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Three fundamental problems in ferroelectricity
a9
Pu: s0022-36!37(!xi)ooo10-8
Pergamon
THREE
FUNDAMENTAL
PROBLEMS
1. Phys. Chem Solids Vol57. No. 10, pp. 1439-1443, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. AU rights resewed 0022-3697/96 915.00 + 0.00
IN FERROELECTRICITY
J. F. SCOTT University of New South Wales, Sydney, NSCO 2052, Australia (Received IS January 1995;accepted 14 June 1995)
Abstract-We review briefly the phenomena of optical bistability in ferroelectrics, the reversible oxidationreduction processes at electrode interfaces in ferroelectric thin films, and the question of critical exponents in ferroelectrics near tricritical points. Keywork A. optical materials, D. ferroelectricity, D. electrical conductivity, D. dielectric properties, D. phase transitions.
1. OPTICAL BISTABILITY Optical bistability in ferroelectrics was discovered by the present author and his coworkers in 1992 [l]. Of course, earlier optical bistability had been studied in great detail in liquid, solid, and gaseous media. Of particular interest has been the observation of perioddoubling and the approach to chaos. One of the best and most comprehensive reviews is that of Flytzanis [2], which covers the period 1969-1984 (114 references); a good earlier discussion of the basic theory is given in the text on lasers by Sargent et al. [3]. In the present paper our interest is in temporal oscillations in the spot size and interference rings of self-focussed beams; these spatial ring patterns were first predicted by Wagner et al. [4] and studied in detail recently in ferroelectrics near T, [5-81. There are two aims of the present summary: on the theoretical side we would like to address the precise classification of data in terms of well known nonlinear equations. There is a curious lack of precision in the published theoretical literature concerning this point: Sargent et al. [3] refer to the optical bistability in passively coupled Fabry-Perot interferometers alternately (on the same page) as characterized by LotkaVolterra equations and by Van der Pol equations. Unfortunately, these two equations are not equivalent. Flytzanis refers to these phenomena [2] in terms of the Dulling equation, which is equivalent to neither the Lotka-Volterra equations nor the Van der Pol equations. Finally, we and others [9, lo] have found that cavity mode competition in lasers or interferometers, including the low-finesse “interferometer” produced by ceramic or crystalline specimens that simply have flat, parallel faces, may be better described in some ways by the May-Leonard
Equations [ll, 121, which are not equivalent to any of the aforementioned. In addition to these macroscopic descriptions via well studied nonlinear differential equations, there are many idiosyncratic microscopic models invented for individual systems, including liquid crystals [ 131. To distinguish among these models one needs accurate data concerning switching event times, dependence upon laser power and other control parameters, and detailed transient time dependences; generally not all these kinds of data have been available for each system of experimental interest. Earlier experimental studies on PMN ceramics [14, 151 have recently been extended to single crystals in our group at RMIT. On the experimental side the first need in analyzing data is to show that the effects observed are thermal focussing and not photorefraction. This is not a completely trivial undertaking. Since the effects are often studied in oxide ferroelectrics where both phenomena can occur, it is important to be able to distinguish between them. In lithium niobate thermal lensing was first observed by Akhmanov et al. [I 61and correctly interpreted by them. It may have been rediscovered in the same material, together with optical bistability, by Lemeshko and Obukhovskii [17], but this time was interpreted as four-wave, phase-matched photorefractive phenomena. We believe that the interpretation given by Lemeshko and Obukhovskii is qualitatively incorrect; the dependence of the intensities they observe as a function of laser wavelengths (440,488, and 628 nm) is far more likely to arise from the different photo-optic coefficient [dn/dT]/n at those wavelengths, which increases rapidly with decreasing wavelength, than to any subtle change in phase-matching conditions. Their observation that the effect in lithium niobate is observed only for
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J. F. SCOTT
laser wave vector along the polar axis z is perhaps also not due to phase matching conditions, but due merely to the fact that dn,/dT is very different from dn,/dT in this strongly uniaxial crystal. Moreover, phase matching at an angle as small as 1.5 degrees [17] is unlikely for four-wave mixing geometries. In our studies the quantitative dependence of the phenomena on air pressure or air flow rate past the specimens shows unambiguously that the observed effects are purely thermal and not photorefractive. Secondary reflections at relatively large angles can be shown to depend in a simple way upon sample length and are therefore also not to be confused with large-angle phase-matched four-wave mixing. Finally, the dependence of the phenomena on incident angle agrees with thermal lensing and not with phase matched processes. The use of single crystals of PMN has permitted observation of spatial patterns involving eight or nine distinct interference maxima in nearly circular configurations around the incident beam [3], in contrast to the rather diffuse transmission reported previously [13-151 for ceramic PMN. The observed data are very similar to those seen in single-crystal barium sodium niobate (BNN) or strontium barium niobate (SBN) [ 181;however, the temporal pattern of eight or nine aperiodic switching events with increasingly long separations is unchanged from the observations in PMN ceramic specimens. It is useful to note that ferroelectric specimens used for thermal lens experiments usually have flat, polished, parallel faces and hence behave as FabryPeriot interferometers of low finesse. A typical finesse [4-7, 14, 151is 1.3. The transmitted light intensity Zr (as a fraction of the incident intensity) through a Febry-Perot, neglecting absorption, is Zr/Zi = [l + csin*(A/2)]-‘,
(1)
where c = 4r*(l - r*)-*
(2)
and r is the reflectivity; here A is the phase shift in the medium within the Fabry-Perot and is a function of local heating in our case, or applied electric field [19, 201. A also depends upon the angle of incidence 8 between laser beam and the normal to the film surface A = (47rnd cos e)/X,
(3)
where n and d are film index of refraction and thickness, and X the laser wavelength. The constant c in eqn (2) is related to finesse F as c = 4F2/7r2,
(4)
which for our F = 1.3 situation is c = 0.68. Since c is less than unity, eqn (1) can be expressed to lowest order as IT/Ii = [l - csin*(A/2)],
(5)
which is the usual expression for a low-finesse FabryPerot. In the present case A will be an explicit function of temperature and hence incident laser power, so that a transcendental equation results: Zr = Zi{1 - (4F2/7r2) sin*[A(Zl)/2]}.
