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An investigation of the ferroelectric properties of barium titanate Nasmyth, Patrick Walden 1952

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AN INVESTIGATION OP. THE FERROELECTRIC PROPERTIES OF BARIUM. TITANATE by PATRICK.WALDEN NASMYTH A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of PHYSIOS fly hi- i r We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF .ARBS Members THE UNIVERSITY OF BRITISH November 1951 of the' Department of PHYSICS COLUMBIA ABSTRACT The theory of ferroelectricity, and in particular the ferroelectric behavior of barium titanate, are discussed and qualitative agreement is obtained with experimental results. Limitations of the theory are pointed out. Experimental procedure used to manufacture samples of polycrystaline barium titanate ceramic, and to investigate their ferroelectric properties is outlined. The phenomena of permanent polarization and hysteresis are investigated in the ferroelectric temperature range, and are discussed. An electric equivalent of the Barkhausen effect i s observed. ACKNOWLEDGEMENT The research reported i n this thesis was supported in part by a grant from the Defence Research Board of Canada. I am indebted to Dr. A.J.Dekker i n particular, and to the other members of the Department of Physics of the University of British Columbia for their suggestions and assistance during the course of the research. P.W.Nasmyth . November 195i TABLE-OF CONTENTS Page I INTRODUCTION 1 II FERROELECTRIGITI. 3 1. Analogy to Theory of Ferromagnetiem 4 2. Theory of Mason and Matthias 12 III EXPERIMENTAL PROCEDURE 17; 1. Preparation of Samples 17 2. Polarization 19, 5. External Field 20 4. Hysteresis Loops ' 21 IV EXPERIMENTAL RESULTS • 22 . 1. Composition 22 2. Polarizatioh 24 5. External Field 26 4. Hysteresis 28 5. "Barkhausen" ;: Effect 52 V CONCLUSIONS 35-REFERENCES $6 APPENDIX A - Auxiliary Equipment 37 1. Furnace 37 2. High Voltage Supply/ 37 LIST OF ILLUSTRATIONS Figure Facing Page 1 Polarization by External Field 5 2 l / " ^ versus 1'. 5 3 Condition for Spontaneous Polarization 7 4; BatiO^ Lattice 8 5 Ionic Spacing in EaTiOj Lattice 8 6 (deleted!) 7 7 Barium Titanate Samples 18 8 Sample holder 18 9 Device for measuring External Field 18 10 Circuit to show,Hysteresis: Loop 21 11 Hysteresis Loop 25 12 Polarization Process 26 13 Variation of Hysteresis Loop with Temperature 29 14 Distortion of Hysteresis Loop 30 15 Potential Minima in BaliOj Lattice. 30 16 Action of Ti Ion 30 17 Circuit to show Electrical Equivalent, of Barkhausen Effect 32 18 Pulses caused by " Barkhausen" : Effect 33 J.1'9 High Voltage Power Supply 37; I INTRODUCTION It is known that in the temperature range between about o o . . 0 and 120 C a single crystal of barium titanate (BaTiO^) consists of domains which develop a spontaneous electric polarization. Polycrystal-line barium titanate ceramic also consists of a large number of random-ly oriented polarized domains which can, to a degree be brought into alignment along a chosen axis by the application of a high DC f i e l d along this axis. Once aligned in this way, there is a tendency for the domains to remain in alignment, with a resulting remanent polarization; and under these conditions the material exhibits both ferroelectric and piezoelectric properties. During the past few years much study has been given to these properties of barium titanate, and to i t s related characteristic of extremely high dielectric constant. In this thesis i t is not attemp-ted to review the f i e l d , but only those aspects are discussed with which we are directly concerned. The purpose of the present investigation is three-fold: (^ a) To learn the procedure of manufacturing polarized elements of barium titanate. (2) (b) To reproduce the experimental results of other workers; to inves-tigate in more detail -the phenomena of spontaneous polarization, ferro-electri c i t y , and piezoelectricity; and to consider the use of barium titanate ceramic as the active element of an electro-mechanical trans-ducer. (c) To attempt a theoretical explanation of the phenomena involved. (?) II FERROELECTRICITY In 1912 Debye postulated the existence of permanent elec-t r i c dipoles i n molecules which have no centre of symmetry. Such dipoles he argued, are bound to arise when two atoms of different electron af f i n i t i e s combine, the electron clouds being displaced towards the nucleus having the higher binding energy. If we place such a material in an electric f i e l d , the dipoles w i l l obviously tend to align themselves in the direction of the f i e l d . It can be shown, however, by the method f i r s t used by Langevin for the case of magnetic dipoles, that unattain-ably high fields would be required to produce an appreciable.degree of alignment'. Many years earlier, i n I8y0, Mosotti had developed an ex-pression for the locally acting f i e l d within a polarized dielectric, but despite the implications of a combination of these two ideas, i t was not until quite recently that the phenomenon of ferroelectricity, involving the 8pontaneouBlalignment of electric dipoles, was discovered, or even suspected. We shall f i r s t outline the Debye-Langevin theory, and de— 1 velop i t to the point where we can predict such a spontaneous alignment. (4) 1. Analogy to Theory of Ferromagnitism Consider a group of N electric dipoles of momentyc*. in the absence of any external f i e l d . The dipoles w i l l be randomly oriented owing to their thermal agitation, and the average dipole moment,^ , of the group w i l l be zero. Now i f we apply an external f i e l d , E , i t can easily be shown that the potential energy of each dipole, due to E, is given by ^ £f i c V 9 • </& yu£(\-<i#i^ (1) where & is the angle between^, and . There w i l l be a tendency for the dipoles to align themselves i n the direction of the applied f i e l d , i.e. to reduce 0 , and therefore the potential energy to a minimum, but