2900-121 -T Report of Project MICHIGAN POLARIZATION, MICROWAVE DISPERSION, AND LOSS IN HIGH-PERMITTIVITY FERROELECTRICS HOWARD DIAMOND January 1960 THE U N I V E R S I T Y OF M I C H I G A N Ann Arbor, Michigan

DISTRIBUTION OF REPORTS Distribution control of Project MICHIGAN Reports has been delegated by the U. S. Army Signal Corps to: Commanding Officer U. S. Army Liaison Group Project MICHIGAN Willow Run Laboratories Ypsilanti, Michigan It is requested that information or inquiry concerning distribution of reports be addressed accordingly. The work reported herein was carried on by the Willow Run Laboratories for the U. S. Army Signal Corps under Project MICHIGAN, Contract No. DA-36039 SC-78801. The experimental work was initially supported by the U. S. Air Force Office of Scientific Research and The National Science Foundation. University contract administration is provided to the Willow Run Laboratories through The University of Michigan Research Institute.

WI LLOW RU N LABORAT O R I ES 2900-121-T PREFACE Project MICHIGAN is a continuing research and development program for advancing the Army' s long-range combat-surveillance and target-acquisition capabilities. The program is carried out by a full-time Willow Run Laboratories staff of specialists in the fields of physics, engineering, mathematics, and psychology, by members of the teaching faculty, by graduate students, and by other research groups and laboratories of The University of Michigan. The emphasis of the Project is upon basic and applied research in radar, infrared, acoustics, seismics, information processing and display, navigation and guidance for aerial platforms, and systems concepts. Particular attention is given to all-weather, long-range, high-resolution sensory and location techniques, and to evaluations of systems and equipments both through simulation and by means of laboratory and field tests. Project MICHIGAN was established at The University of Michigan in 1953. It is sponsored by the U. S. Army Combat Surveillance Agency of the U. S. Army Signal Corps. The Project constitutes a major portion of the diversified program of research conducted by the Willow Run Laboratories in order to make available to government and industry the resources of The University of Michigan and to broaden the educational opportunities for students in the scientific and engineering disciplines. Progress and results described in reports are continually reassessed by Project MICHIGAN. Comments and suggestions from readers are invited. Robert L. Hess Technical Director Project MICHIGAN

W I LLOW RU N LABORATORI ES 2900-121 -T FOREWORD Large-scale electronic systems in the past decade have made increasing demands on the circuit designer to produce better high-frequency amplifier devices, especially of the low-noise microwave type. Recent techniques have emphasized the development of the so-called solid-state amplifiers, of which the ruby maser is an example. Many investigators in this country have explored the possibility of applying semiconductor diodes, ferrites, and ferroelectrics to parametric amplifiers, and some limited success has already been obtained with the diodes. Success withthe ferroelectrics has been somewhat marginal, partially because presently available materials appear to dissipate too much energy for application at microwave frequencies. In order for more rapid progress to be made in improving ferroelectric materials, a more thorough understanding of the nonlinear processes and loss mechanisms for these substances is required. Moreover, there is a need for more data on the microwave dielectric properties of the ferroelectric materials. It is the purpose of this investigation to obtain such microwave data and also to study the mechanisms of the change in permittivity with electric field in certain types of ferroelectrics. It is this later property which is fundamental in the parametric devices. This investigation as a doctoral thesis was initially suggested by Dr. H. W. Welch, formerly of the Electrical Engineering Department at The University of Michigan. The problem as originally stated was to investigate the rather pronounced microwave dispersion of the permittivity observed in BaTiO3 ceramics. (This is shown in Fig. 5 of this report. ) Upon considering the possible reasons for this effect, the question of the role of ferroelectric domains was raised. Subsequent investigations, both experimental and analytical, indicated that the nonlinear processes involving the large-field permittivity in ferroelectric ceramics could not be made consistent with the idea of domain growth or alignment. The work on a theory of the nonlinearity of this class of ferroelectrics was expanded, and consequently what originally appeared to be a side issue became a rather extensive research activity. This investigation constitutes Sec. 1 and 2 of the report. Professor Gunnar Hok, who served as chairman of the doctoral committee, contributed active support and helpful discussions during this work. Dr. Betzalel Avitzur aided in programming an IBM 650 electronic computer for numerically

2900-121 -T THE UNIVERSITY OF MICHIGAN evaluating some of the integrals which appear in Sec. 3. Miss Viola Chang deserves special mention for her invaluable aid in the preparation of ceramics and single crystals used for the experiments, and also for her help in reducing the microwave data. The many discussions with Miss Chang served as a basis for formulating several of the experiments on the ferroelectric ceramics. Vi

WI LLOW RU N LA BORATORI ES 2900-121 -T CONTENTS Preface..................... iii Foreword v List of Figures..viii List of Tables..x Abstract.. 1 1. Introduction............... 2 1. 1. Objectives............ 2 1. 2. Review of Some Ferroelectric Properties.. 3 1. 3. The Classical Theory of Dielectrics..... 9 1. 4. Mechanisms of Dielectric Loss and Their Relationship to Polarization. 18 1.5. Summary.21 2. The Polarizability and Losses in the Perovskite Structure 24 2. 1. Thermodynamic Approach. 24 2. 2. Relationship between the Free and Clamped Coefficients in the Free-Energy, Isothermal, and Adiabatic Conditions. 33 2. 3. Some Polycrystalline Properties. 41 3. A Theory of Large-Field Nonlinearity in the Perovskite Ferroelectrics..46 3. 1. Postulates and General Properties 46 3. 2. The Zero-Field Case. 50 3. 3. The Large-Field Case.55 4. Dispersion Phenomena for High-Permittivity Dielectrics 64 4. 1. Low-Frequency Dynamic Spectra. 64 4. 2. Microwave Properties.71 4. 3. Losses and Dispersion. 85 5. Conclusions.88 Appendix..91 References..94 Distribution List..96

2900-121 -T THE UNIVERSITY OF MICHIGAN FIGURES 1. Crystal Structures of BaTiO3................ 4 2. Effect of Temperature on the Lattice Dimensions and Permittivity of BaTiO3.. 5 3. Definitions Relating to Ferroelectric Hysteresis Loop. 8 4. Schematic of Contributions to Permittivity in a Dielectric.23 5. The Small-Field Permittivity vs. Frequency for Ceramic BaTiO3...... 23 6. Free Energy vs. Polarization with Temperature as a Parameter.... 27 7. The P vs. E Characteristic for a Free BaTiO3 Crystal Based on Devonshire's Free-Energy Function.. 29 8. The P-E Hysteresis Loop Plotter.... 32 9. Double Hysteresis Loops in a c-Oriented BaTiO3 Crystal..33 10. Curie Temperature as a Function of Field.33 11. Automatic Recording Unit for Measuring cA vs. E and T.......42 12. cTE Surface for a Typical Ba 0.65Sr0. 35TiO3 Ceramic.43 13. cTE Surface for a Pure BaTiO3 Ceramic............... 44 14. Relative Capacitance vs. Electric Field............ 45 15. Idealized Thermal Behavior of a Ferroelectric Crystallite.......47 16. The Static Capacity vs. Field Butterfly Loop for a Ba0. 65Sr0 35TiO3 Ferroelectric Ceramic at 25C...48 17. Double Hysteresis Loops for a Ba Sr TiO Ceramic at Various Temperatures.......49 18. 60-Cycle Hysteresis Loops for a Ba0. 65Sr0. 35TiO3 Ceramic.49 19. The Average Characteristic of a Ceramic Grain and the Permittivity Ellipsoid at Zero Field.......... 51 20. Theoretical Values of Permittivity vs. Temperature.56 21. Comparison of Experimental and Theoretical Permittivities..... 56 22. Peak Permittivity vs. the Parameter a.57 23. Theoretical Parallel Permittivity vs. Temperature and Electric Field (for a = 31).. 62 24. Theoretical Transverse Permittivity vs. Temperature and Electric Field (for a = 31)........................... 63 25. Theoretical and Experimental Relative Tunability for Ba0. 65Sr0. 35TiO3 63 26. Theoretical Parallel Permittivity vs. Temperature and Electric Field (for a = 20).......................... 64 27. Dynamic Dielectric Spectrometer.... 67 28. BaTiO3 Single Crystals................ 69 29. Dynamic Spectra for a and c Plates of BaTiO3 Single Crystals (T = 25~C)... 69 30. Dynamic Spectra of Free and Clamped BaTiO3 Single Crystals (T = 25~C)... 70 viii

WI LLOW RUN LABORATORIES 2900-121 -T F I G U R E S (Continued) 31. Spectra of BaTiO3 Ceramics at Room Temperature......70 32. Experimental Arrangement for UHF and Microwave Measurements.72 33. Dielectric Sample Holder for Line.72 34. Frequency Data for Crystal of BaTiO3 c Plate (at 230C).77 35. Frequency Data for Crystal of BaTiO3 a Plate (at 230C).77 36. E vs. Frequency for a BaTiO3 + 0. 5% Fe 203 Ceramic....... 79 37. Q vs. Frequency for a BaTiO3 + 0. 5% Fe203 Ceramic.... 79 38. Data for Hot-Pressed BaTiO3 Ceramic (at 230C).80 39. BaTiO3 Ceramic with Excess TiO2............. 80 40. Data for Hot-Pressed Cd 2Nb207 (at 23C).82 41. c' and Q vs. Frequency for Ba 0.65Sr 0.35TiO3 Ceramic.82 42. Dielectric Constants vs. Frequency for PbTiO3.83 43. Dielectric Constants vs. Frequency for Pb0.35Sr0. 65TiO3 83 44. Tuning Data for Ba 0.65Sr0.3 TiO Ceramic...84 0. 65 0. 35 3 3..... 45. EA and Loss Data for a Typical Ba0.65 Sr0.35TiO3 Ceramic.86 TABL ES I. Some Dielectric Properties of BaTiO3-Like Ferroelectrics 9 II. Dielectric Constants for Various Dielectrics...........15 III. Some Recent Evaluations of the Coefficients in Devonshire's Free-Energy Function..26 IV. Some Values of Elastic and Electrostrictive Coefficients for BaTiO3 Single Crystals............ 37 3X ix

W I LLOW RU N LABORATORI ES 2900-121-T ABSTRACT The purpose of this study is to investigate, by analysis of experimental data, polarization, dispersion, and loss processes in high-permittivity ferroelectrics. The study proceeds from the observation that, according to published data, polycrystalline BaTiO3 and similar compositions show strong relaxation in permittivity at microwave frequencies. This phenomenon is shown to be generally related to polarization and loss properties of the dielectric. Processes of particular interest in accounting for the microwave relaxation are: (a) domain effects, (b) possible dipolar or molecular polarization, and (c) mechanical effects produced by piezoelectric and electrostrictive coupling. In connection with category (a), measurements were made of hysteresis properties and experiments were conducted on ceramic samples, in which the permittivity was measured parallel and transverse to strong electric fields. It is concluded that domains are immobile in the ceramic grains. In order to account for nonlinearity in the permittivity under the action of an electric field, a phenomenological theory is postulated. Each grain in the ceramic is presumed to be described by a free-energy function of the form given by Devonshire for BaTiO3 single crystals. The free-energy function takes into account a transition between a ferroelectric and a nonferroelectric state at a critical temperature (the Curie temperature). For a polycrystalline ferroelectric the transition temperatures of the crystallites are assumed to be given by a Gaussian distribution about some operating temperature. The domains are presumed immobile in those grains in the distribution which are ferroelectric. The free-energy function predicts that a ferroelectric state will be induced by the field in those grains which are not ferroelectric, and it is assumed that the polar axis is induced along the field direction. This model, when analyzed, leads to theoretical results which correlate remarkably well with experimental data. In connection with category (b), measurements were made of the permittivities of single crystals of BaTiO3 at microwave frequencies with the polar axis both parallel and transverse to the signal fields. In contrast with the results for ceramics, the single crystals showed no relaxation at microwave frequencies. In connection with category (c), mechanical effects on the permittivity were analyzed. These effects are important because at sufficiently high frequencies a dielectric may behave as if it were mechanically clamped. Single crystals and 1

2900-121-T THE UNIVERSITY OF MICHIGAN polycrystalline samples have been measured on a dielectric spectrometer over a frequency range from 0. 5 mc to 200 mc. Dimensional resonnances were observed at the low-frequency end of the spectrum. Above frequencies where dimensional resonances are significant, the permittivity was reduced to about one half of that at the low-frequency limit. This result is consistent with predictions based on Devonshire' s free-energy function. It is further shown that a loss mechanism can result in a significant dielectric relaxation. The losses are considered to result from mechanical interactions at domain walls and grain boundaries. The conclusions that may be drawn from this work are as follows. (1) The field sensitivity of the incremental permittivity of polycrystalline ferroelectrics is not a domain effect, but results from an induced ferroelectric state in nonferroelectric grains. (2) Nonlinear permittivity under electric fields must necessarily be accompanied by substantial thermal sensitivity. (3) Orientation of the domains does not contribute to microwave dispersion. (4) Molecular or dipolar relaxation does not occur at microwave frequencies for BaTiO3 single crystals. Therefore, this is not a mechanism for dispersion in the ceramic. (5) Dielectric loss as well as piezoelectric and electrostrictive activity would result in microwave dispersion in high-permittivity ferroelectrics. INTRODUCTION 1. 1. OBJECTIVES The objectives of this investigation are twofold: (1) to investigate both experimentally and theoretically the mechanism by which the small-signal permittivity changes with electric field and temperatures; the analysis is carried out using essentially a macroscopic model; and (2) to investigate the microwave relaxation in the permittivity which is observed for several ferroelectric ceramics. The question of a dielectric loss mechanism is also considered with regard to an explanation for the observed microwave relaxation. Attention is centered on those ferroelectrics of the high-permittivity type - typically the so-called perovskite class of ferroelectrics of which BaTiO is the most familiar member.

W I LLOW RU N LABORATORI ES 2900-121-T The terms introduced in the present subsection are defined and clarified in the succeeding subsections. The reader already familiar with these terms and with the general concepts of the classical dielectric theories may proceed immediately to Sec. 2 of this report. 1. 2. REVIEW OF SOME FERROELECTRIC PROPERTIES Ferroelectricity is defined as that property of certain dielectric materials of exhibiting a spontaneous electric polarization in the absence of an external field. In addition, there is a further condition that must be fulfilled if a given dielectric is to be regarded as a ferroelectric, and this condition is that the direction of spontaneous polarization may be reversed upon the application of an external field of sufficient intensity and reversed polarity. Some classes of dielectrics, such as pyroelectrics and the so-called "electrets, " show a spontaneous polarization, but this polarization cannot be changed or switched by an external electric field; i. e., the polarization is in a sense "frozen in"; therefore, these materials are not classified as ferroelectrics. As a consequence of the property of the polarization direction switching under the influence of a suitable electric field, the ferroelectrics display a dielectric hysteresis loop in polarization, P vs. electric field E. This effect, together with the property in most of these substances of having a rather large permittivity, has resulted in the coining of the term "ferroelectricity" to describe the analogue to some of the properties of magnetic substances (i. e., ferromagnetic properties). Ferroelectricity was first observed in Rochelle salt (NaKC4H406. 4H20) as early as 1921 by Valasek (Ref. 1). The discovery, by independent investigators in the U. S. (Ref. 2) and U. S. S. R. (Ref. 3) during World War II, of the remarkable dielectric properties of the ferroelectric, barium titanate (BaTiO ), stimulated the active research seen in this field today. BaTiO3 belongs to the class of ferroelectrics known as the perovskites, since all members of this class have the crystal structure of the mineral perovskite (CaTiO3). The crystal structure for the members of this class is shown in Fig. 1 with the definitions of the crystallographic directions relative to the polarization direction shown on the figure (it will be necessary to refer constantly to these definitions in the course of this report). It is important to note that the barium titanate structure is the simplest of any known ferroelectric, and this imparts to this material an especially significant role in the development of theories of ferroelectricity. In addition to the existence of dielectric hysteresis loops and the large permittivities mentioned above, there are some further properties of ferroelectrics which The structure of WO3 is even simpler and consists of the BaTiO3-type lattice with the Ba sites vacant. However, WO3 is actually an antiferroelectric; that is, the structure has alternate lines or cells of opposite spontaneous polarization. The net result is cancellation of any externally observable polarization. This is similar to the antiferromagnetic state of some magnetic materials. 3

2900-121 -T THE UNIVERSITY OF MICHIGAN have to be taken into account in this study. All ferroelectric materials contain at least one ion in an asymmetric position in the crystal lattice, and, since these dielectrics are also polar, they are consequently strongly piezoelectric. (In fact, one of the largest technological applications of BaTiO3-type ceramics is the utilization of the piezoelectric effects in transducer elements such as phono pickups, submarine detectors, etc. ) Even in the symmetric nonferroelectric phase where permittivities are rather large but the crystal is not piezoelectric, one would expect large values of electrostriction, since large polarizability in an ionic lattice implies large displacements of ions, and electrostriction and piezoelectric effects thus would be salient factors in an analysis of dielectric losses in these ferroelectrics. Due to the piezoelectric effect in the ferroelectric state, the permittivity for a free crystal is expected to be considerably different from that for a mechanically clamped crystal. For fields of frequency much higher than the mechanical resonant modes, the crystal would then behave as if it were effectively mechanically "clamped. ":OI — C c a axis B /' I..-' I Lo~~~~~~~~~~~~~~~~o X io x oTi i o Ba b6Ti = + 0. 06 Angstrom * Ti 60 = - 0. 09 Angstrom o O 60II = -0. 06 Angstrom The Crystal Structure of BaTiO and Similar The c Crystalographic Axis is the Polar Axis Perovskite-Type Ferroelectrics in the Cubic Schematic of Shift in Ionic Positions of (Nonferroelectric) Phase BaTiO3 in the Tetragonal (Ferroelectric) Phase (After Neutron Diffraction Data of Frazer, Danner, and Pepinsky, Ref.. 4) FIG. 1. CRYSTAL STRUCTURES OF BaTiO3 Figure 2 shows the effects of temperature on the lattice dimensions in the perovskite-like structure of BaTiO3 and also on the values of permittivity along specific crystalline axes. Comparison of the two curves shows a rather remarkable effect. The tetragonal distortion from the perfect cubic structure amounts to only about 1% at room temperature. That is, the difference between the c-dimension and the a-dimension at room temperature is only 0. 04 angstrom, with a unit-cell dimension of about 4 angstroms. At the same time, the isotropic permittivity observed in the cubic state becomes in the ferroelectric tetragonal phase extremely

W I LLOW RU N LA BO RATO R I ES 2900-121-T anisotropic, the permittivity in the c-direction being only of the order of 100 while the permittivity in the a-direction (i. e., transverse to the polar axis) is about 4000. And this happens with only a 1% distortion of the unit cell from the nonferroelectric phase, o 4.030 - a: 4.020 ac 4.010 -7, 4.000 w 3.990ORTHORHOMW if3 sic H ale TETRAGONAL t- 3.980 H-" HBL~ 3.970 - 150 -100 -50 0 50 100 150 TEMPERATURE ( ~ C) (a) -OXl03 - 8 oxo3 -J It! I ( I I II WLL~r, a AXIS - 200 -160 -120 - 80 -40 0 40 80 120 160 TEMPERATURE (~ C) (b) FIG. 2. EFFECT OF TEMPERATURE ON THE LATTICE DIMENSIONS AND PERMITTIVITY OF BaTiO3. (a) Lattice dimensions (after Kay and Vousden, Ref. 5). (b) Dielectric constants of a single crystal (after Merz, Ref. 6 and 7). The transition from tetragonal ferroelectric to cubic nonferroelectric at 1200C is called the Curie point as for magnetic materials. The dielectric constant above 1200C follows a Curie-Weiss law. One further property of the ferroelectrics that must be noted is that a given crystal tends to break up into many domains. A domain is defined as a region within a ferroelectric crystal in which the direction of the spontaneous polarization is the same throughout. This behavior is similar to that of the ferromagnetic materials; however, there are some very striking differences between the ferromagnetic and ferroelectric domains. Since there are no isolated magnetic poles, ferromagnetic domains tend to form in such a way that they provide paths of 5

2900-121 -T THE UNIVERSITY OF MICHIGAN closed flux. Both 900 and 1800 walls exist, and the transition from one wall orientation to another takes place over hundreds or thousands of lattice constants. In ferroelectric domains, isolated charges are possible, and the domains do not necessarily have to form closed flux linkages. The domains tend to be thin and needle-like with both 900 and 1800 walls possible. The ferroelectric domain walls are only one or two lattice constants thick. In the above discussion we have considered some properties of the ferroelectrics, especially in regard to those characteristics in which they differ from the usual dielectric materials. These properties certainly have to be taken into account when dealing with the polarization or permittivity of the ferroelectrics, for both small-field and large-field effects. Indeed, the question arises as to just how one can define a permittivity for these materials when not only is the relationship between D (the electric displacement) and E (the electric field) strongly nonlinear, but the D is not a single-valued function of E because of the dielectric hysteresis. As has been pointed out by Jaynes (Ref. 8), one has to be extremely careful to define what is meant by the term linearity when time -varying fields are applied. Also, the notions of nonlinearity and dispersion have to be distinguished. In the following paragraphs, the definitions of these concepts essential to this study are established. Since BaTiO3 is anisotropic, D is related to E via a tensor permittivity. D. = E..E., (1. 1) 1 1J j where the convention of summation over repeated indices is used. For the case of BaTiO3 in the tetragonal ferroelectric state, as for other perovskite-like dielectrics in corresponding states, the e = c2 E, and 3 = e, the off-diagonal terms being zero. This is referred 11. 22 a 33 c to the crystal axes defined in Fig. 1. From classical theory the polarization is related to the field 2 via P = D - E = (E - 1)E E. (1. 2) r o It is often convenient to define P in terms of the field directly through the electric susceptibility, X. Thus the relationship corresponding to Eq. (1. 1) between polarization and field is P.i =X (1. 3) where X=(E... - l)E ~E....E, (1.4) Xij = (~ij - 1)Eij 0c..~ In the notation in this section we use E as the relative permittivity and E as the permittivity of vacuum. 6

W I LLOW RU N LA BO R A T R I ES 2900-121-T for Ec.. >> 1. One takes the point of view here that, at least for the study of crystalline lattice, (i. e., from the so-called microscopic viewpoint), the variables P (the dipole moment per unit volume) and E are the most convenient to use and are open to direct physical interpretation.3 Now the relationships expressed by Eq. (1. 1) and (1. 3) have meaning for the case where E is time dependent only if P or D is related to E linearly. The linearity requirement is usually expressed in the frequency domain as follows: a dielectric is linear if a unique E(w) or X(o) exists such that D(w) = E() E(O) or P(w) = X(w)E(w). (1. 5) That is, given a cyclic field E(w) at some frequency ao, there is some constant E for this value of frequency wo such that D will be proportional to E(w ) and contain only wo for its frequency component. This does not require that E or X be independent of frequency for linear dielectrics. Whenever E = E(w) or X = X(S), dielectric dispersion is said to exist in the medium; this can be taken as the definition of dispersion. Equations (1. 5) imply the principle of linear superposition in the time domain. That is, if El(t) produces a Dl(t) and E2(t) produces a D2(t), then El(t) + E2(t) produces DI(t) + D2(t). The proof that this property follows from Eq. (1. 5) is left to the Appendix, where it is also shown that a necessary condition is that E(w) or X(w) be independent of the field. This last result is more or less expected by intuition. The fact that D(w) and E(w) are simply related through E(w) does not mean that a similarly simple relationship must exist between D(t) and E(t), for by considering the Fourier integrals C icot E(t) = E(o)te do, (1. 6) _o0 where 1 C~(~1~ -itox E(w) = 2 E(x)e dx, (1. 7) -00 and 00oo P(t) = P(w)ei d, (1. 8) -00 we can derive a general relationship between P(t) and E(t) by substituting P(w) = X(w)E(w) into Eq (1. 8). We get Accordingly, we note that the Devonshire theory of ferroelectrics, a macroscopic thermodynamic theory, uses the variables P and E rather than D and E. P serves as the intensive thermodynamic variable and E as the extensive one. P can be visualized as a measure of atomic "stretching" or distortion upon the application of the field E. 7

