Infrared damping divergence in quartz

Sohd State Commumcatrons, Vol. 17, pp. 861-865,

1975

Pergamon Press

Pruned m Great Brrtam

INFRARED DAMPING DIVERGENCE IN QUARTZ F Gervars and B Prnou Centre de Recherches sur la Physique des Hautes Temperatures, C N R.S., 45045 Orleans, France and D Billard U&R

de Sciences Fondamentales et Apphquees de l’Umversru5 d’Orldans, France (Recewed 23 May 1975 by A A Mmadudm)

The profiles of the temperature dependence of TO and LO AZ-symmetry mode dampmg m quartz are calculated by takmg mto account the actual effect of temperature on the one-phonon densrty of states The mcrease of the populatron factors of the soft phonons when then frequency falls down to low levels creates, for all other modes, a dampmg drvergence whrch agrees wrth that observed expenmentally in the vicmrty of the a+ phase transrtton

The purpose of thrs paper 1s to show that dampmg of any mode can drverge at the approach of a phase transitron within the framework of the exrstmg nncroscopic theory of weakly anharmomc crystals,**’

IF THE LIFETIlvIE of a phonon mode 1sassumed to be hrmted by anhannomc three-phonon couplmg only (cubic potent@, then srmple perturbatron theory’ and more elaborated treatments based on Green function method**’ predict a hnear dependence of dampmg on temperature above the Debye temperature and a temperature-mdependent dampmg m the hrmt T + 0. These expectations have been venfied m a number of compounds, for example m A1203 over a wrde range of frequencies and temperature 4v6 On the other hand, the hnewrdth of certam soft modes 1sknown to depart from a hnear behavror above the Debye temperature m such a way that even the mcorporatron of hrgherorder anharmomc contnbutrons cannot account for the results Such a drvergence of the soft mode hnewrdth 1s observed at the approach of a structural phase transrtron from below m compounds such as PbTrOa ,6 Gd2(Mo04)s,’ cu-Si023 or Pbr,B&TiOs @Besrdes, recent results” have drawn attentron on the damping drvergence of several other optic modes of quartz m the vrcmrty of the (Y-P phase transrtron whereas hnewrdths have been understood in term of cubic anhannomcity only at low temperatures l1

lf

(i) There exrsts soft phonon mode(s) and (or) soft phonon branch(es); (h) The consrdered mode can couple to the soft mode(s) through anharmomc cubic potentral wrth sufficient probabrlity and strength The reason for the drvergence 1sbeheved to be the considerable mcrease of the soft phonon population n(0,)

= [exp (M&J)

- I]”

(1)

when therr frequency S2, falls down to zero or merely to suffidently low level. Tlus softening provrdes a contniutron to dampmg winch mcreases raprdly at the approach of the transrtron temperature, m additron to the usual contnbutron due to three-phonon couphng of other phonon modes, which depends hnearly on temperature above the Debye temperature

861

INFRARED DAMPING DIVERGENCE IN QUARTZ

862

1 ANHARMONIC MODEL In spite of the increase of the dampmg constant yl at the transrtron temperature T,, the ratio y,/sl, remams lower than 7% for all Infrared m6des m quartz so that perturbatron theory may be used to evaluate the dependence of dampmg on temperature The lowestorder rmagmary part of the phonon self-energy of an optical mode (k = 0) ISof the form2*3 rJ@)

=

$$&

Iv'3)(oJ, kill,

--kh)iz

112

x

{[n(or)

+ 2[n(w*)

+

n(w2)

+

116(w- 01 - 4

- n(wz)lS(w + a1 -

02)). (2)

The first part of equation (2) corresponds to summatron processes (w = w1 + w2) and the second part to difference processes (W = w2 - o t ). Damprng F,(w) evaluated at the resonance frequency Sz, is related to the mfrared lmewrdth rf by [29FW/ol

w -52j = rj

(3) In the present case of quartz, the frequency shrfts are beheved to be essentially due to quastharmomc effects,1°*12 that IS the rearrangement of the atom equrhbnum posrtrons at the approach of the transrtron temperature Thus, purely anharmonic contnbutrons to the frequencies are likely weak wrth respect to the slufts due to pure volume effect and thrs mdmates that postive and negative anhamomc contnbutrons counterbalance each other. Terms are renormahzed to account for rearrangement effect so that the harmonic frequencies or m equation (2) are replaced by the quasrharmonrc frequencies thus assumed to be nearly equal to the observed frequencres In all what follows, the factor two of equatron (3) ISmcorporated mto F, (a,) and consequently no drstinctlon between 7, and I’, (a,) wrll be made m thts paper Due to the structure of quartz, the calculatrons of F, (!Jj) would be of great complexrty Some srmphfymg assumptions wrll be made to facrhtate computatrons The k-dependence of the phonon frequencres wrll not be taken mto account Thrs approxrmatron is fully lustrfied in quartz because most optrcal phonon branches are nearly flats so that phonon modes tend to reduce to Emstern oscrllators Anyhow, the neglect of wave-vector conservatron can be consrdered as a reasonable approximation 111crystals

