4962 NICKI OW, WAKABAYASHI, AND SMITH

used for the constants e& and oL, were based on theshapes of dispersion surfaces for the v~ and VL

branches that were calculated from the force-con-stant-model fit to our data. The maximum intensityoccurs at a frequency higher than vp mainly for two

rather obvious reasons. First, even for a singlecrystal of graphite, the dispersion surfaces for VL,

and v& are very steep so that the resolution functionsamples an appreciable number of modes havingfrequencies higher than v, . Second, because of theorientation randomness, for any frequency v greaterthan vp there is a v& = v mode having a wave vectorwhich falls within the resolution (from an origin z )and having a favorable polarization. The long in-tensity tail at high frequencies is due mainly to thelatter effect.

In carrying out this calculation we have fixed theparamete rs specifying the resolution function at those

values which are appropriate for a measurement of vp

=1.35 THz, i.e. , we have not taken into account,for example, the dependence of the resolution onk for v& vo. Thus the relative intensity calculatedfor v appreciably greater than vp is subject to someerror. However, the excellent agreement found

between the calculated peak shape and the experi-mental results (see Fig. 2) for v near vo indicatesthat this error and those arising from the assump-tions mentioned above are probably insignificantfor our purposes.

The calculations for the constant-E scans of theLO phonons near the (111) ring were carried outsimilarly, but with the exception that the intensitywas computed for the resolution function centeredon differing points along the y axis with v fixed.These calculations are also in very good agreementwith the measurements as shown in Fig. 5.

*Research sponsored by the U. S. Atomic Energy Com-mission under contract with Union Carbide Corp.

J. A. Krumhansl and H. Brooks, J. Chem. Phys. 21,1663 (1953); J. C. Bowman and J. A. Krumhansl, J.Phys. Chem. Solids 6, 367 (1958).

2K. Komatsu, J. Phys. Soc. Japan 10, 346 (1955); J.Phys. Chem. Solids 6, 381 (1958).

A. Yoshimori and Y. Kitano, J. Phys. Soc. Japan 11,352 (1956).

James A. Young and Juan U. Koppel, J. Chem. Phys.42, 357 (1965).

'G. Dolling and B. N. Brockhouse, Phys. Rev. 128,1120 (1962).

J. C. Gylden Houmann and R. M. Nicklow, Phys. Rev.B 1, 3943 (1970).

R. M. ¹icklow, N. Wakabayashi, and P. R. Vijayara-ghavan, Phys. Rev. B 3, 1229 (1971).

~L. J. Raubenheimer and G. Gilat, Phys. Rev. 157,586 {1967).

~J. L. Warren, J. L. Yarnell, G. Dolling, and R. A.Cowley, Phys. Rev. 158, 805 (1967).

F. Fuinstra and J. L. Koenig, Bull. Am. Phys. Soc.15, 296 (1970).

~lE. S. Seldin, in Proceedings of Ninth Biennial Con-ference on Carbon, Chestnut Hi/E, Massachusetts, 1969(Defense Ceramic Information Center, Columbus, Ohio,1969) p. 59.

W. DeSorbo and G. E. Nichols, J. Phys. Chem.Solids 6, 352 (1958); W. DeSorbo and W. W. Tyler,General Electric Technical Report No. RL-1202, 1954(unpublished); see also Ref. 4.

M. J. Cooper and R. Nathans, Acta Cryst. 23, 357(1967).

PH YSIC A L REVIEW B VOLUME 5, NUMBER 12 15 JUNE 1972

Theory of Critical-Fluctuation-Enhanced Raman Scattering by Phonons in NH48r

J. B. SokoloffDepartment of Physics, Northeastern University, Boston, Massachusetts 02115

(Received 28 June 1971)

The hypothesis due to Wang and Fleury that the Raman scattering from the 56-cm ~ phononin NH4Br is enhanced by critical fluctuations in the neighborhood of the transition temperatureis derived from a theory of Raman scattering from phonons in a system with partially dis-ordered force constants, by making some reasonable assumptions about the behavior of thepolarizability as a function of ammonium-ion orientation.