(6)
The microscopic physics involved in thermal lensing for such geometries is buried in the function A(Z1). Since this is a thermal process, there is a non-negligible temporal lag. The standard thermal focussing model describes this in detail; generally a rather opaque integral equation results. In the present paper it may be useful to describe this in a somewhat oversimplified pedagogical way: In any thermal focussing theory there is a thermal relaxation term [21] Zi = Zi(O)[l -g(l
+b/t)-‘1,
(7)
even in the absence of a Fabry-Perot cavity. Let us insert this expression in eqn (6) and assume for simplicity that the phase shift A is linear in intensity and temperature. The important result is that following the initial incident laser intensity, the transmitted intensity now is proportional to sin* Q(t), with @(t) = am,[l
- g(l + b/t)-‘].
(8)
Here the maximum phase shift depends upon the thermal gradient and hence the laser intensity. Equation (8) is pedagogically useful in two ways: first, it yields a finite number of zeros in sin* a(t) and does not oscillate without cessation (unlike the theory of refs [ 12, 221); this number of oscillations depends simply upon the thermal gradient, and although the phase shift can be as large [4-71 as 400x in our work, that is not infinite and implies a maximum of cc 400 oscillations. Second, the oscillation is not periodic in time; rather, it has an oscillation frequency that decreases linearly with time (as 1 - gt/b), as observed in our experiments, for short times t K b. That is, frequency decreases linearly in t, so that the change in periods varies as f* (if frequency were independent of t, switching events would be linear in t). This quadratic dependence has been reported earlier, and it was noted [21] that this aspect of the data mimics the results of May-Leonard equations. For long times the oscillation slows down only gradually as l/r; i.e. frequency varies as [l - g + gb/t]. Note, however, that depending upon the relative magnitudes of parameters ‘P,,,
Fundamental problems in ferroelectricity and g, oscillation may cease before one leaves the quadratic regime (i.e. the phase shift becomes less than Awhile t is still less than b); this would occur for “low” laser powers, which may include all data published to date (limited by optical damage). This simple equation may also give a possible explanation of the apparent period-multiplication always observed [19] in PMN around the n = 9 switching event (and therefore not a Feigenbaum-like bifurcation en route to chaos). In eqn (8) a qualitative change (to a non-quadratic regime) in switching time t, vs n arises simply from the time t reaching a value of approximately b and is thus the same for each PMN experiment, with no random characteristic. As recently shown by Zheng et al. [23] the regenerative oscillations appear to have period-multiplying “bifurcations” at t = b. The data previously interpreted as possible period multiplication are consequently viewed as more likely to arise from an equation of the form given in eqn (8) above and unrelated to Feigenbaum bifurcations. No complete theory of aperiodic optical bistability exists at present. Although the data in PMN resemble those of a Van der Pol oscillator near the limit-cycle asymptote in phase space, O’Sullivan et al. have recently shown [24] why the phenomena observed in PMN, SBN, and BNN cannot be interpreted as Van der Pol, or Lotka-Volterra, or May-Leonard systems. As initially pointed out, the difficulty in the data is the extreme asymmetry in the intensity Z(t) vs time near switching events: The leading edge is far more abrupt than the trailing edge. This asymmetry is reversed from that in the Van der Pol equations; and those equations are not time-reversal invariant (changing a control parameter from positive to negative is equivalent to reversing t and --t, but it also makes the solutions to the equation unstable). We have been unable to find any macroscopic equations that reproduce the asymmetry of the switching intensity Z(r) and also mimic the observed aperiodicity, but microscopic theories in two levels of approximation (three-dimensional, one-dimensional, numerical; analytic-both ignoring longitudinal temperature gradients) reproduce [20, 211 both the shape and temporal sequence of the bistable laser intensity Z(r).
