I'HYSICA L RE VIEW B VOLUME ll, NUMBER 6 15 MARCH

Vibrational properties of amorphous Si and Ge~

R. Alben and D. WeaireDepartment of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520

J. E. Smith, Jr. and M. H. BrodskyIBM Thomas J. Watson Research Center, Yorktown Heights, ¹wYork 10598

(Received 10 September 1974)

Calculations of the vibrational density of states and the Raman and infrared spectra have been

performed for random-network, microcrystalline, and polymorph structures of Si and Ge. Thepolymorphs considered include Si III, Ge III, and two clathrate structures. The calculations are basedon simple semiempirical forms for interatomic interactions and Raman and infrared activities. Theresults for some representations of the random network compare favorably with experimentalmeasurements on amorphous Si and Ge. The apparent similarity of the vibrational densities of states ofamorphous Si and Ge to those of diamond cubic Si and Ge is explained by a study of the form of the

density of states for nearest-neighbor central forces. There is an interesting relationship to a simple

tight-binding theory of the electronic density of states in this limit. The variations of infrared and

Raman activities in different parts of the spectrum are discussed. Numerical calculations for energy-loss

spectra in neutron scattering are also presented. A simple model explains the oscillation of intensity

between the high- and low-frequency parts of the spectrum as the scattering vector increases.

I. INTRODUCTION

In this paper we gather together the availableexperimental data on the vibrational properties ofamorphous Si and Ge and present a theoreticalanalysis of most of the major features of the ob-servations as well as some predictions for neutronscattering. The theory is of a semiempirical na-ture. It would clearly be most inappropriate toattempt for amorphous semiconductors what is al-ready a difficult undertaking for their crystallinecounterparts, namely, the calculation from firstprinciples of vibrational eigenstates, band struc-ture, and hence the Raman and infrared crosssections which depend upon them. We choose in-stead to use a simple force-constant prescriptionfor the calculation of the vibrational spectrum andexpress the matrix elements for Raman and infra-red intensities in terms of contributions from in-dividual bonds, involving few adjustable parameters.

In addition to such a scheme for the calculationof spectra, one must adopt a specific model forthe structure of amorphous Si and Ge. Indeed,current uncertainties regarding the details of thisstructure provided much of the motivation for thisstudy, since it was hoped that vibrational proper-ties might offer a further means of discriminatingbetween rival structural models. All that is rea-sonably certain, on the basis of x-ray diffractionevidence, ' is that amorphous Si and Ge are tetra-hedrally bonded, with bond lengths equal to thoseof their crystalline forms to within about I/p anda modest spread of bond angles about the idealvalue of the order of ten deg. Beyond this, onecan only say that the diffraction data are persua-sive of the validity of the random-network model

of Polk2 and others, but not conclusive. In particu-lar, Rudee and Howie have suggested on the basisof other evidence that the structure of amorphousSi and Ge is of a microcrystalline nature, the mi-crocrystals being of the wurtzite structure, andthey claim such a model can be compatible withthe x-ray data.

In most of our calculations we employ represen-tations of the continuous-random-network structure.Since there is no unique set of coordinates whichdefines such a structure, we have done calculationson a number of network models, including those ofHenderson' and Polk, as well as models which weourselves have constructed. We feel that the de-gree of success achieved lends further support tothe network structure. We have investigated somesimple microcrystallite models and our calcula-tions will show that it would be necessary to addfurther refinements (such as large internal strains)to these in order to achieve the same degree ofagreement with experiment.

In view of the uncertainties regarding the stuc-ture of amorphous Si and Ge, it is fortunate thatthere exist well-defined metastable crystallinestructures of Si and Ge which have large unit cells.These are the so-called BC-8 and ST-12 struc-tures. ' They have already served to test ideasregarding electronic properties of the amor-phous forms, and they may similarly serve to testour approach to the vibrational problem. A pre-liminary calculation of this type has been reportedand compared with Raman data for Si III (BC-8)and Ge III (ST-12). We will present here addition-al calculations for the BC-8 and ST-12 structures,as well as results for certain clathrate structures'

11

ALBEN, %EAIRE, SMITH, JR. , AND BRQDSKY

with even larger unit cells. Experimental data onSi-Na compounds in which the silicon atoms formthese structures may soon be available.

Finally, we present predictions for inelasticneutron scattering in amorphous Si and Ge. Inthe limit of large momentum transfer, this pro-vides a direct measurement of the vibrational den-sity of states without the complication of anyweighting due to matrix elements as in the Ramanand infrared experiments. For smaller momentumtransfer we find an oscillation of scattering inten-sity between low- and high-energy loss as the mag-nitude of the momentum transfer changes by aboutl. 5 A ~ (for Ge). This behavior is well explainedin terms of the character of modes at the upperand lower ends of the spectrum, and it is ratherdifferent from that predicted for a polycrystal.

%e shall begin with a short review of past andprevailing theoretical ideas and a summary of theavailable experimental data.

II. REVIEW OF THEORY AND EXPERIMENT

A. Theoretical background

Our intention here is not to give a full review oftheoretical ideas in this area, but merely to setthe scene for the results of the following sectionsby mentioning some relevant previous work. Re-cent reviews have been given by Bell,"Dean, 'I ucovsky, "and Bottger. "

In many amorphous solids there is short-rangeorder (usually meaning nearest-neighbor coordi-nation only) similar to that of crystalline phasesof the same composition. In such cases the vibra-tional spectrum is found to strongly resemble thatof the crystal. The finding of Smith et al. ' thatthis is true for amorphous Si and Ge (see Sec. IIB)is thus typical of many amorphous solids. Mostof these are much more complicated than Si or Ge,and it is difficult, in general, to pursue the matterfurther except to say that it is indicative of theshort range of the dominant interatomic forces.The short range of such interactions would seem,however, to be a necessary rather than a sufficientcondition for the invariance of major features ofthe spectrum, and the nature of the modes respon-sible for the various features must be studied togive an adequate explanation of their lack of depen-dence on structure. The classic analysis of thiskind was that undertaken by Bell and Dean ~' forSiOp. They performed extensive numerical calcu-lations for random-network models and associatedeach of the various peaks on the vibrational spec-trum with modes of a particular local character,in terms of the bending and stretching of bonds.Even SiO&, however, is a comparatively complexsystem, and the picture evolved by Bell and Deanis necessarily less tidy than that given below for

Si and Ge. The comparative simplicity of the latterand the well-developed theory of electronic andvibrational properties of their crystalline phasesmake them ideal prototypes for the theoreticalstudy of amorphous solids in general.

The numerical calculations of Bell and Deanused special techniques" to find the eigenvaluespectrum, rather than direct diagonalization ofthe dynamical matrix. This enabled them to dealwith rather large random-network clusters of sev-eral hundred atoms. However, the advantages ofthis procedure are somewhat diminished when oneconsiders the problem of estimation of Raman andinfrared intensities, and also of identifying andprojecting out boundary-related effects. These re-quire the calculation of eigenvectors which are notdirectly given by the most efficient methods forcomputing eigenvalues. Bell and Dean thus calcu-lated only small samples of eigenveciors, and thiswas found not to be entirely satisfactory, since theintensities did not vary smoothly from one eigen-vector to the next. It is for these reasons that wehave chosen to simply diagonalize dynamical ma-trices, representing relatively smaller modelsthan those of Bell and Dean, but with more sophis-ticated boundary conditions. A lthough the develop-ment of more efficient procedures to treat largermodels along the lines suggested by Bell and Deanis, in the long run, desirable, we believe that atpresent more can be gained from studying smallerclusters with better boundary conditions.