2900-121 -T THE UNIVERSITY OF MICHIGAN mll~00 1 P(t) = f X(wo)E (o)eit dw = jX(w)eiWt JE(t)e -iwt dt d (1. 9) -00 j0 - as a general formula relating P(t) and E(t). It is noted that if no dispersion is present, i. e., if X is independent of frequency, then Eq. (1. 9) becomes: P(t) = XE(t). (1. 10) This means that the polarization will follow the applied field only to the extent that E or X are frequency independent. This is expected, because in order for P(t) to be precisely the same function as E(t), all Fourier (frequency) components in one must appear in the other and in the same proportions. For the case where dispersion is present, as it is for many highpermittivity ferroelectrics, there is no simple algebraic relationship between P(t) and E(t). Definitions of the small-signal susceptibility, spontaneous polarization, coercive field, etc., and how these are related to the hysteresis loop are shown in Fig. 3. AP AE X= the small-signal susceptibility " i (eA- 1)o E = the coercive field c P = the spontaneous polarization s P = remanent polarization r FIG. 3. DEFINITIONS RELATING TO FERROELECTRIC HYSTERESIS LOOP Table I, a schematic, may aid in clarifying some of the more prominent dielectric properties of the materials under investigation in the frequency-field domain. This schematic is divided into four regions, based on the magnitudes of the electric fields and on the frequencies involved. The divisions are made arbitrarily to coincide with some physical process in a single crystal of the ferroelectric. Thus, we take as a small field an electric field which is 8

W I LLOW RUN LABORATORIES 2900 -121 —T much smaller than the coercive field of the crystal (i. e., AE~<< E for a small field). Under c such conditions, domain nucleation and growth effects should not exist nor contribute to the small-signal or incremental polarizability of the material. The division between low frequency and high frequency is made on the basis of whether or not the crystal behaves as if it were mechanically clamped. Frequencies in excess of the elastic dimensional resonances are considered high frequencies. Frequencies in the neighborhood of the piezoelectric modes and lower are considered low frequencies. For BaTiO single crystals, E is of the order of 3 c 1000 volts/cm, and in most crystals the highest elastic resonance frequency observed is of the order of 15 mcs (Ref. 9), so that one may use about 100 volts/cm and 50 mcs as the dividing points. These figures vary from crystal to crystal, of course, but they give a rough idea of what constitutes large and small quantities in field and frequency for the purposes of this study. TABLE I. SOME DIELECTRIC PROPERTIES OF BaTiO3-LIKE FERROELECTRICS Low Frequencies, Small Fields High Frequencies, Small Fields Incremental P and E are taken as Incremental values of P and E are 4irAP taken as linearly related via linearly related via e E E 4p Piezoelectric effects observed. /A AE' Domain nucleation and growth not Piezoelectric effects clamped out. expected. No domain effects expected. Low Frequency, Large Fields High Frequencies, Large Fields Major hysteresis loops. No unique Properties largely unknown. relationship between P and E. Domain nucleation and growth Some heating by cyclic fields. not expected on the basis of Large piezoelectric effects. present knowledge. Domain nucleation and growth Dielectric heating expected. effects. Some nonlinearity in e expected. 4 1. 3. THE CLASSICAL THEORY OF DIELECTRICS The classical theory of dielectrics will be used as the starting point in the analysis of the small-field polarizability in the perovskite-like ferroelectrics. This theory, therefore, will be very briefly reviewed at this point. If a homogeneous dielectric material is placed in a uniform electric field, the molecules or atoms become polarized. By this it is meant, for example, that a physical charge separation occurs due to a separation of the center of gravity As a matter of convenience the cgs system of units, the mostuseful for the thermodynamic analysis, will be used for the analysis in the succeeding sections. On the other hand, when data are reported, they will be displayed and discussed in terms of the familiar practical units. 9

2900-121 -T THE UNIVERSITY OF MICHIGAN of positive (the nucleus) and negative (the electrons) charge within an atom, or that a polar molecule is oriented. The polarization vector, P, is really a measure of the charge separation in that it is defined as the electric dipole moment per unit volume of the material. The dipole moment is given by ~'p~~~~~~ = E e.r~ ~(1 11) 1 i 1 1 where the sum is taken over all charges ei, and Fi is the position vector. In a substance which is electrically neutral, ei = 0, (1. 12) so that the expression for the dipole moment, pu, is independent of the origin taken for the position vector r.. The field effective in polarizing an atom at a lattice point in a dielectric is not necessarily the same as the applied macroscopic field. This results from the'fact that as the applied field polarizes the dielectric, the dipoles thus created give rise to fields which in turn affect the effective field at any point in the dielectric. In addition, if the dielectric placed in the field is of certain configurations, say an ellipsoid, the induced charges appearing at the surfaces give rise to fields which tend to oppose the applied field. This results in the so-called depolarization field. The net result of these effects is in general an electric field far different from that applied, especially in the case of the high-permittivity ferroelectrics. The resultant field at a point within the dielectric is called the local field. One of the earliest calculations of the local field was worked out by Lorentz (Ref. 10). His method is to assume that at sufficient distances from the point at which one wishes to derive the effective field, the medium is treated in a completely macroscopic way, that is, as a continuous medium. In a spherical region or cavity about the point in question, the medium is assumed to consist of its particular atomic structure. In this manner the local field is broken up into a sum of the contributions of fields external to the spherical cavity and the fields due to the induced dipoles within the cavity. El = E +E., (1. 13) e 1 where Eg is the local or effective field, E is the external contribution, and Ei is the internal contribution. The external contribution is the sum of the effects of the applied field, E, the depolarization field, and the fields produced by the surface charges on the walls of the cavity. The internal contribution is the field felt at the center of the cavity as a result of all the dipoles within the cavity. For the case where the dielectric is placed between electrodes such that the depolarization field is zero, one obtains the expression for the famous Lorentz field: 10

W I LLOW RUN LABORATORIES 2900-121 -T - 2 4v - y3. Fjr.-r r. E + -3 P + (1. 14) /Q 3 5 i r.i where E is the applied field, P is the polarization, I. is the dipole moment of each dipole within the cavity, and r is its position vector relative to the origin at the center of the cavity. The summation in the above is over all dipoles in the cavity except the one taken at the origin. For substances having cubic structure such as NaCl and for those with random structure such as a gas or nonpolar liquid, the above reduces to - - 4r - E = E +- 3 P. (1. 15) The perovskite structure does not have cubic symmetry for each ion so that an evaluation of the Lorentz field proceeds with Eq. (1. 14). Equation (1. 2) with Eq. (1. 15) becomes (E + 2) 3 where eo is the static (zero-frequency) permittivity. This shows that a very large "amplification" is possible in the Lorentz local field if a high-permittivity dielectric is considered. Viewed conversely, a favorable local field could cause a large polarization which in turn enhances the local field. Thus one can visualize a situation occurring under favorable circumstances in which the material as a whole can become spontaneously polarized, i. e., ferroelectric. The Lorentz field leads to the Clausius-Mosotti formula EN.ca. X = i (1. 17) E 4 N 3 i 1 1 where N.i is the number of atoms of the i-th species per cubic centimeter and a.i is the polarizability5 of the atoms of the i-th species. This seems to correlate well with experimental data for gases and some liquids. Its application to solids has not been particularly successful, especially for those containing polar molecules as part of their structure. The Lorentz formula was derived under the assumption that the induced dipoles in the medium are parallel. This would be expected of a symmetrical structure at sufficiently low temperatures, but not of a polar substance at elevated temperatures. Moreover, the Lorentz field leads to predictions of ferroelectricity far too easily, for when the denominator of Eq. (1. 17) approaches zero, Polarizability is defined as the induced dipole moment per atom per unit local field. 11

2900 -121 -T THE UNIVERSITY OF MICHIGAN the susceptibility becomes greater and greater, that is, the amplification in local field becomes stronger. An analysis of the Lorentz field results in a prediction of ferroelectricity in such substances as water and other polar liquids above their boiling points - clearly an impossible situation. It can be seen that the Lorentz local field is too strong a correction to the applied field and that further modifications have to be made. There are two objections to the use of the Lorentz field. The first is that although in principle the Lorentz correction may be used to compute the local field for an array of permanent dipoles, provided that these dipoles are all aligned along the macroscopic polarization direction, in practice, it does not work successfully because the thermal agitation of the dipoles disturbs their alignment. The same is true of induced dipoles. The second objection is that if a given dipole is rotated out of alignment from the polarization direction it has the tendency to pull neighboring dipoles out of alignment as well because of coupling between the dipolar fields. In order to meet the objections mentioned above, Onsager (Ref. 11) has developed an expression for the local field based on the following model. The dipole in the material is replaced by a small uniformly polarized sphere which is free to rotate in a cavity cut out of the gross media of fixed uniform polarization. By the solution of an electrostatic boundary value problem, the resultant field in the neighborhood of the dipole is found as the superposition of the uniform field of the medium and the field of the sphere which may be rotated to some angle 0, with respect to the polar direction of the rest of the material. The interaction energy V of a dipole in a field E is given by V = - E = - CiEcos 0, (1. 18) and from the Boltzmann distribution one has for the average value of cos 0, V cos 0 e dQ cos 0 = (1. 19) kTd2 where the integration is carried over the solid angle ~ = 47. If the integration of Eq. (1. 19) is carried out, one obtains the well-known Langevin function: /~~__ E (1 20) cos 0 = coth kT) 13kT (for iE << kT). Since the polarizability is given by 12

W I LLOW RU N LA BO RATO R I ES 2900-121 -T ~ Cos Q cu cos (1. 21) the dipolar contribution to the polarizability is 2 ad.p. 3kT(1. 22) dp PI 3kT' The result of an analysis of Onsagers' model using the dipolar contribution to polarizability, Eq. (1. 22) is that the local field 4 7rP E E + 2 +1'- (1. 23) loc. 2E + 1and E -84| 1 + 2 + 3 + + N2) (1. 24) 2 2 242 4 =+kT 9kT 2 2 This result6 shows that there is no critical point for the dielectric constant. Consequently, the exact onset of ferroelectricity cannot be determined from the Onsager model. If one were to write, after Jaynes (Ref. 12), the expressions for the local fields in the form Ee = E + 1P, (1. 25) the factor 3 takes on the character of a "general Lorentz factor. " For the Lorentz fields, 4w 4r 3 =, and for the Onsager model, 1 = + 1 as is seen from Eq. (1. 23). From this it is 3 2E +1 evident that the Onsager field is always smaller than the Lorentz field with the former approaching the latter as E — 1. Whereas the Lorentz field turned out to provide an excessive correction to the applied field, the Onsager field represents an insufficient correction, since it results in the prediction that a material will never become ferroelectric. Calculations of the local fields in BaTiO3 have been carried out by Slater (Ref. 13) and Triebwasser (Ref. 14), but both have based their calculations on the validity of the Lorentz fields. All the theories for internal fields are beset with great difficulties, and the above examples are to be considered as rather rough idealizations. The study of dielectric properties and the onset of ferroelectricity is very important to an understanding of the nature of chemical bonding. On the other hand, the dielectric behavior depends, as shown here, very strongly on the nature of the local fields, the derivation 6The result given here assumes no displacement polarizability. If the latter is taken into account, one obtains E ~Eo + (E,+ 2)2 427kN +T 13

2900-121 -T THE UNIVERSITY OF MICHIGAN of which assumes some knowledge of the chemical bond. The local fields and chemical binding play the central roles in the theory of dielectrics. Further discussion on this topic is adequately available in the literature. From the chemical-bond point of view, the contributions to the total polarizability or susceptibility is usually taken as a linear superposition of electronic, ionic, and dipolar terms. That is, one writes Xtot Xe +Xi+Xd (1. 26) In this expression the Xe is the susceptibility due to distortions in the electronic shells as a result of the action of the field-, Xi is the displacement of one ion against another, for example, in an ionic crystal, and Xd P. is the contribution from the alignment of dipoles in polar substances. Strictly speaking, these terms are not completely independent. For example, in an ionic crystal such as NaCl, as the ions shift with an applied field, one would expect a distortion of the electronic shells as a result of the ionic displacement. For this reason one often finds the first two terms in Eq. (1. 26) grouped together as the total induced susceptibility, as distinct from the susceptibility due to rotations of permanent dipole moments, Xd.. Whether each of the terms in Eq. (1. 26) contributes to the total susceptibility depends on the chemical and crystalline composition of the substance in question. The ionic structure of NaC1, for example, would not be expected to contribute a dipolar term, i. e., Edp. = 0. An examination of Eq. (1. 26) term by term will show in greater detail how each contribution affects the total susceptibility in the light of classical theory. For Xe, the electronic susceptibility, experimental evidence indicates the validity of the Clausius-Mosotti law and the Lorentz field for those dielectrics whose major contribution to the susceptibility is electronic in origin. This is the case, for example, in nonpolar gases and covalent solids. According to classical theory, an electron bound elastically to a nucleus has the properties of a harmonic oscillator with a resonant frequency wo = I/R/m7, where R is the restoring force constant and m is the mass of the electron. This results in an electronic polarizability which is given classically by 2 a = 2 (1. 27) e 2 2 t -w 0 For most dielectrics the resonant frequencies wo are in the optical frequency range or higher. For dielectrics in the optical region and for frequencies sufficiently removed from absorption or resonance, the relationship between the index of refraction and the dielectric constant is given by Maxwell' s formula: nr = e. 14

W I L LOW RU N LABORATORI ES 2900-121 -T The notation E will be used to indicate the dielectric constant at the highest frequency of interest. It is thus possible to measure the index of refraction of certain dielectrics and from this information to determine the contribution of electronic displacement to the dielectric constant. Table II shows the dielectric constants of various types of dielectrics for three frequency regions, the low-frequency or audio range, the high-frequency or microwave range, and finally the optical frequencies. The compositions shown in the table represent different types of chemical binding. Diamond and sulfur are typical of the covalent bond, and for these materials the atomic nuclei are tightly bound in their equilibrium positions by strongly directed electron bonds. It is seen that almost all of the polarizability of these solids results from electronic displacements.7 On the other hand, NaCl and LiF, which are ionic salts, have, in addition to electronic displacements, polarizability as a result of harmonic displacements of the ions in the electric field. For the case of a polar liquid such as H 20 there is, in addition to the ionic and electronic polarization described above, the possibility TABLE II. DIELECTRIC CONSTANTS FOR VARIOUS DIELECTRICS Hiqu0 C S NaCl LiF TiO TiO BaTiO BaTiO liquid (diamond) audio 81 5.7 4. 1 5. 9 9.0 170 86 200 4000, wave 35* 5. 5 4. 1 5. 9 9. 0 170 86 100** 2000*:* 1.78 5.8 4.0 2.4 1.9 8.5 6.9 5.5 5.8 of 2% - 100% 100% 40% 20% 5% 8% 5% 0. 3% total lXoo of 1% - 100% 97% 29% 11% 4. 4% 7% 4. 5% 0. 2% total ]I and I indicate, respectively, parallel and perpendicular to the optical or polar axis. *The microwave behavior in water is a relaxation phenomenon. This figure is at 250C and 2. 4 x 1010 cps. *"These values are taken at frequencies higher than that for elastic resonances. The differences between eaudio and emicrowave appear largely as a result of mechanical clamping. 7At extremely high frequencies, e. g., in the optical range, only the electrons follow the field. Atomic nuclei and molecules cannot follow the optical fields because of their inertia. 15

2900-121-T THE UNIVERSITY OF MICHIGAN of alignment of the dipolar molecule H20 with the electric field. In water, the low-frequency permittivity of 81 is mostly a result of the dipolar mechanism. It is further seen that as far as small fields are concerned, the permittivities in TiO2 and BaTiO3 have even smaller contributions from the electronic polarizability than the ionic salts. Theoretical expressions for the ionic polarizability have been developed by Born (Ref. 15) and others (e. g., Frohlich, Ref. 16) based on vibrations and normal modes in ionic lattices. For a simple one-dimensional crystal containing two types of ions of mass m and M and charges de, Born obtains the equations p e 2E 1+ e E 3( M)' (1. 28) 2w a 0 from which e2 1 e ( - + 1 (1. 29) i 2 3 m M (1.29) 6w a 0 or 2o /E= 2 +e (1. 30) 2 3 m M w a 0 where AE is the difference between the optical dielectric constant and the dielectric constant due to ionic displacements, wc is the infrared absorption frequency, and a is the nearest 0 neighbor distance in the one-dimensional crystal. The development of the Born equations above neglects the effect of the local fields. If the parameters for NaCl are substituted in the above equations, the theoretical value of AE = 2. 7 is obtained as compared to that observed of about 3. 4. This is a difference of approximately 30%. It is interesting to note the effect of the local field on the Born equations above. Consider E = E + 3P as we have done in Eq. (1. 25) where 0 E here is the applied external field, E is the local field, and 3 is the generalized "Lorentz factor. " If this is substituted into Eq. (1. 28) one may easily obtain an expression analogous to Eq. (1. 30). We will call the quantities on the right-hand side of Eq. (1. 30) identically (Ae), 0 that is, the ionic permittivity, under the assumption that the local field is the same as the applied field. Thus we obtain: (/E) (A. = (1. 31) (aE) 1- 3 16

W I L LOW RU N LA BO RATO R I ES 2900-121 -T 4w If we use the (A\E) 2. 7 and 3 = - for the Lorentz field, then AE = 27, an order of magnitude 4w too large! On the other hand, if 3 = 2 + 1as for the case of the Onsager field where we take E = Es, the static permittivity, which is 5. 9, then we obtain aE = 3. 4, exactly the value obtained via experiment. The above is a rather striking example of the overcompensation which results from the Lorentz field. As the reader may recall, the basic assumptions of the derivation by Lorentz were that the induced dipoles all had the same constant moment and that there were to be no thermal agitation effects. The Onsager field takes these effects and the close-range dipole-dipole interactions into account. Although Onsager' s derivation proceeded with the assumption of permanent dipoles, one would expect similar behavior for the case of a lattice of induced dipoles under the action of thermal agitation - especially in the static or infinite wavelength limit where the disorientation of one dipole will evidently affect its neighbors. A recent paper by Toupin and Lax (Ref. 17) shows that the Onsager result is valid in a lattice of dipoles which are in part induced and in part permanent. The treatment of ionic polarizability here is in reality quite crude but is intended to illustrate the concepts and methods of attack. The theory of ionic polarizability plays a very important role in infrared spectra, or more precisely, the converse may be said. Extensive treatments of the subject abound in the literature. The dipolar or orientational contribution to the electric susceptibility was first introduced in connection with the Onsager field, and this led to Eq. (1. 24). This equation gives the electric susceptibility due to dipoles as: N/A2 Xd - 3kT (1. 32) with the notation being the same as in Eq. (1. 24). These results were derived in a general way for dipoles which were free to rotate. If one were to view the situation, with Kittel, Fr6hlich and others, of a solid polar crystal in which the dipoles had but two orientations - either parallel or antiparallel to the applied electric field, then a result similar to Eq. (1. 32) holds. If a and b are the parallel and antiparallel sites, respectively, and n and nb are the I a b number of dipoles in each site, one obtains the following via the Boltzmann distribution: I E N kT n = e a 2 and HE N kT nb=-e and normalization requires N + nb = N, i. e., the total number of dipoles per unit volume. Thus the excess aligned with the field is 17

2900-121-T THE UNIVERSITY OF MICHIGAN N {T kT HtE n -nb -e =N' a b2 kT' forkT<< 1. Now n - nb is the net number of dipoles per unit volume so the p(n - nb) is the kT a b a b total dipole moment per unit volume or P. Thus P N (1.33) - (1. 33) d. p. E kT' a polar susceptibility three times as great as for the freely rotating molecule but of the same general form. 1. 4. MECHANISMS OF DIELECTRIC LOSS AND THEIR RELATIONSHIP TO POLARIZATION Whenever a dielectric medium is polarized, work must be done. To the extent that the work done in polarizing a substance can never be completely reclaimed, the dielectric is said to have losses. The source of the energy may be the battery which charges a capacitor and thus polarizes the dielectric, or some electromagnetic wave such as a microwave field propagating in a dielectric medium. In the preceding discussion of polarization, the total susceptibility of a dielectric was taken as a superposition of several components - ionic, electronic, and dipolar. It seems natural to treat the question of dielectric losses in a similar manner. That is, it may be said that each particular contribution to the total polarizability has associated with it a specific loss mechanism. Again, it must be understood that the electronic and ionic components are not strictly independent. Compared with the abundance of literature on dielectric polarization there is surprisingly little analysis available on dielectric losses. Most of the analysis is phenomenological or macroscopic in approach. One can conceive of at least five distinct ways in which a dielectric material may exhibit losses. These are: (1) Acoustical Loss. All dielectrics show some degree of electrostriction and one could expect, for the most part, that the greater the polarizabilities, the greater would be the electrostriction. This should be even more evident for small fields in the ferroelectrics which are piezoelectric; i. e., the mechanical extension is a linear function of field in contrast to the quadratic extension for the electrostrictive substances. As a consequence of electrostrictive or piezoelectric "pumping" of the atmosphere or environment, an acoustical wave might be initiated. The energy in this wave is lost from the dielectric system. (2) Joule Heat Loss. Because no dielectric is a perfect insulator, there will be an i R loss due to the conduction. The magnitude of this conduction depends on the type of dielectric, the number of carriers available, etc., but one would expect this type of loss to be restricted to fairly low frequencies for most dielectrics. 18