2. FATIGUE MJWHANISMS IN FERROELECTRIC MEMORIES
It has been observed by several independent groups that ferroelectric thin tilm capacitors made of lead zirconate-titanate (PZT) on Pt electrodes “fatigue”that is, the amount of switched charge measured with each voltage reversal decreases. Many different mechanisms have been invoked to explain this phenomenon, such as stress, domain wall pinning by
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defects, oxygen vacancy diffusion, etc. I believe it is primarily due to charge injection. The phenomenon is absent or greatly reduced if iridium oxide or ruthenium oxide is substituted for platinum. And it does not occur (or is greatly reduced) in “poor quality” films in which the spontaneous polarization is cu 2-5 &/cm2 (as opposed to “good” tilms in which it is 20-40 &/crn2). Here we emphasize that both observations can be explained as due to charge injection, using values known from the published literature from the nominally unrelated field of medical electrodes. The microelectronics literature does not contain an explanation of why iridium oxide electrodes eliminate fatigue, but oddly enough, the biomedical literature does [25-281. Unlike platinum, iridium (and ruthenium) electrodes can be used in oxygen-containing ambients (e.g. in the human body) because they readily form metastable oxides that can reversably oxidize and reduce as voltage is cycled. This prevents detrimental charge injection that might destroy cell tissue in a body or destroy capacitors in a DRAM (dynamic random access memory) in a computer. It is known from the biomedical literature that an upper limit of cu 20&/cm2 of injected charge causes irreversible damage to the electrode interface in Pt electrodes. What has not been recognized previously is that this injected charge limit obtains independent of whether the charge injection is due to real d.c. current or to the displacement current of a switched (ferroelectric) capacitor. But the recognition that these injected currents are equivalent readily explains why “poor” PZT capacitors with low spontaneous polarization values do not fatigue: With a total charge density injected of 2Pr = 4-lO#cm’, they never reach the “Robblee threshold” of irreversible damage of cu 20 $/cm’. By comparison, charge injection begins to cause immediate degradation in a “good” PZT film with switched charge 2Pr = 40-80, well above Robblee’s criterion. Thus, what had apparently been empirical rules in ferroelectric microelectronics (“bad PZT does not fatigue,” and “iridium oxide electrodes on PZT eliminate fatigue”) have very simple electrochemical bases. However, in the spirit of the present conference, considerable work needs to be done to test this hypothesis in very quantitative ways. Since the DRAM market is $8 billion annually, there might be some justification for funding such careful studies.
3. CRITICAL EXPONENTS NEAR TRICRITICAL POINTS
Ferroelectrics are generally thought to exhibit mean-field (Landau) exponents for various parameters near their Curie Temperatures because of their long-range Coulombic interactions. Indeed
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J. F. SCOTT
near the few perfectly second-order transitions in ferroelectrics such as triglycine sulfate (TGS), the exponents cr, p, ~,6 that characterize the divergences of, respectively, specific heat, spontaneous polarization, isothermal susceptibility, and the D(E) or P(E) response along the critical isotherm indeed satisfy the Landau predictions of 0 (step discontinuity or logarithmic divergence), l/2, 1, and 3. In this sense ferroelectrics are both simple and uninteresting. However, many real ferroelectrics have slightly discontinuous transitions at their Curie temperatures (or other transitions above or below T,). Some are quite close to tricritical points, where the transitions go from first- to second-order. In this case it is useful to apply thermodynamics as well as Landau theory to the data. Three key thermodynamic inequalities are (Y’+ p(1 + 6) 2 2
(Griffiths [29])
CV’ + 2p + 7’ > 2
(Rushbrooke
Y’ 2 p(a - 1)
(Widom [3 l])
[30])
(9) (10) (11)
where primes denote values below T,. It is well known [32] that the Landau mean-field values at tricritical points, (viz. (Y= Q’ = 0; p = l/4; y’ = 1; 6 = 5) violate the Grifliths and Rushbrooke inequalities. The purpose of the present section of this paper is to point out that of the many ferroelectrics close to tricritical, none have had all four critical exponents measured accurately to show that Landau theory fails. Careful efforts to measure some of the critical exponents have been made in ferroelectric KDP (KHIPO,), BaMnF4, TSCC (Q-is-sarcosine calcium chloride), BazNaNbSOld (BNN) and ferroelastics of the Lap5014 family [33-361. In TSCC Petzelt et al. [371suggest some non-meanfield results, but Scott and Chen [38] find 28 - y’ = +0.005 f 0.020. The suspicion is that a! and (Y’are experimentally not zero in these systems, but are approximately l/2. Reese found [39] Q’ = 0.50 in KDP; and Scott et al. [40] f&d (Y= 0.54 in BaMnF4. In K2Se04 LopezEcharri [41] finds (Y= (Y’= 0.40. These values all violate Landau theory but are fully compatible with scaling theory, which as is well known, assumes that the critical exponents are not independent parameters but are interrelated in terms of only two independent parameters aE and 4, that describe how displacement vector D (or polarization P) depend upon applied field E and temperature T. For ferroelectrics I/a, = (S + 1)p = 3/2 (whence aT = 2/3) aa = 6/(6 + 1) = 5/6. From such assumptions aT = l/2.
