PROCEEDINGS OF THE IRE

Nonlinear Dielectric Materials*E. T. JAYNES, t SENIOR MEMBER, IRE

Summary-The following is a brief description of nonlinear di-electrics from the standpoint of the fundamental physics involved.Specific available materials and technical applications are consideredonly by way of illustration of general properties. It is shown that,although dielectric nonlinearity and ferroelectricity are quite differ-ent phenomena, the fact that ferroelectrics have high dielectric con-stants makes them the most likely materials to exhibit a high degreeof nonlinearity at electric field strengths safely below the breakdownvalues.

INTRODUCTIONHE MACROSCOPIC electromagnetic properties

T of matter are commonly described by specifyingthe current density J, electric displacement D,

and magnetic induction B as functions of the electricand magnetic "intensities" E and H. In the majority ofsubstances these relations are found to be linear, leadingto the definitions of the conductivity, dielectric con-stant, and permeability tensors. The meaning of theword "linear" must be made more precise as soon as weconsider time-varying fields; in particular, we must becareful to distinguish between nonlinearity and dis-persion. The simple statement that D(t) is proportionalto E(t) will not do; by common usage it is understoodthat linearity is concerned with such a proportionalityin the frequency domain rather than in the time do-main. Thus, consider a Fourier integral representationof the fields:

E(t) =f E(w)eitd&d00 ()

D(i) D(co)eiwtdw

By linearity we mean that the material is characterizedby some unique function e(X) such that

D(X) = e(w)E(w). (2)

This corresponds in general to no simple relation be-tween D(t) and E(t). Thus, a linear dielectric materialis characterized intuitively by the following conditions:

1. No frequencies are present in D(t) which not arepresent in E(t).

2. If El(t) produces Di(t), which we write com-pactly as El--Di and E2-*D2, then (aiEl+a2E2)->(a1D1+a2D2), with similar definitions for linearconductors and magnetic media.

It is possible to have one but not both of these con-ditions satisfied: for example, consider the correspondingmagnetic properties of water placed in a strong but in-homogenous magnetic field. For applied frequencies

* Original manuscript received by the IRE, October 14, 1955.t Microwave Laboratory, Stanford University, Stanford, Calif.

within the range of the proton Larmor frequencies, con-dition 1 is satisfied but not the superposition condi-tion 2.1

Strictly speaking, all such linear laws may be regardedas approximations for several different reasons. In thefirst place, they would presumably fail in any materialsubstance at sufficiently high field strengths; i.e., break-down or saturation effects would occur. Secondly, therereally are no unique relations between the above vectorssince in the work of the highest accuracy one would al-ways expect to find that, for example, the electric dis-placement does not depend only on the electric field, butalso on every other physical condition of the materialsuch as temperature, state of stress and strain, degree ofillumination, and even the entire past history of thespecimen. Thus, even if it should be found that D isexactly proportional to E at constant temperature, thefact that the dielectric constant so defined is a functionof temperature gives rise to electrocaloric effects, inwhich a sudden change in electric field produces achange in temperature. Thus, the linearity or non-linearity of a substance could depend on its degree ofthermal contact with its surroundings; i.e., on whetherit is operated under isothermal or adiabatic conditions.Similarly, if a dielectric is linear under conditions ofconstant stress, it might not be so under conditions ofconstant strain. In practice, however, these are usuallyextremely small effects.

Finally there are more esoteric examples providedby modern physical theories, according to which evena perfect vacuum should have nonlinear properties.In quantum electrodynamics one finds that the phe-nomenon of scattering of light by light (a violation ofcondition 2) should occur due to the formation andsubsequent annihilation of electron-positron pairs;2the cross-section for this process is, however, so smallthat experimental confirmation is not to be thoughtof. Another effect is predicted by General Relativity;an electromagnetic field contains energy and there-fore mass. This produces a gravitational field whichcan in turn deflect a light beam, again in violationof condition 2. Once again, we do not expect any ex-perimental confirmation in the laboratory! In order tobe extremely cautious about the experimental situation,however, we note that electrical measurements of thehighest accuracy are never performed with intensefields; if appreciable deviations from Maxwell's equa-tions did occur in free space at field strengths in excess

I F. Bloch, Phys. Rev., vol. 70, p. 460; 1946.A. Bloom, Phys. Rev., vol. 98, p. 1105; 1955.20. Halpern, Phys. Rev., vol. 44, p. 855; 1934.H. Euler and B. Kockel, Naturwiss., vol. 23, p. 246; 1935.