W I LLOW RUN LABORATORIES 2900-121 -T (3) Frictional Loss. Along grain boundaries, domain walls, fissures, etc., one may assume relative motion of one part of the dielectric with respect to another part as a consequence of electrostrictive or piezoelectric activity. The total loss from friction would be a function of the net wall area. In isotropic ceramics, for example, one would expect this loss to be much lower than for a ceramic composed of nonisotropic grains, because in the former the electrostriction affects each grain in the same way; thus there would be little relative motion of one grain against another. For the nonisotropic media, there will be a distribution of orientations of grains, and each orientation has in general a different electrostrictive or piezoelectric coefficient; therefore, comparatively large relative motions of one grain (or domain) against its neighbor would be expected. (4) Radiative Effects. The polarization of the dielectric as a whole under the influence of a high-frequency field can be considered an oscillating dipole. If a dimension of the dielectric approaches one-half wavelength, then a coherent electromagnetic radiation from the dielectric may result. (5) Molecular Loss. The effects listed in items (1) to (4) above are macroscopic in nature. The molecular loss contribution is taken from a more microscopic viewpoint. The atoms, ions, and molecules in a dielectric are in a state of thermal agitation. For an ionic lattice there are certain characteristic modes or frequencies in the infrared which represent vibrations of the ions or groups of ions against one another. A polarizing field or wave in the dielectric in the interaction with the ions can excite these infrared vibrations and therefore lose energy from the original fields. Similarly, a polar molecule can be thought of an continually interacting or colliding with its surroundings and energy can be exchanged as a result of these interactions. Thus all the interactions which lead to losses occurring on the molecular or microscopic scale during electronic, ionic, or molecular polarization can be grouped under the general heading of "molecular loss." Specific models will be formulated for the above mechanisms of loss in Sec. 4, where the particular ferroelectrics are considered. For now, we will examine the relationship between the polarizability and loss. With the macroscopic approach the loss is usually taken into account by defining a complex permittivity: E = E' - iE", (1. 34) with E" being proportional to the dielectric loss and tan 6 = defining the loss tangent. A complex frequency variable is defined such that p _ - + iwto, with r and o being the real and imaginary components of the complex frequency, respectively. If E is then considered as an analytic function of a complex frequency variable C + iW, then it follows from the theory of complex variables that E' (ar, t) and E"(o-, t) are not independent of one another in the frequency plane. Fr6hlich (Ref. 16) arrives at the same conclusion through the use of superposition and the convolution integral. He defines a decay function a(t) that takes into account the inertia of the polarization within a dielectric upon the sudden application or remroval of an electric field. The a relates the D and E of a dielectric in the following way on the sudden removal of the field E, at the time t = u: 19

2900-121-T THE UNIVERSITY OF MICHIGAN D(t - u) = a(t - u)E(u), for t > u, and ca(t - u) -' 0 as t - o. By an integration of the above for the case of sinusoidally periodic fields Fr6hlich writes: 00 oo E'(w) - E =fa(X) cos wXdX, (1. 35) O0 ET(W) a (X) sin wX dX, (1. 36) 0 and since the a(X) is a parameter appearing in both Eq. (1. 35) and (1. 36) it may be eliminated, upon which: oo 00 E'(t) -E2 E"(m) -d (37) and oo E t(L) = J (E'U) - E) 2 2 d/. (1. 38) O p -w 0 P Li Here E, has the meaning of the permittivity at the highest frequency of interest, and for the 2 present case this will be the optical values, i. e., E,0 - n. In order for the derivation to be valid it is necessary to assume that D and E are linearly related - an approximation that must be made for very small fields in ferroelectrics. These equations indicate that once either the losses or the real part of the permittivity are specified as functions of frequency, the other part must be completely specified. These equations will be used in the analysis of the high-frequency behavior of the ferroelectrics. Equations (1. 35) and (1. 36) are particularly useful when considering molecular models of polarization, since the term a(t) can be derived on the basis of a statistical analysis. The a(t), therefore, provides a link between the microscopic and macroscopic phenomena in the dielectric. If in a particular dielectric the ca(t) is exponential in character, that is, if upon the sudden removal or application of the field, the time average of the polarization decays or grows exponentially, then Eq. (1. 35) and 1. 36) or (1. 37) and 1. 38) lead to the famous Debye equations. Let a(t) = ke - -, where k is some constant and T is a time constant. Using this form of c(t) in Eq. (1. 35) and 1. 36) gives 20

W I LLOW RU N LABORATORI ES 2900-121-T oo 0 1 l+w T and oo E (O) = ke /T sin ct dt = k 2 (1. 40) Equations (1. 39) and (1. 40) may be written in a slightly different form by noting that if t = 0 in Eq. (1. 39) one obtains E' - = kT, where E' is the static value of the permittivity. Thus S S we arrive at the Debye equations for dielectric relaxation in the following form: E E s E' () - Eco 2 2 (1. 41) l+w T and (E' - E0)WT () 2 2 (1. 42) 1+w T It may easily be shown from electrical network analysis that the following equivalent circuit gives the same type of equations: withC = C' - jC" E -C' R" R C" C* CO C T - RC E - C +COO k -- R 1. 5. SUMMARY In the preceding sections some of the properties of the high-permittivity ferroelectrics were reviewed, with primary attention on BaTiO3. General concepts of the local field and polarizability for a dielectric were introduced, and polarizability was further examined in connection with electronic, ionic, and dipolar individual contributions. It was demonstrated that, under the assumption of a linear dielectric, the real and imaginary parts of the complex permittivity are not independent of each other. The ferroelectrics are distinctly nonlinear with regard to large fields. The exact mechanism of this large-field nonlinearity is still being investigated in 21

2900-121 -T T1HE UNIVERSITY OF MICHIGAN several laboratories, and at this time it is not completely understood. However, for small time-dependent fields this study takes the point of view that the substances dealt with are linear dielectrics, the permittivity of which is "set" by the instantaneous value of the large field. This study is basically concerned with the small-field polarizability. Small fields are considered those whose magnitudes are much smaller than the coercive fields of the particular ferroelectric crystal. The question of the onset of ferroelectricity in dielectrics is not a primary concern here. The approach is rather: given a particular ferroelectric or highpermittivity substance, what may be said of the small-signal polarizability and losses as a function of biasing fields and of frequency? Experimentally, the various contributions to the polarizability are separated by measurements of the permittivity as a function of frequency over very wide frequency ranges. As the frequency is increased, the various polarization processes, one at a time, no longer follow the time-varying field because of the inertia of the dipoles or ions, etc. (Fig. 4). The dipolar part of the spectrum, as can be seen in Fig. 4, shows a relaxation spectrum rather than the resonance absorption exhibited by the ionic or electronic mechanisms. This is explained on the basis that the dipoles have no restoring forces in the same sense that ions or electrons have in the crystalline lattice. Soon after the discovery of the properties of BaTiO3 and other isomorphic high-permittivity ferroelectrics, experiments of the type described above were initiated in Great Britian by Powles and Jackson (Ref. 18) and in this countrybyVon Hippel and his co-workers (Ref. 19). These measurements mainly covered the low-frequency to microwave range and the data were taken for polycrystalline (i. e., ceramic) samples of BaTiO3. The data have been smoothed and reported by Kittel in his textbook (Ref. 20) and the results are shown in Fig. 5. One notices a rather marked relaxation of the permittivity in the two decades between 108 and 1010 cps. In this frequency range there is a drop in permittivity of about one order of magnitude. To date, two explanations of this effect have been put forward. Kittel, in discussing the data in his textbook, suggests that the relaxation is caused by the inertia of ferroelectric domain boundaries, and Von Hippel has suggested that elastic resonances in the ceramic are the cause. The first explanation seems untenable, first, because the fields under which the relaxation was presumably measured were very much smaller than the coercive field of the crystal so that significant domain motion would not take place, and second because the ferroelectric domain, according to present knowledge, does not have the gradual wall transition that is ascribed to magnetic materials so that the concept of a wall inertia here has questionable meaning. Moreover, and this should be the coup de grace to both explanations, the relaxation has been reported by Von Hippel (Ref. 21) and others (e. g., Sharpe and Brockus, Ref. 22) for ceramic ferroelectrics at temperatures higher than their Curie point where the ceramic is not ferroelectric, domains do not exist, and the material is not piezoelectric. 22

W I LLOW RU N LA BORATORI ES 2900-121 -T 2000 1500 Es —-- DIPOLAR i IONIC000'opt EL~ —` —-T —--- -',500 r.f1.AND MICROWAVE INFRA-RED OPTICAL X RAY FREQUENCY -. * FIG. 4. SCHEMATIC OF CONTRIBUTIONS TO PERMIT- 10 10 10o 10 10 11 TIVITY IN A DIELECTRIC FREQUENCY (CPS) FIG. 5. THE SMALL-FIELD PERThere are certainly other explanations possible for the MITTIVITY VS. FREQUENCY FOR observed relaxation, two of which will be considered, name- CERAMIC BaTiO3. From the data by Powles and Jackson (Ref. 18) and ly, the possibility of a molecular relaxation of the dipolar Von Hippel (Ref. 19)- smoothed by type described in the preceding sections, and the possibility Kittel (Ref. 20). of a relaxation primarily caused by a loss mechanism. That is, since real and imaginary parts of the permittivity are not independent, anything which produces a significant loss term must also affect the real part. To arrive at an explanation, it was thought desirable first to attempt to reproduce experimentally the observed high-frequency behavior cited by Kittel for ceramics and then to examine the highfrequency behavior for single crystals of BaTiO3 with the high-frequency fields oriented both parallel and perpendicular to the polar axis. If the reported relaxation is basically a molecular phenomenon, one could expect that with the single crystal the relaxation would be extremely sharp and well defined, perhaps even being a resonance phenomenon. The type of model chosen for an analysis of the polycrystalline material would then depend on the results of the latter experiment. In the experiments performed, the single crystals did not exhibit any substantial relaxation at frequencies up to 4000 mcs. Therefore, since no particular molecular phenomenon need be invoked, it was found most convenient to base an analysis on a phenomenological or thermodynamic framework. 23

2900-121 -T THE UNIVERSITY OF MICHIGAN 2 The POLARIZABILITY and LOSSES in the PEROVSKITE STRUCTURE 2. 1. THERMODYNAMIC APPROACH One may study dielectrics with either a macroscopic phenomenological approach or with a model theory or microscopic approach. No new basic information is provided by the former approach, but rather various relationships are derived among the different parameters of the system in question. A thermodynamic "equation of state" is postulated, and from this equation and the laws of thermodynamics, the behavior of the system is derived. Provided that the fundamental equation of state is "correct, " the success of this method is guaranteed. It is the task of a model or microscopic theory or direct experiment to provide the equation of state. The method of applying thermodynamics to dielectrics is to write the combined forms of the first and second laws. Assuming reversibility for small fields and for the lossless case, TdS = dU - E dP +X. dx., (2. 1) m m 1 1 where the summation convention is used for repeated indices, S is the entropy, U is the internal energy, E is the externally applied field, P the polarization, and Xi and xi are the stress and strain components, respectively. Then the thermodynamic potentials used are defined as A = U - TS, the Helmholtz free energy, G = H - TS, the Gibbs free energy, (2. 2) H = U + X.x. - E P, the enthalpy. 1 m m According to the phenomenological theory presented by Devonshire (Ref. 23), one assumes that it is possible to express the Gibbs or Helmholtz free energy as a power-series expansion in the polarization and stresses or strains. Also, from Eq. (2. 1) and (2. 2) it can be seen that dA = E dP - X. dx. - SdT, m m 1 1 and (2. 3) dG = P dE + x. dX. - SdT. m m 1 1 For the particular case of the stress-free crystal with one direction of polarization (as with BaTiO3 in the tetragonal ferroelectric state), Devonshire writes: 1 2 1 4 1 6 A(P,T) = A' +-a'P +- b'P +-c'P +..., (2. 4) 24

W I LLOW RU N L A BORATORI ES 2900-121 -T where Al, a', b', and c' are empirically determined constants which may be, and in fact are, temperature dependent. The primes indicate that the constants are for a free crystal. The expansion is taken in only the even powers of P. since the free energy is independent of the algebraic sign of the polarization. It then follows that E =(i)+bPa+ +c'P +.... (2. 5) For zero electric field, if P7 and higher powers are neglected, 3 5 a'P +b''P+ c'P = 0, in which P = 0 is one solution, as in the paraelectric state. However, P need not be zero and is in fact finite in the ferroelectric state. For this case, 2 -b' ~ b' - 4a'c' -b' b' = 2' (2.6) 2c' 2c' The constant a' can be interpreted as the inverse isothermal susceptibility at P = 0 by noting from Eq. (2. 5) that I8E 1l 2 4 — = a + +c'P +.. 5c + (2. 7) \P T X If one considers temperatures above TO, the Curie temperature, where there is no spontaneous polarization, the limit for the small-signal susceptibility from Eq. (2. 7) is 1 C XIa' = T-T (2.8) a' T - T' The last term is the Curie-Weiss law and is determined experimentally. This in turn allows the evaluation of a'. C is the Curie constant. Several workers have made measurements in order to determine the constants a'b'c' and C. A tabulation of the more recent values is shown in Table III. It is noticed that the latest value of b' is temperature dependent. The freeenergy function of Devonshire allows the prediction, at least qualitatively, of most of the observed experimental data for BaTiO3 crystals. Some criticism has been raised as to the use of the same function on both sides of Curie transition. The justification of its use, however, is that the free-energy function does in fact allow a prediction of the results determined experimentally for BaTiO3 in both the ferroelectric and nonferroelectric states. This point is also tied in with the question as to whether the phase transition at 1200C, the Curie point, is a first- or second-order thermodynamic transition. In a first-order transition, the free-energy function is continuous, but its derivative with temperature (the entropy) has a discontinuity at the transition. For a second-order 25

2900-121 -T THE UNIVERSITY OF MICHIGAN effect, both the free energy and the entropy are continuous, but the derivative of the entropy (which is proportional to the heat capacity) is discontinuous. Experiments indicate that BaTiO3 undergoes a phase change of the first kind at 1200C but with such a small change in entropy that it practically behaves as if the change were of the second order. This may account for the success of the use of a smooth free-energy function on both sides of the upper Curie temperature. TABLE III. SOME RECENT EVALUATIONS OF THE COEFFICIENTS IN DEVONSHIRE'S FREE-ENERGY FUNCTION [ Values taken from table given by Devonshire (Ref. 23)] c a' b' c' Worker T-T 4 0 -12 -22 10 -1. 1 x 10 2.8 x 10 Merz (Ref. 24) 10 - T - 12 Devonshire (Ref. 10 4 -1. 2 x 10 3.5 x 10 23) and recal10 culated by Merz (Ref. 24) T-T 4 T 0 -12 -22 1i. 4 x 10 -0. 68 x 10 2.3 x 10 Merz (Ref. 24) 1.4x 10 T-T 4 0 -15 -23 2.7 x 10 -4.5 x 10 (T2 - T) 9 x 10 Huibregtse 2. 7 x 10 and Young (Ref. 25) To = 1100C T2 = 1750C It is instructive to examine the free-energy function in Eq. (2. 4) for various values of the coefficients a', b', and c'. The value of a' = T -TO is determined experimentally. Stability demands that A > 0 as P -> oo, so that c' > 0 is a necessary condition. The coefficient aP b' may be greater than or less than zero. Figure 6 shows a schematic of A vs. P for varying values of T (and therefore for varying a') for the cases where b' > 0 and for b' < 0. The system is in equilibrium under the condition that the free energy is a minimum. For the freeenergy function in which b' > 0, it is seen that the minimum occurs at P = 0 for temperatures greater than or equal to the critical temperature TO. As the temperature is lowered below To} the minimum smoothly and continuously shifts to finite values of P, corresponding to the 26

W I LLOW RUN LA BO RATO R I ES 2900-121 -T onset of ferroelectricity. Since for b' > 0 there is no discontinuity in P and hence no discontinuity in the entropy, the case in which b' > 0 represents a second-order transition. For b' < 0, the minimum in A jumps discontinuously from P = 0 for T > TO to some finite value of P. Hence b' < 0 represents a first-order transition. A I FOR b > 0 P FOR b < 0 FIG. 6. FREE ENERGY VS. POLARIZATION WITH TEMPERATURE AS A PARAMETER The experimental evidence to date is that BaTiO3 undergoes a first-order transition at its upper Curie point. It may easily be shown that the correct expression for the spontaneous polarization in this case is given by Eq. (2. 6) in which the term in parenthesis is taken with the minus sign. Thus in the neighborhood of To, 2 -b' a' P =-+ s c' b' forT < T, and - c 2 s 27

2900-121 -T THE UNIVERSITY OF MICHIGAN for T > T where T is the Curie temperature, which is not necessarily equal to T for a c c 0 first-order transition. An effect which is predicted from the free-energy function and which is of importance in analyzing the behavior of the ceramic dielectrics, is the shift in transition temperature with an applied field. This can be shown in the following way. The stability requirement demands that at a minimum in free energy a A> 2 > 0 aP From Eq. (2. 5) one has a 2A aE 2 4 = - a' + 3b' P + 5ctP > 0 (2. 9) 2 aP for stability. If one plots (Ref.. 24) P vs. E out of Eq. (2. 5), the hysteresis behavior of the single crystal becomes evident and this behavior is shown in Fig. 7. A value of P for a given aE E is unstable for which a < 0 and a discontinuous change in P takes place (along the dotted aE lines) until one has again a-> 0. The free-energy function predicts double hysteresis loops for T > T, and these have actually been observed by Merz (Ref. 24) and by Diamond and Orr (Ref. 26). These double hysteresis loops come about as a result of the dependence of the coefficient a' on the temperature. For T > To the a' term in Eq. (2. 9) is positive and the slope aE aof the polarization-field characteristic of Fig. 7 is also positive at E = 0. This means that at E = 0 the material is paraelectric, since P = 0 at E = 0 is a stable solution to Eq. (2. 5). Now as E is increased at temperatures T > TO, the second term in Eq. (2. 9), i. e., 3b'P becomes increasingly important. Since b' has a value which is negative, a field is soon aE reached at which -P becomes negative. At this value of E the polarization jumps discontinuously to a new value for which a- is positive and hence stable. This leads to the double hysteresis loops shown in Fig. 7 for T > TO. The double hysteresis loop actually amounts to a shift in the Curie point to higher temperatures with applied field since the single-valued part of the P(E) characteristic is paraelectric and the double-valued part is ferroelectric. It is also interesting to note that at sufficiently high temperatures the crystal will not become ferroelectric even under the strongest of electric fields. This amounts to a change from a first-order transition to a second-order one. This can only be possible if the coefficient b' is a function of temperature as has been found by Huibregtse and Young (see Table III). Knowledge of the rate at which the Curie point shifts with electric field will be required in a later section and may be estimated from Eq. (2. 5). In order to carry out the derivation, the following assumptions will be made. 28

W I LLOW R U N LA BO RATO R I ES 2900-121 -T (a) The double hysteresis loops represent a ferroelectric state induced by the field. (b) The field at which the ferroelectric state is induced is taken to be the field halfway between the sides of the double loop. The temperature associated with this particular loop is the modified Curie temperature. (c) This field represents an inflection point on the P vs. E characteristic. P --- T - To X 1 a b1 4 i 1 6 T - T A J A =- a P +- P c1 o 2 4 6 a - 10 J TA 1 1 3 1 5 FG,= E = a P +b P +c P -12 b -10 a A 1 1 2 1 4 1 -22 - - a + 3b P + 5c P c 3 x 10 p2 T - 1100C FIG. 7. THE P VS. E CHARACTERISTIC FOR A FREE BaTiO3 CRYSTAL BASED ON DEVONSHIRE'S FREE-ENERGY FUNCTION. [After Merz (Ref. 24).] Thus Eq. (2. 5) is rewritten as 3 5 E = a'P +b'P + c'P and the inflection points are - 6b'bP + 20c'P= 0. aP The inflection at P = 0 is of no concern here, so that the above becomes 2 3b' + 10c'P = 0, or P2 3b' (2. 10) for the point of inflection. Recall that b' has a negative value for a first-order transition. 29

2900-121 -T THE UNIVERSITY OF MICHIGAN T - To Substituting Eq. (2. 10) into Eq. (2. 5) gives the dependence of a' = on E. That is, for 3b' 2 convenience, let - f_ =, so that 10c' 3 5 E = a' + b'Q + c' and E b'P3'l,5 a' = - Substituting for a' results in C(E -b'3 5 T T -Q - (2. 11) 0 Here T" is the temperature associated with the inflection point at E and is the modified Curie point. T is the temperature to which-vs. temperature is extrapolated in the Curie-Weiss 0 X law. This is not the same as the observed zero-field Curie temperature. For E = 0 and P 4 0 the inflection occurs on the P-axis. For this case one has the temperature at which the ferroelectric hysteresis loop just begins to show a double character. This is the observed zerofield Curie temperature, T. Therefore, from Eq. (2. 11) with E = 0 the following is obtained: (- 2 4 3b 9b' (2. 12) T T C(-bVQ c c l — /. (2. 12) Numerically, " 13 )2 24 7_ 4 = x(1. 1) x 10 x 0. 7 c To 10 -220C. c 10 x 2.8 x 10 T is found experimentally to be 1100C, so that one expects the Curie transition at 1200C - just where it is observed for BaTiO3! By substituting Eq. (2. 12) into Eq. (2. 11) one obtains the relationship between the zero-field Curie temperature, the "new" Curie temperature, and the field. That is, CE T''- T = ~0 + T - T Hence C E T -' (2. 13) 0 -3b' This last result is rather important, and we will make use of it later. It shows that the shift in Curie point is a linear function of E. Numerically, using Merz's values of C = 10, -22 -12 AT* -3 c' = 2.8 x 10, b' = -1. 1 x 10, one obtains AE 10 ~C/volt/cm. If the temperature30

W I L LOW RU N LA BO R A T R I ES 2900-121-T dependent value of b' (Huibregtse and Young's value in Table III) were to be used, the result (Eq. 2. 13) would be slightly more complicated. One notes from this relationship that the shift in Curie temperature with field occurs only for b' < 0 and c' > 0, i. e., for a first-order transition. Experimentally, the shift in Curie temperature may be measured from the double hysteresis loops. By means of the apparatus shown schematically in Fig. 8 one may display these hysteresis loops at various temperatures. The specimen is placed in a temperature bath of transformer oil and excited by a 60-cycle, 1000-volt power transformer. The x-input to the oscilloscope receives a suitable part of the driving field of the divider R1R2. The polarizing current i, which flows through the specimen, is integrated by the capacitor C, producing an output voltage proportional to the polarization. If P is the polarization in coulombs per square centimeter and A is the electroded area of the specimen in square centimeters, then p = - idt, and since V idt, then P = V. A The above enables one to calibrate the oscilloscope for polarization. Figure 9 shows a series of double hysteresis loops which have been traced off the oscillograms for a BaTiO3 single crystal. The zero-field Curie temperature seems about 50 lower than most of the reported data for single crystals. This may be due to impurities in the crystal.8 Figure 10 shows the shift in Curie temperature with field, the data being taken from the photographs of Fig. 9. The measured shift in this experiment is 1. 2 x 10 ~C/volt/cm, consistent with the theoretical value predicted by Eq. (2. 13). To recapitulate the preceding section, it has been shown that by starting with the freeenergy function for BaTiO3 postulated by Devonshire, it is possible to predict a shift in Curie point with an applied field. An experimental determination of this shift has been made, and the observed value is in fair agreement with that derived from the thermodynamic analysis. This result will be used in a later section in order to explain the observed large-field nonlinearity in the perovskite-type ferroelectrics on the basis of an "induced ferroelectricity. " The shift in Curie temperature upward with field implies such an induced ferroelectricity. 8The crystals were prepared by the writer in the Solid-State Devices Laboratory at The University of Michigan by growing them out of a high-temperature melt. 31

2900-121-T THE UNIVERSITY OF MICHIGAN THERMOMETER TO VARIAC VACUUM FLASK TRANSFORMER OIL TEST SPECIMEN XFMR R3 R1 ~~~VA 3x33 OK Zw.' C J1.5p, f 33K >> 600V CRO X Y O0FIG. 8. THE P-E HYSTERESIS LOOP PLOTTER 32

