Mat. Res. Bull. Vol. 6, pp. 923-930, 1971. Pergamon Press, Inc. Printed in the United States.

POWDER RAMAN SPECTRA: APPLICATION TO DISpLACIVE FERROELECTRICS

Gerald Burns IBM Thomas J. Watson Research Center, Yorktown Heights, N. Y. 10598

(Received July 6, 1971)

ABSTRACT Powder Raman measurements have recently been used to study the lattice vibrational modes of ferroelectrics. This technique is discussed and reviewed. The technique should be applicable to a number of other areas of research.

Introduction

The discovery of ferroelectricity in BaTiO 3 by von Hippel and co-workers

(i) and a group in the U.S.S.R. (2) ushered in a new era in the field of ferro-

electricity. Previous to this work two types of ferroelectric crystals were

known: Rochelle salt (KNaC4H406"4H20) and KDP (KH2P04) and isomorphous mater-

ials (3). This small number of crystals led workers to consider the ferro-

electricity a curiosity. However, BaTi03, with the perovskite crystal structure,

and its large number of isomorphic compounds showed that ferroelectricity could

not only exist in many materials, it could also exist in relatively simple

materials.

These realizations led to considerable further work. Devonshire (4) in

1949 showed how the various macroscopic properties of these perovskite materials

could be related to one another in a phenomenological theory. In 1950, Slater

(5) showed why the perovskite crystal structure should be favorable for ferro-

electricity. This was done by showing that there is a very large internal field

correction (Lorentz correction) for Ti-O-Ti... linear chains. In 1959, Cochran

* Partially supported by the Army Research Office, Durham, N. C.

923

924 DISPLAGIVE FERROELEGTRICS Vol. 6, No. I0

(6) pointed out the connection between ferroelectricity and lattice dynamics.

This was done via the Lyddane-Sachs-Teller relationship,

3 LO i g_o= , (1)

g~ i=l \~TO.-/ 1

where go is the low frequency clamped dielectric constant, g is the optical

dielectric constant (the square of the index of refraction), and ~L0 and ~TO

are the frequencies of the longitudinal and transverse optic vibrational modes

of the lattice, for which there are three infrared active pairs, i = 1,2,3.

From Eq. (i) it is clear that as the transition temperature, T is approached c'

and g becomes very large there most be a compensating change on the right side o

of the equation. It can be shown that ~L0 is not expected to increase, while

~TO could easily become very small. Several subsequent experiments (7) in the

high temperature paraelectric (T > Tc) phase of SrTi03 showed that indeed there

is a "soft" transverse mode that approaches zero frequency as ~T02~(T-To ) as

g increase in a Curie-Weiss manner go~(T-To)-l'-w o

Studies centered around Eq. (i) in the ferroelectric phase (T < T ) have c

proved difficult. For the perovskites the degeneracy of the soft mode is re-

duced as the temperature is reduced below T and Eq. (I) can be written sepa- c

rarely for modes with polarizations parallel and perpendicular to the ferro-

electric c-axis (6). 1

Consider the general cubic perovskite AB03 with space group O h and one

formula unit per unit cell. There are 3 x 5-3 optic modes. For long wave

length or small k, these 12 optic modes can be classified according to the

irreducible representations of the point group O h . They transform as the

3Tlu + T2u irreducible representations. The T2u mode is not infrared or Raman

active and thus is called a silent mode. The Tlu modes are infrared active and

long-range Coulomb forces will split such a mode into a longitudinal and trans-

verse branch. The longitudinal mode has the electric field and polarization

(the separation of positive and negative charge) of the mode parallel to the

propagation direction k, while the transverse mode has its polarization perpen-

dicular to k, and mLO > ~TO" Figure la defines the angles that describe the

direction of propagation of k. Figure ib shows the angular dependence of the

modes and degeneracies. As can be seen~ the silent T2u mode is threefold de-

generate while the infrared active Tlu mode is split by the long-range Coulomb

forces into a doubly degenerate transverse part and a singly degenerate longi-

tudinal part.

Vol. 6, No. I0 DISPL_ACIVE FERROELECTRICS 9Z5

x

oJ

1"2.