(12) (13)
one finds that Q’ = 2 - l/
It is very hard to measure CYor d at high temperatures (such as the 840 K transition in BNN) via conventional specific heat techniques. However, using the Pippard relationship [42-441 one can instead measure changes in longitudinal sound velocity, which is often easier. Care must be taken, however, in that (Y(specific heat) = cy (sound velocity) - 2(# - l), (14) where 4 is a cross-over exponent equal to unity in mean field [45, 461. I end this paper by encouraging experimentalists to measure CX,@,~,6 all for the same slightly first-order ferroelectric and thereby establish at least one textbook case where Landau theory is shown to fail for ferroelectrics. Presumably this will also establish the first clear case where scaling theory works for ferroelectrics. In the absence of such a complete set of measurements on at least one tricritical material, our ferroelectrics texts will be rather incomplete, and our knowledge of the “fundamental properties of ferroelectrics” will not be as fundamental as we might wish. Acknowledgements-1 thank Dr Richard O’Sullivan and MS Xiping Zheng for helpful discussions concerning optical bistability in ferroelectrics.
REFERENCES 1. Chen T, Sheih S.-J., Scott J. F. and Bhalla A., Appl. Phys. Letf. 60,332 (1992). 2. Flytzanis C., Bistability, instability, and chaos in passive nonlinear optical systems. In Nonlinear Phenomena in Solids-Modern Topics (Edited by M. Borissov), pp. 268-314. World Sciknti&, Singapdre (1985). 3. Sargent M.. Scullv M. 0. and Lamb W. E., Laser Oofics, pp.-45-51; 123-127. Addison-Wesley; New York (1974); see also Scott J. F., Phase Transitions in Lasers with Saturable Absorbers. In Frontiers of Nonequilibrium Statistical Physics (Edited by G. T. Moore and M. 0. Scully, NATO AS1 Series Vol. B-135), pp. 465472. Plenum Press, New York (1986). 4. Wagner W. G., Haus H. A. and Marburger J. H., Phys. Rev. 175,256 (1968); see also Gagnon L. J. Opt.Sot. Am. B7,1098 (1990) for the “exact” solutions for the steadystate self-focussing problem in cylindrical geometries. 5. Scott J. F., Sheih S.-J., Furer K. R., Clark N. A., Oliver W. F. and Lee S. A., Phys. Rev. B41,9330 (1990). 6. Scott J. F. and Sheih S.-J., J. Phys. C: Condens. Mater. 2, 8553 (1990). 1. Chen T., Sheih S.-J. and Scott J. F., Phys. Rev. B43,815 (1991). 8. Scott J. F., Sheih S.-J. and Chen T., Ferroelectrics 117,21 (1991). 9. Benkert C. and Anderson D. Z., Phys. Rev. A44,4633 (1991). 10. ChenT., Scott J. F. and Phillipson P. E., Integ. Ferroelec. 3,377 (1993), 5, 1 (1994). 11. Mav R. M. and Leonard W. J.. SIAM J. aml. -_ Math. 2% 243<1975). 12. Philliuson P. E.. Schuster P. and Johnston R. G.. SIAM J. appt. Math. & 541 (1985). 13. Khoo I. C., Yan P. Y., Liu T. H., Shepard S. and Hou
Fundamental problems in ferroelectricity J. Y., Phys. Rev. A29, 2756 (1984); Vasnetsov M. V., JETPLetr. SO,480 (1990); Cheung M.-M., Durbin S. D. and Shen Y. R., optics kett. 8, 39 (1983). 14. Scott J. F. and Chen T.. Intee. Ferroelec. 1.71 (1992). 15. Chen T. and Scott J. F.1 Integ. Ferroelec. 3; 69 (1993). 16. Akhmanov S. A., Krindlach D. P., Sukhorukov A. P. and Khokhlov R. V., Pis’ma Zh. Eksp. Tear. Fiz. 6,509 (1967) [translation: JETP Left. 6, 38 (1967)]; IEEE J. Quanr. Electron. QE-4, 568 (1968). 17. Lemeshko V. V. and Obukhovskii V. V., Pis’ma Zh. Tekh. Fiz. 11, 1389 (1985) [translation: Sov. Tech. Phys. Lett. 11, 573 (1985)]. 18. Chen T., Sheih S.-J., Scott J. F. and Chen H., Ferroelectrics 120, 115 (1991). 19. Chen T. and Scott J. F., Ferroelectrics (1993); Chen T., Scott J. F. and Phillipson P. E., Integ. Ferroelec. 3, 337 (1993), 5, 1 (1994).