WI LLOW RU N LABORATORI ES 2900-121-T 120~ E 1180 1150C 1160C ~ 111~ 85~C 2 ~ 1170C!10o 0 5 I0 15 20 E (volts per mil) ______ FIG. 10. CURIE TEMPERATURE ASA 10,00 C FUNCTION OF FIELD 118~ C 2. 2. RELATIONSHIP BETWEEN THE FREE AND CLAMPED COEFFICIENTS IN THE FREE-ENERGY, ISOTHERMAL, AND ADIABATIC CONDITIONS The values for a', b', c', c, TO, and Tc are all taken under the assumption of zero 10 C 119~ C stresses in Eq. (2.3). This essentially means F IG. 9. DOUBLE HYSTERESIS LOOPS IN A that the crystal is allowed to strain freely in c-ORIENTED BaTiO3 CRYSTAL compliance with an electric field. Experimentally, at sufficiently low frequencies these conditions are met. As will be shown via dynamic spectra in Sec. 3, the crystals of BaTiO3 become effectively clamped at frequencies 7 on the order of 10 cps. According to the behavior of these types of ferroelectrics in the uhf and microwave regions, a free-energy function having coefficients obtained under clamped conditions is required. A more general expansion of the Helmholtz free energy containing terms in the crystal strains has been expressed by Devonshire for the case of the strained cubic crystal as: A = A +2a" P1 + P2 + P3 + + 4ll 1 + P2 + P3 (2. 14) +4bl2 (22 P2 P3 + P P2 + P P3 (Continued on page 34) 33

2900-121 -T THE UNIVERSIT Y OF MICHIGAN 2 11 1 + 12 (x23 1x3 +1 1 2) (Continued from p. 33) 1 /2 2 + 44 + 5 + 6(2. 14) 2244 4) q11x 1 P P2 x3 ) q12 [1 (P2 [ P 32) +x2 (3 +12) x3 21 + P2 + q44 (x4P2P3 + x5P3P1 +x6P1P2) +.. terms in p6 etc. In the above, a", b" are defined as the dielectric coefficients for zero strain, analogous to the a' and b' defined for zero stress. The subscripts 1, 2, and 3 on P and x indicate, respectively, the x, y, and z components of the polarization and strain. The components x4, X5, and x6 are shear strains. The q's are electrostrictive coefficients, and the c's represent the elastic coefficients corresponding to the Young's modulus. Gibbs free-energy function, for the stress-free crystal, may be put into a form analogous to Eq. (2. 14) by replacing in Eq. (2. 14) the double primes by primes and the strains x by the stresses X. This will also involve different elastic coefficients obtained by replacing the c's by new coefficients s, and the q's by Q. With the free-energy function stated in the form above, one treats the crystal in the ferroelectric state as if it were a cubic isotropic crystal which has been strained under the influence of its own spontaneous polarization or "self field." Under these conditions P -> P in the expression for the free energy. The function as stated in Eq. (2. 14) is consistent with the usual equations relating stress to strains and polarization (see also Eq. 2. 3): (Ai)P X ci ijxi qijkPjPk P,T Because of the symmetry between the two a - axes in the tetragonal structure of BaTiO3, c12 =13; c23 C32 0. Devonshire further shortens the notation for qijk to read 9qll - qll q122 - q12 q133 - ql3 - q12 q123 = q321 = q231 =0, etc., 34

W I LLOW RU N LA BORATORI ES 2900-121 -T and P P. 1 s It is therefore possible to derive piezoelectric coefficients from the electrostrictive coefficients for P = P in Eq. (2. 14), viz: 1 s =x1= c x + ~C12(X2 + X3) + q 1 12 (2 3 \ i/P, T and (ax> b =2 P ( a P 1=2l 11 1 Here bll is a piezoelectric coefficient and may be evaluated from a knowledge of the value of q11 and by allowing P1 -> P. Devonshire relates the coefficients for a free crystal to those of the clamped crystal by noting from Eq. (2. 3) that (-)A = X. and setting Xi = 0, obtaining: T-T at = a" =10 2 q1(C + c) + 4qqc - 2q2 c bt 2 11 2 2q111 12 1 11 11 (Cll + 2c 12)(c 11-c12 (2. 15) 2 2, ql C 12 2q12 11 -12 (-cll 2 1 12 b b2t - C 2)(c + 2c ) 12 12 (c 11 -c(12 11 12 The derivation of this result is given in the Appendix. These relationships are of importance in that they enable one to relate the behavior of the clamped crystal to that of a free crystal. For example, the relationship between the free and clamped permittivity may be shown. Let us examine this relationship for a BaTiO3 crystal viewed along its c-axis. It has already been noted from Eq. (2. 5) that -E a'P + b' p+ ( )X, 1 1 11 1 where P2 = P3 = 0 (i. e., there is no polarization along the a-axis). One may also write from the free-energy function Sp ) = E =a"P + b" P + Plx 1x 1 111''' 35

2900-121 -T THE UNIVERSITY OF MICHIGAN (E and since(-p )1 -,c it follows that 11 1q2(c + c )-4q q c +22c 2 2 1 1 6q 112 12 2 c c 1 The latter formula indicates that the small-signal (i. e., for P -> 0) permittivities for the free and clamped crystal above the Curie temperature are the same. For temperatures below the Curie temperature P P, the spontaneous polarization, and a significant difference in the permittivities is expected. If the crystal is viewed with a small signal along its a-axis but is polarized in the c-direction, the free and clamped permittivities are found as before from Eq. (2. 14) viz: __A - E a"P2 +b' P b P P P2 + b'" P 2P \P 2P, X, T 2 2 12 3P 2 1 1 2 + 2qllx 2P2 + 2q12(x+ x3)P2.. 1 2 2 12 1 3)P2 For x = O and P3 = 0, I-~- = a +b" I P +3b" P (a2/ Xa 1 2 11 2 For the small-signal limit, P2 0, and X =a" + b'2 2) (2. 18) XaITa Similarly for the free crystal: 1, 2\ _= a' + b2 (P1) + (2. 19) Xa, 12 Letting Pi = P. the spontaneous polarization along the c-axis, and letting the induced polarization along an a-axis be P with P = 0, one obtains with the aid of Eq. (2. 15): 36

W I LLOW RU N LA BORATORI ES 2900-121-T 2 P,(2. 20) 11 1 12 2c11 ) 12 ll 2 Xa, Xa, (Cll c12 12) (c Ill+ 2c12). Here, as for the case for the c-axis, when the spontaneous polarization goes to zero, the free and clamped small-signal permittivities are the same. In a recent paper, Berlincourt and Jaffe (Ref. 27) have presented some experimental evaluations of the elastic and electrostrictive constants for BaTiO3 single crystals. These data are presented in terms of the elastic compliances si. rather than in the modulii c... Since one set of coefficients forms the inverse matrix of the other set, it is not difficult to compute the c's from Berlincourt and Jaffe's values for s. The electrostrictive coupling constants q and Q are defined by Eq. (2. 21). Xh =hiXi + qhke PkP and (2. 21) x. = -s..X. + Q. P P 1 1j imn m n Some of these constants are shown in Table IV. TABLE IV. SOME VALUES OF ELASTIC AND ELECTROSTRICTIVE COEFFICIENTS FOR BaTiO3 SINGLE CRYSTALS Constant Value in esu Investigator -12 sl 0. 725 x 10 Berlincourt and Jaffe (Ref. 27) -12 s12 -0. 315 x 10 Berlincourt and Jaffe (Ref. 27) -12 s13 = S23 -0. 326 x 10 Berlincourt and Jaffe (Ref. 27) s33 1. 08 x 10-12 Berlincourt and Jaffe (Ref. 27) 3312 3. 1 x 10 Mean value computed from two values given by Devonshire (Ref. 23) -12 Q12 -1. 23 x 10 Mean value computed from two values given by Devonshire (Ref. 23) 12 C 2.4 x 10 Mean value computed from above data, C~~~12 911 ~1. 7 x 101 Mean value computed from above data. q 1 1 3. 2 Mean value computed from above data. 9~2 ~ 0. 28 Mean value computed from above data. 37

2900-121 -T THE UNIVERSITY OF MICHIGAN From these equations one has Qimn ih hmn' and (2. 22) qhmn C hi imn By substituting the coefficients from Table IV into Eq. (2. 16) one obtains for the cdirection: 2 cX 6 2 11 11+12) q111212 1+ 2c cx 4w cXs (Cl1 + 2c )(c - c ) Ecx 1 cX s L (C 1+2c12 11 12 (2. 23) 1 + 6 168 x 0.64 [(3. 2)2(4. 1) - 4(3. 2)(0. 28)(1. 7) + 2(2. 8)2 (2. 4)1 4wT 102 (58)(0. 7) 10 or cx where ecX is the permittivity at zero stress and e is the permittivity at zero strain. That cX cx is, as the crystal is clamped, the polar-direction dielectric constant is reduced by a factor of 1. 5. The experimental results cited in Sec. 3 show about a twofold reduction in permittivity. This is not a bad correlation considering that the values for the Q (from which the q were calculated) given by Devonshire are fairly crude. Applying the same constants to Eq. (2. 20) yields for the crystallographic a-direction: P2 2 - 2q qc - q 2(c - 2c aX aX +qllc12 2q11 12cll1 12 11 12 = 1+. 1 e 2 r (c1 - c 2)(cll + 2c2) ax 11 12 11 12 or (2. 24) eaX 0.64 2900 (3.2)2 x (1. 7)- (6. 4)(0. 28)(3. 2) +(0. 28)2(1) 1+ x''. ax 2 102 (58)(0. 7) ax 10 aX ax_ -2. C ax This result is almost identical with the observed data. The following conclusion may be drawn. Relaxation of permittivity with frequency in the BaTiO -like crystals or ceramics due to mechanical clamping will not reduce the permittivity by more than a factor of about two. 38

WI LLOW RU N LA BORATORI ES 2900 -121 -T This result is especially interesting in view of the data cited by Kittel, reproduced in Fig. 5, where a reduction in permittivity by a factor of ten or more is noted within a frequency range of 100: 1. It was this remarkable graph which motivated the experiments on the single crystals in order to: (1) determine whether the results of Eq. (2. 21) and (2. 22) are experimentally verified; and (2) follow up this investigation by pursuing the question that if the clamping is indeed limited by the factor 2, then can the data of Fig. 5 be correct? A positive answer to (1) and (2) would definitely rule out an explanation of the microwave dielectric relaxation based solely on the clamping concept. This issue will be discussed after the experiments and the experimental results have been described. It is now of interest to consider the difference between the isothermal and adiabatic permittivities. In all of the thermodynamic development up to this point, the dielectric suscepti(52A (52Aor bilities have come out of terms such asor 2. That is, the susceptibility is ap xT SaP xT taken at some fixed value of temperature while the other thermodynamic parameters are being varied. This condition strictly holds for static dielectric measurements only. For alternating fields it is the entropy which is constant, i. e., the measurement of susceptibility is taken under adiabatic conditions. The isothermal and adiabatic susceptibilities can be related to one another in the same sense that the free and clamped values are related. A consideration of the case of a free crystal will show this connection. The susceptibility is found via the free-energy function as X....a.A =. a! + 3b'P..., (2. 25) T P X,T the subscript T being used to indicate the isothermal condition. If the electric field is considered to be a function of polarization and temperature, that is, E = E(P, T), then dE 3 dP + (T) dT, T P from which (SE ~ = ~ (2>E /+T\ (aP) / aE +(P a/TJ EPP)S (2. 26) ~apS aT P S The subscript S indicates an isentropic process. From the definition of the Gibbs free energy one has ( aGT = -S and () = -P TE T from which a2G S aS. T = _(aS) (2 27) aTaP aPOT a- S aE P 39

2900-121 -T THE UNIVERSITY OF MICHIGAN Substituting this into Eq. (2. 26) one obtains \_E \E \ Ka )a a)T (+ aT)KaS )p* (2. 28) Now aE \ /aE \ aT Pas p T P p and -)pT' aE 1 where Cp is the heat capacity at constant polarization. Since ap =-, one may then write P ap X Eq. (2.28) as: /aE ) T -= -: T + P. (2. 29) XS XT C From the free-energy function, E = ) = aP + b'P +..; and P O ~_T- T0- P aT p P -- K P aPT 104 (2. 30) if b' is not temperature dependent. Therefore, Eq. (2. 30) with Eq. (2. 29) gives the result that 2 1 1 P T X..XT 8 (2. 31) XS XT 10 C P The heat capacity of barium titanate is of the order of 0. 9 cal/cm3/deg. = 4 x 107 ergs/cm3/deg. 4 Taking P = PS = 0. 78 x 10 esu and T = 4000K, one has 1 1, ~0.6 x 10 x 4 x 10 -6 ~XT 8 7 6x10 XS XT 10 x 4 x 10 or in terms of e 1 1_ 6 x106 510-7 ---— x 10 6- 5 x 10 cS T 4r a difference which is clearly so small that it makes no practical difference in our measurements of E. 40

WI LLOW RU N LA BORATORI ES 2900-121 -T This section concludes with a summary. In order to ascertain whether the reported strong microwave relaxation effect in BaTiO3 ceramic could be explained on the basis of clamping, it was considered desirable to determine the theoretical magnitude of this effect based on the free-energy function of Devonshire. The calculations have shown that by clamping a BaTiO3 crystal the dielectric constant is reduced roughly to one half of the "free" value. Experiments to be described in Sec. 4 completely confirm this behavior. It was further recognized that isothermal and adiabatic measurements of the permittivity do not in general lead to the same results. An estimate of this effect in BaTiO3 has been made, and it was shown that for this case the difference between the adiabatic and isothermal permittivities is negligible. 2.3. SOME POLYCRYSTALLINE PROPERTIES For many of the applications of ferroelectrics to devices which employ the nonlinearity in the permittivity with field, single crystals are unsuitable. In the first place, the single crystals are rather difficult to make, and in the second place, the properties of the crystal are somewhat different from those of a polycrystalline ceramic - especially if the ceramic contains additives. The high-permittivity perovskite mixed ceramics such as Ba Sr TiO3 and x 1-x 3 Pb SrlX TiO3, for example, exhibit a rather strong dependence of the incremental permittivity on electric field and temperature over a comparatively wide temperature range (compared, that is, to the single-crystal properties). The basic mechanism for the change or switching of polarization in the single crystal has been shown by many investigations to be essentially a domain growth and alignment process. However, an attempt to extend the idea of domain alignment and growth to the polycrystalline dielectric leads to some contradictions which shall be presently discussed. It is therefore of considerable practical interest to examine the mechanism of polarization for the polycrystalline ferroelectrics in light of the theory developed up to this point. First, however, it is well to review some of the properties of ceramic Ba Sr TiO3 which will serve as a model for all the mixed ceramics of this class x l-x 3 of ferroelectrics. In the polycrystalline ferroelectrics it is often necessary to consider the temperature at which the dielectric exhibits maximum permittivity. This is in contrast with the single crystal in which the Curie point is also the point of maximum permittivity as well as the transition between the ferroelectric and nonferroelectric state. For the ceramic this transition takes place gradually over a wide temperature range. It was discovered very early by several investigators (Ref. 28 and 29) that for cation substitution in the Ba site of BaTiO3, the effect is for cations, such as Sr, Ca, Mg, Fe, Sn, Pb, 9The value of x, of course, ranges from 0 to 1. 41

2900-121-T THE UNIVERSITY OF MICHIGAN and others, to lower or raise the Curie temperature. Thus it is possible to have a mixed crystal or ceramic ferroelectric with a room-temperature Curie point by the inclusion of the proper amount of additive. The "proper" amount for Ba Sr TiO3 is x = 0. 65. x 1-x 3 Extensive experimental data have been taken in our laboratories on Ba-Sr-TiO3 and isomorphic ferroelectrics with regard to the electric field and temperature sensitivity. In order to facilitate the handling of a large volume of routine data it was arranged to have the incremental permittivity at a fixed frequency (10, 000 cps) plotted automatically as the temperature and electric field were varied. The experimental arrangement is shown in the form of a block diagram in Fig. 11. In this diagram it is seen that a small signal is fed through the specimen and a resistor R2 whose resistance is much smaller than the reactance of the specimen. The signal voltage across R2 will thus be proportional to the capacitance and therefore to the permittivity of the specimen. This voltage is rectified, amplified, and placed on the Y-input of a Mosely X-Y recorder. The biasing electric field is supplied through resistance R1 by a motordriven, high-voltage, variable power supply, the output voltage of which is sampled and fed onto the X-terminal of the X-Y recorder. With this apparatus the incremental or small-signal permittivity is plotted automatically vs. electric field with the temperature as a parameter. X-Y RECORDER i I I I I DRUM PEN INPUT INPUT MOTOR DRIV- 0,000 CYCLEMPLIIER EN D.C. SUPPL SIGNAL GENER TO-PROVIDE AND DETECTOR FIELD.01 VOLTS TEMP. MEASURING THERMOCOUPLE FIGc11,AUOAI RECORDINGOR THERMOMETER EA(ET) R _-l _ 0DEWAR FLASK SPECIMEN_ A OIL BATH HEATING AND — COOLING COIL FIG. 11. AUTOMATIC RECORDING UNIT FOR MEASURING eA VS. E AND T Typical results for a Ba0 65Sr0. 35TiO3 ceramic composition are shown in Fig. 12 where 42

W I LLOW RU N LA BO RATO R I ES 2900-121 -T vs. the electric field and temperature as the independent variables. It is evident that the Curie region is extremely broad for the zero-field case and becomes even further spread out in temperature as an electric field is applied; consequently, the temperature of peak permittivity becomes difficult to measure. Nevertheless, a very definite shift in the peak toward higher temperatures is observed and has the value of about 1. 4 x 10-3 ~C/volt/cm, which is roughly that which is measured for a BaTiO3 single crystal. This behavior is to be compared with that of the pure BaTiOG ceramic shown in Fig. 13 and that of the BaTiO3 single crystal shown for zero-bias field in Fig. 2. The pure ceramic shows a rather sharp Curie peak, and what is extremely significant, there is very little dependence of permittivity on the biasing field in the room-temperature region. This constitutes experimental evidence that ferroelectric domains are not being aligned under the action of the field, since any alignment in BaTiO3 (which has such large anisotropy in its permittivities) would certainly appear as a marked decrease in the incremental permittivity. Further evidence is noted in the data shown in Fig. 12 for the mixed ceramic. It is seen that on the lowtemperature side of the Curie peak, as the temperature is decreased, the large-field nonlinearity is decreasing, despite the fact that the number of domains is presumably increasing (i. e., the material has more grains becoming ferroelectric as the temperature is lowered). 6000 5000 4000 3000 2000 FIG. 12. eTE SURFACE FOR A TYPICAL Ba0. 65 T0.35i3 43,o ~ ~ ~ ~~~~~~fo~~~6

2900-121-T THE UNIVERSITY OF MICHIGAN 00 5000 Z I00 is.O0. 44 O00o regard to the large electric fields. The question is then raised: what is the explanation for thermal energy the domains become mobile and then align themselves with the field. A test for such a postulate can be made by the following experiment. If the incremental permittivity verse to the electric field, if domains were being aligned. An experiment of this type has been performed" with a Ba0Sr0TIO ceramic composition, and the results are shown in i. 65 0. 35 3 reduced with field. It is therefore quite evident that domains are not being aligned with the 44

W I LLOW RU N LA BO RATO R I ES 2900-121-T _I-IJ -0 3 6 9 12 15 18 2t ELECTRIC FIELD (kv cm) FIG. 14. RELATIVE CAPACITANCE VS. ELECTRIC FIELD. Measured transverse (I) and parallel (11) to the field for a Ba Sr TiO ceramic. The question, therefore, of the cause of the large-field nonlinearity in these mixed ceramics is still unanswered. It is at this point that we introduce the notion of an induced ferroelectric state in grains or regions which are essentially nonferroelectric in the absence of the applied field. This is the same type of effect that is observed as a shift in the Curie point of a single crystal and which is predicted from the thermodynamics and observed experimentally (cf. Fig. 9 and 10). In the ceramic or polycrystalline dielectrics, for temperatures in the vicinity of the peak permittivity, one assumes that there is a distribution of Curie points in the different crystallites and therefore that there are a significant number of grains which are actually nonferroelectric under zero-field conditions. The reason for the distribution of Curie points in the different crystallites may be taken as a result of the original processing of the polycrystalline material. For example, since the percentage of strontium in BaTiO3 influences the Curie point, one could expect in preparing a Ba-Sr-TiO3 ceramic by first mixing the components and then sintering, that the grains thus formed would not all have precisely the same composition and therefore that a distribution of Curie points would result. In fact, in preparing samples for the experiments cited here, it was noticed that the longer and harder (higher temperatures) that the samples of (BaSr)TiO3 were sintered, the lower in temperature the peak permittivity was, indicating that the amount of Sr entering the BaTiO3 lattice by diffusion was, of course, a function of firing time and temperature. Naturally, a distribution of percentage of Sr among the grains is expected for this situation. Also, it is possible to have a distribution 45

2900-121 -T THE UNIVERSITY OF MICHIGAN as a result of crystallite strains. The consequences of the postulate of induced ferroelectricity in the polycrystalline material will now be further explored. 3 A THEORY of LARGE-FIELD NONLINEARITY in the PEROVSKITE FERROELECTRICS 3. 1. POSTULATES AND GENERAL PROPERTIES In this section we again specify the case of Ba Sr TiO3 ferroelectrics, although the x 1-x 3 results could be qualitatively the same for any ferroelectric whose free-energy function allows the prediction of the shift of the Curie transition to higher temperatures with electric field. To illustrate qualitatively the consequences of our postulate of induced ferroelectricity in the mixed ceramic as the source of large-field nonlinearity, consider the idealization of the thermal behavior of the single crystal shown in Fig. 2. This is presented schematically in Fig. 15: The over-all behavior of the polycrystalline sample is then to be considered an average over many such crystallites taking into account some distribution of the Curie temperatures, T The idealization shown in Fig. 15 leads one intuitively to the following qualitative conclusions for a material operated in the vicinity of its Curie transition. (1) The incremental permittivity would first increase for small fields and then decrease (Fig. 15c). This has been verified experimentally, and the data are shown in Fig. 16 as a butterfly loop of sample capacity vs. field. (2) The transverse (e in Fig. 15c) and the parallel (c in Fig. 15c) permittivities would slightly increase for small fields and then both decrease. From the figure it is seen that one could expect that the transverse permittivity would not decrease to the same extent as the parallel permittivity. This explains the data of Fig. 14 which could not be made consistent with domain orientation, as has been previously noted. (3) The rather peculiarly shaped butterfly loop shown in Fig. 16 and predicted by the idealThe effect is actually observed as can be seen from the data shown in Fig. 17 and 18. Since these conclusions are based on the postulate of an induced ferroelectric state, one can expect that the hysteresis loops would be thinner than for a material in which domains were being moved about. Also, the constricted part of the loop would become longer as the temperature is increased, as can be deduced from Fig. 15. All these properties are experimentally confirmed by the butterfly and hysteresis loops for the Ba 6Sr0 3TiO3 ceramic shown in Fig 16, 17, and 18. The hysteresis loop data are traced from oscillogramrws made with the circuit of Fig. 8. 46

W I LLOW RU N LA BO RA T O R I ES 2900-121 -T NO FIELD, E=O Ec I Tr T WITH FIELD, E E~4~'C (b) i T. T VARIATION OF EA WITH FIELD AT TEMPERITURE Tr TRANSVERSE TO FIELD Ec PARALLEL WITH FIELD (C) E FIG. 15. IDEALIZED THERMAL BEHAVIOR OF A FERROELECTRIC CRYSTALLITE. Based on data of Fig. 2 and the free-energy function. T is taken as the "room temperature" or operating temperature. r 47

