Infrared spectra and second-harmonic generation in barium strontium titanate and lead zirconate-titanate thin films: “Polaron” artifacts

JOURNAL OF APPLIED PHYSICS VOLUME 94, NUMBER 5 1 SEPTEMBER 2003

Infrared spectra and second-harmonic generation in barium strontiumtitanate and lead zirconate-titanate thin films: ‘‘Polaron’’ artifacts

J. F. Scott, a),b) A. Q. Jiang,a) S. A. T. Redfern, Ming Zhang, and M. Dawbera)

Department of Earth Sciences, Cambridge University, Cambridge CB2 3EQ, United Kingdom

~Received 26 February 2003; accepted 9 June 2003!

We report infrared absorption spectroscopy and second-harmonic generation data for bariumstrontium titanate~BST! and lead zirconate-titanate~PZT! ceramic films in the 300025500 cm21

region. Second-harmonic generation experiments give temperature dependences in accord withoxygen vacancy cluster theory@S. A. Prosandeev, Sov. Phys. JETP83, 747~1996!; S. A. Prosandeev,V. S. Vikhnin, and S. Kapphan, Integr. Ferroelectr.32, 1047~2001!; J. Phys. Condens. Matter14,4407~2002!#. A percolation model of vacancy ordering is discussed. The present work shows thatearlier data interpreted as polaron spectra in these films were actually artifacts due to interference inthe optical apparatus used@B. Guettler, U. Bismayer, P. Groves, and E. Salje, Semicond. Sci.Technol.10, 245 ~1950#; more recent ‘‘polaron’’ spectra in WO3 films may also be [email protected], A. Azens, and G. A. Niklassson, J. Appl. Phys.90, 1860~2001!#. Numerical estimates ofthe polaron massm** 516me in SrTiO3 and BST help prove that dielectric data in strontiumtitanate interpreted as bipolarons@A. Levstik et al., Appl. Phys. Lett.81, 4046 ~2002!# are alsoartifacts. © 2003 American Institute of Physics.@DOI: 10.1063/1.1596715#

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I. INTRODUCTION

The role of polarons in strontium titanate (SrTiO3) andits isomorphic mixed crystals with Ba-substitution@bariumstrontium titanate~BST!# is of interest in understanding itlow-temperature properties, including the fact that(SrTiO3) has a relatively high superconducting temperat(;0.25 K) in the form of a slightly oxygen-deficientn-typesuperconductor (;n51019cm23), and thatTc does not varymonotonically with n. This has suggested that it is notBardeen–Cooper–Schrieffer system and that bipolaronsplay a role in the superconductivity. The notion that biplarons may be the ground state for superconductivity remcontentious, especially in the case of YBCO.1,2

In addition, the role of oxygen vacancies in ferroelectperovskite oxides is thought to be important in fatigue3–6 andin long-term degradation7–13 ~time-dependent dielectricbreakdown! when these materials are used in thin-film foras switching devices~memories!; the latter effect is equallyimportant in high-dielectric paraelectrics such as B(BaxSr12xTiO3). Earlier studies on lead zirconate titana(PbZrxTi12xO3 or PZT! showed strong infrared absorption14

in the 4000 cm21 region, attributed~incorrectly, as we showlater! to self-trapped polarons at oxygen vacancy sites;no studies were done as functions of vacancy concentraor temperature. In the present article, we do both and sthat these spectra are artifacts. Thus, in the first part ofarticle, we devote considerable space to a careful refutaof polaron interpretations of infrared data on ferroelectfilms. In the second part of the article, we report~nominallyforbidden! second-harmonic generation~SHG! in bothdonor- and acceptor-doped BST as functions of vacancy c

a!Also at: Symetrix Center for Ferroics.b!Electronic mail: [email protected]

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centration and temperature; the SHG intensity dataI (T) sat-isfy the oxygen-vacancy-cluster model of Prosandeevetal.15,16

II. INFRARED DATA

A. Sample preparation and experimental setup

The doped and undoped Ba0.7Sr0.3TiO3 thin films weregrown on 4 in. platinum-coated Si wafers via chemical sotion deposition, using a propionate-based solution. Forther processing details, see Hasenkoxet al.17 The dopantswere added to be compensational: i.e., Mn was added toon Ti sites and La on Ba/Sr sites, taking into accountcharge compensation by cation vacancies. The single-pBST films were;300 nm thick and exhibited a polycrystaline structure with columnar morphology. A BST thin filmsample and several PZT samples from J. Cuchiaro at Sytrix were also studied as comparisons.

The relationship between macroscopic high-temperaelectrical characteristics, oxygen vacancies in undopedacceptor-doped SrTiO3 and BaTiO3 crystals and bulk ceramics has been explained by the point defect chemistry mointroduced by Smyth and co-workers.18 The most detailedsurvey of donor-doped bulk systems was given by MoosHardtl.19 Modifications to the conventional defect chemistmodel have been proposed by Nowotny and Rekas.20 Theimpact of grain boundaries on the defect chemistryperovskite-type titanates has been revealed by Hagenbeetal. as well as M. Vollmannet al.,21 while the phase separation and formation of extended defects on perovskite sfaces has been investigated in detail by Szotet al.22

For titanate thin films, the first comprehensive study hbeen published by Hofmanet al.23 Recently, it has beenfound by Ohly24 that the high-temperature properties of BSthin films differ distinctively from the bulk properties an

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3334 J. Appl. Phys., Vol. 94, No. 5, 1 September 2003 Scott et al.

indicate that the conventional defect chemistry model phaps may not apply to thin films.17 ~Note, that Ohly’s work iscompatible with our x-ray photoemission~XPS! data,25 andit may explain the observations by Shanthi and Sarma, todiscussed in a later section, that each oxygen vacancSrTiO32x contributes not 2.0 electrons to the conductiband but forx.9% contributes only 0.5e.!

The mobility of electronic carriers in alkaline earth titaates has been studied by Choiet al.26 and by Moos andHardtl.27

In the present article, we complement those studiesdetermining the effect of donor/acceptor doping and teperature on oxygen vacancies microscopically through tSHG intensity.

Bruker IFS 113v and Bruker IFS 66v spectrometers wused to record infrared spectra. The 113v spectrometerused for low-temperature measurements, whereas a mscope attached to the 66v spectrometer was used to mespectra at room temperature and at high temperatures.bar lamps, Hg Cd Te detectors and KBr beam splitters wchosen for measurements in between 650 and 6000 c21

and tungsten lamps and CaF2 beam splitters were chosen fomeasurement between 4000 and 8500 cm21. For high-temperature measurements, the samples were heatedAr atmosphere in a heating stage which was coupled wquartz windows and a water-cooling system. Lotemperature spectra were measured with the sammounted inside a closed-cycle Leybold cryostat equipwith KRS5 windows. The substrates~double-polished Si wa-fers! and Au mirrors were used as references. The expmental results showed that the use of the Si wafer ormirror did not lead to significant spectral variations in tnear-infrared region. Single-beam reflectance spectra ofreferences and the samples were recorded with an instrumtal resolution of 4 cm21 in normal ~for the microscope! ornear normal~for the 113v spectrometer! incident condition,and the resultant spectra were calculated wlog10(I sample/I reference), whereI is the single-beam reflectancintensity.

B. General spectral characteristics

1. Line shapes and peak energies expected forpolarons in perovskite oxides

The absorption spectrums(v) for small polarons is ex-pected to be symmetric~unlike asymmetric absorption peakfor large polarons! and approximately Gaussian. Howeveour spectral data~Fig. 1! agree with neither those shapes nfor those calculated for large polarons. The only basisconcluding that the original bulk strontium titanate infrardata were ‘‘small’’ polarons was that the absolute absorbameasured by Baer was too small28 by a factor of35 to fit the‘‘large polaron’’ model of Gurevichet al.,29 even includingthe correction factors of Feynmanet al.30 However, evenReik,31 while concluding that strontium titanate is describby a small polaron model, emphasizes that this materialresents a ‘‘marginal case’’ between large and small. And bSchirmer32 and Kudinovet al.33 stress that the high-energtail has ‘‘prevented complete interpretation of free small p


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laron absorptions.’’ For the polaron coupling coefficient apropriate to BST, Emin calculated34 the absorption positionwidth, and shape. The absorption peak lies nearhv52Wp ,whereWp is defined as the polaron energy; and its half widat half height is typically a few timeshv0 , wherehv0 is theenergy of a characteristic phonon in the material~to be dis-cussed later!. Guettleret al.14 and Reik and Heese35 give theabsorption as a function of frequencyv by a slightly differ-ent approximation:

s~v!5~s0 /v!exp$2~hv2WD24WH!2/~8WHhv0!%,~1a!