this tendency w i l l be counteracted by thermal motion, and i f we assume equipartition of energy, we can use the Boltzmann distribution function to calculate the number of dipoles, per unit volume having an angle 6 with E. We get . • • . ('-co*®)Ar , where k is Boltzmann's constant, T is the absolute temperature, do* is an element of solid angle, A is a proportionality factor. We can put d<^> - m 5<VP cl& a^-cC I. ^ cos&.fie^ Yki'. ws**e<to (2) rt bo face paqp (5) ft f \ e y • st~e 4e =. number of dipoles per unit volume. Then / - — =• —=i e / k ' . ^ -i Putting (2 =y—~ an* integrating; we get the, result that — a -a _ e + e i = a a - i Slotting the ratio of f i e l d energy to thermal energy, Q_ = » as abcissar and the: ratio of the average dipole moment to the individual dipole moment,/^ , as ordinate, we get. the curve shown in. Fig.1, which, for small a ( i . e ^ E * * * 0 approximates to a/5« Now take, for example, yU. - 1 x. 10"^ esu cm.,,' of. the order of magnitude we would expect, and require a very low value, of. 0.01 tor/~ , corres-ponding to a = 0.0J. We would require di 1200 esu / cm. or. 5.6 x 10^ volts / cm. at 20°C. r This derivation can be considered correct only for completely free-dipoles, as approximated in the case of a gas, and i t is seen that the polarization w i l l be extremely low for ordinary (6) temperatures and permissible f i e l d s . In a solid, however, the interact-ion between each dipole and i t s neighboring dipoles must be considered. It is apparent that the effect w i l l be to increase the f i e l d w i t h i n the medium i n the way suggested by Mosotti. The following theory,is outlined as an analogy to the Curie-Weiss theory of ferromagnetism. Let us represent the f i e l d within our dielectric by £i - € + X P where P is the polarization per unit volume produced by the applied f i e l d , E, and is given by ^yyZT , and \. is an unknown factor defining the contribution by P to the total internal f i e l d . From ( j ) * _ P rt-lL I w h e r e / - > ~ X H ^ L ' i — - - <?olU a- - J- h e r e <2i = and again for low f i e l d or high temperature^**,(ti <^ L l ) we get or Now % — . must have the dimensions of temperature. Then say " X 3 3 T-TX /VC~ 3k(T--T~c) - i — _ /v^ i" (7) and y(_ may be a function of T, E, and the history of the sample i n the case of a ferroelectric material. Now plot ' a g a i n s t I as i n Fig.2, and the result is a straight line of slope ^ (< o - (10) So far we have used the approximation (6) for the case of high temperature or low f i e l d . For normal temperatures we must use (5) where k I Solving for P and dividing by v\ Now (Fig.j) plot ^ ^ a g a i n s t <Z-C from. (11) and from (5). From (5) we get a curve similar to Fig.l with an. i n i t i a l slope of 1/5. From ( l l ) we get a series of straight lines of constant T. Taking f i r s t the case for E = 0, we recognize immediately, the p o s 8 i b i l i t y ; o f spontan-eous polarization^below a c r i t i c a l temperature> T Q. The i T 2 T Q line i s tangent to the curve of (5)'at the origin. At any temperature below T G, the value of the predicted spontaneous polarization is given bytthe inter-cept of the two curves,, as shown. There are, of course, two intercepts in each case, but a consideration of the free energy in the two possible, stable conditions indicated shows, a much higher probability for the upper one, and i n every case we may neglect the intercept at the origin.. Above T Q the curves do not intercect and we expect no polarization. (8) The slope of the T_ line i n Fig. 3 is " • ^ (12) Therefore T~ 2^  . 1 and We arrive at the same value as before for the high temperature case, ( 9 ) . From (10) and (12) we now get A - ( 1 5 ) which can be evaluated experimentally from a knowledge of the dielectric constant, € , above the c r i t i c a l or Curie temperature, and the known relation that <f-/ A . " TTTr-(14) Let us now consider i n particular barium titanate (BaTiO^ ) the simplest of the known ferroelectric materials in it s crystal struc-ture. The unit c e l l of the barium titanate la t t i c e is of the "pervos-kite" form, consisting of a single titanium ion (T1<"'"M) surrounded symmetrically by eight barium ions (Ba*"*") on the corners of a cube, and six oxygen ions (0~ ) centered on the faces of the cube, (see. Fig.4). . By X-ray study the edge of the cube has been found to be yery^nearly 4 R. The radius of an oxygen ion is approximately 1 . J 2 £ and of a Mtaniumeion 0 . 6 4 R . It is seen (Fig.^) that the titanium ion has a freedom of move-1 (9) ment of at least 0.86 A3 in any direction without coming i n classical contact with another ion. Techniques have recently been developed by which single crystals of barium titanate can be grown up to several millimeters in size, and i t has been demonstrated by a number of workers 1»2) that the barium titanate lattice is a pure cubic structure above a c r i t i c a l temp-erature of about 120°0, which we have already referred to as the Curie temperature. As the temperature is lowered through this Curie point, the lattiee stretches along one of its cubic axes into a tetragonal font. In the tetragonal phase, X-ray study ^ shows the titanium ion to be dis-placed from the centre of the la t t i c e , in the direction of one of the oxygen ions, by a distance which varies with temperature and reduces to zero at the Curie point. At 20°0 the displacement is approximately 0.16 A3. Under these conditions each unit c e l l of the lat t i c e becomes a permanent electric dipole, and as soon as a number of adjacent dipoles become lined up i n the same direction, whether by an external f i e l d or thermal agitation, an internal f i e l d of the Mosotti type is set up, which spreads as more dipoles f a l l into lin e . Thus the entire crystal would develop a spontaneous polarization, P, per unit volume, and under controlled conditions crystals have been made to do so. Under normal c i r -cumatances, however, due to