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PROCEEDINGS OF TIIE IRE

of 105 v/cm or 105 oersted, they would almost certainlylhave escaped experimental detection thus far.3

In spite of the above considerations, the vast majorityof substances is found to he linear to a high degree ofaccuracy at all field strengths commonly attained, hencewe denote as "nonlinear" only those substances in whichthere are substantial and easily demonstrable effectsarising from violation of conditions 1 or 2 above, oc-curring at easily produced field strengths. Each suchsubstance is potentially capable of important technicalapplications, since by violation of condition 1 we gener-ate harmonics and beat frequencies, while violation ofcondition 2 enables one to modulate one signal by thepresence of another.

In the case of conductivity and permeability, sub-stances which are nonlinear in the above sense havelong been known; thyrite and, to a certain extent,electrolytic cells and various solid-state devices, suchas rectifiers, may be regarded as media with nonlinearconductivity (more correctly as circuit elements withnonlinear conductance), while the hysteresis and satura-tion effects in ferromagnetic materials respresent greatnonlinearity. By contrast, nonlinear dielectric materi-als, although not entirely unknown in the past, havenot until recently been available in forms of interest toelectrical engineers. Probably the earliest known exam-ple of dielectric nonlinearity was the discovery, over80 years ago, of the Kerr electro-optical effect.4 Glassand many liquids, in particular nitrobenzene, developoptical birefringence in fairly strong electric fields;thus the dielectric constant at optical frequencies varieswith a low-frequency electric field in violation of thesuperposition condition. An example of a nionlinearcapacitance at microwave frequencies is provided bycrystal rectifiers, particularly the germanium welded-contact variety,5 in which the barrier capacitance variesstrongly with bias voltage. By far the most importantnonlinear dielectrics, however, are the ferroelectriccrystals or ceramics.

FERROELECTRICS

From a phenomenological point of view, ferroelec-tricity may be defined as the electric analog of ferro-magnetism, and the fundamental criterion of ferro-electricity is the existence, at certain temperatures, ofhysteresis between D and E. The static electric displace-ment then depends not only on the applied electricfield but also on the past history, in such a way thatwith sufficiently slow periodic variation of E we ob-

I There is, of course, indirect evidence associated with atomictheory suggesting that the laws of electrostatics hold at field strengthsfar beyond this limit; for example, the fact that the wavelengths ofthe spectral lines of hydrogen can be calculated to great accuracy onthe assumption that the electric field of the nucleus is a coulombfield. However, this could hardly be called an electrical measurement.

4M. Born, "Optik," p. 365, J. Springer, Berlin; 1933.5 H. C. Torrey and C. A. Whitmer, "Crystal Rectifiers," Chap. 13,

M.I.T. Radiation Laboratory series No. 15; McGraw-Hill Book Co.,Inc., New York; 1948.

tain a D-E hysteresis loop exactly like the familiarB-H curves of ferromagnetic materials. Above a cer-tain temperature T,, called the Curie point, this hys-teresis disappears, but the relation between D and Emay remain appreciably nonlinear tip to temperaturesfar above TC. Since oscillograms illustrating theseeffects have been published recently in this journal,6they will not be repeated here. In the neighborhoodof the Curie point more complicated phenomena aresometimes found.7

Several distinct classes of ferroelectrics, with widelydifferent chemical composition and crystal structure,are now known. The first to be discovered was Rochellesalt (sodium potassium tartrate tetrahydrate), widelyused for its piezoelectric properties.8 This substanceappears to be unique in that it possesses two Curietemperatures (-18°C and +230C) and is ferroelectricbetween them. Other ferroelectric tartrates9l0 showonly a single Curie point. Mueller" has shown that theelectrical, mechanical, and thermal properties ofRochelle salt can be correlated very satisfactorily by asingle thermodynamic free-energy function valid onboth sides of the Curie points. This is important notonly from the standpoint of economy of description,but it indicates that the ferroelectric-"paralectric"phase transition at the Curie points is probably not avery drastic rearrangement from a molecular point ofview as is the case in many phase transitions, for exam-ple, that between diamond and graphite."2 This coni-clusion seems well established also for the other classes"3of ferroelectrics.

