqrt(CD) //Now must fit peaks to each new wave Variable V_fitOptions = 4 // suppress progress window Variable V_FitError = 0 // prevent abort on error Make /N=2000 /O roo t:$(FolderName):FitData:$("fit_field" + num2str(k+1)) Wave FD = root:$(FolderName):F itData:$("fit_field" + num2str(k+1)) //Fit with double AsymPearson 7 Equal strain approximation FuncFit /L=2000 /N/Q DoubleAsymP7s _EqualStrain W_coef CD /D/X=d /W=STDV /C=T_Constraints Wave TF = $("fit_field" + num2str(k+1)) FD = TF KillWaves $("fit_field" + num2str(k+1)) Wave WC = W_coef Wave WS = W_sigma // save the coefficients Ba[k] = WC[0] BS[k] = WC[1] C1[k] = WC[2] C2[k] = WC[3] amp2[k] = WC[4] AL1[k] = WC[5] AR1[k] = WC[6] AL2[k] = WC[7] AR2[k] = WC[8] mL1[k] = WC[9] mR1[k] = WC[10] mL2[k] = WC[11] mR2[k] = WC[12] ErrBa[k] = WS[0] ErrBS[k] = WS[1] ErrC1[k] = WS[2] ErrC2[k] = WS[3] Erramp2[k] = WS[4] ErrAL1[k] = WS[5] PAGE 183 183 ErrAR1[k] = WS[6] ErrAL2[k] = WS[7] ErrAR2[k] = WS[8] ErrmL1[k] = WS[9] ErrmR1[k] = WS[10] ErrmL2[k] = WS[11] ErrmR2[k] = WS[12] //Make component waves in the components folder componentpath = "root:" + FolderName + ":PeakComponents" MakeCompAsymDoubleP7s_ESfolder(WC, d[0], d[numpnts(d)1], "field" + num2str(k+1), componentpath) Wave P1 = root:$(foldername):peak components:$( "field" + num2str(k+1) + "Peak1") Wave P2 = root:$(foldername):peak components:$( "field" + num2str(k+1) + "Peak2") Wave RX = root:$(foldername):peak components:$( "field" + num2str(k+1) + "reconX") //*****Need to set the scal e of these waves before these are comparable to the calculated values NA1[k] = areaXY(RX, P1, (WC[2] (W C[3] WC[2])), (WC[2] +(WC[3] WC[2]))) // Area 1 Numeric (integrated intensity) NA2[k] = areaXY(RX, P2, (WC[3] (W C[3] WC[2])), (W C[3] + (WC[3] WC[2]))) // Area 2 Numeric (integrated intensity) RI[k] = NA1[k]/NA2[k] //ratio of integrated intensities EndFor End PAGE 184 184 Igor code for fitting doublet pe aks and recalculating in tensities for the deconvoluted peaks. This code is called for by the code for calculating rati o of integrated intensities of diffraction peaks constituting a doublet, such as 002 and 200 for a tetragonal sample, for different time periods during application of cyclic electr ic fields of different amplitudes #pragma rtGlobals=1 // Use modern global access method. // this function fit doubl et peaks to a asymmetr ic pearson VII function Function DoubleAsymP7s_EqualStrain(w, x) : FitFunc wave w variable x variable result1 variable result2 variable result3 variable result4 variable result5 variable y01 //y0 value calculated for peak 2 assuming there is equal strain between the 002 and 200 domains // W_coef[0] = 500 //Base // W_coef[1] = 200 // Background Step : Now changed to slope // W_coef[2] = 57.55//Center1 // W_coef[3] = 58.2 //Center2 // W_coef[4] = 2000 // Y0 Value peak 2 // W_coef[5] = 9000 // Area Left Peak 1 // W_coef[6] = 7000 // Area Right Peak 1 // W_coef[7] = 4000 // Area Left Peak 2 // W_coef[8] = 3000 // Area Right Peak 2 // W_coef[9] = 5 // m variable Left Peak 1 // W_coef[10] = 5 //m variable Right Peak 1 // W_coef[11] = 8 //m variable Left Peak 2 // W_coef[12] = 8 //m variable Right Peak 2 y01 = w[4]*((w[7]*gamma(w[11]0.5) /gamma(w[11])) (w [8]*gamma(w[12]0.5)/gamma(w[12]))) / ((w[6]*gamma(w[10]0.5)/gamma(w[10])) (w[5]*gamma(w[9]0.5)/gamma(w[9]))) if (x PAGE 185 185 if (x>w[2]) //right hand side of peak 1 result2= y01* (1+((xw [2])^2)/((w[6])^2))^(w[10]) else result2=0 endif if (x PAGE 186 186 reconX[i] = startx + i*((endx startx)/499) Background[i] = w[0] + w[1]*x if (reconX[i] PAGE 187 187 y01 = w[4]*((w[7]*gamma(w[11]0.5) /gamma(w[11])) (w [8]*gamma(w[12]0.5)/gamma(w[12]))) / ((w[6]*gamma(w[10]0.5)/gamma(w[10])) (w[5]*gamma(w[9]0.5)/gamma(w[9]))) For (i=0;i<500;i+=1) reconX[i] = startx + i*((endx startx)/499) Background[i] = w[0] + w[1]*x if (reconX[i] PAGE 188 188 Igor code to input integrated intensities