impurities, or to other factors not f u l l y understood, the crystal usually breaks up into polarized domains with their polarization vectors, P, oriented more or less at random to eachr-other so that the resultant overall polarization of the crystal is very (10) low. Much study 2'3>4, and o t h e r s ) h a B b e e n g i v e n t o t h e f o m a t i o n and behavior of these ferroelectric domains, but since the present work is concerned with polycrystalline barium titanate ceramic rather than with single crystals,, we are not directly interested here, a l -though we shall have reason to mention the subject again later. There are two other phase changes at approximately 5° Gand -70°0 to which some study has been given 5*6) w e sh a].l confine ourselves to the tetragonal phase and the temperature region of the cubic phase immediately above the Curie point. Now, applying the foregoing theory to barium titanate we find experimentally that Tfc l i e s between 120 and 125° C, and we ;have the empirical relation developed by Jonker and van Santen ^ ) . These experimental values lead to a value for A of 0.05. Then from (12) we get 3 k To and from X-ray measurement we have a = 4 x 10 cm. where a is,the dimension of the unit c e l l of the l a t t i c e . Thus by calculation V\ = 1.56 x 10 2 2 and yt»* = l4.4 x 10" 1 8. If we take the displacement of the titanium ion as 0.16 at room temperature as observed by X-ray diffraction, and assume that, the oxygen ion moves an equal distance to meet i t , the dipole moment f (11) would be 6 x 4.8 x I O - 1 0 x 0.16 x 10"8 = 4.6 x I O - 1 8 = 4.6 debye units. We wimA4 see that our theoretical value fory^. is too large by a factor of 5* We would, in fact, expect i t to be too large since we have attributed the entire polarization to the displacement of the ions, whereas i t is actually due, in part, to electronic and atomic polarization. However, we would hardly expect the difference to be as great as indicated. In order to correct for this error, we would have to know, i n detail, the configuration of the f i e l d within the l a t t i c e , and: even in such a comparatively simple lattice^ as barium titanate, the mathematics involved turn out to be quite formidable. Either assumptions and approximations must be made which probably render the result useless, or else the mathematics become so involved as to be prohibitive. Continuing, i f we assume a l l dipoles to be directed in one direction, we obtain from our theory a saturation polarization 5) Hulm has experimentally measured the spontaneous polarization, S, of a single crystal of barium titanate at 20- G (12) S a 16 x IO"0" coulombs / cm.2 Now, i f on Fig. J, we plot the line corresponding to 20°C we read a P value of 0.60 for , which, for zero external field, is the ratio of spontaneous to saturation polarization. Thus our theory-gives us A A S = 0.60 x 75 x 10~° = 45 x 10~° coulombs/cm.2 which is greater by a factor of almost J than the experimental value. This difference is attributed, at least in part to the effect mentioned above and to the fact that there is no apparent justification for assuming that our value for A , calculated from data observed above the Curie temperature, should hold below the Curie temperature with a modified crystal structure. 2. Theory of Mason and Matthias A somewhat more elaborate theory is due to Mason and 8) Matthias who consider a model of the barium titanate lattice as shown in Fig. 15. It is assumed that the titanium ion forms a co-valent bond with one of the oxygen ions and is displaced from the centre of the lattice in that direction. Above the Curie temperature, the thermal energy: is sufficient to cause any one of the six positions to be equally: probable and, from X-ray measurement, the cell appears to be cubic. Below the Curie temperature the thermal energy is no longer sufficient to cause each position to be equally probable and the displacements of adjacent ions tend to litie up spontaneously as: (13) previously, described. A dipole moment develops in the direction of displacement, and the crystal becomes ferroelectric,, and,, since the unit cell has now lost its centre of symmetry,., i t also becomes piezo-electric. The axis in the direction of the displacement: becomes longer, and the crystal assumes its tetragonal form. Suppose,, now, that, a l l the minima of Eig. 15 have initially the same value... Then apply an external field, E, along the Z-axis, say. An internal field will be produced, as before, given by where now we consider P to be made up of P e due to electronic and atomic polarization and due to the displacement of the titanium ions. Thus £. * (£ +. \ +. tO^ or K = . - \l (15) G * A f t 1 where is the polarizability per unit volume due to a l l causes except the titanium dipoles. X can be determined from the dielec-tric constant,. €, measured at low fields and at very low or very high temperature where Pd is not effective. Under these conditions 4tp • £ " e -i-M Thus ~ - s -K.— and from a measured value of 550 for <?( o ' •« /'+ \(^7.a) (16) (14) The internal f i e l d causes an increase i n potential energy at minimum 2 equal to _ /^+.\P \ where Ax = A>% & , in which 4e » charge on Td ion / r 0 = displacement from centre of la t t i c e , and a similar decrease at 1. The potentials of the other four minima remain unchanged. We then apply the Boltzmann distribution function in the usual way and put.. - O^yx where K ( and Y\x are the numbers of titanium ions per unit volume in. minima 1 and 2"respec-tively. Thus we derive the relation f , /r-*\PITT U 7 ) Nov? consider the condition for, spontaneous polarization. Setting E^";= 0 and a » <~r- — - (18) l - kT aP equation (17) becomes P is different from zero only : - x+ ~* # Cl9) i n which — —  for values of A ^ - 5 . Eor Q c a n have positive:.