Ferroelectricity: The Fundamentals Collection

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Ferroelectricity The Fundamentals Collection

Edited by Julio A. Conzalo and BasilioJimknez

WILEYVCH

WILEY-VCH Verlag CmbH & Co. KCaA

This Page Intentionally Left Blank

Ferroelectricity The Fundamentals Collection

Edited by julio A. Gonzalo, BasilioJirntnez

This Page Intentionally Left Blank

Editors:

Julio A. Conzalo Depattainento de Fisica de Materialrs. U A M , Madrid Barilia jirndnez Research Professor. ICMM-CSIC. Madrid

All hooks piiblished by Wiley-VCH are carefully produced. Nevertheless, authors, rditors, and publisher do not warrant the iiifbririation contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for

British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by Die Deutsche Bibliothek

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the internet at http://dnb.ddb.de.

0 2005 Wiley-VCH Verlag GinbH & Co KGaA, Weinheim All rights reserved (including those of translation in other languages). No part of this book inay be reproduced in any forin by photoprinting, microfilm, or any other means - nor transmitted or translated into a machine language without written permission from the publishers. Registered namcs, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Typesetting Manuela Treindl, Laaber Printing Strauss GmbH, Morlenbach Bookbinding Groi3buchbinderei J . Schaffer

GmbH & Co. KG, Grunstadt ISBN-13: 978-3-527-4486-5 ISBN-lo: 3-527-40486-4

Ferroelectricity The Fundamentals Collection

Edited by Julio A. Conzalo and BasilioJimknez

WILEYVCH

WILEY-VCH Verlag CmbH & Co. KCaA

I"

Contents

1

Foreword I 6.Jimdnez, J. A. Conzalo

2

Selected Early Work 1921-1961

2.1

Piezo-electric Activity of Rochelle Salt Under Various Conditions

5 7

J. Valasek

2.2

Rochelle Salt as a Dielectric

21

C. B. Sawyer, C. H. Tower

2.3

A New Seignette-electric Substance

26

C. Busch, P. Scherrer

2.4

Theoretical Model for Explaining the Ferroelectric Effect in Barium Titanate W P. Mason, 8. T: Matthias

2.5

Theory of Barium Titanate

42

A. F. Devonshire

2.6

The Lorentz Correction in Barium Titanate

66

J. C. S/ater

2.7 2.8

Phase Transitions in Solid Solutions of PbZrO, and PbTiO, C. Shirane, A. Takeda Dielectric Constant in Perovskite Type Crystals

80

87

J. H. Barrett 2.9

Ferroelectricity of Glycine Sulfate 90 6.T. Matthias, C. E. Mi//er,J. P. Rerneika

2.10

Switching Mechanism in Triglycine Sulfate and Other Ferroelectrics E. Fatuzzo, W.J. M e n

2.11

Crystal Stability and the Theory of Ferroelectricity

99

W. Cochran

f:rrroelectnrity: The Fimdamentals Collection. Edited by Julio A. Gonzalo and Basilio Jimenez Copyright 0 2005 WILEY-VCH Vcrlag G m b H b; Co. KGaA, Wctnhrini ISBN: 3-527-40486-4

91

27

VI

I

Contents

3

Ferroelectrics 1966-2001: An Overview R. Blinc

3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.6 3.7

Introduction 105 Statistical Indicators I06 Phase Transitions and Critical Phenomena 108 Ferroelectric Liquid Crystals I13 Ferroelectric Thin Films I15 Crystalline Films I15 Ferroelectric Memories and Integrated Ferroelectrics 116 Two-dimensional Ferroelectricity in Crystalline Films I18 Freely Suspended Ferroelectric Smectic Thin Films I19 Dipolar Glasses and Relaxors 121 Incommensurate (IC) Systems 125

4

Phase Transitions in Ferroelectrics: Some Historical and Other Remarks V. L. Ginzburg

5

Theory of Ferroelectrics R. E. Cohen

6

Ferroelectric Ceramics: History and Technology C. H. Haertling

7

Tenth International Meeting on Ferroelectricity: IMF-10, a Jubilee Meeting

I05

137

155

W. Kleeman

7.1 7.2 7.3 7.4

131

A Touch of History 181 A Wealth of Science 182 A Glimpse at Spain, at Madrid and at Spanish Life and Culture Summary 185

184

181

I’

1 Foreword 13.Jimtnez, J. A. Conzalo

The prehistory of ferroelectricity, or rather, the early history of ferroelectricity, begins, as pointed out by Prof. Sidney B. Lang (BenGurion University, Beer Sheva, Israel) with the early-recordedobservation of pyroelectricity more than twenty-three centuries ago. The Greek author Teophrastus wrote that the mineral “lyngourion”(probablytourmaline) showed the property of being able to attract little bits ofwood. This property must have had something to do with the heating/ cooling of the mineral in question. More recently, in the 18thcentury, investigations of the phenomenon of pyroelectricity made a significant contribution to early researches in electrostatics. In the following century this contributions were extended to other researches in mineralogy,thermodynamics, crystal physics,etc. Piezoelectriceffects were intimately connected also with the discovery of piezoelectricity in 1880 by the Curie brothers in France, and forty years later,with the discovery of ferroelectricity in 1921 by Valasek in the US. In the second half of the 20‘’ century research in pyroelectric, piezoelectric and ferroelectric effects of materials has flourished and has found a very large number of varied applications. Amongst the thirty-two crystal classes (point groups), as it is well known, eleven

of them are characterized by having a center of symmetry, and, being centrosymmetric they cannot have a polar character, cannot be ferroelectrics. The application of an external electric field will produce displacements of the positively and negatively charged atoms in the unit cell. The resulting strain of the cell will be the same upon reversal of the electric field, showing its electrosctrictive character. The twenty-one remaining crystal classes lack a center of symmetry, they may have one or more polar axes, and (with the exception of the cubic class 4.32)show piezoelectric effects, i.e. the application of pressure causes an electric current to flow in a -certain direction, and the application of a contrary tensile stress gives rise to a flow of charge in the opposite direction. If an electric field is applied the crystal will be stretched. This makes piezoelectric crystals extremely versatile as electromechanic transducers. Such modern new tools as the Tunneling Microscope and the Atomic Force Microscope, which make possible the investigation of actual microscopic processes in crystal surfaces are allowed just by the availability of excellent piezoelectric materials. Of the twenty piezoelectric classes, as it is well known, ten have a single polar axis and they are spontaneously polarized. This

Ferroeleclricity: The Fundamentals Collection. Edited by lulio A. Gonzalo and Basilio j i m h e z Copyright 0 2005 WlLEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40486-4

2

I spontaneous polarization is temperature I foreword

dependent, resulting in the pyroelectric effect. All ferroelectric crystals belong to one or another pyroelectricclass and have the property that an external field can reverse their polarization. Sometimes, however, the field must be extremely high to achieve in practice polarization reversal. There are many analogies between ferroelectricity and ferromagnetism as pointed out by Valasek, but there are also important differences, which set apart one physical phenomenon and the other, as might be expected. There are several criteria to classify ferroelectric crystals and the main ones were established early in the game. For many years the phenomenon of ferroelectricity was known to occur only in Rochelle Salt. Later potassium dihydrogen phosphate and a number of its isomorphs were recognized as ferroelectrics. Then during the Second World War, barium titanate, the prototype of many oxide ferroelectric perovskites to come was discovered. After this discovery and due to efforts of Matthias, Pepinsky, Smolenski and others, the number offerroelectric materials increased in a spectacular way. The need of a classification scheme for ferroelectricswas recognized and several criteria were proposed: (i) Chemical classification: ferroelectrics were classified in two groups: hydrogen-bonded (KH,PO,, or KDP, triglycine sulfate, or TGS, etc) and double oxides (BaTiO,, KNbO,, Cd,Nb,O,, etc). (ii) Number of allowed directions for the spontaneous polarization: uniaxial ferroelectrics,like triglycine sulfate or Rochelle Salt, and multiaxial ferroelectrics, polarizable along several crystallographic directions, which become equivalent in the high temperature paraelectric phase, like BaTiO,, and many perovskites. (iii) Ferroelectrics with a non-centrosymmetric high tempera-

ture phase (like KDP, which is piezoelectric above T,, the ferroelectric Curie temperature) and ferroelectrics with a centrosymmetric high temperature phase (likeBaTiO, and many other perovskites, TGS and its isomorphs, etc). (iv) Order-disorder ferroelectrics, in which permanent dipoles which are randomly oriented above the transition temperature become spontaneously ordered below T, (potassium dihydrogen phosphate, triglycine sulfate) and displacive ferroelectrics, such as many perovskites, in which reorientable dipoles in the paraelectric high temperature phase are not clearly recognizable. (Recentinvestigations however tend to soften the strict separation between both groups, i.e. between the order-disorder and the so called displacive ferroelectrics, stressing the coexistence in practice of order-disorder and displacive features in many or even in most ferroelectrics. It is pointed out that in a rigid lattice, almost undeformable dipoles may show a predominantly orderdisorder transition, while in a more deformable lattice, deformable dipoles, in which the constituent ions show clear temperature dependent displacements, can be taken as a displacive transition). Usually typical order-disorder transitions show Curie constants of the order of C = lo3 K while typical displacive transitions show much larger Curie constants, with C = los K. An effective field approach to ferroelectrictransitions (which may be generalized to include deformable dipoles) gives C in terms of the number of unit dipoles per unit volume (N) , the elementary dipole moment (p) and Boltzmann's constant (k,) as C = 4 R N p2/kB This suggests larger values for N and p in the ferroelectrics usually considered as prototypically displacive like BaTiO,. The history of the theoretical understanding of the phenomenon of ferroelectricity is one of slow progress over the years in the both fronts, the microscopic structural, lat-

1 Foreword - 6. Jimknez,I. A. Conzalo 13

tice dynamical front, and, to a lesser extent, also in the thermodynamic, phenomenological front. The crystal structure ofthe first known ferroelectric,Rochelle Salt, is so complicated that the chances of success of a microscopic theory, first attempted by Kurchatov (1933),along a line similar to that of the theory of Weiss for ferromagnetism, were very small. The statistical theory of Slater (1941),in which the arrangement of dipolar units (H,PO,) is linked by H-bonds, depicting KDP, was somewhat more successful, but only qualitatively. About twenty years later, Blinc would introduce quantum tunneling to describe the huge isotope effect in this ferroelectric. But it was the discovery by Wul and Goldman of ferroelectricity in BaTiO, (1945, 1946) followed by other perovshtes, such as KNbO, and KtaO, (Matthias 1949), LiNbO, and LiTaO, (Matthias and Remeika, 1949 and PbTiO, (Shirane, Hoshima and Suzuki, 1950), as well as the almost simultaneous discovery of antiferroelectricity in ZrTiO,, which shifted attention to the theoretical description of ferroelectric and antiferroelectric phenomena in a really much simpler structure, the perovskite structure, much more amenable to microscopic investigation. Mason and Matthias (1950) did propose a simple microscopic model to describe Ferroelectricity in BaTiO,, in which the small Ti4+ion surrounded by six oxygen ions is depicted as being in an off center six fold potential well minimum, giving rise to a unit dipole, capable of undergoing order-disorder changes under the influence of both thermal energy and the electrostatic energy of the effective local field in which the cooperative influence of all other neighboring unit dipoles predominates. This model was then strongly criticized by J a p e s , using arguments apparently plausible but, not too well founded and it was used latter as starting point for improved statistical models.

Independently, Slater (1950) pointed that the ferroelectric behavior of BaTiO, could be caused by long-range dipolar forces (via the Lorentz local effective field) competing with local short-range forces. Latter this provided the basic framework to describe the phase transition in displacivetype ferroelectrics. Mueller (1940),and later Ginzburg (1945, 1949) and Devonshire (1949, 1951, 1954) were the first to propose macroscopic, thermodynamical theories of ferroelectricity. Devonshire’s theory, which described in detail the successive phase transitions in BaTiO,, (from cubic to tetragonal, from tetragonal to orthorhombic and from orthorhombic to rhombohedral) became the paradigm of a phenomenological theory for ferroelectricity, and has remained so through several decades. In 1960 Anderson and Cochran recasted the microscopic theory in terms of crystal lattice dynamics, and predicted successfully the existence of “soft-modes”,later observed by neutron, infrared and Raman scattering. A selection of representative papers from Valasek to Cochran is given in the first section of this book. Then from the first historic International Meeting on Ferroelectricity in 1966, one can follow better the ups and downs of research in ferroelectric materials and ferroelectric transitions, through the series of International and European Meetings of which the last one was, for the moment, the 10* International Meeting on Ferroelectricity, held in Madrid in September 3 to 7, 2001. The Organizing Committee was happy to count on such distinguished speakers as Profs. Blinc, Ginzburg, Miiller for the Opening Session, which was inaugurated by the Minister of Education, Culture and Sports of Spain Dr. Pilar del Castillo,accompanied by the Chancellors of the main Universities in Madrid.

4

I

1 Foreword

To understand the impressive developments which have taken place since the discovery of Rochelle salt and then Barium Titanate to our days it is necessary to take account of the great variety and large amount of materials discovered since then presenting ferroelectricproperties, from the oxides and inorganic salts to the polymers, liquid crystals and biological tissues. This large variety of materials has made possible numerous applications of linear and nonlinear dielectric properties so that ferroelectrics constitute now a large section in the field of advanced functional materials with an important share in the market. The possibility of presenting the ferroelectric material in various forms: mono and polycrystalline, meso and nano structured, massive and thin a m , allows the use offerroelectrics in the most diverse technology applications,from devices to transform high power energy to integration in silicon electronic devices in the new technologies. In this respect it is not an exaggeration to say

that ferroelectric materials make up now a strong and well defined chapter of the Science and Technology of Materials. At the same time the ease of preparation has allowed that, not only in advanced countries but in developing countries also, competitive scientific research in these materials has been possible contributing substantially to their technologic and social development. The importance of ferroelectric materials is well attested by the series of current scientific meetings: the ISAF in the USA, the ECADP in Europe, the monographic ones regularly held in Japan and Russia. In all of them the number of contributors grows steadily. All of this foretells a promising future for Ferroelectricity and ferroelectric materials. Madrid, September 2004 ] d i o A. Gonzalo

Basilio jimtfnez

UAM, Madrid

CSIC, Madrid

2 Selected Early Work 1921-1 961 J. Valasek C. B. Sawyer, C. H. Tower G. Busch, P. Scherrer W. P. Mason, B. T. Matthias A. F. Devonshire J. C. Slater G. Shirane, A. Takeda J. H. Barrett B. T. Matthias, C. E. Miller,J. P. Remeika E. Fatuzzo, W.J. M e n W. Cochran

Ferroelectricity: The Fundamentals Collection. Edited by lulio A. Conzalo and B a s h Jimknez Copyright 0 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40486-4

This Page Intentionally Left Blank

2.7 Piezo-electric Activity of Rochelle Salt Under Various Conditions - J.Valasek 17

478

J . VALASEK.

SECOND

[SERIES.

PIEZO-ELECTRIC ACTIVITY OF ROCHELLE SALT UNDER VARIOUS CONDITIONS. BY J. VALASEK. SYNOPSIS.

Electrical Properlies of Rochelle Salt Crystal are analogous t o the magnetic properties of iron, the dielectric displacement D and polarization P varying with the electric field E in the same general manner as B and I vary with H for iron, and showing a n electric hysteresis with loops distorted by a n amount corresponding t o the permanent polarization PO, whose value is about 30 e.s.u./cm.s but varies for different crystals. The dielectric constant ( K = d D / d E ) was measured from - 70' to 30' C. and found to be surprisingly large, increasing from about 50 a t - 70' t o a maximum of about 1,000near 0 ' . The modulus of piezo-electric activity for shearing stresses ( 6 ) varies with temperature, - 70' l o 40° C . , in a very similar manner. increasing from less than I O - ~ a t - 70° to a maximum of about 10-4 at 0'. T h e ratio 6 / K varied wilh the electrode malerial, being greater for tin foil than for mercury electrodes. The difference may be due to the alcohol used in shellacking the tin-foilelectrodes on. There are other indications t h a t 6 and K are related. The uarialion of B with humidity is such a s can be accounted for by the decrease in the dielectric constant of the outer layer as a result of dehydration. The change of polarization produced by pressure as measured by the change in the hysteresis loop agrees with the value found directly from the piezo-electric response, as required by Lord Kelvin's theory. Alsofatigue efecls on S produced by temporarily applied fields are traceable to fatigue in the polarization. The electrical conductivity below 45' is less than 5 X I O - ~ rnhos/cm.s b u t from 43' to 57' increases rapidly to 5 X 10-(. Optical Properlies of Rochelle Salt as Calculated from the Natural Pola7ization.Assuming only one electron is displaced the nalural period corresponds to a wave0 ' . the observed length of 4.2 @ and the speci/ic rotation for sodium light comes out 1 value being 2 2 O . 1 .

ECENTLY' the writer described some experiments on the dielectric and piezo-electric properties of Rochelle salt, which were made for the purpose of correlating and explaining the effects observed chiefly by Cady and by Anderson. The plates used were cut with faces perpendicular to the il axis and with edges a t 45' with the and c axes. The present paper is a continuation of the work, the variations in the electrical properties having been studied more extensively. T h e apparatus and method of observation have been already described in the paper referred to above. The niorc important results obtained at that time can be summarized as follows: In the case of Rochelle salt the dielectric displacement D, electric intensity E , and polarization P behave in a manner analogous to B, H , and I in the case of magnetism. Rochelle salt shows an electric 1

J. Valasek, PHYS. REV.( 2 ) , XVII, p. 475.

8

I

2 Selected Early Work 1921-1 961 2.;Ix.]

PZEZO-ELECTRIC

A C T I V I T Y OF ROCHELLE S A L T S .

479

hysteresis in P analogous to the magnetic hysteresis in the case of iron, the loops however being distorted by an amount corresponding to the permanent polaiization of the crystal in the natural state. This point of view is very effective in accounting for many of the peculiarities observed. In an electric field the piezo-electric activity has a maximum for a definite value of the field and decreases to a small value in both directions. The position of the maximum corresponds to the greatest rate of change of polarization with electric field in the case of the condenser experiments. In fact if force and electric field are equivalent in changing the piezoelectric polarization then the response for a given force in various applied fields must necessarily give curves of the same general nature as curves of aPIdE or aDlaE against E. I t is permissible to interchange D and P in most cases because of the large dielectric constant of Rochelle salt. RELATIONBETWEEN POLARIZATION AND PIEZO-ELECTRIC ACTIVITY. The activity of a piezo-electric crystal is intimately related to the natural polarization. According to Lord Kelvin this natural moment is masked by surface charges so that the crystal appears to be uncharged. This polarization or piezo-electric moment can be measured independently of the charges on the electrodes, through the distortion of the hysteresis loop. The center A of the loop is found by a consideration of symmetry and may be assumed to represent the condition of no polarization. If the natural condition of polarization is assumed to be half way between the two branches of the loop a t zero field then the value of the permanent polarization P Ois proportional to A B , Fig. I. There being

Fig. 1.

no field applied, the equation for the work done per unit charge carried through the condenser is: 4"

(+ - Po)t

= 0,

2.7 Piezo-electric Activity of Rochelle Salt Under Various Conditions - J. Valasek 19

480

J . VALASEK.

SECOND

[SERIES.

so that

P”F’ Qo where Qo is the apparent average permanent charge a t zero field given by Calculation gives the

A B (Fig. I) and where S is the area of the plate. value: 30 e.s.v./cm2.

According to Lord Kelvin’s theory an applied stress will change this polarization so as to create free charges on the electrodes. A force of 250 grams applied to the crystal should consequently shift the loop by an amount equivalent to the piezo-electric response for 250 grams. When this experiment was performed another, but more unsymmetrical loop, was obtained. The change in polarization by the loop method was I 14 e.s.u./cm2 while the piezo-electric response amounted to 121 e.s.u ./cm? The value of Po obtained from the hysteresis loops is only approximate because of the assumptions involved in its determination. I t cannot, moreover, be fixed definitely enough to be put down as a physical constant of Rochelle salt because it varies with different specimens, besides changing with temperature, pressure and fatigue. The valuePo = 30 e.s.u./cm.2 is thought to be a representative value and is checked by other measurements. The writer would not be surprised, however, to find other specimens giving several times this value. The change in polarization due to pressure however is derived by a differential method eliminating much of the uncertainty in measurements on one loop. T h e result i n this case should be fairly definite, as indeed it seems to be. Piezo-electric activity depends on both the crystalline structure and on the polarization. I t is greatest for a polarization somewhat larger than normal and decreases in both directions for changes in this quantity, the polarization being changed by appIying an electric field. I t has been shown by the writer that this relation between activity and applied field is approximately like that of the derivative dD/aE of the curve relating the dielectric displacement D and the electric field E of the crystal used as a condenser. Since this latter relation is in the form of a hysteresis loop it follows that the activity is also a double-valued function of the applied field depending on the direction of variation of the field. A curve illustrating this effect is reproduced in Fig. 2. The readings were taken in as short a time as possible to eliminate fatigue. These curves show that the piezo-electric response a t zero field depends on the previous electrical treatment of the crystal. The latter fact has also been noted by W. G. Cady in the report previously referred to.

10

I

2 Selected Early Work 1921-1961

E::Ix.]

PIEZO-ELECTRIC A C T I V I T Y OF ROCHELLE SALTS.

481

This after-effect does not persist very long but dies off exponentially with the time. The piezo-electric response or ballistic throw of the galvanometer for 2 5 0 grams has been observed to return to half value in I minute and to normal in over 20 minutes after fields of 150 volts have been applied for 3 minutes previously. There is a much greater aftereffect in the direction of increased activity.

Fig. 2.

A corresponding dielectric effect is indicated by the double value of the condenser charge at zero field in the hysteresis loops. This is clearly due to a fatigue in the polarization and it also dies off exponentially with time. Herein is probably found the explanation of the “storage battery effect” described by W. G. Cady who observed that after applying a field of IOO volts for some time there was, on removal of the field, a small current which decreased gradually and flowed from the crystal as from a miniature storage battery. The piezo-electric fatigue may well be a direct result of the fatigue in the polarization, as there seems to be a close relation between piezoelectric activity and polarization. I t appears that the activity is approximately proportional to the rate of change of polarization with applied field and hence proportional to the dielectric constant. An examination of the temperature variation of the two quantities leads to this conclusion. It is further confirmed as regards field variation by the fact that the relation of activity to applied field is like aDfaE vs. E where aD/aE is merely the instantaneous value of the dielectric constant K . As an approximation we can write the piezo-electric modulus 6 proportional to K : 6 = A.K.

2.1 Piezo-electric Activity of Rochelle Salt Under Various Conditions - J.Valasek

J . VALASEK.

If this equation were exact A would be a fundamental piezo-electric constant of the substance, being of the order of I X IO-’ between - 20’ C. and 20’ C. At some temperatures and for some exceptional specimens the relation does not seem to be so simple.

+

EFFECTOF MOISTUREON PIEZO-ELECTRIC PROPERTIES. In order to investigate the effect of dryness on the activity of Rochelle salt, some phosphorus pentoxide was enclosed in the chamber containing the crystal. The crystal soon started to dehydrate and after a few days was covered by a white coating. T h e piezo-electric throw for a load of 250 grams continually diminished. When the response was tested a t different fields a more interesting fact was observed. Besides t h e decrease in response, the maxima were displaced along the field axis into a condition of greater polarization. This is shown by Fig. 3, the curves

being taken after the lapse of the following times: (6) I day, (c) 3 days, ( d ) 12 days. The decrease in the maxima and also their displacement is in the same direction as, and may be entirely due to, the effect of different dielectric properties of the crystal and of the dehydrated layer. In other words the presence of a layer of inactive dielectric of relatively low specific inductive capacity will diminish the charge on the plates due to the polarization of the central active layer, and thus decrease the piezoelectric response. I t will also diminish the effective field across the active layer making it necessary to increase the potential difference

I”

12

I

2 Selected Early Work 1927-7967

483

PZEZO-ELECTRIC ACTZVZTY O F ROCZfELLE SALTS.

' No. O L ' 5.

between the plates to produce the same field across the inner layer, thus shifting the position of maximum activity. The effects due to uniform layers can be readily calculated. Let P o be the polarization produced in the middle layer by pressure, let P 1 and PZ be the electrically induced polarizations in the dielectrics I and 2 respectively (Fig. 4). Since the dielectric displacement is solenoidal we have:

D'

= El

+ 4nP1+

4nP0 = E Z

+ 4nPz = 4nu.

Since

PI

=

(KI

-I)E~

and

Pz =

(KZ

- I)EZ

we can write

D'

Fig. 4.

+-

= K ~ E4sP0 ~ = KZEZ.

The difference of potential between the plates is zero so that, replacing Po by u O : o = Es(d - t ) Elk

+ -4"cr (d - t ) + 9(u - crop, K2

Ki

giving :

This gives us a relation between the piezo-electric response a t zero field of the crystal with the dry shell and of the same crystal before it dried. The assumption is made that the elasticity of the shell is equal to that of the crystal so that a given total force produces the same polarization in the crystalline portion. The position of the maximum will be changed to another value of total potential difference on the crystal. Let V be the total potential drop and V' be the drop across the crystalline part. When there is no dehydrated layer present V = V' = dE, where E is the field strength in the dielectric. When there is a layer of uniform thickness (d - t)/z on both faces then

V = (d

- ,)El'

+ tE',

where E" and El are the field strengths in the dielectrics 2 and tively. The dielectric displacement

D

= K~E= " K ~ E-/-' 4?rPo,

I

respec-

2. I Piezo-electric Activity of Rochelle Salt Under Various Conditions - J. Valasek 113

484

J. VALASEK.

SSCOXO

[SERIES.

is solenoidal, and we can eliminate E" from equations above and write:

Since the last term is small compared to the rest of the expression, this gives :

The following quantities were measured and substituted in these equations. ~1 = 1000, d = 0.22 cm., K2 = 180, t = 0.14. The quantity t is an average obtained by breaking the crystal in several places and it is probably not very accurate because of the irregularity of the outer layer. We should, however, get a rough check on the plausibility of the proposed explanation. We find that

0: = 0.24 and that

Q

E'

= 0.38.

While the values of Q'/Q and E'IE from the maxima of curves a and d of Fig. 3 are respectively 0.39 and 0.33. The agreement is not as good as could be desired even after making allowance for the difficulties in estimating t. Possibly there is a true humidity effect with respect to piezo-electric activity but the above shows, a t least, that it is quite small.

PIEZO-ELECTRIC ACTIVITYAND TEMPERATURE. In order to investigate the variation of activity with temperature, the chamber holding the crystal was immersed in COZ snow. After everything was thoroughly cooled and a t - 75" C., the chamber was allowed to heat up. Above - 35' C. an electric heater was used. It was wound on a glass jar and insulated from the crystal chamber by a felt jacket. This jar was immersed in a n oil bath to steady the heating rate while the felt eliminated any rapid changes of heating of the crystal. The current was gradually increased so as to keep the rate of heating uniformly between 10 'i a n d I O C. per minute so as to eliminate thermoelastic stresses. The temperature was measured by means of a copper-constantan thermocouple directly soldered to an electrode on the crystal. In this way the actual temperature of the crystal itself was measured.

14

I

2 Selected Early Work 1921-1961

Zp'".]

PIEZO-ELECTRIC: ACTIVITY OR ROCHELLE SALTS.

4%

When the piezo-electric response or galvanometer throw for 250 grams was measured a t the various temperatures for the first specimen the curve of Fig. 5 was obtained. This was duplicated to check the second

Fig. 5 .

maximum. At - 70" C. the piezo-electric activity is comparatively negligible. As the temperature is raised slowly the activity stays small 30" C. is reached. At - 20" C.it is rising very rapidly, reaching until a maximum a t about 0" C. I t decreases again but at 23" C. comes to a small but'sharp maximum from which it diminishes slowly, becoming very 50" C. The magnitude of the second maximum varies with small at the temperature a t which heating begins. This second maximum was found in the case of two crystal plates provided with tinfoil electrodes attached by shellac. Three other crystals were prepared with electrodes of mercury held against the crystal by two rectangular cups attached by wax. The thermocouple wires were immersed in the mercury. None of the crystals so prepared gave the second maximum. Moreover, none of them were as active as those used above. The variation of piezo-electric response of these specimens is shown in Fig. 6. The increase a t 30° to - 15"C. and the decrease at 20" C. to 30" C. are remarkably consistent. Between - 15" and 20" C., however, they each show different characteristics. These mercury electrode crystals seemed to give more constant results than the crystals with tinfoil electrodes attached by shellac. I t was then suspected that the increased response of the crystal with

-

+

+ +

+

-

2.7 Piezo-electric Activity of Rochelle Salt Under Various Conditions

- J. Valasek

115

486

SECOXD

J . VALASEK.

SERIES.

tinfoil electrodes and the presence of the second maximum was in some way due to the penetration into the crystal of the alcohol solvent of shellac. Accordingly one of the crystals originally with mercury electrodes was provided with the other type. As soon as the shellac was sufficiently dry, Curve b, Fig. 7 was obtained, the response originally

-70 -60

-S#

-t4

-JO

-8b -/O q6UR-C

7.

0

/o

20

J0

+x

T€,rf-K447UN

Jlnr CRIJTAL - pwfcwur TncArnEffT

Fig. 7.

having followed Curve a. The sensitivity increased seven-fold a t some temperatures but there was no second maximum. In two weeks the characteristics had changed to those shown in Curve c, seemingly checking the suspicion that alcohol was responsible for the second maximum and for the increased sensitivity. I t would be interesting to use alcohol cup electrodes and investigate the continuous effect of alcohol soaking into the crystal. T:ie effect is probably not chemical.

16

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2 Selected Early Work 1921-1961 'OL.

No. 5.

PIEZO-ELECTRIC A C T I V I T Y OF ROCHELLE SALTS.

387

In a paper on the piezo-electric effect on Rochelle salt, A. M. Nicolson' describes a method for desiccating the crystals by soaking in alcohol and heating, thus making them stronger and more sensitive. T h e writer is certain, from his work on the subject, that complete desiccation will make the crystal entirely inactive. In the above method, apparently, the treatment is not prolonged enough to completely dehydrate more than a shell around the crystal and its effectiveness may be connected with the penetration of the alcohol into the crystal. T h e heating a t 40' C. may also be effective in allowing a rearrangement and recrystallization of some of the molecules or groups not properly oriented. W. G. Cady, as well as the writer, has observed that heating the crystal will usually increase its sensitivity permanently, although sometimes the reverse is true. An interesting side-light on the temperature variation of piezo-electric activity is offered by a study of curves like t h a t of Fig. 2 but at different temperatures. They show that the effect of temperature is not so much to change the piezo-electric activity a s to shift the position of the maximum from one value of the field to another. This is probably connected with the variation of the dielectric constant with temperature which will be taken up presently. The charging throws of the crystal used as a condenser show variations similar to those of the activity except that they do not tend to zero but to a constant value a t the lower temperatures. The crystals giving the second maximum on the piezo-electric curve show a similar peculiarity here. The crystals with the mercury electrodes give more regular curves. A t 20' to 25' C. the crystals of both types begin to conduct, causing a steady drift of the galvanometer. Experiments seem to indicate that Ohm's law holds a t least approximately. T h e conduction was a t first thought to be electrolytic because of the manner in which Rochelle salt melts. Instead of real melting it appears t h a t the crystal dissolves in its water of crystallization which is set free at 55' C. The desiccated crystal, however, decomposes into a tarry product and emits heavy white fumes above 150' C., without melting. The dehydrated crystal also begins to conduct above 20' C., making it probable t h a t electronic and not electrolytic co-nduction is observed. Measurements of conductivity were made on the natural crystal a t various temperatures up to its liquefying point. T h e values obtained after the conductivity was sufficiently large to use a Wheatstone bridge are as follows: ' A . M. Nicolson, Proc. Am. Inst. Ele. Eng., Vol. 38. p. 1315 (1919).

2. I Piezo-electric Activity of Rochelle Salt Under Various Conditions - J.Valasek 117 SECOND

. I . VALASEK.

488

[SERlES.

TABLE I. Temperature. Coadqctivity. Less than 43' C... . . . . . . . . . . . . . . . . ..Leu3 than 0.5 X 10-8 mhos/crn3. 43 .................... 0.5 x 10-8 45 .................... 1.0 x 10-8 47 .................... 0.3 x 10-7 .................... 49 0.5 x 10-7 51 .................... 0.5 x 10-0 53 .................... 0.6 x 10-4 .................... 1.7 x 10-4 54 57 .................... 5.0 x 10-4 Greater than 57 .................... 5.0 x 10-4

A t temperatures below 20' the dry crystal is a fairly good insulator ~ oo C. The having a specific conductivity of 5 X I O - ' ~ r n h ~ s / c m .at conductivity decreases slightly a t still lower temperatures. In all these measurements the surfaces were thoroughly dried by the presence of phosphorus pentoxide in the crystal chamber.

TABLE11.

_ _

-. -

-~ .__ ~

Dielectric Constant.

Temp. Cent.

I --

-70 . . . . . . . . . . -50 . . . . . . . . . . -30 . . . . . . . . . . -20 . . . . . . . . . .

-10.. . . . . . . . . 0.......... 10 . . . . . . . . . . 20 . . . . . . . . . . 30 . . . . . . . . . .

l A

Piezoelectric Modulus. B

_ I _ _ _

71 85 140 386

1,100 688 423

42

1 ~

146 $0 252 924

928 645 i 146 Conduction conimences

I

0.041 X lo-' 0.068 " 0.41 '' 5.4 " 18.9 " 22.9 I' 18.9 "

13.5 2.2 0.41

" "

''

0.017 x 10" 0.017 0.065 1.08 6.07 6.75 7.42 8.10 1.08 0.41

'I

" " " " " " 'I

"

Table 11. gives values of the dielectric constants for a field changing from o to 880 volts/cm. and of the piezo-electric modulus 814 for shearing stresses of 2 2 0 grams/cm.2. The modulus &4 is defined by the relation given by Voigt ul = - &4YZwhere ul is the surface density of charge and Y,is the shearing stress producing it. The given values are thought to be the most representative in each case. They are subdivided into two classes, according to whether the electrodes were tinfoil attached by shellac (column A ) or mercury in direct contact with the crystal (column B ) . The former method is the most convenient to use in general practice, but the latter is thought to give more exactly the properties of Rochelle salt in the direction of the & axis. The dielectric

18

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2 Selected Early Work 1921-1961 ‘OL‘ 5. No.

PIEZO-ELECTRIC A C T I V I T Y O F ROCHELLE SALTS.

489

constants are surprisingly large, a fact noticed by Pockelsl who supposes that this is due to “internal conductivity.” The writer however has measured separately the conductivity a t these temperatures and is of the opinion that this is a true dielectric constant arising from polarization of the dielectric, and for this reason being so closely related to the piezoelectric effect. Because of the relatively low specific inductive capacity of the desiccated crystal it is thought that the high specific inductive capacity of Rochelle salt is partly due to the water molecule.

MAGNETICANALOGY. There seems to be a strong analogy between the behavior of Rochelle salt as a dielectric possessing hysteresis and having an exceptionally large dielectric constant, and the phenomena of ferromagnetism. Accordingly some of the features of Weiss’s theory of magnetism may find their counterpart in the phenomena in Rochelle salt. Weiss2 plots the susceptibility against the reciprocal of the absolute temperature and finds that the curve may be represented by a succession of straight lines. He interprets the sudden changes in slope as due to changes in the number of magnetons. If the data of Figs. 8a and 12 are plotted against the reciprocal of the absolute temperature one likewise gets what may be considered to be a succession of straight lines. Actually however there occur rounded corners where the curves suddenly change direction. This may be due to slight non-uniformity in heating which occurred in spite of the precautions taken. I t is considered that the straight portions are a t least as definite as those shown in Weiss’s paper. The most abrupt 20’ C. These may accordingly be changes are a t - 20’ C. and a t considered as the “Curie points” in Rochelle salt.

+

RELATIONTO OPTICALPROPERTIES. Some of the optical properties of Rochelle salt can be at least approximately found from the electricaI data given. In the course of this calculation it is of course necessary to introduce some assumptions notably as to the nature of the permanent polarization. If one knew just how the permanent polarization was ptoduced he could find the free period of this mechanism. The data needed are the force per unit displacement f and the mass m of that part of the molecule. The wave-length corresponding to the free period is given by the expression

c

being the velocity of light. 1

Pockels, Lehrbuch der Krystaloptik, p. 508.

* P. Weiss. J. de Physique, Vol. I . p. 968 (1911).

2.1 Piezo-electric Activity ofRochelle Salt Under Various Conditions - J . Valasek 119

J . VALASEK.

490

SECOND

[SERIES.

Let us assume for simplicity that only one electron is involved in the creation of the permanent moment. T h e quantityf can be derived from the value of the permanent moment Po = 30 e.s.u./cm.2 and from the displacement of the electron producing it. Since the force on an electron inside a dielectiic of polarization Po is roughly equal to QPoelthe expression for the wave-length may be put in the form:

x

=

2*c

dg*

The value of d , the displacement of the electron, can be found as follows: Taking 30 e.s.u./cm.2 as the natural polarization, the moment per molecule is obtained by dividing by the number of molecules per c.c., the result being 7.1 x I O - ~e.s.u. ~ I n each of these molecules there are 140 electrons, this being the sum of the atomic numbers of the constituent elements in Rochelle salt. If we suppose as above t h a t only one of these is effective in producing the piezo-electric moment, its displacement from the center of force of the rest of the molecule will be d = 2.7 x IO-" cm. I t would of course be more reasonable to suppose that a t least several of the electrons are displaced by different amounts, and to the extent that we do this the value calculated above becomes smaller. IJsing these values of POand d for the simple case treated above we find for the wave-length the value: X = 4.2 p .

Coblentz2 shows the presence of fairly strong absorption in this region of the infra-red. This may, however, be due to the water of crystallization and not to the cause cited above. These two possibilities should be distinguishable experimentally because the character of the absorption due to these electrons should change greatly with the temperature, as the piezo-electric elasticity or force per unit displacement of the electrons changes. The natural period as found above should be the same as that involved in rotatory dispersion formulz, since both the piezo-electric effect and optical rotation are due to an unsymmetrical or twisted structure of the molecule. J. J. Thornson3 gives an approximate formula for the specific rotation, namely: e2dp2 P

=

m

1

H. A. Lorentz, Theory of Electrons, p. 306.

2

W. W. Coblentz, Infra Red Spectra, Vol. J. J. Thornson. Phil. Mag., Der.. 1920.

2,

'

p. 38.

20

I

2 Selected Early Work 1927-1 967

"".I

"OL. No. 5.

PIEZO-ELECTRIC ACTIVITY OF ROCHELLE SALTS.

491

in which e is the charge of the electron, c is the velocity of light and M and m are the masses of the molecule and of the electron, d is the radius of the molecule, fi is the free period, and n is the frequency for which p is to be calculated. Using the value of p derived from piezo-electricdata we find for sodium light the specific rotation of the order of magnitude of IO', the tables giving 22.1'. Considering the fact that so little is known of the structure of the Rochelle salt molecule, the approximation is fair. The writer is indebted to Professor W. F. G. Swann, who initiated this research and gave many helpful suggestions, and to Dr. W. R. Whitney, Director of the Research Laboratory of the General Electric Company, whose presentation of two beautiful crystals made the work possible. PHYSICAL 'LABORATORY, UNIVERSITY OF MINNESOTA.

2.2 Rochelle Salt as a Dielectric - C. B. Sawyer, C. H. Tower

FEBRUARY I , 1930

PHYSICALREVIEW

VOLUME 35

ROCHELLE SALT AS A DIELECTRIC BY C. B. SAWYER A N D C. H. TOWER THEB R ~ ~ LABORATORIES, SH CLEVELAND (Received November 6, 1929) ABSTRACT Both saturation and hysteresis appear in Braun tube oscillograms made a t various temperatures with a condenser whose dielectric consists of Rochelle salt slabs cut perpendicular to the a-axis. The dielectric constant for such slabs may reach a value of 18,000. Curves are also given, showing the variation in mechanical and electrical saturation with temperature. These correspond in only a general way to the piezoelectric constant’s variation with temperature. Certain marked peculiarities are noted in the resulting mechanical deformation when Rochelle salt is excited with alternating potentials. Clear Rochelle salt half-crystals have been produced up t o forty-five centimeters in length.

THE

remarkable physical properties of Rochelle salt, the most piezoelectric active of all crystalline substances, have been reported by other authors.’ Comparatively small plates and few crystals were used in their determinations. Work a t this laboratory has been carried on for a number of years on Rochelle salt with a view towards commercialization. I t has, therefore, been necessary to produce large clear crystals in quantity. Clear, flawless half-crystals are grown up to 45 cm in length and 2 kg in weight. The dielectric strength and insulation value of plates from such crystals is very high. Many hundreds of plates (mostly perpendicular to the a-axis of the crystal) have been produced and their electrical properties measured. Thus a Rochelle salt plate 4.75 mm thick shows a dielectric constant of 18,000 when tested a t 15OC at 60 volts 60 cycles alternating current. The highest previously reported value which has come to the attention of the authors is about 1380.2 An air condenser of area and capacity equal to that of the crystal plate would have a plate separation of only 0.00475 mm, if 1380 be taken as the crystal dielectric constant; and 0.000262 mm (0.0001”) if 18,000 be taken. I t is thus evident that a comparatively thin layer of cement or dehydrated Rochelle salt between the body of the crystal and the foil electrode will introduce a very large error in the determination of the dielectric constant. A n y adhesive such as balsam in xylol, Japan Gold size, or beeswax dissolved in benzol with a small addition of rosin, may be used in dilute solution for atttaching the foil. I t is important, subsequently, to 1 Frayne, Phys. Rev. 21,348 (1923); Isley, Phys. Rev. 24,569 (1924); Laureyand Morgan, J. Am. Chem. Soc. 46, 2192-6, (1924); Pockels, Encyklopadie der Math. Wiss. Vol. 5, Part 2; Valasek, I’hys. Rev. 17,475 (1921); 19,478 (1922); 20,639 (1922) and 24,560 (1924). Voigt, Lehrbuch der Kristallphysik, Chap. 8, Leipsig (1910). 2 Valasek, Phys. Rev. 19, 488 (1922).

269

I

21

22

I

2 Selected Early Work 1927-7967

270

C. B. SAWYER A N D C.

H. TOWER

rub down the foil very thoroughly to bring it as close to the crystal surface as possible. A series of Braun tube oscillograms was obtained. For this purpose, and for all other results reported in this paper, a crystal plate wasemployed measuring about 8.5X5.5X0.5 cm, cut with its plane perpendicular to the

m7+

7

130 volts

60 cycles

I

1

I

I

I

Fig. 1. Schematic connection of Braun tube. Capacity of crystal plate 0.004 t o 0.2 M f ; of C 0.7 Mf. R110.45 rnegohms; R,==3180 ohms; R,==31800 ohms.

a-axis, and its long edges parallel to the c-axis. All measurements and oscillograms were carried out with 60 cycle current from the power tines. All vertical deflections are on the same scale as in Fig. 2. Fig. 1 shows the connections employed with the Braun tube for obtaining crystal oscillograms. At the left is a resistance acting as a voltage divider. To the right is the crystal plate under test, connected in series with a con-

1161 v. per cm 387 v. per cm Restrained Fig. 2. Oscillograms of crystal plate, free and restrained. Temperature 15" C; frequency 60 cycles per 8ec.

denser giving voltages proportional to the charge on the crystal. The resulting oscillograms, such as shown in Fig. 2, have ordinates proportional to crystal charge and abscissae proportional to applied voltage. Fig. 2 shows comparative oscillograms of a plate when entirely free, and of the same plate restrained by cementing it between two thick aluminum

2.2 Rochelle Salt as a Dielectric

- C.

B. Sawyer, C. H. Tower 123

ROCHELLE SALT A S A DIELECTRIC

271

plates, thus very largely precluding mechanical motion due to piezo-activity. The left-hand vertical pair is for 387 peak volts per cm ; the right-hand vertical pair is for 1161 peak volts per cm. The upper pair is unrestrained; the lower pair is restrained. Dielectric constants calculated from these oscillograms show exceedingly interesting and suggestive values: from the lefthand oscillogram (restrained plate) about 430;from the left-hand oscillogram (free plate) in the saturated range about 330; from the same oscillogram for a complete cycle, excluding saturation range 10,500; and for maximum instantaneous value not less than 200,000. Such enormous values of the dielectric constant in connection with less efficient foiling of the crystal plates, may account in part for the previously observed storage battery effect. Supplementary tests indicate that little

15" 0" -8" Fig. 3. Hysteresis and saturation of Rochelle salt plate 11. Potential gradient in dielectric 387 (peak) volts per cm; frequency 60 cycles per wc.

change in the value of the dielectric constant is to be looked for as a result of improvement in foiling. I n these supplementary tests, electrodes of saturated Rochelle salt solution were used and results did not differ significantly from those obtained from a carefully foiled plate. Moreover these large values of the dielectric constant of Rochelle salt have been observed in many hundreds of plates of various dimensions from many different crystals. Determination of the dielectric constant was usually made by applying 112 volts of 60 cycles alternating potential to the free crystal plate and noting the resulting current. In addition, circuit resonance and condenser substitution methods served to check this first method, all three giving results in substantial agreement. Fig. 3 comprises a series of comparative oscillograms made from the same crystal plate a t different temperatures as indicated. Proceeding from top

24

I

2 Selected Early Work 19221-7961

272

c. B .

s.4 W Y E H A N D

c. €1. TOII’ER

left to bottom right it is evident that as the temperature is decreased both the voltages and charges required for saturation greatly increase. So also do the areas of the hysteresis loop. Here again the method of applying the foil electrodes to the crystal isof great importance as the shape and area of the loop will vary somewhat with this factor. All of the oscillograms were made with the greatest care. A second crystal plate gave results identical with the first. Two other plates of the same dimensions as before but with their long edges cut a t 45’ to the c-asis, showed no essential differences in the derived oscillograms. Though no special humidity precautions were observed, the resistance of the plates, at 100 volts constant potential, never fell below many megohms.

Fig. 4. Temperature variation of saturation effect and piezoelectric constant.

If a standard plate-its long edge being cut parallel to c-axis-is electrified with an alternating potential, it will be deformed and such deformation can be observed and measured conveniently with a microscope. For the results shown in Fig. 4, one short edge of the plate was cemented to a large lead block and various values of 60 cycle potential were applied to the electrodes. Th e alternating motion produced under these conditions lies i n the plane of the plate and is perpendicular to the c-axis. T h e relation between total deformation and electrification is shown for various temperatures. Saturation is again in evidence and saturation values again increase greatly with decrease in temperature. Keeping close pace with it is the voltage required to produce saturation. But i t is very noteworthy t hat considerable voltage must be applied before the crystal shows appreciable deformation.

2.2 Rochelle Salt as a Dielectric - C. 6. Sawyer, C. H. Tower 125

ROCHELLE SALT A S A DIELECTRIC

273

Fig. 5 shows the close relationship existing at different temperatures between: Ist, volts per cm required for mechanical saturation; 2nd, the energy loss per cubic centimeter per cycle; 3rd, the charge per cubic centimeter required for electrical saturation. Though not shown in this figure, these curves are followed closely by those of the voltage required for electrical saturation and of the deformation a t mechanical saturation. N o determina-

Fig. 5. Properties of Rochelle salt at various temperatures.

tions of the piezoelectric constant were made, but Valasek’s3 most recent curve of t h e temperature variation of the piezo-electric constant is included for the sake o f comparing temperature variation of this property with those of the others. I t has been the great privilege of the authors to carry on this work begun under the very able leadership of the late Charles F. Brush, Jr.

3

Valasek, Science \‘ol. LX\’ No. 1679, p. 235 (1927)

26

I

2 Selected Early Work 15)21-1961

2.3 A New Seignette-electric Substance

C.Busch, P. Scherrer, Natuuisss. 23, 737 [7 935)

pared to those of the Seignette salt but the main behaviour is completely analogue. A purely qualitative test of the piezoelectric activity until liquid air temperature using the method of GIEBE and SCHEIBE gave for the magnitude of the piezoelectric module d,, a parallel behaviour to tj3,. Moreover, it was proved by means of DEBYESCHERRER pictures' at room temperature and at liquid air temperature that there were no relevant structural changes at the higher Curie point. The crystals were obtained from a saturated solution of KH2P0,at its boiling point by slowly lowering the temperature. The investigated objects were small plates of about 1 cm2in surface and 1 mm thickness, Aluminium foils were glued on these plates as electrodes. The dielectric constant measurements were performed in a capacity bridge and the temperature was estimated by a copper-constantan thermocouple. Further investigations, i.e. field intensity dependence and temperature behaviour of the piezoelectricity in KH,PO, and in the isomorphs NH,H,PO,, KH,AsO, and NH,H,AsO, are in preparation. The measurements below liquid air temperature were performed in Charlottenburg, because in Zurich there was no liquid Hydrogen at our disposal. We would like to thank especially Herr Prof. STARK, president of the Physikalisch-Technischen Reichsanstalt in Berlin, for his friendly collaboration, he provided us with the necessary quantity of liquid Hydrogen. In the same way we feel us obliged to thank Herr Prof. WESTPHAL and specially Herr Dr. J. ENGL for providing us a working place in the Physical Institute of the Technische Hochschule in Charlottenburg.

The anomalous physical and the technically very interesting dielectric and piezoelectric properties of the Seignette salt have been until now only observed in mixed crystals of Potassium-Sodium-tartrate and Potassium-Ammonium-tartrate. In order to clarify more deeply the so called Seignette-electric properties, it would be very valuable to know more substances showing analogue electric properties but having, if possible, a much simpler structure than that of the Seignette salt. Among others, the question whether water of crystallization is necessary or not for the existence of these well known anomalies is of special interest. Systematic investigations based on theoretical models have led now to a new material which dielectric constant shows a similar temperature behaviour to that of the Seignette salt. It is the primary Potassium Phosphate (KH,PO,), it forms tetragonalscalenohedric crystals and has no water of crystallization.The main dielectric constant 6,, of this material is very interesting, which is measured along the crystallographic caxis. Fig. 1 shows its temperature dependence as obtained in measurements with 50periods alternating current and a field intensity of about 1000 Vlcm. At room temperature 6,, has a value of about 30, increases rapidly from -So", reaches about 130" a maximum value of about 155, that will be retained until about -190", at about -200" follows a steep decrease to a value of about 7. Similarly to Seignette salt, two Curie points, at about -130" and -195" can be Zurich, Physikalisches Institut der Technidetected. Both points are marked as el and schen Hochschule, 2Gh August 1935 e2 in the Fig. The absolute values of the dielectric constant are clearly smaller com- G . Busch, P. Scherrer 1. Made by the candidate rer. nat. (PhD student) 1. Frei. English version: Dr. rer. nat. Rafael J . Jimknez Riob6o

2.4 Theoretical Modelfor Explaining the Ferroelectric Efect in Barium Titanate - W. P. Mason, 6. T. Matthias 127 PHYSICAL

REVIEW

VOLllME

74.

NUMBER

11

DECEMBER

1.

1948

Theoretical Model for Explaining the Ferroelectric Effect in Barium Titanate W. 1'. MASONA N D B. T. MATTHIAS Bell Telephone Laboralmics, Murray Hill, New Jersey (Received August 26, 1948) In order to explain the properties of a barium titanate single domain crystal, a previous theory of the ferroelectric effect in rochelle salt has been extended to the three-dimensional structure of barium titanate. This involves six equilibrium positions and results in significant differences from the single bond type of structure of rochelle salt. The theoretical features considered a r e a calculation of the spontaneous polarization as a function of temperature, the dielectric constants along the a = y and C = Z axes as a function of temperature, the relaxation of the dielectric constant at high frequencies, and the hysteresis loops. All of these features are explained by the three-dimensional model considered here.

IN

a previous paper,' a theoretical explanation was given for the ferroelectric effect in rochelle salt, which depended on the motion of a hydrogen nucleus between the two equilibrium positions of a hydrogen bond. I t is the purpose of this paper to show that the principal features of the barium titanate single domain crystal can be explained by a n extension of this model to the threedimensional structure of barium titanate involving six equilibrium positions. I. EXPERIMENTAL DATA

Barium titanate above the transition temperature of 120°C has the cubic cell shown by Fig. 1. The barium atoms occupy the corners of the cell, the oxygens the face-centered positions, while the titanium is usually pictured as being in the center of the cell. As a matter of fact, it probably makes 3 covalent bond with one of the face-centered

0 - BARIUM

- OXVLLN

-

uNlT CELL FOO BARIUM TITANATE

TlTANlUY

ASOVL 120 C

FIG.1. Unit cell for barium titanate.

' W. P. Mason, Phys.

Rev. 72, 854 (1947).

oxygens and is displaced in the direction of that oxygen by about 0.16AZ from the center of the cell. Above 120°C the thermal energy is sufficient to cause any one of the six positions to be equally probable and the cell appears to be cubic from x-ray measurements. Below 120°C thermal energy is no longer sufficient to cause any position to be equally probable, and most of the molecules in a given region or domain line up along one of the six directions, a dipole moment develops in that direction and the crystal becomes ferroelectric. The axis along which the titanium has been displaced becomes larger than the other two, as shown by the x-ray measurements of Miss Megawa (as shown by Fig. 2) and the crystal changes from cubic to tetragonal form. The dielectric measurements of multicrystalline ceramics, multi-domain crystals, and single domain crystals all show the presence of a ferroelectric material below 120°C. Dielectric displacement-electric field curves occur in the form of hysteresis loops. The dielectric constant at low field strengths for multicrystal ceramics,' as shown by Fig. 3, rises to a high value at the temperature of 120°C. Above 120 degrees, the dielectric constant follows a Curie-Weiss law approximately, and the dielectric constant decreases inversely as the difference between the tempera8 This value for the displacement of the titanium atom from the center of the unit cell has recently been measured by x-ray methods by Gordon Danielson, Phys. Rev. 74,

986 .. (1948).

'H: D:'Megaw, Proc. Roy. Soc. 189, 261-283 (1947). ' Von Hippel, Breckenridge, Chesley, and T i m , Ind. Eng. Chem. 38, 1097-1109 (19%).

1622

28

I

2 Selected Early Work 1921-1967 FERROELECTRIC

EFFECT IN BARIUM

ture and the Curie temperature or c = co+

CI(T- To) t

TITANATE

1623

Data on Unit Cell Axea of Barium Titanale an a Function d Temperature (Data from H. D. Meraw).

(1)

where co is the constant dielectric constant for temperatures much higher than the Curie temperature. C is a constant, T the temperature, and TOthe Curie temperature. Below the Curie temperature the dielectric constant decreases from its high value to a value of about 350 near absolute zero. The steady decrease is interrupted a t two temperatures 10°C and -7OOC. A t these temperatures no discontinuities occur in the axis length and hence these points cannot be associated with a change in dipole moment and hence with the position of the titanium nucleus. I t has been suggested by Matthias and von Hippel' that these are due to a change from octahedral bonding of the titanium atom to a hybrid type of bonding which may become more probable at the lower temperature. Since this does not involve a n apFIG.2. Cell dimensions as a function of temperature. preciable change in the position of the titanium nucleus, this appears to be a reasonable suggestion. As the result is a small second-order change (unpublished).' A t 23.7-centimeter wave-lengths, in the dielectric constant, it is neglected in the the former found a dielectric constant and tans of theory presented here. r=1250 t o 1420, t a n s i 0 . 2 , (2) The dielectric constant for multidomain crys- while at 1.25 centimeters Yager found a dielectric tals is not too different from those for the multicrystalline ceramics. Figure 4 shows the measurements of Matthias and von Hippel' for the a and c axes. The dielectric constant along the a axis is higher than that along the G axis. The lowering of the Curie point is probably caused by the impurities introduced. By introducing larger z amounts of mineralizers, single domain crystals s of a relatively large size have recently been grown, and these show a very marked difference between the dielectric constants along the two axes. As shown by Fig. 5, the dielectric constant along the c axis is less than that for a ceramic material. When the dielectric constant along the a axis is measured over a frequency range, a relaxation TCUURATURE IN bCCRCCS C E N T l G l l C f occurs at about 15 megacycles and the dielectric FIG. 3. Dielectric constant of barium titanate ceramic as constant drops t o about 1200 or less, as shown by a function of temperature. Fig. 6. A similar relaxation in the dielectric con' The dielectric constants of barium titanate ceramics stant of the ceramic occurs at about loocycles as have recently been measured at 1.5 megacycles and 9450 shown by the measurements of Nash8 and Yager megacycles over a temperature range from 20'C to 160°C 0

'B. T. Matthias and

A. von Hippel, Phys. Rev. 73,

1378-1384 (1948). D. E. Nash, r., J. Exper. Theor. Phys. Acad. Sci. U.S.S.R. 17, 537 h941).

by J. G . Powles of Imperial College of Science and Technology. The results are described in a note Sent to Nature. From the variation of the relaxation frequency with temperature, one can calculate that the achvation energy is 3.65 kilocalories r mole in fair agreement with the value found in Eq. (63r

2.4 Theoretical Modelfor Explainingthe Ferroelectric Effect in Barium Titanate - W. P. Mason, B. T. Matthias 129

1624

W . P . M A S O N A N D B . T. M A T T H I A S

7000

boo0

c

5000

t' z do00 U U

5 3000 c Y

w 0

2000

1000

0

-1bO

-120

-80 -40 0 40 TEYCCRlTURE IN DECREES C

80

120

FIL 4.

Dielectric constants for the two crystallographic axes for multi-domain crystals of barium titanate.

constant of approximately c=250 to 320, tan6k0.70.

(3)

From these measurements it can be calculated that the dielectric constant has a relaxation frequency of about 6.2 X loocycles. The relaxation of the dielectric constant a t these frequencies shows definitely that the high dielectric constant is due to a temperature movable dipole rather than a high dielectric constant of the type due to the near vanishing of the factor (1 - f l y ) in the dielectric equation r-1

-=-

4*

N I / N 2=eBIkT

(4)

where y is the polarizability and fl the Lorentz fartor, since the polarizability y due to electrons, ions and atoms should not vary with frequency up to the infra-red frequencies. Hence, a temperature variable dipole of the type discussed in the next section is required to give a relaxation frequency as low as 15 megacycles.

(5)

where E is the potential difference between well 2 and well 1, k is Boltzmann's constant and T the absolute temperature. Suppose now that all the minima of Fig. 7 have initially the same potential, which is set equal to zero. Then if we apply a field E. in the z direction, a polarization P. in this direction results. This polarization causes an internal field F of the Lorentz type given by the equation

F=E+BP

y

1-fly'

1 to position 2 directly across the unit cell, the form of the potential barrier may be as shown by Fig. 8 in which AU represents the height of the potential curve at the center with respect to that at the minima. If the nucleus went directly from position 1 to position 3, it would in general have to cross a higher potential barrier than AU, but equilibrium between the two positions can be established by the nucleus jumping to a position slightly to one side of the center in the direction 3 and hence it is thought that the potential barrier determining the relaxation frequency for a 1 to 3 jump will not be much higher than for a 1 to 2 jump, namely A U. For low frequencies, i.e., for frequencies well under the relaxation frequency, equilibrium values can be calculated by using Boltzmann's principle that the equilibrium ratios of numbers of nuclei in two potential wells are in the ratio

(6)

where fl is 4r/3 for an isotropic material but will be much less than this when the titanium nucleus comes close to the oxygen atom. The total polarization consists of a part P. due to electrons and atoms and a part Pd due t o the dipole caused by the displacement of the titanium nucleus from,

11. SPONTANEOUS POLARIZATION AND

DIELECTRIC CONSTANT UNDER EQUILIBRIUM CONDITIONS

The model considered here is the one shown by Fig. 7. Here there are six potential minima in the direction of the six oxygens which are displaced a distance 6 from the center of the unit cell. If the titanium nucleus is taken from a position such as

FIG.5. Dielectric constants for a single domain crystal.