for di fferent electric field amplitudes into a single wave #pragma rtGlobals=1 // Use modern global access method. function inputarea(wave0) // th is function inputs integrated in tensities of doublet peaks for different amplitudes of applied electric fields in a single wave string wave0 variable i, j string foldername foldername = getdatafolder (0) make /N=(16,2) root:$(fol dername):eta:$("areapos") make /N=(16,2) root:$(fol dername):eta:$("areaneg") wave apos = root:$(foldername):eta:$("areapos") wave aneg = root:$(foldername):eta:$("areaneg") wave posp1= root:$(foldername):positive:Coeficients:$("Numerical Area Peak 1") wave posp2= root:$(foldername):positive:Coeficients:$("Numerical Area Peak 2") wave negp1= root:$(foldername):negative:Coeficients:$("Numerical Area Peak 1") wave negp2= root:$(foldername):negative:Coeficients:$("Numerical Area Peak 2") for (i=0;i<16;i+=1) apos[i][0]= posp1[i] apos[i][1]= posp2[i] aneg[i][0]= negp1[i] aneg[i][1]= negp2[i] endfor end PAGE 189 189 Igor code for calculating etas for differe nt amplitudes of applied electric fields #pragma rtGlobals=1 // Use modern global access method. function calceta(wave1,wave2) // this function calc ulates eta values for different amplitudes of applied electric fields wave wave1,wave2 variable i,p1,p2 // p1 is peak 002 and p2 is peak 200 p1=323.038 // integrated inte nsities of the specific p eak for unpoled sample p2=753.087 make /N=16 $("etadiff") make /N=16 $("etapos") make /N=16 $("etaneg") wave epos = $("etapos") wave eneg = $("etaneg") wave ediff = $("etadiff") for (i=0;i<16;i+=1) epos[i] = ((wave1[i][0]/p1)/((wave1 [i][0]/p1)+2*(wave1[i][1]/p2)))(1/3) eneg[i] = ((wave2[i][0]/p1)/((wave2 [i][0]/p1)+2*(wave2[i][1]/p2)))(1/3) ediff[i] = epos[i] eneg[i] endfor end PAGE 190 190 Igor code for fitting a single peak, such as 111 for a tetragonal sample, for different amplitudes of applied electric fields. The function for f itting used here is Asymmetric Pearson VII. #pragma rtGlobals=1 // Use modern global access method. Function MakeAndFit111f(W2DMat) // this function fit a single pe ak such as 111 for different amplitudes of applied electric fields Wave W2DMat variable i, j,field string FolderName field = Dimsize(W2DMat, 1) FolderName = GetDataFolder(0) NewDataFolder /O root:$(FolderName):StandardDevs NewDataFolder /O root:$(FolderName):CalibratedData NewDataFolder /O root:$(FolderName):FitData NewDataFolder /O root:$(FolderName):Coeficients NewDataFolder /O root:$(FolderName):RunSpecs Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Baselines") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Amplitude") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Centre") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("WidthLeft") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("mLeft") Make/D/N=(field) /O root:$(Folde rName):Coeficients :$("WidthRight") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("mRight") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("Slope") Make/D/N=(field) /O root:$(FolderN ame):Coeficients:$("ErrBaselines") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrAmplitude") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrCentre") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrWidthLeft") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrmLeft") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrWidthRight") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrmRight") Make/D/N=(field) /O root:$(FolderName):Coeficients:$("ErrSlope") Make/D/N = 8 /O W_Coef //New waves to hold fitted data Make/D/N = 8 /O W_Sigma //Errors wave PAGE 191 191 Make /N=36 /O root:$ (FolderName):RunSpecs:tt111 Wave Ba = root:$(FolderNam e):Coeficients:$("Baselines") Wave Am = root:$(FolderNam e):Coeficients:$("Amplitude") Wave Ce = root:$(FolderNam e):Coeficients:$("Centre") Wave WL = root:$(FolderNam e):Coeficients:$("WidthLeft") Wave