-or negative values between zero and one, corresponding to spontaneous polarization along the positive or negative Z-axis. Here, of course,, i n the absence of external f i e l d , the Z-axis may be chosen in any one of the possible six directions. (15) From (18) we get T^ . » ^- • '(jZ^) w h i c h i e / the same, expression we had before with the additional factor - — r ~ r / - A o to. correct for electronic and atomic polarization. Y is now eliminated from a knowledge of £ 0 ^ and, \ is evaluated,, as before, by experimental measurement of £ above For barium titanaterwe get. A. = 0.124 which leads to a value for the dipole moment of yU. = if.344x IO - 1 8- esu cm which agrees quite well with,the experimental value of 4.6; x.. 10" as derived from X^ray data.-As before we get. a value for saturation polarization For 27^ 0JMason and Matthias calculate — — => 0 . 0 0 = — Then S = 61,000 esu = 20.5 x IO""0"•-'coulombs/cm.2 which is greater by a factor of approximately 2 than their experimental value of 11.8 x 10" 6. In correcting for the atomic and electronic polariz-ation, this theory: approaches more .closely to the' true situation. However the two limitations remain that.' ' (a) There is no apparent reason to expect the value, A, to be:, the same above and below the Ourie temperature. There i s , however,, no (16) simple means of obtaining a value for \ in the temperature range below U Q . (b)) The value of A would be expected to vary considerably, depending on the configuration of changes in the immediate neigh-borhood of the point in question and will vary from.one point of the lattice to another. Owing to the complexity of the mathematics, however, we are forced to assume a constant value. ? . • • • (17)) III: EXPERIMENTAL PROCEDURE.. 1. Preparation of Samples After some experimentation, a manufacturing procedure very similar to that used by von Hippel and his co-workers at the 9) Laboratory for Insulation Research, M.I.T>. , was adopted. The dry barium titanate powder, obtained from the Titanium Manufacturing; Company, is pressed in a steel mold into cakes 1 inch in diameter and approximately 3/l6 inch thick. Pressures varying from 1000 to 20,000 psi were tried and i t was found that, over this range, results are almost independent of pressure except that at very low pressures the cakes are weaker mechanically and difficult to handle without damage, and at pressures above about 6000 psi the powder tends to compact diBContinuously (as a snow-ball packs) and cleavage faces develop which are indiscernable at the time but: show up after firing. A pressure of 5000 psi was used for most of the samples made. The cakes are thgn stacked 5 o r 6" high with platinum f o i l separators in a small crucible. The following firing cycle was found to be most satisfactory and was used throughout the ex-(18) periments: (a) Increase temperature at a rate- of 100°© per hour from room temperature to 1350°G"i (b) ) Hold at 1550°0 for six hours. (c) Qool at the cooling rate of the furnace. It takes about 43 hours to reach.a temperature 10°0 above room temperature. Prepared in this way the samples are homogeneous and vitreous and have very good mechanical properties. Some of the f i r s t samples prepared had very small blow holea throughout their volume. These, however, were entirely eliminated i n later batches by s i f t i n g the powder through a 200 mesh per inch screen directly into the mold. In f i r i n g the samples shrink to about 7/8 inches in diameter. The two faces are ground f l a t and the thickness is reduced to 4 mm. with carborundum powder in water on a plate glass surface. The faces are then polished.in a similar manner with very fine a l -undum powder. Electrodes are applied to one or both faces by painting with DuPont silver paste No. 4351 which is baked on i n another short heating cycle at.600°G. A l/32 inch wide strip around the edge is l e f t unpainted i n order to reduce the tendency to. break down elec-t r i c a l l y over the surface during the polarizing process. Flexible leads about 6 inches long are soldered to the silver electrodes. (See Fig. 7) :.o~: (19) 2. Polarization The samples are polarized i n a direction perpendicular to their f l a t faces by the application of a high DC f i e l d in that direction. The power supply used to supply the DC f i e l d is described i n Appendix A. A simple device (Fig. 8),was made to hold the sample between two bakelite discs with a light pressure applied by springs,, while the polarizing voltage is applied between, the electrodes. In the case of the samples with only one silve r electrode, one of the bakelite discs is replaced by one of brass, ground and polished to a smooth f l a t surface to make good contact with the surface of the sample. At., f i r s t some d i f f i c u l t y was experienced with sparking over the edges or electrical.* breakdown through the sample at. voltages:, of about 5000 volts. With the following precautions, however, the voltage can be successfully raised to"16,000 or a f i e l d strength of 40,000 volts per centimeter without breakdown: (a) The entire assembly, including holder and sample is placed i n a bath of transformer o i l . (b) Before applying the voltage the sample is heated in the o i l bath to 120°C for 10 or 15 minutes i n order to remove any moisture which might be absorbed by the sample. The o i l bath is also used as a means of controlling the temperature of the sample during polarization. (20) Details and variations of the polarization procedure w i l l be discussed under " Results" ::. 