Ammonium bromide crystallizes in a CsCl struc-ture lattice below 457. 7 K. ' Since the ammoniumion is of tetrahedral symmetry, it has two possi-ble orientations in its cubic environment. Belowa temperature of 235 K, the ammonium ions inrows parallel to one of the crystallographic axes(let us call it the z axis) order such that all the

ions in a given row are oriented the same way andions in neighboring rows have opposite orienta-tion. ' Wang and Fleury found that above thistransition temperature, in addition to a first-orderHaman spectrum reflecting the single-phonon den-sity of states because of the breakdown of f conser-vation, there is a fairly well-defined phonon peak

TH EORY OF CRITICAL - F LUCT UATION- ENHANC E D. . . 4963

at 56 cm ' whose intensity decreases as one movesin either direction away from the transition tem-perature. The behavior of the intensity of thismode was attributed to short-range order. 3 Theydid not, however, discuss the mechanism for this.In this article, such a mechanism will be discussedusing a simple model of first-order Raman scatter-ing from phonons in a system with disordered forceconstants.

Let us assume, following the discussion of Love-luck and Sokoloff, that ammonium bromide can betreated as a system with force constants betweenammonium ions depending on the orientation of theammonium ions. Then the equation of motion of theone-phonon Green's function can be written as

jB» '

= &„6„„6„, (1)where co is the phonon frequency, z, p, y label unitcells, i, j, l label atoms within a unit cell, and g,x', y label polarization of the phonons. HereH'~ ~, represents the part of the force constantsdependent on orientation and H '~~, denotes the partnot dependent on orientation. Following the notationof Edwards and Loveluck, we may write

o a' (I'Hof»~ ~ ». Q ting t, i ~C e ~e» g»' C

where 0; cr' denote the orientation of the ammoniumion in the unit cell, the restriction i =j= 1 meansthat this term is only nonzero for ammonium ions(the ammonium is called ion 1), and T',~ 8 „.is theorder-dependent force constant. Here, C' is 1 ifthe ammonium ion in the oth unit cell has orienta-tion 0; and it is zero otherwise. We now may iter-ate Eq. (1) to obtain

G= G +G BG +G BG BG+ ~ ~

in matrix notation. Here

(AG)= (A)G +(AG B)G + ~ ~ ~ (8)

Since the dispersion curves found by neutron dif-fraction7 in the ordered (tetrahedral-symmetry)state of ammonium chloride look very much likewhat they would be for a cubic-symmetry crystal,the orientation-dependent part of the force con-stants between ammonium ions appears to be smallcompared to the rest of the force constants. Theyshould also be small in ammonium bromide. Hence,to a good approximation we can keep just the firstterm in Eq. (8). Using this approximation andEqs. (6)-(8), we find that the orientation-dependentpart of the Raman-scattering cross section is pro-portional to

Z P„'P„', (C' C~~ )ImG (9)eely'

tives of the polarizability tensor in Eq. (6) would

be independent of the indices n and P, as in Ref.6.] Let us assume in this system that we may write

', =ZP„'C'.~+a» ty

if i refers to an ammonium ion (i.e. , i =1), and

BP/Bu', „ is independent of n if i refers to a bromideion; i.e. , we have assumed that the change in thepolarizability caused by displacing one ammoniumion depends only on the orientation of that one ion.This can be understood as follows: For a givenpair of tensor indices (i. e. , photon polarizationindices) and a, given phonon polarization x, BP/Bu, „will in general depend on the orientation of thetetrahedral symmetry ammonium ion. We are as-suming here that the influence of neighboring am-monium-ion orientation on BP/Bu is negligible.

In order to evaluate the average in Eq. (6), letus consider the average (AG ), where A= C' C8 .Using Eq. (3), we have

G = (&o 1 —H —(H')) (4)Now we Fourier transform Eq. (9) to give

where 1 denotes the three Kronecker p's in Eq.(1) and ( . ) denotes a. thermal average over allammonium-ion configurations in the lattice. Also,we have

B,",,„,=6,~6, ,(H,'„',„.—&H,"„'„,&) . (6)

Following Born and Huang, s we may write theRaman-scattering cross section as proportional to

ImG'~ ~.

where P is the polarizability tensor and u' „denotesa displacement in the g direction of the ith ion inthe Mh unit cell. Tensor indices on P have beendropped for notational simplicity. [If the crystalhad complete translational symmetry, the deriva-

Z P„'P„', 2 S-'(k)imG", '(-k, ~), (10)

where

8mr' ($) Q e( 1 (R~-s8) (Cs Ca'a

and Go", (k, &o) is the Fourier-transformed unper-turbed Green's function. Because of short-rangeorder, the pair correlation function S" (k) shouldpeak (e.g. , show an Ornstein-Zernike form) abovethe transition temperature at wave vector k =Q,where Q = (v/a)(1, 1, 0) and a is the spacing betweennear-neighbor ammonium ions. Thus, just abovethe transition temperature, there should be strongRarnan scattering from phonons near the point Min the CsC1 Brillouin zone [i.e., k= (v/a)(1, 1, 0)],