20. Sheih S.-J., Chen T., Beale P. D. and Scott J. F., Ferroelectricss 123, 1 (1991). 21. Gordon J. P., Leite R. C. C., Moore R. S., Port0 S. P. S. and Whinnery J. R., J. appl. Phys. 36,3 (1965). 22. Vasnetsov M. V., JETP Left. So, 480 (1990). 23. Xiping Zheng, O’Sullivan R. A., Scott J. F., Ye G. and S&mid H. Iiteg. Ferroelec. 9,225 (1995). 24. O’Sullivan R. A.. Scott J. F. and Xinine .- Zhena.-, InteE. I
Ferroelec. 9,215 i1995). 25. Robblee L. S., Mangaudis W. J., et al. Muter. Res. Sot. Symp. Proc. 55,303 (1986).
26. Robblee L. S., US Pat. No. 4,677,989. 27. Robblee L. S., US Pat. No. 4,717,581. 28. Robblee. L. S. and Cogan S. F., Metals for medical electrodes. In Encyclopedia of Materials Science &
29. 30.
31. 32.
33. 34.
35.
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Engineering, Suppl. Vol. 1, (Edited by R. W. Chan). Pergamon Press, Oxford (1988). Gr&ths R. B., J. Chem. Phys..43, 1958 (1965). Rushbrooke G. S.. J. Chem. Phvs. 39.842 (19631. Widom B., J. Cheh. Phys. 41, f633 (i964).‘ ’ Stanley H. E., Introduction to Phase Transitions and Critical Phenomena, Chap. 10. Oxford University Press, Oxford (1971). Errandonea G., Phys. Rev. B21,5221 (1980). Qu X. X., Xu X. Y., Xia H. and Zhang X. K., J. Phys. Condens. Muter. 2. 7077 (1990): Ou X. X.. Xia M.. Xu X. Y., Zhang W., Zhang X. K. and Wang G. Y ., J. ihys. Condens. Mater. 2,55 (1990). Chen T. and Guangyang Hong, Chin Phys. 7, 422
(1987). 36. Ting Chen and Scott J. F., Phys. Rev. B40,
8978 (1989). 37. Petzelt J., Volkov A. A., Goncharov Yu. G., Albers J. and Klopperpieper A., SolidState Commun. 73,5 (1983). 38. Scott J. F. and Ting Chen, Phase Transitions 32, 235 (1991). 39. Reese W., Solid State Commun. 7,969 (1969). 40. Scott J. F., Habbal F. and Hidaka M., Phys. Rev. B25,
1805 (1982). 41. Lopez-Echarri A., Tell0 M. J. and Gili P., Solid State Commun. 36,102l (1980). 42. Pippard A. B., Phil. Mug. 1,473 (1956). 43. Garland C. W., J. Chem. Phys. 45,1005 (1964). 44. Janovec V., J. Chem. Phys. 45, 1874 (1966). 45. Fossum J. O., J. Phys. C 18,553l (1985). 46. Strukov B. A., Taraskin S. A., Minaeva K. A. and Fedorikhin V. A., Ferroelectrics 25,399 (1980).
Pu: s0022-36!37(!xi)ooo10-8
Pergamon
THREE
FUNDAMENTAL
PROBLEMS
1. Phys. Chem Solids Vol57. No. 10, pp. 1439-1443, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. AU rights resewed 0022-3697/96 915.00 + 0.00
IN FERROELECTRICITY
J. F. SCOTT University of New South Wales, Sydney, NSCO 2052, Australia (Received IS January 1995;accepted 14 June 1995)
Abstract-We review briefly the phenomena of optical bistability in ferroelectrics, the reversible oxidationreduction processes at electrode interfaces in ferroelectric thin films, and the question of critical exponents in ferroelectrics near tricritical points. Keywork A. optical materials, D. ferroelectricity, D. electrical conductivity, D. dielectric properties, D. phase transitions.
