INELASTIC LIGHT SCATTERING Proceedings of the 1979 US-Japan Seminar held at Santa Monica, California, USA 22-25 January 1979

Editors

E. BURSTEIN University of Pennsylvania, USA

and

H. KAWAMURA Kuansei Gakuin University, Japan

PERGAMON PRESS OXFORD · NEW YORK · TORONTO · SYDNEY · PARIS · FRANKFURT

U.K.

U.S.A.

CANADA

AUSTRALIA

FRANCE

FEDERAL REPUBLIC OF GERMANY

Pergamon Press Ltd., Headington Hill Hall, Oxford 0 X 3 OBW, England Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. Pergamon of Canada, Suite 104, 150 Consumers Road, Willowdale, Ontario M2J 1P9, Canada Pergamon Press (Aust.) Pty. Ltd., P.O. Box 544, Potts Point, N.S.W. 2011, Australia Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France Pergamon Press GmbH, 6242 Kronberg-Taunus, Pferdstrasse 1, Federal Republic of Germany

Copyright © 1980 Pergamon Press Ltd. All Rights Reserved. No part of this publication may he reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. First edition 1980

British Library Cataloguing in Publication Data

Inelastic light scattering. 1. Light - Scattering - Congresses I. Burstein, E II. Kawamura, H 535\4 QC427.4 79-42761 ISBN 0-08-025425-X

Published as a special issue of the journal Solid State Communications Volume 32, Number 1 and supplied to subscribers as part of their subscription.

Printed in Great Britain by A. Wheaton Et Co Ltd., Exeter

PARTICIPANTS OF THE U.S. - JAPAN SEMINAR

U . S . PARTICIPANTS

P r o f . E B u r s t e i n (US C o o r d i n a t o r )

Depar tment of P h y s i c s U n i v e r s i t y of P e n n s y l v a n i a

Prof. J L Birman Department of Physics College of the City of New York, CUNY

Dr G Burns I.B.M. Thomas J Watson Research Center

Prof. P M Champion Department of Chemistry Cornell University

Dr P A Fleury Bell Telephone Laboratories Murray Hill

Prof. R W Hellwarth Department of Physics University of Southern California

Prof. R M Hexter Department of Chemistry University of Minnesotta

Prof. M V Klein Department of Physics University of Illinois

Prof. W Kohn Department of Physics University of California at San Diego

Prof. P Pershan Department of Physics Harvard University

Prof. Y R Shen Department of Physics University of California

Prof. S Ushioda Department of Physics University of California

U.S. OBSERVERS Prof. H Z Cummins Department of Physics City College of New York, CUNY

Prof. L M Falicov Department of Physics University of California

Dr J M Friedman Bell Telephone Laboratories Murray Hill

Prof. B I Halperin Department of Physics Harvard University

Professor M Levinson Department of Physics University of Southern California

Dr G Lucovsky Xerox Palo Alto Research Center Palo Alto

Prof. D L Mills Department of Physics University of California

Prof. D Nelson Department of Physics Harvard University

Dr R Nemanich Xerox Palo Alto Research Center Palo Alto

Prof. S Solin Department of Physics University of Chicago

Dr J M Worlock Bell Telephone Laboratories Holmdel

Dr P Y Yu I.B.M. Thomas J Watson Research Center

111

IV JAPANESE PARTICIPANTS Prof. Hajimu Kawamura (Japanese Coordinator) Department of Physics Kuansei Gakuin University

Prof. Ryogo Kubo Department of Physics University to Tokyo

Prof. Masayasu Ueta Department of Physics Tohoku University

Prof. Yutaka Toyozawa Institute for Solid State Physics University of Tokyo

Prof. Akiyoshi Mitsuishi Department of Applied Physics Osaka University

Prof. Takashi Kushida Department of Physics Osaka University

Prof. Terutaro Nakamura Institute for Solid State Physics University of Tokyo

JAPANESE OBSERVERS

Prof. Ei-ichi Hanamura Department of Engineering Physics University ot Tokyo

Prof. Kazuo Murase Department of Physics Osaka University

THIRD COUNTRY NATIONAL Prof. M M Moskovitz Department of Chemistry University of Toronto

Solid State Communications, Vol. 32, p. v. Pergamon Press Ltd. 1979. Printed in Great Britain.

EDITORIAL

Solid State Communications will be pleased to publish the Proceedings of Conferences that deal with recent theoretical and experimental developments in the solid state sciences, whose papers meet our normal standards. In particular the papers should be "newsworthy", make a significant contribution to the field, and solid state scientists should "be glad to have seen it" (the quotes are from our Referee Report Form which appears at the back of the first issue of each volume). The Proceedings of the US - Japan Seminar on Inelastic Light Scattering is the first of what we hope to be a continuing series.

Solid State Communications, Vol. 32, p. vii. Pergamon Press Ltd. 1979. Printed in Great Britain.

PREFACE

The Joint US—Japan Seminar on Inelastic Light Scattering which was held in Santa Monica, California on 22 to 25 January 1979 is one of a continuing series of Seminars on "Current Developments in Science" which are jointly sponsored by the United States National Science Foundation and the Japan Society for the Promotion of Science as part of the United States - Japan Cooperative Science Program. These joint Seminars provide a medium for personal interactions between theorists and experimentalists from the two countries. Over the past decade the inelastic scattering of light, e.g. Raman and Brillouin scattering and related luminesce, has been employed as a "probe" of unparalleled power and versatility, for the study of condensed matter and macromolecules . Lasers, combined with increasingly sophisticated experimental techniques have enabled one to study remarkably diverse phenomena in condensed matter physics, chemistry and biology, whose time scales range

-15 from 1 second to 10 seconds, Never­theless, fundamental questions remain regarding the basic interaction of light with matter in these phenomena. The aim of the US and Japanese Coordina­tors of the Joint Seminar on Inelastic Light Scattering was to organize a Program which would deal with these questions ^ad which would iocus on

recent important theoretical and experimental developments that reflect the complementarity of Japanese and US efforts in this important field. The topics covered by the papers presented at the Seminar and printed in these Proceedings, include Resonant Raman Scattering and Luminescence, Light Scattering under Intense Illumination, Resonant Brillouin Scattering and Non-Local Optics, Enhanced Raman Scattering by Molecules Adsorbed in Metals, Inelastic Light Scattering in Superionic Conductors and in Glasses, Raman Scattering by Soft Modes in IV-VI Compound Semi­conductor and Ferroelectrics» and Central Peaks in Inelastic Light Scattering at Structural Phase Transitions. The Coordinators of the Joint Seminar on Inelastic Light Scattering take this opportunity, on behalf of all the participants in the joint seminar, to thank the US National Science Foundation and the Japanese Society for the Promotion of Science for their sponsorship of the Joint Seminar. We also wish to express our appreciation to our colleagues for the valuable advice that they provided regarding the Program of the Joint Seminar and for their whole hearted support in i-mplemeirting it.

E. Burstein H. Kawamura

Splid State Communications, Vol.32, pp.1-6. Pergamon Press Ltd. 1979. Printed in Great Britain.

A STOCHASTIC THEORY OF SECOND ORDER OPTICAL PROCESSES

R. Kubo Department of Physics, University of Tokyo

T. Takagahara and E. Hanamura Department of Applied Physics, University of Tokyo

Hongo, Bunkyoku, Tokyo.

Perturbations acting on a radiative system in its excited states affect the coherence property of a second order optical response of the system. In order to understand general features of such effects, perturbations are considered as a stochastic Markoffian process. A general formalism is developed for calculations of spectral as well as temporal responses. In the static limit and also in the extremely fast limit of perturbation, only the coherent Raman or Rayleigh scattering is present, but in general intermediate cases the incoherent luminescence process coexists. The theory can cover the whole range from the static to the fast limit without appealing to the ordinary sort of perturbational calculation. It is illustrated for a three-level model, a four level model and for an optical center coupled with a quantum Brownian oscillator.

1. GENERAL FORMULATIONJ We illustrate first the general formulation by a simple three level system, S, interacting with its environment, R, and the photon field. The quantum states of S are designated by A, B, and C and the photon numbers by n19 n2, ... . The Hamiltonian of the total system is written as

H = H + HP +

where

H = H„ + «E + H' (1)

is the Hamiltonian of the system S+R, H1 being the interaction of S with R, Hp the Hamiltonian of the photon field and V is the interaction of S with photons,

and combines C with B. In a second order pro­cess starting from the initial state:

| a > = | A, ηχ + 1, n2 > with the energy

a = A + ω-, and ending at the final state:

C, ηχ, n2 + 1 > |c > with the energy

c = C + ω2, a photon ω is absorbed and a photon ω2 is emit­ted. In the intermediate state, S is in B and photon numbers are either ( n 1 , n 2 ) o r ( n 1 + l ,

+ 1) We ignore the latter state since we are interested in the near resonance condition:

V = V* + V~ + V* + V~ +

The interaction V~ creates or annihilates a photon Mi and combines the state A with B. Likewise V~ creates or annihilates a photon ω2

P(t) = [ dtj λο± | dt'f W •Ό J0 J0 J0

A + ω1 ^ B, or C + ω2 ^ B. In the initial state, the environment R is as­sumed to be in equilibrium represented by the density matrix OR. A simple perturbation cal­culation gives the probability P(t) to find the system in the final state at time t as

-ί(Η+ω )(t-t ) -iH(t,-t0) Tr <c|e |c><c|v |b><b|e

Ί-«9/ -ί(Η+ω )tp 1 * |b><b|V~|a><a|e ^ ^|a> (2)

i(H+Q> )t' iH(t«-t») ί(Η+ω )(t-t') p <a|e ^|a><a|V_ |b><b|e χ * |b><b|vl|c><c|e |c>, R -L d.

1

2 A STOCHASTIC THEORY OF SECOND ORDER OPTICAL PROCESSES Vol. 32, No. 1

where TrR is taken over the states of R. In this expression, propagation of the states is governed by the Hamiltonian (l). More rigorous­ly we could have included an appropriate part of the total Hamiltonian in a damping-theoretical manner. Instead of doing so, we introduce an effective damping constant into the intermediate state (also into the initial and final states if necessary). Namely the propagators in the intermediate states in Eq. (2) are replaced by

^(iH^KvV^UH^Mti-t·) _ This artifice saves divergence at the resonance and takes account of damping which brings the system out of the states under consideration.

For large t, P(t) becomes proportional to t and defines the transition probability rate W, which is given by an expression of the form:

Γτ1 W = ■8τάτΓ*τΊΠ

J0 1 J 0 dT2(l+II+III+ c.c. (3) lim e

s-KVO Here I, II and III are decompositions of the propagator under the integral sign of Eq. (2) as shown in Fig. 1 according to the time ordering of propagation to the left and to the right indicated by two parallel arrows. The four time variables in (2) are renamed as 0, τ,, τ,, τ in the increasing order.

Li>

for the transition rate. By the relation: 1 lim

s+0 . . / r = ffö(a-c) - ,

s+i(a-c) a-c the first term and its c.c. give a pure Raman spectrum, except the factor |V1| |V2| ,

2 6(Α-σ+ω1-ω2) (6) Raman 2,_ . x γ^ΙΒ-Α-ο^) and a negative term. As is expected from the conservation of energy, the latter is cancel­led by the sum of (II), (III) and c.c, which amounts to

YbtY2+(b-a)2]-1[Y2+(c-b)2]-1

In the presence of interaction between S and R, this cancellation is no longer complete and generally there arises a continuous spectrum for the emitted light, which corresponds to the luminescence as we see in the following. Our stochastic theory consists in replacing the Hamiltonian (l) by a stochastic Hamiltonian H(t) which represents random modulations of quantal evolution of S by the interaction of S with R. The motion of R is incorporated in a stochastic evolution of H(t). This is an extension of the stochastic theory of line shape problem which

Fig. 1. Decomposed propagators in a second order optical process. Three typical diagrams are drawn according to the chronological order of time variables, t.. , tp, t', and t'.

If there is no interaction between S and R, the propagators are simply given as IQ = eacpt-iy^iib-cJJd-^Mia-cJi^-Tg)

-(yb+i(a-b))x2] ,

II0 = exp[-(Yb+i(b-c))(x-T1)-2Yb(T1-T2)

-(Yb+i(a-b))T2] ,

III0 = expL-(Yb+i(c-b))(T-T1)-2Yb(x1-T2) -(Yb+i(a-b)h2], (U)

which yield, according to Eq. (3),

w = s^[v^b-c) s+i^-c> v^a-b ) (I)

1 1 1 Yb+i(b-c) 2Yb Yb+i(a-b)

+ 1 1 1 Yb+i(c-b) 2γ^ Y^+i(a-b)

+ c.c]

has been developed by one of the present au­thors? The propagators in the expression (5) are then replaced by stochastic operators which describe the evolution of the system S. Al­though the method is rather straightforward, analytical calculations can be made only in simple cases, so that we have made model calcu­lations for a few typical examples, which are discussed in the following.

2. A THREE LEVEL SYSTEM? The simplest example is a three level system in which the excited state B is modulated in a random fashion. Its energy is assumed to be a stochastic process,

'b 'b

(I)

(II)

(III)

(5)

b(t) = b + b'(t), with b'(t)=0, with the mean square fluctuation,

Δ = <V(t)2>1/2

and the correlation time, τ = 1/Ί . m m

If the modulation is very slowly changing time, namely if the parameter

(7)

(8)

in

Vol. 32, No. 1 A STOCHASTIC THEORY OF SECOND ORDER OPTICAL PROCESSES 3 α = Δτ m

is very large, the modulation is almost static, and so only the pure Raman emission is present. The modulation affects its intensity only slight­ly. On the other hand, if the modulation is faster with larger γ , it causes the dephasing effect in the quantal evolution of S in the intermediate state producing an extra damping,

y' = Λ * . as is well known in spin resonance problems. This adds to γ, making the effective damping,

Ύ = Yb + Υ' to replace γ^ in the expression (5)» except γ^ in the factor 1/2*^ in the terms (II) and (III). The reason for this unbalance is that the cor­responding propagation is for the pair of the state vectors Ib(t)> and <b(t)|and so the phase factor is cancelled. In this narrowing condi­tion of the modulation, we find the simple result:

1 W = Tro(a-c)

X y2+(b-a)2

' V ^ ' N W A M 2

(Raman)

(Luminescence).

Note that the second term consists of Lorentzian factors representing absorption and emission. This shows that the unbalanced effect of dephas­ing caused by the random adiabatic modulation gives rise to the incoherent luminescence, which disappears in the limit of very fast modulation, where the effect is completely averaged out and there remains only the coherent Raman scattering.

In order to see how the thing goes from one limit to the other, we show a calculation for a two-state jump model of the modulation b'(t). We assume b'(t) takes only two values ±Δ in a random way. This model can be treated analyti­cally. But we show here only examples of nu­merical results in Figure 2 for a particular set of model parameters, γ, Δ, and the incident photon energy measured by the off-resonance,

Δα^ = ω1 - b + A , as depicted in the figure and for three differ­ent values of the modulation rate ym. The curves are the intensity distribution of the emitted light plotted in the logarithmic scale against the off-resonance energy,

Δω2 = b + C. Besides these continuous spectra, there exists a delta-function peak at the Raman frequency Δω =0.5, which is not shown in the figure. Its intensity changes with ym but not so much. We see in the figure that for a rather slow modu­lation with ym = 0.01 there are a broadened Raman peak and two luminescence peaks corre­sponding to emissions from two possible excited levels b ± Δ. The former is a Raman side band caused by energy exchange with R. This becomes flatter as ym increases and becomes unnotice-able. Luminescence peaks are broadened, merge into a single peak and get narrowed with in-

Two-State Jump Model

ie>-

HB(t)

creasing process.

a well-known motional narrowing

Fig. 2. Emission spectra of a three-level system with adiabatic random modulation of a two-state jump model. A delta-shaped Raman line at Δω2 = 0.5 is not shown. The level scheme is depicted in the figure.

3. OFF-DIAGONAL MODULATION IN A FOUR LEVEL MODEL? Another simple but illuminating example is a four level system which has two excited levels, B and B2· Perturbation by R is re­presented by the Hamiltonian:

Βχ ß(t)

l«(t) B2 The initial state A is combined with Βχ by Vx and the final state C with B2 by V2. So the second order process is induced by the presence of the off-diagonal modulation Ω(ΐ). If it is static, mixing of the states B1 and B2 connects A and C producing pure Raman scattering. If it is very fast, everything is averaged out and there is no second order process. In general there are Raman and luminescent processes co­existent. The off-diagonal perturbation modu­lates adiabatically the excited levels on the one hand and on the other hand it causes non-adiabatic transition between them. The effect of the former is essentially the same as the previous three level case. The latter effect is closer to luminescence in the usual sense.

A model calculation was made for a two-state jump model which assumes two values ±Δ as before. Figure 3 illustrates the results for the parameters as depicted in the figure. In

4 A STOCHASTIC THEORY OF SECOND ORDER OPTICAL PROCESSES Vol. 32, No. 1

Off-Diagonal Modulation(Two-StateJumpModel) Δωρ0.3

Fig. 3. Emission spectra of a four-level system with off-diagonal random modulation in a two-state jump model. The level scheme is depicted in the figure. The level |bx> is optically connected only to |a> and |b2> only tol c>.

slow modulation cases, a broadened Raman peak appears at the center accompanied by two lumi­nescence peaks corresponding to transitions from two excited levels. As the modulation rate becomes faster, there occur broadening and narrowing. k. AN OPTICAL CENTER COUPLED WITH A BROWNIAN OSCILLATOR. In a condensed matter, an optical center may be coupled with vibrational modes of surrounding atoms. As a model, we consider a pair of electronic states, g and e, which are coupled with a localized vibrational mode with a proper frequency ω0. The oscillator is cou­pled with other modes of the environment and makes a Brownian motion. For simplicity the frequency ω is assumed to be the same for g-and e-states, but the origin of oscillation is shifted so that the vibrational quantum number can change by any amount. Second order process­es are possible from an initial state |g, N^> to a final state |g, N^> via intermediate states |e, N'>. The quantum Brownian motion of the oscillator is treated by a method more or less standard in the laser theory. The formulation of the section 1 is extended easily for this more complex system.

Figure h shows a model calculation of emis­sion spectra. The incident photon is at an off-resonance, Δω1 = ω1+Ε(έ)-Ε(β) = 2.5 (in the unit of ü)Q).

Thus the peak at Δω2= ü)2+E(g)-E(e) = 2.5 is the Rayleigh line and those at Δω2 = 0.5 + integers are Raman lines. Peaks at ω2 = integers corres­pond to luminescence emissions from the excited states |e, Nf> to the ground states |g, Nf>. Δ is the shift of the origin of the local oscil­lator for the states e and g, γ^ is the damping

Raman Spectrum of

Localized ElectrorrPhonon System

Δω0

Fig. k. Emission spectra of an optical center coupled with a Brownian local oscillator. The emission intensity is plotted in an arbitrary scale. That for the case B = 0.01 should be multiplied by ten. The frequency or energy is measured in units of the vibration frequency of the oscillator. The Rayleigh line at Δω2 = 2.5 is not shown here.

Vol. 32, No. 1 A STOCHASTIC THEORY OF SECOND ORDER OPTICAL PROCESSES 5

as was introduced in section 1, B corresponds to the previous ym and is a time constant for the Brownian motion, and the Boltzmann factor exp[ -Hu)0/kBT] governs the thermal distribution of the initial states |g, Ni>. Values of these parameters are depicted in the figure. For slow Brownian motion of the oscillator with B = 0.01, the coherent Rayleigh and Raman lines are domi­nant and luminescence emission is weak. The intensity scale for this case is reduced by ten in the figure. For a faster motion with B = 0.1, the Rayleigh and Raman lines are broadened and luminescence lines are enhanced. For even faster motion with B = 1.0, the spectral struc­ture is almost washed out leaving a big peak around Δω2 = 0 and few other peaks. There ex­ists, however, in all cases a sharp delta-func­tion Rayleigh line, which is not shown in the figure, with almost invariable intensities (in relative magnitudes, 2.06, 2.15 and 1.U8 for B = 0.01, 0.1 and 1.0, respectively). Lumines­

cence for slower Brownian motion is hot since vibration in the excited state does not achieve thermal equilibrium before light is emitted. The equilibrium is nearly attained in the case B = 0.1.

5. TRANSIENT RESPONSE. So. far we considered stationary responses to continuous steady exci­tation. Now we turn to non-stationary transient responses to an incident light pulse which is represented by

V(t) = e-iWltE(t) ,

with an envelope function:

E(t) = -±r f1" euVu)du. » —loo

For a square envelope with duration T, we have <|>(u) = (1 - e"uT)/u.

The transition probability P(t), Eq. (2) is cal­culated for this pulse excitation and the emis­sion intensity is obtained from dP(t)/dt as a function of time t after the start of the pulse? Here we discuss an application of the method to the previous model of an optical center coupled with a Brownian oscillator?

Figure 5 shows an example of such a model calculation. The parameters correspond to those used for Fig. k. The rate constant B is chosen as 0.1 for which Fig. k showed well-defined Raman and luminescence responses. One differ­ence from the previous case is that here we as­sumed a radiative damping constant γβ = 0.01 for

the ground state g, whereas in the previous case we assumed yg = 0 and ye = γ^ = 0.1. The unit of time scale is the period of the local oscil­lator. The pulse starts from t = 0 and is switched off at t = 250. At the early stage t « γ"1 or B"1 after switching on the pulse, the incident photon frequency has an uncertainty of f , which is reflected on the broad emission spectrum at the initial stage. Then the Rayleigh and Raman responses grow with wavy structures of the form,

sinCAo^ - Δω2 ± ηω 0 Η/(Δω - Δω2 ± na)Q) or

Time Resolved and Frequency Analysed Spectrum

Δ=1.0 B=0.1 Ye=0.1 Yg=0.01 Δω, = 2.5

400

300

* Δω-,

Fig. 5. Transient spectral response of an optical center coupled with a Brownian oscil­lator to a pulse excitation. The frequency or energy is measured in units of the vibration frequency of the oscillator. The incident pulse has an envelope of a rectangular form and is switched off at t = 250. The spectral intensity is plotted in arbitrary units. The intensity should be reduced by ten and thousand for the case of t = 300 and t = 1*00, respectively.

6 A STOCHASTIC THEORY OF SECOND ORDER OPTICAL PROCESSES Vol. 32, No. 1

cos(Au), - Δω? ± nü)Q)t. The luminescence lines at integer points of Δω2 also become noticeable (t=100). At a later time, say t=200 £ γ" , the spectrum develops to a structure almost similar to that in the sta­tionary case. The strong peak at Δω2 = 2.5 is the Rayleigh line broadened by the damping Yg. After the pulse is switched off at t=250, the Raman and Rayleigh responses decay out within the phase memory time which corresponds to the spectral widths of these lines. On the other hand, the luminescence emission decays with a different rate constant, which is the sum of the radiative damping and the vibrational re­laxation rate in the excited states. In the intermediate ranges of time after switching off

REFERENCES

1. TAKAGAHARA, T., HANAMURA, E., and KUBO, R., J. Phys. Soc. Japan k3_, 802 (1977). KUBO, R., TAKAGAHARA, T., and HANAMURA, E. , Lecture Note in Physics Vol. 57, ed. M. Ueta and Y. Nishina, (Springer Verlag, Heidelberg, 1976) p. 304.

2. KUBO, R., in Fluctuation* Relaxation and Resonance in Magnetic Systems* ed. D. ter Haar, (Oliver and Boyd, Edinburgh, 1962) p. 23. KUBO, R., in Stochastic Process in Chemical Physics, ed. K. Shuler (John Wiley & Sons, I969) p. 101.

3. Ref. 1. Also HUBER, D.L., Phys. Rev. 158, 8U3 (1967), 170, hlQ (1968), 178, 93 (1969), Bl, 3409 (1970).

k. TAKAGAHARA, T., HANAMURA, E., and KUBO, R., J. Phys. Soc. Japan j+3, 8ll (1977). 5. TAKAGAHARA, T., HANAMURA, E., and KUBO, R., J. Phys. Soc. Japan hk_, 728 (1978). 6. TAKAGAHARA, T., HANAMURA, E., and KUBO, R., J. Phys. Soc. Japan 1*3, 1522 (1977). 7. TAKAGAHARA, T., HANAMURA, E., and KUBO, R., J. Phys. Soc. Japan M^, 7^2 (1978). 8. HIZHNYAKOV, V.V., and TEHVER, I.Y., Phys. Status Solidi 2^, 755 (1967), TEHVER, I.Y. and

HIZHNYAKOV, V.V., Soviet Phys-JETP U2_, 305 (1976). MUKAMEL, S., BEN-REUVEN, A., and JORTNER, J., Phys. Rev. A12, 9 7 (1975), J. Chem. Phys. 6US 3971 (1976). BEN-REUVEN, A., JORTNER, J., KLEIN, L., and MUKAMEL, S., Phys. Rev. A13, 1^02 (1976). T0Y0ZAWA, Y., J. Phys. Soc. Japan 1*1, U00 (1976). KOTANI, A. and T0Y0ZAWA, Y., J. Phys. Soc. Japan 1+1, 1699 (1976). KUSUN0KI, M., Progr. theor. Phys. 55., 692 (1976), 6θ_, 71 (1978).

the pulse, thermalization of vibration is in­complete and hot luminescence is produced.

6. CONCLUSION. Our basic problem is the ques­tion how a coherent quantum process is inter­rupted by perturbations to open channels to in­coherent quantum processes. We treated this problem for the case of second order optical processes considering the perturbation as sto­chastic processes. This method has the advan­tage to make a non-perturbational treatment possible. The models we discussed here are simple but are illuminating to make us under­stand the general features of the problem. The method is in many ways complementary to other approaches developed in recent years!

I Solid State Communications, Vol.32, pp.7-12. Pergamon Press Ltd. 19 79. Printed in Great Britain.

ON THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING P.M. Champion, G.M. Korenowski and A.C. Albrecht

Chemistry Department, Cornell University Ithaca, New York 14853

Resonance Raman excitation profiles of totally symmetric vibrational modes are investigated using a model that analytically includes the complete subspace of Franck-Condon active vibrations associated with each intermediate electronic state. This model is used to fit data obtained in resonance with the Soret band of cytochrome c. The ex­citation profiles are asymmetric and peak distinctively to the blue of the Soret absorption maximum. Large damping factors and/or in-homogeneous site distributions by themselves cannot account for the observed data. The theoretical results imply that the excited state lifetime associated with the Soret band of ferrocytochrome c has a lower limit on the order of 50 fs and that, compared to the ferrous form, the ferric cytochrome has a larger x-y splitting and a shorter lifetime.

In contrast to the large multidimensional heme system, we also present the results of a simple model calculation applicable to smal­ler molecules. A two dimensional subspace is explored, where Raman Franck-Condon (RFC) and Franck-Condon (FC) factors are calculated for different potential energy surface parameters. Under certain conditions the RFC based scattering profiles of one vibration are strongly coupled to the FC based behaviour of the other vibration. Rather complex profiles are then predicted even for the simple two-dimensional case.

The vibronic theory of Raman scattering nat­urally identifies two basic sources of scattering. These have been termed "A-term" and "B-term" act­ivity'. The activity associated with an A-term involves resonances with individual zero-order electronic states and derives intensity from Raman Franck-Condon (RFC) overlaps arising from shifts in the excited state equilibrium position and/or changes in the excited state potential energy sur­face. Raman activity associated with modes invol­ved in vibronic mixing of two zero order electron­ic states is classified as B-term activity. The amplitude of the B-term modes depends upon the product of the transition dipole moments of the electronic states being mixed, while the ampli­tude of the A-term modes depends upon the square of the transition moment of the individual elect­ronic state in resonance. In the following dis­cussion, we focus on the resonance Raman scatter­ing intensity associated with A-term activity, paying special attention to the multidimensional nuclear subspace that must be included in the sum over the resonant electronic states.

In order to demonstrate the influence of this subspace on the theoretical Raman excitation pro­file, we attempt to fit data obtained in resonance with the intense near-ultraviolet "Soret" absorp­tion band of cytochrome c2. Neglect of the nuc­

lear coordinate space generates theoretical exci­tation profiles and absorption bands that are highly symmetric and that do not compare well with the data; inclusion of this subspace, however, re­sults in very good agreement with the data. In contrast to the large multidimensional heme chro-mophore of cytochrome c, we also present the re­sults of a simple model calculation applicable to smaller molecules^.

We begin our discussion with the well-known1 expression for the total light scattered by the mn'th transition in a molecule, averaged over all orientations, into a Raman band centered at freq­uency v:

The quantity I0 is the incident power flux(W/cm2), (e2/nc) is the fine structure constant and a represents the p,ath (p,a=x,y,z) component οτ the molecular polarizability tensor. The subscripts in eq. (1) refer to the initial and final states of the molecule (i.e. hv=hv0-(En-Em); v0=incident laser frequency). The molecular polarizability tensor is often expressed as a sum over virtual molecular eigenstates' and, if the excitation fre­quency is near resonance with a set of molecular states |e> faidth, re) we can write:

Research supported through the National Institutes of Health AM20379 and the Materials Science Center of Cornell University. Present address Chemistry Department, Columbia University, New York, New York 10027.

7

THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING Vol. 32, No. 1

/ 0 ) = v <m|Mrr|exelMpln> ^ W m , n 1 Ee-Em-hv0-ire

(2) Within the zeroth order Born-Oppenheimer approx­imation the molecular eigenstate is written as a product of functions in electronic and nuclear subspaces. The latter is factored into the eigen­state of the Raman active mode, |v>, and the eigenstate, |v'>, reflecting the remaining multi­dimensional nuclear coordinate space.

Since we limit our discussion to (RFC al­lowed) A-term scattering, we evaluate the elec­tronic eigenfunctions at equilibrium nuclear con­figurations. Thus, for fundamental (Δν=1) scat­tering from the cold (T=0K, Em=0) ground state we can rewrite eq.(2) as2:

(V}o,i = Σ Σ , ev v Δ + ε , ev v' 1Γ evv'

In eq.(3) we have split the sum over intermediate molecular states into two parts. The first part contains the usual sum over electronic and Raman vibrational states; the second part contains the sum over excitations of the remaining modes in the intermediate state. Here we also associate an energy, εγ·, with each configuration of the re­maining (ν') vibrations. We set εν'=0 when |v'>=|0·>. The quantity Δθ ν is simply the elec­tronic plus Raman vibrational energy in the in­termediate state, minus the incident laser energy:

Jev = E° + v(hvR) ηνΛ (4) where v is the occupation number of the Raman mode in the intermediate state and VR is the excited state Raman frequency. We now reduce eq.(3) to a final functional form:

(a Ι η Τ Σ Σ KpCT(e,v) |<0'|ν'>|2 / s* («ρσ'Ο,Ι ~ e¥v' Δθ ν + εν. - ire (5)

where we let revv' be independent of vibronic

where ρ(ε) is the density of states in v' space. The summation over v' space is now completed by performing the integral over ε from 0 to «. Thus we have:

(αρσ)θ,1=ν; ev Kp<?(e?v)FPVU)pQV(£)d£ Δβν + ε - ire (7)

where we have been careful to note how the aver­age FC factors and densities of states in v' space, in general, can depend on the virtual par­ent states |e°)|v>.

The product of FC factors with the density of states function in eq.(7) is quite similar to the product one might describe for constructing

( M a W M p W <0ΐν><νΐΊ> i<Q'iv>r (3)

an absorption band except that the Raman mode under study has been removed for special treat­ment in eq.(7) and is being summed over expli­citly. This fact allows us to make some reason­able assumptions about the functional form of F e v U ) p e v U ) . For the multidimensional heme sys­tem a truncated Lorentzian appears to work quite well in fitting the absorption band shape2. We let the function maximum be at ε=ε0 above the parent state zero-point energy and let the HWHM of the Lorentzian equal δ. The Lorentzian is truncated because there are no states below the zero-point energy. We assume that the same Lor­entzian applies for each term In the Σ . Eq.(7) then becomes:

^°\>y»~\vPa^K Ae%-ire(e-4^ (8)

where Ν(δ,ε0) is the normalization factor for this function and is equal to the area of the truncated Lorentzian. Evaluation of the inte­grals2 yields the final result:

(V}0.1 Σ Kpo(e,v) e v

state. It is important to emphasize at this point that |v'> is multidimensional and that any given energy in this subspace is many-fold degenerate, if only accidentally. We also note that in many treatments the role of the |v'> subspace is elim­inated by setting <0'|v'> = δο',ν* yielding a simple sum over molecular states:

«■„>ο,ι=ί.,£Ώ^- (6) ev Aev - ire where, as before, the Kpa(e,v) are products of electronic moments (Μσ)αο Ρο(ΜΛ)ρθ αο and RFC factors, <0|vxv|l>. 9 ' P *9

We next propose to complete the summation over v' space in eq.(5) by integration. We pro­ceed by fixing εν' at a value ε and sum over only those v' corresponding to states in v1 space hav­ing energy between ε and ε+dε. This sum is re­placed by the average of all Franck-Condon (FC) factors for these states |<0'|v'>|2=F'(ε), times the number of states in this interval, ρ(ε)αε,

ε2+δ2

o (Δ +ε -ir )(Δ +ε -ι'Γ ) + δ ν ev o e'v ev o e;

(9)

where ee is the phase angle for the pole at -Aev+ire and θ+ is the phase angle for the pole at ε0+ιδ. Thus, to asses the effects of the multidimensional subspace on the Raman excita­tion profiles, we need only specify the Lorentz­ian function (via ε0 and δ). In the case of the nearly x-y degenerate heme chromophore we have four terms in the Σ which are simply related2. These terms can be calculated and summed easily with a small computer. The expression for the absorption intensity is quite analogous to eq.(9) and can be calculated in a similar fashion2 so that we are required to fit both the Raman ex­citation profile and the absorption band shape with the same model parameters.

The results of some model calculations for various fitting parameters are shown in figures 1 and 2. Figure 1 contains examples where the non-Raman subspace has been neglected; in this case highly symmetric profiles are predicted. If the subspace is included as outlined above,

V o l . 3 2 , N o . 1 THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING

>-H ω z Ld \-Z

LU >

LÜ or

JU r = 5cm-' W O .

16.5 19.0 21.5 24.0 26.5 ENERGY (kK)

29.0 31.5

Figure 1: Theoretical excitation profiles as a function of the homogeneous broadening parameter, re. Other parameters used in the calculation of these curves are: E°= 23800 cm"1, ES-Eg= 100 cm"1, hvn= 1362 cm"'. The curves are displaced and norm-ma li zed for ease of viewing.

>-

■z. Ld

LU > _l LÜ

r = 200cmH

€0=600cmH

8 = 800 cm'1

r=5cm-' €0=600cm~' 8 =800 cm"1

rs5cm"' €0= 600 cm'1

8 = 1600 cm"1

r=5cm_l

8 * 800 cm"' 16.5 19.0 21.5 24.0 26.5 29.0

ENERGY (kK) 31.5

Figure 2: Theoretical excitation profiles as a function of the Lorentzian distribution function described in the text. Other parameters used in the calculation are the same as figure 1. The curves are displaced and normalized for ease of viewing.

the calculated profiles are asymmetric and skew­ed distinctively to the blue as shown in fig.(2). Figures 3 and 4 contain experimental results as well as theoretical fits to the absorption and excitation profiles of cytochrome c2. The model

parameters used in the fitting are found in Table I. The good agreement between theory and experiment makes it evident that the complete vibrational subspace needs to be included in the theoretical calculations. Moreover, in large Table I: Parameters

E° X

Fe2+ 23800 Fe3+ 23600

by Lx 100 400

Used in hvR

1362 1374

Calcu r 50

300

lations εο

400 400

(cm"1) 6

800 800

molecules like heme, it appears that a substan­tial amount of the absorption linewidth may be due to the large number of vibrational modes, each having a weak FC structure. This view contrasts with the idea that the Soret absorp­tion band is broadened due to an extremely fast (=5 fs) electronic relaxation and/or phase changing process. We wish to stress the fact that the lifetime broadening mechanism results in absorption bands that are symmetric in shape whereas the density-of-states-FC broadening mechanism results in the observed asymmetric band shapes. Phonon broadening of optical spec­tra is also observed in solid state systems* and in many cases where the zero-phonon line is not resolved, one also observes smooth, asymmetric absorption band shapes.

In smaller molecules, however, the FC structure is often resolved due to the presence of fewer, more active, modes. In ordeg to dem­onstrate this case, model calculations have been performed for A-type resonance scattering throughout a single electronic transition in which only two vibrations are both FC and RFC active. The two vibrational frequencies in the ground state (excited state) are taken to be 950 cm"1 (900 cm-1) and 1300 cm"1 (1250 cm"1). Thus, they each suffer a decrease in force con­stant upon electronic excitation. The remaining parameters are the displacements (AQ'S) of the potential energy surfaces along each of the two normal coordinates upon electronic excitation. These are given characteristic values for either "small" or "large" displacements corresponding, in absorption, to a very short FC progression (0-0 strong, 0-1 weak) or an extensive FC prog­ression, respectively. Each individual vibra-tional-electronic transition is assigned a homogeneous band width, re, of about 5 cm"'. However, to simulate librational and/or low fre­quency sequence broadening5, the individual Lorentzian bands are given a Gaussian distribu­tion, resulting in band widths of about 400 cm"' for every vibronic transition built from the two high frequency vibrations under explicit con­sideration. Both the complete absorption (FC) band and the resonance Raman profile for each vibration are computed for all four combinations of "large" and "small" potential energy displace­ment parameters. (This involves explicit compu­tation of the various FC and RFC integrals and the appropriate summation in vibrational space indicated in eq.(3)). The most striking result is the complex nature of the Raman profile pre­dicted for a weakly FC active vibration when the second vibration is strongly FC active. Unlike in a simple one dimensional problem, in this two dimensional case the structure in the profile does not reflect in any obvious way the size of

10 THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING Vol. 32, No. 1

4.8

4.2

INTE

NSI

TY

b Φ

LU > 2.4 !5 ÜJ

* 1.8

1.2

0.6

0

-

"~

-

) J

Δ \

A Δ \ 1 1 Δ ^ ^

1 1 Α Δ

\ Ml Γ / Δ 1

1 M \ I

1f* \ u

Γ Δ ι I 16.5 19.0 21.5 24.0 26.5

ENERGY (kK) 29.0 31.5

Figure 3: Experimental excitation profile of the 1362 cm"1 mode of ferrocytochrome c. The solid dark line is the measured absorption spectrum. The thin solid lines are the result of a theoretical calcu­lation using the parameters in Table I and assume a Boltzmann dis­tribution of ground state energies (T=200 cm"1). The thin upper curve is the calculated Soret absorption band, displaced for ease of viewing. The measured absorption bands in figs.(3) and (4) are scaled to represent the relative concentration of cytochrome present in each sample. The excitation profile data have the v4 dependence factored out and are corrected for monochromator response and reabsorption effects.

Vol. 32, No. 1 THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING 11

16.5 19.0 21.5 24.0 26.5 29.0 31.5 ENERGY (kK)

Figure 4: Experimental profile of the 1374 cm"1 mode of ferric cytochrome c. The solid dark line is the measured absorption spectrum. The thin solid lines are the result of a theoretical calculation using the parameters in Table I and assume a Boltzmann distribution of ground state energies. The upper curve is the calculated Soret absorption band, displaced for ease of viewing. The theoretical excitation profiles in figs.(3) and (4) are not scaled independently(i.e. the weaker scattering in fig. (4) is predicted using the larger value of re.

12 THE VIBRONIC THEORY OF RESONANCE RAMAN SCATTERING Vol. 32 , No. 1

z Lü

UJ >

A (absorption)

B (Raman)

16.0 18.0 20.0 22.0 24.0 26.0 ENERGY (kK)

28.0

Figure 5: The model vibrational-electronic absorption band (top) and the predicted resonance Raman scattering profile (bottom) for the case where the Raman scattered vibration (950 cm"1) is "weakly" (AQ=0.05(amu)*/2 A) Franck-Condon ac tive and a second vibration (1300 cm""1) is "strongly" FC active (AQ=0.3 (amu) 1' 2 Ä) . See the text and especially reference 3 for more details.

the scattered vibrational quantum. In fact it more nearly reveals the presence of the second FC active vibration (the one not being viewed by Raman scattering), but even then not in any reg­ular fashion. We present in figure 5 just one

example ( of the very many possible combinations) which emphasizes how in only two dimensions the resonance Raman profile of a given vibration may be quite complex, provided other vibrations are FC active in the same electronic transition.

J. Tang and A. C. Albrecht, in Raman Spect-roscopy Vol. 2, H.A. Szymanski Ed.,Plenum Press New York (1970).

P.M. Champion and A. C. Albrecht, Journal of Chemical Physics (submitted).

G. M. Korenowski, Ph. D. Thesis, Cornell University (1979) unpublished.

REFERENCES 4a. R.H. Silsbee, in Optical Properties of Sol­

ids, Sol Nudelman and S. Mitra Eds,Plenum Press, New York (1969)

b. D. B. Fitchen, in Physics of Color Centers, W. B. Fowler,Ed., Academic, New York (1968).

5. G. M. Korenowski and A. C. Albrecht,Chemical Physics 00, 0000(1979).

)Sol id State Communications, V o l . 3 2 , pp .13-18 . Pergamon Press Ltd. 1979. Printed i n Great B r i t a i n .

BISTABILITY AND ANOMALIES IN ABSORPTION AND RESONANCE SCATTERING OF INTENSE LIGHT

Y. Toyozawa The Institute for Solid State Physics, The University of Tokyo

Minato-ku, Tokyo 106, Japan

Under intense resonant light, the number density n of elementary excitations with well defined low energy threshold (e.g. excitons) can have more than one stationa­ry solutions, provided that the absorption lineshape depends sensitively upon n. A variety of optical anoma­lies appear since the absorption, emission and scattering spectra are multivalued functions of the intensity and the frequency of incident light. A variation principle on the evolution of this open system, analogous to the second law of thermodynamics, is derived by which one can judge not only the local stability but also the global stability of the multiple stationary states. Bistable domains may be formed as a transient stage, with their boundary moving so as to expand the more stable domain, towards the global stability.

1. Introduction In resonance light scattering, the

electronic system is subject not only to virtual, but also to real transitions to its excited states. As the result, the secondary radiation consists of spectral components with varying degree of correlation to the incident light, depending on the extent to which the relaxation has been completed before emitting the radiation J"1* However, in a certain limiting situation — the so-called rapid modulation case, the spec­tra consist of only two components characteristic of the two extreme situ­ations: (i)Rayleigh line due to elastic scattering without any relaxation of the system and (ii)the ordinary lumines­cence from the fully relaxed excited states which keeps no correlation with the incident light. The integrated intensities of the components (i) and (ii) have the ratio of dephasing time Γ"1 to (twice) the depopulation time γ"1, of the excited states, reflecting the duration times for the correlated and uncorrelated scattering processes. An extension of this rule to the quasi-rapid modulation case in which Γ depends on the frequency ω of incident light has also been made on a certain model system? We shall be concerned here with a spectral instability under intense light originating from the ω-dependence of Γ.

Under intense resonant light, the population n of the excited electronic states will reach high enough value to

influence the width Γ of the absorption spectra through the interaction between excited electrons. The change in absorption coefficient will in turn affect the generation rate of excited electrons. The stationary value of n, determined self-consistently from the balance of generation and depopulation (γ) rates, is a complicated function of ω and I , which can even be multivalued in certain situation. Such a possibil­ity was pointed out in a previous paper? However, the argument on the resulting instabilities of optical spectra was still tentative, since the general principle was missing which governs the relative stabilities between different branches of multivalued spectra. The purpose of the present work is to provide such a principle, and to study the spectral instabilities on a more sound basis.

Incidentally, the problem as formu­lated here belongs to the simplest type of bistability in non-linear non-equilibrium states for which one can unambiguously derive the variation principle — an analogue of the princi­ple of least free energy or greatest entropy for the stability of equilibrium states.