2900-121 -T THE UNIVERSITY OF MICHIGAN 400 390 —- _ wlO 380 370 360 350 340 -250 -200 -150 -100 - 50 0 50 100 150 200 250 ( I VOLT/MIL=.4KV/C M ) FIELD (volts per 10 mils) FIG. 16. THE STATIC CAPACITY VS. FIELD BUTTERFLY LOOP FOR A Ba 0.65Sr0. 35TiO3 FERROELECTRIC CERAMIC AT 250C. 1 volt/mil = 0. 4 kv/cm. (4) The application of strong fields to the polycrystalline sample would be expected to have two effects. First, as larger and larger fields are applied, more and more of the grains become ferroelectric and therefore have reduced permittivities (cf. Fig. 15), so that the resultant permittivity of the sample as a whole is reduced. Those grains that have not yet become ferroelectric under the influence of the field have rather low Curie temperatures in the distribution, and as more and more of the grains become ferroelectric under the influence of stronger and stronger electric fields, the distribution becomes increasingly weighted toward higher temperatures by those high-permittivity grains which have not yet become ferroelectric. Thus a second effect is expected as a result of the applied field, namely, a shift in the peak permittivity toward higher temperatures. Both properties are shown in the experimental data already cited in Fig. 12. The evidence, therefore, for an explanation of the large-field nonlinearity based on an induced effect rather than domain orientation seems quite strong. In order to carry out an examination of the process of induced ferroelectricity in ceramics analytically, a model based on the following postulates is considered for a polycrystalline sample. (1) A Gaussian distribution of Curie temperatures, T, is taken about some temperature, T. Only the cubic to tetragonal transition is to be considered. 48

W I LLOW RUN LA BO R A T O R I ES 2900-121 - T P P 10 C 35 C (a) XI A' ~~~P 150C 450C 260C 560C (b) FIG. 18. 60-CYCLE HYSTERESIS LOOPS FOR A Ba0 65Sr0 35TiO3 CERAMIC. (a) FIG. 17. DOUBLE HYSTERESIS LOOPS FOR Maximum field is 20 kv/cm. Note that the Ba0. 65Sr 0. 35T CERAMIC AT VARIOUS3 loop is extremely thin. (b) With expanded TEMPERATURES oscilloscope gain in the zero-field region. Note that there is no remanent polarization at E = 0, which implies that domains are not being switched by the field in this region. (2) For the zero-field case, each grain is presumed to have a permittivity given by a Curie-Weiss law above its Curie temperature, T, and some appropriate constant average c value below T (cf. Fig. 15). The average value taken for the constant part of the permittivity would lie between c and e c a (3) When the tetragonal state is induced by an electric field, the c-axis tends to be aligned with the field. (4) The shift in Curie temperature with field of any one grain is taken in accordance with the results of the thermodynamical arguments of Sec. 2. 1. 49

2900-121 -T THE UNIVERSITY OF MICHIGAN The model thus chosen undoubtedly is fairly rough. However, it should contain enough of the essential features of the material to allow at least qualitative agreement with the experimental properties outlined above. 3.2. THE ZERO-FIELD CASE First, it is necessary to determine a value for the average permittivity for the constant part of the characteristic for the ceramic grains (Fig. 19). This is to be considered as the mean value of permittivity taken out of the permittivity tensor ellipsoid. That is, from what is meant by the mean, 2 im r i e(o, 0) do do f=0 0=0 c = (3. 1) av 2 w r do jd0 0 0 In Fig. 19, it is assumed that the ellipsoid becomes a sphere in the Curie-Weiss region, i. e., the grain in the region T > T is cubic and e = e. For T < T. e is computed from Eq. (3. 1) with the measured values of e and c for a single crystal at temperatures slightly less than a c T. For the material as a whole, T is taken as a Gaussian distribution about some approc c priately chosen temperature T. The permittivity e (<, 0) is associated with the equation of the ellipsoid shown by Fig. 19, i. e., 2 2 2 x y z + + = 1, 2 2 2 a a c or in polar form: 2 2 2 2 2 2 e sin 0 cos e sin 0 sin 2 e cos 0 +~ + = 1. (3.2) 2 2 2 a a c Then Eq. (3. 2) may be written in the form: 1 C e(0, ) 1 c _, (3. 3) 2 2 2 2 cos 0 sin 0 Y1 -k sin 0 2 2 c a 2 2 2 where k = 1- C /C c a 50

WI LLOW RU N LABORATORIES 2900-121-T I Is C,e b Y Tr Tc T z T= Tr ea fa Eboc FOR PEROVSKITES IN THE X TETRAGONAL STATE FIG. 19. THE AVERAGE CHARACTERISTIC OF A CERAMIC GRAIN AND THE PERMITTIVITY ELLIPSOID AT ZERO FIELD By substituting Eq. (3. 3) into Eq. (3. 1) one obtains 27r 1T 7r /2 d 2~ d a 27 r=0o =0 - k sin2 Js - -k sin 0 The latter expression is in the form of Legendre's integral of the first kind (an elliptic integral) and may be evaluated from the tables. From the experimental data for a single crystal of BaTiO3 (shown in Fig. 2b) in the tetragonal form are taken: e = 4000, a e = 400, 51

2900-121 -T THE UNIVERSITY OF MICHIGAN and k = 1- = 0. 999. Applying this to Eq. (3. 4) and evaluating the integral from the tables, one obtains e Z 940, av in excellent agreement with the experimental results for polycrystalline tetragonal BaTiO3 (cf. Fig. 13 for T < 1200C). Assuming a model based on a collection of grains whose transition temperatures T are 22 c given according to a Gaussian distribution with a variance y about some temperature T, and that the dielectric behavior of any one grain is similar to that shown in Fig. 19, it is then possible, in principle at least, to compute the zero-field permittivity as a function of temperature. This is done out of the expression for the mean in a Gaussian distribution: (T -T ) c r 2 e(T, T0)e dT 0 c e(T) = 0 (3. 5) (T -T ) c r oo 2 e dT j c 0 E for T < T where E(T, T 4C for T > T T-T c 0 In this expression, T is some chosen temperature about which the distribution is taken. This can be controlled experimentally, for example, by cation substitutions in the perovskite structure (Sr or Pb substituted for Ba in BaTiO3 will lower or raise, respectively, the temperature 3. T ). The a determines the degree of spreading of the distribution and is another factor under experimental control. The permittivity e of any given grain is assumed to be e for T < T av - c and T T for T > T where C is the Curie constant. It is important to point out again that TO - T c 0 is not the same at T. The latter factor, T, is the actually observed Curie transition temc c perature. TO is about 100 lower in temperature than T, depending on the crystal. The fact 0 c that TO $ T results from the nature of the first-order transition at the Curie temperature 0 c (cf. Sec. 2. 1). The evaluation of the integrals in Eq. (3. 5) proceeds as follows. First, consider the denominator, which is the normalizing factor. 52

W I LLOW RU N LABORATORIES 2900.-121.-T (T -T )2 c r ao o D - e dT, (3. 6) 0 and for convenience let x- T - T, (3.7) C r so that 2 x D J e dx. (3.8) -T r Now T is the temperature of maximum for the distribution and is taken as about 3000K, i. e., r in the room-temperature range. On the other hand, a would be of the order of about 100. That is, experimentally the variance of transitions about the Curie peak is, very roughly, ~100. Therefore, one has: x -900.2, order of e e x = -T r It is clear then that extending the lower limit of the integral of Eq. (3. 8) to - oadds practically nothing to the integral. In this instance Eq. (3. 8) becomes 2 x oo 2 D=J e ar dx= a1/. (3.9) -00 Let us now consider the numerator of Eq. (3. 5). (T -T )2 C r ~~00 2 a d N -e(T, T)e dT, (3. 10) c 0 and defining x as before, N becomes 2 x - -00 2 N = E (x,T)e dx, (3. 11) -T r 53

2900-121 -T THE UNIVERSITY OF MICHIGAN c for T < T where c(TO0 T) = av for T 0 4'_C for T >T T - c 47-C 1. 7 x 10 It is more convenient to define E(T0o T) in terms of the variable x. Also, To is strictly related to T by way of the free-energy function for the crystal, but for the sake of simplicity we take T - T = 10. This is consistent with the results of Sec. 2. 1, where it was shown c 0 that the difference between T and TO would be about 100C. The value of the constant 4irC = 5 c 1.7 x 10 is taken from the recent experimental results of Drougard and Young (Ref. 30) for a very pure BaTiO3 single crystal. Since x = T - T, one can write: c r e for x = T - T avT r l + 10 x for x < T- T of Eq. (3. 11) into two separate intervals, viz: 2 2 -xTC ) dX + c X (3. 13) T, - T-T +10 -x r -T r T-T r r The first integral in Eq. (3. 13) is calsled N1, and the second integral N2. Considering N2, one has: 2o 2 X X T-T r - 2' ~ / The first integral is a standard form and13) is callfound in the tables as the con jugater2 Considerr functin on. Thus, N2(T) ) e rfc ( (3. 15) 42 aL 2\a/j= 2av Thi inegrl s astadar frm nd s oun inthetalesas heconugae ero fuctin.Thus,

W I LLOW RU N LABORATORI ES 2900-121 -T which may be evaluated from the tables.ll To evaluate N consider: 1 2 T-T x r The integrand in Eq. (3. 16) will always be finite for the limits set on the integral, since the denominator can only be zero for x = T - T + 10, and x must be less than or equal to T - T by the condition of the upper limit. The actual integration of Eq. (3. 16) was carried out on an IBM 650 electronic digital computer. The machine program consisted of a numerical procedure using Simpson's Rule with various values of ac and temperature as parameters. For convenience, T was taken as 3000K (i. e., room temperature), although the results would be qualitatively the same for another value of T r Results have been compiled for a = 2, 10, 20, 31, and 50, and for temperatures in the range of 2650K to 4000K. These results are plotted in Fig. 20 for a = 10, 20, and 50. The curves for a = 20 and 50 are very similar to those for commercial so-called K6000 and K3000 ceramic compositions. The theoretical permittivity vs. temperature curve for a = 31 is compared in Fig. 21 with the experimental data of Fig. 12 for the zero-field case. The temperature scale of the theoretical curve is shifted so that the peak permittivities occur at the same temperature for both the experimental and the theoretical data. This is permissible since the theoretical peak is set by the parameter T, which is an adjustable parameter. That is to say, T does not determine the shape of the curve but only sets the temperature of the peak permittivity. The results given in Fig. 21 show a truly remarkable fit between the theoretical and experimental data. (This result inspires a certain confidence in the theory. ) However, a better test of the theory would be if it can be used to predict the experimentally observed decrease in permittivity under the influence of strong electric fields such as is shown in Fig. 12. A plot of the peak permittivities as a function of the parameter a, was made as a result of the work on the digital computer and is shown in Fig. 22. 3.3. THE LARGE-FIELD CASE In the previous section, results were derived for the zero-field case. The effect of an electric field on the dielectric properties of polycrystalline ferroelectrics is now to be considered. The following assumptions are made in carrying out the derivations. 2 T-T /T-T\- T r 2 Cf: the tables of Janke-Emde, p. 23, where E2 ( ar) =-3 e di, a function which is tabulated. 55

2900-121 -T THE UNIVERSITY OF MICHIGAN 9000 Based on: et loo 8000 - do \ (T -T )2 C r dT 00 d 2 e 7100u -O~ -.20 I~aY (T)= c(T, T )e ~ 6000 j 0'00(i ____ / for various values of the parameter a. 5000.50 4000 --— r —-— V l......... 1.- / / ~. 2000 _ _ _ _~~~.................... j _ _ _ _ _ _ _ l t - -,ooo — J-J...........- - t~~~~~~ 200C -4- -- - - I - --- ----- j 1o000 270 80 290 300 310o 320 330 340 350 36 370 380 390 TEMPERATURE ( K) FIG. 20. THEORETICAL VALUES OF PERMITTIVITY VS. TEMPERATURE 700I 6000' _ _ 5000 I~~ I SOOG ~~~ j -'% — _ _~~~~~~~~~~~~~~~ _ _.........._.._ 4000 <-s — T-,,. /~~~~~~~~~~~~~~~~~' l..... - 3000, ii AIk, _ I I i:~~~~~~~~~~~~~~~~~ __~~~~~~~~~~~~~~ ___ ____ __ ___- _ —--- __ __ - - -- ~~~[1000 ] 1_: I __~ ~~~~~~ ~~~~~~~~~~~~~~~. -. 4, --- i- - --! - -- - - - ----- 1O 10 20 30 40 50 60 70 80 90 I00 TEMPERATURE ( C) FIG. 21. COMPARISON OF EXPERIMENTAL AND THEORETICAL PERMITTIVITIES 56

WI LLOW RU N LA BO RATO R I ES 2900-121-T ll ~~~~~~lol~~ ~c FIG. 22. PEAK PERMITTIVITY VS. THE PARAMETER a arguments given in Sec. 2. 1 which resulted in Eq. (2. 13). 57 4 1 o /

2900-121-T THE UNIVERSITY OF MICHIGAN - (T* —E-Tr) c r _0 2 e(T', E)e dT; C C 0 F(T, E) = 2 2 (3. 17) c r 0o 2 Jw e dT" C 0 with the parameters defined as before. In accordance with the linear shift in Curie point with an applied field (cf. Eq. 2. 13) one may write T"' = T + -yE and (3. 18) T = T + 10, where T* is the modified Curie point, T is the zero-field Curie point, and y is a parameter C C determined theoretically (cf., Eq. 2. 13) or experimentally (cf., Fig. 10). For the experimental results shown in Fig. 10, one has = 1. 3~/kv/cm; (3. 19) also, 47rC T - T for T > T' 0 C ( for I[ fields e = F c forT <T<T (3. 20) e for fields c- - c e for T < T. av c The latter equation requires some explanation. For any grain in which T < T, e = E on the c av basis of the assumption that the domains are frozen in the ferroelectric grains. If T < T < T'" C- -C for a given grain, the field has induced a ferroelectric state in the grain. Since the induced polar axis is presumed to be aligned with the field, one takes e = e parallel to the field, and e = E if the grain is viewed normal to the field. For T > T' in a grain, a Curie-Weiss law is a 4f47rC 4C c presumed to hold, i.e., T It is rather interesting that the CurieT - TO T - T' + 10- yE Weiss law is field dependent and that at T = T*, where e = e c max 4irC e - (3. 21) max yE + 10' 58

W I LLOW RU N LA BO RATO R I ES 2900-121-T which is also field dependent. These properties may be shown by the free-energy function in the following way. Considering Eq. (2. 4) and (2. 7) for the free energy and susceptibility, one has 1 2 4 a a'+3b'P + 5c'P +.; X also, from Eq. (2.5), 3 5 E = a'P+bP + c'P + Then X may be written E 1 X:l — 2 (3. 22) a' + b'P + 5c'P +. and P = f(E) from Eq. (2. 5), with T-T T 4 a = C-10; C -12 b'- -10 and -22 c'- 10 By substituting P — from Eq. (2. 5) into Eq. (3. 22), one has 4v _z at2 3b'+ 3bE) (3. 23) at +~ Ia! +3 E a' or Eq. (3. 23) may be written: e —4r 4 +.). (3. 24) As a matter of convenience, leading to a rough approximation, only the first term will be considered, even for fields as large as 20 kv/cm. That is, 4 T T= T1500C at 20 kv/cm 1 C 104 c a - T - 250 for at T - To 40 T = 10~C, 0 To 1100C, and (~= x 10, and 59

2900-121 -T THE UNIVERSITY OF MICHIGAN 2 -12 2 bE2 = -3 x 10 1 x (70 kv/cm). Thus the second term of Eq. (2. 24) is 3b'E 9 -12 3 _ 4 x 10 x 3 x 10 x 5 x 10 at - -60, as compared with the value for the first term of 250. This amounts to a 20% correction to the peak permittivity. To the extent that the first approximation is valid, - 4irC/T - TO for T > T' and the property in Eq. (3. 21) follows. In actual practice, it is just as easy to program c the computer for the expression of Eq. (3. 22) or Eq. (3. 24) as it is for the first approximation. The integration of Eq. (3. 17) proceeds in a manner similar to that employed for the zerofield case. One lets x- T* - T - yE, c r so that Eq. (3. 17) and (3. 20) become, respectively, x o00 2 e(x, T, E)e dx -T - yE e(T, E) = 2 (3. 25) aoo 2 e dx -T -yE r with for x< (T - T - yE) T-x-T + 10 r r re for || fields e(x, TE) = c for T - T > x > T-T - yE (3. 26) a for J fields r e for x > T - T. av r The denominator of Eq. (3. 25) becomes, as before, 2 2 i e a dx-~ r. (3. 27) T - ~E 60

W I LLOW RU N LABORATORI ES 2900-121 -T The numerator is broken up into three integrals, each of whose limits correspond to the intervals of Eq. (3. 27). Thus: 2 2 2 2 c(x, T, E)e dx E e dx (3. 28) -T -yE T-T r r 2 2 x x E T-T T-T — yE ar 2 r 24C + or e dx + T-ex-Td dx T - x T + 10 e ac i T-T -yE -T - yE r r N1 + N2 + N3, respectively. The first term on the right in Eq. (3. 28) is, as for the zero-field case, the complementary error function N a _ I - E _ (3 29) 1 2 -av 2 a The second term, N2, is obviously a difference of two error functions: N or E 2 2 2 2 with e used for parallel fields or e for transverse fields. These terms are easily evaluated from tables of error functions for specific values of the parameters, a, T, E, etc. The last term in Eq. (3. 28), 2 x T-T -yE r 47rC a N - = T- x- T + 10 e dx, (3. 30) -T - yE r is the same as the integral of Eq. (3. 16) except for the limits of integration. Thus, with the same computer program as before, with limits of integration set by the values of the electric field, E, one may numerically evaluate Eq. (3. 30). The results fora = 31, and for various values of parallel electric fields are shown in Fig. 23. These results are to be compared with the experimental data for a commercially 61

2900-121 -T THE UNIVERSITY OF MICHIGAN prepared Ba0 65Sr0. 35TiO3 ceramic shown in Fig. 12. The agreement is both qualitatively and quantitatively excellent. Figure 24 shows the theoretical values of permittivity vs. temperature and field, for transverse fields. Theoretical "tuning" data for the temperature of zero-field peak permittivity are presented for both parallel and transverse fields in Fig. 25, and these results are in qualitative agreement with data for the parallel and transverse field experiment cited in Fig. 14. Figure 26 shows the theoretical cTE characteristic for the case where a = 20. Here, where the peak permittivity is large, for temperatures slightly higher than the temperature of peak permittivity, the dielectric constant exhibits a slight increase with field before finally decreasing. This is consistent with the experimental data of Fig. 16, 17, and 18. 800ooo -- 7000 -____ - T__ I For Tr =300 7000. 6000 _; IKV/CM TEMPERATURE ( K) I000 270 280 290 300 310 320 330 340 350 360 370 380 TEMPERATURE (OK) FIG. 23. THEORETICAL PARALLEL PERMITTIVITY VS. TEMPERATURE AND ELECTRIC FIELD (FOR a = 31) The results of the theory outlined in the two preceding sections, and the excellent agreement of this theory with experimental data, indicates that the nonlinear property of ferroelectrics need not be based on a domain process. One of the many interesting implications of this result is that certain high-frequency and microwave large-signal applications of the ferroelectrics may be feasible if domain orientation and growth time constants are not involved. 62

WI LLOW RU N LABORATORI ES 2900-121-T 7000.~..-........For Tr 300 700t................. 6000 270 280 290 300 310 320 330 340 350 360 370 380 390 TEMPERATURE ( ~ K ) FIG. 24. THEORETICAL TRANSVERSE PERMITTIVITY VS. TEMPERATURE AND ELECTRIC FIELD (FOR 500 = 31) 6 2 ~....1 2 4000.1~E=IOKV/CM... EE ~~~~~~~~~~~3000~ ~63 200. 1000 270 280 290 300 310 320 3 34 1 2 36 37 380;90 TEMPERATURE (" K) FIG. 24. THEORETICAL TRANSVERSEXPERMITTIVITY VS. TEMPERATURE AND ELECTRIC FIELD (FOR a 050351).5..

2900-121-T THE UNIVERSITY OF MICHIGAN For Tr = 300 7000... _.. —-i... 600 0 300 310 320 0 340 30 360 370 380 E= KV 4 50004. 1. LOW-FREQUENCY DYNAMIC SPECTRA 000 00 3000 Ti and 0I along the polar direction (Fig. 1). If a relaxation due to the molecular group were ~~~~~~~64 ~ DISPERSION PHENOMENA for HIGH-PERMITTIVITY DIELECTRICS 4. 1. LOW-FREQUENCY DYNAMIC SPECTRA data on Permittivity with frequency for a dielectric serve as a means of separating mechanisms of polarization. The results cited by Kittel (Ref. 20), which are shown in Fig. 5, for ceramic BaTiO3 show almost a tenfold decrease in permittivity over a frequency range of 0. 1 mc to the Hilbert Transform relations of Eq. (1.37). Ferroelectric domain processes are being would not expect the domains to move about under very weak signal fields.