In this regard it might be noted that prior to thework of Bell and Dean, most interpretations of ob-served spectra had been based on the study of iso-lated clusters such as Si04 (see, e.g. , Gaskell'6).This is a highly questionable approach in any sys-tem which cannot be divided into strongly boundmolecules which interact weakly. Only very re-cently has there been an effort to incorporate intothis point of view a proper allowance for the inter-actions of the molecular unit with its environ-ment '3'~7

The k =0 selection rule restricts Raman and in-frared spectra to a series of sharp lines for crys-tals, while for amorphous solids the absence ofperiodicity allows all vibrational eigenvectors tocontribute to the spectrum; so the Raman and in-frared spectra both extend over the entire vibra-tional spectrum. Of course, this statement doesnot imply that the weighting of different parts ofthis spectrum need be similar. In practice, it isoften the case that the infrared and Raman spectraare only moderately distorted by this weighting.Shuker and Gammon'8 asserted that theoretical con-siderations would lead one to expect a roughly con-stant weighting within each of the bands of which theRaman spectrum is composed. However, subse-quent developments have not given much support

VIBRATIONA L PROPERTIES OF AMORPHOUS Si AND Ge 2273

to this idea. In particular; as mentioned in Sec.VIB, the lowest-frequency bands have a strikingvariation of intensity in both Raman and infraredspectra. (See also discussion by Whalley andBertie. '

)Hass estimated infrared intensities for SiO~

on the basis of a point-charge model, and Bell andDean have made some preliminary calculations ofthe same sort, as well as calculations of Ramanintensities for SiO& using a bond-polarizability mod-el analogous to the one used here. We might notehere that, while the choice of the appropriatezeroth-order approximation for inf rared intensitiesis an obvious one in most cases, being given sim-ply by a point-charge model, in the case of a homo-polar solid such as Si or Ge this gives zero. Hereit is necessary to invent a more elaborate model,which we undertake in Sec. IIIC.

The key to several of the interpretations of theexperimental and numerical results which we shallgive is the consideration of a simplified model inwhich all but the nearest-neighbor central forcesare neglected and nearest neighbors are assumedto have exact tetrahedral symmetry. At first, thismight seem an unreasonable model for a covalentsemiconductor, since tetrahedrally bonded struc-tures cannot even be stable with respect to suchforces. Nevertheless, it does appear to be truethat the central forces are the strongest forcesat work in the determination of vibrational eigen-states. It has proved very useful, at every stage,to develop a clear picture based on these simplifi-cations, which is then perturbed by the considera-tion of the effects of the smaller noncentral forcesand distortions from local tetrahedral symmetry.

B. Experimental background

We know of at least five different experimentalmethods which have given information on part orall of the vibration density of states of amorphoussilicon and germanium. These methods are Ramanscattering, "' """infrared absorption ' ' or re-flection, low-energy electron tunneling, ~' high-energy electron energy loss, ' and neutron scat-tering. ' The most comprehensive data, as regardsthe variety of samples studied, have been obtainedby Raman and infrared spectroscopy. We shallreview the results of each of the above types ofvibrational spectroscopy on amorphous Si and Geas well as the results of studies on Si and Ge poly-morphs. "6'7

I. Raman scattering

Smith and his co-workers" ' were the firstto obtain Raman spectra of amorphous Si and Geas well as a number of related III-V amorphoussemiconductors. A direct trace of a typical low-temperature (27-K) spectrum of amorphous Si isshown in Fig. 1. A first-order Stokes Ramanspectrum is proportional to the factor n(~, T)+1,where n(v, T) is the Bose-Einstein distributionat temperature T for vibrational energy (d. Theanti-Stokes spectrum is proportional simply ton(&u, T), and so vanishes as the distribution func-tion goes to zero. At 27 K and (d&100 cmn (v, . T)«1 (as can be seen by the absence of theanti-Stokes spectrum), and the Stokes spectrumis independent of the distribution function. Thespectrum shown therefore differs from the vibra-tional density of states only by the factor 1/~ dis-

ANTI-STOKES

200 100I I I I I I I . I I I I j I . I I

300 400 500 600 700 800 900 IOOO I 100 I 200ENERGY ( crn I)

FIG. 1. Raman spectrum of amorphous Si at 27 K. The material was prepared by high-energy ion bombardment. In-strumental resolution is 10 cm . The spectrum was excited by 0.24 W of 488-nm light and the measured peak height at480 cm is 35 photons/sec {Ref. 21).

A R JR BRO

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2275VIBRATIONAL PROPERTIES 0F AMORPHOUS Si AND Ge

0.9C)~~ o.e-

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l00 200I I

500 400ENERGY (cm }}

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500I

600 700

I I I I t

200 300 400 500 600 700 800ENERGY (c~ ')

FIG. 3. {a) Transmit-tance of 37 p, m of a-Si onboth sides of a wedgedhigh-resistivity crystal-line Si substrate relativeto a matched uncoatedsubstrate. (b) Transmit-tance {solid line) of 25 pmof a-Ge on both sides of awedged high-resistivitycrystalline Si substraterelative to a matched un-coated substrate. Thedashed curve is an esti-mate of the interferencefringes in the transmit-tance {see Ref. 29).

spectrum wou e ald b that of the vibrational densityof states. Figure 2(b) gives equivalent results

846 ' The results of Wihl et al.for a.morphous e.~ ~

in ood agreement with those shown in Fag.Th lower parts of Figs. 2&aj an, scrystalline densities of states, as well all as broadened

h '' It' l th tth g 1versions of them.f the reduced Raman spectra are g&venfeatures o e r

by the crystalline density of states, apprltho h the higher-frequency modesbroadened, al oug

l stron-of the re uced d Raman spectra are relative y sger.

2. Infrared speetroseopy

The infrared absorption spectra, of Brodsky andLurio are a so s ow1 shown in Figs. 2(a) and 2(b).There is awe - ell-d fined peak in the rniddle of es ectrum w ere es h th Raman results show only ahint of structure; in addzhon, hhere are low- andhi h-energy peaks as in the Raman spectrum.zg -ene

f the experimental difficultiesHowever, because o e ein t e a,r-ih f -'Dfrared region, the relative strengt sof the three peaks are not known precise y.believe the Si data to be more reliable, the rea-

AL BEN, tIVE AIRE, SMITH, JR. , AND BRODSKY

sons for which are easily seen from the raw dataof Fig. 3. The complicating effects of multiplereflection interference effects and low absorptionin the thin-film samples distort the Ge spectrum.The Si data are clearer mainly because of thickersamples, the higher frequency range, and a fortu-itous masking out of the fringes because of thick-ness variations. In Fig. 4 we see that the ab-sorption measurements of Prettl et al. and thereflectance measurements of Stimets et al. showrelative intensities of the three peaks of amorphousGe different from the results of Ref. 29. In theabsorption spectrum (Fig. 4) of Prettl et alPB thetwo lower-frequency peaks are barely discernibleas shoulders in the high-frequency peak. In con-trast to this, the ref lectivity data of Stimets et al.have been analyzed to give an absorption spectrum(Fig. 4) with the three distinct peaks, but the twolower-frequency peaks are relatively more en-hanced than in the Brodsky-Lurio results. Forthe sake of comparison with our calculations be-low, we shall use the amorphous Si data of Fig. 2.However, the lowest-frequency peak should beregarded with some caution, in light of recent mi-crowave loss measurements '"' which suggest thatmechanisms other than lattice vibrations might beimportant for low frequencies.

The total strength of the infrared absorption inamorphous Ge and Si is substantial compared tothe usual lattice absorption of the closely relatedcrystalline III-V compounds. The integrated ab-sorption strength, expressed as 6&, the contribu-tion to the low-frequency dielectric constant, isabout 0. 5 for amorphous Si and 0. 3 for amorphousGe. Typical values for crystalline III-V's are

I.O

I

TAL0 at q=O

& Ib

I

IOOI

200ENERGY (cm ~)

I

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300 400

FIG. 5. Tunneling density of states of amorphous Geas deduced by Ladan and Zylbersztejn (Ref. 31). Here0. is the tunneling conductivity at the voltage V= S~/e.

3. Tunneling spectroscopy

I adan and Zylberstejn"'" have deduced the vi-brational spectrum of amorphous Ge from phonon-enhanced electron tunneling through a Ge barrier.Their samples were evaporated onto 77 K sub-strates, but apparently exposed to ambient tem-peratures before being cooled to 4. 2 K for mea-surement. As seen in Fig. 5, their results arequalitatively similar to the reduced Raman spec-trum. We have not made any calculations of tun-neling spectra.

4. Electron-energy-loss spectroscopy

around 4e =2. Although some workers 8'3 haveattributed the disorder-induced absorption in amor-phous III-V's to a redistribution of the crystallineoscillator strength, it is clear that in the case ofSi and Ge, where there is no first-order oscillatorstrength for the crystal, the spectrum cannot beviewed as a broadening or redistribution of process-es allowed in the crystal.

0.8—

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Schroder and Geiger' have reported that high-resolution (5-meV) detection of the energy loss of25-keV electrons can show features of the twophonon spectra of amorphous Si and Ge. Morerecently Schroder34 reported that the single-phonondensity of states is also observable by similartechniques. We make no effort to interpret theseresults.