2. Nonlinear Rate Equation For simplicity and definiteness,

we shall hereafter consider the system of excitons in insulating crystal, with number density n, which are generated

13

14 RESONANCE SCATTERING OF INTENSE LIGHT Vol. 32, No. 1 by absorbing incident light, spread with diffusion constant D, and annihi­lated with rate constant γ through radiative and/or non-radiative process­es. With appropriate normalization of light intensity I , the rate equation for n can be written as

3n/at -DV2n = IF(u),n) -yn=G(n). (1) The nonlinearity comes into our problem through the n-dependence of the absorp­tion coefficient F(o),n). Neglecting the saturation effect, we assume the n-dependence to come solely from that of the spectral lineshape, namely, from the n-dependence of the self-energy: Δ(ω,η) +ίΓ(ω,η) of the optically allowed exciton (K = 0). The normalized line-shape, which is given by a pseudo-Lorentzian form7

F(u),n) =τΤ1Γ[(ω Δ) 2 + Γ2]· (2) is asymmetric, usually with high energy tail: the broadening Γ(ω,η) rises rapidly as ω increases beyond the^renor-malized exciton energy E0 = E0 + Δ(E0), in accordance with the rising density of states of the (renormalized) exciton band ER (we assume the normal dispersion :^K-^E0 ) , as shown schematically in Fig.l. For vanishing n, the exciton-phonon and exciton-impurity scatterings contribute to the matrix elements, whereas exciton-exciton collisions become important with increasing n .

relations, is negative around E0 because of the ω-dependence of Γ mentioned above, and its magnitude increases with n (red shift). 3. Homogeneous Stationary Solutions

with Instability Taking into account the (ω,η)-

dependences of Γ and Δ in the lineshape formula (2), we showed in ref. 6 that the homogeneous stationary solution η(ω,Ι) of eq. (1), and hence the absorp­tion spectra F(u),n(a),I)) = (γ/Ι)η(ω,Ι) for a fixed value of I , can be multi­valued function of ω within a certain region of (ω,Ι), depending sensitively upon the functional form of F(o),n). The multivalued absorption spectra were shown by curve (c) of Fig.l in the previous paper (Piand P3 in that Figure should be corrected as P3 and Pi, respectively) and are reproduced in Fig.2, curve (F) of the present paper.

It was also suggested that the domains with locally stable solutions ni and n3 (the multivalued solutions are numbered by the suffix i =1,2,3 in the increasing order of ni) may be formed as a transient stage. To allow for such situation, we have introduced the diffusion term in eq. (1) of the present paper.

com ω τ ω Μ

Fig.2. Absorption spectra F(o),n(u),I)) and the cross-section of Rayleigh scat­tering R(ü),n(u),I)) as functions of ω, for a fixed value of the light intensity I . Thick line represents the stable branch, thin line the metastable branch, and broken line the unstable branch.

Fig.l. The broadening Γ(ω,η) of the exciton absorption spectra as a function of photon energy ω and number density n.

The energy shift Δ(ω,η), connected with Γ(ω,η) through the Kramers-Kronig

4. Global Stability The irreversibility, inherent in

kinetic equation (1), can be brought into a more transparent form of evolu­tion criterion8

(d/dt) | £[n(r,t)]dr = - |on/8t)2dr = 0, (3)

Vol. 32, No. 1 RESONANCE SCATTERING OF INTENSE LIGHT 15 where

<Un(r,t)] EL(n(r,t)) + (D/2) (Vn(r,t))2, rn (4)

L(n) Ξ- G(n)dn i Ω

= γη2/2 -I fn F(oa,n)dn

'0 (5)

Eq. (3) is obtained by multiplying (1) with 3n/3t, integrating over the whole volume of the crystal, and considering the boundary condition that the gradi­ent of n normal to the surface should vanish in the absence of surface source and sink.

The inequality (3) indicates the direction of evolution: the functional Φ = /<£. [n]dr always decreases with time except in the stationary states. The most stable stationary state to which the system finally approaches is that in which Φ takes the lowest possible value (global stability).

For the local stability test, we calculate the differential of Φ up to the second order in the fluctuation 6n(r,t):

6Φ= I dr6n[(3L/3n) -DV2n]

+ | dr|<5n[(32L/3n2) -DV2]6n. (6) The first order differential vanishes for stationary states (compare (1) and (5)), as it should. Of particular interest are the homogeneous stationary states ni , which are the extremum point of L(n). They are locally stable or unstable against any small fluctuation <Sn(r,t) = n(r,t) - ni according as 32L/3n| £ 0 . This is evident from the sign of the second differential in (6) as well as from the kinetic equation (1) linear­ized in 6n(r,t)§

Among the locally stable states, the one with the lowest value of L^ is the stable state and others metastable, since Φ is never-increasing function. Namely, Φ not only serves as a Liapunov function9 around each locally stable states, but also provides the criterion for the global stability. L and Φ will be called respectively local and global potentials.

In Fig.3, we showed schematically the local potential L(n) and its deriva­tives of our system, by reading - 3L/3n = G = IF - γη from Fig. 2 of ref. 6.

5. Bistability and Domain Boundary To substantiate the above statement

on the local and global stabilities, we consider a moving domain boundary of soliton type1^11 n(x - vt) such that η(ξ) tends to the locally stable stationary solutions ni and Π3 as ξ + + °° , as shown schematically in Fig.4. Eq.(1) then reduces to the ordinary differen­tial equation

Fig.3. Local potential L and its derivatives as functions of number density n. - 3L/3n = G = IF - γη repre­sents the net generation rate, ni and n3 represent stable or metastable solutions, and n2 unstable solution.

i = x-vt

Fig. 4. Domain boundary solution n(x-vt) of eq. (1).

Ό(άζη/άξζ) +v(dnAU) - (3L/3n) =0. (7) If n and ξ are reinterpreted as posi­tion and time, respectively, (7) repre­sents the classical equation of motion for a particle with mass D, subject to the potential - L and the viscous force with coefficient v . Multiplying (7)

16 RESONANCE SCATTERING OF INTENSE LIGHT Vol. 32, No. 1 with dn/αξ, integrating over ( - » < ξ < + °° ) with initial and final conditions (n = ni and n3, respectively, and άη/άζ = 0 for both), one obtains

J +co (άη/άξ)2άζ =L 3 -Li . (8) — CO

This indicates that the domain with lower L value expands, taking over the other domain.

The standing domain boundary (v = 0, ξ = x) is realized only when Li =L 3. In this case, eq.(7) has an integral of motion:

| (dn/dx)2 -L(n) =-Li , (9) which can be further integrated to give the standing domain boundary solution nB(x) in an implicit form:

2 1 rnB(x) -A ( £ ) 2 x= dn[Li -L(n)] 2, (10)

Jn 2 with the origin of x chosen so as nB(0) = n2 (see Fig.4). Since L is of the order of γη2, the thickness of domain boundary is of the order of ZD/γ, which is the diffusion length of excitons during their lifetime. Similarly, the velocity of domain boundary when Li L 3 turns out to be of the order of /ϋγ", which represents the average velocity of the diffusing front during the life­time.

6. Analogies to Classical/Quantum Mechanics and Thermodynamics The above analogy to classical

mechanics can be traced to the expres­sion (4), which turns out to be the Lagrangian for a classical particle according to the reinterpretation of n, x( =ξ) and - L(n) as mentioned before. Correspondingly, the variation principle for Φ turns out to be the principle of least action in classical mechanics. This formal analogy implies that in more complicated systems, the spatial insta­bility for pattern formation on the one hand, and the temporal instability for chemical oscillation on the other hand, can be treated from a unified view­point.

While the first order differential in (6) is related to classical mechanics as mentioned above, the second order differential include the Schrödinger operator with potential 32L/3n2 taken as a function of r through n(r). If the two domains ni and n3 are coexistent with standing boundary (Li = L 3 ) , the nodeless function dnB(x)/dx (see Fig.4) is the eigenfunction of the Schrödinger operator with the lowest eigenvalue 0, as is easily confirmed with (9). Namely, the standing domain boundary solution is stable against any fluctu­ation 6n(r), except that it is neutral against the translation: 6n(r) = (dnB/dx) •<5x.

Incidentally, the equipotential condition Li =L 3 for the standing bound­ary can be written as

rn3 G(n)dn = 0. (11)

•'ni This means that the areas of the shaded regions below and above the abscissa in Fig.3(b) whould be equal. The net generation rate G(n) is negative in the region x < 0 (n<n 2), tending to decrease n so as to push the boundary towards the n3 region. This is conterbalanced by the opposite driving force from the other side x > 0 (n > n2) where G(n) is positive, through the diffusion term. (11) is analogous to the Maxwell's equiareal rule for the gas-liquid equi­librium (n-► volume, G -»■ pressure, stationary + equilibrium) . Similar argument was made on the bistability in current-voltage characteristics I2

The analogies similar to those described above have been discussed in different ways for a variety of systems by a number of authors. We refer here only to Haken's review article13and Graham's monograph}h where one can also find comprehensive references.

As can be seen from (4) and (5), the global potential Φ represents the "work" necessary to build up "order" ( n and Vn) against the decay (γ) and diffusion (D) which destroy it, the work done from outside (IF) being counted by minus sign. In the linear regime with small n , Φ resembles conceptually the entropy production!5 For finite n, however, the n-dependence of F(u),n) gives rise to a deviation therefrom which led to the instability beyond the threshold.

On the other hand, the existence of such a potential as a criterion for global stability is limited to one variable (n) case. In the case of many variables, the subsidiary conditions such as detailed balance14 have to be satisfied for the existence of the poten­tial. In fact, we have implicitly made such an assumption in setting up the rate equation (1) with only one order parameter n. For a more complete description of exciton gas, one would have to specify the distribution of excitons in momentum space as well. Moreover, the source and sink of excit­ons are more or less concentrated around K = 0 for various reasons not mentioned here. We have ignored all these facts, and described the exciton gas with n alone on the implicit assumption that mutually colliding excitons are in their own thermal equilibrium determined uniquely by n .

We have confined ourselves to the deterministic aspect of the evolution, disregarding (except in the stability tests) the fluctuation with its distri­bution function. The latter plays important role near the bifurcation points where the instability appears or

Vol. 32, No. 1 RESONANCE SCATTERING OF INTENSE LIGHT 17 disappears, as has been studied in various systems including lasers13*14 and chemical reactions I6 In our system, there are two bifurcation points in the (ω,Ι) plane.

7. Anomalies in the Optical Spectra

As was mentioned in earlier sec­tions, the resonance light scattering as well as absorption spectra depend sensi­tively on the relaxation in the excited states. In particular, the cross-section by Rayleigh scattering of incident light with frequency ω is given, apart from unimportant factor, by1'1*

R(Ü)) =i düi F (ώ) [ώ ίγ/2] (12)

For intense light, one has to put F(ü),n(u),I)), which can be multivalued in ω but not in ώ (variable of integra­tion) . (The absence of bar on the first argument ω of F in ref.6 is a mere misprint). R((D) for fixed I is shown schematically by curve (R) in Fig.2, which is the reproduction of curve (b) in Fig.4 of ref.6. The remarkable reduction in the central region and on the high energy side, with convolution opposite to the absorption spectra in the instability region, is due to the resonance enhancement of n and Γ. One can also reproduce the spectra with the use of eq. (3.6) in ref.5.

Let us now study the bistability in the absorption spectra and the Rayleigh cross-section spectra on the basis of the arguments in previous sections. If the light intensity is uniform through­out the crystal, the homogeneous stationary solution ni of eq. (1) with the lowest value of L(ni) will be realized. As one increases ω for fixed I , the stable state jumps from branch 1 to branch 3 (Pi + P 3 and Ri + R 3 in Fig. 2) at the point ω = ω«τ(Ι) where the equi-areal rule (11) (see also Fig.3(b)) is satisfied. The discontinuity line ω = ωτ(Ι) on the (ω,I)-plane starts and ends at the two bifurcation points mentioned in the end of the precedent section. The number density n , the absorption coefficient F . the dephasing rate Γ and

the luminescence intensity rise abruptly, whereas the Rayleigh cross-section drops.

In a crystal with thickness larger than the absorption length, however, the light intensity I is a decreasing func­tion of depth from the illuminated surface, the domains of high (n3) and low (ni) densities may be formed near and away from that surface. As inci­dence intensity I 0 or ω exceeds the transition point, the high density domain sets in on the illuminated surface, gradually grows penetrating deeper into the crystal and finally reaches the other side. The absorption coefficient, as measured by transmission, will not show any significant disconti­nuity, since it is the average over the two domains with thickness varying continuously with ω or I . On the other hand, the intensity of the Rayleigh line scattered backword or forward may exhibit this discontinuity more signifi­cantly if the thin region near the relevant surface is mainly responsible for the scattered light. The existence of two domains will be seen most directly in the luminescence spectra since the emission bands from them are different in peak position as well as in width.

As for the available evidence (or at least a promising candidate) for the instability described here, we refer to the recent experiment on the two photon resonance Raman scattering by excitonic molecules in CuCl by Mita and Ueta17 and a brief discussion on it in ref.6, leaving more detailed arguments to later publication

In conclusion, we have considered various anomalies in the absorption, scattering and luminescence spectra associated with population instability under intense light. The intensity ratio of Rayleigh scattering to lumines­cence, which is related to the dephasing time, changes abruptly at the transition point. The optical bistability described here presents a simple but instructive example of non-linear non-equilibrium systems for which the variation principle for the global stability is estabilished in unambiguous way.

REFERENCES

1. HIZHNYAKOV V. & TEHVER I., Phys. Status Solidi 21., 755 (1967). 2. HUBERD.L., Phys. Rev. 170., 418 (1968). 3. KUBO R., TAKAGAHARA T. & HANAMURA E., Proc. 1975 Oji Seminar on Physics of

Highly Excited States in Solids, ed. UETA M. & NISHINA Y. (Springer-Verlag, 1976) p.304 ; see also KUBO R. et al., this issue.

4. TOYOZAWA Y., KOTANI A. & SUMI A., J. Phys. Soc. Japan £2, 1495 (1977). 5. KOTANI A. & TOYOZAWA Y., J. Phys. Soc. Japan 4JL, 1699(1976). 6. TOYOZAWA Y., Solid State Commun. 28., 533 (1978) 7. TOYOZAWA Y., Progr. Theor. Phys. 27., 89 (1962). 8. GLANSDORFF P. & PRIGOGINE I., Thermodynamic Theory of Structure, Stability

and Fluctuations (Wiley-Interscience, 1971) Chap. ΊΧ . 9. LASALLE J. & LEFSCHETZ S., Stability by Liapunov's Direct Method, New York

1961.

18 RESONANCE SCATTERING OF INTENSE LIGHT Vol. 32, No. 1 10. DAVIDSON R., Methods in Nonlinear Plasma Theory (Academic Press, New York,

1972). 11. NITZAN A., ORTOLEVA P. & ROSS J., Faraday Symp. Chem. Soc. £, 241 (1974). 12. TAKEYAMA K. & KITAHARA K., J. Phys. Soc. Japan Ü , 125 (1975). 13. HAKEN H., Rev. Mod. Phys. £2, 67 (1975). 14. GRAHAM R., Springer Tracts Mod. Phys. 66. (1973) p.l. 15. DE GROOT S.R. & MAZUR P., Non-Equilibrium Thermodynamics (North-Holland,

1969) . 16. NITZAN A., Phys. Rev. All, 1513 (1978). 17. MITA T. & UETA M., Solid State Commun. 22, 1463 (1978); see also UETA M.,

this issue.

Solid State Communications, Vol.32, pp.19-24. Pergamon Press Ltd. 1979. Printed in Great Britain.

RESONANT RAMAN SCATTERING AND LUMINESCENCE DUE TO EXCITONIC MOLECULE E. Hanamura and T. Takagahara

Department of Applied Physics, Faculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113 JAPAN

Competitive behavior of two channels of resonant Raman scattering and luminescence is theoretically discussed for giant two-photon excitation of excitonic molecule. Time-integrated and time-resolved emission spectra of the excitonic molecule excited resonantly by a short pulse are described in terms of the stochastic theory of intermediate state interaction. Particularly we propose the time-resolved spectrum in which the frequency-time uncertainty due to the finite observation time is incorporated. This theory is shown to explain the observed characteristics of the emission spectra, and the relaxation constants of the excitonic molecule in the intermediate state and the exciton in the final state are determined from comparison between the observed and the calculated spectra.

1. Introduction

An excitonic molecule is a bound state of two single excitons. This excitonic molecule was for the first time observed in the emission spectrum of CuCl heavily excited through the.. band-to-band transition by Mysyrowicz et al· and successively by Ueta and his coworkers.2 The electron to hole mass ratio in this material is very small such as 0.05 so that the internal motion of this excitonic molecule is understood by analogy with that of a hydrogen molecule. The mass ratio, however, is usually of an order of one in many semiconductors and there arises a question whether the excitonic molecule is stable or not depending upon the value of the electron to hole mass ratio. In 1972, we could prove by variational calculation that the exci­tonic molecule is stable for any values of the electron and hole masses.3 Triggered by this theoretical result, the excitonic molecules have been observed in many semiconductors, e.g., in CdS, CdSe and ΖηθΛ In these experiments, the excitonic molecule was formed from two elec­trons and two holes excited in band to band transitions by transferring extra energy to the lattice system. They observed the light emitt­ed when the excitonic molecule radiatively an­nihilated. The genuine study of the excitonic molecule started after proposal of direct and effective creation of the excitonic molecule by giant two-photon absorption.

2. Giant Two-Photon Absorption due to Excitonic Molecule

A crystal is usually transparent to radia­tion field with its frequency less than the exciton band. The excitonic molecule can be directly and very effectively created by irra­diating the crystal by radiation field with reasonable intensity and with tunable frequency by half the molecular binding energy lower than the exciton band. The absorption coefficient becomes as large as that of the single exciton

and is by an order of 10 larger than that of the ordinary two-photon absorption. This ex­tremely strong absorption comes from the mutual enhancement due to the giant oscillator strength and the resonance effect. This giant oscillator strength is due to the fact that in the second photon absorption process which causes the tran­sition from the intermediate state to the final state of an excitonic molecule created, one can excite any valence electron within the large molecular radius around the virtually created first exciton to make an excitonic molecule. The situation is quite different from that which occurs in ordinary two-photon absorption due to band-to-band transition. There, the same electron which is excited into the intermediate state has to interact again with the second pho­ton. The further enhancement comes from the resonant effect. This is due to the smallness of the energy denominator in the expression of the second order perturbation with respect to the electron-radiation interaction V,. One year after this proposal, Gale and Mysyrowicz ob­served this giant two-photon absorption due to the excitonic molecule in the excitation spec­trum and confirmed the expected large oscillator strength. This was observed also in the absorp­tion spectrum7 as a sharp absorption line at the expected position.

The most rapid decay process of the excit­onic molecule created by this giant two-photon absorption is the radiative process in which an electron-hole pair is radiatively annihilated and the other electron-hole pair remains as a transverse or a longitudinal exciton. The for­mer process may be considered as two-polariton scattering because two incident light as well as both the emitted photon and the remaining trans­verse exciton behave as two polaritons inside the crystal.0 Here arises a serious problem whether this emitted light is considered as luminescence or Raman scattering. The excitonic molecule in CuCl is the most suitable subject to study this relationship because 1) the crystal is very transparent to the radiation fields of

19

20 LUMINESCENCE DUE TO EXCITONIC MOLECULE Vol. 32, No. 1

emitted lights and 2) the excitonic molecule has the strong and sharp absorption spectrum due to the giant two-photon excitation which depends on the excitation power. This makes it possible to use the excitation power as another freedom to study the competitive behavior of two chan­nels.

3. Third Order Optical Process

In order to answer the problem of two chan­nels, we study the third order optical process in which two incident photons of ω- are absorbed and an excitonic molecule is created, and sub­sequently a photon of ω« is emitted leaving be­hind a single exciton in the final state e. This latter matrix element is denoted by V2. The probability P(t) of observing the emitted light ω? at time t is given by taking trace of the product of this probability amplitude i['(3)(t) and its conjugate ψ(3)(0* with re­spect to the reservoir coordinate R; P(t)-TrR[*(3)(t)pR*(3)(t)*], (1)

Ψ( 3 ) ω fdtlfldt2P > 0 j 0 > 0

dt0

x <e|exp{-i(t-t1)H}V2exp{-i(t1-t2)H} x v^exp{-i(t2-t3)H}V^exp{-it3H}|g>. (2)

The state propagates to the left and to the right starting from the ground state |g>p <g| at

time 0 and reaches the final state|e>at time t on theHboth ends. We consider the excitonic molecule in the intermediate state and the sin­gle exciton in the final state as the relevant system, and other excitons and excitonic mole­cules created already as well as phonons and imperfections as reservoirs. The interaction between the system and the reservoirs is assumed to be Gaussian and Markoffian process. The giant two-photon absorption process due to for­mation of an excitonic molecule is completed in much shorter time than any involved relaxation times. Therefore we may replace Viexp{-i(t«-t~) H}V~ in eq.(2) by Wmg=-i(V1)ml(V^Jlg/(Wi - ω χ ) J

times 6(t9-to). In evaluation of P(?), we em­ploy the stochastic theory of intermediate state interaction.9 The product of propagators to the left and to the right for a common interval τ, which appears in the expression for the third order optical process, is simplified in the fast modulation limit as Tr^pR{<a|exp{-iTH}|e><d|exp {ixH}|b>} = <Sac6dbexp{-i(A)abT-YabT}. Here the effect of the reservoirs can be described by a relaxation constant Yab= Yab+T^ra+^b)» which is composed of elastic and inelastic collisions γ ^ and decay rates Ta and Γ^ of levels a and b, respectively. For the diagonal component a = b, Yaa (~Ya+ra) i s a s u m o f inelastic collision rate γ| and decay rate Ta.

The diagrams in fig.l are useful not only in calculating the emission spectrum but also in understanding the physics involved. The upper line describes propagation of the state to the

2ω, ^.Ji ( i )

-ex ·

xmg Y* J J— m —L * e x

eg fjt xme \

* - q — * - » i

( i i )

t -ex-

^:1 \

(iii)

2ω,

τ9

' me

■ e x -

m

\

*-ex

ω2

Fig.l. Decomposed diagrams of the third order optical process via an excitonic molecule. Three types of contributions depend on time orderings of the interaction with the radia­tion field, g, m and e denote the initial ground state, the intermediate state with an excitonic molecule and the final exciton state, respectively. The wavy lines re­present photon propagation.

Vol. 32, No. 1 LUMINESCENCE DUE TO EXCITONIC MOLCULE 21 left-hand side and the lower does that to the right-hand side in the density matrix expression. Then we have three kinds of diagrams according to the chronological order among times of two-photon absorption and subsequent emission in the upper line and those in the lower one. For ex­ample, in the time interval τ, in the i-diagram, an excitonic molecule is created in the upper line and it is in the ground state in the lower line, so that the effect of its environment is taken into account by the relaxation constant Ymg as a result of frequency modulation on that energy difference as well as radiative decay. Corresponding to three energy differences, we have three relaxation constants Yme» Yme an(* Yeg. In the time interval i£ in ii- and iii-diagrams, the excitonic molecule is really cre­ated so that the radiative decay and inelastic collision contribute to the decay constant ym. Therefore these two diagrams describe the lumi­nescence process while i-diagram contains the Raman scattering component because this repre­sents the correlation between two polarization operators. 1. Under stationary irradiation by a mono­chromatic laser light with ω.., the emission spectrum Έ(2ω±, (J^) is calculated as t + °° limit of P(t)/t as F(2CO-I,Ü)O) =lim P(t)/t

γ γ mg 'me 2 2 2 2 (2ωΊ-ω ) +ymn (ω9-ω ) +γ 1 mg mg z me me

γ {1+(2ωη-ω )(ω0-ω )/γ γ } -rJ-|- e g 1 m g 2 me//Tmg'me Ύ 'm 2 2 (2ωΊ-ω9-ω ) +γ 1 2 eg eg (2ω--ω9-ω ){(2ω -ω)/γ -(ω9-ω)/γ } 1 I eg 1 mg mg I me me ·.

2 2 (2ωη-ω0-ω ) +γ 1 2 eg7 'eg (3) The first term in the angular parentheses came from ii- and iii-diagrams in fig.l and describes the luminescence spectrum. This is also realiz­ed from the fact that the peak frequency u^ due to this term is independent of ω^. On the other hand, the second and third terms originated from i-diagram and contain the Raman component as read from that the emission peak α^ due to these two terms shifts with change of 2ω^. The ab­sorption spectrum due to giant two-photon exci­tation of an excitonic molecule is represented by the first factor of eq.(3) beside unimpor­tant constant. Therefore γ is determined by full width at half the maximum in the absorption spectrum. yeg is fixed by half width at half the maximum of the Raman component and yme is determined by half width at half the maximum of the luminescence component in the emission spec­trum. The last remaining decay constant γ will be fixed from the ratio of Raman peak to lumi­nescence peak as ί ΐ + ^ ω ^ ω ^ ^ / γ ^ γ ^ / γ ^ versus 2/ym. 2. In real experiments, they use laser pulse with some distribution of carrier frequency as the excitation source. When the intensity dis­tribution of two-photon excitation frequency 2ωι has, e.g., the Gaussian form f (2ω^) = 1//2τΓχ a^exp [-(2ωι-2ω^)2/2σ^], the observed emission spectrum is represented by the following super­position:

ioo d(2Wl)f(2Wl)F(2W ,ω ). (4)

— 00 3. Furthermore when the incident pulse has the Gaussian envelope in time: exp[-(t02/2)2] and the emission spectrum is integrated in time, the emission spectrum is given by P(~) = Tr[ψ(3) ( t - ) p j ( 3 ) (t—)*], where m

*(3)(t)=-±-\ «1(2(0. γπσ) _οο λ

2ω,-2ωΊ 9 )exp[-( Lo V]

/ —00 ■/ —00

^icu^+iü)^ -iHCt-tJ dt2e <e|e

-iH(t -t ) -iHt9 x V^e L — Z' We |g>.

Then the spectrum is shown to have the same form as eq.(4) with the deviation σ = /σ^ + Q'I .

When a value of σ is less than the relaxa­tion constantsγ and γ , the Raman and lumi­nescence lines are clearly resolved. On the other hand, the Raman line is smeared out and only the luminescence line is observable for the much larger value of σ than Ymg and yeg. These facts were already observed.' Ueta and Mita^O observed coexistence of the Raman and lumines­cence lines due to the excitonic molecule in the time-integrated emission spectrum under the giant two-photon excitation by the narrow band laser pulse. The relaxation and decay constants are uniquely determined and eq.(4) gives the emission spectrum which describes whole of the ob­served characteristics as shown in fig. 2 . In the second experiment, the spectral width of ex­citation light was increased by an order of mag­nitude. Then the Raman line broadens by the same order of magnitude while the luminescence line is still sharp because it is independent of the spectral width of the incident light and is determined by yme. These observed characteris­tics can be represented as shown in fig.3 by the calculated spectra in terms of the same relaxa­tion and decay constants in fig.2. It is noted that another broad emission line is observed in the experiment and that this is assigned to the secondary (and higher) step emission of the hot excitonic molecules formed from two single exci-tons left behind in the first (and higher) step emission processes.

4. Transient Emission Spectrum

The competitive behavior of the Raman and luminescence channels on time coordinate is studied by the time"resolved emission spectrum after the pulse excitation. Let us assume the observation system open between T and Τ+ΔΤ. The probability amplitude that the system falls into the final state in this time interval is given by Δ ψ ( 3 ) ( Τ )

(Τ+ΔΤ rt - 2 ί ω ρ 2 + ί ω 2 ί 1 ' d t 2 e ( t 2 ) e

-iH(T+AT-t ) - i H ( t - t ) - i H t 9

x e V2e We |

22 LUMINESCENCE DUE TO EXCITONIC MOLECULE Vol. 32, No. 1

where ε (t2)= fd(2Wl)erp[-i(2Wl-2WJ)t2]

0 2 2 x 1/Jno βχρ[-(2ω--2ω1) /σ ] ,

Relative Excitation

M, Ymg=0.086mev Yme=(X152

Energy(2ujrUJ JYeg =0.066 f 0.157 m7v ^Ym =0.05 9 0.215 h 0.314

FWHM of incident light =0.053 mev

1.0 0.5 0.0 Relative Photon Energy (mev)

Relative L Ymg=0.086mev Excitation Yme=0.152 Energy(2uuruj ) Yeg =0.066 f0.632mevmgYm=0.05 90.885 FWHM of incident M.839 light =0.4 mev i 2.908

3.0 2.0 1.0 0.0 Relative Photon Energy (mev)

Fig.3. Calculated emission spectra via exciton-ic molecule with recoil of a longitudinal exciton under wide band excitation (FWHM = 0.4 meV).

Fig.2. Calculated emission spectra under the giant two-photon excitation of an excitonic molecule with recoil of a longitudinal ex­citon under narrow band excitation (FWHM-0.053 meV). The origin of the abscissa cor­responds to M£-luminescence line.

and this describes the incident pulse corres­ponding to the giant two-photon excitation of an excitonic molecule. Here both the effects of carrier frequency distribution and of pulse ex­citation are taken into account by the deviation σ. Under the assumption that the amplitude of the emitted radiation is proportional to the probability amplitude of finding the final state of the total system in the time interval T and Τ + ΔΤ, we define the emitted light intensity by

I(T) = -^Tr [Δψ(3) (Τ)ρκΔψ(3) (Τ)*]. (5)

In our previous paper, we defined provisionally the time-resolved emission spectrum in terms of ι|Λ3)(0 in eq.(2) as

Kt) ^τΓ[ψ(3 )(Ορκψ( 3 )(0*]. (6) We may say that the definition of I(t) and I(t) come from the particle and the wave pictures for the emitted light, respectively. Unfortu­nately, we can not guarantee the pojgitiveness of I(t). As to the new definition I(t), this is free from that drawback of I(t) and the fre­

quency-time uncertainty due to the finite ob­servation time is suitably incorporated. For the detailed expression I(T), see the paper. Under the excitation by rectangular pulse 0 < tn < τ, the Raman component disappears suddenly after the pulse is switched off (τ<Τ), while the luminescence component decays with the radiative life time l/"Ym· This is in contrast with the result in the previous theory that the Raman component survives for the order of the phase relaxation time. We show one model cal­culation in fig.4 under the Gaussian pulse ex­citation at t = 0 with the half time width 6.04 (meV)~l. The emission intensity becomes to be concentrated on the Raman line initially. After the pulse peak is over, the emission peak shifts smoothly in time from the position of the Raman line to that of the luminescence line. We have also the case in which both lines persist for all the time region depending upon the relative value of off-resonance (2ωι-Ü^ ) and relaxation constants.

Segawa et al. observed the time dependence of the emission peak intensity under resonant excitation of the excitonic molecule by the giant two-phonon absorption, and they resolved the Raman and luminescence components according to the population dynamics.*2 The calculated time dependence of both the components is shown in fig.5. Here the Raman intensity is shown to follow the same time dependence as the incident pulse while the luminescence component decays

Vol. 32, No. 1 LUMINESCENCE DUE TO EXCITONIC MOLECULE 23

Ymg=ai5 Yme=025mev Yeg =0.1 Y m =0.1 FWHM of incident light=0.5mev off resonance energy of incident light = -06 mev Pulse Duration Time=25ps Duration Time of Observation=25 ps

t=10

Pulse Envelope

=-10(mev_l) 10 Relative Photon Energy (mev)

F i g . 4 . Time-resolved emiss ion spectra under e x c i t a t i o n by a Gaussian p u l s e . The time coordinate i s drawn in u n i t s of (n 4.136 p s .

e V ) " 1 ^

Ymq=0.3 Ym e=0.5 mev Veg=0.169 Ym=0.02 FWHM of incident light = 0.5mev Pulse Duration Time = 25 ps Duration Time of Observation

=25 ps

TimeCmev )

Fig.5. Decay profile of Μγ line under just reso­nant excitation of excitonic molecule. The total emission intensity is decomposed into Raman and luminescence components as shown by dashed line.

with the radiative life time l/ym. The relaxa­tion and decay constants have been determined from the time-integrated emission spectrum. Then the time dependence of fig.5 represents the observed features very well.

5. Discussions

The competitive aspects of the Raman and lumines­cence channels are studied both in the emission spec­trum as a function of frequency and in the time dependence of emitted light intensity. The re­

laxation constants used in figs. 2 and 3 have been determined in the weak excitation case but they increase as the excitation power increases. This is because the collisionsof the excitonic molecule in the intermediate state and the sin­gle exciton in the final state with other highly excited molecules contribute to the relaxation constants γ_Λ, ymo and γ . The increase of ymcr mg me eg ' m & with increase of excitation power was observed as broadening of the giant two-photon absorption spectrum due to the excitonic molecule.7 The ratio of the peak value of the Raman line to

24 LUMINESCENCE DUE TO EXCITONIC MOLECULE Vol. 32, No. 1 that of the luminescence line is given approxi­mately by 1/γβδ{1+(2ω1-ωπ18)2/γΙΙ18γΙΙ1θ} versus 2/ym and this ratio is expected to decrease as the excitation power increases.

We could determine the values of relaxa­tion and decay constants by comparison between the observed and calculated emission spectra. These values look to be of an order of reasona­ble magnitude. The calculated spectra catch also the characteristic features observed by Segawa et Ά1.±Δ both for the ML and MT process­es leaving behind the longitudinal and trans­verse excitons, respectively. It is noted that the Raman and luminescence lines have the same order of magnitude for the ML process, while the Raman line in the MT process is much

stronger than the luminescence line. This is clear because the peak value of the Raman line is approximately proportional to the reciprocal of Yeg and the value of yeg is much smaller for the MT process than for the ML process. This is understood as follows: In the ML process, the inelastic scattering of the longitudinal exciton into the transverse one plays an important role in determining the value of γ , while in the M-p process the reverse process is quenched at low temperature. At present, almost all experiment­al facts have been analyzed in terms of our sto­chastic theory in the fast modulation limit. It will be more interesting to look for effects beyond the damping theory.

References

A.Mysyrowicz, J.B.Grun, R.Levy, A.Bivas and S.Nikitine: Physics Letter _26A (1968)615. H.Souma, T.Goto, T.Ohta and M.Ueta: J. Phys. Soc. Japan 9 (1970)697. O.Akimoto and E.Hanamura: J. Phys. Soc. Japan J33 (1972)1537. See, e.g., E.Hanamura, in Optical Properties of Solid, ed. by B.O.Seraphin (North-Holland Amsterdam, 1976) ch.3. E.Hanamura: Solid State Commun. Y2L (1973) 951. G.M.Gale and A.Mysyrowicz: Phys. Rev. Letters 2 (1974)727.

10. 11. 12.

M.Ueta and N.Nagasawa: Lecture Notes in Physics .57 (Springer-Verlag, 1976)p.l. T.Itoh and T.Suzuki: J. Phys. Soc. Japan 45 (1978)1939. T.Takagahara, E.Hanamura and R.Kubo: J. Phys. Soc. Japan 43 (1977)802, 811, 1522; ibid. _44 (1978)728, 742. T.Mita and M.Ueta: Solid State Commun. 27 (1978)1463. E.Hanamura and T.Takagahara: to be published in J. Phys. Soc. Japan. Y.Segawa, Y.Aoyagi, O.Nakagawa, K.Azuma and S.Namba: Solid State Commun. 27 (1978)785.

Solid State Communications, Vol.32, pp.25-28. Pergamon Press Ltd. 1979. Printed in Great Britain.

RECENT DEVELOPMENTS IN NON-LOCAL OPTICS

Joseph L. Birman Department of Physics, City College, CUNY, New York, N.Y. 10031

Dramatic optical effects can occur in non-local, or spatially dispersive, media when both the wave-vector ancj. frequency dependence of the linear non-local dielectric coefficient e(k,kT, ω-ω1) play a role. In this paper we explore four classes of phenomena related to optical non-locality occurring near an exciton resonance in a semiconductor. These are: 1) Resonant Inelastic (Brdllouin and Raman) Scattering predicted by Zeyher, Brenig, and Birman and now a very active area of Light Scattering; 2) Transient Optical Phenomena: the precursors predicted by Birman and Frankel,especially the new Exciton Precursor have not yet been found; 3) Lateral Beam Shift and Selective Internal Reflection: new effects predicted by Birman and Pattanayak have not yet been identified: they can probe surface structure: 4) Gyrotropic-non-local effects, which can arise due to linear wave-vector effects at a resonance, give novel results for propagation and scattering of circularly polarized light,

I. Introduction Non-local, or spatially dispersive, optical

phenomena arise in frequency regimes where both the wave vector, and frequency (Jej endence of the dielectric response function eC^k1; ω-ωτ) plays an important role.l The underlying physics generally involves coupled, or "moving" excita­tions created by the radiation. A simple yet very fruitful "model system" involves phenomena at optical frequencies near an exciton resonance in a semiconductor like CdS, GaAs or the like. One consequence of non-locality is the existence of "anomalous", or extra modes propagating for laser frequencies ω > ω« where ωρ is the longi­tudinal exciton frequency. The "extra" modes are mixed exciton-photon polaritons. In a bounded medium the propagating "physical polari-ton" is in general a superposition of three waves; each with its own dispersion, and with frequency dependent mixing coefficients determ­ined by the Maxwell Boundary Conditions and the Additional Boundary Conditions. In the simplest geometry: normal incidence of plane polarized monochromatic radiation from vacuum upon the plane boundary (vacuum/non-local medium) the propagating exciton-photon polariton in the medium consists of two coupled transverse plane waves.

To be specific we recall an explicit model which has received considerable currency for exciton-polaritons and which remains useful, despite some serious limitations: the so-called "dielectric approximation". Represent the exci­ton as a "crystal oscillator" with energy ω = ω + bq2 where ft = 1, b - 1/2 M, M is the exciton mass; thus exciton motion (kinetic energy) is included via the finite exciton mass. Linear response theory gives the non-local, linear

("longitudinal") susceptibility in general as X(r,r',u>) = χ0 o(r-r') + (e/ω)2 £ {(<0|j(r)|q> <q|j(rr)|0>)/(ü3q + ω + in) + h.c?} with <0|j(?)|q> the current operator matrix element between states |Q>, and |q>.Near the bare exciton resonance frequency ω * ωη the "dielectric approximation" takes the form:

X(k, ω) = xf + F/(ω02 + bk2 - ω2 ΙωΓ).

(1) In a bounded medium this form is evidently incorrect since Fourier Transformation of (1) produces a spatially translationally invariant susceptibility: x(r-r*) which is inconsistent with broken translational symmetry : a boundary present. Physical quantities which determine, or could be determined from, χ(ζ,ω) are: χη: background susceptibility; F the oscillator strength (or current matrix elements); ω^ the "bare" exciton frequency; M the exciton mass and Γ a damping coefficient. In a non-local medium wj.th dielectric coefficient e(k,ü)) * 1 + 47rx(k,a)) plane waves obey Maxwell's equations; £he dj.spersjJ.on equation for plane waves is k = k0^ e(k,ü)) where k_ = (ω/c) is the vacuum wave number of the wave: Ordinary (local) dis­persion theory is recovered if M ■+ °°. Since the dispersion equation is of fourth degree two sets of waves propagate, corresponding to right and lef£ running plane waves with wave numbers k-(ω), k2(ω). The presence of a boundary determines the "physical mode" structure which is (at given frequency ω): incident plus reflected waves in vacuum, joined to the correct linear combination of the two "polarization" plane waves: P. and P2· The physical polariton can be represented by the total polarization

Supported in part by ARO-DAAG29-79-G-004Q; NSF-DMR78-12399 and BHE-PSC CUNY - RF 12343.

25

26 RECENT DEVELOPMENTS IN NON-LOCAL OPTICS Vol. 32, No. 1

TOT J A. (ω) exp i [k. (ω) z - ut] j-l j j (2)

where the Fresnel factors A,(ω) give the relative amplitudes of the waves P- and P„. The boundary conditions which determine the A (ω) are the Maxwell Boundary Conditions (continuity of n x E and n x H) and the Additional Boundary Conditions. It has been shown that to each functional form of dielectric coefficient corresponds a specific abc, so the electrodynamic problem is fully determined once the dielectric function is given. If the dielectric approximation (1) is assumed to hold right up to the plane crystal-vacuum boundary (J) there corresponds the abc:

(n-L - Ύ)

where n,

(n9 " Ύ) 0 (at I) (3)

*i/C "1 ~ i/o* n2 " k2/k0 a r e t h e r e indices (at i) of tne two modes: E . E_ electric field amplitudes of waves k., and k« at

fractive ar£ the

frequency ω and at the boundary £> an<* ^γ(ω) - k where (k+)~1 is the range of the non-local part of the kernal χ(£,ω).2

To emphasize the distinction between dif­ferent functional forms of ε($,ω) and their associated abc we give some other abc which have been used. Thus, vanishing of exciton-polariton polarization

(P1 " X0 V + (P2 " X0 V = ° (at Σ> <4> corresponds to a non-translationally invariant susceptibility suitable for Fresnel excitons.^ Vanishing of exciton-poTariton flux

n1(P1 - χ0 Εχ) + n2(P2 - Χ ρ Ε2) (at £) (5)

corresponds to a non-translationally invariant susceptibility suitable for Wannier excitons.2

It is abundantly clear that any quantita­tive theory of physical properties or inter­actions involving exciton-polaritons requires the correct abc for that system, in order to precisely define initial/final state of the "physical polaritons"; i.e. the correct compo­sition of each in terms of the constituent waves k and k .

To recapitulate: the exciton-polariton excitations in an insulator for frequencies near exciton resonance have a rich structure involving a dual coupling: a) of exciton to photon yielding several polariton plane waves and, b) of the external (incident plus reflected) photon to the internal coupled polariton waves by means of the boundary J, and the boundary conditions. Theory and experiment must be con­cerned with analyzing the structure and dynamics or interactions of these polaritons. To this end we shall briefly discuss four classes of phenomena related to the non-locality.

II. Resonant Inelastic (Brillouin, Raman Scat­tering

3 Zeyher, Brenig and Birman first gave a quantitative theory of inelastic (Brillouin)

scattering based on the non-local, polariton, picture and the multi-mode description of exciton polaritons. The kinematics of inelastic scattering is simple. Let the functions ω (k) be the dispersion equations of the physical exciton-polaritons: the index j accounts for multibranch effect for finite mass. An inelas­tic scattering event can occur from ω, (k) -»· ωτ ,(kf) if the frequency shift Δω = ^ω (£) -

-> ->· -J ->■ ->■ ω1 T(kf) and pseudomomentum transfer q = k1 -k satisfy the dispersion equation of some other physical crystal excitation Aü)(q). For example for Resonant Brillouin Scattering (RBS) Aü)(q)=c|q| c is a sound velocity; for Resonant Raman Scat­tering (RRS) Δω(α) = ω : essentially independent of q. The dispersion of the quasiparticle pro­duced "tunes" the scattering process. Branch indices jTjf may or may not correspond. Most of the practical cases require "backward" scatter­ing so 1tT = -|kr|. Certain predictions of this model can give dramatic evidence of the existence of extra branches in the physical ωρ; and the onset of new channels for inelastic scattering. The extra modes yield a Brillouin "octet" instead of the familiar Brillouin "doublet" which occurs for OL « ω«. Since the initial experimental verification of these predictions^ additional work has revealed (for ω^ well above ω«) multi-phonon, multistep Brillouin and mixed Brillouin-Raman Scattering. Forbidden scattering has been observed also.

In addition to kinematic effects, the ZBB theory quantitatively predicted the frequency dependence of the RBS cross section, and the line shape of the Brillouin scattered radiation. The theory of ZBB compared quantitatively the predicted results due to use of either of the three abc (or corresponding 8(k,&0 and demon­strated the possibility of distinguishing the applicable abc, and thus determining the struc­ture of the physical polariton mode.

Since another paper in this Symposium will discuss the current experimental develop­ments in RBS, show how exciton mass and polari­ton branch disperion above resonance can be determined from RBS as well as compare the theory with the present experiments, we defer further discussion to that paper.

III. Transient Optics An entirely different manifestation of

non-locality can be probed by investigating linear transient optical response - such as pre­cursors. Birman and Frankel" extended the Sommerfeld-Brillouin classical precursor theory to non-local media. As a result of the existence of new polariton modes, new "Exciton Precursors" are predicted to arise if a temporally chopped laser excitation ω < ω. is applied. Changes in signal velocity occur for ω < ωγ < ω« . These effects have not yet been observed. They can provide added information about polariton dispersion and the dynamics of the "build-up" of reflected/refracted fields and also relate to the Time Dependent Extinction Theorem discussed by Pattanayak and Birman7.

