Powder raman spectra: Application to displacive ferroelectrics

Mat. Res. Bull. Vol. 6, pp. 923-930, in the United States.

POWDER RAMAN SPECTRA:

1971.

Pergamon

Press, Inc.

Printed

APPLICATION TO DISpLACIVE FERROELECTRICS

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

(Received

July 6,

10598

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. electricity. known:

(2) ushered in a new era in the field of ferro-

Previous to this work two types of ferroelectric crystals were

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 ferroelectricity.

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

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(6) pointed out the connection between ferroelectricity and lattice dynamics. This was done via the Lyddane-Sachs-Teller

3

relationship,

LO i

g_o=

,

(1)

i=l \ ~ T O . -1 /

g~

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

g

(the square of the index of refraction),

is the optical 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 and g

is approached c' 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. high temperature paraelectric

Several subsequent experiments

(7) in the

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

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

o

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

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 reduced as the temperature is reduced below T rarely for modes with polarizations

and Eq. (I) can be written sepac parallel and perpendicular to the ferro-

electric c-axis (6). Consider the general cubic perovskite AB03 with space group O h formula unit per unit cell.

There are 3 x 5-3 optic modes.

1

and one

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 .

3Tlu + T2u irreducible representations. active and thus is called a silent mode.

They transform as the

The T2u mode is not infrared or Raman The Tlu modes are infrared active and

long-range Coulomb forces will split such a mode into a longitudinal and transverse 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 perpendicular to k, and mLO > ~TO" direction of propagation of k. modes and degeneracies.

Figure la defines the angles that describe the Figure ib shows the angular dependence of the

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 longitudinal part.

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oJ 3X

1"2.

co

IX

TI.(LO)

2X

=Y TI.(TO)

E(TO,LI

^

x

o

e

(a)

o

g(o])

e

(b)

(c)

(d)

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 crystals.

(Actually

these modes by the Raman technique

these modes are not Raman active but they could be made

active by the application

of an external

electric

the direction of k, the same frequencies membered

that in a Stokes-Raman

to designate

in single

radiation,

k=k.

--1

=

a3.

l

It should be re-

if the subscripts

energy require the k and w of the vibrational

--

Then, no matter what

would be obtained.

experiment,

incident and scattered

field.)

conservation modes to

i and s are used

of momentum and

be:

-k

IS

- w . s

(2)

The laser that is used to provide the incident light for the Raman experiment and the monochrometer

are usually fixed in the laboratory.

direction of k in the laboratory

is fixed.

Thus, the

For the cubic perovskites,

matter what the direction of k, the same frequencies would be found. could ask why use single crystals sample to determine

the frequencies

by the organic chemist. can be used whenever

at all?

no One

Why not use a powder or ceramic

of the modes?

In fact,

this is often done

As can be seen in Fig. ib, indeed a ceramic sample

the modes are not infrared active or if the crystal has

cubic symmetry. Many cubic perovskites

become ferroelectric-tetragonal

below Tc (Oh

C4vl).

Then the mode pattern becomes more complicated.

happens

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

i

+

Figure ic shows what Group theory

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predicts that Tlu ÷ A I + E.

Vol. 6, No. 10

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 parallel 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. Fig. ic.

These results are shown in the lower part of

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 while this mode is observed by Raman scattering,

(I0).

Thus,

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.

statement.

This is only a partially true

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

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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 is excellent. E(LO).

(i0).

As can be seen, the agreement

For example, notice the weak powder Raman line corresponding to

This line is also very weak in the single crystal measurements.

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

The

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

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I

I

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I

I

I

I

I

V D

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

,,, ,,,

23°C

, I000

I 800

I

I

I

600

I

I

400

I

I

200

0

ENERGY SHIFT (cm "1) FIG. 2 Measured powder Raman data for PbTiO 3. The arrows show the measured single crystal results. 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. ficult to obtain.

However, single crystals are often dif-

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

The We

have also discussed the results of Raman measurements at various temperatures which include measurements of the soft mode in the ceramic systems and (Pb,Ba)TiO 3 (15).

(Pb,Sr)TiO 3

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).

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Thus, one can see that powder Raman techniques can be used eyen when the modes are highly anisotropic. knowledge about the spectra.

In general, it is very helpful to have some 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. References i.

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.

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). paper on this subject is in preparation.

ii.

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

12.

M. H. Cohen and F. Reif, in Solid State Physics, Vol. 5. Seitz and D. Turnbull. Academic Press, New York (1957). and T. J. Rowland, Acta Met. ~, 731 (1953). G. Burns, J. 2048 (1961), and Phys. Rev. 135, A479 (1964). B. Bleaney Rubbins, Proc. Phys. Soc. (London) 77, 103 (1961). W. H. 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).

Stater, Phys. Rev. 78, 748 (1950).

An extended

Edited by F. N. Bloembergen Appl. Phys. 32, and R. S. Jones, Jr., T.

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Vol. 6, No.

15.

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

16.

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

17.

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

18.

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

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