30

I

2 Selected Early Work 1921-1961

FERROELECTRIC

EFFECT IN BARIUM

TITANrZTE

162.5

the mid-position of the unit cell. The dipole moment introduced by this change is

(7

p = 4e6,

since the valence of the titanium is 4 for the structure, e is the electronic charge, and 6 the distance the titanium nucleus moves in going from the center of the unit cell to the equilibrium position. An addition to the dipole may also result from a displacement of the oxygen in the direction of the titanium. The electronic and atomic polarization exerted will be proportional to the local field F, so that

L P O T C N T I A L MINIMA

x

FIG. 7. Theoretical model for barium titanate, showing positions of oxygens and potential minima for the titanium nucleus.

about 350. hence 1

1-87

or

where y is the polarizability per unit volume due to all polarization except that of the titanium dipoles. The polarizability y can be determined from the dielectric constant co measured a t verv low or very high temperatures, for since (zo

- ) 1/4u = P B / E = y FIE

(9)

and for Pd suppressed, F = E / ( l -By), hence

B

- l+-(to-

1) = 1+0(27.8).

4r

(11)

This internal field caused by the applied field E , causes a decrease in the potential at the minima 1 and an increase in the potential a t 2 equal, respectively, to

u*=+(--),. Ez +PPz 1-Br

The potentials for the other four wells are unchanged by this field and hence,

us= u4= us= us=0.

(131

By Boltzmann’s principle (Eq. (S)), the relative number of nuclei in the six potential wells, all expressed relative to N s are The dielectric constant c near absolute zero is

R4 0 z

035

b 01 3

Na= N , = Ng= Ns.

02

Y

0001

aoi

01 10 10 loo ~RLWLNCY IN MLCACYCLLS rcn SLCONO

0

ioao

FIG.6. Dielectric constant of an axis as a function of frequency.

s

Then, since the total number of nuclei is equal to N where N is the number per cubic centimeter, we have

2.4 Theoretical Modelfor Explaining the Ferroelectric Efect in Barium Titanate - W. P. Mason, 6. T.Matthias

1626

\V. F. M A S O K A N D B . T. h I . A T T H I . 4 S I

!

/k

DIST.INCE

3 5

2 4

6

\’

. I

DISTANCE

2 3

POTENTIAL WELL NUMBERS

P,

_=_

WELL NUMBERS

from the center of the cell.

Substituting in the values from Eqs. (14) we have

N

The polarization of a dipole nature excited along the Z axis will be then

P,= (NI - N ? ) p /( 1 -07) - NIJsinh[(E= +@PJ 2 +coshC(E.+PPJ/(1 -07)

I d k I’ Icrlk T

‘

(19)

Examining this equation, we see that PJNp will have a solution different from zero only if A is equal to 3 or greater. I f A is greater than 3, P J N k can have a positive or negative value lying between zero and 1. This represents a spontaneous polarization along the positive or negative 2 axis due to the internal field generated by charge displacements of the titanium nuclei from the central position. In general any one of the oxygen atoms can be considered as lying along the 2 axis and only chance determines in which direction the spontaneous polarization occurs. I f we solve for PJNp as a function of A , the relation shown by Fig. 9 results. This is a very much larger increase of P,/Np with increase in A than occurs for a single bond of the hydrogen bond type which is determined by an equation of the type PJNp = tanh(AP,/Np). (20) The relative increase for this type is shown by the dashed line of Fig. 9 for the same percentage increase in A. Some confirmation for this sudden increase in polarization is obtained from the cell dimensions shown by Fig. 2. T h e changes in cell dimension, which are independent of the direction of polarization along the 2 axis, can be regarded as due to the electrostrictive effect in barium titanate. The electrostrictive effect for the barium titanate ceramic has been investigated in a previous papers and i t is there shown

(17)

All the equilibrium values of spontaneous polarization, coercive fields, dielectric constants, etc. can be determined from this equation. Let us first consider the condition for spontaneous polarization and the ferroelectric effect. This can be obtained by setting E , equal to zero and determining the conditions for which the polarization P, is different from zero. Setting E , equal to zero and introducing the substitution

A =C0N1~’/(1-0r)ll/kT.

sinh(AP,,‘Np)

NIJ 2+cosh(AP,/Np)’

3.4,5,6 3.4.1.6 1.2.5.6 I .2.5.6 ? .2.3.4

FIG.8. I’otential distribution as D function of distance

-

31

Equation (17) becomes

POTEIlTIAI

POTENTIAL

I

(18)

FIG.9. Theoretical curve for ratio of spontaneous polarization P . to the total polarization Nu a s a function of the factor A . 8 W : P . , F s o n , “Electrostrictiveeffect in hariuni titanate ceramics, Phys. Rev. (to be published).

I

32 2 Selected Early Work 1921-1961

FERROELECTRIC EFFECT IN BARIUM

that the ceramic has an increase in thickness and a decrease in radial dimension given by the strain equations

S33=Qll(Ps)*;

+Q12(Ps)'

40

50

(21)

S11=S22=

where

20

(cm2/stat coulomb)2; (cm2/stat coulomb)2.

Qll =6.9 X QI2= -2.15 X

I0

While the value of Q11/Q12 is not exactly equal to -2 for the ceramic, a guide to the spontaneous polarization is obtained from these values. At 2OoC, S33 the longitudinal thickness strain is equal to 6.7 X lomawhile the radial thickness strain is equal to S11= 4 2 = -3.3 X lo-* from the measurements of Fig. 2. With these values and the electrostrictive constants of Eq. (21), the indicated spontaneous polarization for the two effects is

P,= 31,500

1627

TITANATE

stat coulomb cmz coulomb

= 10.5 x lo+--

(low),

cm2

(22)

stat coulomb

P, = 39,000

Oo TLYPERATURL IN

00

FIG.10. Measured

DLCRELI C

spontaneous polarization as a function of the temperature.

temperature is evident, and this agrees qualitatively with that shown b y Fig. 9. T o find if the spontaneously generated polarization agrees quantitatively with that calculated from Eq. (19) we have to evaluate A and p by other methods. One method for doing this is to measure the dielectric constants at low field strengths as a function of temperature. The calculated value can be obtained from Eq. (17) by dividing the polarization P. into the spontaneous part PS and a very small alternating part P@iuL. T h e applied field EpivL is assumed very small and hence we have

cm2

=12.9XlO-'-

coulomb cm2

(E,+OPo)eiwf+OPs (radial).

stat coulomb P, = 3 5 , 2 5 0 ~ cm2

160, 126 (1947).

1 -Pr

coulomb *

cm2

(23)

-

(BPS )

(Ez+OP~)ec'];T -sinh

+cosh[

This value agrees quite well with that measured electrically by means of the hysteresis loops. For this value Matthias and von Hippel' find a value 12 X 10-6 coulombjcm2 while Hulme finds a value 16 X lo-' coulomb/cm2. This calibration allows one to obtain the spontaneous polarization as a function of temperature, and this is shown plotted by Fig. 10. The very sudden rise in spontaneous polarization just below the Curie

SF.H u h , Nature

P

= sin h

Taking the average of these

= 1 1.7 X lo-'-

,3-

sinh

(E.+/3Po)eiwfp (1 - P r W

1-07

APs cosh-+sinh-.

P

-

APs

kT

(24)

NP

NP

Similarly, cosh[

(E,+pPo)ciwf+8PS] ;T

A Ps

-=cash-

1-81!

+[(E.fBPo)ejwt] -sinh-. 1-&

NP

P

APs

kT

Nfi

(25)

Inserting Eqs. (24) and (25) in (17) and solving for the constant and time variable parts, we ob-

2.4 Theoretical Modelfor Explaining the FerroeleGtric Efect in Barium Jitanate

- W.

I

P. Mason, 6. T. Matthias

33

1628

W . P . MASON A N D B . T . M A T T H I A S

tain Eq. (19) for the constant part, and for the time variable part we have

Poeju0' (E.+pPo)ejYt Np2

(1-Br)kT

cubic and all directions equivalent, i t is thought that the best values for C a n d TOwill be obtained from a dense ceramic piece. From the dielectric constant above 12OOC of Fig. 3, we obtain the values

2 cosh(APs/Np)+l

C= 40,000; 2'0 = 393°K

C2 + ~ o ~ h ( A P s l ~ d ] *

Solving for Po,multiplying by 4r and adding the dielectric constant for electrons and atoms, the dielectric constant for the x axis becomes

(30)

and from low temperature measurements Q

= 350.

(31)

Taking the ratio of C / T , of Eq. (29) we find j3= (4rTo/C)=0.124

(32)

upon inserting the experimental values. Then, since the number of dipoles per cubic centimeter (as determined from the size of the unit cell) is N = 1.56X lon ; k = 1.38X lo-", we have

C= 40,000 = or

Above the Curie point, the spontaneous polarization PSdisappears and this equation reduces to (4rA c*= co+-=co+-

/a)

3-A

c T-To

(28)

upon introducing the value of A from Eqs. (18) and ( t l ) ,where

4x(1.56X 1022)pz[1+0.124(350/4r)] 3X1.38X10-*6 p=4.34X lo-".

(33)

This value of p agrees fairly well with the value one would obtain from the recent x-ray observations that the titanium atom is displaced b y 0.16A from the center of the unit cell. If the oxygen atom moves an equal distance to meet it (which could not be determined by x-ray observations), the dipole moment would be (4e+ 2e) (0.16 X lo-*) = 6X4.8 X 10-lo X 0.16 X loF8=4.6 X

(34)

If all the dipoles pointed in one direction, the total polarization would be N p = 1.56X10MX4.34X10-18=67,500e.s.u. = 22.5 X coulomb cm2. (35)

The single domain crystals have so many impurities in them to prevent the breaking up of the crystal into multi-domains that they do not revert to a cubic crystal above the Curie point. This is shown by the different dielectric constant for the two directions above the Curie point. The same is true to a lesser extent for the multidomain crystals, but the ceramic pieces show a pronounced maximum and a Curie region above 120°C, much in agreement with Eq. (28). Since above the Curie temperature the crystal becomes

The measured value of approximately 35,500 e.s.u. is 53 percent of this. If all the quantities entering Eq. (18) for A were independent of temperature except T,the absolute temperature, the value of A for 27°C = 300 K would be 3.94. and from Fig. 9 the theoretical value of the polarization P8/Np should be 0.90, rather than the measured value of 0.53, which corresponds to a value of A = 3.090. This result indicates t h a t some of the quantities in the expression for A decrease as the temperature is lowered. A similar result is also required for the variation of dielec-

34

I

2 Selected Early Work 1927-1961

FERROELECTRIC EFFECT IN BARIUM TITANATE

1629

0.1 4 tric constant with temperature. A value of a. A=3.090, Ps/Np=0.53, and j3 set at 0.096 (in g" 0.12 order to give a value of A = 3.090), and all the P 2 010 other quantities unchanged, results in a dielectric 7. constant of 1390 which agrees well with the 3 000 dielectric constant for a ceramic or for a multiL 0 u 0.08 domain barium titanate crystal. The variation 3 5 0 04 may be ascribed t o 8 or to co because the measured 200 300 TLYCCRATYRC IN DZCRLLS LOSOLUTE temperature expansion coefficients indicate that N a n d p should be relatively constant. From the FIG.1 1 . Value of Lorentz factors @I and @ I for z = c and y = a axes as a function of temperature. x-ray data of Fig. 2 it is seen that from 12OOC to OOC, N should increase by 0.15 percent. Since the titanium atom is tightly bound to the oxygen, the much larger than that along the c axis. To dedistance between the center of oxygen and termine the dielectric along the a axis, according titanium should not change appreciably because to the model shown by Fig. 7, with a field applied of temperature contraction, and hence p also will along the Y axis, and a spontaneous polarization not change much with temperature. The value of occurring along 2,the potentials for all six wells co, however, may be different for the a and G axes are since the a axis decreases while the c axis in81Psr u2=-. creases. Hence, cc may be smaller and ea larger u l = 1 -81v 1-417 than co. The Lorentz factor 8, also, may vary considerably depending on the condition of the C&+82P,lr &+82P, (36) surrounding electrical charge configurations. For U3= ; u4=(--).; (1 -827) 1 -827 isotropic conditions, the theoretical value is 4a/3=4.19. For the case of the titanium surUS= US=o. rounded closely by the oxygens the experimental value is only 0.124. As the temperature is de- We assume that B2 along the Y axis may be creased, all the oxygen atoms come closer to- different from 81,along the 2 axis. Applying the N ) and gether and hence a decrease in a is to be expected. Boltzmann principle and relating N1, N2, Assuming all the variation due to 8 , the values to N, to N6= N8 we find agree with the dielectric constant measurements are shown plotted by Fig. 11. With these values of 8 (assuming all the other quantities in A are independent of the temperature), A can be evaluated as a function of temperature and the theoretical values of spontaneous polarization can be determined from Eq. (19). These are shown plotted by the dashed line of Fig. 10, and these agree closely with those determined from the electrostrictive effect. Hence, two independent sets of data are satisfied by the 8-curve. Y

-(!YE);

III. DIELECTRIC CONSTANT ALONG a AXIS

Measurements for the dielectric constants along the a axis for single domain crystals show that the dielectric constant along this axis is very

Since

+

NI Nz +Ns+ Nd+ Ng+ Na= N we find for Ns, the value

(38)

2.4 Theoretical Modelfor Explaining the Ferroelectric Efect in Barium Titanate - W. P. Mason, B. T. Matthias

1630

W. P . MASON A N D 9 . T . MAT'THIAS

Inserting the value of Na, Nd, and

P, = (Na- N 4 ) p = -~

I

35

Ns in the expression for the polarization along the Y axis, we have NPsinhC(E, +B2Pu)/( 1 -B n ) IrlkT

_

C 1+cash C (Pips)/ ( 1 - BIT)I P /T+coshC ~ (G+PzPA/(1-

_ 827)

_'

I P T/I~

~ (40)

To determine the dielectric constant along Y for small fields, we can replace

+

Eu BZP"

Eu +PZPU (41)

'Then

where A=

(G); NP'PI [ ,

= NP'PI 1 +

B I ( ~ )

1.

(43)

Solving for the ratio of Puto E,, multiplying by 4 r , and adding 6 the dielectric constant due to other sources than the dipole moment, the dielectric constant along y becomes

Now, since the crystal becomes tetragonal due to the distortion caused by the electrostrictive effect, y may increase along the a axis and cause t o to become larger. As before, however, we assume all variation to occur in 82 and write 1 -= 1-027

+--. 41r- 1) /%(to

1

(45)

Inserting this value in Eq. (44)for the dielectric constant

At the Curie temperature where the crystal changes from tetragonal form to cubic form the value of 82 must be equal to 81 and hence the dielectric constant along the Y axis will have a Curie temperature at the same temperature as the one along the 2 axis. For other temperatures, 02 will not, in general, equal 81 on account of the shift in charge due t o the electrostrictive effect. One might expect, however, that the shift in charge might to a first approximation produce

additive effects and that, in general 2P2+PI

= 3B,

(47)

where is the Lorentz factor for the cubic crystal. The factor of 2 is used for 02 since the charge along the X and Y axis is only half that along the 2 axis. According to Eq. (46) the very high dielectric constants along Y shown on Fig. 5 have t o be accounted for by the near vanishing of the de-

36

I

2 Selected Early Work 1921-1 911

FERROELECTRIC EFFECT IN BARIUM

1631

TITANATE

nominator of Eq. (46). The values of fit to make Hence, the rate of change of the polarization the denominator vanish, with the experimentally along the Z axis is determined values of f i 1 are shown by the dashed d(N1- N ~ P line of Fig. 11. These values would agree with the dP, -=-above speculation if the average value of 0 fell off dt dt with temperature according t o the dotdash line. = -N1(2a1,z+a1.a + ~ , 4 + a i . s + a 1 . d ~ Another verification of the near vanishing of the numerator is the very low value of the relaxation +N2(2at I + ~ z .3 + m . 4 + ~ .s + w . SIP frequency for the dielectric constant along the Y axis, shown by Fig. 6. As shown by the next +Nap(ar, 1 -as. 2) +N4p(ar, I - a4.z) section, this can be accounted for by the same +Nsr(aa1-as.z)+Nsr(as.1-as.z). (51) potential barrier for both Y and 2 directions, provided that the denominator of Eq. (27) for the When a field is applied along the 2 axis, the podielectric constant along the c=Z axis is about tential minimum U 1is lowered, and Uz raised by 100 times as large as that of Eq. (46) for the Y amounts shown by Eq. (12). Hence, axis. IV. RELAXATION FREQUENCIES FOR THE DIELECTRIC CONSTANTS

T o determine the high frequency behavior of the dielectric constants that is predicted b y the model of Fig. 7, one can no longer use the Boltzmann equilibrium relation of Eq. (5) t o determine the relative number of titanium nuclei in the various potential wells. Instead, one has to relate the time rate of change of the number in a given potential well t o the probability of transition for a given time from one potential well to another. a L 2 the probability of a nucleus in well 1 jumping to well 2 per unit time is, according to Eyring's reaction rate theory, a1,

= (kT/h)e-Au/kT

(48)

where h is Planck's constant, k Boltzmann's constant, and AU the difference between the maximum height of the potential barrier and the potential of well 1. The time rate of change of the number N 1 of nuclei in wells of type 1 is obviously dNi -=

dt

- N I ( ~z I+ w, ,

3+ai. 4 + ~ . s + w .

a)

+ N z ~ 1zf N s a a , ifN4a4, I +N6as 1 + N w

I.

(49)

Similarly, dNa _=_

dl

N d a , I +m,3 +ax

4+az,

S+W.

+Nisi, r+Nia,. z+Niac.

6)

2

+N6ah2+Neaa2.

(50)

(E. 1- B n

By the discussion of Section 11, it appears that the highest potential barrier in going from 1 to the 3,. 4, 5 or 6 potential wells is also nearly AU. Hence, ffl, Z = f f l . 3 = a I . 4 =

(53)

ffl.6 = f f l . ( I .

Also, az,1 =at,a =az. 4 = az. 6 =

(54)

az, 6.

In going from potential wells 3, 4, 5 or 6 to any of the other wells, the highest potential barrier is A U , since these minima are not changed by a field along Z and hence

kT aa, n=a4,,=as,,=as..=-e-Au'kT

t

(55)

11

where n has all values from 1 to 6 except the one which makes the second index equal to the first. Therefore, introducing thesevalues in Eq. ( S l ) , the time rate of change of polarization along the Z axis becomes, for a simple harmonic field,

2.4 Theoretical Modelfor Explaining the Ferroelectric Efect in Barium Titanate - W. P. Mason, 6. T. Matthias

1632

W . P.

M A S O N A N D B. T . M A T T H I A S

If w is zero, this reduces to the Boltzrnann condition for determining the ratio of N2/N1. Since we are dealing only with infinitesimal fields, the sum of Nt and NI can be taken equal to their equilibrium values given by Eq. (16). Since Eq. (56) can be written in the form

I

37

this becomes

6k T coshC(E8+BiPJ/(1 - P l y )

I d kT

NI.~ sinh C(E.+BIP,) / ( 1 -P n 1I d k T

=[

--P.]

2 +coshC(&+BiPA/(1

-Pir)]rlkT

(58) Introducing the relations of Eqs. (24) and (25) and solving for the time variable parts of the polarization, P O ,noting that

E

+BJ's

- ( N -N*) ~ c & ( ~ ) Y ] ,

~-PIY

(57)

kT

we find for the dielectric constant as a function of frequency, the equation

4rA 2 cosh(APs/Np) +1 e, = co

x(

2+cosh(APs/Np)

_

+--

2 cosh(APs/Np)+l 2 +cosh(A P a / N p )

______ .

2+cosh(APs/Np)

)+( cosh(APs/NF)

-eAU

(59)

Ik f

When the last term in the denominator equals the sum of the other two, the dipole dielectric constant has equal resistance and reactance values and the corresponding frequency is the relaxation frequency. 'This frequency fa is given by 6kTe-AUlkT fa =

2 ~ h

[

c o s h z ( 1-

+

A (2 cash (A PslN1.1) 1 ) (60)

( 2 +cosh(APslNd)'

For 27°C =300"K, we found A =3.090; Ps/Np=0.53. Introducing these values and the values

k=1.38X10-16; T = 3 0 0 ; h=6.56, lo-?', we find for fa, the value

f ~ 1.6 = X 101ze-Au~kT. From the data of Eqs. (2) and (3), the relaxation frequency of a ceramic (which probably coincides with that for the c axis direction) is 6.2XlO' cycles. From this one obtains a value for the potential maximum' of eAUlkT = 260 ; A U = 3.35 kilocalories per mole. This value represents the amount of energy to remove the titanium nucleus from its equilibrium position to a position in the center of the barium titanate unit cell. The data of Fig. 6 show that the dielectric constant along the a axis is relaxed at a frequency of about 15 megacycles at room temperature. Applying the same process to calculating the dielectric constant along the a axis, one finds

1

S"=CO+

2 +cash- --

_______ .

(64)

38

I

2 Selected Early Work 1921-1961

FERKOELECTKIC EFFECT IN

BARIUM

TITANATE

1633

To obtain a dielectric constant of 150,000 at 27°C = 300'K, the real part of the denominator has to be 0.0028. Hence, the indicated relaxation frequency for this temperature is 2 +COS~-

6kT

A P s ABz 1+(1%/4*)(60-1) --

Nfi

j0=--&UlkT-

(

81 1+(81/4*)(eo-l) [2 +cash (APslNp)]

A Ps

)I coshx

Introducing the numerical values, = 645 or

A Li= 3.0 kilocalories.

l h u s the indicated activation energy for going from the 1, 2 wells to the 3, 4, 5 or 6 wells is only slightly higher than that between opposite wells such as 1 and 2. This calculation also checks the facts that it is the near vanishing of the denominator of Eq. (64)that causes the very high dielectric constant along the a or X = Y axes. V. COERCIVE FIELDS ALONG a AND c AXES

'The coercive fields along the a and c crystallographic axes and the interaction between a field along c and a polarization generated along a can be calculated from Eqs. (36) and (40), giving the polarizations along the c =Z direction and the a = Y direction. In terms of complete fields and polarizations along the two directions these equations become

p

=-

Nfi sinhC(E,+BJ'A/ (1 -an) I d k T

' 1 +cosh[(Ea+BZA/(1 -Blr)]a/kT+~o~hC(Eu+B*P")/(1 -82~)IfilkT'

From these two equations and the constants evaluated previously, the coercive fields for the two directions can be approximately calculated.

FIG.12. Method for obtaining spontaneous polarization and co-

ercive field.

B

w

3 I

(66)

The calculations show that i t takes considerably more of a negative field along 2 to reverse the sign of a domain along Z t h a n it does to change

2.4 Theoretical Modelfor Explaining the Ferroelectric EfJkct in Barium Titanate - W. P. Mason, B. T. Matthias

1634

W.

P. MASON A N D B. T . M A T T H I A S

I

39

will cause the polarization to increase from 35,600 stat coulombs/cm* to 41,500 stat coulombs, an increase of 16 percent. This agrees quite well with the increase measured by HulmO who found an increase of about 13 percent for this case. If we put on a negative voltage along the axis the ratio P J N p will decrease steadily until the difference between the left hand side of Eq. (69) and PJNp reaches a maximum. This occurs for P,/Np=0.405, and it requires a negative field of

E,= 74 e.s.u./cm

= 22,200 volts/cm.

(70)

This is the theoretical field strength to switch the direction of a domain along one direction of 2 to FIG.13. Hysteresis loop showing relation between polarization along Z and field along Z for a single domain barium that along the other. Single domain crystals have titanate crystal. been observed to switch at around this value of field strength. the direction from Z to Y. To show this, let us A true single domain crystal, however, will assume that no field or polarization exist along Y. have a hysteresis loop for a considerably smaller Then Eq. (66) can be written in the form field strength than this. For such a crystal a typical field strength polarization curve is as P. sinhC(AEz/BNd +(APz/Np)l -= * (68) shown by Fig. 13. When the voltage is in the direction of the spontaneous polarization, the N P 2 +cash C (AEIBNp) ( Ap1/Nrc)1 curve has a tail toward the right hand side that is Now, since AEz/BNp is going to be a very small considerably different from the rounded relation quantity for any field that can be applied, this on the left hand side. This dissymmetrical type of can be written as curve occurs down to field strengths of the order of 1000 volts per centimeter and appears to result sinh ( AP./Np) from the fact that on the application of a nega2' +cosh(AP./Np) tive field along Z,parts of the domain can be spontaneously polarized along Y.To see that this PI A E, 2 cosh(AP,/Np) 1 =__is possible one can examine the conditions for Np BNp (2+cosh(AP./Np))' spontaneous polarization along Y given by Eq. (67). Here we set E, equal t o zero and solve for I f the applied field E. is zero, this equation re- the conditions that will give a finite value of P, in duces to that for the spontaneous polarization. the presence of a field E., and a spontaneous If we plot the left hand side of Eq. (69) as a polarization P,. 'The onset of P, will be defunction of PJNp (assuming A =3.090 for room termined when P, approaches zero, and thus we temperature) the curve of Fig. 12 results. The can replace the hyperbolic sinh by the argument, left hand side is larger than the right, up to a value of P,/Np=0.534 when the two are equal, and the hyperbolic cosh by unity. Then the and this reoresents the theoretical value of equations to solve are spontaneous polarization for no applied field. I f p, (ro2)1/(1 - h Y ) ( P l k T P , -=--the applied field is positive, a larger ratio of (71) Np ~ + C O ~ ~ C ( E , + P I P , ) / ( ~ - P ~ ~ ) ] ~ / ~ T ' P , / N p is required to satisfy Eq. (69). Since a t room temperature, A =3.090; Np=67,100 e.s.u.; this reduces to the B = 0.096 ; cosh(AP./Np) = 2.68, the coefficient If multiplying E , is 1.24X10-'. I t takes, then, a Apz) 1 + [ B z ( ~ o - 1)/4r] very high field to increase sensibly P./Np. For p, 2+cosh=NP 81 1+[S1(60-1)/4r] example. a field of 30,000 volts per cm = 100 e.s.u.,

+

[

+

Ez=oi

(

I

40 2 Selected Early Work 7927-7967 FERROELECTRIC EFFECT IN BARIUM TITANATE

163s

The difference between the left hand side and the right hand side is the denominator of Eq. (44)for the dielectric constant along the Y axis. This denominator is small (about 0.0028 for room temperature) but is always positive, hence no spontaneous polarization can exist along Y as long ae there is no static field -E,. For the addition of a static field, Eq. (71) takes the form

A positive field E. in the same direction as P . makes the left hand side still larger than the right, and no possibility exists for polarization along Y. If, however, a negative field E. is applied, the left hand side can be made equal or less than the right hand side, and spontaneous polarization can exist along Y. Since AE./BINp is a small quantity, this equation can be written in the form

Since for room temperature the numerator is equal to 0.0028, the denominator to 2.53, the field E. to cause a domain to switch to the Y direction is 0.0028 0.108 X 67,100 E,=X ~ 2 . e.s.u./cm=780 6 volts/cm, 2.53 3.090

(74)

which is considerably less than the voltage required to shift a domain along 2. The question arises as to why the whole domain does not go over in the Y direction. This appears to be owing to the fact that when parts of the large domains change direction, they exert an Ey field on the remainder of the domain that is still directed along 2.Then the term cosh[(E,+B#,)/(l -&y)] x ( p / k T ) can no longer be replaced by unity, and the equation for the field to produce a spontaneous polarization along Y becomes

AE.

-= BlNP

tl

sinh (AP./Np)

and the field E, becomes larger. There is no definite saturation for the effect which accounts for the rounded shape of the left side of the hysteresis loop of Fig. 13. When a positive E. voltage is applied, all the Y domains revert back to the 2 direction, which accounts for the taillike shape of the right hand side of the curve of Fig. 13. When a field is applied along Y, the relation between P,, and E, is very linear and shows no hysteresis effects up to a field strength of 300 volts per centimeter, at which field the crystal usually breaks down because of the high conduc-

(75)

tivity along the a axis. Up to that voltage, no domain shift in the Y direction has occurred. T o obtain the field for the shift requires that both Eqs. (66) and (67) shall be solved simultaneously for the P, and P . polarizations and this is not attempted here. VI. SPBCIFIC HEAT ANOMALY OF BARIUM TITANATE

The specific heat anomaly of barium titanate ceramics for the 120°C transition has been measured by Harwood, Popper and Rushman,lo and Nature 160,58 (1948).

2.4 Theoretical Modelfor Explaining the Ferroelectric Efect in Barium Titanate - W. P. Mason, 6. T. Matthias

1636

W. P. MASON A N D B. T. MATTHIAS

Blattner and Merz." The former obtain a value of 0.14 cal./gram whereas the latter obtain 0.2 cal./gram. I t has been shown by Mueller'* that the specific heat anomaly is related to the spontaneous polarization by the equation

Q =S/2P.2

(76)

where Q is the specific heat anomaly in ergs/cc, i3 is the Lorentz factor and P. the spontaneous "Helv. Phvs. Acta. Vol. XXI. Fasciculus Tertius el Quartus (1948). "Annals New York Academy of Sciences, Vol. XL (Art. 5), page 353.

I

41

polarization. Since the specific heat anomaly was the integrated increase from about 100°C to a temperature above the Curie temperature we have from Fig. 10, that P,=27,OOO c.g.s. units of charge per square cm. Q, the specific heat anomaly, is 0.2 cal./gram= 1.2 cal./cc=S x 107 ergs/cc. This gives a value of 0 determined by the specific heat anomaly of

0 = 0.138

(77) which agrees reasonably well with the value given in Eq. (32), obtained from dielectric measurements.

42

I

2 Selected Early Work 1921-19Gl

SCVI. T l ~ o r gof Barium Titanate.-Part

I.

By A. F. DEVOKSHIRE, H. H. Wills Physical Laboratory, University of Bristol *. [Reaeivod July 26, l949.1

SUMMARY. Tine theory of the dielectric and crystallographic properties of barium titanate is considered. By espanding the free energy as a function of polarization and strain and making reasonable assumptioils about the coefficients, it is found possiblo to ncwunt for the various crystal transitiong. Calculations are made of the dielectric constants, crystttl at,rtins, internaJ energy, and self polarization RS functions of teinperaturo. Finally relation8 are obtained between the coefficients in the free energy and the ionic force con&nts. These two wed to estimate ~0111eof the coeficients which are not complotely deteriuinad by cxperinientd data. $1.

INTRODUCTION.

Iw the Inst few yeare t h e l~ropci'tiesof a iiuinber of substances I-xiioww as ferroelect,ricsor seigiiette-electricshave been initch studied. At present three groups of them ml)strzIices are known, t y j & ~ members l of the tllree gronps being Roclielle mlt,, potamiuni tliliydroge~~ pliosp2iate, and Liwium titnilate. All t hest! snbsttuiccs I ~ R V C crrtr~iiiproperties in coiiiiiicrii. At sufficiently high temj~cratiirestheir lircq)(~tiesare norninl, tliotrgli tile tlielectxic const..~.~lt~s are 11~11aily ~ . n t J liigli. l~ -4s the t ~ ~ n l ~ ~falls i ~ t 1 1 ~ the; dielectrio coiistniit incrc.:ms ;~nclmarlies t~ 1)CaIi ttt i t transition tempertbI,iu'e. At tliiri t~ciiiperatmc~ t liere i R EL c11~iige oS crystal Sorni to one of lower xymiiictry. Below t l h teniperatwe each crgstn I Ixenks up iiit o doninins nntl there is Clem evidence tlmat3 these c h i n i n s we polarized. The substance shows the lwopevt ias of hystere& aacl sntjnration thwt one woulrl espcct froni snoli n s t m o t iirr. There may he lower

* Cnnimaniaated 1,y tlir Xutllor.

-

2 5 Theory ofBanurn Titanate - A F. Devonshlre

Theory of Barium Titanate.

1041

transition temperatures. Rochelle salt bccomes norinal again at still lower temperatures, and barium titanate has two lower transition tenipratures at which there are further changes of crystal form. The crystal changes are always small, the shears involved being usually less than a degree. There are also small specific heat changes at the trailsition temperature. In this paper we shall consider only the most recently discovered poup of ferroelectrics, the third, and in particular barium titanate. This is the only known pure subHtancc in the group, though solid solutions of barium titanate with l a d or strontium titanate show similax properties. \f'e shall first review t,lieexperimental evidencc on tho crystal structure, specific heat, saturation polarization and dielectric constant for small fields. We shall then discuss the tlieory of the substance, first on a phenonienological basis and then in terms of a molecular model. This will include a discussion of theories already put forward. We shall not consider the dielectric constant at large fields, nor any of the timedependent phenomena, such as hysteresis or the dependence of dielectric constant on frequency of field. We hope to deal with some of these in a later paper. $2. CRYSTALSTRUCTURE

Atwve the transition temperature barium titanate BaTiO,, has a cubic structure. The barium ions lie at the corners of a cubic lattice, the titanium ions at the body centres, and the oxygen ions at the face centres. Below 120" ('. it W ~ shown S by Megaw (1946) that the substance becomes tctragonttl ; one of the axes (usually taken to be the c-axis) becomes lengthened, and the other two shortened. The axial lengths as a funotion of temperature are shown in fig. 1. It will be seen that the change appears to set in rather abruptly. This was verified by Harwood, Popper and Rushman (1947), who showed that in a given crystallite c/o changed discontinuously from 1 to 1.005. There is a range of a few degrees, however, in which the substance is a mixture of cubic and tetragonal forms. Optical studies by Kay (1948) Matthinu and Yon Hippel (1948) and Bliittncr, Kanzig and Merz (1 949) show that each crystal has broken up into a number of domains. In the simplest ty-pc the domains aro arranged in the way shown in fig. 2. The domain extends from one face to 21 parallel one and the directions of the tetragond axe3 Lire as shown. Yore recent optical and X-ray studies by Kay, Vousden and \Vellard (1949) show that there is a. further t,runsition at about - 1 0 ' (;. Below this temperattire the crystal is orthorhombic. At the trmsition the c-par'tiiietcr shortens slightly and the a-parameter increases so t h t h t the two become cqual and there is :L shear of about 14' in the M plane. The polar axis, which was formerly in the c-direction is now along a diagonal in the ca plane. The above authors also report that there is a further transition at about, -70°C. Below this temperature the crystal i R probably rhombohedra1 with the polar axis along the [ I 1 11 direction.

I

43

I

44 2 Selected Early Work 1921-19Gl

1042

Dr.A. F.Devonshire on the Fig. 1.

Lattice spacing of BaTiO, as a function of temperature (Mepaw 1946). Fig. 2.

2.5 Theory ofBariurn Titanate - A . F. Devonshire

I

45

I t has been found by Wul (1946) and also by Harwood, Popper and 1<usiunsn (1947) that there ia a huiup in the specifio heat C I ~ Win the miglibourhood of 120' C. Blattaer, Kanzig and Merz (1049) found A Iiuiup in the neighbourliood both of 120" C. and of 0' U. The additional specific lie&, is only a siiiall fraction of the normal specific heat so it is difficult to separate the two, but Blsttner, etc. found thst the t o t d additional heat ww about447 cals.jiiiole at the higher transition, and about 1(i calls./iiiale at the lower trrtmitioii. Tlie other authors' resulte ctre iivt sttttect in their papers, but estimting very roughly from their ctirl-es the total dditional heat ap1mm to be about 20 or 30 crtls./niole. Thero is only a linlitecl amount of eviclence on the spontaneous polerizatioii. A field whioh is strong euough to orientate d l the domaiiw in the s ~ n i edirection will protliice con.ritlernlrle induod polarization since the cloinaiiis are tlieniselves highly lmlttrjzthle. I t ir, difficult, therefore, to xepar,ztr the spontoneousfrom the indticecl polarization. For sufiiciently strong fieltls, liowever, the polariz$tioii becomes n Iiiietw function of fielcl strength, nnd h,v est,rirpolatiiig back to zero field Hidin (19.15) estimated that, the spontaneuus 1mlnrizntio1i nt rooin teuiper&m wtw rsbout 16 microcoulombs per mi.$. Mattliiw and Von Hippel (1!!48) wtiinated it t o be about 12 iiiicrocoulombslcm.'. Unfor%unt&tuly we do not know the spontsueous polarization &FJ s function of tenipar&ure, but Hulm (1947) lius given the t o t d polarization for largo Eel& as ttfiinction of temperature. h tilo temperature falls it rises rather ra,pidly in the neigblioiwliood of 120' C. and then incrtrasos slowly. The mcasuremttnta (la not, go below Oo ('. $4. ~)IELECTI~IC* CONSTANTFOR SMALL FIELDS. Numorons measurements of tlie dielectric conshiit for small fidh have born iniKIcJ, usunlly witli alternating ourmiit. Papm ky varioris Russian authors litive been aliiimarizert by Wul (l!)M). Jackson and &xldisli (l!J45)and Kiislininn and Strivens (1!+4O) hsve stu&ecl the effect of varying the ctmpositioii. Von Hippel, Rrockanriclge, Clherjley and Tkza (1946) l m ~ ~pibli~hetl r many nimsuremantij for varioiw compositions, fielcl strengths, nnd frequencies. All the aiitlhors8grm that! above the liigliest. trtmsition t.einpcrntnr@the dielectric cowt$nnto b e p n Cnrie-Weim lnw : that is A e= -

T-T,,+%

wliert+cu uiay ur limy not, be zero and A i8 very large, about 105 clegrsw. Below this teinperttture the dielectric con&mt drops to about 1000-WNt, and as the tein1)erst:ire deci.ea8es renmim constant or dowlg clecrwes. There are arii~ll1maIis at, lower temperatures. These almost. certainly correspond to the lower crystal tramitionr;. All the above iueavurements were instie with pc)wders. Now, above the highest transition temnptmture the siibstance h a g complete cubic ywiimetry and therefore A single dielectric coilstrtnt :

46

I

2 Selected Early Work 1921-1961

1044

Dr. A. I?. Devonshire on the

but below this temperature the dielectric constants along the different axes will be different and experiments on powders will only give a mean value. Matthias and Von Hippel (1 948) have carried out experiments on single crystals ; but even a single crystal may contain several differently orientated domains. However, they made measurements on a crystal in which all the domains had been made parallel by applying a strong electric field, and they found a dielectric constant of 500 along the c-axis and 1700 along the a-axis. Mason and Matthim made measurements on eingle domain crystals and found dielectric constants along the a-axis Fig. 3.

T "C

Dielectria constant of RaTiO, single domain crystals (Merz 1949).

of the order of los. Merz, however, found more moderate values. His results are shown in fig. 3. The discontinuities at the lower tranaition temperatures are very well marked. $5.

THEORIES.

Suggestions put forward to account for the behaviour of barium titanate are usually based on the assumption that the titanium ion can move rather easily. Miss Megaw (1946) has pointed out that if the ions are all assumed to have the Goldschmidt radii they cannot be fitted together exactly t o make a cubic structure but that they have a certain amount of free space. Rushman and Strivens (1946) have developed this idea by suggesting that the titanium ions are slightly displaced from the, symmetrical position and therefore form dipoles which can rotate. I t is known that such a set of clipoles have a Curie temperature above which they rotate freely, but below which they are parallel to one another. This latter &ate, of coiirse, corresponds t o the ferroelectric region.

2 5 Theory of8arium Titanat8 - A F Devonshire

I

47

Theory of Buriunz ‘I’iianak.

1045

Mason and Natthias have developed in considerable detail a rather different model. They assume that the titanium ion has six equilibrium positions slight’ly displaced in the axial directions from the symnietrical one. In the cubic state tho ions occupy the positions at random. In the tetragonal state they occupy mainly positions along on0 axis. The transition from cubic to tetragonal symmetry therefore corresponds to a transit ion from a disordered to an ordered state. By assuming suitable values for the ionic shift, the polarizability of the rest of the material, and the Lorcntz factors, they have accounted for the dielectric constant in the cubic region, and the two dielectric constants in the tetragonal region. They have itlso considered hysteresis and frequency effects. The greater part of this paper considers the theory of barium titanate in a phenomenological way. We expand the free energy in terms of the strains and polarization of the crystal, use certain properties of t h e crptal to determine the coefficients, and then predict other properties. R’esults obtained in this way are of course, independent of any atomic model. This inetliocl has been applied to Rochelle salt with considerable success by Alueller ( I W O ) , but owing to the very different symmetry of barium titanate we cannot make any direct use of his results. Mason (1948) has applied this method to the electrostrictive effect in barium titanate; (:insburg (1946) has applied i t to the highest transition, but only a liniited aiiioiint of experimental evidence was available when his paper was published. \Ve lave been able to show that by assuming reasonable values for tho coefficients wc can explain t,he succcssivo transitions through the cubic, tetrsgonal, orthorhombric and rhombohedra1 forms. In determining the coefficients we use the observed transition temperatures, the value of the dielectric constant in the cubic region, and the observed striiin and saturation polarization at a single temperature in the tetragonal region. \Ve are then able to predict the various dielectric constants in the tetragonal, orthorhombic and rhombohedra1 regions. We also predict values for the strain and saturation polarization in these regions. It, is not possible to verify d l these predictions a8 many of these quantities have not yet been observed. Finally we consider an atomic model for barium t itmate. Following tlie method used by Born for ionic crystals we trext the ions mainly as point centres of force, though also taking into account their polarizttbility. After checking t lie force constants by calculating tho intertttoniic distance we makc a n rstimatc of the elastic constants. In conjunction with the results ;tIreatly obtained this enables us to calculatc tlie electrostrictive constants. \Vc then attempt to crtlculate direct ly the field in whicli eachion moves, tlieotlicrions heinginthosymmetrical positions. Thesecalculations cleilrly int1ic:tte that the stable position of tho titanium ion is the syninietricnl one. It seems just possible that the oxygen ions might have two mispimetrical equilibrium positions displaced towards the nearest Ti ions, h i t the calculntions are not acauratc cnough to make a definite predict ion. \Se lial-e, however, assumed thiit an ion moving individually has only one posit ion of equilibrium, and that the spontaneous polarization

48

I

2 Selected Early Work 1921-1 961

1046

Dr. A. F. Devonshire on the

is caused by the Lorentz field, in other words, it is D cooperative effect, since this field only exists when ions of one sign move together. The Lorentz field requires some consideration since it plays an important part in the theory of ferroelectrias. When a body becomes polarized an ion or dipole in the body will experience a force due to the polarization of the rest of the body. We may assume this force to be proportional to the polarization and put it equal to ,8P where ,8 is the Lorentz factor. Owing to the slow fall off with distance of the dipole force it is a " long range " one, that is dipoles in distant parts of the body have an appreciable effect. Hence for an insulated body fi is dependent on external shape and reaches its maximum value for a needle or plate polarized parallel to a long axis. But for an uninsulated body p always has this value whatever the external shape. A maximum value of /3 corresponds to a minimum polarization energy since this is -4fiP per unit volume, and hence the uninsulateci body will always collect surface charges in suoh a way w to make 6 a maximum, say fi,,,. For point dipoles in a cubic or random array /Im can be shown to be 4 q 3 (Fowler 1936). If the array is regular but not cubio flm will vary somewhat with direotion though its mean value will still bc 4$3. If the dipoles are not point dipoles &, will have a different value. Mason and Matthias have found it necesssry to assume that /3m is slightly different for the tetragonal and the other directions and also varies slowly with the temperature. These assumptions are reasonable, but the values they have found it necessary to assume for Is, are near 0.10. This differs considerably from the theoretical value of 4.19 even taking into account the fact that the dipoles are far'fmm being point dipoles. Physically it mean8 that the cooperativo effect between the dipoles is rather small. We have assumed that p,,, haa the value h / 3 , and that the Lorentz field approximately balances the short range restoring force. By taking into account thermal vibrations, including anharmonic terms, we are able to show that the restoring force increases with temperature, and hence explain the existence of a transition temperature. For the properties we are dealing with in this paper there is no important difference between the predictions of the two models. The Mason Matthim model would noed a little modification to account for the lower transitions, but this could certainly be done. There am bigger differences in other phenomena, but these will be dealt with in a later paper.

5 6. PHENOMENOLOGICAL THEORY : IXTRODUCMON.

We shall consider the substance as a strained cubic crystal. All the changes from cubic symmetry are small, so this is quite legitimate. We shall use the notation given by Cady in his " Piezoelectricity ", as far as possible. For barium titanate, however, it is necessary to consider higher order terms in the free energy than any used by Cady, so we shall have to introduce some new notation. Now the free energy of a crystal can be expressed in several different forms. We can take as our independent variables polarization and stress,

2.5 Jheory of0ariurn Titanate - A . F. Devonshire

T h y of Barium Titanate.

1047

polarization and strain, field and stress or field and strain. We shall start by expressing the free energy as a function of polarization and stress with the stresses equated to zero. We then have

+ + +45; {p"+ p;:+p:} +t

A =+X'(PE p; P,"}

+

(PEP! pqp:+ ep;> +g'(q+P;+p:}. * * (6.1)

1

5;2

The zero of free energy is taken to be that of the unpolarized, unstressed crystal. All the terms of the second and fourth orders are included and some of the terms of the sixth order. We shall find it necessary to take into account all these terms t o account for the behaviour of the crystal. The derivatives of A with respect to P, give the field-components for the free, that is unstressed, crystal. Hence we have

E, =x'P,

+4; IP!+&&',(Pi +P,")+{'Pi. . . .

(6.2)

In the absence of a field the right-hand side of (0.2) must be zero and for stability A must be a minimum. We shall find that we can account satisfactorily for the observed facts are positive, is negative, and x' is a if wo assume that 5' and decreasing function of temperature which passes through a zero value in the neighbourhood of the upper transition temperature. Since x' is the reciprocal susceptibility for zero polrtrization we can obtain its value from experiment directly above the upper transition temperature where there is zero polarization for zero field. W e find, in fact, that it is a linear decreasing function of temperature which, if extrapolated, passed through zero a little below the transition temperature. With these assumptions A will clearly be always positive for sufliciently large x', that is, at sufficiently high temperatures. The minimum value of A will then correspond t o 7~x0 poiarizlztion. For x' small or negative, however, the minimum value of A will correspond to a finite polarization. The second order term is independent of direction. But for a given resultant polarization the fourth order torm has its minima along the axes and the sixth order term has its minima, along the diagonal directions. Hence as the temperature falls and the magnitude of the polarization incresses the direction will be likely to change from an axial to a diagonal one. For zero field, equation (6.2) and the similar equations become

P,=O, or 5'P~+E;,P,2+E;,(P~+p,2)+X'=O, P,=O, or 5'Pi+S;,Pi+E;r(P2,+P:) x'=O, P, -0, or CPf ,P,"+&,(PE+Pt) x' =O.

+

+

+

1

..

(6.3)

J

There are four sets of solutions of these equations which may correspond to minima of A, namely P, =P,= P, =0, P, =P, =0, ('pf 4; lP,' x' =0,

+

+

P,=% py=pz,5 ' ~ : + ( 4 ; , + 5 ; ~ ~ ~ , 2 + ~ ' = 0( ,c ) pz=p,=pr, 5'Pf+ (6;1 25;dP;t X'"0, (4

+

. .

(6.4)

I

49

50

I

2 Selected Early Work 1921-1961

1048

Dr. A. F.Ilevonshire OR the

with, of courw, tho corresponding solutions obtained by interchanging P,, P,, and P,. The corresponding values of A are given by A=O, A==i5'P,S+)t;,pt+lttP,2, A= t(5;1+ 5;21p: X'Pi, A=B t'P,"+i(6; 1$-2&)P,J+~x'P~. The necessary conditions for A to be a minimum are

+

am+

(a)

(6) (c)

1

1I . .

(6.5)

(d) J

x'+t;2Pw,

Bs;,+5tP:>o,

(6)

x1+25;,p:>0,

91;1+5'p:>o,

(c)

d.t;z+t'p:>o.

).(4 (

. . .

(6.6)

It ia possible for A t o have more than one minimum, and we then have to determine which is the least. If we plot the values of A given by (6.4) and (6.5) a~ a funotion of x', taking C' and f' to be constant then we find that for x' positive and large enough the least minimum of A is zero, but as x' decreases the least minimum is successively given by equations (b), (c) and (d). When the minimum given by (b) is equal t o that given by (a) we have

......

(6.7)

. . . . . . .

(6.8)

$f;'~+)&P$+&x'=O,

and from (6.4 b) we have ~'P',ft;,P;+X'=o.

Let us denote by xk the value of x' which satisfies these equations and by P, the corresponding value of P,. Then we have 5'=3x;/P& (;,=-4x;/P;.

. . . . . . . . . (6.9)

. . . . . . p..I

(6.10)

If we assume that 5' and me c o n e b t s , independent of tem$erature, and therefore always given by (6.9) and (&lo),and if we also put

P;=ZP:,

. . . . . x'=txL, . . . . . . . . t;,=-~&, . . . . . . . P:=xP;,

P;=yP:,

(6.11)

(6.12) (6.13)

then equations (6.4) and (6.5) become x=y=z=o, x=Y=O, 3z2-4z+t=0, x=O, y=z, 3z2+4(-1fa)~+l=0, z=y=z,

3z2+4(- 1+2a)z+t=0,

(c) (d)

.

.

(0.14)

2.5 Theory of6anum Jitanate - A. F Devonshtre

Theory of Barium Titanute.

and

1049,

A d , A.= X,)'y' ( 3 t 3 4 hft) ,

1

A

I

+

,

~

xo'p21 ,Iz3 +:!(-l+~)~"zzt},

. . . .

(6.15)

The equations are now in a convcnient form for calculatmion,since they are now in the form of a relation between two variables z and t with a single parameter a. The other constants of the equation enter only 8s scale factors. The variable t is related t o the temperat.ure, and the relation is probably approximately linear. Me' have seen this is the case cxprimcntally for t positive, since x' is proportional t o t . I n fig. 4 we plot z and A/x;P: as functions o f t for CL equal to 1.2. As already stated the minimum value of A is given in turn by equations (a), ( b ) , (c) and (d). The value of z will therefore change discontinuously at three transition temperatures as z is given in turn by equations (a), (b), (c) and (d). The polarization, w~hich was originally zero, will in turn point along a cube edge, a face diagonal, and (t body diagonal. The corresponding effect on the crystal symmetry will be to change it from the original cubic successively to tetragonal, orthorhombic and rhombohedral. In the Schonflies notation the symmetry of the crystal will he in turn O h , CnVand CJv. $ 7 . PIIEXOMESOLOGICAL THEORY : DIELECTRIC COSSTAXT.

tn the high temperature region where the crystal has cubic symmetry the dielectric constant is independent of direction and is given by €==

1 +47r/x',

. . . . . . . .

(7.1)

but in the temporature regions where the crystal has less symmetry the dielectric constant is no longer independent of direction and hats to be descrihetl by a tensor. It is, in prrtctice, more convenient t o work in terms of the susccptibility qrs arid the reciprocal suscept ildity xrs. For small fields and polarization they are defined by the relations

. . . . . . . . . . . . . . . . . and art3 connected by the equation

. . . . . . . . (7.3). d is t'he determinant of the x ' s and S, is the minor of xrsin that q,=xrSiA,

where determinant. \Than the relation between field and polarization is no longer Linear we define x by the equation Xsr'Xrs'

W PA -,= aE 2=: ap,

ap,

aP,ar,'

*

. .

, (7.4)

I

51

52

I

2 Selected Early Work 19221-19Gl

Dr. A. F. Uevonahire on the

1050

Fig. 4.

-. .--*

'.*%,

p.'.

rhombohedraI -21.0

-240

ortborhombic 20.0

-16.0

-12.0

I--

ktra4d -8.0 -4.0

\

cubic

0.0

BaTiO, as a funotion of temperature (theoretical).

!Free energy and polariestion of

2.5 Theory of6ariurn Titanate - A . F. Devonshire

Theory of Barium Titanate.

1051

Hence from (6.1) we have

with similar expressions for we have

xuz=xZL.=

Using the equatione (6.11) to (6.13)

x*u=o,

xi@ f4az1,

x x z =XU” =

xu=

xrz etc.

~ ; , (t 122

x y z = xrr=

+I~z‘)),

xzu=o*

}

xly =xtz =0, xyz= x64w xzz= xw =xu= x;l(t -122+8az+ 15z?), xyz=

xuz=xzy =x3=.

(6)

I , (d)

x=tx:;, SER.

7,VOL. 40, NO. 3 0 9 . 4 C T . 1949.

4E

I

53

54

I

2 Selected Early Work 1321-1961

1052

l)r. A.

F. Devonshire on the

and experimentally xis a linear function of temperature. We can therefore reasonably assume that 1 is a linear function of temperature throughout, and since by definition it is 1 at the upper transition temperature, which we shall call T,, we can conveniently write

. . . . . .

t=(T-To)/(L"L'-TO),

(7.10)

where Tois a parameter to be determined. Since T, is the temperature at which t and therefore x vanishes wo can determine it by extrapolating x to zero. Unfortunately it is only slightly less than T,, and since the transition is spread over a few degrees this means that T,--To is rather uncertain. From the measurements of Harwood, Popper and Rushman (1947) we estimate that Tois 118' C. and T,--To about 10' C,,making To 128" C. Since x=txC~=x~(T-To)i(T,-T~),. . . . . (7.1 1 ) we can determine x;/(T, -To)from the observed slop of x plotted against temperature. The measurements of different obsorvers agree fairly well and we find that X ; / ( ~ , - ~ T , ) =10-4(degrt~~)-5 ~.~~ (7.12)

. . . .

and hence if we take TI-To equal to 10°, then x;=1*0x 10-3.

. . . . . . .

(7.13)

l', can be determined by making the calculated saturation polarization agree with the observed value at a given temperature. If we take P, to be 10 microcoul~mbs~cni.~, then from (6.1 I ) , (6.14) and (7.10) we can show that the saturation polarization is about 16 niicrocoulo~iib~~cm.~ at room temperature in agreement with Hulm's result (1947). CL is best determined by choosing it 80 that one of the lower transition temperatures agrees with the observed value. We find, in fact, that, if we take a to be 1.2 both transition temperatures agree fairly well with the observed values. If we substitute the above values in (6.9) ,(&lo)and (6.13) we find that 5;,=-4*4

X

&=5*3X

. . . . . . . . . . . . . .

(7.14) (7.15)

and 5'=3*7 x lo-".

. . . . . . .

(7.16)

We can now plot the principal dielectric constants as a function of temperature. The dieleotria constants in the principal directions are relatcd to the reciprocal susceptibilities by the relation €=

1 +47rx-'.

The results are shown in fig. 6. If we compare them with the results of Merz given in fig. 3 we see thaC the theoretical curves have the same form,

2 5 Theory of Barium Titanate - A F Devonshire

Thcorg of Barium Titamate.

I

1033

but there is no exact numerical agreement. In the tetragoid region anti c,. I n the orthorhotnbic region E , and cC are 110 longer principal dielthctric constants ; and the crystal is no longer a single domain. However, E, must be ri nimn of ebb and eYy, ;tIltl ca iiiiist be a mean of all three dielectric constants. Xn the rhomhoheclral cast? we should expect

our cU and crz correaponcl t,o hlerz’s

c, =E , = &(Exr

+2€,,).

However, Merz found c,, to be much larger than eC. This may be the result of domain boiindaries shifting under the application of a field. This ~vouldnot affect t,he polarization in the c-direction since, presumably all dninainn have a positive component of polarization in that direction $8

Fig. 5.

-170-c -13o-c

-9O.c

-SOT

-m*c

30%

70-c

llO%

150%

. 2

19O.c

Principal dielectric constants of BaTiO, (theoretical).

the crystal had in the tetregonal region. On the other hand, the coinp n s n t u in the a-direction would be randomly positive or negative, so a shift in boundary with fields niight alter the net polarization and heiico increase e,.

0 8. PHENOMENOLOGICAL THEORY : HICAT OF POLARIZATION. The polarization will be accompanied by a chsngc of internal energy, which will be given by the usual forniula

E = = = T ( g ) -A,

. . . . . .

.

(8.1)

P

where A is given by (G.1).

Now A depends on temperature partly through L

:F 2

55

56

I

2 Selected Early Work 1921-1961

1054

Dr.A. F.Devonshire on t?w

the polsrimtion, so we can write

and hence for zero field

(g)p (g) =

. . . . . . . .

(8.4)"

PIp

Now we have assumed all the ooeffioients to be independent of temperature except x', which is given by x'=x~(T-T,J/(T,-TJ.

. . . . .

(8.6)

Hence

40

60.44'

T inkd c"ccpy (r.l%/mob

,201

2.5 Theory ofBariurn Titanate - A. F. Devonshire

I

All the terms of order pl or lower order are given, where it has been assumed that the strain is of the order of the square of the polarization. This will shortly be verified. Terms not given vanish because of the crystal symmetry. The cornponentu of field and stress are given by the equations

If we assume that the six components of stress are zero, then we get six relations between strain and polarization, namely

1

' ~ = ~ ~ 1 ~ , + ~ 1 2 ~ ~ y + ~ 2 ~ + ~ ~ 1 ~ ~ + 8 1 ~ ~ ~ ; + ~ , 2 ~ ,

. . . . . . . . . . . . .

0 =C( 4 Y a s 844P&.

. . . . .

t

. . .

(9.3

J

If we solve these equations for the strain in terms of the polarization we find that

If wc substitute these values for the strain in equation (9.1) then we get an expression for the free energy in terms of polarization for zero stress, that is, we obtain equation (6.1) except for terms of higher order than P4. Comparing the coefficients of P2 etc., we find that XI',

1

From (9.4) we can see that the strains are proportional t o the square of the polarization as already stated.

57

58

I

2 Selected Early Work 1927-1961

Dr. A. .!I Devonuhire on tb

1056

Tn the tomImraturc ranges where the crystal ie respectively cubic, tetragonal, orthorhombic and rhombohedra1 equations (8.4) become

.

Y

,..z -

=?

-.- r!,-- -~542’~Jc14.

.

. .

(9.6)

J

Froin the observed stminu and polarizationu we can get some inforinat ion about the constants. ‘rho most convenient quantity t o use is the difference between the strains along the c and *-axes, that is ze-xx, which is given h?r

. . . . .

z,-s,=P~(y12-!/11)/(Cll-clp),

(!).T)t

in cither the tetragonal or the orthorhombic rcgion. This is about 0.01 at room temperature when P, is about 16 nlicrocouloinbs. Hence we have ~ ] 2 - y I l ~ = ( c I * - c I ~ ) %10-’2. 4x

. . . .

.

(9.8)

Similarly in the tetragonal region we have for the volume expansion 2, 4 - Y U t Z Z =€(:’

--y

11

-2g,,)/ (GI 1 +2c, 2).

-

*

*

(9.9)

The volume expanrJion is more difficult to determine since we have to estimate the unstrained volunie by ext’rapolation from the cubic region. However, w e b v c approximately - ( ~ , , + 2 ~ 1 ~ ) I ; : ( c , l f 9 ~ , , ) X o ~ N X10-13.

. . .

(!j.lO).

The shear yz just below t.he second transition temperature is about 14’. Hence we h a w -y,,-c4,x2*7x

In the orthorhombic region we have

+ +

10-12.

. . . . . .

+

(9.11)

.22 Y, 2, =21)2(-Y, 1 -2Y12 1; (c11 2 G l Z ) . * (9,12 )I Hence in all regions the strains can be evaluated in terms of P: and the ratios (gla-gll)/(cl1-c12) and (al f 2 g 1 2 ) i ( c , +2c,2) , given empirically

2 5 Theory $Banurn Titanate - A F Devonshire

159

Theory of Barizcm Titunate.

1057

by (9.8) and (9.10). The calculated strains are given in fig. 7. A term representing the thermal expansion is added SO that the figure shows the total variation of axial length with temperature. By coilsidering the observccl piezoeleotric resonances Mason ( 1 ! ~ 8 ) lias been able to calculate the y's and c's indq~ndent~ly for the ceramic. These quantities might, howe\-er, be very different, for the single crystal, so WT are not able to make use of his results. Fig. 7. .034

0

I,;ittice spacing of BaTi03 relative to bpaciny r r t IW0 ('. ( t lieoretical).

$10. MODELTHEORY.

Our picttwc of barium titanatc is that it is a n ionic crystal, :~ndwe shall follow Born's treatment of ~ 1 1 0 1crystals ~ a3 given by Fowler ( 1 9:W), The ions arc regarded as point centres of forw, and the potentirtl energy of t w o ions with charges and e2 n t a distance T apart is assumed to have the form *--pr-6+

Xr-

9.