5. Inelastic-neutron-scattering spectroscopy

0.00

I i I

1 00 200 300 400ENERGY(cm I)

FIG. 4. Comparison of the absorption constant vswave number for amorphous Ge as determined by threedifferent groups. Solid line: Brodsky and Lurio, Ref.29, em~=166 cm ~; short-dashed line: Stimets et al. ,Ref. 30, m~=164 cm; dash-dotted line: Prettl et al.. .Ref. 28, m~ = 910 cm

Axe et al. ' have preliminary results for theelastic-neutron-scattering spectrum for the low-frequency branch of the amorphous Ge. The re-sults show that the low-frequency peak in the den-sity of states of amorphous Ge is slightly broaderand shifted to lower frequency compared with thetransverse accoustic (TA) peak of the crystallineform. As we show below, the neutron technique

VIBRATIONA L PROPERTIES OF AMORPHOUS Si AND Ge 2277

should be a good measure of the entire vibrationaldensity of states, particularly if proper accountis taken of the dependence of the spectrum on mo-mentum transfer. However, our treatment doesnot give any explanation for the observed smallshift of the TA peak.

6. Polymorphs and elathrates

There exist several crystalline polymorphs ofSi and Ge. These phases are generally obtainableby various high-pressure and -temperature cycles. 'Several of the polymorphs are metastable at am-bient conditions after the completion of the pres-sure-temperature processing. In addition, thereare complicated cagelike forms of Si with smallamounts of Na which have clathrate crystal struc-tures. '0 Raman spectra of Ge III (ST-12), Silll(BC-S), and wurtzite Si (2H-4) have been report-ed ' 6' and analyzed with a model similar to thatused below for the clathrates. We have tried toobserve Raman scattering from Si clathrates andhave yet to succeed.

III. INGREDIENTS OF A THEORY

A. Structure

Our numerical calculations are, in essence,simply normal-mode calculations for various mod-els intended to represent amorphous Si and Ge. A

prerequisite for the calculation is then a set of co-ordinates defining the positions of all the atoms inthe model. It is desirable that the model containmany atoms in order adequately to represent anonrepeating structure, but it is also desirablethat the dynamical matrix, which is of the orderof three times the number of atoms (n) be smallenough to be conveniently diagonalized. The gainsin resolution, volume-to-surface ratio, and limit-ing wavelength all vary as fractional powers of ~,while the calculation cost tends to increase as ~'for large n. These conditions limited the size ofmodels to about 90 atoms, with relatively littleto be gained from considerable increase in costand/or computational complexity for larger num-

bers of atoms. Since simple clusters of 90 atomstend to have about 60% of the atoms with at leastone surface bond, boundary conditions are quiteimportant. This is especially true in calculationsof infrared intensities, as will be discussed inSec. IIIC.

We have done calculations on three types of rep-resentation of the continuous random network.These are hand-built periodic "Henderson" mod-els, computer-generated periodic-defect models,and hand-built relaxed-cluster "Polk" modelswith quasiperiodic boundary conditions. It mightbe rioted that the computer-generated cluster mod-els with relatively low distortions and no dangling

bonds have only very recently become available. "It would be most desirable if low-distortion peri-odic models could be generated by computer, butthis goal has yet to be achieved. (See, however,Ref. 40. )

The first type of model is constructed by fillinga cube, which would contain 64 atoms in a diamond-cubic arrangement, with tetrahedral units connect-ed in such a, way that (i) a, substantial number offivefold rings of bonds are present, (ii) all atomswith bonds protruding through the surface can beconnected with other atoms in the structure dis-placed by unit vectors parallel to the cube edges,and (iii) distortions from tetrahedral bonding ateach site are as small as possible. These modelscan therefore be considered as repeat'ng "rystalstructures with large unit cells. Henderson hasbuilt such a model with 61 atoms and an rms angu-lar distortion of 12.5'. We have built anothersuch structure with 62 atoms with an distortionof 14.9'. These models have much greater angu-lar distortion than the V. 1' of the Polk model be-cause the severity of the constraint of periodicboundary conditions requires considerable relaxa-tion of the requirement of small distortions fromtetrahedral symmetry.

The second type of model is derived from a dia-mond-cubic structure by a computer procedurewhich creates defects containing fivefold rings ofbonds. In this procedure an atom is selected atrandom and removed. Those atoms which had beenneighbors of the removed atom are bonded to oneanother. This reduces the number of sixfold ringsof bonds, creates fivefold rings, and also disordersthe structure in some sense. At each stage in theprocess tests are made to ensure that no fourfoldrings of bonds are created. When the desired num-ber of defects have been created, the structureis relaxed in such a way as to minimize the Keat-ing" elastic energy. Structures created in thisway have relatively large distortions and are con-sequently of little direct use, but it is interestingto compare the results which arise from them withthose of other models.

The third type of random-network model is derivedfrom the tetrahedrally bonded cluster describedby Polk. These clusters have a good deal lessdistortion from tetrahedral bonding than do theperiodic models. They present the problem of thetreatment of surface effects. Bell and Dean '

used "fixed" and "free" boundary conditions. Thefollowing quasiperiodic boundary conditions seemspreferable. Each atom with a surface bond isbonded to another such atom displaced by a vectorchosen to minimize the deviations from tetrahedralbonding. Pairs of bonds to be so connected arechosen so that fourfold rings of bonds do not occur.In this way we have created models with very low

ALBEN, WEAIRE, SMITH, JR. , AND BRQDSKY

TABLE I. Summary of structural properties of mod-els.

IOO—

50—

Model

616258Polk~

3.8%4 3%3.5%

1%

(~2g) i /2

12.3'14.9'12.5'

70

fivefoldrings/atom

0.410.420.380.38

sixfoldrings/atom

0. 900. 851.070. 90

058-

Does not include distortions associated with quasiperi-odic boundary bonds. Typical boundary-bond angulardistortions are 15'—20'. Boundary-bond distance distor-tions are constrained to be zero.

100—

50—

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UJ

~ IOO-OCQ

M

o 50

1

0POLK

IOO—

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0a- Ge EXPERIMENT

IOO—

50—

00 I 2 3

DISTANCE (BOND LENGTHS)

FIG. 6. Radial distribution functions for some of themodel structures used in the calculations of vibrationalspectra. "61-atom" is a periodic hand-built computer-refined structure due to Henderson; '$2-atom" is a sim-ilar structure built as part of this work; "58-atom" is aperiodic defect structure derived from diamond cubic;"Polk" refers to low-distortion nonperiodic models ofthe type studied by Polk. (See Table I for distortionsand ring statistics of these models).

interior distortion, and distortions at the surfaceonly somewhat higher than for periodic models.For properties which are particularly sensitive tothe distortion, which include infrared absorption,it is still desirable to count only contributions fromthe internal part of the cluster (see Sec. IIIC).,

A comparison of some of the models for whichvibrational properties were computed is given inFig. 6, where we show radial distribution func-tions, and in Table I, where we indicate bond lengthand angle distortions and ring statistics for rep-resentatives of the three types of structure.

B. Force constants

The phonon dispersion relations of the crystal-line group-IV semiconductors have been investi-gated with extraordinary thoroughness by meansof neutron scattering, both on and off the princi-pal symmetry directions. Various semiempiricalmodels have been used to interpolate and interpretthe data. In all of these, nearest-neighbor centralforces are the largest forces involved. If smallershort-range forces which resist angular distortionsare also included, one obtains a qualitatively sat-isfactory spectrum. If one demands a very highdegree of agreement with the experimental data. ,it may be necessary to include quite distant inter-actions, as Herman showed. This should not,however, be allowed to obscure the essential sim-plicity of the phonon dispersion relations in thesematerials.