TI.(LO)

3X

IX

2X =Y TI.(TO )

o e o e

(a) (b) (c) (d)

c o

E(TO,LI ^

g(o])

FIG. 1

a) Coordinate defining the direction of k. b) m vs 0 for a cubic crystal

with O h symmetry, c) ~ vs O for a tetragonal crystal with C4v symmetry. d) The shape function for the modes in Fig. ic.

Suppose one were to measure these modes by the Raman technique in single

crystals. (Actually these modes are not Raman active but they could be made

active by the application of an external electric field.) Then, no matter what

the direction of k, the same frequencies would be obtained. It should be re-

membered that in a Stokes-Raman experiment, if the subscripts i and s are used

to designate incident and scattered radiation, conservation of momentum and

energy require the k and w of the vibrational modes to be:

k=k. -k -- --1 IS

= a3 . - w . (2) l s

The laser that is used to provide the incident light for the Raman exper-

iment and the monochrometer are usually fixed in the laboratory. Thus, the

direction of k in the laboratory is fixed. For the cubic perovskites, no

matter what the direction of k, the same frequencies would be found. One

could ask why use single crystals at all? Why not use a powder or ceramic

sample to determine the frequencies of the modes? In fact, this is often done

by the organic chemist. As can be seen in Fig. ib, indeed a ceramic sample

can be used whenever the modes are not infrared active or if the crystal has

cubic symmetry.

i Many cubic perovskites become ferroelectric-tetragonal below Tc (Oh +

C4vl). Then the mode pattern becomes more complicated. Figure ic shows what

happens to the TIu(LO) and TIu(TO) modes when the symmetry is C4v. Group theory

926 D I S P L A C I V E F E R R O E L E C T R I C S Vol. 6, No. 10

predicts that Tlu ÷ A I + E. Figure ic shows how these modes vary with angle,

Consider the doubly degenerate TIu(TO) branch. In C4V symmetry when k is in

the x,y-plane (e = z/2) the transverse normal modes have a polarization paral-

lel to the z-axis, AI(TO) , and parallel to the x,y-plane, E(TO). With ~ along

the z-axis (8 = 0) the transverse normal modes have a polarization in x,y-plane

and are degenerate E(TO) modes. These results are shown in the lower part of

Fig. ic. For k different from these principal directions (0 < e < ~/2) the

mode with a polarization in the x,y-plane will still be an E(TO) mode as shown

in the figure. However, the mode with its polarization in the z,k-plane will

have a frequency that varies with angle as shown. This mode is called a quasi-

mode and varies from E(TO) to AI(TO) as e increases from 0 to ~/2 as shown in

Fig. ic. The longitudinal modes AI(LO) and E(LO) are also shown for k along

the principal axes as well as a quasimode for 0 < 8 < n/2. It is important to

realize that in general the AI(LO) could also connect to the AI(TO). However,

the labels AI(TO) and E(LO) for 8 = n/2 must be changed because the quasimode

curves cannot cross. The situation (8) sketched in Fig. ic where the TO, A I

and E curves are connected and similarly to LO, A I and E curves are connected,

resembles what is called the electrostatic approximation (8). Actually no

approximation is needed to determine the quasimodes at all angles, if all the

A 1 and E mode frequencies are known (9). Inversely the quasimode equations can

be used to determine the principal axis mode, i.e., A I and E, as has been done

in PbTiO 3 in several cases (I0).

Lastly, we comment that the T2u mode will have B 1 + E character in C4v.

In principle, the E mode can be infrared active to give a TO and LO part as

discussed above. However, measurements in PbTiO 3 have shown that the infrared

strength of this mode is so weak that no splitting can be observed (I0). Thus,

while this mode is observed by Raman scattering, it has no measurable angular

dependence in the ferroelectric phase and it is not shown in Fig. ic.