The second term represents the \'an der Wads attraction, and the third term the repulsive forces. 'Che potential energy of BaTiO, per iinit cell is then given by the expression

&R)=

7 + -49. la2

3-31(XrI.$-X,,11+ 3Xo0)

1 2.:30(Xo~,+Xoo)

+

(1
It9

ti.07h,,eI. li.Oiho., f-(K, -r (2R/l/3)9 2)*

- 4-?0(p1 .r +pnR- 11:6

-

8.7 I p I'

(2R/2/3)6

where we Iinve assumed the iinit cell to be cubic with an edge of length It,

I

60 2 Selected Early Work 7921-1967

Dr.A. F.Devonahire on the

1068

and the charges on the Ba, Ti and 0 ions to be 2r,46 and -26 respectively, --c being the electronic charge. The valuee of ABB, Aoo, AoB, PBB, pOB, and hohave been given by Fowler (1936). We have used his methods to estimtlte hoT and pOT,and the contributions of the terms containing and /LBT are negligible. Tne completeset of values used hTT, ABT, is given in Table I.

TABLEI. Force constants between ions in c . ~ . Runits. . hgg AOB A00

AOT

82.7 x 99.0 x

113.5x lo-" 15.6x

P])B

239.0 x

poB

162.0 x 10-60 135.0x 10-" 31.3 x IW"

poo

If we c h o w H; so that 4(R) is a minimum this will give us the side of the unit cell at zero temporaturn. Using the force constants from Table I. we find that H. ie 4.04 A. as compared with the extrapolated observed value of 3.99 A. This suggests that our force constants are reasonably accurate, and we can now use equation (10.1) to estimate the compressibility. The change in potential energy of the unit cell when we alter R by a small amount AH. is &(LIR)~+"(R), and hence the change per unit volume is +(L~R)~+"(R)R-*. But the change in free energy is fc,,(z~+y~+22,)+C12(~~~+YyZ,+Z~*),

where x ~ = ? / ~ = zdR/R. ~=

If wc neglect the difference between free and potential energy these two quantities can be equated, and we have Putting in the values for the force constants from Tablc I. we have cll+2c1,= 1.2 x 10 dynes/cm.2.

. . . .

(10.3)

As our amumption that the ions are point oentres of force can only be ~t rough approximation to the truth we shall not attempt to calculate c,, and cI2 separately but shall assume c , ~to be equal to 2c,,. This is equivalent to be taking Poisson's ratio to be which is not far from tho value for most substances. This makes c11=6*0x 10'2 dyneslcm'., . (10.4) C1,=3.0X 1012 dynes/cm*.,

a,

and hence from (10.8) and lO.IO), g11+2g,,=9-69 =-11*2, 61I gia =- 0.8.

i.

1.

I

...

. . . . . . (10.6)

2.5 Theory ofBarium Titanate - A. F. Devonshire

Theory of Boriurn Titanate.

1059

If we assume central forces between the ions c44 should have the same value as cI2. This makes grr=--8. Substituting the above values in equations (9.5) we get =6;;

-70 x 1O-I2,

and &=[f;+20*8x

10-12-221*3

x

x Putting in the values of 6; and from (7.14) and (7.15) this gives -4.4 x 10-'*+70 X 10-"=65*0 X lo-'' ' . . (10.6) 5;;=5.3 x 10-1'- 5.5 X 10-''= -0.2 X 10-'2. =6;;+5*6

}

We shall now examine the field in which each ion moves. \Ve assume all the ions except one to be fixed, and then it is a straightforward matter to calculate the field in which that ion moves in terms of the force constants. If the ion is displaced a distance r from its symmetrical position then its potential energy can be expanded in the form

@a(@,4)+r4W, $61, where 8 nnd # are polar coordinates referred to some suitable axis. For barium nnd titanium, which are symmetrically placed, a is independent of 0 and 4 and is given by aB= 1.2 x lob ergsjcm.'2, aT 4-7 x 105 ergs-cm.*. The contribution to a, and uT from the short range repulsive forces, which is positive, is several times larger than the contribution from the electrostatic nnd Van der Waal's forces. Thus aB and aT are bound to be positive. The case is different for tho oxygen ion. This lies in a much less symmetrical field, and if we take the initial line to be the line joining the 0 ion to a neighbouring Ti ion we find that ao=04 x 105[(7.4cos20+44 sin26)-arR-31964(cosa 8+a sin28)] orgs/cm.*, (10.7) where x ia tho polarizability of the oxygen ion, and H, is the edge of the unit cell. The term in a is there because if the ion is displaced there will be an electrostatic field acting on it propoItiona1 to its displacement. This will polarize the ion and there will be a lowing of potential energy proportional to the square of the displacement. There is no similar term for the barium or titanium ions because, owing to the symmetrical arrangement of the ions round them, the electrostatic field acting on a displaced ion is proportional to the cube of the displacement. Various values have been given for a. A value of about 2.4 x 10-24 gives the correct value for the refractive index of CaTiO,, after allowing €or the polarizability of the Ca and Ti ions. This value makes a, about zero along 6=0, and positive for other directions. The value of Q given above, however, is only A mean value and is likely to be different for different

. . .

I

62

I

2 Selected Early Work 1921-1961

Dr. A..!I Devonshire on the

lOfi0

directions since the oxygen ion must be very distorted. Hence we cannot predict with certainty whether a, will be always positive or may have negative values in certain directions. In the latter case, of course, the symmetrical position of the oxygen ion will be unstable, and there will be positions of equilibrium off centre. These are likely to be in the direction 6=0, that is, in the direction of the neighbouring Ti ions. We must now consider the relation between the constants a, 6, etc., and the free energy. This involves studying the thermal vibrations of the substance. \Ye assume that the vibrations of each ion are independent. Let u# +a& tu# 6& b,y4 b,t4 Z6,2y222 21id2x24 26,,,xZy2

+

+ +

+

+

be the potential energy of an ion when displaced t o a point (x, y, z ) relative to the syminetrical position as origin, the substance being iinpolarized. The above expression includes all terms of lower order than the sixth. The clisplacd ion is equivalent t o a dipole which we assume to have components ( y 2 , yuy, y 3 ) where the y’s are constants of the order of the ionic charge. Now if the substance is polarized them will be a force acting on the ion having components (/3Pz,/3PU,PPJ, here ,8 is the Imentz factor, and hence an additional term in the potentid energy of magnitude -P(YPP.r

+,YU,+

YZZ PA.

Then the partition function f for II single ion is given by

- . . . ]/kT} d z d y dz.

-26&*

. . .

( I 0.8)

We shall assume that bkT/a2is small. Then terms in b2 can be neglected and we haw b z z d . . +26,,y2z2+.

..

kT -‘a?2-

kT If we now put

.=@y,P,ux-’+s’,

then

*

‘ I d z dy dz.

. .

. . . . . .

(10.9) (10.10)

2.5 Theory ofBariurn Titanate - A. F. Devonshire

Theory of Barium l'itunczte.

1061:

The free energy per unit voluriie is given by

x

-;v~rz-ioyf-t~B(P~+P:-Y~),

. . .

(io.iq

whrre S is the number of uiiit cells per unit, volume, ancl the sirnimatioti is u w r the differentvioos i n the. uiiit cell. The t-eriii in €" arises because in summing over the unit cells we have reckoned twice over the equilibriurn potmtinl energy of ttrc ions, which is -&f(P:+P%+P'f). Hence w e have

If wu cottipitre this expression with cquat$ion(!), 1 ) we

tiat e

A s before, thc summation is over the different ions in the unit cell, and when this sunimation is carried out the quantities involved will be the sanie for d l axes (or pilire of axee) owing to thc crystal symmetry. It is possible to muko rough estimates of the 0's and U'S in terms of the. force constants in the way already described, and we find that the term involving T in (LO.14) is s n d compared with the other terms at, constant temperatures. Xow we have already found that x" has t o vanish at tt temperature To,so that approximat'ely 4

1=ppzY-". a,

. . . . . . .

(10.17)

Now froin (9.1) we see that the coefficients y,, ttnd g,, ineasiire the variation of ix" witli strain. R'ow as we have already statetl the third

I

63

I

64 2 selected Early Work 1921-1911

1062

On the The0y of Barium Titmate.

term in X I ' is small and 8, being the Lorentz factor, will not depend much on strain, and hence we have approximately

and

where R is the side of the unit cell. Now a, deomses a8 about the tenth power of R, and yz will not vary much with 1%. Hence from (10.20) and (10.17) we have 91 1 2gJ 2 M -5/3 ,-20, . . . . (10.21) if we take 8, the Lorentz factor, to have the value 47713. This differs from the value given in (10.5) by a factor of 2. In view of the approximate nature of both calculations this is however, not surprising. From (10.19) we see that the small value of gI2given in (10.5) means that u, is little changed by a strain in tho y-direction. This is what we should expect. Again we found that x" was given by an equation of the form

+

.

x'= x ~ ( -To)/( T TI-To),

XI'=

under conditions of zero stress, that is x$(T, -To)is the rate of variation of xff with temperature, when the stress is zero. Hence

axif

=-

+a(y,,+2g1*),

. . . . . .

(10.22)

FT where LY. is the coeffioientof thermal expansion and the partial differentiation with respect to T is for constant strain. This follows because gll and g12give the variation with strain. Henco from (11.14) it follows that

The thermal expansion coefficient a can be obtained from the X-ray data above the highest transition temperature and is about Hence, ~inceg,,+~,,isepproximately- 10, the value of a(gtl+2gJ2) is approximately -lo-', and shoe the left-hand side is 104 the first term on the right-hand side must be 2 x 10-4. It oan be verified that it is of this order. Phyaically the whole expression in (10.23) gives the rate of decrease of polarizability with temperature. The second term gives the indimt effect of temperature change through the medium of lattice expansion. It corresponds t o an i n m s e in polarizability, as is physically obvious since lattice expansion must dlow the ions to move more easily. The fist term gives the direct effect of thermal vibrations. We see that they muse the polarizability t o dearease, and this term is comparable with, though larger than, the other.

2.5 Theory ofBariurn Titanate - A . F. Devonshire 165

Ova the Theory of the High Pramre Helium Discharge.

1063

This work is part of a project carried out with the financial assistance of the Electrical Research Association, and is published with their permission. I3EFERESCES. BLATTSER, KAXZIG, and MERZ,1949, Helv. Phy.9. Actu, 22,35. CADY,Piezoelectricity (!Sew* York : McGraw-Hill & Co.). FOWLER, 1936, Sta'tati&icalMechanics (2nd ed.), (Oxford : University Press). GINSBERG,1946, J . Picys. G.S.S.R., 10, 107. HAHWOOD, POPPER, and RWHMAX, 1947, A'atttre, 160, 59. HnLni, 1947. Suture, 160, 127. KAY,1948, dcta Crynt, I,229. KAY, VOVSDES, and WELLABD,1949, Nature, 163, 637. JACKSOX,and KEDDISH, 1945, Il'ature, 158, 717. MASOS, 1918, Phy.9. Rev., 74, 1134. MASOS, and MATTHIAS, 1948, P h p . Rev., 74, 1622. MATTHIAS,and vos HIPPEL,1948, P h p . Rev., 78, 1378. MEGAW,1946, Tram. Faraday Soc., 4 2 ~224. , MERZ, 1!)49, Yhys. Rev.,75, 687. MCELLER, 1940, Phy8. Rev., 57, 829 ; 58, 565, 805. and STRIVENS, 1946, Tram. Faraday Soc., 42a, 231. RUSHMAN, VON HIPPEL,BROCKEBRIDOE, CHELYEY,and TISZA,1946, I d . & Eng. Chem., 38, 1097. Wiit, 1946, J . P h p . U.S.S.R., 10, 95.

66

I

2 Selected Early Work 19221-1961

PHYSICAL

REVIEW

JI'NI:

1.5.

I D S 0

The Lorentz Correction in Barium Titanate J. C . SLATER .lasrachuselb fnsldirlc of Techndoqy. Cuinbrid~e,.Itassachusel/s* (Received March 13, 1950) It is assumed, following Devonshire, that the ferroelectric behavior of RaTiOI arises because of the Lorentz correction, leading to a vanishing term in the denominator of the expression for dielectric constant. If the polarizahility varies slowly with temperature, the temperature variation of dielectric constant follows. This temperature variation is assumed to come from that part of the polarization resulting from the displacement of theTi ion, in a field whose potential energy has fourth-power as well as second-power terms in the displacement. The main object of this paper is to compute the Imentz correction exactly, not assuming spherical symmetry, hut taking account of the precise crystal structure. When this is done, it is found that the Ti ions, and those oxygen ions which are in the same line with them, the line being parallel to the electric field. exert very strong fields on each other, the resulting local

field a t the Ti ion being much greater than if computed on the assumption of spherical symmetry. This enhanced field makes it clear that even a relatively small ionic polarizability for the Ti ions will be enough to lead to ferroelectricity. The polarization of the Ti ions is however an essential feature of the theory; if they are not polarized, the Lorentz correction is profoundly modified. leading almost exactly to the value given by the approximate theory assuming spherical symmetry, and not resulting in ferroelectricity. Detailed formulas are given for comparison of the present theory with Devonshire's results, so that the present methods can be incorporated in his treatment of the effect of elastic strain energy on the stability of the various phases below the Curie point.

I. INTRODUCTION

point a t 118°C the crystal is cubic. A unit cell holds a HE ferroelectric properties of barium titanate Ba ion a t each corner of the cube, an 0 ion at the center have aroused much interest in recent years, of each face, and a Ti ion a t the center of the cube. The particularly because its structure is so simple that there permanent polarization in the range between 118" and is good hope of understanding it fairly completely j 0 C is along a 100 direction in the crystal. There is a Enough progress has been made very recently so that slight mechanical deformation associated with this we begin to have a satisfactorily consistent theory of its polarization: the crystal axis in the direction of the behavior, but this thecry has hardly advanced far polarization slightly expands, whereas the two axes enough so that direct numerical comparisons with a t right angles to it contract, the amount of mechanical experiment can be made with complete success. The deformation being proportional to the square of the present paper fills in one missing gap in this theoretical polarization, so that the crystal becomes tetragonal treatment. To explain what this gap is, we give in the in this range. In the range between 5" and -70°C present introductory section a discussion of some as- the polarization is along a 110 direction; here again pects of the present state of understanding of the the crystal stretches along the axis of polarization and shrinks a t right angles, producing an orthorhombic problem. Barium titanate has a ferroelectric Curie point at symmetry. Below -70°C the polarization is along a about 118°C. Above that point, the dielectric constant 111 direction, again with crystal deformation, reis of the form [constant/(T- T J ] , where the constant sulting now in rhombohedral symmetry. The phase is very large, of the order of 150,000"K, and where T changes from tetragonal to orthorhombic, and from is the temperature, T , the Curie temperature.' The orthorhombic to rhornbohedral, are changes of the dielectric constant even of a single crystal does not first order, with latent heats. It is not quite certain literally become infinite a t the Curie point ; the curve experimentally whether the change from cubic to is rounded off somewhat, but the Curie law holds to tetragonal is of the first order or is a second-order transiwithin a very few degrees of the Curie point, so that tion of the lambda-point variety; if it is the latter, there are very large dielectric constants. Below the however, the increase of polarization just below the Curie point: there is a permanent polarization and the transition point occurs much more rapidly than would dielectric constant decreases from the very high values be expected from elementary theories of lambda-point close to the Curie point, but remains high, of the order transitions. At the various transitions, the free energies of the two of several thousand, around room temperature. At phases in equilibrium with each other must be equal. about 5°C there is a phase change, and a t about - TO'C another phase change, the dielectric constant becoming Since these phases correspond to different polarizations, very large near each phase change, and permanent it is clear that it requires a negligible external effect polarization persisting in each phase. The nature of the (that is, a negligible electric field) to shift from one phase changes is well understood. Above the Curie phase to the other, or to modify the magnitude or *This work was assisted in part by the Signal ('orps. the OKR, direction of the polarization. This is the qualitative explanation of the very large dielectric constant near and the Air Materiel Command. 1 S. Roberts, Phys. Rev. 75,989 (1949); 76, 1215 (1949). the transition temperatures, and it explains as well the * P.W. Forsbergh, Jr., Phys. Rev. 76, 1187 (1949) gives an ex- directional dependence of dielectric effect. Thus near tensive set of references relating to the phase transition.

T

748

2.6 The Lorentz Correction in Barium Jitanate - J. C. Slater

LORENTZ CORRECTION I N B A R I U M TITANATE

the transition between tetragonal and orthorhombic form, where the spontaneous polarization is shifting from the 100 to the 110 direction, it is particularly easy LO rotate the polarization from the one direction to the other, so that the dielectric constant related to polarizing the crystal a t right angles L o the existing spontaneous polarization becomes very large, though that associated with changing the magnitude of the polarization, without change of direction, is not unusually large. With this sketch of some of the observed facts in mind, we naturally ask what is responsible for the abnormally large polarization of the crystal. There are in general two types of possible polarization : electronic polarization, or distortion of the electronic structure of one of the ions, and ionic polarization, arising from the displacement of the ions as a whole. They can be distinguished on the basis of the frequency dependence: i n the optical part of the spectrum, where the frequency is too high for whole ions to follow the oscillations, only the electronic polarization remains. The dielectric constant of BaTiOt as found from the index of refraction in the visible spectrum is no larger than one would expect from the electronic polarizabilities of its constituents, and is3 in fact about (2.40)?=5.76.Thus it is clear that the abnormality in the low frequency dielectric constant arises from ionic displacement. It is generally considered that it is the Ti ion which is responsible for the effect. The reason is that the Ti ion is surrounded octahedrally by six oxygens; the Ba ions, being rather large, stretch the structure enough so that the hole in the middle of the octahedron occupied by the Ti is slightly too large for that ion, if we compute using conventional ionic radii ; and as a result the Ti ion is rather loosely held, with a small restoring force, or large ionic polarizahility.‘ This seems to be the qualitative reason for the effect, but it is clear that the ionic pdarizability arising in this way is not larger in order of magnitude than that usually found. T o explain the enormous effect which this has on the dielectric constant, it is usually to invoke the Lorentz correction in the theory of the Clausius-Mossotti formula. According to the Lorentz correction, we assume that the field acting on a dipole to polarize it is not really E, but is E+P/3co (in rationalized m.k.s. units) or E+4*P/3 (in non-rationalized Gaussian units). Let US assume that the dipole moment induced in a volume v is a times the field, where LY is the polarizability. We then have P = (a/v)(E+P/3so), so that the dielectric constant is P K=

1+-=

tnE

alelkv

l+-

1 - a/3cOV

(1)

in rationalized m.k.s. units; here, as elsewhere throughout the paper, we obtain the corresponding formula in W. J. hferz, Phys. Kev. 76, 1221 (19491. See, for instance, G . H. Jonker and J. H. van Santen, Science 109, 632 (1949). 3 4

I

67

749

non-rationalized Gaussian units by replacing towherever it appears by 1/(4r). From (1) we see that the dielectric constant becomes infinite when a/3eoV (in m.k.s. units) or 4 4 3 v (in Gaussian units) becomes unity. This is the so-called “47r/3 catastrophe.” The ferroelectricity of barium titanate, then, can arise from a quite h i t e value of the ionic polarizability, and does not demand infinite polarizabilities. We next ask why the dielectric constant shows the temperature dependence which it does, and why it has a Curie point. We can explain this if we assume that the quantity &coV shows a slow decrease with temperature, being slightly greater than unity a t absolute zero, and going through the value unity a t a certain temperature T,, which then must be the Curie temperature. Thus if we have

cr/3tov= l-C(T-T,),

(2)

where C is a constant, we find a t once that K=

-2+[3/C(T-

TJ].

(3)

This simple hypothesis, then, is enough to result in a Curie law for the dielectric constant; in the region we are interested in, the second term is very large compared to the additive constant, -2, which can be neglected. If, then, the constant in Curie’s law is known to be of the order of 15O,O0Oo, C is of the order of 2X1W6, indicating a very slow change of the quantity (2) with temperature. What physical mechanism, we may ask, is responsible for this slow decrease of a/v with temperature? The first suggestion, which has been made by certain writers,“ is that the change arises from the obvious increase of volume with temperature, coming from the thermal expansion. The volume coefficient of expansion of BaTiO, is about 3>( lo-“, so that this alone is more than capable of explaining the whole effect. But this neglects an important fact: as the volume increases, the atoms get farther apart, the repulsive forces between them get smaller, the elastic restoring forces consequently decrease, and the ionic polarizability increases. In fact, there is evidence that a increases faster than v, under these conditions. Thus it is found that either application of pressure: or substitution‘ of smaller ions for Ba, resulting in a decrease of volume at constant temperature, results in a decrease of the Curie temperature. That is, a/v is smaller at a given temperature, for a smaller a, so that we have to go to a lower temperature for the quantity a/3cov to become equal to unity. This argument shows that an increase of volume at constant temperature will make a/v increase, rather than decrease. Since we have seen that a / v nevertheless decreases with increasing temperature, there must be an additional influence making LY decrease with increasing temperature, a t constant volume. Such a possible effect isof course known. The familiar Langevin theory

‘For instance, Jonker and van Santen, reference 4. W. J. Merz, Phys.Rev. 78, 52 (1950).

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J . C . SL.4'I'ER

of rotating permanent dipoles leads to an average polarization inversely proportional to the absolute temperature. Such a temperature dependence leads to a temperature coefficient C equal to l/Tc, very much larger than we are looking for. The rotating dipole is not the only model leading to this temperature dependence. We get it whenever we have an ion which can move in a region of arbitrary shape and size in which the potential energy is constant, surrounded by infinitely high walls; that is, when we have an ion in a potential well. The rotating dipole is the case in which the available region for motion is a thin spherical shell. Other shapes of potential wells which have been discussed are those in which the ion has several alternative locations allowed to it, all with identical energy.' Any such model, then, will lead to the same temperature coefficient C, of the order of 100 times as great as the value we are looking for, provided we assume the ordinary Lorentz correction. Such a model is probably approximately correct for rochelle salt,8 in which C is in fact not far from l/Tc; but not for BaTiOa. At the other extreme is an ion held by a linear restoring force; in this case the polarizability a is independent of temperature, so long as the volume, and hence the force constant, remains constant. Between these two extremes, we should be able to find a model capable of explaining the facts, and in fact we can. If we start with a linear restoring force, but modify the potential by adding a term in the potential energy in the fourth power of the displacement, as well as in the square, we introduce, as Devonshire9 has shown, a linear decrease of polarizability with temperature whose coefficient C is proportional to the coefficient of the fourth-power term in the potential energy. A reasonable value of this fourth-power term leads to a value of C of the order of magnitude of that observed. I t seems highly probable that this is the explanation of the observed temperature variation, and hence of the occurrence of Curie's law. A difficulty arises when we use this model, however. The same fourth-power term in the potential energy of the ion leads not only to a temperature dependence of the polarizability (that is, to a temperature dependence of the coefficient of the term proportional to P,where P is the polarization, in the free energy), but leads also to a fourth-power term in the free energy, a term proportional to P4. Such a term is essential in calculating the spontaneous polarization below the Curie point. If we calculate the spontaneous polarization from this term, however, choosing the coefficient to describe the temperature dependence

of polarizability correctly, we find that the polarization

increases much more slowly below the Curie point than is observed. This discrepancy between observation and theory is removed, as Devonshire has shown, by including the electromechanical terms in the free energy, the ones which lead to the piezoelectric effect and electrostriction. When the crystal is allowed to distort itself, in the tetragonal phase below the Curie point, the free energy is decreased by a term proportional to the square of the elastic strain. We have already seen that this strain is proportional to the square of the polarization; thus the effect of the electromechanical coupling is to introduce into the free energy a term, with negative coefficient, proportional to P*. The effect of this term is to make the crystal polarize much more rapidly below the Curie point; in fact, as Devonshire has shown, it is easy for the term to be great enough to lead to a first-order phase change at the Curie point. This mechanical effect does not alter the temperature dependence of the polarizability, however, and hence leaves Curie's law unaffected. Devonshire has shown that, by a complete expansion of the free energy in powers of the polarization and strain, one can arrive a t constants for the various coefficients which lead to a rather complete understanding of the successive changes from one type of polarization to another, as the temperature goes down. This argument seems convincing, and in its broad outlines it is. However, there is a feature missing in it, which we shall supply in the present paper. We have not taken account of the contribution to the polarizability made by the electronic polarization of the various ions of the crystal. The optical value of the dielectric constant shows that the value of a/3c0v arising from the electronic polarization is about 0.61. Thus a further contribution of only about 0.39 must be made to this quantity by the ionic polarizability of the Ti ion, in order to produce ferroelectricity. This fact was entirely neglected in the treatment of Devonshire mentioned above, the most elaborate treatment of the problem to date. Not only this, hut our formula (l), on which our discussion has been based, is correct only if the Lorentz correction in its simple form is valid. But it has been pointed out by several writers+'O that there are large deviations from the Lorentz formula in the BaTiOJ crystal. If the polarization is along a 100 direction, a third of the oxygens will fall into linear arrays, in the same lines in which the Ti ions are located; these we shall call oxygens of type a. The other two-thirds, which we shall call type b, do not fall in line with the 'Ti We now find that in the presence of the Ti ions, ' W. P. Mason and B. T. Matthias, Phys. Rev. 74, 1622 (1948). ions. These writers e lain the temperature variation hy use of an the type a oxygens are much more strongly polarized artificially low v3ue for the Lorentz correction which, in the light than the type b ions. Furthermore, these type a oxygens o f the present paper, seems to have no physical justification. are very close to the Ti ions, and are oriented in such a W. P. Mason, Phys. Rev. 12, 854 (1947). A. F. Devonshire, Phil. Mag. (Series 7) 40, 1040 (1949). Such way as to be able to polarize the Ti ions with maximum calculations regarding the effect of a fourth-power term in the effect. The result is that the Ti ions are really in a field

energy were made independently by P. W. Anderson, Phys. Rev.

78, 341 (1950), and hy the present writer, before the work of Devonshire came to their attention.

'OH. (1949).

I;.

Kay and P. Voustlen, Phil. Mag. (Series 7 ) 40, 1019

2.G The Lorentz Correction in Barium Titanate - J. C. Slater 1.0 I< E K 'I'z

c o K I< E c'r I o N

169

IS BARIUM TITANATE

75 1

much greater than the Lorentz value E+P/dco. The potential energy of the ions in this local field is tield is, so to speak, enhanced at the location of the Ti -q(Es+E,y+E,z)= -qE.r, where r is the radius ion by a considerable factor. Thus, in order to result in vector. We shall now proceed to treat this system by statisferroelectricity, we do not need nearly as great an ionic polarizability on the part of the Ti as we should tical mechanics. We shall operate entirely with classical otherwise suppose. In fact, we shall find that the quan- statistical mechanics; this should be suitable at the tity a / 3 c 0 v arising from the ionic polarizability of the Ti temperatures concerned in the BaTi03 problem, though need be only 0.06 in order to produce ferroelectricity, our results would need correction at fairly low temperarather than the value 0.39 mentioned above. The tures. Then the first step in handling the problem statisperovskite structure met in the ferroelectric form of tically is to compute the partition function, Z . If we UaTiOo is thus particularly favorable for the production were dealing with a single polarizable ion, this would of ferroelectricity. I t has already been pointed out" that be (2rmkT/h2)tw, where m is the mass of the ion, and the other polymorphic forms of BaTiOa, which do not w = J exp(-++qE. r)/kTdv. (4) have these rows of alternating Ti and 0 ions, do not have this favorable predisposition toward ferroelec- If our system contains N identical ions, the whole tricity, and in fact they show quite ordinary dielectric partition function is the value above, raised to the Nth behavior. power, divided by N !; or, using the Stirling formula for The main point of the present paper, then, is to work the factorial, we have out the Lorentz correction in detail, and to show how Z = [(e/Nh3)(2rmkT) 43"~". it is to he incorporated into calculations of the dielectric constant and the free energy of the type made by The free energy A E is then given by the equation Devonshire. As a preliminary to this, we take up in the A E = -kT 1nZ. That is, next section the calculation of the free energy of the .4 E = - NkT In[ (e/.Vh3)(2rmkT)'1- n'kT lnw. crystal, as arising from the displacement of the Ti ion in the field of its neighbors, including a fourth-power term, This free energy is expressed as a function of T and E ; following to a considerable extent the calculations of it is for this reason that we have denoted it by the subDevonshire. We adopt a rather different statistical script E. The entropy is given b y - ( ~ A E / ~ T )and E, method, however, which seems more straightforward hence is than his. Then we pass on to the treatment of the Lorentz correction, taking account of the actual crystal S= N k ln[(e/.~ha)(2rmkT)f]+ (3/2):Vk+!Vk lnw + N k T ( a lnw/aT)E. structure of BaTi03. By differentiation of (4), the last term may be re11. THE FREE ENERGY OF Ti IONS writ ten IN A NON-LINEAR FIELD NkT( a Inw/d T )E = iV( - qE . r)/ T , In our first discussion we shall treat the Ti ions as where (4-qE.r) signifies the average value of this being independent of each other; later we shall consider quantity over the distribution given by the Boltzmann their interactions. Let the potential energy of an ion a t position ( x , y, z) (measured from the position of equi- factor exp[(-++yE. r)/kT]. The total dipole moment T , so librium of the ion) in the absence of an external field of the distribution is given by - ( ~ A E / ~ E Jetc., that its x-component is be +(x, y, z). When it is displaced to position x, y, z, let (5) the resulting dipole moment have components (qx, yy, (Moment),=.VkT(d lnw/aE,) T = Nqx, q z ) , q being the effective charge on the ion. We shall where again the last form comes b y differentiating (4). assume + to have cubic symmetry, on account of the The internal energy U is given by U = A E+ TS ; thus symmetry of the field surrounding the Ti ions. In this it is respect we introduce greater specialization than does U = (3/2)iVkT+iV(+-yE. I), Ikvonshire, who assumes a general potential energy. That assumption is necessary if, for instance, we are the sum of the kinetic energy as given by equipartition, interested in the displacement of an oxygen ion, since and the mean potential energy of the ions as displaced these are not in positions of cubic symmetry in the by the field E. In all these formulas if we are dealing lattice, hut it is not necessary for the Ti ions, which we with unit volume of the material, so that N becomes the assume make the principal contribution to the ionic number of ions per unit volume, our expressions give the polarization. Now let there be a local electric field E, values of free energy, entropy, moment, and so on, of components (Ez,E,, E J , acting on the ion. (We per unit volume. Thus in particular the moment per postpone until later the question as to how this depends unit volume, given by (S), is the polarization vector P. If the volume per ion, or the volume of unit cell, is 11, on the external applied field.) Then the additional . as in the preceding section, we then have the number of ' 1 K. I). Hurlmik and H. 1'. Evans. l r . . Acta Crve. 1. 330 (1948); H. T. Evans, J I . , and K. D. Burbink, J. Chern: Phis. 16, atoms per unit volume, or N , equal to l/v, so that we can express the quantities properly by replacing .V by 1/11. 634 (1948).

+

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2 Selected Early Work 1921-1961 J.

C. S L A T E R

If we are dealing with unit volume, we then have S= - (aAE/aT)E, P,= - (aA B/BE.)T. Then we have at once

( a s / a E , )T = ( a ~ ~ a r ) (6) analogous to the Maxwell relations of thermodynamics. This shows us that if the polarization depends on temperature, the entropy must depend on the field, or polarization. If, for instance, we assume that the polarizability depends on temperature as in Eq. ( 2 ) , we then find from (6) that a t the Curie point

S =So - 3 3COCE=So- CP2/6~0, where SOis the value of entropy when E or Pis zero. We therefore see that if C is small, the change of entropy with polarization is also small, so that there is only a small increase of entropy in going from the ferroelectric state at low temperature to the unpolarized state a t high temperature. The free energy A E which we have computed, like the Helmholtz free energy in ordinary problems of statistical mechanics, is the one which is simplest to find, from the partition function. More convenient for most purposes, however, is a free energy A P , analogous to the Gibbs free energy, expressed in terms of the polarization and temperature, rather than in terms of the field and temperature. This is defined b y the relation

For the case of spherical symmetry, bl will equal b2, and $I will reduce to ar'+W. We must now insert this expression into (4), and compute the partition function. This cannot be carried out exactly, and we therefore use series expansion methods, treating b1 and b2 as small quantities, and disregarding all terms of higher powers than the first in these quantities. We expand the quantity -$+qE.r appearing in the exponent of (4) in power series in the coordinates about the point where it is a minimum, and it then becomes the sum of a term independent of the b's, and a term linear in the b's. We expand the exponential function of this linear term in power series, retaining only the term in the series linear in the b's. The integration then becomes straightforward, and we find

1

w= ( r k T / a ) *exp(q2E/4akT) 1 -q(3b1+2b2)(RT/az)

+2b2( E,2E2+ E,2Ez2+ E?E,2)]

t

.

This agrees with the result of Devonshire [reference 9, Eq. (10.11)],provided we consider the differences in notation. The free energy A E is then and is to be expressed as a function of P and T . That is, it is A P = A E + E . P , if we are dealing with unit volume of material. When expressed in terms of the proper variables, it has the properties that

s= - ( a ~ ~ / a ~E.=(aAp/aP,)T. )~,

A E = -NkT In[(~/Nh~)(rkT)~(2rn/a)~] - NqZl?/4a

+-3N(kT)2 (3bi+ 2b2)+ NkT(3b,+ 2bz)$l?/4a3 4a2

(7)

I n many ways the most convenient way to approach the problem is to find A p as a function of P and T ; we shall discuss the results later. We note that in the absence of an electric field E , the free energy A p will have a minimum value as far as changes of polarization are concerned, a t constant temperature. Corresponding to this, A P will be unchanged when there is a change of phase, involving change of polarization, a t constant temperature, so that the equality of the values of A l . for the two phases is the condition for equilibrium. I t is this free energy A p which Devonshire considers in his paper, but his method of finding it is less straightforward. We now have the necessary statistical and thermodynamic background for our calculations of free energy. Following Devonshire, but modifying his results for the special case of cubic symmetry, we shall assume that the potential energy of a displaced ion is d x , y, e)= a(9+yl+z2)+bl(x'+yl+z0

+2b2(3z2+ 99+22y2).

+2bz(E:E>+E?E?+

EfE,Z)].

From this free energy, we can use our general equations to calculate various other quantities. Thus we find S= Nk ln[(e/Nha)(rkT)a(2m/a)~ ]+3Nk

3 NPT -- ---(3b1+262) -Nk(3bi+2bz)$'l?/'ka, 2 a2

rs=R+&[

(8)

1- kT(3b172br)]

2a

N# - - E , [ ~ I E = ~ + ~ Z ( E ~ ~ + E(9) ,~], 4a'

U = 3.\'kT- Nq2E/4a+-[bi(Ez4+Eyl+ N# 16a4

Ex')

2.6 The Lorentz Correction in Barium Titanate - J.C. Slater

1 +--[b,(P,’+P,”+P,’) XSy4

+2b2(P,’P,”+P,’P,”+US!’,’)1.

(10)

.4t this point cur treatment has differed significantly from that of Devonshire (who introduces the Lorentz correction before making these steps), so that (10) does not agree with Devonshire’s Eq. (10.13) (which, in addition, contains several misprints). Let us now call attention to a number of aspects of these results. From (Y), giving the polarization as a function of the field, we see, as we mentioned in the preceding section, that the polarization decreases linearly with temperature, for constant local field. I t is this decrease of polarizability with temperature which we have seen in Eqs. (2) and (3) to be responsible for the temperature dependence of the dielectric constant in the neighborhood of the Curie point. In order to get a small temperature dependence of polarization, we clearly need only assume a small value of the quantity (3b1+2b2), determining the non-linear behavior of the law of force. We shall mturn later to the necessary value of this quantity, and show that it is reasonable. Associated with this decrease of polarizability with temperature is a decrease of entropy, as given in (8),with electric field; we immediately verify that Eq. ( 6) is satisficd. I t is interesting to look into the physical reason for this change of entropy with polarization, for by (6) this will then tell us why the polarizability decreases with temperature. Let us consider the entropy for a linear oscillator, in which the b’s are zero. From (8),we have in this case

I

71

force constant for this new position of equilibrium, increasing the vibration frequency about that point, and hence decreasing the entropy. I t is this effect which results in the term in E , in (8). At the same time, as the temperature goes up, the particle spends more time in a region of high amplitude, where the restoring force constant is larger; this results in the decrease of entropy with temperature described by the term - ( S k 2 T / a 2 ) 3 ( 3 b ~ + 2 b 2 ) /in 2 (8). I n this way we understand the effect of the fourth-order terms on the entropy, and hence on the polarizability. I t is interesting to see that the effect is very simple and elementary, and quite different from any order-disorder explanation of entropy change, such as is invoked in some other phenomena leading to lambda-points. III. THE NATURE OF THE LORENTZ CORRECTION

In Section 11, we assumed that the various ‘Ti ions vibrated quite independently of each other. This assumption is incorrect, since actually they exert forces on each other, rather than having an external force acting on them which depends only on their displacement. This situation is well known from the theory of the specific heat of solids. We know that if we treat the vibrations of the atoms separately, we arrive at a specific heat theory of the Einstein type. On the other hand, if we consider their interactions, we must introduce normal coordinates, describing the various standing waves which can be set up in the crystal, and arrive a t a specific heat theory of the Debye type. If there are several types of atoms or ions, then as Born and others have shown, there will be various branches to the vibration spectrum, some being of the so-called acoustical type, and having frequencies extending down to acoustical ranges, while others are of the so-called optical type, with frequencies in the infra-red. These optical vibrations are the ones in which the ions of d s e r e n t type oscillate in opposite directions ; they are connected with the residual rays or Res/slrulzZen. In our case, the S= :Yk In[(e/.Vh3) (rkT)d(2m/a)*]+3iVk. vibrations of the Ti ions with respect to the rest of the Let us particularly observe the dependence on a : S lattice will be of this optical type. And in general it is contains a term - (3/2)rVk h a . This dependence is fun- found that the spectrum of frequencies in an optical damental, and has a simple explanation. For a large branch is not widely spread out, so that the correct restoring force constant a, the particle a t a given energy, theory is not greatly different from an Einstein-type or temperature, is confined to a small volume; but we theory, in which all frequencies are assumed to be know that the smaller the effective volume available identical. For this reason, in the present treatment we to a particle, the smaller is its entropy. Or alternatively, shall not consider the interactions of the ions, but for a large value of a, the frequency of oscillation of the shall treat them as being independent. The writer has particle will be large, its stationary states as given by been informed by Dr. P. A. Anderson, of the Bell the quantum condition will be far apart, and a t a given Telephone Laboratories, that he is working on a more temperature there will be fewer occupied states than elaborate treatment, in which the normal coordinates for a smaller value of a ; this again leads to small en- are properly introduced. tropy. Now let us return to our case of the non-linear This simple discussion is correct in some respects; restoring force, where the b’s are different from zero, but it neglects a very important aspect of the problem, and positive. If we polarize the ion by a local field, SO the Lorentz correction. The ordinary theory of elastic that its position of equilibrium is displaced from the vibrations of the ions deals with short-range forces, origin of our (2,y, z ) coordinates, the fourth-power only interactions between nearest neighbors ordinarily terms will have the effect of increasing the restoring being considered. When the ions b y their displacements

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produce electric dipoles throughout the interior of the solid, however, these dipolesproduce long-range electric forcea acting on each dipole of the system, and these dipole forces must be considered separately. We can make a first approximation to a theory including these Lorentz forces if we proceed in the following way. We introduce as coordinates representing the motion of the Ti ions first the coordinate of the average displacement, or center of gravity, of all the ions; then, as additional coordinates, the displacements of the separate ions from the displaced position which they would take up if the whole Ti lattice were displaced bodily. The displacement of the whole lattice will be equivalent to the polarization produced by the ions, and there will be a term in the energy coming from the Lorentz interaction of the field produced by the resulting dipoles, with the dipoles themselves. There will be further terms in the energy arising from the displacements of the separate ions from this average position, and these will be treated essentially as in the preceding section. This is admittedly only a partially satisfactory way of handling the complete problem of the interactions of the displaced ions, including their electrostatic interactions, but it should give a satisfactory first approximation. Let us see how it would work out, if the Lorentz assumption of spherical symmetry were correct. In that case, if there were a polarization P, the local field acting on a given ion would be E+P/3eo. If the ions were displaced in such a way as to produce the polarization P, each ion would make a negative contribution to the electrostatic energy, since it is displaced in the direction of the field P/3c0.The total electrostatic energy per unit volume arising in this way will be -P/6s, the factor 4 (leading to - P / 6 ~ 0 rather than - p / 3 € 0 ) arising because we are really dealing with interactions of forces between pairs of dipoles, and we must count each pair only once, not twice. We can then take care of thebrentz correction by supplementing the free energy A p of (10) by this correction term -P/6e0,where we are dealing specifically with unit volume of material. We can now take thii revised free energy, and inquire what is the polarization as a function of field arising from it, and consequently what effect the Lorentz correction has on the dielectric constant. From (7) we find

This agrees with the assumption leading to Eq. ( l ) , and when we remember that N = l/o, we see that the polarizability a is given by

showing a linear decrease with temperature proportional to the b's, qualitatively agreeing with ( 2 ) . Thus our addition of the correction term -P/6c0 has correctly taken care of the Lorentz correction. We note that the entropy expressed as a function of the P's, by (7), is unaffected by the presence or absence of the Lorentz correction; it is

S=h'k In[ (e/NK)(rkT)3(2m/a)i]+3NK

This shows the quadratic decrease of entropy with polarization, proportional to the b's, which we have already discussed. We can discuss the spontaneous polarization below the lambda-point from the expression (11). Let the applied field E be zero. Then we have various possibilities for spontaneous polarization. Let us consider the simple case where P , is different from zero, P" and P , equal to zero. Then we have from ( 1 1 )

The spontaneous polarization will be zero at the Curie point ;hence we have iV$/(6ro~)= l + [ k T c ( 3 b l + 2 b , ) / ~ ~ ] .

(14)

We may then rewrite (13), correct to linear terms in the b's, in the form (PP)/eo=3Nk(3bi+2bz)(T,-T)/bi.

(15)

This indicates that P is proportional to (To-T)+below the Curie point, the usual type of dependence for a lambda-point theory. If the actual transition is a lambda-point transition rather than a phase change of the first order, however, the temperature dependence of polarization given by (15) is much too slow.Thusif we have something not far from spherical symmetry for the T i ion, so that b, and b2 are comparable in size, and (15) reduces to 15 NR(T.--T), comparison with obIf we disregard the cubic term, to get the dielectric served curves shows that if the real variation of polarizabehavior above the Curie point, in a small &Id, and tion with temperature is to be fitted to a curve of the solve for P,, treating the b's as small quantities, we have form ( W ) , the coefficient must be something like 60 times as large as that given above. This situation, as we indicated in the introductory section, can be corrected by considering elastic distortion d t.he crystal, as we shall describe in a later section.

2.6 The Lorentz Correction in Barium Titanate - I. C. Slater

L 0 R E N T Z C O I t R E C T I0 N

IFi B A R I U M T I T . \ N A T E

IV. THE LORENTZ CORRECTION FOR THE ACTUAL CRYSTAL

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that the ordinary Lorentz correction 4rP/3 (in Gaussian units) is correct not only at a lattice point, but In the preceding section, we have considered the at a point in the center of the unit cell. Thus this way in which the Lorentz correction affects the free ordinary correction is valid for the actions of the Ba energy, and hence the expression for dielectricconstant. and Ti ions on themselves and on each other. On the Now we are ready to ask how this argument is modified other hand, for interactions between the oxygens and if we take account of the polarizability of the other the other ions, the ordinary correction is entirely inions in the crystal. In the present section we shall deal correct. Thus if we ask for the action of the Ti ions on only with the polarization above the Curie point, where those oxygens which lie along the same line parallel to the polarization is proportional to the field. In the next the polarization (or z axis), which we have previously section we extend the argument to a calculation of the called oxygens of type a, we find that the Lorentz coreffect of the Lorentz correction on the free energy, so rection is really (30.080+4~/3)P, or approximately 8.2 times the ordinary value 4uP/3. The action of the as to treat the problem below the Curie point. In the BaTiOs crystal, each unit cell contains five oxygens of type a on the Ti ions contains this same ions. Let us assume that the crystal is polarized along greatly enhanced Lorentz factor. I t is this fact which the z axis; symmetry demands that each of the ions we shall find operating to make the field a t the position be polarized along the z axis. Let us then write the total of the Ti ions so much greater than we should get from polarization in the crystal as the sum of five terms: the a simple Lorentz argument, as we mentioned in the polarizations arising from each type of ion. When we introductory section. Let us now write the polarization of the Ti ions as consider the whole crystal, each of these types of ions , of the Ba ions as PB., that of the oxygens of will form a simple cubic lattice. We then visualize for P T ~that each type of ion a simple cubic lattice of equal dipoles, type a as PO., and that associated with the two lattices and we wish to find the field of these dipoles at one of of oxygens of type b as P O b l and POb.2. By symmetry, we the lattice points. I t is well known that the field of a shall find that P O b l = P O b 1 , but it will simplify our prosimple cubic lattice of dipoles at one of its lattice points cedure to keep them separate. These polarizations are is P/3eo, where P is the polarization resulting from this assumed to arise from electronic polarization, except , we shall sometimes assume lattice alone. Thus, for instance, if we had only the in the case of P T ~where polarization arising from the displacement of the Ti that it is the sum of that arising from electronic and ions, as we were considering in the preceding section, ionic polarizations. Then from the relations (16) we the Lorentz correction as treated in that section would can write the effective or local fields acting on each type be correct. However, the field of a simple cubic lattice of ion to polarize it. We shall write these fields in the of dipoles is not equal to P/3c0 at most points within m.k.s. system; to get the corresponding formulas in the the unit cell. Thus for instance the field exerted by the Gaussian system, we replace € 0 by 1 / ( 4 ~ ) as , usual. We lattice of Ti ions is not equal to P/3co a t the position of shall then want to use the quantities 8.668/4r and one of the oxygen ions, but is quite a difTerent amount. 30.080/4r. We shall abbreviate these as What we must do is to examine the field actually exerted at the position of each ion, by the latticesof the 8.668/4r= 0.690= p , 30.080/4r= 2.394= q. (17) dipoles of all types of ions. Fortunately the necessary calculations have been Then we have: performed in papers by McKeehan'? and by Luttinger and Tisza.ls Luttinger and Tisza give the field exerted Field on Ti=E+(l/co)[gP*i+)PBa +(q+4)PO~+(--fq+))(PObl+POb?)]. at various points through the lattice by a lattice of dipoles whose polarization is unity, polarized along the Field on Ba=E+(l/ro)[SPTi+aPBn (-p+f)POa+ (3p+))(PObl+POb2)1. z direction. They express their results in the nonrationalized Gaussian units, and the fields which they Field on Oa=E+(l/fO)[(qf))PTi+(-p+f)PB. ++PO.+ (fPf#)(PObl+ P O b Z ) ] . give are those in addition to the value 4rP/3. Thus they find that the additional field at a point (E, t), <) Field on obl=E+(l/f0)[(-3q+f)PTi (jP+))PBa+ ()$+!j)POo+fPObl in the unit cell, where (, t), are the ratios of the dis+(-f+f)POb?](18) placement from a lattice point to the lattice spacing, are given by expressions S&, t), <), where The equation for the field a t o h ? is like that at O b 2 , with subscripts 1 and 2 interchanged. We observe, as we S,(O, 0,0)=S,(+, 3, + ) = O pointed out above, the very large factor q++ by which S,(O, +,4)=4.334, S&, +,0)=-8.668, S.(+, 0,0)=-15.040, S,(O, 0, f)=30.080. (16) the Ti ions and the type a oxygen ions interact on each other. Let us see what these mean. The first statements show We can now set up simultaneous equations, stating I* L. W. McKeehan. Phys. Rev. 43, 913 (1933);72, 78 (1947). that the polarization resulting from each type of ion 12 J. M.Luttinger and L. Tisza, Phys. Rev. 70, 954 (19461; 72, equals the corresponding polarizability per unit volume, 257 (1947).

+

<

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74 2 Sdected Early Work 1921-7%1

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C. S L A T E R

multiplied by the appropriate local field." Let the cubic lattice. Let us now ask what is the situation with polarizabilities of Ti, Ba, and 0 be UTI, UB., and ao, the actual numerical values for BaTiOr, in the optical respectively; L Y Twill ~ be assumed to be the electronic spectrum, 80 that we disregard the ionic polarieability polariibility, and will be specifically augmented by the of the Ti. As values for the electronic polarkabilities of ionic polarizability when we are dealing with low fre- the various ions, we take the following values: quencies. Then we have crn/Co(m.k.s.) = = ~ W ( U T I (Gawian)=2.34XlP cc ar~~-24.42X1O-U cc, ao/cp30.OX 1 F cc. (21) PTI= ( a ~ i / V ) ( & e l d On Ti), PB. = (aBJv)(field on Ba), The values for Ti and Ba were supplied by P. W. PO. = (ao/v)(field on Oo), Anderson and W. Shockley of the Bell Telephone P O b l - (uo/v)(field on Obd, Laboratories; they come from unpublished work inPobi= (uo/w)(field On oh) (19) volving an intercomparison of the refractivities of a where the fields are as given in (18). We at once sce by good many ~rystals.'~The value for oxygen is chosen inspection that P O b I = P O b ¶ , 50 that we have four simul- to give the correct value for the optical dielectric contaneous equations for the four unknowns PTI,PB., stant, which except for a small deviation very close to Ph, P-1. These equations can be solved by determi- the Curie point is approximately (2.40)2=5.f6. This nants for the polarizations, as functions of E. We can procedure is in accordance with the findings of the also solve for the dielectric constant, which we get from group a t the Bell Telephone Laboratories, who find the sum of all the polaritations, divided by the field. that the oxygen ion appears to show diflerent polariiIn expressing these quantities, we encounter the com- bilities in compounds of merent structure." We take binations aTI/QV, and similar expressions for the other the lattice spacing to be 4.00X 10-8 cm, a value which polarizabfities. We abbmviate these as XTi, etc., so is correct at a temperature in the required range (we that by definition we have X T I = U T ~ C(inO V m.k.s. neglect the thermal expansion at this point), so that u units), or 4mTJV (in Gaussian units), with similar is 6 4 x l W cc. Then we have expressions for XB., XO.In terms of these quantities, x~i=o.o365, X~.=0.382, X0=0.470. (22) the solutions of (19) prove to be When we substitute these values, and p and q as given in (17), into Eqs. (20), we find that we get the correct value 5.76 for the dielectric constant. The associated value of ( ~ - 1 ) / ( ~ + 2 )is 0.613. If the ordinary hrentz correction were applicable, this would be equal to )(XTI+XB.)+XO, which is 0.609. In other words, in thii case of the optical refractivities, the crdinary Lorentz formula gives a very good approximation to the truth. The reason for this is undoubtedly that XTIis small, XB. is of the same order of magnitude as Xo, 80 that we are not very far from the special case XTI=O,XB.=XO,in which we have pointed out that (20)reduces exactly to the Lorentz caae. It is also interesting to calculate the polarkations of the various types of ions in this high frequency case. When we substitute. we find In Eqs. (20) we have solved our problem of setting PTI/(€& = (XTI/A)(1.262)=3.3ox~1, up the correct Lorentz factors for the BaTiO, structure. PBJ(c&) = (XaJA)(0.921)= 2.41Xa, Let us now examine some of the implications of these Po,,/(eoE) = (Xo/A) (1.174)=3.08X0, formulas. It can be verified that in spite of their comPosil(~oE)E (Xo/A)(0.9%)-2.47Xo. (23) plication they reduce to the exact Lorentz expression in two important cases: Grst, when XTIis zero, and XB. If the ordinary Lorentz correction were applicable, we equals XO;second, when XO is zero. The first case is should find these numerical coefficients in each case to one in which the oxygen and barium ions, assumed be equal to (~+2)/3=2.59. We notice that they are of identical for that special case, together form a face- the same order of magnitude; but there is a diatinct centered cubic lattice; the latter that in which the tendency for the Ti and the type a oxygen ions to be barium and titanium together form a body-centered polarized more than this value, and for the others to be polarized somewhat less. We shall find in the low fre"Similss equations have ban set u for thh QK by J. M. Richardson, formerly of the Bell Te/&hone LaborarcrieS, in quency m, where the Ti ie polarized much more, unpublirhcd work, but the rerultr were not arriad sr far as in the that this tendency in greatly enhanced. -t work. See J. M.Rlcbardwn end W. Shoddcy, Phys. Rev. "W.Shockley Phya. Rev. 70,105(A) (1946). FO, 105 (1946). See .Lo . H.M h t e n and W. Opchowtki, Phynica's Grav. 14, 545 (~94%).

UW.ShocLley; Phys. RSV. 73,1273 (1948).

2.6 The Lorentz Correction in Barium Titanate - J.C. Slater

L O R E N T Z C O R R E C T I O N I N B A R I U M TITAN.-\TE

I

75

757

Now let us pass on to the problem of the static as compared with the value 0.186 which we find from dielectric constant, where the Ti ions can polarize by our more correct theory. The elementary theory of the ionic displacement. Let us assume that the total Imentz correction thus is so far wrong that the ionic polarizability of the titanium is the sum of the value polarizability of the Ti ion needs to be only 0.186/1.173 a function of =0.160 times as great, or about one-sixth as great, as given in (21), and another quantity a ~ ; , temperature which is given by (12). Let the correspond- if the elementary theory held. Let us now examine how ing value of X be XT[= CYTi’/fOv. We shall then have to this very large enhancement of the effect of ionic disin place of X T in ~ Eqs. (20). In that placement comes about. use .YT~+XT~’ case we see at once from (20) that the dielectric constant We can understand the situation if we examine the ran be written in the form relative polarizations of the various types of ions. We can use (20) to find these, now using XT~+XTLIinCl+C&Ti’ 62 (cl/cd -(cZ/C4) of X T for ~ the Ti ion. In this case, for ferroelecstead K-l=-=-+ tricity, the determinant A’, obtained by using X ~ i f X ~ i l (.:I+c4xTi’ 1.4 1+ (CdCdXTi’ instead of X T ~is, zero, so that a t the Curie point the polarization per unit field is infinite. We can still where evaluate the brackets in (20), however, and these will c1= ~ T i + ~ ~ B ~ + 3 x O + ~ x T i x O still give the relative magnitudes of the polarizations 3px02- (3/2) (#+ q)2xTixBwy0 of the various types of ions. We then find - $(3q- p ) * X ~ i X o 84X~,.Y0’ ~cz = 1+pxo- (3/2)(p+q)?XB,S0-i(3q- p)2Xo’ P T i / (f OE) = C (XTi+ XT[>/A’]( 1.262)y P ~ a / ( d )= (x~a/A’)(-o.047), c 3 = l from Eq. ( 2 0 j , Pool(&) = (xo/A’)(0.939), PObl/(&) = (XO/A‘)(0.046). including terms in - Y T ~but not in ST: These values are strikingly different from those of (23). G I = -5(ff+ (3/2)q2)Xo+6(3q-P)~Xo* +3(#fq)2XBaX0. (24) Only the relative values now have significance; and we see that the polarization is almost entirely contributed In case X T ~S , B XO ~ , are all zero, so that the only by the Ti ions, and the type a oxygen ions. The Ba and polarization comes from the ionic displacement, we type b oxygen ions are hardly polarized a t all. In fact, have c1=0, c2=c3=1, c 4 = - f , and we have at once when we take account of the polarizabilities, we find the values given in the introductory section, that the Ti ions contribute about 37 percent of the total polarization (of which about 31 percent comes from ionic displacement, six percent from electronic polarization), the type a oxygen ions about 59 percent, the type b oxygen ions about six percent, and the Ba in our present notation. However, when we put in the ions about two percent in the reverse direction. We , from (22), determined from the thus see the evidence of the effect which we discussed values of X T ~etc., optical behavior of the material, the situation is entirely in the introductory section: the Ti ions, by their different; we find polarization, polarize the type a oxygen ions; these in turn act back on the Ti ions; and the net result is to c I = 1.834, c2= -5.892, c3=0.385, c 4 = -2.076, K = 3.84+ 1.93/(1- 5.39X1-i’). (26) build up the polarization of both types of ions, resulting in linear chains of dipoles all pointing in the same This important result (26) shows that the effect of direction, the positive end of one to the negative end the polarization of the other ions is greatly to enhance of the other, and producing spontaneous polarization the effect of the ionic displacement of the Ti ions in below the Curie point. producing ferroelectricity. Thus from (25) we see that Now that we have found the influence of the polarizif these other ions were not helping the polarization, ability of the other ions on the dielectric constant, we we should get ferroelectricity only when XTI)=3. On can combine Eq. (26) with an assumption similar to (2), the other hand, from (26), we get the same result when to investigate Curie’s law. I t is clear that, to make the 5.39xTi’= 1, or when X~i’=0.186.This indicates en: denominator of (26) vanish at the Curie point, we must hancement of the effect of the ionic displacement by a assume that factor of approximately 16. Some such enhancement, 5.39XTi’=S.39[aTi’/(too)]= l-C(T-Tc). (27) of course, would arise from the ordinary Lorentz correction. If this correction were applicable in its simple If we insert this value in (26), we find that form, we should get ferroelectricity when K = 3.84+ 1.93/C(T- T J , (28) f ( X ~ i +XTO+ X B.) X o = 1. as a substitute for (3). It is interesting to find that, in We already know that )(XTi+XBa)+xO=0.609. spite of the large change in the general situation proThus we should require that X~I)=3(1-0.609)=1.173, duced by the electronic polarizability, still (28) is not

+

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2 Selected Early Work 1921-1961

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. c.

s I, .i ‘r I.:

very different from ( 3 ) , so that the qualitative discussion given in the introductory section regarding the temperature coefficient of the ionic polariaability of the Ti ion is still correct. The constant C, however, is now seen to be about 1.3XlOW, instead of the value 2X 10-& given in the introductory section. We may now compare (27) with (12), the ionic polarizability as determined from our molecular model, and ask whether the constants of (12) have reasonable values. Comparing them, we have approximately

5.3Yg2/(2aea~~)= 1, C= [k(3bl+2b1)];a?.

(29)

If we combine these equations, we have (3bl+ 2b,)/a = (5.39Cg2)/( 2 4 ) .

(30)

If we assume that the charge 9 is n times the electronic charge (where n is 4 if the Ti ion is quadruply charged), and use the value of C given above, and v=64X lo-?‘ cc =64X 10-30m3,theright sideof (30) becomes 1.15X 1ff0n2 (m.k.s. units). We can judge whether this value is reasonable or not, by recalling that our expansion of the energy, which in the case of spherical symmetry can be written ar2+br‘+. . ., is really the beginning of a power series, which is bound to diverge, or have a singularity, when the Ti ion gets very close to one of the oxygen ions of the octahedron. This divergence will arise because all terms of the series have the same order of magnitude. Thus we should expect that a t a distance Y where we have a divergence, a y 2 and biA should be of the same order of magnitude, or r2=4/b, where for this crude calculation we may set bl=bz=b. Then b/a= (1.15/5)X1ff0n2=2.30X1019n2, and i2=1/2.30 X 10-1g/n2, I = (2.l/n)XlG-%= 2.l/n angstroms. This is certainly of the right order of magnitude for interatomic distances, showing that the value of b which we have found necessary to explain the Curie constant is of a reasonable magnitude. Since the actual distance between the Ti ion and its neighboring oxygen ion is just 2A, this crude argument suggests that it might be more likely that n should be approximately unity than 4, which we should have with a quadruply charged ion. From (29) we can also estimate the value of the constant a. If we assume again that q equals n electronic charges, we find that a= 7.6n2 ev/(angstron$. That is to say, the energy of the displaced ion, displaced a distance of 1A from its position of equilibrium, would be 7.6n2 volts. This again is of a reasonable order of magnitude, 1A being halfway to the oxygen ion. Here again n = 1 would be more reasonable than n= 4, which would lead to an energy of about 120 volts at a distance of 1.4. Without more detailed study of the interionic forces, however, it is hardly possible to estimate how great the expected restoring forces should be. It is, of course, not inconsistent with the structure of BaTiOs to assume a smaller value of n than 4,which would correspond to the strictly ionic compound. Thus if the oxygen ions on the average were singly rather than doubly charged, the Ti ion would have to have a single

IZ

positive charge. Some evidence as to the charge carried by the Ti ion should eventually become available, when it is known by x-ray measurements exactly how much the Ti ions displace in the spontaneously polarized condition. The magnitude of the spontaneous polarization is known, being equal to about 16X 1 P coulomb/ cm2. We have seen that we may expect about 31 percent of this, or 5 X 1 C 6 , to come from ionic displacement. With a charge of n electronic charges on each ’I’i ion, this would correspond to a displacement of 0 . 2 0 , ’ ) t angstroms. There is some x-ray evidence for a displacement of about l7 0.16A, suggesting n = 1 ; but also some evidence for a smaller displacement,18 suggesting a larger value of n. These questions must await further experimental information. V. THE LORENTZ CORRECTION AND THE FREE ENERGY

In Section IV we have considered only linear terms in the force acting to polarize the various ions. Now we shall pass to the more general problem, where we include higher power terms as in Section 11, but treat the Lorentz correction properly as in Section IV. We shall find it convenient to work backward, starting from the equations for the electric field in terms of the polarization, ending up by integrating these expressions to get a formula for the free energy. First we note that by differentiating (10) with respect to P , we get the z component of the field polarizing the Ti ion. We have

where we have now written the polarization arising from ionic displacement of the Ti ion as PT;. The field acting here is of course the local field, not the external field; in Section I1 we were not taking account of the difference between these two fields. But now, using results of Section IV, we know how to find this local field: it is set up in Eq. (18). I n interpreting that set of equations, we must now distinguish between that part of the polarization of the Ti ions arising from electronic polarization, and that from ionic displacement. We shall call the first part of the polarization P T ~the , second part PT:, so that Eqs. (18) are now where it appears by to be modified by replacing P T ~ P T ~ + P TAs ~ ’ .before, we let X T equal ~ aTi/C02’, where C Y T ~is the electronic polarizability of the Ti ion. The corresponding relation giving the ionic polarizability must now be given by (31), where we recall that the E , appearing in that equation is the local field acting on the T i ion. We may then rewrite Eqs. (18) and (19), modi“Danielson, Matthias, and Kichardson, Phys. Rev. 74, 986 (1948). Kay, Wellard, and Vousden, Nature 163, 636 (1949).