If simple forces resisting angular distortionsare to be incorporated in a force-constant schemethere is some arbitrariness in the choice of theform of such forces. The prescription of Keating 'is to use a term in the potential energy of the form

l(ul ul~) r ~-(~)+(ut-u ) r (~))' (&)i6 ~ gg&}

to represent such bond-bending forces. In addi-tion, there is the central-force term, which maybe written

VIBRATIONA L PROPERTIES OF AMORPHOUS Si AND Ge 22'79

where the expression in square brackets is thecompression Cn(l) of the bond 4 of atom /. Here(and throughout this paper) e. and P are bond-stretching and bond-bending force constants, re-spectively; the sums are on atoms E a,nd theirnearest neighbors 6; rn(l) is the unit vector fromthe equilibrium position of atom l to that of itsneighbor lb, ; u, and u«are the displacement vec-tors of these atoms. Equation (1) may be derived,in the harmonic approximation, from the assump-tion of a potential-energy term which depends onlyon the sealer product of nearest-neighbor vec-tors. ' The alternative "valence-force" model 'is derived from the replacement of this scalarproduct by the corresponding interbond angle. Yetanother choice is that of the Born model, 46 in whichthe potential energy is

PBorn g [(~& ~& ) .P (t)]&ur u

Born Born g (u u )24

Somewhat confusingly, n and P are also conven-tionally used here, but their meanings are not thesame as in the Keating model —hence our designa, -tion "Born. "

A. Priori, there is not much to be said in favorof any of these except that the first two, being ro-tationally invariant, avoid certain possible pitfallsinherent in the Born model. Martin4 has arguedin favor of the Keating prescription, on the groundsthat the identity which it implies, relating the threeelastic constants of the diamond-cubic structure,is close to being satisfied by the experimental data.The Keating model is also attractive in tha, t it isslightly more economical for computation than thevalence-force model. We have chosen this pre-scription for the calculations presented here butdo not wish to belabor the distinctions between ita,nd the others, which are of little significance atthe level of detail with which we are concerned.

By the same token the details of the phonon dis-persion relations, such as the flattening of the TAnodes near the zone boundary, which have beenmuch discussed in terms of more forces of longerrange, such as are incorporated in the shell mod-el ' and the bond charge model, ' as well as exten-sions of the above formulations, will not concernus here.

However, the existence of such forces must beborne in mind in the interpretation of' our results.As well as influencing the shape of the spectrumand the character of the modes directly they mayalso be reflected in the structure itself. Most

model structures have been refined by the minimi-zation of the energy without regard to these moredistant, forces. It may be argued, for instance,that the small width of the second-nearest-neigh-bor peak in the RDF of the Polk model, relativeto the experimental RDF, is due to the neglect ofsuch forces.

C. Matrix elements

Having calculated the vibrationa, l modes of a givenstructure with suitable force constants, it remainsto assign infrared and Raman activities to thesemodes.

Both activities may be written as complicatedintegrals over the electronic sta, tes of the valenceand conduction bands. "' However, even in crys-tals where the calculation of such electronic statesis relatively straightforward, such a first-princi-ples calculation is a formidable undertaking. InIn the case of diamond cubic Si and Ge, for whichthe k =0 TO modes are Raman active (and not in-frared active), only Swanson and Maradudin" haveattempted such a calculation to date.

The expressions which we will use to calculateRaman and infrared activities are based on a, local-ized point of view, in which these quantities (orrather, the transition probabilities associated withthem) are written as sums of local contributions.Short-range order and local symmetry can then beused to restrict the allowed forms to a few possi-bilities, which may be weighted with adjustablecoefficients. Presumably the Wannier-functionformalism is the key to the relation of such a pic-ture to that which involves integrals of matrixelements of extended wave functions. While onemay appeal to this in principle, it is difficult tofollow through in practice, and we appeal ratherto the general success of theories based on thelocal picture, as practiced by Phillips' and othersin recent years, as a justification for such an ap-proach. A resort to the band picture might be nec-essary if, for instance, any significant dependenceof the Raman spectrum on the exciting frequencywere to be found. To date, no such dependence hasbeen reported. It should be noted, however, thatthe frequencies which have been used are ratherclose together.

The transition probability whose square deter-mines infrared activity is proportiona, l to the dipolemoment M associated with the vibrational mode.If we write the displacement of atom / in a givenvibrational mode as u, then, to first order, thedipole moment is a linear function of u, . In keep-ing with our above remarks, we may try to ap-proximate this function by a, sum over all of thenearest-neighbor bonds of the system. However,if it is further assumed that each bond can be treat-

2280 ALBEN, WEAIRE, SMITH, JR. , AND BRODSKY

FIG. 7. Mechanism for infrared activity. During avibration, bond charge moves from extended to com-pressed bonds, resulting in a local electric-dipole mo-ment M. The local moments cancel for modes of thediamond cubic structure, but they do not cancel in amor-phous structures.

ed as having inversion symmetry about its center,as is reasonable for a homopolar material, andthat a rigid translation of the bond can give no di-pole moment, such a description will give identi-cally zero for the induced dipole moment. Tosee this, consider the dipole moment associatedwith a given bond, which, by the second of theabove conditions, may be written

M= T (u, -uz).

Here u., and u~ are the displacement vectors of thetwo atoms joined by the bond, in some given vibra-tional mode. Under inversion, the vector uponwhich T operates does not change sign, but Mmust do so, and this is incompatible with the invari-ance of T, unless T —= 0. This is the same argu-ment as that applied by Lax and Burstein' '" tothe diamond-cubic structure, but is here appliedto a single bond. See also Zallen" and Chen andZ allen. '6

Clearly it is necessary to consider contributionsfrom more than one bond at a time, if a nonzeroinfrared activity is to be obtained. The obviouschoice is take pairs of neighboring bonds. Denotingeach atom by / and its nearest neighbors by l~, wewrite

part of a fully bonded cluster, as is desirable insome cases where the surface termination is notfully satisfactory, we obtain large contributionsfrom the surface, due chiefly to certain termswhich identically cancel when summed over a fullybonded system. These terms are precisely theones which are set aside when (5) is reexpressedas (Al).

We have not explored any further forms of theinfrared activity, since (5) gives quite good resultsby itself.

The Raman activity of a mode is given by theinduced polarizability. In this case, therefore,we are dealing with a second-order tensor whichis a linear function of the displacements u&. Againwe write this as a sum of contributions from in-dividual bonds, imposing invariance with respectto translations and inversion. Even if we furtherrestrict the range of possibilities by the imposi-tion of cylindrical symmetry about the bond, sothat each bond is to be treated precisely as a homo-polar diatomic molecule, there remain three in-dependent forms for the polarizability:

1 ZIr~(I)r~(I) —-' I lui r~(I),

+2 g LZ (rE (I) ul + ul rA (I)1 3 I l ul rk(I )ld,

(7)

os = Z I ur ' r~(I)pl4

where I is the unit dyadic. By collecting termsreferring to pairs of neighbors, we may note thatthe first and third expressions depend only on thebond compressions C~(l).

It is difficult to attribute relative degrees ofimportance to these expressions a pro~i since,as we noted in the Introduction, the first-principlesexpression for Raman activity is so complicated.However, there is one important distinction be-tween the first expression and the other two. Thelatter vanish in the case of perfectly symmetrictetrahedral bonding, since in that case

Qr, (l) =0. (9)

(5)

This formula may be interpreted as representingthe dipole moment due to the transfer of chargefrom extended to compressed bonds (see Fig. 7).It may be reexpressed in various other forms. Oneof these, which is given in Appendix A, is especial-ly convenient for the qualitative understanding of(5) and also for certain calculations. This isbecause if we attempt to use (5) for an isolated

The first expression, on the other hand, has aquadratic dependence on the bond vectors r~(l)and hence does not vanish. If (9) were to bevery small in a given structure, clearly the ex-pressions (7) and (8) would be expected to contrib-ute little to Raman activity. However, for ran-dom-network structures of interest, the quantity(9) typically has magnitude 0.2 times the nearest-neighbor distance, and n& and n, cannot be ne-glected on grounds of approximate symmetry.

VIBRATIONAL PROPERTIES OF AMORPHOUS Si AND Qe 2281

z'LLI

bJCLXLLJ

O

I

O

I

CU

DENSITY OFSTATES

'INFRARED RAMAN

I I I r r r t I I I I II I r I I I I I

I I I I I I r I r t I I I I I I I

I I I I r I I I I I I I II I t I I I I I I it I I l I I 1 I I I I I I I I III I I I li

I I I I I I I I I l I I I I I l I I l l I I I I I I I I / I I j I l I l I I I l fI I I I I I

I I t I I I I I I I I II ~ I I I r I 1 I r I I I I I I I I I I I 1 I I r r « ~ « I r I I I I I I I I I I I I I I

I I I I I I

I I I I t I I l I I I l I I l I I I I r I lrl I I I l I

0 200 400 600 0I

200 400 600 0ENERGY {cm I

200 400 600

I J]) I I I I I I ' I I I I I I I I I I r I t I I I I I ' I I I I I I I I Ir I « t rr t I I I r I » I I l ) I ««« I I l l I I[] t r I

FIG. 8. Theoreticalresults for density ofstates, infrared absorp-tion, and Raman scatter-ing for 61- and 62-atomperiodic models comparedwith the experimentaldata of Fig. 1 on amor-phous silicon. The infra-red absorption mechanismused in the theory is thatof Eq. (5). The Ramanmechanism is given inEqs. (6)-(8) with the rel-ative weights described inthe text. The position ofthe maximum frequencymode for diamond cubic(the Raman-active TOmode) is indicated by thedashed line on the Ramangraphs; the position ofthe density-of-states min-imum near the center ofthe spectrum for diamondcubic with the force con-stants used in these cal-culations is indicated bythe arrow at 380 cm ~ onthe infrared-theory re-sults. Spectra are formedby adding weighted Lor-entzian contributions fromk=0 modes and are nor-malized to the same max-imum value.