In order to describe the correlation function inmore detail, let us introduce an Ising-model-like

J. B. SOKOI OF F

parameter S to describe the NH4' ion orientationon site n. The parameter 8 will have the value+1 for one of the orientaticns and —1 for the other.Then we get

C' =1+crS

where o =s 1 (describing the two different orienta-tions). Using this notation, we may write

(C' Cz )=1+a(S )+&1'( SB) +os'( S SN) . (11)

The correlation function (S,S,) is just the type ofcorrelation function occurring in an Ising-modelantiferromagnet. Using our new notation, we find

the relation

Z P'„P„' (C' C ) = (P,'+P, )'+(P;+P„)(P„' P,)-x(S +S )+(P„'-P„) (S S ) . (12)

Since in the cubic-symmetry disordered phase thereis no strong zone-center first-order Raman effect,we have

or decreases as the transition temperature is ap-proached from below. If we assume that the 56-cm 'mode becomes the 67 cm ' at low temperatures be-cause they both have the same polarization selec-tion rules (an interpretation which differs fromthat of Wang and Fleury), the intensity of the peakfirst decreases and then increases as we go down intemperature below the transition temperature. Inneutron scattering from antiferromagnets the in-tensity is generally observed to simply increase.Thus the fluctuation term [term of the form of Eq.(16)t must be larger in NH4Br than in typical anti-ferromagnets. Other interpretations which involvecoupling of modes' have also been suggested for thedecrease in intensity below the transition tempera-ture.

In order to better understand the observed behav-ior of the Raman scattering, we consider theGreen's function G further. If we write Eq. (4)in greater detail, we obtain the following equationof motion for G:

=Z P„'C:= .'(P;+P,-) =0- (Is} Z(&o 6 8—H, ~-Z (C' CB) T',t )Gq„——6 „. (18)

(since C"= —,' in the cubic phase). Hence we are

left with the relationship

Z P„'P," (C'C", )=(P,'-P;) (S S,) . (14)tyiy

Since both S" (k) and Go,'P(-k, ~) have either cubicor tetragonal symmetry, only diagonal elementsin the gg' indices contribute, and hence the Ramancross section in Eq. (10}is given by

Z(P;-P„-)'Zg(k)G'" (-k, (u), (i6)x k

where

g(k)=Z e'i''8 8» (S S ) .

For a system undergoing an order-disorder tran-sition g(jt) behaves like

1(16)

near wave vector Q, where 0, approaches zero asthe transition temperature is approached. Belowthe transition temperature there exists, in additionto a term of this form, a term proportional to

m'6(k -Q),where m is the order parameter (i.e. , differencein number of NH4' ions of each orientation in agiven row of NH4' ions parallel to the c axis —i.e. ,the analog of sublattice magnetization of an anti-ferromagnet). This term decreases as the transi-tion temperature is approached from below sincem decreases. The relative weights of these twoterms will determine whether the intensity of thepeak in question in the Raman spectrum increases

We have suppressed polarization indices for sim-plicity (i. e. , all terms are assumed to be matricesin these indices). The order-dependent force-con-stant matrix in the above equation is given by

T q (C CB )=Z T q ((I+oS )(1+o'Sq) )ea'

=E T,'+(T",—T,)

x ((S,)+ (SB))+(T' —T ')(S —S~)

+(T~t +T~a —T~q —T~'8) (S„Sq ) . (19)

Following the assumptions of Ref. 4, we takeT 8

= T 8 and T 8= T 8 in the disordered phase.

Then the only terms which survive are

Z T,+ 2(T"- T' ) (S S,) . (20)(ye'

Actually, it was shown in Ref. 4 that for force con-stants only depending on the orientations of the ionswhich they connect (e.g. , force constants found bydifferentiating the exchange in the Ising or Heisen-berg model of magnetism with respect to lattice co-ordinates), T 8= T=~ and T', = T ~ for etP evenin the ordered phase. For the special model offorce constants only connecting nearest-neighborNH~'ions, we can further show, using the well-known sum rule

Ts + g Taa'Ca 0R Qi e8 8o' 8/o

and the fact that T'=, = T 8, that T" = TFourier transforming Eq. (18) using the above

relationships gives

[uP —8'(k) -Z T-'(k) —sZ[T"(k') T' (k')]-ay' f.i

THEORY OF CRITICAL-FLUCTUATION-ENHANCED. . . 4965

xg(k —k')jG (k, k")=5f"