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2.6 The Lorentz Correction in Barium Titanate - J. C . Slater L O R E N'1'Z

C O It K E C ' r I 0 N

BARIUM

TITANAI'E

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Lorentz correction term is 3X 1.69/0.309= 16.4 times as important as would be given by the simple Lorentz theory. This is the same enhancement factor which we have already met in our discussion of Eq. (26), only now expressed in somewhat different language. In the process of setting up the solution (33), we have had to solve separately for the various polarizations PB*, etc. We shall not give the separate formulas for these quantities, but we shall give the formula for the total polarization P,in terms of P T ~ 'We . find that

=--[

2asol',~,' l+-(3b1+2b2)] kT

h'q2

IN

az

:(lJLT,+PzTL'+ P r d + (q++VJz"&

C"/i,+

+

+(if+f)(Prot,i+ P r o d = P r ~ a j X ~ a , (q$!?)(PrTi+PzTi')+( - p + i ) P z B a

&r+

+ilJzO:t+ €"&+

(fp+ $ ) ( ~ ' z O b l + ~ ' r O b " ) =

PzOa/xO,

( - $q+ S)(PrTi+PrT,')

+(if'+i)PzBaf +fPZObl+

(apff)PzOu

( -p + W * O b z =

PZOblIXO.

62

X [ ~ ~ P = T ~ ~ ? ~ ~ ~ ( P ~ T I ' ~ + P I-T P rITI;L . )I

!$(PrTi+PzTi'+PrBa)f ( - p + j ) P z O ,

€&z+

I+

(-~~+~)(P,OI~I+~~O~~)=P~TI/.~T~I

CS

We are particularly interested in the relation in the case of spontaneous polarization, when E=O, but P and PT,' are not. I n that case, we may combine with (33),and find P z = PIT:[ (CZc3- clc4)/C3c6]. But we can prove, by straightforward algebra, that czc3- c1c4= 652

(32)

There is an equation similar to the last one for I ' z O b Z . We can now solve all equations except the first for P r ~ iP, = BP,o., ~ , P r O b l , PZobzin terms of P=T{and E,. We then substitute these values into the first equation of the set (32), and this becomes an equation relating E , and P T ~ 'containing , now the higher power terms from that equation. When we carry out these steps, which can be simplified by comparison with the similar prohlem involved in finding the first equation of (20), the resulting equation is

Thus the equation above becomes

Pz= ( C 6 / C 3 ) Y r T t ' , (34) under conditions of spontaneous polarization. When we insert numerical values, we find P,= P,~,'/0.309 = 3 . 2 4 P , ~ , ' ,as we have mentioned in earlier paragraphs. We readily verify that above the Curie point, where we are interested in the dielectric constant, this same relation still holds to a first order of accuracy, agreeing with the results of Section IV. We can then use (34) in connection with (33), to express the field in terms of P, rather than P T ~ 'and , find

where and ( J and c 4 are as given in (24). In case the electronic polarizabilities of all the various ions are zero, we have c 3 = c 6 = I, Q = -5, and hence the only effect of the Lorentz correction in (33) is to subtract the term PT,'/3€0 from the right-hand side, or to substitute a local field E+P/3eo for the field E, as we should expect, and as we saw in Eq. ( 1 1 ) . However, when we take account of the electronic polarizability, there are two changes: the Lorentz correction tcrm /'/3c0 is changed from - $ to c4/c6, and the other term is multiplied by a factor a / a . When we insert t h e numerical values which we have been using. we find that C3/cb=0.3@, c 4 / c b = - 1.69. I n other wbrds, the

This expression allows us to integrate Eq. (7), taking account of (lo), to get the free energy A P . We have -4 p = -Y k T In[ (e/iVh3)( r k T ) y 2 m / a )33

+[ N ( kT)'/az]: (36

I+

262)

+2b?(P,2P?+ Pi2PL2+Pi2P:)]+-

c3c4

I*

.

662

2s0

(36)

78

760

I

2 Selected Early Work 1927-1967 J . C. SLATEK

If we use (34), and express this in terms of PTI',the expression (with the exception of the last term) reduces to the same value (10) which we found earlier (when we remember that the P appearing in (10) is just the ionic polarization of the Ti ions, which we are now calling P T ~ ' )The . last term, however, is not the same as the Lorentz correction function in Section 111. That was - P ~ i ' ~ / 6 ewhereas 0, our term (c3c4/ct)(p1/2e~) can be We verify from (24) that written as (c,/c,)(P~i'~/2eo). 0 the case where the only this reduces to - P ~ i ' ~ / 6 cfor polarization is that of the Ti ion, thus verifying our result; but for the actual case, it is 16.4 times as great, in agreement with previous statements concerning the effect of the correct treatment of the Lorentz correction. Expression (36) is the one which we should use for discussing relations between theory and experiment. We may define the Curie point as the point where the term in P goes to zero. That is,

- ( C I / C B ) ( A ' ~ ~ / ~ E1+kTC(3b1+2b2)/d, OU)=

(37)

to be compared with (14), the formula derived from the simple Lorentz theory, in which c4/c3 is replaced by -4. If we substitute (37) in (36), we can rewrite the term in In in the form (ca/c6)2(ap/1v$)

( k ( T - Tc)/a')(3b1-k2bd.

We can also use (37) to rewrite the other terms of (36) in alternative forms; we can disregard the terms of (37) involving the b's in terms which are small of the first order. When we do this, we find the alternative formula in place of (36), A I' = -NkT In[(e/A7h3)( ~ k T ) ~ ( 2 43 ~/a) (,V(kT)'/~'))t(3b1+2b2) -t( C 4 / ~ 0 G 6 ) ' ( ~ ~ ' / 4 ~ ~ ) ~ T~ (~~) -( 3 h +2 h ) (~,/to~,)'(NQ1/16~')[bt(Pd+ P:+Pz') 2bZ(P,2P,?+ P.PPZ+ P2P,2)].

+ +

+

(38)

This form of the expression is chosen to make comparison easy with Devonshire's formulas. We get agreement with the terms linear in the b's, in Devonshire's Eq. (10.13) (correcting for certain obvious misprints in his equation), if we set his 0 (Lorentz factor) equal to

p= - c r / ( r o c ~ )= - (4rcr)/c6(Gaussian units) =5.09(4r/3). (39)

VI. FREE ENERGY AND ELASTIC STRAIN

The free energy as we have computed it disregards the elastic energy; it is calculated on the assumption that the crystal does not deform when it polarizes. As we have pointed out earlier, this does not lead to the correct formula for the polarization below the Curie point. If we set E,=O in (3.9, solve for P, (assuming that P,= P,=O), and use (37), we have

to

Lac4

as compared with (15), where we have the factor 3 in place of -c&'/c3c4. We find - C ~ ~ / C ~ C ( = 1.93; in other words, the relation between polarization and ternperature is not very different on the correct treatment from the value given in the elementary theory, so that the disagreement between this function and experiment is as bad as that discussed in Section 111. We have already mentioned that, as pointed out by Devonshire, this disagreement is removed by supplementing A P, as given in (38), by additional terms in the elastic strains, and terms involving both strains and polarization components, which therefore are responsible for the piezoelectric effect. The partial derivative of this free energy with respect to one of the strain components then gives the related stress component; if we set the stresses equal to zero we get in this way a set of equations from which we can solve for the various strains under conditions of vanishing stress. We substitute these values in the expression for free energy, and have finally a free energy as a function of polarization components, for the condition of vanishing stress. We find that the resulting correction terms are of the fourth-power in the polarization, hence modifying the fourth-power terms in (38), and changing completely the relation (41), and thus the polarization below the Curie point. Since our argument a t this point follows exactly that of Devonshire, we merely refer the reader to his paper, particularly his Eq. (9.1), giving the free energy in terms of strains and polarizations. To facilitate comparison with his results, we give formulas, in our notation, for various coefficients appearing in his paper, his Eqs. (10.14), (10.15), (10.16):

That is, for these terms, we must use a Lorentz correction 5.09 times as great as given by the elementary theory. On the other hand, in the terms independent of the b's, leading to the value of the Curie temperature, we get agreement with Devonshire's values if we use a value of fl equal to p= -c4/(e0c3) = - (4rc4)/cr(Gaussian units) = 16.4(47/3), (40) thereby bringing about agreement between our Eq. (36) Devonshire has made some attempt to compare the and Devonshire's (10.17). In other words, no single modified Lorentz factor will take care of the whole iiumerical predictions of the theory with experiment, and it is obvious that the present paper, making very correct Lorentz treatment, as we have developed it.

2.6 The Lorentz Correction in Barium Titanate - J. C. Slater EVAPORATION OF 71NC AND ZINC OXIDE

large changes in some of the coefficients, will invalidate most of that comparison. We shall not attempt a t this time, however, to make an alternative comparison with experiment. There are two ways in which this comparison can be made. First, the observed Curie temperature permits an evaluation of q2/a, and the temperature dependence of the dielectric constant gives us (3b,+2bn) la?. We have already considered the resulting values of these constants, and have shown them to be of reasonable order of magnitude. To check them better we should have to have an elaborate study of the structure of the crystal from the standpoint of atomic theory, in order to be able to compute the interionic forces. Devonshire has given such a theory in a rather elementary way, but the writer believes that a more elaborate treatment would be necessary before the results could have great pretensions to accuracy. On the other hand, the behavior of the polarization below the Curie point gives information about the coefficients of the fourth-power terms (and sixthI'ower terms, which Devonshire also has to introduce),

I

79

761

but modified by the contributions resulting from the elastic strain. If the elastic and piezoelectric constants of single crystals were known accurately enough, we could evaluate these modifications, and then we should be able to find the b's independently from the fourthorder terms, as well as from the second-order terms, and hence have a valuable internal check of the theory. Unfortunately, these elastic and piezoelectric constants are not known su5ciently well. It is to be hoped that future experiments will supply this missing information. I n the meantime, it can be said that there does not seem to be anything about the present form of the theory which does not have a good chance of agreeing with experiment, when better experimental data are available. The writer is greatly indebted to his colleague Professor A. von Hippel for stimulating his interest in the problem, and for useful discussions; and to Drs. W. Shockley and P. W. Anderson of the Bell Telephone Laboratories, for valuable exchange of information regarding the work on the subject of those Laboratories.

80

I

2 Selected Early Work 1927-1961

JOURNAL

OF THE PHYSICAL SOCIETY OF JAPANVOL

7, NO 1, JAN-FEE, 1952

Phase Transitions in Solid Solutbn~of l?bZrOs and PbTiOs (I) small concennatione of PbTiOa By Gen SHlRA6E and Akitsu TAKEDA lbkyo Inslitule of Tsehnology. Oh-okayanur. Talcgo (Received Marell 2s. 1961) The dielectric, ealcjrimctric and dilntnmetric mwumments have bee11 made on the solid aolutiom F’b(Zr-Ti)Os which contain d l amounts o(: PbTiOa lam than 10%. E d d a the ordinary a r b pobt nesr 220°C, the existence of another transition, for example 140W in Pb(Zr95-Tib)08, waa confirmed. It seem8 reasOnable to interpret these two transitions phase changa from punelectric state tu termelectric one and further to antifermektrio one. Whereas the uppr trannition temperature ia marly coIlrtant for tlm concentrations, tlm lower trmition temperature increasta with decreasing Ti mncf!ntratbn snd both temperatma seem to coincide at pure Pbzlos. Therw resulds suggest tlmt Pb7dx m a y be antiforroe!ectric below its Curie point 220°C.

curie point previously reported by Waku and Hori through their permittivity measurements. In part I these phase transitions in solid solutions near the pure lead zirconate side shall be reported. Investigations on the whole range of d i d solutions, especially those on the crystal structure will be reported in part 11.

5 1. Introduction Whereas it is certain that lead titanate’) is a fermlectric of BaTG t y e ) , lead zirconate is not at all a ferrq&ctric but may surely be antifemelectric, for recent investigations3Jhave established many essential differences between the two materials all Wig in support of the above statement. Moreover, though the both are belonging to a tetragonal modification of perovskite structure,4Jthe signs of tetragonality of them are indeed in the opposite sense, namely c/u= 0.988 for lead zirconate while C / U = 1.053 for lead titanate. It is worth investigating whether these two characteristic contrasts, dielectric and structural ones, do imply any inner relationship to each other. And it may be desirable to inquire whether some peculiar properties are to be found if solid solutions were to be formed between the two. Fortunately Waku and HoriS) found in 1949 that these two materials really form solid solutions. Rut since they believed at that time the components are both ferroelectrics. they paid no special attention to the most interesting region, tliat is the very regio? wherein the tetragondlity of tlte niixed crystnl npproaclics unity. Now we have carried out detailed invesligatioris on specific heRt, thcnnal expansion and crystal structure as well as dielectric properties over the whole concentration range of this binary system,and we found in solid solutions conraining small amounts of PbTiOa the existence of another transition besides the ordinary

2. Permittivity Measurement. The specimens were prepared from reagentgrade PbO, ZrOy and TiO,. The ingredients were mixed in the desired proportions and then sinkre3 at about 1200°C after preliminary calcination. No special precaution, such as deso ribed by R ~ b e r t s , ~was ) paid for preventing the evaporation of lead oxide during the sbtering process. The dielectric test samples, 0.5cm’ in area and 0.1cm in thickness, were coated on both sides with a conductive silver paint, which was then fired at about 600°C. The capacity of these specimens was measured by a resonance method at a frequency of lMC/sec and a field strength of about 10V/cm. Fig. 1 shows the dielectric constant versus rising temperature curves for PbZrOn,PbiZr95Ti5)Q and Pb(Zr90-Ti10)Oa. With increasing I%Ti03 concentration, the Curie point of these solid solutions at first slightIy decreases from 220°C of pure PbZrO?’)oJ and then increases towards 490°C for pure PbTiO2). In Pb(2r97.5-Ti2.5)O3, another small peak was observed at 180T basides the sharp maximum at the Curie point. This small peak corresponds to another transition as shown in the 6

2.7 Phase Transitions in Solid Solutions ofPbZr0, and PbTiO,

-8-

6

,yx)

.

LOf L \ b I ~ W A (0

RLh-LdJa

‘.‘Aw J

J

-W

I

I

81

fi:

polarization from theae fiw. Fig. 4 shows the maximum polarization PIW the r e m a n e m polarization P, and the spontaneous polarization P. a t rising temperature. The residual polarization above the Curie point may be accounted for by the conduction term a t these relatively high temperatures. So we assume that the broken line shown in the figure can be interpreted as a superfluous effed due spontaneous

(P #

L

170°C

(Vol. 7,

2. Though these hyetereeis loop do not show sutficient saturation at this kld strength of 17KV/cnr, yet we can roughly estimate the

Q)

0

Shurane, A. Takeda

k A k h TAKWA ~

2mo

R”nq

- G.

-w(,-(>

“1000

Fi;, t’ 1 [yntcrclrria Iwqw ( d 1’b(Zr9&l‘i5)02 15,m+~ =: 17 ~W/CUI.

53. Hysteresis Loopa. Dielectric properties of these solid solutions were also studied from the temperaturedependcnce of hysteresis loops using a method sliglitly modified from Sawyer and Tower’s”. All specimens are of nearly the same dimensions, I.0cm3in area and 0.1 cm in thickness. Meaw e m e n t was made in a liquid paraffin which Waa bzing stirred VigorouSl~in order to obtain

ai;

riaiii:

ttllll~x?r:lLclN!.

It must be noticed that a smiill pciik at 140°C i n the permittivity curve is just corresponding to‘ the appearance of the spontaneous polnrization at this point. This result is to be mnipsreJ with the case of pure PbZrO.,’), of which the hysteresis loops are almost always linear helow the Curie mint. The hysteresis loops of Pb(ZflO-TiLO)O, show the ferroelectric c h x a cteristic even a t mom tmperature (Fig. 3). This fact suggests that the lower transition may

82

I

2 Selected Early Work 1927-1961 7

19.52)

mw

m a

llO0a

Fig S Hptermis loops d Pb(7&Ti10)0~.

$4.

Ln~=20/KVm.

Efftct of D.C. Biasing Field an the Permittivity.

ICI

2m

IY)

~9

Tugrraturr W )

A study was made of the effect of a D.C. biasing fmld of 10KV/cm upon the dielectric constant of Pb(Zr95-Ti5)Ox with the reeult shown in Fig. 6. Above the lower transition t e m p r a t m of 14OoC, remarkable decrease of permittivity waa observed and at the same time the Curie point of 215'C was raised by about 5OC. These phenomena are quite similar to those f o d m BaTiGa)9), and, combined with the data of the spontaneous polarization, show therefore that the intermediate phase of this material between 140°C and 215OC is surely a ferroelectric one.

Fig. 4. Maxiinurn polarization PO,rernaormce p l a r h t i o n P, and spontnnecw polarimtlon. P, d Pb(Zr%-Ti5)OJ at rising tampcrature.

L Y)

I

I0

I

I50 Temprdurr

I

m (.II

I

zn

Fig. 6. SponLenmrr plnrkmrtion of Pb (LrUi~Ti6)0, a t viiryinx toinwature, l1eatin3 and cooling rnte bcinx about I°C/rnin. OOCUT below t h w o o m temperature for this

*ante.

On the other hand, below 140"C, the only effect of biasing fEld is to shift this transition point towards lower temperature by about 5°C.

2.7 Phase Transitions in Solid Solutions ofPbZr0, and PbJiO, - G. Shurane, A. Takeda

a

8 Akhm TAKEDA

Glen 6-

Fig. 7 b w s the dielectric constant ofP b Z a I Pb(Zr95-Ti5)Q and F'b(Zr9o-TilO)Cb measured at mom temperature (15°C) and at 1MC/ em versus the biasing voltage. obeervd decrease of permittivity in Pb(Zr90-TilO)O~ is in accordance with the hysteresis loops of this substance shown in Fig. 3.

-I

I

I

1

\\

U)

sI

(Vol. 1,

.-3

0

uc.C;s&..e

15

plA \xr/cm\

Fig. 7. PerrniLtiviLy of PbZrO,,

20

Pb

(~r~~-i-Tij)OlAndPb(%rDO TilU) 0) vmtm D.C. b i i n g voltage.

5 6 . Specific Heat.

____---

the jump of spontaneous polarization. With the observed value of dTc/dE and the transition energies, we get for the change of spontaneous polarization APs= 10 microcoulomb/cmzat 140°C and dPs=-15 rniaocoulomb/m* at 215°C. These results are in tolerable agreement with direct observation on the oscillograph and suggest' that the spontaneous polarization surely be absent below 140°C. Thus the above experiments have shown that in the lowest phase below I4O0C, Pb(Zr95-Ti5) 0, may not have spontaneous polarization at if ,ii were trulf t h e phase should necessarily b e eithq paraelectric or antiferroelectric. Now as will be shown in the phase diagram (Fig. 13), this lowest phase has to be coitinuously connected to the phase of pure PbZrO,below its Curie point, and this, clearly, mest inevitably be either ferroelectric or antiferroelectric. Hence it seems reasonable 10 assume that the lowest phase of abovesolid solutions, and conscqriently also that of Pb%rO;r itself, may belong to the category of antiferrocleclrics, namely, in which each dipole is certainly subject to some restraint but its alignment never amounts to a net polarization as a whole. Fig. 9, shows the specific heat curve of Pb(Zr97.5-Ti2.5jO:,, in which the lower transition was'observed at 180°C in the permittivitv measurement. As might have been expected, two peaks were really observed in this curve, and the transition energies are estimated at 130 cal/mole (at 180'C) and 160 cal/rnole (at

In order to obtain the more detailed information concerning the two phase transitions in F'b(Zfi5-Ti5)O3, we have measured the specific heat of this specimen. Use is made of an adiabatic calorimeter of Nagasaki-Takagi typelo), which is an iinprovcmmt of Sykes' onc. I'nwdered specimen of about 25 g, contained in a d a s s vessel of 2.5 R. wits heated by ;I r a t c of about Z"C/min. Specific heat versus rising temperature curve of this specimen is shown in Fig. 8. This cllrve shows a small hump at about 140 C besides a sharp maximum at 215"C, and the transition e m v i e s are estimated respectively at 83 c:II/ mole and 180 cal/mole. If we assume tentatively these transitions are of the first order, the Clnpeymn-Clausius equation should be available at these points, i.e. -dTc/dE= Tc-APS/L were TC denotes the 22OOC).

83

1 :..A

-u

-1

I

84

I

2 Selected Early Work 19221-19621

Phars

1952)

Il).cmriPionr

'sub Y)

t

1

I

in Wfi-TtY, (0

9

I J

Fig. 9. Bpeoiba best d Pb(W.S-TIB.S)@

4 6. Dielectric Reddual Polarization. Though above ercperiments. espcially the observation on an millograph, have &own that the spontaneous polarization of Pb(ZS-Ti5)Oa really disappearsbelow W C , leading u to the amurnption of antiferrmlactridty of the bw-t pbak, yet the similnt-phenomw WaJd be obeerved if, alternatively, the coercive &Id were to inmeam auddsntJr at llo°C leaving the muinsic p&rizatim mchmgd. To uiticize the latter poesibility amn directly, we have carried out the following experiments. It is well known that BaTi4 ceramicscan be permancntiy polarized by an adequate input of high D.C.tieldo). So, for the m s a u e p e n t afthe residual p o & i d Pb(Z1S-Ti5)oa. we have& the 8ame specimen as that for the obeervaLian of hyatereaia loope. At 160°c, which is in ferroelectric region, both electrodes of the specimen were at first oonnected to a biasiag D.C. voltage of 10KV/onfor 5 minutea and then to a galvammter drcuit. After an initial discharging cwnmt had disappeared, we raised the temperature at a coDletant rate and found a h a r p maximum of discharging current at the Curie point, as shown achematically in Fig. 10(a). Thb phenomenon is quite similar to that obeerved in barium titanate ceramics.") After the same polarizing treatment at 160°C, now the temperature was lowered at the same rate as in heating. We observed,' in this case bo. a discharging current shown in Fig. 1O(b) a h a rather &it marhum at W a c . which -da to the h e r tramition point on -(Fig. 5). What is to be noticed hsn in tbe time mlegral of discharging currant d'hmting is nearly tk-mme order of magni-

4 t

I

1

ia

19

Tns.dmvr\l)

Fig. 10. &hematk &m ktg and -kbg

I

Z

W

'

of dieohsrgCarrenb d

Pb(zrsJn6pawharthetampastme is r a i d or amled by R

-

coastont mtc.

(a),

(b) dischsrgiw current

afkr polariing beatmetit at

160°C. (a> ahorbing and diac h a m anrrmt without pols-

r i h g timatmat.

In the c o r n of above experiments, we found the following intenstine phencunenon. Once the specimen is highly polarized in ferroelectric region, it begins to show an absorbing and discharging currents as shown in Fig. 10 [ c ) , even after it is heated until above the Curie point. The currents are however not IarKe and b m e the less pronounced a8 the more frequently m the heat treatment repeated. This Phenomenon mumably m a y be due to som mall internal strain perhaps made up initially by a polarizing field and Wing afterward until above the Curie point. If it is cooled again,

::/

2.7 Phase Transitions in Solid Solutions ofPbZr0, and PbTiO, - G. Shurane, A. Takeda

~~~kndtopdruitstimepaciman bo,,@@,~dbsalion -wmitatim .

anrJIlsianhsmtyetbssnM,iteeane that,the olreerved curienta in fieun (c) am iWWdQWdY-

totanprahMdsri-

*

v&vu of the valua of Epanmmm ~laiiGathn dmwn in Fig. 5.

I7. ! r i w m d Ihrpmdon. Linear thermal expansion of l'b(Z195-Ti5)ol wan measured at tempaatureafram -150'Cto m*c, and the d o u s Mlume ChangeB of thia epecimen give a further support for the

d i d i t y of the aseumptiOn pmpoaed above. Ditometric teat samples were cut out from ceramic disks and have cylindrical form of 3 an in length and 0.5 an in diametar. Werential dilatation of this spedmen agah& eili tubewas magnihdabout1OOOtimeaaslarge by IneaIm of an optical hvsr. Fa.11 &owe (Z--h)/has a functim of tanmature, Ghen I0 ie the length of ebe epecimenatOOC. Heating and eodig rate is about I"C/min. Thia figure aham large anomaha volume dmngm at about 140% and 215°C. Large thenna hyetereais on abling i e to be cunpared with the cdcrea p o d i Meresin of spontaneous polarization (Fig. 5).

-Joe

0

Kx)

200

w,

TlnpratutcvCI

Fig. 11.

Ltnear tbermal expamion of w

-comparieon of

this result with the volume changes obeerved in PbZr41) and J3aTiOl*2)~~) cbdyebowathatthesignsofvolumechanges at the two tramitions are quite consistent with the aermned eharaaeristh of the three phasea

y.u.4 '

&XO,

1 -

0 .

I

I

85

86

I

-

2 Selected Early Work 1921-1961 1952)

ma38 l).lluibionr

Pb(*nfi

(0

11

turmsre tbrn thaw, of orones by .bout lV-WC. So we think that if we had been -a to avail ouralves of ZrO, of high purity all this experiment, ,we would be to o b t h all phase boundaries higher than thoee in Fq. 13 each by about l(P-15"C. In conclusion,we wish to express our sincere thanks to Profasor Y. Takagi for his kind guidance in the cou~geof this research and t8 Mr. C. Kanzaki for taking troubles of chemical analysis of pure ingredient. We are aiso grateful to Mr.J. Hori for stimulatingour intereat in this problem and to Mr. E. Sawaguchi for his helpful diacusibns. References pig.

d the P b m ptJriotsgs(sm.

IS. Rhas

border line may +hap

be between 7-7.5 in

Ti amcentratloo at3hs room temperahwe, for a study of thsdielectricpropsties of PbCZ192.5Ti7.6)oI shown that it ie already ferroslectric at room tanperatwe. An X-ray study of the CIyBtal atructurs of this system performed at rooan temperature aIs0 dram a line of danarcation of Werent phaees at t h e crmpoeitioaa. Detaile Of X-ray study will be d d in part 11. For the study of t b m eolid solutions, ape m is the puity of Zro,, cially of i w h i c h d l y o c a t a i n e si and Ti as impuritioa. Spectroscopic analysis of our ZIQ* chows the existence of a few percents of Si as impurity. Effectof thin impurity on the phase d m was examiued from a preliminary etudy of dielectric properties of PbZzQ and Pb(Zr95-Ti5) 01 prepared from 2x0, of high purity which contained as impuritb only small amounts of Ti(0.21%), Si(O.o9%] and Fe(O.W%J The phase tramitiom obeerved in these specimezle are completely of the same nature as described above except that all the transition tempera-

1) Q.Bhir~u~eaodS. Houhiao, J. Phga. Bac. Japan. 6 (1961) 266. 2) Sea, for imtanoe. A vm Eippl, R e v . Mod. P b . 22 (1860) 221. 8) E. &m@, G. 8hSr8ne and Y. Taaei J. Pbp. 8oa 6 (1961) 885. 4) ED. Ma34w. P m Phjn. 8oa. W o n . 68 (1946) 19s. 6) W ~ L Usod J. ~ o r i reported , at the a m miiteator ths PieaOelectr10 Mataid, Tokyo, Mny 1,

a

1850.

6) 8. Rakata, J. Am. &ram. h., SS(1960)63. 7) 0. Smwyex and 0. T o m , Php. hv., 35

(laso)

289.

8 ) 8. Bober& Phys: Rev., 71 (1947) 890. 9) LE (kor& A.T. Dermieon. XM. N m h and R. wbiddrnsto * n,Namra,163(1940)6S&. A 10) 8. N@I and Y. Takagl, J. App. Phys. Japan, 17 (1948) 1C4. 11) E. 8swagwhi pDa T. Akioh, J. P b p SOC. Japm, 4 (lQ49) 117. IS) 8. Sawn& and Q. Bhiraqe, J. Pbp. Boo. Jspaa, 4 (lQ49) 62. 13) Q. 8hlram ard A. Tatada, J. Phga. SOC. Japan, 6 (1961) 128.

2.8 Dielectric Constant in Perovskite Type Crystals - J. H. Barrett 187 PHYSICAL REVIEW

VOLUME 86.

NUMBER

APRIL

1

1.

1952

Dielectric Constant in Perovskite Type Crystals JOHN H. BARREIT* Rice Instilute, Houston, Texas (Received December 26, 1951)

Slater’s theory of the dielectric constant in BaTiOa has been extended by treating the ionic polarizability quantum mechanically instead of classically. This leads to an expression for the dielectric constant which is good at all temperatures and shows a deviation from the Curie-Weiss law at low temperatures. The theory is applied to SrTiOa and to KTaOi above its transition at 132°K.

INTRODUCTION

LATER has proposed a theory’ to explain the dielectric behavior of BaTiOa, a t least above its Curie temperature. His model treats the crystal as though the Ba and 0 ions were fixed in position and as though each Ti ion acted like an independent harmonic oscillator with small additional anharmonic terms. The Ti ion is supposed to be very loosely bound, and the only interaction between these oscillators is through the electric field. The system was then treated by statistical mechanics in a completely classical fashion. I t seemed desirable to find also the results of a quantum-mechanical treatment, particularly a t low temperatures.

S

I. DERIVATION OF THE QUANTUMMECHANICAL FORMULA

Following Slater, let the potential energy of the Ti ion in an electric field be given by #J=

~(.”+r2+22)+b1(3C4+Y4+24)+2b2(.”Y2+.”z~+y~z*)

-q(xEz+~Eu+zEJ. (1) Let the part of independent of the b’s be called #J1 and the part containing the b’s be called & The #J2 will be treated as a perturbation. Using #Jl as the potential energy in the Hamiltonian operator, the problem is just that of a simple harmonic oscillator in an electric field, and the energy levels are given by #J

Wlnrn0=(I+m+n+$)hv-q2,?P/4a. (2) These energy levels are degenerate, so that the first-order perturbations to the energy levels must be obtained by solving a series of determinants. For any group of levels having the same unperturbed energy, the sum of the energy perturbations is equal to the sum of the diagonal matrix elements. Assuming that the energy perturbations are small with respect to kT, one can use the approximation exp(-x) = 1 --x in computing the partition function. The justification of this approximation will be considered in Sec. 111.Let the first-order perturbation energies be denoted by Wlmnl and the diagonal matrix elements by (ZmnI&?IZmn).Then, using the approximation for the exponential given above, it can be seen that the diagonal matrix elements may be used 1

Holder of Gen. Elec. Charles A. Coffin fellowship. J. C. Slater, Phys. Rev. 78, 748 (1950).

in place of the true first-order energy perturbations to compute the partition function. The diagonal matrix elements are given by

(CmnI $2 I Zmn) =b1((3/4PL)[(21~+21+1)+ (2m2+2m+ 1 ) +(2nz+2n+1)l+(342/4u2P)C(2C+ 1)E3 (2m+ 1)E,2+ (2n+ 1)E21 (qY16u4)CE2+ Ei+E,41) +2b~((1/4P2)[(21+1)(2m+l)+ (214- 1)(2n+ 1) (2m+1)(2n+ 1)1+(q2/8a28)C(21+ 1)(E,2+EdL)

+ + +

+(zm+l)(E,2+E,2)+(2n+l)(E,2+E,2)1

+(g4/16u4)CE2E,2+E2E,2+E,2E,21),

(3)

where P=2a/hv. Let B be the partition function for a single oscillator, and let Z = z N / N ! be the partition function for a system of N oscillators. The polarization ) T , A E = -KT InZ. is given by Pi= - ( ~ A E / ~ E ,where Now one obtains for (YT~’, the ionic polarizability resulting from the Ti ion, ( l / N ) ( P x / & ) = (q2/2a)[1- ((3b1+2bz)h/2a2) Xcoth(hv/2kT)]. (4)

(YT~’=

It should be noted that this formula is applicable only above the Curie temperature in the unpolarized or nonferroelectric state. The above treatment will reduce to Slater’s if kr>>hv. 11. APPLICABILITY OF THE QUANTUMMECHANICAL FORMULA

In order to see how K depends on XT~’=(YT~’/QV for perovskite crystals in general, where v is the volume of the unit cell, one must consider the Lorentz correction as Slater has done in his paper. This will be done here in less specific fashion. Because of the linearity of a set of equations such as his Eqs. (18) and (19) relating the polarization to the field, one will get, K

=A

+B/(1-DXT~‘).

(5)

Slater’s Eq. (26) is a particular example of this relation for BaTiO,. The values of A , B, and D will depend on the values of the electronic polarizabilities of the various atoms in the lattice and on the geometry of the lattice. ) be the only temperature dependent The X T ~will quantity in K. 118

88

I

2 Selected Early Work 7927-1967 DIELECTRIC CONSTANT IN

119

PEROVSKITE CRYSTALS

well. From the value of K near O'K, T1 can be computed to be about 60°K. This explains why the dielectric M constant is observed experimentally to begin to deviate K= from a Curie-Weiss law around 50°K. #Ticoth(T1/2T)-To Another ferroelectric to which this might be expected where (6) hv 2&ovB to apply is KTaOs above its transition to the ferroelecTi=-, M= tric state. Hulm, Matthias, and Lon? have reported the k qz(3b1+2bz)kD' behavior of the dielectric constant of this compound a t low temperatures. The behavior in this case is like that of curve (c) in Fig. 1 down to the transition temperature To= 2a8sv (Z-1). q2(3b1+2bJkD 2aco~ of 13.2"K. The graph given in their article can be fitted fairly well by Eq. (6), but exactly how well The behavior of K as given in Eq. (6) is shown in Fig. 1. cannot be determined merely by taking values from For T>>T1,+TIcoth(TJ2T) asymptotically approaches the graph.t T and (6) becomes a Curie-Weiss law. As T approaches I t also seems of interest that the behavior of O'K, K will approach M/(+T,-To). Roughly, it may LiT1C1HnOs.H20, reported recently by Matthias and be said that Tl is the dividing point between the low Hulm,' follows something of the nature of Eq. (6) above temperature region where quantum effects are im- its transition to the ferroelectric state. Of course the portant and K deviates from a Curie-Weiss law and the theory given here could not be expected to fit this case high temperature region where a classical approximation because of the entirely diilerent crystal structure. and a Curie-Weiss law are good. If a material undergoes However, any model having discrete energy levels will a transition to the ferroelectric state a t a temperature cause K to level off near absolute zero, because the above T1,the quantum effect will be unnoticed. system settles into the ground state and thereby loses Now we shall see how this theory applies to BaTi08. its temperature dependence. Using Slater's value of a= 7.6na ev/Az= 1.2X lo%* joules/mz and m as the mass of the Ti atom gives III. JUSTIFICATION OF THE APPROXIMATION TO THE EXPONENTIAL T1=410n degrees absolute, where n is now the number of electronic charges on the T i ion. Since the CurieI n Sec. I exp[(Zmn( q5zlZmn)/kT] was approximated Weiss law holds in BaTiO, down to the transition as l-[(lmn1q5*(Zmn)/RT]. The validity of this must temperature at about 390°K, 12 cannot be much above 1. be examined after the values of certain constants are Consider the values of the constants in Eq. (6) in determined. Assuming bl= bz= b and using the formula the case of BaTiOa. Experimentally M = 1.5X106, T0=39O0K, and Slater has computed] B=1.93. This gives 2ascov/[q2(3bi+ 2bz)kDI- lo6, (7) (Dq2/2~co~-1)-4X lo4. Combining (4) and (5) and noting that A is negligible,

The last quantity can be said to be zero, so that a=Dqz/2eov as Slater has given in his Eq. (29) with D = 5.39. Now it seems reasonable that the quantities considered just above, excepting TO,will be of the same order of magnitude for other compounds of the same structure as BaTi03. The Tocan vary quite drastically because it is determined by the difference of two very nearly equal quantities. Nevertheless, it will still be expected that the latter order of magnitude relationship given in (7) will not be exceeded and that a is still given by a= D$/2eov. In particular SrTiO, might be expected to behave in the same manner as BaTiO3 with a different TO.Hulm2 has reported the dielectric constant of SrTiOs to follow a Curie-Weiss law down to about 50°K. He found M=8.4Xl(r, To=-35°K. Below 50°K the dielectric constant falls below the Curie-Weiss law and approaches a value of 1300 in the neighborhood of O'K. A curve of the form (a) in Fig. 1 seems to fit these facts quite f

J. K. Hulm, Proc. Phys. Soc. (London) A63, 1184 (1950).

I

T1

T

FIG.1. Curve a : To= -0.25T1;Curve b : To=O; Curve G: To=0.25T1.

* Hulm, Matthias, and Long, Phys. Rev. 79, 885 (1949).

t Note d c d in proof:-The author has recently obtained by private communication precise measurements of the dielectric constants of KTaOt from Hulm, Matthias, and Long and of SrTiO, from J. F. Youngblood. Equation (6) fits the data quite well both for SrTiOs and for KTaOI above 13.2'K. The value of TIobtained by fitting Youngblood's data is about 100°K. However, this higher value does not appreciably affect any of the results of this paper. 'B. T. Matthias and J. K. Hulm, Phys. Rev. 82, 108 (1951).

2.8 Dielectric Constant in Perovskite Type Crystals - J. H. Barrett 189

120

JOHN H. BARRETT

IV. DISCUSSION for a and the definition of M given below Eq. (6), b/a2=B/5Mk. Now it can be seen from Eq. (3) that We may expect the quantities involved in the theory the validity of the approximation in question depends -excepting To-to be roughly of the same magnitude on b/TkT<
90

I

2 Selected Early Work 1921-19G1

that in the transition region of ammonium chloride. At the same time, the absorption spectrum due to the SO, tetrahedra shows 6. T. Matthias, C. E. Millecj. P. Remeika a similar rearrangement. One could thereBell Telephone Laboratories, Murray Hill, fore assume that an ordinary ammonium Newjersey, Phys. Rev. 104, 849 (1956) transition acts as a trigger to induce a moment in the SO, groups. The mechanism would be thus strongly reminiscent of the (ReceivedAugust 24,1956) one causing ferroelectricity in KH,PO, as described by Slater [GI. This point of view We have discovered that glycine sulfate, suggests a formal similarity among the (CH,NH,COOH),H,SO, [l],and its isomor- ferroelectric sulfates thus far encountered phous selenate are ferroelectric. The Curie regardless of crystal structure or water of point of the sulfate is 47 "C and that of the crystallization. The role of ammonium is selenate is 22 "C. As in all ferroelectrics played by guanidinium in GASH, by methwhich are not cubic above their Curie points, ylamine or urea in some alums, and by glythere is only one ferroelectric axis. For gly- cine in the material here reported. cine sulfate at room temperature the sponWe want to thank A. N. Holden for his taneous polarization is 2.2 . lo4 coul/cm2 critical reading of this paper. and the coercive field is 220 v/cm. It is tempting to regard this result as confirmation of a general mechanism operative References in such ferroelectric materials as ammonium sulfate [2], and with it the same for 1 NICKLPS,Compt. rend. trav. chim. 1849. the guanidine aluminum sulfate and its 2 MATTHIAS,B.T., REMEIKA.J. P., Phys. Rev. isomorphs (GASH) [3] and some of the 1956,103, 262. 3 HOLDEN,MATTHIAS,MERZ,REMEIKA,Phys. alums [4]. Rev. 1955,98,546. In the infrared absorption of ammonium 4 PEPINSKY, JONA, SHIRANE,Phys. Rev. 1956, sulfate [5]there are two characteristic fea102, 1181. tures below its Curie point. The 3.3 p band 5 POHLMAN,R., 2. Physik 1932, 79, 394. due to the NHd shows a change similar to 6 SLATER,J . , J Chem. Phys. 1941,9,16. 2.9 Ferroelectricityof Clycine Sulfate

2.70 Switching Mechanism in Trglycine Sulfate and Other Ferroelectrics- E. Fatuzzo, W. J. Merz P H Y S I C A I,

R E V I I< W

VOI.UMIi

116,

NIJMBER

1

OCTOBER

1.

I

91 I959

Switching Mechanism in Triglycine Sulfate and Other Ferroelectrics E. FAT~ZZO AND W. J. MERZ Labwalorics RCA Limiled, Zurich, Switzerland (Received January 9, 1959;revised manuscript received June 19, 1959) The reversal of the spontaneous polarization in a ferroelectric crystal is governed by two mechanisms: the nucleation of new domains and the growth of these domains by domain wall motion. We have investigated the switching properties of triglycinc sulfate (TGS) as a function of applied electric field, temperature, and thickness of the samples. I t is proposed that at low fields nucleation is the slower inechanisni and hence dominates the switching process while at high fields domain wall motion determines the rate of switching. The former process leads to an exponential dependence of switching time on applied electric field and the latter to a linear dependence. A model for the nucleation and domain wall motion is treated mathematically and is compared with experimental observations. The shape of the switching current pulse was found to yield much information. 'The shape depends strongly on the applied electric field and is correlated with the nucleation time as well a s the domain wall motion time. The asymmetry of the pulse increases with decreasing field and can be associated with the interaction between domains and domain nuclei. This interaction in various ferroelectrics is discussed and its relation to the switching is considered.

1.

INTRODUCTION

NUMBER of investigators have dealt with the problem of determining the mechanism by which the spontaneous polarization in a ferroelectric material is reversed. I\lerz,l.*Little: Wieder,' Landauer-YoungI)rougard,6*EChynoweth,' Miller,* and Burfoote have examined this process in BaTiOa while similar studies have been conducted on Rochelle salt (RS) by WiedeP

A

W. J. Merz Phys. Rev. 95,690(1954). W. J. Merz: J. Appl. Phys. 27,938 (1956). E.A. Little Phys. Rev. 98,978(1955). ' H.H.Wiedkr, Phys. Rev. 99,1161 (1955). 61.andauer, Young, and Drougard, J. Appl. Phys. 27, 752 ( ' 19.56). R: Landauer J. Appl. Phys. 28 227 (1957). ' A. G.Chyndweth Phys. Rev. h 0 , 1316 (1958). R. C. Miller, Phyb. Rev. 111, 736 (1958). J. C.Burfoot, Proc. Phys. SOC.(London) 73,641 (1959). loH.H.Wieder, Phys. Rev. 110,29 (1958). 1

on guanidine aluminum sulfate hexahydrate (GASH) by Pruttonll and WiedeP and on triglycine sulfate (TGS)by Pulvari and Ruebler.ls From these studies it is generally concluded that two steps are involved in the switching process: nucleation of ferroelectric domains at the surface of the sample followed by growth of these domains through the crystal by domain wall motion. I t is assumed, on the basis of phenomenological arguments, that nucleation of domains is a statistical process while the domain wall motion is assumed to be controlled by a kind of viscous drag. In this paper we shall discuss the switching properties of triglycine sulfate (TGS), a ferroelectric discovered l1 l2

l3

M. Prutton, Proc. Phys. Soc. (London) B70, 1064 (1957). H.H.Wieder, Proc. Inst. Radio Engrs. 45, 1094 (1957). C.1'. Pulvari and W. Kuebler, J. Appl. Phys. 29,1742 (1958).

92

62

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2 Selected Early Work 79221-7967

E. FATUZZO A N D W. J . MERZ

field, while the switched charge remains constant. This charge is equal to : I/

Q.= 2P,u= t A , , ~ j , (1) where P, is the spontaneous polarization, u is the electrode area and j is a factor of the order of 0.5 to 1.0, which allows for the shape of the i.-t curve. If f remains constant,..,i and 1/t, show exactly the same dependence on the magnitude of the field.

ts------

FIG. 1. Observed switching current I,U C T ~ Wtime I in a ferroelectric crystal. Symmetry m is arbitrarily defined as t'/t".

2.1. Field Dependence of the Switching Time in

Triglycine Sulfate (TGS) by Matthias, Miller, and Remeika" and whose dielectric In Fig. 2, l / t r is plotted as a function of applied properties were investigated by Hoshino, Mitsui, Jona, field E. The curve can be subdivided into 3 parts: a Pepinsky,16 and by Pulvari and KueblerJa TGS possesses a low coercive field and a relatively large first part (I),which is curved, a second part (II), which spontaneous polarization which makes it highly is less curved and a third part (111) which is linear and whose extrapolation passes through the origin or very suitable for these investigations. close to it. Between parts I1 and I11 there occurs an inflection point (see Pruttonl8 on the polarization 2. EXPERIMENTS The switching in TGS was studied by the conventional technique in which one measures the current i. which flows through a series resistor to the crystal electrodes as a function of time t. First, an electrical square pulse is used to align all the dipoles in one direction, then a second pulse of opposite polarity is used to measure the switching current as function of time. The behavior of current with time is shown in Fig. 1. The quantities of interest are: i,.,, the maximum switching current; t., the switching time; t', the rise time; t", the decay time; and m=t'/t", the symmetry of the pulse, as indicated in the figure. Three of these t,, and m, are sufficient to define all parameters, imnx, five quantities. All five variables are functions of the applied field, the temperature, the geometry and the history of the sample. For a given temperature the switching time t, decreases and the maximum switching current i,, increases with increasing applied electric

FIG. 2. Reciprocal switching time 1/1. applied field E in TGS.

uer5u(

0.6

0.5 0.4

0.3 0.2 0.1

0 I

1.5

2

25 3 3.5

5

7,s

10

FIG.3. Symmetry m of the switching pulse DCTSUS electric field E in TGS.

reversal process in ferroelectrics). In region 111 there is apparently a linear dependence of 1/1, on E: l/t,= RE. (2) Unfortunately the available voltage range in part I11 is too small to permit establishing this relation unambiguously. Part I, on the other hand, can be fitted over 4 decades by an exponential law (see Merz' in BaTiOs) of the form: W t J = (l/td exp(-a/E), (3) where t o and a are constants. 2.2. Shape of the Switching Pulse in

I4 Matthias, Miller, and Remeika, Ph s. Rev. 104, 849 (1956). Hoshino, Mitsui, Jona, and Pepinsly, Phys. Rev. 107, 1255 (1957).

TGS

An investigation of the shape of the switching current as a function of applied field shows that the symmetry m = t'/t" depends strongly on the applied field as shown in Fig. 3. At a field strength of lo4 In

H. Prutton, J. Brit. Inst. Radio Engrs. 19,93 (1959) ; Fig. 3.

2.10 Switching Mechanism in Trglycine Sulfate and Other Ferroelectrics - E. Fatuzzo, W. J. Merz

S W I TCH I N G M EC H A N I SM

IN TRIGLYCINE SULFATE

I

93

63

v/cm, m is of the order of 0.5. With decreasing field the symmetry is reduced and for very low fields m reaches a value ma which is low but not zero. The dependence of the “rise time” t’ on the field E in the low-field region is shown in Fig. 4. It depends quadradically on E except a t extremely low fields where an exponential behavior can be observed. An exponential behavior is expected for very low fields since mo is then constant and 1, depends exponentially on the field. 2.3. Thickness Dependence of the

Switching Time For field strengths larger than lo4 v/cm a quadratic dependence of the switching time on thickness of the sample is observed, (Fig. 5). In this experiment a voltage pulse of 500 v was applied to samples of thicknesses between 0.005 cm and 0.05 cm corresponding to fields of lo4to loKv/cm. For field strengths smaller than lo4 v/cm no simple relation between switching time and thickness could be found.

2.4. Switching in Steps at Low Fields

If a pulse of duration greater than the switching time is applied to a ferroelectric crystal, it is switched Ipul 1-I

1001

-yrT--,

0 5‘ I - - H i i L p C l l d 05 I 2 10

completely and a normal current pulse is observed [Fig. 6(a)]. On the other hand, if voltage pulses much shorter than 1, are applied there is no net reversal of polarization even upon successive unidirectional switching. The switching transient appears as shown in Fig. 6(b). There is a critical pulse length t* a t which the crystal begins to switch and if a series of pulses each longer than 1* is applied, the polarization can completely be reversed [Fig. 6(c)]. It is interesting to note that in this case the shape of the individual small current pulses fit together and add up to the normal pulse except for the initial peaks A , B, C, D, E , F. The critical time t* depends on the amplitude of the applied pulse: it is a very small fraction of la a t very low fields and almost equal to 1, for fields of about 20 kv/cm (see also Zen’iti e l a1.l’).

2.5. Heating Experiments In order to distinguish between processes which take place a t the crystal surface and those occurring in the ”Zen’iti, Husimi, and Kataoka, J. Phys. SOC. Japan 13, 661 (1958) ; Fig. 3.

FIG.5. Switching time 1. ve~su.rthickness d of the sample. Applied voltage = 500 v.

bulk, the influence of inhomogeneous heating of the sample was investigated. A small heater was placed in front of the electrode of a TGS sample for a time short enough to influence the temperature of the surface only. I t is anticipated that the surface temperature primarily determines the nucleation rate, whereas the temperature of the bulk affects primarily the domain wall motion. At low switching fields (<-lo4 v/cm) surface heating has a large influence on the length of the switching pulse, but less on its rise time. With high switching fields the surface heating has a strong effect on the rise time of the switching transient but not on the decay time (and hence the change in pulse length is not so pronounced as in the former case). At low fields with heating times of the order of a few seconds during which only the surface is warmed, the pulse becomes narrower and higher. Upon cooling

i-I.

Ef

iFIG. 6. Full and partial switching current i. ve~swtime I in TGS. The (b) length of the individual pulse is i.

94

64

I

2 Selected Early Work 1921-1961

E. F A T U Z Z O A N D W. J . M E R Z

the pulse returns to its original shape passing through the same intermediate stages as appeared during warming. If, on the other hand, the crystal is warmed for a period of the order of 30 minutes a pronounced difference is observed between the heating pulse shape and that occurring during cooling. During heating the pulse initially becomes narrower and higher as previously and upon continued heating, the pulse becomes so narrow as to be nearly unobservable. Upon cooling the decay time rapidly returns to its initial value while the rise time remains unobservable. After about 15 minutes of cooling the pulse is once more back to its original shape. This asymmetry between heating and cooling is interpreted to be a consequence of nonuniform heating in which the bulk remains cool while the surface warms quickly, and nonuniform cooling during which the bulk remains warm while the surface cools quickly. Further discussion of these phenomena will be found below. 3. DISCUSSION OF FIELD AND THICKNESS DEPENDENCE OF THE SWITCHING TIME

K , 10, and a of Eqs. (6) and (7), one can fit the low-field range of the experimental curve with Eq. (7) and the high-field range with Eq. (6), as is shown in Fig. 7. Since l/t,= l/(tn+ld), one can conclude that the switching time t. is determined principally by the slower of the two mechanisms (nucleation or domain wall motion). At low fields (region I of Figs. 2 and 7) the rate of nucleation is low so that the switching is primarily governed by the nucleation (l,>>!d) which then leads to an exponential law for the switching time [Eqs. (4) and (7)]. On the other hand, we have to assume that a t high fields (region I11 of Figs. 2 and 7) the rate of nucleation is extremely large so that the switching time is primarily determined by the velocity This then leads to the of the domain walls (Id>!,). linear dependence of l/t, on E, Eqs. (4) and (6), as observed experimentally. Since it appears that a t high fields ld>>l,, then t , must deviate from the behavior described in Eq. (7) at fields higher than 15 kv/cm and must decrease much faster than described by this equation. With this assumption, the fit with the experimental results is very good (Fig. 7).

In the introduction a two-step process was proposed for the switching in ferroelectrics: nucleation of domains and domain wall motion. The experimental results just discussed will be interpreted on the basis of this model. Let us define the “nucleation time 1,” as the time necessary to form all nuclei, from the first to the last, and the “domain wall motion time, Id” the time necessary for one domain to grow through the sample. The total switching time can then be approximated by tEL+td.

(4

We assume that the domain wall motion can be described by v=d/td=pE=pI/’/d, (5) where d is the distance the wall travels and p is the mobility of the domain wall. In Eq. (5) the coercive field strength for domain wall motion is neglected. This assumption agrees with the experimental results. From Eq. (5) it follows that l/td=

pE/d= KE=pV/d2,

(6)

with K = p / d . On the other hand, we assume in our model that the nucleation of new domains is governed by a statistical law, in which at low jields, the probability of forming new domains depends exponentially on the applied field in the following*way: Pn=$o

FIG. 7. Reciprocal switching time I[/*, domain wall motion time l/k, and nucleation time l/6. uclsus applied field E for TGS.

exp(-culE),

and hence (l/ln) = (l/to) exp(--a/E). (7) This particular dependence of nucleation time on electric field fits our experimental results on TGS as will be shown, and further this form is the same as that found for BaTiO3.l By proper choice of the parameters

0

Also the quadratic thickness dependence of the switching time at high fields (E> lo4 v/cm) as shown in Fig. 5 fits our model [see Eq. (6)]. At this point, however, it is important to distinguish between domain wall motion in the forward and sidewise directions. The fact that we do observe a quadratic dependence on crystal thickness indicates that in TGS the forward motion is very much faster than any sidewise motion, under the assumption that the mobility I.( is not field dependent. Before we can discuss other experimental results we have to develop our model further, which will be done now. 4. FURTHER DEVELOPMENT OF THE

PROPOSED MODEL

Some of the properties of the nucleation time I,,, domain wall motion time td, rise time 1’ and decay time I” are summarized in Tables I and 11. I t can be concluded from the relationships described in Tables I and

2.10 Switching Mechanism in Jriglycine Sufate and Other Ferroelectrics- E. Fatuzzo, W. J. Men

SWITCHING MECHANISM

I1 that it is not unreasonable to attempt to correlate the shape of the switching pulse with the two switching mechanisms by identifying 1" with 1, and 1' with f d . It is difficult, however, to understand why the rise time, which precedes the decay time, should be due to domain wall motion, which must follow nucleation. For the reversal of polarization we consider three states for each nucleus: A+lJ-+C

( A ) latent nucleus, not formed yet; number=n,; ( B ) nucleus formed; number= n z ; (C) nucleus grown through the sample; number= n3. 4.1. Transition

TABLE I. Relationship between 4 and t,,.

_____

- -~

Intermediate fiplds

Low fields

h<
TRIGLYCINE SULFATE TABLE

Intermediate fields

t'<
__

95

65

11. Properties of 1' and 1".

Low fielda

t,i%"=to

I

f'g'l"

High fields

could not he measured

exp(u/E) ~~

is made (the pulse is applied at t=O) and represents the time a t which a particular domain is formed. The quantity q is a function of the interval (1-7) and represents the contributions to the total current due to the growth of a single domain. The total current is then

(23)- (C)

For the transition ( B ) - ( C ) we assume that (a) all the nuclei start growing as soon as they are formed; (b) they grow primarily in the forward direction; (c) they all take the same time, t d , to grow through the crystal and (d) the contribution of each growing domain to the observed current is not constant during its growing process. The contribution to the observed current can vary with the position of the growing domain because first, the velocity can be a function of position and _._ _

IN

fdg!,.

t.Et.=to exp(u/E)

~~

High fielda

4 >>L t.gfd=

(KE)-l

__ ~

second the rate of change of the neutralizing charge on the electrode area (which is the measured current) can also depend on position. The latter point is a consequence for example of a conical shaped domain growing with uniform velocity through the crystal. Although alternative models can be employed to describe these experiments, the one suggested above is the simplest one which fits the experimental evidence. If we denote the total number of nuclei, latent, formed and growing, and already grown, by

N = nl+nz+n3, and the fraction already formed by

we can write for the number d n of domains created between the time T and T+&

di, = iV (dvjdt)b,qd7,

where qo and r Zare constants and (1-7) is equal to zero a t the time when the nucleus is formed. By integration of Eq. (13) it is seen that the charge collected on the electrode due to the growth of a single domain increases during growth according to an exponential law. Hence it is necessary to make an arbitrary definition of t d which for our purposes is chosen as td=

2.372.

(14)

Combining Eqs. (12) and (13), we have

(8)

(9)

dn= N(dv/dl)-,dr. Their contribution to the current a t the time 1 is

We assume, like Landauer, Young, and Drougard6 that q first increases and then decreases. One can explain the decreasing part of 9, because when the apex of the conical domain reaches the opposite electrode its contribution to the current will fall rapidly as the angle between the domain wall and the plane of the electrode approaches 90" (Fig. 8). If it is assumed that the decreasing tail of the switching pulse of a single domain carries most of the charge (which would correspond to the observed shape of the Barkhausen pulse'), then it suffices to consider only that part of 7 corresponding to this tail. Since a rapidly decreasing function of any arbitrary form lead to essentially the same conclusion we have taken, for simplicity, the following relationship : I ) = I )exp[O (1- 7)/721, (13)

From this we can calculate the shape of the switching current once we know the rate of the creation of new nuclei. By differentiation of Eq. (9) we obtain Ndv/dl= -dnl/dt,

(10) (11)

where the time 1 is the time a t which an observation

FIG. 8. Figure demonstrating the growth of domains through the sample.

1 1;

.&, A

.* *

+

+

B

96

66

I

2 Selected Early Work 7927-7967

E. FATUZZO AND W. J. MERZ

which with Eq. (15) leads to

obtained by differentiating Eq. (23) and equating it to zero. With Eqs. (14) and (18) we obtain

t’=To evaluate Eq. (17) it is necessary to calculate dnl/dl from the transition ( A )- ( B ) . We have assumed that this transition is governed by a statistical law. The statistical law implies that infinite time is necessary to form all the nuclei. We therefore redefine the nucleation time 1, as the time necessary to form W% of the nuclei. In the case that pl= const= 1/7] (see below)

1,=2.371. (18) The probability p l of the transition ( A ) to ( B ) is in general a function of nl, n2,na, and 1, where only two of the n’s are independent (Eq. 8). It is furthermore assumed that p1is independent of time. The fact that p l is a function of n2 and n3 can be interpreted as an interaction between domains and nuclei; the number of growing and already grown domains influences the probability of the formation of new nuclei. As a first approximation it can be assumed that this interaction is nearly the same whether the existing domain is small or has already grown through the sample. That is, we express p1 in the form (19)

and since n2+n3=N-nll we can write pl as a function of nl alone, i.e., (20) dnl/nl= - pl(nl)dl, with the boundary condition nl(0) = N .

nl= N exp(-t/rl),

- (dnl/dl)+,=

(N/TI)exp(-r/7J.

(21) (22)

Substituting Eq. (22) into Eq. (17) gives i,(l)

= io[exp(-

1/72)

- exp(-t/~l)],

5. DISCUSSION

5.1. Shape of the Switching Pulse

Equation (23) which describes the switching time under the condition of p l being constant is shown in Fig. 9. The switching pulse is approximated by the difference of two exponentials with the time constants 71 and 7 2 . The shorter of the two time constants appears as the rise time and the larger as the total switching time regardless of which happens first. (The rise time is here defined as the time required to obtain 90% of the maximum current.) Thus we can write [see Eqs. (4), (14), and (18)]: (a) at low fields (M>/,!)

which shows the inverse square dependence of 1’ on field E as observed experimentally (Fig. 4). On approaching very small electric fields the point is reached where the switching is determined by the rate of nucleation alone. If no interaction between domains and nuclei occurs, the current then becomes i,= const (dvldt) = const exp( - t / n ) ,

As demonstrated below, the simplifying assumption that $1 is constant results fortunately in an expression which accounts for a major fraction of the experimental results. This assumption implies that there is no nucleus-domain interaction. Assume p1=1/71 to be constant, then Eq. (20) can easily be integrated yielding which leads to

( 24)

~

2.3 1/tn- l / l d

4.2. Transition ( A ) - ( B )

p,=j(n2+na),

1 In(tdl1”)

(26)

which means that I’=O and hence the pulse symmetry

m = t‘/t’’ should also be zero. Experimentally, however, one observes that m approaches a finite value no which is different from zero. It is then apparent that one cannot neglect entirely the interaction between domains and nuclei. Nucleus formation could be either hindered or enhanced by the depolarizing effects of the adjacent domains. Further, the presence of a favorably oriented domain could enhance the probability of forming a nucleus (if the shape allows it), since fewer domain walls would be required. From the present model and experiments it appears that enhancement is predominant. This

(23)

with TON 20=

71(1/71-1/72)‘ This expression is shown in Fig. 9 and will be discussed in the next section. The maximum of the current occurs at a time equal to the “rise time” 1’ that can be

FIG.9. Calculated switching current i, versus time 1 in TGS.

2.70 Switching Mechanism in Jriglycine SuFate and Other Ferroeledcs - E. Fatuzzo, W. J. Merz SW1TCH I NG

MECHANISM

enhancement of nucleation adjacent to an already formed domain gives the appearance of a sidewise domain wall motion. This is to be distinguished from a true sidewise motion in which the walls move in steps of a single row of dipoles at a time. The present experiments do not decide between these two phenomena and only the former is under consideration. If the nucleations are aided by existing domains, then p l increases with w and from Eq. (9) $1 decreases with increasing nl. Taking Eq. (9) into account, one ran write Eq. (20) as iV(dv/dl) = n l p ~ ( a J , (27) where pl(n1) is an increasing function with decreasing nl. Thus, the rate of nucleation shows a maximum because, at t=O, dw/dl starts increasing since p , ( n l ) is increasing. However, a t a time lo’ when the number of latent domains nl becomes small dw/dt must decrease. We define lo’’ as the time it takes for dw/dt to decrease from its maximum value to 10% of it. The ratio t { / t { ‘ can therefore be taken as a measure of interaction between nuclei and domains because if the interaction is large then to’ becomes large and to” small. At low electric fields where the current is proportional to the rate of nucleation [Eq. (26)] the ratio t~’/tollis identical with mo so that mo is not only a measure of the symmetry of the switching pulse but also a measure of the interaction between the nuclei and existing domains. The smaller mo, the smaller the interaction. For TGS the presence of nucleus-domain interaction was revealed by the finite experimental value for mS‘0.05.

5.2. Partial Switching Phenomena

In Ser. 2.4 a series of experiments were described in which pulses shorter than the switching time were applied to the crystals and the corresponding switching transients observed. In the case where i
IN TRIGLYCINE SULFATE

I

97

67

FIG. 10. Figure demonstrating the shape of the current pulses at incomplete switching. The rate of nucleation d u / d l is plotted versus time.

the charge switched with each successive pulse increases initially and then decreases just as in the total switching pulse [Fig. 6(a)], the domains which have remained polarized increase the probability of forming new nuclei as is assumed in the present model. Further, the initial peak on each transient becomes progressively larger (these peaks, A , B , C, D,E , F , are very much larger than those caused by stray capacitance). This is interpreted in the following manner. The domains progress through the crystal in the form of cones until the apices reach the opposite electrode. At this point the walls become perpendicular to the electrode surfaces and the domains assume a more cylindrical shape. If the field is switched off before all the domains have attained this cylindrical configuration, those in which the angle between the wall and the surface is appreciably less than !No,even if the apices of the cones have reached the opposite electrode, will collapse. At the low fields used in these experiments the switching current is determined by the rate of nucleation only and if we denote with t 1 the time it takes for the domains to reach the other surface and with tz the time it takes them to open up, it follows from Fig. 10 that the second shaded area represents nearly the current which flows when the corresponding pulse is applied whereas the first shaded area represents the peak which is due to the collapsed domains of the previous pulse. At the later stages [Fig. lO(a)] the first shaded area is much larger than in the earlier stages [Fig. 10(b)], which explains why the peaks E and F are larger than peaks B and C . 5.3. Heating Experiments

We have assumed in 2.5 that the heating of the surface accelerates the nucleation of new domains whereas the heating of the bulk accelerates the domain wall motion. Furthermore, we concluded that the rise time of the switching pulse a t low fields is determined by the domain wall motion time and at high fields by the nucleation time. The contrary is true for the decay time. Thus heating the surface should affect the decay time of the pulse when low fields are applied and the rise time of it when high fields are applied.

98

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2 Selected Early Work 1921-1961

E . FATUZZO A N D W. J. MERZ

TABLE 111. Values of 1120 and Pa for different materials. ~-

agreement with our observations of a large ma. The quantity ma for a number of ferroelectric materials is Material tn o P I (r coiil/cm*) shown in Table 111. If one compares the values of mo GASH < 1/50 0.35 with the values of the spontaneous polarization P, one TGS 1/20 2.2 finds that mo and thus the interaction between domains US 1/10 0.24 and nuclei increases with P. although, of course Thiourea estimated by Goldsmith' 3.2 LiH,(SeOs)r 1 15.0 domain-nucleus interaction cannot depend on P . alone. BaTiOa I from Merzb 26.0 The only exception in this table seems to be Rochelle salt. G . J. Goldsmith (private commiinicatian). b See reference 1. Experiments of Husimi and Kataokam on BaTiO, can be interpreted on the basis of our "switching in If the whole crystal is heated and then the heater is parts" results with TGS. Applying pulses of 2 kv/cm removed, one expects that the surface cools considerably they observed a switching time of the order of 10 psec. while the bulk retains the heat, so that the domain wall With pulses of 1 psec duration the polarization could motion is faster than at room' temperature, while the be reversed; however, with pulses of 0.5 psec duration nucleation rate is the same. Hence during cooling the it was not possible to do so with a reasonable number of rise time of the pulse remains much shorter than at pulses. These results can be explained by assuming that room temperature. These observations are consistent the domain wall motion times at fields of 2 kv/cm is with the proposed model. of the order of 0.5 psec. If pulses shorter than 0.5 psec are applied the growing domains collapse upon removal 6. COMPARISON WITH OTHER FERROELECTRIC of the field. With the value for the thickness given MATERIALS by the authors one can deduce a forward mobility In BaTiO3 one finds an exponential law [Eq. (O)] of 14 cmz/v sec. This value is much higher than the one up to 50 kv/cm.' At low fields this material switches in formerly estimated' by calculating the mobility from steps like TGS when many pulses are applied. Using the slope of the linear part of the 1/1. ve~susE curves. our model we thus conclude that in BaTiOs the switch- Because up to fields of 20 kv/cm the switching in ing is very strongly dominated by nucleation, even BaTiOa js controlled by the nucleation, the linear part more so than in TGS. Furthermore, one finds in BaTiOl in the 1/1, versus E curve is not due to domain wall a very symmetrical switching pulse which is practically motion as was assumed before. The linear part due to independent of applied field, that is m=mo=l. This domain wall motion would appear at much higher fields. Recent experiments by Stadler2' demonstrate that at behavior is explained by our model by assuming a very very high fields BaTiOa shows the following field strong nucleus-domain interaction. Some recent experiments by MillersJ8 using the strength dependence of the switching time: etching techniq~e'~ show that the domains in BaTi03 I/t#= kEa. do grow sidewise. He found that the sidewise wall velocity varies exponentially with field, according to This behavior cannot be explained by our model Eq. (3). I t is rather difficult to interpret this result in without making additional assumptions. terms of a normal sidewise motion, whose velocity 7. ACKNOWLEDGMENTS should be expected to be linear with field. I t can, We would like to thank Dr. R. Nitsche and Miss S. however, be interpreted according to our model by assuming a continuous formation of new nuclei along Zingg for the preparation of the crystals and E. Bank1 the edges of a switched region. This, however, indicates for some technical assistance. We also would like to the presence of a strong nucleus-domain interaction in thank Dr. G. J. Goldsmith for many helpful discussions.

-_