Vol 17,No 7

havmg a large number of phonon branches Then after renormahzatron of the frequencies of the phonons which couple to the mode J, damprng IS rewntten 111 the form

- nG&+ WI 1

(4)

where &S&J ISthe one-phonon density of states, and S&,the sti arguments and consequently p(X2J are temperature-dependent A ngourous treatment would require accountmg for a possrble temperature dependence of the Vt3)(O~, kr ~~ - kr Jo) potentials 111 equation (2) due to atom rearrangement effects But we feel that the dependence of dampmg on temperature essentially follows from the thermal population factors and thus neglect the former effect, consrstent wrth the sun of the quantum treatment whrch yields equation (2) The strength of the anharmomc couplmg 1s contamed m the constant factor I;(w,,O). When the Vt3)(01, krJr - kr J2) potentials are extracted from the sum over I, they are tmphcrtely assumed to have the same magnitude for all possible combmatrons of phonon J, and ~2 with the phonon J The constant Fs(o#) serves as an adlustable parameter to fit the data. Inspectron of equation (4) shows that the profile rj(fi,, 2YFj(c9, 0) 1scalculated w&out any flttrng parameter AN approxunate one-phonon densrty of states has been constructed, rt IS a composite of the optrcal data u)~14neutron datas-16 avrnlable at low frequent and ke structure of the Bnlloum zone calculated by Elcombe u The temperature dependence of a large amount of data is known at the cntrcal pomts I’ and M Most optical frequencies are nearly temperaturemdependent, vrz: la,(O) - S&(1000 K)l/!+(O) < 5%, wlule certam other phonon modes or branches are strongly dependent on temperature (1) The A r-symmetry soft mode Thts mode 1s decoupled from the two-phonon state17 for the calculatron of the one-phonon density of states and consequently its frequency is taken to vary as s2, = A(To’- T)‘j (n) The two lowest “acoustic” phonon branches m the [@O] direction 15*16

(m) The two TO lowest-frequency modes of A2 symmetry ‘*

Vol 17,No 7

INFRARED DAMPING DIVERGENCE IN QUARTZ

I

863

I

910K 60 c

TO - 1071 cm-l

*

I

P I

LO - 1229 cm-f

300 K

0t 0

500

_

1000

200

1500

600 TEMPERATURE (K)

l

w (cm-l)

FIG. 1 The approxunate one-phonon density of states used m the present calculations. The hatched part 1s due to temperature-dependent phonons The nonhatched part corresponds to phonons that are considered as mdependent of temperature

FIG. 3 Temperature dependence of calculated (full curves) and expenmental (symbols) mfrared mode dampmg. The dotted curve 1s the contribution of difference processes Thrs contributron is obviously found to be zero for the 1229 cm-’ mode which 1s the most energetic of the crystal I

30-

I

TO - 777 cm-l

LO- 790 cm-l I

TO- 495 cm-l LO- 551 cm-l

d

I

20-

P

i

4

:

Pj-

lo-

,

I

4 r’&

: o-

_I----

30-

TO -

363 cm-’

LO -

387 cm-l

8 01

c

20-

lC200

600 TEMPERATURE (K)

1,

la00

FIG 2 A compmson of observed mode dampmg (circles are TO, tnangles are LO modes) wrth profiles (full curves) calculated wrth the ald of equation (4) Dotted curves are the calculated contnbutrons of dtfference processes The approxnnate one-phonon dennty of states we have used 1s shown 111Rg 1, at two temperatures correspondmg to cxand 0 phases of quartz

o-

200

600 TEMPERATURE (K)

T, -

II lo

FIG 4 A comparison of observed mode dampmg (circles) with calculated profiles (dotted curves are difference processes, full curves are the sum of both summation and difference contributions) No slgmficant devlatlon of neither experunental nor calculated TO and LO dampmg 1s found owmg to the proxumty of their energy levels