where WD is the difference in energy between empty afilled states,v0 is the phonon energy, andWH is the polaronhopping energy, approximately equal to half the polarbinding energyWp , and h is Planck’s constant.~Note thatthis is a three-parameter fit, plus a scale factor.! In Eq. ~1a!,the term 4WH in the numerator is a function of assumedimensionality and, hence, number of nearest neighbors.@3D# tetrahedrally coordinated system~four nearest neigh-bors! Schirmer has shown32 it is 16/3 and in two-dimensionahexagonal arrays that it is six rather than four; it is eightthe Holstein two-well system. Dimensionality effects haalso been considered by others.36,37Alexandrovet al.38 havereplaced the denominator by a termh2 which yields Eq.~1!only in the limit of extremely small polarons, buth is notseparately and independently measurable. And Eminargued39 that the termhv0 in the denominator in Eq.~1! isactually the thermal average of the energy of the phonwith the strongest polaron coupling coefficient, which

FIG. 1. Infrared data at 293 K@fitted to a Eq. 1~a!# for strontium titanatecrystal, showing the strong phonon peak~see Refs. 46 and 47! at1280 cm21; and for BST ceramic film, showing transmission interferenc


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3335J. Appl. Phys., Vol. 94, No. 5, 1 September 2003 Scott et al.

^n(T)11/2&hv0 and is approximated at low temperaturesthe zero-point energyhv0/2. This introduces an importanfactor of approximately 2 in the fitting of data, and so flow- T(kT!hv0), we rewrite Eq.~1a! as

@3D#:s~v!5~s0 /v!exp$2~hv2WD24WH!2/

~8^n11/2&WHhv0!%

5~s0 /v!exp$2~hv2WD24WH!2/

~4WHhv0!%; ~1b!

and in the most general case, the denominator of the enential in Eq.~1b! may be given by the Bose factor of phonon occupancyn11/2& as

8 WHhv0^n11/2&

58WHhv0$1/21@exp~hv0 /kT!21#21%; ~1c!

We emphasize here that the experimental temperature dedence for linewidths in the feature in Fig. 1 do not satiseither Eqs.~1a! or ~1c!; that is, the width is neither temperature independent, nor does it vary as rapidly askT or as theBose factor in Eq.~1c!. This is the first indication that thespectra in Fig. 1, or Ref. 14, are not due to polarons.

For @2D# the formula equivalent to Eq.~1a! is

@2D#:s~v!5~s0 /v!exp$2~hv2WD

26WH!2/~8WHhv0!%, ~1d!

and for

@1D#:s~v!5~s0 /v!exp$2~hv2WD

28WH!2/~8WHhv0!%, ~1e!

for a two-well one-dimensional system. In general, assuma dimensionality lower than@3D# does not improve orworsen the fit to experimental spectra; it merely yieldsdifferent value of polaron energy~half the peak energy, or athird, etc.!.

The spectral shape for absorption from large polaronquite different; it is asymmetric, non-Gaussian, and givabove a thresholdhv53Wp by

s~v!5~s0 /v!~kR!3@11~kR!2#24, ~1f!

wherehk5@2m*( hv23Wp)#1/2. This curve@Fig. 1~b!# hasa very steep slope on the low-frequency side and differs nligibly for @3D# and @2D# large polarons, unlike the smalpolaron case. Note that this formula does not containphonon energy.

Let us digress also to consider what the phononquencyv0 is: The Froehlich interaction between electroand phonons is Coulombic and involves only longitudinoptical ~LO! phonons of the long wavelength. For a cryswith more than two atoms per primitive cell, such as strotium titanate or BST, there will be several LO phonobranches~three in the case of SrTiO3). However, in thisfamily of materials, one branch~with transverse opticabranch near 110 cm21 in SrTiO3 or BST at ambient temperatures! has by far the strongest oscillator strength~polaroncoupling coefficienta54.5). The frequency of the LOphonons in strontium titanate or BST associated with this


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is 815 cm21 ~despite the fact that two weakly infrared-activLO/transverse optical~TO! pairs lie between the energies110 and 815 cm21). Thus, the first question is whetherv0 inEq. ~1! is 117 ~TO! or 815 cm21 ~LO! or some weightedaverage of all branches and all wave vectorsq. The conven-tional polaron theory of Landau or Feynman originally moeled carriers interacting only with long-wavelength Lphonons LO(q50), but beginning with the Holstein modelshort-range electron–lattice interaction was invoked simto the deformation potential in covalent crystals. Eagles40 forrutile and Salje41 for WO32x have used rather highfrequency phonons. Reik31 used^v&5692 cm21 for SrTiO3 ,corresponding in his notation toh^v&b/251.7 where b51/kT. This is the average of the three LO phonon energweighted by the polaron coupling coefficienta for each,from Barker: viz.,a50.2 for v1(LO)5170 cm21; a51.2for v2(LO)5473 cm21; and a54.5 for v3(LO)5773 cm21. But others~e.g., Guettleret al. for PZT! haveinserted very low TO or local-mode energies in these expsions and not LO frequencies. Perhaps the best~most self-consistent! work is that of Calvaniet al.42 who used thelocal-mode frequencies they measured independently, arifrom oxygen vacancies in the same cuprate samples usemeasure the polaron absorption. Almost all early theoretpublications on polarons ignore the LO/TO difference, alowith wave vector dispersion, apparently erroneously thinkthat both are small. Sincea54.5 in SrTiO3 is based on theLO frequency of 815 cm21 anda varies asv0

25/2, this is animportant point. To usea54.5 but a value ofv0 differingfrom 815 cm21 is not self-consistent. In TiO2 , the importantv0 is evaluated asv05v(LO)5807 cm21 ~Eagles!,40 and in«-WO3, Salje finds41 v05560 cm21 for polarons andv0

5742 cm21 for bipolarons, both values approximating thknown oxygen-stretching phonon energy range. We arguthe present article that in Eq.~1! the value ofv0 that wouldbe required to fit data for strontium titanate, BST or PZdoes not correspond to any known optical phonon brav~TO; q50), and that in particular the strongly Froehiccoupled LO phonon at 815 cm21 plays no particular role.Note that the system dimensionality affects the numbernearest neighbors, for the cubic BST structure, with six neest neighbors for@3D# hopping; four for@2D#; two for @1D#;and only one for Holstein’s original two-site model. In facHolstein in his early work43 pointed out a difference of32in the polaron bandwidth for the@3D# model of Yamashitaand Kurosawa,44 compared with his@1D# model, though itwas by no means apparent that this arose from dimensioity arguments. In his original fit of polaron data on strontiutitanate,45 Barker avoided these problems by simply fittindata to a Gaussian whose peak position and linewidth wboth free parameters, subject to no constraints at all. Injudgement, this is not sufficient to establish even thatspectra are due to polarons~as opposed to point defects od2d absorption bands, or midgap states due to oxygencancy clusters!.

None of our least-squares fits to the equations descrherein yield a characteristic phonon energy correspondinknownq50 phonons nor to the one-phonon density of sta


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peak of approximately 120 cm21 ~from the two-phonon Ra-man spectra!.46

The form of Eq.~1! is appropriate when the width of thrigid–ion ~electron band calculated ignoring ion relaxatio!polaron band is greater thanWH ; in this case, Firsov showethat the width of the absorption line is nearly temperatindependent.47 Generally, however, polaron absorptiowidths will depend upon temperature. In the opposite limin which the rigid–ion bandwidth is much less thanWp , thetemperature dependence of the polaron absorption peakwidth is given explicitly, in the present notation, by Bogmolov et al.48 and also by Alexandrovet al.49 as

s~v!5~s0 /v!exp$2~hv22Wp!2/8WpkT%, ~2!

~herek is Boltzmann’s constant! but Eq.~2! is valid accord-ing to Bogomolov only above 600 K~i.e., half the DebyetemperatureQ/2) and, therefore, not useful in fitting ouT-dependent data. This form derives from Eq.~1c!. Note thatin contrast to Eq.~1a! and ~1b! this is only a one-parametefit plus a scale factor; in general, it is impossible to fit eperimental data for most materials to this equation, becaWp determines the peak position and also the linewidththe polaron absorption, which in reality are independent,the experimental polaron width increases much more slothan linearly withT. @Unfortunately, this key Eq.~2! is mis-printed and is dimensionally incorrect in the review by Autin and Mott.#50 No equivalentT-dependent low-temperaturexpression is known to us. Bogomolovet al. also estimatebound polaron energies in TiO2 ~and related compounds witTiO6 octahedra such as BST! as

s~peak!50.7 eV55600 cm21, ~3!

in reasonable agreement with the peak~which we will showto be an artifact! in present experiments, or those of Ref. 1The theoretical estimate of the polaron absorption peakergy from Appel51 is ~present notation!:

WD14WH50.357a2hvLO55900 cm21 ~4!