Another class of ferroelectrics is represented by thesalt potassium dihydrogen phosphate, KH2PO4 andother substances of similar chemical composition andcrystal structure (i.e., the alkali and ammonium phos-phates and arsenates)."4 Although they have foundapplications based on their electro-optical properties,the fact that their Curie points are at liquid-air temper-atures limits their usefulness as nonlinear dielectrics inthe purely electrical sense. KH2PO4 is at present uniquein that the molecular mechanism of its properties (dif-ferent arrangements of hydrogen bonds) is probably

6 W. P. Mason and R. F. Wick, "Ferroelectrics and the dielectricamplifier," PROC. IRE, vol. 42, pp. 1606-1620; November, 1954.

7 W. J. Merz, Phys. Rev., vol. 91, p. 513; 1953.8 W. G. Cady, "Piezoelectricity," McGraw-Hill Book Co., Inc.,

New York, 1946.9 W. J. Merz, Phys. Rev., vol. 82, p. 562; 1950, vol. 83, pp. 226,

656; 1951.1' B. T. Matthias and J. K. Hulm, Phys. Rev., vol. 82, pp. 108;

1951."1 H. Mueller, Phys. Rev., vol. 47, p. 175; 1935, vol. 57, pp. 829-

842; 1940, vol. 58, pp. 565-805; 1941, Zeit. Krist., vol. 99, p. 122,1938, Ann. N. Y. Acad. Sci., vol. 40, p. 321; 1940.

12 R. Smoluchowski, et al., "Phase Transformations in Solids,"John Wiley & Sons, New York; 1951.

13 H. R. Danner and R. Pepinsky, Phys. Rev., vol. 99, p. 1215;1955.

14 G. Busch and P. Scherrer, Naturwiss., vol. 23, p. 737; 1935.G. Busch, Helv. Phys. Acta., vol. 11, p. 269, 1938.C. C. Stephenson and J. G. Hooley, Phys. Rev., vol. 56, p. 121;

1939.W. Bantle and P. Scherrer, Nature, vol. 143, p. 980; 1939.J. and K. Mendelssohn, Nature, vol. 144, p. 595, 1939.

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understood with greater certainty than for any otherferroelectric. 15"16The third class of ferroelectrics,"7 represented by the

cubic form of barium titanate, BaTiO3 and severalsimilar substances, is at the present time the mostinteresting from a theoretical standpoint and the mostuseful in terms of applications. The Curie point ofBaTiO3 is at 120°C, and at room temperature the mostnearly perfect crystals grown to date7 exhibit hysteresisloops with coercive force as low as 600 v/cm and aspontaneous polarization of about 26 microcoulombs/cm2, very much higher than in the first two classes offerroelectrics. The dielectric properties of BaTiO3 singlecrystals above the Curie point have been measured byDrougard, Landauer, and Young,'8 with results thatmay be summarized as follows. For crystals that arefree to distort in accordance with their electrostrictiveproperties (i.e., under conditions of zero stress), thebehavior is given within experimental error by the free-energy function

F(P, T) = Fo(T) + A(T)P2 + B(T)P4, (3)

where P is the dielectric polarization, Fo the free energyat zero polarization (which is irrelevant for dielectricproperties, although it largely determines the specificheat of the material), and A, B are linear functions oftemperature, given by the empirical equations

A = 3.8 X 10-5 (T - 105)B = 4.5 X 10-1' (T - 175),

which are in cgs units, with the temperature in degreescentigrade. The electric field is, from thermodynamics,

E = aFIaP = 2AP + 4BP3. (5)

Therefore, the incremental (small signal) dielectricconstant is

e= 1 + 4ir(9P/lE) = 1 + 47r/(2A + 12BP)247rA2/(2A3 + 3BE2), (6)

the approximation being valid at field strengths forwhich the cubic term in (5) is small compared to thelinear one. Since B is negative in the temperature rangewhere these experiments were performed (119°C to150°C), we have the rather surprising result thatapplication of a biasing field increases the dielectricconstant of the crystal. This phenomenon is seenclearly in the oscillograms of Merz,7 and may be shownby thermodynamic arguments'9 to be connected withthe fact that the crystals exhibit a first-order transitionat the Curie point; i.e., as the temperature is lowered,

15 J. C. Slater, Jour. Chem. Phys., vol. 9, p. 16; 1941.16 C. C. Stephenson and J. G. Hooley, Jour. Am. Chem. Soc., vol.66, p. 1937; 1944.