+

~~~~~

~

~~

18R. 19

C. Miller and A. Savage, Phys. Rev. 112, 755 (1958).

J. A. Hwton and W. J. Merz, Phys. Rev. 98, 409 (1955).

Husimi and K. Kataoka, J. Appl. Phys. 29,1247 (1958). H. L. Stadler, J. Appl. Phys. 29, 1485 (1958).

10 K. zL

2. I I Crystal Stability and the Theory of Ferroelectricity - W. Cochran VOLUME3, NUMBER 9

PHYSICAL REVIEW LETTERS

The mechanism of dislocation relaxation Proposed by Seeger' does not seem to offer an explanation for the apparent absence of peaks in bcc metals. A principal difference between dislocations in fcc and bcc metals is that for the latter a dissociation of complete dislocations into symmetric partials i s not possible. The insensitivity of the peaks to impurity content indicates that intrinsic point defects such as vacancies may be important. These facts suggest a new mechanism based upon the thermally acttvated motion of symmetric paired partial dislocations between vacancy-pinning points; minimum energy configurations a r e those for which the vacancy i s immediately above either partial. It is in these positions that its interaction with the dilatational stress field of the partial will be strongest. The activation energy i s determined by the energy required for the transition (by

I

99

NOVEMBER 1, 1959

glide of the dislocation pair) between the two stable states. This model will be discussed in detail in a forthcoming publication.* We wish to acknowledge the support and encouragement of Professor A. W. Lawson throughout the course of this work. The zone-refined iron was obtained through the courtesy of J. W. Halley, K. K. Fetters, and G. W. Rengstorff; it was purified by the latter under the project supported by the American Iron and Steel Institute. 'P. G. Bordoni, J. Acoust. SOC. Am.

26, 495 (1954).

'D. H. Niblett and J. Wilks, Phil. Mag. 2, 1427 (1957)* 'H. L. Caswell, J. Appl. Phya. 29, 1210 (1958). 'D. 0. Thompson and D. K . Holmes, J. Appl. Phys. 30, 525 (1959). -5A. Seeger, Phil. Mag. 1,651 (1956). 'L. J. Bruner (to be published).

CRYSTAL STABILITY AND THE THEORY OF FERROELECTRICITY W. Cochran* Atomic Energy of Canada, Limited, Chalk River, Ontario, Canada (Received August 7, 1959; revised manuscript received October 1, 1959)

The condition that a crystal should be stable for all small deformations i s that all the normal modes should have real frequencies.' The limit of stability against a particular mode of vibration i s approached as the corresponding frequency approaches zero. In what follows we show that there is reason to believe that a ferroelectric transition, at least in certain crystals, is a s sociated with such an instability or near -instability. The condition for a ferroelectric transition i s therefore a problem in lattice dynamics, and when it i s treated a s such new insight into the problem is obtained. We illustrate this by means of a simple example, in which it is shown that it is possible for a diatomic cubic crystal to exhibit properties which a r e remarkably similar to those of barium titanate. A shell model for the ions in an alkali halide crystal, proposed by Dick and Overhauser,* has been found to give a satisfactory explanation of the dielectric properties of alkali halides'j' and of the lattice dynamics of sodium iodide.' A similar model accounts quite well for the observed relation between frequency w (=2nv) and wave vector ( q = 2 n / h ) of certain of the normal modes of germanium.' The theory of Woods, 412

Cochran, and Brockhouse,' based on this model, gives the following expressions for the frequencies of the transverse optic (T.O.) and longitudinal optic (L.O.) modes of wave vector zero in a diatomic cubic crystal:

Here j~ i s the reduced mass of the ions, E the high-frequency dielectric constant (the square of the optical refractive index), and u the volume of the unit cell. 2 %i s the effective ionic charge, while R,,jc is the restoring "short-range" force on any one atom when the two lattices carrying the nuclei of different type a r e displaced a small relative distance ii. Primed quantities depend explicitly on the parameters of the shell model. In a crystal such as sodium iodide the two quantities on the right of Eq. (1) a r e of the same order of magnitude, but the t e r m R,' which a r i s e s from the short-range interaction i s about twice as great as the other, which a r i s e s from the Coulomb interaction. Let us consider the

100

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2 Selected Early Work 7927-1967

PHYSICAL REVIEW LETTERS

VOLUME3, NUMBER 9

situation that would arise if their difference approached zero. It i s found from the theory that wT may approach zero without the crystal necessarily becoming unstable against other vibration modes; indeed all transverse optic modes for which 4 is not close to zero may retain quite usual frequencies. The quantities which appear on the right of Eq. (1) may be temperature dependent, as the lattice vibrations are in practice not completely harmonic, so that near T = T, we may postulate

where 7 is a temperature coefficient and T, is the temperature at which the crystal would become unstable. It has been shown by Lyddane, Sachs, and Teller' that w

L

'/WT'

= EO/E,

(4)

where co is the static dielectric constant. Combining these equations, one finds

NOVEMBER 1, 1959

phase. When the short-range potential is given explicitly by

V = ! R '(u ' + u ' + u ')+iB(u '+u ' + u ') R

O

x

y

z

X

Y

Z

+$Bt(u" + u ' + u ") X

+ iB"(u

X

Y

Z

'u ' + u 'u Y

y

e

'+u2 'u%'),

with B C O , Bland B">O, the crystal makes a first-order transition to a tetragonal phase at a temperature exceeding T,. Equation (10) r e fers to the "free" crystal; the potential V, cannot therefore be used as it stands to give wT in a noncubic phase, since, for the lattice vibrations, the crystal is "clamped" by its inertia. At the cubic-tetragonal transition the atoms become displaced a relative amount uo= (3 1 B I /4B')",

along a crystallographic axis, say [OOl]. The minimum T.O. frequency i s reached just before the transition, and is given by

Expressions for the spontaneous polarization as

a function of temperature, etc., are also obso that a Curie-Weiss law is followed with a Curie constant

C =(c +2)'(Z1e)*/9uR,,'y z ( c +2)/4ny.

-

-

(8)

The condition c0 00 i s thus OT' 0. The ionic polarizability of one unit cell is found to be given by ai=(Z'e)a/R0 ',

(7)

while the electronic polarizability is as usual given by 4nae/3tJ

=(E

- l)/(E

+2).

(8)

The condition wT'=O is therefore the same as (9) so that the terms "instability" and "polarizability catastrophe" are synonymous in this instance. If one now postulates that the short-range potential between the two Bravais lattices Is neither precisely harmonic nor precisely isotropic for comparatively large displacements, it can be shown that the crystal may become spontaneously polarized without becoming completely unstable. Instead it makes a transition to another

tained in terms of the above atomic parameters. As the temperature decreases still further the crystal makes a second transition in which the atoms become relatively displaced along [Oll], followed by a third which leaves them displaced along [ l l l ] . These results will be understood when it is pointed out that Eqs. (3) and (10) lead eventually to an expression for the free energy C, of the unstressed crystal which i s almost identical with that postulated by Devonshire.8 The theory has been extended to apply to antiferroelectric transitions in diatomic crystals, and to ferroelectric transitions in other cubic crystals, including barium titanate. The equations which apply to the latter are greatly simplified by assuming that in the T.O. mode of lowest frequency (there are three T.O. modes in all), or in a static field, the framework of oxygen atoms is not distorted. The crystal structure a n a l y ~ e s ~of, ' ~tetragonal barium titanate and lead titanate support this assumption. The dielectric properties of barium titanate, and the movements of barium and titanium atoms relative to the oxygen octahedron, may be accounted for by assuming a temperature dependence of certain atomic parameters analogous to that given by Eq. (3), and a short-range po413

2.1 I Crystal Stability and the Theory of Ferroelectricity

VOLUME3, N U M B E R 9

PHYSICAL REVIEW LETTERS

tential for relative movement of titanium and oxygen atoms given by Eq. (10). A very small departure from a harmonic potential is found to be sufficient to account for the dielectric properties, and,the numerical values required for other atomic parameters are physically reasonable. The equation corresponding to Eq. (4), applicable to a perovskite-type crystal, is found to be ( w2 w3 04 ) L" ( w 2 w3 (L'4 ) 7 z = € 0 /€.

(11)

- W.

Cochran

I

lo'

NOVEMBER 1 , 1959

quency range should yield interesting results. The statement that the problem of the onset of ferroelectric properties in any cr y st al is a problem in lattice dynamics is probably correct, but a t present there is little prospect of detailed application of the theory of lattice dynamics to low-symmetry or disordered crystals. I a m indebted to a number of colleagues, especially Dr. B. N. Brockhouse, for helpful comments on this work.

(This result does not depend on the assumption *On leave f r o m Crystallographic Laboratory, Cavenof an undistorted oxygen framework.) The infr a r e d absorption frequencies ( ( L ' and ~ ) ~( w ~ ) ~ dish Laboratory, Cambridge, England. lM. Born and K. Huang, Dynamical Theory of Cryshave been measured by Last," and are in no way tal Lattices (Oxford University P r e s s , Oxford, 1954). unusual. It follows from the theory that ( C L ~ ) ~ ' 'B. G. Dick and A. W. Overhauser, Phys. Rev. 112, should be proportional to ( 7 - T c ) in the cubic 90 (1958). phase, and should reach a n abnormally low value 3J. E . Hanlon and A. W. Lawson, Phys. Rev. 113, 472 (1959). estimated as u = 2 o r 3 x 10" cps just before the 4W.Cochran, Phil. Mag. (to be published). first transition. This frequency should split ap'Woods, Cochran, and Brockhouse, Bull. A m . Phys. preciably in the tetragonal phase, and each f r e Sac. &, 246 (1959). quency in this phase should vary inversely as 'W. Cochran, Phys. Rev. Letters 2, 495 (1959). the square root of the corresponding "clamped" 'Lyddane, Sachs, and Teller, Phys. Rev. g,673 dielectric constant. Although relaxation of the (1941). *A. F. Devonshire, Advances i n Physics, edited by dielectric constant of barium titanate has been N . F. Mott (Taylor and Francis, LM., London, 1954), reported for frequencies of the order 10" cps, Vol. 3, p. 85. Benedict and Durand" found that when a single 'Shirane, Danner, and Pepinsky, Phys. Rev. 105, crystal i s used, the dielectric constant of the 856 (1957). cubic phase i s the s am e as the static value up to 'OShirane, Pepinsky, and Frazer, Phys. Rev. 91, u = 2 . 4 x 10" cps, the limit of their experiment. 1179 (1955). The resonance frequency predicted h e r e lies in "J. T. Last, Phys. Rev. 105, 1740 (1957). the millimeter wavelength range. A study of the '*T.S. Benedict and J. L. Durand, Phys. Rev. 109, 1091 (1958). properties of barium titanate in this difficult fre-

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3 Ferroelectrics 1966-2001 : An Overview R. Blinc

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3 Ferroeledrics 1966-2001 : An Overview R. Blinc J. Stefan Institute, Ljubbana, Slovenia

The number of publications and scientific meetings in the field of ferroelectricity has been increasing exponentially in the last 30 years. In addition to the classical fields of phase transitions and critical phenomena four new sub fields have emerged: a)ferroelectric liquid crystals. b) thinfilms and integrated ferroelectrics which led to the discovery of 2 D ferroelectricity and the development of non-volatile ferroelectric random access memory, c) dipolarglassesand relaxors as well as d) incommensurate systems. Quantum effects have been observed as well. Keywords: Ferroelectricity, Liquid Crystals, Thin Films, Dipolar Glasses, Relaxors, Incommensurate Systems, Critical Phenomena

3.1 Introduction

In this review ofthe development ofthe field of ferroelectric research in the period 19662001 1 shall deal with two main points: Statistical indicators measuring the quantitative growth of this field in the last 30 years

Breakthroughs that have been achieved in the understanding of this field and the new physics and new fields that have emerged. In this connection one should mention that in addition to the classical fields of soft modes, phase transitions and criticalphenomena a number of new fields are being studied at present: The traditional wisdom was that ferroelectricity like ferromagnetism can occur only in the crystalline solid state. In contrast ferroelectricity and antijirroelectricity have been shown to occur in liquid crystals and liquid crystalline polymers as well. This gave rise to an active new research field with many applications. The miniaturization requirements of nanoengineering forferroelectric memories with high information density have reopened the question how the ferroelectric propcrtics of a system are changed if the dimensions reach the nanometer region. The old question of how many unit cells are necessary so that ferroelectricity which is a collective phenomenon, does not disappear, thus became important not only for physics but also for technology.

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3 Ferroelectrics 79662007:An Overview

These investigations led to the discovery of two-dimensional (2D) ferroelectricity in polymeric thinfilms as well as to the construction of non-volatile ferroelectric random access (NV-FRAM) memories and the emergence of the field of integrated ferroelect rics. Other mayor developments are in a periodic and disordered systems:

subclass of ferroelectrics, namely relaxors which similarly as glasses exhibit no symmetry breaking phase transition but where a ferroelectric state can be easily induced by an electric field E > EL. Relaxors exhibit giant electrostrictive and dielectric coupling and a huge dispersion of dielectric properties and are rather important for electromechanical applications.

/ncommensurate ferroelectrics, where the Whereas the leading idea of the Prague periodicity of the modulation wave and meeting was the soft mode instability and the periodicity of the basic lattice cannot the fact that many properties of ferroelectric be expressed as a ratio of two rational phase transitions can be described within numbers so there is no translational pe- the framework of lattice dynamics, the asriodicity in spite of perfect long range or- pects discussed in Madrid are much more der, allowed for the study of phasons, diverse. Though phase transitions physics is still a hot topic, newly emerging fields and amplitudons and solitons. Substitutionally disordered random solid applications are catching an increasing solutions of ferroelectric and antiferro- amount of interest [I]. electric systems were found to lead to a new state, the dipolar glass phase, where the low temperature state may be charac- 3.2 terized by an order parameter function Statistical Indicators instead of a single order parameter. If we have in addition to substitutional The number of publications in the field of disorder and competing interactions also ferroelectrics has in the last 30 years incharge disorder such as in perovskite solid creased exponentially (Figure 3.1). Accordsolutions we obtain another important ing to the data-base of the Institute of Sci-

in the field of 1970 and 2001.

3.2 Statistical indicators - R. Blinc

entific Information in Philadelphia, USA (which goes back to 1970) this number has increased from 200 on file in 1970 to about 20000 in 2001. I n the eighties and nineties a number of new-subfields have emerged

such an incommensurate systems, dipolar glasses, relaxors and ferroelectric liquid crystals. In the last period the growth was particularly visible in applied fields such as integrated ferroelectrics and thin films.

Table 3.1 Meetings on ferroelectricity.

/S/F ( 1 titernational Symposium on Integrated Ferroelectrics):

lSAF

(Intcrnational Symposium on Applied Ferroelectrics): 1" 1971 New York, USA; 2"" 1975 New Mexico, USA; 3'" 1979 Minnesota, U S A 4"' 1983 Maryland, USA; 5'" 1986 Pennsylvania, USA; 6"' 1988 Zurich, Switzerland; 7''' 1990 Illinois, USA; 8'" 1992 South Carolina, USA: 9'" 1994 Pennsylvania, USA; 10'" 1996 Brunswick, N J , USA; 11 "' 1998 Montreux, Switzerland; 12 "I 2000 Honolulu, USA.

1" 1989 Colorado, USA; 2'Id 1990 Monterey, USA; 3Id 1991 Colorado, USA; 4'"1992 Monterey, USA; 5'" 1993 Colorado, USA;

6'" 1994 Montcrey, USA; 7'" 1995 Colorado, USA; 8'"1996 Tempe, USA; 9"' 1997 New Mexico, USA; 10"' 1998 Monterey, USA; 11'I' 1999 Colorado, USA: 12 111 2000 Aachen, Germany; 13 'I' 2001 Colorado. USA. /MF

EMF

(International Meeting on Ferroelectricity):

(European Meeting on Ferroelectricity): 1" 1969 Saarbriicken, Germany; 2""1971 Dijon, France; 3'd 1975 Zurich, Switzerland; 4j" 1979 Portoroi, Slovenia; 5"' 1983 Milaga, Spain; 6'" 1987 Poznan, Poland; 7'" 1991 Dijon, France; 8''' 1995 Nijmegen, Netherlands; 9"' 1999 Prague, Czech Republic.

1" 1966 Prague. Czechoslovakia; 2""1969 Kyoto, Japan; 3"l 1973 Edinburgh, UK; 4'" 1977 Leningtad, USSR; 5'" 1981 Pcnnsylvania, USA; 6'" 1985 Kobc, Japan; 7"' 1989 Saarbriicken, Germany; 8'"1993 Maryland, USA; 9"' 1997 Seoul. Korea; 10"'2001 Madrid, Spain. FLC

(Intcrnational Symposium Crystals) :

ECAPD 011 Ferroelectric

1" 1987 Bordeaux, France; 2"' 1989 Gotcborg. Swcdcn; 3"' 1991 Colorado, USA; 4'" 1993 Tokyo, japan; 5'" 1995 Cambridge, UK; 6"' 1997 Brest, France; 7'" 1999 Dartnstadt, Gerniany; 8''' 2001 Washington, USA.

Liquid

(European Corifcrcnce on Applications of Polar Dielectrics): 1" 1988 Zurich, Switzerland; 2"" 1992 London, UK; 3'" 1996 Bled, Slovenia: 4I'' 1998 Montreux, Switzerland; S'"2000 Jurmah, Latvia.

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3 Ferroelectrics 196&2001: An Overview

Whereas the I M F in Prague in 1966 was The form of the singularity of thermodythe first conference of its kind, there are at namic quantities at the phase transition present at least seven kinds of conferences point in uniaxial ferroelectries with direcdevoted to various aspects of ferroelectric tion dependent interactions has been deterresearch: International Meetings on Ferro- mined [Sl]. In view of long range dipolar electricity (IMF) from 1966, European Meet- fields the results differ from the phenoings on Ferroelectricity (EMF) from 1969, menological theory by logarithmic correcInternational Symposia on Applied Ferro- tions, 1nlT- TJ. The influence of quenched electrics (ISAF) from 1971, European Con- local randomfields (RF) on phase transitions ferences on Applications of Polar Dielectrics has been investigated by Imry and Ma [52]. (ECAPD) from 1988, International Symposia on Ferroelectric Liquid Crystals (FLC) Table 3.2 Major developments in phase transitions from 1987, International Symposia on In- and critical phenomena. tegrated Ferroelectrics (ISIF)from 1989 and 1937-1971: Exact solutions of six vertex spin Asian conferences on Ferroelectricity. One model in a field (Lieb) and eight vertex model should also not forget the successful series in zero field (Sutherland, Fan and Wo). of Electroceramics conferences. The places 1968-1974: Improper ferroelectrics like and dates of the conferences held so far are gadolinium molybdate where P i s not the primary order parameter (Indenbom, Levanyuk, collected in the Table 3.1. Sannikov, Dvorak, Pytte, Aizu).

1975:Influence of quenched local random 3.3 Phase Transitions and Critical Phenomena

In this classical field of ferroelectric research there have been so many important advances and discoveries in the last 30 years that only a few can be mentioned. The Landau theory of the dielectric anomalies in improper ferroelectrics like gadolinium molybdate, where the phase transition order parameter is not the polarization, has been worked out by Levanyuk and Sannikov [45].These systems where the lattice instability does not occur at the center of the Brillouin zone were predicted by Indenbom [46] and investigated independently also by Dvorak [47],Pytte [48]and Aizu 1491, The understanding of the influence of defects on the anomalies near phase transitions has been another important development [SO]. It has been shown that defects provoke anomalies similar to those due to critical fluctuations.

fields on phase transitions (Imry, Ma), 11. Order phase transition destroyed by RF in Heisenberg, XY and other systems with continuous symmetry for any D and Ising systems for D < 3.

1988: Influence of defects on anomalies near phase transtions (Levanyuk, Sigov) provoke anomalies similar to critical phenomena. 1970-1994 Effects of hydrostatic pressure (Samara, Peercy, Nelmes) on FE phasc transition. Effects of uniaxial pressure (Gonzalo).

1987:Microscopic understanding of FE phase transitions & oxygen polarizability (Biltz, Benedek, Bussmann-Holder).

1980:Quantum paraelectrics K,.,Li,TaO

,,

Sr,-,Nb,TiO, and quantum fluctuations in zero dimensional H-bond systems (K,H(S04),: T, = 100 K) and deuteron glasses.

1980: Development of new techniques (2D NMR, EPR, ENDOR, NQR) for local structure of ferroelectrics (Miller, Borsa, Rigamonti, Dalal, Blinc. Seliger. Michel).

1990-2001: Developments in I R spectroscopy (Volkov, Kozlov, Petzelt, Grigas) & HyperRaman spectroscopy (Vogt) of FE and FE thin films.

3.3 Phase Transitions and Critical Phenomena - R. Blinc

They showed that a second order phase transition will be destroyed by arbitrarily weak RF in all Heisenberg, X-Y or other systems with a continuous order parameter symmetry and in king systems for d < 3. The experimental situation was reviewed by W. Kleemann [53]. The search for static and dynamic critical exponents of ferroelectric phase transitions was started by the suggestion of Elliot [54] that pseudo-spin models may give a good indications of the critical behavior to be expected close to T,by exploiting the analogies with the king model. A huge amount of data has collected so far. The study of the effects of hydrostatic pressure on ferroelectric phase transitions has given many new insights into transition mechanisms and was pioneered by the work of Samara [55,56].Uniaxial pressure effects were also studied by a number of groups (see, for instance Ref. [57]). The use of synchrotron radiation and high j u x nuclear reactors has led to a much more precise determination of the structural changes at ferroelectric phase transitions as well as phonon instabilities (See, for instance the papers [%]). One should also mention the developments in infrared soft mode spectroscopy (A. A. Volkov, G. V. Kozlov, J . Petzelt) which was recently applied also to thin films [59].There have been many advances as well in Raman and hyper-Raman spectroscopy [60].Microcave dielectric spectroscopy measurements in particular have shown a cross-over from displacive to order-disorder behavior on approaching T, in SbSI and other systems [61]. A significant contribution to a better understanding of phase transition mechanism has been also provided by local techniques such as N M R and N Q R as well as EPR and E N D O R . (For review see Ref. [62,63])Here one should mention the I7O NQR studies of H-bonded ferroelectries which allowed

I

and accurate determination of the proton and deuteron freeze-out in the double well 0 - H - 0 potentials and the use of the isotropic part of the chemical shift tensor for the study of the order-disorder vs. displacive character of phase transitions by N. S. Dalal [64] (Figure 3.2, 3.3 and 3.4), the ENDOR studies of the local phase transitions near paramagnetic defects of the same author, the work of Bonera, Borsa and Rigamonti on perovskite transitions, the work of D. Michel on betaines and the EPR work on critical phenomena in SrTiO, and other perovskites as well as H-bonded ferroelectries by K. A. Miiller. One should also note the observation of the dynamic symmetry breaking in KSCN and RbSCN antiferroelectrics by ’“K and X7RbN M R [65]. A special contribution to the field of quantum efects in ferroelectrics is the observation of incoherent deuteron tunneling in Rb,_,(ND4),D2P04 deuteron glasses at low temperatures by 2D deuteron exchange N M R as well as by deuteron and Rb T, measurements [66]. The deuteron and Rb relaxation peaks change from a thermally activated regime to a temperature-independent 0-D-0 deuteron tunneling regime at low temperature (Figure 3.5). Another important contribution is connected with quantum efects in “zero-dimensional” systems like K3H(S04)2 and Rb,H(SO,), where the hydrogens are disordered in double minimum potentials [67, 681. Whereas the deuterated crystals undergo an antiferroelectric phase transition, protonated analogues remain paraelectric down to the lowest temperatures investigated. N M R and IR measurements [69]have shown that this is not due to the geometrical isotope effect i.e. the fact that the proton moves to the center of the H-bond. The absence ofa phase transition in the protonated isomorphs is thus due to zero point quantum fluctuations or a tunneling splitting

109

110

I

3 Ferroelectrics 19GGZOal: An Overview I

1'11

Lo

-I 138

T(K:

148

0.6 0.5

lo

4

3

I28

0.4

T(K)

148

138

Figure 3.2 Low resolution NQR data showing order-disorder features for RbH2P0,.

b VL =

c IBb cp = $ (0. Bo) = 25"

B 5 w

109.3 MHZ

20O-

-20; -40 43-

I

1

I

1

I

. . _._..-._.-- ....

-

191.0

Ia

190.8

-

190.7

-

0

f

<

,90.61

,

I

I

I

,

0 - lovr temperature A - high

I

,

phsse

temperam phase I

I

, ,

Figure 3.4 Coexistence of order-disorder and displacive features in hydrogen-bonded solids example o f squaric acid. Averaged high resolution "C N MR peakpo~itionofH ~SQasafunctionof

3.3 Phase Transitions and Critical Phenomena - R. Blinc

ol

9

Wl”ORADP42

a

Wi1RADP-50 W,”RAOP-50

0.OM

0

I00

100

900

400

500

600

IO O W (K-’) Figure 3.5 Temperature dependence of the inverse ”Rb % --f -% spin lattice relaxation rates in normal and deuterated RADP and DRADP. The temperature independence of the relaxation at low temperatures shows the effect of incoherent proton respectively deuteron tunneling.

larger than the interaction energy. These quantum effects are smaller in the deuterated compound due to the larger mass of the deuteron. Other quantum paraelectrics studied are KTaO, and SrTiO, where however a polar low temperature phase can be easily induced by doping with small amounts ofpolar impurities e.g. K,-,h,TaO, or KTa,-,vNb,xO,[53]. There has been significant progress in exact solutions of spin models of ferroelectrics and antiferroelectrics following the work of Slater [70] who suggested that KH,PO, type ferroelectrics can be represented by ice-type “six vertex” models. Lieb‘s solution ofthe ice-type six vertex models [71] was described as the most significant new exact result since the work of Berlin and Kac (1954) on the spherical model in the pioneering work of Onsager (1944)on the twodimensional lsing model. Sutherland [72] and Fan and Wu 1731 introduced the square-

lattice eight vertex model which has been solved in zero field. The field has been reviewed by Baxter [74]. In this connection it is interesting to note that a recent 2D ” P exchange N M R study of Rb,,,(ND,),, 5D,P0, showed the validity of the Slater ice rules and at the same time explained the symmetry properties of these systems which are in apparent contradiction with the above “six-vertex’’rules. Unpaired Takagi H3P0, groups invert the bias of all bonds they pass resulting in a switching between the six vertex configurations. This in turn produces a time averaged H-bond network and symmetric double well type 0 - H - 0 potentials [75]. There have been also many advances in the microscopic understanding of ferroelectricity. In this connection one should mention first of all the work by Bilz, Benedek and A. Bussmann-Holder [7G]on the oxygen polarizability and its role in ferroelectrics.

I ”’

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3 Ferroelectrics 19GG2001: An Overview

Figure 3.6 2D exchange NMR in DRADA. (a) Ratio of the cross peaks to diagonal peaks as a function of mixing time. (b) Intra-bond 0-D..O ts O..D-0 deuteron exchange time as a function of inverse temperature.

Figure 3.7 2D exchange NMR 0-D..O deuteron spectrum in DRADA with x = 0.32 at J = 40 K for the three different mixing times. The fact that the cross peaks at long mixing times become as strong as the diagonal peaks demonstrates that the 0-D..O H-bond potential is of the symmetric double minimum type on this timescale.

3.4 Ferroelectric Liquid Crystals - R. Blinc

The collective orientational order excitations of ferroelectric liquid crystals are characterized by a paraelectric (Sm A) phase soft Ferroelectricity was traditionally believed to mode which splits below T,(Figure 3.9) into occur only in crystals. Whereas an isotropic amplitudon and phason modes. The phason liquid can be optically active, it cannot be is here the zero frequency Goldstone mode ferroelectric. This is because the optical ac- which tries to restore the broken symmetry tivity coefficient is a pseudo-scalar, whereas of the Sm A phase. The existence of gapless the polarization is a polar vector. Table 3.3 Goldstone mode in FE liquid crystals is due shows the major developments in ferro- to the fact that here a continuous symmetry electric liquid crystals. In 1975 R. B. Meyer group is spontaneously broken at T, whereas et al. discovered ferroelectricity in the tilted in solid ferroelectrics a discrete group is brochiral smectic C“ (Sm C;’<)liquid crystalline ken. When an external magnetic field is apphase [2].The technological importance of plied perpendicularly to the helical axis of a these materials was realized after the disFE liquid crystal with Ax > 0, the molecules covery offast electrooptic switching in ferrotend to align into the field direction. This electric (FE) liquid crystals by N. A. Clark and S. T. Lagenvall in 1980 [3]. The search will distort the originally “smooth” helical for liquid crystals with a large spontaneous arrangement of the Sm C“ phase and inpolarization led to the discovery of antiferro- duce a “soliton-like” structure for H < H,. electricity (AFE) and intermediate ferri- As a result of the breaking of the continuelectric and incommensurate phases in liq- ous helical translational symmetry and the uid crystals by Chandani et al. in 1989 [4]. corresponding periodic perturbation of the The structure of FE and AFE liquid crystals potential, experienced by the phason, we have Brillouin zones and a splitting of the is schematically shows in Figure 3.8. phason dispersion bond into acoustic and optic-like branches for Hi H,. In the unTable 3.3 Major developments in ferroelectric wound Sm CQ phase for H < H , the transliquid crystals. lational symmetry remains the same and the 1975: Discovery of FE liquid crystals by Mcyer point symmetry changes from D, into C,. (Tilted chiral smectic C phase is ferroelectric). Therefore no continuous symmetry group 1980: Surface stabilized FE LC displays for fast is broken at the Sm A - Sm CQ transition electro-optic switching (Clark, Lagcnvall). and no gapless phason exists in the Sm C“ 1989: Antiferroelectricity in SmC, phases phase for H > H, (61. discovered by Chandani. Giant Electrostriction in F E Liquid-Crystal1975-1976: Soft mode in FE LC (Blitic). line Elastomers has recently been discovered 1980-1984: Atnplitudon and gapless pliason in (Figure 3.10) [7]. FE LC predicted by Blitic and Zeks and Mechanisms for converting electrical enexperimentally discovered by Muscvic. ergy into mechanical energy are essential 1984: Freely susprrided FE smectic thin films for nanoscale transducers, sensors, actua(Heinekamp) up to N = 2 (2D ferroelectricity). tors, motors, pumps, artificial muscles and 1990: Soliton-like distorsions in F E LC i n m icrorobots. external fields (Muscvic). Nanometer scale actuation has been so far 2001: Giant elcctro-striction in ferroelectric mainly achieved by the linear piezo-electric liquid crystalline elastomers (Lehmari, Kremer). effect in quartz and “smart” PZT ceramics.

3.4 Ferroelectric Liquid Crystals

I

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3 Ferroelectrics 19662001: An Overview o f 0

0 0 0 0

0 0 0 0 0

Figure 3.8 Structure o f ferroelectric (a) and antiferroelectric (b) liquid crystals. The temperature dependence o f the spontaneous polarization o f DOBAMBC is shown in (c).

t Figure 3.9 (a) The first observation o f the slowing down o f the overdamped soft mode in the paraelectric Sm A phase and Its splitting into the amplitudon (T;') and gapless phason modes (~5')in the ferroelectric Sm C* phase. (b) The first observation o f the phason dispersion in the Sm C* phase o f DOBAMBC at J,- J = 0.1 K [S].

Table 3.4 The achievable strains in various electrostrictor materials. Material

Electrostrictiue strain (at electricfield strength E)

Frrely suspended FLCE film

4% latcral strain (1.5 M V in-')

P(VDF-TrFE): electron-irradiated poly(viny1idene fluoride-trifluoroethylene) copolymer

4% (150 M V m - ' )

PMN,, ,PT,,

0.15% (1.0 M V . m ')

lead magnesium niobate-lead titanate ceramics

PBLG: monomolecular, grafted layer o f poly-y-L-glutamate

;'

=Ah/h,

0.005% (300 M V . m ')

3.5 Ferroelectric Thin Films - R. Blinc

Figure 3.10 Giant Electrostriction in FE Liquid-Crystalline Elastomers. The view o f tilting of the liquid crystalline molecules due t o the electroclinic effect. Each layer shrinks by up to 4% in an external field.

The achievablc strains are of the order of 0.1 %.

Ultrathin ferroelectric liquid crystalline (polysiloxane) polymers yield a 4% strain at only 1.5 MV/m via the electroclinic electrostrictive effect - tilting of the liquid crystalline molecules in response to an applied electric field - resulting in a reduction of the thickness of the smectic layers. The effect reaches a maximum at the paraelectricferroelectric Sm A + Sm C" transition.

3.5 Ferroelectric Thin Films 3.5.1 Crystalline Films

The traditional wisdom for ferroelectic thin films has been that for film thicknesses smaller than 100 n m the depolarisation field will destroy any switchable polarization making small particles or thin films nonferroelectric [8,91. The general prediction was that the transition temperature T, will decrease with decreasing size and ferroelectricity will vanish below a minimum critical thickness. Recent thin film experiments [lo] have however shown that switchable ferroelectric films can be made down to 0.9 nm for a crystalline Langmuir - Blodgett deposited ran-

I

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I dom copolymer of vinylidene fluoride with 3 Ferroelectrics 196&2001: An Overview

trifluoroethylene, P(VDF - TrF 70 : 30) on graphite. The minimum thickness is just two monolayers. The critical size for small spherical lead zirconate-titanate (PZT) particles was calculated to be 25 A [Ill. Similarly it has been reported that switchable ferroelectric films can be made of PZT down to 3 or 4 nm [12] in disagreement with earlier theories but in agreement with recent theoretical calculations. This shows that finite-size effects impose no practical limitation on thin film ferroelectric memory capacitors though for some designs tunneling currents may become too large. The lateral width of the memory element as well does not represent a limitation at present as no change in the coercive field has been observed when the lateral size of the PZT cell was decreased from 1 p to 0.1 p (Figure 3.11).The voltage necessary for polarization reversal in such a 100nm x 100nm x100 nm cell is in the technologically accessible range of 5 V. The corresponding hysteresis curve has been measured via the atomic force microscope (AFM) in the piezoelectric mode [ 131.

4

--

-ID

-20

-10

0

10

20

Jo

SPmW Figure 3.11 Hysteresis loops of 1 prn (solid squares) and 100 nrn (empty circles) large PZT cells measured via AFM microscopy in the piezoelectric mode 1131

Nano-phase ferroelectric cells for Gbit non-volatdememories thus seem to be a real possibility. It should be mentioned that finite size effects such as depressions of T, and reductions in P, have indeed been observed in some nanocrystals as small as 250 A in diameter [14].It is however known that at such sizes this is not an electrostatic phenomenon but seems to be due to surface strains or to inhomogeneity effects [15].One should also stress that the local polarization P as a function of the depth z may actually increase as the surface is approached - as in the case of PZT - leading to an increase in T, as the thickness d decreases. Alternatively it may decrease resulting in a depression of T, with decreasing d. Both effects have been observed by now. This can be described by the fact that the extrapolation length can have either sign leading to an increase of a decrease of P at the surface. 3.5.2 Ferroelectric Memories and Integrated Ferroelectrics

Whereas 30 years ago ferroelectric memories were not considered to be feasible, work on non-volatile ferroelectric random access memories (FERAM) is now going on at various Japanese companies such as Matshusita/Panasonic (Figure 3.12), Sony, Hitachi, and Toshiba as well as at Samsung in Korea and Siemens in Germany. Non-volatile memories are permanently keeping the stored information and make the periodic rejuvenation of the information used in dynamic random access memories (DRAMS) unnecessary. FERAMs are at present better than magnetic MRAMs because ofcross-talk problems (Table 3.5) [16]. The first commercially successful integration of a FE film on a semiconductor integrated circuit (Ba,Sr,-,TiO, on GaAs) was

3.5 Ferroeledric Thin Films - R. Blinc Table 3.5 FERAM features. Non volatility (no need for batteries). Fast WRITE operation < 100 ns. Potential for DRAM Density.

NOW: Kbit Class (Smart Cards) Substitute for EPROM. FUTURE: 1 M b i t and 1 Gbit Class

3

Figure 3.12 Road map o f integrated ferroelectric technology (FERAM business plan from Matsushita/Panasonic).

achieved in 1988 by Scott, Aranjo and McMillan (see Ref. [17]and references therein). This was a low loss amplifier integrated circuit for 800 MHz - 2.3 CHz operation used for mobile digital telephones. These BST/Ga chips were 50 times smaller than their predecessors and reached production level of 270 million in 2001. In 1993 fatiguefree SrBi,Ta,O, (SBT) films were invented and in 1997 used in “smart” credit cards. More than one million were produced the first year (Figure 3.13).

for SRAM/DRAM/Flash.

In 1998 the first 16 Kbit ferroelectric RAMS were made by Panasonic and used in 8-bit Si microprocessors. Most FERAMs use a one transistor-one capacitor cell that requires that the FE capacitor be switched and its displacement curve compared for a READ operation and subsequent reset. Another possibility is a FET with a ferroelectric film as a gate (Figure 3.14).The source-drain current of the FET is quite different for +P and -P states of the FE gate. This cell can be thus read simply by monitoring the cur-

Figure 3.13 Market impact of FE “smart cards”.

Figure 3.14 Transistor - ferroelectric FET (gateoxide o f a MOS transistor is replaced by a FE thin film).

I

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3 Ferroelectrics 19662001: An Overview Table 3.6 Major developments in integrated

ferroelectrics. 1988: Scott et al.: First commercially successful

integration of a FE film on a semiconductor integrated circuit (Ba,Sr,-,TiO, - abbreviated as BST - on GaAs) for 800 MHz - 2.36 GHz operation used in digital mobile phone.

Efforts are made to fabricate < 100 n m ferroelectric capacitors for Gbit-density non-volatile memories. 3.5.3 Two-dimensional Ferroelectricity in Crystalline Films

2001: BST/Ga chips are 50 times smaller than their predecessors and reached a production level of 270 million $ in 2001. Found in most mobile phones.

In the above mentioned P(VDF-TrFE 70 : 30) crystalline films [lo] (Figure 3.15) the first order ferroelectric transition tem1995: Fatique-free SrBi,Ta,O, (SBT) films perature T, at 108 “C on heating and 77 “C invented (Aranjo, McMillan, Scott-Nature April on cooling was found to be nearly equal to 19951. the bulk value, even in films as thin as 10 A. 1997: SBT films used in “ferroelectric” smart In view of the absence of strong finite size cards (no telephone authorization needed, can effects when the thickness is decreased from be read at a distance of 30 cm). 150 A to 10 A the above films must be con1998: First 16 Kbit ferroelectric random access sidered to be two-dimensional ferroelectrics memory (PZT) made by Samsung, Panasonic and Fujitsu. (Figure 3.16), i.e., the ferroelectric state is 1999: FERAMs “embeded”inside a microhere generated only within the plane of the processor (Panasonic, Symctri with SBT). film. 2000: 4 Mbit FERAMs [Samsung (PZT), PanaThe inter-plane coupling is thus rather sonic (SBT)]Gbit FERAMs being developed. weak and the system can be described by an anisotropic king model. The occurence 2001: 2D ferroelectricity in Langmuir-Blodgett deposited vinylidene fluoride trifluoroethylene of two dimensional ferroelectricity in polyP(VDF-TrFe 70 : 30) thin films (Bune). meric thin films thus explains the absence of a critical thickness below which ferrorent without switching. The FE gate has to electricity would vanish. Fluctuations in two be switched by a voltage pulse only when dimensions are not expected to destroy the ordering as in the case of an isotropic Heiwriting in a new bit. The present state of art are 4 MHz PZT senberg ferromagnet because of the anisoFE-RAMS from Samsung and Matsushita. tropy of the coupling.

Figure 3.15 (a) PVDF - TrFe 70 : 30 copolymer chain. (b) Atomic resolution STM image.

Figure 3.16 (1) The temperature dependence of the pyroelectric current and spontaneous polarization for different numbers o f monolayers: (a) 30, (b) 5, (c) 2. (2) The voltage dependence o f the pyroelectric current at T = 25 "C for different numbers o f layers. The saturation is good for 5-monolayer (b) and 30-monolayer films (a) and incomplete for 2-monolayer films (c). Ferroelectric switching i s obtained at 25 "C in all films and > 50% o f the maximum expected polarization is achieved in all films.

The low temperature second order phase transition at T, 2 20 "C in P(VDF-TrFE) filrns of 30 monolayers or less can be described as a surface layer transition controlled by the interaction with the substrate or the top electrode.

3.5.4 Freely Suspended Ferroelectric Smectic Thin Films

Freely suspended smectic films represent an ideal, defect free system where the crossover from the 3D to the 2D X-Y universality class can be studied as the number N of smectic layers decreases (Figure 3.17). The interior layers are still Sm A-like, whereas the two surface layers already show

'*O

I

3 Ferroelectrics 19662001: An Overview

I

OETE*

Srn C* like +

I

YEW FRAME

Sm A like -+

Sm C* likc +-

Figure 3.17 Free standing smectic film

Figure 3.18 Smectic layers in a free standing tenlayer film above the bulk transition temperature. The interior layers are still Sm A like, whereas the two surface layers already show a finite polarization, characteristic for the Sm C* phase.

a finite tilt and finite polarization character- ers. This phenomenon is similar to surfaceistic for the ferroelectric Sm C" phase (Fig induced ferromagnetism in thin ferroure 3.18). Ferroelectricity is here surface in- magnetic films. duced like in PZT. It is interesting to note that in the bulk The temperature dependence of the tilt the Sm A - Sm C" phase transition is of 1'' angle for different numbers of layers from order, whereas it is of second order in the N = 2 to N = 10 are shown in Figure 3.19 N = 2 layer system. By decreasing the film together with the bulk results [18].The two thickness, the conjugate field penetrates exterior layers undergo a phase transition deep enough into the sample to drive the into the ferroelectric state at much higher phase transition close to the critical point temperatures than the interior bulk-likelay- similarly as in the ( Ec, T,) case.

Figure 3.19 The temperature dependence o f the tilt angle (which is proportional to the spontaneous polarizations) for different numbers of layers o f freely suspended ferroelectric liquid crystal DOBAMBC. The solid curves are fits to a discrete mean field theory [18].

3.6 Dipolar Classes and Relaxon - R. Blinc

competing FE and AFE interactions and random fields lead to a random freeze-out of the deuterons into one of the two possible In contrast to ferroelectrics, dipolar glasses sites in the 0 - D - - 0 bonds (Figure 3.21) and relaxors show no macroscopic space [19,20].These systems may provide a congroup symmetry brealung at any tempera- ceptual link between spin glasses where ture. The key feature of all dipolar glasses at least for king systems - an equilibrium and relaxors is a “rough” energy landscape phase transition is believed to exist and in order parameter or phase space (Figure structural glasses where the situation is less 3.20). This is rather different from the clear. In contrast with magnetic spin glasses smooth single or two-valley type free energy where we deal with random bond type insurface of homogeneous ferroelectrics. To teractions, proton and deuteron glasses are what extent can this picture and the prob- characterized by random bonds and random lem of the nature of the glass transition be fields. The king random bond - random put into a quantitative statistical-mechanics field (RBRF) model for these systems was description is one of the great open prob- proposed in 1987 [21]. Since the variance of lems of condensed matter physics. Whereas the random field A acts as an effective orit is clear that in these multi-relaxational dering field, the corresponding Edwardssystems a dynamic transition takes place Anderson order parameter qEA should be characterized by a near freezing-out of the non-zero at all temperatures. There is howlongest relaxation time and a splitting be- ever still a “glass” transition into a “nontween the field cooled and zero field cooled ergodic” DG phase at the Almeida-Thouless linear dielectric susceptibilities, the exist- (A‘I) line TdA) due to replica symmetry ence of an equilibrium glass transition has breaking [22]. not yet been definitely established. This is The basic problem in deuteron glasses is so in spite of the near divergence of the non- how to measure the Edwards-Anderson linear susceptibility which however shifts order parameter qEA as there is no macroand tends to disappear with decreasing fre- scopic conjugate field attached to it. The quency. answer is provided by local techniques such One of the most active fields have been as NMR, NQR and EPR. The first experiproton and deuteron (DG) glasses, i.e. mixed mental determination of qEA(T) in D U D P solid solutions of FE and AFE crystals such by N M R has been reported in 1989 (Figure as Rb,-, (ND,), D,PO, (DRADP) where 3.22) [23]. 3.6 Dipolar Glasses and Relaxors

Homogeneous ferroelectric

Ft

FA

-P

L

P

(PA Deuteron pseudospin glass

Figure 3.20 Free energy surface in a homogeneous ferroelectric and a deuteron pseudospin glass.

I

I

122 3 Ferroeledn‘cs 19662001:An Overview

0

0.2 xpe

0.4

I

0.6

X-

0.8

I

Figure 3.21 Phase diagram of the ferroelectric solid solution Rb,,(ND4),D2P04.

Table 3.7 Some developments in dipolar glasses

and relaxors.

1954 Systems with diffuse phase transitions (Smolenski, Isupov). 1970:Relaxors (Smolenski). 1982-1984: Proton glasses (Courtens, Schmidt).

1987:Random bond - random field Ising model of proton and deuteron. 1989: First direct measurement of Edwards Anderson order parameter and local polarization distribution function in D-RADP. 56 (Blinc,Zalar, Pirc). 1991:Field-cooledand zero field cooled susceptibility in deuteron glasses (Levstik, Filipic, Kutnajk, Pirc). 1992 Dielectric non-linearities in relaxors (Colla, Levstik, Tagantsev, Glazounov). 1992 Random field induced domain states of relaxors PMN (Westphal, Kleemann, Clinch&, Tagantsev, Glazounov). 1998 Morphotropic phase boundary theory of giant susceptibility and electromechanical response (Ishibashi). 1999 Spherical random bond-random field model of relaxors (Pirc, Blinc). Determination of Edwards-Andersonorder parameter and local polarization distribution functions of relaxors (Zdar, Blinc, Pirc). 2001: Determination of the intermediate

monodinic phase at the morphotropic phase boundary in PZT (Noheda, Shirane).

-1.5

-1

a:,

op

05

I

IS

Figure 3.22 (a) Temperature dependence of q,, as determined from the second moment of the ”Rb % + - % N M R spectra in Rbo55(ND4)044D2P04. The dotted line represents the fit to the random bond-random field model with A/J2 = 0.35, Tg= J/K = 90 K, whereas the dashed line represents the best fit for the pure random field model where Jg = 0 K and A1I2/k = 68 K. The solid line represents the fit to the pure random bond model with Jg = 151 K and A?\”2/k= 0 K. (b) Temperature dependence of the local polarization distribution function W(p) for Rbo56(ND4)044D2P04where A/J2= 0.35 [23].

The measured qEA(Tj curve shows that the king RBRF model gives the best fit to the N M R experimental data and allows for a determination of the random bond and random field variances. In the same publication the first measurement of the local polarization distribution function w(~) for DRADP has been reported (Figure 3.22). The situation is quite different in perovskite relaxors, i.e. in perovskite solid solutions like PbMgl,3Nb,,,03 (PMN) with substitutional and charge disorder [24].

3.6 Dipolar Classes and Relaxors - R. Blinc

Whereas the two position 0-D-0hydrogen bonds represent the basic reorientable dipoles in deuteron glasses, the basic ingredient of relaxors are locally ferroelectric polar nanodomains [25] which vary in size and orientation nearly continuously. In contrast to dipolar glasses a ferroelectricphase can be induced in relaxors with relatively modest electric fields E > E,, resulting in a giant, frequency dependent, dielectric anomaly and huge electromechanical coupling. To describe this situation, the spherical random bond - random field (SRBRF) model has been introduced where the pseudo-spin is proportional to the polar cluster dipole moment (Figure 3.23) [26,27].The order parameter field jj = (3) is here quasi-continuous and not discrete as in deuteron glasses. This model properly describes the N M R lineshapes (Figure 3.24) and the local polarization distribution function W(jj)as well the temperature dependence of the EdwardsAnderson order parameter [28] (Figure 3.25) and the dielectric non-linearity [29]. Dielectric non-linearities in relaxors have been investigated by a number of groups and a large difference between the “ac”and “dc” effects has been found (See,for instance Ref. [30]).A dynamic SRBRF model was pro-

s

posed [31] in 2001. The coupling between polar nanodomains has been suggested to be mediated by soft TO phonon modes [32]. Whereas in perovskite type relaxors random bonds dominate random fields (similarly as in dipolar glasses), the opposite is true in uniaxial SBN-type relaxors which can be described by the random king model [33,34]. The importance of random fields is also seen in polarization switching experiments in PMN - PT [35]where the AVRAMI equation fails for the long time tails. For E > E, the polarization takes 8 decades of time to saturate demonstrating the extremely broad relaxation time distribution (see also Ref. [30,36]). Finally one should mention that the theory of the giant dielectric and electromechanical response near the morphotropic phase boundary, which is important for all piezo-electric transducers and other applications of relaxors has been worked out by Ishibashi [37]. Whereas the morphotropic phase boundary was previously regarded as the boundary separating the rhombohedra1 and tetragonal regions of the PZT phase diagram it has been recently established that this boundary corresponds in fact to the nearly vertical line separating the rhombo-

Figure 3.23 Schematic phase diagram of relaxors according to the SRBRF model. The temperature dependence of the Edwards-Anderson glass order parameter 9 is as well shown for various values of the random field variance A.

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3 Ferroelectrics 19662001: An Overview

Quadratic c a s e

Linear case

T = 400 K

I

-400

200

-200

400

-400

'

-200

I

.

0

I

'

200

400

V - V ~( k H z )

V - V ~( k H Z )

Figure 3.24 "Nb X -+ -% NMR lineshapes in a PMN single crystal for different orientations of the external field with respect to the crystal axes. Both the linear case where the NMR resonance frequency is a linear function of the local polarization and the quadratic case where the orientation is such that the linear term is absent and the NMR resonance frequency is quadratic in the local polarization, are shown. Note characteristic singularity and asymmetry lineshape in the quadratic case.

300

SRBRF model

200 N= A

f " 100-

0 -5,O

-2.5

0.0

2,5

5.0

0

P.

100

200 300 T (K)

400

Figure 3.25 Local polarization distribution function W&) and temperature dependence of the Edwards-Andersonorder parameter derived from 93NbNMR in PMN. The solid line shows the theoretical fit for the SRBRF model.

hedral and monoclinic phases for 0.46 I x < 0.51 (Figure 3.26). The applicationof an electricfield induces the rotation ofthe polar axis and monoclinic

distortions which are the origin of the high piezo-electric response [38]. The pressuretemperature phase diagram of relaxors has been determined by Samara [39].

3 7 Incommensurate (IC) Systems - R. Blinc

a)

b)

PbZr,UTi,O,

IUlX

0

Figure 3.26 (a) The p,/p,-dependences of the dielectric susceptibilities. The ordinate shows la1 x but not x itself. The inset shows the corresponding phase diagram where C, T, and R indicate cubic, tetragonal and rhombohedra1 phases, respectively [37] (b) New PZT phase diagram arround the morphotropic phase boundary 1381

3.7

Table 3.8 Major developments in incommensurate

Incommensurate (IC) Systems

systems.