A brief resume of the theory may be use­ful at this point. Assume the simplest geometry:

Vol. 32, No. 1 RECENT DEVELOPMENTS IN NON-LOCAL OPTICS 27 normal incidence from vacuum of a linearly polarized monochromatic laser at OL upon a bounded non-local medium with plane surface J at z=0. Let the time varying amplitude of the incident wave be represented at z=0+, t=0 by a cut-off laser of amplitude :

f (0 , t) = ^sin c^t 0(t) (6) The total field amplitude in the medium at (z,t) can then be represented by generalizing Sommer­feld' s complex integral representation for the local case. We have

f(z,t) = Re 2π /

ρτοτ ( ω ) (üMt^) du) (7)

where P i s given above in ( 2 ). The entire space- ana time-development of the amplitude is contained in this integral. To extract the time development (for fixed z) asymptotic analysis has been used (saddle point method), and the detailed results are given elsewhere. We may represent the results qualitatively as

f(z,t) - I fa(z,t) 6(t-ta) + fL(t) 6(t-tL)

(8)

where a - SP, BP, EP stand for three partially overlapping packets of precursors: "Sommerfeld Precursor" (SP) which contains "high" frequency information (ω > ωη) from the photon-like polariton dispersion Qurve (inner branch); the TTBrillouin Precursor" (BP) which contains "low" frequency information (ω < c O from the inner branch dispersion curve; and the "Exciton Pre­cursor" (EP) which probes the "high" frequency (ω > ωη) dispersion of the "outer" or additional (exciton-like) branch. The actual forms of f (zt) are given elsewhere: generally they are Bessel (or Airy) functions of (t-t ) where t is an appropriate ''latency" or delay-time. The laser "signal" fL(t) arrives last, after the precursors, and with a delay given by the "signal arrival" delay time t . Actual numeri­cal values of the delay times, relative ampli­tudes of the precursors and the like depend on the particular non-local medium. The precursor scenario in CdS (a non-optimum material) occurs in the range of 0.1-10 psc which is within present day capability. Other materials may well have much more favorable (longer) t .

An analysis of transient reflectivity from a non-local medium reveals additional interesting possibilities. Since reflectivity probes the region ωρ < ω < ω between transverse and longitudinal "bare exciton" frequencies much information can be obtained regarding surface­like (evanescent) polaritons which "live" in this frequency region. The structure of the precursors gives detailed information about these excitations.7 An analysis of transient transmittivity from a thin non-local slab has been recently given and indicates the possible utility of that geometry.7

IV. Lateral Beam Shift and Selective Total Internal Reflection

When a plane polarized physical polariton (coupled wave) travelling in a non-local medium impinges on a plane boundary with a local medium, "selective total internal reflection" (STIR) phenomena may arise. Each wave satisfies its own Snell's Law, so one may have STIR if one or more of the constituent waves is at or above its proper critical angle Θ . Each wave is pre­dicted to be"laterally °^ displaced" as in the usual theory : in addition the coupling produces a displacement of one wave at the critical angle of the other; also novel interference pheno­mena in the evanescent waves at the boundary. Interesting surface-related phenomena may be revealed by investigation of these phenomena. A mathematical analysis of these effects with estimates of the magnitude of lateral shifts and other interference effects is in progress and will be reported elsewhere. The physical situation requires the assumption of an incident laser beam (which we take to have Gaussian pro­file across its plane front). The Gaussian beam, in the non-local medium, is decomposed into physical polariton modes i.e. coupled waves - each physical polariton characterized by a mode wave number. The reflected waves are regrouped into the reflected beam which reveals the displacement and new relative amplitude structure vis-a-vis its composition. Polarization effects (e.g. use of circularly polarized physical polaritons of σ and π radiation) will produce a transverse beam shift.

V. Gyrotropic - Spatially Dispersive Phenomena

Linear finite-wave vector effects in e(k,oo) may arise due to symmetry breaking in isotropic crystal classes as well as in natur­ally optically active crystals.1 When both linear and quadratic effects are present some very unusual phenomena may result. Consequences for physical polariton structure, inelastic scattering of polaritons, and boundary value phenomena are now in course of investigation. Using a formalism based upon an inverse dielec­tric function with linear wave vector present, we can justify the constitutive relation:

? + (4ττγ(ω)/ε, ) χΡ = α(ω)Ε - γ(ω) χ Ε b (9)

where spatial dispersion-gyrotropy are related to the second terms on right and left sides of this equation: if γ ■ 0, local non-gyrotropic optics results. Here γ(ω), α(ω), and Gf are fully determined from the given inverse dielec­tric function e~l(k,U)). Among the consequences of this constitutive relation is an unusual structure of the physical polariton field. The wave number of propagating modes satisfies a sixth degree equation which admits a set of three "right running" waves: two are left (right) circularly polarized, the third is right (left) circularly polarized which alternative holds depends on the relative signs of γ and R. For "left running" waves, interchange left or right. As in the earlier discussion, for a bounded medium, the full set of boundary conditions determines the correct linear combination which is the physical polariton. Elsewhere we shall discuss the abc appropriate to this situation, as well as results of calculation of reflec­tivity and scattering process.

28 RECENT DEVELOPl^ENTS IN NON-LOCAL OPTICS Vol. 32, No. 1 VI. Conclusion plasmas,... and waiting for new effects to be

-. ^ ^ -i JJ J i discovered. Non-locality, or spatial dispersion, is

alive and well, and living in condensed matter,

REFERENCES

1. The early literature (pre-1966) is reviewed in AGRANOVICH, V.M. and GINZBURG, V.L. "Spatial Dispersion in Crystal Optics" (John Wiley, 1966). This remains a useful reference. A more recent collection of articles is in "Polaritons" ed by BURSTEIN, E. and DE MARTINI, F. (Pergamon Press 1974).

2. BIRMAN, J.L. and SEIN, J.J., Physical Review ]B6, 2482 (1972); ZEYHER, R., BIRMAN, J.L., BRENIG, W. Physical Review B£, 4613 (1972); TING, C.S., FRANKEL, M.J., BIRMAN, J.L., Solid State Communications 17, 1285 (1975). See also AGARWAL, G.S., PATTANAYAK, D.N., WOLF, E., Physical Review BIO, 1447 (1974); MARADUDIN, A.A. and MILLS, D.L., Physical Review B7_, 2787 (1973).

3. ZEYHER, R., BRENIG, W., and BIRMAN, J.L., Physical Review B6, 4617 (1972). 4. WEISBRUCH, C , ULBRICH, R., Physical Review Letters J3£, 865 (1977);

WINTERLING, G., KOTELES, E., Solid State Communications ^3, 95 (1977); and private communications; YU, P. (to be published) and private communications.

5. YU, P., Solid State Communications (this volume). 6. BIRMAN, J.L., and FRANKEL, M.J., Optics Communications J3, 303 (1975);

FRANKEL, M.J., and BIRMAN, J.L., Physical Review A15, 2000 (1977). 7. PATTANAYAK, D.N., BIRMAN, J.L., Optics Communications 25^ 281 (1978);

BIRMAN, J.L., FRANKEL, M.J., in "Coherence and Quantum Optics IV (1978) ed. MANDEL, L. WOLF, E. (Plenum Publishing Co., New York) pages 197-207? T).L. Johnson, Phys. Rev. Lett.(1978)

8. PATTANAYAK, D.N., BIRMAN, J.L., (to be published). 9. ARTMANN, K., Annals of Physics (b), _2, 88 (1948);

LOTSCH, H., Journal of the Optical Society of America J5£, 551 (1968); HOROWITZ, B.K., and TAMIR, T., Journal of the Optical Society of America 61, 586 (1971).

10. PATTANAYAK, D.N. and BIRMAN, J.L. (to be published).

Solid State Communications, Vol.32, pp.29-32. Pergamon Press Ltd. 1979. Printed in Great Britain.

BRILLOUIN SCATTERING O F EXCITON POLARITONS A N D ADDITIONAL BOUNDARY CONDITIONS

P e t e r Y. Yu

IBM Thomas J . Watson R e s e a r c h Center Yorktown Heights , New York 10598

The impor tance of exciton damping, Γ , on the Bri l louin sca t te r ing efficiencies is pointed out. Bri l louin sca t te r ing efficiencies have been calculated as a function of Γ using two different additional boundary conditions (ABC). F o r Γ l a r g e r than some value ?c , a s a s sumed in an e a r l i e r calculat ion of Brenig , Zeyher and Bi rman , the dependence of the sca t te r ing efficiencies on incident photon energy is r a the r insensi t ive to the ABC used. However, for Γ smal l compared with Γ the sca t te r ing efficiencies calculated with differ­ent ABC's show quali tat ive di f ferences .

The theory of resonant Bri l louin s c a t t e r ­ing of exciton polar i tons was f i rs t p resen ted by Brenig, Zeyher and Bi rman. These authors showed how the f requencies , linewidths and s ca t ­tering efficiencies of the Bril louin peaks depend on the incident photon energy. In pa r t i cu la r they pointed out that the var ia t ion of the sca t te r ing efficiency with photon energy is sensi t ive to the so-cal led additional boundary conditions (ABC). These ABC's a r e n e c e s s a r y for descr ibing the e lec t rodynamics at the surface of a spat ial ly d ispers ive medium. As an example, Brenig et al . calculated the scat ter ing efficiencies of longitudinal acoustic phononsinCdS for th ree different ABC's . However, the scat ter ing effic­iencies they obtained show mainly quanti tat ive differences. The purpose of this a r t i c l e is to point out the impor tance of the exciton damping p a r a m e t e r Γ in compar ing scat ter ing efficien­c ies calculated with different ABC's . As pointed out by Tait there is a c r i t i ca l value, Γ€ , for each ma te r i a l and the polari ton effect is signif­icant only when Γ is sma l l e r than ^ c . In such c a s e s I find quali tat ive differences between the scat ter ing efficiencies calcula ted with differ­ent ABC's .

Let us a s s u m e an exper imenta l geomet ry as shown in Fig . 1. A radiat ion field E^ is incident normal ly on a spatially d i s p e r s i v e , s e m i -infinite medium. The die lec t r ic function of the medium is a s sumed to be given by the e x p r e s s -

.4

VACUUM

E j _ ( k j ^ ) _

Er (-kj,cüj)

Es(ks,oüs)

SPATIALLY DISPERSIVE MEDIUM €(k,u)

E'o

LA

Fig. 1 Schematic rep±-esentation of a resonant Bri l louin exper iment using a semi- inf ini te , spa t i a l ly -d i spe r s ive medium. Broken, solid and wavy l ines denote respec t ive ly photons, polar i tons and longitudinal acoustic phonons. The l a rge rectangle r e p r e s e n t s polar i ton-LA phonon in terac t ion .

where ε 0 is the background, local d ie lec t r ic constant; ft ω^ and ft ω,*, a r e respec t ive ly the longitudinal and t r a n s v e r s e exciton energ ies ; and M and Γ a r e respec t ive ly the effective m a s s and damping p a r a m e t e r of the exciton. In genera l Eq. (1) has two complex solutions,

ε ( ^ ω ) / ε 0 = c Z k Z A 2 / 2 Z . , . Z j * e 0 * l + ( u L - a ) T ) / ( u ) T +

5iL M

i ωΓ ) (1)

29

30 BRILLOUIN SCATTERING OF EXCITON POLARITONS Vol. 32, No. 1

kj and k^ for a given ω, corresponding to .two propagating modes which are called usually the upper and lower polaritons. Inside the medium the polaritons are scattered by longit­udinal acoustic ( LA ) phonons. In this article I shall consider only the case where a lower polariton ( E^ ) is backscattered into another lower polariton state ( E^) which then produces a scattered photon outside the medium.

The calculation of the scattering efficien­cy, η , can be decomposed into two parts: *» -> (a) the evaluation of the transmission coefficients of the incident and scattered photons at the sample surface; and (b) the calculation of the scattering cross section of the polaritons by LA phonons inside the sample using perturbation theory. By noting that the ,exciton-part l of the polariton couples more strongly to LA phonons than the 'photon-part1, one obtains the expres s ­ion: 1» 5

while ABC3 is given by: '

N i dft 8ττ2ΚΔ \ c / v i V s ( Ki+Ks)

In Eq. (2) N^ is the number of incident photons per unit time; dN s /df i is the number of scatt­ered photons per unit time per unit solid angle; ω3 is the scattered photon frequency; v, x and P denote respectively the velocity of energy transport, the attenuation coefficient and exciton polarization of the polariton; while subscripts i and s refer to the incident and scattered polarit­ons respectively. |Mjf | denotes the exciton-LA phonon ( deformation potential ) matrix element, n p h is the LA phonon occupation number and c is the speed of light. Equation (2) i s very similar to the polariton scattering efficien­cy of optical phonons given in Ref. 5. It differs from the expression obtained by Brenig et al. * with regard to the definition of the velocity of energy transport. In this article I shall use the definition given by Tait. 3

The quantities in Eq. (2) which are sensitive to the ABC are the exciton polarizat­ions Ρ^ and P s of the incident and scattered polaritons. Although various forms of ABC's have been proposed in the l i teratures, 2 two ABC's have been studied more extensively than others. Following the notations of Brenig et al. l I shall denote these two ABC's as ABC 1 and ABC3. ABC 1 was first proposed by Pekar and has the form:

ABC1: ( ηχ2 - e 0 ) E 1 + ( n 2

2 - ε 0 ) Ε 2 = 0, (3)

ABC 3: — + n 2 " ne = 0 (4)

rij, n2 arid ne are defined respectively as ckj/ω,

ck2/o) and (c/ω ) [( ω2 - ω 2 + ΐω Γ) / {ti ω τ / Μ ) r .

ÜD α:

^ -5

-20 -10 0 10 I 20

üJj-cuT (cm'1)

Fig. 2 The LA phonon scattering efficiencies in CdS plotted on a logarithmic scale as a function of frequency. The curves have been calculated using Eq. (2) with ABC1 (solid line) and ABC3 (broken curves) and three different values of ? / 2TTC (c) 10 c m - 1 .

(a) 1 ■1 (b) 4 •1 and

Vol. 32, No. 1 BRILLOUIN SCATTERING OF EXCITON POLARITONS 31

Using Eq. (2) and these two ABC's , I have calculated the scat ter ing efficiencies of the LA phonon in CdS for different values of Γ. The resu l t s for Γ/ 2nc = 1, 4 and 10 c m " a r e shown in Fig. 2. The incident and sca t t e red photons a r e a s sumed to have propagation d i r e c t ­ions and polar izat ions all perpendicular to the c r y s t a l ' s c - ax i s . The following p a r a m e t e r s deduced exper imenta l ly ° were used in the c a l ­culation: ( i»>L - ω χ ) / 2 τ ο = 15 c m " , eQ = 9. 1, M= 0.94 t imes free e lec t ron m a s s , and LA phonon velocity equal to 4 .27 x 10^ cm s e c ~ l . My re su l t s for v/Zirc = 10 cm~* a r e s imi la r to those of Brenig et a l . * In this case the s c a t t e r ­ing efficiencies for ABCl and ABC3 show only quantitative differences . When Γ/2πχ is dec reased to 4 cm ,the sca t ter ing efficiencies change quite d ras t i ca l ly as shown in F ig . 2(b); however, the r e su l t s for both ABC's a r e stil l s imi la r . Differences show up when T/Zirc is dec reased to 1 cm . The sca t te r ing effic­iency for ABC3 now exhibits three s t ruc tu re s : a shoulder at ω· = ccp, a peak at ω8 = ω-τ and a peak at ω · = ω_. while the curve for ABCl shows one broad maximum around ω .

These results can be unders tood in t e r m s of a ' c r i t i ca l value ' , Γ , defined by Tait as :

r c = 2 Γ2η ω τ2 ( CÜL- Ü , T ) / ( M C Z ) J 1 / 2 . (5)

It was pointed out by Tait ^ that the na ture of the solutions of Eq. (l) changes d ras t i ca l ly between

r < f c a nd r > r c . Only for Γ < Tc a r e the solut-

ionscoupled photon-exciton modes that can be cons idered as po la r i tons . When Γ is l a r g e r than r

c the solutions to Eq. (1) r e semble m o r e the uncoupled photon and exciton. In other words for Γ > Tc the medium is not spatially d i spe rs ive and ABC's a r e no longer nece s sa ry . In caa»e of CdS the value of Γ obtained from

1 C

Eq. (5) is 3.7 c m " 1 . Thus it is not surpr i s ing that the two curves in F ig . 2(c) a r e quite s imi la r and fu r the rmore the two peaks at ω·= ιο-ρ and ω = ω -p a r e a s soc ia ted with the ' b a r e ' exciton.

In Fig . 2(b) Γ is approximate ly equal to Γ and although the polar i ton effect is evident in that the ' b a r e 1 exciton resonances a r e absent, the two curves for ABCl and ABC3 a r e still s imi l a r . The differences between the s c a t t e r ­ing efficiencies in F ig . 2(c) ( v < Γ ) can be t r aced to differences in the exciton polar izat ions obtained with ABCl and ABC3. The exciton polar iza t ion calculated with ABCl peaks strongly at the longitudinal exciton energy fiait with a weaker and b roade r maximum around η ω ^ . With ABC3 the exciton polar iza t ion shows a s t ronger peak at ηω ·ρ than ηωτ · F u r t h e r m o r e the peak at ΐίω ^ i n c r e a s e s rapidly with dec rea se in Γ .

In conclusion I have shown that the B r i l l -ouin sca t te r ing efficiencies of exciton polar i tons a r e sensi t ive to the ABC only for exciton damp­ing smal l compared to Γ . This is impor tant because so far resonant Bril louin scat ter ing has been repor ted in seve ra l semiconductors and in all c a s e s the valuesof exciton damping e s t ima t ­ed from the Bri l louin spec t r a a r e sma l l e r than

Γ . Thus in pr inciple the appropr ia te ABC can be de te rmined by compar ing the theore t ica l and exper imenta l sca t te r ing efficiencies. Such a compar i son is being made for CdS.

REFERENCES

3.

W. Brenig, R. Zeyher , & J. L. B i r m a n , Phys . Rev. B 6 , 46 17 (1972). R, Zeyher , J . L. Bi rman, & W. Brenig, Phys . Rev. B 6, 46 13 (1972). More r ecen t r e fe rences can be found in G. Mukhopadhyay, Solid S ta te Commun. 2£, 277 (1978). W. C. Tait, Phys . Rev. B 5,648 (1972).

J . J . Hopfield, & D. G. Thomas , Phys . Rev. _U2,563 (1963). Eo Burs te in , D. L. Mi l l s , A. Pinczuk, & S. Ushioda, P h y s . Rev. Le t t e r s 2£, 348 (1969); and ibid 22 ,913 (1969). S.I . P e k a r , Sovi et P h y s . J E T P 6, 785 (1958).

BRILLOUIN SCATTERING OF

G. S. A g a r w a l , D. N. Pat tanayak , & E . Wolf, P h y s . R e v . B 10, 1447 (1974) ; J . L B i r m a n , & J. J. Se in , P h y s . R e v . B 6 , 2482 (1972) . G. Winter l ing , & E . K o t e l e s , P r o c e e d i n g s of Internat ional C o n f e r e n c e on L a t t i c e D y n a m i c s , ed. by M. B a l k a n s k i ( F l a m a r i o n

EXCITON POLARITONS V o l . 3 2 , No . 1

Sciences, Par i s , 1978) p. 170; P. Y. Yu, & F. Evangelisti, unpublished.

9. R.G. Ulbrich, & C. Weisbuch, Phys. Rev. Letters 38, 865 (1977); G. Winterling, & E.S . Kote les . , Solid State Commun. 23, 95(1977); C. Hermann, & P. Y. Yu, Solid State Commun. 28, 313 (1978).

32

7.

8.

Solid State Communications, Vol.32, pp.33-38. Pergamon Press Ltd. 1979. Printed in Great Britain.

INTERFERENCE ANALOGY OF RAMAN-LUMINESCENCE PROBLEM AND EXPERIMENTAL DISCRIMINATION BY POLARIZATION CORRELATION

Takashi Kushida Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan

Light scattering and luminescence under resonant optical excitation are shown to correspond to an interference fringe and broad background, respectively, in an optical interference experiment, if suitable defini­tions for the above two processes are adopted. It is proposed to discriminate between these two by a polarization measurement which is analoguous to an interference experiment. The results of such an experiment in ZnSe and ZnTe are also reported.

The distinction between light scattering and luminescence often becomes difficult in the case of resonant optical excitation, and controversial arguments have been reported on the relation between these two processes under such a condi­tion. In the present paper, we study this problem using definitions based on the coherence of the process. Similarity between this problem and an interference experiment is discussed and a method to distinguish these two processes from the measurements of the polarization character­istics is considered. This method is applied to the secondary emissions in ZnSe and ZnTe under the excitation above the band gap.

Though various definitions of the light scattering and optically excited luminescence have been used in the literature, we adopt very intuitive definitions that the light scattering is a single coherent quantum process, while the luminescence consists of independent optical absorption and emission. For example, as pointed out by Heitler, we cannot tell whether the atom is in the ground state or in the excited state in the case of resonant monochromatic light excitation of a two-level atom, when the atom is undisturbed. In this case, the state realized must be described as the coherent superposition of the two states, and the absorption-emission sequence should be regarded as a single coherent process. On the contrary, if the atom is dis­turbed during the absorption-emission cycle, these two are considered to be separate processes. The former coherent second-order optical process can be called light scattering, while the latter incoherent process luminescence.

In order to extend these concepts, let us consider that a second-order optical process occurs in a system as a multi-step transition via energy-conserving intermediate states and that this system makes random interaction with the environment. Though each step of the second-order optical process consists of a transition to an intermediate state and a transition from that state to another, these two transitions must be considered as a single connected process when the system does not suffer from random perturbation during these transitions. It is possible to associate the creation and destruction of an oscillating dipole to the transition to an inter­mediate state and that from that state to another,

respectively. Then, these two transitions should be regarded as a single coherent process when there exists phase correlation of the oscillating dipole between the moments of its creation and destruction. When this condition is satisfied in every step of the transition, the second-order optical process is a single coherent process, which we call light scattering.

Since the degree of phase correlation can be known from an interference experiment, we easily see a close relation between our problem and an optical interference experiment. For example, what we measure in Young's experiment is the phase correlation of the light beams emitted from the source at different times. A similarity between Young's experiment and resonance flou-rescence experiment discussed by Heitler3 has already been pointed out.2

For the stationary light, the pattern on the screen in Young's experiment is expressed as4

2νϊ" h + 12 + Ί Ι 2 Γ<τ)σο3[θ(τ)], (1) where 1^ and I2 are the intensities in the case only one slit is open. The degree of coherence Γ and the phase difference Θ are the absolute value and the argument of the following normalized correlation function

Γ 1 2 (τ ) = < 2 i 0 7 f <<|>s<t + T)<|>g( t ) > (2) 1 2 '

where φ3 is the optical field at the source and τ is the difference of the times to measure the phase correlation. The angle brackets denote average over times long compared to the coherence time of the source. The intensity of eq.(1) can be separated into the coherent and incoherent parts as follows,

i = ( l -D d 1 + i 2 ) + Γ(Ι +1 +2/ΓΤ cos0) . (3) From our definition, we may consider that the coherent part corresponds to scattering, while the incoherent part to luminescence. No lumines­cence appears when there exists definite phase relation, while all the secondary emission is ascribed to luminescence when there exist no phase correlation. In between, the secondary emission can be described as the sum of the scattering and luminescence.

What we observe in the actual secondary emission experiment is the mean of the above intensity over τ. The degree of coherence Γ

33

34 INTERFERENCE ANALOGY OF RAMAN-LUMINESCENCE PROBLEM Vol. 32, No. 1

decreases with τ, while the observation time of the secondary emission is limited by the lifetime of the intermediate excited state. Therefore, the mean degree of coherence T becomes finite, and the intensity ratio_between luminescence and scattering is given by Γ/(1 - Γ).

For simplicity, let us consider the fast modulation limiting case, where Γ(τ) is consider­ed to decrease exponentially with a time constant T'. In this case, the relaxations due to the interaction between the system and environment can be classified into two; one is the population relaxation which shortens the lifetime of a quantum state, while the other is the dephasing relaxation which broadens the energy of a quantum state but does not affect its lifetime. The time constant T' is identified as the reciprocal of the dephasing relaxation rate in the intermediate state concerned. Namely, 2n~/T' is the homogene­ous energy width (FWHM) of the intermediate state when its lifetime Tm is infinite. Then, Γ is obtained as T'/(2Tm + T') = T2/2Tm, which gives the intensity ratio between the luminescence and scattering as 2Tm/T', i.e., as the ratio between the dephasing and amplitude relaxation rates of the intermediate state. '

It should be mentioned that our definitions of light scattering and luminescence are not necessarily the same as those in the literature. For example, ShenO made a decomposition of the second-order optical process into two considering that the emission intensity should be propor­tional to the population in the resonant excited state in the case of luminescence but not in the case of scattering. What he did corresponds to the separation of eq.(1) into the phase independ­ent term (1^ + I2) and the rest. This clearly explains why the rate of the scattering in his definition becomes negative in some frequency region. Hanamura's expressions for the lumines­cence and Raman scattering terms in the case of resonant two-photon excitation of the excitonic molecule are completely the same as those obtained by Shen6 in the stationary case. When the relaxation in the initial state of the tran­sition can be neglected, which is the case for the exciton problem treated by Hanamura, the integrated resonance Raman intensity vanishes. This is again understood well if we consider that his resonance Raman scattering corresponds to the third term in the right hand side of eq.(1).

An analysis of the response of a three-level system to the resonant monochromatic exciting light in the fast modulation limiting case shows that, in our definition, the secondary emission frequency is correlated with the exciting light frequency in the case of scattering but not in the case of luminescence, and also that the emis­sion spectral width is independent of the width of the intermediate state in the case of scat­tering but not in the case of luminescence. As for the transient response, the decay time of the emission intensity can be shorter than the life­time of the intermediate state in the case of scattering. This is because this coherent process is associated with the off-diagonal element of the atomic polarization, whose decay time is determined by the rate of the transverse relaxation (I/T2) which includes both amplitude and dephasing relaxations and accordingly can be shorter than the longitudinal relaxation time of the diagonal element. Thus, the features of the

luminescence and light scattering obtained from our definitions are smoothly connected with those of the ordinary luminescence and usual off-resonance light scattering.

It should be pointed out, however, that the separation as in eq.(3) does not mean that the secondary emission components due to light scat­tering and luminescence can always be separable on account of different physical properties such as spectral shape, time behavior and so on. Further, it is not always possible to separate the experimental system into the system and perturbing environment without ambiguity. Only when this is possible, as in the case where the energy of the quantum concerned in the multi-step transition is larger than the homogeneous energy width of the intermediate state, the above interference analogy can be applied and the dis­tinction between light scattering and lumines­cence is physically meaningful, at least as a concept. There often exist cases where it is possible to regard the material in problem to consist of a simple multi-level system and the environment which perturbs the system weakly. In such a case, we can separate the secondary emission into light scattering and luminescence, whose relative magnitude of the contribution is determined by the ratio of the amplitude and dephasing relaxation rates in the intermediate state.

Now, let us consider the experimental dis­crimination between light scattering and lumines­cence under resonant optical excitation. Since the scattering corresponds to the interference fringe while the luminescence to the broad back­ground when an interference analogy can be applied, it should be possible to distinguish these two processes by an experiment analoguous to an interference experiment. Usually, we use the spectral property and transient response of the secondary emission for the discrimination. Namely, when a second-order optical process is a single coherent process, the emission frequency is independent of the energy position and width of the intermediate state and is correlated with the incident light frequency, the emission spec­tral width can be narrower than the natural width, and the emission decay time can be shorter than the lifetime of the intermediate state. These are all the direct consequence of the interfer­ence effect. In Heitler experiment, for example, the secondary emission should have natural width even in the case of undisturbed atom because the population is distributed in the resonant excited state, but in fact the emission cancels out on account of an interference effect in the frequen­cy region other than the exciting light frequency.

The degree of phase correlation between two differently polarized lights can be determined from the measurement of the polarization charac­teristics on their superposition. For example, let us consider the superposition of right and left circularly polarized lights with the same frequency and intensity. If there exists a definite phase relation between the two beams, the superposed light should be linearly polarized. On the contrary, if there is no phase correlation, the superposed light should be unpolarized. This experiment can be regarded as an interference experiment and it is possible to consider an analoguous secondary emission experiment.

V o l . 3 2 , N o . 1 INTERFERENCE ANALOGY OF RAMAN-LUMINESCENCE PROBLEM 35 Suppose that the secondary emission is cir­

cularly polarized under circularly polarized light excitation. The emission is observed in the forward or backward direction along the ex­citing light and the material is assumed to be isotropic around the optic axis. Then, the secondary emission should be ascribed to the scattering if it is linearly polarized under linearly polarized light excitation. This is because this result indicates that there exists a definite phase relation between the secondary emissions under σ and σ~ excitations when the exciting σ and σ~ lights are phase correlated with each other. One exceptional case is that the atom or the center suffers from dephasing collisions which are completely isotropic around the optic axis. Such a case is not likely in actual systems.

The secondary emission is generally only partially polarized even under polarized light excitation. Further, the total secondary emis­sion intensities I+ and I" under σ+ and σ~ light excitations are not necessarily the same with each other. Even in such a case, because the sum of the polarized lights is still polarized pro­vided there exists a definite phase relation, the degree of polarization (the ratio of the polar­ized part to the total intensity) P of the emis­sion under linearly polarized light should be given by the weighted mean of the degrees of polarization P+ and P" under σ+ and σ~ light excitations as P = (P+I+ + P"I~)/(I+ + I"), if the polarized parts of the emissions under σ+ and σ~ excitations are phase correlated with each other in the case of linearly polarized light excitation. Therefore, if this relation holds, we may be able to conclude that at least the polarized part of the secondary emission is due to light scattering.

able intensity of multiple LO lines cannot be explained on the basis of the resonance Raman scattering mechanism.

We compared the temperature dependence of the intensity, spectral position and width of the Raman-like lines and exciton luminescence lines in ZnTe excited with an Ar laser above the band gap.11 The decrease of the intensities of the Raman-like lines with temperature rise was observed, which was very similar to the tempera­ture quenching of the luminescence. However, in the case that the photon energy of the secondary emission is larger than the free exciton energy, where the strong temperature dependence was observed, this dependence was found to agree with that calculated from the dependence of the energy denominator of the Raman tensor on the emitted photon energy. Therefore, the temperature dependence of the Raman-like lines is concluded to be explained well by the variation of the resonance condition due to the thermal shift of the exciton energy, and accordingly to be con­sistent with the Raman scattering mechanism.

In order to clarify the origin of these Raman-like lines, polarization measurements were made on the secondary radiation of ZnSe emitted in the backward direction parallel to the normal­ly incident exciting light within the solid angle of 0.03 sr. under the band-to-band excitation of an Ar laser. To avoid depolarization due to the dewar window and crystal surface, measurements were made at room temperature on a cleaved sur­face (110) of the sample. Figure 1 shows the secondary emission spectra under the excitation of circularly polarized light. The spectra as well as the intensity did not change when σ+ and

This type of polarization correlation method to distinguish between light scattering and luminescence is superior, in some case, to the methods to use spectral properties, transient response and so on, because it can be applied well even when the excited state concerned has a very short lifetime and also when the final state or the intermediate state concerned is very broad energetically.

As an example, we apply the above method to the Raman-like multiple LO lines observed in ZnSe and ZnTe under the band-to-band excitation. Under monochromatic light excitation above the band gap, II-VI semiconductors show a series of narrow emission lines separated from the exciting light by multiples of the LO phonon energy. The spectral width and the intensity of a higher order line are not very different from those for the first order line, and resonance enhancement is observed when the line position comes close to the exciton energy. First, these lines were interpreted in terms of resonance Raman scatter­ing.8 On the other hand, from the sample depend­ence of the secondary emission spectra of CdS, Gross and his coworkers^ concluded that these Raman-like lines should be ascribed to lumines­cence due to hot excitons. Further, these authors-LO observed strong temperature quenching for the 2LO line in CdSe, and mentioned that such a temperature dependence as well as the compar-

600 400 200 AV(crrf')

Fig.l The spectra of the left (a) and right (b) circularly polarized emissions of ZnSe at 300K under the excitation of right cir­cularly polarized light at λ = 457.9nm. - -1 -1

Δν = λ0 L - λ .

36 INTERFERENCE ANALOGY OF RAMAN-LUMINESCENCE PROBLEM Vol. 32, No. 1

σ~ were exchanged. The spectrum consists of a broad band around the band gap and two narrow lines. The broad band is ascribed to the ordi­nary luminescence, because its energy position depends on the crystal temperature but is inde­pendent of the exciting photon energy. On the other hand, the energy separations of the narrow lines from the exciting laser line coincide with the 1L0 and 2L0 phonon energies regardless of the crystal temperature. These 1L0 and 2L0 Raman­like lines are predominantly left circularly polarized under right circularly polarized light excitation, while are right circularly polarized under left circularly polarized light excitation. This indicates that the angular momentum is almost conserved in these second-order optical processes. For the broad emission band, on the other hand, the σ* and 0~~ emission components are essentially the same under the excitation of circularly polarized light.

The secondary emission spectra under linear­ly polarized light excitation are shown in Fig.2.

600 400 200 0 _600 400 200 AV(cm-')

case, we can consider that the system consists of the polariton states with energies ;hu)£ — ntTo o (n = 0,1,2) , where "η~ω and η"α 0 are the energies of the exciting photon and LO phonon, respective­ly, and that the phonon field, thermal radiation field and crystal imperfections including the surface and impurities play the role of the perturbing environment.

Fig.2 The emission spectra of ZnSe at 300K under the linearly polarized exciting light at 457.9nm. The electric vector of the emission is perpendicular (a) and parallel (b) to [001], while the exciting light is polarized along [001].

800 600 400 200 AV(cnrr')

Fig.3 The secondary emission spectrum (a), and the difference spectra (b and c) of ZnTe at 77K under the 514.5nm excitation. (b) : difference between O and cr emission components under the σ exciting light, and (c): that between x and y emission components under the x-polarized exciting light.

These spectra did not depend on the relation between the directions of the polarization of the exciting light and the crystal axis. We notice that the linear polarization memory is almost retained for the Raman-like lines while not for the broad luminescence band. This result indi­cates, as discussed above, that the phases of the σ and σ~ Raman-like emission components are well correlated with each other provided the exciting σ and σ~ lights are phase correlated with each other. Therefore, we conclude that these lines are due to resonance Raman scattering. In this

In Fig.3, rne ordinary secondary emission spectrum of ZnTe at 77K is compared with the dif­ference spectrum between the σ4" and σ" emission components under circularly polarized light exci­tation and also that between the parallel and perpendicular emission components under linearly polarized light excitation.12 The polarization of the LO lines is correlated with that of the incident light both circularly and linearly, whereas the broad band with a dip, which is due to the exciton polariton luminescence, 3 does not show linear polarization correlation, though cir­cular polarization memory remains considerably for this band. Because the degrees of polariza­tion (-0.6) under linearly and circularly polar­ized exciting lights almost coincide with each

Vol. 32, No. 1 INTERFERENCE ANALOGY OF RAMAN-LUMINESCENCE PROBLEM 37

other for the LO lines, we conclude that at least the polarized parts of these emissions are due to light scattering. This agrees with our conclusion derived from the dependence of the circular polarization memory of these emissions on the applied transverse magnetic field.13 The result that the linear polarization memory is quite different from the circular polarization memory for the broad exciton band is reasonable because this band is due to incoherent lumines­cence process. The finite circular polarization memory in this case only means rather slow relax­ation rate of the angular momentum in the relevant polariton state.

The circular polarization memory for the 1L0 line of ZnSe was measured on the (110) surface (forbidden configuration) using exciting light in the transparent spectral region near the ab­sorption edge. The degree of circular polari­zation defined as pc = (L+ - L_)/(L+ + L_), where L+ (L_) is the intensity of the σ+ (σ~) emission component under the excitation of σ~ light, was found to change from a negative value in the off-resonance region to a positive one very near the resonance to the exciton level. An experiment in the allowed configuration on the (001) surface gave a similar result.15 Further, in this case, the degree of linear polarization defined as P& = (Lx ~ Ly)/(LX + L y ) i where Lx and Lv are the emission intensities in the z(xx)z and z(xy)z configurations,16 respectively, with x//[100], y//[010] and z//[001], was found to vary almost in parallel with pc as a function of the exciting light frequency.15 Although small difference was observed between p^ and pc, it is ascribed to the depolarization effect of the polished sample surface, because p£ = pc is expected for our case of Raman scattering in cubic ZnSe.

The negative values of pc and p£ in the off-resonance region are explained by the deformation potential-type electron phonon interaction, while the positive values near the resonance condition are by the dominant contribution of the intraband Fröhlich interaction of Wannier exciton with an LO phonon.17 This type of Fröhlich interaction is considered to be dominant also in the case of the excitation above the band gap, because the degrees of both circular and linear polarizations observed are always positive for this case and also because the Raman-like lines show enhance­ment for the resonance with the exciton level. Thus, it is concluded that the multiple LO lines in ZnSe and ZnTe observed under the band-to-band excitation should not be classified as lumines­cence due to hot excitons but as resonance Raman scattering by intraband Fröhlich interaction.

Bonnot et al.18 measured the excitation spectra for the emission at the energy of the lowest exciton "η"ωΘΧ in CdS under linearly and circularly polarized light excitations. They found that the excitation bands which appear at the energies of ΐιωθχ + ηΐΐω^ (η = 1,2,..) have both linear and circular polarization correla­tions with the emission. On the other hand, Permogorov et al.19 measured the degrees of linear and circular polarizations for the Raman­like lines in pure and Ni-doped CdS crystals and also in CdSe under linearly and circularly pola­rized exciting lights. They observed that the

Raman-like lines conserve the linear and circular polarization memory considerably. Although these experiments have been discussed from the view­point of hot exciton luminescence, these phenom­ena of the appearance of the LO-phonon replica are considered to be essentially the same as the multi-phonon resonance Raman scattering reported by several investigators. Further, it is prob­able that the Raman scattering in our definition play a significant role in these second-order optical processes, because the reported values of pc and p£ are close to each other for some lines. However, in order to determine whether the Raman scattering is dominant or not by applying our method, more detailed polarization measurements seem to be needed. Namely, it may be necessary to determine the degree of polarization P instead of p£ and pc by making the directions of observa­tion and excitation collinear.

Finally, it should be noted that the fre­quency correlation between the exciting and emit­ted lights and also the circular or linear pola­rization memory by itself cannot be always used as the criterion for the discrimination between light scattering and luminescence. For example, in the case of calcium metaphosphate glass doped with Eu3+ ions, the secondary emission appearing at the same frequency as the exciting light at room temperature was found to retain the linear polarization memory (p^~0.5). In this case, however, the dephasing relaxation time T'~3ps is much shorter than the lifetime (~0.9ms) and the emission is easily identified as due to luminescence. '^0 & polarization measurement revealed that the circular polarization memory is essentially zero for this emission.

Acknowledgements - The author thanks Prof.Y.Oka and Dr.S.Kinoshita for helpful discussions. The work was supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education.

REFERENCES

1. See, for example, R.Y.Shen, Physical Review B 1-4, 1772 (1976); J.R.Solin and H.Merkelo, Physical Review B 14, 1775 (1976) ; and also references cited in reference 2.

2. T.Kushida, Technical Report of ISSP (The University of Tokyo) A773 (1976).

3. W.Heitler, The Quantum Theory of Radiation (Oxford, 1957) 3rd. ed. p.196.

4. M.Born and E.Wolf, Principles of Optics (Pergamon Press, 1975) 5th ed., Chap.10.

5. T.Kushida, E.Takushi and Y.Oka, Journal of Luminescence 12/13, 723 (1976).

6. R.Y.Shen, Physical Review B 9^, 622 (1974). 7. E.Hanamura, Journal of Luminescence 12/13,

119 (1976). 8. R.C.C.Leite and S.P.S.Porto, Physical Review

Letters Γ7, 10 (1966); R.C.C.Leite, J.F.Scott and T.C.Damen, Physical Review Letters 22, 780 (1969); M.V.Klein and S.P.S.Porto, Physical Review Letters 22^, 782 (1969) .

9. E.F.Gross, S.A.Permogorov, V.V.Travnikov and A.V.Selkin, Fizika Tverdogo Tela \A_, 1388 (1972) [Soviet Physics-Solid State 14, 1193 (1972)].

38 INTERFERENCE ANALOGY OF RAMAN-LUMINESCENCE PROBLEM Vol. 32, No. 1 10. E.Gross, S.Permogorov, Ya.Morozenko, and

B.Kharlamov, Physica Status Solidi (b) 59, 551 (1973).

11. M.Kwietniak, Y.Oka and T.Kushida, Journal of the Physical Society of Japan ^4, 558 (1978).

12. Y.Oka and T.Kushida, Proc. Intern. Conf. on Luminescence, Paris (1978) (to be published).

13. Y.Oka and T.Kushida, Solid State Communica­tions 27, 1367 (1978).

14. Y.Oka and T.Kushida, Nuovo Cimento 39B, 483 (1977).

15. Y.Masumoto and T.Kushida, (unpublished).

16. B.Tell, T.C.Damen, and S.P.S.Porto, Physical Review 144, 771 (1966).

17. R.M.Martin and T.C.Damen, Physical Review Letters ,26, 86 (1971) .

18. A.Bonnot, R.Planel, and C.Benoit a la Guillaume, Physical Review, B 9 690 (1974).

19. S.A.Permogorov, Ya.V.Morozenko, and B.A.Kazennov, Fizika Tverdogo Tela Γ7, 2970 (1975) [Soviet Physics-Solid State Γ7, 1974 (1976).

20. T.Kushida and E.Takushi, Physical Review B 12, 824 (1975).

Sol id State Communications, V o l . 3 2 , pp .39 -42 . Pergamon Press Ltd. 1979. Pr inted i n Great B r i t a i n .

The Nature of the Raman Damping Constant and Its Effect Upon Resonant Light Scattering From Molecular Systems in the Weak Field Limit

J. M. Friedman

Bell Laboratories Murray Hill, New Jersey 07974

ABSTRACT

With the use of a time dependent tetradic operator scattering formalism, a comparison is made between resonant light scattering (RLS) generated by monochromatic and pulsed exci­tations. This comparison reveals that the Raman damping constant is the reciprocal of the total dephasing time associated with time dependent ensembles. A discussion is given of the effect of the lifetime of both the excited resonant intermediate state (Tj) and the phase rela­tionship between pairs of contributing states (T2*) upon the coherent and incoherent makeup of the RLS. The question of resonance Raman versus resonant fluorescence is again reex-amined.

1. INTRODUCTION There have been several studies both experimental1-5 and

theoretical6-25 relating to the question of resonance Raman scatter­ing (RRS) versus resonant fluorescence (RF). From such studies it is clear that the character of resonant light scattering (RLS) is inti­mately associated with the nature of the damping constant that appears in expressions for the RRS cross section. Despite the fun­damental importance of the damping constant in the understanding of RLS, the damping constant is frequently treated as an adjustable parameter used in fitting Raman excitation profiles to vibronic theories with little attention paid to its origin. There have been only a very few experimental studies4,5,26,27 which focus on the rela­tionship between RLS and the damping constant. Since the under­standing of the interplay between RLS and the damping constant should increase the information about molecular systems obtainable from conventional RRS experiments, we present in this paper a systematic theoretical development of the nature of the Raman damping constant and its connection to the question of RRS versus RF.

2. THE ISOLATED MOLECULE

The evolution in time of the system consisting of photons interacting with a molecule can be viewed in terms of evolving amplitudes or products of amplitudes which define the elements of a density matrix. From scattering theory7 the amplitude at time t, of the state s, resulting from the interactions of a monochromatic photon of energy Ep with a molecule is:

<s |*( t )> - e"i(Ep"Es)t<s|G(Ep)V|g,lEp> (1)

where the energy of the molecular ground state g is taken to be zero, V is a perturbation operator which is typically the electric dipole moment operator and G(EP) is the Green's operator which contains the full dynamics of the interaction. Alternatively25,22 the element ρΜ-, of the density matrix at time t resulting from the evo­lution of the initial matrix element Pgg(t=—°°) is given by:

pe.(t) - <s|^(t)><</f(t)|s'> β « s s ' | 0 ( t ) , ^ ( O » = e " i ( A " V ) t « s s ' | G ( A ) V | g g » (2)

where « w ' H ' s X s l , | g , g » - |g, lE p><g>WL Δ . - Ep-EpS

ΔΜ· - Es~Es' a n d G a n d v a r e t h e superoperator28 30 (tetradic operator) equivalents of the dyadic operators G and V respectively. If we consider a Raman process in which the final product state |g',lEf> refers to the molecule in a state g' associated with a

scattered photon of energy Ef, then the photon counting rate (PCR) for the RLS which derived from either 1. or 2. can be expressed in terms of the T matrix or the tetratic T matrix respec­tively:

PCR=-^ |<g ' ,E f |^ ( t )> | 2 = (3)

PCR = -~Pg'g'(0 - - i « g ' , g ' | T ( 0 ) I g g » (4)

In both instances the derivation is accomplished by allowing the width associated with final state to go to zero as the last mathemati­cal step. Equations 1 through 4 provide the framework for under­standing the resonant scattering processes in terms of relaxation and dephasing processes. Although 1 and 3 are mathematically more tractable than the tetratic equivalents, the latter is more suited for analysis of coherence because the coherent properties of RLS are contained within the time dependence of the coupled amplitudes which define the elements of density matrix.