~wherea is the polaron coupling coefficient andh is Planck’sconstant; note especially thatvLO is not v0) and is also infairly good agreement with the present experiment.~Fig. 1!.@There is an algebraic error in Ref. 45, Eq.~4.11!, p. 213;Appel gives our Eq.~4! incorrectly as, in his notation,J2E050.1395a hv0 rather than J23E050.357a hv0 ,where 2J is the polaron bandwidth andE0,0.]

@We emphasize the agreement between energy estimin Eqs.~3! and~4! above and the spectral features in Fig.or Ref. 14 because these were the important ‘‘red herrinthat produced erroneous interpretations of the data.#

Note that if we usea54.5, we must use52,53 vLO

5815 cm21, since the former was derived by Barker seconsistently from the latter for each LO phonon mode as

a5~2pe2/hvLO2 !~2me vLO!1/2~Z/«`!. ~5!

The high-frequency tail, emphasized in Fig. 1~b!, shouldhave been an early give away that these data are not dupolarons, but unfortunately another possibility has beencussed for SrTiO3 by Reik,31 who suggested two explanations: ~1! That the optical phonons in strontium titanate a


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highly dispersive, and~2! that a nonadiabatic version oEagles’ theory54 would explain the tail. We show that the rereason is that the spectra in Fig. 1 and Ref. 14 are artifdue to interferences in the 300 nm films used, which unftunately give an apparent peak in absorption~actually a dipin transmission! near energies predicted for polarons.

2. Linewidths and self-consistent values ofgÄWp ÕvLO : False clues

It has been of theoretical interest to examine the ratiopolaron energiesWp and phonon frequenciesv0 . For PZT,Guettler et al.14 fitted v05130 cm21, WD50, and 4WH

54080 cm21, whereWp52WH . However, in early theoriesthe phonon which dominates polaron interaction must belong-wavelengthq50 LO phonon, and as just discussethis lies52,53at 815 cm21 in SrTiO3 and in Sr-rich BST.~Thisis very nearly the same value ofv0 as found in WO3 andother perovskites by Salje and Hopman.!55 And so, we havea similar question regarding the ratiog as to whether it isv0

or vLO which entersg. We note that the ratio of parameteg5Wp /v0515.7 for PZT from Guettleret al. fails to satisfy~by 33) all known theoretical models. That is, Ref. 14 is nself-consistent regarding peak energies and linewidths.strontium titanate, if we examine the experimental ratiog5Wp /v0 for the valuea54.5, we must use the valuvLO5815 cm21 used by Barker to derive that value ofa.That is,g has been calculated theoretically as a functiona, and a was calculated as a function ofvLO ; hence, onemust calculateg self-consistently with this value ofvLO .The ratio~neglectingWD) of

g5Wp /vLO52400 cm21/815 cm2153, ~6!

for the feature in Fig. 1 is plausible; this was a second ‘‘rherring’’ in assigning interference features to polarons in povskites such as PZT and strontium titanate. Austin aMott50 have noted that this ratio is generally of orderg55.0 and for some models is 4.0, and Bogomolovet al. findg5Wp /vLO54.0 in TiO2 ; for values ofa54.5, eight dif-ferent theoretical estimates56–63 give Wp /vLO54.5 to 4.9.

Finally, we can estimate the number density of trappelectrons in our BST films from the polaron peak positiousing Barker’s strontium titanate data;46 for a peak near4200 cm21, we estimaten as a few times 1020cm23. This isin accord with other estimates of near-surface vacancy ccentration for ABO3 perovskites.64–67This is another plausi-bility argument that contributed to the misidentification ofspectral artifact in Ref. 14.

3. Vacancy –concentration dependence

a. Of polaron absorption intensities: Another false clu.Our experimental data show very little change in absolabsorption intensity as the number of carriersn is varied viadonor or acceptor doping. At first, this seems unreasonafor polarons, since the intensity should be linearly proptional to n, but as Barker46,53 has pointed out, the imaginarpart of the dielectric susceptibility is given by

«95cn/vLO2 , ~7!


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wherec is a constant, and due to coupling between plasmand LO phonons,vLO

2 increases approximately linearly witn. For v.2000 cm21 Barker finds experimental confirmation of his theoretical prediction that

lim ~n/vLO2 !5n/n5constant asn→`. ~8!

b. Of absorption peak energies. The four expressionsgiven in Eqs. ~1!–~4! all generally agree: 4900 cm21

620%, theory and experiment, for polaron absorption penergy. We note also that there is a simpler relationswhich can be used to estimate the polaron absorption penergy for polarons that are not too small: One-halfFroehlich polaron coupling coefficienta is68 the number ofLO phonons dragged by each electron, and sincephonons involved have energy,52,53 815 cm21 and the po-laron coupling coefficient in SrTiO3 ~and hence, BST! isknown53 to be a54.5 for the infrared mode of strongeoscillator strength, the product of these two numbers giverough estimate of the polaron energyWp as

Wp5a vLO/250.534.53815 cm2151834 cm21, ~9a!

and peak absorption energy at

hv52Wp53668 cm21, ~9b!

in reasonable agreement with experiment14 ~the slightlyhigher experimental value could conceivably involve sobipolaron contribution to the spectra, as discussed for peskite oxides by Salje41 and by Kostur and Allen!.69 Varyingthe oxygen-vacancy concentration of BST by donor dopwith La ~0.7%! or acceptor doping with Mn~0.7%! gavesmall but reproducible changes in the intensity, shape,peak position in the absorption spectra~Fig. 2! near4000 cm21; we know now that these were simply duechanges in index of refraction and hence to the interferewavelength peak in the films studied. The conductivity atV of the La-doped specimens was 1026 A/cm2; for the Mn-doped specimens, 231028 A/cm2; and for the undoped, 331027 A/cm2. In our donor-doped specimen, a 2.5% shiftpeak position from 4150 to 4050 cm21 was observed.@Ashift of the same sign and magnitude in polaron absorppeak frequency with donor or acceptor doping was alsofortunately observed for TiO2 by Bogomolov and Mermin.70

FIG. 2. Near-infrared transmission for 0.7% donor doping~La! and 0.7%acceptor doping~Mn! at 292 K.


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The profile~and width! of the BST peak and that in PZT aralmost identical, which is expected in a simple oxygevacancy polaron model.#71–74

4. Polaron mass m ** in strontium titanate and bariumstrontium titanate

Despite the fact that much of the present article isvoted to explaining why prior data arenot due to polarons,the theoretical considerations we provide about polamass, etc., may prove useful in other contexts.1,2

We note that the effective~band! mass75,76m* in SrTiO3

is ;5.1me and, hence, the polaron mass is56

m** 5m* ~120.0008a2!/~120.167a10.0034a2!

53.1m* 516me , ~10!

whereme is the free electron mass. This value is in excelleaccord with recent values from dielectric measurementsa model.2 ~This value assumesa54.5, which, in turn, wasbased on an LO phonon energy of 815 cm21.) The polaronmassm** in PZT was previously estimated14 as 3m*, usingthe Froehlich approximation, in very close agreement wthe value 3.1m* for BST in Eq. ~10!. ~The strong couplinglimit of Landau and Pekar77 gives an even larger estimate om** 5m* a4/4859 m*; however, SrTiO3 and BST are notin this very-small-polaron limit.! Austin and Mott havepointed out thatm** .10me is sufficient to prevent the for-mation of a cryogenic metallic state50 in SrTiO3 , and wenote that reduced strontium titanate does not always becmetallic; it can remain semiconducting down to mK temperatures~although it becomes superconducting!.78–81 Thisalso supports the numerical value of 16me in Eq. ~10!. Notethat some band structure models of strontium titanate hboth light and heavy conduction electrons. Thus, it is apossible that the mass we calculate is for the heavy electrwhich could form polarons, whereas a metallic conductarises from the light electrons; this is the point of viewGonget al., to be discussed later. The polaron coupling cefficient a54.5 is relatively large in SrTiO3 , producing asmall or intermediate polaron~larger than the radius Kostuand Allen69 calculate for the typical perovskite polaron o50% of the lattice constanta0) in which the Rice–Sneddonmodel82 is more accurate than the Landau–Pekar76 treatmentof the Froehlich Hamiltonian. We note that accordingDevreeseet al., a54.5 is sufficient to form a bipolaronground state in@2D# but not @3D#. Therefore,@2D# localiza-tion of oxygen vacancies may be of paramount importancbipolaron stability in these materials.