IV. NUMERICAL RESULTS

A. Results for hand-built periodic-random-network models

Figure 8 shows the results of a calculation usingHenderson's periodic 61-atom random-networkmodel and our 62-atom model and the theory ofSec. III compared with experimental data for Si.The force constants used are n =0.475@10dyn/cm, P/n =0.2. The choice of P/n is some-what arbitrary, in view of the simplicity of thisforce-constant model. Values in the range 0.15-0. 35 are consistent with various aspects of thephonon dispersion relations for the dia, mond-cubicstructure. Keating's values, which were fitted toelastic constants, should not be assumed to haveany universal validity.

The above value of 8/o was chosen at an earlystage in this investigation and was not variedthereafter. The position of the lowest peak in thespectrum depends on P in an essentially linearmanner (as we shall show in Sec. VIA). R is cleartha, t the theoretical. peaks could be brought intoslightly better agreement with the experimental

ones by a small adjustment of P/n, but this wouldhardly add much to our understanding. Havingfixed P/n, the value of It'. was fixed by fitting thezone-center TO frequency calculated for the dia-mond-cubic structure (indicated by a, dashed verti-cal line in Fig. 8) to experiment. Another signifi-cant frequency, as will be discussed below, cor-responds to the minimum in the LA-LO density ofstates for the cubic structure. This is indicatedby the arrow in the result for the infrared spectrumof the 61-atom model.

The weighting of the various contributions tothe Raman activity was determined as follows.The third mechanism is associated with a depolari-zation ratio of zero. The fact that the observedspectrum was found to have a similar shape in theH~ and ZV ""' configurationsa"" with a depolari-zation ratio of 0.8a 0. 1 then precludes any signifi-cant contribution from 0. 3, and it was accordinglygiven zero weighting. This leaves (apart fromthe over-all scale) only the ratio of the coefficientsof e, and e& to be determined. By itself, e, pro-vides a reasonable description of the upper half ofthe spectrum, but gives too little activity in thelower half. The inclusion of e& remedies this de-

ALBEN, WEAIRE, SMITH, JR. , AND BRODSKY

DENSITY OF STATESI

INFRARED

O

I

(0

O

I

FIG. 9. Theoreticaldensity-of-states, infra-red, and Baman resultsfor defect models. Themaximum bond-length dis-tortion for the 58-atommodel is 10%. For the55-atom model it is 30%.Infrared and Bamanmechanisms are the sameas for Fig. 8.

0 200 400 600 0 200 400 600 0 200 400 600ENERGY (cm )

ficiency. The ratio 3. :3 for the weighting of thetwo expressions was used in the calculations ofFig. 8.

In Sec. VI we shall give an interpretation ofthese and other results, i.e. , an explanation ofhow these results arise from the assumed formsfor force constants and activities. They play aprimary role in our interpretation of the experi-mental data, .

B. Other random networks

In Fig. 9 we show results for density-of-states,infrared, and Raman spectra for defect models.

These spectra show certain sharp structures (e.g. ,the splitting of the lowest-frequency peak) whichare associated with k=0 modes of 64-atom diamond-cubic cell from which the models were derived.Nonetheless, it appears that there is a third peakin the center of the spectrum and a correspondingenhancement of the ir scattering from this peak,as for the hand-built model.

In Fig. 10 we show results for quasiperiodicmodels. There is a, difficulty in obtaining mean-ingful ir spectra for such models, since a fewstrongly scattering regions with large distortionscan dominate the spectrum. (The ir contribution

O

I

CO

OI—

I

g)

200 400 600 0 200 400ENERGY (cm i)

600 0 200 400 600

DENSITY OF STATES INFRARED RAMAN[TT i FIG. 10. Theoretical

density-of-states, inf ra-red, and Baman resultsfor 85- and 92-atomquasiperiodic Polk mod-els. These models havevery low interior distor-tions, but large effectiveangular distortions at theboundaries. The Bamanmechanism for these cal-culations was taken as asum of the mechanisms ofEqs. (6)-(8) with equalcoefficients. Only the HHspectrum is shown. Theinfrared mechanism isthe same as for Fig. 8.Spectra are formed fromcontributions of the nor-mal modes of the finiteclusters.

VIBBATIONAI PROPERTIES OF AMORPHOUS Si AND Ge

0

DENSITY OF STATES INFRARED

xi

400 600 0

x

200 400ENERGY (cm I)

I I II

I I I) I I I

t I I II

I I II

I I I/ I I Ill II I I II

I I II I I II I I I

RAMANI I j

I I I

II I

tI I I

II I &

jI I I

j I I If

' I I

(I I

I

x, )

Wl—I

IIjl

I

I

I

I

I

I

"r, . ;i. . . ~

600 0 KG 400

FIG. 11. Theoreticaldensity-of-states, inf ra-red, and Raman resultsfor a microcrystallitemodel consisting of astrain-f ree sphericalcluster of the wurtzitestructure containing 64atoms. Calculations forfixed (connected points)and free (Ã's) boundaryconditions are shown.The Raman mechanism isthe same as for Fig. 8.

from a given atom is proportional to its devia, tionfrom tetrahedral symmetry. See Appendix A. }%e attempted to minimize these effects by weight-ing the contributions of modes to the spectra (andthe density of states) by their projection on the 36fully bonded interior atoms in the cluster, i.e, ,by the sum of the squares of the components u,associated with these atoms. This reduces theeffects of boundary-condition-induced modes. How-ever, this was judged still to be unsatisfactory,and calculations mere therefore performed in whichonly the interior bonds contributed to the infraredmatrix element. This is done by using the alter-native form (Al) and summing only over interioratoms. [Use of the original form (6) would leadto difficulties in this case. ]

Qf significance is the relative sharpness of theuppermost peak of the density-of-states and Ramanspectra compared with the periodic models. Thisappears to be associated with the lower angulardistortion for the interior bonds of this model.The experimental results, interestingly enough,seem to be in better agreement with the more dis-torted models.

C. Microcrystals

Microcrystallite models have been proposed toexplain certain aspects of electron diffraction onamorphous Ge. A microcrystal presumably is aregion of well-ordered material separated by anidentifiable boundary from other mell-orderedregions. (By contrast, the random network wouldbe regarded as structurally homogeneous. } Be-cause of the boundaries, a small microerystallitewould not rigorously obey the usual k selectionrule in Raman scattering or infra, red absorption.To investigate this we examined spectra, fromapproximately spherical 64-atom samples of dia-mond or wurtzite-structure models. These mod-els represent quite small microcrystals and havemany surface bonds. Even so, the Raman scatter-ing seems to indicate a very sharp crystallike TOpeak (see Fig. 11). To broaden this peak, it ap-

pears thai internal strains comparable with thosein the random-network model would be necessary.

Note that, according to the arguments of Ap-pendix A, the infrared activity calculated here fora microcrystal arises entirely from surface con-tributions.

D. Polymorphs

TABLE II. Calculated spectra for the 136-atom clath-rate (silicon). Lines with relatively large intensity areunderlined.

