Apparent soft mode linewidth divergence in PbTiO3

Solid State Communications,

Vol. 10, pp. 311-314, 1972.

Pergamon Press.

Printed in Great Britain

A P P A R E N T SOFT MODE LINEWIDTH DIVERGENCE IN PbTiOs B.D. Silverman IBM Research Laboratory, San J o s e , California, 95114

(Received 12 October 1971 by A.A. Maradudin)

The apparent soft mode linewidth divergence observed in Raman scattering from PbTiOs in the vicinity of the transition temperature is explained by an enhancement of scattering processes that are allowed due to finite acoustic phonon lifetimes.

quency can result in closer tuning of this frequency to the separation of the T A - L A branches and therefore to greater soft mode damping. The resulting frequency dependent damping constant also provides an absorption peak centered at zero frequency. Such a central peak, resulting from a frequency dependent damping constant, has been previously d i s c u s s e d in connection with thermal relaxation and Brillouin scattering in liquids. 4

AN A P P A R E N T divergence of the soft mode damping constant has been recently observed in PbTiO3 as the temperature approaches the transition temperature a s s o c i a t e d with the ferro- to paraelectric phase transition. 1 The purpose of the present paper is to point out that this behavior could arise from a frequency dependent damping constant that reflects the enhancement of scattering p r o c e s s e s involving finite acoustic phonon lifetimes. As the soft mode frequency approaches values that are comparable to transverse acoustic-longitudinal acoustic phonon splittings in the crystal, there is an increase in soft mode scattering involving the annihilation of a t r a n s v e r s e acoustic phonon with the creation of a longitudinal acoustic phonon. Such explanation requires consideration of finite phonon lifetimes since, over the range of measurement, the soft mode frequency is greater than the lifetime broadened separation between the transverse and longitudinal acoustic branches, l The proposed explanation is similar to the interpretation of the observed temperature dependence of the ultrasonic attenuation of a longitudinal sound wave in quartz, s since the damping proposed for both c a s e s would vanish if the acoustic branches are not assumed to be lifetime broadened. For the ultrasonic c a s e , however, the increase in attenuation with temperature is essentially attributed to a d e c r e a s e in thermal phonon lifetime with increasing temperature. For the present c a s e , the temperature variation of the soft mode fre-

The contribution to the susceptibility from the soft mode can be written X,(~.,) -,. [~.,~ - ~

+ 2~_o(A(~ ) - i r ( ~ ) ) ] - '

(1) where A(aJ) and F(aJ) are the real and imaginary s e l f energy contributions to the soft mode frequency, a~o, and arise from the interaction of the soft mode with all other phonons in the crystal. We have assumed that the lowest fourth order frequency independent contribution to A(c~) has already been incorporated into a; o and that ~=o "~ (To

-

T).

(2)

In what follows, attention will be focussed on F(w), the imaginary contribution to the soft mode phonon s e l f energy since this d e s c r i b e s the damping of the soft mode. The lowest order contribution to F(w) from scattering p r o c e s s e s involving three phonons can be written

311

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LINEWIDTH DIVERGENCE IN PbTiOa

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k o k, k 2

(/11- /12)

~

~2--(Tc-T)

; T
lo ./, J~ ("JL k = CLk

[ (co~

F - ~o, - co) ~ + r ' "

-

1"=

( c ~ - cot + co) = +

]

F2/

C'°Tk = CTk o

k

J

(3)

For small deviations from harmonicity one expects this lowest order scattering to be predominantly responsible for the damping of the soft mode. To allow for effects resulting from finite phonon lifetimes, the Lorentzian function with a finite width has been used in equation (3) in place of the 8 function specifying energy conservation. r is an average spectral width or the reciprocal of an average phonon lifetime. In effect, therefore, all p r o c e s s e s that satisfy pseudomomentum conservation are allowed to a greater or l e s s e r extent, dependent upon the lifetimes of the phonons involved as well as the energy exchanged during the process. Since the structure is pseudocubic and becomes more nearly cubic as the transition temperature is approached, we will assume that all scattering p r o c e s s e s not allowed by symmetry in the cubic phase make a negligible contribution to the damping of the soft mode in the tetragonal phase. We therefore consider the proc e s s illustrated in Fig. 1. An isotropic dispersion relation of the following form will be assumed for the soft mode optical branch (.D~ =

~0,2

+

a2 k 2

(4)

The acoustic branches will be assumed isotropic and dispersionless. As shown in Fig. 1, the proc e s s we consider is one for which the soft mode phonon of wave vector k0 c o a l e s c e s with a transv e r s e acoustic phonon of wave vector k t with the creation of a longitudinal acoustic phono/1 of wave vector kz. The pho/1on energy transfer for the process can be written (oo~ + a 2k2o3/2 + c T k ,

-* c L (k2o + k~ +

2k ok~ c o s 0 ) t'2 . (s)

O is the angle between the propagation vector of the soft mode and phonon 1. For sufficiently high soft mode phonon frequency, there is an energy e x c e s s for the process, the energy of the two annihilated phono/1s being greater than the energy

~'0{O) + "~"l(T) : ~'2(L)

o(0)

~/

FIG. 1. Three phonon scattering p r o c e s s e s . of the created phonon. As the temperature approaches the transition temperature, T c , the soft mode frequency d e c r e a s e s and hence the energy e x c e s s decreases. At a temperature sufficiently close to Tc such that the soft mode frequency just becomes equal to the maximum separation between the transverse and longitudinal branch, only collinear scattering p r o c e s s e s will be allowed for infinite phonon lifetime. For finite r , however, scattering p r o c e s s e s are allowed in a cone about the forward direction. Performing the integration over angle in equation (3) leads to k 2 fl"

F(co) "~- f dk

koc L

t "~ In

(co- Ack - kocr) 2 + F 2 (co- Ack + kocr) 2 + F 2

2Fko c r } + (co + Cr k)tan-t ( ( c ~ - A c k ) 2 - k~c~r + F 2) with Ac = CL - c r .