17 von Hippel, Breckenridge, Chesley, and Tisza, Jour. Ind. Eng.Chem., vol. 38, p. 1097; 1946.

18 M. E. Drougard, R. Landauer, and D. R. Young, Phys. Rev.,vol. 98, p. 1010; 1955.

'9 E. T. Jaynes, "Ferroelectricity," Princeton University Press,Princeton, N. J., Chap. 3; 1953.

the spontaneous polarization j'umps discontinuouslyfrom zero to a finite value (about 18 microcoulombs/cm2). Some of the first crystals grown, which wereless perfect, exhibited a second-order transition and, asrequired by thermodynamics, a positive sign of B.20It must be remembered, however, that this increase dueto bias occurs only under conditions of zero stress, andtherefore can be seen only at sufficiently low frequencies,below all of the mechanical vibration modes of thecrystal. Measurements made at Stanford Universityby Mr. V. Varenhorst at frequencies of 20, 40, and 120mcs showed in all cases a decrease in dielectric constantwith bias voltage. The results were complicated bytemperature hysteresis effects which persist above theCurie point and are not understood, but in a typicalcase at 40 mcs and 130°C, application of a biasingfield of 1700 v/cm lowered the dielectric constant from6,700 to 6,000.Growth of good single crystals of BaTiO3 is still a

difficult and costly art, and most of its applications todate have involved the ceramic material, often withvarious added impurities. The ceramic also exhibitsdielectric nonlinearity, a typical result6 being a decreaseof dielectric constant with biasing field such that afield of 10 kv/cm lowers e from 1,400 to 1,100, while afield of 30 kv/cm lowers it to 700. Similar results havebeen found at Stanford University, with an interestingadditional qualitative observation that the loss tangentof a ceramic at radio frequencies may be lowered sub-stantially by application of a biasing field of a fewkv/cm. Further data on properties of ceramics havebeen given by von Hippel.2'Many details concerning the physical properties of

BaTiO3 have been omitted here; in particular the phe-nomena of domain formation and motion which areessential to an understanding of the properties of singlecrystals below the Curie point. These have been de-scribed by Forsbergh,22 Merz,23 and Little.24 A recentdiscussion of the theory of ferroelectrics has been givenby Devonshire.25

THEORY OF DIELECTRICS

It might be supposed that with modern knowledgeof the properties of atoms and molecules, it would bea straightforward matter to calculate the dielectricconstant of any material of known composition fromfirst principles. Unfortunately, this turns out to be anextremely complicated problem on which little progresshas been made; only in the case of gases where the di-electric constant is very close to unity and the similarcase of dilute solutions of polar molecules in nonpolarliquids can one claim quantitative success. Although it

20 W. J. Merz, Phys. Rev., vol. 76, p. 1221; 1949.21 A. von Hippel, "Dielectric Materials and Applications," John

Wiley & Sons, New York; 1954.22 P. W. Forsbergh, Jr., Phys. Rev., vol. 76, p. 1187; 1949.23W. J. Merz, Phys. Rev., vol. 88, p. 421,1952;vol.95, p. 690; 1954.24 E. A. Little, Phys. Rev., vol. 98, p. 978; 1955.25 A. F. Devonshire, Phil. Mag. Suppl., vol. 3, p. 85; 1954.