With the results shown in Fig. ic it might appear that a ceramic can no

longer be used to determine the modes in a Raman spectra. All the modes be-

tween AI(TO) and E(TO) would appear, as would all the modes between AI(LO ) and

E(LO). Thus, the spectrum would be smeared out. This is only a partially true

statement. The results of a powder Raman experiment on a ferroelectric ceramic

would be spread in frequency between the expected limit but in a predictable

way (ii). It must be remembered that in a Raman experiment a monochrometer is

swept through a range of m with a slit opening corresponding to a certain d~.

Whenever dm/d8 is zero there will he many more properly oriented crystallites

in the ceramic sample than when dm/d0 is large (see Fig. ic). Note that dm/de

Vol. 6, No. I0 DISPLACIVE FERROELECTRICS 9Z7

is zero at just the positions corresponding to what is normally called the modes

of the lattice, i.e., A I and E, TO and LO modes. Considering the crystallites

to be uniformly distributed in solid angle (sin 0 d~)(d0) the line shape for

the axial symmetry case of C4v is:

g(~) = Isin 0 d~ S(~), (3)

where S(m) is the Raman scattering efficiency for a particular mode (8) which

is a smooth function of m. Equation (3), and its generalization to lower sym-

metry cases (12), will result in infinite peaks (assuming zero line width) or

a step discontinuity in the powder Raman spectrum whenever m(O,~) exhibits

extrema, unless S goes to zero. Figure id shows the result from Eq. (3). The

details of the shape of the quasimode spectrum are not important for the result

in Fig. id. Only the fact that d~/dO is zero at the principle axis lattice

modes is important. Of course, a finite line width of the Raman lines will

broaden the peaks and step discontinuities. As can be seen in Fig. id, the

E(TO) mode can be expected to appear with a width comparable to that observed

in single crystals since the line comes from a 0-independent mode as can be

seen in Fig. ic. The other modes can be expected to be broader.

One is tempted to say at this point that although the results will be

smeared in a predictable way, they will be spread in frequency, thus at any

given frequency the intensity will be greatly reduced from single crystal

results. Furthermore, the scattering volume for a ceramic should be consider-

ably reduced from that obtainable in a single crystal since most of the Raman

scattered scattered intensity comes from particles in the uppermost surface

layers of ceramic. However, the scattering efficiency from many of the ferro-

electric perovskites is very large and easily observable.

The mesurements shown here were made with what is now called "conventional"

Raman techniques. A 70 milliwatt He-Ne laser, 1-meter double monochrometer,

ITT FW-130 star tracker photomultiplier, and dc detection were used. Figure 2

shows a typical result for the powder Raman spectra of a ceramic of PbTiO 3.

The arrows indicate the positions of the single crystal results obtained by

standard right-angle Raman measurements (i0). As can be seen, the agreement

is excellent. For example, notice the weak powder Raman line corresponding to

E(LO). This line is also very weak in the single crystal measurements. The

region between = 650 and 500 cm -I shows very clearly the quasimodes. All the

modes that were observed in the single crystal results were also observed in

the powder Raman results. However, single crystals have the advantage that

9Z8 DISPLACIVE FERROELECTRICS Vol. 6, No. I0

I I I

V D

23°C

, I I000 8 0 0

Measured powder Raman data for PbTiO 3. crystal results.

I I I I I I

,,,~ + ~,,, + ,,, ,,, ,,,

I I I I I I I 600 400 200

ENERGY SHIFT (cm "1) FIG. 2

The arrows show the measured single

0

the quasimodes can be measured and the results used to obtain some of the

modes that cannot be directly observed.

Besides the above-mentioned advantage, single crystals allow polarization

techniques to be used to determine the symmetry character and transverse and

longitudinal properties of the modes. However, single crystals are often dif-

ficult to obtain. Ceramics afford the advantage that they can be made easily

for a large variety of substances. We have reported (ii) room temperature

powder Raman studies of the Pb(Ti,Zr)O 3 system for which it is difficult to

grow single crystals. We have shown that one way to study BaTi03, which has

broad and difficult to understand spectra, is to measure (Pb,Ba)TiO 3 (13). The

results are in excellent agreement with polariton results in BaTiO 3 (14). We

have also discussed the results of Raman measurements at various temperatures

which include measurements of the soft mode in the ceramic systems (Pb,Sr)TiO 3

and (Pb,Ba)TiO 3 (15). The solidus region of the Li20-Nb205 phase diagram,

which contains LiNb03, has been determined using the powder Raman method (16).