1. OPTICAL BISTABILITY Optical bistability in ferroelectrics was discovered by the present author and his coworkers in 1992 [l]. Of course, earlier optical bistability had been studied in great detail in liquid, solid, and gaseous media. Of particular interest has been the observation of perioddoubling and the approach to chaos. One of the best and most comprehensive reviews is that of Flytzanis [2], which covers the period 1969-1984 (114 references); a good earlier discussion of the basic theory is given in the text on lasers by Sargent et al. [3]. In the present paper our interest is in temporal oscillations in the spot size and interference rings of self-focussed beams; these spatial ring patterns were first predicted by Wagner et al. [4] and studied in detail recently in ferroelectrics near T, [5-81. There are two aims of the present summary: on the theoretical side we would like to address the precise classification of data in terms of well known nonlinear equations. There is a curious lack of precision in the published theoretical literature concerning this point: Sargent et al. [3] refer to the optical bistability in passively coupled Fabry-Perot interferometers alternately (on the same page) as characterized by LotkaVolterra equations and by Van der Pol equations. Unfortunately, these two equations are not equivalent. Flytzanis refers to these phenomena [2] in terms of the Dulling equation, which is equivalent to neither the Lotka-Volterra equations nor the Van der Pol equations. Finally, we and others [9, lo] have found that cavity mode competition in lasers or interferometers, including the low-finesse “interferometer” produced by ceramic or crystalline specimens that simply have flat, parallel faces, may be better described in some ways by the May-Leonard
Equations [ll, 121, which are not equivalent to any of the aforementioned. In addition to these macroscopic descriptions via well studied nonlinear differential equations, there are many idiosyncratic microscopic models invented for individual systems, including liquid crystals [ 131. To distinguish among these models one needs accurate data concerning switching event times, dependence upon laser power and other control parameters, and detailed transient time dependences; generally not all these kinds of data have been available for each system of experimental interest. Earlier experimental studies on PMN ceramics [14, 151 have recently been extended to single crystals in our group at RMIT. On the experimental side the first need in analyzing data is to show that the effects observed are thermal focussing and not photorefraction. This is not a completely trivial undertaking. Since the effects are often studied in oxide ferroelectrics where both phenomena can occur, it is important to be able to distinguish between them. In lithium niobate thermal lensing was first observed by Akhmanov et al. [I 61and correctly interpreted by them. It may have been rediscovered in the same material, together with optical bistability, by Lemeshko and Obukhovskii [17], but this time was interpreted as four-wave, phase-matched photorefractive phenomena. We believe that the interpretation given by Lemeshko and Obukhovskii is qualitatively incorrect; the dependence of the intensities they observe as a function of laser wavelengths (440,488, and 628 nm) is far more likely to arise from the different photo-optic coefficient [dn/dT]/n at those wavelengths, which increases rapidly with decreasing wavelength, than to any subtle change in phase-matching conditions. Their observation that the effect in lithium niobate is observed only for
1439
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J. F. SCOTT
laser wave vector along the polar axis z is perhaps also not due to phase matching conditions, but due merely to the fact that dn,/dT is very different from dn,/dT in this strongly uniaxial crystal. Moreover, phase matching at an angle as small as 1.5 degrees [17] is unlikely for four-wave mixing geometries. In our studies the quantitative dependence of the phenomena on air pressure or air flow rate past the specimens shows unambiguously that the observed effects are purely thermal and not photorefractive. Secondary reflections at relatively large angles can be shown to depend in a simple way upon sample length and are therefore also not to be confused with large-angle phase-matched four-wave mixing. Finally, the dependence of the phenomena on incident angle agrees with thermal lensing and not with phase matched processes. The use of single crystals of PMN has permitted observation of spatial patterns involving eight or nine distinct interference maxima in nearly circular configurations around the incident beam [3], in contrast to the rather diffuse transmission reported previously [13-151 for ceramic PMN. The observed data are very similar to those seen in single-crystal barium sodium niobate (BNN) or strontium barium niobate (SBN) [ 181;however, the temporal pattern of eight or nine aperiodic switching events with increasingly long separations is unchanged from the observations in PMN ceramic specimens. It is useful to note that ferroelectric specimens used for thermal lens experiments usually have flat, polished, parallel faces and hence behave as FabryPeriot interferometers of low finesse. A typical finesse [4-7, 14, 151is 1.3. The transmitted light intensity Zr (as a fraction of the incident intensity) through a Febry-Perot, neglecting absorption, is Zr/Zi = [l + csin*(A/2)]-‘,
(1)
where c = 4r*(l - r*)-*
(2)
and r is the reflectivity; here A is the phase shift in the medium within the Fabry-Perot and is a function of local heating in our case, or applied electric field [19, 201. A also depends upon the angle of incidence 8 between laser beam and the normal to the film surface A = (47rnd cos e)/X,
(3)
where n and d are film index of refraction and thickness, and X the laser wavelength. The constant c in eqn (2) is related to finesse F as c = 4F2/7r2,
(4)
which for our F = 1.3 situation is c = 0.68. Since c is less than unity, eqn (1) can be expressed to lowest order as IT/Ii = [l - csin*(A/2)],
(5)
which is the usual expression for a low-finesse FabryPerot. In the present case A will be an explicit function of temperature and hence incident laser power, so that a transcendental equation results: Zr = Zi{1 - (4F2/7r2) sin*[A(Zl)/2]}.