Very recently, our polaron mass value of 16 has helpshow that the bipolaron model of Levstiket al.83 for dielec-tric anomalies at 37 K cannot be correct in strontium titanaRef. 83 is now known to be due to artifacts~series resistancein the dielectric bridge!.

5. Temperature dependence

The temperature dependence of our absorption spectshown in Figs. 3 and 4. IncreasingT moves the absorption


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peak to lower frequencies in BST. The shift is rather smThis disagrees with the small polaron case consideredReik ~stronglyT dependent!.

Rather, more important than the temperature shift ofinfrared peak position is the fact that the spectral linewiwidens with increasing temperature. A narrowing of widunusual in solid-state physics, was predicted in the origpolaron theory of Yamashita and Kurosawa,44 who find spec-tral linewidth Dv given by

Dv5c1 exp~2A/kT!, ~11!

FIG. 3. Apparent absorption spectra in BST at elevated temperat(290 K,T,655 K). Traces are displaced along the ordinate axis by atrary amounts for clarity.

FIG. 4. Apparent absorption spectra in BST at cryogenic temperat(18 K,T,245 K); sample from J. Cuchiaro at Symetrix Corp. Ordinateabsorption in arbitrary units.


l.y

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wherec1 is a temperature independent constant andA is acharacteristic energy; this dependence was also emphaby Reik.31 However, our data disagree qualitatively with E~11! and quantitatively with Eq.~2!.

According to standard defect chemistry theory, owould expect that substitution of La13 for Sr12 would resultin oxygen wave function distortion and Ti13 -like 3d statesin the gap, which increase conductivity. This absencealso been observed by Higuchiet al.84,85 for Mn and Nbsubstitutional for Ti in SrTiO3 single crystals and suggesthat much of the La did not substitute for Sr in our BSsamples. It may have accumulated at grain boundaries inform of La2O3. This shows the pitfalls of naively applyingdefect chemistry models to fine grained ceramic films. SHofman et al.23 for a discussion of this point. Finally, wenote that our experiments on strontium bismuth tanta~SBT! thin films revealed no polarons in the infrared spectas with pure SrTiO3

An additional complication to interpreting the dataFig. 1 is that the broad peak lies very near in frequencythat of the characteristic O—H hydroxyl vibrations, which inthe gas phase are always around 3700 cm21. It is known thathydrogen always goes into SrTiO3 as hydroxyl ions, attach-ing to one of the Ti—O bonds to form Ti—OH @we noteparenthetically that it is possible to get 100% hydroxyl ocupation in a stable perovskite,86 e.g., bernalite, Fe(OH)3].We carried out measurements to high~650 K! and low ~20K! temperatures, with the results shown in Fig. 4. By goito low temperatures, we had hoped to see other features,as the disappearance of any polaron peak~it disappears at 10K in rutile!49 or the infrared absorption of sharp-line Ospectra superimposed on the broad peak, but the line3652 and 3756 cm21 are merely surface water. However,hydroxl-structure perovskites such as Fe(OH)3 , the O—Hmode is shifted86 substantially, from 3700 cm21 to lower en-ergies near 3100 cm21, and we tentatively interpret thebroader absorption in Fig. 1 near 3100 cm21 as due to OHclusters. Not shown in our spectra are the lines at lowfrequencies; in addition to seeing the well-known infrareactive phonons of BST, we see the strong Si—O and Ti—Osubstrate lines at;1100 and 600 cm21 respectively, identi-fied by Craciun and Singh,87 in our BST.

6. Relationship to resistivities

It has been claimed, that in rutile, there is an empirirelationship88 between the peak absorption coefficieI (max) due to polarons and the electrical resistivityr(;3 ohm cm) of the samples. Near 300 K, it is given by

I ~max! cm21 r~V cm!55.33102 V, ~12!

and is only 20% less at 700 K. We do not think that trelationship in Eq.~12! can be justified for any wide-bandgap oxide, such as SrTiO3 , BST, or rutile, since none othose materials exhibit ohmic conduction~they are Schottkyor Frenkel–Poole conductors at voltages less than a few vand Fowler–Nordheim above that, except at cryogenic teperatures!. Therefore, one cannotdefinea resistivity in ohmcm for them in order to test Eq.~12!. No relationship similarto Eq. ~12! is expected even if the material were ohmic.

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7. Final disproof of the polaron interpretation

By varying the thickness of the thin-film samples stuied, we find that the broad feature in Fig. 1~and in PZTsamples! varies exactly as predicted for thin-film transmision interference, and its shape corresponds closely topredicted interference shape. In addition, if a wider rangewavelengths is measured~Fig. 5!, additional interferencesare observed. Hence, if one does wish to look for polaronferroelectric thin films, these interferences must be shifout of the wavelength region of interest by carefully seleing film thicknesses. The aforementioned rather protracaccount is designed to prevent other investigators frstudying artifacts. Relatively, few publications in solid-staphysics describe failures: We simply failed to measurelarons in BST or PZT films.

III. SECOND-HARMONIC GENERATION DATA

A. Experimental setup

The intensity of optical frequency-doubled light is usally several orders of magnitude less than that of the funmental wave. For this reason, a high-intensity high-powlaser light source is needed if optical SHG signals are todetected from illuminated crystals. We used aQ-switchedNd–YAG laser with a high peak power and low jitter as tfundamental light source, with a 20 ns pulse width and 2mJ energy pulse energy. This equates to a peak pulse pof around 10 MW, in a beam of 6 mm diameter, or arou3.53107 W cm22. Measurements of SrTiO3 were taken intransmission geometry, with the beam focused to a diamof around 100mm. The beam was attenuated by passingthrough beam splitters in order to avoid sample damage.

FIG. 5. Infrared transmission peak energies vsT in BST film.


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laser was triggered using an external trigger from a concomputer. The sample was held on anx2y stage, positionedvertically with the laser beam passing horizontally throughThe stepper motor controllers for thex2y stage were polledand set using the data acquisition and control computer, sthat the beam could be rastered across the sample.

The light passing through the sample was filtered,remove the fundamental wave, using a high-power interence beam splitter. A narrow band-pass~532 nm! filter wasthen used to exclude background light, and frequendoubled light ~532 nm! was measured using a lowbackground photomultiplier. In addition to the light mesured at the sample, the fundamental beam was split befalling on the sample, and part passed through a second sdard quartz sample, filtered in the same way as that atsample, and passed to a photomultiplier, from which a msure of the incident intensity was made. Thus, the intensitthe sample was measured with respect to the intensityreference, for each shot of the laser. The signal fromphotomultipliers was passed to an analogue-to-digital cverter board on the computer via low-noise preamps, givseveral decades of range to a high sensitivity. Further filare placed along the optic axis of the instrument to exclulight from the laser flash lamp, the furnace, and interfereof frequency-doubled light created at the sample wfrequency-doubled light created at the quartz reference.data acquisition and control software worked through a sply operated graphical interface, and incorporated puchecking for each laser pulse measured. Checks were mto ensure that only pulses for which the reference and sampulse peak maxima coincide and for which the onset ofreference and sample peaks coincident in the time domwere accepted.

A polished ~100!-oriented single crystal of SrTiO3 wasused to measure the spatial dependence of the frequedoubled intensity of light, shown in Fig. 6. Furthetemperature-dependent measurements of a BST thinwere made in reflection geometry with an unfocused beimpinging upon a sample attached to the cold finger oHe-circulating cryostat. Data were collected between 20270 K and the temperature-dependent SHG intensityshown in Fig. 7.

B. Vacancy–concentration dependence

In Prosandeev’s model,15,16 the SHG intensity should beproportional to oxygen-vacancy concentration, with thean-satzthat the vacancies form correlated extended defects; weach oxygen vacancy is associated a local dipole, andcorrelation length describing the orientation of nearbypoles is the parameter of interest. The SHG intensity is ofdescribed as proportional to ‘‘cluster size’’,^S(T)&; how-ever, this term is misleading since^S(T)& refers to the ex-pectation value of the polarization of the vacancy dipo~hence, correlation length!, and is not closely related to thgeometric size of the cluster. In particular,^S(T)& and,hence, the SHG intensity both become vanishingly smalT→`. The SHG intensity at 77 K is expected to be twice


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strong as at ambient and, at 4 K, is expected to be nearlyorder of magnitude larger than at room temperature.