The PCR for RLS in the isolated molecule limit when there is a single intermediate resonant state, s, is easily shown to be the sin­gle following term:

Iv < l 2 l v I2

PCR = constants x g f δ (Eg+Ep-Eg-Ef) (5)

where Ags = Eg+Ep-Es and Es includes appropriate level shifts. It follows from Eq. 5 that the spectra of the RLS generated by mono­chromatic excitation consists of monochromatic Raman spectral lines. This monochromaticity reflects the fact that the energy con­serving system is the photon plus molecule. The excitation profile for these Raman lines follows a Lorentzian line shape (same as the resonant absorption) which has a full width at half height of r s assuming the ground state has a zero width.

The damping constant can be related to both the time depen­dence of the resonant state s and the dilution of the oscillator strength of s over a range of energies. The population of s within the superpositions of states generated by the monochromatic excita­tion is |<s|i/f(t)>|2 or equivalently pss(t). Using either 1 or 2 we find that the steady state population is:

which reveals that the oscillator strength associated with the g—'s transition is diluted over Ts. If the excitation consists of a short

39

40 THE NATURE OF THE RAMAN DAMPING CONSTANT IN THE WEAK FIELD LIMIT Vol. 32, No. 1

pulse we find that for an excitation pulse that is a delta function in time, the time dependence for the resonant state population is:

Pss(0= |Vsg|2e"IV (7) which indicates that the resonant state population decays away in time with a decay constant Ts which is the reciprocal of the lifetime of s (optical Tj time, transverse relaxation time). Examination of the structure of r s reveals that it originates from the coupling of s to radiative and non-radiative continua or quasi-continua that are effectively at 0° K.

Since Ts is the rate constant for the coupling of s to the various available continua, the relationship between the duration of the excitation pulse and ΐ γ 1 will determine which states are coupled during the excitation. Monochromatic photons, which are pulses of infinite duration, create a steady state superposition of all the cou­pled states. This superposition is an eigenstate of the closed sys­tem. A short pulse ( « ΐ γ 1 ) on the other hand actually generates the state s which is not an eigenstate of the system. The s state population then irreversibly decays by virtue of the coupling to the continua. The conditions under which most RRS experiments are conducted, typically approximate the exact eigenstate limit.

The superposition of states that comprise the exact eigenstate of the system consisting of monochromatic photon and molecule (0°K) has the form:

*(t) = lg,lEp> +C 1 e- i ( ^ ) t | s> +C2e-i(A*Hg',lEf> + (8)

This is a coherent superposition in that the amplitudes associated with the Raman active states, i.e. |g,lEp>,|s>, |g' , lEf>, maintain a fixed phase relationship over time. Consequently monochromati-cally excited RRS in the limit of an isolated 0°K molecule is a coherent process where the coherence refers to the phasing between the amplitudes associated with the eigenstate of the iso­lated system. Using Eq. 2, it is found that the off diagonal ele­ments of the density matrix i.e. psg(0 continue to oscillate indefinitely at Ags which is indicative of an infinitely long dephasing time (t2 = oo). In the pulsed experiment however, we find that the oscillations of psg(t) associated with the coherent superpositions of amplitudes decay are exp- s (T2 = 2TS = — ) provided the

* s pulse duration is short compared to ΐγ1 . This dephasing occurs because the pulsed excitation generates a distribution of psg each having a slightly different oscillation frequency. In contrast, mono­chromatic excitation results in a single psg which can be thought of as a single Fourier component of the lineshape of the resonant absorption. Because there is no loss phase between amplitudes for the monochromatic excitation, the contribution to the scattering process of two or more intermediate states will result in construc­tive or destructive interference31'32 from the contributing intermedi­ate states. In the pulsed experiments the phase relationship between the amplitudes of the two resonant states decays as exp— (rs+rs')t/2 a pss,(t), so that manifestations of interference effects are of finite duration.

ponents, the fluctuation generates a distribution of density matrix elements such as psg in much the same way as the pulsed photon excitation. As with the pulsed excitation, there is now a loss of phase memory between the coupled amplitudes. It is assumed that the spread in Fourier components of the phonon wavepacket is greater than that associated with any resonant or near resonant psg (impact approximation) then using Eq. 2, the population, pss(t), of the resonant state s is now given by:

Pss(0 : lv„ 2Γ, Δ 2 +Γ 2 ) Γ5

(10)

lv„ Δ 2 +Γ 2 1 +

Γ* 1 sg

Ts/2 where rsg = TJ2 + r*g, is the reciprocal of the dephasing time, T2, and r*g, which results from the bath induced fluctuation of s with respect to g, is the reciprocal of the pure dephasing time, T2*. A comparison between Equations 10 and 6 indicates that the interac­tion with the bath results in an additional contribution to Pss(t) which is proportional to the ratio of the bath induced pure dephas­ing time to the dephasing time of the pulse excited isolated system. Similarly, the PCR of monochromatically excited RLS for the single intermediate state is now given by:

IVsgPlVsg'l PCR = (Ag

2+rs2g)

δ(Δ ') + Γ2 1 sg Ts/2 -ra Δ2- +Γ 2

^g ' s" 1 sg,

(11)

The bath induced second term is associated with spectral lines that, in contrast to the Raman lines, originate at Es and have a spectral width (hwhh) of rsg. It follows that this resonance fluorescence scattering associated with the second term originates from the full distribution of psg type Fourier components and is consequently incoherent. By integrating over the energy of the emitted photon we find that the integrated intensity for this incoherent contribution

is proportional to Γ* 1 sg ^s/2

It follows that the second term in Eq. 10 is

the Γ *g induced incoherent population of s. Both the coherent and incoherent scattering have excitation profiles that follow the line shape of the resonant absorption. The width of the excitation profile Tsg (Raman damping constant) differs from that of the iso­lated molecule by the pure dephasing term. For a given value of the damping constant the Raman (coherent) PCR will be fixed; however, if there is a sizeable contribution from the pure dephas­ing, the signal to noise for the Raman scattering will be affected since intensity will have been transferred to the incoherent RLS via t* damping which can be expected to contribute a relatively broad background to the spectrum.

If there are two intermediate states, s and &', then the PCR for the RLS is in first order a composite of the following pathways for the evolution of elements of the density matrix:

, , / g S \ , / ^ \ g ' s ' ^ ^ s ' g '

3. THE CONDENSED PHASE Monochromatically excited RLS from an isolated molecule is a

single coherent process. When the molecule is in thermal contact with a non 0°K heat bath the energy conseving system is no longer the molecular plus photon and additional potentially incoherent contributions to the RLS can be expected. One way to view the thermal interaction is to consider the effect upon the RLS of colli­sions between the photon molecular system and pseudo particles such as phonons which represent the unit excitations of the bath. Phonon scattering, for example, can result in loss or gain of energy by the photon molecular system. A disparity in the interaction between the various photon coupled states and the incident pho­non, results in a change in the relative energy spacing of the cou­pled levels. This shifting is equivalent to a phonon induced transi­tion from one Fourier component to another. If the phonon is a wavepacket of finite duration then this interaction produces a fluctuation in the energy separation between the coupled states. If the phonon wavepacket contains a sufficient spread in Fourier com-

S > g g r vsg>

,*K

^>gg

ggc '-ssOg g g ^ ^ > s s \ ^ g s \ ,s~ gg^ yss\ gg'C ^ ' δ \

^ g ' s ^ ^gS' where the double letter refer to the indices of the elements of the

Vol. 32, No. 1 THE NATURE OF THE RAMAN DAMPING CONSTANT IN THE WEAK FIELD LIMIT 41

density matrix. RRS arises only from the first two diagrams while the resonant fluorescence is derived from all these diagrams. Evaluation of the PCR using Eq. 4 reveals that as with the isolated molecule the RRS is expected to manifest interference effects resulting from the cross terms between the s and s' contribution to the cross section. These cross terms persist only when the phasing between the g and g' amplitudes remains identical for the s and s' contribution during the course of the scattering even.22 For exam­ple if the initial state g consists of the product of the molecular ground state, a photon of energy Ep and a phonon of energy Ek, then in order for there to be coherent cross terms in the cross sec­tion of RLS the energy of the final state amplitude arising from the s intermediate state must be identical to that from the s' intermedi­ate state. That is if as a result of a phonon scattering process the s and s' levels fluctuate in energy then the cross terms retain an explicit phonon energy dependence which when integrated over the spread in phonon energies results in the loss of these cross terms as t—'oo. Consequently, the cross section for RRS represents, in effect, the probability that the photon scattering event occurs without the contributing levels experiencing a heat bath induced fluctuation in energy. When s and s' fluctuate with respect both to each other as well as to g, the resulting PCR for the incoherent resonant fluorescence is devoid of cross terms. In a pulsed experi­ment, the rapid preparation of a coherent superposition of s and s' results in a beating phenomenon in the temporal evolution of the resonance re-emission which decays away with a decay constant Γ = r S g+ rs'g + rs*' where in the absence of cross population effects™'21 between s and s', T*s is the reciprocal of the pure dephasing time associated with the fluctuation of s with respect to s'. If the spectrum of the RLS were monitored on a time scale that in short relative to Γ- 1 , then interference effects between the s and s' contributions should be detectable.

4. CONCLUSION In general, the damping constant associated with RRS for reso­

nance with an intermediate state s, is given by rgs = -y- + Γ8*

where Γ8 is the reciprocal of T b the transverse relaxation time asso­ciated with the s state population and Γ^ is the reciprocal of the pure dephasing time T2* associated with the dephasing of the gs dipole. For monochromatic excitations, partitioning of RLS into coherent and incoherent components is determined by the ratio of

Γ rs* to -r1. In the absence of pure dephasing process, all the RLS is RRS and the damping is determined by irreversible relaxation processes that comprise Ts. At finite temperatures where the pure dephasing is typically picoseconds or less in condensed phase sys­tems, the the only systems that will exhibit RRS with reasonable signal to noise levels are those with T* times that are comparable to the short T2* times. When RLS is generated by a short excitation pulse, coherence is lost over time even in the isolated molecule limit. The PCR for the RLS is proportional to the population of the resonant state s, which decays as e s . The initial phase rela­tionship or coherence between the ground state g and s decays as e sg. Interference effects between off resonant contributions and the resonant contribution to the RLS are co-temporal with the exci­tation pulse since the off resonant contributions have been shown 1,6-8 to follow the pulse in time. Interference effects, such as beat-

—r .t ing, between two resonant states, s and s', will decay as E ffl

where rss< = Tsg + r s g + Γ*8- and Γ8*< is the pure dephasing between s and s'. Consequently a system where the resonant states are long lived relative to the exciting pulse and to the pure dephasing times rsg> Ts*g afld r*s·, will radiate coherent resonant re-emission over a time scale that is short relative to the total re-emission lifetime. In many condensed phase systems the coherent resonant scattering will follow the excitation pulse like the off resonant scattering but in general it has the decay characteristics of the off diagonal ele­ments of the density matrix which can result in the coherent RRS having a finite lifetime (Γ~υ.

REFERENCES

[1] P. F. Williams, D. L. Rousseau and S.. H. Dworetsky, Physi­cal Review Letters 32, 196(1974).

[2] D. L. Rousseau, G. D. Patterson and P. F. Williams, Physi­cal Review Letters 34, 1306 (1975); 37, 1441 (1975).

[3] J. L. Carlsten, A. Szoke and N. G. Raymer, Physical Review A 75, 1029 (1977).

[4] J. M. Friedman and D. L. Rousseau, Chemical Physics Letters 55, 488 (1978).

[5] R. M. Hochstrasser and C. A. Nyi, Journal of Chemical Phy­sics 70, 1112 (1979).

[6] J. O. Berg, (A. Langhoff and G. W. Robinson, Chemical Physics Letters 29, 305 (1974).

[7] J. M. Friedman and R. M. Hochstrasser, Chemical Physics 6, 155 (1974).

[8] S. Mukamel and J. Jortner, Journal of Chemical Physics 62, 3609 (1975).

[9] W. Heitler, Quantum Theory of Radiation (Oxford University, Oxford, 1954).

[10] S. Mukamel, A. Ben-Reuven and J. Jortner, Physical Review A 12, 947 (1975).

[11] D. L. Huber, Physical Rev. 158, 843 (1967); 170, 418 (1968); 178, 93 (1969); 178, 392 (1969).

[12] A. Omont, E. W. Smith and J. Cooper, Astrophysical Journal 175, 185 (1972); 182, 283 (1973).

[13] V. Hizhnayakov and I. Tehver, Physica Status Solidi 21, 755 (1977).

[14] T. Takagahara, E. Hanamura and R. Kubo, Journal Physical Society of Japan 43, 802; 811; 1522 (1977); 44, 728; 742 (1978).

[15] Y. R. Shen, Physical Review B9, 622 (1974).

[16] R. M. Hochstrasser and F. A. Novak, Chemical Physics Letters 41, 407 (1976); 48, 1 (1977); 53, 3(1978).

[17] K. Rebane and P. Saari, Journal of Luminescence 16, 223 (1978).

[18] S. Mukamel and A. Nitzan, Journal of Chemical Physics, 66, 2462 (1977).

[19] Y. Toyozawa, Journal of the Physical Society of Japan, 41, 400 (1976).

[20] A. Kotani and Y. Toyozawa, J. Phys. Soc. Japan 41, 1699 (1976).

[21] A. Nitzan, submitted for publication.

[22] J. M. Friedman, submitted for publication.

[23] F. Novak, J. Friedman and R. M. Hochstrasser, in Laser and Coherence Spectroscopy, edited by J. Steinfeld (Plenum, New York, 1978).

[24] D. L. Rousseau, J. M. Friedman and P. F. Williams in Topics in Applied Physics edited by A. Weber (Springer-Verlag, Ber­lin, Heidelberg, New York, 1979).

42 THE NATURE OF THE RAMAN DAMPING

[25] J. M. Friedman in Advances in Laser Chemistry, edited by A. H. Zewail (Springer Series in Chemical Physics, Springer-Verlag, New York, 1978).

[26] J. M. Friedman, D. L. Rousseau and F. Adar, Proceeding of the National Academy of Science 74, 2607 (1977).

[27] J. M. Friedman, D. L. Rousseau and V. E. Bondybey, manuscript in preparation.

[28] A. Ben-Reuven and S. Mukamel, Journal of Physics A, 8, 1313 (1975).

CONSTANT IN THE WEAK FIELD LIMIT Vol. 32, No. 1

[29] A. Ben-Reuven, in Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice (Wiley, New York, 1975), Vol 23, p. 235.

[30] K. E. Jones and A. H. Zewail, in Advances in Laser Chemis­try, edited by A. H. Zewail (Springer Series in Chemical Phy­sics, Springer-Verlag, New York, 1978).

[31] A. Nitzan and J. Jortner, Journal of Chemical Physics, 57, 2870 (1972).

[32] J. M. Friedman and R. M. Hochstrasser, Chemical Physics Letters, 32, 414 (1975).

I Solid State Communications, Vol.32, pp.43-49. Pergamon Press Ltd..1979. Printed in Great Britain.

RESONANCE RAMAN SCATTERING AND LUMINESCENCE UNDER TWO-PHOTON EXCITATION

M. Ueta, T. Mita and T. Itoh

Department of Physics, Faculty of Science, Tohoku University, Sendai, 980 Japan

Two-photon forward Raman scattering in CuCl is the four polariton process; two incident polaritons ti(ji)0(k0) are scattered into a pair of lower branch polaritons having different energies via excitonic molecule state at k=2k0. Scattering into lower and upper branch polaritons can take place via excitonic molecule with small k<k0. At resonant excitation, the secondary emission consists of Raman and luminescence components. From the line shapes of luminescence bands, the relaxation of resonantly created excitonic molecules is discussed.

1. Introduction

Two-photon resonance Raman scattering in which the excitonic molecule and exciton states are the intermediate and final states, respec­tively, has been found in our laboratory in CuCl [1-2] and successively in CuBr [3], CdS [4] and ZnSe [5]. The two-photon absorption for the direct generation of excitonic molecule has a so-called giant oscillator strength [6] and its peak energy is half the binding energy of exci­tonic molecule less than the free exciton energy. The giant two-photon absorption, GTA, in CuCl and CdS is the transition to the excitonic mole­cule of Γι symmetry. In CuBr and ZnSe, three GTA bands are found into the Γι, Γ3 and Γ5 sym­metry states.

In CuCl and CuBr, emission bands, generally called M, due to excitonic molecules generated by the inellastic collision between hot excitons have line shapes showing their Maxwell distribu­tion and the emission process is the radiative annihilation with leaving an exciton behind [7]. The two-photon Raman scattering is resonantly enhanced in GTA and Raman spectra are found in the M band region. The final states are the longitudinal and transverse excitons in CuCl and the triplet exciton is also one of them in CuBr and ZnSe.

In CdS and ZnSe the line shape of the M emission is rather complex, so that objections have been raised against the conclusion that the emission is attributed to the excitonic molecule. However, we found the two-photon resonance Raman in both crystals in the M band region. Trans­verse and mixed mode excitons were left in the scattering process in CdS. However, we do not reject the possibility of the overlap of other emissions in M band region due to free electron« hole plasma or exciton-exciton and exciton-free carriers interactions. The present talk is mainly focused to the Raman and luminescence ex­cited in GTA in CuCl and the relaxation of reso­nantly excited excitonic molecules is discussed.

2. Giant Two-Photon Absorption Band in CuCl

Figure 1 shows two-photon absorption into excitonic molecule in CuCl measured with using two laser beams having broad and narrow energy band width; both of them are pumped by an N2 pulsed laser. The broad band laser was used for the probe light of measuring the transmission of crystal. The energy of the excitonic molecule, which is twice the energy of the GTA, is obtained to be 6.372eV from the combined energy of the narrow laser and the transmission minimum in broad laser. The GTA is also measured at 3.1861eV with using a single beam of intense laser as probe light. The line shape is sym­metric when the laser intensity is weak and with increasing intensity, the line width is broadened especially to high energy side. The half width varied from 0.15meV to -lmeV, as seen in Fig.l of ref.[l]. The energy difference between GTA and transverse exciton, 3.203eV, is 17meV, thus the binding energy of the excitonic molecules is 34meV.

3. Two-Photon Resonance Raman Scattering of CuCl

3-1. Backward Scattering

The two-photon resonance Raman scattering, TRRS, by the excitation into the region around GTA with using dye laser of 0.2meV half width is shown in Fig.2. The scattering is expressed as;

ηω„ = 2ηω ine E (1) With corresponding to two exciton states, trans­verse and longitudinal ones, the Raman lines are called the Mj* and ML

R. When the crystal is ex­cited into the GTA peak, hü)0(k0)=3.1861eV, which is called hereafter the resonant excitation, the scattering is much enhanced. Broad bands in 3.163eV and 3.168eV regions are due to the emis­sion of excitonic molecules in Maxwell distribu-

43

44 RESONANCE RAMAN SCATTERING AND LUMINESCENCE Vol. 32, No. 1

CuCl(1.6K)

Two Photon Absorption (ω,* ω4=2ωβ)

3.195 3.190 Photon Energy (eV)

A-

3.1797 eV

GO*

3.180

F i g . l Giant two-photon absorpt ion , GTA, in CuCl measured with simultaneous e x c i t a t i o n of two l a s e r beams, having narrow and broad energy band widths . The absorption band i s ob­served in a minimum i n s i d e the spectrum of broad band l a s e r .

CuCl (16K)

Raman Spectra

3-1856

31853

3.170 3.165 3.160 3.170 Photon Energy (eV)

3.165

Dye Laser

3.160

Fig. 2 Two-photon resonance Raman spectra, MrR and MmR, via ex-citonic molecule of CuCl in backward scattering configu­ration.

Vol. 32, No. 1 RESONANCE RAMAN SCATTERING AND LUMINESCENCE 45

tion. Fig.3 shows comparison between GTA, Raman yield and excitation spectrum of the broad band. The yield of Raman scattering drops very rapidly when the excitation energy enters into the high energy side of the GTA peak. While in the high energy side of the GTA, the excitation of broad emission is also effective. This fact suggests that the asymmetric broadening of GTA is ascribed to the collision between excitonic molecules.

In resonant excitation, the sharp line has been found to consist of Raman and luminescence

as mentioned later. The luminescence component is named as ML° and MT°. For the resonant ex­citation, the excitonic molecule is created really or virtually with k=2kQ and the final state exciton is left with k~3k0 due to the momentum conservation, as shown by an inset in Fig.4. The energy dependence of exciton on k is small in 3k0 region so that the ML

R and MTR

show no detectable shift with change of scat­tering angle in backward scattering configura­tion. 3-2. Forward Scattering

In the forward scattering configuration, the exciton is left with k~k0. The longitudi­nal exciton has a negligibly small energy change in a region 0<k<2kQ. In fact the difference of kinetic energy with k=2kQ and k=kQ is smaller than 0.05meV with using the excitonic mass

M * m + nL = 4.6m [8], ex e n o

On the other hand the transverse exciton is con­sidered to be an excitonic polariton [9] and its energy has a large k-dependence. Thus the energy of the M-pR depends on the internal scat­

tering angle a [10]. Experimental results are as follows [11]. In Fig.4 scattering configu­rations with resonant excitation are shown. Two laser beams a) and b) which are split from a frequency tunable dye laser pumped by an N2 pulsed laser, incident on a thin crystal of ~20μ thickness with almost the same angle 0=65°. The emission was measured from the direction normal to the illuminated surface. When one of two beams, a) is shut off, the spectrum corre­sponds to that for the backward scattering and

Broad band Excitation Spectrum

shows strong M LR , M T

R and weak V lines in addi­tion to Ii emission of a bound exciton, as shown by curve b ) . On the other hand the forward scat­tering spectrum obtained by turning off the beam b) shows additional lines named LEP, HEP, HEP1, LEP(II) and UP(II) in curve a ) . The M L

R and M TR

lines in forward scattering spectrum have no de­tectable peak shift from those in the backward scr Bering. The LEP(II) and UP(II) increase in­tensities when the crystal is excited with two beams. In the two beam excitation, excitonic molecules are created really or virtually with smaller k which is varied in a range 0<k<ko, depending on the incident angle Θ. With 0*65°, the excitonic molecule is created at 0.67ko.

3-2-1. LEP, HEP and HEP1 Lines

LEP and HEP lines change peak energy with a. Both lines coincide with excitation laser light when a=0, but with increasing a, the HEP line shifts toward high energy side and the LEP line does to low energy side and approaches the MrjR for a=180°. The appearance of the M»jiR line in curve a) is due to the backward scattering of excitation laser reflected from the crystal rear surface.

Two Photon Absorption (Absorbance)

Raman Intensity (Energy Dependence)

3.185 3.186 3.187 PHOTON ENERGY(eV)-

3.188

Fig.3 Comparison between giant two-photon absorption band, Raman scattering efficiencies and excitation spectrum of broad emission in CuCl.

46 RESONANCE RAMAN SCATTERING AND LUMINESCENCE Vol. 32, No. 1

m ω o

m z

D Φ

I 3.210 3.200 3J90 3.180 3170

PHOTON ENERGY (eV) 3.160

Fig.4 Raman spectra in CuCl resonantly excited in the giant two= photon absorption peak with laser beam a ) ; curve a ) , and with laser beam b ) ; curve b ) . Various scattering processes into the lower and upper branches of excitonic polariton are schematically shown in the inset.

The peak energies of the LEP, HEP and HEP' lines are plotted with respect to a in Fig.5. From the energy and wave vector conservation and the polariton dispersion given by

1/2 1/2

k(o>) = \ ωχ - ω / \ ωτ, - ω /

,(2)

the relation between the energy of scattered polariton and a is calculated as shown by a solid line in the figure. In eq.2) ΐιω and "ηω are the energies of the longitudinal and trans­verse Ζ3:(Γ7 x Γ6) exciton, at k=0. ίιω , and ηω , are those of the Ζι,2:(Γ8 χ Γ6) exciton, and are given to be 3.287eV and 3.269eV, respec­tively [12]. The following parameters are found to be best fitted.

it is mentioned that the LEP and HEP observed in a given direction are not the true partner, because they are scattered in different direc­tions with satisfying the momentum conservation.

Among the LEP, HEP and excitation laser light, the LEP travels through the crystal with fastest group velocity given by the derivative of dispersion curve with respect to k and the HEP does with slowest velocity. With resonantly excitation of a 25ps pulsed laser, it has been found that the LEP pulse comes out of crystal at first, followed by the excitation laser pulse and successively by the HEP pulse [13]. Further the polariton-polariton scattering has been found to take place only in a thin layer less than 60μ thickness under the illuminated surface.

3-2-2. LEP(II) and UP(II) Lines

ü)L = 3.2082eV, ωχ = 3.2027eV.

With resonant excitation, a photon of the LEP is scattered and an exciton is left at po­lariton state. The polariton comes out of the crystal as a photon of the HEP as schematically shown in Fig.4. The excitation photon is also a polariton in the crystal, so that the forward scattering is considered to be the polariton= polariton scattering; two polaritons of the same energy and wave vector are scattered to two po­laritons of different energies and wave vectors and the scattering is resonantly enhanced via excitonic molecule state. The LEP and HEP lines are one solution as a pair and HEP1 and M, for the back scattering are the another one. However,

The LEP(II) is due to the scattering to the lower branch of the polariton state from exci­tonic molecule generated by two laser beams with combined wave vector k=0.67ko. Its peak energy can be less or larger than the LEP depending on a. UP(II) line locates at the energy region of the upper branch polariton. With increasing a, UP(II) line approaches the longitudinal exciton energy, ηω^=3.2082eV and increases its peak in­tensity. The shift with respect to a is almost the same as that expected for the upper branch polariton generated by the process shown in Fig.4. The lower branch polariton of the pair to the UP(II) is not observed in this scattering configuration.

Vol. 32, No. 1 RESONANCE RAMAN SCATTERING AND LUMINESCENCE 47

ÜJ

O X Q.

3.20

ΪΒ 3.19 tu

2 3.18

3.17

CuQ (16K) EXCITATION

AT 3.1861 eV

· · - · - . . · · - · · · « ·_

10 20 30 40 INTERNAL SCATTERING ANGLE CX(deg)

Fig.5 Plot of Raman line energies versus internal scattering angle a, in the forward scattering configuration. Solid curve: calculated from polariton dispersion.

3-3. Luminescence of Upper Branch Polariton Called V [14]

The peak energy of the V line, 3.208eV, is equal to that of the bottom of the upper branch polariton. The line shape has close relation with M ^ and broad side band. When the broad band is weak compared with sharp MLR line the V line is also very sharp. On the other hand when IL accompanies the side band, which is in the case of intense laser excitation, the V line has also a side band at high energy side. The energy sum of the V and M-R line, 3.164eV, is just twice as much as that of the excitation laser, 3.186eV. Thus the V line is the mirror image of the M^ in energy as well as in line shape with respect to the excitation laser. Taking into account these facts together with momentum conservation, the V line is considered to be an emission of upper branch polariton from its bottom region through the decomposition of excitonic molecules at k~0.9ko.

The pair to the V line is a lower branch polariton, the energy of which is equal to that of the M^. From the excitation spectrum, it is found that the effective laser energy for en­hancing the V line is limited only in peak region of the GTA. Thus the excitonic molecules really created with k=2k0 relax in k-space and a con­siderable part of them populates at k~0.9ko. During the excitation at GTA with laser of nano second pulse, excitons are generated by the VLR

Raman scattering and they reform excitonic mole­cules which are responsible for the broad side band of Maxwell distribution. The M ^ and side broad emission induce the transition of excitonic molecules populated at k~0.9ko to the bottom region of the upper branch polariton. Thus, the V line has line shape of mirror image of the M^ and side band.

The following suggestion is proposed that excitonic molecules created with resonant exci­tation have two kinds of distribution; one is the

Maxwell distribution, the effective temperature of which is higher than the lattice and another is non equilibrium distirubion due to the rapid relaxation from 2kQ, probably localized in a range from ~2kQ to Γ point.

From the forward scattering experiment, the polariton dispersion curve is determined precisely. By using the polariton dispersion curve, the energy and width of the Raman line can be calculated if the energy and spectral width of the incident laser light and observa­tion direction are given. Further, the infor­mation of the relaxation of excitonic molecules in k-space can be obtained by knowing line shape of the M, 0 and M^° luminescence as mentioned later.

4. Relaxation of Excitonic Molecules Generated by Resonant Excitation

When the excitation energy falls well in­side the two-photon absorption band, excitonic molecules are really created at 2k0 and the luminescence called 1 ° is found at 3.164eV to­gether with M ^ as reported previously by the use of narrow energy band laser such as 0.06meV [15]. Picosecond laser excitation showed that the luminescence was separable from Raman through the time resolution of rise and decay of emis­sion [16].

However, it is not well understood whether the excitonic molecules keep their wave vector 2k0 during the radiative decay or relax along their parabolic energy band, and if they relax, what kind of distribution is realized in the k= space. We suggested previously that the relaxa­tion took place in such a way to loose their wave vectors and spread along dispersion curve, since the M O had line width of 0.3meV but the MJ 5 lu­minescence was too broad to be detected.

However, in order to discuss the relaxation from line shape of luminescence, care must be

48 RESONANCE RAMAN SCATTERING AND LUMINESCENCE Vol. 32, No. 1

paid to avoid various effects on emission caused by the stimulation, overlap of broad Maxwell type band and reabsorption by the optical conversion from exciton state to the excitonic molecule state. Further the picking up of luminescences emitted to various directions must be minimized. In general, the crystal shows brighter emission from edges which is due to the internal reflec­tion of emission scattered to various directions and stimulated. By shutting off emission from crystal edges and with using weaker excitation, the above mentioned unfavorable effects are al­most avoided.

Fig.6 shows emission spectra thus obtained in nearly the backward scattering configuration with dye laser of 0.07meV spectral width. The excitation energy was finely changed across the resonant energy in curves from a) to d ) . In this case the GTA band had 0.2meV half width as shown in the figure.

Sharp lines of 0.13meV half width which shift with the change of excitation energy are the Mr£R and ML

R. In pure backward scattering, the M^R should not appear because the polariza^ tion vector of the scattered polariton must be parallel to that of exciton left behind. In the figure, the M L

R is much weaker than the Mr^, in­dicating that observation is really made in nearly pure backward scattering.

The wider bands which do not change peak energy are ascribed to the Μτ° and M^° (broken curve) luminescence. The ML<5 has 0.3meV half width as reported previously and the M-p° is found this time with -lmeV half width. The total in­

tensity (area) of the M^,0 line is larger than that of the M L° line.

These line shapes provide the information about the distribution of the excitonic mole­cule in k-space. Excitonic molecules having wave vector much larger than 2kQ give the similar line shape for both the M L° and Mrj,0 luminescence at lower energy part. On the other hand, those of wave vector around and smaller than 2k0 con­tribute to the line shape of high energy side of the Mrj,° line to make it broadened on account of the polariton effect, while for the M L ° line it is made to have sharp cut toward high energy side.

From various model calculations including the polariton effect and the geometrical selec­tion rule, excitonic molecules spread in a range of several times of kQ around a center, somewhat smaller than 2kQ, is found to explain both the line shapes and relative intensities of the M L° and M T° luminescence. The conclusion is in agree­ment with that proposed in the V emission pro­cess mentioned in 3-3. The more detailed studies including the angular dependence of the emission spectra are expected to clarify the problem.

The relaxation is considered to depend on k, with which excitonic molecules are resonantly created. The Γι excitonic molecules are shown to be created at k=0 with excitation of two circu­larly polarized light having resonant energy but opposite wave vector, kQ, and -k0 [17]. The transition to excitonic molecule with one of these light is forbidden so that, Raman lines from k=2kQ shown in Fig.4 do not appear at all, and Raman or luminescence from only k=0 is ob-

ai85 3.187<eV)

GTA Dye Laser

Excitation Energy

a)

b)

0

d)

3.18620 (eV)

3.18617

3.18610 (Resonance)

3.18593

3.170 3.165 Photon Enegy (eV)

3.160

Fig. 6 Coexistence of luminescence ML° and Mp 0 (broken curve) and Raman M L

R and M TR at resonant excitation. The profiles of

excitation dye laser and giant two-photon absorption band are also shown.

Vol. 32, No. 1 RESONANCE RAMAN SCATTERING AND LUMINESCENCE 49

served. The M^0 is observed at 3.164eV but the Mf° has the same energy as that of excitation light, so that it is not distinguished from Rayleigh scattered light of excitation laser.

On the other hand when two beams have slight­ly different energies u)i(ki) and ü)2(-k2), with satisfying no)i+hü)2=2x3.1861eV, the excitonic molecule is created at very small wave vector, k=ki-k2. With hu)i=3.1835eV, ηω2=3.1886βν, the excitonic molecules are created at k=0.05kQ. The ML° has 0.3meV and the MT° is found at 3.1869eV with 0.6meV half width. From the band width of the Mp0 luminescence, the excitonic molecules are found not to disperse broadly but remain in a localized region of ±0.02kQ around 0.05ko [18]. Thus, the relaxation of excitonic molecules is found to depend on k.

It will be important to investigate the k-dependence of relaxation in more detail. With two circularly polarized laser having energies, ωι=ω0+όω and 0)2-ωο-όω, the k vector of resonantly

1. NAGASAWA N., MITA T. & UETA M., J. Phys. Soc. Japan 41, 929 (1976).

2. UETA M., J. Luminescence 18/19 (1979). 3. NASU Y., KOIZUMI S., NAGASAWA N. & UETA M.,

J. Phys. Soc. Japan 41, 715 (1976). 4. NOZUE Y., ITOH T. & UETA M., J. Phys. Soc.

Japan 44, 1305 (1978). 5. ITOH M., NOZUE Y., ITOH T., UETA M., SATOH S.

& IGAKI K., J. Luminescence 18/19 (1979). 6. HANAMURA E., J. Phys. Soc. Japan 3£, 1506, 1516

(1975). 7. Review: UETA M. & NAGASAWA N., Springer Lecture

Note in Physics 5>7, 1 (1976). 8. CHONG I.Y., GOTO T. & UETA M., J. Phys. Soc.

Japan 34, 693 (1973). 9. HOPFIELD J.J., Phys. Rev. 112, 1555 (1958). 10. INOUE M. & HANAMURA E., J. Phys. Soc. Japan

41, 1273 (1976).

excited molecules is varied at will by changing όω. The M«£° luminescence can be observed at energy position between two laser light. If we could find the variation of the ML° and Mj 0 lumi­nescence line shapes with k-K), it would be very valuable for the discussion of the relaxation in connection with the band width of excitonic mole­cule state. It has been shown that if one uses a dye laser system having narrower energy width, ~0.02meV, the line shape variations are expected to be observed.

Acknowledgment - The authors are grateful to Prof. Y. Toyozawa and Prof. E. Hanamura for the many discussions. The model calculation of spread of excitonic molecules by one of our co= workers, Y. Nozue is appreciated. The present works are supported by the Grant-in-Aid for the Scientific Research from the Ministry of Educa­tion.

HENNEBERGER F. & VOIGT J., Phys. Status Solidi (b) J76, 313 (1976).

11. ITOH T. & SUZUKI T., J. Phys. Soc. Japan 45 , 1939 (1978).

12. STAUDE W., Phys. Status Solidi (b) 43, 367 (1971).

13. SEGAWA Y., AOYAGI Y., AZUMA K. & NAMBA S., Solid State Commun. 2jJ, 853 (1978).

14. SUZUKI T. & ITOH T., Submitted to J. Phys. Soc. Japan for publication.

15. MITA T. & UETA M., Solid State Commun. _27, 1463 (1978).

16. MASUMOTO Y., SHIONOYA S. & TANAKA Y., Solid State Commun. ,27, 1117 (1978).

17. NAGASAWA N., MITA T. & UETA M., J. Phys. Soc. Japan 45, 713 (1978).

18. NAGASAWA N., MITA T. & UETA M., Submitted to J. Phys. Soc. Japan for publication.

REFERENCES

)Solid State Communications, Vol.32, pp.51-54. Pergamon Press Ltd. 1979. Printed in Great Britain.

RESONANT EXCITATION OF BOUND EXCITON LUMINESCENCE IN GaAsn P ALLOYS* -x x D. J. WolfordT and B. G. Streetman

Coordinated Science Laboratory and Department of Electrical Engineering University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Shui Lai and M. V. Klein Department of Physics and Materials Research Laboratory

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Bound exciton luminescence in GaAs^_xPx alloys is sufficiently broadened by alloy disorder that little information can be gained about electronic energies and phonon couplings. By resonantly exciting narrow exciton components with a dye laser we have recovered sharp luminescence features like those in GaP. Detailed results are given for x=0.50 where broad­ening is greatest. Excitons bound to group VI donors show phonon coupl­ing characteristic of an effective mass level at the X]_ conduction band minimum, whereas nitrogen traps show phonon assistance characteristic of a deep level—stronger coupling to phonons throughout much of "fc-space and more zero phonon broadening. We identify phonons and verify zero-phonon luminescence lineshape for both cases.

Introduction

The recombination of electrons and holes at radiative centers in III-V compounds has both intrinsic interest and applications to devices. Alloying allows one to tune the band gap to use­ful spectral regions. In GaAs1_xPx alloys with xk0.46 the gap is indirect, and the luminescence is qualitatively similar to that of GaP except for randomness of the group V element and except for deeper binding of excitons to nitrogen cen­ters.2» The spectra are sufficiently broadened by alloy fluctuations that only partial informa­tion can be gained about electronic and vibra-tional energies and exciton-phonon couplings. We have resonantly excited narrow components of the inhomogeneously-broadened zero-phonon line of donor-or nitrogen-bound excitons with a dye laser for x values between 0.50 and 0.96. Sharp fea­tures like those in GaP have been recovered. Results will be given here for x=0.50. Resonance excitation was previously done in GaP:N and GaP:Bi.6

Figure 1 shows relevant energies of zero-phonon luminescence peaks in the alloys. The D^ level from excitons bound to neutral donors closely follows the free exciton, thus implying that the bound exciton is effective-mass-like with electron properties derived from the X· valleys. The Nx exciton level of isolated N sharply contrasts with this behavior. Instead,

*Supported by the Joint Services Electronics Program (U.S. Army, U.S. Navy, U.S. Air Force) under contract DAAG-29-78-C-0016 and by the U.S. National Science Foundation under contract DMR 77-04382. It has used facilities in the Materials Research Laboratory supported by NSF contract DMR 77-23999. tPresent Address: IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598.

— i 1 1 1 Γ"

GaASi-χΡχ'. N (n-type) h 5°K

?5

GaAs

N Luminescence Donor Luminescence Free Exciton Edge EgX

0.5 x

1.0 GaP

Fig. 1 Composition diagram indicating the Νχ and Np emission peaks for excitons bound to isolated nitrogen centers, the peaks of the Dj and D^ emission lines for excitons bound to donors, the free exciton edge Ε^ , and the assumed Γ , L , and X. band edges.

X

51

52 RESONANT EXCITATION OF BOUND EXCITON LUMINESCENCE Vol. 32, No. 1 as x is reduced, it rapidly and progressively deepens from its shallow A-exciton origin in GaP. This suggests that the NX(A) electron is not effective-mass-like, a point of view adopted in recent theories of the N center.

This distinction is reinforced by the dif­ferent nature of the phonon-assistance observed in the luminescence at donors and nitrogen in GaP. Donors couple to all phonons at the X-point, with LAX strongest,5 consistent with the electron being at X^.7 Phonon assistance in GaP:N involves a continuous spectrum of phonons ending at LAX, LO and TO phonons at Γ, and, most strongly, an optical phonon between LO^ and TO^ in energy. (See Fig. 3 below.) This illustrates that the electron's wave function is more extended in k-space than that of the donor-bound exciton. Resonance excitation allows for the first time a study of these effects in a mixed crystal.

Results for Donors

The top curve in Fig. 2 shows ordinary photoluminescence (PL) of Se-doped GaAs rP c excited at 13 meV above the free exciton (Eg x

X). The shaded peak (DX) is essentially an envelope of zero-phonon lines inhomogeneously broadened due to fluctuations in local alloy composition; to a first approximation the emission curve is also the absorption curve.5

The laser energy is lowered to coincide with part of the main PL peak in the lower curves of Fig. 2. The low energy peak M is downshifted by 2.5 meV. It shifts down in energy under uniaxial stress, suggesting an electronic origin, and is currently being studied in detail. Besides M the lower curves in Fig. 2 show other sharp peaks that move with the laser energy. They are due to absorption into a component of the Dx line fol­lowed by phonon-assisted emission without relaxa­tion (A-Ep). The effect is equivalent to in-resonance Raman scattering. The sharp features have maximum intensity when the laser energy is at the D^ peak, as shown in the middle curve. This curve has a peak at a shift of 22.4 meV, which we associate with the 23 meV "local mode" reported in GaP:Te and GaP:Se.5 Then there is a cluster consisting of a strong peak at 30.6 meV, two sharp peaks at 32.2 and 33.5, and a stronger peak at 35.3. The latter peak and one at 48.2 dominate the spectrum in the near and off reso­nance regions (e.g., lowest curve). These two peaks are due to Raman scattering from the ternary lattice and are assigned to GaAs-like and GaP-like LO modes. They have the parallel polar­ization characteristic of "forbidden" resonance Raman scattering of LO phonons via the Fröhlich coupling mechanism 12 The peak at 33.5 meV is also seen off-resonance and, like the low energy shoulder on the Lo£ peak, is ascribed to TO modes.

The 30.6 meV peak is assigned to the X- longitudinal phonon [X-^Ga)] in which the Ga atoms move and the others are at rest. This mode is created when an electron at the X^c conduc­tion band minimum recombines via the r^c con­duction band minimum.7 The spectrum near 30.6 meV has a sharp cutoff at LAX and lower energy asymmetry arising from phonon dispersion and is similar to that found for GaP donor luminescence.5 The peak agrees with a broader one seen in free exciton emission. 3 In the virtual crystal

2.02 Energy (eV)

2.04 2.06 2.08 i 1 1 1 1 ' — r

GaAs!-xPx : Se, x-0.50 ηο κ 4 .3χ10 1 6 στΓ 3

5°K

6100 6000 Wavelengtn (Ä)

5900

Fig. 2 Secondary radiation of Se-doped GaAs^P^ upon excitation with 5883-Ä light from a dye laser (upper curve) and with successively longer wavelengths (dashed lines of lower curves). Numbers in parenthesis are energy downshift in meV. Estimated zero-phonon line (hatched) has a halfwidth of 11.2 meV.

AX approximation this X-^Ga) mode1ύ would be LA for 0.11<x<Ll.OO and LOX for x<0.11. The X3 mode of the As sublattice [X3(As)] is LAX for GaAs. It would cross the X. (Ga) mode at x=0.11 and track to a localized As gap mode in GaP as x ->1, but under a no-crossing assumption for longitu­dinal modes,ltf the X^(Ga) and Xß(As) modes would hybridize when their energies are close, giving weak two-mode behavior for the X^(Ga) density of states near x=0.5. We interpret the 32.2 meV peak as the weak upper component. At x=0.65 it splits more from the lower energy component and is less intense; it is almost unobservable at x=0.77. These trends are consistent with the hybridization picture. We see a TAX peak near 13 meV like that observed in donor-doped GaP for x>0.90, but not for x=0.50. This phonon requires electron recombination via the high-lying Tl5c band,7 a rare process at this value of x; Xi(Ga) phonon creation via the nearby T^c edge is more likely.

Vol. 32, No. 1 RESONANT EXCITATION OF BOUND EXCITON LUMINESCENCE 53

Results for Nitrogen

Photoluminescence data for N in GaP and in alloys up to x=0.75 are shown in Fig. 3. The upper curve gives the GaP A-line spectrum already discussed briefly. Note the continuous spectrum in the acoustic region: a maximum at 28.5 meV (below the LAX-cutoff), and the 13.0 meV peak coinciding with TAX. These features rather closely match the calculated acoustic phonon density of states for GaP.llf The "X" peak at 48.5 meV is the strongest in the optical region and has been variously attributed to infrared combination bands4 L0L,5 and L0X.16

Alloying inhomogeneously broadens the A line (Νχ for x<1.0) and its phonon replicas. This rapidly reduces spectral information so that by x=0.83 only LO sidebands of the Νχ peak are distinct. The TA and LA participation (arrows) and the originating Νχ lineshape and intensity (hatched) were estimated by a Huang-Rhys emission linefitting scheme (results shown dotted).17 Both alloy disorder and Nx localization are greatest near mid-alloy.17 Hence, broadening is most pro­nounced for x^0.50, (top PL spectrum of Fig. 4); here, only a hint of phonon structure remains, and the zero-phonon line is indistinct.