(6)
The microscopic physics involved in thermal lensing for such geometries is buried in the function A(Z1). Since this is a thermal process, there is a non-negligible temporal lag. The standard thermal focussing model describes this in detail; generally a rather opaque integral equation results. In the present paper it may be useful to describe this in a somewhat oversimplified pedagogical way: In any thermal focussing theory there is a thermal relaxation term [21] Zi = Zi(O)[l -g(l
+b/t)-‘1,
(7)
even in the absence of a Fabry-Perot cavity. Let us insert this expression in eqn (6) and assume for simplicity that the phase shift A is linear in intensity and temperature. The important result is that following the initial incident laser intensity, the transmitted intensity now is proportional to sin* Q(t), with @(t) = am,[l
- g(l + b/t)-‘].
(8)
Here the maximum phase shift depends upon the thermal gradient and hence the laser intensity. Equation (8) is pedagogically useful in two ways: first, it yields a finite number of zeros in sin* a(t) and does not oscillate without cessation (unlike the theory of refs [ 12, 221); this number of oscillations depends simply upon the thermal gradient, and although the phase shift can be as large [4-71 as 400x in our work, that is not infinite and implies a maximum of cc 400 oscillations. Second, the oscillation is not periodic in time; rather, it has an oscillation frequency that decreases linearly with time (as 1 - gt/b), as observed in our experiments, for short times t K b. That is, frequency decreases linearly in t, so that the change in periods varies as f* (if frequency were independent of t, switching events would be linear in t). This quadratic dependence has been reported earlier, and it was noted [21] that this aspect of the data mimics the results of May-Leonard equations. For long times the oscillation slows down only gradually as l/r; i.e. frequency varies as [l - g + gb/t]. Note, however, that depending upon the relative magnitudes of parameters ‘P,,,
Fundamental problems in ferroelectricity and g, oscillation may cease before one leaves the quadratic regime (i.e. the phase shift becomes less than Awhile t is still less than b); this would occur for “low” laser powers, which may include all data published to date (limited by optical damage). This simple equation may also give a possible explanation of the apparent period-multiplication always observed [19] in PMN around the n = 9 switching event (and therefore not a Feigenbaum-like bifurcation en route to chaos). In eqn (8) a qualitative change (to a non-quadratic regime) in switching time t, vs n arises simply from the time t reaching a value of approximately b and is thus the same for each PMN experiment, with no random characteristic. As recently shown by Zheng et al. [23] the regenerative oscillations appear to have period-multiplying “bifurcations” at t = b. The data previously interpreted as possible period multiplication are consequently viewed as more likely to arise from an equation of the form given in eqn (8) above and unrelated to Feigenbaum bifurcations. No complete theory of aperiodic optical bistability exists at present. Although the data in PMN resemble those of a Van der Pol oscillator near the limit-cycle asymptote in phase space, O’Sullivan et al. have recently shown [24] why the phenomena observed in PMN, SBN, and BNN cannot be interpreted as Van der Pol, or Lotka-Volterra, or May-Leonard systems. As initially pointed out, the difficulty in the data is the extreme asymmetry in the intensity Z(t) vs time near switching events: The leading edge is far more abrupt than the trailing edge. This asymmetry is reversed from that in the Van der Pol equations; and those equations are not time-reversal invariant (changing a control parameter from positive to negative is equivalent to reversing t and --t, but it also makes the solutions to the equation unstable). We have been unable to find any macroscopic equations that reproduce the asymmetry of the switching intensity Z(r) and also mimic the observed aperiodicity, but microscopic theories in two levels of approximation (three-dimensional, one-dimensional, numerical; analytic-both ignoring longitudinal temperature gradients) reproduce [20, 211 both the shape and temporal sequence of the bistable laser intensity Z(r).
2. FATIGUE MJWHANISMS IN FERROELECTRIC MEMORIES
It has been observed by several independent groups that ferroelectric thin tilm capacitors made of lead zirconate-titanate (PZT) on Pt electrodes “fatigue”that is, the amount of switched charge measured with each voltage reversal decreases. Many different mechanisms have been invoked to explain this phenomenon, such as stress, domain wall pinning by
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defects, oxygen vacancy diffusion, etc. I believe it is primarily due to charge injection. The phenomenon is absent or greatly reduced if iridium oxide or ruthenium oxide is substituted for platinum. And it does not occur (or is greatly reduced) in “poor quality” films in which the spontaneous polarization is cu 2-5 &/cm2 (as opposed to “good” tilms in which it is 20-40 &/crn2). Here we emphasize that both observations can be explained as due to charge injection, using values known from the published literature from the nominally unrelated field of medical electrodes. The microelectronics literature does not contain an explanation of why iridium oxide electrodes eliminate fatigue, but oddly enough, the biomedical literature does [25-281. Unlike platinum, iridium (and ruthenium) electrodes can be used in oxygen-containing ambients (e.g. in the human body) because they readily form metastable oxides that can reversably oxidize and reduce as voltage is cycled. This prevents detrimental charge injection that might destroy cell tissue in a body or destroy capacitors in a DRAM (dynamic random access memory) in a computer. It is known from the biomedical literature that an upper limit of cu 20&/cm2 of injected charge causes irreversible damage to the electrode interface in Pt electrodes. What has not been recognized previously is that this injected charge limit obtains independent of whether the charge injection is due to real d.c. current or to the displacement current of a switched (ferroelectric) capacitor. But the recognition that these injected currents are equivalent readily explains why “poor” PZT capacitors with low spontaneous polarization values do not fatigue: With a total charge density injected of 2Pr = 4-lO#cm’, they never reach the “Robblee threshold” of irreversible damage of cu 20 $/cm’. By comparison, charge injection begins to cause immediate degradation in a “good” PZT film with switched charge 2Pr = 40-80, well above Robblee’s criterion. Thus, what had apparently been empirical rules in ferroelectric microelectronics (“bad PZT does not fatigue,” and “iridium oxide electrodes on PZT eliminate fatigue”) have very simple electrochemical bases. However, in the spirit of the present conference, considerable work needs to be done to test this hypothesis in very quantitative ways. Since the DRAM market is $8 billion annually, there might be some justification for funding such careful studies.