3.. External Field The external f i e l d of the polarized sample is measured qualitatively in the following way. A sample with only one plated electrode is placed in a horizontal position with i t s exposed face up. A mechanical device (Fig. 9) is used to raise and lower a brass electrode from contact with the sample to a maximum spacing of about 3 cm. where, in a l l cases, the field.has dropped to an unmeasurably low value. The two electrodes are connected through a b a l l i s t i c gal-vanometer or an electrometer, and deflections are observed as the spacing between the removable electrode and the sample is varied. See Fig. 12(b) and (c). Results obtained by this method have not been so use-f u l as had been hoped. Quantitative measurements have been quite impossible due to leakage of surface charges, probably through a thin layer of absorbed moisture on the surface of the sample, and due to neutralization of the surface charges by ions in the a i r . To obtain reliable quantitative data i n this way, the sample, should be enclosed i n an evacuated chamber, and this has not been attempted. However, some qualitative remarks are made under " "Results" :i. O 6C? /v a n a c 3ok rvt p ie <?3 5 rvic 0.5 t t<y o r e to ( 2 1 ) 4. Hysteresis Loops. The. circuit, of Fig. 10 is used to display the ferro--electric hysteresis loop of the samples under varying conditions, on the screen of an-oscilloscope. Some of the loops: have been photo-graphed and are reproduced in Fig. 15. For these experiments, the sample is supported in an o i l bath in the same way as for polarization, and, as well as preventing breakdown over the surface of the sample, the o i l bath again acts as a means of controlling the temperature of the sample. By means of the.variac, the voltage across the sample can be varied from zero to about, 6000 voits RMS at the line frequency of 60 c/s. In Fig. 10, the resistance divider, R].R2* provides a linear horizontal sweep, proportional to the applied voltage. The capacitive divider, C i C x , provides a vertical deflection proportional to the displacement, of the sample.. Gj_ is a paper capacitor and is assumed to be linear over the range of voltages involved. serves only to prevent:.a static charge from building up on the upper vertical deflection plate. The resistance, R, was added to damp out shocked oscillations i n the secondary of the transformer.- Various values up to 50*000. ohms are used under different conditions. The necessity for this addition w i l l be explained later. (22) IV. N EXPERIMENTAL RESULTS:. 1. Composition It is knovm 9) that small quantities of other titanates i n dilute solution i n barium titanate have pronounced effects on the dielectric constant, the Curie temperature, the degree of polarization, and other characteristics. Most substances are found to lower the Curie point and decrease the dielectric loss, making the resulting mixture more suitable as a dielectric material than pure barium titanate. Lead titanate, on the other hand, is found to raise the Curie point and increase the area of the hysteresis loop with a resulting higher remanence and higher coercive force. The remanence, is also found to be more truly permanent than i n the case of pure barium titanate. For use as a piezoelectric material, i t i s , of course, desirable to obtain as high and as permanent' a remanence as possible, since the piezoelectric coefficients depend directly on the remanent., polarization. To this end some experiments were d o T t e with barium titanate. A reasonable homogeneous mix of the two powders was obtained-by placing them in a small ball-mill for several hours. (23)) . Only two mixtures - 2.3% and 10% lead were t r i e d and compared with pure barium titanate. Samples containing 2.5% lead titanate showed both remanence and coercive force greater by a factor of about 2 than;samples of pure barium titanate under equivalent conditions.. They are found to saturate at.a lower f i e l d and to require a shorter-polarization, time. The remanent p o l a r i z a t i o n also shows a greater permanence than;in the case of pure barium titanate.: At the end of the maximum period of observation - about 6 weeks - the p o l a r i z a t i o n of the pure barium titanate i s s t i l l observed to decline at a very slow rate, whereas the 2.5% mixture appears to have reached a constant, value. A.much longer period of observation w i l l be required however before a statement can be made about.the long, term permanence of pol a r i z a t i o n . The 10%imixture does not appear to be s i g n i f i c a n t l y -d ifferent i n characteristics from the 2.5% mix, and i t . i s concluded that: the percentage of lead titanate i s not c r i t i c a l within, quite broad l i m i t s . However it.does show a higher conductivity and lower d i e l e c t r i c strength above the Curie point and f o r this reason most of the experimentation was done^with: pure barium titanate and a 2.5% solution of lead t i t a n a t e . No further experimentation was,, carried out i n t h i s directions (24) 2. Polarization In general, three polarizing procedures were tried. (a) . The polarizing voltage of up to 16,000 volts was merely applied across the sample at room: temperature and maintained- for periods varying up to one. hour. The resulting polarization was found to be very low and from the results of later observations i t is believed that in order to reach anything approaching saturation polarization, in this way, the voltage would have to be maintained for days, at least. It appears that, before the spontaneously polarized domains,, of which the crystals are composed, will align themselves to any degree with an applied field, they must f i r s t be "loosened" by heating or by application of a high AO voltage. (b) The polarizing voltage is applied at room temperature and is maintained at about 10 Kv while the temperature of the sample is slowly raised to above the Gurie temperature and lowered again. While above the Curie point, the voltage should not be raised above this value because at highu temperatures the conductivity of the material increases and its dielectric strength decreases and there is danger of electric breakdown. When the temperature has been lowered again, however, the voltage may be increases to 15 or 16 Kv for a short time before the end of the polarizing cycle. The best results, i.e. the highest and most permanent, remanence, have been produced by this method. (c) An AC voltage of J-4 Kv is applied to the sample and the hys-teresis loop is observed. The sample