C. Temperature dependence

Prosandeevet al.15,16 also give an expression~Eq. 13!for SHG intensityI (T) as a function of temperature89,90

I ~T!5I 0@Ts coth~Ts /T!2T0#21/21I B , ~13!

whereTs is a saturation temperature~known to be between35 K and 38 K from the temperature at which the dielectsusceptibility flattens out upon cooling!. The only adjustableparameter in Eq.~13! is T0 , which in isomorphic KTaO3 is0.48Ts ;15 so we take T0517 K as a startingvalue in theleast-squares fit of the present work. The constant IB in Eq.(13) is not in Prosandeev’s single-crystal theory but wasadded by us to represent the temperature-independent con-tribution of grain boundaries(and surfaces) in ceramic BSTspecimens.

Saljeet al.91 first introduced this saturation term involving @Ts coth (Ts/T)2T0#

21 in a different context. Kleemannet al.92,93 later used this expression for SrTiO3 :Ca. In thelatter model, the saturation temperature was assumed tsomewhat smaller and given byTs5TE/2, wherekB TE is theenergy of an Einstein oscillator corresponding to the expmentally measured soft optical mode of long-wavelengthq50 (TE534 K in SrTiO3; 23 K in Ca:SrTiO3).

Equation~13! is tested in Fig. 7 for both pure SrTiO3

from Kapphanet al.89 and our BST data, and the single uknown parameter Ts is an adjustable parameter fitted to tdata; the known value in KTaO3 T0517 K ~the saturationtemperature of the dipole pseudospins! agrees with valuesknown independently for SrTiO3 from both dielectric dataand theoretical models.90 Mueller94 has used exactly thesame value ofTs538 K for SrTiO3 as we have, and Dec an

FIG. 6. SHG data~293 K! for a SrTiO3 crystal. Coordinates are real spac(x, y), SHG intensity~z!.


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Kleeman92,93 have argued that althoughT0 might be muchsmaller than 38 K~nearly zero! in pure SrTiO3 , it will be ofthe order of 20 K in SrTiO3 mixed with CaTiO3 ~or perhapsBaTiO3). Thus, all independent sources support our vafor this parameter for BST. The fit is very sensitive to tratio Ts /T0 .

IV. PERCOLATION OF POLARIZATION

Prosandeev’s work15,16suggests that the ordering of oxygen vacancies in ABO3 perovskites should give rise tophase transition which is first order and of a percolationture, rather than displacive~it is not clear that this is the 23231 superlattice@actually 51/2351/231] self-ordering ac-tually observed in SrTiO3). To test this hypothesis, we havfitted the activation enthalpy dataH(x) for oxygen–vacancymotion from Steinsviket al.95,96 The results are shown inFig. 8. The exponent fitting the dependence ofH upon va-

FIG. 7. ~a! SHG intensity vs temperature for strontium titanate reducedhydrogen~diamond symbols! or vacuum~triangles!. Solid curves are our fits@to Eq. ~12!# with Ts538 K and T0537 K ~H reduced! and T0530 K~vacuum reduced! andI B50. ~b! SHG intensity in BST vs temperature~293nm MOD film with no top electrode!. Solid curve is@a fit to Eq.~12! with#Ts538 K, T0525 K, andI B contributing 2.0 units of intensity. The anomalies at 77 K and 273 K are due to liquification of nitrogen and freezingwater on the sample surfaces and are intentionally shown as examplartifacts that can occur.


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cancy concentrationx is g51.7860.10, which agrees withinthe small (65%) experimental uncertainty with the valu7/4 expected for a cubic lattice,97 that is,

H5H0 ~xc2x!g, ~14!

with xc57% ~i.e., SrTiO2.80) taken as the vacancy concetration at which 23231 superlattices self-organize, as evdenced by bright planes in the high-resolution electroncrographs ~HREM!; this adds support to Prosandeevgeneral idea ofpolarzation percolation.

In a different context, it is clear from the Kelvin-probwork of Nowotny and Rekas98 that there exists a great difference between the oxygen–vacancy concentration neasurface and in the bulk of barium titanate on Pt electro~and devices made from related compounds such as BST!.This is manifested in the fact that the work function chanversus oxygen partial pressure varies98 with exponent 2.560.3 for surfaces~Kelvin-probe measurements! but 4.460.4 for bulk. Those authors interpret their results as edence for a ‘‘segregation boundary layer’’ which they sugest may involve adsorbed species, primarily chemisoroxygen. They infer that there is a strong gradient in oxygvacancy concentration in these ABO3 perovskites, as in ouPZT fatigue model.1–3

An important addition to Prosandeev’s model is thesight from Becerroet al.99 ~and earlier work by Grenieretal.!100–102 that the oxygen–vacancy clusters in perovskiare not isotropic but, in fact, develop along@101# directionsand, at higher vacancy concentrations, along planes. Hever, this may be true only in Fe-doped SrTiO3 or CaTiO3

and not in reduced SrTiO3 , because Gonget al.103 find onlyvery short-range ordering in the latter~and a slight prefer-ence for vacancies at one site in the tetragonal phase oduced SrTiO3 ; note that the room-temperature tetragonphase of reduced SrTiO3 is not the same phase as unreducSrTiO3 exhibits below 105 K, but one without a doubled-uncell!. Most recently Marionet al.99 have shown that large(3100) electrical conductivity increases occur at 1300 K7% vacancy-concentration CaTiO3 . The known phase dia

FIG. 8. Graph of activation enthalpy~see Ref. 98! for displacement ofoxygen vacancies in strontium titanate vs vacancy concentration, shopercolation exponent~see Ref. 99! of g57/4 near a critical vacancy concentration of 7%.


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gram shows that this is not due to a displacive structuphase transition; and they conclude that it is a percolatransition from a low-conductivity ordered-vacancy state blow 1300 K to a disordered ionic conductor. Janovec~privatecommunication! has pointed out that the low-energy~un-charged! domain walls in PZT are along@110# directions,parallel to the chains of oxygen vacancies inferredBecerroet al.; this supports our model of domain wall pinning by ordered vacancies.3,5

V. RUTHERFORD BACKSCATTERING DATA

Rutherford backscattering spectroscopy data for Bhave been shown elsewhere104 which yield a peak hydrogenconcentration of 150 ppm~0.15 at.%!; the concentration isnot uniform but is maximum in a layer 147 nm from thferroelectric surface, i.e., midway through the 300 nm thfilm. Concentrations in PZT and SBT are lower and pecloser to the surface. Hydrogen is known not to substitinterstitially in perovskite oxides but to form hydroxl ionsoxygen sites. The strongly detrimental effects of even smamounts (,4%) of H2 on PZT and SBT were detailed verrecently104 and include the fact that in the presence ofPt-electroded SBT partially decomposes into elemental~as reported by Hartmannet al.!104 and PZT partially decom-poses at 823 K to yield 0.3% Pb.

VI. CONCLUSIONS: RELATIONSHIP BETWEENSECOND-HARMONIC GENERATION AND INFRAREDDATA

The present spectroscopic study does not confirmpresence of strong polaron coupling in BST or strontiutitanate films. The assignment and quantitative analysisambiguous even in the early ‘‘classic’’ infrared studiesBarker or Reik on bulk material~Klinger105 has remarkedthat d↔d Ti13 transitions also occur in this same 0.6–0eV energy region, but Bogomolovet al.106 and Austin andMott50 have pointed out that the absorption in TiO2 ~or BST!is 503 too large for that interpretation, on the basis of coparisons between TiO2 and Ti2O3). It had originally beenthought106 that small polarons were much more likelyTiO2 than in SrTiO3 , based upon the anisotropic manvalley band structure model of the latter material by Kaand Leyendecker;107 however, that model is no longeviewed as even qualitatively correct.108,109The conclusion ofAustin and Mott that TiO2 is qualitatively different fromSrTiO3 in this respect is disproved by the uv reflectivispectra of both as obtained by Cardona110 and by Kurtz.111

~The argument of Austin and Mott was based on the assution that there are nod electrons in TiO2 , but that there are instoichiometric SrTiO3 . We emphasize that pure SrTiO3 has ad0 configuration and is an insulator by virtue of its banstructure, not a correlated Mott-type insulator.! We see~Fig1! no polaron absorption whatsoever in pure~unreduced!strontium titanate.