(6)

This expression clearly illustrates a number of significant points. First, as the photon lifetime, "r, becomes infinite (F -. 0), the damping goes to zero since the process does not conserve energy. If, however ( c o - A c k t ) 2 - c2L k~ = 0, there is a finite contribution to r(co) for infinite ~r, since energy conserving p r o c e s s e s are possible If ( ( c o - Ack~) 2 - c~ ko)''~ T << 1 for any wave vector in the zone, then all acoustic phonons will participate in damping the soft mode. If we let the soft mode wave vector, ko, go to zero, and assume that the mode is u/1derdamped

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LINEWIDTH D I V E R G E N C E

we can write I"(~) "~"

IN PbTiOs |

"

dk ( ~ -2k~F A¢k) 2

The imaginary part of the susceptibility divided by frequency has been plotted in Fig. 2. To obtain this, the T A - L A splitting has been neglected compared with s o b mode frequency. F, the spectral broadening has been retained in the denominator of the integrand appearing in equation (7). It is s e e n that as the temperature approaches the transition temperature, the soft mode absorption decreases in frequency and broadens as we have discussed. It is to be noted that there is a 'central peak' or an absorption peaked about zero frequency. A similar absorption has been previously discussed in connection

.......

Absorption 8s I Function Of Frequerc¥ --PMImetrized in Teml[m'~nu~

(7)

Since the absorption is then relatively sharply peaked, one can replace (~ by the value of the frequency at the line center, a~o. For values of the soft mode frequency that are significantly larger than the maximum T A - L A splitting one finds that r(c~) -,. F/aJ~. The soft mode damping constant will then appear to diverge as the reciprocal of the square of the soft mode frequency as T -, To. Divergent behavior has recently been observed by Raman scattering measurements in the tetragonal phase of PbTiOs.1 It is therefore proposed that this behavior can be understood as resulting from an increase in allowed scattering processes as the soft mode frequency decreases. Performing the integral over wave vector as indicated in equation (7) does not remove the apparent divergence for soft mode frequencies greater but approaching the maximum T A - L A splitting. The detailed temperature dependence of the damping constant for frequencies comparable to the T A LA splitting will, however, be given by performing the integral over wave vector for a model which treats the dispersion of the branches and coupling to the soft mode in a more realistic manner. For such frequencies one would expect the mode to become strongly damped, the damping constant no longer exhibiting divergent behavior. The ir~crease in damping constant at T -, Tc has been observed to temperatures such that the ratio of the soft mode frequency to maximum T A - L A splitting at the zone edge for phonons polarized along the spontaneous polarization direction is approximately equal to 3/2.

313

T I >T2>T3 >'1"4

l

FIG. 2. X " / ~ vs. a~. with light scattering in liquids 4 as well as for noncentrosymmetric ferroelectric materials, s It is interesting to note that critical scattering apparently plays no role in the proposed explanation. It has been assumed that the phonons responsible for soft mode damping do not reflect the onset of the transition, i.e., their frequencies and lifetimes are assumed to be temperature independent as T -, To. Soft mode critical scattering from modes adjacent on the optical branch for which these assumptions are not valid will introduce additional temperature dependent terms into the expression for the damping constant as T-, To. There are other possible soft mode scattering processes besides the ones discussed. Those involving an acoustic and optic phonon will be detuned as T -, Tc, the soft mode frequency decreasing. Others involving only optical phonons will have their strengths reduced due to reduced phonon occupation probability. The purpose of the present note is however not to discuss the relative importance of the various scattering proc e s s e s , but to point out that one of the lowest order processes can indeed result in the apparent divergence of the soft mode damping constant as

T-,To. Acknowledgements - I would like to thank G. Burns for bringing this problem to my attention and for many stimulating discussions. I am indebted to E. P y t t e for constructive criticism and to P. Fleury and S. Shapiro fc~ pointing out the relevance of the central peak discussed by Mountain to the present work.

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LINEWIDTH DIVERGENCE IN PbTi03

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REFERENCES 1.

BURNS G. and SCOTT B.A., Phys. Rev. Lett. 25, 167 (1970).

2.

SHIRANE G., AXE J.D., HARADA J., and REMEIKA J.P., Phys. Rev. B2, 155 (1970).

3.

KAWASAKIK., Prog. theor. Phys. (Japan) 26, 795 (1961). SILVERMAN B.D., Prog. theor. Phys. (Japan) 39, 245 (1968) and references therein.

4.

MOUNTAIN R.D., J. Res. Natl. Bur. Std. 70A, 207 (1966).

5.

COWLEYR.A., Proc. Second Int. Meeting on Ferroelectrici~, 1969, J. Phys. Soc. (Japan) 28,239 (1970).

Die in Raman Streuung an PbTiO3 auftretende Divergenz der 'soft mode' Linienbreite wird mit dem Einsatz hydrodynamischer Streuung erklart.