ML = root:$(FolderN ame):Coeficients:$("mLeft") Wave WR = root:$(FolderName) :Coeficients:$("WidthRight") Wave MR = root:$(FolderNam e):Coeficients:$("mRight") Wave S = root:$(FolderNam e):Coeficients:$("Slope") Wave ErrBa = root:$(FolderName) :Coeficients:$("ErrBaselines") Wave ErrAm = root:$(FolderName):Coeficients:$("ErrAmplitude") Wave ErrCe = root:$(FolderNam e):Coeficients:$("ErrCentre") Wave ErrWL = root:$(FolderName):Coeficients:$("ErrWidthLeft") Wave ErrML = root:$(FolderNam e):Coeficients:$("ErrMixLeft") Wave ErrWR = root:$(FolderName):Coeficients:$("ErrWidthRight") Wave ErrMR = root:$(FolderN ame):Coeficients:$("mRight") Wave ErrS = root:$(FolderNam e):Coeficients:$("ErrSlope") Wave d= root:$(FolderName):tt111 i=0 For(i=0; i PAGE 192 192 Make /N=36 /O root:$(Folde rName):StandardDevs:$("Time" + num2str(50*i+100)) wave STDV = root:$(FolderName):StandardDevs:$("Time" + num2str(50*i+100)) //input data For(j=0;j<36 ;j+=1) CD[j] = W2DMat[j][i] EndFor STDV = 1/Sqrt(CD) Variable V_fitOptions = 4 // suppress progress window Variable V_FitError = 0 // prevent abort on error Make /N=2000 /O root:$(FolderName):FitData:$("fit_Field" + num2str(50*i+100)) Wave FD = root:$(Fol derName):FitData: $("fit_Field" + num2str(50*i+100)) FuncFit /L=2000 /N/Q AssymP7Elderton W_coef CD /D/X=d /W=STDV /C=T_Constraints Wave TF = $("fit_Field" + num2str(50*i+100)) FD = TF KillWaves $("fit_Field" + num2str(50*i+100)) Wave WC = W_coef Wave WS = W_sigma Ba[i] = W_coef[0] Am[i] = W_coef[2] Ce[i] = W_coef[7] WL[i] = W_coef[3] ML[i] = W_coef[5] WR[i] = W_coef[4] MR[i] = W_coef[6] PAGE 193 193 S[i] = W_coef[1] ErrBa[i] = W_sigma[0] ErrAm[i] = W_sigma[2] ErrCe[i] = W_sigma[7] ErrWL[i] = W_sigma[3] ErrML[i] = W_sigma[5] ErrWR[i] = W_sigma[4] ErrMR[i] = W_sigma[6] ErrS[i] = W_sigma[1] Endfor End PAGE 194 194 Igor code for fitting single peak s. 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[109] http://physics.nist.gov/PhysRe fData/XrayMassC oef/tab3.html. PAGE 200 200 BIOGRAPHICAL SKETCH Abhijit Pra manick was born on October of 1980 in Toofanganj, a little known small town in India. Over time he traveled extensively and lived in different places in India. He received a Bachelor of Engineering in Metallurgy from the National Institute of Technology (NIT), Rourkela, India in July of 2002. He then moved to the Indian Institute of Science (IISc) located at Bangalore, a south Indian city of technologi cal prominence. He enjoyed the city life of Bangalore and the academic environment of th e campus in IISc. He received a Master of Engineering in Metallurgy from IISc in 2004. He then joined KLATencor Corporation as an Applications Development Engineer. As part of hi s job, he traveled to countries including the USA, Singapore, China and South Korea. As much as he liked traveli ng, he was eventually getting tired of such a mobile lifestyle. He left his job in A ugust 2005 and joined the Department of Materials Science and Engineer ing at the University of Florida (UF) as a PhD student. He joined Dr. Jacob Jones research group in su mmer of 2006. He thoroughly enjoyed the exciting and cosmopolitan student life at UF. His last ye ar as a PhD student has been quite eventful. During this time, he traveled across the globe performed his experiments and analyses, and contributed to numerous scientif ic publications. He received his PhD from UF in the spring of 2009 and immediately started as a postdoctoral research er at the New York State College of Ceramics at Alfred University, situated at Alfred, NY. Apart from his general education in scien ce and technology, he ha s often wondered about the meaning of all existences and of life in particular. He contemplated the viewpoints of science, religions and philosophies, but could never convince himself en tirely. Though the truth has eluded him so far, he remains hopeful in figuring it out someday. 