Energy(cm"~)

Relative Intensities

Type Infrared

137142194232236278279321390401418435452491492499504505507510510517

~e

~2eAgegF)Age

F2

F2eAgeg

F2&e

0.0030.003

8.907

0.315

0.6150.012

0.051

0.0000.000

0.000

0.000

0.002

0.036

1.149

0.006l.442

1.1160.0000.176

0.0060.020

0.000

0.018

0.000

0.000

0.000

0.0000.002

0.DOS

0.0000.000

0.182

0.032

0.015

In Tables II-V me give the mode identificationand intensities for the infrared and the three Bamanmechanisms of Elle. (6)-(8) for the BC-8, ST-12,136-atom, and 46™atomclathrate structures.(The numbers refer to the number of atoms percubic cell. ) The Raman mechanisms are given

2284 ALBEN, WEAIRE, SMITH, JR. , AND BRODSKY

DENSlTY OF STATES lNFRARED RAMAN

—II—I

LLI

K

O

O

CO

200» tli»li

600 0 200 400ENERGY (ce-i)

I I I I I l~ I LL i I I

600 0 200 400l

600

FIG. 12. Theoretical density-of-states (of 138 k=0 modes), infrared, and Raman spectra for a clathrate structurewith 46 silicon atoms in the unit cell. Mechanisms are the same as for Fig. 8.

separately, since it is not at all clear that theweighting of various terms used in Sec. VA shouldapply to these structures. Both infrared andRaman intensities are divided by the number ofatoms in the unit cell and multiplied by a, constantwhich is the same for all structures. It might

TABLE III. Calculated spectra for 46-atom clathrate(silicon). Lines with relatively large intensity are under-lined.

be noted that in these structures there is a dis-tinct tendency for the Raman intensities to be high-est for the highest-frequency modes, while their intensity is highest in center of spectrum. Thisbehavior is similar to that found for random-net-work results, we give in Fig. 12 a spectrum ofHaman and infrared intensity as well as densityof k =0 modes for the 46-atom clathrate.

E. Other force constants

We have also done calculations on random-net-work models with Born-model force constants. Re-sults for the Henderson 61-atom model are shownin Fig. 13, where it may be seen that there is no

really significant difference from the Keating force-constant results, except for expected slight shiftsin the relative positions of the various peaks.

Energy(cm ')

120153168201206226231240312317332355398435

450455488493496497502506509513523536

Type Inf rared

0.001

Ee

F(

FFiu

FeAge

Age

F2e

F2e

Age

F&u

Ee

0.042

0.0026.627

14.114

4.971

0.801

0.015

0.105

Relative intensities

0.0000.000

0.0080.003

0.0000.000

0.0020.030

0.0040.003

0.0120. 000

0.0000.002

0.0030.010

0.0180.020

0.4320.5760.5461.587

0.0030.0180.0060.003

0.574 0.002

0.124 0.002

0.411

0. 020

0.024

V. INTERPRETATION OF EXPERIMENTALAND NUMERICAL RESULTS

TABLE IV. Calculated spectra for Si III (BC-8).Lines with relatively large intensity are underlined.

Energy(cm ~) Type Infrared

Relative intensities

200228439440454480.529

F2

FtaAgeF2+e

0.039

110.274

0.000

0.003

3.621l.18

0.063

0. 147

0. 1000.206

2.327

A. Density of states

We sha, ll begin an interpretation of the experi-mental and theoretical results of the previous sec-

VIBRATIONAL PROPERTIES OF AMORPHOUS Si AND Ge

H6l MODEL WITH BORN FORCES

DENSITY OF STATES INFRARED RAMAN((I I! I(/ I( I

II(&/ II I

/

I!!((!I[(I I | I I!/ I &!/ Il([((I /( II[

I III

I I Ii (I(I ((!/(I I/

I l (III!([IlI ~

0(( I I I I (1( I (

200 400FNERGY (crn )

lIil600 0 200 400

l .

600

FIG. 13. Theoretical results for density-of-states, infrared, and Raman spectra for the 61-atom model with Born(rather than Keating) force constants. Here we have taken Pjnz ~-—0.6.

tions with an examination of the relationship be-tween structure and the vibrational density of statesN(~), and give an explanation of the similarity ofN(&u) for the amorphous and crystalline phases,discussing not just the two prominent peaks, butalso some more subtle structure in the center ofthe spectrum.

Figure 14 shows a histogram of the vibrationaldensity of states for diamond-cubic Si derivedfrom the Keating model with the same c. and P

parameters used for previously discussed struc-tures. This simple model is known to be slightlyinaccurate in the region of the TA peak, ' but thislevel of detail hardly concerns us here.

TABLE V. Calculated spectra for Ge III (ST-12).Lines with relatively large intensity are underlined.

Relative intensitiesTO

Energy{cm ) Type B

7097

101104108115122129167184204209218245250252255259274293299303304315316

B(

A2

Ag

82EA(BgA2

B2Ag

B(B(A2

B2

A2EA)BgBg

79.6740.6380.249

0.4770.050

0.0040.0306. 8220.065

0.0060.0460.0010.0070.0090.915

0.0940.0891.3330.030

0.0900.5080.0060.000

0.0010.000 0.0103.765

0.0020.000

0.007 0.0000.0000.000

0.0010.214

0.0640.0000.0680.1430.000

1.026

1.319

0.0000.0160.0010.030 0.3540.0000.000

0.0010.0000.0120.046

0.0040.007 0.0520.0000.000

LA LO

0I

0.5~~~max

1.0

FIG. 14. Density of states for the Keating model withP/e =0.2 for the diamond cubic structure. Histogram isformed from frequencies for 4000 randomly chosen kpoints.

2286 AL BEN, Vf E AIRE, SMIT H, JR. , AND BRODSKY

TA TO

LA LO

200 400ENERGY (cm-I)

600

FIG. 15. Density of states for the Keating model with

p/n =0 for the diamond cubic structure. 6 functions at~ = 0 and Sn are represented by finite blocks.

The most prominent features of the density ofstates are the peaks at low and high frequency,arising from bands which are conventionally la-beled TA and TO, respectively. The terms "trans--verse, " "longitudinal, " "acoustic, " and "optical, "as usually defined, are only meaningful in a peri-odic system. It is therefore inappropriate to usethem in an amorphous system. One must insteadconcentrate on the bond-bending or stretchingcharacter of the modes, as Bell and Dean have em-phasized, "' if one wishes to develop a classifica-tion of modes which may have relevance to theamorphous case. One way of seeing the essentialcharacter of the modes is to take the limit in whichthe forces resisting bond bending (i.e. , changesin interbond angles) go to zero, P/n -0 in the no-tation of Sec. IV. The appropriate values of P/nfor a description of the observed density of statesare sufficiently sma, ll that this limit does not takeus very far from reality. Figure 15 shows thenature of the density of states of the diamond-cubicstructure in this limit. (We shall shortly explainthe manner in which this was obtained, preferringfor the moment to emphasize qualitative featureswhile postponing related mathematics. ) The TAand TO bands are completely flat and hence give6 functions in N(u&). The TA modes are now at~ =0. The existence of N zero-frequency modes,where N is the number of atoms, is easily demon-strated since the system has 3N degrees of free-dom, and the requirement of constant bond lengths(and thus no change of elastic energy, if P/n =0)constitutes 2N linear constraints; so there are 3N—2Nsuch modes. These are pure bending modes.The origin of the second 6 function is more obscure,since it is necessary to invoke tetrahedral sym-metry to demonstrate its existence, as we shallshortly describe. However, examples of bothkinds of modes can easily be set up on a closedring of bonds, as shown in Fig. 16.

Note that the simple arguments given above re-

garding the existence of zero-frequency modes andthe explicit construction given in Fig. 16, do notdepend on the details of the diamond-cubic struc-ture. All that is necessary for the existence of thefirst 5 function is fourfold coordination, while forthe second one tetrahedral symmetry is also nec-essary. The lack of dependence of these featureson the topology of the structure is expressed in thefollowing theorem, which also bears on the formof the rest of the density of states.

Theorem. The density of states for the case ofequal central forces (P/n =0 in the case of Keatingmodel), expressed as a function of m&u, has thefollowing form for a homopolar solid with perfect-ly symmetrical tetrahedral bonds. There is a 6function containing one state per atom at zero, an-other at 8n, and a band bounded by these values.Furthermore, this band is given by the spectrumof the simple Hamiltonian

(10)

which operates on (scalar) basis functions I l),one of which is associated with each site. s~

A proof of this theorem has been outlined by%eaire and Alben. " A more economical versionwas given by Alben et al. , the details of whichare contained in Appendix B. The essential step

FIG. 16. Examples of localized modes, at frequencies~~ =0 and Se, which may be formed on any closed ringof bonds if there is perfect local tetrahedral symmetryand central nearest-neighbor forces. The displacementvectors are in the plane of the bonds which belong to thering and perpendicular to their bisector.

VIBRATIONAI PROPERTIES OF AMORPHOUS Si AND Ge 2287

TABLE VI. Corresponding approximations for theelectronic and vibrational problems.

