PHYSICAL REVIE% B VOLUME 15, NUMBER 9

Phonon self-energy in Lira03 and LiNb03

1 MA Y 1977

Franqois GervaisCentre de Recherches sur la Physique des Hautes Temperatures, Centre National de la Recherche Scientifique, 45045 Orleans, France

Jean-Louis ServoinU.ER. de Sciences Fondamentales et Appliquees de 1'Universite d'Orleans and Centre de Recherches sur la Physique des

Hautes Temperatures, Centre National de la Recherche Scientifique, 45045 Orleans, France

(Received 28 October 1976)

Infrared reflection spectra (extraordinary ray) of lithium niobate and lithium tantalate in the temperature

range 300—1300 K exhibit a feature which cannot be described correctly with an additional classical oscillator

and which is understood as a phonon self-energy efFect. Comparison is made with room-temperature Raman

data where two additional peaks in the same spectral region have been assigned to a fundamental polar

phonon mode activated by a distortion of the structure.

In a recent work, Penna and co-workers' re-ported the observation of a number of peaks inthe A, -type Raman spectrum of lithium tantalateat room temperature and this number is greaterthan expected from a group-theory analysis basedon recent determination of the crystal structure,viz. , four A, modes. Since their total number ofmodes is consistent with that predicted by an anal-ysis involving the group C„Penna et al. ' assignedthe nine peaks to fundamental phonons and assertedthat the structure of LiTaQ, is slightly distortedso that the symmetry group would be C, insteadof C,„. Among the five additional modes, threeare found above 300 cm ' near 460, 660, and 750cm '. A, -type modes in LiTaQ, are polar, henceinfrared active. The present paper reports theanalysis of the extraordinary reflectivity spectrumin the range 275-1000 cm ' which also exhibitsfeatures —but not necessarily "peaks, " as will beemphasized —at two of these three frequencies,viz. , -450 and 660 cm '. Results at high tempera-ture in LiTaO, and the isomorphous compoundLiNbO, indicate the occurrence of same effect andwill be also reported. Due to the fortuitous sen-sitivity of the ref lectivity technique to the dis-persion found near 460 cm ', we will concentrateon this point. Penna et al. report the existenceof a polar phonon mode the TQ and LQ frequenciesof which are 458 and 465 cm ', respectively, witha damping of 13 cm '.

The reflectivity spectrum of LiTaO, at roomtemperature is shown in Fig. 1. Our results agreewell with those of Barker et al. ' in the range 400-1000 cm '. Concerning the band 200-400 cm ',the reflectivity found at the top is a few percenthigher than that reported in Ref. 2. Results athigher temperature, but still in the ferroelectricphase, are shown in Figs. 2 and 3 for LiTaQ, andLiNbO, . Barker and co-workers' were able to

fit their room-temperature reflection spectrumwith the usual dielectric function model expressedas a sum of classical oscillator contributions

A.JTO

TO —~ +&& TO+

We reported recently' that attempts to fit LiNbO,2, spectra with the model equation (1) are worseand worse with the increase of the temperatureat which the spectrum was recorded, while the fitis rather correct at room temperature as shownpreviously by Barker and Loudon. ' The fact that

LiTa03- A) 300 K

)0.2-

ULU

0.1—

~l ~ ~~ ~ ~

~ 0 ~

I I

&00 Cto

FREQUENCY (cm ")

FIG. 1. Comparison of experimental reflectivity datawith curves calculated with (a) the four-parameter modelequation (2), (b) the same model with the additionaloscillator reported by Penna et al. (Ref. 1), (c) with astrongly damped polar mode at 460 cm (TO-LO splitt-ing higher than that of Penna et al . (Ref. 1) with a damp-ing - 100 cm ), (d) with phonon self-energy terms ofthe form shown in Fig. 4 acting on the TO mode near600 cm ' [Eq. (4)]. All calculated curves are mergedapart from the region of the minimum and fit the experi-mental data satisfactorily as shown in the inset.