864

INFRARED DAMPING DIVERGENCE IN QUARTZ 2. RESULTS

2 1 crphase Dampmg of TO and LO As modes calculated with thrs model and deduced from mfrared reflectrvrty measurements1o are compared m Ftgs 2,3 and 4 as a functron of temperature. The vtbratronal modes are labelled with the atd of then frequency at room temperature. The profres obtamed wrth the present simple model agree farrly well wrth the expenmental damping divergences, m the a phase In Fig. 2, dampmg of two modes separated by only 55 cm-’ drsplay rather dtfferent experimental behavtors. Thrs is just what the present model yrelds smce the calculated profile of the TO mode damping IS found to drverge sensrtrvely more than that of the LO mode thus stnkmgly mdrcatmg quantrtative agreement between theory and expenment The contriiution of hfference processes IS plotted separately m Rgs. 2,3 and 4 Thrs contnbuuon 1s obvtously zero for the 1229 cm-’ mode which IS the most energetic among all modes. Difference processes are found to be much less important than summatron processes, apart from the 1071 cm-’ mode The dampmg divergence of the latter mode 1s only caused by dtfference processes owmg to the exrstence of a gap m the regmn 850-1050 cm-’ where no phonon branch takes place (Fig. 1) so that no anharmomc couplmg with soft phonons through additive processes can occur Tlus gap added to a very low dens@ of phonon states m the range 600-680 cm-’ also explams the low contribution of difference processes to the dampmg of low- and mtermedrate-frequency modes Besides, the average of the thermal population factors of the phonons wluch mteract wtth mode J vra drfference processes is smaller than that of summation processes 2 2 Pphase The 363 and 777 cm-’ modes are of B2 symmetry m the 0 phase Consequently they lose then polar character and are no longer observed m an mfrared expenment 18 The A, mode wluch softens m the a phase is of B, symmetry m Pquartz Neutron measurements= mdrcate that Its frequency mcreases according to the form s2,’ = a(T- To) Unfortunately, the pnnaple of the measurements did not provtde the magnitude of a An estimate of tlus coefficient can be

Vol. 17, No 7

made from the calculated mode frequency,ls nonetheless. Then the one-phonon densrty of states 1sknown smce, apart from the soft mode &scussed hereon, the mode frequencies wluch are temperature-dependent m the a phase, are found to be practically constant m the /3phase “*M The dampmg profiles of the 55 1,107 1 and 1229 cm-’ modes are found to fit the data satisfactorily m the temperature range 300-1000 K mchrdmg both phases, and mamly to account for the peak in the vrcmity of the transition temperature (Figs. 2 and 3) Concemmg the 495 cm-’ mode, the parameter I’/(oj, 0) found m aquartz should be reduced by 20% m the /3phase to obtam a correct adjustment of the profile to the data Tlus corresponds to a renormahzatron of the magmtude of the cubtc potential from one phase to another It is worth notmg that the frequency of thts mode m the 0 phase 1sreduced by 13% with respect to its value at T = 0, that 1sthe largest shift among the eight modes That there 1sno need of renormahzation of the anharmomc constants for the three other modes gtves even more support to the present model 3 CONCLUSION A simple calculation shows that the divergence of the dampmg profiles of TO and LO AZ modes m the vrcmrty of the a-p phase transrtron of quartz can be described wtthm the framework of exrstmg anharmomc theones by only takmg mto account the actual dependence of the one-phonon density of states on temperature It 1s thus not necessary to mvoke any mechamsm other than what is present m usual crystals,3m particular exphcit incorporation of the order parameter 1snot required It 1s to be emphasrzed that contrary to soft modes, for example the E-symmetry soft phonon m PbTiOs, whose frequency rapidly moves towards zero at the approach of the phase transltron,6 the frequency of most of the A 2 modes studied presently rs nearly temperature-mdependent or only weakly dependent on temperature, while that of the 363 and 495 cm-’ modes do not change by more than 13% between room temperature and 1000 K Herman et al. 2o consider that the two-phonon density of states m F’bTr03 1s temperature-independent, and explam the damping divergence of the soft E mode by an approach towards a two-phonon peak (so that a hrgh dampmg is expected at the crossmg), when the

Vol 17,No 7

INFRARED DAMPING DIVERGENCE IN QUARTZ

865

m a quantrty whose mcrease wrth temperature is much more rapid than a lmear one, as long as p($ * &) keeps a sufficrent value.

phonon frequency moves towards zero with temperature or eqtnvalently when the polanton frequency decreases. The present srtuatron is somewhat drfferent smce A2 modes do not move as do soft phonon modes. Due to the movement of soft branches mto the very-lowiequency regon at the approach of the phase tranntron emperature, the one-phonon density of states we&ted y the Bose-Emstem factor (n(!&) - kBT/MZ, (when SZ2, Q kBT) mcreases rapidly wrth the decrease of the rft phonon frequency, at constant temperature) thus creases consrderably m this regron. Consequently the mbmed two-phonon dens&es of states p(&&(S2, * SL,)weighted by thermal populatron terms, that essentrally reduce to kBT/iXl,, contribute to dampmg

The profiles of the E-symmetry mode dampmg calculated on the same bases fit the data obtamed by mfrared reflectrvrty equally well Careful measurements of the temperature dependence of the Raman hnewrdths are bemg performed to complete the present results

Acknowledgement - Numerical computatrons have been performed at the G R.I., Orlt%ns

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