IC systems represent a class of materials where the spontaneous polarization varies in space, P = P(r) with a periodicity which is incommensurate to that of the basic lattice. Though the incommensurate (IC) phase in NaNO, - existing over 1.5 "C -has been discovered already in 1961by Tanaisaki and the intermediate IC phase in thiourea was found in 1963by Futama and Chiba the rapid growth of this field started in 1977 by the discovery of the IC structure of K,Se04 and the study of its IC dynamics in 1980 by Axe, Shirane and Iizumi 140, 411. The use of superspace groups for the description of IC phases has been introduced by Janner and Janssen in 1976 [42].Between 1977 and 1980 the Landau theory for IC systems was developed by Levanyuk, Dvorak, Ishibashi, Bruce and Cowley and ANNNI spin models were proposed by Selke and Siems. The local structure of IC phases has been illu-

1961:Discovery of intermediate IC phase in NaNO, over 1.5 OC (Tanaisaki, Phenomenological theory by Gesi). 1970:Discovery of IC phases in thiourea (Chiba). 1976: Superspace description of IC phases (Janner,Jansen, de Wolf). 1977 Discovery and IC dynamics in K,SeO, (Axe, Shirane, Iizumi). 1977-1980 landau theory of IC phases (Levanyuk, Dvorak, Ishibashi, Bruce, Cowley). 1997-1980 Many experimental findings by NMR. NQR, EPR (Blinc, Peterson, Milia, Aleksandrova) and neutron scattering (Currat, Quilichini, Dolino). 1982:Discovery of explanation of 3q IC phase in quartz (Aslanyan, Levanyuk, Dolino). 1985: Discovery of quasy-crystals (Schechtman). 1990 IC phase in high T, superconductor. 1998 39 solitons discovered in proustite (Apih et al.).

I

125

I minated by NMR, NQR ( B h cet al., Peters-

126 3 Ferroelectrics 79662001:An Overview

YAMUMOTO,T., Integr. Ferroelectrics 1996,12, 161. T., AHN,C. H., TRISCONE, J. M., Appl. 12 TYBELL, Phys. Lett. 1999,75,856. M., HARNAGEQ C., ERFURTH,W., 13 ALEXE, V., Appl. Phys. 2000, HESSE,D., GOESDE, A 70,1. 14 ISHIKAWA,K., YOSHIKAWA,R., OKADA,N., Phys. Rev. 1988,837,5852. M., MAKINO,Y., Ferroel. Lett. 1998, 15 TANAKA, 24,13. 16 SCOTT,J.F., Ferroelectric Memories, Springer Verlag, Berlin, 1999. 17 Scorn, I. F., Ferroelectrics 2000, 236,247. i a HEINEKAMP, S., PELCOVITS, R. A., FONTES,E., YI,E.. CHEN,PINDAK,R., MEYER, R. B., Phys. Rev. Lett. 1984,52, 1017. 19 COURTENS, E., ]. Phys. (Paris) Lett. 1982,43, V. H., et al., Phys. Rev. 1984, L199 ; SCHMIDT, 830, 2795. 20 SCHMIDT, V. H.,et d., Phys. Rev. 1984,830, 2795. B., BLINC,R., Phys. Rev. 1987, 21 PIRC,R., TADI/~; 836,8607. 22 LEVSTIK,A., FILIPIE, C., KUTNJAK, 2.. LEVSTIK, References I., PIRC,R., T A D I8~.. BLINC, R., Phys. Rev. Lett. 1991,66,2368. W., Phys. Rev. Lett. 1959,3,412. J., PIRC,R., TAD14 B., 23 BLINC,R.,DOLINSEK, 1 COCHRAN, B., KIND, R., LIECHTI, O., Phys. Rev. 2 MEYER,R. B., LIEBERT, L., KELLER, P., ]. de Phys. Lett. 1975,36, L69. Lett. 1989,63,2248. 3 CIARK,N. A., LAGERWALL,S. T., Appl. Phys. 24 SMOLENSKI, G.A.,]. Phys.Soc.]ap. 1970,28, 26. Lett. 1980,36,899. 4 CHANDANI, A. D. L., GORECKO, E., OUCHI, Y., 25 CROSS,L. E., Ferroelectrics 1987,76,241; ibidem 1994,15,305. TAKEZOE, H., FUKUDA, A.. rap. J. Appl. Phys. 26 BLINC,R., et al., Phys. Rev. Lett. 1999,83,424. 1989,28, L1261. 27 PIRC,R.,BLINC,R., Phys. Rev. 1999, 5 MUSEVIE,I., BLINC,R., ~ E K S B., , FILIPIE,C., EOPIE,M., SEPPEN, A,, WYDER, P.,LEVANYUK, 860,13470. A,, Phys. Rev. Lett. 1988,60,1530. 28 BLINC,R., et al., Phys. Rev. 2000,861,253. I., BLINC,R., OEKS, B., The Physics of 29 KUTNJAK, Z., et d.,Phys. Rev. 1999,B59,294. 6 MUSEVI~, Ferroelectric and Antferroelectric Liquid Crystals, 30 TAGANTSEV, A. K., GLAZOUNOV, A. E., J. of the Korean Phys. SOC.1998,32,951. World Scientific 2000. c., V., Phys. Rev. 31 PIRC,R.,BLINC,R., BOBNAQ 7 LEHMANN,w., S K U P I N , H., TOLKSDORF, 2001,863,054203. GEBHARD, E., ZENTEL, R., KRWGEQ P., LOSCHE,M., KREMER,F., Nature 2001,410, P.M., PARK,S. E., SHIRANE, G., 32 GEHRING, Phys. Rev. Lett. 2000,84, 5216. 447. B. D., Sol. State 33 KLEEMAN,W., et al., in Fundamental Physics a BATRA,I. P., SILVERMAN, ofFerroekctrics 2000 (Ed. COHEN,R. E.), Cornrnun. 1971,11,291:TILLEY, D. R., OEKS, AIP Conf. Proc. 2000,535,26. B., Sol. State Cornrnun. 1984,49,823. 34 KLEEMAN,W., Phase Trans. 1998,65, 141. 9 TILLEY, D. R.,OEKS, B., Sol. State Commun. D.,private communication. 35 VIEHIAND, 1984,49,823. 10 BUNE,A. v., FRIDKIN, v. M..DUCHARME, s., 36 WESTPHAL, v., KLEEMAN,w., GLINCHUK, M. D., Phys. Rev. Lett. 1992,68,847. BLINOV, L. M., PALTO,S. P., SOROKIN, A. V., 37 ISHIBASHI,Y.,IWATA,M.,]pn.]. Appl. Phys. YUDIN, S. G., ZLATKIN, A., Nature 1998,391, 1998.37.L985; 1999,38,1454. 874.

son et al., Milia et al., Alexandrova et al.) and neutron scattering (Currat, Quilichini, Dolino).Systems with one (lq),two (2q)and three (3q) independent modulation waves were observed.The modulation waves were found to be either of the plane-wave type or were soliton-like. Phason and amplitudon modes were observed in the plane wave regime both by NMR and NQR as well as by neutron scattering. (For references see Ref. [43]) 3q-type solitons were found by "As NQR in proustite [44] together with a lq-3q transition. In 1985 a new family of IC structures namely quasicrystals were observed by Shechtman et al. In 1990 the IC phase in high T, superconductors was found.

11

References - R. Blinc

Cox,D. E., SHIRANE, G.,Guo, R.. JONES, B.. CROSS,L. E., Phys. Rev. 2001,

38 NOHEDA, B.,

39 40 41 42 43

44 45 46 47

48 49 50

51 52 53

54

55

56 57

58

863, 2501. SAMARA, G. A,, Phys. Rev. Lett. 1996, 77, 314; 59 see also Phys. Rev. 2001, B61, 3889. 60 IIZUMI, M., AXE, J. D., SHIRANE, G., SHIMAOKA, K.. Phys. Rev. 1977, 815,4392. AXE,J. D., IIZUMI, M., SHIRANE, G.,Phys. Rev. 61 1980, 822, 3408. J A N N E R , A,, J A N S S E N , T., Phys. Rev. 1977, 815, 62 643. Incommensurate Phases in Dielectrics, Vol. I and 11, Elscvicr Science Publishers, 1986 (edited by BLINC, R., LEVANYUK,A. P.). 63 APIH,T., ct al., Phys. Rev. Lett. 1998, 80, 2225. LFVANYUK,A. P., SANNIKOV, D. G., ETF 1968, 55, 256. 64 INDENBOM, V. L., Kristallograjiya 1960, 5, 115. DVORAK,V., Ferroelectrics 1974, 7, 1; Phys. Stat. 65 Sol. (b) 1971, 45, 147. 66 PYITE,E., Solid State Commun. 1970, 8, 2101. AIZU,K., J. Phys. Sac. Jap. 1970, 28, 717. LEVANYUK,A. P., SIGOV, A. S., Defects and 67 Structural Phase Transitions, Gordon and Breach, New York, 1988. 68 LARKIN. A. I., KHMELNITSKII, D. E., Zh. Eksp. Teor. Fiz. (.ETF) 1969, 56, 2087. 69 IMRY, I., MA, S. K., Phys. Rev. Lett. 1975,35,1399. KLEEMANN,W., et al., Int. J. ofMod. Phys. 70 1993, 7, 2469. 71 ELLIOT,R. J., I N STRUCTURAL PHASETKANSITIONS 72 A N D SOFTMODES,EDITORS SAMUELSEN, E. J., ANDERSEN,E., FEDER,J., Universitetsforslaget, 73 Oslo, 1971. 74 SAMARA, G.A,, J. Phys. SOC.Jap. 1970, 28, Suppl. 399. 75 PEEREY, P. S., SAMARA, G. A,, Phys. Rev. 1972, 86, 2748. 76 KORALEWSKI,M., NOHEDA, B., GLESIAS,T., GONZALO, J. A., Phase Transitions 1994, 47, 77.

.

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NELMES,R. J., E.G. LOCKWOOD, D. J., OHNO, N., NELMES, R. J., AREND,H.,Jap. J. Appl. Phys. 1983, 24, Suppl. 24-2, p. 510). PETZELT, J., OSTAPCHUK, T., Ferroelectrics 2001, 249, 81. VOGT. H., Jap.J. Appl. Phys. 1985, 24, Suppl. 24-2, p. 112. GRIGAS, J., et d.,Jap.J. Appl. Phys. 1985, 24, Suppl. 24-2, p. 525. BLINC, R., in Magnetic Resoname ofphase Transtions (Eds. OWENS, F. J., POOLE, C. P., FARACH,H. A.), Academic Press, London, 1979. Local Properties at Phase Transitions (Ed. MULLER,K. A.), North Holland, Amsterdam, 1976. DALAL,N. S., KLYMACHYOV,A., BUSSMANNHOLDER, A., Phys. Rev. Lett. 1998, 81, 5924. BLINC, R., et al., Phys. Rev. 1991, 843, 569. DOLINSEK, J., AREON,D.. LIAR, B., PIRC,R., BLINC,R., KIND,R., Phys. Rev. 1996, 854, R6811. GESI,R., J. Phys. Soc. Jpn. 1980, 48, 886; 1981, 50. 3185. NODA,Y., KASATANI, H., WATANABE, Y., TERAUCHI, H., J. Phys. Soc. Jpn. 1992, 61, 905. MIKAC,U., et al., Phys. Rev. 1999, B59, 11293; Phys. Rev. 2000, B61, 197. SIATER,J., J. Chern. Phys. 1941, 9, 16. LIEB,E. H., Phys. Rev. 1967, 162, 162; Phys. Rev. Lett. 18, 1046; 1967, 19, 108. SUTHERLAND, B., J. Math. Phys. 1970, 1 1 , 3183. FAN,C., Wu, F. Y., Phys. Rev. 1970, 82, 723. BAX~ER, R. J., Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982. KIND, R., BLINC,R., LIAR, B., et al., to be published. BILZ, H., BENEDEK,G.,BUSSMANN-HOLDER, A., Phys. Rev. 1987, B354840; ibid. 1989, 839, 9214.

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4 Phase Transitions in Ferroelectrics V. L. Cinzburg

Ferroelectricity: % Fundamentals Collection. Edited by JulioA. Gonzalo and Basilio Jimknez Copyright 0 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40486.4

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4 Phase Transitions in Ferroelectrics: Some Historical and Other Remarks’) V. 1. Cinzburg P. N. Lebedev Physical Institute, Russian Acad. Sci., Moscow, Russia

The talk is devoted to comments on the his- tion for the present conference. However, tory of creation of the theory of ferroelectric when invited to do this, I felt I wished to phenomena. It is primarily concerned with attend an International conference on ferrothe work of V. L. Ginzburg and other Rus- electricity at least once in my life. The point sian physicists in this field. The topics is that I first took interest in ferroelectric touched upon are: (i) application of the phenomena in 1945 and published a numLandau theory of phase transitions to ferro- ber of papers [2-71 on this subject, but not a electrics (ii) elaboration of the “soft mode” single time did I have an opportunity to parconcept (iii) relation between the ferro- ticipate in such conferences. And when inelectricity and the high-temperature super- vited to the 6th such conference held in conductivity (iv) determination of the lim- Kyoto (Japan) in 1985, I submitted a paper its of applicability of the Landau theory of [8]mainly devoted to my own works [2-71. phase transitions (v) revealing of special Such were the conditions of life and work cases of ferroelectricity: improper and in- for most physicists in the Soviet Union in commensurate ferroelectrics, ferrotoroics. the years of “cold war”. To somehow illustrate the conditions of our work, I shall 1. My talk is devoted to comments on the mention the following fact. There was such history of creation of the theory of ferro- a phrase in my paper [8]: “I cannot attend electric phenomena. I shall touch upon re- the present Meeting and most probably will sults, mostly of phenomenological nature, not be able to in future”. The editor of the which are now well-known and are eluci- Russian version discarded this phrase. I am dated in many books (I shall mention here happy to have lived to the downfall of the at least the excellent monograph by M. Lines Communist regime nearly 10 years ago. and A. Glass [l]).Therefore, the question Now people in Russia have freedom of immediately arises of why I have chosen speech and migration. I mention it because such a subject for the talk. Answering this the participants of the conference are mostly question, I shall note, first,that I myselfhave young people and they should not forget had no intention to offer any communica- about the conditions of life under totalitarian (Fascist or Communist) regime and, if 1J Report prepared for the 10‘’ International Meeting on Ferroelectricity (Madrid, Spain, September 3-7,2001). needed, should defend democracy.

132

I

4 Phase Transitions in Ferroelectrics: Some Historical and Other Remarks

I would like to note that I am against priority disputes and generally forwarding the questions of priority. Everything is clear in this sphere from corresponding publications, especially at the contemporary level of information exchange.That is why, mentioning some of my works in report [8] and below in this paper can only be justified by the fact that they were written rather long ago when Soviet physicist were unable to publish their works freely (suffice it to say that the Journal of Physics USSR stopped being published in 1947, and when it became practically impossible to publish papers in English during several years). I have no priority claims to anybody.2) It is time, however, to proceed to the subject matter (I shall make use of my paper [8], but repetitions are inevitable).

era1 and transparent. Clarity and generality were reached as a result of investigation of barium titanate (BaTiO,). In 1944 Wul and Goldman found [15] that barium titanate ceramics showed a high dielectric permitivity E which varied strongly with temperature and had a rather high maximum at T = 400 K. The polycrystalline nature of the samples and shortage of data veiled the fact that they were dealing with a new ferroelectric (paper [15]ends with a remark that the available “facts, as well as the composition and structure of barium titanate do not allow it to be included in the group of ferroelectrics”).However, I understood that BaTiO, is a ferroelectric [2] and applied the Landau theory [14,1G]to describe its behaviour; as the order parameter I chose electric polarisation P. It should be noted that the Landau theory of phase transitions is the mean (self-consistent)field theory, and in particularly simple cases it reduces to the Van-der-Waalsand other constructions that had been well-known for a long time. The power of the Landau theory rests on a consistent use of the symmetry laws and the automatism of its application.This was just reflected in paper [2],which was submitted for publication on July 31, 1945. In this paper, the thermodynamic potential (more precisely, its density) was written in the f01-m:))

2. The discovery of ferroelectricity on an example of Rochelle salt may, although rather conditionally,be referred to 1920 [9] (see [l]). Much was clarified [lo]within the decade to follow, but as far as I know, it was only in 1937 that Jaffe hypothesised the occurrence of phase transitions at Curie points (at 8, = 255 K and O2 = 297 K) in Rochelle salt [111. In the ferroelectricregion (i.e., at a temperature Tlying between 8, and e,) the crystal is monoclinic, and outside this region it is orthorhombic.Further, a phenomenologicaltheory of the behaviour 0 = @ , + a P 2 +P- P 4 + ~ P 6 - E P , (1) the Rochelle salt was developed in [12,13]. 2 G Since this salt is a complicated object and the general approach to the phase transition where E is the electric field strength and the theory, known as the Landau theory [14],was coefficients a,p, and y depend on the temnot used, papers [12, 131 were not very gen- perature T. 2)

the basis of some agreement on juridical collaboraUnfortunately,various statements are sometimes far tion with the USSR, the USA government used my from reality. That is why I permit myselfto note that testimony, with reference to paper (21, to reject the although the author of several hundred papers and a claim ofpayment for some patents exploiting BaTiO,. number of innovations, I have never applied for any patents. 1 shall recall the fact, mentioned also in [S], 3) To be absolutely precise, in [2] expression (1) with y = 0 was written down. Later, however, it was found that somewhere in the fifties I gave evidence in court necessary to add also the term 1p6 in case the tranintheUSSR,upontheorderfromthe USA,concemsition was close to the ticriticappoint Oh. ing use of piezoelectric barium titanate sensors. On

4 Phase Transitions in Ferroelectrics: Some Historical and Other Remarks - V. L. Cinzburg

The decomposition (1)is valid, generally speaking, only in the vicinity of the transition point 8; we mean second-order transitions or first-order transitions that are close to the tricritical point (this point was earlier termed the critical Curie point; at this point, i.e., at T = 8,, the coefficient p(e,,) is equal to zero). Far from the tricritical point 8,, in the case of a second-order transition one may put y = 0. In this case

>o T=8

and at T > 8 in equilibrium we have a paraelectric phase (i.e., P(T) = 0), while at T c 0 spontaneous polarization occurs and

The jump of the specific heat is (4)

Taking into account that in equilibrium

a@ - = 0 , we obtain E = 2 a P + 2 p P‘.In a dP E - 1 weak field we have P = Po + -E and, 411

accordingly, 2x a(T)=-,T>0; E - 1

a(T)= --,

II

E - 1

T i 0.

(5)

Here in E(‘T) a certain term c0 not connected with the transition is in fact neglected; assuming also that E >> 1,by virtue of (2) we obtain

E(T)= -

II

aQ(0 - T )

,T<0.

Thus, we arrive at the Curie-Weiss law and to the so-called “law of two” according to which E (T> e) = 2 for the same value of 18 - TI. In paper [2]we presented a number of other results, in particular, those near first-order transitions close to second-order transitions; moreover, some experimental data concerning BaTiO, and KH,PO, type ferroelectrics were discussed in [2]. It was also noted that above the Curie point BaTiO, showed, of course, no piezoeffect. But in the ferroelectric phase it must be observed, and paper [2] pointed to the corresponding possibilities for tetragonal or orthorhombic pyroelectric (polar) phases (i.e.,at T c 8; what was this phase in BaTiO, was then unknown). I do not want to dwell here in more detail on the content of paper [2] because it was published in English as well. 3. The main shortage of paper [2]was that if treated, in fact, the one-dimensional case alone where spontaneous polarization Po was aligned in only one direction. This approach is pertinent in the consideration of the properties of Rochelle salt or KH,PO, type substances possessing preferred axes already outside the ferroelectric phase. In the case of some ceramics, i.e., a polycrystal, for which the ferroelectric properties of BaTiO, were discovered [15],my consideration [2]with a single order parameter 13 was also natural. But in application to BaTiO, single crystals a more general approach with vector parameter 13 was needed. How to do it within the Landau theory is quite clear (see, for example, [16, 171). Why I did not do it at once can only be explained by the fact that my interest in this problem was insufficient and I was occupied also by quite different problems. But after the appearance of new experimental data [ 18-20] I eventually did it [3,4], although regretfully with some delay (paper [3] was submitted for

I

133

134

I publication on July 7,

4 Phase Transitions in Ferroelectrics: Some Historical a1od Other Remarks

1948 but could not already be published in English). Specifically, in [3,4]I used the thermodynamic potential (D = (Do + a(Pi

+%(Pi

+ Py' + P:)

+ Py" + P,")

+p2 (P2Py' + P; 1 +-s,, 2

P:

(o",+ o'w

+ s12 (oxxoyy+ ~

+ Py' P:)

Devonshire (21-231. The potential used in those papers differed from (7) only by addi1

tion ofthe term - 6' (P: + Py" + P,") . Mean6 while, when writing terms of the order of 9,one should use the general expression (P2 = P; + Py' + P,")

@,(P)

1

= GY1 p6 + Y2 [Pi(Py' +

+ o',,)

This was later done by Kholodenko and Shirobokov [24]. Incidentally, in [21]Devon(7) shire referred to my paper [2],and therefore he might have paid attention to the Landau - Y1 (oxxP,"+ oyyPy'+ .zzp:, theory, but failed to do it and hence obtained - Y2 [ o x x (P," + Pi?) only a particular result. Paper [21]was submitted for publication on July 26,1949, i.e., + on (P,"+ P i ) + crzz (Pi+ P,")] a year later than (31. At the same time, as - 2 Y3 (oqPxPy + oxzpxpz has been said, in ref. [21] the role of @-order terms was at least partially allowed for, +o y z ~ y ~, ) ( E X P X + EyPy + E Z P Z ) , and the experimental data, in particular, where = ( 9 ) = {P', Py, P,} is the polari- paper (251,were also used and taken into zation vector (the order parameter) and oik account much wider than in [3]. So, I do not is the stress tensor. in the least want to diminish the value of Devonshire's works. + Ek 'k) , and (7) In equilibrium Ei = apl, implies the relation for Ei, Pi, and oik. At 4. Quite a noticeable role in the understandT < 8, when a < 0, solutions are possible that ing of the mechanism of ferroelectric and correspond tetragonal and rhombohedra1 some other transitions has been played by symmetries. Furthermore, in [3] the coeffi- the so-called "soft mode" concept. It concients fll and f12 were so chosen that the ab- sisted in the following. Under second-order solute minimum of (D corresponded to the transitions and first-order transitions close tetragonal structure in accord with the ex- to the tricritical point, the frequency on one perimental data [18].I shall not present for- or several normal crystallattice modes tends mulas here, for they coincide with those to zero or strongly decreases. For real crysavailable in all modern courses (see 11,171). tals, however, the picture may be very comUnfortunately, I do not remember why I did plicated. The "soft mode" concept has set in not include in [3,4],unlike [2],terms of the gradually as a result of a number of experiorder of pb. Therefore, solutionscorrespond- mental and theoretical studies. It would be ing to the orthorhombic phase were not interesting to see a comprehensive analysis found and first-ordertransitions close to the of the history of this question. I can only tricritical point were not considered. This make some comments on this issue. As far shortage was partially offset in papers by as I know, Landsberg and Mandelstam [26]

+Ts" 1

(ov 2

x x ~ ,+, oyy%z)

+ UiZ+ 0;)

4 Phase Transitions in Ferroelectrics: Some Historical and Other Remarks - V. L. Ginzburg

were the first to pay attention to the “soft mode” in 1929 during the study of combinational (Raman) scattering of light in quartz near the a q p transition. It turned out that the line of 207 cm-’ (at room temperature) extends and smears out with rising temperature, and disappears altogether in p-quartz (i.e., at T > 8 = 846 K). Later on it became clear [27-291 that as the temperature grows, the frequency of this line falls sharply. These facts were taken into account by A. P. Levanyuk and me in the paper [30] published in 1960 and devoted to the spectral composition of light scattered near second-order phase transition points. In that paper we spoke of the “soft mode”, i.e., the disappearance of the frequency of the oscillation responsible for the transition, as a well-known concept (the corresponding quotation is presented also in [S]). I would not like, however, to dwell in more detail on light scattering and “soft modes”, the more so as 1 can refer in this connection to the reviews [31, 321. Now I shall notice that the “soft mode” appeared explicitly already in papers [3, 41, that is, in 1949. Since papers ($41 were not translated into English, I shall present, although it has already been done in [8], a rather long quotation: “Let us also consider the dispersion of the dielectric constant of barium titanate. This problem is much more complicated than the one considered above because it does not admit a purely thermodynamical treatment. It can, in fact, be solved only through the lattice mode investigation. Some statements and dispersion estimates can however be made without such consideration. For simplicity we shall restrict ourselves to the case of a field aligned parallel to a spontaneous moment, i.e., along the corresponding axis of the cube”.Next, comparing the static expression 2 a P + 2 p, P, = E (here P is polarization along the indicated axis) with the equation of motion for an anharmonic oscillator

m [ + ~ t + k & + s =&e ~ E,

I

(9)

we come to the equation (for the field E = E~ eiwt)

p P+ v P + a P

E ’ + p1 p 3 = 0

P = e N c , p=-

elcer

2

m

v=-

2e2 N ’

z 2e2N

’

(10)

From this, for the dielectric constant E we obtain the expressions 2n:

E =

a

+ i o v - o2p , T > q 2n:

E =

-2a+iwv-w

2

p

,T
(11)

From (11)and (3) it is clear that the eigenfrequency mi of the mode in question near a second-order transition is as follows:

Obviously, at the transition point o,= 0, and wi is precisely the “soft mode”. In [3] I present the estimate of the dispersion, which agrees qualitatively with the measurements, [33] published a year before. It should be noted that in [3, 41 expressions (12) were not written. It seems to me that they follow from (11)in an obvious way, provided one knows a usual formula on the type A . But may be I &=Em+ 2 2 0 0 - 0

-iyw

am wrong and the consideration of the soft mode in [3,4] was indeed somewhat vague (“obscurely worded, even with benefit of hindsight”, as was noted by Cochran [34]). I wrote about it in some more detail in [8].4) 4)

Note that in the English version of paper [8]the editor changed without my knowledge the phrase in my polemics with Cochran.

135

4 Phase Transitions in Ferroelectrics: Some Historicaland Other Remarks

Here, I s h d restrict myself to the remark that in any case I do not agree with the opinion (see [l], Sec. 1.1 and [35]) that the “soft mode” concept was “introduced by Anderson [3G] and Cochran [37] in 1960” (this is a quotation from [35]). The papers of Cochran [37,38] are indeed valuable and more extensive than the corresponding material in [3,4], but they appeared 10 years lather. As concerns the paper by Anderson [lo], its citations are “adaptedby repetition”. As was justly noted by Cochran 134) in respect of the report by Anderson [36], “contrary to what many who have cited this paper believe, it is not at all primarily about soft modes!” To make it dearer, I shall add that the book in which Anderson’s paper [36] appeared was issued only in the Russian language, and I am sure that those who cited it had never seen the book (for some more details see [S]).

10 KURCHATOV,I. V., Segnetoelectriki(Ferroekxtrics).

11 12 13

14 1s

16

17

18 19 20 21 22 23 24 25 26

References LINES,M. E., GLASSA. M., Principles and Applications of Ferroelectrics and Related Materials. Clarendon Press, Oxford, 1977 [Russ. transl.,Mir,Moscow, 19811. V. L., ZhETF 1945,15,739; GINZBURG, Journ. Phys. USSR 1946,10, 107. V. L., ZhETF1949,19, 36. GINZBURG, GINZBURG, V. L., Usp. Fiz. Nauk 1949,38, 490. GINZBURG, V. L., Fiz. Tverd. Tela 1960,2, 2031; Sov. Phys. Solid State 1960,2, 1824. GINZBURG, V. L., SOBYANIN, A. A., Fir. Tverd. Tela 1983,25,2017. GINZBURG, V. L., LEVANYUK,A. P., SOBYANIN, A. A., Ferroebctrics 1986,73, 171. P. N., Lebedev Physics GINZBURG, V. L., TRUDY, Institute 1987,180, 3; Ferroelectrics 1987,76, 3. VALASEK, J., Phys. Rev. 1920,15, 577; 1921,17, 475.

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32 33 34 35 36

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M. GTTI, 1933. Le champs moleculaire dans le dielectriques. Paris, Herman, 1936. JAFFE, H., Phys. Rev. 1937,51,43;1938,53, 917. MUELLEQ H., Phys. Rev. 1940,57,829 1940, 58,805. CADY,W., Piezoelectricity. N. Y. London, 1946. Russian transl. M. IL, 1949. LANDAU,L. D., ZhETF 7 1937,19,627. WUL,B. M., GOLDMAN, I. M., Dokl. Akad. Nauk S S S R 1945,46, 139. LANDAU, L. D., LIFSHITZ, E. M., Stutisticheskaya Fizika (Statistical Physics) Part 1 M. Nauka, 1995. LANDAU,L. D., LIFSHITZ, E. M., Efectrodinamika Sploshnykh Sred (Electrodynamics of Continuous Media) Sec. 19 M. Nauka, 1992. MEGAW,H. D., Proc. Phys. SOC.1946,58, 133. ROBERTS,S., Phys. Rev. 1947,71, 890. RZHANOV,A. V., Usp. Fiz. Nauk 1949,38, 461. DEVONSHIRE, A.. Phil. Mag. 1949,40, 1040. DEVONSHIRE, A., Phil. Mag. 1951,42, 1065. DEVONSHIRE, A., Adv. Phys. 1954,3.85. M. YA., KHOLODENKO, L. P., SHIROBOKOV, ZhETF 1951,21,1237, 1250. KEY, H. F.. Phil. Mag. 1949,40,1019. LANDSBERG,G.S., MANDELSHTAM, L. I., 2s. Phys. 1929,58, 250. NEY, M. J., Zs. Phys. 1931,68,554. NADUNGADI,T. M., Proc. Indian Acad. Sci. 1940,A1 1,85. NARAYANASWAMY, P. K., Proc. Indian h a d . Sci. 1947.26, 521. GINZBURG, V. L., LEVANYUK, A. P., ZhETF 1968,39, 192; Sov. Phys. JETP 1961,12, 138. GINZBURG, V. L., LEVANYUK,A. P., SOBYANIN, A. A., Phys. Rev. 1980,57,151; this paper was also published in: Light Scattering Near Phase Transitions (Eds. CUMMINS, H. Z., LEVANYUK, L. P.), Amsterdam, North-Holland, 1983. KRISHNAN,R. S., Ferroelectrics 1981,35, 9. J. G.,Nature 1948,162, 614. POWLES, COCHRAN, W., Ferroelectrics 1981,35, 3. COWLEY, R. A., Ferroelectrics 1984,53, 27. ANDERSON, P. V., Paper in the book ”Fizika Dielektrikov” (Physicsofoielectrics) (Ed. SKANAVI, G.I.), M. Izd. Akad. Nauk SSSR, 1960. COCHRAN, W., Adv. Phys. 1960. 9, 387. COCHRAN, W., Phys. Rev. Lett. 1959,3,412.

5 Theory of Ferroelectrics R. E. Cohen

Ferroelectricity: The Fundamentals Collection. Edited by Julio A. Gonzalo and Basilio Jirnbnez Copyright 0 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40486.4

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Theory of ferroelectrics: A vision for the next decade and beyond Ronald E. Cohen Carnegie Institution. of Washington 5251 Broad Branch Rd., N . W. Washington, D. C. 20015

Abstract

In the last, ten years significant advances have been made in our understanding and ability to compute and predict properties of ferroelectrics and piezoelectrics using fiiridaiiiental physics. Phase diagrams, electromechanical and elastic properties, arid effects of defects and surfaces are now amenable to computation. Most importantly, new techniques have been developed, and new understanding of the meaning of polarizat,ion in dielectrics has been developed. Prospects for the future are discussed. Key words: ferroelect,rics;elect,ronic st,ruct,ure;piezoelectricity; piezoelectrics; first,-principles; phase diagrams; surfaces, defects

1 Introduction

About ten years ago a concerted effort began t o apply modern first-principles band structure methods t o the p r o b l e m of ferroelectricity, with a vision t h a t e1ectroinec:lianical properties could perhaps one day b e designed coniputationally [l].This effort has been at, least as successful as hoped, and a range of problems that we didn’t, even realize existed at the time have been identified and solved. Understanding elect,roniechanical response has turned out t o be a rich problem, both from a fundamental condensed matter physics perspective, as well as from the applied physics and even engineering perspectives. There is iiiuch expanded int,erest, in the field now t,hat a new class of rria,terials have been discovered, single crystal piezoelectrics with ultra-high electromechanical couplings such as PhIIN-PT (PbMgi Nbz 0 3 - P b T i 0 3 ) which will revolutionize 3 3 fields ranging from sonar arid hydrophones t,o medical ultrasonic imaging [2]. T h e current st,ate of the field, as will be discussed below, is that, we now can conipute all of the parameters related t o elect,romechanical coupling, particularly t,he piezoelectric constants, for pure, ordered, single crystal phases [3,4]. J. Phys. Chem. Solids, Vol. 61 (2000)

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are that quite reliable results can also be obtained as functions of temperature [ 5 ] .The problems now being studied are how to treat disordered solid solutions, and to understand the underlying physics behind the observed material behavior-why is one material better than another. In the future, using the computational techniques developed, and the insights obtained from theory and experiment, we will be able to predict material properties of as-yet unsynthesized complex materials. The current state of the field and future prospects are outlined here. This is not intended as a comprehensive review, but should be a useful introduction to the field of first-principles studies of ferroelectrics and piezoelectrics. The reason to use first-principles methods is that (1) one is not reliant on parameterized theories or (2) on fitting possible inaccurate experiments, (3) one has access to the underlying potential surfaces, (4)can clearly see the origin of observed behavior, and ( 5 ) they can be applied to hypothetical or not yet synthesized materials, or (6) for temperature, pressure or compositions for which data are not available. By “first-principles,” we mean that experimental data are not used to constrain parameters; rather one starts from the fundamental interactions among electrons and nuclei. Most of the first-principles methods that have been applied to ferroelectrics are based on the density functional theory (DFT) [6] and some are based on Hartree-Fock theory. The DFT states that the ground state properties of a system are given by the charge density, and Kohn and Sham [7] showed how to compute the charge density and energy self-consistently, using an effective exchange-correlation potential (Vzc) that accounts for the quantum mechanical interactions between electrons. The local density approximation (LDA) takes the exchange-correlation potential from the uniform electron gas at the density for each point in the material. The Generalized Gradient Approximation (GGA) includes the effects of local gradients in the density [8]. Given a form for the exchange correlation energy E,,[p], one can find the self-consistent charge density and compute the energy for any arrangement of nuclei (atoms). From the energies, zero temperature phase diagrams, phonon frequencies, and elastic constants can be computed. In the frozen phonon method, for example, one displaces atoms and computes the change in energy as a function of displacement, from which the potential surface for an instability, or phonon frequencies, can be derived. In principle one can also compute finite temperature properties, for example using Monte Carlo or molecular dynamics methods. In practice, self-consistent methods are extremely computationally demanding, especially for large systems, so instead one can find effective potentials or Hamiltonians fit to the first-principles results to obtain finite temperature properties. Properties of ferroelectrics are extremely sensitive to volume (pressure), which can cause problems since the present first-principles methods are not perfect, and small errors in volume (typically several percent, or more in some cases) 2

5 Theory ofkrroelectrics - R. E. Cohen

can result, in large errors in computed ferroelectric properties. The GGA is generally an improvement over the LDA, but still many computations for ferroelectrics so far have been done at the experimental lattice. The GGA gives reasonable c/o, at the experimental volume. For PbTi03 at the experimental volume (427.27 bohr3), for example, wc found c/a equal to 1.12 (90% high) for LDA (Hedin-Lundqvist), 1.09 for LDA (Wigner), and 1.073 for GGA (PBE), compared to the experimental value of 1.063. However, the volume is still too large in GGA and c/a is thus overestimated. A new form for the exchange and correla.t,ioncalled the Weighted Density Approximation (WDA) shows much promise, and gives excellent results for the lattice constants and energetics [9],so t,here is hope that in the future all properties of ferroelectrics could be computed truly ab i ~ i i t i uwithout recourse to the experimental volume.

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2.1

Milestones

Tota,l energy and electronic structure

At first it was unknown whet,her first-principles methods within the LDA would even give ferroelectric ground states where they should. If fact, in the initial computations for BaTi03 (read “initial” as over one year!) the author’s LAPW computations did not give a ferroelectric ground state due to a minor bug t1ia.t apparently did not effect other systems! After this problem was fixed, the ferroelectric instability was found at the experiment, volume, though it, was extremely sensitive t,o pressure. The initial questions then after finding the instability were (1) what, is the underlying cause of the ferroelectric instability arid (2) why do different similar materials behave differently; particularly why does BaTi03 show a series of phase transitions from cubic, to tetragonal, to orthorhornbic, and to rhombohedra1 with decreasing temperature whereas PbTiOs has a single phase transition to tetragonal? The potential surface was initially mapped out, using the LAPW method, and the charge density and electronic st,ructure were analyzed [lo-121. PbTi03 showed a much deeper well wheii thc tetragonal strain was included, whereas in BaTi03 the rhombohedral phase had the lowest, energy. Thus the t,etragonal strain is responsible for the tetragonal ground state in PbTiO3. Analysis of the charge densities and densities of states showed t,hat the ferroelectric instability is due t o hybridization between the 0 2p states and the Ti 3d states, and if the 3d variat,ional freedom was removed from the problem, the ferroelectric instability vanished [lo].In BaTiOs, the Ba is quite ionic and spherical, whereas the Pb in PbTi03 is very riot spherical in the ferroelectric phase, and polarization of the P b helps stabilize the large st,rairi arid tJhe tetragonal ground state in PbTi03. Thus thc following picture emerged. All Coulomb lattices are unstable with

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stabilize crystals with respect to off-center displacements. In perovskite like BaTi03 and PbTi03, the 0 2p states strongly hybridize wit,h the do cation, in this case Ti4+,reducing the short-range repulsions thus allowing off-cent,er displacements. Without strain, or when strain effects are minor, the lowest energy off-center displacements are along the (111) directions. Thus the ground state is rhombohedral. As temperature is raised, the off-center displacements disorder over two directions, giving an average orthorhombic symmetry, then there is a disordering over four directions giving tetragonal symmetry, and final a disordered cubic phase at high temperatures. The same features are found in BaTi03 and KNb03 [13,14]. One should not overemphasize this orderdisorder picture too much, however. In the ideal order disorder picture, the atoms are in off-center sites at all temperatures, and there should be a large configurational entropy change at the ordering phase transitions, which is not, observed. Furthermore, in the ideal disorder picture, the local potential for displacing a single B-site cation such as Ti or Nb is unstable, but calculations show that the diagonal, self-force constant, is positive, and so there would be a restoring force for displacing a single cation from its ideal position in the perfect cubic perovskite, and only when groups of ions are moved (for example in a I?-point displacement) is there a multiple well potential surface [15]. It is clear that the polarizability of P b plays a special role in ferroelectrics. PbTiOy has a very large strain (6%) where tetragonal BaTi03 has only a 1% strain. The ground state of PbZrOs is antiferroelectric and it has a complex phase diagram [16], whereas BaZrOs is cubic [17]. PbZr03 has a complex structure at low temperatures, and it was structural total energy relaxations within the LDA that clarified the crystal structure [16]. Thus first-principles theory has the power now to find problems with experimental determinations of complex structures. Another example of theory finding problems with experimental analysis is LiNbO3. In the LiNb03 structure there are chains of oxygen octahedra with Li ions shared between two octahedra. Based on neutron scatter experiments, it was suggested that the Li atoms hopped between two octahedra at high temperatures in the high temperature phase, and ordered to occupied one set of octahedra at low temperatures in the polar phase [18.]. Analysis of experimental data suggested that the underlying potential surface had a triple well structure for Li displacements, and all models considered the Li motions to the important physics driving the ferroelectricity. Inbar and Cohen performed LAPW frozen phonon computations and found a quite different picture [19]. They found no triple well, and they found that the double well for Li displacements was quite shallow, and the double well for oxygen displacements much deeper. The deepest wells were found for coupled displacements of Li and 0. They found that the ferroelectric instabilities in LiNb03 and LiTaOBare quite similar to the instabilities in the oxide perovskites, and that a primary driving

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5 Theory ofFenoelectrics - R. E. Cohen

force is the do configuration of the Nb5+ and Ta5+ ions, and hybridization with the surrounding oxygens that allows the do cation to go off-center. This sets up local fields that drive or order the Li off-centering. Though static total energy computations are very powerful, and further examples will be given below, they are limited in the ability to find phonon or instability eigenvectors or behavior for arbitrary wavevector. Advances that give the linear response for arbitrary wavevector are discussed next.

2.2 First-principles lattice dynamics-linear response

Using conventional methods, first-principles methods with the LDA or GGA scale as N3, where N is the size (say number of atoms) of the system. This makes it prohibitive to study arbitrary wavevectors. For small wavevectors a very large supercell would be required. In linear response computations that problem is circumvented, and one can compute the response of the system to small perturbations with arbitrary wavevector with approximately the same cost as the primitive cell computation. Yu and Krakauer developed a linear response LAPW code and applied it to phonon dispersion in KNb03 [13,20]. They found the ferroelectric instability to be dispersive in certain directions, indicating correlations in displacements in real space. In other words, if the unstable band were flat, it wouldn’t matter how the atoms were displaced in one cell relative to the next. Analysis of the instability dispersion showed that there should be chain-like correlations, consistent with streaking observed in x-ray diffraction [21]. Linear response methods also allow computation of Born effective charges (discussed below) and the dielectric constant. A number of different ferroelectric perovskites have now been studied using linear response methods. The unstable modes of KNb03 and BaTi03 show very similar dispersion, consistent with the same sequence of phase transitions for the two. The soft branch has primarily B-ion character. PbTi03 shows a much more flat soft branch, indicating that correlations between cells are less important than in KNb03 or BaTi03. The soft modes also have appreciable A-ion character (Pb) in PbTi03 compared with KNb03 or BaTi03. PbZr03 is similar to PbTi03 in that the soft branch remains unstable for any wavevector, but the zone boundary R- and M-point instabilities are deeper than the I?point instability, consistent with the fact that the ground state in PbZr03 is antiferroelectric, and PbTi03 is ferroelectric.

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Effective Hamiltonians-finite temperature properties

Although self-consistent methods are very powerful, they generally cannot provide anything but static, essentially zero temperature, properties, at least for the complex systems we are discussing. Ferroelectric phase transitions, however, occur with increasing temperature from one ferroelectric state to another, or to the high temperature paraelectric state. Dielectric and piezoelectric properties are strongly dependent on temperature, especially around the phase transitions. Thus we want techniques that can give finite temperature behavior. One approach is to use a Gordon-Kim based model and Molecular Dynamics or Monte Carlo [22,23],but these models have not been successful so far to compute finite temperature properties for ferroelectrics. An effective Hamiltonian approach has, however, been very successful [24,25]. One develops a symmetrized expansion of the free energy in terms of local mode coordinates, effective charges, and strains. Only the important local modes are used, and thus the number is degrees of freedom is reduced from the full atomic Haniiltonian. Local modes are important if they are related to the soft mode, and the choice of local modes is key to the success of the method. A rigorous formulation of local modes as Wannier-like functions has been developed [26]. Though the effective Hamiltonian method contains all of the physics to give qualitatively correct results, one might question whether it would be quantitatively accurate since it neglects the higher frequencies modes, and does not even try to estimate their average self-consistent effects on the ferroelectric behavior. One such example is the importance of thermal expansivity, which is not included in the effective Hamiltonian models so far. Since ferroelectricity is very sensitive to volume, one would think that neglect of thermal expansivity might lead to serious errors. This might be partly responsible for the shifts in predicted transition temperatures relative to experiment. So far, however, it appears, that the effective Hamiltonian models are very accurate, and they are the only game in town for finite temperature properties of ferroelectrics at the present time. Since the underlying first-principles calculations are also not perfect, it is surprising and encouraging that such high accuracy can be obtained.

2.4 2.4.1

Physics of Polarization Polarization in Periodic Boundary Conditions

Contrary to intuition and many textbook discussions, polarization cannot be determined directly from the self-consistent charge density or changes in the charge density alone within periodic boundary conditions. Rather one must consider changes in polarization as current flow, and this can be determined

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from the phases of the wave functions. King-Smith and Vanderbilt showed how this can be computed using a Berry’s phase approach [27]. A more intuitive approach is the realization that polarization changes can also be considered as the charge transport on displacing Wannier functions [as].Practical calculations generally use the Berry’s phase approach, and the development of this technique was a key advance in the ability to compute electromechanical properties. A review of the theory of polarization is given by Resta [29]. Born effective charges are defined as the change in polarization with displacement of a nucleus, = dPp/du;,, for the change in ? moving ion i; note that the effective charge is a tensor. The effective charges are not necessarily equal to the nominal ionic charges, and can be quite different. In oxide ferroelectrics they tend to be greatly enhanced for certain directions, with values up to several times the nominal values. The enhancement of the effective charges is due to strong covalent hybridization [30-321. Effective charges have been predicted for a number of ferroelectric systems now. They are accessible to experimental determination, for example by optical measurements of LO-TO splitting. Very large values were experimentally obtained for BaTi03 many years ago [33], but there was no theoretical basis for understanding the results for many years. It would be of great interest for experimentalists to measure these values for a variety of ferroelectrics and compare with the theoretical predictions. Piezoelectric constants can also be determined theoretically from changes in polarization with respect to strain, as discussed below, using the Berry’s phase technique.

2.4.2 Density polarization theory

Another area of much discussion is in even more fundamental problems of polarization under Kohn-Sham theory. It was argued that DFT did not properly account for electronic polarization, and implied that previous calculations were incorrect, and that the gap problem and low dielectric constants in LDA were due to this failure [34]. Mazin and Cohen [35] questioned this, and showed that there is no problem with established methods such as the LDA. The situation has been clarified [36,37] and it appears that to obtain the proper polarization in exact Kohn-Sham theory (which no one knows how to do for real bulk systems of interest, anyway) one would have to consider the polarization as well as the charge density. Realistic estimates of this effect suggest it may be very small (3%, [38]). Regardless of the theoretical importance of density polarization in the exact Kohn-Sham theory, it is questionable if it is solely responsible for the gap problem and LDA dielectric constants. The gap problem varies considerably from one material to another, as do dielectric constants. For example, although the LDA high frequency dielectric constant 7

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study of a set of 11 materials gives an r.m.s. error of 8% from experiment for em using LDA [40]. It is not at all clear than there is a uniform cause for these deviations.

2.5 Piezoelectricity The electromechanical response important in transducer applications is given by the piezoelectric constants, or by the figure of merit of the electromechanical coupling factors. Piezoelectric constants can be written in different ways and one can transform from one set to the other using thermodynamic transformations [41].The set most easily computed are the piezoelectric strain coefficients eijk = (de/a&jk).=,, the change in polarization with strain at zero applied field, usually reduced to the Voigt form E ~ j, = 1...6. Since the polarization can now be computed using the Berry’s phase approach, one can compute the polarization versus strain and obtain the derivatives for the piezoelectric constants numerically. It is important then to allow the atomic coordinates to displace as functions of strain, because this is in fact the major contribution to the piezoelectric response. The piezoelectric constant can be written as

where the first term, e$k, is the change of polarization with homogeneous strain, that is no atomic relaxations, and the second is from the displacements of the ions in response to strain, carrying effective charges 2. This was first carried through for PbTi03 [3]. Ferroelectric PbTi03 is tetragonal, and there are three independent piezoelectric constants which can be found by three strains, tetragonal, orthorhombic, and a monoclinic strain. Two methods can be used, the direct method, that is finding the polarization versus strain, relaxing the internal coordinates at each strain, and using the effective charges as shown above. Both methods give the same final results, but the two step procedure also illustrates the relative importance of the two terms. First one must find the optimum minimum energy structure at the volume of interest, in our case, the experimental zero pressure volume. Then, for example for ea3(using Voigt notation), one varies the c-axis length, and computes the change in polarization, giving ek3. One finds the effective changes by displacing 8

5 Theory ofFerroelectrics

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R. E. Cohen

cach ion separately, and computing the change in polarization in each case, and then finds how the at,oms relax as a function of strain. Finally one can compute the polarization at the final relaxed strained lattice. This gives two coniputatio~isof e33, which we checked are identical, and give 3.23 C/m2. The homogeiieous contribution is -0.88, and the internal relaxations give 4.11, so the t,otal is dominated by the at,omic relaxations. Similarly, we found eI5=3. 15 C/m2 and ~ ~ ~ = - 0C/m2. . 9 3 These are the proper moduli that do not include homogeneous deformations of the original spontaneous polarization, which are not observable. As pointed out later in this volume, the straightforward way to extract, the proper piezoelectric response is t,o compute differences in Berry's phases, and convert. to piezoel ric: constants only in the last, step [42]. The coniputcd moduli were very close to some experimental results, but there is a wide spread between different, experiments even for PbTi03, indicating the importance of ohtaining bet,ter values of experimental piezoelectric moduli iinder well-defined clectrical boundary conditions in order to test, theory.

We also computed e13 for two ordered structures of PZT50/50, near the morphotropic phase boundary and obtained values significantly lower than experiniental measurements for PZT ceramics (single crystals were never made successfully for PZT) extrapolated to low temperatures [4]. This lead us to toncliide either that (1) extrinsic effects are very important even at low tempcratures in PZT and/or (2) that the single crystal e33does not dominate the piezoelectric response in the poled direction in PZT. Using an effective Hamiltonian and Monte Carlo, Garcia and Vanderbilt studied piczoelectric response of BaTi03 as a function of applied field and teriiperature [43].They obtained good agreement with the temperature dependence of the piczoclectric constants, and found a field induced rhombohedra1 to tetragonal pliase transition at very large fields. Rabe and Cockayne used an effective Hamiltonian for PbTi03 and found d33 as a function of temperature using hlonte Carlo, also finding good agreement with the experimental temperature dcpendencc, which peaks strongly at the ferroelectric phase transition [5].

2.6 Solid solutions

hlost, useful piezoelect,rics are solid solutions rather than pure ordered compounds. This allows their properties to be tuned to meet engineering specifications. But, more than that, niariy complex solid solution ferroelectrics, for example, the new single crystal piezoelectrics with giant electromechanical coupling, like PMN-PT, have much enhanced properties relative to the pure compounds. Even the eridiriernber PMN is complex, with ions of different valence, Mg" and Nb5+ on the same crystallographic site. There has been some work to study the orderiiig energetics in these systems. In PST-PT, for ex9

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relative energetics of ordering, and to find an order-disorder phase diagram basic on the Cluster Variation Method (CVM) [44]. More recently, self-consistent methods have been used along with Monte Carlo [45]. Even simple Madelung sums have been shown to illustrate some of the important behavior in these systems [46]. Only limited work so far has studied how ordering (short-range and long-range) interacts with ferroelectricity and piezoelectricity. Computations show that different ordering schemes can drastically affect piezoelectric response in one case [47],but how general this is not yet known.

2.7 Defects, domain boundaries, and surfaces Real crystals are not perfect and infinite, and defects, domain boundaries, and surfaces play important roles in ferroelectrics. Park and Chadi studied oxygen vacancies in PbTi03, which may be important in fatigue. Domains have been studied using an effective Hamiltonian, and it was found that 180" domain boundaries are very sharp, and even one unit cell away the structure looks like bulk [48] Large scale LAPW computations were performed on periodic BaTi03 slabs. The computations showed that just one unit cell away from the surface, the charge density is indistinguishable from bulk BaTi03. There are two types of surfaces with either T i 0 2 or BaO terminations. On the Ti02 surface the dangling Ti-bond relaxes back onto the surface, but atomic relaxations are quite small. A surface state was found in the band structure. LAPW computations for symmetrically terminated surfaces were straightforward to interpret, and using results from slabs with two BaO and two Ti02 surfaces, an average unrelaxed surface energy of 920 erg/cm2 was determined by comparing with the bulk BaTi03 energy [49]. Using plane waves and pseudopotentials, Padilla and Vanderbilt found an average relaxed surface energy of 1260 erg/cm2 [50]. Hartree-Fock calculations using Gaussian basis sets gave 1690 erg/cm2 (unrelaxed) [51]. The differences in basis sets as well as HartreeFock versus LDA may be responsible for the differences. The effect of periodic slabs versus isolated slabs, as used in the Hartree-Fock study, amount t o only about 3% in the average surface energy for symmetric slabs. Clearly these are exploratory studies, and we cannot confidently state the exact surface energy for BaTi03 except all agree it is high, which explains the difficulty of of cleavage in BaTi03 and the rough surfaces that form on fracture. In the LAPW study, periodic asymmetric and polar slabs were also studied [49]. As discussed there, asymmetric or polar periodic slabs produce artificial potential gradients across the slab which is like an external applied field. This is an undesirable effect that makes it difficult t o interpret the results. Nevertheless, it was interesting that the polar slab has a higher energy than the ideal slab, due to the unscreened depolarization field. The field is large enough

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to cause the slab to be metallic with densities of states overlapping from one side of the slab to the other. It is much less metallic than would be the case in the rigid band picture, however, with oxygen density of states piling up at the Ferini level. This is due to the fact that 0'- really wants to be closed shell, and not metallic, wen on surfaces. In the pseudopotential study, symmetric slabs were studied, but a polarizat,ion parallel to the slab was also considered [50]. Thc isolated slab Hartree Fock study is the most straightforward to interpret, and they found results similar to bulk BaTiOy. The iise of isolated, rather than periodic, slabs, greatly facilities interpretation of results for asymmetric or polar slabs [51]. Padilla and Vanderbilt also studied SrTi03 surfaces and found a very similar surface energy of 1360 erg/cm2 compared with 1260 in BaTiOs. In SrTi03, they found without relaxation a value 14% higher, very close to the estimate of 12% for t,lie surface relaxation energy in BaTi03 of Cohen [49].

3

The Future: A Vision

We see that in the last ten years significant advances have been made in computing properties and understanding origins of ferroelectric behavior. We now have the ability to predict ferroelectric instabilities, electromechanical coupling, phonon dispersion and optical spectroscopy, elasticity, pressure behavior, order-disorder, and some surface and defect properties of structurally and chemically relatively simple pure compounds. Now the challenge is to address problems in more complex solid solutions, some that are structurally and/or chemically heterogeneous and some with frequency dependent properties.

3.1 Relaxor systems

In relaxor systenis the dielectric response has a broad peak as a function of t,emperature, rather than a sharp peak in a normal ferroelectric, and a frequency dependent response. The origin of this behavior is still controversial, but, is most, likely due to heterovalent disorder. It will be a challenge to compute this behavior from first-principles, but models can be parametrized using firstprinciples resultq as in the effective Hamiltonian models discussed above, and t,heri Inoleciilar dynamics or Monte Carlo simulations could be used to simulate st,ill rather small disordered systems. To simulate much larger, mesoscopic, systems, it may be necessary to parametrize models for interactions between nanoregions, leaving the atomic domain all toget,her. But this may be possible using only ab initio results.

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5 Theory offerroelectrics

New piezoelectrics

The discovery of new single crystal piezoelectrics with huge electromechanical coupling [2,52,53] are revolutionizing the field of piezoelectric transducers. Theory did not play a role in discovering these materials, but it, may play an important role in showing how they work and helping design better materials. The new materials are rhombohedral, with a morphotropic phase boundary to a tetragonal phase nearby. The large response is d33, which is an elongation along the cubic (001) direction, and parallel to the applied field. The rhombohedral phase though has polarizations locally about (111). As the the field is increased, the numbers of domains with polarizations that project in the (0,0,-1) direction decrease, and there is a large piezoelectric strain induced as the local polarizations are slowly rotated up towards the (001) direction. At a critical field, a field induced phase transition to true tetragonal symmetry occurs, and the piezoelectric response changes and becomes more normal. An idea of why the strain is so large in these materials can be understood by considering PbTi03. PbTi03 is tetragonal, with a large 6% strain at zero pressure and low temperatures. There is no rhombohedral phase stable in PbTiO3 at zero or positive pressures, but if there was, one would see a small strain in that phase, and one would see a giant strain as one went from the rhombohedral to tetragonal phases. One can think of phases like PMN-PT as chemically engineered “rhombohedral PbTi03.” It is clear that the polarization of the Pb2+ ion is key to this large strain behavior [12], though a thorough understanding of exactly how the Pb drives the strain has not yet been fully elucidated

3.3 Materials by design

The goal of this research program is to lead to the ability to computationally design useful materials. We are clearly on that path, but it will be a few years before one can practically hope to routinely design materials by computer. On the other hand, it is also largely a matter of having the right good idea. We understood the role of P b in the strain of PbTi03 some years ago, and if we siiiart enough to think of trying to make “rhombohedra1 PbTi03” we might have found this large strain effect before experiments. However, theoretical developments were just a few years behind to realistically do this, because the first first-principles computation of piezoelectric constants in pure tetragonal PbTiO3 were only done this last year, and it hasn’t been that long since the Berry’s phase approach to computing polarization was discovered. As we make further advances, however, in the next revolution perhaps theory will lead the way rather than follow. 12

5 Theory ofFerroeleGtrics- R. E. Cohen

Acknowledgements This work was supported by ONR grant N00014-92-J-1019. I would like to thank T. Egaini, H. FL~, H. Krakauer, I. hlazin, K. Rabe, R. Resta, G. SaghiSzabo, aiid D. Vanderbilt for helpful discussions. Computations were perforined on tlic Cray .J90- 16/4096 at t h e Geophysical Laboratory, supported by NSF EAR-9512627, The Keck Foundation, and the Carriegie Institution of Washington.

References L.L. Boyer, R.E. Cohen, H. Krakauer and W.A. Smith, First principles calcuhtions for ferroelect,rics - A vision, Ferroelec. 111 (1990) 1-7.

T.R. Shrout, S. Park, C.A. Randall, J.P. Shepard, L.B. Hackenberger, D.J. Pickrell and W.S. Hackenberger, Recent advances in piezoelectric materials, Proc. SPIE - Int. Soc. Opt. Eng. (USA),SPIE-Int. Soc. Opt. Eng, (Adelaide, SA, Australia, 1997).

G. Saghi-Szabo, R.E. Coheri, and H. Krakauer, First-principles study of piezoelect,ricit,yin PbTi03, Phys. Rev. Lett. 80 (1998) 4321-4324. G. Saglii-Szabo, R.E. Cohen, and H. Krakauer, First-principles study of piezoelect,ricit,yiii t,etragonal PbTiOy and PbZr -1Ti -1 0 3 , Phys. Rev. B , in press 2

2

(1999).

K.M. Rabe and E. Cockayne, Temperature dependent, dielectric and piezoelectric response of ferroelectrics froni first principles, in: R.E. Cohen, ed., First-principles Calculations for Ferroelectrics: Fifth Williamsburg Workshop, AIP, 1998) 61-70. P. Hohenberg and W. Kohn, Inhoniogeneous elect,ron gas, Phys. Rev. 136 (1964) 864-871.

W. Kohn and L..J. Sham, Self-consistent equations including exchange and correlation effect,s. Phys. Rev. A 140 (1965) 1133-1140.

.J .P. Perdew. K. Burke and hl. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865-3868. 1.1. hlazin aiid D.J. Singh, Weighted density functionals for ferroelect,ric matc:ria,ls, in: R.E. Cohen, ed., First-Principles Calculations for Ferroelectrics, (American Iiistit,ut,eof Physics, 1998) 251-264. R.E. Cohen aiid H. Krakauer, Lattice dynamics and origin of ferroelectricity i n BaTi0:j: Linearized augmented plane wave total energy calculations, Phys. Rev. B 42 (1990) 6416-6423.