W I L OW RU N LA BO RATO R I ES 2900-121 -T a signal field transverse to the polar direction. As will be shown in the succeeding sections, experiments on single crystals at uhf and microwave frequencies have failed to reveal any substantial dispersion behavior other than effects due to dimensional resonances, and therefore molecular relaxation is not a likely mechanism for the observed dispersion effects in the polycrystalline samples. In Sec. 2t..2 the relationship between the permittivity of a BaTiO3 crystal which is free to strain and that for a mechanically clamped crystal is derived. It has been shown that the free permittivity is roughly twice that of the clamped value, both for the "a" and the "c" crystallographic directions. For frequencies higher than the dimensional resonant modes for a given crystal, one would expect that the substance would behave as if it were mechanically constrained. Therefore, if the dispersion observed for ceramic BaTiO3 were caused solely by a clamping effect, one would expect a reduction in the permittivity by a factor of about two. It has been noted in Sec. 1. 3 that for a linear dielectric the real and imaginary parts of a complex permittivity are not independent if one part can be specified as a function of frequency. Although ferroelectrics of the perovskite type are strikingly nonlinear with regard to large fields, the point of view has been adopted here that for very small signal fields the incremental permittivity is linear. The incremental signal fields used in the experiments are of the order of 10 volts/cm as compared with coercive fields of about 10 volts/cm. If one were to assume large frictional losses at, say, grain boundaries or domain interfaces in a polycrystalline aggregate, then these losses would lead to a relaxation spectrum which could easily account for a tenfold reduction in the real permittivity. For high-permittivity substances such as the perovskites, one may expect large electrostrictive and piezoelectric effects as a consequence of the very large polarizability, and hence it would not be unreasonable for large frictional losses to exist in these ferroelectric ceramics. In order to examine further the notions presented above, experiments covering from low frequencies to the microwave region have been designed. The substances investigated include both single crystal and ceramic BaTiO3, ceramic Ba 0.65Sr0. 35TiO3, Pb 0. 40Sr0. 60Ti3, and Cd2Nb207. Because measurements were made over a very wide frequency range, it was necessary to employ three separate experimental techniques. For frequencies from 0.5 mc to 150 mc a special low-frequency dielectric spectrometer has been constructed. In the range from 20 me to 250 me a Boonton Corporation Type 190-A high-frequency Q-meter was employed. A slotted coaxial transmission-line technique was used in the frequency region from 300 mc to 4000 mc. In this section the low-frequency spectra will be considered. For piezoelectric BaTiO3, the dimensional resonances are expected to occur in the 0. 5-inc to 150-inc range. In order to best display permittivity vs. frequency in a region where rather sharp resonant peaks are expected, a dynamic method of obtaining data must be employed. To 65

2900-121 -T THE UNIVERSITY OF MICHIGAN - i this end, a low-frequency dielectric spectrometer has been constructed. The block diagram for this instrument is shown in Fig. 27. The operation of the unit depends upon mechanically sweeping a series of General Radio Corporation unit oscillators with a General Radio Type 1750-A drive motor. The output of the oscillator is sampled with a crystal diode, D1 in the figure, and the rectified voltage is used to control an electronically regulated oscillator power supply (General Radio Type 1263-A) which adjusts the plate voltage of the swept oscillator in such a way that a constant output is assured. The oscillator output is placed across two capacitors in series, C and C. C is the sample and C is a calibrated variable capacitor. The value of C is much greater than that of C, i. e., its reactance is smaller, so that the voltage x appearing across C is approximately proportional to the value of C. The high-frequency voltage appearing across C is rectified by a crystal diode D2 and filtered by C2. The resultant d-c, which is proportional to the value of C, is placed on the Y terminals of an oscilloscope. A potentiometer, which is placed on the drive shaft of the same motor that sweeps the unit oscillator, provides the X drive for the oscilloscope. The oscilloscope trace is photographed with the aid of a Polaroid Land camera attachment. While this method of displaying dynamic spectra is extremely simple, it has nonetheless proven to be very effective. What is actually measured with this method is the magnitude of Cv, which is proportional to the magnitude of the complex permittivity. However, for substances whose Q is of the order of 10 or greater,l2 the magnitude of the permittivity is approximately equal to the real part. The spectrometer has been calibrated through the use of rutile (TiO2) standards which were in turn calibrated on a Q-meter. The ceramic samples are prepared in the following manner. A thin section, 0. 5 mm thick, of the ceramic is sliced out of a larger bar or rod by means of a diamond cutting wheel. The faces of this section are painted with a silver conducting paste. The unit is then baked in a furnace at 6500C for one-half hour in order to remove the organic binders from the silver conducting paint. The ceramic section is then sliced into squares, 1 mm on a side, with a diamond wheel. The small ceramic plate thus formed has electrodes on two faces. The samples are held in a sample holder between a pair of phosphor bronze spring clips which are adjusted to provide only enough pressure to hold the sample in place. This is done in order to minimize mechanical loading. The basic ceramic stock is prepared in the laboratory by mixing the constituent oxide powders with a binder, such as methyl cellulose, and a stearate lubricant. Water is then added to form a slurry which is ball-milled for 8 hr. The mix is then dried in a drying oven at 850C. The dry powder is screened through a 100-mesh screen and pressed into cylindrical -shape witha hydraulic press at pressures from 5000 to 10,000 psi. The cylinders are then'2The Q is defined as = t Im(Z) where Z is the complex impedance. 66

W I LLOW RU N LABORATORI ES 2900-121 -T fired at a temperature which depends on the composition; for the BaTiO3-SrTiO3 series the firing is done at 13500C for 2 hr. All the ceramics have been fired in a Harper 5-kw glowbar furnace, which is provided with an automatic temperature control and programer. MECHANICALLY MOTOR DRIVE SWEPT ENTIOMETER E ICTRONICAL CONTROLLED OSCILLATOR FIG. 27. DYNAMIC DIELECTRIC SPECTROMETER Preparation of the single crystals of BaTiO3 is somewhat more difficult than that for ceramics. The crystals are formed by a method discovered by J. P. Remeika of the Bell Telephone Laboratories. In this method, BaTiO3 powder is mixed with potassium fluoride in a covered platinum crucible which is placed in a furnace at 14000C and held at this temperature for 10 hr. This allows a quantity of the barium titanate to dissolve in the fluoride flux. At the end of the 10-hr hold period the temperature is lowered at a rate of about 300C per hour until a temperature of 9500C is reached. The furnace is then allowed to cool rapidly at its natural rate. The crystals precipitate out of the molten flux in the form of thin platelike crystals, roughly 1 cm on a side and about 0. 2 mm thick. At first an attempt was made to cut these crystals into smaller squares with a diamond wheel using the method employed for ceramics, but this was unsuccessful. A method later adopted consists of first determining the directions of the atomic planes by observing the orientation of the domains with the aid of a polarizing microscope. (The domains tend to line up 67

2900-121 -T THE UNIVERSITY OF MICHIGAN are untwinned and single domain, an electrical polarizing technique is employed. Polarizing is accomplished by placing the cleaved sample on a piece of electrically conducting glass to which one terminal of a 300-volt battery is attached. The sample on the conducting glass is placed under a polarizing microscope, and while one observes the crystal through the microscope, a probe, which is attached to the other battery terminal, is touched onto the crystal at those areas in which domains are seen to exist. As this is done, the domains flip perpendicularly to the plane of the plate until the whole crystal becomes essentially a single domain. Copper electrodes are then applied to the appropriate sample faces by means of vacuum evaporation. If the electrodes are placed parallel to the direction of the spontaneous polarization, the sample is called a c plate; and if the electrodes are perpendicular to the spontaneous polarization, one has an a plate. This is shown in Fig. 28. One measures e for an a plate and a c for a c plate. Typical results are shown for BaTiO3 single crystal a and c plates in Fig. 29, 30, and 31. Figure 29 shows data taken in the region 0. 5 mc to 150 mc. The resonant peaks observed are dimensional resonances, and the frequencies and character of the peaks vary widely from crystal to crystal. The resonances can be substantially diminished or even made to disappear by mechanically clamping the crystal. The particular modes of vibration for the resonances have not been specifically identified in these spectra, since it is the purpose of these data to determine the reduction in permittivity as a result of increasing the frequency beyond the range of dimensional resonances. It has been found that the ratio of low-frequency permittivity to high-frequency permittivity varies from crystal to crystal. Most a crystals have ratios from 1. 2:1 to about 2:1, averaging around 1. 5:1, and most of the c crystals have ratios of about 2:1. This is in complete agreement with the theory developed in Sec. 2. 2 where the relationship between the free and clamped permittivities is derived. The theoretical ratios were shown to be 1. 5:1 and 2:1 for the a and c crystals, respectively. Figure 30 shows spectra for other BaTiO3 crystals for the frequency range from 0. 5 mc to 150 mc. The effect of clamping on reducing the mechanical resonances is shown. In Fig. 31 a BaTiO3 ceramic spectrum is displayed both with a biased sample and an unbiased sample. Originally, in the absence of an electric field, no resonance is observed. When a biasing field is applied, a dimensional resonance appears and in fact remains after the field is removed. The latter phenomenon is due to the remanent polarization of the BaTiO3 sample (i. e., it has become "poled"). The ratio of low-frequency to high-frequency permittivity for the ceramic in this case is about 1. 2:1. The conclusion to be drawn from this section is that a reduction in permittivity in BaTiO3 due to mechanical clamping or dimensional resonance effects would, at most, be of the order of a ratio of 2:1. The ratio for ceramics measured on the dielectric spectrometer is closer to 1. 2:1. Therefore, as a result of both theory and experiment, it would seem that the tenfold 68

Cu diII~71 3000ELIECTRODES' Is /__562000- Z-~~- ~ ~ a ~ e iar,I S z, II II I~~~~~~~~~0 -i i.5.5 5 15 50 FREQUENCY (mcs) a Plate a Plate 240_ ~-'Is~-q / 120-0..5 1.5 5 15 c Plate FREQUENCY (mcs) FIG. 28. BaTiO3 SINGLE CRYSTALS. a- and c-oriented BaTiO3 single crystals showing orientation of spontaneous polarization direction as FIG. 29. DYNAMIC SPECTRA FOR a AND c PLATES OF BaTiO3 compared to electrodes. SINGLE CRYSTALS (T = 250C) s~~0 <1~~~~~~~ 0%

C) )3 o lO 0-O 1000 1120 __ 500 - 560- 00-0 0 20 ~ "" "J.5 1.0 2.0 5 0 0 50 60 100 150.5 1.5 5 15 50 200 FREQUENCY (mcs) FREQUENCY ( mcs FREQUENCY (mcs) No Field c Plate, Free 2000- II 1000 _ 560 - 0-.5.0 2.0 5 o 20 50 60 100 150. 5 50 /5 [.5 5 15 50 FREQUENCY (mcs) FREQUENCY (mcs ) Field of 10 kv/cm Applied C a Plate, FreeZ 2000 560 U i000- --- 15 0.0 5 b 20 60 0 Io 50.5.5 5 FREQUENCY (mcs) FREQUENCY (mcs)' a Plate, Clamped After Original Field Is Removed FIG. 30. DYNAMIC SPECTRA OF FREE AND FIG. 31. SPECTRA OF BaTiO3 CERAMICS AT CLAMPED BaTiO3 SINGLE CRYSTALS (T = 250C) ROOM TEMPERATURE I zT

W I LLOW RU N LA BO RATO R I ES 2900-121-T reduction in permittivity for a BaTiO3 ceramic as reported by Kittel (see Fig. 5) cannot be explained on the basis of dimensional resonances and clamping effects alone. 4.2. MICROWAVE PROPERTIES Measurements have been made of permittivity vs. frequency in the microwave region for both ceramic and single crystals of ferroelectrics. A slotted transmission line and probe technique has been employed for the measurements. The schematic of the experimental arrangement is shown in Fig. 32. The ferroelectric specimen is treated as a lumped element terminating a transmission line. Appropriate corrections are necessary for the effects of fringing fields at the sample and for discontinuities introduced by the sample holder. The sample holder is diagrammed in Fig. 33. An additional correction is required to take into account wave propagation within the specimen. Since permittivities of the perovskite ferroelectrics are of the order of thousands, even small samples can have an appreciable fraction of a wavelength within the medium at the higher frequencies. The sample holder is calibrated in the following manner. First at a given frequency a copper "specimen" of the same dimensions as the sample is used to short-circuit the line and a reference nul in the standing wave pattern on the slotted section is established. The line is then open-circuited by removing the copper short and replacing the empty sample holder, and the open-circuit reference nul is established. Then a rutile (TiO2) standard of known capacitance is placed in the sample holder and another reference nul is found. The above reference positions are presumed lossless since the standing wave ratios are very much greater than for the case of a real sample. (The voltage standing wave ratio for the open and short-circuited line is of the order of several hundred as compared to tens for the samples measured). From the above data the following equivalent circuit is deduced for the sample holder. Rx' —'! Tcx Holder Sample The values for L and C are remarkably constant over a very wide frequency range (500 mc to -9 -13 3000 mc) and are numerically: L~ 0. 9 x 10 henry and C = 0. 2 x 10 farad. In actually reducing the data, the slight variations in these values at a given frequency point are taken into account. R and C, respectively, are the lossy part and the reactive part of the sample. x x The correction for the propagation effect within the sample is dependent on the dimensions and shape of the sample as well as upon the permittivities. It is therefore necessary to develop a correction formula which allows these factors to enter as parameters. The crystals 71

2900-121-T THE UNIVERSITY OF MICHIGAN and ceramic samples used in the experiments were of rectangular cross section so that the correction formula involves solving the wave equation under boundary conditions imposed by this geometry. Jaynes and Varenhorst (Ref. 31) have obtained a correction factor for a onedimensional sample (infinitely wide sheet), but these results are not applicable to the problem under consideration. A summary of the analysis leading to the correction factor for a rectangular parallelepiped and a cylindrical disc follows. D.C. BIAS BLOCKING CAPACITOR I MEGOHM PROE SIGNAL SLOTTED LINE SAMPLE GENERATOR METER PROBE SIGNAL GENERATOR ATTENUATOI COAXIAL SLOTTED LINE SAMPLE H.P.TYPE 805 S HOLDER FIG. 32. EXPERIMENTAL ARRANGEMENT FOR UHF AND MICROWAVE MEASUREMENTS TYPE N CONNECTOR TYPE N SNORT CIRCUIT CONNECTOR WITH CENTER PIN DRILLED OUT TEFLON SUPPORT SAMPLE CRYSTAL TEIERMOCOUPLE 3\S L I D I N G ~ SLIDING PIN CONTACT FIG. 33. DIELECTRIC SAMPLE HOLDER FOR LINE 72

W I LLOW RU N LA BORATORI ES 2900-121-T The electric field in the sample must, at all frequencies, satisfy the wave equation: 2 2 V E+ k2E = 0, (4. 1) where k w= w /e, and the factor dit is understood. The mks system of units is employed. In general e is a tensor; that is, 0 0 e = (011 22 E= 0 E22 0 e33 The dielectric constant takes this form for the case where a principal axis coordinate system is used for a tetragonal single crystal. For the case of an isotropic ceramic one takes e C22 = e33. The coordinate system for the rectangular sample is shown below. x a CZ y The z-direction corresponds to the axial dimension of the coaxial line. The metal electrodes are at the planes z = 0 and z = c and the following assumptions are made. (a) In the neighborhood of, and within the sample, the electric field is entirely z directed. (b) The wavelength in air is much greater than the length of the sample c. (The sample lengths used were 1 mm or less. ) (c) The sample is considered lossless to a first approximation. Assumption (a) appears reasonable on the basis that parallel plane electrodes are employed on a high-permittivity sample. Therefore, most of the flux would be inside the sample and parallel to the z-axis. Jaynes and Varenhorst have shown in the paper noted above (Ref. 31) that the cavity resonant modes within the dielectric that are associated with the z-direction are not excited when assumption (b) is valid. Assumption (c) is not expected to lead to difficulty because of the relative independence of the dielectric constant measurement upon the loss tangent in this method. If this proves not to be the case, a first-order correction can be made to account for the effect of loss in the sample. With these assumptions, 73

2900-121 -T THE UNIVERSITY OF MICHIGAN E(0, y)= E(a, y)= E(c, 0) = E(x, b) E1, (4. 2) and 2 2 E = E, k L McE, (4. 3) z where E1 is the applied field at the sample boundaries. The apparent value of the dielectric constant, ap, is given by Cap (4. 4) ap wAE where A is the cross-sectional area of the sample and I is the total current (displacement) flowing through it, that is, I= |TJdA= fwEE(x, y)dxdy. (4.5) The solutions of Eq. (4. 1), E(x, y), and Eq. (4. 4) and (4. 5) give the desired relationship between e, and E. ap To obtain the solution to the differential equation most conveniently, the following transformations are made: E-E - E'; u =-. (4. 6) 1 a Then the wave equation restated becomes: 7r E'aE 2 27 + 2 +,(E'+ Ei(E ) +E= 0, (4. 7) a y with the boundary conditions E'(O, y) = E'(7r, y) = E'(u, 0) = E'(u, b) = 0, (4. 8) and E' = E'(u, y). The solution proceeds by taking a finite Fourier sine transformation13 of Eq. (4. 7) and (4. 8) on u for the interval 0 = u = r. This leads to a transformed differential equation: d e(n, y)-a e(n, y) + O = 0,9) dy 13See, for example, R. V. Churchill, Modern Operational Mathematics, McGraw-Hill, 1944, Chap. X. 74

W I-LLOW RU N LABORATORI ES 2900-121 -T where 22 c2 n k n= 1, 2, 3. a = k2E1S{1}, S{1} = and 2r e(n, y)= S{E(u y)}= E(u, y) sin nudu. 0 The differential Eq. (4. 9) has the solution: e(n,y) = c e + c2e + 2 (4. 10) where -ab 3 e - 1 1 2 ab -ab a e - e and -ab - e - 1 C2 2 ab -ab a e - e The inverse transformation is given by E2 1 e(n, y) sin nu, (4. 11) ir n=1 which, when combined with Eq. (4. 4), (4. 5), and 4. 6) yields b a Cap ab E 1 7r n= a11 Cap ab E1 {.| S7n-l (e(n, y) sin + E1 )dx dy. (4. 12) 0 0 From Eq. (4. 10) there results, F1 4k2 1(1)n 2 abtn = 1 + 1 - tanh (4. 13) 7r a n where a2 - k, and n = 1, 2, 3,. a 2 2 k = w pu, 75

2900-121 -T THE UNIVERSITY OF MICHIGAN and where a and b are the cross-sectional dimensions of the sample. This latter formula relates the apparent value of permittivity (c ) with the "true" value, E. In the derivation it was ap assumed that the media was lossless. If the use of the formula is limited to about 10% corrections or less, and if there is about a 10% error in the correction formula itself due to the assumption of lossless media, the total error would be of the order of 1%. If a sample is in the form of a disc of radius a, with the coordinate system as shown below, the calculation of the correction is very much simpler than for the rectangular case. Z The sample is again assumed to be immersed in a uniform z-directed field, E1, and the other assumptions used for the previous case are again applied. The wave equation in cylindrical coordinates, assuming no angular dependence of E, is a2E aE z 1 z 2 Z +1 z + k2E =, (4. 14) 2 r ar z ar which, for positive and real k, has the solution, E =E J (kr), (4. 15) where E = E (a) = E J (ka), 1 z o o and J (kr) is a zeroth order Bessel function of argument kr. Applying the same arguments as in the previous case, a I = 27rE we | rJ (kr)dr, (4. 16) o o o and J (ka) 2 1 E E — (4. 17) ap ka J (ka)' 0 which is the desired correction formula for the disc-shaped sample. Equation (4. 17) can be expanded in a series, which may be a more useful form: e = e1+ ~a a + 48] (4 18) ap 8 48 76

W I LLOW RUN LABORATORIES 2900-121 -T Typical of the results for measurements on the single crystals in the uhf and microwavefrequency regions are shown in Fig. 34 and 35. Figure 34 shows the data of c' and Ec" for a single crystal c plate of BaTiO3 for the frequency range of 20 mc to 4000 mc. Figure 35 displays similar data but for a single crystal a plate. The results for both the a- and c-oriented crystals show that there is a decrease in c' at the lower frequency end of the spectrum but that the permittivity in the 400-mnc to 4000-mnc range is relatively constant. The decrease in 200' ='T - i "1 tan 6 - ~100 20Q 40 100 200 40 1000 2000 4000 10 E' III I 0 10 20 40 100 200 400 1000 2000 4000 10.000 FREQUENCY (mcs) FIG. 34. FREQUENCY DATA FOR CRYSTAL OF BaTiO3 c PLATE (AT 230C) 3000 —....- ----. C = C - i" 2500 E. tan FIG. 35. FREQUENCY DATA FOR CRYSTAL OF BaTiO3 a PLATE (AT 230C) 77 I::1 500.....100 77

2900-121 -T THE UNIVERSITY OF MICHIGAN the low-frequency region may possibly be due to the same type of dimensional effects as have been observed in the dynamic spectra, but which are not resolved here because the data are taken point by point. This would further explain the relatively large scatter of the data points in the low-frequency region. The important thing from the point of view of the purposes of this experiment is that no strong relaxation or resonance effect has been observed for the BaTiO3 single crystals at microwave frequencies (up to 4000 mc for this experiment). A total of eight crystals have been measured and each crystal exhibits data similar to those displayed in Fig. 34 and 35. A very recent measurement of the permittivity of single crystals of BaTiO3 at 24, 000 mc and 56, 000 me has been reported by Benedict and Durand (Ref. 32) with the same conclusion drawn, namely, there is no substantial relaxation of the permittivity of BaTiO3 single crystals at microwave frequencies (i. e., to 56, 000 mc). On the other hand, the data for the ceramics does show a tendency for dispersion at microwave frequencies. Figures 36 and 37 show microwave data for a BaTiO3 ceramic composition containing a small (0. 5%) Fe203 additive. The Fe203 is added as a fluxing agent, enabling one to obtain a high-density ceramic with a moderate firing schedule (13500C for two hours). The results of the measurement are compared in Fig. 36 with those cited by Kittel in Fig. 5, and it is seen that these data do show a dispersion at high frequencies but that this is not as pronounced as the data shown by Kittel. Similar results are shown for other BaTiO3 ceramic preparations such as is shown in Fig. 38 and 39. It is not clear why the initial permittivity of the hot-pressed ceramic14 (Fig. 38) is as high as 2000, a value twice as great as is usually encountered for BaTiO3 ceramic. The answer may be that the hot-pressing technique produces an unusually high-density ceramic or that hot pressing produces a large number of a domains (high-permittivity direction) in the plane of the pressed disc. The hot-pressing method of preparation consists of firing the ceramic under a pressure of about 5000 psi for one-half hour. The heating is accomplished with the aid of an induction heater with a metal sleeve about a ceramic die. The data for hot-pressed Cd2Nb207 ferroelectric ceramic is shown in Fig. 40. It is noted that the decrease in the real permittivity with frequency is very slight to 3000 me and that the losses are relatively low (Q = E'/E" = 25 at 3000 mc). The permittivity and loss for a typical Ba 0.65Sr0. 35TiO3 ceramic are presented in Fig. 41 and the dispersion appears very similar to that of the pure BaTiO3 ceramic. The data for PbTiO3 and Pb 0.35Sr0. 65TiO3 ceramics for frequencies between 1800 and 4000 mcs have been very kindly supplied by C. B. Sharpe and 14The hot-pressed sample for this experiment and samples of Cd Nb 07 ceramic were supplied through the kind offices of Dr. Alex deBrettville of the Signal Corps Engineering Laboratories, Fort Monmouth, New Jersey. 78

WILLOW RUN LABORATORIES 2900-121-T 1400.. i~..~ ~ ~ ~4.........i f "'... —L'r4, t, ) i' 1200.. E' ON BoTiO3. 5%Fe2O0 1000 800 --.......... E' 600 ___ __ __ _ __ _ - t- ----- - 8~ ~ ~ ~ ~~~~\\ 400 200 -_ _ _10 20 40 100 200 600 1000 2000 4000 10,000 FREQUENCY (mcs) FIG. 36. C VS. FREQUENCY FOR A BaTiO3 + 0.5% Fe2O3 CERAMIC 70 \ 60 50 Q (tan 6) 40 Q 30 20 10 o10 20 40 100 200 400 1000 2000 4000 icooo FREQUENCY (mcs) FIG. 37. Q VS. FREQUENCY FOR A BaTiO3 +0.5% Fe203 CERAMIC 79

2900-121-T THE UNIVERSITY OF MICHIGAN 3000 E 2000 _____- _____ t ____ _ __ _ _ __ _ __ _ _______~tan 60 2000 10 20 40 100 200 400 1000 2000 4000 10,000 FREQUENCY (mcs) FIG. 38. DATA FOR HOT-PRESSED BaTiO3 CERAMIC (AT 230C) 1500 " 10 20 50 20- 500 1000 TAN s= — 500 I _-"10!0 20. 0 I00 200 4 00 I000 2000 4 000 I,000 FREQUENCY (mcs) FIG. 39. BaTiO3 CERAMIC WITH EXCESS TiO2. e and Q vs. frequency (230C) 80