Electronic density of states

Realistic Hamiltonian (difficult)

Simple tight-binding Hamiltonians

Allow only short-range interactions

Keep iteration parameters Vf andV2 fixed at the same valueseverywhere

Vibrational density of states

Realistic Hamiltonian (difficult)

Simple force-constant Hamiltonians

Allow only short-range forces;in fact, only central nearest-neighbor force constant ct

Keep ct fixed and treat all bondangles as cos" (3)

Equivalent problem:simple Hamiltonianone basis functionper site (plusadditional gfunctions in thedensity of states)

is the projection of the -displacement vector u, alongeach bond l4. The projections thus defined forma new basis set of four (scalar) functions per atom.The analogy with tight-binding theory is at onceapparent (four sp orbitals per atom in this case).However, the four projections of u, along the bondsl4 are not independent. For the case of tetrahe-dral symmetry, the required constraint is simplythat these projections add to zero. Translatingthis into tight-binding language, we have the condi-tion for purely P-like wave functions. Thus byforcing wave functions to be P-like (which is ac-complished by making the weighting of an appro-priate part of the tight-binding Hamiltonian go toinfinity and discarding the s-like part of the spec-trum) we can make a precise analogy with the formu-lation of simple tight-binding theory by Thorpe andWeaire. ' The above theorem then follows fromthe "one-band-two-band" theorem of the latterauthors, which relates the band structure for sim-ple tight-binding Hamiltonians to that of (ll) to-gether with various 5 functions. The two 5 func-tions correspond, respectively, to p-antibondingand p-bonding functions.

It is amusing that these two formalisms should,when peeled down to bare essentials, reduce to thestudy of the same Hamiltonian. The steps involvedare indicated in outline in Table VI.

The above theorem is exact only for perfectlysymmetrical tetrahedral bonding, such as in thecase of diamond cubic (and was indeed used to pro-duce Fig. 15 from the one-band spectrum60). How-ever, we may expect it to apply in an approximatesense to structures which are only close to thiscondition, such as the random network. Remem-ber that the w =0 5 function would still exist in thiscase, according to the argument given earlier inthis section, but the other one would be broadenedby departures from exact tetrahedral symmetry.In addition, of course, both peaks have a finite

I I I] ( I I

tI I I

i I I I i I I I i I I I i I I

t

I

I

I

1

=0.05A

I ~i

I

200 400ENERGY (cm ')

600

FIG. 17. Density of states N((d) for the 61-atom peri-odic model for P/a. = 0.15 and 0.05, illustrating thechange in the spectrum as P 0.

width for finite P.The picture evolved thus far is as follows. Any

perfectly tetrahedral structure should have sharppeaks at low and high frequencies in the sameplaces as the diamond-cubic structure, but if thetetrahedral bonding is not perfectly symmetric theupper one will have an additional broadening. Therest of the spectrum (which we shall discuss be-low) is contained in the region between the twopeaks.

These conclusions are in good agreement withthe results of Sec. V. In particular, we noticethat the upper peak does broaden as distortions areincreased in the structural models used for calcu-lations.

A further calculation which may aid in under-standing the above remarks is shown in Fig. 17.This shows that as P- 0 the lower peak becomessharper, being roughly proportional to P in width,while the upper one remains much the same, sincemost of its width arises from distortions (see alsoFig. 2 of Ref. 58). The energy of the lower peakalso tends towards zero as P-0. Incidentally, theasymmetry of the effects of distortion on the twopeaks is somewhat disguised by the linear scale for& which is traditionally used in these plots. Thenatural scale for a theoretical analysis is one pro-portional to the frequency squaI ed (because this isthe eigenvalue of the secular equation). Plotted onsuch a scale, the density of states for the diamond-cubic structure becomes exactly symmetric aboutits center for any force-constant scheme, such asthat of the Born model, which involves only near-est-neighbor forces.

2288 ALBEN, %ZAIRE, SMITH, JR. , AND BRODSKY

6I ATOM - PERIODIC 62 ATOM- PERIODIC

I

2I

0I

-2I

0

I

-2I

0

85 ATOM-CLUSTER

20I ATOM -CLUSTER

I

-2I

0

92 ATOM- CLUSTER

FIG. 18. One-banddensity of states for 61,62, 85, 92, and 201-atommodels. His tograms givethe distribution of eigen-values for k = 0 modes for61- and 62-atom periodicmodels and for the finitenumber of modes of the85-, 92-, and 201-atomcluster models. The 201-atom cluster result isprobably representativeof the (unbounded) Polkcontinuous-random-net-work model.

I

-2I

0

The central part of the density of states is notindependent of topology, the particular form shownin Fig. 15 being for the diamond-cubic and related 'structures only. According to the theorem dis-cussed above and the work of Thorpe and Weaire,this part of the density of states is closely related[through the one-band Hamiltonian (10)] to the low-er half of the valence band of electronic states.The structure dependence of the latter has been ofsome interest. Thorpe and Weaire predicted thatthe two-peaked structure found in the lower half ofthe valence band of the diamond-cubic phase of Sior Ge would be replaced by a single broad peak orplateau in the amorphous phase. This was subse-quently confirmed by x-ray emission, 6 by x-rayphotoemission spectroscopy (XPS),~4 and by uv pho-toemission, 65 but the force of the original argumentwas somewhat diminished by numerical calcula-tions of the density of states66 of the one-band Ham-iltonian. These showed that the topological disor-der of the Polk model is not, after all, quite suffi-cient to give a featureless spectrum, but rather

one with a reduced, but still quite pronounced, dipin the center (see Fig. 18 for one-band results forvarious random-network structures). The explana-tion of the experimental results is therefore notconclusive, as was originally thought. It is inter-esting to ask if significant structure can be dis-cerned in the corresponding part of the vibrationaldensity of states. Does either theory or experi-ment for the amorphous phase contain the dip foundbetween LA and LO peaks in the density of statesof the crystal'? The answer in both cases appearsto be yes. This cannot be stated unequivocally onthe basis of the experimental data alone since, ifwe examine the infrared data, it is not possible tosay whether the dip which precedes the highest-frequency peak corresponds to the LA-LO splitting,which is what we have in mind, or the Lo- TO split-ting. However, if we examine the numerical re-sults of Sec. V, which agree rather well with theexperimental data, we can decide this, since thedip occurs almost exactly at LA-LQ dip frequencyof the corresponding crystal spectrum. Also, al-

VIBRATIONAL PROPERTIES OF AMORPHOUS Si AND Ge 2289

most exactly half of the modes lie below the dip inwhich we are interested. This test strongly sup-ports the former of the two possibilities consideredabove. Thus the three peaks in the density of statesindicated by the observed ir and Rarnan spectra areconsidered to correspond, respectively, to modesof local character similar to the TA, LA, and LO+ TO bands for diamond cubic, the LO and TO hav-ing merged because of the effects of distortions(see above). This unfortunately piles paradox uponparadox, since it is a result opposite to that whichis found in the corresponding electronic measure-ments discussed above. A potential explanation,however, may be provided by matrix-element ef-fects. As we shall see, these amplify consider-ably the effect of the middle peak inthe ir spectrum.It is possible that other effects suppressthealreadydimi. nished (compared to the crystal) structure ex-pected on the basis of the one-band results for theelectronic problem.

B. Matrix-element effects

All of the immediately preceding analysis is re-lated to the vibrational density of states. We shallnow consider how this is modulated by matrix-ele-ment effects to give the Raman and infrared spec-tra. We cannot claim to give any final explanationshere because of the semiempirical approach usedin the calculations, but nevertheless it would appearthat much can be learned from them.