Polarons could, in principle, dominate the electrictransport properties of BST under many conditions of prtical interest. We note however that51 ‘‘the mean free path forelectrons in a polar crystal is determined from the band m

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m* and not the polaron massm**, since the electron–phonon interaction is short range and unscreened.’’ The Sdata and the activation enthalpy data of Steinsvik showoxygen vacancy clustering is important and that Prosaeev’s model of polarization percolation is appropriatethese ferroelectric oxides; we not only confirm Prosandeemodel in quantitative detail for SHG intensity, but we shothat Steinsvik’s activation enthalpies fit the expected 7/4 pcolation exponent. We note that evidence for oxygevacancy self-ordering into planar structures has very recealso been obtained in Bi2Ti4O11 ~which consists of naturalayers of Bi4Ti3O12 and TiO2); 112 in this system, the oxygenvacancies order within the TiO2 planes.

The relationship between the SHG data on oxygen ctering and the infrared data, interpreted in terms of polarois manifested by the recent studies of Crandleset al.113 onhighly reduced strontium titanate. They carried out infrarstudies of SrTiO3 for carrier concentrations from 1018 to1021cm23 and observed that the midinfrared reflectance n4000 cm21 is temperature independent, as we observe hfor BST. Based upon that, they conclude that, contraryearlier interpretations by Barker,46 by Lee et al.,114 and byReik and Heese,115 these arenot polaron spectra. They alsreject the interpretation by Calvaniet al.116 of intervalleycarrier scattering. They conclude that the midinfrared speare probably due to oxygen clustering, which Shanthi aSarma117 have shown to produce midgap states, althouthey also conclude that the plasmon is weaklylocalized.Thus, even in bulk, the interpretation of polaron spectramains moot.

We end this article with mention of a puzzling coincdence: As Shanthi and Sarma discuss, based upon themeasurements of Gonget al.,103 for SrTiO32x each oxygenvacancy contributes only 1.0 conduction electron contrarythe 2e expected from defect chemistry models, forx,962%; and forx.9% contributes only 0.5e. ~By compari-son, substituting La for Sr contributes exactly 1.0e, as ex-pected from defect chemistry.!118 This unexplained thresholdat which the defect chemistry changes qualitatively havalue remarkably similar to the levelx5762% at whichSteinsvik’s data show self-ordering of oxygen vacanciThis unexpected coincidence has not been noted previoin literature and has a suggested explanation.119 Perhaps oxy-gen vacancies above;8% each contribute 0.5 electron tthe conduction band plus 1.5 that remain trapped as alaron at the vacancy site. The hypothesis of Gonget al. isthat this ‘‘filling factor’’ of ;0.5e per oxygen vacancy arisefrom the light and heavy electron conduction bands in strtium titanate, and that only;25% of the electrons liberateby oxygen vacancies wind up in the light electron banhence, the other 75% contribute negligibly to conductioncause of their very heavy mass. A relevant model with decalized states that become localized by impurities has bdiscussed for oxides by Alexandrovet al.120

We emphasize that oxygen vacancy clustering in thmaterials is a surface or near-surface phenomenon, not aproperty.121 The kinetics of oxygen vacancies and stoichioetry changes at surfaces in SrTiO3 have been analyzed verrecently by Maieret al.121 Thus, despite a number of caref


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studies of oxygen–vacancy behavior in strontiutitanate,103,121–123further work will be required on thicknesdependence. Work by Maglione124 uses polarons to explainthe low-temperature dielectric and ultrasonic loanomalies125 in strontium titanate. Based upon earlier studof relaxation,126 he provides a ‘‘quantum polaron’’ theorconnecting polarons and oxygen–vacancy clustering.though no explicit connection to Prosandeev’s model is pvided, this may be made through the earlier work of Vumeister and Glinchuk.127

Notable is the fact that Maglione invokes polaron copling with the lowest frequencytransverseoptical phonon~TO!—not the LO. This is the ‘‘soft mode.’’128 But Magli-one’s model of dielectric loss at cryogenic temperaturesfers from that of Scott129 or of Kityk et al.130 Other veryrecent studies of self-trapped electrons in perovskites inclthe theoretical models of Kotominet al.131,132which explainthe green luminescence in KTaO3.

Recent results from a theoretical investigation by Donerberg and Birkholz133 show unambiguously that the electronic ground state of oxygen vacancies V0

1 in BaTiO3 is ofeg symmetry, as predicted by Eglitiset al.,134 whereas theelectron paramagnetic resonance spectra of Ti13, 135,136 aswell, show that it ist2g . This implies that the EPR of Ti13

polarons trapped near oxygen vacancies do not recomwith the oxygen-vacancy ground state—an unexpectedunexplained wrinkle which merits additional study. Perhaof even greater importance, Schirmeret al. ~in press! haveshown that BaTiO3 exhibits a mixture of small Andersonlocalized polarons and intermediate polarons is detected

We emphasize in closing that, in addition to artifacts thcan arise from optical studies of thin films, intrinsic diffeences should also be present between polarons in thin fiand in bulk. As emphasized by Xiet al.137 and Petzeltetal.,138 the defect density and strain in films of pure strontiutitanate reveal very different soft-mode behavior in films adifferent roles for oxygen vacancies. These differencshould be studied further via film spectroscopy, now thatpitfalls are understood.

ACKNOWLEDGMENTS

Work at Cambridge was supported by EPSRC. Thethors thank J. Cuchiaro at Symetrix and R. Waser at Aacfor some BST and PZT samples, A. J. Hartmann forunpublished XPS data on La-doped strontium titanate,D. Emin, E. K. H. Salje, O. F. Schirmer, and W. R. Datars fhelpful discussions.

1B. K. Chakraverty, J. Ranninger, and D. Feinberg, Phys. Rev. Lett.81, 433~1998!.

2A. Alexandrov, Phys. Rev. Lett.82, 2620~1999!.3M. Dawber and J. F. Scott, Appl. Phys. Lett.76, 1060 ~2000!; ibid. 76,3655 ~2000!; ibid. 76, 3801~2000!.

4U. Robels, J. H. Calderwood, and G. Arlt, J. Appl. Phys.77, 4002~1995!.5J. F. Scott and B. Pouligny, J. Appl. Phys.64, 1547~1988!.6J. F. Scottet al., Integr. Ferroelectr.4, 85 ~1994!.7R. Waser, T. Baiatu, and K. Hardtl, J. Am. Ceram. Soc.73, 1645~1990!.8R. Waser, T. Baiatu, and K. Hardtl, J. Am. Ceram. Soc.73, 1654~1990!.9R. Waser, T. Baiatu, and K. Hardtl, J. Am. Ceram. Soc.73, 1663~1990!.

10R. Plumlee, Sandia National Laboratories Report No. SC-RR-67-~1967, unpublished!.


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11R. Gerson and T. C. Marshall, J. Appl. Phys.30, 1650~1959!.12J. F. Scott,Science and Technology of Electroceramic Thin Films, edited

by R. Waser and O. Auciello~Kluwer, Dordrecht, 1995!, p. 249.13H. M. Duiker and P. D. Beale, Phys. Rev. B41, 490 ~1990!.14B. Guettler, U. Bismayer, P. Groves, and E. Salje, Semicond. Sci. Tech

10, 245 ~1995!.15S. A. Prosandeev, Sov. Phys. JETP83, 747~1996!; A percolation model in

this context was also discussed by H. Qian and L. A. Bursill, Int. J. MPhys. B10, 2027~1996!

16S. A. Prosandeev, V. S. Vikhnin, and S. Kapphan, Integr. Ferroelectr.32,1047 ~2001!; J. Phys.: Condens. Matter14, 4407~2002!.

17U. Hasenkox, S. Hoffmann, and R. Waser, J. Sol-Gel Sci. Tech.12, 67~1998!; S. Hoffmann and R. Waser, J. Eur. Ceram. Soc.19, 1339~1999!.

18~a! N.-H. Chan, R. K. Sharma, and D. M. Symth, J. Am. Ceram. Soc.64,556 ~1981!; D. M. Smyth, Annu. Rev. Mater. Sci. 329~1985!, and refer-ences cited therein.~b! A different interpretation is given by F. D. Morri-son, D. C. Sinclair, and A. R. West, J. Am. Ceram. Soc.84, 474 ~2001!.

19R. Moos and K. H. Ha¨rdtl, J. Am. Ceram. Soc.80, 2549 ~1997!, andreferences cited therein.