W I LLOW RUN LABORATORIES 2900-121-T G. Brockus (Ref. 22) and are shown in Fig. 42 and 43, respectively. Here again, no unusual dispersive effects are observed. The particular composition Pb 0.35Sr0. 65TiO3 was chosen because this ceramic has its peak permittivity in the room-temperature region and is therefore similar to the Ba 0.65Sr0.35TiO3ceramic. 0. 65 0. 35 3 In Fig. 44, data are presented for the variation in the 1000-mc permittivity and Q as a function of a d-c biasing field for a Ba 0.65Sr0. 35TiO3 ceramic. It is noted that the decrease in permittivity (parallel permittivity, in this case) is the same as for lower frequencies (Fig. 14). An extremely interesting feature of these data is the increase in Q as a d-c field is applied. This does not mean, however, that the absolute losses necessarily decrease, since, if even the approximate validity of the Debye equations (1. 41) and (1. 42) may be assumed, one may write: 2 2 r (E - e )WT wt f >> In terms of an equivalent electrical circuit the Debye equations amount to: C* —- -- T C (E) where C' = C' - jC" and e — >C 00 00 - - RC (4. 20) e' ->C + C S 00 Thus Eq. (4. 19) becomes: C' C + C (1- w R C C" -, = 2 -RC (4. 21) wRC for C > > Co. Therefore, if C alone would decrease by the application of a field (as it would for a Ba0 65Sr0. 35TiO3 ceramic, (Sec. 3), then Q increases inversely as C decreases, and the dissipative element R need not change with field in order to effect the increase in Q. This general behavior in Q is noted from Fig. 44. Much better data on this inverse behavior in Q 81

2900-121-T THE UNIVERSITY OF MICHIGAN I E = C - iCE " 1 tan 6 - 600 --, LT; I _ __ __ 60 1~~ ~~~~~~~ ~~- -n............................... I i I I I i I I i I, i 300 L30 10 20 40 100 200 400 1000 2000 4000 10,000 FREQUENCY (mcs) FIG. 40. DATA FOR HOT-PRESSED Cd2Nb2O7 (AT 230C) 6000_ 60 5000 5 0 4000 i _ _ _ _ _ _ _ 40 3000 30 Q 2000 _ 420 I ~~~~~~ — ~-TP 1000 10 0 _ _ _ _ _ _ _ _ _I _ _10 10 20 50 100 200 500 1000 2000 5000 10,000 FREQUENCY (mcs) FIG. 41.' AND Q VS. FREQUENCY FOR Ba0 65SrOT 35Ti03 CERAMIC 82

WI LLOW RU N LABORATORI ES 2900-121-T 80 70 I- I I z 60 e ) Z - 0 C.) -J w 0 0 1.0.0 0 0 4.0 FREQUENCY (kmc) FIG. 42. DIELECTRIC CONSTANTS VS. FREQUENCY FOR PbTiO3. Bias field 0; ambient temperature = 260C. [After Sharpe and Brockus (Ref. 22). ] 5000 03 40 30 00 w w -J I 000 1.0 2.0 4.0 FREQUENCY (kmc) 83

2900-121 -T THE UNIVERSITY OF MICHIGAN 6000 12 5000 10 4000 000 MC. DATA 8 2000 _______ 4 0 5 l0 15 20 25 D-C ELECTRIC FIELD (kvY/cm ) FIG. 44. TUNING DATA FOR Ba. 65Sr0. 35TiO3 CERAMIC. el MEASURED WITH 1000-minc signal. are obtained at lower frequencies where more accurate data are obtainable for this and similar ceramics. The data for the a-oriented and c-oriented single crystals of BaTiO3 fail to show microwave relaxation; therefore, on the basis of these data it is concluded that atomic or molecular relaxation does not exist at microwave frequencies. The explanation of the observed microwave relaxation of the permittivities for polycrystalline samples rests on: (a) dimensional resonances which, due to the small sizes of the crystallites making up the ceramics, would occur in the microwave range; (b) domain growth or orientation time constants; and (c) an apparent relaxation produced by a loss mechanism. Dimensional resonances would not be able to occur within the crystallites of a ceramic, since these would undoubtedly be effectively clamped by the bulk of the material. The effect of clamping has been discussed in previous sections, and Fig. 30 shows experimentally that, for a clamped crystal, the dimensional resonances are reduced or even disappear. It has also been shown in Sec. 3 that domain orientation or growth does not substantially contribute to the incremental permittivity, so that it would be unreasonable to postulate that the permittivity is reduced at microwave frequencies by domain-motion time constants when the domains do not even contribute to the permittivity. On the other hand, since all materials are to some extent electrostrictive, and for ferroelectrics they are piezoelectric, the mechanical compliance of the material to the signal fields would result in frictional effects at grain and domain boundaries. Thus domains may enter into the relaxation indirectly, i. e., by means of a loss mechanism. 84

W I LLOW RU N LA BO RATO R I ES 2900-121-T This type of loss and its effect on the high-frequency properties of these ferroelectrics is to be discussed in the following section. 4. 3. LOSSES AND DISPERSION In previous sections it was shown that the Debye equations lead to a particularly simple equivalent circuit and that expressions for Q = ( 6) are given in Eq. (4. 19) or (4. 20). It was also mentioned in regard to dispersion effects in dielectrics of the high-permittivity type, that in addition to effects such as molecular relaxation and dimensional resonances, dispersion may be a direct consequence of a loss mechanism. In Sec. 3 it has been shown that excellent correlation with various experimental data for permittivities is obtained if one assumes that domain motion and orientation do not contribute to the susceptibility. Thus, domain orientation and growth processes have been ruled out as a direct relaxation mechanism, but domain effects, due to the existence of domain walls, may very well enter into a mechanism of dielectric loss, which in turn would produce the observed relaxation. A direct molecular mechanism of dielectric relaxation is being ruled out, at least to frequencies of 56, 000 mc, on the basis of the experimental results with the single crystals. In a recent paper (January 1959) Lewis (Ref. 33) has investigated energy-loss processes in ferroelectric ceramics, particularly that of pure BaTiO3, and has concluded that although domains do not enter into the small-field polarization they do play a role in the loss mechanism. I have arrived at the same conclusion about the polarization, and in fact found it necessary to assume no domain motion at all in deriving the theoretical models of Sec. 3. Lewis attributes the small-field losses to a so called "microhysteresis, " which he feels is due to "... the switching of individual atomic moments in successive lattice planes traversed by the moving boundary. " With this I do not agree. Any "switching of atomic moments" in a BaTiO3 ceramic amounts to, in the tetragonal state (cf. Fig. 1 and 2) moving an ion, say Ti +4, from one potential minimum to another. This would involve an activation energy of the order of AE = kAT, where k is Boltzman's constant and AT is the difference between the Curie temperature and room temperature at which the measurement is assumed to be made. For AT = 1000K, E 0. 009 electron-volts. +4 If we consider the Ti ion in a double potential well, then the distance between minima is of the order of 2 angstroms and Ti has four electronic charges. Thus to attain an activation energy of 0. 009 volts, an electric field of the order of 0. 009 ___volts E- - 10 volts/cm -8 2x10 cm 85

2900-121 -T THE UNIVERSITY OF MICHIGAN is required. This figure is very rough, of course, but even if one allows several orders of magnitude for local-field "amplification" the resultant field is very much larger than the small-signal fields. This is one of the reasons why the theory of "microhysteresis" based on a local switching effect appears to be a rather untelable explanation for the high losses observed in the ferroelectrics. Moreover, ferroelectrics are known to be somewhat lossy at temperatures above their Curie points where domains are assumed not to exist. Some experimental evidence does show a reduction in the losses as a ferroelectric is heated through its Curie point, but even at the reduced value, the losses are still somewhat high. Typical data for a ceramic are shown in Fig. 45 where the signal frequency is 1 mcs. A possible explanation for this behavior is to be discussed presently. 50001 1 _ \k 1 r 1 I 300 10 W 1 0 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE ( C) FIG. 45. EA AND LOSS DATA FOR A TYPICAL Ba0 655r0 35TiO3 CERAMIC One of the most fundamental characteristics of ferroelectrics is their piezoelectric property. By this it is meant that a field will produce a linear strain in a single crystal or domain, and if the field is reversed, a strain of opposite sign will result (e. g., if a field in the direction of the polar axis produces an extension, a field in the opposite direction would cause a compression). This is to be distinguished from electrostriction, which is a quadratic effect. In electrostriction the material has an extension along the field direction, and this extension is independent of the sign of the field. Due to the quadratic behavior, electrostrictive effects would be much smaller than piezoelectric effects at small fields. Although it is felt that domain motion of the type in which atomic moments are switched is not a particularly good model for dielectric losses for the reasons cited above, it is not unreasonable to assume that domains make themselves felt in the piezoelectric or electrostric86

W I LLOW RU N LABORATORI ES 2900-121-T tive interactions (effects which are completely ignored by Lewis's paper). To illustrate, consider the example of a 1800 domain wall under the influence of an applied field. This example is shown in the schematic below. PS'1 p - I E -, P"S I -* P8 I 1800 wall in the absence of a field 180~ wall in the presence of a parallel applied field, E0 In the absence of an external field, the two domains have opposite directions of spontaneous polarization, and a 1800 wall is formed between them. When an external field is applied, the piezoelectric interaction causes a linear extension in the domain which has its spontaneous polarization parallel with the field. The domain in which the spontaneous polarization is antiparallel undergoes a compression. Thus at a 1800 domain wall there would be a rather large relative motion of one set of atomic planes over another whenever an electric field is applied, so that a loss mechanism involving what amounts to "wall friction" may be conceived. This mechanism does not require an actual switching of atomic moments. Since there would also be, to a smaller extent, relative motion at 900 walls in the presence of a field, these types of domain walls would also affect the loss. In addition, there would be the possibility of relative motion of the ceramic grain boundaries due to electrostriction, so that this frictional type loss could contribute to the dielectric losses even above the Curie temperatures. In presuming a piezoelectric and electrostrictive loss mechanism in the ferroelectrics, it is assumed that the frequency is sufficiently high that joule heat losses due to conduction may be ignored. On the other hand, the frequency must be sufficiently low that the wavelengths are not of the order of magnitude of sample dimensions (a few millimeters), so that radiation does not take place from the sample. Frequencies of the order of 10 to 10 cps would be the rough range of validity for these assumptions. Good quantitative data for an attempt to verify any particular loss model are, to date, lacking. It is considered to be a major part of the future research in ferroelectrics at this laboratory to develop models of dielectric loss and to attempt quantitative experimental tests of these models. The concept of piezoelectric and electrostrictive activity being responsible for the dielectric losses can be examined only qualitatively in the light of our present data. An observation of Q vs. temperature for a typical Ba 0.65Sr0. 35TiO3 ceramic shown in Fig. 45 shows that as the temperature is increased through the peak-permittivity range, Q is increasing. This would be consistent with the concept that as the individual ceramic grains become heated they undergo a ferroelectric to paraelectric transition, and the domains disappear. Thus piezoelectric activity ceases above the Curie temperature of a given grain. It 87

2900-121 -T THE UNIVERSITY OF MICHIGAN is also noted that the Q shows an apparent decrease at the high-temperature limit. This may be due to the fact that the BaTiO 3-type ceramics become very conductive at elevated temperatures, so that the high-temperature reduction in Q may be due to conduction losses (the Q vs. temperature data were taken at 1 mc). If one compares the data for Cd2Nd207 (Fig. 40), which is nonferroelectric at room temperature, with that of the BaTiO3 ceramics (Fig. 38, 39, and 41), which are ferroelectric at room temperature, it is noted that the losses are somewhat smaller in the Cd2Nd207. An even more interesting observation can be made by comparing the loss behavior of a BaTiO3 single crystal with that of the ceramic. It is seen that the losses in the crystals, which have been prepared as single domain and hence contain no domain walls, actually decrease with frequency, whereas the data for the BaTiO3 ceramic, which presumably contains a large number of domains, shows increasing losses with frequency. It is recalled that at the low-frequency end of the spectrum, BaTiO3 single crystals undergo rather striking dimensional resonances. As the frequency is increased, the dimensional effects cease, and since there are no domains in the crystal, the losses would be expected to decrease. On the other hand, the ceramic contains many domains. It is not necessary to assume dimensional resonances for the individual domains; even highly damped relative motion of the domains in accordance with their piezoelectric behavior would be sufficient to produce losses which would increase with frequency. The ideas set forth in this section are only qualitative, to be sure. It remains for a future research program to produce a more formal loss model and resultant analysis. Very carefully controlled experiments leading to quantitative data on the number, sizes and types of domains and their possible effects on dielectric losses are required. Such a large-scale experimental program has been beyond the scope of the present study. However, it has been one of the results of this studyto suggest this particular direction for future research. 5 CONCLUSIONS The conclusions stated in this section will be itemized concisely. For a more detailed discussion of a particular point, the reader is referred to the preceding sections in the text. On the basis of the experimental and analytical investigations of the polarization, microwave dispersion, and losses in high-permittivity ferroelectrics, especially those of the BaTiO3 type, the following conclusions may be drawn. (1) According to the experimental work described in Sec. 4, little or no microwave dispersion has been observed for the permittivity of either a- or c-oriented barium titanate single crystals. On the other hand, experiments on high-permittivity ceramics, both of the pure barium titanate and mixed composition types, show that there is a definite microwave relax88

W I LLOW RU N LABORATORI ES 2900-121 -T ation effect for most of the ceramics. Since this effect was absent in the barium titanate single crystals, it is concluded that a polar or molecular mechanism for relaxation is not present. It is further concluded that the observed dispersion for the ceramic composition must be due either to a mechanism of dielectric loss, which is characteristic of ceramic grain and domain properties, or due to a reduction in the permittivity as a result of mechanical clamping of the ceramic grains at microwave frequencies. It has been shown analytically and with highfrequency spectra that this latter effect could account for only a twofold reduction in permittivity through the microwave region. (2) The ceramic dispersion observed in the experiments is not as strong as that reported by Kittel (Fig. 5 and 36). In order to explain the extremely large reduction in permittivity over two decades in frequency as reported by Kittel, one must adopt a point of view that dielectric losses play a significant role in the microwave relaxation effect in high-permittivity ferroelectrics. These losses may come about as a result of piezoelectric and electrostrictive action at domain boundaries and at the interfaces of ceramic grains. (3) It has been shown that if domain alignment, orientation, and growth are assumed responsible for the nonlinearity in the permittivity of the ferroelectrics, then one is led to a conflict with the experimental data. In order to describe analytically the observed nonlinear effects, a phenomenological model has been postulated. This model presumes that the domains are "frozen" in the ceramic grains which are ferroelectric and are not moved about by the electric field. The ceramic material is thought to consist of a collection of grains, each of which has a Curie temperature which is given by a Gaussian distribution about some appropriately chosen temperature. In this way the ceramic composition, at the temperature about which the Gaussian distribution is taken, has half of its grains in a paraelectric (nonferroelectric) state. The electric field acts on the nonferroelectric grains in such a way as to induce a ferroelectric state in them. This effect is consistent with the predictions of a freeenergy function postulated by Devonshire. The theory developed on the basis of this model leads to results which correlate remarkably well with the experimental data. (4) On the basis of the model described above, all nonlinear effects in the barium titanatetype ferroelectrics must be accompanied by strong thermal sensitivity; therefore, the present activity in some circles in seeking a ferroelectric of the barium titanate or perovskite class which is sensitive to electric fields but which is insentitive to temperature changes would appear to be unfruitful. This is simply because the free-energy function demands that the thermal and electric-field effects go hand in hand. (5) Since excellent correlation between experimental and theoretical data has resulted from the assumption of ferroelectric domains being "frozen in" with regard to large electric fields, it is doubtful whether domain effects could contribute to the small signal incremental 89

2900 —121 -T THE UNIVERSITY OF MICHIGAN permittivities at high frequencies. The large value of permittivity observed in the perovskite ferroelectrics could then be thought of as coming about as a result of the action of the local field in the perovskite structure. Calculations on the local field in BaTiO3 by Slater (Ref. 13) and Triebwasser (Ref. 14) seem to bear this out. In order to account for the large anisotropy in the permittivities for the perovskites in the ferroelectric state, it is interesting to speculate about the possibility of a directed bond between ions in the structure such as Ti and OI (Fig. 1). For such a case the large transverse permittivity could be thought of as resulting from torsional motion of the directed bond pairs, Ti-OI. Although measurements on single crystals of BaTiO3 at frequencies up to 56, 000 mc have failed to indicate such a polar association, this does not mean that one cannot exist. It may well be that millimeter-wave spectra would reveal this mechanism. Investigations along these lines are to be left for a future program. 90

W I LLOW RU N LABORATORI ES 2900-121-T APPENDIX A. 1. This section of the Appendix presents a proof that superposition in the time domain follows from the statement of linearity in the following form: D(c) = e(w)E(w) or P(w) = X(o)E(w), where E or X has a unique value at a given frequency. By the Fourier integral, let D (t) = D (w)eidw e()E ()ei, A. 1) 1 d, 1 -00oo -o00 - iwt where E 1( = t)e dt. (A. 2) -o00 When Eq. (A. 2) is substituted into Eq. (A. 1), one can say that given an El(t) one can predict the D (t). Similarly, given an E 2(t) one gets D 2(t) via D (t) = e(w)E2()e dw, (A. 3) 2 2 -o0 where E2(w) = 2 E2(t)et dt. (A. 4) -00 Superposition in the time domain means that D (t) = Dl(t) + D (t) is produced by E3(t) = E (t) + E2(t). By adding Eq. (A. 1) and (A. 3) one gets: 31 22 D3(t) = Dl(t) + D2(t) = e(w)E+ o)E2()eit dw, (A. 5) ~or -D3(t) = | i()eit [E (w)+E2(w) )] dw. -00O 91

2900-121 -T THE UNIVERSITY OF MICHIGAN But by the property of Eq. (A. 2) and (A. 4) one may write E = E () + E 2(W) = I E (t) + E (t) e dt. (A. 6) 3 1 21 2irJ l J Thus by substituting El(w) + E2(w) = E3() into Eq. (A. 5) and interpreting property (A. l)(using the subscript 3), one gets the result that D 3(t) = Dl(t) + D2(t) is produced by E3(t) = El(t) + E2 (t). It is important to note that this result would not follow if the e were a function of the electric field, since e could then not have been factored from E (w) and E2(w) in Eq. (A. 5). A. 2. This section of the Appendix presents a derivation of the relationship between the free and the clamped coefficients in Devonshire's free-energy function for BaTiO3. Let the Helmholtz free energy for the crystal at some' temperature be given by the following expansion in terms of the polarizations (Pi) and strains (x.). - 122 22 222 2 1 A = A +-2a" (P1 +2P2 + P3 4 + 4) bA=A -a" IP +P +P ~- IP +P + PP 2 12 2 1 3 2 3 2 11 1 2 3 22 2p201 i 2 2 12( 1 2 x1X3 + X2X3) +2 c44 4 5 6 (A. 7) + q11 x1P 1 + x2P2 + x3P3 ) + q12 x ( 2 +P 2) +x2 (p1 +P3 3 (P1+ P2)] +q (X P P +xPP +x P P )+ 44(x4 2 3 x5P 1P3 x6 1P2) Here the double primes indicate that the coefficients are for the clamped crystal. Subscripts 1, 2, and 3 refer to the coordinates x, y, and z. The strains with subscripts 4, 5, and 6 are shears. For the cubic to tetragonal transition there is only one polar axis, although strains may exist along the three crystallographic axes. The assumption may be made that no shear strains exist for this geometry. Therefore, we set P2 P3 x4 x5 = x = 0. Also by the geometry of the crystal there are two equivalent a-axes in the tetragonal state so aA that we take x2 = x3 along the a-axis. Setting the stresses Xi = - equal to zero, one has 1 8x. C11x1 + 2c12X2 + q11P1 10, 92

W I LLOW RU N LA BO RATO R I ES 2900-121 -T and (A. 8) CX2 + C 12(x+2) +q2P1 = 0. ll 2 12 1 2 12 1 From these equations one may solve for the strains xl and x2 in terms of the polarization and electrostrictive coefficients, obtaining (2q12 q 11 ) c 12 q11c 11 2 2 x = P -cP 1 (C1 + 2c 12)(c -11c ) 1 1' and (A. 9) l11C12 12 c11 2 2 x = = P X2 x3 = (c + 2c )(c -c2) 1 1 11 12 11 12 where a and 1 are defined as a matter of convenience. Substituting these values for the strains into Eq. (A. 7) gives the free energy the form: A = A + aP -b P +-c (a + 2f )P 0 2 1 4 11 1 2 11 1 (A. 10) 2 4 4 4 +c12(2aM + f)P1 + qll aP1+ 2q12fP14 On the other hand, it has already been noted in Eq. (2. 4) that the free energy expressed in terms of the "free" coefficients is written as 1 a 2 1 4 A=A + -a'P +-b' P +.... (A. 11) 0 2 1 4 11 1 Equating the coefficients on the same powers of P1 in Eq.(A. 10) and (A. 11) yields T- T a' = al' 4-, 10 and 2 2 bq =bc +2 2qc + c (A. 12) etc., for the higher-order coefficients. This gives the relationship between the free and the clamped coefficients in the Helmholtz free energy. 93

2900-121 -T THE UNIVERSITY OF MICHIGAN REFERENCES 1. Valasek, J., Phys. Rev., 1921, Vol. 17, p. 475. 2. Von Hippel, A., et al., Cambridge, Mass., Massachusetts Institute of Technology, Laboratory for Insulation Research, NDRC Report 14-300, 1944, and NDRC Report 14-540, 1945. 3. Wul, B., J. Phys. (USSR), 1946, Vol. 10, p. 95. 4. Frazer, B. C., Danner, H., and Pepinsky, R., Phys. Rev., Oct. 1955, Vol. 100. 5. Kay, H. F., and Vousden, P., Phil. Mag., 1949, Vol. 40. 6. Merz, W. J., Phys. Rev., 1949, Vol. 76, p. 1221. 7. Merz, W. J., Phys. Rev., 1952, Vol. 88, p. 421. 8. Jaynes, E. T., Proc. IRE, 1955, Vol. 43, No. 12. 9. Butler, T. W., Quarterly Progress Report No. 18, Ann Arbor, Mich., The University of Michigan Research Institute, July 1958. 10. Lorentz, H. A., The Theory of Electrons, Tuebner, 1909. 11. Onsager, L., J. Am. Chem. Soc., 1936, Vol. 58. 12. Jaynes, E. T., Ferroelectricity, Princeton, N. J., Princeton University Press, 1953. 13. Slater, J. C., Phys. Rev., 1950, Vol. 78, p. 748. 14. Triebwasser, S., J. Phys. and Chem. Solids, 1957, Vol. 3, No. 1 and 2. 15. Born, M., and Von Karman, T., Physik Z., 1912, Vol. 13. 16. Frohlich, H., The Theory of Dielectrics, London, Oxford University Press, 1958, second edition. 17. Toupin, R. A., and Lax, M., J. Chem. Phys., 1957, Vol. 27, No. 2. 18. Powles, J. G., Jackson, W., Proc. Inst. Elec. Engrs. (London), 1949, Vol. 96, p. 383. 19. Von Hippel, A., Rev. Mod. Phys., 1950, Vol. 22, p. 221. 20. Kittel, C., Introduction to Solid State Physics, New York, Wiley, 1956, 2nd ed. 21. Von Hippel, A., Dielectrics and Waves, New York, Wiley, 1954. 22. Sharpe, C. B., and Brockus, G., Microwave Properties of Ferroelectric Materials, Ann Arbor, Mich., The University of Michigan Research Institute, March 1959, Final Report. 23. Devonshire, A. F., "Theory of Ferroelectrics, " Advances in Phys., April 1954, Vol. 3, No. 4. 24. Merz, W. J., Phys. Rev., 1953, Vol. 91, p. 513. 25. Huibregtse, E. J., and Young, D. R., Phys. Rev., 1956, Vol. 103, p. 1705. 26. Diamond, H., and Orr, L. W., Ann Arbor, Mich., The University of Michigan, Engineering Research Institute, 1954, Technical Report No. 31. 27. Berlincourt, D., and Jaffe, H., Phys. Rev., 1958, Vol. 109, No. 1. 94