4532

I'HONON SELF-ENERG Y IN LiTa03 AND LiNb03 4538

I—

Q2-C3LU

LLLLI

LiTa03 A) 690 K

0.1—

300 500 voo gooaI I I I I

400 440 480

FREQUENCY (crn " )

FIG. 2. Same as legend to Fig. 1.

not only dampings are considered as constant butalso same damping for TO and LO modes is im-plicitly assumed in the case of a single reflectionband as emphasized in Ref. 5, constitutes a weak-ness of the classical dispersion theory equation (1).Same TO and LQ dampings may be a good ap-proximation when the reflection band is narrowand then when nearly same phonon decays areexpected but there is no reason for an identitybetween yTp and yLp when the LO energy is farabove the TO one, that is when the reflection bandis wide. Both infrared and Raman results"'indicate differences between TO and LQ dampingsof adjacent modes in these compounds. For thesereasons, me tried to fit reflectivity data with theaid of the factorized form of the dielectric func-tion"

the data are higher than the calculated curve.Since the width near the minimum reaches at least30 cm ', a lack of resolution cannot be invokedto explain the discrepancy. The situation is re-versed with respect to that reported by Barkerand Hopfield" concerning BaTiQ„SrTiO„andKTaO, where the experimental data are below thecurve calculated with Eq. (1), at a similar re-flectivity minimum. As a result, any attempt tocouple modes according to their theory yieldsworse fits. ' Actually the failure of both modelsequations (1) and (2) is not surprising becausea Kramers-Kronig analysis reveals an asymmetri-cal line shape for both TQ near 600-cm ' and LQnear 400-cm ' phonon modes. This asymmetry ismore or less pronounced, depending on compoundand temperature (results for LiNbO, in Ref. 3).Equations (1) and (2), which imply constant damp-ings, are thus not expected to correctly describeasymmetrical phonon lines that may reflect fre-quency -dependent damping functions.

We have calculated the frequencyAependent reald~(&u) and imaginary I'(~) parts of the phonon self-energy of the TQ mode near 600 cm '. AKramers-Kronig analysis has been combined withEq. (2), and constant TO and LO frequencies anddampings which yield a good description of thereflection spectrum outside the range of the min-imum at 420 cm '. The result is shown as dotsin Fig. 4 for LiNbO, at 1175 K, for example. Apeak is clearly visible in the y function at -460cm '. Based on Green's-function approach' Mara-dudin and Fein" have established the form of thelowest-order phonon self-energy. Let us rewritetheir result in the form

Lp (d + zy&gp400

Gap —(d + 'LygTp40(2)

This model has several advantages which have beenthe subject of recentpapers, ' which we mill merelyoutline. The TO and LO frequencies are approxi-mately positioned at the highest-slope points ofthe low- and high-frequency edges of a reflectionband, respectively. A preliminary dressing tothe data fit is thus readily achieved as soon asTO and LO frequencies are positioned. DampingsyTp and y~p impose the slopes at these edges,respectively. The number of runs used to obtainagreement with the data turns out to be muchless than that when Eq. (1) is used. In addition,fits are generally better near LO frequencieswithout inclusion of additional secondary oscilla-tors which often palliate the weakness of the clas-sical dielectric function. Fits of the model equa-tion (2) to the LiTaO, and LiNbO, data are satis-factory at any frequency and temperature apartfrom the reflectivity minimum near 420 cm ' where

Q3-

I)o 02-IJJ

LLLUlZ

0.1-

BXP .

3oo 5oo 7oo

440 480

FREQUENCY (cm-' )

FIG. 3. Same as legend to Fig. 1.

FRANQOIS GERVAIS AND JEAN-LOUIS SERVOIN 15

d~, (~)+ fz', ((g) = Q Ii'"'(0 k, s» -kva) I'1/112

(S&+R2+ 1

X(d —97j —QP2 —lI

2(n, -n, )+

(d+ CO

(3)

liNb03- A) )175 K3

-80-- 100--120~

oooo c

3I50

50- TO RESONANCE

I

300 500 700

FREQUENCY (cm-~)

FIG. 4. Experi. mental phonon self-energy terms(dots) of the 572-cm ' TO mode. Full curves areself-energy functions calculated with the model Eq. {4)which are found to fit the experimental reflection spec-trum as shown in Fig. 3.

where symbols have their standard meaning and

where a finite linewidth I' hasbeen conferred to theoriginal 5 functions. If an intense two-phonon peakexists at the frequency w„„=&, ~ ~„and if thephonon self-energy functions are assumed to beflat up to the limits of influence of the mode 6japart from this peak, an oversimplification ofE|1. (3}yields

&(u(&u)+ —,'iy((u) = &(u, + ,'iy, +g'/((u ——(u„„ iI'), (4)

where g has the dimension of a frequency. Equa-tion (4) is fully consistent with the experimentalresult shown in Fig. 4.