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is easy to describe in words the mathematical proce-dure that would give a rigorous treatment, in practiceone must make from the start drastic simplificationswhich make it difficult to interpret whatever agreementor disagreement with experiment is found. The basicreason for this is that the properties of a crystal are notmerely the properties of a single molecule, multipliedby the number of molecules present, but the interactionsbetween them are an essential part of the picture, andthese are never taken into account in a correct way.From the point of view of quantum mechanics, we wouldhave to say that it is not meaningful in any preciseway to speak of the states and behavior of individualmolecules or atoms, but the only really correct attitudeis the global one in which we enumerate the possiblequantum states (wave functions) of the crystal as awhole. Almost without exception, however, theoreticaltreatments of dielectric properties of solids have beenbased on the concepts developed by Clausius andMosotti about 100 years ago. Here one regards a solidas composed of a large number of polarizable objects(in various cases the mechanism of polarization may bethought of as distortion of electronic distributions ofatoms or ions, motion of ions, or rotation of molecularaggregates having a permanent dipole moment) eachwith polarizability a, so that each object, in an electricfield F, develops a dipole moment

M-=aF. (7)

The field F is not, however, the same as the macro-scopic applied field E; because of the interaction of thepolarizable objects with each other, there is an addition-al term commonly taken as proportional to the netpolarization, with a proportionality constant /, knownas the Lorentz factor.

F=E +OP. (8)

Lorentz showed that if the polarizable objects are ar-ranged in a cubic or random array, and each main-tains the same constant dipole moment (no thermalagitation effects), i would have the value 47r/3. If thereare N of these polarizable objects per unit volume, thepolarization is

P = NaF = Na(E + OP),so that the dielectric susceptiblity becomes

X = P/E = Na/(1 - Nac). (9)Introducing the dielectric constant

E= 1 + 47rx, (10)

and assuming the Lorentz value / = 4ir/3,the well-known Clausius-Mosotti formula

we arrive at

e-1 4irNae+ 2 3

(11)

which is presented in some textbooks as if it were a

rigorous relation.

Many refinements of this treatment have been made,and a very complete account of them may be found inthe recent book of B6ttcher.26 They have led to someimprovement in agreement with experiment but notto any appreciably deeper understanding, because thebasic concepts remain the local polarizability and localfield F, which in modern theory no longer have a precisemeaning. Nevertheless, this classical treatment con-tains enough of the truth to be very useful in giving aqualitative understanding of dielectrics, provided cer-tain precautions are observed. In the first place, (11)cannot be used when the polarizability is due to freelyrotating permanent dipoles (i.e., polar molecules) ex-cept in the case of high dilution when it reduces to

- 1 = 4rNa << 1.

From statistical mechanics, one can calculate the polar-izability of a rotating dipole of moment M, with theresult a=M2/3kT, with k Boltzmann's constant and Tthe temperature in degrees Kelvin. Eq. (11) predictsan infinite dielectric constant, i.e., ferroelectricity, whenNao/> 1, so that this should occur at sufficiently lowtemperatures for any substance with rotating dipoles.However, if we insert the numerical values, we find thatmany polar substances should be ferroelectric at tem-peratures far above their boiling points! This is thefamous "4ir/3 catastrophe," which was resolved byOnsager27 with the observation that strong correlationsbetween the motions of nearby dipoles reduce the effec-tive Lorentz factor; the results of his approximatetreatment are obtained if we formally replace /3 in theabove equations by 47r/(2e+ 1); the opposite conclusionis then obtained that ferroelectricity does not occurunless the polarizability becomes infinite.The fact that ferroelectricity is so easy to "explain"

when one uses poor mathematical approximations haslong plagued theoreticians and has delayed any reliableunderstanding of the true cause of ferroeelctricity. Forexample, Rochelle salt was for many years treated as anassembly of rotating dipoles exhibiting the catastropheof (11). Another example of a model which predicts fer-roelectricity as a result of poor approximation is an arrayof harmonic oscillators interacting with each other. Suchan oscillator with a particle of charge e, mass m, andresonant frequency w has a temperature-independentpolarizability of e2/mW2, and therefore according to theabove equations one can always produce a ferroelectricarray by making the resonant frequency sufficientlysmall. However, this model is so simple that it can betreated rigorously; an orthogonal transformation ofcoordinates enables one to find the states of the arrayas a whole, and it is found when the problem is treatedcorrectly that ferroelectricity cannot occur unless thepolarizability of a single oscillator becomes infinite. We