Using the fact that g(k) is rigorously equal io5 (k —Q) at zero temperature, we see that Ga(k, k")is not mixed with G (k+Q, k"). This occurs be-cause the term involving the correlation functionin Eq. (18) is a convolution of a function with G~.This implies that within the approximation usedhere for the force constants, zone-boundary and

zone-center phonons are not mixed. From the dis-cussion given previously in this paper, the Raman-scattering cross section is proportional to Im G (Q,

Q) at zero temperature, whereas the optical ab-sorption is proportional to ImGO(0, 0). Thus Ramanscattering should measure modes at the point M,whereas optical absorption should measure modesat T. Such a difference in the measured frequen-cies is borne out experimentally'; the optical ab-sorption only shows a peak at 153 cm ' which is notobserved in the Raman. Since this can be explainedby symmetry, however, the evidence is not con-clusive.

One question that arises is why the intensity ofcertain peaks shows great temperature dependencewhereas the intensity of other peaks does not. Forexample, the 134-cm mode shows little tempera-ture dependence in its intensity. A possible ex-planation for this is that there probably would occura peak in the Raman spectrum of the completely dis-ordered state at about 130 cm ' anyway. It wasshown in Ref. 4 that the Raman spectrum of thecompletely disordered state is shaped like the one-phonon density of states. Thus, if a peak occurredin the one-phonon density of states at about 130cm ', there should be little change in its scatteringintensity as the short-range order disappears abovethe transition temperature (i.e. , the system be-comes completely randomly disordered). At 56cm ', on the other hand, there is probably no peakin the density of states, and hence the observationof such a peak in the Raman spectrum depends onthe existence of order with periodicity Q. Inciden-tally, the 76-cm ' mode appearing in the crossedpolarization scattering appears to also show aslight narrowing and increase in intensity justbelow the transition temperature, just as the56-cm ' mode. Thus, the 76-cm ' peak might alsocome from points near the point M. It has lowerintensity, however, probably for reasons havingto do with the fact that the density of phonon statesnear 76 cm is lower than at 56 cm '.

Since G (Q, Q) occurs in the expression for thelow-temperature cross section, and since it isjust Green's functions with wave vectors near Qwhich are enhanced by the correlation function justabove the transition temperature, we expect the56- and 76-cm ' peaks to have the same polariza-tion selection rules above as below the transition

temperature (i.e. , the 56-cm ' mode will appearonly in the uncrossed configuration and the 76-cm 'mode only in the crossed configuration).

In Eqs. (16) and (IV) we have assumed that oneaxis is picked out as the c axis, and thus there is aunique Q. This would be true if the crystal wereunder strain which caused it to always order withthe same c axis. It is clear that there will alwaysbe local strains which will favor a particular axisin a local region of the crystal; below the transi-tion temperature, such regions will become do-mains of a particular Q. Thus our procedure iscorrect if we understand the average in the correla-tion function to be only over a region of the crystalthat is small compared to macroscopic dimensionsbut large compared to microscopic dimensions.The resulting Green's function and hence scatteringintensity must then be averaged over all possibledirections of Q in order to find the Raman crosssection for a finite crystal.

By keeping only the term proportional to G inEq. (8), we have neglected broadening of the phononlines due to disorder, but this effect is higherorder in the ratio of H' and H& and does not affectthe basic picture qualitatively if this ratio is fairlysmall.

In conclusion, we have shown how the Wang-Fleury picture for NH4Br (i.e. , that in the vicinityof the critical temperature, the Raman scatteringfrom phonons near some point at the Brillouin-zoneboundary is enhanced by critical fluctuations) canbe derived from the Born-Huang formulation ofthe problem of Raman scattering by phonons. Theenhancement of the cross section of the phonons atpoint M follows quite naturally from the Born-Huang expression for the cross section after makinga reasonable assumption about the dependence ofthe polarizability tensor on ammonium-ion orienta-tion.

If the assumption of Eq. (V) were not made, wewould obtain a correlation function with more thantwo c's in Eq. (9). Since the sites n and p in Eq.(S) must still be correlated in the manner suggestedin the discussion following Eq. (10), the conclusionsof this paper should still be correct.