3. CRITICAL EXPONENTS NEAR TRICRITICAL POINTS
Ferroelectrics are generally thought to exhibit mean-field (Landau) exponents for various parameters near their Curie Temperatures because of their long-range Coulombic interactions. Indeed
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J. F. SCOTT
near the few perfectly second-order transitions in ferroelectrics such as triglycine sulfate (TGS), the exponents cr, p, ~,6 that characterize the divergences of, respectively, specific heat, spontaneous polarization, isothermal susceptibility, and the D(E) or P(E) response along the critical isotherm indeed satisfy the Landau predictions of 0 (step discontinuity or logarithmic divergence), l/2, 1, and 3. In this sense ferroelectrics are both simple and uninteresting. However, many real ferroelectrics have slightly discontinuous transitions at their Curie temperatures (or other transitions above or below T,). Some are quite close to tricritical points, where the transitions go from first- to second-order. In this case it is useful to apply thermodynamics as well as Landau theory to the data. Three key thermodynamic inequalities are (Y’+ p(1 + 6) 2 2
(Griffiths [29])
CV’ + 2p + 7’ > 2
(Rushbrooke
Y’ 2 p(a - 1)
(Widom [3 l])
[30])
(9) (10) (11)
where primes denote values below T,. It is well known [32] that the Landau mean-field values at tricritical points, (viz. (Y= Q’ = 0; p = l/4; y’ = 1; 6 = 5) violate the Grifliths and Rushbrooke inequalities. The purpose of the present section of this paper is to point out that of the many ferroelectrics close to tricritical, none have had all four critical exponents measured accurately to show that Landau theory fails. Careful efforts to measure some of the critical exponents have been made in ferroelectric KDP (KHIPO,), BaMnF4, TSCC (Q-is-sarcosine calcium chloride), BazNaNbSOld (BNN) and ferroelastics of the Lap5014 family [33-361. In TSCC Petzelt et al. [371suggest some non-meanfield results, but Scott and Chen [38] find 28 - y’ = +0.005 f 0.020. The suspicion is that a! and (Y’are experimentally not zero in these systems, but are approximately l/2. Reese found [39] Q’ = 0.50 in KDP; and Scott et al. [40] f&d (Y= 0.54 in BaMnF4. In K2Se04 LopezEcharri [41] finds (Y= (Y’= 0.40. These values all violate Landau theory but are fully compatible with scaling theory, which as is well known, assumes that the critical exponents are not independent parameters but are interrelated in terms of only two independent parameters aE and 4, that describe how displacement vector D (or polarization P) depend upon applied field E and temperature T. For ferroelectrics I/a, = (S + 1)p = 3/2 (whence aT = 2/3) aa = 6/(6 + 1) = 5/6. From such assumptions aT = l/2.
(12) (13)
one finds that Q’ = 2 - l/
It is very hard to measure CYor d at high temperatures (such as the 840 K transition in BNN) via conventional specific heat techniques. However, using the Pippard relationship [42-441 one can instead measure changes in longitudinal sound velocity, which is often easier. Care must be taken, however, in that (Y(specific heat) = cy (sound velocity) - 2(# - l), (14) where 4 is a cross-over exponent equal to unity in mean field [45, 461. I end this paper by encouraging experimentalists to measure CX,@,~,6 all for the same slightly first-order ferroelectric and thereby establish at least one textbook case where Landau theory is shown to fail for ferroelectrics. Presumably this will also establish the first clear case where scaling theory works for ferroelectrics. In the absence of such a complete set of measurements on at least one tricritical material, our ferroelectrics texts will be rather incomplete, and our knowledge of the “fundamental properties of ferroelectrics” will not be as fundamental as we might wish. Acknowledgements-1 thank Dr Richard O’Sullivan and MS Xiping Zheng for helpful discussions concerning optical bistability in ferroelectrics.
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