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R.E. Cohen and H. Krakauer, Electronic structure studies of the differences in ferroelectric behavior of BaTi03 and PbTi03, Ferroelec. 136 (1992) 65-84. R.E. Cohen, Origin of ferroelectricity in oxide ferroelectrics and the difference in ferroelectric behavior of BaTi03 and PbTi03, Nature 358 (1992) 136-138. R. Yu and H. Krakauer, First-Principles Determination of Chain-Structure Instability in KNb03, Phys. Rev. Lett. 74 (1995) 4067-4070. D. Singh, Local density and generalized gradient approximation studies of KNb03 and BaTi03, Ferroelec. 164 (1995) 143-152. P. Ghosez, E. Coackayne, U.V. Waghmare and K.M. Rabe, Lattice dynamics of BaTi03, PbTiO3, and PbZr03: A comparative first-principles study, Phys. Rev. B in press (1999) D.J. Singh, Structure and energetics of Antiferroelectric PbZrO3, Phys. Rev. B 52 (1995) 12559-12563. R.D. King-Smith and D. Vanderbilt, First-principles investigation of ferroelectricity in perovskite compounds, Phys. Rev. B 49 (1994) 5828-5844. H.J. Bakker, S. Hunsche and H. Kurz, Quantum-mechanical description of the ferroelectric phase transition in LiTaO3, Phys. Rev. B 48 (1993) 9331-9335. H.J. Bakker, S. Hunsche and H. Kurz, Coherent phonon polaritons as probes of anharmonic phonons in ferroelectrics, Rev. Mod. Phys. 70 (1998) 523-536. I. Inbar and R.E. Cohen, Comparison of the electronic structures and energetics of LiTaO3 and LiNb03, Phys. Rev. B 53 (1994) 1193-1204. R. Yu and H. Krakauer, Linear-response calculations within the linearized augmented plane-wave method, Phys. Rev. B 49 (1994) 4467-4477. R. Comes, M. Lambert and A. Guinier, The chain structure of BaTiOs and KNb03, Solid State Comm. 6 (1968) 715-719.

Z. Gong and R.E. Cohen, Molecular dynamics study of PbTi03 using nonempirical potentials, Ferroelec. 136 (1992) 113-124. L.L. Boyer, H.T. Stokes and M.J. Mehl, Application of a Kohn-Sham-like formulation of the self-consistent atomic deformation model, Ferroelec. 194 (1997) 173-186.

W. Zhong, D. Vanderbilt and K.M. Rabe, Phase transitions in BaTi03 from first principles, Phys. Rev. Lett. 73 (1994) 1861-1864. W. Zhong, D. Vanderbilt and K.M. Rabe, First-principles theory of ferroelectric phase transitions for perovskites: The case of BaTiO3, Phys. Rev. B 52 (1996) 6301-6312. K.M. Rabe and U.V. Waghmare, Localized basis for effective lattice Hamiltonians: Lattice Wannier functions, Phys. Rev. B 52 (1996) 13236-13246.

14

5 Theory ofFerroeledrics

-

R. E. Cohen

[27] R.D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B 47 (1993) 1651-1654. [28] D. Vanderbilt and R.D. King-Smith, E1ect)ricpolarization as a bulk quantity and its relation to surface charge, Phys. Rev. B 48 (1993) 4442-4455. [29] R. Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach, Rev. Mod. Phys. 66 (1994) 899-915. [30] R. Resta, M. Posternak and A. Baldereschi, Towards a quantum theory of polarization in ferroelectrics, Phys. Rev. Lett. 70 (1993) 1010-1013. [31]

M.Post,ernak, R. Resta and A. Baldereschi, Role of covalent bonding in the polarization of perovskite oxides: the case of KNb03, Phys. Rev. B 50 (1994) 8911-8914.

[32] W. Zhong, R.D. Kingsmith and D. Vanderbilt, Giant LO-TO splittings in perovskite ferroelectrics, Phys. Rev. Lett. 72 (1994) 3618-3621. [33] .J.D. Axe, Apparent ionic charges and vibrational eigenmodes of BaTi03 and other perovskites, Phys. Rev. 157 (1967) 429-435. [34] G. Gonze, P. Ghosez and R.W. Godby, Density-Polarization Functional Theory of the Response of a Periodic Insulating Solid to an Electric Field, Phys. Rev. Lett. 74 (1995) 4035-4038. [35] 1.1. Mazin and R.E. Cohen, Notes on the static dielectric response function in the density functional theory, Ferroelec. 194 (1997) 263-270.

[36] G. Ortiz, I. Souza and R.M. Martin, Exchange-correlation hole in polarized insulators: Implications for the microscopic functional theory of dielectrics, Phys. Rev. Lett. 80 (1998) 353-356. [37] D. Vanderbilt, Nonlocality of Kohn-Sham exchange-correlation fields in dielectrics, Phys. Rev. Lett. 79 (1997) 3996-3969. [38] X. Gonze, P. Ghosez and R.W. Godby, Density-functional theory of polar insulators, Phys. Rev. Lett. 78 (1997) 294-297. (391 P. Ghosez, J.P. Michenaud and X. Gonze, The physics of dynamical atomic charges: t,he case of AB03 compounds, Phys. Rev. B 58 (1998) 6224-6240. [40] F. Bernardini and V. Fiorentini, Electronic dielectric constants of insulators calculated by t,he polarization method, Phys. Rev. B 58 (1998) 15292-15295. [41] M.E. Lines and A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon Press, Oxford, 1977). [42] D. Vanderbilt, this volume (1999). [43] A. Garcia and D. Vanderbilt, Electromechanical behavior of BaTi03 from first principles, Appl. Phys. Lett. 72 (1998) 2981-2983.

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5 Theory offerroelectrics

Burton and R.E. Cohen, Non-empirical calculation of the Pb(Sc0.5Ta0.5)03-PbTi03 quasibinary phase diagram, Phys. Rev. B 52 (1995) 792-797.

[45] B.P. Burton, this volume (1999). [46] L. Bellaiche and D. Vanderbilt, Electrostatic model of atomic ordering in complex perovskite alloys, Phys. Rev. Lett. 81 (1998) 1318-1321. [47] E. Cockayne and K.M. Rabe, Enhancement of piezoelectricit,y in a mixed ferroelectric, Phys. Rev. B 57 (1998) R13973-R13976. [48] J. Padilla, W. Zhong and D. Vanderbilt, First-principles investigation of 180 (degrees) domain walls in BaTi03, Phys. Rev. B 53 (1996) R5969-R5973.

[49] R.E. Cohen, Surface effects in ferroelectrics: Periodic slab computations for BaTi03, Ferroelec. 194 (1997) 323-342. [50] J. Padilla and D. Vanderbilt, Ab initio study of BaTiO3 surfaces, Phys. Rev. B 56 (1997) 1625-1631. [51] L. Fu, E. Yashenko, L. Resca and R. Resta, Hartree-Fock studies of the ferroelectric perovskites, in: R.E. Cohen, ed., First-Principles Calculations for Ferroelectrics: Fij?h Williumsburg Workshop, (AIP, 1998) 107-117. [52] S.E. Park and T.R.. Shrout, Relaxor based ferroelectric single crystals for electromechanical actuators, Materials Reseurch Innovations 1 (1997) 20-25. [53] K. Uchino, High electromechanical coupling piezoelectrics: relaxor and normal ferroelectric solid solutions, Solid State Ion. Diffus. React. (Netherlands), Elsevier, (Chiba, Japan, 1997).

16

6 Ferroelectric Ceramics: History and Technology C.H. Haertling

Ferroelectricity T)le Fundamentals Colkction. Edited by Julio A. Gonzalo and Basilio Jimknez Copyright 0 2W5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40486-4

This Page Intentionally Left Blank

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Ferroelectric Ceramics: History and Technology Gene H.Hacrtlhig'." t>epartmrnt of Cvaniic and Marcrials I'nginrcnng. Clrniwn liniversily, (:lrmwn. South Carolina 296344WX' Ferroelectric ceramkc were horn In the early I y 4 & with the discovery of the phenomenon of f e ~ l e c t r i c l t yas the source of the unusunlly high dielectric constant in ceramic barium titanate capacitors. Since that time, they have been the heart and soul of several multlbllllon dollar indnstries, ranging from high-dlelertric-mnstant capadtore to later development$ in piezoelectric t r a d u c e r s , positive temperature coerncient devices, and electmoptic Ugbl vulves. Materials based on two composltional systems, barium Utanale and lead zirconate tltanate, have dominated the neld thmughout their history. The more recent developmeots In the fleld of f em l ect r i c ceramics, such s9 ntedlcd ultrasonic composites, high-displaeemnt pkzoelectric actuators (Moonis, RAINBOWS),photostrklors, and thin and thick f i b s for piezoelectric and intepted-circuit spplicatiom have sewed to kcep the industry young nmidst its grodng maturity. Various ceramic formulations, their form (bulk, films). fabrication. functlon (properties), and future are described in relation to their ferroelectnc nature and speciflc areas of application.

s

1. Introduction

of ferroelectricity in single-crystal matenals . ' (Rochelle salt) in 1921 and its subsequent extension into the realm of polycrystalline ceramics (barium titanate, BaTiO,) during the early to mid-l9jos. thew has been a continuous mccessioo of new materials and technology developments that have led to a significant number of industrial and commercial applications that can be directly crcdited to this most unusual phenomenon. Among these applicationsare bighdielectric-consmnt capacitors. piezclelectric sonar and ultratwf the discovery

centennia

sonic transducers. radio and conununication filters, pyrcrlecmic security surveillance devices. medical diagnostic transducers. stereo tweeters, buzzers. gas ignitors. positive temperature coefficient(PTC) sensors and switches. ultrasonic motors, electrooptic light valves. thin-film capacitors. and f e r n electric thin-film memories. (electrically The history of the discovery of f-lectricity switchable spnntaneous polarization) is a lascinating one that extends as far back as the mid-160Us when Rochelle salt (sodium potassium tartrate tetrahydrate)was first prepared by E l k Seignette in La Rochelle. France. for medicinal purposes. However. it wm approximately 200 years later before this water-soluble. crystalline material would be investigated for its pyroclectric (thermal-polar) properties. another halr century More its piezoelectric (stress-polar) properties would he uncovered. and finally another 40 years would pass helore frrroelectricity (a hypothetical but yet unproved property ofsolids at the turn of the 20th century) would be first discovered by Joseph Valasek in this same material.' Rochelle salt war a popular material in these initial studies. because it was readily available and easily grown as large single crystals of excellent optical quality. but its water solubility eventually led to iu disuse in later years. Seven1 excellent papers on the history of fermelectricity have been written, and the reader is referred to these for many of the details.z-" This paper is intended to cover only the highlights of fcrroelectric ceramics and cannot hope to treat all of its diverse aspects. In this regard. only personalities nnd companies involved in the early history ure specifically mentioned, although it is clearly recognized that, since then, there have been many excellent individuals and institutionsthat have been involved in the research. development, and application of these very interesting materials. (1) ChrOtIO~ogicdHislory of Ferroclecnic MattriaLv A chronological listing of many of the more notable specific events in the history of ferroelecvic olaterials is given in Table I. Because this article emphasizes a comprehensive review o f lerroelwuic (FE)polycrystalline ceramics from a materials p>intof view. timeline events involving compmitioos, processing, labrication techniques, ptopdes. pa~ents.and applications are all included in Table 1. whereas the specific5 involving femlectric single crystals and the development of the phenomenological basis for the fermelectric phenomenon are

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6 fernelectric Ceramics: History and Technology Jr>wiiulof rhr Aniericun Ceramic S c ~ r i ~ ~ - H u e r f l i i ~ g

79R

Table 1. Solable Events in che H l o n of

1824 1R80

Pyroeleccricity discvvered in Rochelle salt Piezoelectricitydiscovered in Rochelle salt. quanr. and other minerals

1912

Ferroelectricitytint proposed as property of solids

I921

Femlecmcity discwered in Rwhelle salt

1935

Fcmlectricity discovered in KH,PO, BaTiO, high-K (21200) capacitors developed Femxlectricity dircovmd in AB0,-type pivskite BaTiO BaTiO, rk rted ar useful piem annsducer, Pat. No. 2 4RS60 Phenomenologicd theory of BaTiO, i n t n d u d LiNb03 and LiTaO,, reported as FE

lY4l 1944 194s 1949 I949

1951

1952 19.53 19.54 1955

I955 19.55

1957 19SY 1961

1%1 I964 1964 1967 I969

I969

1970 1971

I973 1977 1978 1980 1981 I983 1991

1992 IVY3

1997

C o m p of nntifenoclectricity introduced PLT reported as FE solid-solution system. phase diagram esublished PhNb,O, reponed as FE PZT reported as useful piezo trnnrducer. Pat. No. 2 708 244 P k crreit in BaTiO, reported Chemical co recipitation of FE materials introduced' Alkali niobatea reported as FE BaTiO, barrier layer capacitors developed PZT SA and 5H MPB-type piem compositions. Pat. No. 2 91 I 370 Lattice dynamics theory for FE materials. soft moden inVoduced PMN relwor materials reported Oxygenlatmosphere sintering for F E s developed FE semiconductor (PTC)devices developed Optical and I30 properties of hot-pressed FE ceramics reported Terms "fmuic" and '+fetmelilsticity" introduced Optical Vnspanncy achieved in hot-pressed PLZT ceranircs P U T compositional phase dia gm established. Pa.No. 3 666 666 Useful W O propenier reported for PUT. Pat. No. 3 737 21 I Oxygenlatmmpheresintering of Pl.ZT 1 0 full transparency FE thin films developed Engineered (connectivitydesigned) FE composites developed Electrmtrictive relwor PMN devices developed. Pat. No. 5 345 139 Sol-gel whniques developed for the preparation of

FE films

Photostrictiveeffects reponed in PZT and PLZT Mocmie piezo flextensional devices developed. Pat. No. 4 Y9Y 819 RNNBOW piem hending actuators developed. Pat. No.5471721 Inlegration of FE films to silicon technology. Pat. No. 5 038 323 Relaxor single-crystal materials developed Tor piero transducers

research and development of tmnspmnl cleccrooptic lead lan(4) thanum zirconate titanate (PUT) cerdmics the late 1%. the engineered ferroelectric compsites of the late 1970s,(51 the development of Icod miignesium niohaie (PMN) relaxor ceramics and the use of sol-gel techniques far the preparation of fenwlectric films in the I9Hh. (6) the strain-amplified ac~ n t m of the ciuly 199Os, and (7)the current integnted ferrwlectric films 011 silicon. Many of the items listed in Table I an' described in detail in separate sectiona throughout thc poper. (2) Birth of Fernelectric Ceramics The story of the discovery of ferroelectricity and pieroelectricity in ceramic materials is equally fascinating and began in the early lWs under a cloud of secrecy. becnuse World War I1 wps under way. Spurred on by the pressing need for higlwrdielectricsonstant capacitors than could he obcained ftuin steatite. mica. Ti02, MgTiO,. and CaTiO,, ( K S 100). unpublished work by Thurnauer' and Wainer and SolonionXfinnly eslabliahed BaTiO, as a new type of ceramic capucitor with K > 1100. Near the end of World War 11, in the mid-IYWs. publications began to appear in the open Iitemture, and it became evident that concurrent work on BaTiO, as a highdielectric-constant material had been conducted by several countries. including the United States. United Kingdom. USSR, and Japan. Shortly t h e d t e r . in 1935 and 1946. the work of Wul and Goldman" in the USSR and von Hippel's group'" at the Massachusetts Institute of Technology established that Ihc source oT the high dielectric constant in BaTiO,% emanated from itu ferroelectric properties. Work on sinplecrystal BaTiO, subsequently corroborated lhese findings. The knowledge of the femlectric naturr of ceramic BaTiO, pruved to be invaluable when it was discovered by Gray" (in 1945)that un external electric field could orient the domains within the grains, thus producing a ceramic material that acted very similar to a single crystal possessing hoth femlectric and piezoelectric pWpertieS. This electrical aligning. or "poling" process a$ it has come to be called, was thus correctly identilied as the key w turning an inert ceramic into an electmnwchanicdly active materid with a multilude of industrial and commercial uses. This u'w a most stanling discovery. because the prevailing opinion was that ceramics could not be piezoelectrically active, hecause the sintered and randomly oricnted crystalliter would. on the whole. cancel out each other. This

Abbreviations Used Fernelectric Marurials P2T Lead zirconate titanate P U T Lead lanthanum zirconate timate PMN Lead magnesium niobate IT Lead titanate PZN [mad zinc niobate PSZT Lead staiinate zirconate titanate PZ Lead zirconate BST Bariirm strontium titanate SBT Strontium bismuth titanate

Othen E AFE PE

S W not created in detail. The time period is from the early 1800s to the present (1999). involving eventS from the early work on singlecrystal Rochelle sail to (1 1 the binh of femxlectric ce-

ramics in the 194Or, ( 2 ) the development of lead zirconale titanate (ET)piezoelectric ceraniics in the intd-1950s. (3) the

Vol. 82. No. 4

PTC NTC MLC BLC

MPB

Ferroelechic Antiferroelectric Piuiteleccric Slisloop ferroelecuic Positive temperature coefficient Negative temperature cmfficient Multilayn capacitor Barrier layer capacitor Morphotrophic phase hiundary

6 Fenoclectric Ccmmics: History and Tcchno/ogy- G. H. Haertling

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Ferroelectric cemics: Hisloq and T ~ C h / o # y

799

proved not to he the caw for kmwlectric mystallites. becauu they could he p@rmanentlyaligned or reoriented in an electric field. soniewhdt analogous to magnetic alignment in perniancnt magnets. Thus. as puinted out hy Jaffe" in his excellent treatise on piezoelectric ceramics. thc three fundamental steps that were critical to the understanding of ferrcwlecuicity and pieznelectricity in ceramics werc ( I ) the discovery of the unusunlly high dielectric constant of BaTiO,, (2) the discovery that the twigin of the high dielectricconstant was due to its fernelectric (permanmi internal dipole moment) nuture. thus ushering in a new class of t'erroelectrics (the simple oxygen octahedral ABO, group). and (3) the discovery of the electrical poling pnxess that aligns the internal dipoles of the crystallites (domains) within the ceramic and causes it to act very similar to a single crystal. For more details on the history offerrcwlecuic ceramics, the reader is referred to several excellent puhIications.'.'-ls (3) Bask for Piezoelectric@ in Solidx Piczcwlectricity. a p~)penypossessed by a select group of materials. was discovered in 1880 by Jacques and Piem Curie dunng their systematic study of the effect of pressure on the generation of electrical charge by crystals, such as quanz, zincblende. arid tourmaline. The name "piezo" is derived from the Greek.meaning "to press:" hence. piezoelectricily is the generation of electricity as a result of a mechanical pressure. Cady'" defines piezorlcctricity as "electric polarization pmduced by mechanical strain in crystals belonging to cenoin clasher. the polarization being proportional to the strain and ch&ngingsign wich it." An understanding of' the concept of piezoelectricity in solids begins with an understanding of the internal S~NC~IUC of the materid: for purposes here, consider a single erystallite. This crystrllite has a definite chemical composition and, hence. is made up of ions (atoms with positive or negative charge) that are constrained to occupy positions in a specific repeating relationship to each other, thus building up the srm~tureor lattice of the crystal. The smallest repeating unit of the lattice is called the unit cell, und the specific symmetry possessed by lhe unit cell detennines whether it is possible for piezoelectricity to exist in the crystal. Furthennore, the symmetry of a crystal's internal structure is dle&d in the symmetry of its external propenies (Neumann's principle). In The elemenLv of symmetry hat are utilized by crfstallographen to define symmetry ahout a point in space. e.g., the central point of a unit cell. are (1) a center of symmetry. (2) axes of rotation. (3) mimx planes, and (4) combinations of these. All crystals can be divided into 32 different classes or p i n t p u p s utilizing these symmeay elements. as shown in Fig. I . These 32 point groups are subdivisions of seven basic crystal systems that are. in order of ascending symmeuy. uiclinic. monoclinic. orlhorhombic. tetragonal, rhombohedra1 (trigonal), hexagonal. and cubic. Of the 32 point groups, 21 clusses are noncentrosymmeaic (a necessary condition for piemelectricity to exist) and 20 of these are piezoeldc. One class. although lacking a center of rynunetry. is not piczoelectric because of other comhined symmetry elements. A lack of a center of syinnieny is all.imponmt for the presence of piezoelectricity when one considers that a honiogemous stress is cenvosymmetric and cannot produce an unsymmemc result. such as a vector-quantity-like polarization, unless the material lrcks a center of symmetry. whereby a net movement of the pcnitivc md negurive ions with respect to each other (as. a result of the stress) produces sleciric dipoles. i.e.. polarization. Furthermore. for those materials that are piezoelecmc but not ferrwlectric ke.. they do not pcwess spontaneous polarization). the stress itself is the only means by which the dipoles are generated. For piezoelectricity. the effect is linear and reversihle. and the magnitude of che polarization is dependent on Ihe magnitude of the stress and the sign of the charge produced is dependent on the type of stress (tensile or compressive).

I

Ftg. 1. Ininlatiomhip of piezoelectric and subpnapr on rhe buris of syninietry. (4) Piezmkci7icity in Ferroclrchie Ceramics As mentioned previously, Ihe poling process is the critical

element in being able to utilize the piezoelectric effect in a ferroelectricceramic. Without poling, the ceramic is inactive. even though ench one of tbe individual crystallites is p i e m electric itself. With poling, however, the ceramic becomes extremely useful, provided that it is not heated above its Curie temperature (Tc). where it loses its polarization and all of the onenution of the polarization produced by the poling process.i7 Two effects we operative in piezoelectric crystals. in general. and in krroelcctric ceramics. in particular. The direct effect (designated as a generator) is identified with the phenomenon whereby electrical charge (polarization) is generated from a mechanical stress, whereas the converse effect (designated as a motor) is associated with the mechanical movement generated by the application of an electrical field. Both of these effects are illustrated in Rg. 2 as caftoans for easy grasp of the principles. The hasic equations that describe thess two effects in regard to ehctric nnd elastic properties are.13 D = d E + t?E

s = JET+ dE

(generator)

(1)

(motor) (2) where D is the dielectric displacement (consider it equd to polarizirion), T the stress, E the electric field, S Ihe strain, d il piezcelecmc coefficient. s the material compliance (inverse of modulus of elasticity), and e the dielectric constant (permittivity). The super scrip^^ indicate a quantity held constant: in the case of er, the,slrens is held constant, which means that the piuoelecuic element is mechanically unconstrained, and. in the c&s of #, the electric field is held constant. which means the electrodes on the element are shortcd together. Equations (1) and (3). in matrix f a actually describe R set of equations that relate these properties along different orientations of the material. Because of the detailed nature of the many equations

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6 Ferroclcctric Ceramics: History and Technology Journul of thc Antmrican Ceramic Socicn-Hacniing

no0

GENERATOR

Vol. 82.

No. 4

MOTOR

Fig. 2. Piezoeluaic effsll: in fmoelenric ceramics.

involved, the reader i s refemd to several sources on the subject.'3.'R-mSuffice it to say that, because this is a piewlearic solid. 9 s . ( I ) and (2) relate given propenies, such as electric displacement (polarization) and strain to both the mechanical and ekctrical states of the material. F u r t h m . these prop d i e s are directional quantiticci, and. hence. they ore usually specified with subscripts to identify the conditionsunder which they are determined, e.g., 4 , indicates thnt this piemlecaic coefficient relaces to the generation of polarization (direct eff s t ) in the elecnodes perpmdicular to the 3 or venical d i m tion and to the SIICSS mechanically applied in the I or lateral direction; d,3 indicates the polarization generated in the 3 diraction when rhe stress is applied in the 3 direction. Typical relationships for this coefficient are: D3 = dJ3

(directeffw?)

(3)

S, = d,,E,

(conveneeffect) (4) where the d cmfficients are numnically equal in both equations. Thc d cOemcients are usually expressed as X ~ WC/N ' ~ for the dirix? effect and xIO-'~ mN for rhe converse effect. d coefficients are desirable for those materials that are High utilized in mdonal or vibrational devices. such as sonar and sounders. In addition to the d coefficients. open-circuit g coefficients are also used to evaluate piezoelectric ceramics for their ability to gemate large mounts of voltnp per unit of input stress. The g c o ~ t a n is t relaud to the d constant via the relationship g=-

d

Keo where K is the relative dielectric constant and E, the pemittivity of free space (8.854 x I F h ) . Thus. a high g constant is possible for a given d coefficient if the matcrial has a low K. High-g-cunstant ceramics arc uvually femnlecvically hard mrrerials that do not switch their polarization readily and posses. Iowa K values. They 8n used in deviccr such as portable gas ignitors and patio lighters. The piezoelectric coupling factor (e.g., &33, k31, and kp) is a convenient and dircct measurement of the overall strength of thc electromechanical effect. i.e., the ability of the c m i c vansducer to convert one form of energy to another. It is defined as the square mot of the ratio of energy output in electrical form 10 the total mechanical mrgy input (dim? effect). or the square root of thc ratio of the energy available in mechanical form to the total electrical energy input (converse effect). Because the conversion of elecvical to mechanical en-

ergy (or vice versa) is always incomplete, k is always lest than unity. Commonly used as a figure-of-merit for pieuwlectncs. the higher k values are most desirable and constantly sought after in new malcrials. For ceramics, k, is a typical measurement used in comparing materialsvalues ranging from 0.35 for BaTiO, to as high as 0.72 for PLZT. AIl of Ihe propenics mentioned here may be nalized in a pic7.oelecoiC cemmic. which is. in rcnlity. a poled femlectric ceramic merial. During the pm~cssof poling, there is a small expansion of the material along the poling axis and a slight contraction in both directionsperpendicular to it. The strength of the poling field. often in combination with elevated tcmperature, is an important factor in determining the extent of alignment and. hence, the resulting pmpenies. Alignment is never complete: however, depending on the type of crystal srmctun involved. the thoroughness of poling can be quite high. ranging from 83% for the temgonal phase to 86% for the rhombohebl phase, and to 91% for h e onhorhombic phase. when compared with single-domain. single-crystal values. Because all ceramic bodies am macroscopically isotropic in the "as-sintercs' condition and must be poled to render them useful as piezoelectric materials. they are all femlectric as well as piezoelectric. (5) &xis far Femekctricity ia Ceramics Figure I shows that there are 10 crystal c l a s ~ sout of a possible 20 that arc designated as pymelecuic. This group of mataials possesses the unusual characteristic of k i n g pennanenUy poldzed within a given temperature range. Unlike the more general piezoelearic e l a s ~ that s produce a polarization under s m , the pyroektrics develop this polarization sponuu~ouslyand form permanent dipoles in the structure. This polarization also changes with tempernture-hence, the term pyroelceuicity. Pyroelecaic crystllls. such as tourmaline and wumite, are often called polar materials. thus referring to the unique polar axis existing within h e lattice. The t a d dipole moment varies with tempm~rc,leading to a change in sign for the current flowing out of a short-circuited crystal. A subgroup of the S ~ ~ ~ ~ ~ I I C polarized O U S ~ Y pynxlectrics is u very special category of materials known as femlmtrics. Similar to pymelectrics, ma~erialsin this group possess spontaneous dipoles: however, unlike pymlecuics. thew dipolcr ore rwmible by an electric field of some magnitude less than the dielecvic breakdown of the material itself. Thus. the two conditions necessary in P materia1 to classify it as u fenoelectrk am (1) the existence of spontaneous polarization and (2) u demonstraled reorienting of the polariWti0n. Four types of ceramic femlecmcs are also given in Fig. I

6 Ferroelectric Ceramics: History and Technology - C. H. Haertling

I

16'

April 1994

Frrrocleclric Ceramics: History and Technology

as wbcategorier of the gcneral group of ferroelectric materials. with typical examples representing the type bawd on its unitcell structure: 11) thc tungsten-bronze group. (2) the oxygen octahedral group. 1.1) the pyrochlore group. and (4) the bismuth layer-structure gn,up. Of these. the second group (ABO, perovskite type) is hy lar the most important category. KOnomically. llie families of compositions listed (BaTiO,. PZT. P U T . PT (lead titanate). PMMN. and (Na.K)NbO,) represent the bulk of the ferroelectric ccramics manufactured in the world today. A typical ABO, unit-cell structure is givcn in Fig. 3 . For example. the PLZT unit cell consists of a corner-linked network of oxygen octahedra with Zr'' and Ti" ions occupying sites ( B sites) within the octahedral cage and the P v * and h" ions situuled in the interstices (A sites) created by the linked octrahedra. As a result of the different valency between Ph" and LA,,' some of the A sites and B sites we vacant (referred to as vacancies) to mainlain electrical neutrality in the stntcture. When an electric field is applied to this unit cell, the Ti" or Zr'' ion moves to n new position along the direction of the applied field. Because the crynallite and. hence. the unit cell is randomly oriented and the ions are constrained to move only along certain crystallographic directions of the unit cell. it is most often the case that an individual ionic movement only closely approximates an alignment with the electric field. Howewr. when this ionic movcment does occur. it leads to a mixroscopic change in the diniensions of the unit cell and the ceramic as a whole. The dimensional change can be as large a$ a few tenths of a percent elongation in the direction of the field and approximately one-half that amount in the other two orthogonal directions. The original random orientation of the domain polariiation vectors (virgin condition) ran be restored by heating the material ahove its Tc..This process is known R5 thermal &poling. Also shown in Fig. 3 is the revenibility of the polarization caused by the displacement of the central Ti"+ or ZY* ion. llisplacement is illustrated here as occurring along the c axis in

F$. 3. Peerovskiie ABO, unit cell for PZT or PLTT. illusimting 180" prilurization reversal for lwo of the six possible polariration slates produced by displacement of the central cation in ihe iocragonal plwe.

80 I

a tetrogonal stNctufc. although it should he undemwod that it can also occur along the orthogonal a or h axes as well. The views of "polarization up" and "polarization down" (represeitting 180" polarization reversal) show two of the six possihle

permanent polarization positions. When many of rhese unit cells. which ore adjacent to each other. switch in like i n m . this is r e f e d to as domain reorientation or suzitching. The homogeneous areas of the material with the same polarization orientation ure referred to as domains. with domain walls existing between mas of unlike polarization mienlation. There exists in tetmgonal materials both W (strain-pruducing doniaiiis on switching) and 180" domains (nonstrain-producing domains), whereas the strainproducing entities in rhombohedra1 malerials are 71" and 100' doniains with the 1x0" domains remaining BS nonstrain prcducing. Macroscopic changes occur in the dimensions of the material when strain-producing domains are switched. Because af the empirical naNTe of determining lhe reversihility of the dipoles (as detected by a hysteresis loop measurement). one cannot predict the existence of fcrroeiectricity in a new material with much accuracy. However. the basis for Ihe existence of femlecmcity rests primarily on structural and symmetry considerations.The special relationship of ferroclectrics as a subgroup of piezoelectrics (Fig. I ) infers that "all ferroelemrics (poled) are piezoelectric, but not all pieznelectrics are ferroelectric." The Current nuinher of fenoelectrics is in the thousands when one includes the ninny wrnnric solidsolution compositions. Femlectrics are no longer the great '.accident of nature.' that they were once thought to he in thc 1920s and 1930s. (6) EIectrosnieriOn in Fernelectric Centmies

Electrostriclion is another ekctromechanical effect that exists in fermelecmc manics. In electrostriction.the sign of the deformation that occurs with an electric field is independent of the polarity of the field and is proportional to even powers of the field. In pieroelwtriciry. the deformation is linear with respect to the applied field and changes sign when the field i.s reversed. This means in practical teams that electrostriction produrns an expadon in most nuterids in the dinctioit of thc field regardless of its polarity. and this expansion relaxes back to zero when the field is removed. The corresponding equations are S = mE2

(in term$ of electric field)

(6)

S = QP'

(in terms of polarization)

(7)

where P is the poiari~~tion and !ti and Q the corrcsponding electrcrrtric~ivecoefficients. Similar 10 piezoelectricity, elecbnsfriction deals with vector quantities. and, hence. appropriate suhscripts must he uwd. Although electrostriction is a general property of all dielccuic materials. whether they are crystalline, amorphous. polar, or cenuosymmcaic. it can be particularly large in ferroelectric materials just above their T,, where an electric field can enforce the energetically unstable ferrnelwtric phase. More commonly. this effect is utilized to good advantage in rctlruor materials. such as PMN. FZN (lead zinc niohate). and PU T. where the Tc is not sharp but rather is spread out over a moderale temperature range, thus allowing for a reasonable ternperaturr range of operation for devices made from them. Electrostrictive materials can be operated either in the electrostricrive mode (as staled above) or in the field-hiased piewelectric mode. In the latter case. a dc electric field bias is appIied to the material to induce u lkrcxlectric p)l whereupon thc material act.. as a normal piezoelectri as the field is applied. and. the stronger the field. the higher the piezoelectric effect until saturation sets in. The relationships relating the resulting piezoelectricd coefficient to the induced polarktion and the dielectric permittivity are

4,= ZQ,iP,%,

(8)

162

I

6 Fcmlcctnc Ceramics: History and Technology Journol of rhr Ainerkair Ceramic Socicr)-ffaenling

852

where Q,,and QI2 are h e longitudinal and transverse elmmtrictive coefficients. respectively. Thus, a large dielectric constant and a high polarization are required to produce a large induccd piezoelectric coefficient.These coefficients are essentially independent of Some advantages for elecmstricton over conventional piemleevics are ( I ) hysteresis in the main-field dependence is minimal or negligible in a selected temperaturc range, (2) realized deformation is more stable and comparable to the best piezoelectric ceramics. and (3) no poling is q u i d . "hehe advantages. however. are balanced by the disadvanuges of ( I ) a limited usable temperature range due to the strong tmperat u n dependence of the electrostrictive effect and (2) especially small deformation at low fields. beesuse elcctrostriction is very nearly a quadratic function at low electric fields, which then usually necessitates higher operating voltages to achieve mcderate deflection. Although the actual mechanism or mechanims leding to the large electrustrictive effects is not fully understood. it is generally believed that the elecmstrictive effect in these marerials is due to the field-advated c o a k e n c e of micropolar regions to macrodomains of the parent famcketric.'*z Consequently, the mechanism is essentially the fame as che nonpolar. cubic prototype of the fenoclectric phase undergoing a momentary phase transformation to the ferroelectric phase while under the influence of an electric ficld F~ud~ermore. because a higher applied electric field leads to a larger m g nitude of the induced ferroelectric polarizatim and strain. one is able to use this effect for achieving voltagedependent dielectric, piemleceic. pyroelectric, and electroopric properties in ceramics. Longitudinal strains (IS high as 0.1% in PMN m d 0.3% for P U T ( L a Z R i = 9/65/35) have been reported for these elecmstrictive materials. Elechostriction-like effects are also evident in nonpolar. antiferroelectric (antipolar adjacent unit cells) materials that undergo a phase change from ancifermelecvic (AFE)to fenuelectric when a sufficientlyhigh electric field is applied. Although this effect is very abrupl (occurring suddenty at a SFeCifie voltage) in a material such as lead stannate zirconate titanale (PSZT),in another ceramic. such as PLZT, the change is "washed out" over a range of voltages; i.e.. the former is more digital, and the latter is more analog. LMlgitudinal strains BS high as 0.8% have been reported in these antiferroelecuic

Fig. 4.

Vol. 82. No. 4

17) Elscnwptic Efffierr The electmopticpropertien of PLZT malerials are intimately related to their ferroelectric pmpucies. Consequently. varying the ferroelectricpolarization with an electric field. such as in a hystemis loop. also prcduces a change in the optical properties of the ceramic. Moreover. the magnitude of the observed elecmoptic effect is dependent on both the strength and direction of the electric field. PLZT ceramics display optically uniaxial properties on a microscopic scale. and also on a macroscopic scale when polarized or activated wi!h an electric field. Thae is one unique symmey axis in uniaxial crystals, the optic axis (colinear with the ferroelectric polari7ation vector in ceramic F'LZT), which posses= optical properties different from the other two orthogonal axes. That is, light traveling in a direction along the optical axis and vibrating in a direction perpendicular to it possessn an index of refraction (&,) different from light traveling in a direction 90" to the optic axis and vibrating parallel to it (nJ. The absolute difference between the two indexes is defined as the bireftingence: i.e., n, - N~ = An. On a macroscopic wale, An is equal to zero before poling and has some finite value after poling, depending on the composition and the degret of poling. With relaxor materials, 4n is not permanent hut exists only as long as the electric field is present." A typical setup for determining the behavior of clectrooptic ceramics is given in Fig. 4. Linearly polarized white light. on entering the electrically energized ceramic, is split in two orthogonally vibrating white light components (represented in Pig.4 by red and green waves), whose vibration directions are defined by the crystallographicaxes of the crystallitesacting as one optical entity: in this case, the oaes arc defined by the direction of the electric field. Because of the different refractive indices, n, and no, the propagation velocity of the. twci components is different within the material and results in a phase shift known as optical retardation. The total retardation (r)is a function of both A~Iand the optical path length (0. according to the relationship

r=h

t

where I is the ceramic thickness along the optical path. In the case of white light (average wavelength of0.55 pm). as shown in Fig. 4, when suMcient voltage is applied to the caarnic, a half-wave retardation is achieved for one.component relative to the other. The net result is one of rotating the vibration direction of the polariztd light by 90°, thus allowing it to be uanb-

Rmic rerup fw evoluatinp elwrrmptic .;hutterlmnlulatorcharacteristics (open condition shown)

0061 I!'+

164

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6 Femlectric Ceramics: History and Tiihnology

XI34

Journul <$the Amenran Ctrumir Scn.icp-Huerrling

counteract the IlaNral p-type conductivity of the PZT and, thus. increase the electrical resistivity of the materials hy at least 3 orders of magnitude. The donors are usually compensated by A-site vacancies. These addirivcs (and vacancies)enhance domain reorientation: ceramics produced with these additives are characterized by square hysteresis loops, low coercive fields, high remanent polarization, high dielectric constants. maximum coupling factors, higher dielectric loss. high mechanical compliance. orid reduced aging. Off-valent uccepfors.such as Fe" replacing ZP' or Ti4*. arc compensated by oxygen vacancies and usually have only limited solubility in the lattice. Domain rmrientation is limind, and. hence, ceramics with acceptor additives nre characterized by poorly developed hysteresis loops. lower dielectric constants. low dielectric losses. low compliances, and higher aging rates. Isoidenr additives. such as Ba2* or S$+ replacing PV+or Sn'+ replacing Z P OT Ti'+. in which the substitutingion is of the same valency and appmximately the same size as the replaced ion. usually produce inhibited domain reorientation and poorly developed hysteresis loops. Other properties include lower dielectric loss. low compliance. and higher aging rates. Dopants are usunlly added in concenbations of 5 3 at.%. Modifiers arc substituted into the original PZT composition as solid-solution constituents in concentrations of 2 5 at.% The most common examples of modifier systems ure (Pb,La) (Zr.Ti)O,, (Pb,Sr)(Zr.Ti)O,. (Ph.Ba)(Zr.Ti)03. Pb(Zr,Ti.Sn)O,. (Pb,La)TiO,, and Pb(Mg.Nb)Q1-PbZrO,PbTiO,. although. in actuality. there are many of these leadcontaining, solid-solution systems." One system that embraces

Vol. KZ. Nu, 4

i

al compositional aspects of the dielectric. piezmlectric. pyroe ctric. ferroelectric, and electrooptic ceramics is the PLZT skstem.3aFigure 5 shows the PLZT system with the parrnt FAT

phw diagram. Several areas on the diagram arc color coded for easy identification: ( I ) the fcrroelcctric tetragonal and rhombhedral phases are shown in orange. (21 the onhorhonibic antiferroelectric phiw in purple. (3) the cubic paraelectric ( P E nonferroelectric) phases in white, (4) the morphtrtropic phase boundary (MPB) in magenta. ( 5 ) the pyroelectric applicotion areas near PbTrO, in blue. ( 6 ) the economically important MPB compositions that embrace almost all of the triinsduccr applications in green, (7) the conipositional area for AFE-to-FE, enforced-phase devices in gray. and (8) specific cwmpositions in these regions in yellow. Figure 5 shows that the effect of ndding lanthanum to the PZT system is ( I ) one of maintaining extensive solid solution throughout the system and (1) one of decreusing the stability of the femrlecmc phases in favw of the pardelectric and antifermelectric phaws. as indicated hy the red line. which shows the reduction of the Tc with increasing lanthmuni. At a 6W.75 ratio of PUPT (where PZ is lead zirconate. PhZrO,). a concentration of 9.0%lanthanum (designated as 9/65/35) is sufficient to d u c e the temperature of the stable ferroelectric polarization 10 slightly below mom temperature. nsultirig in a materinl that is nonfmoelettric and cubic in its virgin state. The cross-hatched area existing along the FE-PE phase boundary denotes a region of diffuse. metaslilhle nlaxor phase\ that can he electrically induced to a fmoelectric phase. Materials within this region exhibit a quadratic strain and elecrrwpoc behavior.

G Ferroelectric Ceramics: History and Technology - C. H. Haertling Ferrorlecmc C r m m i r ~HI rlorv und Trchiiolu~v

'The solubility of lanthanum in the PZT lattice is a function of cornprisition and is related directly to the amount of PT present. The compositional dependence of the solubility limit is indicated by the dashed line adjaceni to the mixed-phase region (double cross-hatched area) in Fig. 5. For the two end-member compositicin.i. PL and PT,thcse limits are 4 and 32 at.% respecti\vly. The soluhility limits for intermediate compositions are proponional to their Zrmi ratios. Modification of the PZT system hy the addition of lanthanum sesquioxide has a marked beneficial effect on severdl of properties of the material. such as inemsed square. ness of the hysteresis loop, decreased coercive field. increa.wd dielectnr constant. maximum coupling coefficients. increased mechanical compliance. and enhanccd optical transparency. Thc optical transparency was discovered in the late 1960s as a result of an in-depth study of various additives to the PZT syhicin." Kesulrs from this work indicate that La". as a chemical mtdifier. is unique among the off-valent additives in producing transparency. The reason for this behavior is not fully understood, however. it is known that lanthanum is. to a large en!ent. effective k c a u x of its hgh solubility in the oxygen octrahedral SINCIIIR. thus producing an extensive series of single-phase. solid-solution compositions. The mechanism IS believed to be one of lowering the distomon of the unit cell. thereby reducing the optical anisotropy of the unit cell and. at the same time. promoting uniform grain growth and densificatlon of ii single-phace, pore-free microstructure. Electrooptic compositions in the P U T phase diagram are generally divided into three application areas: (1) nonmernory quadratic. ( 2 ) memory. and (3) linear. As mentioned previou.ilq. the quadratic materials are located along the E - P E phase boundary. principally in the cross-hatched m a . Memory compositions having stable. electrically switchable polarisation and optical states are largely lc~atedin the ferroelectricrhombohedral phuse region, and the linear materials possessing nonswitching. linear strain. and electrooptic effects m confined to the area encompassing the tetragonal phase. (3) PMN Ceramics Although the study of relaxor materials began in the early IYhOs with work on single-cr)istal Ph(Mg,,,Nb,,,)O, (PMN) mutcnnls"' and continued i n the mid-1960s with PMN as one of the triaxial components in the PZ-PT-PMN solid-solution system.'' more-recent work in the early 1980s with PMNhated rcla.xorceramics h a led to their successful a plication as and highhigh-strain t o . I % I electrostrictive Jielectric-constant(>25 OOO) capaciton." The phase diagram for this sytcin IS given in Fig. 6. The most popular specific comp&tion in this \)siem is Ph(Mg,,lNh2,,)0,j-U. IPbTiO,. which is PMN containing 10% PT.thus increasing the T,,,(the tenipernturr of maximum dieleclric constant for relaxors.

Mole%PT Fin. 6. Phnsc rlispzm of clie PMX-Pf solid-soltitton system."

I

165

Xt1.s

equivalent to 7' for n o d ferroeleccrics)of PhlN to -40°C. For this compohon. the temperature of polarization loss (7;,) is -10°C: hencc. the material is a relaxor at room temperature 125°C).An addition of -28% PT causes the material io revert to a normal f e m l e c v i c tetragonal phi= with T, = 13o"C. Unlike PZT and P U T . PMN ceramics are somewhat difficult to prepare in a phase-pure condition. Several method\ of powder preparation have been developed wer the years to rcditce the undesirahle pyrochlore phase to a bare iiiiiiiiiium.'i but the process that has met with consistent succcss i\ the so-called columbite precursor method.42 In this technique. M y 0 and Nb,O, are first reacted 10 form thc columhitt:stmcttire. MgNh,O,, which is then reacted with PhO and TiO, to fonn the PMN-PI' compositions (4) Fernelectric F i h 'I'hc 1970s and 19x0s witnessed the emergence of thin and thick films (both ferroelectric and nonferrwlectric1 as an imponant category of materials that was brought about hy the maturing of laser and transistor technologies (e.g.. optical fihers, integrated optics. microelectromechanical systems. microprocessors, and computers) and promises to be the springhoard for the age of integration beyond the 1990s into Ute next century.43" New materials developnienr during this time perid was one of form he.. from bulk to films) rather than composition. Almost all of the current compositions that are used in the fabrication of films had their beginnings i n tlic hulk mated\. Examples of these include BaTtO,, h a n k wonrium T . titanate. PZT.PLZT. PNZT(Nb). PSZT(Sn), PBZTIBa). I bismuth titanate. lithium niobate. barium strontium niobate. strontium barium tantalate, and potassium niobate. Thus. one can say (at least for the present time) that 'an adequate nuniber of ferrnclectric compositions now exist and are being prtduccd as good-quality. po1ycr)stalline thin and thick filnis by a vanety of forming methods. These films will form the h i s for the development of new structures and devices well beyond the turn of the century. 111.

Processing

( I ) Powder Preparation Femlectric cernmics are uaditionally made from powders formulated from individual oxides: however, the newer electrooptic macerials and some of the PTC ceramics utilize chemical c o p r e ~ i p i t a t i o n ~or~hydmthermal'* ~' techniques. The processing method that one selects to prepare the powders depends, to a large extent. on cost, but even more important is the end application. Understandably, electmoptic ceramics require higher-purity. more-homogeneous. and higher-rractibity powders than do the pieimle~lrcceramics. because inhomogeneitier can be detected optieally much more easily than elec~rically.As a result. different powder process techniques have evolved in the two cases. Piezoelecvic ceramics continue to he prepared from the most economical, rnixed-oxide (MO) process?' whereas the optical ceramics utilize specially developed chemical coprecipitatim (CP) processes" iiivolving liquid inorganic or organometallic precursors. Although not yet fully achiesed. the trendq in this area are toward the development of one unified process that meets the objectives of both types of material,. There is a commonality in these objectives. heeause the more recent piezoelectric devices dentand higher-quality inaterial (essentially zero porosity ), and the electrooptic\ require a more economical process. A flowsheet describing the essential sldps for both thc !vlO and CP processes is giwn in Fig. 7. There are many sieps lhri are common to both methods. The essential differences between the iwo methods occur in the powder forming and densification stages. In the MO methods. this very simply consists of wet milling (sluny form) the individual oxides or other compounds. such as the carbonates or nitrates that decompose io the oxides during calcining (a high-temperature solid-state

166

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6 Ferroelectric Ceramics: History and Technology Jovmnl oj nhc Amtni-un C'ernmtc P~ciep-Hurnliri~

X M

Compositlon Mwed Oxides

Copredpilation

Mixing

Blending

I Calcining

[ Cold P y i n g ]

Flg. 7. Flow she* for pmcersing of piezoelectric and elcctrooptic ceramics.

chemical reaction) at 800"-900"C.In the CP method. the stming materials are usually solutions that are mutually soluble in each other, thus producing an atomically homogeneous precursor solution that is precipitated while blending. Becauw the particle sizes of the CP powders are usually much finer than the M O powders, (0.034.1 pm vs I pm). the CP powders am more reactive and are calcined at a lower temperature. -500°C for I h. Ball milling of the calcined material is necessary for both types of powders to produce the required chemical and optical homogeneity. This is a very critical step in the process. because ton little milling doc9 not produce the necessary homogeneity. and over milling increaws the likelihood of contamination leading to optical scattering. A common practice is to use a plastic-lined mill with high-density media (alumina or zirconia balls) and a nonpolar. nonflammable milling liquid. such as trichlor~xthyleneor freon TF. for the e l c e t ~ t i cmaterials: however. distilled water is a better liquid for pieurlcctrics from a cost and environmentally preferred standpoint.Depending on the particular powder characteristics, milling times may vary from 2 to 16 h. The milled pqwders are then thoroughly dried. mechanically broken up. homogenized in a V-blender. and mired for further prucessing.

(2) Fuming and Firing (Dens@nioa) There are a variety of forming methods that have been developed over the years that have been successfully used in compacting the powders to a specific form or shape prior to densification. Cold pressing in a steel mold is, perhaps. the oldest and most econoniical of these methods and. thus. is given in Fig. 7: there are. however. several more methods. including extrusion. slip casting. tape casting. roll compaction. screen printing. and injection molding. All of these techniques

vo1. 82. F i l l . 1

are cumntly uscd with excellent Nuccess. and the choice of one over the other is urually made o n the basis of cost and convenience for the end application. For details on these techniques it is suggested that the reader consult various excellent text\ on the s ~ b j e c t . ~ ~ ~ ~ ~ ~ ~ In addition t o composition and powder prepmation. densilication of the powder into a pore-free. fully dense ceramic element is the third area of pmessing that is extremely critical to achieving a high-quality product. The flowsheet in Fig. 7 shows two methods. i.e.. conventional sinteriiig and hot pressing. Of these two. sintering is. by far, the oldest and tnoct economical method of consolidation. but it has its liini~ations when it conies to achieving full density. Full density is rarely achieved with conventional sintering of Cerroelectric ccramics unless special techniques are used toassist the sintering process during tiring. An example of this is the use of an oxygen atmosphere for sintering lead-containing ceramics. such as YLT and PUT." With air atmosphere only. densities of -96% of theoretical can be achieved, but with an oxygen atmosphere. this value can approach 99%. Another example is the use of excess PbQ during sintering to compensate for PM) loss (volatilizaion) as well as providing for higher dmsification rates via liquid-phase sintering. When both of these techniques are used. bulk densities approaching IOO'A can be achieved. as evidenced by the high optical m s p m c y ohtained in P U T 9/6SM sintered ceramics.'2 Typicnl sintering conditions for conventional PZT are 1250OC for 5 h with flowing oxygen and 60 h for transparent PU T. Although oxygen-atmosphere sintering can and does pmduce fully dense nnd transparent ceramics when the proper pnxedures are used. there continues to exist a problem with this process in consistently achieving high optical transparency. On the other hand. hot pressing is a viable method of producing fully dense ceramics. and its worth has becn proved over scveral decades of experience. Slugs of PLZT as laree a* 150 mm (6 in.) in diameter and 25 mni ( I in.) in thickness are regularly hot-pressed to full density and high transparency. Typical hot-pressingconditions are 1250°C for I6 h at I4 MW (2"M psi). Other densification methods that have proved to be successful for ferrwlertric ceramics in more w e n t years are ( I ) hot isostatic pressing. (2) vacuum sintcring. and (3) a two-step process of sintering and then hot isostatic pressing. The twostep technique involving pmsintering was developed to eliminate the need for a cladding enclosure in the final gas isostatic hot pressing step. After densification,the final steps involved in the processing of Iennelecvic ceramics (Fig. 7) m ( I ) slicing of the *lug. (2) lapping of the slices. (3) polishing of the plates for electronpric elements. (4) eleermding. and (5) evaluation of the pans for further assembly to componenb. Some typical samples of bat-pressed and sintered PLZ? and PZT ceramics are shown in Fig. 8 with thick (12 pm) filnis on sapphire (round suhstrate) and glass (rectangular suhrtrate). The transparent pan on the "sintered ceramic" label is a fully dense. oxygen-sintered. P U T 9/65/35 ceramic.

(3)RAINBOW Pmcessing The latest development in the processing of hulk matenals consists of the high-temperature chemical reduction of high lead-containing femwlectric wafers to produce strain-amplified wafer actuators called RAINBOWS. tin acronym for reduced and internally biased oxide wafer." More specifically. this technology involves ihe local reduction of one surface of it ceramic wafer. thereby achieving an anisotropic. stress-bia.red. dome or saddle-shapedconfiguration with significant tnternal tensile and compressive stresses that act to amplify the axial motion of the wafer and also increase thc overall strength 0 1 the material. After reduction. the flat wafer changes its shape 10 one that resemhles a contact lens. This is believed to be duc 10 ( I ) the reduction in the volume of the reduced layer Ilrrpel) metallic lead) compared to the unreduced material. (21the dif-

Fig. 8. Typical examples of PIZ" aiid PZT ceramics and films.

lerciitial thermal contraction between the reduced and unreduwd Iayen on ccxiling to room temperature. and 13) the volume change in going from the paraelectric to ferrnelectricstate at its Tc. Typical steps for the RAINBOW process involve placing a flat wafer on a graphite block. insening the block and wafer into a fumace preheated to 975OC. leaving it there for 1 h and then removing it for cooling to room temperature in -45 min. The net result is a monolithic (monomorph) smcture consisting of an unreduced piemlectric (ferrnelectric) layer that is highly stressed. primarily in compression. and an electrically conducting reduced layer that produces the stre.w.Thew internal stresses have been shown to be instrumental in achieving unusually high axial bending displacement as well as enhanced load-bearing (4) Thin- and Thick-Film Processing The various techniques that are currently available for the fabrication of thin films are noticeably more varied in ype and in snphistication than a couple of decades ago. Better equipment and more advanced techniques have. undoubtedly. led to higher-quality film and may be a primary factor in the now rnutine achievement of ferrnelectricicy in films as thin as 0. I pm and as thick as 22 pni prepared by a selection of different methods. Tahle II lists the major methods prewntly used to prnduce ferroclectric films. Reviews of the details concerning most of these techniques are given in previous

In general, there are two major categories of deposition techniques Jor films: ( I ) physical vapor deposition and (2) chemical prncesses involving chemical vapor deposition (CVD)and chemical solution depnsition. The first of these requires a vacuum to obtain a sufficient flux of atoms or ions capable of depnsiting onto a substrate. where= the wnnd usually does

Tahle 11. Thin. and Thick-Film Deposilion Technique rnywal v a p l Jeplsluon

Sputtering (rf magnetron. dc. ion beiun) Evapnrution (e-hcani. resistance. molecular heam cpiiaxy) Laser ablation ( 7 r l U ! < . l "DplH

dcpnrluon 1CYI)I

MOCVI) lmetal.+irganic CVDI PECVD Iplavna-enhancrd CVD) 1.KVD (low-pwswre CVDl rhcrnBsai umul depont,un Sol-gel (s(ilution-Eelationj MOI) (metalloorgoiiicdcmmpnsition) Mrlr wlmion dcprsrtion

L.PE (liuuid-ohaw coilaxvl

not; thus, one can roughly identify 1he.se two categories as vacuum and nonv:rcuum techniques. respectively. 'headvantages of the vacuum methods arc ( I ) dry purity and cleanliness. (3) conipitibility integmted-circuit (IC) processing. and ( film growth: however. these arc offset by disadvantages. such as ( I ) slow deposition rates. ( 3difficult stoichionietry control in ferrnelectric mulucomponent systems. where evaporation or sputtering rilles differ considernhly, (3) high-ternprature postdeposition aniaal. often required Cot crystdlization. aiid (4) high-capital equipnient acquisition and maintenance co!,ts. The chemical techniques are usually characterized by ( I ) higher deposition rates, 12) good stoichiometry control. (3) large area. pinhole-free films. and (4) lower initial equipment costs. These advantages. especially in the CBSC of CVD and its many varia~ions.would seem to preclude the use of vacuum methnds; however. the limited availability and toxicity of some of the ferrnelectric precursors have posed some prnblems for this method. Comhining the advantages of excellent compritional control, spin-onhpray-on/dip-co;lting capability. and very lnw equipment costs. the wet chemical solution deposition techniques (sol-gel nnd MOD)already have been quite successful and extensively used in producing thin and thick films of PZT. PLZT, .wd many other materials. The ready availability. low cost, and water solubility of many of the precursors for the wet chemical methods have also significantly contributed to their pnpularity. Examples of some MOD acetate precursor solutions in the PZT system are shown in Fig. 9. Some of the suhstrntes that have been sucredully used for the deposition of films include silicon. platinized silicon. sapphire. magnesia, strontium titanate. silver foil. lithium niohte. gallium orsenide. fused silica. zirconia, and glass.

IV. Properties ( I ) hiicmstrucrm Conipnsitions in the PLZT qslein and to some degree in the PZT system, whether piezoelectric or electmoptic. arc characterized by a highly uniform microstructure cunsisting of rnndomly oriented grains (crystnllites)tightly bonded together. An example of such a microsmcture is given in Figs. I M A ) and (B) for PLZT 9/65/35 The sample was polished and thermally etched at I IS0"C for 1 h. As is typical for most PLZT hotpressed microstructures, little or no entrnpped porosity is evident. This is due to the influence of the external pressure during hot pressing, which aids in pore removal while the niatericil is in a themhemically active state at elevated temperatures;.In actuality. some small amount of porosity exists i n all the materials whether hot-pressed or atmosphere sintered: however. l t is more iniponant in the electrnnptic than in the pieznelectric materials. because porosity in the 0.5-5 pm diameter range is quite effcctive in scattering light. Figure 10 also shows that the microsmcture is very uniform. with an average gmin sire for this sample of -8 pni. Pieznelecvic ceramics usually posses grain sizes in the range of 2-6 pm. whereas the clectrooptics c o w a wider range from 2 to 10 pm. depending on their intended application. A uniform grain size is a highly desirable feature fmm the standpoint of performance.

Flg. 10. Typical mkmrmCNres of (A) hot-preucd elefmopucP U T 9/65/35, (I31 h o t - p r e d clsvoopllc PlZT 9/65/35 a higber magnification. ICI ha-pressed PLZT I 2 / 4 0 / ~in mnsmined light. md tDJ chenucslly elchcd PLZT 7/65/35 in reflected light

Dumain pattems. i.e.. regions of uniform and homogeneous spontnneous polarization within a grain or between several grains, also can be revealed in the micmsvucturr of a f e w electric memory ceramic when transmitted light or reflective light and chemical etching techniques arc used. Examples of these domain pattems are shown in Figs. IqC)and (D),respectively, for materials with an average grain size of 8 pm. Figure I O K ) shows that a p d ~ m i n a n ~ofe 90" domains is evident in the tetragonal PLZT IUW60 cemmic, whereas the domains of rhombohedral PIZT 7/65/35 in Fig. IND) arc mostly 180". Ihe domains in 7/65/35 show up as a bilevel structure. because one end of the electric dipole chemically etches faster than the oppositeend. Distinctive fearurcs here are ( I ) absence of etched grain boundaries because of the fully dense structure and ( 2 ) bridging of grain boundaries by domains. indicating little disorder at the boundaries, (these pre. dominantly 180" domains arc -3 p n x IS pm in size). No domains are observed in the microsvucturr of the 9/65/35 material in Fig. IO(A), becausc this polished &on was rhennally etched and not chemically etched: it., it was ecched at. 1150°C. where domains do not exist. (2) EIccrnrol pr0P.rti.s (A) Dielccrrc Properties: Becnuse almost all of the useful properties of fenoelecvic ceramics are related in some manner to their response with an electric field. the electrical hehavior of these materials is important to their successful

application in dielectric. piemlcftric. pyrcelectric. or electrooptic devices. Ferroeiectricsare. in general, characterized by ( 1 ) higher dielectric constants (200- 1 0 M)O) than ordinary insulating substances (5-100). making them useful as capacitor and energy-storage materials. (2) relatively low diekcnic loss (0.1%-7%). (3) high specific electrical resistivity (>LOg3 fl. cm). (4) moderate dielectric breakdown (100-120 kV/cm for bulk and 500-800 kVlcm for thin films). and ( 5 ) nonlinear electrical, elcctmmechanicnl,and electroopic behavior. Not all of these properties arc optimized and realized in a given material of chemical composition,and. heme. a variety of ceramic materials are manufactured and arc available from several different companies Lhroughout the world. A summary of typical properties for selected compositions is given in Table III. Small-signnl (1 W )relative dielectric constam values for several rloeted compositions are given in Table 111. They range from a low of 225 for lead niohate to a high of 24 OOO for PMN-F'T (WIO).Values for the PZT and PLZT compositions are intermediate. ranging from 1300 for PZT-4 (a hard. A-sitesuhrtituted piezo material) to 5700 for a phase-boundary. relaxor P U T material:The loss tangents (tan 8 ) vary in value from 0.4% to 6% for the various ceramics, and. in general. the lawer loss lactors arc associated with the lower dielectric constants. IS) Hysteresis Zopst The hysteresis loop (polarization versus electric field) is the single most imponant measurement that can be made on a femlectric cenunic when charncteriz-

6 Fernelectric Ceramics: History and Technology - C.H. Haertling April 1999

Ferroelectric Ceramics: Hiscon. und Technology

-

09 v: c.