As suggested by Wang, 3 neutron diffraction shouldreveal whether there actually is a 56- and a 76-cm 'mode near the point M, as suggested in this paper.Although no modes of such low frequency at thepoint M are observed in neutron work on ND4C1, 7

they may still be observed in NH~Br and ND4Br,since the phonon frequencies in NH4Br and ND4Brare generally lower than in NH4Cl and ND4Cl be-cause of the mass difference between the Cl andBr ions. Since the model used here is not specificto NH4Br, it may also be possible to observe en-hancement of the Raman scattering from phononsin some antiferromagnetic crystals.

4966 J. B. SOKOLOFF

ACKNOWLEDGMENTS

The author would like to thank J. Loveluck, C. H.

Perry, P. A. Fleury, and C. H. Wang for valuablediscussions. He would also like to thank C. H. Wang

for revealing some of his unpublished results.

lC. H. Perry and R. P. Lowndes, J. Chem. Phys.~51 3648 (1969).

2C. H. Wang and P. A. Fleury, in Light ScatteringSpectra of Solids, edited by G. B. Wright (Springer,New York, 1968), pp. 651-663.

C. H. Wang, Phys. Rev. Letters 26, 1226 (1971).J. M. Loveluck and J. B. Sokoloff (unpublished).S. F. Edwards and J. M. Loveluck, J. Phys. C

Suppl. 3, S261 (1970).

6M. Born and K. Huang, Dynamical Theory of CrystalLattices (Oxford U. P. , London, 1954), p. 369.

7H. C. Teh, B. N. Brockhouse, and G. A. DeWitt,Phys. Letters 29A, 694 (1969); Phys. Rev. B 3, 2733(1971).

R. Brout, Phase Transitions (Benjamin, New York,1965), pp. 22-29.

L. Corliss (private communication).' C. H. Wang (unpublished).

PHYSICAL REVIEW B VOLUME 5, NUMBER 12 15 JUNE 1972

Excitonic Polarons in Molecular Crystals*

P. M. Chaikin, A. F. Garito, and A. J. HeegerDepartment of Physics and Laboratory for Research on the Structure of Matter,

University of Pennsylvania, Philadelphia, Pennsylvania 19104(Received 2 December 1971)

Band narrowing and band shifts due to excitonic-polaron formation are considered for anelectronically polarizable medium. Experimental evidence of such effects in molecular crys-tals is cited. The central result is that the electron-exciton interaction in the strong-couplingregime leads not only to an indirect attractive electron-electron interaction, but also becauseof small-polaron formation leads to severe band narrowing. This band narrowing tends to favorthe insulating state. The implications of the excitonic polaron for the metal-insulator transition,organic semiconductors, and the question of the possible existence of superconductivity in or-ganic molecular crystals are discussed.

Since Landau first considered the interaction ofa charge carrier with a lattice polarization, thecoupled electron-lattice or polaron problem hasbeen of continued interest in the study of solids. 'Of particular importance is the small-polaronlimit, relevant when the binding energy of the elec-tron and the induced lattice polarization is suf ficient-ly larger than the electronic energy bandwidth thatthe lattice distortion is localized to the immediatevicinity of the electron. Since in this regime theelectron must carry the compact lattice polarizationas it moves from site to site, small-polaron forma-tion leads to significant band narrowing as well assizable band shifts. 3

However, the total polarization in a solid consistsof two parts, the lattice polarization and the electron-ic polarization. In this paper we wish to point outthat the electronic polarizability of molecules inthe vicinity of a given electron in a molecular crys-tal leads to small-polaron effects with importantexperimental consequences. Since the electronicpolarizability of the molecules in question can beviewed as arising from virtual excitation of molecu-lar excitons (propagating molecular excited statesor Frenkel excitons), the electron together with

its tightly bound polarization cloud will be denotedas an excitonic polaron. The excitonic or electron-ic polaron was first considered by Toyozawa.The Toyozawa theory is directly analogous to con-ventional polaron theory in the weak-coupling limitwhere a small effective-mass enhancement results.However, excitonic-polaron effects should be par-ticularly important in crystals containing hetero-cyclic hydrocarbons, since these molecules areknown to have large n-electron polarizabilities.Moreover, the narrow bandwidths associated withthe relatively weak intermolecular transfer integralsfound typically in molecular crystals suggest thatthe strong-coupling limit is appropriate and theexcitonic polaron will be "small" in the conventionalsense described above.

As a model we consider a molecular crystal con-sisting of two perfect rigid-lattice subsystems: anarrow electron energy band containing conductionelectrons and a separate subsystem of polarizablemolecules whose electrons are paired in cr and pmolecular orbitals. The elementary excitationsof this polarizable subsystem are propagating ex-citons. For simplicity we consider only a singleexciton band.