Like donor broadening, Νχ broadening can be eliminated with monochromatic resonance excita-

-200 -100 0 Energy Relative to Νχ Zero-Phonon Line (meV)

x under illumination above the band gap x>0.75. Dotted curve i s the resu l t of a

Fig. 3 Photoluminescence spectra of N-doped GaAs-, P. for l>x_ calculation giving the zero-phonon lineshape (hatched) of halfwidth W and prominent phonons (arrows).16

tion. For \L=6034 and 6140 A, Fig. 4 shows Nx luminescence overlaid by sharp structures with photon energies that overlap our estimated zero phonon linewidth. Since the laser energy is still above the latter, we interpret these structures

Fig. 4 Secondary radiation of N-doped GaAs ,-P c (NN<1016 cm-3) upon excitation with 5888-Ä light from a dye laser (upper curve) and with succes­sively longer wavelengths (dashed lines). Numbers in parenthesis are energy downshifts in meV. Gaussian zero-phonon line (hatched) is deduced from emission linefits.17

as phonon-assisted absorption followed by photon emission (Ap-E). For the next two curves (XL=6240 and 6342 Ä) the laser is in resonance with the zero phonon line, and we observe both A-Ep and Ap-E. Note the continuous acoustic participation characteristic of nitrogen (as with top curve in Fig. 3) and the combination and overtone scat­tering. The lowest curve is due mostly to ordi­nary RS; this shows that the hatched lineshape adequately represents the Νχ distribution.

Phonon energies are labelled near the peaks in Fig. 4. We have followed the peaks with x to pure GaP. The sharp optical modes at Γ behave like those seen with the donor. The other peaks are broader, more like those of a density of states and track with x to pure GaP as follows: 11.5 meV+13.1 meV, 24.3+28.0 meV, 35.6+33.7 meV, and 45.6+48.5 meV("X"). We attribute this latter mode to high density of states of the host lattice from a minimum in the <110> L0 branch near the K-point. Energy agreement with fits to two-phonon Raman scattering is good;15 this peakfs energy is too low for L0L5 or L0:'k.15 Supporting

54 RESONANT EXCITATION OF BOUND EXCITON LUMINESCENCE Vol. 32, No. 1

this assignment, our data show that LOp (or "X") can be directly traced to disorder-induced Raman structure of the alloy seen far off-resonance between TOf and Lof.11 This is consistent with a high density of near-K-point phonons. The large LOp width as compared to LOp (Fig. 4) reflects the LO dispersion near K.15

As a vibration from near the K zone boundary, LOp ("X") would involve mainly P-atom motion and would produce deformations around Ga atoms. We suggest that the relatively strong Νχ(Α) coupling to this mode occurs because the electron occupies Ga near-neighbor sites with high probability. This is supported by recent theories of the N-trap.18

Conclusions

Under resonance excitation of excitons bound to group VI donors one observes phonon-assisted luminescence lines characteristic of an effective mass level at the X-, conduction band minimum. LAX is the most prominent phonon, but weak two-mode behavior is suggested for X-i(Ga) vibrations. The two L0r modes are also present via the

Fröhlich coupling mechanism and are the only strong modes remaining below resonance. ΤΑ^ is absent away from GaP, suggesting that coupling to this phonon in the ternary alloy may be band-structure-dependent .

The exciton bound to nitrogen is not an effective mass level. Its strongest coupling is thought to come from LO phonons from near the K zone boundary, and to TA and LA phonons perhaps from the same region. This is consistent with electrons found well away from the X- minima in κ-space, occupying gallium sites with high proba­bility. Because of the relatively strong coupl­ing, multiple order and combination spectra of these phonons are seen. L0r phonons are also observed via Fröhlich coupling. The zero-phonon luminescence lineshape has been verified by the resonance excitation.

Acknowledgment—We thank Monsanto Corp. for the crystals studied in this work and for assistance in carrying out this research.

11. N. D. Strahm and A. L. McWhorter, Light scattering in solids, edited by G. B. Wright (Springer, N.Y., 1968), p. 455.

12. R. M. Martin, Physical Review B4, 3676 (1971). 13. A. Onton and L. M. Foster, Journal of Applied

Physics 43, 5084 (1972). 14. L. Genzel and W. Bauhofer, Zeitschrift für

Physik B2j>, 13 (1976). 15. R. M. Hoff and J. C. Irwin, Canadian Journal

of Physics 51, 63 (1973). 16. J. L. Merz, R. A. Faulkner, and P. J. Dean,

Physical Review 188, 1228 (1969). 17. D. J. Wolford, W. Y. Hsu, B. G. Streetman,

and J. D. Dow, Intl. Conf. on Luminescence (Paris, 1978).

18. J. D. Dow and P. Vogl, private communication; M. Jaros, private communication.

1. A. A. Bergh and P. J. Dean, Light-Emitting Diodes. (Clarendon Press, Oxford, 1976).

2. D. J. Wolford, R. E. Anderson, and B. G. Streetman, Journal of Applied Physics 48, 2422 (1977); and references therein.

3. W. Y. Hsu, J. D. Dow, D. J. Wolford, and B. G. Streetman, Physical Review B16, 1597 (1977); and references therein.

4. D. G. Thomas and H. H. Hopfield, Physical Review 150, 680 (1976).

5. P. J. Dean, Physical Review Jj^7, 655 (1967). 6. M. Gundersen and W. L. Faust, Journal of

Applied Physics 44, 376 (1973). 7. T. N. Morgan, Physical Review Letters _21,

819 (1968). 8. M. V. Klein, Physical Review B8I, 919 (1973). 9. J. L. Yarnell, J. L. Warren, R. G. Wenzel,

and P. J. Dean, Symp. on neutron inelastic scatt., Vol. 1 (IAEA, Vienna, 1968), p. 301.

10. G. Dolling and J. J. T. Waugh, Proc. Intl. Conf. on Lattice Dynamics, edited by R. F. Wallis (Pergamon, N.Y., 1963), p. 19.

REFERENCES

I Solid State Communications, Vol.32, pp.55-57. Pergamon Press Ltd. 1979. Printed in Great Britain.

ENHANCED RAMAN INTENSITY OF MOLECULES ADSOREED ON METAL SURFACES EXPERIMENTS AND THEORY

R. M. Hexter Department of Chemistry university of-Minnesota Minneapolis, MN 55^55

The results of AES, SIMS and SEM studies of silver surfaces which have been electrochemically oxidized and reduced in aqueous chloride solutions show that the processing yields a highly purified layer of metallic silver with a monolayer of chloride. The Raman peak reported at 2^0 cm" for such surfaces is attributed to a surface chloride vibration, an assignment supported by lattice-sum calculations that include image charge effects.

In a seminal theoretical investigation, Philpott recognized the possibility that the interaction of excited molecular electronic states with surface plasmon modes could affect Raman transitions of molecules near metal sur­faces. He also suggested the possibility of observing resonance Raman spectra as a result of the interaction between molecular excited states and the normally non-radiative branch of the sur­face plasmon dispersion curve of a free-electron metal.1 Simultaneously, several laboratories observed the apparent enhancement of the Raman spectrum of pyridine, by a factor of at least 1θ\ when it was adsorbed at an electrochemically reformed silver surface.2»3 Since then, a number of other molecules or molecular ions adsorbed at metal surfaces have been detected by Raman spectroscopy. These include several substituted pyridines and other nitrogen heterocyclics,3 as well as I2,4 CN-,5»6 C03

=,5 and SOT' adsorbed from solution and, most importantly, CO adsorbed from the gas phase on silver films deposited in UHV.8 In the last case, adsorbed CO could be de­tected with as little as 10 Langmuirs exposure to CO. Because the Raman scattering cross section of CO is so small, this observation demonstrates both the great sensitivity of the technique, as well as the fact that the enhancement phenomenon does not only depend upon electrochemical rough­ening of the substrate.

A number of theoretical models have been suggested for the enhancement phenomenon. These fall into five categories. For each model, the enhancement is supposed to be due to:

1. The existence of a large electric field gradient (M.05 V/cm), such as those which have been applied to semiconductors and dielectric crystals in order to study electric field-modulated Raman spectra.3

2. Modulation of the electronic polarizahil-ity of the metal surface by a vibrating adsorbate, analogous to the Mclntyre-Aspnes theory of electroreflectance. > °

3. Development of a polarizability-induced dipole due to both the applied electro­magnetic field (ordinary Raman effect) and the image field due to the dipole

induced in the metal, sometimes called the self-polarization effect.11»12

4. Pre-resonant or resonant excitation of conduction electron resonance by adsorbate-covered metal "bumps".13

5. Coupling of the excited states of an ad­sorbate molecule to the non-radiative branch of the surface plasmon dispersion curve, the latter states thus serving as a continuum of intermediate states in the Raman process.l > 1 1 +

In evaluating these models, the following experimental observations must be accounted for:

a. the extent of the enhancement itself, i.e., by a factor of _> 10".

b. The dependence of the enhancement on the wavelength of the exciting light (ap­proximately as ω~2).

c. Only physisorption is required; electro­chemical roughening or pretreatment is not generally necessary.

d. In the case of electrochemical roughen­ing or pre-treatment, the extent of the enhancement depends on the amount of charge transferred.15

While Models (3), (4) and (5) can quali­tatively account for the enhancement phenomenon itself [Criterion (a)], Models (1) and (2) can­not, since either the model does not suggest the source of the enhanced intensity, as with Model (1), or else it fails to account for the enhancement of a Raman-active, infrared inactive mode of the adsorbate, as with Model (2). Model (3) predicts a dependence of the enhancement on the separation of the adsorbed molecule from the surface of the form I Ί ^ (1 - a/4R3)"2, where rel a is the polarizability of the adsorbate. Since a ^ R3, enhancement is predicted. On the other hand, as R -> 0, I rel 0, which is clearly incorrect. Model (4) satisfies Criterion (b) only if it is supposed that the system consists of metal "bumps" suspended in a dielectric medium. Our experiments have shown, however, that the Raman scattering systems under dis­cussion are virtually pure—the adsorbates are present only in trace quantities.16 Models (3)

55

56 ENHANCED RAMAN INTENSITY OF MOLECULES ADSORBED ON METAL SURFACES Vol. 32, No. 1

and (5) account for the observed dependence of the enhanced intensity on the frequency of the exciting line.17»18 Model (3) is a macroscopic theory while Model (5) is microscopic; they may be equivalent to each other.

We have developed Model (5) in consider­able detail.14 In addition, we have derived surface selection rules for Raman spectra of either a single molecule adsorbed at a metal surface or for a monolayer of such molecules, viewed as a two-dimensional crystal. These selection rules can be derived from those pre­viously stated for optical processes at a metal surface,19'20 but we have derived them quite generally by the first use in molecular spectro-scopy of the charge conjugation operator. We have also made the first use in molecular spectroscopy of Shubnikov Type II ("grey") groups.21 Certain difficulties with the use of ordinary concepts of depolarization at metal surfaces were discussed, as well as the demon­stration that, unlike infrared absorption-reflection spectroscopy, Raman spectra at metal surfaces can be carried out at relatively small angles of incidence (i.e., close to normal incidence).

Elsewhere we have reported detailed AES, SIMS, SEM and electrochemical studies of the reformed silver surface.16 These studies demonstrate that electrochemically reformed silver is in a state of extra-ordinary chemi­cal purity. It is primarily a pure silver surface, quite free of oxygen, sulfur, cyanide and other typical impurities. Some chloride is present as AgCl, resulting from the simul­taneous exposure of the silver surface to oxygen and aqueous chloride. However, depth profiling shows that the AgCl is primarily confined to the surface of the reformed silver.

The most significant result of this research is that we have been able to demon­strate that surface chloride is most likely responsible for the observation of a Raman line at 240 cm"1 that has been observed by all workers who have studied the Raman enhancement of pyridine at a silver electrode, following an electrochemical cycling of the electrode in aqueous chloride.3»17 The same line, which is very strong (5 x 104 counts/sec), has been ob­served on a polycrystalline silver surface which has simply been etched in aqueous KC1.18 We have been able to show that the frequency of the Ag-Cl stretching mode due to a single monolayer Cl~ on a Ag surface should be of this magnitude, significantly higher than the analogous mode in bulk AgCl (103 cm"1).22»23 The observation of a line at 226 cm""1 in the Raman spectrum of CN~ adsorbed on silver has been similarly assigned to the Ag-C stretching mode.6 We are currently studying clean Al surfaces to determine if Raman lines due to adsorbed oxygen can be detected. Our experiments demonstrate that the Al-0

stretching mode can be easily detected. The detailed calculation which demonstrates

that the Raman line at 240 cm"1 of a silver sur­face etched in aqueous KC1 is due to adsorbed chloride depends parametrically on coverage?2»23 We have previously shown the Raman spectroscopy of metal surfaces can provide structural and orientational information concerning adsorbed molecules.14 The structural information derives from deductions of point symmetry and internal mode force constant evaluations. Orientational information comes from agreement with selection rule predictions based upon assumptions of par­ticular orientations, as well as from measure­ment of depolarization factors. In addition to the parametric evaluation of coverage, in the case of unit coverage it may be possible to use high resolution to discern the effects of two-dimensional crystallography—that is, lattice reconstruction.14 Comparison of the results with those obtained using LEED on the same sys­tem will be of interest.*4 This aspect of these studies may be most interesting, as surface re­construction may play a role in passivation of a metal surface.

The fact that the trace ions can be de­tected iii situ, on a metal in contact with aqueous electrolyte, gives Raman spectroscopy a number of advantages over all particle spectro-scopies, in which the sample must be withdrawn from the electrolyte, washed and dried thorough­ly, and then studied under UHV conditions. In carrying out these operations, it is possible if not probable that the nature of the surface is changed. None of the particle spectroscopies can directly yield structural information, and all of them are damaging to the sample in some respect.

These advantages of Raman spectroscopy are not unique to electrolytic systems. While gen­erally thought to be a weak effect, enhancement at a metal surface lifts it into a regime of great practicality.

It can be used in_ situ, without the need for transferring the sample to a UHV chamber.

Because the substrates are excellent heat sinks, the technique is non­destructive.

Because of intensity enhancement by the metal surface, the trace adsorbate mole­cule itself—not a fragment of it or a secondary particle—is the analytical probe. • Because of surface selection rules, it is a probe of surface symmetry and struc­ture.

Few other spectroscopies have comparable potential for the study of corrosion of metal surfaces, and the coupling of Raman spectro­scopy with SIMS, as we have shown, is of even greater value.16

REFERENCES

1. PHILPOTT, M.R. , Journal of Chemical Physics, 62 , 1812 (1975). 2. FLEISCHMANN, M., HENDRA, P.J. and McQUILLAN, A.J., Chemical Physics Letters, 2b_, 163 (1974). 3. JEANMAIRE, D.L. and VAN DUYNE, R.P., Journal of Electroanalytical Chemistry, _84, 1 (1977). 4. COONEY, R.P., REID, E.S., HENDRA, P.J. and FLEISCHMANN, M., Journal of American Chemical

Society, 9, 2002 (1977). 5. OTTO, A., Surface Science, 75> 392 (1978).

Vol. 32, No. 1 ENHANCED RAMAN INTENSITY OF MOLECULES ADSORBED ON METAL SURFACES 57

6. FURTAK, T. E., Solid State Communications, _28, 903 (1978). 7. COONEY, R.P., REID, E.S., FLEISCHMANN, M. and HENDRA, P.J., Journal of Chemical Society

Faraday Transactions I, _73» 1 6 9 (1977). 8. WOOD, T.H. and KLEIN, M.V., Journal of Vacuum Science and Technology, March/April (1974). 9. OTTO, A., Proceedings of the International Conference on Vibrational Spectroscopy of

Adsorbed Layers, Jiilich, June, 1978. 10. McINTYRE, J.D.E. in Advances in Electrochemistry and Electrochemical Engineering, 9, 61

(1973). 11. KING, F.W., VAN DUYNE, R.P. and SCHATZ, G.C., Journal of Chemical Physics, 9, 4472 (1978). 12. EFRIMA, S. and METIU, H., Journal of Chemical Physics (to be published). 13. MOSKOVITS, M., Journal of Chemical Physics, 69, 4159 (1978). 14. HEXTER, R.M. and ALBRECHT, M.G., Spectrochimica Acta 35A [3] (1979). 15. PETTINGER, B. and WENNING, U., Chemical Physics Letters, 56, 253 (1978). 16. EVANS, J.F., ALBRECHT, M.G., ULLEVIG, D. and HEXTER, R.M., submitted to Journal of

Electroanalytical Chemistry. 17. CREIGHTON, J.A., ALBRECHT, M.G., HESTER, R.E. and MATTHEW, J.A.D., Chemical Physics

Letters, 55, 55 (1978). 18. PETTINGER, B., WENNING, U. and KOLB, D.M., Berichte der Bunsengesellschaft, to be

published. 19. PEARCE, H.A. and SHEPPARD, N., Surface Science, 5^, 205 (.1976). 20. IBACH, H., HOPSTER, H. and SEXTON, B., Applications of Surface Science 1_, 1 (1977). 21. CRACKNELL, A.P., Thin Solid Films, 2A_, 279 (1974). 22. DELANAYE, F., LUCAS, A.A. and MAHAN, G.D., Proceedings of the 7th International Vacuum

Congress and 3rd. International Conference on Solid Surfaces (Vienna, 1977), R. Dobrozemsky, et^ al_9 eds., pp. 477-480.

23. NICHOLS, H.F. and HEXTER, R.M., in preparation. 24. For a discussion of preliminary findings of LEED applied to surface oxygen on Al, see

S. B. M. Hagström, R. Z. Bachrach, R. S. Bauer and S. A. Floodström, Physica Scripta, 16, 414 (1977). See also J. H. Charig, Applied Physics Letters, 10, 139 (1967).

I Solid State Communications, Vol.32, pp.59-62. Pergamon Press Ltd. 1979. Printed in Great Britain.

ENHANCED RAMAN SCATTERING BY MOLECULES ADSORBED ON ELECTRODES—A THEORETICAL MODEL

M. Moskovits

Lash Miller Laboratories and Erindale College, University of Toronto, Toronto, Canada M5S 1A1

The unusually large Raman signals originating from molecules adsorbed on electrode surfaces are postulated to arise from resonance enhancement by electronic processes at the rough metal-electrolyte surface. By model­ling this transition region between metal and electrolyte with a collec­tion of metallic spheres surrounded by adsorbate and ambient medium on top of a flat metal substrate, satisfactory agreement is obtained between measured and calculated spectra of both AR/R and the excitation function of the Raman signals.

Greatly enhanced Raman scattering from pyri-dine and CN~ adsorbed on electrochemically rough­ened silver electrodes has been reported by several groups.1 More recently CO adsorbed on evaporated silver was also found to produce unusually large Raman signals.2

Recently I proposed^ that these large sig­nals arise from resonance enhancement of the adsorbate vibrations by the optical conduction resonance of the two-dimensional colloid-like layer formed by microscopic roughness on the silver surface. By modelling the rough surface as a layer of metal spheres lying on a flat metal substrate with adsorbate molecules and ambient medium filling in the remaining spaces, a reson­ance absorption (called optical conduction reson­ance by Marton4) related to surface plasma ab­sorption is predicted to occur in the layer of spheres. The excitation function reported by Creighton e£ al.lc was satisfactorily reproduced when the optical conduction resonance frequency was substituted into the formula given by Al-brecht and Hutley5 for the intensity of Raman lines as a function of excitation wavelength applicable to pre-resonance enhancement.

More recent measurements by Pettinger t al.6 of the dependence of the Raman intensity associ­ated with the same ring vibrations of pyridine upon excitation wavelength, although indicating the same trend as those of Creighton et^ _al. , show that Albrecht and Hutleyfs expression does not reproduce the measured data well. This is quite understandable, since the latter is meant to apply to systems which possess one or two well-defined absorption bands tailing into the excita­tion region while the line shape of the conduc­tion resonance does not abide adequately by this condition. The shape of the excitation function will therefore be better simulated by the total absorption spectrum of the rough metal layer. In addition to the Raman excitation spectrum of two vibrations of pyridine adsorbed on silver, Pettinger e_t _al_. report the spectrum of p-polarized AR/R for the same system. Both the excitation spectrum and the spectrum of AR/R may be shown to be correctly prediced by the model proposed in ref. 3.

(AR/R)p for a system consisting of metal spheres of radius R atop a flat metal surface may be calculated following a change in adsorb­ate concentration using the expression given in ref. 7,

h AR/R = ( δ π / λ ^ ο β φ ^ !

Im-ΑΎ.

( 1 - ε 3 / ε ι ) l-e3AYn/eiAYt

cot2<J>i-ei/e3

(1)

where λ, φ^ ε^ and ε3 a re , respectively the vacuum wavelength, the angle of incidence, and the complex d ie l ec t r i c constants of the incident medium and the metal subs t ra te . The quant i t ies Δγη and Ayt are given by

Δγ = 4πΔ(ηα)/{[1+2νυ+(8π/3)(η°α0/3)] n [l+2vU+(8Tr/3)(na/a)]}

and

Δγ = 4πΔ(ηα)/{[1-νυ-(4π/3)(η°α°/3)] n [ l -vU-(W3)(na/a) ]>

in which v is the metallic fraction of the vol­ume of the layer which consists of metal spheres (bumps), adsorbate and ambient medium; a = 4R/3, R being the mean radius of the metal spheres; Δ(ηα) = ηα - η°α° in which n and n° are the numbers of molecules in the roughness layer per unit area after and before adsorption while a and a0 are their mean polarizabilities. The quantity U is related to the effective com­plex dielectric function of the metal comprising the spheres as follows:

U (ε3*-ε1)/(ε3*+2ε1)

ε3 is equal to ε3 in the limit of very large spheres (i.e. when R is large). When R is comparable to or smaller than the mean free path of the electrons in the spheres, a correction must be applied to ε3 to take into account the

59

60 ENHANCED RAMAN SCATTERING BY MOLECULES ADSORBED ON ELECTRODES Vol. 32, No. 1

decrease in the conductivity of the metal result­ing from electron scattering at the boundaries of the spheres. This we accomplish as in ref. 7. £3 is written as a sum of interband terms, ε^ , and a free electron term (the convention ε = ε1 - ielf is used throughout):

^3 2 / (ω 2 - ίω/τ) + l

Likewise, 83 may be wri t ten b

£3 = ε3 ω 2/(ω2-1ω/τ ) + 1 P

where τ , the new electron relaxation time, i s related to τ , the relaxation time in the bulk, via l / τ* = 1/τ + 4VF/3R, VF being the Fermi velo­c i ty . Since τ and ωρ are obtainable from the long wavelength behavior of the complex d ie lec­t r i c constant of the metal and VF i s known for the noble metals, AR/R may be calculated from eqn. 1 and subsequent equations, provided that the following four parameters are specified: v, R, na and n°a°.

Pett ingerTs AR/R data were f i t t ed to eqn. 1 using a l eas t squares routine and the opt ical constants reported by Johnson and Christy for s i l v e r . 8 Vp, τ and ωρ were set at 1.38 x 108 cm

sec and 1.387 x 1016 sec"1 1.35 10 -m respectively. The results are shown in Fig. 1. The best fit was obtained with v = 0.84, R = 25.4 Ä, an = 3.06 x 10"9 molee cm and a°n° = 4.19 x 10~g molec cm. Note that a decrease in an was found following adsorption of pyridine. The value of R indicates that the roughness which brings about this effect is submicroscopic and not the large hills and valleys which are apparent after subjecting the Ag electrode to many successive oxidation-reduction cycles. Thus the large enhanced Raman signals arise from al­most undetectable micro roughness. Indeed, large effects have been reported even after one oxi­dation-reduction cycle and before any large-scale roughening was observed.

The an values obtained are reasonable. (If n is of the order of 1015 molecules cm"2, a would be of the order of 3 Ä3. For comparison, the polarizabilities of water and Cl" are 1.5 and 3.6 IK respectively.) The absorption spec­trum10 of the layer consisting of the metal bumps, adsorbate and ambient medium11 can now be calculated from the imaginary part of the com­plex dielectric function of the layer. The latter is given by7 (εΓ = 1 + yt/a) in which yt = (3avU+4Tma)/[l-vU-(47r/3)(na/a)]. The absorp­tion spectrum, -Im(sr), is plotted in Fig. 2, scaled so as to bring it into the range of the Raman intensity of the 1008 cm"1 vibration of pyridine normalized with respect to v4,6 reported by Pettinger.

As is apparent from Figs. 1 and 2, the rough metal model yields satisfactory agreement between calculated and observed AR/R and to the extent that the excitation function of resonance-enhanced Raman lines is expected to follow the absorption spectrum of the substance,12 the agreement between the observed and calculated versions of the latter quantity is also good.

For other values of the parameters which enter into εΓ, the model predicts that for v values more commonly encountered in deposited films (0.5-0.7) a maximum develops in -Im^r) which comes at an excitation wavelength (AeT

o: \ or <

5 0 0 600 700 800 WAVELENGTH (nm)

Fig. 1: (AR/R) versus wavelength for Ag P

electrode due to adsorption of pyridine. Angle of incidence =66°. Circles are measured values taken from the work of Pettinger et al.6

Line was calculated using Eqn. 1 and parameters given in text.

700 nm and 550 nm, respectively, for v = 0.75 and 0.67. -Im(er) is found to decrease in the excitation wavelength range between 500 and 700 nm and to increase sharply between 700 and 800 nm with increasing R. Increasing an decreases -lm(er) slightly below 600 and sharply above it. At an = 3 x 10~8 molec cm, -Ιπι(εΓ) is almost independent of λθ, possibly explaining the con­tradictory statements one encounters concerning deviation of the enhancement from v4.

Recently Van Duynell+ proposed a mechanism for the enhancement which involves taking the polarizability of the system to be that of the adsorbed molecular dipole together with its image dipole. Efrima and Metiu15 present a more complete analysis of this mechanism. The overall polarizability is shown to have a pole and hence to display resonant behavior. The existence of such a resonance even for a flat surface was first pointed out by Dignam and Fedyk,18 who indicated the relation between this process .and surface plasma resonance. The mechanisms proposed by Van Duyen and Metiu and

Vol. 32, No. 1 ENHANCED RAMAN SCATTERING BY MOLECULES ADSORBED ON ELECTRODES 61

500 600 700 WAVELENGTH (nm)

800

Fig. 2: Circles are measured6 Raman intensities of the 1008 cm"1 band of pyridine adsorbed on Ag electrode as a function of excitation wave­length. The v4 dependence has been factored out. Line is the calculated absorption spectrum ot the rough metal layer, adsorbate and ambient medium [-Im(e )] multiplied by 1.58 to bring the values into the range of the measured excitation function.

that herein proposed have an underlying connec­tion. If Metiufs analysis were to be repeated for adsorption onto a sphere and if many adsorbate-covered spheres were to be brought close together so that their surface plasma resonances coupled to yield the conduction resonance, one would pass from MetiuTs mechanism to the present one. Despite the formal similar­ity, differences between the two cases would manifest themselves experimentally. For exam­ple, the flat surface resonance occurs at a higher frequency than does conduction reson­ance, with consequences in the excitation spec­trum and the wavelength region in which substan­tial enhancement occurs. The polarization pro­perties are also expected to differ. More importantly, the "image dipole resonance" occurs only in the presence of adsorbate and is marked­ly dependent on the distance of the molecule from the surface while the conduction resonance occurs even in the absence of the adsorbate. It is clear, however, that spheres on a surface and a flat surface are two extremes, with reality between.

Acknowledgements—Financial support from The National Research Council of Canada, The Atkin­son Foundation and Imperial Oil is gratefully acknowledged. The author is indebted to Dr. Pettinger for sending him his data prior to publication.

References

Pyridine (a) M. Fleischmann, P.J. Hendra and A.J.

McQuillan. Chemical Physics Letters 26;. 163 (1974).

(b) R.P. Van Duyne. Journal de Physique (Paris) 05: 239 (1977).

(c) J.A. Creighton, M.G. Albrecht, R.E. Hes­ter and J.A. Matthew. Chemical Physics Letters 55.: 55 (1978).

CN" (d) A. Otto. Surface Science^: L392 (1978). (e) T.E. Furtak, Solid State Communications

(in press). Thomas H. Wood and M.V. Klein. Journal of Vacuum Science and Technology (in press); D. Dilella, R. Lipson, P. McBreen and M. Moskovits (in preparation).

M. Moskovits. Journal of Chemical Physics 69_: 4159 (1978). J.P. Marton and J.R. Lemon. Physical Reviews _4: 271 (1971). A.C. Albrecht and M.C. Hutley. Journal of Chemical Physics 55 : 4438 (1971). B. Pettinger, U. Wennig and D.M. Kolb (private communication). M.J. Dignam and M. Moskovits. Journal of the Chemical Society, Faraday Transactions II 69: 65 (1973). P.B. Johnson and R.W. Christy. Physical Re­views B6: 4370 (1972). B. Pettinger and U. Wennig. Chemical Physics Letters .56: 253 (1978).

62 ENHANCED RAMAN SCATTERING BY MOLECULES ADSORBED ON ELECTRODES Vol. 32, No. 1

10. The quantity -Im(er) is actually the dielec­tric loss function and not the absorption spec­trum which goes approximately at -Im(er)/X. Quantum Mechanical Theories of resonance Raman indicate, however, that the Raman intensity will eo as (l/λ)4 [<f [ i><i| g>]2/ [ (ü)gi-Q)L)2 + Γ 2 ]. 1 ^ Since the λ 4 term is factored out of the function plotted in Fig. 2, it is the quan­tity (<f|i><i|g>)2 which determines the inten­sity and this quantity is approximately propor­tional to -Im(er). (The term (ü)gi-ü)L)2 is normally smaller than Γ2 in true resonance Raman. Its wavelength dependence will there­fore not be important.)

11. The refractive index of the ambient medium ni i *■

= ε ^ was taken to be 1.33, that of water. It

is not entirely clear that this choice is justi­fied.

12. R.J.H. Clark in Advances in Infrared and Raman Spectroscopy _1: 143 (1975), R.J.H. Clark and R.E. Hester, eds., Heyden (London).

13. D.L. Rousseau and P.F. Williams. Journal of Chemical Physics £4: 3519 (1976); J. Tang and A.C. Albrecht in Raman Spectroscopy, H.A. Szymanski, ed. (Plenum, New York, 1970), Vol. 2, p. 33.

14. F.W. King, R.P. Van Duyne and G.C. Schatz. Journal of Chemical Physics , 4472 (1978) .

15. S. Efrima and H. Metiu. Journal of Chemical Physics (in press), communicated privately.

16. M.J. Dignam and J. Fedyk. Journal de Phy­sique (Paris) C5, 57 (1977).

(Solid State Communications, Vol.32, pp.63-66. Pergamon Press Ltd. 1979. Printed in Great Britain.

GIANT RAMAN SCATTERING BY PYRIDINE AND CN~ ADSORBED ON SILVER* C. Y. Chen, E. Burstein and S. Lundquist

Physics Department and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia PA 19104

The giant RS by pyridine and CN~ on Ag is accompanied by a strong RS continuum which is attributed to inelastic light scattering by charge carrier-excitations· The enhanced RS by the adsorbed molecules and by the charge carrier-excitations are attributed to surface roughness en­hanced EM fields at the metal surface resulting from the excitation of transverse collective electron-excitations and surface-EM modes, and to surface roughness-induced radiative-excitation and radiative-recom­bination of particle-hole pairs.

The enhanced Raman scattering (RS) in the visible by adsorbed molecules on metal surfaces has been observed for a variety of molecular species )e.g. pyridine and other nitrogen hetero-cyclics, crystal violet, CN~ SCN COo", Cl" etc.) deposited electrochemically and chemically from solution onto various metals (e.g. Ag, Cu, and Pt) and, more recently, for CO vapor depo­sited onto Ag in high vacuum 1"7. The enhance­ment of the RS cross-section is particularly pro­nounced (10^ to 10^) for pyridine and CN"~ ad­sorbed on Ag. In the case of crystal violet and methyl orange which absorb in the visible, the resonance-enhanced RS is further enhanced when the molecules are adsorbed on Ag.

It is now generally agreed that the inter­action of the incident radiation with the elec­tronic excitations of the metal(which then couple with the electronic and vibrational excitations of the adsorbed molecules^ or with the electronic excitations of the metal-adsorbed molecule com­plex, play an important role in the enhancement of the RS cross-section of the adsorbed molecules. The RS tensor for the adsorbed molecules can be viewed macroscopically as involving contributions of the form (dxA/öQj)Qj, (dxs/dQjQ* and Φ χ ^ / Ö Q ^ Q A , where Q.» is the atomic displacement coor­dinate of the vibrating molecule, χ^ and \g are the electric susceptibilities of the adsorbed molecules (adsorbate) and metal (substräte),and XS-A *-s t n e "mutual" electric susceptibility of the substrate-adsorbate complex. (OXA/ÖQJ)QJ corresponds to the "direct RS by the adsorbed molecule; (dxs/dQ.)Q4 represents the modulation of the electric susceptibility of the metal by the vibration mode of the adsorbed molecule; and (°*XA-s/dQj)Qj represents the modulation of the mutual electric susceptibility of the metal-mole­cule complex. The goal of theory and experiment is to ascertain the microscopic processes that account for the giant RS cross-section of the adsorbed molecules.

As shown by Jeanmaire and Van Duyne2 and by

Albrecht et al the strong enhancement of the RS by pyridine, deposited electrochemically onto a Ag electrode from a aqueous solution containing pyridine and KC1, appears after a single electro­chemical oxidation reduction (i.e. double-poten­tial step) cycle. Pettinger and Wennig° have shown, moreover, that a detectable RS by pyridine adsorbed on a single crystal epitaxial film of Ag can be observed when the charge-transfer in the anodization process corresponds to only 0.02 of a Ag monolayer, and that the RS intensity increases with the amount of charge-transfer to a maximum when the charge-transfer for a (111) surface is 7 layers of Ag and then decreases. The RS intensity at the maximum is ~10^ greater than that observed when the charge-transfer is equivalent to 0.02 monolayer. Scanning electron microscope and Auger electron spectroscopy studies9,10 indicate that the electrochemical processing of the Ag electrode serves a) to "clean" the surface and b) to produce surface roughness on a submicroscopic scale» Pettinger et al11, on the other hand, find on the basis of of electro-reflectance measurements that there is no evidence of any surface roughness induced-ex­citation of surface plasmons by p-polarized EM radiation even after significant anodization charge-transfer. They do find that the RS in­tensities of the totally symmetric modes of the adsorbed pyridine at 1008 cm'* and 1037 cm*l, normalized to an ufi dependence, increase about one order of magnitude when the wavelength of the incident radiation is increased from 457nm to 647 nm. They also report that the relative reflectance change versus wavelength Δ R/R versus λ ) curve exhibits a broad minimum at 750 nm which they suggest may be due to optical absorption by a Ag-pyridine-Cl" complex.

The enhanced RS by electrochemically depos­ited pyridine on a Ag electrode is not very sen­sitive to the polarization of the incident or scattered radiation, nor to the angles of icidence or scattering10. We find this to be true aldo

* Research supported in part by ARO-Durham and by NSF through the University of Pennsylvania Materials Research Laboratory.

63

64 GIANT RAMAN SCATTERING BY PYRIDINE AND CN Vol. 32, No. 1

100 h

10 CpS

loo y

2000 1500 1000

103cps

cm"

=t= 4000 3500 cm 3000 -1

PYRIDINE ON Ag (50mW,5l45A)

JL 500 200

_L 2500 2000

Fig. 1 Raman spectrum of pyridine adsorbed on an evaporated A film after an anodization charge-transfer equivalent to **40 monolayers of Ag. The spectrum was obtained using 50 mW of 5145 Jt excitation.

for the enhanced RS by CN" deposited electrochem-ically on Ag. We also find, as was observed for pyridine, that the RS by CN" increases with an­odization charge-transfer to a maximum, followed by a decrease. Furthermore, as was noted by Otto for CN" chemically deposited on Ag (e.g. by immer­sion in an alkaline KCN solution which etches the the Ag surface) and measured in air1, we find that the enhanced RS by pyridine and CN" deposited electrochemically on Ag is accompanied by a strong RS "continuum" which extends beyond 4000 cm"1 (Fig. 1). We find, moreover, that the RS continuum also increases with anodization charge-transfer to a maximum at essentially the same charge-transfer

o

Έ

O O

<

a: >-CL

UVJ

80

60

40

20

PYRIDINE ON QAg (50mW,5l45A ) y

-

-

" / ,

my

1 i 1

A

A

-

1 10 20 30 40

RS CONTINUUM AT - 1 0 0 7 c m " 1 ( x 1 0 3 c p s )

Fig 2 Plot of the intensity of the pyridine peal at 1007 cm"1 versus the intensity of the Raman scattering continuum at ~1007 cnT1 with increasing anodization charge-transfer up to an equivalent of ,M50 monolayers of Ag

at which the enhanced RS by the adsorbed molecules is a maximum (Fig. 2). The RS continuum, like the enhanced RS by the adsorbed molecules, is relatively insensitive to the polarization of the incident and scattered radiation and to the angles of incidence and scattering. The RS continuum that accompanies the enhanced RS by the adsorbed molecules is similar in character to the RS continuum which is exhibited by electrochemically roughened Ag surfaces in the absence of adsorbed molecules. It is accordingly attributed to the inelastic light scattering by charge carrier-ex­citations in the metal (e.g. by electron-hole pair excitations) via (ρ·Α) processes (Fig. 3) which are made possible by the breakdown in mom­entum conservation caused by the sub-microscopic ( ΙΟΟΑ) surface roughness." The strong correl­ation between the enhanced RS by pyridine and CN~ adsorbed on Ag and the enhanced RS continuum in­dicates the liklihood that the microscopic mech­anisms for the two phenomena have important features in common.

o>s,k s »"s ouj.kj

*F-Hr ωδ,Ι<δ

k Γ

Fig. 3 Schematic diagram showing the radiative-excitation and radiative-excitation transitions that are involved in the "two step" (p-A)2 mecha­nism for surface roughness-induced Raman scatter­ing by particle-hole pair excitations in metals.

Vol. 32, No. 1 GIANT RAMAN SCATTERING BY PYRIDINE AND CN 65 We wish to report also that we have made a

concerted effort to observe the RS by pyridine and CN" on smooth evaporated Ag films, without success. In order to avoid any roughening of the surface that would result from the etching of the surface by the electrolyte, the Ag films were maintained at a cathodic bias from the moment of immersion into the electrolyte. The cathodic bias was initially increased in order to reduce any oxide or sulfide layer that may have formed during transit, and then decreased. No detectable sur­face RS by pyridine or CN" was observed under these conditions. These results reinforce the conclusion that the sub-microscopic surface roughness that is introduced by the electrochem­ical processing is essential for the observation of an enhanced RS by the adsorbed molecules.

Sub-microscopic surface roughness may play several key roles in the RS by the adsorbed mole­cules and in the RS by charge carrier-excitations in the metal:

i) It may make possible the adsorption (i.e. chemisorption) of the molecules to the metal surface.

ii) It leads to a breakdown in the conserva­tion of momentum and thereby to an enhanced radi­ative-excitation and -recombination of particle-hole pairs in the surface region of the metal^.

iii) It leads to the appearance of transverse collective-excitation resonances of the charge carriers (also termed "conduction electron res­onances") of the type that occur in thin Ag island (e.g. aggregate) films, which can be excited at frequencies in the visible and near infrared by both s- and p-polarized radiation^.

MoscovitslG has recently proposed that the enhanced RS by adsorbed pyridine on a Ag electrode is a direct result of the excitation of the conduction electron resonances of sub-microscopic "bumps" on the surface of the electrochemically processed Ag. He suggests moreover that the con­duction electron resonances are responsible for the minimum in the AR/R versus λ spectrum and for the wavelength dependence of the RS peaks of pyridine on Ag reported by Pettinger et al 11 However, Moscovits does not address the specific question as to how the excitation of the conduc­tion electron resonances is transfered to (and back from) the adsorbed molecules.

With regard to the role played by the con­duction electron resonances of the electrochem­ically roughened electrode surface, we note that the increase in absorption in the visible and near infrared that occurs when the resonances are ex­cited by EM radiation is in part due to a reso­nant increase in the EM field in the surface region of the metal and in part to the surface roughness-induced excitation of particle-hole pairs,i.e. to increased dielectric loss in the metal. Thus the surface roughness of the metal electrode a) en­hances the EM field of the incident radiation at the adsorbed molecules and the emission of scattered radiation via the conduction electron resonances21, and when there is appreciable surface roughness (i.e. large anodization charge-transfer) also via surface EM modes, and b) enhances the radiative-excitation and -recombination of particle-hole pairs in the surface region of the metal. We note also that the fact that the conduction electron resonances are excited by both s- and p- polarized EM radiation accounts for the insensitivity of

the enhanced RS by the molecules and enhanced RS continuum to the polarization of the incident and scattered radiation and to the angles of incidence and scattering.

A number of theoretical models have been pro­posed for the giant RS by pyridine and CN" mole­cules on Ag:

i) Otto 12 has proposed that the reflectivity of the metal is modulated by the local electric field set up by the vibrating charges of the ad­sorbed molecules by mechanisms similar to those that play a role in electro-reflectance.

ii) King et al 17 propose that the electric field at the adsorbed molecules is greatly increased by the superposition of the local field of the image dipoles induced in the metal by the electronic ex­citations of the molecules, and that the electronic polarizability of the molecules is thereby greatly increased.

iii) Efrima and Metiul° invoke the interaction of the electronic excitations of the adsorbed mole­cules with the collective and single particle exci­tations of the charge carriers in the metal, an extension of the model proposed by Philpottl^ which considers explicitly the interaction with surface plasmons. The interactions lead to a broadening and shifting of the frequencies of the electronic exci­tations of the molecules and, thereby, to an enhance­ment of the matrix elements for the "direct" RS by the molecules.

iv) Burstein et al-^ suggest that the RS pro­cesses involve surface roughness-enhanced excita­tions of particle-hole pairs in the surface region of the metal, which interact with the electronic and vibrational excitations of the adsorbed mole­cules, and charge-transfer excitations of the metal-adsorbed molecule complex.

The enhancement of the EM field at the ad­sorbed molecules by the excitation of the conduc­tion electron resonances of the roughened metal surface will obviously contribute to the enhance­ment of the RS by the adsorbed molecules in each of the theoretical models. On the other hand, the role played by the surface roughness-induced radia­tive-excitation and -recombination of particle-hole pairs will be different in each of the models.

In closing we mention briefly some recent results:

Tsang and Kirtley^l have observed enhanced RS by monolayerg of 4-pyridine carboxaldehyde and of other molecules with carboxy groups,but not benzoic acid, chemisorbed on the insulating oxide of a metal-insulator-metal tunneling structure in which Ag and Cu were used as the overlayer metals. They find that the RS by the molecules is further enhanced when the tunneling structures are put down on substrates with rough surfaces.

In our own experiments, we have observed strong RS by iso-nicotinic acid (4-carboxy pyridine) and by benzoic acid chemisorbed from aqueous solution onto thin (10 to 100 Ä) Ag island films as sub­strate, which is comparable to that observed for pyridine adsorbed on an electrochemically processed Ag electrode. When the Ag island films are used as an overlayer on the same molecules chemisorbed on glass, we observe a strongly enhanced RS for iso-nicotinic acid but none for benzoic acid. Thus the close proximity of the molecule to the metal surface is per se not sufficient for the observa­tion of enhanced RS by the molecule. These re­sults, and their implications regarding the mecha-

66 GIANT RAMAN SCATTERING BY PYRIDINE AND CN' Vol. 32, No. 1

nisms for the giant RS by adsorbed molecules, will adsorption of molecules onto the metal surface be discussed more fully in a subsequent publication^ We note at this time that the use of metal island films as substrates, or as overlayers, for the ob­servation of RS by adsorbed molecules provides one with a simple technique for studying the chemistry of the metal (e.g. Ag) surface and specifically the

Acknowledgements - We wish to acknowledge valu­able discussions with A. Otto, J. Kirtley, W. Plummer, G. Ritchie, and J. Tsang and in par­ticular the valuable exchange of information and discussions with R. P. Van Duyne.