is run through a temperature p •to face page %y (25) cycle as before until a maximum value of remanence is observed. The AG; is then removed and a W. polarizing voltage of perhaps 15 Kv, immediately applied. In the cases tried the DG voltage was l e f t on for only about 5 minutes and results were not so good as i n ( b j ) above. A longer application of the polarizing voltage might produce better results, because, as observed i n a number of other experiments,, i t appears that once the dipoles have become "used" to behaving in one fashion, i t takes minutes and sometimes hours to adapt themselves to any sudden.change. No further investigation was carried out. A study of the phenomena involved here could, i t s e l f , be the subject of an entire paper., By projecting backwards the f i n a l slope of the hysteresis loop at saturation, as i n Fig. 11, from the equivalent vertical de-flection voltage, v, of the P intercept, P B, and from the c i r c u i t constants, of Fig. 10., and the area, a, of the electrodes on the sample we get the expression P s = G-^ v/a microcoulombs/cm.2 where is in microfarads, v is in volts a is- in cm.2 In this way we measure a "saturation" polarization of 5>5 x 10"^ coulombs/cm.2 at a f i e l d of 20 Kv/cm. after heat treatment. This &e no y/eG ipie *~ 1 T T + + + 7 c ? / e c f r o d e is tic- C a^l ^ /a/i o ot e "fcr or £~fec i r o ^ ^ e r to) ( b ) Cc) Fx'j o /2 Fb(oxln<i.i c'o rt ir?o oess (26) is in close agreement with,Hulm's: ^) value.of 5»3 x 10"^ for poly-crystalline ceramic. As. Hulm points, outi; however,; i t is.probable that.a ceramic: sample is s t i l l far from truly/saturated at this field.. The 2»5%' lead titanate mixture saturates more easily and gives a value of about 8 x 10 : .coulombs/cm. under the same conditions,, (This seems reasonable compared to the value reported by a number of. workers of about 16 x 10" for a single domain crystal.) 3. External.. Field; The external.field of the samples is determined as described under "Experimental Procedure"". For reasons previously stated, no reliable- quantitative results have been obtained, but. the following remarks are made. We first:polarize the sample as in Fig. 12(a), and we assume that the dipoles align themselves as shown.. Removing the polarizing voltage and connecting a ballistic galvanometer as in Fig. 12(b) we measure a steady discharge current (electron flow) of the order of 0.01 x. 10"^ amperes: in the direction of arrow #2. This current decreases exponentially to an unmeasureable value over a period of 2 or 3 days, and is assumed to be due to a relaxation of some, of the dipoles. As the polarization of the sample decreases, some of the induced charge oh the electrodes is released and/ippears as a discharge current in the external circuit. How we remove the upper electrode completely out, of ( 2 7 ) range of the external field of the sample. At: f i r s t i t was. expected that the induced charges on the electrodes would, be released and would neutralize each other through.the external circuit, causing a current, again in the direction of arrow # 2 . A current in the direction of arrow #1 was expected when.the electrode was replaced. As observed, however, these last two deflections actually occur in the opposite directions. ^ Hie theories advanced to explain similar phen-omena in wax: electrets cannot be applied here because,, in most, cases,; they assume a migration of ions in the molten wax under the influence of'the polarizing field. In our case the ions are tightly bound, in the lattice and cannot be assumed to migrate at,a rate fast.enough to explain the observed phenomena. The following explanation is offered. Under the influence of the high intensity polarizing field, some electrons may; "soak" into the dielectric: from the negative: electrode and become trapped in the lattice, forming a negative, sur-face charge. A similar positive surface charge may be formed on the other side, and in the interior the dipoles would s t i l l be oriented as expected. Now when the polarizing voltage is removed these.sur-face charges remain and induce an external field in opposition toe. the field of the dipoles. If we assume the field of the surface charges to be greater than the field of the dipoles (Fig. 12(c)) the directions of the deflections would then be as observed. (28) The only piece of experimental evidence in support: of this theory is that, the removal of, the surface layers of the sample decreases the observed external field. This was. done by grinding off the faces after polarization. On: one occasion, also,, three pieces of titanate were polarized in series. Then assuming a l l of the sur-face charges-to be on the two outside pieces, it. was thought: that the centre piece alone might: give,deflections in the opposite directions.. This was. actually not observed, but: the deflections caused by the centre piece were unobservably small, while its polarization was ap-parently just as high as that.of the outside pieces which-produced high deflections. This indicates that the phenomenon is at least partly one of surface charges, and i t may be that, owing to the finite conductivity of the titanate, the charge penetrates, with reduced intensity,., into the interior of the sample. In the final steady state, reached some days after polarization,, i t is assumed that.this charge reaches an equilibrium with the:field of the dipoles. Under these conditions then, the observed external field should be a relative measure of the polarization, and from measurement; of piezoelectric, voltages,, i t appears that this is approximately true. 4. Hysteresis The remarks concerning the size and shape of the : hysteresis loops observed for different-mixtures andunder different, conditions will not be repeated here. (29) P Eig. 1J illustrates the variation with temperature^ of the hysteresis loop of a new sample containing 2.5$:lead titanate, at a constant peak AG) field of 12,000 volts per centimeter.. The