For transparent ceramics it has been shown that polarization selection rules

can be used in a manner similar to single crystals (17). Also, electric field

induced Raman measurements should be obtainable from ceramics for T > T as C

they are in single crystals (18).

Vol. 6, No. I0 DISPLACIVE FERROELECTRICS 9Z9

Thus, one can see that powder Raman techniques can be used eyen when the

modes are highly anisotropic. In general, it is very helpful to have some

knowledge about the spectra. For example, here all the polycrystalline solid

solution systems mentioned contain the end member PbTiO 3 which has a strong

underdamped spectra. However, there should be many other areas where these

ideas can be applied.

It is a pleasure to acknowledge useful and enjoyable collaboration with

B. A. Scott in much of the work discussed here.

i.

References

A. von Hippel and co-workers, N.D.R.C. Rep. No. 300 (August 1944). See A. von Hippel, R. G. Breckenridge, F. G. Chesley, and L. Tisza, Ind. Eng. Chem. 38, 1097 (1946).

2. B. Wul, Nature 157, 808 (1946).

3. (a) F. Jona and G. Shirane, Ferroelectric Crystals. MacMillan, New York (1962). (b) E. Fattuzzo and W. J. Merz, Ferroelectricity. John Wiley & Sons, New York (1967). (c) W. Kaenzig, in Solid State Physics, Vol. 4. Edited by F. Seitz and D. Turnbull. Academic Press, New York (1957).

4. A. F. Devonshire, Phil. Mag. 40, 1040 (1949); Phil. Mag. 42, 1065 (1951); A~van. Phys. 3-, 85 (1954).

5. J.C. Stater, Phys. Rev. 78, 748 (1950).

6. W. Cochran, Phys, Rev. Letters 3--, 412 (1959); Advan. Phys. 9, 387 (1960); Advan. Phys. iO, 401 (1961).

7. A.S. Barker, Jr. and M. Tinkham, Phys. Rev. 125, 1527 (1962). R.A. Cowley, Phys. Rev. Letters 9, 159 (1962) and Phys. Rev. 134, A981 (1964).

8. R. Loudon, Advan. Phys. 13, 423 (1964).

9. L. Merten, Phys. Status Solidi 25, 125 (1968) and Z. Naturforsch. 22a, 359 (1967).

i0. G. Burns and B. A. Scott, Phys. Rev. Letters 25, 169 (1970). An extended paper on this subject is in preparation.

ii.

12.

G. Burns and B. A. Scott, ibid., 25, 1191 (1970).

M. H. Cohen and F. Reif, in Solid State Physics, Vol. 5. Edited by F. Seitz and D. Turnbull. Academic Press, New York (1957). N. Bloembergen and T. J. Rowland, Acta Met. ~, 731 (1953). G. Burns, J. Appl. Phys. 32, 2048 (1961), and Phys. Rev. 135, A479 (1964). B. Bleaney and R. S. Rubbins, Proc. Phys. Soc. (London) 77, 103 (1961). W. H. Jones, Jr., T. P. Graham and R. G. Barnes, Phys. Rev. 132, 1898 (1963).

13. G. Burns and B. A. Scott, Bull. Am. Phys. Soc. 16, 415 (1971), and Solid State Comm., to be published Z May 1971.

14. A. Pinczuk, E. Burstein, and S. Ushioda, Solid State Comm. ~, 139 (1969).

930 DISPLACIVE FERROELECTRICS Vol. 6, No. I0

17.

18.

15.

16.

G. Burns, Bull. Am. Phys. Soc. 16, 415 (!971) and to be published.

B. A. Scott and G. Burns, Bull. Am. Ceram. Soc. 50, 467 (1971) and to be published.

W. J. Brya, Phys. Rev. Letters 26, 1114 (1971).

P. A. Fleury and J. M. Worloch, Phys. Rev. 174, 613 (1968).