REFERENCES

10,

11, 12,

M. Fleischmann, P. J. Hendra and A. J. McQuillan, Chem. Phys. Lett. 2£, 123 (1974). D. L. Jeanmaire and R. P. Van Duyne, J. Electroanal Chem. 84, 1 (1977). M.G. Albrecht and J. A. Creighton, J. Am. Chem. Soc. 99, 5215 (1977). A. Otto, Surf. Sei. 75, 392 (1978). T. E. Furtak, Solid State Commun. 2£, 903 (1978) . R. M. Hexter, Solid State Commun. (this issue) T. H. Wood and M. V. Klein, J Vac. Sei. and Tech. March/April (1979). B. Pettinger and V. Wenning, Chem. Phy. Lett. 56, 253 (1978). M. G. Albrecht, J. E. Evans and J. A. Creighton, Surf. Sei. (in press). R. P. Van Duyne, "Chemical and Biological Applications of Lasers" Vol. 4, Ch. 5, C. B. Moore ed (1978). B. Pettinger, V. Wenning and D. M. Kolb, Ber. Bunsenges Phys. Chem. 82., 1326 (1978). A. Otto Proc. Conf. on Vibrations in Adsorbed Layers, Julich Germany, 1978.

13. It is possible that the RS continuum actually corresponds to "hot luminescence", i.e. to particle-hole pair recombination radiation.

14. E. Burstein, Y. J. Chen, C. Y. Chen, S. Lundquist and E. Tosatti, Solid State Commun. 29, 567 (1979).

15. See for example, S. Yoshida, T. Yamaguchi and A. Kinbara, J. Opt. Soc. Am. 62, 1415 (1972); and J. P. Marton and J. R. Lemon, Phys. Rev. B 4, 271 (1971).

16. M. Moskovits, J. Chem. Phys. 69 4159 (1978) and Solid State Commun. (this issue).

17. F. W. King, P. P. Van Duyne and G. C. Schatz, J. Chem. Phys. 69, 4472 (1978).

18. S. Efrima and H. Metiu, J. Chem. Phys. 60, 59 (1978).

19. M. Plilpott, J. Chem. Phys. 62, 1812 (1975). 20. E. Burstein, C. Y. Chen and S. Lundquist,

(to be published). 21. J. C. Tsang and J. Kirtley, Solid State

Commun. (in press). 22 C. Y. Chen, I. Davoli and E. Burstein (to be

published).

1.

2.

3.

4.5.

6.7.

8.

9.

Solid State Communications, Vol.32, ρρ^67-70. Pergamon Press Ltd. 1979. Printed in Great Britain.

LIQUID-LIKE RAMAN SCATTERING BY SUPERIONIC MATERIALS

S, Ushioda Department of Physics, University of California

Irvine, California 92717

M. J. Delaney Department of Physics, University of Illinois

Urbana, Illinois 61820

The Raman spectrum of a superionic conductor, α-AgI, is compared with the spectra of the melt of AgC£ and AgBr, in order to determine the responsible Raman processes. We conclude that because of the strong anharmonicity and high translational disorder in superionic conductors, spectra analyses based on the concept of harmonic lattice are unsuit­able. A dynamical picture of ions similar to that of liquids appears to be more useful than the phonon concepts.

The so called "superionic conductors" are a group of ionic crystals which exhibit unusu­ally high ionic conductivity at temperatures substantially below the melting point.(1) The conductivity of these materials are comparable to that of liquid electrolytes {~1(Ω.αη)-1} which is about 10^ - 106 times the conductivity of ordinary ionic crystals at similar tempera­tures. The unusually high conductivity results from a high degree of ionic disorder and very low activation energy (~0.1 eV) for the mobile ions. From the structural studies it has been known that there are many available sites for mobile species of ions in the lattice and that the vibration amplitudes of the ions are unus­ually large.

(2) Thus one can envision a very

interesting situation in terms of lattice dynamics in superionic conductors. Since the lattice vibrations have very large amplitudes and indeed some ions actually move from site to site, one expects a highly anharmonic lat­tice. Moreover, the lattice is highly disor­dered due to the migration of mobile ions and one expects the usual conservation rules for the wave vectors to be violated.

In order to investigate the basic features of the dynamics of ions in superionic conduc­tors, we have focussed our attention on Agl which is one of the simplest members of super­ionic conductors and studied the Raman spectra of the compound at varied temperatures. At room temperature and atmospheric pressure, Agl has the wurtzite structure (3-phase) and it transforms into the superionic phase (a-phase) at 147°C via a first order phase transition. In the α-phase the iodine ions form a rigid bcc structure, while the silver ions move among many possible sites in the iodine sublattice^ Early x-ray studies suggested that the silver sublattice is in a liquid-like state permeating the rigid iodine sublattice.(?) Entropy data shown in Table 1 comparing the entropy changes upon ß-KX transition in Agl and melting in AgCJl and AgBr suggest the same physical picture;

,(3)

i.e. the entropy change at the 3-*a transition is comparable to that at the melting transitions

Table 1. Entropy Changes (cal/mole.deg)

β/α Solid/Liquid TM Agl AgCl AgBr

3.48 2.7 3.1 4.2

535°C 455°C 434°C

in Agl, AgC& and AgBr. Thus it may be reason^ able to look upon the 3"^ transition as melting of the silver sublattice. This physical pic­ture led us to make a comparison study of the Raman spectrum of α-AgI with the melt spectra of AgCÄ and AgBrWin order to understand the origin of the observed Raman spectrum of a-AgI,(5) In this paper we will concentrate on the discussion and interpretation of the light scattering mechanisms underlying the Raman spec­trum of a-Agl.

Fig. 1 shows the Raman spectra of α-AgI at two temperatures. There is very little change with temperature once the crystal is in the α-phase. The spectrum consists of a broad wing starting at the zero frequency shift and a noticeable shoulder at -110 cm~^ with a weak tail extending to about 240 cm-1 . The region about 110 cm~l where the weak shoulder is seen is the frequency range of the zone center opti­cal phonons of the 3-phase. The sharp rise below about 15 cm~l is mostly due to leakage of the elastically scattered light. This low frequency region has been studied in detail by Winterling et al.(6) All the authors(5) agree on the raw Raman spectrum of α-AgI. Different views arise in interpreting the data.

On the basis of a simplified model calcula­tion of a disordered lattice, Alben and Burns(7) concluded that the observed Raman spectrum re-

67

68 LIQUID-LIKE RAMAN SCATTERING BY SUPERIONIC MATERIALS Vol. 32, No. 1

50 100 FREQUENCY (cm")

Figure 1. Raman spectra of ct-Agl at two temperatures.

fleets the one phonon density of states. They agree that the wave vector conservation rule in inelastic light scattering processes is relaxed due to translational disorder and consequently first order Raman scattering becomes allowed from all points of the Brillouin zone. They compare the model-based one phonon density of states with the observed Raman spectrum and conclude that the experimental data can be ex­plained as the projection of the one phonon density of states by first order Raman scatter­ing processes.

We were lead to a different interpretation of the observed spectrum through a comparison of the α-AgI spectrum with the Raman spectra of molten salts of AgC£ and AgBr as well as the solid spectra of these compounds. Fig. 2 com­pares the spectra of α-AgI (solid) and the melt of AgC£ and AgBr. The apparent similarity of "the α-AgI spectrum to the melt spectra of AgC& and AgBr is very striking. A more interesting fact is revealed when the frequency axes are scaled using the different masses of the hal-ides. It was found that the three spectra of Fig. 2 coincide with each other if the frequency axes are scaled by the inverse square root of the halide masses and the intensity scale is adjusted appropriatelyW. Another interesting fact is that the melt spectra of AgCÄ and AgBr shown in Fig. 2 emerge from the respective spec­tra of the solids continuously across the melting point. No sudden change takes place as the crystalline AgCJl and AgBr melt into the liquid state. Since AgCÄ and AgBr have the rock salt structure, their Raman spectra in the solid phase is completely second order scattering, apart from a possible small addition of defect induced first order scattering. We can rule out

the possibility that the high temperature solid spectra of AgCÜ and AgBr contain a large amount of defect induced first order scattering, be­cause the high temperature spectra develops gradually from the low temperature (room temper­

ed

Z

> a: < en

GO

<

(/>

Γ RAMAN SPECTRA |

Sample la b

Ic

Agl AgBr

AgCl

Phase Solid ( a )

melt melt

Temp. 1 250 C

^ 4 4 5 C ^ 4 6 5 C|

(c) ■ s v s w ^ . . . ^ · ^ . /

•••V-. (b)

\ (a)

Figure 2 .

100 200 300 4 0 0 FREQUENCY (cm"')

Raman spectra of ot-Agl (solid) and the melt of AgCl and AgBr.

V o l . 3 2 , No. 1 LIQUID-LIKE RAMAN SCATTERING BY SUPERIONIC MATERIALS 69 ature and below) spectra without any part of the spectra showing an exponential growth of the intensity with temperature. Since the defect (mostly Frenkel defects) concentration grows exponentially with temperature, one expects the intensity of the defect induced first order scattering to grow exponentially also with rising temperature. Thus we are led to the notion that the melt spectra of AgC£ and AgBr are closely connected to the second order Raman spectra of the solid phase and consequently to the two phonon density of states rather than the one phonon density of states. A similar obser­vation was reported or rare gas solids where the second order spectrum of the solid phase trans­formed into the liquid phase spectrum continu­ously across the melting point.

(8) From the fact that the spectrum of a-Agl

(solid) can be made to coincide with the melt spectra of AgCÄ and AgBr by simple frequency scaling, we infer that the underlying scattering mechanisms are essentially similar, and that the dynamics of the ions in α-AgI is similar to the dynamics of the ions in the liquid state. It is interesting to note that the appropriate frequen­cy scaling ratios are the square roots of the halide masses and not the reduced masses of the unit cell. This indicates that the observed spectra in Fig. 2 mostly reflect the motions of halide ions rather than the combined motions of silver and halide ions. The large polarizabil-ities of the halide ions compared with that of the silver ions must be responsible for this mass scaling behavior of the Raman spectra.

At the beginning of this paper we suggested that one should expect difficulties with the usual harmonic approximation, because of strong anharmonicity and a high degree of disorder in superionic conductors. However, so far we have discussed the Raman spectra of α-AgI and the melt spectra of AgC& and AgBr in the phonon lan­guage; i.e. in the harmonic approximation of lattice dynamics. Recently, some theories of light scattering in superionic conductors not based on the harmonic approximation have been proposed. The earliest theory of Raman Scatter­ing by mobile ions was proposed by Klein(8), and theories focussing on the low frequency

region (Brillouin scattering) were reported by Huberman and Martin(9)and by Subbaswamy(lO). More recently, Geisel and coworkers(H)pub­lished a series of papers on the theory of Raman scattering spectra of superionic conductors. Their theory is based on a model of mobile ions which oscillate in a sinusoidal potential well and migrate from site to site under the influ­ence of stochastic forces from the vibrating stationary sublattice. They obtain the dynamic structure factor for light scattering without using the harmonic approximation. Thus their theoretical Raman spectrum contains first and second order as well as higher order Raman pro­cesses. Although this theory is based on a one dimensional model, the calculated spectra re­flect the general features of the observed spec­trum quite well. The low frequency region is dominated by contributions from the diffusing motions of the mobile ions, while the vibration-al component incorporated in the starting equa­tion of motion reproduces the shoulder at ~110 cm""1 in α-AgI. By adjusting the friction constant (damping) and the potential height of the model, Geisel was able to simulate the Raman spectrum of a-Agl. (11)

Since the dynamics of the ionic motions is so anharmonic in superionic conductors, it is perhaps not very meaningful to discuss the Raman spectra in terms of one phonon and two phonon processes which are based on the concepts of essentially harmonic lattices. Now we believe that we should focus on developing more realis­tic (e.g. three dimensional with lattice struc­tures) models along the lines started by Geisel and others (ID.

Acknowledgement - We gratefully acknowledge support by the National Science Foundation for this research. One of us (S.U.) would like to thank Yamada Foundation for financial assistance and the Institute for Solid State Physics, Tokyo University for hospitality during the writing stage of this work.

REFERENCES

1. See for example: a) Superionic Conductors, ed. by G. D. Mahan and W. L. Roth (Plenum Press, New York, 1976), b) Fast Ion Transport in Solids, Solid State Batteries and Devices, ed. by W. Van Gool, (North Holland, Amsterdam, 1973), and c) Physics of Superionic Conductors, ed. by M. B. Salamon, (Springer, Berlin, 1979).

2. S. HOSHINA, J. Phys. Soc. Jpn. _U, 315 (1957), and L. W. STROCK, Z. Physik, Chem. B25, 411 (1934) and B24, 22 (1934).

3. For a good review on Agl, see K. Funke, Prog. Solid State Chem. _11» 345 (1976). 4. M. J. DELANEY and S. USHIODA, Phys. Rev. B16, 1410 (1977). 5. R. C. HANSON, T. A. FJELDLY and H. D. HOCKHEIMER, Phys. Stat. Solid, (b) 20» 567 (1975),

G. BURNS, F. H. DACOL and M. W. SHAFER, Solid State Comm. _19, 291 (1976), and M. J. DELANEY and S. USHIODA, Solid State Comm. 19, 297 (1976).

6. G. WINTERLING, W. SENN, M. GRIMSDITCH and R. KATIYAR, Lattice Dynamics, ed. by M. Balkanski, (Flammarion, Paris, 1977), p. 553.

70 LIQUID-LIKE RAMAN SCATTERING BY SUPERIONIC MATERIALS Vol. 32, No. 1

7. R. ALBEN and G. BURNS, Phys. Rev. B16, 3746 (1977). 8. M. V. KLEIN, Light Scattering in Solids, ed. by M. Balkanski et al (Flammarion, Paris,

1976), p.351. 9. B. A. HUBERMAN and R. M. MARTIN, Phys. Rev. B13, 1498 (1976). 10. K. R. SUBBASWAMY, Solid State Comm. 19, 1157 (1976) and 21, 371 (1977). 11. T. GEISEL, Solid State Comm. ^4, 155 (1977); Physics of Superionic Conductors, ed. by

M. B. Salamon, (Springer, Berlin, 1979), Ch. 8; W. Dietrich, T. Geisel, and I. Peschel, Z. Physik B29, 5 (1978) and references therein.

Solid State Communications, Vol.32, pp.71-74. Pergamon Press Ltd. 19 79. Printed in Great Britain.

LATTICE DYNAMICS OF SIMPLE SUPERIONIC CONDUCTORS

Gerald Burns* F. H. Dacol*

R. Albent

*IBM Thomas J. Watson Research Center Yorktown Heights, New York 10598

^General Electric Research and Development Center Schenectady, New York 12301

ABSTRACT

Using harmonic lattice dynamics we are able to calculate 1st order Raman results in the superionic conducting phases of Agl and Cul. We find that much of the calculated response is allowed because of the structural disorder that occurs in these materials. The evidence for the 1st order nature of the experimental results is given and the calculation is compared to experi­ment. It is also pointed out that Raman experiments (and calculations) can be performed on oriented single crystals of these disordered cubic materials yielding response that transforms as the A, E, and T irreducible representations of the cubic point groups.

If one assumes an ion in a well with barrier height E, oscillating at a frequency J>, and capable of jumping over the barrier to a similar well a distance d away, then the dc ionic conductivity, σ1, can be related to some of the material properties by2

σ ~ (pd2/kT) exp ( -E /kT) (1)

This expression implies that σ and the phonons are intimately connected.

Several groups have studied phonons in many types of superionic conductors 1. Here we shall only discuss results for the simple binary materials Agl and Cul. Figure 1 shows experimental Raman re­sults for these two materials in their high tempera­ture crystal structures. The room temperature crys­tal structures for these two salts are the rather simi­lar hexagonal wurtzite and cubic zincblende struc­tures respectively. The transverse optic (TO) pho­nons at the zone center occurs at 106 and 126 cm"1

for Agl and Cul respectively. At 147°C Agl trans­forms into its high temperature α-phase while Cul transforms into a hexagonal phase at 369°C and then at 407°C into its high temperature α'-phase. The a and a' structures, which can be seen in Fig. 2, do not appear to be similar. In the Agl α-phase the I-ions are at the body center lattice points, space group Im3m (Ojj9), and the two Ag-ions per cell are ran­domly distributed over the twelve 12d-sites of this

T 1 1 1 f r

Fig. 1 The experimental and calculated reduced Raman response in the high temperature α-AgI and α'-CuI phases.

71

72 LATTICE DYNAMICS OF SIMPLE SUPERIONIC CONDUCTORS V o l . 3 2 , N o . ]

Im3m-0u • 6b sites ° 12d sites Δ 24h sites

ϊ

5 - 2 Fm3m-Oh or F43m-Td

Fig. 2 The α-AgI (on the left) and α'-CuI (on the right) crystal structure. The I-ions are on the body centered and face centered positions (large open circles and solid circles) respectively.

space group.3 In the Cul α'-phase the I-ions are at the face centered lattice points, space group Fm3m (Oh5), and the four Cu-ions are probably randomly distributed among the eight 8c-sites of this space group. Thus, both of these superionic conducting structures are disordered.

In spite of these different crystal structures there is one very important similarity: in both struc­tures the positive ions are tetrahedrally coordinated to four I-ions. This local coordination is the same as in the wurtzite and zincblende room temperature structures and we believe this to be very important in understanding and explaining, both experimentally4»5

and theoretically6, most of the spectral properties in the phonon region of the spectrum. The experimen­tal results for the high temperature phases of the two salts are rather similar in that they both show a peak at the TO frequency of their room temperature structures when the reduced Raman intensity is plotted.7 As noted previously4 this can be used to show that the Ag-ions in Agl (and Cu-ions in Cul) have tetrahedral coordination in the a-phase (in the α'-phase). This result was not apparent before the Raman work and has since been confirmed by x-ray, EXAFS and neutron experiments.1 In addition, both salts have a shoulder, in the response (Fig. 1) « 30 and 40 cm"1 respectively at approximately the tran­sverse acoustic (TA) zone edge phonon frequencies of the room temperature structures. Thus, despite the structural differences of the materials, there are striking similarities in the observed spectra which we attribute to disorder.

We have performed lattice dynamic calculations,6 in both the room temperature and high temperature structures of these salts, taking disorder

into account. The calculations use harmonic lattice dynamics to find the 1st order Raman response, but the usual k-selection rule is broken by the disorder. While other workers have suggested that the re­sponse is mostly 2nd order,8 we believe the response to be primarily 1st order in the a and α'-phase. A summary of our arguments is given below.

(1) The atoms are not at centers of symme­try, therefore the modes can be similtaneously infra­red and Raman active. (For glasses and amorphous semiconductors that are tetrahedrally bonded one observes 1st order Raman spectra as determined by the proper scaling of the intensity with Bose-Einstein factor.9 Infrared response is certainly al­lowed for the ionic salts Agl and Cul since there are two oppositely charged atoms per primative unit cell and the reduced Raman response is quite similar to infrared results.4)

(2) In Agl, in going from the wurtzite phase (where 1st order Raman lines are allowed) to the α-phase, there is very little change in the Raman spectrum for ω > 80 cm-1. In fact there is a slight increase in intensity,4 while if the spectrum in the high temperature phase were 2nd order one would expect a decrease in intensity.

(3) When the Raman spectra in α-phase Agl are compared at two widely differing temperatures (147 and 402°C) and scaled according to what is expected for the temperature dependence of 1st or­der Raman scattering, the results are in good agree­ment with each other.4

(4) Above 4 kbars of hydrostatic pressure at room temperature Agl has a phase transition to the NaCl structure (fully ordered). In this structure, where each ion is at a center of symmetry, one ex-

Vol. 32, No. 1 LATTICE DYNAMICS OF SIMPLE SUPERIONIC CONDUCTORS 73

4.0r

_ 3.0h

Ί Γ — DATA IN NaCI PHASE-Hanson, Fjeldjyö Hocheimer|

— a-PHASE

Fig. 3

100 150 ENERGY SHIFT (cm"')

Reduced Raman response of the data of Hanson et al.10 of Agl in the NaCI structure (obtained at 5 kbars).

250

pects no 1st order but only a 2nd order Raman scat­tering. Indeed a weak spectrum is observed.10 The reduced result is shown in Fig. 3. The peak at 68 cm-1 is just what one would calculate for the TO modes for Agl in octrahedral coordination by using ω τ ο vs reduced mass plots.4 (TO modes are lower in frequency for 6-fold then for 4-fold coordination). The features at approximately 2 and 3 times the TO frequency are what would be expected for 2nd order spectra.

(5) Although the above reasons are all based on experimental observations, we note there is sepa­rate theoretical evidence. First, taking the ordered room temperature Agl wurtzite structure and allow­ing some of the ions to move slightly off their equi­librium positions gives this structure a degree of dis­order. When this is done and the Raman scattering calculated6 one finds that the k-selection rule is bro­ken and a response at frequencies lower than the TO

— HH ■--HV

d,,d2

and d?

a'-Agl

30

20

10

40 80 120 160

dj and d3

(no depol. 20 bond rotation) (0

20 15 10 5

H 3 0 d3

(HH only) 20 10

0

02 and d3 (no depol.

bond stretch)

0 40 80 120 160 ENERGY SHIFT (cm"')

Fig. 4 Calculated reduced Raman response, for the various possibilities as noted, for a-Agl (and for Agl if it were in the α'-crystal structure).

74 LATTICE DYNAMICS OF SIMPLE SUPERIONIC CONDUCTORS Vol. 32, No. 1

mode appears. The magnitude of this response is about 20 to 3 0 % of the spectral intensity at the TO frequency, which agrees with what one observes. Second, when the same calculations are applied to the two different disordered crystal structures of Agl and Cul the response shown by the solid line in Fig. 1 is obtained. Again, about the right ratio of the low frequency to TO frequency response is obtained. The low frequency shoulder is a direct manifestation of acoustic modes having observable Raman activity because of the breakdown of the k-selection rule due to disorder.

For the above reasons we believe that the observed spectral response in the high temperature phase2 of these superionic conductors is primarily 1st order. Of course, there are undoubtedly some 2nd order effects but they are not the primary response. However, the broad intensity above the TO frequen­cies, where there are no vibrational modes, must be 2nd order in origin.

There still is a theoretical problem associated with the Raman calculation. For this calculation we need an expression for the linear change of polariza-bility, A, with atomic displacement, u. The general form for the yß element of the A tensor is

v ^ = = 2 U D i . yß (2)

where i runs over all the atoms and a runs over the three directions x, y, and z, and the third rank tensor Dj depends on atom i and its environment. The sim­plest form of Dj, the one we use, involves only the nearst neighbor unit vectors but still requires three parameters d j , U2 and d3. This is

+ d2^2; «yrij + J d a | 3 rij rij rij rij >

The primed sum runs over the nearest neighbors of atom i and r - 0 is the projection of the unit vector r-on the a direction. The parameter di measures the depolarized scattering due to bond stretching, the 02 parameter measures the depolarized scattering due to bond rotation, and the d3 parameter measures the polarized scattering due to bond stretching.6 We compute the scattering for these three cases by set­ting each d in turn to unity while the other two are set equal to zero. Then we can display the results in an arbitrary way depending on what we believe are the polarized or non-polarized, etc. contributions. Previously5»6 we have more heavily weighted the polarized contribution because the experimental re­sults show that it is stronger then the depolarized spectra. For example, the calculated results in Fig. 1 represent Raman spectra that are dj + (02 + d3>/2.

However, one can, for these cubic structures, determine the Raman response that transforms as the A, E, and T irreducible representations of the cubic point groups.5 Experiments on oriented single crys­tals in the a or α'-phase could be used to determine these responses. Such experiments have not as yet been done. However, experiments of the polarized, HH, and depolarized, HV, radiation have been performed.10»11 Figure 4 shows the calculated results for Agl in the α-phase (and the results of Agl in the α'-phase) for various possible values of d's. As can be seen the HH results are always similar but there are differences in the HV. Detailed comparisons of these and similar results for the copper salts should prove interesting. Also, as mentioned, experiments on oriented single crystals should be possible and the results can be compared to calculations.

(3)

3. 4.

10. 11.

REFERENCES For a summary of the field see: (a) "Solid Electrolytes" edited by HagenmuUer and W. Van Gool (Academic Press, 1978); (b) "Superionic Conductors," edited by G. D. Mahan and W. Roth (Plenum Press, 1976); (c) K. Funke, Prog, in Sol. State Chem. 11, 345 (1976. For example see, C. Kittel, Introduction to Solid State Physics (John Wiley and Sons) the chapter on diffusion. See reference 1, 4 or L. W. Strock, Z. Phys. Chem. B25, 411 (1934) and B31, 132 (1936). G. Burns, F. H. Dacol and M. W. Shafer, Phys. Rev. B16, 1416 (1977), and Solid State Commun. 19, 291 (1976). G. Burns, F. H. Dacol, M. W. Shafer and R. Alben, Solid State Commun. 24, 753 (1977). A more complete paper is in preparation. R. Alben and G. Burns, Phys. Rev. B16, 3746 (1977). The reduced Raman intensity, Irecj, is defined in terms of the measured intensity, lRam, as Ire(j, = ioI R a m / ( l + n) where n is the Bose-Einstein factor n = [exp (h<o/kT)-l]_1. M. J. Delaney and S. Ushioda, Phys. Rev. B16, 1410 (1977). J. E. Smith, Jr., M. H. Brodsky, B. L. Crowder, M. I. Nathan and A. Pinczuk, Phys. Rev. Lett. 26, 642 (1971). R. Alben, D. Weaire, J. E. Smith, Jr. and M. H. Brodsky, Phys. Rev. B l l , 2271 (1975), M. H. Brodsky in "Topics in Applied Physics," edited by M. Cardona (Springer-Verlag, Berlin, 1975) p. 205. R. C. Hanson, T. A. Fjeldly and H. D. Hockheimer, Phys. Status Solida 70, 567 (1975). R. Nemanich, J. C. Mikkelsen, Jr. and R. M. Marten, this conference and the September 1978 International Semiconductor Conference.

3. 4.

1.

2.

3.4.

5.

6.7.

8.9.

(Solid State Communications, Vol.32, pp.75-78. Pergamon Press Ltd. 1979. Printed in Great Britain.

LIGHT SCATTERING STUDY OF FERROELASTICS Hg2Br2

M. Daimon, S. Nakashima and A. Mitsuishi

Department of Applied Physics, Faculty of Engineering, Osaka University 565 Suita, Japan

Brillouin and Raman scattering spectra of the soft modes in Hg2Br2 have been measured using triple pass Fabry-Perot interferometers. The temperature dependence of the coupled soft optical and acoustic modes are analyzed by use of the thermodynamic potential and the equations of motion for the coupled oscillators. These two methods have given the same result for explaining the elastic anomaly of Hg2Br2 crystals below the transition temperature.

INTRODUCTION The elastic anomalies accompanied by

structural phase transitions have recently been investigated in improper ferroelastic materials such as gadolinium molybdatel»^ and mercurous halides3"5. in these investigations softening of the elastic constants in the ferroelastic phase has been explained in terms of inter­action (coupling) between soft optical and acoustic modes. Since both of the modes can be observed in Hg2Cl2 and Hg2Br2 by light scattering techniques, these materials are quite suitable candidates to examine the coupling mechanism. The variation of the soft phonon frequencies with temperature reflects the coupling mechanism and accurate measure­ments of their temperature dependence are de­sirable.

According to Barta^ the phase transition is due to the instability of the transverse acoustic phonon at the X point on the zone boundary of the Brillouin zone in the tetrago­nal phase. The calculation following the group theoretical method by Laverncic and Shigenari" also shows that the soft mode above the tran­sition temperature belongs to the 14 represen­tation at X point.

More recently Raman and Brillouin spectra of Hg2Br2 were observed using triple-pass Fabry-Perot interferometers^.

In this paper we shall describe briefly the experimental results on Hg2Br2 and discuss the analyses by use of the thermodynamic po­tential and Lagrangian for the coupled oscilla­tors. Our purpose is to explain selfcon­sistently the behaviors of both the soft opti­cal and acoustic modes.

EXPERIMENTAL Raman scattering spectra of the soft opti­

cal mode below the transition temperature are shown in Fig.l. As the Raman peaks were well separated from Rayleigh line at all the temper­atures except just below the transition temper­ature, the peak positions could precisely be determined with an accuracy of 0.1 cm~l. The small peak observed at the position of 2.7 cm~l is due to the strong higher-frequency Raman

band at ~37 cm which could not be rejected by an interference filter.

Figure 2 shows the frequency and the line-width of the soft optical mode as a function of the temperature. The linewidth is less than 1 cm" except in the vicinity of the transition temperature. As is shown later the soft acoustic mode is not overdamped. 1^X2 (X=C1 and Br) may be the first examples in which underdamped soft acoustic and optical modes are

1 1 1 1 1 1

Η 9 2Β Γ 2

1 1 1 fl n il

Γ

N\ A A

\ 1 , , 1 , 1

1 ' 1 1 1 :

il

VwJ30.3K

W^i3 A-8 K

H1.5K

, , , , 1 5 (cnT1) 10

Fig. 1: Raman spectra of the soft optical mode in the ferroelastic phase.

75

76 LIGHT SCATTERING STUDY OF FERROELASTICS Hg2Br2 Vol. 32, No. 1

100 110 120 130 RO 150

TEMPERATURE (K)

Fig. 2: The frequency and the halfwidth of the soft optical mode as a function of temperature.

observed. Our Raman measurement showed that the soft

optical mode behaves as Fig. 3: Brillouin spectra of the acoustic phonons propagating along [100] direction in the ferroelastic phase.

u>R(T) = 2.17(T0-T) Ύ/2 (cm"1) (1)

where T0 - 146.1 K and γ/2 = 0.33±0.01. This result is consistent with Barta's result3 but contradicts to the recent result by Benoit et al.7 who mentioned that the soft optical mode varies as (TQ-T)1'2. However, the experimental conditions were not described in their paper.

Figure 3 shows the Brillouin spectra taken with a right angled scattering geometry, when the incident wave vector k^ is along [101] and the phonon wave vector q is parallel to [100]. The elastic constant C56 in the ferroelastic phase was estimated from this experimental ar­rangement. Though the principal axes rotate around the C-axis by π/4 or 3π/4 at the phase transition, the definition of C66 In the ferro­elastic phase used here is based on the axes of the paraelastic phase. The temperature de­pendence of C£6 is fitted to the form

including the sixth order term of the order parameter and (b) the equations of motion for the coupled oscillators.

DISCUSSIONS (a) Thermodynamic potential approach

The potential function in the ferro­elastic phase can be written in the form as^

Φ - Ψο + *int (3)

BQ^ + x DQ-Λ + £ GQ6 + I a*

Ψίη FXAQ2+fx2Q2 2 ^

(4)

(5)

C66(T) = C66(0) -(Τ0-ΤΓ

(2)

where 6 = 0.37±0.05. It is noted that the temperature at which the elastic constant C66 reaches zero (T β 144.5 K) does not coincide with the temperature at which the soft optical mode frequency becomes zero.

We shall analyze above experimental re­sults by use of (a) the thermodynamic potential

where Q is the order parameter and X^ is the spontaneous strain. The equilibrium conditions 9t|>/9Q = 9ψ/3Χ^ = 0 are imposed on ψ, where the interaction terms are omitted. The frequency of the soft optical mode is given by

UR(T) = - —*-m 9Q2

(6)

Vol. 32, No. 1 LIGHT SCATTERING STUDY OF FERROELASTICS Hg2Br2 77 where m is the effective mass.

0 The elastic constant Cfö is given by^

C66(T) 9X6

,_JL± )A" (7) 6 9Q

When we omit the higher-order coupling term KX2Q2/2 and assume that the coefficient D is negligibly small, the fitting of the elastic constant and the frequency of the soft optical mode to the data is excellent, if we take Q = (TQ-T;&and 3=1/6.

It is known that Rushbrooke inequality must hold for the critical exponents.

a + 23 + γ > 2 (8)

where a is the exponent of the specific heat and 6 = γ - 23· Our values of 3 = 1/6 and γ = 2/3 lead the inequality to a >. 1. Such large value of a has not yet been reported in other materials. The measurement of the spe­cific heat is required to examine this problem.

Hochli and Scott1 showed that in quartz the temperature dependence of the order pa­rameter associated with the transition was. described by the relation η - ηχ = A(T*-T) , where η is the order parameter, and ηχ, A, T* and 3 are parameters which they fitted to their data. They found that 3 = 0.34±0.02 gave the best fit to the data over a relatively large temperature range below the transition tempera­ture. Banda et al.12 suggested that the ap­pearance of such non classical exponent 3 could be explained in the framework of classi­cal Landau theory. If their approach is ap­plied to our case and it is assumed that the coefficient B in Eq.(4) is proportional to T0-T, we are able to find the best fit pa­rameters to explain the behavior of the soft optical phonon. However, in this case the elastic constant C55 does not vary with temper­ature and shows a step-wise change at the tran­sition temperature. The thermodynamic po­tential including up to the fourth order term of the order parameter in Eq.(4) leads to Ü)R(T) OC (TO_T)1/2 and also the step-wise change in C66 a t t n e transition temperature.

When the higher order term of the coupling, KX2Q2/2, is included in the potential function, the elastic constant varies with temperature below the transition temperature. However, we could not find the best fit pa­rameters to interpret its behavior over the observed temperature range below T0.

If the sixth order term of the order pa­rameter is omitted in the thermodynamic po­tential, the step-wise change of the elastic constant would always be expected, even if the order parameter vaires as Q α (TQ-T)1' . The experimental observation contradicts to this expectation. This fact indicates that the sixth order term of the order parameter is necessary to interpret consistently the behaviors of both of the soft optical and acoustic modes in Hg2Br2

on the basis of the thermodynamic potential.

(b) Coupled oscillator model As is mentioned in the previous section,

the coupling of the strain with order parameter plays an important role in the temperature vari­ation of the elastic constant. Dynamical ap-porach for such coupled system has been per­formed by several authors^3»!4. Rehwald^ pointed out that the coupling energies are classified into three groups (a) AXQ, (b) AXQ2 and (c) AX2Q, where X is the strain and Q is the order parameter coupled with strain. Hg2Br2 is thought to correspond to the case (b). If we let u denote the displacement coordinate for the acoustic mode (so that X = 8η/8ξ, where ξ is used for the Cartesian coordinate), the kinetic and potential energy densities for the acoustic mode are14 : T = pu^/2 and U = 0(3"α/8ξ)2/2 where p is mass density and C is the elastic constant. The kinetic and potential energy densities for the optical mode are T0p = mQ2/2 and Uop = KQ2/2, respectively. In the case of Hg2X2 the coupling energy is given by Uc = Aidu/dZ)^2. Lagrangian for the coupled system is therefore expressed by

1 ? 1 ? L = j puz + j mQz

w3u,J A(^)Q

2 α(8ξ} fKQ2

(9)

The equations of motion are obtained from the Lagrangefs equation, where harmonic wave solutions of the form u = u0 εχρ[ί^ξ-ωί) ] etc. are sought. The results are

(0)2 - ω2)Χ - I M Q2 = o a p

(ω2 - ω2 + ) Q = 0 ix m (10)

where ω = K/m and ω2 = q2C/p. The solution in the vicinity of ω is given by

2 2 4A2q2 Q2

a pm ω£ (11)

where we use the condition, ω^»ω . If we assume, following the results in ref.

4, that Q2 « (TQ-T)1/3 and ω^ α Q and rewrite Eq.(ll) by use of the elastic constant C, we obtain

C(T) (To-T)1/3

(12)

This equation describes well the observed be­havior of the elastic constant below the tran­sition temperature, which is given by Eq.(2).

The mean field theory of Banda et al. was not adequate to explain at least the behavior

78 LIGHT SCATTERING STUDY OF FERROELASTICS Hg2Br2 Vol. 32, No. 1 of the elastic constant in Hg2Br2· The soft optical phonon which has a temperature de­pendence of ü)R(T) = ACTQ-T)1'^ has been ob­served in SbSll", which has a chain like structure like Hg2X2» Scott1' suggested that

its behavior would be explained by Banda's ap­proach. The measurement of the acoustic phonon in this material will shed a light on this problem.

REFERENCES

DVORAK, V., phys. stat. sol. (b) 45 , 147 (1971) HOCHLI, U.T., Phys. Rev. B6, 1814 (1972) BARTA, C , KAPLYANSKII, A.A., KULAKOV, V.V. , MALKIN, B.Z. and MALKOV, YU.F., Sov. Phys. JETP. 43, 744 (1976) DAIMON, M., NAKASHIMA, S., KOMATSUBARA, S. and MITSUISHI, A., Solid State Commun. 2£, 815 (1978) AN, CAO XUAN, HAURET, G. and CHAPELLE, J.P., Solid State Commun. ^4·, 443 (1977) LAVRENCIC, B.B. and SHIGENARI, T., Solid State Commun. 13, 1329 (1973) BENOIT, J.P., AN, CAO XUAN, LUSPIN, Y., CHAPELLE, J.P. and LEFEBVRE, J., J. Phys. C. 11, L721 (1978) AIZU, K., J . Phys. Soc. Jpn. 313, 1390 (1972) SLONCZEWSKI, J.C. and THOMAS, H., Phys. Rev. Bl, 3599 (1970)

10. STANLEY, J.E., Introduction to Phase Transitions and Critical Phenomena (Clarendon Press, Oxford) 1971, P.167

11. HÖCHLI, U.T. and SCOTT, J.F., Phys. Rev. Lett. 26, 1627 (1971)

12. BANDA, E.J.K.B., CRAVEN, R.A., PARKS, R.D., HORN, P.M. and BLUME, M., Solid State Commun. 37, 11 (1975)

13. PYTTE, E., Structural Phase Transitions and Soft Modes ed. by E.J. Samuelsen et al. (Universtetforlaget, Oslo) 1971, P.151

14. REESE, R.J., FRITZ, I.J. and CUMMINS, H.H., Phys. Rev. B^, 4165 (1973)

15. REHWALD, W., Adv. Phys. 22 , 721 (1973) 16. HARBEKE, G., STEIGMEIER, E.F. and WEHNER,

R.K., Solid State Commun. _8, 1765 (1970) 17. SCOTT, J.F. Rev. Mod. Phys. 46, 83 (1974)

1.

2.3.

4.

5 •

6.

7•

8.

9.

Solid State Communications, Vol.32, pp.79-82. Pergamon Press Ltd. 1979. Printed in Great Britain.

Light Scattering from Correlated Ion Fluctuations in Ionic Conductors

R. J. Nemanich, Richard M. Martin and J. C. Mikkelsen, Jr. Xerox Palo Alto Research Center

Palo Alto, California 94304

Inelastic light scattering spectra of Cul and Agl in the a and melt phases are reported and shown to involve two depolarized Lorentzian components centered at zero frequency. The narrow component is interpreted in terms of ionic motions and we propose that the observed depolarized scattering is caused by correlated configurations of the mobile ions.

The class of solids often called "superionic" conductors are crystalline materials that exhibit anomalously high ionic diffusion constants1'3 comparable to those of liquids. Many recent studies have attempted to infer the microscopic atomic motions which determine the diffusion mechanisms.1"4 Inelastic light scattering experiments are particularly useful because they permit study of dynamic fluctuations on the time scale of the atomic motions. The technique is well-established in studies of molecular diffusion and relaxation modes in liquids5 and recently has been used to study ionic motion in Agl6 and RbAg^.7 Analyses of the spectra to date have been considered only in the context of non-interacting ions diffusing in a lattifce.8"11 In the present paper we report spectra of Cul and Agl in the "superionic" « phase and in the melt. We show that the results presented here can be explained by a more complete description of the ionic motions which includes correlations among the interacting mobile atoms. The a phase of Agl (stable from 148°C to 555°C) her, a bcc lattice of I anions with Ag cations distributed over four-fold coordinated distorted "tetrahedral" sites with six such sites per Ag ion.4 The a phase of Cul (407°C to 600°C) has an fee lattice of I ions. The Cu ions are distributed predominantly in the tetrahedral sites with a small occupation (-10%) of octahedral sites,4 with two and one sites per Cu ion respectively. An important difference between the two crystals is that the cation sites in «-Agl are highly distorted tetrahedra whereas in «-Cul the cation sites have regular tetrahedral or octahedral symmetry. This description in terms of the average distribution of the mobile ions on regular lattices is sufficient if the ions do not interact. However, it has recently been shown that interactions among ions of the same species are important12 and a complete description of the system requires specification of correlations among the mobile ions. We discuss below the nature of the expected interactions and the relation of the dynamics of the correlated configurations of the mobile ions to the light scattering spectrum. The experimental observations reported here involve inelastic light scattering which can be described5'13 in

terms of correlation functions of fluctuations in the dielectric tensor 6ey(r,t) Sij(q.co) = | d r dr' dt e ^ O - i « ^ ö q j (r,t) öeij (r,0)- ,(1)

where q and ω are the differences in wave vectors and frequencies and ij the polarization indices of the incident and scattered light. Low frequency fluctuations in the dielectric function are caused by dynamic changes in local atomic configurations. If we let a(r,t) be a generalized configuration coordinate and öqj(r,t) = Ayafrt), then S(q,w) is given by13»14

Sijfa,*) = Ay2 < aq2 > f(q,w) (2) where < aq2 > is the mean square fluctuation and f(q,co) is normalized response function. For relaxing or diffusing systems which obey linear rate equations, the form of f(q,<o) is a central peak Lorentzian5

Hqf«) = r(q)/*[w2 + P2(q)] (3)

corresponding to an exponential time decay of a dielectric fluctuation. In general, there will be a sum of such Lorentzians. This is qualitatively different from scattering involving harmonic motions which yield peaks at finite frequencies and S(q,w) -+ 0 as ω -* 0.14»15

For all spectra reported here, the scattering is obtained from isotropic polycrystalline or liquid samples. In this case there are two independent measurable spectral intensities, I j ^ and l±\\ , where ± and II denote polarizations relative to the scattering plane. The contribution of the fluctuations to the specific polarization spectrum can be determined by angular averages over the components of the tensor Sy.5»16 The result is that in an optically isotropic medium foy can be divided into a totally isotropic component 5ejj = öij(Tröe)/3 which gives polarized scattering, i.e., I^JJ = 0, and the traceless remainder which gives completely depolarized scattering with Ijjj = (3/4) I j ^ . Two simple cases merit mention. Density fluctuations of isotropic systems have öey - öey = 0 and therefore give polarized scattering. In contrast, fluctuations with symmetry which belongs to a degenerate representation must be traceless and the scattering is completely

79

80 LIGHT SCATTERING FROM

depolarized. The second case is analogous to rotational fluctuations of anisotropic molecules.5

Raman or inelastic light scattering spectra of Agl and Cul were obtained from zone refined and triply-sublimed polycrystalline samples respectively. The samples were contained in quartz ampoules, and a backscattering geometry was employed where the scattered light was collected at ~20° from the incident

H 0\-ω U 1 1 1 1 1.