upper limit of the ferroelectric: range is clearly observed at., about 120°G. The fact that the hysteresis curve appears to open out. again above this point is misleading. This is not a true hysteresis loop, but is caused by a phase shift introduced in the C^GX network of Fig. 10 by greatly increased^conductivity of the sample at:high tem-peratures. Et. is this phase shift,also, which makes the slope of the., resulting ellipse continue to decrease above the Gurie point. Actually this slope should remain constant, and equal to the slope of the f i r s t curve of Eig. 15> and. to the slope of the saturation portion of the; other curves where only the electronic and atomic polarization is effective and the dipoles do!notcontribute appreciably. The necessity for heat treatment in the polarizing cycle is clearly.seen by comparing: the curves at equivalent temperatures before and after the heating cycle. Actually the init i a l shape can be made to approach the final shape without heating, but this requires unpleasantly high fields with danger of dielectric breakdown,, and long polarizing time. Ah: interesting effect was observed! and is illustrated in.Pig. l 4 . It was found that, at some temperatures and at some value8 of applied field, the small hook seen near the tips of the hysteresis loop in Fig. l4(a), and expanded in (b), appeared. In an effort to explain its appearance, the wave-form of the applied. 2T fe> feee p a g e 3& (?o) voltage (across AB: in Fig. 10) was. checked and the form corresponding: to Fig. l4'£a);and (b) is shown: in (c). (Note: The fact that the upper and lower halves of the cycle are not:symmetrical is due to a non-linearity in the oscilloscope sweep,., and is not significant). The following explanation, is offered on a simplified, and purely classical basis. Consider two of the six potential minima assumed to existtwithin the unit c e l l of the barium titanate lattice (Fig. 15) to be in line with the applied f i e l d , and,; for the present", neglect the other four. We represent a section through these two minima graphically in Eig. 16, and assume the. titanium ion to be at.position 1 as the AC) f i e l d increases: through zero:in the direction indicated. Now we associate a displacement current with the horizontal motion of the titanium ion. In a region around position 1 the motion of the ion is comparatively unhindered by the internal f i e l d of the lattice and i t is free.to move.under the action of the applied f i e l d . During this part of the cycle the displacement.current is large. At 2 the motion of the ion is slowed down by the central potential hump, and the corresponding:;current is small. Having reached 3 with the applied field, s t i l l increasing, the ion w i l l jump to position 4 with.a resulting high pulse of current.of very short, duration. The ion w i l l then continue to move in the direction of 5-with a corres-ponding small current: until the peak of the applied f i e l d is reached. Now the transformer used to provide the alternating (3D ' field is of very high impedance and has poor regulation. As the load (i.e. the real current iri the external circuit, corresponding: to the displacement, current in the sample) increases,, the output, voltage of the transformer decreases, and vice versa. The output wave-form of the transformer on no load! (shown dashed on Eig.. l4(c))) is a reasonably good sine wave,, and i t is recognized immediately, that the distorted form is exactly as one would expect from the process described in the previous paragraph. The corresponding numbers on Eig. l4(a) and (c) and Eig. 16 identify corresponding parts of the cycle. The hook in the hysteresis loop is thus identified with the? sudden.transition of a large group of titanium ions from, one potential minimumi to the other. In: Eig. l4(c))the peak of the distorted wave projecting;; above the undistorted form at 3? is a n overshoot caused by the inductance of: the transformer. Attempts, so far, to calculate the height: of the central potential hump by the field at which this transition takes place, have resulted; in answers: which are too low by at least one order of magnitude. However the observed phenomena do appear to f i t the pro-posed theory-very nicely, in.a qualitative way, and further investigation of the effect mighttprove a useful means of increasing our knowledge off the process of permanent polarization.. O W V W N A c £ < J U I L/Q '^n T o / - Bar (?2) 5. " Barkhausen" Effect During the course of investigations i t was observed that on close inspectiontthe steep portions of the hysteresis, curve, marked ABiand GDI on Fig. 11, were of a different!: texture on the os-cilloscope screen than the rest of the loop. No deflections were observed, but these portions of the curve appeared to sparkle, while the remainder was quite solid. It was recognized immediately that, this is exactly the part of the cycle where the Barkhausen effect is observed in ferromagnetic materials, and that this might be.the electric equivalent, which, so far as the writer is aware, has not previously been observed in ferroelectric materials., i • In order to check this possibility.,, apparatus was set up as shown in block form in Fig. 17. A sample: containing 2.5% lead titanate was used.. The amplifier has a low cut-off at about. 