20J. Nowotny and M. Rekas, Ceram. Int.20, 213 ~1994!; ibid. 20, 217~1994!; ibid. 20, 225 ~1994!; Ceram. Int.20, 237 ~1994!; ibid. 20, 251~1994!; ibid. 20, 257 ~1994!; ibid. 20, 265 ~1994!.

21M. Vollmann, R. Hagenbeck, and R. Waser, J. Am. Ceram. Soc.80, 2301~1997!; R. Hagenbeck and R. Waser, Acta Mater.48, 797 ~2000!, andreferences cited therein.

22K. Szot and W. Speier, Phys. Rev. B60, 5909~1999!, and references citedtherein; M. Wierschem, T. Lindemann, R. Pankrath, and S. E. KappFerroelectrics264, 315 ~2001!.

23W. Hofman, S. Hoffmann, and R. Waser, Thin Solid Films305, 66 ~1997!.24C. Ohly, S. Hoffmann-Eifert, K. Szot, and R. Waser, Integr. Ferroele

33, 363 ~2001!; J. Eur. Ceram. Soc.21, 1673~2001!.25M. Dawber, J. F. Scott, and A. J. Hartmann, J. Eur. Ceram. Soc.21, 1633

~2001!.26G. M. Choi, H. L. Tuller, and D. Goldschmidt, Phys. Rev. B34, 6972

~1986!.27R. Moos, S. Schoellhammer, and K. H. Ha¨rdtl, Appl. Phys. A: Mater. Sci.

Process.65, 291 ~1997!.28W. S. Baer, Phys. Rev.144, 734 ~1966!.29V. L. Gurevich, I. G. Lang, and Y. A. Firsov, Sov. Phys. Solid State4, 918

~1962!.30R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M. Platzman, Ph

Rev.127, 1004~1962!.31H. G. Reik, Z. Phys.203, 346 ~1967!.32O. F. Schirmer, Z. Phys.B24, 235 ~1976!.33E. K. Kudinov, D. N. Mirlin, and Y. A. Firsov, Sov. Phys. Solid State11,

2257 ~1970!.34D. Emin, Polarons and Bipolarons in High-Tc Superconductors and Re

lated Materials, edited by E. Salje, A. Alexandrov, and W. Liang~Cam-bridge University Press, Cambridge, UK, 1995!, pp. 80–109.

35J. P. Falck, A. Levy, M. A. Kastner, and R. J. Birgeneau, Phys. Rev. B48,4043 ~1993!.

36V. V. Kabanov and O. Y. Mashtakov, Phys. Rev. B7, 6060~1993!.37D. Emin, Phys. Today35, 34 ~1982!.38A. S. Alexandrov, V. V. Kabanov, and D. K. Ray, Phys. Rev. B49, 9915

~1994!.39D. Emin, Phys. Rev. B48, 13691~1993!.40D. M. Eagles, J. Phys. Chem. Solids25, 1243~1964!.41E. K. H. Salje,Polarons and Bipolarons in High-Tc Superconductors and

Related Materials,edited by E. Saljeet al. ~Cambridge Univ. Press, Cambridge, UK. p. 110.

42P. Calvaniet al., Ibid., p. 110.43T. Holstein, Ann. Phys.8, 343 ~1959!.44V. Cataudella, G. De Filippis, and G. Iadonisi, Phys. Rev. B62, 1496

~2000!; J. Yamashita and T. Kurosawa, Phys. Chem. Solids5, 34 ~1958!.45A. S. Barker,Optical Properties and Electronic Structure of Metals an

Alloys, edited by F. Abeles~North–Holland, Amsterdam, 1966!, p. 452.46R. A. Cowley, Phys. Rev.134, A981 ~1964!; J. G. Skinner and W. G.

Nilsen, J. Chem. Phys.48, 2240~1968!.47Y. A. Firsov, Sov. Phys. Solid State10, 1950~1968!.48V. N. Bogomolov, E. K. Kudinov, and Y. A. Firsov, Sov. Phys. Solid Sta

9, 2502~1968!.49A. S. Alexandrov, V. V. Kabanov, and D. K. Ray, J. Am. Ceram. Soc.80,

218 ~1997!.50I. G. Austin and N. F. Mott, Adv. Phys.18, 41 ~1969!; note that the


ol.

.

n,

r.

s.

Austin–Mott model gives a polaron peak athv52Ep, whereas that of P.Gerthsen, E. Kauer, and H. G. Reik, 1966Festkorperprobleme, edited byF. Sauer Braunschweig, 1966!, p. 1, gives a peak athv54Ep, whereEp isthe polaron energy!; see O. F. Schirmer and E. Salje, J. Phys. C13, L1067~1980! on this point.

51~a! J. Appel,Solid State Physics, edited by F. Seitz, D. Turnbull, and HEhrenreich~Academic, New York, 1968!, p. 204; for discussion on whyn* and notm** dominates electrical transport, see~b! J. F. Scott, Ferro-electrics232, 25 ~1999!.

52C. H. Perry, J. H. Fertel, and T. F. McNelly, J. Chem. Phys.47, 1619~1967!.

53A. S. Barker, Phys. Rev.145, 391 ~1966!.54D. M. Eagles, Phys. Rev.145, 645 ~1966!.55E. K. H. Salje and G. Hopman, Philos. Mag. B43, 105 ~1981!.56H. Froehlich, H. Pelzer, and S. Ziemau, Philos. Mag.41, 221 ~1951!.57M. Gurari, Philos. Mag.44, 329 ~1953!.58E. Haga, Prog. Theor. Phys.~Kyoto! 11, 449 ~1954!.59T. Lee, F. Low, and D. Pines, Phys. Rev.90, 297 ~1953!.60T. Lee and D. Pines, Phys. Rev.92, 883 ~1955!.61E. P. Gross, Phys. Rev.100, 1571~1955!.62R. P. Feynman, Phys. Rev.97, 660 ~1955!.63T. D. Schultz, Phys. Rev.116, 526 ~1959!.64J. F. Scott,Ferroelectric Memories, Advanced Microelectronics Vol. 3

~Springer, Heidelberg, 2000!, p. 92.65Z. Wu and M. Sayer,Proc. ISAF~IEEE, New York, 1992!, p. 244.66T. Miharaet al., Integr. Ferroelectr.1, 269 ~1992!.67T. Miharaet al., Nikkei Electron.581, 94 ~1993!.68H. Froehlich, H. Pelzer, and S. Zienau, Philos. Mag.41, 221 ~1950!.69V. N. Kostur and P. B. Allen, Phys. Rev. B56, 3105~1997!.70V. N. Bogomolov and D. N. Mirlin, Phys. Status Solidi27, 443 ~1968!.71E. K. H. Salje, B. Wruck, and S. Marais, Ferroelectrics124, 185 ~1991!.72O. F. Schirmer and E. Salje, Solid State Commun.33, 333 ~1980!.73E. Salje and B. Guettler, Philos. Mag. B50, 602 ~1984!.74B. Guettler, E. Salje, and S. Ghose, Phys. Chem. Miner.16, 606 ~1989!.75F. C. Brown, I. G. Austen, and N. F. Mott, Adv. Phys.18, 41 ~1969!.76C. Kittel, Introduction to Solid State Physics, 5th ed.~Wiley, New York,

1976!, p. 312.77L. D. Landau and S. I. Pekar, Zh. Eksp. Teor. Fiz.18, 419 ~1948!; V. V.

Kabanov and D. K. Ray, Phys. Rev. B52, 13021~1995!.78M. L. Cohen,Superconductivity, edited by R. D. Parks,~Marcel Dekker,

New York, 1969!, Vol. 2, pp. 615-663; note that the occurrence of supconductivity at;0.3 K is interpreted as a superconducting semiconducand not as evidence of a metal, although many authors maintainSrTiO3 becomes metallic forn.1019 cm23.

79C. S. Koonce, M. L. Cohen, J. F. Schooley, W. R. Hosler, and E.Pfeiffer, Phys. Rev.163, 380 ~1967!.

80H. Suzuki, H. Bando, Y. Ootuka, I. H. Innoue, T. Yamamoto, K. Takhashi, and Y. Nishihara, J. Phys. Soc. Jpn.65, 1529~1996!.