809

ing its electrical behavior. This loop is very similar to the magnetic loop (magnetization versus magnetic field) one obtains from a ferromagneticmaterial: the very name " f e m l e c tric" has been appropriated from rhis similarity. even though there is no fern, i.e.. iron constituent, in ferroelecuics as a major component. Hysteresis loops come in all sizes and shapes, and, similar to a fingerprint. identify the materid in H very special way. Therelore. one should become familiar with such a measurement. Although early workers in the field of fcnoelectrics utilizd a dynamic (60 Hz) measurement with a Sawyer-Tower circuit and an oscilloscope d o u t . more-recent work usually has been done with a single-pulse or dc (-0. I Hr)Sawyer-Tower measurement using an X-Y plotter or computer readout." Typical hysteresis loops obtained from various femlecmc ceramic materials are illustrated in Fig. 11: (A) a linear tracing from a BaTiO, capacitor. (B) a highly nonlinear loop from a low-eoercive-field (soft)memory fcnoclecuic such as is found in the rhombohedral region of the FZT phase diagram, (C) a narrow. nonlinear loop obtained from a slim-loop fermelearic (SFE)quadratic relaxor that is located in the F E P E boundary region of the PL2X system. and (D)a double loop that typically is obtained from a nonmemory antifemxlecuic material in the PSZT system. The antifemelectric materials are essentially nonpolar. nonfemlectric ceramics that reven to a ferroelectric slate when subjected to a sufficiently high electric field. Outwardly, they differ from the SFE relaxor materials in that (1) the dielectric cons tan^.. usually are lower. (2) higher electric fields are usually required to induce the ferroelectric state, and (3) the onset of the fcrroelectric state and rhe return of the antifermelectric state are usually fairly abrupt. thus giving the loop an appearance of two subloops that arc positively and negatively biased. These characteristicsare shown in loop (D)in Fig. 1 1. A considerableamount of information can be obtained from a hysteresis loop. Figure 1 I also shows that ( I ) the loop in (B) mveals that the material has memory, whereas the loop in (C) indicates no memory. (2) high remanent polarization (P.) nlates to high internal polarizability. strain, electromechanical coupling. and electmptic activity. (3) for a given material. the switching field (E,) is an indication of the grain size for a given material he.. lower E, means larger grain size and higher E, means smaller grain size), (4) a high degree of loop squareness usually indicates betw homogeneity and uniformity of grain size, ( 5 ) an off-cenered loop from the zero voltage point (the loop is usually centered symmetrically amund zero voltage) indicates some degree of internal electrical bias that may be caused by internal space charge andlor aging, (6) the sharpness

Fig. 11. Typical hyrlereris loops from Y~VIWS kmrlecbic ccrmicb: (A) BaTiO, capacilcf, ( 8 ) soh (easily swishable) FZT. (C)PLZT 8.616935 relaxor, and (D)PSZT antifmoeleclric material.

170

I

6 Fernelectric Ceramics: History ond Technology

XI0

Journal of rhr American Crnirnir Socirn-Hacnling

of the loop tips indicates a high electrical resistivity (>LO" fl.cm). (7)high induced polarization in relaxor matcrials indicates high electmtriction strain and high elecmopuc coeficients. (R) the slope of the P-E loop at any point along the loop IS equal to the large-signal dielectric constant, (9) the opening up of the loop of a SFE relaxor material can indicate nonohmic contact between the e l e c d c s and the ccramic. and (10) a ruddcn large change in "apparent" polarization is usually a result of incipient dielectric breakdown. Remanent polarizations for most of the Icad-mntaining ferroelefnics typicnlly vary from 30 to 40 pC/cm*. whereas the coercive fields v q over quite a wide range. from -2 kV/cm to near electrical breakdown I- I25 kV/cm), depending on the type of dopants and moctificrs added. The strains asmciated with two of these materials (i.e.. ferroelectric and SFE) on traversing their hysteresis Imps are given in Fig. 12. In the ferroclectric caw. the switching strain accompanying the polarization nversal pmccss results in the familiar "butterfly" loop, with the rcmanent stmin state in the center of the loop (point 0).Positive voltage then results in a longitudinal expansion of the ceramic. whercas Q negative voltage (less than the coercive field) nsulu in a longitudinal contrncuon. This is known as the linear strain effect in piezoelectric materials and does not involve domain switching. For the SFE relaxor case, there is no remanent strain when the electric field is not applied, because. in his caw. the rest position of the ion is in the ccntcr of the unit ceU. However, when ihe field is applied. ionic movement (polarization) and strain occur simultaneously. both being dependent upon the strength of the field. Because the sign of the strain produced (positive for elongation) is the same regardless of the polarity of tbe field. this is the elecvostrictiveeffect mentioned previously. tC) Pirxwlrrtrir and &/ectnistricriue Pmpmies; Compositions within the PZT and P U T system possess some d h e highest elecmmechanical coupling mfticients anninable in ceramic mal~rials.Somc typical values of k tlz. d,l, and g33far rhesc materials an! given in Table If1with BaTiO) and the niobaterr. Maximum values of &,, (0.72) and tfI3(710 x IFt2C N arc found in the soft (easily switchabk) P U T composition 7/60/40.This composition is located within h e morphotropic phase-boundary region Separating the fermelectric rhombohedrnl and tetragml phases. Over the yean, there his been considerable speculation concerning the reasons for this maximum in coupling at the MPB.wb' These may be sumnhuized a\ being due to ( I ) the existence of a mixture of phases at the boundary. (2) a concurrent maximum in dielectric

. &.

-4

Vol. $2. No. 1

constant at the MPB. (3) a Iiiger number of rwrienubk polarization directions existing in the MPB mixed-phase region. and (4) a maximum in mechanical compliance in the boundary region. permitting maximum domain reorientation without physically cracking. Also included in Table 111 are some Iypical elcctraslrictiveQ and m values for representative compositions. Tible 111 shows that most QI1 coefficients arc in a rather nmow rdngc of 0.010-0.022 m4/C2. as are the Q,:coefficients in a range of 0.008-0.012 m4/C2.Also given the Q vnlucs of two PLZT ferroelectric compositions (7/65/35 and 8/65/35) for comparison. pointing out the observation that the Q coefficients art: similar in magnitude regardless of the ferrwlecuic or nonrerroelecmc nature of the material. Thi5 is because the Q cocffi cient relates the resulting strdin to the electrically induced polarization. regardless of whether w material has permanen1 polarization. The m coefficients. on the ocher hand, relate the htruin to the electric field; hence, their valucs v q more widely. ranging m2N2. from 1.7 Y IO-'" to 1 I .7x ( D J Pyrrickcfrir Properties: Although the pymelectric effect in crysmlline materials has been known for many centuries. it has been within only the last four drcsdes that this cffect hac been studied in ferrocleclric As mentioned previously. this effect occurs in polar materials and is manifested in a change in polrcriution a\ a function of tempenrure. This results in a reduction of the bound charge required for compensation of the reduced dipole moment on increasing temperature and vice versa on decreasing temperaturn: thus. the change in voltage on the material's electrodes is a measure of the change in the muterial's polarization due 10 absorbed thermal energy. A common figured-merit for pyrorlectricr is

P

FOM = cfK tan

w2

(13)

where p is the pyroclecmc charge ccxFficient. c the spccific heat. and tan 6 the dielectric loss mngcnc. Maximizing the performance of a material then involves selecting a ceramic with a high pyinelecuic coefficient and low specific heat. dielectric constant. and dielecmc loss factor. This is difficult to achieve in a given material. and. most often. i b performanceis limited by the dielectric loss, which is reflected in a poor signal-to-noise rntio. Two families of cerimics have dominated this area of endeavor: PLT and BST (barium strontium timate) nlaterinl.*. However PLZT and PMN are aka considered viable candidates. The former two materials are considered femrlectric thnmal dclectors (absorbed energy generating the tempernturedependent change in polnrization). w h e w . the latter two. as well as EST. can be Lunsidercd dielwuic bolometers (cleclricnlly induced. temperatureaependent change in dielectricconstant niak.rialst.O*Ceramics. in many cascs. are considered better choices for thermal imaging applications than crysulline mamiids with higher pymeleeuic cocffcienLsbecause of their lower cost, availability. ease of processing, and good stability. These marerials in bulk and thin-film forms are used is commercial products for lawenforcement, night rurveillwe. and .wcuritv applications. (El Optical and &kctrooptic Properties: Unlike the PZT cewnics and other ferroelectric mtcrials that are opnque. the most outstanding feature of the PLZT materials is their high optical tmnslucency and uanspnncy. Optical trdnsparrir) is both a function of the concentrationof lanthanum and the ZrtTi ratio with a maximum in lranspilrcncy occwring along the FEPE phasc houndivy and beyond. until mixed phases produce opacity (!tee Fig. 5). For example. the 65/35 ZrlTi ritio canlpositioas am most transparent in the lanthanum range rrom R to 16 at.%. whereas the lorn0 compositions arc simikarly transparent in the 22% to 284 range. A typical transmission curve for a 9/65/35 composition ib

6 Ferroclectn'c Ceramics: History and Technology - C.H. Haertling

then

111

V.

Fig. 13. Tlie material is highly absorbing helow 0.37

pn. w tiich ir the commonly ;iccepted value for the onset of high ahwrption in the hulk material. For thin films. this u l u e 1. closer to (1.35 pii. A fairly consrant opticid transmission of -0Y; wcurs throughnut the h h l c spcrctnini from 0.5 fin1 to tlic nsdr infriirrd iit 6.5 p i (\cc inset). Beyond this. nhsorption ;,pin hrgiiir to tiikr place. and. at I?pin. the material IS.once ain. fully iihwrhing. The liigh-\urlace-rcflectilin I ~ s s e \ 3 l ( i h r I\W wrl;ue';i rhown in lig. 13 :a ii function of tlie gli index nf rcfriictii)n ( 1 1 - 13 I uf [he PLZT.

h u r common tjlx, oScIcctro~)pticeffect\ have heen iound to he opcr;ttivc in Ierrorlectric rnatenal~in general and In iic, in p;inicular t I ) quadratic. Kerr. and hirefrint l i depolarizntion aonriiemory scattering. ( 3 ) lin-

. and hirefringent effects. and 14)niemor).

scattering. The t w ( i t y c s uttliic rclaxor-type. 'N65/35nialeriah w t h Iiiieaily polari~edlight: the third type uses a high coercive field. tctrag~inal.mernnq material. such a\ 12/40/60. with polariied light; and tlie fourth type comnionly uses a low coercivc field. Ihoriih~hedral.memory material. such as 7/65/35. and doe\ not use polariiers. hut. rather. relics on the variable-angle \tattering of light from diffcrent pliuiLed areas to achieve a \patiall) varying iniagc in the ceruinic, Contrast ratios as high as 3OWI can he attained with polarized light. whereas thesc :otio\ are limited to <W/I for \chcmes involving nonpolarized. icattercd light. Specilic properties of the more-common PLZT electrooptic compositions are lirted in 'Tahlc 111. PlLT iniiterials tire ulso known to pohsess many special photosensitive phenomena that are directly linked to their nticrostructural. cheniiciil. electronic. ;mi optical prnperties. including ( I ) photoconductivity. (7)photovolt4c proprrtles. ( 3 ) photo-nwstcd domain switching. (4I ion-implantationenhanced phoio\ensitivity. ( 5 t photochromic effects. ( 6 ) phoiomcchanicsl (photostrictive) behavior. (7)photorrfracrivc effects. and (XI plioroexcitcd space charge phenom~na.~"." Although materials with such a multitude of propenics and q x i a l effect\ hold promise for mnny new applications for the future. i t shcruld nlso be remenibcrrd that these same eHbcts can. and olten do. limit their application

100

a, u K

60

m 2z .-

40

F

t-

2c

0 I

Appliuitiuns

-

t f

h

5

171

The applications lor ferrorlectric ceraniics are nianilold and prrvasive. covering all areas of our workplaces. homes. and autoniohiler. Similar 10 mo>t materials. the successlul applicar i m of these piezoelectric. pyroelectric. ferrnelcctric. electrnstrictive, and clectrooptic ceramics and films are highly Jependent on the relative ease with which they can he iidiiptcd to iiseful and reliahlr. devices. This is. to a great extent. the rc.ason that they bavc been s(r successful over the years in linding an increasing nuniber of applications. Their \iniplicity. compact iiw. low co11. and high reliahility are very attractive fc;iiiircs ti) the design engineer. Many generul category npplic;itions f o r bulk and film electnxeraniics are given in Fig. 14 As indicated in Fig. I?. some of thcse applications are more appropriate for hulk materials, some for films, and some for huth hulk and lilnis. Although there always will bt: a Jc.mnnd for hulk devices. it i s cenainly obvious that the trend in the industry i s toward film devices. The reasons lor this include (I)lower operating voltages. (21 si2.e and weight conipntibility uith integration trends. ( 3 ) hettci processing conipatihility \\ it11 rillcon technology. (4) case d' fahrication. and (51 Itiucr cost\ through tntegrdtion. ti) Capacitors One category of applic:itions for ferrixlectric-type matcrialk is that of high-dielectric-constant capacitor\. particularly MI,(:s. MLCs are cxlremely important to our everyday lives i n that they are essential to a11 of our currently prcduccd electronic components. and. as such, they constitute i t significant portion of the multibillion dollar electronic ceramic\ business as a whole. Most ceramic capacitors we, in reality. highdielectric-constant ferrnrlectric compositions which have their ferrorlecuic (hysteresis Itwp) properties suppressed with suitable chemical dopants while reraining a high dielectric const:int o \ w a broad temperature range. BaTiO, was hislorically h e first composition used for high-diclec.lric-constant capacitors. and i t (or its variants) remains the industry standard: however. lead-containing relaxon such as PMN and PLN are making I n tune with ever-shrinking electronic compmests in

Reflection Losses

80

I

xi I

Fcrriwlcclric Cprumicr: H i w r v ~ i n d'I'iY'/rnclhk,fi.l

/

PLZT 9/65/35 0 375 m m Thick

Fie. 14. Applications of hulk and film ceramic electronic m i a l s . this age of integration, capacitor techniques have trended loward ( I ) more-sophisticated tape-casting pmcedures, (2) surface-mount W s , and (3) tired layer thicknesses appmaching 4 pm. MLCs. 0.5 mm x Imm and several hundred layers thick, am now produced with capacitances of several microfarads. Tape-casting methods arc now reaching their practical limit. and thin-film deposition techniques an being explored. Typical applications include gmeral-use discmte and MLCs. voltagevariable capacitors. and energy-storage capacitors."*,*O By far the largest majority of applications for electro-active materials occurs in the orea of piezoelectric ceramics. In this category. the ceramics are usually poled once at the factory, and no polarization norientation takes place after that throughout the life of the device. Tbse devices can be divided into four different groups, as given in Fig. IS.Two of these groups are as mentioned previously. i.e., motors and generators: however, the third category involves the use of combined motor and generator functions in one device, and the fourth category includes devices operated at higher frequencies. i.e., at resonance. Because of the more-recent interest in electrically biased electrostrictive devices that act as electricallytunable piezoelectric components. some of the specific applications in Fig. 15 are also now being developed with eltcvastrictive materials. Examples of ceramics that arc utilized in a vpriety of piezoelectric and electrosuiaive applications. both large and small. are shown in Fig. 16. Figure IS also shows that he number of applications for piezoelectrics as motors is quite numemus. This is particularly m e for the whole family of micro- and macro-pinOelectric actuators. The micro-devices are considered to bc mOse that utilize the basic piezoelectric strain of the ceramic (mcasumi in micrometers), whereas the macro-devices we those that use a displacement-amplifyingmechanism w enhance the fundamental strain (masurd in millimeters). This is explained more thoroughly in Table IV, which lists dl of the current ctramic actuator technologies and includes some of their important characteristics. Table IV shows that a variety of direct extensional configurations, composite flextensional srru~turcs, and bending-mode devices are used to achieve a mechanical output. Maximum smss generation (40MPa) or loading capability i s noted for all of the direct extensional devices. including the piemelectrics, electrostrictors, and antiferroelectrics: however,

their strain (displacement) is limiled to 4S% or lens. On the other hand. maximum displacement of several tens of percent can be achieved with displacement-amplifyingmeans, such as composite (flextensional sm~ctu~e, Moonie) or bender (unimorph. bimorph. RAINBOW)shuctures. but this is usually accomplished at the expense of considerably less force generation, greater complexily. w d higher cost. In most cases, the aCNatOrS are operated with electric fields 4 0 kV/cm for longevity and reliability. However, even such modest fields can result in rather high voltages (>lo00 V) if the actuator is relatively thick ( V = El); thus. the multilayer technology devcloped for capacitors is often used to reduce the operating vdtage below I 0 0 V. Although unimorph and bimorph structures have been SUE-

Fig. 1s. Piezoelectricand clecrrostncfive applications for ferroelectric ceramics.

174

I

6 Ferroelectric Ceramics: History and Technology

Monolithic ( d 3 ,mode)

4 E 3 V

D

Morulliiluc td,, mx*l

40

Eapansion

P

0.m

40

Conmion

P

4.1s

40

Expansion

E

n.2x

40

Contraction

E

-0.09

A

0.50

Expansion

A

0.08

Contraction

PE

. + Monolithic (sll mode)

D4

Q

V

Monolithic (sI? mode)

. + Monolithic (sI

4IIC

D

D

Composite structum td,, mcuic) (flextunstonall

4

v

40

8 9 V

10

-1.0

I'nitnrnph (bender, Bimcirph 1 kndcr)

Kamhiw nxmomtxph (berukr)

4 6 %

D

modc \ia explosive shock wuvcs or projective impact. uscful few hundred kilovolls or kiloamps lasting lor many microvcMfds can k obuined.These oncshac supplies have found ntany USLS in miliwry applications. "m

ClCCtnCdl pulses of u

ywer

(3) Composifes Piezcxlecmc coniposites represent one of the latest technologies developed for engineering the lac1 hit of high performance from a pierwlccuic transducer. When nu delikntcly intmduco a second phase in a material. conncaivily of che phases is a critical parameter. There are 10 connectivity paltrms possibk in a twa-phiru solid, ranging from 0-0 (unconnected three-dimensional checkerboard pattern) to 3-3 (interpenelrat-

ing pattern in which both phases am three-dimensionally selfconnected). Some of lhrse connectivity panems nrc panicularly well wited for decoupling the longitudinal and t r a n a v e ~

picznekctric effects. such that materials with significantlyenhanced (up w a factor of 100 or more) pianlecuic pmpcrtiea arc possible. Moreover. a ceramic-polymer comptwite offers distinct advantages, such &E a wide rnnge of acoustic impedance matching. b W bandwidth. low elecnieirl losses. and, for medical ultrasound applicatianr. send-receivc capahiliiy in a m p a a package. Considerahk mgincaing ingenuity has been demonstrated in designing. fabricating. and packaging the many types of diphasic s t ~ ~ c t ~ pMajor c z . applications include hydrophones. sen5ors. and mcdicnl ulmtsonics.L1'"

G Ferroelectric Ceramics: History and Technology - C . H. Haertling Apnl I Y W

Fc~rror~l~~nric (.'rrumics: IKrtory o ~ Trrhnokrgy d

I

81.3

FIR. 16. Vanety of ferroclccuic ceranlicn used in picmelcctnc and electmstrictivr applications. such a( sonar. accelemnielers, actuators. and Sensow (Photograph counery of ED0 Western.)

cessfully applied to many devices over the past four decades. their inability to extend the force-displacement envelope of perfumiance has led to a search for new actuator technologies. One mch device developed in the early 1990s is the Mnonie$11named hecause of its crescent-shaped, shallow cavities on the interior surfaces of the end caps (see Table IV), which are bonded to a conventionally electrnded pirznclcctric ceramic disk. When the ceramic is activated electrically, the shallow cavities permit h e end caps to flex, thus convening and amplifying the radial displacement of the ceramic into a large axial motion at the center of the end cups. Advantages of the Moonie include (1) a factor of 10 enhancement of the longitudinal displacement. (2) an unusually large d , , coefficient exceeding 2500 pCIN. and (3) enhanced hydrostatic respon~e,'~~''Recent improvements in the basic Moonie design have resulted in an element called a Cymbal. a device that possesses more-flexible end caps. resulting i n higher displacement." Applications include tranrduccr arrays, medical imaging transducers. and hydrophones. Another device recently developed ti] increase the fnrcedicplaceinent performance of a piezoelectric actwdtor is the RAINBOW. In its most basic sense. a RAINBOW can be thought of as a premessed, axial-mode bender similar in operation to the more conventional unimorph Ixnder. Unlike the unimorph and Moonie. which are composite svuctures, the RAINBOW is a monolithic monomorph that IS produced from a conventinnnal. high-lead-containing piezoelectric ceramic by means of a high-temperature. chemical reduction reaction. As inentioned previously. this process produces significant intrrnal compressive and tensile stresses that are instrumental in achieving it5 unusually high displacement characteristics. Displacements a3 high RS 0.25 mm fnr a 32 inm diameter x 0.5 mi thick wafer have been achieved for these devices in a dome

mode of operation while suatairiing loads of 1 kg. Marinrum displacements (if > I mm can be achieved wilh wafers (32 nini diameter), thinner than 0.25 mi when operating in a saddle mode. Prototypes of RAINBOW pumps, speakers, optical deflectors. vibratory feeders. relays. hydrophones. switches. platform levelers. Sensor and actuator arrays. and toys have been demonstrated: however. no commercial products have been yet k e n prodr~ced.~'.'~ Sonie examples of these different types of' piezoelectric devices are included in Fig. 17. A novel type of bimorph application of somewhat recent vintage is the optoniechanical (photostrictive) actuator. The photostrictive behavior is a result of a combined photovolktic effect (wherein light prcxluces a voltage in the ceramic) and a piezoelectric effect (wherein this voltage pmduces a strain in the material via the converse piezoelectric effect). PLZT ceramics with donor-type doping exhibit large photosuictive effects when irradiated with high-energy. near-ultraviolet light. A bimorph configurarion with no connecting wires has heeii used to demonstrate prototypes of a photo-driven relay and a remntc micro-walking device, and a photophone of the future has been

envisioned.".'" (2) Explosive-ro-Eleemd Transducers (EETs) Studies on the stress-induced depding of ferroelecrric ccntmics were initialed in the mid- 1950s. which culiuinatid i i i the development uf one-shut power supplies that iiiade use o f this effect. This depnling behavior is optimum (1.e.. iriaxioiuiii oirtpur in the shortest period of dine) for ferroelectric coinpo sitions located along the houndary between the polar ferroelrr tric phase and the nonplnr antiferroelectric phase. wch shown in the gray m a of the phase diagrerri uf Fig. 5. Althiwgh this depoling does cur somewhat more slowly undei hydrostatic pressure. when it is accnmplished in an extremely last

173

Fig. 18. C'innmcrci;tl and military applications of PLLT elcctrwiplic cc'ra~nic\:rsianing at upper nphi and guing c l a h w u EEU-I@ ~ flyers goggles ilhoiograph counesy of Sandin Nauonal I.ahoraiorics.), RI-R cockpit viewing p n (Photograph counery of Sandin Saiional 1.abora11rries.1.film wnicn tl'hoiogmph cciunc\) of LYI'.).offwi image \crier il'hoiogrnph counesy of Steigcr.1. and premier image enhancement s)stcnl (Photograph i.oune,) of Exmian Krxlak. 1

iiuns. revstance degradation. and nicmory imprint (persistent memory. i 2 . the revawnce t o \witch out of a given memory
their succer\.

Vl.

Future Prospecls

Prewnt ni;irkct irriid\ c o ~ t t ~ i ~tou cshow that the future fur lerroclectnc ceramic> IS hright ant1 cuntinuer to gel even hrighter ar the tramition i s made from pawive to electrically a c m e " w a r t " and "verj snlafl" materials. In this reg&. a m a n material senses a change in the environment and. using an cxtemal feedhick control system. inakes a useful response. I\ in a ccimhined \ensor/ocriiator ceramic A very sman malerial mires ii change i n the env~r~~nnient and responds by reacting aiid tuning (self-oontrolling~one or more of i t s propenies to irptiiiiire itr hchawir. An example of the sman iype FI a piemelecti-ic cer;miic and of the very smim type i s a nonlinear. e l c c t r ~ i m i o t i \ c relitxur. Multifunctionitlit) i s ;I key concept o i tl~cssroiiterial\ thai will he exploitctl with all the ingrnuit) that design engineer\ c:m mu\ter." I n the future. inure and inure application\ iur nonlinear. electri~stri~iive relaxor niatenals. rucli a\ PMN and PLZT. will miergc a\ thc rclriitles\ drive toward niiniatunmtioii and integrdiiun continuex Indeed. this wry trend will also encouray' more inateri;il\ rewnrch efforts to develop hctter ferroelectric ;ind t'lertn~rtrictiveceramic\. .A\ niche ;ipplicationa heconic morc prevalent in the future.

composites and displacement-amplifsing techtiiquc:a and inatcrials will proliferate in a continuing effort to widen tlie forcedisplacement envelope of performance. There device>. t w . w i l l hecome smarter and smarter a s the applications dcmand. Urought on by the need for higherapacity nicniuricr. expanded data processing capability. and marter device.;. the direction set a few y e m ngo for fcmxlectric films i s expected 10 continue and hroadeit in scope Thin- and thick-lilm teclinologies alike will also share in thc current trend toward coniposiie and gmiled .truciures with .;pecificolly engineered. :ind often unique. propenies. .Multiply deposited layers of different materials. or graded layer5 of the wine inatcrial. are mrw achiebahle with m i s t conventional film depisition pmccsse\ on a micro scale, and this will he more commonplace i n the future on a nanu scale.xn')I Bccause thin- and thick-film technulugies generally h not limit. hut rather enhance. the portfolio o f materials to be used in venous applications. it is expected that a viiricty uf maicriiih w i l l continue to be studied. but there will he a. narrowing do.c\n to fewer senou'i candidates of known behavior in order to bring the devices i n development t i ] the marketplace. Undouhtedl). HST. PZT. PLZT. PMN. and SBT are destined to he leciding candidate\ in thk arena. Keprding f i l i r i depo~itiiintcc1iiiiipc\. ;it thtr htage in thc de\~c.lopruentCJI' the filnis. i t is v e r j drl'liculr io .judge which lilni deposition technique will emerge ;1\ tlie lawrite: howvcr. hecoure scvernl niethtdr have heeii u\ed wccessfully. i t i s moqt likely that several methud\ will uiri iw. md a specilic nierhcd selected will he dictated hy C I ~ S I :uiJ the applicaticm.

mented transmissive and reflective Several coinmercial and military applications tor PLZT shutters and linear gate arrays are given in Fig. 18.

Fig. 17. F~;amplesof FZT,PLZT. and PMN piczoelecuic w d clwtrnsuictive dcviccr: tstllning at upper right and going clockwise) Mcb ~"ola tweeter. Triangle pm-grill lighter. Yoiomla himorph. Murpta ,ntcrmdiatr frequency reslmators, Morgan Maim ulkasonic c k m r icramic5. Aura RAINBOW cernrnic<.lick PMS Wuatw. fenmleevio lilin memory. K& P 1 . n FlO dev~rc.RAINBOW mouse lo) ac. tuator. Mmnic PCIUUI~HS.R d i o Shack buzzer. und unimorphh.

(4) Eleerrooprics

Since the late I%&. when transparent P U T materials were first developed femxlecuic c e m i c s have been thoroughly rcworched. and their characteristics studied to the point that they have now taken their place alongside single crystals as legitimate candidates for electraptic applications. Compared to single crystals. ferroelectric ceramics an generally. but not always. less transparent, less uniform on a mirraseopic scale hecause of their plycrystalline nature, and somewhat less optiniurn in their electmiplic and hysteretic properties. On thr other hand. eloftnmptic ceramics have several specific characteristics that make them well suited for a variety of electmoptic applications. Among thew characteristics are ( 1 ) small areas ton the order of the grain size. from I to 10 jm) that can be electrically switched independently of other adjacent arras. (2) witched are= that are stahle with time in inemory material\ o r stable with applied elwtric field in nonmemcq materials. (3) light transmission characteristics of the switched areas that depend on the thoroughness and direction of switching or paling. (4) switched anas thac exhibit either electrically variahle light scattering hehaviw or electrically variable birefringence. and 15)ceramics that can he hot-preswd or sintercd in a variety of s17.c)and shapes wth a high degree of optical uniformity on a macrnscopic scale. Thin polished plata of PLZT.such as Y/65/35. when used in conjunction with linearly polarized light. make excellent wideaperture clectrooptic shutters. linear gate arrays. and color filters. Their fast re\ponse (in the low micro.;efond range). thin profile. wide viewing angle. and wide temperature range of opemiion t 4 " C to + R O T ) are highly desirable characteristics for most devices: however. these are offset hy the disadvantage, of I ) low ON-state transmission of - I59 as a result of the polarizers and (2) high operating voltages (-3.50V)that are required to reach a full ON mndition. Despite the low ON-slaw transmission characteristics. high contrwt ratios of 3ooull are caxily achie>ed when high-efficiency polarirers are used. The In,larizer-PIZT configuration usually used in a shutter device i s shown in Fig. 1. Vacuum-depsited o r gmoved, interdigital clectrdes (not shown) on: commonly used to decrease the operating voltage by reducing the gap between thc pntitive and negative finger elam&%.These finger electnxler ;ire thin enough l
(5) Films By far the largest numbn of application\ in ferrwlcctnc ceramics remains awociated with bulk materials. hut a trend toward thin and thick films for some of these applications has recently developad and is steadily increaging in intensity.Aside fmm the obvious advantages. such ns smaller size. less weight. and easier inteption to IC technology. fermelectricfilms offer additional benefits, including (1) lower operating voltage. (2) higher speed. and (3) the ability to fabricate unique micro-level strwtures. Equally important. but not as obvious. is the fact that n m y materials that are difficult. if not impossible. to fahricate to a dense cerwnic as II hulk material arc relatively ensy to produce as films. Moreover. the sintering temperatures of the films are usually hundreds of degrees celsius lower thim that of the bulk. and this often can be the deciding factor in a successful design and appli~ation.~'-'~"~ fhe several most imponant film applications are included in Fig. 11. Here again, mme applications are exclusively designated for films while others are mutually shared with hulk materials. Applications for thick films (2-20 m)include electrooptic and some piezoelectric devices, whereas the remdining applications(i.e.. capacitors. infrared sensors. memories. huffer layers. integrated optics. and antireflection coatings) invalw thin films ranging in thickness from 0.2 to 2 pm.n5The rea$on for the thicker films in the f m r cases is to obtain maximum strain or eleccrogcic output from the films without having to resort to electric fields too near to dielectric breakdown."" This is n a a problem in the latter applications.bccause the films are subjected only to low voltages (40V)or to no voltage in their operational environment. Large-scale manufncturing is presently underway to incorporate fernxlectric films a. storage and bypass capacitors in and other IC circuiuy. In a conventional DRAM (dynamic random-access memory) computer memory application. one SiO, capacitorin in series with every switching transistor; however, as the memories become more dense, the area iaken up hy rhe low-dielectric-consul SiO, capacitor is ~ O Clarge. , and new materials with higher dielectric constant^ must he substituted. A fenoeleclric DRAM (RDRAM) film. because of its much higher dielectric constant (-IW vs 4). occupies much leks wafer area than the normal SiO, capocitor. thus allowing much greater capacity memories to he fabricated on a given silicoii wafer,w A5 ineniories become more dense in the future. the transition to fermclectric films will become a necessity. and operating voltages for these memories will continue to decrease toward I V. At present, BST film capacitors are the t t ~ pcontenders for these applications. The development of ferroclectric random access memo0 (FERRAM) films for nonvolatile memory applications. such a,. computer random-access memories, smart cards, and radiofrequency identification tags. is presently underway itnd ha\ reached modest pmduction levels for specific niche application~. '~ The issues faced in nonvolatile memories are much more challenging than those involving simple capacitors. hecause. in h e nonvolatile case. the two memory states consist of the two polarization states (polarirution up and polarization down in Wg. 3) in the film. one k i n g a 0 and the other k i n g a I . In thin c s e , the f e m l e c m c polarization is const;tntl) beins switched,and this immcdialely raises the issue of switching fatigue. FERRAM films of sewral compositions. including PZT.PLZT. and SrBi;Ta,O, (SBT ih a layer-typeteenoelectric, see Table 11). are' actively being investigated. Among these. SBT is. perhops. the material of choice in regard to fatigue SBT materials exhibiting little t
178

I

G Ferroelectric Ceramics: History and Technology

818

Vol. 82. No. 4

J~~urnrrl of rhc America!! Ceramic Srwier).--HarnhR

Dull.. 21 171 33-65 tlY96). MG H. Haenlmg. "PZT and PLz1' Thick Filnia on Silver. Sspphm. md Silicon": pp. 15-30 in Ccruniic Tranrncuonr. Vnl. X6. Inrgrored Tnin Fzlmr oml .4ndIratr~~n.c Editcd bv R K Pmdv. P. hlonis. A Rarhko. U. Vanhmr. and

Gene Haertling received his R S . degree in ccrunic enpiiieenng from the University of Missouri al Kolla in 1954. His M.S. and Ph.D. degrces. also in ccrnmic cnginecrJnirerbiiy of Illinois in IOh(l and 1961, respectively. F r m and manaperinl p(isition\ at Sandia Natioital I.ahoratories. pd llie firct tran.;parest ierroelecrnc ceramic\. the P1.%7' tlead Innthnnurnzirconate titanarc) materials. which are n o u used in huth military and cnmrnercial applicationr. From 1974 i o 1987, he WLI?I Vicc-Re\ident of the 'reechnical Stafi aiid Manager of the Ceramic Research Group if1 Motorola. Inc.. Albuquerquc. NM. Just prior 10 .joining M(ilorola. he uab prewlrnt of Oploceram. Inc.. ii small entrepreneurial conip;iny hc founded. which was engaged in the development and manul'acture of PIZT elecmriplic cermics. Aflcr hriefl) srrbing 1x1 the fmilcull) :I( 1:niversity of Miw)uri at Kolla from lYX7 io I M X . Dr. Haenling pined the Ccrarnii cnt of Clemnon 1nhervty i n the Bishi)p L)i\tinfuislted Profeswr of (:crurnic 16, While al Clemwn. he drveliiped the special procesh lor pnducing hipliiicnt. piercwlectric ceramic acluatorr known as RAINBOWS. He i s :I nii'iiiadcnty of Engineerr iind a Fellow of ?'lie Arnerican Cer;iinic He ha
nu.

G Ferroelectric Ceramics: History and Technology - G.H. Haertling April IYW

Ferroelecrric

Cerariiics:

Because of their intrillsic dielectnc nature and large number of' inlcrnctive and electrically variable properties. fernelectric c'crarnics are destined to figure proniinently in the fuluw. Bulk. (hick-film. and thin-film fimns of these materials have now proved their worth. and, together, they will C O I I S I I ~ ~aI Istrong ~ portfolio of niaterids for future applications in electronics.

References

28s -9I I1987). ' L E.Cross and R. E Ncunham. "Hiwry of Ferrrrleeirtcr". pp ?XY-S05 in Crr.mucr and Cn 1117a1100. Vnl 111. H,,d~.T?chrtologyCcwmm-t',i.ri, Prcwnr, r u i w r . American Ceramic Sociay. Wcsunrlle, OH. 1987 'Cj BuKh. "llow I Dircovcrd lhc Fenwlecuir. hopcnier of KH,PO,."

t s d

Fe rrwkrnc$. 71.4347 I1987) bJ Fouak. "kmnlocmciry: Remarki on Hirturrcul Aipcch and Pmwnl

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27. 31-34 fIYHOI "M A M c y and P A Davis. "Slandnrd SELECT EIccm~smcl~rc Lead Magnmum Niohslc Actualon for Activc and Adaptive Optical Com~mcnlr." opr Enxnp.29(111 1373-X2(lWn :'I..F. Cmrr. "Relaxor Pemleciric\:' F c r r w l e d r t ~ . 76. ~ , 241-67 (1987). 1. Zhao. Q.M Than& N . Kim. and T. Shnrul. "Elrcuumcchanrrrl ProperiRcliror Fefmclccvic Lcad Mdgnmuni Niohate-lrad Tiunate CcramJpn. I Appl. P h s . . P m I. 34 [I01 S658-63 t l 9 9 S ) :W P Harmer and D. M. Smyth. "Nanosrrucrure. Ikfrel Chemistry. 2nd Rnpcnles o i Rclaxcr Fcfmclec~ncs."ONR Final Rcpt. No NWl14-82-K-01Yli. Izhigh Univerrily. Bclhlchcm. PA. Fcb. IW2. >OR. P. Umdcur. K. GJchigi. P. M. Pruna. und 1 '.R . S h w l . "llllra-High StrainCcramics with Muldplc Fncld-lnduccdPhase T m r i m n r " J. An. Crnim s w . , 7 7 l l l l 3 r 2 - 4 4 (1994). "G H Haenling. "F.lstro-opic C-icr d Dev~cer~'. pp 371492 In Elerrronir Ctmmicr. Fdacd hy L. M Lcvinwrt. Marcel Dckker. Ncu. York. IOR8 % Guudman. "Ceramic Capcilnr Materials": pp, 79 138 in Crromir Muirrtolr /or Elrrrronto. Fdiled hy R. C . B u c h u n . M a n ~ Dckker. l New York.

"R. Mancmk and R. Wmicke. "Cennuc Boundwy-lsycr Caprcilon." bps 7rch R m . 41 I I1112) 338--46(lYK31.

'W C lid1 and I t L Tullcr. "Cmmir Senum. Theor). and Pracucc". pp x for Elcrrn,ntcr. Edited by L. M Levinsnn Mnr. 265-371 tn C ~ m Mmtrialr c d Lkkker. NEW Ymk. IYHR. "I3 Kulwcb. "PrCu Maierixls 'Technology. IYSS-I9RO": pp. 138.54 III Advancer in Ccrdmicr. Vol I, GNW Roundnr) PhrnnmrM m tl*crnmic Crrumics Edwd hy L.M. Lrsmm. Amcncan Ccraniic Soeiety. Wcsierville. OH.

1981 '.ti Slurme. K Sulukt. and A. Taknla. "Phase Transitions m Solid Soluimn\ <,f laad %trwsaic a114 Lead Tiranate. 11," J. Plip. Sor. Jpn.. 7 111 12-18

t195?l

.

"0 Jaffc. H S. Koth. and S. Marzulln. "Pie7cxleclnc Praperlicr 01 Lcad Lirconrte-.l.md Tiranale Solid-Solution Ceramics." J 4ppl Phyr 11 161 RIMIn I l Y Y I . "ti H Haenling. "Reroeuleci"~and Elrctrcmpl~Ccmntc\". pp 13?--2? 111

I

XI7

Hisroq und 7erhnology

Cmimir Morrriulr /or E l u ~ ' ~ m m rIAmd $. h) I'L. E. Cmrr. S J Jang. and R 1% Newnham "lmrpe t:lecimslricIiw Etlcrh an Relanx Femrelccir,c%"turmr~lr*.mrr.23. IR7Y? ( I'l8fli. "S. K u m m and K Uihino. "Rixent Applioaemr o i P M N - B a d Elramtors." Ft'morlerrnrr. 50. 197-202 (1YR1l. *'M. Kahn. I). Hurk5. I. H u m . and W. Schulzc. "Ccramic Capxilor Tech nolngy": pp. 191-274 in Elrcrronic Crmmrn. W i l d b) L M Levinson hl.tr. i e l Dekker. New York. IVMX. " M F Yw. H C Ling, ;tnd W W. Rhtdcr. "Preparatmi and hapcnic\ (11 PhO-MgO-Nb,O, Ceramics Near Ihc PWMg,,.Nb,,iO, Con~pr\iltun:' J Marrr. H a , 4 141 9 3 ( W (19891, '3.I..Swam and T. R. Shroul. "Fnhaainm d P c n Aiie b a d Magnowm Niobate." Mum. Rev. Rull.. 17, 124S-XI 1 IYLLZI. " M Sayer and K ?+mnivPr."Ceramic Thin Films Pahricaitnn and Applications." Scienrr (Wmhingrnn. DCI. 247. 1056 N) (IWJl. -G H Haenling. "Fcmxlccmc Thin Films for Elccvnnic Applicmnn<," J Vnr Sci 7rcbnol. A. 9 131 41620 (19911 "0 Sahuri. "Propcmer of SemiconductweBanum T!maicr." J P h n Tor Jpm. I 4 [ Y ] 1159 74 (IYSYJ. "'0 H Haenling snd C E.lnnd. "Rcccnt Impurcnrcnu in lhe Oplrcal and f:lcctmupiic R o p n i e r of P U T Ceramics." Frmrlrrrric~.3. 269-80 I19721 "K S Mardiyam. "Fine R n i c l r Penwrkire Prws\ing." Am Cwant. .So< Bull.. 63 1.11 SYI-91 (IPWI. ' W 1. Uiuron. "Hydmlhmnrl Synthesir of Advanccd Ccramw Powder,." Am. C W ~SO<. . Bull.. 67 1101lh71-lX ( 1 9 ~ 8 ) 'yW D Kingery (FA). Ccromir felrrrcntmi P r o r r r . ~ ~pp. I 235. .UII Technology Press And Wdcy. New York. 1YtY.l W W . S Ynung, "Multilayer Ceramic Tcchnoluyy". pp 403 24 in Crrr,mt< Murerwlr for E1ecfrontr.r. FA& b) L. M Levtnson. Mmcl Dckker. New York 19x8 'IT F. Muma) and R ti Dungan. "Oaygcn Finng Can Rcplace Hot Pressing for PLT." Crmm I d , 82 (61 74-77 ( 1964) '% S. S n w . " l m p v e m n m in Atmosphere Siniering MTranspnreni P I Z T Ccrmcr." 1. Am Crmm .Sot , 56 (91 479 -RO t 1973). "ti H. Hsmlmg. "Rainhw Ceramics-A New Type of Ulim.lligh Dtr. placcmcnt Actuuior.-AAm. C'crmm Srx. Bull.. 73 111 93-96 (19941 YC. Elw..\alde. I. E Cmsr. and C A. Randall. "SUuctum--hmpcny Rehimiin\ m a Reduced and lnlernilly Uiaxd Oadc Wafer (RAINHOW) Ac1wm)r MaI W ~ . " Jam. reran,. SOL..r) IS1 201b48 ( 1 9 ~ 6 1 . "G, LL "lnflwncr of Inicmd Smss on lhc ElccuomcchanicalPropcnic- of P1.7.T Socrr-Biared (RAINBOW) Ceramic Acnialmi'; Ph.D Di\rcnaiion. Clcmwn Unwcrsay. Clcmron. SC. 19% '"K D Budd. S. K Dcy. and D A Payne. "Sol4iel Pmncewng n i I'hTiO,. Phm,.pzT. and PUTThin him%." Rr. Ctront. r r o r . 36. 107 21 (19x?1 "R W. V a l , "Meldlo-Organic Duumpcaitttm (MOD1 Roccrsrag o i ker. mclecvic~and Elcctnmptic Films A Review." F ~ r r , x + l t ~ r n 102. c ~ , 53-6X ~

(Ilrx)).

'XG. H. Hacnling. "PLZT Thin Films R @ from Acmle I'rrcurwn' firrmrlrrincr. 116. 5 1 4 3 ( I W I ) '"L M. Sheppard."Advancer tn Pnrersmg oflrrrorlrcmc'lhiii f.ilm\."Ant Ctrmn, Sw. Bull., 71 111 8.5-95 (IYJ21 "'R W Schwuru.T J Onyle. S.J I i r k w m i M. R Sinclns. D. Dimm.anJ C.D. Uuchheii. "Sol-Crl Pmcrsing FLT Thin Films. A Rc\,tcw af the Siairof-lhe-An and Rxw O@limrmimn Saaieg~ca."h r q r Ferr,wlrcrr,. 7.259-17 (1995).

"'0Aucicllo. A. 1. Kingon. S. B. KNpanidhi. R Ramcrh. M. de Keii

G.J. Domian$. S. K. Dey. P. V Alluri, R. A. Tuale. R. W. Schwanz. R Junes Jr.. and S. B. Deru. "Elccrrwcramr Thin Films. Pan I:Roccasing." M H S

RuIL. 21 161 2S-.SX (IS%). *?Z.Y. hlcng. U. Kumar. and 1.. F.. Cma. "Elcctramclion in Leud Lanlhanum Zlnonau Tiianate (PlZT) Ccramcs."in Piciwbrrnr and tlrrrrrwfnniw M a u l d ! /or Tronsducrr Applrcmns. ONR Annual Rep. No. N00014-82 K0339. Prnnsylvania State Ihiverrity. Univcrsily Park. PA. May 1985. "It. Banno. "Ptczoelccmc Transducer and Piezoelcctnc Ccramcr": pp 2017-23 ~n Enqrlr~prdtuo/Advan~rdMarrrialr. Edited by R. W. Cahn R i . pamon Rcs. Oxfont. l'.K.. 1995. "V. A. Isupor. "Ropeniesof PMTn.Zr)O, Piuoclwwie Ceramics and ' i n . turn of Their Oneniaiional Diolcctrir Polmzeuion," Zrw Phw -.Solid Sran ( ~ " g i .T,~ET~.I. 10 141 w - o i (19681 "I.. Rcnpuigui. 'Thennodynanuc Thwr) 01 lhc Mixphinmpic Phase Tratisition Tnrupnal-Rhomhoheddral in thr Perovrkiu I~rmlrclrius."S o l d Slrrlr C,,mmU". 11. x2s-28 11972) MS. T.Liu, J U. Heaps. and 0. N. Tuiw. 'TicPynnlecmc Propenier n i ihc Lanthanum-hpul Fmrlwmc PLZT Ceramics," fi~rrrwlrcrrirr,3. ZRI-XS (19721 "'R R Zeyiang. W H. Schr. and K. V. Kichl. "Enhnnccd Pyruelccmc Prop aim at a n-ff. Plwc Tranwinn." fcrmo~lecmrr.11. 355-58 (1976) "'R. W. WQunom. P. C (hhind. und N. M S h o w k s . "Fcnwlcclric Mai e n a l ~fsn 'I'hmnUl IR Dc~eclurs."Frrnlerrnrr. 64. 25147 (1987).

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Ferrorlectrii-it). The, Fundamt,ntals Collet.tion Edited h y Basilio Jimenez, Julio A. Gonzalo Copylight Q 2005 WILEY-VCH Verlag GinbH & Co. KGaA. Welnllelnl I S B N . 3-527-40486-4

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7 Tenth International Meeting on Ferroelectricity: IMF-10, a JubileeMeeting W. Kleeman Laboratoriumfir Angewandte Physik, Universitat Duisburg-Essen, D-47048 Duisburg, Germany

historical evolution and its connection with people. They are the heroes of science - pioneers and shining examples - and very ofThe tenth International Meeting on Ferro- ten provide motivation and enthusiasm to electricity, IMF-10, took place in 2001, the those who are at the beginning of their cafirst year of the third millenium. Spain’s reer, the young scientists. It was a brilliant capital Madrid was elected to host delegates idea to start the main sessions on Monday out of 42 countries throughout the world. with reports on the development of great Although the organizers and chairpersons, scientific ideas by high-ranking speakers. J . A. Gonzalo, B. JimCnez, M. Tell0 and Let me name a few highlights: The development in ferroelectrics reI. Garcia Solt, chose a very traditional place for this meeting, namely a university cam- search and applications between 1966 pus and not a fashionable city hotel, they (IMF-I, Prague) and 2001 (IMF-10,Madrid) managed to make it a real jubilee meeting. was presented by Robert Blinc, who attended It will retain a long-lasting memory owing all meetings from the beginning. He masto a number of outstanding events during tered this enormous task with supreme skill the course of the five busy days, from Sep- and deep insight and knowledge. It must have cost him many weeks of his summer tember 3 to 7, 2001. Celebrating the number ten in a series vacations. His report Ferroelectrics 1966and setting the third milestone in the book 2001: An Overview [l]is a testimony of a of international ferroelectrics history, it was great development and worth reading by tempting to reflect about the development both professionals and newcomers in the of the I M F series. The chairman, Julio field. The history of understanding Phase tranGonzalo, took this opportunity with care and sitions in ferroelectrics (some historical resensuality. He reminded the audience that science is not only the achievement and pre- marks) was delivered with charm and husentation of brand new results at the fore- mor by Vitali L. Ginzburg [2]. With a glimpse fronts of research, but it is also the reflec- onto the incurable incompleteness of infortion on the development of the field, its mation or - more simply - human igno7.1 A Touch of History

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7 Tenth International Meeting on Ferroelectricity: IMF- 70, aJubiIee Meeting

really first proposed Devonshire’s theory and Cochran’s soft mode. Wisely he put his name onto the list of “matter to be discovered” when describing a new class of materials, the ferrotoroics, whose scientific evidence will still be a task for the future generations. In his contribution on Critical phenomena near structural phase transitions since thirty years Nobel Prize Laureate K. Alex Miiller [3] incited an intense discussion on criticality in systems with long-range interactions, an issue which still deserves careful consideration and is presently, again, a matter of controversy. Similarly attractive, in particular to young people, was an attempt to bring into mind science’stop people on Wednesday. I believe there has been common agreement that this extra session on great physicists of the last century has absolutely been worthwhile. A surprisingly vast audience took this occasion and learnt a lot from the carefully researched biographical reports on the Braggs, W. H.B. and W L. B., by A. Michael Glazer [4]),on Peter Debye (by Eric Courtens, [S]),on John Bardeen (by F. Sols, [6])and - extremely touching - on Lev D. Landau. It was a unique and fortunate chance for the audience to listen to the report of V. L. Ginzburg (71, Landau’s close colleague. Finally Stanley L. Jaki reflected on the Relevance ofmaterials science [8], to which ferroelectricity is supposed to belong to a large extent. Similar historical reflections (or tutorials, as introduced at the occasion of IMF-9 in Seoul, Korea,) are strongly recommended to be included in the schedules of forthcoming IMFs. Even when being reduced to nonjubilee size, they are assumed to be extremely motivating for young scientists who risk to lose information on the roots of their passion and profession.

7.2 A Wealth o f Science

The scientific program was well equilibrated and kept the standards of previous meetings. It comprised eight Plenary Talks, 39 Invited Talks, 33 Invited Presentations, 57 Oral Presentations and about 800 Posters. The total number of contributions, 940, finally exceeded that being delivered at previous meetings (IMF-8, Williamsburg: 593; IMF-9, Seoul: 873). Clearly, the bulk of individual work was contained in the large number of posters. However, since the reviewer had no time to browse this enormous amount of information adequately during the meeting, he restricted himself to visit the talks in order to classify their impact onto the most important fields. Within the 129 orally presented contributions - exempting the eight reviews delivered as plenary lectures - the following distribution to different key subjects was observed: Spectroscopy Structure, Domains and Defects Thin Films, Micro- and Nanostructures Materials Phases Transitions Theory Applications

29 23 21 18 17 12 9

This table mirrors the fact that I M F has always primarily been considered as a forum for basic and fundamental research in ferroelectricity and related fields. The wealth of applications,ofwhich the community is very proud, is now rather found in other wellestablished international meetings like ISlF and ISAF. It is understood that these do not compete with IMF, but rather cover the complementary spectrum of applied science. When returning from a conference you are usually asked about its scientific high-

7.2 A Wealth of Science - W. Kleeman

lights. Then you will take a personal choice by adding the caveat that only the fiiture can draw up a real ranking list. This bearing in mind I dare naming my personal hitlist of outstanding papers presented as oral contributions to IMF-10, following the above list of key subjects. It may serve as an approximate overview of the ample field being covered by IMF-10.

Thin Films, Micro- and Nanostructures and Applications

Naoki Wakiya, Tokyo Institute of Technology, japan, reported on the Preparation and properties of nouelferroelectric andferromagnetic array structure thinfilms [12]. In an impressive way a very modern aspect of artificial material development was presented, namely the use of heterolayer structures including ferroelectric and ferromagnetic Spectroscopy thin film systems. They are shown to have Marie-Hiline Lemie-Cailleau, Universite de the potential for new magneto-electronic Rennes, France, presented Photo-induced FET devices and memories with “multiple ferroelectricity at picosecond time scale [9], value recording” abilities. where light-induced polar charge-transfer phases have been evidenced in pump-and- Materials probe experiments at ESRF, Grcnoble. Sub- a) B. A. Strukou, Lomonosov Moscow State sequently to the generating laser pulse a University, Russia, reported on Phase time delayed X-ray beam served as a transitions and dielectric relaxation in dyecrystallographic probe of the metastable ing K D P [13] and showed that non-isoferroelectric phase on a picosecond scale. morphic dye molecule inclusions affect Structure, Domains and Defects a) Vladimir Shur, University of Ekaterinburg, Russia, presented Barkhausen j u m p s during domain wall motion inferroelectrics [lo],where noise components of the switching current during polarization reversal by domain wall motion in gadolinium molybdate and lithium tantalate have been recorded together with a simultaneous visualization of the domain kinetics. b) Within his report on Domain responsefeutures o I S B N : Ce [ll],Peer Lehnen, University of Duisburg, Germany, showed the evidence of random field-induced polar nanodomains in strontium barium niobate. Apart from indirect proofs like second harmonic generation and birefringence precursor tails they were for the first time visualized in real space by high-resolution piezo-electric force microscopy (PFM).

the phase transition behavior of the classic ferroelectric in a unique way and have a significant influence on its dielectric and conductive properties. This is another example of a heterostructure, whose material properties can be tailored in a comparatively simple way. b) T. W. Noh, Seoul National University, Korea, presented A new candidate material f o r use inferroelectric random access memory ( F R A M ) [14], namely lanthanum-doped bismuth-titanate (BLT) thin films for fatigue-free F R A M devices. They have low processing temperatures, large remanence, high resistance against hydrogen damage below 350 “C and excellent retention properties.

Phase Transitions a) Beatriz Noheda, Brookhaven National Laboratory, Upton N. Y., USA, presented a rcvicw on her Low symmetry phases in piezoelectric systems: P Z N - x P T single crys-

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7 Jenth International Meeting on Ferroelectricity: IMF- 70,aJubi/ee Meeting

tal and powder [15]. After the discovery be fast diffusion pathways for impurities of the a monoclinic phase between the and are, hence, preferred sites for extetragonal and the rhombohedra1 ones change reactions. Their wall-induceddisat the morphotropic phase boundaries of tortion extends over several lattice units and diverges when approaching a second PZT and PMN-PT in 1999 the details of this delicate and still disputed event is order phase transition. in continuous careful exploration. It turns out that special care has to be taken when preparing single crystal or pow- 7.3 A Glimpse at Spain, at Madrid dered specimens of the compounds b) A. Onodera, Hokkaido University, Sap- and at Spanish Life and Culture poro, Japan, described Novel ferroelectricity in the 11-VI semiconductor ZnO [ l G ] , As an important ingredient to the success of the meeting, the social and cultural prowhen Zn2+is partially substituted by Li’. This unexpected effect is probably due gram of IMF-10, made it an unforgettable to the change in the electronic distribu- event for all participants. Four show eletion owing to the lack of d-electrons in ments, viz. singing, dancing, dining and Li’ rather than to an ionic size mismatch sightseeing, were chosen in a well-equiliand thus arising off-centrality of the im- brated manner and guaranteed optimum purities. Thin films of Zn0:Li are pre- success and a good time to everybody. The singing group of University students, dicted to have a wide potential for piezo“La tuna”, gave a traditional performance on electric applications. Monday evening and even successfully inTheory cited the audience to join some traditional a) Yoshihiro Ishibashi, Aichi Shukutoku dancing. The traditional Conference Dinner University, Nagakute-cho, Japan, pre- on Thursday night took the delegates and sented a Theory ofthe morphotropic phase their spouses to a large party hall outside boundary [17] within the framework of Madrid. Being served as a delicious dessert, the Landau formalism. He could show the Spanish Dance Group of the Universidad that in the vicinity of the morphotropic Complutense de Madrid offered an exciting phase boundary the electric and the elas- and perfect show of traditional dances. My tic compliances become extremely large, favorites were “La Danza Expanola” by while the coercive field minimizes due Manuel de Falla and the “Fandango de to the system’s instability against trans- Huelua”, a vital Flamenco tradicional. This verse fluctuations. This theory, which evening thus provided a welcome occasion stresses the importance of the near-isot- to get into touch with one of the most reropy of the thermodynamical potential, nowned elements of Spanish culture. is presently checked by ab initio calcula- iMuchas gracias a todas bailadoras y bailadores! tions. Two excursions were offered on Wednesb) E. K. H. Salje, University of Cambridge, UK, gave a report on Mesoscopic structures day afternoon, where bus tours to the hisinferroelastic and co-elastic materials [IS]. torical sites Toledo or Segovia could be choA lot of new information on twin do- sen. I personally preferred the Toledo tour, mains was delivered. Most exciting for since I was extremely eager to return to the solid state physicists: twin patterns can superb city of El Greco, which I last visited

References - W. Kleeman

35 years ago as a young student. And it became a success not only to me, but to all participants. Despite the tremendous early September heat all of us enjoyed a competently organized and well-equilibrated visiting program.

tina and Brazil might be taken as another symbol for the close and fruitful international cooperation in the field of ferroelectricity.

References

7.4 Summary

Apart from personal memories of all the events that happened during the week at Madrid, the lasting testimony of IMF-10 is the proceedings contained in a series of seven issues of Ferroelectrics, Volumes 267273. By collecting the manuscripts of the oral contributions and of 326 selected posters the editors, /. Gonzalo, R./ime‘nezRioboo, F. Montero and L. Pardo published nearly 450 papers in this timely updated “handbook” of ferroelectrics physics. Many of the participants will certainly welcome that it is not only available as a print version but also in electronic form. viz. as a compact disc for easy reading at the computer. IMF-10 will thus be an unforgettable jubilee meeting. The community has to thank lulio Gonzalo, his colleagues and their staff for all of their efforts to make IMF-10 such an outstanding event. Nevertheless, science goes on and forthcoming meetings are already scheduled. The “Jirroelecticians” look forward to the next one, which for the first time will take place in South America, at Iguazu Falls in 2005. The forthcoming meeting, IMF-11, at the border between Argen-

BI I N C . R., Ferroelectrics 2002, 267, 3. GINZBURC, V. L.. Ferroelectrics 2002, 267, 23. 3 Mur L E R , K. A,, Ferroelectrics 2002, 267, 33. 4 GLUER, A. M., Ferroelectrics 2002, 267, 35. 5 COURTFNS. E., Ferroelectrics 2002, 267, 43. 6 SOLS,F., Ferroelectrics 2002, 267, 61. 7 GINZBUKC V., L., Ferroelectrics 2002, 267, 69. 8 J A K I , S. L., Ferroelectrics 2002, 267, 77. 9 COLLET, E., LEM~E-CAILLEAU, M.-H., BURONLECOINTE, M., CAILLEAU, H., WULFF,M., L u n , T., KosllltiAKA, S., MEYER, M., TOUPET, L., RABILLER,P., TECHERT, S., Science 2003, in print. 10 SHUR, V., RUMYAN’I‘SEV,E. L., PEI.EGOV, D. V., KOZHEVNIKOV,V. L., NIKOLAEVA,E. V., SHISHKIN, E. I., C H E R N Y K H , A. P., IVANOV, R. K., Ferroelectrics 2002, 267, 347. 11 L ~ H N E N P., , DEC,J., K L E E M A N N W., , WOIKE, TII.,PANKRATH, R., Ferroelectrics 2002, 268, 1

2

513. 12 13

14

15 16 17 18

WAKIYA, N., BAO,D., SHINOZAKI, K., MIZU.IANI,N., Ferroelectrics 2002, 273, 149. S T R U K O V , B. A., SHAIDSTEIN. 1. V., PAVLOVSKAYA, T. V., GRABOVSKII, S. V., UESU, Y., FUKUNAGA, M., CARMAN, I.., Ferroelectrics 2002, 267, 329. N O H , T. W., KANG, B. S., SEO,S., PARK,B. H., Bu, S. D., YOON, J.-C., Ferroelectrics 2002, 267, 121. NOIIEDA,B., Cox, D. E., SHIKANE, G., Ferroelectrics 2002, 267, 147. ONODERA, A., Ferroelectrics 2002, 267, 131. ISHIBASHI.Y.. Ferroclectrics 2002. 267. 191. SALJL, E. K. H., Ferroelectrics 2002, 267, 113.

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