26 C. J. F. Bottcher, "Theory of Electric Polarisation," ElsevierPress, Amsterdam; 1952.

27 L. Onsager, Jour. Am. Chem. SOc., vol. 58, p. 1486; 1936.

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also note that Slater's theory of KH2PO4, already men-

tioned as probably the best available theory of a ferro-electric, does not make use of electrostatic interactionsbut rather ones of a more direct mechanical nature. Inspite of these and other considerations,28 many peoplebelieve that in barium titanate we have the realiza-tion of the classical polarization catastrophe; in thepresent writer's opinion it is doubtful whether any

existing theory is free of the effects of poor approxima-tions of the above type, and interactions other thanelectrostatic may well be the essential ones.

If, disregarding these precautions, we assume that a

classical type of theory should have at least a qualitativeusefulness, what can be said about the expected occur-

rence of dielectric nonlinearity? According to the aboveequations, this would require that either the polariza-bility or the Lorentz factor be field-dependent. We con-

sider separately the three cases that the polarizationis due to: (a) rotating permanent dipoles; (b) transla-tional motion of ions; or (c) electronic distortion ofatoms or ions.

(a) Rotating Dipoles. The polarizability a= M2/3kTgiven above is an approximation valid only at fieldstrengths F such that MF<<kT. The exact expression,first calculated by Langevin in 1905, is

a = (M/F)L(a) = M2/3kT - M4F2/45k3T3 +

where L(a) = coth a -a-' is the Langevin function anda = MF/kl. From this we find that at a= 1 the polariz-ability is lowered by about 6 per cent, and for a>5,L(a) is essentially equal to unity, so that a variesinversely as the internal field F. Since molecular dipolemoments are of the order of 10-18 esu, we find that atroom temperature an appreciable nonlinearity couldbe expected only for F greater than about 4X 104 esu,

or 1.2 X 107 v/cm. Since we must remember to use theOnsager field for F, the applied field E would have tobe of the same order of magnitude; thus a measurablenonlinearity due to saturation of rotating dipoles couldbe expected only at very low temperatures and intensefield strengths.

(b) Translational Motion of Ions. Here the prospectsare considerably brighter. In many types of crystalsthe size of the lattice is determined by the larger ionsthat have to fit into it, and if small ions are also present,

28 J. M. Luttinger and L. Tisza, Phys. Rev., vol. 70, p. 954, 1946;vol. 72, p. 257; 1947.

they may be free to move in the interstices, throughdistances of an appreciable fraction of an Angstrom,but cannot move further due to contact with thelarger ions. It is seen without any calculation that thisresults in a contribution to the polarization of the crys-tal which saturates rather abruptly at a certain value.KH2PO4 is undoubtedly of this type, with movablehydrogens; other crystals of similar structure, eventhough not ferroelectric, might be expected to shownonlinearity. However, materials with high dielectricconstants should provide the most favorable possibili-ties, since according to the above equations we thenobtain an internal field F which is considerably "ampli-fied" above the applied field E. If appreciable motionoccurs, the Lorentz factors might also vary.

(c) Electronic Distortion. As a simple example, con-sider an atom which has a ground state llo with energyEo, and an excited state /j with energy E, such that thematrix element of the dipole moment operator betweenthem,

Mo,= e f*z+idV

does not vanish.29 Using quantum mechanics and statis-tical mechanics,30 the following formula for polarizabil-ity may be obtained:

Moi 2 tanh aax =

kT a

where

L4(E1- EO))2 + MO, 12F2]1/2a=kT

It is seen that appreciable nonlinearity requires that abe of order unity or greater and that the term in F2must contribute substantially to a. Therefore, sinceMol will typically be of the order of magnitude 10-18 esu,if the two energy levels are sufficiently close togetherthe situation is about the same as in the case of rotatingdipoles. If a high dielectric constant leads to greatinternal field strengths, these conditions might be met,although it appears that the case of movable ions re-mains the most favorable to development of strongnonlinearity.

29 L. I. Schiff, "Quantum Mechanics," McGraw-Hill Book Co.,Inc., New York, sec 25; 1949.

'0 See ref. 19 (pp. 58-60) for a similar calculationi.

1955 1737