First we have attempted to explain the high levelof reflectivity in the minimum at 420 cm"' as aneffect of the additional phonon reported by Pennaet al. '

(QTo = 458, 0 = 465 cm ') although no se-quence "hump and hole" is visible in the reflectionspectrum at 460 cm '. The result is shown inFig. 1. The conclusion is that such a polar phononmode with a splitting of 7 cm ' and a dampingof 13 cm ' as reported by Penna should be clearlyobserved in the reflection spectrum, contraryto experiment. If this oscillator is considerablydamped then the ref lectivity level fall again belowthe data. If one increases the TQ-LQ splittingup to -13 cm ' and if one keeps a high damping(typically 100 cm '), the correct reflectivity level

is then reached but one "sees" a dip in the cal-culated reflectivity which is not observed experi-mentally and thus one does not fit the data cor-rectly. This situation is found more or lessmarked in both lithium niobate and tantalate atany temperature. Conversely, a fit with the fac-torized form of the dieiectric function Eg. (2)where the phonon self-energy of the TQ mode near600 cm ' alone is described with the model Eq.(4} whereas all other TO- and LO-phonon modesare described with constant parameters, allowsthe best agreement with experimental data asshown in Figs. 1-3. The form of the 4&v(&o) and

y(co) functions used is shown in Fig. 4, for exam-ple. The structure of both calculated curves isclearly consistent with the result deduced fromthe experiment. Minor discrepancies are due toinaccuracies of the Kramers-Kronig analysis inthe present conditions together with the simplifica-tions inherent to both models Egs. (2) and (4).

We would like to emphasize that the actual posi-tion of the TQ phonons whose frequencies arebelow 275 cm ' and which are not observed pre-sently (that is essentially the most polar TO modenear 200 cm ') is found to play a negligible rolein the calculation of the spectrum. This is an ad-ditional advantage of the model Eq. (2). The re-flectivity indeed appears essentially determinedby the A o and yTo parameters in the iow-fre-quency half of a band, while it is the LO param-eters that play the essential part in the high-fre-quency half, hence the usefulness of the four-parameter model is derived. Concerning thelowest-frequency TQ mode, any downshift of thefrequency is accompanied by an increase of theoscillator strength as long as LQ frequenciesare fixed. Both effects approximately counter-balance each other so that the contribution of thisTQ mode to the dielectric function remains nearlya constant at frequencies above 300 cm '. John-ston and Kaminow" reported that the 200-cm ' TQmode softens with increasing temperature downto the transition temperature at -890 K. Pennaand co-workers" claimed that they did not ob-serve any softening of this mode although theirown measurements show a considerable decreaseof the asymptote of the lowest-frequency A, polar-iton from 200 cm ' down to 45 cm ' where it seemsto saturate. The softening (although perhaps notdown to zero) of the asymptote of the polariton,which is the position of the A, TQ phonon itself,is confirmed by other authors. '4 But independentof this controversy, it appears that some con-fidence can be attached to the parameters obtainedin the present analysis according to the argumentsabove. To reinforce this assertion, let us notethat the frequencies determined presently agree

15 PHON ON SE LF-EN ERG Y IN LiTaO3 AN D Li N 103 4535

TABLE I. Comparison of Raman data of Ref. 1 with

the TO and LO frequencies associated with the main

infrared bands observed presently at room temperaturein LiTa03.

Penna et al. (Ref. i)This work

LO

40i403

TO

594594

LO

866863

within 0.5% with those given by Penna et al. ' asshown in Table I.