81A. Fujimori et al. Phys. Rev. B46, 9841~1992!.82T. M. Rice and L. Sneddon, Phys. Rev. Lett.47, 689 ~1981!.83A. Levstik et al.Appl. Phys. Lett.81, 4046~2002!; Erratum~in press!.84R. Moos, A. Gnudi, and K. H. Hardtl, J. Appl. Phys.78, 5042~1995!.85T. Higuchi et al., Phys. Rev. B61, 12860~2000!.86C. A. McCammon, A. Pring, H. Keppler, and T. Sharp, Phys. Che

Miner. 22, 11 ~1995!.87V. Craciun and R. K. Singh, Appl. Phys. Lett.76, 1932~2000!.88O. A. Dubovskii and Y. V. Konobeev, Fiz. Tverd. Tela~S.-Peterburg! 7,

946 ~1965!.89H. Granicher, Helv. Phys. Acta29, 211~1956!; W. Prusseit-Elffroth and F.

Schwabl, Appl. Phys. A: Solids Surf.51, 361 ~1990!.90K. Tani, Phys. Lett.25A, 400 ~1967!.91E. K. H. Salje, B. Wruck, and H. Thomas, Z. Phys. B: Condens. Matter82,

399 ~1991!.92J. Dec and W. Kleeman, Solid State Commun.106, 695 ~1998!.93W. Kleemann, J. Dec, and B. Westwanski, Phys. Rev. B58, 8985~1998!.94K. A. Mueller, Jpn. J. Appl. Phys., Part 124, 89 ~1985!.95S. Steinsvik, Ph.D. thesis, Oslo, 1999.96S. Steinsviket al. J. Phys. Chem. Solids58, 969 ~1997!.97R. Balescu,Statistical Dynamics~Imperial College Press, London, 1997!,

p. 238.98J. Nowotny and M. Rekas, Ceram. Int.20, 251 ~1994!.99A. I. Becerro, C. McCammon, F. Lagenhorst, F. Seifert, and R. Ang


https://www.researchgate.net/publication/200030749_Introduction_To_Solid_State_Physics?el=1_x_8&enrichId=rgreq-f4c7037f-7d4a-4f56-b837-76ba0c432da5&enrichSource=Y292ZXJQYWdlOzIzNDkxODk4MjtBUzo5NzMwMTMxMzg4NDE3MkAxNDAwMjA5ODUwMzI5

https://www.researchgate.net/publication/200030749_Introduction_To_Solid_State_Physics?el=1_x_8&enrichId=rgreq-f4c7037f-7d4a-4f56-b837-76ba0c432da5&enrichSource=Y292ZXJQYWdlOzIzNDkxODk4MjtBUzo5NzMwMTMxMzg4NDE3MkAxNDAwMjA5ODUwMzI5

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e

ge

on

V

thplaac

ov

e,

ell

ol

.

ett.

vs-

J.

.

en-

J.

.


Phase Transitions69, 133~1999!; S. Marion, A. I. Becerro, and T. NordbyIonics 5, 385 ~1999!.

100J. C. Grenier, J. Durier, M. Pouchard, and P. Hagenmuller, Mater. RBull. 11, 1219~1976!.

101J. C. Grenier, G. Schiffmacher, P. Caro, M. Pouchard, and P. Hamuller, J. Solid State Chem.20, 365 ~1977!.

102J. C. Grenier, M. Pouchard, and P. Hagenmuller, Struct. Bonding~Berlin!47, 1 ~1981!; these results differ qualitatively from those of Ref. 103reduced but undoped SrTiO3.

103W. Gong, H. Yun, Y. B. Ning, J. E. Greedan, W. R. Datars, and C.Stager, J. Solid State Chem.90, 320 ~1991!; actually the vacancy sitepreference is rather large—~ W. R. Datars @private communication~2001!# reports 12% vacancy in the a–b plane and 10% vacancy ina–c plane. Since the a–b plane has twice as many sites as the a–cthis difference is quite significant for total numbers of vacancies in eplane.

104A. J. Hartmann et al. J. Korean Phys. Soc.32, S1329~1998!; Y. Shi-makawa and Y. Kubo, Appl. Phys. Lett.77, 2590~2000!; ibid. 75, 2839~1999!.

105M. I. Klinger, Phys. Status Solidi27, 479 ~1968!.106V. N. Bogomolov, E. K. Kudinov, S. T. Pavlov, and L. S. Sochawa, S

Phys. Solid State10, 2043~1968!.107A. H. Kahn and A. J. Leyendecker, Phys. Rev.135A, 1321~1964!.108L. F. Mattheiss, Phys. Rev. B6, 4718~1972!.109R. D. King-Smith and D. Vanderbilt, Phys. Rev. B49, 5828~1994!.110M. Cardona, Phys. Rev.140, A651 ~1965!.111S. K. Kurtz, Proc. Int. Meeting Ferroelec., edited by V. Dvorak, A.

Fouskova, and P. Glogar~Publishing House, Czech. Acad. Sci., Pragu1966!, Vol. 1, p. 413.

112A. Q. Jiang, Z. H. Chen, Y. L. Zhou, and G. Z. Yang, Adv. Mater.~Wein-heim, Ger.! ~to be published!.

113D. A. Crandles, B. Nicholas, C. Dreher, C. C. Holmes, A. W. McConnB. P. Clayman, W. H. Gong, and J. E. Greedan, Phys. Rev. B59, 12842~1999!

114C. Lee, J. Yahia, and J. L. Brebner, Phys. Rev. B3, 2525~1971!.115H. G. Reik and D. Heese, J. Phys. Chem. Solidi28, 581 ~1967!116P. Calvani, M. Capizzi, F. Donato, S. Lupi, P. Maselli, and D. Peschiar

Phys. Rev. B47, 8917~1993!.


s.

n-

.

ene,

h

.

,

i,

117N. Shanthi and D. D. Sarma, Phys. Rev. B57, 2153~1998!.118Y. Tokuraet al., Phys. Rev. Lett.70, 2126~1993!.119J. F. Scott,Proceedings of the Conference on Ferroelectrics~St. Charles

University, Marseilles, France, 27 September 2000!.120A. S. Alexandrov, A. M. Bratkovsky, and N. F. Mott, Phys. Rev. Lett.72,

1734 ~1992!.121Y. Aiura et al., Advances in Superconductivity, edited by T. Fujita and Y.

Shiohara~Springer, Tokyo, 1994!, Vol. 6; J. Maier, J. Jamnik, and MLeonhardt, Solid State Ionics129, 25 ~2000!.

122D. D. Sarma, S. R. Barman, H. Kajueter, and G. Kotliar, Europhys. L36, 307 ~1996!.

123Y. Haruyamaet al., Jpn. J. Appl. Phys., Part 132, 543 ~1993!.124M. Maglione,Defects and Surface-induced Effects in Advanced Pero

kites, edited by G. Borstelet al. ~Kluwer, Amsterdam, 2000!, p. 27.125J. F. Scott and H. Ledbetter, Z. Phys. B: Condens. Matter104, 635

~1997!.126M. Maglione, Ferroelectrics176, 1 ~1996!.127B. E. Vugmeister and M. D. Glinchuk, Sov. Phys. JETP52, 482 ~1980!;

Rev. Mod. Phys.62, 993 ~1990!.128P. A. Fleury, J. F. Scott, and J. M. Worlock, Phys. Rev. Lett.21, 16

~1968!.129J. F. Scott, J. Phys.: Condens. Matter11, 8149~1999!.130A. V. Kityk, W. Schranz, P. Sondergeld, D. Havlik, E. K. H. Salje, and

F. Scott, Phys. Rev. B61, 946 ~2000!.131E. A. Kotomin, R. I. Eglitis, and G. Borstel, J. Phys.: Condens. Matter12,

L557 ~2000!.132E. A. Kotomin, R. I. Eglitis, A. V. Postnikov, G. Borstel, and N. E

Christensen, Phys. Rev. B60, 1 ~1999!.133H. Donnerberg and A. Birkholz, J. Phys.: Condens. Matter12, 8239

~2000!.134R. I. Eglitis et al., Phys. Rev. B56, 8599~1997!.135R. Scharfschwerdt, A. Mazur, O. F. Schirmer, H. Hesse, and S. M

dricks, Phys. Rev. B54, 15284~1996!.136S Koehne, O. F. Schirmer, H. Hesse, T. W. Kool, and V. Vikhnin,

Supercond.12, 193 ~1999!.137A. A. Sirenko, C. Bernhard, A. Golnik, A. M. Clark, J. Hao, W. Si, and X

X. Xi, Nature ~London! 404, 373 ~2000!.138T. Ostapchuket al., Phys. Rev. B66, 235406~2002!.


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Infrared spectra and second-harmonic generation in barium strontium titanate and lead zirconate-titanate thin films: “Polaron” artifacts