1000 c/ssand would therefore differentiate any discontinuities in the polarization curve, and we would see a pulse on the oscilloscope.. This,,in fact, was observed. As the large condenser,. G:, is slowly charged through the high resistance,.. R, pulses begin to appear at a field., of about: 5000 volts per centimeter at room temperature. The amplitudes of the pulses, appear to be randomly distributed over a range, of about 6 to 1. Fig. I8(a),(b) and (c)) is a series of photographs taken as the field was slowly increased. The oscilloscope was allowed to sweep continuously at 60 c/ssand the exposure in each case was l/lO second. (3?)) (The negative overshoot following each pulse is caused by the amplifier,)) It w i l l be noted, that at low fi e l d , small amplitude pulses predominate, and that there.is a very well defined minimum: (Fig. 18(a));. As we increase the f i e l d (Fig. 18(b)) the proportion of large amplitude pulses increases,, artd.it appears that there is ah equally well defined maximum,., although an occasional pulse of much, higher amplitude (usually off screen) was observed but not, caught ih a photograph. (Note: This series of pictures was taken at 65°C.; and the range of amplitudes is somewhat less than at room temperature).. A t . s t i l l higher f i e l d (Fig. 18(c)) the pulse rate increases very abruptly and almost a l l pulses are of the maximum amplitude. Beyond this pointl the amplifier became confused and the resulting picture had the appearance of cir c u i t noise.. Aa w i l l be seen i n Fig. 17, the sample is completely isolated from any sources of noise, in the charging circuit, andAt is believed that this is truly an electrical, equivalent of the Barkhausen effect. It can be further proven that the pulses are not due to extraneous causes by charging the condenser,, Q, and then completely removing;the power supply. The pulses continue as & slowly discharges by leakage. Again, no quantitative investigation has yet been under-taken,., but the following comments are made. The maximum and minimum pulse size seems to indicate sharp limits to the permitted range of domain sizes.. The occasional very high pulse may be caused by an exact coincidence of two or more normal pulses. As the temperature of the sample is raised towards the Curie point, the amplitudes of a l l the pulses decrease, as would be expected from the:fact that the displacement; of the titanium ion is known to decrease, and the pulses also become fewer in number. Also, as the temperature is raised, pulses can be observed, at lower; and lower field,, as would also be expected from the fact that the lattice is known to contract-along i t s tetragonal axisj i.e. i t ap-proaches more closely to cubic, and the height of the potential hump at its centre.-should decrease, while at the. same time the thermal energy of the titanium ion increases and i t requires less additional energy to raise i t over the hump. Above the Curie temperature, where, by the theory, the permanent dipoles and, therefore, the ferroelectric; domains have;; ceased to exist,, only a very few very small amplitude pulses are seen, and i t is probable that i f the sample were maintained at a high temperature for an extended length of time, these also would disappear. (35) W CONCLUSIONS It is d i f f i c u l t to state specific conclusions at this stage, since the results reported in t h i B thesis are mainly descriptive and are not easily condensed. It appears, however, that the parallel be-tween ferroelectricity and ferromagnetism can be carried even farther than in the theory outlined earlier, to include an electric equivalent of the Barkhausen effect, and i t . i s believed that valuable information on the ferroelectric behavior of barium titanate could be gained by further study of the distortion of the hysteresis loop attributed to the action of the titanium ions, and of the " Barkhausen " effect. Very l i t t l e has been said i n the foregoing pages about the original intention of investigating the use of barium titanate as the active element of an electro-mechanical transducer. The investigations, however, indicate that i t should be at least as good as other known piezoelectric materials for this purpose, provided the long term perm-anence of i t s remanent polarization can be demonstrated, and i t has, of course, the obvious advantage that i t can be moulded as a ceramic into any desired form and polarized in any desired direction. (36) REFERENCES 1) H. Megaw Nature l ^ L , 484 (19^5) 2) P.W. Forsberg (Jr) Phys. Rev, J6 , 1187 (1949) 3) C.C., Danielson, B.T.. Matthias, J.M. Richardson Phys.Rev. J4. , 986: (1943) 4) W.J. Merz Phys. Rev. TJL* 687 (19^9) 5) J.K. Hulm Nature 160 , 127 (1947) 6) H.F. Kay, P. 7»busden P h i l . Mag. (7) 4<D » 1 0 1 ? (1949) 7) C H . Jonker, J.H. van Santen. Science 10£ , 632 (1949) 8) W.P. Mason, B.T. Matthias Phys. Rev. ] ± 1 6 2 2 (1948) 9) A. von Hippel (and others) Ind. and Eng. Chem. 3j3 , 1097 (1946) 10) A. Gemant R.S.I. 11 , 65 (l94o). 11) F. Gutmann Rev.Mod.Phys. 20 , 457 (1948) 2 SI 10 k 500 • A W 0.001 O.lyut f 10k' 1^ s Sealed in oil h 0 o 0-1500 r- ! ^ 1 1 X _ < —< 6 6 20 Ki F i t } %Jf€ 19 (37) APPENDIX A - Auxiliary Equipment 1. Furnace The furnace used for f i r i n g the samples consists of a 2" I.D. alundum tube 18" long, wound with a platinum heating element. The tube is set in a vertical position and packed i n a thermally i n -sulating material. During operation the ends of the tube are sealed with firebrick. The furnace temperature is controlled by a platinum thermocouple in contact with the alundum tube and coupled to a Wheelco Pbtentiotrol, Model 2J24l. A maximum temperature of about 1500°C can be reached safely,, and, with careful calibration the chamber temperature can be controlled to within plus or minus 2°0. 2. High Voltage Supply The c i r c u i t of the high voltage supply, used for polarization of the sample, is shown in Fig. 19. By a combination of the two controls (the grid-leak resistance of the oscillator and the tuning of the tank circuit) a range of voltages from 2 to 20 Kv can be obtained. The only remarkable feature of the ci r c u i t is the r . f . transformer which is of a common type often used in television receivers and designed for an output of 3 or 4 Kv. It was found that this could (38) be raised slightly but that r . f . breakdown occurred at 5 or 6 Kv. The transformer was therefore sealed in insulating o i l in a lucite container, and has been successfully operated at 12 Kv. A conven-tional voltage doubler is used to rectify and increase this to 20 Kv/. 

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