It should be pointed out that the consideration offrequency-dependent parameters for the 594-cm 'TO phonon, thus accounting for its asymmetri-cal line shape, also confers a reverse asymmetri-cal shape to the adjacent 403 cm ' LO phonon, con-sistent with the experiment. ' This is readily un-derstood by inspection of Eq. (11) in Ref. 5. y»and pz o dampings indeed are nothing more thanthe damping function evaluated at the Q«and+o frequencies, notwithstanding the values thatboth «u(~) and y(&u) functions take between both

frequencies, especially in the vicinity of a peakin the two-phonon density of states in the presentcase.

Several tries have shown that the dip near 660cm ' may be described indifferently either withan additional oscillator as done by Barker et al. 'or by the model equation (4). But if the first de-scription is choosen, then a TO-LO splitting ofat least 5 cm ' associated with a damping of -60cm ' is necessary to account for the dip in thereflectivity. Raman data indicate no TO-LO split-ting higher than 1 cm ' if any, and the same peakappears in both A, and E spectra (the same dipis observed in the E infrared spectrum as well).Consequently, it seems to us that a descr;ptionof the dip based on the phonon self-energy mech-anism also should be preferred.

Our results in the range 400-600 cm ' seem tohave several important consequences. First, no

dip assignable to a polar mode is found in the re-flectivity near 460 cm ', whereas the data are con-sistent with a peak in the two-phonon density ofstates which contributes to the reflectivity viaan anharmonic mechanism involving the frequen-cy-dependent phonon self -energy of the neighborTO mode and which consequently creates asym-metric phonon lines. This situation prevails inboth LiTaO, and LiNbO, compounds at any tem-perature in the range 300-1300 K. The dependenceof the phonon self-energy on temperature, es-pecially in the vicinity of the phase transition willbe presented in a subsequent paper. T'hus ratherlarge anharmonicities are found in both crystals.One may then wonder whether the additional struc-ture observed in the Raman spectrum' actuallycould be two-phonon peaks instead of fundamen-tal phonons activated by a lowering of symmetry.

'The second point concerns infrared ref lectivityspectroscopy. It is not physically clear, especial-ly in the transparent regime, "whether a hump inthe several-phonon density of states is observedin the infrared absorption as a polar nth-ordermode with a finite oscillator strength or via phononself-energy terms of the fundamental modes. Anexample where the reflectivity technique is sen-sitive enough to conclude in favor of the secondmechanism is presently found. Ab initio calcula-tions limited to single-polar-mode crystals alsosupport the physics of the phonon self-energy bycomparison with experimental infrared spectra. "

ACKNOW'LEDGMENTS

Numerical calculations have been performed atthe Centre Interuniversitaire de Calail de la RegionCentre, Orleans. Discussions with Dr. B. Piriouand P. Alain are gratefully acknowledged.

'A. F. Penna, A. Chaves, P. da R. Andrade, and S. P. S.Porto, Phys. Rev. B 13, 4907 (1976).

2A. S. Barker, A. A. Ballman, and J. A. Ditzenberger,Phys. Rev. B 2, 4233 (1970).

J. L. Servoin and F. Gervais, High Temp. High Press-ure (to be published).

A. S. Barker and R. Loudon, Phys. Rev. 158, 433(1967}.

F. Gervais and B. Piriou, J. Phys. C 7, 2374 (1974) ~

N. Nippus, Z. Naturforsch. 31A, 231 (1976).Inspection of the calculation procedure of Barker andLoudon in Ref. 4 shows that they determined the imag-inary part of the zeroes of the dielectric function thatthey mistook for the linewidth at half-height of the LOpeak in their comparison with Raman results, whereasactually this is 2yLo, consistent with Raman linewidths.

D. W. Berreman and F. C. Unterwald, Phys. Rev. 174,791 (1968).

~F. Gervais and B. Piriou, Phys. Rev. B 10, 1642(1974); 11, 3944 (1975) ~

' A. S. Barker and J.J. Hopfield, Phys. Rev. 135,A1732 (1964).

' A. A. Maradudin and A. E. Fein, Phys. Rev. 128,2589 (1962).W. D. Johnston and I. P. Kaminow, Phys. Rev. 168,1045 (1968).

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FRANQOIS GERVAIS AND JEAN-LOUIS SERVOIN 15

72 (1971);M. Rokni, L. S. Wall, E. Amzallag, and T. S.Chang, Solid State Commun. 10, 103 (1972).

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