W I LLOW RU N LA BO RATO R I ES 2900-121 -T 28. Shelton, G. R., Bunting, E. N., and Kopell, L., Dielectrics for Ceramic Components, National Bureau of Standards, July 1953, Report No. 4. 29. High Dielectric Ceramics, Niagara Falls, N. Y., Titanium Alloy Manufacturing Co., 1950. 30. Drougard, M. E., and Young, D. R., Phys. Rev., 1954, Vol. 95, p. 1152. 31. Jaynes, E. T., and Varenhorst, V., Stanford, Calif., Stanford University Microwave Laboratory, Technical Report No. 287. 32. Benedict, T. S., and Durand, J. L., Phys. Rev., February 1958, Vol. 109, No. 4. 33. Lewis, B., Proc. Phys. Soc. (London), January 1959, Vol. 73. 95

2900-121 -T THE UNIVERSITY OF MICHIGAN DISTRIBUTION LIST 6, PROJECT MICHIGAN REPORTS 1 January 1960- Effective Date Copies Addressee Copies - Addressee 1 Office, Chief of Research and Development 2 Assistant Commandant, U. S. Army Artillery and Missile School Department of the Army Fort Sill, Oklahoma Washington 25, D. C. 3 Assistant Commandant, U. S. Army Air Defense School ATTN: Army Research Office Fort Bliss, Texas 1 Office, Assistant Chief of Staff for Intelligence Department of the Army, Washington 25, D. C. Fort Belvoir, Virginia Fort Belvoir, Virginia ATTN: Chief, Combat Developments /G-2 Air Branch ATTN: Combat Developments Group 1 Commanding General, U. S. Continental Army Command Fort Monroe,. Virginia 1 Commandant, U. S. ArmySignal School Fort Monroe, Virginia Fort Monmouth, New Jersey ATTN: ATSWD -G ATTN: SIGFM/SC-DO 2 Commanding General, U. S. ArmyCombat Surveillance Agency 1 Commandant U. S. Army Aviation School 1124 N. Highland Street, Arlington 1, Virginia Fort Rucker, Alabama 1 Chief, Research and Development Division 3 President, U. S. A~rmy Artillery Board Office of the Chief Signal Officer Fort Sill Oklahoma Department of the Army, Washington 25, D. C. 1 President, U. S. Army Air Defense Board 25 Commanding Officer U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey 1 President, U. S. ArmyAirborne and Electronics Board ATTN: SIGFM/EL-DR Fort Bragg, North Carolina 1 Commanding General, U. S. Army Electronic Proving Ground 1 Commanding Officer, U. S. Army Signal Electronic Research Unit Fort Huachuca, Arizona PostOffice Box 205, Mountain View, California ATTN: Technical Library 2 Office of Naval Operations 1 Office of the Director, Defense Research and Engineering Department of the Navy, Washington 25, D. C. Technical Library, Department of Defense (1) ATTN: OP-37 Washington 25, D. C. (1) ATTN: Op-07T 1 Chief of Engineers, Department of the Army Washington 25, D. C. 1 Office of Naval Research, Department of the Navy 17th and Constitution Avenue, N. W. ATTN: Research and Development Division Washington 25, D. C. 1 Office, Chief of Ordnance, Research and Development Division ATTN: Code 463 Department of the Army, Washington 25, D. C. ATTN: ORDTB, Research and Special Projects 2 Director, U. S. Naval Research Laboratory Washington 25, D. C. 1 Commanding General ATTN: Code 2027 Quartermaster Research and Engineering Command, U. S. Army Natick, Massachusetts 1 Commanding Officer, U. S. Navy Ordnance Laboratory 2 Chief, U. S. Army Security Agency Cornona, California Arlington Hall Station, Arlington 12, Virginia ATTN: Library 2 Commander, Army Rocket and Guided Missile Agency Redstone Arsenal, Alabama 1 Commanding Officer and Director, U. S. Navy Electronics Laboratory San Diego 52, California ATTN: Technical Library, ORDXR-OTL ATTN: Library 4 Director, U. S. Army Engineer Research and Development Laboratories Fort Belvoir, Virginia. 4 Department of the Air Force (1) ATTN: Chief, Topographic Engineer Department Headquarters, USAF, Washington 25, D. C. (2) ATTN: Chief, Electrical Engineering Department (1) ATTN: AFOIN-lB1 (1) ATTN: Technical Documents Center (1) ATTN: AFOAC-E/A (1) ATTN: AFDRD 1 Commanding General (1) ATTN: AFDRD (1) ATTN: Directorate of Requirements U. S. Army Combat Development Experimentation Center Fort Ord, California 3 Commander in Chief, Headquarters, Strategic Air Command 1 Commandant, U. S. Army Command and General Staff College Offutt Air Force Base, Nebraska Fort Leavenworth, Kansas (1) ATTN: DINC ATTN: Archives (2) ATTN: DORQP 96

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AD Div. 14/2 UNCLASSIFIED AD Div. 14/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystals - Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystalsPOLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis POLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystals - HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystalsRept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization Rept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization tables, 33 refs. 3. Ferroelectric crystals - tables, 33 refs. 3. Ferroelectric crystals(Rept. no. 2900-121-T) Dispersion (Rept. no. 2900-121-T) Dispersion (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystals- (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystalsProperties Properties The purpose of this study is to investigate, by analysis of ex- S roeectic t The porpose of this stody is to investigate, by analysis of ex- Froe perimental data, polarization, dispersion, and loss processes perimental data, polarization, dispersion, and loss processes Processe in high-permittivity ferroelectrics. The study proceeds from Processes in high-permittivity ferroelectrics. The study proceeds from y. 6. Ceramic materials -. 6. Ceramic materials — the observation that, according to published data, polycrystal- the observation that, according to published data, polycrystalline HaTiO3 and similar compositions show strong relaxation Dielectric properties and similar compositions show strong relaxation Dieli line BaT'03 7. Ceramic materials - 3 7. Ceramic materialsin permittivity at microwave frequencies. This phenomenon in permittivity at microwave frequencies. This phenomenon Pola is shown to be generally related to polarization and loss prop- Polarizat is shown to be generally related to polarization and loss prop- I Title: ~~~~~~~~~~~~~~~~~~~Tte rjc IHGNerties of the dielectric, Processes of particular interest in erties of thedielectric.ProcessesofparticularinterestinII.IDiamondPHowardMICHIGANe rties of the dielectric. Processes of particular interest in II. Diamo accounting for the microwave relaxation are: (a) domain ef- III Diamond, Howard accounting for the microwave relaxation are: (a) domain efI~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ ~~II. U.SDAm igal op I.U.Amyoinal Cowrps fects, (b) possible dipolar or molecular polarization, and (c) fects, (b) possible dipolar or molecular polarization, and (c) IV. U. S. mechanical effects produced by piezoelectric and electro- S.ir For ce mechanical effects produced by piezoelectric and electro- Sciei Scientific Research Strctiveculng.(vr.TeNtional Scenceac strictive coupling. (over) V. The National Science Foundation Foundation VI. Contract DA-36-039 VI. Contract DA-36-039 SC-78801 SC-78801 Armed Services Armed Services Technical Information Agency Technical Information Agency UNCLASSIFIED UNCLASSIFIED AD Div. 14/2 UNCLASSIFIED AD Div. 14/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystals - Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystalsPOLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis POLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystals - HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystalsRept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization Rept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization tables, 33 refs. 3. Ferroelectric crystals — tables, 33 refs. 3. Ferroelectric crystals(Rept. no. 2900-121-T) Dispersion (Rept. no. 2900-121-T) Dispersion (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystals — (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroetectric crystalsThe purpose of this study is to investigate, by analysis of ex- Properties The purpose of this study is to investigate, by analysis of ex- Prop perimental data, polarization, dispersion, and loss processes Proeses perimental data, polarization, dispersion, and loss processes 5.Pro in high-permittivity ferroelectrics. The study proceeds from 6 rac in high-permittivity ferroelectrics. The study proceeds from 6 rac the observation that, according to published data, polycrystal- Dieletic proeries the observation that, according to published data, polycrystal- 6.Diel c e line BaTiO3 and similar compositions show strong relaxation tieleric materies line BaTiO3 and similar compositions show strong relaxation in permittivity at microwave frequencies. This phenomenon Polarization -in permittivity at microwave frequencies. This phenomenon Plrat is shown to be generally related to polarization and loss prop- ITle roject is shown to be generally related to polarization and loss prop- ITle erties of the dielectric. Processes of particular interest in I. accounting for the microwave relaxation are: (a) domain ef- II. Diamond, Howard erties fth dielcicoPve relaxation are: (a) domain ef- II. III U S. rmy SgnalCorp accontin forthe mcrowve rlaxaton ae: () dominDe- a IonDiamndwowar fects, (b) possible dipolar or molecular polarization, and (c) III. U.S. Army Signal Corps III. IV. U. S.Air ForceOffice offects, (b) possible dipolar or molecular polarization, and (c)IVU.SAiFocOfcef mechanical effects produced by piezoelectric and electro- Sin U.iS.fic Rsrch mechanical effects roduced by piezoelectric and electro- Scientfic, ~~~~~~~~~~~~~~~~~~~~~Scientific Research ScienifcRsah strictive coupling. (over) V. The National Science strictive coupling. (over) V. The National Science Foundation Foundation VI. Contract DA-36-039 VI. Contract DA-36-039 SC-78801 SC-78801 Armed Services Armed Services Technical Information Agency Technical Information Agency UNCLASSIFIED UNCLASSIFIED + + -

AD UNCLASSIFIED AD UNCLASFE The conclusions that may he drawn from this work are as UNITERMS The conclusions that may he drawn from thia work are as UIEM follows. (1) The field sensitivity of the incremental per- Polarization follows. (1) The field sensitivity of the incremental per- Polarizaio mittivity of polycrystalline ferroelectrics ia not a domain Dispersion mittivity of polycrystalline ferroelectrics is not a domain Disperaio effect, hut results from an induced ferroelectric state in Lospoesseffect, hut results from an induced ferroelectric state inLosp nonferroetectric grains. (2) Nonlinear permittivity under Polycrystalline nonferroelectric grains. (2) Nonlinear permittivity under Polycrysaln electric fields must necessarily he accomplished hy suh- BTOelectric fields must necessarily he accomplished hy suh-. stantial thermal sensitivity. (3) Orientation of the domains Reaxtion stantial thermal sensitivity. (3) Orientation of the domains BaTiO3; does not contrihute to microwave dispersion. (4) Molecular Permittivit does not contrihute to microwave dispersion. (4) Molecular Permitivit or dipolar relaxation does not occur at microwave fre- Frequency or dipolar relaxation does not occur at microwave fre- Feu quencies for BaTiO3 single crystals. Therefore, this is Microwave quencies for BaTiO3 single crystals. Therefore, this is Firequnc not a mechanism for dispersion in the ceramic. (I) Di- Dielectric not a mechanism for dispersion in the ceramic. (I) Di- Dielectri electric loss as well as piezoelectric and electrostrictive Doanelectric loss as well as piezoelectric and electrostrictive Dmi activity would result in microwave dispersion in high- Dparactivity would result in microwave dispersion in high-Dioa permittivity ferroelectrics. Mioleuar permittivity ferroelectrics. Mioleuar Electrostrictive Electrosrctv Coupling Coupling Hysteresis Hysteresi Ceramic Ceramic Free energy Free eneg Curie Curie Ferroelectric Ferroeleti Nonferroelectric Nonf errolcti Crystals Crystals UNCLASSIFIED UNCASFE AD UNCLASSIFIED AD UNCASFE The conclusions that may he drawn from this work are as UNITERMS The conclusions that may he drawn from this work are as UIEM follows. (1) The field sensitivity of the incremental per- Polarization follows. (1) The field sensitivity of the incremental per- Polarizato mittivity of polycrystalline ferroelectrics is not a domain Dispersion mittivity of polycrystalline ferroelectrics is not a domain Dispersio effect, hut results from an induced ferroelectric state in Lospoesseffect, hut results from an induced ferroelectric state inLosp noneroelcticgrains. 2)Nnierpermittivity udrPolycrystalline nonferroelectric gan.2)Nlierpermitivity udrPolycrysaln electric fields must necessarily he accomplished hy suh- BaTiO3eeti ilsms eesrlyb copihdb u-BT0 stantial thermal sensitivity. (3) Orientation of the domains Reaato stantial thermal sensitivity. (3) Orientation of the domains B~O does not contrihute to microwave dispersion. (4) Molecular Permittivit does not contrihute to microwave dispersion. (4) Molecular Permiattvy or dipolar relaxation does not occur at microwave fre- Frequency or dipolar relaxation does not occur at microwave fre- Frequenc quencies for BaTiO3 single crystals. Therefore, this is Microwave quencies for RaTiO3 single crystals. Therefore, this is Microwav not a mechanism for dispersion in the ceramic. (I) Di- Deeti not a mechanism for dispersion in the ceramic. (I) Di- Deeti electric loss as well as piezoelectric and electrostrictive Domincti electric loss as well as piezoelectric and electrostrictive Dominct activity would result in microwave dispersion in high- Diomarn activity would result in microwave dispersion in high- Diomarn permittivity ferroelectrics. Mioleuar permittivity ferroelectrics. Mioleuar Etectrostrictive Electrostitv Coupling Coupling Hysteresis Hysteresi C eramic Ceramic Free energy Free enrg Curie Curie Ferroelectric Ferroeleti Nonf erroelectric Noriferroeeti Crystals Crystals UNCLASSIFIED UNCASFE

-H ++ AD Div. 14/2 UNCLASSIFIED AD Div. 14/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystals - Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystalsPOLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis POLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystals- HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystalsRept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization Rept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization tables, 33 refs. 3. Ferroelectric crystals — tables, 33 refs. 3. Ferroelectric crystals(Rept. no. 2900-121-T) Dispersion (Rept. no. 2900-121-T) Dispersion (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystals — (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystalsThe purpose of this study is to investigate, by analysis of ex- Properties The purpose of this study is to investigate, by analysis of ex- Prop The purpose of this stody ~~~~~5. -.Ferroelectric crystals — perimental data, polarization, dispersion, and loss processes Ferroelectric crystals perimental data, polarization, dispersion, and loss processes in high-permittivity ferroelectrics. The study proceeds from Processes in high-permittivity ferroelectrics. The study proceeds from Proc the observation that, according to published data, polycrystal 6. Ceramic materials - the observation that, according to published data polycrystal- 6. Ceram line HaTiO3 and similar compositions show strong relaxation Dielectric properties and similar compositions show strong rDielecti op line BaT'03 7. ~~~~~~~~~Ceramic materials - line BT0 togreaxi atin7Cerals in permittivity at microwave frequencies. This phenomenon Cericatias-in permittivity at microwave frequencies. This phenomenon Cera Polarization Polarization is shown to be generally related to polarization and loss prop- is shown to be generally related to polarization and loss prop- I. ~~~~~~~~~~~~~~~~~~~~~~~~~~~I. Title: Project MICHIGAN erties of the dielectric. Processes of particular interest in II Tit:ojec MICHGANerties of the dielectric. Processes of particular interest in II. Diamo accounting for the microwave relaxation are: (a) domain ef- IIU.DiAmod Howar accounting for the microwave relaxation are: (a) domain ef- III. U. S. fects, (b) possible dipolar or molecular polarization, and (c) III U. S. Army Sinlcor fects, (b) possible dipolar or molecular polarization, and (c) IV. U. S. fec,~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~ V (b). posibedpar ForcmoeOfcef.UlS.Air polriztin andf(c)o mechanical effects produced by piezoelectric and electro- IV ieU.iS.cAReForc mechanical effects produced by piezoelectric and electro- Scie strictive coupling. SinfcRearhstrictive coupling.SceticRsah (over) V. The National Science (over) V. The National Science Foundation Foundation VI. Contract DA-36-039 VI. Contract DA-36-039 SC-78801 SC-78801 Armed Services Armed Services Technical Information Agency Technical Information Agency UNCLASSIFIED UNCLASSIFIED AD Div. 14/2 UNCLASSIFIED AD Div. 14/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystals - Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Ferroelectric crystalsPOLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis POLARIZATION, MICROWAVE DISPERSION, AND LOSS IN Analysis HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystals - HIGH-PERMITTIVITY FERROELECTRICS by Howard Diamond. 2. Ferroelectric crystals - Rept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization Rept. of Project MICHIGAN. Jan 60. 95 p. incl. illus., Polarization tables, 33 refs. 3. Ferroelectric crystals — tables, 33 refs. 3. Ferroelectric crystals(Rept. no. 2900-121-T) Dispersion (Rept. no. 2900-121-T) Dispersion (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystals — (Contract DA-36-039 SC-78801) Unclassified report 4. Ferroelectric crystalsProperties Propete The purpose of this study is to investigate, by analysis of ex- The purpose of this study is to investigate, by analysis of exS. Ferroelectric crystals - S. Ferroelectric crystals — perimental data, polarization, dispersion, and loss processes perimental data, polarization, dispersion, and loss processes in high-permittivity ferroelectrics. The study proceeds from Processes in high-permittivity ferroelectrics. The study proceeds from Proc the observation that, according to published data, polycrystal6 Ceramic materials - the observation that, according to published data, polycrystal- 9. Cerai thne observaTion ta n d similar compositions show sta, polycytao- Dielectric properties line BaTiO3 and similar compositions show strong relaxation Cer line maTei3 7. Ceramic materials - Dielecti in permittivity at microwave frequencies. This phenomenon in permittivity at microwave frequencies. This phenomenon ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Polarization Plrzto is shown to be generally related to polarization and loss prop- is shown to be generally related to polarization and loss properties of the dielectric. Processes of particular interest in I. Title: Project MICHIGAN erties of the dielectric. Processes of particular interest in I. Title: accounting for the microwave relaxation are: (a) domain ef- II. Diamond, Howard accounting for the microwave relaxation are: (a) domain ef- II. Diam, fects, (b) possible dipolar or molecular polarization, and (c) III. U. S. Army Signal Corps fec, (b) possible dipolar or molecular polarization, and (c) III. U. S. Army Signal Corps IV, U.S. Air Force Office of IV. U.S. Air Force Office of mechanical effects produced by piezoelectric and electro- ScientSfic Rsrch mechanical effects produced by piezoelectric and electro- Sc Scientific Research ScientfcRsac strictive coupling. (over) V. The National Science strictive coupling. (over) V. The National Science Foundation Foundation VI. Contract DA-36-039 VI. Contract DA-36-039 SC-78801 SC-,7801 Armed Services Armed Services Technical Information Agency Technical Information Agency UNCLASSIFIED UNCLASSIFIED + ++

AD UNCLASSIFIED AD UNCLASSIFIED The conclusions that may be drawn from this work are as UNITERMS The conclusions that may be drawn from this work are as UNITERMS follows. (1) The field sensitivity of the incremental per- Polarization follows. (1) The field sensitivity of the incremental per- Polarizati mittivity of polycrystalline ferroelectrics is not a domain Dispersion mittivity of polycrystalline ferroelectrics is not a domain Dispersion effect, but results from an induced ferroelectric state in sproe effect, but results from an induced ferroelectric state in spro Loss processes fl gan.~udrLoss processes nonferroelectric grains. (2) Nonlinear permittivity under Polycrystalline nonferroelectric grains. (2) Nonlinear permittivity underPolycrystae electric fields must necessarily be accomplished by sub- BaTiO electric fields ust necessarily be accomplished by sub- BT stantial thermal sensitivity. (3) Orientation of the domains Ra stantial thermal sensitivity. (3) Orientation of the domains a 3 does not contribute to microwave dispersion. (4) Molecular Permittivity does not contribute to microwave dispersion. (4) Molecular Permlttivity or dipolar relaxation does not occur at microwave fre- r y or dipolar relaxation does not occur at microwave fre-it quencies for BaTiO3 single crystals. Therefore, this is Frequency quencies for BaTiO3 single crystals. Therefore, this is Frequency ~~~~~~~~~~~~~~~~~~~~~~~~~~MicrowaveMirwv not a mechanism for dispersion in the ceramic. (5) Di- Dicri not a mechanism for dispersion in the ceramic. (5) Di-Mlcrowave electric loss as well as piezoelectric and electrostrictive Dleacirc electric loss as well as piezoelectric and electrostrictive Dielectric activity would result in microwave dispersion in high- Domain activity would result in microwave dispersion in high- Domain permittivity ferroelectrics. Mipolecular permittivityferroelectrics. Dpolar Molecular Molecular Electrostrictive Electrostrictive Coupling Coupling Hysteresis Hysteresis C Ceramic Ceramic Free energy Free energy c Curie Curie _ _ Ferroelectric Ferroelectric:: Nonferroelectric Nonferroelectric Crystals Crystals _! UNCLASSIFIED UNCLASSIFIED i +. cav ___ AD UNCLASSIFIED AD UNCLASSIFIED C The conclusions that may be drawn from this work are as UNITERMS The conclusions that may be drawn from this work are as UNITERMS follows. (1) The field sensitivity of the incremental per- Polarization follows. (1) The field sensitivity of the incremental per- Polarizatio mittivity of polycrystalline ferroelectrics is not a domain Dispersion mittivity of polycrystalline ferroelectrics is not a domain Dispersion effect, but results from an induced ferroelectric state in Los processes effect, but results from an induced ferroelectric state in pro Loss processes Loss processes nonferroelectric grains. (2) Nonlinear permittivity under Polycrystalline nonferroelectric grains. (2) Nonlinear permittivity under ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~PolycrystallinePoyrsaln electric fields must necessarily be accomplished by sub- BTO electric fields must necessarily be accomplished by sub- BT stantial thermal sensitivity. (3) Orientation of the domains BaTiO3 stantial thermal sensitivity. (3) Orientation of the domainsa 3 does not contribute to microwave dispersion. (4) Molecular Relaxation does not contrib ute to microwave dispersion. (4) Molecular Relaxation or dipolar relaxation does not occur at microwave fre- or dipolar relaxarmittiviton does not occur at microwave fre -Permittivit Frequency Frequency quencies for BaTiO3 single crystals. Therefore, this is Microwave quencies for BaTiO3 single crystals. Therefore, this is Microwave not a mechanism for dispersion in the ceramic. (5) Di- Dielectric not a mechanism for dispersion in the ceramic. (5) Di- Dielectric electric loss as well as piezoelectric and electrostrictive D electric loss as well as piezoelectric and electrostrictiveDoa activity would result in microwave dispersion in high- Domain activity would result in microwave dispersion in high- Dpomar permittivity ferroelectrics. Dipolecular permittivity ferroelectrics. Molecular Molecular Molecular Electrostrictive Electrostrictive Coupling Coupling Hysteresis Hysteresis Ceramic Ceramic Free energy Free energy Curie Curie Ferroelectric Ferroelectric Nonferroelectric Nonferroelectric Crystals Crystals UNCLASSIFIED UNCLASSIFIED