0 10 20 30 40 50

FREQUENCY CcnrO

Fig. i The low frequency inelastic light scattering spectra of «-Agl, α-CuI and Agl melt The filled circles are the data, and the solid line through the solid circles is a two Lorentzian fit to the data. The lower solid line in each spectrum is the contribution due to the narrow component The dashed line represents the contribution due to the elastically scattered light

light. The polarization geometries employed were I j j | and I||||. (In this geometry I|||| is essentially equivalent to I_L j_.) To obtain good light penetration the red and ir lines (6471Ä, 7525Ä and 7993Ä) of a krypton ion laser were used. Small spectral slit widths and a triple grating monochromator allowed measuring spectra to

Table 1. The narrow (rn) and broad (]\) light scattering spectra and the hopping measurements.20

Material Phase Temperature (°C) Cul a 535 Cul Melt 650 Agl « 162 Agl Melt 570

CORRELATED ION FLUCTUATIONS Vol. 32, No. 1 as low as 2 cm-1. The contribution due to elastically scattered light was determined by comparison with the scattered light from a roughed aluminum flat We have measured the full Raman spectra of the Cul and Agl samples in the «-phase and melt. The a-phase spectra for high ω are essentially equivalent to those previously reported.17"19 In addition the high frequency spectra of the melts are similar to those of the «-phase. In all cases a polarized mode is observed at ~120 cm"1, and the low frequency spectrum is depolarized. This study is not concerned with the higher frequency modes, but it is important to note that the polarization characteristics of. these features show that light is not depolarized by any optical characteristics of the sample or the experimental system. The low frequency portions of the spectra are essentially completely depolarized and the J_|| spectra are shown in Fig. 1 for «-Agl, «-Cul, and Agl melt The spectra of «-Agl are essentially equivalent to those reported by Winterling, et alß It is clear from Fig. 1 that each low frequency spectrum contains a narrow and a broader component. We have fit each of our spectra with two Lorentzians centered at zero frequency, as was done in Ref. 6. The fits are also shown in Fig. 1 and the widths of the Lorentzian components of the « and melt phases of Agl and Cul are summarized in Table 1. The spectra for Agl and RbAg^ previously have been interpreted in terms of models in which only site polarizability variations cause the scattering: the interactions between mobile ions are considered as negligible, and the site symmetry is determined by the fixed ions. This is a configuration fluctuation involving a single mobile ion and the lattice of fixed ions. Huberman and Martin8 and Klein9 analyzed the low frequency scattering in terms of a hopping model. More recently, Geisel10 and Dieterich, et Ö/.11 have carried out more extensive analyses of ionic motion in which the entire frequency range can be described. In the appropriate limits this model, as well as the hopping models, yield an unshifted component in the spectrum with a width Γ = f(6D/l2) where D is the diffusion constant and 1, the jump length. Here f is a factor ~1 which depends upon the geometry11 and which relates the dielectric fluctuation rate to the ionic hopping rate τ"1 = 6D/12. As shown in Table I, the width of the narrow Lorentzian feature is in general agreement with this characteristic rate.20 In this model the polarization characteristics are dependent on the site symmetries. Because the tetrahedral sites are distorted in «-Agl, the scattering from hops between sites corresponds to rotational fluctuations of anisotropic molecules and leads to depolarized scattering.8^9'11 On the other hand, in «-Cul, the

linewidths observed in the low frequency rate (V1) determined from conductivity

Γη (cm"1) Pb (cm"1) τ"1 (cm-1) 3 . ± 1 45 ± 8 1.3 4. ± 1 35 ± 8 3 . ± 1 30 ± 8 2 .1 6. ± 1 25 ± 8 5 .2

Vol. 32, No. 1 LIGHT SCATTERING FROM CORRELATED ION FLUCTUATIONS 81 tetrahedral and octahedral sites are not distorted. If the sites have such high symmetry, site fluctuations give only polarized scattering. Therefore, this model cannot account for the depolarized scattering observed in α-CuI and there is no obvious relation to the melts of both materials. Interactions between the mobile ions can cause qualitative changes in the ionic fluctuations and the resulting light scattering. The case of interacting charged ions diffusing in a continuum is well understood and known to be important in scattering from liquids.5 If there were no interactions the densitv fluctuations give a very narrow peak with r = Dq* where q is the light wavevector transfer. Including interactions leads to a qualitatively different central peak5»21 with r ~ Dq0

2 ~ 4Wc 0 , where q0 is the inverse of the Debye-Huckel screening length. The physical picture of the ionic fluctuations in the interacting system is that of highly correlated motions of ions screened by neighboring ions with a characteristic dielectric relaxation time 4πσ/ε0. In the continuum the resulting light scattering is weak5»21

(~q2/q02) and polarized (because it is assumed to

couple only to ion density fluctuations) and therefore cannot explain the present results. The observed depolarized low frequency scattering can be understood by considering the local site symmetries of the mobile ions taking into account interactions with the other ions in the disordered crystal. The local symmetries are not determined solely by the symmetry of the rigid 1 sublattice but also by the positions of the other mobile ions. Consider, for example, the affect of a Coulomb interaction ~e2Al between two Cu atoms occupying neighboring tetrahedral sites which are separated by 1 = 3.1 A. If we take e. ~10 the interaction energy is ~0.5 eV. The energy is minimized not by having each Cu at the centers of the tetrahedra but rather with each Cu ion displaced by x = e2/cl2K. Here K is the force constant for displacing the Cu ion with respect to the I tetrahedron. If K is approximated by Μωτο

2, when ωτο is the infrared active optic mode -120 cm-1 in α-CuI, then the displacement is x ~ 0.07A. This is comparable to the root mean square thermal displacement of this optic motion, i.e., (kT/mcoTo2)1/2 ~ 0.06Ä. For the nearest neighbor pairs, there are three configurations, namely the orientations along the cubic axes. The resulting fluctuations of these pairs transform as a non-degenerate l'i and a doubly degenerate Γ3+ representations. The former corresponds to the total density of pairs and gives polarized scattering, the latter, with differences in densities of the different types of pairs, giving completely depolarized scattering. The decay rate of the pair fluctuation is approximately the sum of the hopping rates of the two atoms r ~ 2/τ. As shown in Table I the width Γ is approximately 2/τ where τ is

1. See articles in Superionic Conductors, ed. by G. D. Mahan and W. L. Roth (Plenum Press, New York, 1976).

inferred from the conductivity. A more quantitative comparison requires an extensive analysis of the correlations which determine both τ"1 and the relation of Γ to τ'1. More distant neighbor pairwise interactions would also be possible. These would give similar relaxation times but in general would belong to different symmetry representations. Of course, more complicated many body Cu-I configurations could also be present. The scattering we have described is analogous to collision induced scattering which is well known in liquids and gases.5 In this case a "collision" corresponds to the formation of a nearest-neighbor pair and Γ"1 represents the "collison" time. The proposed interactions between the mobile ions can be observed in other experiments. For example, low symmetry configurations necessarily cause quadrupole fields at the nuclei and contribute to the spin-lattice relaxation rate Tf1. NMR measurements have been carried out in Cul22 where it was argued that the fluctuating quadrupole fields at a Cu nucleus were generated by neighboring Cu ions. Nuclear quadrupole relaxation and depolarized light scattering therefore provide complementary measurements of the same ionic fluctuations. The spectra for the melts are qualitatively the same as in the a phases. As shown in Table I the width of the narrow component increases by < 2 upon melting. This is in qualitative agreement with structural studies,4

high ω Raman features,23 and NMR data,22 all of which lead to the conclusion that local atomic order and the dynamics are very similar in the a and melt phases. In this paper we have discussed the dynamic aspects relating to the narrow low frequency component of the light scattering spectrum, but little has been said about the broader component. This feature is also depolarized and observed in both the « and liquid phases. A straightforward explanation is that the feature is due to relaxation after or during the diffusive motions. In a jump diffusion model of the ionic motion in the solid phase, the width of the broad component would be related directly to the inverse of the hopping time. The data suggests a similar phenomena in the liquid. We have reported low frequency depolarized light scattering spectra for α-CuI, «-Agl, and their melts and have proposed thai a contribution to the scattering is caused by fluctuations of correlated configurations of mobile ions. The correlation is expected from simple estimates of ion-ion interactions and the picture which emerges is one of ionic quasiparticles-highly correlated configurations of an ion screened by the neighboring ions in a manner similar to the Debye-Huckel screening in a continuum, but modified by the constraints of the underlying lattice structure.

Acknowledgement~We gratefully acknowledge helpful discussions with J. B. Boyce, T. Geisel, and S. A. Solin.

2. First Ion Transport in Solids, Solid State Batteries and Devices, ed. by W. Von Gool (North Holland, Amsterdam, 1973).

References

82 LIGHT SCATTERING FROM CORRELATED ION FLUCTUATIONS Vol. 32, No. 1

3. K. Funke, Prog, in Solid State Chem. 11, 345 (1976).

4. J. B. Boyce and T. M. Hayes, Topics in Applied Physics, ed. by M. B. Salamon (Springer-Verlag, New York, in press).

5. B. J. Berne and R. Pecora, Dynamic Light Scattering (John Wiley and Sons, New York, 1976).

6. G. Winterling, W. Senn, M. Grimsditch and R. Katiyas, Lattice Dynamics, ed. by M. Balkanski (Flammarion, Paris, 1977). p. 553.

7. R. A. Field, D. A. Gallagher and M. V. Klein, Phys. Rev. B18, 2995 (1978).

8. B. A. Huberman and R. M. Martin, Phys. F?v. B13, 1498 (1976).

9. M. V. Klein, in Light Scattering in Solids, ed. by M. Balkanski, R. C. C. Leite and S. P. S. Porto, (Flammarion, Paris, 1976) p. 351.

10. T. Geisel, Solid State Commun. 24,155 (1977) and Lattice Dynamics, ed. by M. Balkanski (Flammarion, Paris, 1977) p. 549.

11. W. Dieterich, T. Geisel and I. Peschel, Z. Physik B29, 5 (1978).

12. W. Schommers, Phys. Rev. Letters 38, 1536 (1977); and P. Vashista and A. Rahman, Phys. Rev. Letters 40, 1337 (1978).

13. L. D. Landau and I. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), Vol. 8, p. 390.

14. P. C. Martin, Measurements and Correlation Functions (Gordon and Breach, New York, 1968).

15. R. J. Nemanich, Phys. Rev. B16, 1655 (1977). 16. G. Hertzberg, Molecular Spectra and Molecular

Structure, Vol. 2 (Van Nostrand, New York, 1945), p. 246.

17. G. Burns, F. H. Dacol, M. W. Shafer and R. Alben, Solid State Commun. 24, 753 (1977).

18. M. J. Delaney and S. Ushioda, Solid State Commun. 19, 297 (1976).

19. G. Burns, F. H. Dacol and M. W. Shafer, Solid State Commun. 19, 291 (1976).

20. We have calculated τ"1 from available conductivity.3 The hopping length 1 was chosen as the distance between nearest neighbor tetrahedral sites4 and the same length was used for the Agl melt. In addition in all cases including the melt, it was assumed that only the Ag or Cu ions contributed to the conductivity.

21. R. Zeyher, Z. Phys. B31, 127 (1978). 22. J. B. Boyce and B. A. Huberman, Solid State

Commun. 21, 31 (1977). 23. R. J. Nemanich and J. C. Mikkelsen, Jr., in Proa

of the Nth Int. Conf on the Physics of Semiconductors, Edinburgh 1978 (in press).

Solid State Communications, Vol.32, pp.83-84. Pergamon Press Ltd. 1979. Printed in Great Britain.

RAMAN SCATTERING AND BOND STRUCTURE IN CHALCOGENIDE GLASSES (Se-Ge SYSTEM) H. Kawamura and M. Matsumura

School of Science, Kwansei Gakuin University, 1-1-155 Uegahara, Nishinomiya, 662 Japan

The Raman Scattering from amorphous Se-Ge is studied in the whole range of germanium content up to 80%. The result suggests an appearance of three-fold coordinated bond between Se and Ge above the germanium con-content of 40%.

The Raman Scattering from selenium-germa­nium glasses have been studied by several au­thors·1- '2'3. However, they are confined within the range of germanium content of less than 40%. Lucovsky et al. discussed the structure of the charcogenide glasses from these experiments in terms of the random arragements of the two-fold coordinated bond of charcogen and tetrahedral bond of germaniuml,2'3'4. We have extended the range of germanium content up to 80% using vac­uum deposited specimens. The experiments give an evidence of three-fold coordinated bond around selenium atom as well as around germanium atom for the specimens with germanium content of 50% or more.

The specimens were prepared by means of the vacuum deposition of powdered ingot onto quartz plates from tungsten boat. The ingots were pre­pared by melt and quench of the weighted high purity elemental powder in silica tube. In order to obtain the amorphous materials with germanium content of more than 40%, the sub­strate had to be cooled down to 140K. The Raman spectra were obtained at room temperature in back scattering configuration with argon-ion laser operated at 5145A with 100 mW. We used cylindrical lens, in order to reduce the energy density falling onto the surface of the specimen.

To obtain the "approximate" density of vibra-tional states for amorphous solid, the observ­ed Raman intensities were multiplied by factor

ω(ωί-ω)"4[1-βχρ(-ϋω/Μ)] for Stokes side, and factor

ω(ω±-ω)~4 [expCiiü)/kT)-l] for anti-Stokes spectra under the assumption of constant Raman tensor^. Here, ω and ωι are the phonon and laser frequencies, respectively, and T is the temperature of the specimen. The reduced Raman spectra at Stokes side and anti-Stokes side agree well with T=400K, indicating that temperature of the specimen increases by about 100C due to the laser irradiation.

The reduced Raman spectra for amorphous Sei-xGex with differnt germanium content x are shown in Fig. 1. Below x = 1/3, the spectra are similar with those obtained by the previous authorsl>3. While band A at 250-260 1/cm which is ascribed to the vibration of Se-Se bond de-

150 200 250 300 RAMAN SHIFT (cm-·)

350

Fig. 1. Reduced Raman spectra of Se-Ge glasses for different Ge content.

creases with the increase of germanium content x, band B at 202 1/cm and band C at 218 1/cm increase with x. These bands were ascribed to the vibration of GeSe4 molecule1'3' . We found new band D at 315 1/cm whose intensity increases with the increase of germanium content. Band E at 281 1/cm which appears in the range of large content of germanium will be related with the phonon density of states for amorphous germani­um6. Above 40% of germanium fraction, we can observe a series of bands at 174 1/cm (band F), 157 1/cm (G), 145 1/cm (H), 125 1/cm (I), 110 1/cm (J) and 95 1/cm (K).

Fig. 2 shows the relative values of the "approximate" density of vibrational states which are measured from the area of bands in the reduced spectra. Although the separation of the broad structure around 250-300 1/cm is

83

84 RAMAN SCATTERING AND BOND STRUCTURE Vol. 32, No. 1

5e,.xGex

Fig. 2

ATOMIC FRACTION. Ge

Relative intensities of bands as functions of Ge content.

delicate, this figure indicates the general trend of the relative variations of the density of vibrational states with the fractional change of germanium atoms. The new band D which in­creases with x and decreases above x=0.33 may be associated with the vibration of Ge-Se bond. We have had no idea about the mode of this vibration at this stage. The total number of density of bands F, G, H, I, J and K grows above x =0.4 and becomes maximum at x=0.5, indicating that these structures are associated with Ge-Se bond with equal coordination numbers. In fact, we observe some similarities with Raman and infrared spectra of crystalline GeSe of orthorohmbic structure7'8 as well as with the infrared spectrum of amorphous GeSe9. These fine structure may arouse a suspicion of the crystalization of the specimen. However,

we have obtained only broad bands in x-ray diffraction without any structure which suggests an evidence of crystalization. The crystalline GeSe is constructed from three-fold coordinated bonds supported by three p-electrons each from Ge"" and Se+. It may be reasonable to suppose that the same kind of bonbing is operating in amorphous state to support "one to one" binding between selenium and germanium.

In Fig. 3, the bond structure for amorphous Se-Ge are shown schematically. Below 33% of Ge, the linear chains of selenium atoms cross at germanium atoms with tetrahedral bonding.

Se,_xGex

x < 1/3 x > 1/3

Fig. 3 Schematic diagram of bond structure for amorphous Se-Ge.

Above 33% of Ge, Ge-Se bond of three-fold co­ordination as well as Ge-Ge bond becomes pre­dominate. There will be mismatches or dangling bonds between these three-fold bonds and two-fold bonds of Se or four-fold bond of Ge. These "imperfections" will give rise to the segregation of GeSe of three-fold bonding from the host glass. This might be the reason why the glass can not be obtained with the melt-quench method above 40% of germanium. Acknowledgement-Raman experiments were performed at Research Laboratory, Matsusita Electronics Ltd..

REFERENCES

1. TRONC, P., BENSCUSSAN, M., BRENAC, A. and SEBENNE, C., Phys. Rev.B 8, 5947 (1973) 2. LUCOVSKY, G., NEMANICH, R.J., SOLIN, S.A. and KEEZER, R.C, Solid State Ccmmun. 17

1567 (1975) 3. NEMANICH, R.J., SOLIN, S.A. and LUCOVSKY, G., Solid State Conmun. 21, 273 (1977) 4. Lucovsky, G., GALEENER, F.L., KEEZER, R.C, GEILS, R.H. and SIX, H.A., Phys. Rev.B

10, 5134 (1974) 5. KOBLISKA, R.J. and SOLIN, S.A., Phys. Rev.B 8,, 756 (1973) 6. WIHL, M., CARDONA, M. and TAUCH, J., J. Non-Crystalline Solid 8/10, 172 (1972) 7. CHANDRASEKHAR, H.R. and ZWICK, U., Solid State Commun. 18, 1509 (1976) 8. SIAPKAS, D . I . , KYRIAKDS, D.S . and EOCMMXJ, N.A. , S o l i d S t a t e Ccmnun. 19, 765 (1976)

9. CHAMERLAIN, J.M., SIRBEGOVIC, S.S. and NIKDLIC, P.M., J. Phys. C: Solid State Physics 7, L 150 (1974)

i Solid State Communications, Vol.32, pp.85-88. Pergamon Press Ltd. 1979. Printed in Great Britain.

CONJECTURE ON THE EFFECT OF SMALL ANHARMONICITY ON VIBRATIONAL MODES OF GLASS

R.W. Hellwarth

Electronics Sciences Laboratory, University of Southern California, Los Angeles, CA 90007, USA

We conjecture here that, unlike for crystal lattice modes, some nearly harmonic vibrational modes of a glass lattice may have their (thermally populated) energy levels shifted (from equal spacing) more than they are broadened, both the shifting and broadening arising from small anharmonic terms in the lattice potential. This conjecture could be verified by observing either (a) saturating infrared absorption, (b) an infrared photon echo, or (c) altered infrared absorption (possibly even gain) of a probe beam in the presence of a second pump beam.

1. Introduction We put forward here a conjecture about

the quantum vibrational states of a glass which, if true, would mean that certain in­frared and optical properties are quite different from what has been previously assumed. Certain scattering data and physical arguments suggest that many of the nearly harmonic (not tunneling) lattice vi­brations of a glass may have their energy levels shifted (from equal spacing) more than they are broadened, both the shifting and broadening arising from the small an-harmonicities in the interionic force con­stants. This is in contradistinction to the situation familiar in crystals where the anharmonicities in the lattice forces broaden the excited vibrational levels more than they shift them.

With infrared absorption or Raman scat­tering, a single vibration of a crystal may be probed (because of "crystal-momentum" conservation or "wavevector matching"). These vibrations are always observed to exhibit a single broadened peak, for any distribution of population among the vibra­tional levels, verifying that the crystal lattice levels are broadened more than they are shifted. Unfortunately, in a glass, lack of translational symmetry prevents probing a single vibrational excitation of the lattice by the above techniques, and no direct evidence for the response function of a single vibration in a glass exists as yet. However, our own and others' optical Brillouin-scattering studies, over a range of temperatures, of the linewidths and frequen­cies of transverse and longitudinal acoustic phonons in glasses has revealed a qualita­tive difference in the temperature behavior of phonon damping in a glass and in a crys­tal. (Compare, e.g., references 1 and 2.) This difference may arise because acoustic phonons damp by coupling to vibrations that have the inequality of level broadening and shifting reversed as we described above.

We will not trace here the arguments con­necting phonon damping and vibrational-anhar-monic character of glasses and crystals. Theory suggests that many of the vibrational modes of a glass lattice should be highly localized (i.e., having most of the mode energy distributed among fewer than ten nuclei). It should not be surprising therefore if at least some vibrational levels in glasses are shifted more than they are broadened, as they are often ob­served to be in molecular gases and li­quids. We will sometimes refer to a vibrational mode whose levels are shifted more than they are broadened as having a "molecular" anharmonic character as op­posed to "crystalline" anharmonic character.

In the next section 2, we present a more mathematical description of how this difference in the effects of lattice an­harmonicities may appear in some lattice vibrational modes of a glass structure. We will then propose (in Section 3) three direct experimental tests for the existence of "molecular" anharmonic character in the vibrational modes of a glass sample. We note that a useful infrared device, such as a tunable laser or a saturable absorber, may be realized with a glass, should it prove to have the hypothesized "molecular" anharmonic character. We reiterate that we are concerned here with vibrations which are nearly harmonic oscillators and not with the tunneling motions which are also believed to play a prominent role in glass lattice dynamics, at the low frequency (<20cm 1) portion of the lattice spectrum. 2. Anharmonicity conjecture for glass

The potential energy V of either a crystalline or glass lattice of N ions whose displacements from their average (equilibrium) positions are r. (i=l,2,—N) may be written as a superposition of a har­monic potential V, having the form

85

86 CONJECTURE ON THE EFFECT OF SMALL ANHARMONICITY Vol. 32, No. 1

V, =h^ . .rfc.tr. har i,j=l ι 13 3 CD plus a small anharmonic perturbation V_, which is highej order than quadratic in the displacements r.. (We are neglecting the highly anharmonic interactions which may give jise to tunneling states in glasses.) The c.. are the (tensor) harmonic force constants. " It is customary to rewrite (1) in terms of vibrational mode coordinates q (a=l,2,—3N) and their natural frequencies

V, =^Σ ω 2q 2 har a a na (2) The potential (2) in a quantum mechanical Hamiltonian gives rise to 3N uncoupled harmonic oscillators, each of which has energy levels n }ίω (n =0,1,2,---«») and eigenstates |n >. The 3N mode coordinates q are linear functions of the nuclear dis­placements r^:

q =Σ.ΰ .·?.ΛΓ Ma 1 ai 1 1 (3) where m. is the mass of the i ion and the U . are the usual unitary matrices.

If the anharmonic correction V , to (1) is not too large, its effect may be described as shifting and broadening the energies n Ηω of the normal vibrational

& α α modes (and also causing mode excitations to scatter each other). This is true for ei­ther a glass or a crystal. In a crystal it is expected from theory that the level width Γ(η ) of the n quantum level of mode a (i.e., of the state with n phonons in mode a) is much larger than the shift ΔΕ(η ) in 01 4 the energy of the same vibrational state. This has been repeatedly verified experi­mentally by infrared absorption and Raman scattering. Because of crystal-plus-photon wavevector conservation, both of these tech­niques can probe the response of a single crystal lattice mode, having a particular wavevector and polarization. The response is always observed to be a single broadened resonance peak, even when the excited levels (n >0) are thermally populated so that the response is a superposition of several tran­sitions having n ->n ±1 for several n . 0 a a a

In a glass the interionic potential is also dominated by a harmonic part of the form (1^. However, because the force con­stants c . are now random (rather than peri­odic) £he resulting normal-mode matrix ele­ments U · of (3) are generally small except for labels i referring to ions in a localized cluster. This cluster contains most. of the energy of an excitation of mode a. The theory of the broadening and the shifting of the levels of these localized modes is

formally the same as for the crystal, but the density of final states into which a given localized excitation can decay (via anharmonic coupling) is much smaller than for the unlocalized modes of a crystal. Localized modes can decay only into spa­tially proximate modes. Therefore, even if the anharmonic forces are comparable in glass or crystal (and these anharmonic forces are expected to have significant terms only among near neighbors in either glass or crystal) the broadening of a crystal vibrational level should be general­ly greater than for its counterpart in glass.

Level shifts, on the other hand, ap­pear to be comparable (for comparable force constants) for crystal and glass. In a crystal, level shifts can be estimated as if the unit cells comprised localized, un­coupled modes, rather like the actual lo­calized modes in a glass. For both, the level shifts are expected to be comparable to analogous level shifts in vapor, if in­deed similar molecular units exist in vapor. The possibility clearly exists therefore that, at least for some vibrational modes of a glass, the situation is the reverse of that for crystals; that is, glass modes may exhibit the following inequality

ΔΕ(η )>Γ(η ) , K a K or '

(4)

(5b)

where ΔΕ(η ) is the anharmonic shift in energy J.-1 Ot

of the n vibrational state In > of mode a, 1 a '

and Γ(η ) is the (homogeneous) width of the state's energy. Since the inequality (4) is often obeyed by the energy levels of vibrations of molecules in a vapor, we sometimes refer to this property as "molecular anharmonicity."

If one drives a single glass lattice mode a with a force F cosart one expects to find a steady state response of the form <q (t)>=E~ nReF e"ia)tG (ω) (5a)

Mor J n=0 a nor J K J

W h e r e ΔΕ(η +1)-ΔΕ(η ) ? 9 G Ξ c / { (ω + ^-πτ —) -ω na nor ^ a H J

-2ίω[Γ(ηα)+Γ(ηα+1)]/η}

Unlike for crystal modes where the G (ω) superpose to form a single reso­nance, the glass modes may have distinct resolvable transitions G between each na adjacent pair (n,n+l) of levels because of (4). The relative strength c of each of these lines is nearly that for the harmonic oscillator, and therefore proportional to (n+l)e"3ü**n (l-e_3übt)2 when the mode is in thermal equilibrium at temperature T=H/k$. If kT>)iü) , then the G exhibits a maximum

a na with increasing n.

A plane electromagnetic wave propagating in a glass with a definite wavevector applies a force to nearly every mode, rather than to one or a few modes as in a crystal. The linear response of the electric polarization

Vol. 32, No. 1 CONJECTURE ON THE EFFECT OF SMALL ANHARMONICITY 87 density is a superposition, <*Σ G (ω)Κ , J r r ' η,α nor J or o£ line patterns from many modes a weighted by K . In this superposition, the line structure (5) of individual modes is washed out. Infrared absorption experiments or Raman scattering experiments cannot there­fore determine whether some individual modes in fact exhibit the multiline response (5) with condition (4). However, there are several experimental effects which are sensitive to whether lattice-mode levels are in fact shifted more than they are broadened. We discuss three of these in the next section. 3. Experimental tests

We suggest here three experimental ef­fects, the observation of any one of which would be strong evidence for the hypothesis that some glass vibrational modes have the "molecular" anharmonic character expressed in eqn. 4. These are: (a) a saturating absorption of electromagnetic (e-m) waves by the glass lattice, (b) photon echoes, and (c) a change in absorption for one e-m wave due to a second wave at different fre­quency, beyond what purely thermal effect might occur.

3.1. Saturating absorption. It is well known that the response of an fideal' harmonic gscillator fwhich does not obey (4)) cannot be saturated. That is, the response (5) is essentially independent of the pro­babilities W α for the state In > to be

n ' a occupied. This can be understood from the well known sum rule, true for any oscilla­tor no matter how it is coupled to other modes, that requires / ωΣ ImG (ω)οΙω=π o n nor J

independent of the W (6)

If all G (ω) are nor J

single narrow resonances of the same shape and center frequency, then clearly their sum (the total response) will be constant. Indeed, saturation of crystal lattice ab­sorption has never been observed.

However, if a mode should obey (4) so that the important G (ω) are distinct reso­nances about distinct center frequencies, then changing populations W„a will change the strengths c of the G and change the response at a given frequency ω. The sum (6) is then satisfied by keeping the sum of strengths c essentially constant. Speci­fically, a strong driving force F at the resonant frequency of one G (ω) will tend to equalize the populations of the nth pair of levels and reduce G (ω). This power not dependent reduction in absorption may be observable if it is present in a real glass, provided that background absorption from modes that do not saturate easily is not too high. Heating-induced changes in ab­sorption could be distinguished by their much slower decay times.

3.2. Photon echoes. If the anharmoni-

cities and couplings of an oscillator re­sult in its having the "molecular anharmoni-city" property (4), then it becomes essen­tially an ensemble of two-level systems which can be probed independently of each other. That is, an external force which causes transitions between one pair of levels need not cause transitions between any other pair. It is in such circumstances that "photon echoes" can be observed.? 7 Without rederiving this well known effect, one can imagine how it might be observed in a glass obeying (4). A first pulse of radiation is made to traverse the sample, followed by a second a time t, later. Then after an additional time t, the glass may emit a third pulse. This is provided that the pulses were long enough to excite only a single pair of levels in modes obeying (4) and that there are enough such modes to produce an observable pulse. As the delay time t, is increased, the echo pulse intensity decreases as exp(-2T W H ) where Γ is the total width of the level pair. na r

This places a practical limit on t,. Back­ground absorption also reduces the echo. However, if, in the face of all difficulties, an echo were observed, it would be unequivo­cal evidence of the existence of molecular anharmonicity as expressed by (4) in glass modes.

3.3. Two-wave interactions. From the sum rule (6), it is evident that, if an e-m wave saturates a given component G of a lattice-mode spectrum, then other com­ponents must change in intensity in a man­ner such that the sum in (6) is constant. A second wave probing a second component G may experience thereby altered absorption and refraction. This may be easier to observe than the saturation discussed in 3.1 of a single wave's absorption. For example, the first wave may be chopped and the intermit­tent change in absorption of the second wave be detected by lock-in detection methods. There may also be a signal due to temperature changes. If the observed absorption has a sign (±) opposite that from temperature changes then the evidence is unequivocal. If not, the much slower decay of thermal changes may be used to distinguish them from the anharmonic effect (4).

There exists a more remote but interest­ing possibility that the unequally spaced levels of glass lattice modes can be op­tically pumped in analogy to solid state laser levels so as to produce inversion be­tween one (or more) pair of levels and per­haps even produce thereby net gain. This gain would exist over a broad band and hence might be used to realize a tunable in­frared-to-far-infrared laser oscillator.

This work was supported by the U.S. Department of Energy under UC-Lawrence Livermore Laboratory subcontract 7509105, and by the Air Force of Scientific Research under Grant No. 79-0098.

88 CONJECTURE ON THE EFFECT OF SMALL ANHARMONICITY Vol. 32, No. 1

REFERENCES 1. H.J. McSkimin, J. Appl. Phys. 24, 988 (1953). 2. A.S. Pine, Phys. Rev. 185, 1187 (1969). 3. R.J. Bell, N.F. Bird, and P. Dean, J. Phys. Cl, 299 (1968). 4. Quantum Theory of Solids, C. Kittel (John Wiley and Sons, New York, 1963). 5. T.C. McGill, R.W. Hellwarth, M. Mangir, and H.V. Winston, J. Phys. Chem. Sol. 34,

2105 (1973). 6. R. Hellwarth and M. Mangir, Phys. Rev. BIO, 1635 (1974). 7. I.D. Abella, N.A. Kurnit, and S.R. Hartman, Phys. Rev. 141, 391 (1966).

/Solid State Communications, Vol.32, pp.89-93. Pergamon Press Ltd. 1979. Printed in Great Britain.

RAMAN SCATTERING FROM SOFT Τ0-ΡΗ0Ν0Ν IN IV-VI COMPOUND SEMICONDUCTORS K. Murase and S. Sugai

Department of Physics, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, 560 Japan

The soft T0-phonon modes were investigated below the cubic-rhombohe-dral transition temperature in Pb"|-xGexTe (x=0.05) and SnTe. We observed resolved Raman spectra from ordinary and extraordinary modes. The temperature dependence of the spectra was analyzed in terms of phonon-phonon interactions. We also discussed origins of extra Raman spectra which appear even in the high temperature phase. The energy gain of valence bonding electrons due to lattice distortions was estimated to be M09erg/cm3 by analyzing anomalous temperature behavior of the optical dielectric constant. This is about two order of magni­tude larger than the depth of the well of free energy.

IV-VI compound crystals, PbTe, SnTe and GeTe and their alloys, are well known narrow gap semiconductors which are very useful and important materials for infrared lasers and detectors with tunable wavelength by tempera­tures. These crystals take either a NaCl or a ferroelectric rhombohedral type structure de­pending upon alloy compositions. With increas­ing temperature, a distorted crystal transforms to a NaCl one at a second order or a nearly second order transition temperature Tc.'~3

Recently much efforts have been made to under­stand the lattice instability from microscopic view points.4-9 The structural instability of IV-VI compounds is accompanied by the softening of TO phonon modes near Γ point which has been interpreted by an interband electron-TO phonon interaction model 10-13 with phonon anharmonici­ty. An apparent carrier concentration depend-ence^ of transition temperatures in SnTe has been analyzed based on this model.4>/>8 Para-electric behavior in Pb-j_xSn-|_xTe has been phe-nomenologically understood by the phonon anhar­moni city."

Raman scattering experiments have been made in order to obtain a direct information about the TO modes. In this report, we will show the recent experimental data and discuss the lattice instability problem on the micro­scopic view point. We will also discuss the origin of the additional spectra which appear even in the high temperature phase.

Raman scattering measurements have been carried out on Pbi_xGexTe single crystal ob­tained by a vapor transport method and SnTe single crystals grown in the liquid phase. We used 5145 A Ar ion laser line. The incident angle of the light(^70mW) is about 30°. The penetration depth is about 100 A. Near back scattered light from the sample surface was observed with a slit width of 2.5 cm"1 through an iodine vapor cell to avoid the elastic com­ponent of the scattered light. The surface of the specimen was etched with HBr+5%Br solution to remove mechanical damage. Figure 1 and 2 show typical Raman Stokes spectra above and

below the transition temperature which are nor­malized by the temperature factor Ν(ω)+1. In the low temperature phase, we newly observed resolved TO phonon spectra due to the static distortions. The reduced scattering intensity increases almost linearly with decreasing tem­perature below Tc. The temperature behavior of TO frequencies are shown in Fig, 3 and 4 where insets demonstrate the resolved spectra. From the orientational dependence of the spectra in Pb-j_xGexTe (x=0.05), it is considered that the lower frequency lines correspond to the ordina­ry modes and the higher the extraordinary modes.5>lz^ In SnTe the lower line was so sharp as determined by the slit width. The higher line is very poor in intensity compared with that in Pb-|_xGexTe. Below Tc, the TO phonon frequencies are expressed as:

flü)D (*ωτο) 2. V =4 *ω coth(*W2kT)d0fa), (11

where C-j is the coefficient of the phonon-phonon interaction term and i's indicate a TO phonon wavevector and the polarization. The term ΒΊ· includes an interband electron-TO phonon interaction part as well as a short range repulsive and a long range Coulomb part. In order to f i t the Equation (1) with the ex­perimental data, we used the following parame­ters assuming Debye temperature θ[)=145 Κ (=100 cm"1) as shown in Table 1.

As shown in Fig. 1 and 2, we have observed two distinguished lines at 126 cm"1 and 144 -1 These frequencies are practically un­changed in temperature and the sample we meas­ured. Observed intensity is also constant in temperature. It may be concluded that neither of them are related to the L0 phonon modes. Following the Raman scattering work of mono-layer- thickness oxide and tellurium films on Pb-j_xSnxTe by Cape et al. , there is a large possibility that our twin spectra originate from tellurium and/or tellurium oxide thin layer on the specimen. Unsuccessful surfaces

89

90 RAMAN SCATTERING FROM SOFT TO-PHONON

TABLE 1

Values of Coefficients in Eq. (1)

Vol. 32, No. 1

often showed relatively intense 126 and 144 cm"' lines. Shimada et al.'G reported the ob­servation of similar twin spectra as ours in Pbi xSnxTe. Their peaks are at 131 and 147cm"1 both of which did not change appreciably with temperature and composition. Their observed intensity showed anomalies with temperature. Apart from their explanations of the effects as an enhancement of the LO mode scattering, it might be pointed out that their spectra also come from surface thin layer of tellurium and/

z LU

0 50 100 150 200 FREQUENCY (cm"1)

Fig. 1. Raman spectra in Pb]_xGexTe (x=0.05), (a) above and (b) below the transition tempera­ture. The maximum count rate due to TO modes is ^5/sec.

Pbl-xGexTe

ex-ord. modes

Oi l ) (100)

ord. modes

SnTe

B (cm"2)

2030 1630 795

2170 690

C

0.108 0.0863 0.0424

0.147 0.0468

>-1—

z LU 1— 2

(a) T = 251K

SnTe p = Ux1C

(100)

20 (cm-3)

(b) T=22K

0 50 100 150 200 FREQUENCY (cm""1)

Fig. 2. Raman spectra in SnTe, (a) above and (b) below the transition temperature. The maximum intensity at the TO peak corresponds to ^2/sec.

or tellurium oxide. The intensity anomaly may be connected with such situation as a depletion layer immediately below the thin layer. A similar argument might be applicable to an understanding of experimental data by Brill son et a l . in SnTeJ7 The different behavior of the spectra between others and ours should be due to different surface treatments. The prob­lem is \/ery important in a sense that such a system as a IV-VI compound surface may provide a good subject about an interaction of vibra­tion modes of a thin adsorbed layer with the host crystal lattice and electrons.

In Fig. 2 the remaining spectra above Tc has been tentatively adjusted by a phonon den­sity of states calculated by Cowley et al J 8

I t may be suggested that experimental spectra in this frequency range arise from a vacancy or disorder induced f i rst order scattering.

In order to discuss a more comprehensive picture for the mechanism of the lattice insta­bi l i ty , we need informations both on the lat­tice and the electronic properties.9 I t has been showed that the knowledge of the valence electron polarizability is important which is related to the optical dielectric constant ε^. 13,19 This is predominantly related with an°° average bond energy gap EQ rather than a mini­mum optical gap.2u The temperature dependence of the gap EQ may be decomposed as

EG(T)= , ( 0 ) + ΔΕ-( dynamical) + ΔΕΛί,ε) , ( 2 )

Vol. 32, No. 1 RAMAN SCATTERING FROM SOFT TO-PHONON 91

Fig.

R>1-xGexTe x=005

o:(111) • :(100)

>. 00 l l |

nn (100) A 6-5K

AC \ AJ 0 50 100 150

TEMPERATURE (K)

3. Temperature dependence of TO phonon modes below the transition temperature Tc in Pb-j_xGexTe (x=0.05).

Έ u csi

O Ü15

or < a

> o z LU a UJ or

10

SnTe [ p = 1.A;

I 1 1 1

<1020cm"3

(100)

. 1 .

> . » -in z UJ 1 -z

_J

I 21.2 K w

10 5C (cm-1

1 1 ^ 1 I

A

A

)

50 100 TEMPERATURE (K)

Fig. 4. Temperature dependence of TO modes below Tc in SnTe.

where EQ(0) i s t n e g a p f o r a NaC1 s"tructure at T=0 and u is the^relative displacement of the sublattices and ε the |t£ain tensor induced by the displacement; AEQ(u,e) is a change of the gap due to the static lattice distortions below Tc which can be expanded by these components with an 0n invariant formal By taking into account only the displacement u, the gap EQ(T) with nearly free electron model is expressed as

EG2<T>- -GO E™(T) + 4^2 u2

a (3)

where Ξ is an interband matrix element for a unit displacement and a the lattice constant.

We determined the dielectric constant from the interference fringes for hot wall epitaxi ally grown single crystal films of IV-VI compounds on BaF2(or KC1) substrates with a Fourier spectrometer (Digilab FTS-16AS) in the infrared and the far-infrared region at temper­ature between 4 and 300 K as typically shown in Fig. 5. A bend appears in the temperature be­havior of em near Τς determined by a resistivi­ty anomal v.00 Using Penn's relation em= 1 + (fifip)2/EQ^(T) with a plasma frequency Ωρ due to all the valence electrons n composed of 10 electrons per unit cel l , the temperature be­havior of the gap EQ(T) is derived as shown in Fig. 6, where we took account of the tempera­ture dependence of the lattice constants I»2 2»2 3

for the calculation of Ωρ. The gap EQQ(T) is assumed to be an extrapolated value from the measured gap at the high temperature phase. The deformation potential Ξ is estimated to be 12.5eV for the SnTe specimen and ^30eV for the Pb-|_xGexTe (x ^ 0.07) by Eq.(3) using u values.3,14 One can also estimate an extent of the energy gain of valence bond electrons due to the deformations from the unstable NaCl structure at T=0 using the energy difference AEQ(0)=EG(0) - EG(°) in Fig. 6. The gain W per unit volume is approximately (1/2)ΔΕβ(0)·ηθ^ assuming the number of bonding electrons corre­sponds to an effective number of electrons neff=(3EG/4Ep)*n 2 0 with the Fermi energy EF MOeV). The gain W is estimated to be 8 χ 108

erg/cm^ in SnTe and 109erg/cm3 in Pb-j_xGexTe (x ^ 0.07). The valence electron energy is much larger than the depth (M07erg/cm3 in Pbi_xGexTe (x=0.05)) of the well of the free energy which were expanded with distortion coordinates using the TO phonon mode tempera­ture coefficients and the measured elastic con-

92 RAMAN SCATTERING FROM SOFT TO-PHONON V o l . 3 2 , N o . 1

stantsJ4 The difference is looked upon as the lattice energy loss due to distortions.

In conclusion, we interpreted the tempera­ture dependence of TO mode frequencies by phonon-phonon interactions. It was demon-

56

54

u? h-52 I ; z o u 42

fcta

40

LU

5 38

< Ü 36 o

34

B Q - S B i W SnTe

P = 1-3x102 0cm-3

Pb,_xGexTe χ-0·07 P = A.7x1018cm"3

100 200 TEMPERATURE(K)

300

Fig. 5. Temperature dependence of the optical dielectric constant. Allows indicate the transition temperatures measured by the resis­tivity anomaly.

strated the principal importance of the temper­ature behavior of the average gap to understand the lattice instability. Now it is desirable to investigate the temperature dependences of the coefficients which appears in the elementa­ry excitation procedure in more fundamental way.

£.3U

2.40 > <L·

o ÜJ

£ 230

LU o < cc LU > <

I ! I

1

o o

o o

-· · 8 o - o · · m o o ·

-*■ '

^ -^ " -~- **

— /v/

2.00

S3

"

1 1

o

•

J

1 1

o PbTeG 0 °

8

j ^ ^

'" Pb^GeJe x = 0 0 7

Γ Π Γ ^ α

SnTe P = 1 3 x l 0 2 0

l 1 -

] o j

3 _J Ί

1 1

Ί H

-* "I ~ ~ -"7

□ _ :

— cm~3 Z

-100 200 TEMPERATURE (K)

300

Fig. 6. Average gaps are derived from Fig.5. The temperature coefficients yj£ are 0.64, 0.46 and 0.15 mev/K at high temperatures in PbTe, Pbi_xGexTe (x=0.05) and SnTe, respectively.

Acknowledgment - The authors wish to thank Professor H. Kawamura for constant encourage­ment and stimulating discussions. We wish to thank Dr. Katayama for fruitful discussions. The important contribution of T. Higuchi and T. Fukunaga are gratefully acknowledged. We thank Drs. K.F. Komatsubara, Y. Kato, and K.L.I. Kobayashi for supplying SnTe specimens. We also thank Shimada for stimulating discus­sions orr their Raman spectra. One of the authors(KM) was supported by the Kurata Founda­tion.

REFERENCES

Hohnke D. K., Holloway H. and Kaiser S., Journal of Physics and Chemistry of Solids 33, 2053 (1972). Muldawer L. J., Journal of Nonmetals 1_, 193 (1973). Iizumi M., Hamaguchi Y., Komatsubara K. F. and Kato Y., Journal of Physical Society of Japan 38, 443 (1975). Kobayashi K. L. I., Kato Y., Katayama Y. and Komatsubara K. F., Physical Review Letters 37, 160 (1976). Murase K., Sugai S., Takaoka S. and Katayama S., Proceedings of the 13th Inter­national Conference on the Physics of Semi­conductors, Rome, Aug.30 - Sep.3, 1976 (Ed. Fumi F.G.), 305 (1976).

Kawamura H., Nishikawa S. and Nishi S., ibid, p310. Kawamura H., Murase K., Sugai S., Takaoka S., Nishikawa S., Nishi S. and Katayama S., Proceedings of the International Conference of Lattice Dynamics, Paris, (Ed. Balkanski), 658 (1977). Sugai S., Murase K., Katayama S., Takaoka S., Nishi S. and Kawamura H., Solid State Communications 24, 407 (1977). Murase K., Sugai S., Higuchi T., Takaoka S., Fukunaga T. and Kawamura H., Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh, 4 September 1978, (Ed. Wilson B.L.H.) 437 (1979).

8

1.

2.

3.

4.

5.

6.

7.

8.

9.

Vol. 32, No. 1 RAMAN SCATTERING FROM SOFT TO-PHONON 93 10. Kristoffel N. and Konsin P., Ferroelectrics

£, 3 (1973). 11. Kawamura H.s Katayama S., Takaoka S. and

Hotta S., Solid State Communications ]±9 259 (1974).

12. Natori A., Journal of Physical Society of Japan 41_, 782 (1976).

13. Katayama S. and Kawamura H., Solid State Communications ;21_, 521 (1977).

14. Sugai S., Murase S., Tsuchihira T. and Kawamura H., to be published in Journal of Physical Society of Japan, 1979.

15. Cape J. A., Hale L. G. and Tennant W. E., Surface Science 62, 639 (1977).

16. Shimada T., Kobayashi K. L. I., Katayama Y. and Komatsubara K. F., Physical Review Letters 39, 143 (1977).

17. Brill son L. J., Burstein E. and Muldawer L., Physical Review B9, 1547 (1974).

18. Cowley E. R. and Dabby J. K. and Pawley G. S., Journal of Physics C. 2, 1916 (1969).

19. Lucovsky G., Martin R. M. and Burstein E., Physical Review B4> 1367 (1971).

20. Phillips J. C , Bonds and Bands in Semicon­ductors, (Academic Press, New York and London) 1973.

21. Rehwald W. and Lang G. K., Journal of Physics C. 8., 3287 (1975).

22. Houston B., Strakna R. E. and Belson A. S., Journal of Applied Physics 3?_, 3913 (1968).

23. Novikova S. I. and Shelimova L. E., Soviet Physics, Solid State Z, 2052 (1966).

I Solid State Communications, Vol.32, pp.95-101. Pergamon Press Ltd. 1979. Printed in Great Britain.

LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS

Terutaro Nakamura, Yasunori Tominaga, Masayuki Udagawa, Seiji Kojima and Masaaki Takashige

The Institute for Solid State Physics, The University of Tokyo, Tokyo 106 Japan

A method of analyzing the overdamped polariton spectra have been developed. Polariton dispersion curves of the soft E-mode in BaTiOß have been obtained. By observing polariton peaks of KDP, low-frequency B2(z) response has been concluded to be oscillatory. Pure E-TO plateau response of KDP has been attributed to the proton motion of E-symmetry in x-y plane. The phase transition in L1NH4C4H406·H2O has been identified to be the intrinsic ferroelastic transition, by observing a soft acoustic phonon and by comparing with KDP. High pressure polarized Raman spectra of a single domain BNN crystal have been observed using a diamond anvil cell.

§1 Introduction

Recent light scattering works by authors' group on structural phase transitions are presented. Following topics are included. (1) Raman scattering from polaritons of over-damped soft phonons, examples: BaTiOß, KDP. (2) Brillouin scattering studies on acoustic phonon softening at the "intrinsic ferro­elastic phase transition", example: LAT. (3) High pressure polarized Raman scattering using a diamond anvil device, example: BNN. (4) Peculiar low-lying response in E-spectra from KDP.

§2 Raman Scattering from Polar!tons of Overdamped Soft Phonons

In this section, a method of analyzing the overdamped polariton spectra and results of analysis are presented.1^5)

Polaritons of Overdamped Phonons

The polariton dispersion curve was first obtained from Raman scattering experiments by Henry and Hopfield on GaP. Here, we discuss on the observation of Raman

scattring from polaritons of ferroelectric soft phonons with very high damping constant Γ. Many books describe the polariton dispersion

curve of a damped phonon as a ω vs Re q curve and a ω vs Im q curve. In Raman scattering, however, experimental

condition is that q is real18)and ω is complex, complementary to infrared spectroscopy. Thus we solved the polariton equation^':

(4#-ω2)(-ω2+ 12ωΓ + ω§)

-CB= " ««δ VES - */""υ » - 0 (2.1) using a computer on the condition that q is real and ω is complex. Here, OJQ and 2Γ are frequency and damping constant of a single oscillator»respectively. Remarkable results

were:(l) For higher Γ(Γ/ω0 > 0.5), Re ω vs q curve has a maximum at q - qM; (2) For much higher Γ(Γ/ω0 > 1), Re u)(q) vanishes in a region ς>ς 0 (ς 0 Μ Μ >; (3) Im ü)(q) splits into two in the region q>qn. This is seen in Fig.l which shows the loci or the roots of the polariton equation (1) in the complex plane. The root moves following arrows from the origin 0. From 0 to q„, Re ü)(q) increases with increasing q; beyond qM, Re ü)(q) decreases with increasing q. For Γ/ÜJQ > l(overdamping), the two roots become pure imaginary when q>qQ. The line shape of the Raman spectra of an

overdamped phonon can be computed and bears following features:(1) The lineshape depends on q - value. (2) For q > q„, there is no Stokes and ant1-Stokes peaks Dut response is just a broad peak with the maximum at ω = 0. (3) For q < qM, there are Stokes and anti-Stokes peaks.

Experiments

We established an (experimental method of observing the Stokes and anti-Stokes polariton peaks, that is : (1) Near-forward scattering geometry makes the observation of low q scattering possible.Wave vector selection was made using a stop which has openings for the scattered light having only certain values of scattering angle to pass through. (2) Using an I9-fliter we deduced the elastic scattering by ID-*.

Analysis of Data

(1) It should be noted that if we make three approximations below, we can analyze xlata in a simple way, without using a computer ; Three approximations are : (i) eQ» eoo(valid for high dielectric constant materials), (ii) (c2q2/oi2) » ε«, (valid for the lowest branch of polariton dispersion curve) and (fit) Γ is independent of ω. (2) Definition of the polariton frequency.

95

96 LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS Vol. 32, No. 1

-1000 -800 J £

E v— 3

CO

B

/

: : \ - 2 0

50

-

-

A

^-4 %

[^

«^ .A

\ f?u

20 J £ L

Ü J_ 800

Real ω

1000 <cm

Fig.l Loci of the root of the polariton equation (1.1)in the complex plane.

If the above approximations are taken, the response function can be written as

χ(ς>ω) Λ. ω(5(-ω2 + ω£ + 12ωΓρ) here, ωρ introduced by

* 1 +εοωδ + 2 2 czq^

(2.2)

(2.3)

is called polariton frequency(definition of ωρ) and Γρ given by

Γ = p i + £ο ω| C2q2

(2.4)

is effective damping constant of polariton. Since x(q,ü)) is given by (2.2), we can write X(q,ü)) as

ωιο)2 X ( q' w ) = « Κ ω - ω ι Μ ω - ω2)

where,

ωιω2 = -ω*

ωι +0)2 - ί2Γρ

(2.5)

(2.6)

(2.7)

ωι and ω2 are called complex polariton frequency. (3) Energy dissipation is given by ωχ"(ω). From the Raman scattering experiments, this quantity can be obtained. (4) The peak frequency of the spectra ωχ"(ω) can be shown to be the polariton frequency ou. Thus au can be obtained from experiments.

Example I : BaTiO..

In 90° or 180° Raman scattering spectra from overdamped soft phonons in BaTiOß, no Stokes and anti-Stokes peaks are observed, but only a broad peak centered throughout on zero frequency appears. This broad peak was computer-fitted by an overdamped oscillator by DiDomenico et al1!), i.e. analysis was only indirect. This is no longer true, because, from the

above, if the scattering from the polariton is observed, it should show the Stokes and anti-Stokes peaks. The polariton frequency ω obtained from the

experiments as a function of q was confirmed quantitatively to follow the definition formula of au (2.3), because this formula can be rewritten as

_L 1 εο 0)2 " ω$ "c^F (2.8)

and the experimentally determined values of l/ü)p as a function of 1/q2 is on a straight line.3,4) Now, note that the definition formula of uu

(2.3) is identical to the polariton dispersion relation of the single oscillator model without damping. Hence, it is concluded that, experimentally

determined polariton frequency ω obeys the same dispersion relation as thatPfor an oscillator without damping, although the soft phonon in BaTi03 is highly overdamped. The polariton dispersion curves of soft E-

phonons in BaTi03 are shown in Fig.2.

Example H : KH2P0/

(1) The polariton peak observation was made on ΚΗ2Ρθ45Λ Distinct Stokes-anti Stokes peaks of B2(z) TO-phonon polariton spectrum

Vol. 32, No. 1 LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS 97

50

40

Polariton dispersion relation in BaTiÜ3

h— ω0 = 43οπι-1 (40°C)

•o;0 = 38cm-1(20oC)-

•w0 = 34cm"1(6.5°C)

500 Wave vector

1000 Q (cm"1)

1500

Fig.2 Polariton dispersion curves of soft E-phonons in BaTiOß. The polariton frequency is determined definitely, although the soft phonon is overdamped. We put (Ü)/2TTC) ■> ω , (q/2 ) -> Q.

on acoustic phonon softening at the "intrinsic ferroelastic phase transitions is reported^»?) It must be noted that the terms "ferro-

elasticity" and "ferroelastic phase transition" should be defined independently and should not be confused. First we define the ferroelasticity. A

crystal in which the spontaneous strain exists and switches from one orientational state S to another one S1 under application of an external stress is called a ferroelastic crystal and the characteristic of the ferroelastic crystal is called ferroelasticity.

Now we define a phase transition, of which order parameter is the homogeneous strain, as ferroelastic phase transition. When this transi­tion occurs, spontaneous strain should appear. On such a phase transition, the fluctuation

of the homogeneous strain diverges and elastic instability of lattice takes place. If we grasp this on the basis of the phonon picture, acoustic phonon energy fun decreases and vanishes at the transition point, or elastic stiffness C-H vanishes following

(3.1)

The above definition of the ferroelastic phase transition is clearly described on the basis of Landau theory. Expand the free energy as:

F = -re Pyfl 1 2 , 1 4 , , n . . j-m , - p < ICijx "Ä8* + hn k^ η

+ (3.2)

are observed only for a very small q-value: by the near forward scattering with a scatter­ing angle in the crystal Θ^Ο.2? This result is in accord with the observation of the Raman scattering under high pressure by Peercyl2)who has found a similar curve to ours under pressure> 6 kbar From this experimental result, the character of the broad peak centered throughout on zero frequency of 9U°-scattering B2(z) spectrum of KDP, of which origin is still controversial,is confirmed to be an overdamped phonon, namely oscillatory in nature, in agreement with Peercys experiments.

In the present experiment, more pronounced peak than Peercys is observed. Furthermore, our polariton method shows more directly the phonon character of the low frequency broad response of the B2(T0) spectrum, because in this method only the experimental geometry is specified while under the hydro­static pressure the lattice potential in which ions vibrate is changed. Furthermore, LO B2(z) phonon was observed.

The existence of LO phonon found by this observation provides a strong additive evidence for the oscillatory nature of B2(z) mode in KDP.

§3 Intrinsic Ferroelastic Phase Transition

where x is homogeneous strain, η and ξ are other physical quantities. If only c j among coefficients is temperature dependent as (T - TQ) and other coefficients are temperature independent, then the phase transition which the crystal undergoes is defined as the ferroelastic phase transition, or more definitely "intrinsic" (or"proper") ferroelastic phase transition. The elastic stiffness anomaly is a phenomena

of coramom occurence at structural phase transitions. But in many cases among these phenomena the lattice instability at the phase transition takes place owing to a mechanism different from the ferroelastic phase transition, and the anomaly of CJJ occurs as a result of it. Since the microscopic picture of the intrinsic

ferroelastic phase transition is not not clear, many scientists are skeptic on the existence of the intrinsic ferroelastic phase transition.

We found recently that L i N H ^ H ^ ' ^ O , (LAT) provides the very example of the intrinsic ferroelastic transition, as described below. LAT is ferroelectric below Tc = 98K.

Dielectric susceptibility at low frequency rises sharply at Tc following

XX - Xx + " (3.3)

In this section Brillouin scattering studies

98 LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS Vol. 32, No. 1

with very small Curie-Weiss constant C- 2K. Dielectric susceptibility at 2MHz is temperature idependent clamped dielectric constant χχ. Reciprocals of free and clamped dielectric constants(χχ)-1 and (χχ)-1 vs temperature is shown in Fig.4. Since LAT is piezoelectric in the para-

electric phase, the free energy is given by:

■ -|cpx2 + aPx + γ(χ χ) _ 1Ρ 2 (3.4)

here,c denotes elastic stiffness at constant P, and a is the coupling constant. If the phase transition is intrinsic ferroelastic transition, then

(T - T0) (3.5)

while, if the transition is intrinsic ferro­electric transition, then

(Xx)" 4π (T - τ0)

Temperature dependence of (χχ)"^» ( χ ) ~ ^ , cP, c^, l/k.2 and 1/ad (k: electromechanical coupling factor, d: piezoelectric modulus) at intrinsic ferroelectric transition and at intrinsic ferro­elastic transition is schematically shown in Fig.3 (a) and (b),respectively. The temperature dependence shown in (a) is in complete agreement experimental results on KH2PO4. The temperature dependence of (χx)~l and (χχ)"Ί among (b) is in good agreement with the experimental results on LAT, as seen from Fig.4. The temperature dependence of k~2 and (ad)-l among (b) is also in good agreement with the results on LAT deduced from dielectric measurements.

P E To see how c and c. behave as a function of temperature was investigated by Brillouin scattering experiments. The stiffness constant obtained directly from Brillouin scattering experiments is c ^ and shown in Fig.4. The constant c ^ can be calculated using the relation valid for a piezoelectric crystal as shown in Fig.4.

These figures are completely in agreement with the expected curves of intrinsic ferro­elastic transition (Fig.3 b ) .

Temperature dependence of spontaneous strain was determined by Terauchil3)using χ rays, in agreement with what is expected for the intrinsic ferroelastic phase transition.

Crystals which seem to undergo the ferro­elastic phase transitions so far known are paratellurite TeOj^) pottasium trihydrogen selenite ΚΗβ^βΟβ)?1*) and lanthanum pnenta-phosphate L a P s O ^ ^ i judging from considerable evidences.

§4 High Pressure Raman Scattering

In this section, Raman scattering studies on structural phase transition under high pressure up to 35 kbar are reported*;) The high

To 7c ToTr (a) (b)

Fig.3 Temperature dependence of (χ χ) _ 1 ,(χ Χ) _ 1 ; CP,CE; k"2,(da)-1; (schematic) (a) The order parameter is polarization (case I), (b) The order parameter is strain.

pressure was produced by using a gasketed diamond anvil device. So far, most of Raman scattering measurements using a diamond anvil have been made on powder or on broken pieces of crystals. This is the first observation of the polarized Raman spectra from polished single domain crystal plates of definite orientation, under high pressure produced by a diamond anvil device.

The sample used was Ba2NaNb5015 (BNN) which undergoes successive phase transitions at about 560°C, 300°C and -168°C. The phase HE between 300°C and -168°C (point group C2V - mm2) is a ferroelastic phase sandwitched between phase H (T > 300°C) and phase Ή (Τ >-168°C) of both C^v - 4mm symmetry. Recently we found disappearance of ferroelasticity above 22kbar ajt_ropm temperature, by observing the disap­pearance of the ferroelastic domain structure?« Raman scattering experiments were made on BNN

crystal set in a diamond anvil device. The pressure was determined by the ruby Ri-line fluorescence from the small ruby crystal.

The diamond anvil and the optical path are shown in Fig.5a. The sample was contacted with one of the surfaces of the diamond anvil on

Vol. 32, No. 1 LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS 99

120 140 T (K)

Fig.4 Temperature dependence of (χ*)""1, (χΧ)-1; CP,CE; k"2, (da)"1 of LAT.

which grease was applied which acts as adhesive. Linearly polarized beam of 400 mW at 6471 A

from Kr-ion laser was focused to the center of the sample with beam diameter 40 μ. The back scattering geometry was used and analyzed by a double monochrometer with concave gratings (Jobin Yvan Ramanor HG2S). Fig.5b shows the Raman spectra B-^x) y(xz)y

and B2(y) x(yz)x at room temperature and atmospheric pressure. The ferroelasticity in BNN is reflected on the anisotropy between a-axis and b-axis. The difference between the two spectra Βχ(χ) and B2(y) reveal this anisotropy clearly on the mode at about 32 cm"1. Fig.5c shows the B2 spectra below 140 cm"1 in

the pressure region between 0 kbar to 35 kbar at room temperature. The mode at about 32 cm-1

shows anomalous hardening and a decrease in intensity with increasing pressure, and

Spectrometer

Stop

Rotatable diamond mountplate

Diamond anvils Gasket Moving piston

Cyrindorical bearing

Fig .5 (a) The diamond a n v i l device and the o p t i c a l path for Raman s c a t t e r i n g measurements.

50 100 150 Frequency shift (cm-1)

(b) Low frequency Raman spectra of x(yz)S and y(xz)y at atomospheric pressure.

merging into the strong mode at about 5Ό cm" are observed. The lineshape of the mode at about 50 cm above the transition pressure ptr (y 22 kbar) is similar to that observed in B^ spectrum at 0 kbar. On the other hand, a remarkable pressure

dependence has also been observed in the totally symmetric mode x(zz)x spectra. This dependence must be connected to the deformation of the NbOg octahetra under high pressuie.

100 LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS Vol. 32, No. 1

60 120 Frequency

0 60 120 shift (cm-1)

(c) Pressure dependence of low frequency Raman spectra of x(yz)x

§5 Peculiar Low Lying Response in E-symmetry Raman Spectrum of KDP

In the last section, a short comment is presented on the peculiar low-lying response in E-symmetry Raman spectrum of KDP?-0)

Experiments

The E(x,y) spectra obtained by the x(yz)y (or equivalently by the x(zx)y) geometry have been known to have an anomalous lineshape different from the usual lineshape of the damped harmonic oscillator, that is, to have a low-lying plateau response, extending from the Rayleigh line to ^90 cm"l. Several approaches were tried to explain this anomalous lineshape. Previous analyses were, however, made only on x(yz)y spectrum, which is a mixed mode spectrum. In the present work, the E symmetry pure TO spectrum and the E symmetry pure LO spectrum are obtained separately, at the geometry z(xz)x and x(yz)x,respectively. Comparing these spectra, following aspects can be seen.

(1) In the pure E-TO spectrum, the existence of the platesu response is evidently recognized. (2) In the pure E-LO spectrum the non-existence of the plateau response is recognized. (3) The x(zx)y spectrum is understood to be the single sum of the E-TO response and the E-LO response quantitatively. (4) The plateau response of E-TO spectrum lies only in the low energy region ω < 100 cm~l. (5) The platesu response cannot be fitted by oscillator models. (6) The intensity of the plateau response decreases with

Fig.6 Basis vectors of protons in KDP at room temperature phase for B2 and E symmetry (in dashed rectangles), due to Shur.

decreasing temperature, and vanishes in the low temperature.

Discussions

(1) Since the plateau response is obtained only in E-TO spectrum and completely vanishes in E-LO spectrum, this anomalous lineshape very likely originates from the first order Raman scattering at q=0. (2) The motion of protons in KDP is almost confined in the x-y plane. On the other hand, for crystals of point group D2d(^2m), the irreducible representation of proton motion which gives rise to x- or y- polarization is E(x»y). Accordingly the Raman scattering from the proton motion with polarization fluctuations in the x-y plane must appear in the E-symmetry spectrum. Γ The proton collective motion of the irreducible representation E(x,y) is evidently different from that of the irreducible representation B2(z). The basis vectors of the proton motion of these representations have been obtained by

Vol. 32, No. 1 LIGHT SCATTERING STUDIES ON STRUCTURAL PHASE TRANSITIONS 101 Shuri^ (a) For irreducible representation B2, of which proton motion gives rise to pseudo-spin switching, as seen in Fig.6 B2jthe basis vector of one proton is accompanied by the basis vector of another proton which is anti-parallel to the former, resulting in the cancellation of polarization in x-y plane, (b) For irreducible representation E, as seen in Fig.6 E,the basis vector of one proton in x-y plane is accompanied by the basis vector of another proton which is parallel to the former, therefore, nonzero macroscopic polarization in x-y plane can be produced.) (3) The plateau response is observed only in the high temper­

ature phase and vanishes in the low temperature phase, in which the proton motion is frozen. Terefore, the plateau response is very likely originated from the proton motion. (4) The anomalous profile of E-TO response lies only in the very low energy region, where the proton collective motion must appear. From these arguments, we propose that the

anomalous response in the E-TO spectrum should be assigned to the proton collective motion of the E(x,y) symmetry in the x-y plane. This conclusion is in agreement with that obtained from the mixed mode spectrum by Shigenari and Takagi.19)

References

l)T.Nakamura,L.Laughman and L.W.Davis: Proc. Taormina Conf. on Polaritons,E.Burstein et al ed. p.85 Pergamon (1974)

2)T.Nakamura and Y.Tominaga: Phys. Letters 50A, 5 (1974)

3)Y.Tominaga and T.Nakamura: Solid State Commun. 15, 1193 (1974)

4)Y.Tominaga and T.Nakamura: J.Phys.Soc.Japan J39, 746 (1975)

5)Y.Tominaga and T.Nakamura: Solid State Commun. _27, 1375 (1978)

6)A.Sawada,M.Udagawa and T.Nakamura: Phys.Rev. Letters J39, 829 (1977)

7)M.Udagawa,K.Kohn and T.Nakamura: J.Phys.Soc. 44, 1873 (1978)

8)S.Kojima,T.Nakamura,K.Asaumi,S.Takashige and S.Minomura: Solid State Commun.

9)S.Kojima,K.Asaumi,T.Nakamura and S.Minomura: J.Phys.Soc.Japan 45 1433 (1978)

10)Y.Tominaga,T.Nakamura and M.Udagawa: J.Phys. Soc.Japan ^6, 574 (1979)

ll)M.DiDomenico Jr.,S.P.S.Porto and S.H.Wemple: Phys.Rev.Letters 19 855 (1967); Phys.Rev. 174*522 (1968)

12)P.S.Peercy : Phys.Rev.Letters J31 379 (1973) 13)H.Terauchi,H.Takenaka,N.Matsumori and A.

Sawada: J.Phys.Soc.Japan 44 1751 (1978) 14)P.S.Peercy and I.J.Fritz: Phys.Rev.Leeters

32 466 (1974) 15)N.R.Ivanov,L.A.Shuvalov,G. Schmidt and E.

Shtol'p: Izv.Akad.Nauk SSSR, Ser.fiz. _39 933 (1975) N.R.Ivanov and L.A.Shuvalov: ibid, d 656 (1977)

16)J.C.Toledano,G.Errandonea and J.P.Jaguin: Solid State Commun. _2l0 905 (1976)

17)M.S.Shur: Kristallografiya _11 448 (1966); Soviet Physics-Crystallography 11 394 (1966)

18)Strictly speaking, q is complex in the crystal when light propagates in a crystal. But the imaginary part is very small if a transparent crystal is concerned. On the other hand, the imaginary part of ω of the overdamped phonons is larger than the real part. Therefore,q can be regarded as real.

19)T.Shigenari and Y.Takagi: J.Phys. Soc.Japan 42 1650 (1977)

Solid State Communications, Vol.32, pp.103-109. Pergamon Press Ltd. 1979. Printed in Great Britain.

ANHARMONIC PHONONS AND CENTRAL PEAKS AT STRUCTURAL PHASE TRANSITIONS

P. A. Fleury and K. B. Lyons

Bell Laboratories, Murray Hill, New Jersey 07974

Recent light scattering studies of solids undergoing structural phase transitions clearly demonstrate that the "central peak" phenomenon is not one, but several processes (both static and dynamic) contributing in different degrees to the spectra observed for different types of transition. Our own work on Pb5Ge3On, SrTi03, K2Se04 and BaMnF4 is reviewed and the studies by others on KDP, KH3(Se03)2, KD3(Se03)2 are discussed briefly.

Introduction: The fundamental problem of critical dynamics -

the temporal behavior of fluctuations in the order parameter - has repeatedly been demonstrated to be of greater complexity, in the case of structural phase tran­sitions, than is suggested by the quasiharmonic soft mode theory. In particular the appearance of appreci­able scattered intensity, centered at zero frequency shift, in addition to the inelastic peaks arising from the soft mode is an almost universal occurrence sufficiently close to the phase transition and has been interpreted as the introduction of a second characteristic time into the problem. Distinction among the many proposed mechanisms for these ubiquitous central peaks depends crucially upon obtaining their quantitative spectral signa­tures. Most importantly the distinction of dynamic from static processes can be reliably made only by measurement of spectral width. Finer distinctions among various dynamic mechanisms require determina­tion of dependencies of such widths and line shapes upon wave vector, symmetry, temperature etc. For example such dynamic processes as entropy fluctua-tions(1) (EF) or phonon density fluctuations(2) (PDF) can, via nonlinear coupling to the soft mode (order parameter fluctuations) contribute to singular central peaks. In addition static processes - associated with defects - have been assigned responsibility for strongly singular elastic components in these and other materi­als/3'4'5,6* In all cases we shall regard the static, defect related phenomena which give rise to elastic scattering as extrinsic to the phase transition. This is not to say that we expect their influence on critical dynamics to be trivial or uninteresting, but merely that the body of theoretical and systematic experimental wisdom avail­able on this subject is presently insufficient to warrant detailed discussion here. In what follows we shall regard the elastic central peak then as a noninteracting addition to the quasielastic and inelastic spectrum, assigning static phenomena no role beyond that of pos­sibly smearing out the transition. Observations and A nalysis

Using a resonant reabsorption technique for the removal of the elastic spectral component (Δι/ - 0 ± 300 MHz) we have determined quantita­

tively the linewidths and lineshapes of several dynamic central peaks as well as their interactions with both acoustic and soft optic phonons in materials represent­ing three major classes of structural transition. These classes are distinguished by the critical wave vector qc, for the soft phonon instability and can be labeled a) ferrodistortive ?f=0; b) antiferrodistortive, qc — mn/na\ and c) incommensurate, qc = (m7r/wtf+A), where m and n are integers.

All spectra reported here were excited with linearly polarized 5145A light from a single mode argon laser tuned precisely to the frequency of the strong sharp absorption line in molecular iodine. The scat­tered spectra are viewed through a temperature stabil­ized iodine cell placed in front of the spectrometer (either a pressure scanned tandem Fabry Perot inter­ferometer or a double grating spectrometer), stored in an on-line computer and normalized by a previously stored overall response function containing effects of subsidiary iodine absorption, collection optics and spec­trometer response.(7)

Ferrodistortive PbsGe3On A complete description of our investigations in

lead germanate has been published elsewhere.(8) Here we present only a summary of the salient features. Pb5Ge30H is a uniaxial ferroelectric (Cj) which under­goes a continuous transition to C3A symmetry at Tc - 451K.(15) The Alg soft mode (polarization fluctuations) appears in the x(zz)y scattering geometry (Ps\\z) as an underdamped phonon for T < 390K which follows a>2 — a'(Tc-T) where *' = 5850 GHz2/K.

For (Tc-T) between 55K and 10K the soft mode is overdamped, but continues to slow down according to ω2/Γ5 = b'(Tc-T) where b' = 12 GHz/K. Upon closer approach to Tc this characteristic frequency ceases to change and saturates at ys = ω2/Γ5 = 72 GHz (for (Tc-T) < 6K). Over this same temperature range an additional quasielastic scattering of unusual lineshape grows in intensity. This scattering exhibits a narrowing linewidth which reaches a minimum value of approximately 4 GHz (decon­volved HWHM) at ΓΓ, and obviously couples to the LA

103

104 ANHARMONIC PHONONS AND CENTRAL PEAKS Vol. 32, No. 1

Δι / (GHz)

Fig. 1. Central peak and Brillouin spectra of single domain lead germanate at

various temperatures near Tc = 451K. Data (open circles) are quantita­

tively fitted to the coupled mode theory described in the text by the

solid curves.

Brillouin components. The data (open circles) in Fig. 1 show the coupled acoustic-relaxing soft mode spectra of a single domain sample. The strongly singular elastic central peak has been removed from these spectra by the iodine absorption. The high frequency wing of the overdamped soft mode (ys = 72 GHz) is not discerni­ble in this figure, but is quantitatively fitted by the pro­cedure giving rise to the solid curves in Fig. 1 and which we now describe briefly.

The following formalism can be used to describe the dynamic fluctuations in all the transitions discussed in this paper. The light scattering spectrum, £(ω), for a system of n linearly coupled excitations can be writ-ten:(8)

S(a>) = j - ImZfiFjXu (1)

where for the special case of n = 2:

Xu = Xi

X\X2 Xl2 =

AX\Xl I-A2 (2)

X1X2

F2 is the scattering efficiency and χ,·(ω) the dynamic susceptibility, respectively, for no coupling (A2 = 0). For this discussion we have used (1 = acoustic and 2 = optic mode) X! = [ω2—ω2 + 2/ωΓ!)-1, where ωχ and Y\ are deter­mined from the Brillouin spectrum well above Tc (where A2 = 0). The uncoupled but relaxing soft optic mode response is taken to be χ2(ω) - [ω2-ω2

2 + 2/ωΓ2+/ωδ2τ(1-/'ωτ)-1]-1 The soft mode frequency, ω2, vanishes as (Tc-T)l/2 in the simple soft mode theory. χ2 differs from the simple

Vol. 32, No. 1 ANHARMONIC PHONONS AND CENTRAL PEAKS 105

soft mode behavior as ω2 —* 0 in that: a) the soft mode frequency parameter saturates at ω ^ = ω\ + δ2; and b) a central peak of width τ'Ηω^/ω&ο) appears.

The behavior of S in Eq. (1) when A2 ^ 0 is more complicated, and depends upon τ, δ and FjF2 as well as A. For transitions in which the soft mode is not Raman active above Tc, A is proportional to the order parameter and hence nonzero only for T < Tc. Since Pb5Ge3On is piezoelectric below Tc (but not above) the coupling constant A is proportional to the piezoelectric coefficient, #31 = 1.25xlO5 dyne/stat coul. A2 = q2a]x/pm*\ where m* = 4π/ω2

2€3 €3 is the dielectric constant, p is the density and q the scattering wave vector, the value of A2 at 300K is 25xl06 GHz4, as determined from independent measurements.(8) The values of ω{ = 18.5 GHz, Tl = 0.5 GHz, (F2

2Rex22/^i2Rexn) = 1-8 were determined from spec­tral measurements far from Tc.

Using these parameters, we can account correctly for the small anomalies in the LA phonon velocity and attenuation and the overall lineshape of the central peak and Brillouin spectra at all temperatures except within about 1.5K of Tc. For this calculation, we take T~1 = 29 GHz and δ = 60 GHz as constants near Tc, and use ω2

2 = a'(Tc-T) where a' is given above. Very close to Tc the dynamic central peak ceases to narrow, presumably because of the smearing of Tc over our small (2 x 10~3 mm3) scattering volume. A smearing in Tc of approximately IK is sufficient to account for the dynamic central peak behavior, and is consistent with the gross variation in Tc observed by varying the location of the scattering volume within the sample. Detailed spectral lineshape fits where a spatial distribu­tion of Tc is included as an adjustable parameter are published elsewhere.(8) The additional complications introduced by the spatial variation of Tc have thus far prevented a more quantitative test for the deviations from mean field behavior predicted for a uniaxial dipo­lar system by renormalization group theory. While we have observed clear departure from mean field behavior for the total scattered intensity in the relaxing soft optic mode, a quantitative assessment of this departure would require the theory to be extended to include effects of symmetry breaking defects upon the critical behavior of the uniaxial ferroelectric.

A ntiferrodistortive SrTi03 The well-known, cubic-tetragonal transition at

Tc = 107K in SrTi03 is accompanied by a softening of the Γ25 symmetry phonon at the R point in the cubic Brillouin zone. Below Tc the order parameter fluctua­tions appear as zone center Raman active modes of Ag and Eg symmetry in the polarized and depolarized scattering geometries respectively. As described else­where(9) the polarized quasielastic spectrum is dom­inated by Brillouin scattering from LA phonons and by two dynamic but not detectably singular central peaks. These latter features have been identified as entropy fluctuations and phonon density fluctuations which cou­ple at most extremely weakly to the order parameter. Neither of these features appears in the Eg (depolar­ized) geometry so that a detailed search for a singular dynamic central peak in that geometry has proven fruit­ful.

Figure 2 illustrates its temperature dependent lineshape as well as the evident interaction with the TA Brillouin components. The facts that the maximum central peak intensity occurs below Tc and that the feature cannot be seen above Tc, suggests that we are observing the soft phonon PDF indirectly through their relaxational contribution to the soft mode self energy and the soft mode coupling to the TA phonons. Such

1/1 C ω

ω

ω o

en

-2 -1 0 1 Frequency Sh i f t , cm"1

Fig. 2. Depolarized central peak and Brillouin spectra of SrTi03

near Tc = 107K.

coupling affects not only the relaxing soft mode lineshape but causes anomalous behavior in both the velocity and attenuation of the TA phonons near Tc. We have made careful measurements(9) of the TA Bril­louin frequency shifts and linewidths, as well as the coupled mode lineshapes and intensities as functions of temperature; and have been able to obtain quantitative spectral lineshapes (as shown in Fig. 2) using the cou­pled mode analysis described above.

In SrTi03 the order parameter is the Ti06 octahedron rotation angle, Φ, and has been accurately measured: <f>~~(Tc-T)0-33. The ratio: Α/φ has been determined from the total shift of the Brillouin com­ponents upon traversing Τ0(Δ.ωι/ωχ = 0.12) to be equal to 3100 GHz2/degree. Assuming τ"1 to be tempera­ture independent (r_1 = 15 GHz), we have been able to fit the entire lineshape quantitatively as shown in Fig. 2c with the following values of the above parame­ters: ωχ = 35 GHz, Tx = 0.5 GHz, Γ2 - 100 GHz, (FlRex22/F2RQXn)=0.\2, δ(Γ) = 12.502+Γ-7;) GHz. The value of δ2(7;) = 1.5 JC 104 GHz2 joins smoothly with corresponding values inferred from neutron scattering above Tc. The value of ω2

2 determined from our fit reaches a minimum value of approximately 104 GHz2. This is equivalent to the failure of the dynamic central peak to collapse to zero linewidth and implies that either a second much lower frequency relaxation process con­tributes to χ2(ω) or that the transition is smeared. The order parameter measurements on SrTi03 give no indi­cation that Tc is smeared detectably. The existence of

106 ANHARMONIC PHONONS AND CENTRAL PEAKS Vol. 32, No. 1

a second relaxation in χ2(ω) would produce an addi­tional central peak of intensity comparable to that we have observed. Since we could detect no additional scattering using the iodine absorption cell technique we must conclude that any additional central peak is less than 0.3 GHz in width. Earlier experiments have sug­gested the presence of a very slow response in SrTi03 possibly associated with defects.03,14* The presence of two singular central peaks in SrTi03 (the one which we have observed directly with τ"1 = 15 GHz, and the second which we have inferred indirectly with T"1 < 0.36 GHz) would account for all observations made to date regarding this transition(9).

Incommensurate: K2Se04 and BaMnF4 The third major class of transition we have inves­

tigated is driven by a phonon instability in the prototyp­ical phase whose wave length is not a rational fraction of the parent unit cell dimension. The resulting struc­tural distortion (below 7p is therefore incommensurate with the parent structure. The expected sequence of behaviors for the order parameter fluctuations in such a case is shown schematically in Fig. 3. Of particular interest is the behavior shown in Fig. 3c which has the dispersion curve splitting into an upper branch with

Fig. 3.

0 2/3 ττ/α ττ/α WAVEVECTOR

Schematic behavior of the soft mode dispersion curves

(a) above, (b) at, and (c)-(d) below the transition Γ, to

the incommensurate phase in an extended zone scheme

showing the amplitude (solid) and phase (dotted)

components.

ordinary soft optic mode behavior and a lower branch whose frequency vanishes at qc for all temperatures in the incommensurate phase. The former corresponds to a dynamic amplitude distortion in the order parameter, while the latter describes a phase distortion - and has therefore been dubbed the "phason". If at some lower temperature, Tcomm, the system experiences a transition to a commensurate structure (Δ—K)) the phason will develop a "gap" and become an ordinary soft phonon; much as we are familiar with in SrTi03.

Our studies of both K2Se04 and BaMnF4 were motivated by attempts to observe the phason directly in the incommensurate phase. In K2Se04 the transition to the incommensurate phase at Tt = 129K is described by a soft mode of critical wave vector q* = [.31,0,0].

0 Av(GHZ)

+20

Fig. 4. Polarized Brillouin spectra near Γ, = 129K in K2Se04

Note the dynamic central peak (< 5 GHz) superimposed

on the shifting mixed acoustic mode (7-8 GHz) and the

small but detectable residual stray light component at

Δ* - 0 ± 1 GHz.

As described elsewhere(10) both the phase and ampli­tude modes in K2Se04 have been observed below Tcomm " 93K and followed upward in temperature. The amplitude mode takes no notice of the lockin transition

Vol. 32, No. 1 ANHARMONIC PHONONS AND CENTRAL PEAKS 107

at Tcomm and behaves as an ordinary quasiharmonic mode softening with respect to the incommensurate transition at 7 : ω% - 2β(Τ-Τ) GHz2, with T in °K.

In fact the amplitude mode exhibits quite accu­rately the simple quasiharmonic soft mode behavior, so that the central peak evident in Fig. 4 can be inter-preted(10) as merely the overdamped soft amplitude mode - without the need for invoking any relaxation processes. The acoustic anomaly evident in the 7 GHz phonon of Fig. 4 is accounted for by a simple acoustic-soft optic mode coupling of the type discussed above.

The phase mode, however, becomes progressively broader and weaker as Tcomm is approached from below and cannot be observed above Tcomm. The reasons for this remain obscure. Reference 10 presents some relevant speculations.

BaMnF4 represents a more interesting example of an incommensurate transition. At Tt = 247K a phonon instability at q*c - [.392, .5, .5] drives a second order transition from a C2

12 structure to an incommensurate phase.(11) No lockin transition occurs at lower tempera­tures. Neutron scattering01) and ultrasonic experi­ments0 2) have shown significant evidence for fluctua­tion (nonmean field) effects near Tr Subsequent Bril-louin scattering experiments03) have confirmed the existence of significant dispersions in the transverse acoustic (TA) velocity anomaly at frequencies below approximately 1 GHz. However, no direct phason scattering was reported in Ref. 13.

Quite recently Lyons and Guggenheim04)

reported a singular dynamic central peak in BaMnF4 with several unusual properties:

1) The C. P. intensity is appreciable only in a small and nearly symmetric temperature interval near Γ,.

2) The linewidth does not exhibit singular narrow­ing near Tf.

3) The linewidth is q dependent: Γ = Da1 with D =0.14 ± .02 cm2 sec-1.

4) The C. P. is observable only for q in the be plane, not ϊοτ q\\a.

5) Careful measurement of the LA phonon spec­trum in the same scattering geometry for which the central peak appears reveals a previously-undetected frequency dispersion (approximately 4%) for the c directed velocity anomaly - in the vicinity of 10-20 GHz.

6) As evident in Fig. 5 the narrow central peak sits on top of a broader but strongly temperature dependent "background" (presumably the high fre­quency wing of the soft mode) which extends out to at least the LA phonon peaks at ± 20 GHz.

Observations 1), 4) and 5) support the interpreta­tion that the central peak carries no appreciable scatter­ing strength of its own but is enhanced through its interaction with the c directed LA phonon. Observa­tion 6) suggests the soft mode wing is similarly enhanced; thereby arguing that the central peak enters as a relaxational self-energy in the soft mode response

"I Γ Tc±0.5 K

- H . Η-η„

T C - 2 K

T c - 4 K

-10

T, + 7K

10 -10 Δι / (GHz)

Tc + 4 K

Tc + I K

10

Fig. 5. Dynamic central peak in BaMnF4 near Γ, = 247 K. Note temperature

independence of linewidth and persistence of scattering above Γ,.

108 ANHARMONIC PHONONS AND CENTRAL PEAKS Vol. 32, No. 1

has been attributed to entropy fluctuation scattering, primarily on the basis of its absolute width agreeing fairly well with the value of 62 MHz predicted for right angle scattering in KDP. Neither critical narrowing nor q dependence of the linewidth have been observed thus far;(1) but experiments are continuing.

Elastic Central Peaks Several light scattering experiments have now

been carried out with sufficient frequency resolution to identify a number of singular central peaks whose ori­gins are "static". In two cases, as shown in Table I, quite severe limits have been placed on the frequency domain operationally defining the term elastic. In oth­ers the restrictions are not so severe but, unless a very low frequency dynamic process can be inferred from

TABLE I

Material

Pb5Ge3On

SrTi03

KH2P04 (KDP)

K2Se04

BaMnF4

KH3(Se03)2

K(HxD1_x)3(Se03)2

Qc*

(0,0,0)

(.5,.5,.5)

(0,0,0)

(.31,0,0)

(.392,.5,.5)

(0,0,0)

(0,0,0)

T 1 trans

451K

107K

117K

129K

247K

212K

Tc(x)

Central

Peak

i) Dynamic

iDStatic

i) Dynamic

ii) Dynamic

i) Dynamic

ii) Static

Dynamic

Dynamic

Static

Static

Characteristic

Frequency

(HWHM)

4—30 GHz

< 10 Hz

10—15 GHz

< 300 MHz

50 MHz

< 1 Hz

' 5 GHz

~2GHz

W)

< 200 MHz

< 200 MHz

Mechanism

PDF

Defects

PDF

?

Entropy

Dislocations

(annealable)

Simple overdamped

soft mode

Entropy (?)

Strains

(annealable)

H,D impurity

function, which is in turn linearly coupled to the LA phonon (with Fx » F2). Observation 2) points to a diffusive dynamic process as the origin of the central peak. However, independent measurements of the thermal diffusivity are presently lacking so that a definitive identification of the feature with entropy fluctuations cannot yet be made. The intriguing possi­bility that the central peak in BaMnF4 might arise from an overdamped phason cannot be excluded at this time. Further work centering on an independent measure of thermal diffusivity near Tt is in progress.

The final singular central peak for which a finite linewidth has been observed occurs in ferroelectric KDP, where the thermal diffusivity is known. Seen only within approximately 0.1K of Tc the 50 MHz wide central peak observed by Mermelstein and Cummins(1)

Vol. 32, No. 1 ANHARMONIC PHONONS AND CENTRAL PEAKS 109

spectral analysis as in the case of SrTi03,W we shall nevertheless attribute these to static processes also.

As discussed in Ref. 8 the elastic peak in single domain Pb5Ge3On has an easily observable singular component whose intensity diverges much more strongly near Tc than does the dynamic central peak. Very high resolution interferometry(8), combined with the results of intensity autocorrelation spectroscopy(15)

has shown this feature to be spectrally narrower than approximately 10 Hz. Together with its strong intensity divergence, this suggests static symmetry breaking defects as the origin of this central peak. However, no more definitive statement as to the nature or the den­sity of these defects can presently be made, other than to say that they are not identical to the defects which cause the observed spatial variation of Tc in lead ger-manate.(8).

KDP has also been observed(16) to exhibit an elas­tic singular central peak. A recent suggestion by Cour-tens(17) that frozen out deuterium impurities (present as a naturally occurring isotope of hydrogen) might account for the feature seems to have been proven incorrect. In a series of clever experiments Courtens(5)

has succeeded in annealing away the static KDP peak by judicious heat treatment. The spatial distribution of central peak intensity observed in a rapidly annealed sample argues, in fact, against impurities of any sort as

the defects responsible. Rather "intrinsic" defects such as dislocations are the a more likely cause. Prolonged annealing (18 hours at 140<C!) reduced the singular central peak intensity by nearly a factor of 50 in KDP, and potentially opens the way to very detailed studies of critical dynamics in this material(5) without the res­trictions imposed by the iodine cell technique.

Similar, but less spectacular reduction of the static central peak by annealing (this time at low tempera­tures, -24°C) was achieved earlier in a related material KH3(Se03)2 by Yagi et al.(6) Their observation that the central peak intensity is strongest in the initial stages and decreases upon repeated temperature cycling argues for inhomogeneous strains - introduced by sample preparation - as the origin of this annealable central peak.

These authors have also investigated mixed cry­stals of K(HxD1_Jf)3(Se03)2 and have found elastic cen­tral peaks which cannot be annealed away.(3,4) Thus D's in the x=.95 sample and H's in the nominally pure x=0 sample are thought to be responsible for these singular elastic features. A controlled series of experi­ments, where x is systematically varied and the changes in Tc, soft mode dynamics and hopefully even entropy fluctuations are carefully monitored should provide valuable insight into important questions on phase tran­sitions in defected systems.

REFERENCES

[1.] [2.] [3.] [4.] [5.] [6.] [7.] [8.] [9.] [10.] [11.]

[12.] [13.]

[14.] [15.]

[16.] [17.]

D. Mermelstein and H. F. Cummins, Phys. Rev. B16, 2177, 1977. Rev. Letters, 37, 1088, 1976.

, J. Phys. Soc. Japan, 44, 2009, 1978. , J. Phys. Soc. Japan, 44, 1257, 1978.

1171, 1978. Phys. Rev. Letters, 38, 609, 1977.

M P. A. Fleury and K. B. Lyons, Phys. H. Tanaka, T. Yagi and I. Tatsuzaki, H. Tanaka, T. Yagi and I. Tatsuzaki, E. Courtens, Phys. Rev. Letters, 41, T. Yagi, H. Tanaka and I. Tatsuzaki, K. B. Lyons and P. A. Fleury, J. Appl. Phys., 47, 4898, 1976. K. B. Lyons and P. A. Fleury, Phys. Rev. B. 17, 2403, 1978. K. B. Lyons and P. A. Fleury, Solid State Communications, 23, 477, 1977. P. A. Fleury, S. Chiang and K. B. Lyons, to be published in Solid State Communications. D. E. Cox, S. M. Shapiro, R. A. Cowley, M. Eibschutz and H. J. Guggenheim, to be published in Phys. Rev. B. I. J. Fritz, Phys. Rev. Letters, 35, 1511, 1975. D. W. Bechtle and J. F. Scott, J. Phys. C , 10, L209, 1977. A subsequent report, D. W. Bechtle, J. F. Scott, and D. J. Lockwood, Phys. Rev. B, 18, 6213, 1978, demonstrated qualitatively the existence of a dynamic central component in the vicinity of Tr K. B. Lyons and H. J. Guggenheim, to be published. D. J. Lockwood, J. W. Arthur, W. Taylor, T. J. Hosea, Solid State Communications., 20, 703, 1976. L. N. Durvalsula and R. W. Gammon, Phys. Rev. Letters., 38, 1081, 1977. E. Courtens, Phys. Rev. Letters, 39, 561, 1977.