Again it is fruitful to consider the limit of purelycentral forces. In this limit the mean-square com-pression of bonds divided by the mean-square dis-placement varies as &~ over the entire spectrum.(See Appendix A. ) Now the domi. nant contributionto the Raman tensor z, involves only the compres-sion of each bond. If we neglect the correlationsof compressions on different bonds, we are led toexpect an +' dependence of the Raman activity.(The other contribution a2, which contributesmainly at low frequency, cannot be so written. } Wecan thus attribute most of the over-all increase ofRaman activity with &, apart from the low-frequen-cy region, to the approximate &~ dependence asso-ciated with the mean-square compressionof bonds. 7

For the infrared spectrum a precisely similarargument can be used to relate the activity to themean- square bond compre ssion and hence to ~2,but clearly this is not in keeping with the resultsof Fig. 18. The uppermost peak has a much loweractivity than one would expect on such groundsalone. Indeed, here lies the principal qualitativedifference between the infrared and Raman spectra.In Appendix A we give a proof of a theorem whichexplains the diminished contribution of this peak.It is shov', ~ that in the limit of purely central forcesthe matrix element (5) is of second order in the dis-

VI. PREDICTIONS FOR NEUTRON SCATTERING

Inelastic neutron scattering at high-momentum-transfer vector Q has been advocated as a meansof directly studying the phonon density of states ofpolycrystalline and amorphous elemental solids.In the case of polycrystalline systems, this is justi-fied by an argument to the effect that the experi-

100 200 300 400 500 600ENERGY (cm ')

FIG. 19. Frequency dependence of the average inten-sity per mode for infrared absorption (61-atom model),as determined by dividing infrared absorption by densityof states (see Fig. 8).

tortion from exact tetrahedral symmetry for themodes which contribute to this peak, while it is offirst order for the spectrum as a whole. This ten-dency for the infrared activity to be greatest formodes near the center of the spectrum is seen inFig. 19, where we have plotted the dispersion ofthe infrared activity per mode.

It is sometimes said that the differences betweenthe two spectra can be simply understood by appealto the results for the diamond-cubic structure, forwhich the TO k = 0 modes are Raman active and notinfrared active. This is far from obvious, sincethe latter statements follow simply from crystalsymmetry without regard to the forces or matrixelements involved. The interpretation of the datafor amorphous Si and Ge clearly requires consid-eration of both forces and matrix elements in somedetail. If, for instance, ~~ and ~3 were the domi-nant forms of Raman matrix elements, the calcu-lated Raman spectrum for the random networkwould be greatly changed, but the spectrum for thecrystal would necessarily still consist of the singleTO line.

2290 ALBEN, WEXIRE, SMITH, JR. , AND BRODSKV

ment is equivalent to a spherical averagi. ng of thecross section for a single crystal for all Q havinga given modulus IQI, and these vectors, when re-duced to the first Brillouin zone, will give an ap-proximate average over the entire zone if IQI issufficiently large. ' ' In justifying the method foran amorphous solid, it has usually been picturedas a polycrystal, "but this is not necessary sinceone can argue more directly as follows.

The one-phonon contribution to the cross sectionfor the low spectrum in coherent inelastic scatter-ing is proportional to 3

2

f(Q) g„n (12)

Here 8,. denotes the position of the j th atom andu,". is its displacement vector for the vth vibration-al eigenstate. In a random-network structure, onemay argue that if Q is much greater than 2v timesthe inverse of the spread in nearest-neighbor dis-tance, the phase Q ~ H,. is essentially random and(2) may be replaced by P,. Iu,. ~ Qia, which may inturn be replaced by —,'I/It since the system is iso-tropic and Z, luJI2 is normalized to unity. Thus,in the high-Q limit, one expects the coherent in-

This is a sum over all vibrational eigenstates f(l.The factors entering this expression are a phononoccupation factor, a De bye- Wailer factor, ~' a 5function expressing energy conservation, and aterm which sums the contributions fromallatoms j,

elastic-neutron spectrum to approach the phonondensity of states n(E). In practice, multiple-pho-non processes become important at high Q; sosome compromise i.s necessary. 3' Thus it is im-portant to understand the effects of structural cor-relations on the form of the spectrum for inter-mediate values of Q.

Model calculations should be useful in the designand interpretation of neutron-energy-loss experi-ments. In order to apply (12) to our models, allwe need to do is compute eigenvalues and eigen-vectors and perform the indicated operations. Wepresent results for periodic and quasiperiodic mod-els and compare them with those expected for apolycrystalline powder of diamond-cubic material.

For the periodic model we regard the basic unitas a unit cell of an infinite crystal. In this case theusual crystal selection rules imply that Q must dif-fer from the 0 vector of the modes which contributeby a multiple of a reciprocal-lattice vector. How-ever, in the limit that the unit cell is very large,the Bragg points of the reciprocal lattice fill kspace quite densely. Thus if we restrict attentionto k = 0 modes of the 61-atom model, as we havedone, we may still obtain S(Q, Z) for a fairly densegrid of Q points. In Fig. 20 we show such resultsfor S (Q, Z), which is S(Q, E) without the Debye-Waller and phonon occupation factors. Parametersappropriate to Ge have been used.

From the figure we see that at low Q there is asound-wave-like peak at a frequency proportionalto IQI. As IQI increases, there is an oscillationof intens ity between high- and low -freque ncy partsof the spectrum with a period of about 1.5 A ' su-

Q = I 60A Q = 3.09A Q= Oe65 A

Q = 6.21A Q= 7~90A Q =16e96A

0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300

ENERGY LOSS (cm )

FIG. 20. Calculated neutron energy-loss spectra for different values of momentum transfer Q for the 61-atom net-work model. P(Q, E) is related to the usual dynamic structure factor S(Q, E) as follows: S'(Q, E) =S(Q, E)(([s(~)+ 1]/co}&& ( Q t ) . The vibrational density of states p(E) is shown for comparison. Parameters appropriate for Ge were used.

&IBRATIONAL PROPERTIES OF AMORPHOUS Si AND Ge 2291

Q =' 0.5I A 0=5.06K '

LLI

M

0 loO 200 300 0 l00 200 300 0 IOO 200 300

Q =6.12A Q=7.65A I

0 l00 200 300 0 l00 200 300 0 l00 200 300

Q = 8.57A Q = I2.24K

LLJ

(3(f)

0 l00 200 300 0 l00 200 300 0 lOO 200 300ENERGY LOSS (cm )

l IG. 21. Calculated neutron energy-loss spectra for the 85-atom quasiperiodic model for Ge (see also Fig. 20).

perimposed on a background proportional to thedensity of states. For I Q I greater than 6 A ', thescattering is almost indistinguishable from thedensity of states.

In Fig. 21 we show similar results for an 85-atom quasiperiodic model. In this case the scat-tering is treated as being due to a finite cluster,and with this interpretation we may obtain mean-ingful results for any Q. The results shown arefor Q's along the z spatial axis. The results aresimilar to those for the periodic model. We havea,iso considered Q's along the x direction and ob-tained similar results.

For comparison, we show in Fig. 22 results forscattering from a diamond-cubic polycrystallinepowder. Except for the sharpness of the high-en-ergy peaks in S(Q, E), which follows from the rela-tive sharpness of this peak in the density of states,the behavior resembles that of the amorphousmodel's sound-wave-like behavior followed by in-tensity oscillations with increasing IQI, and even-tually a spectrum proportional to the density ofstates. However, for the polycrystal the intensityoscillations are far more complex than the simplebehavior seen in the results for the amorphous ma-terials.

These results show that inelastic neutron scatter-ing at sufficiently high IQ I (above about 8-9 A ')or averaged over the oscillation period (1.5 A ')should give a quite direct measure of the densityof vibrational states in amorphous materials. Theyalso indicate that oscillations in peak intensity withincreasing IQ I are to be expected because of struc-tural correlations implicit even in random struc-tures. Figure 23 illustrates this oscillation of in-tensity between the TA and TO peaks. Here weplot the ratio of the intensity for the energy losscorresponding to the maxi. mum i.n the upper peakof the density of states to that at the maximum inthe lower peak. As IQI increases the ratio (fortwo different directions of Q) displays quantitative-ly the regular oscillations evident in Figs. 20 and21. This oscillation agrees extremely well with asimple function derived (with no adjustable parame-ters) from the following approximationsvz: (i)Modes in these two peaks are given the exact "pure-bonding" or "pure-antibonding" character discussedin Sec. VIA, which arises from tetrahedral sym-metry and the dominance of central forces. (ii)All other correlations are neglected. The functionwhich gives the ratio of the weights of the lowerand upper peaks is derived in Appendix C. With

2292 ALBEN, WE AIRE, SMITH, JR. , AND BRODSKY

Q=O.44A IQ = 1.55A Q = 5.1 I A

LLJ

V)

t—, 10 100 200 500

I t

IOO 200 500I 'll

200 500

Q = 4.44A Q = 5.99k Q = 7.55A

LLJ

V)

0 l00 200 300I

0 100 200 500I 1 I

l00 200 500

Q =9.IOA Q = I? OOA

LJJ

(3(fl

0I

IOO 200 500 0 100

J

200 500I

0 l00 200 500

FIG. 22. Calculated neutron energy-loss spectra for a diamond cubic structure polycrystalline Ge powder. Histo-grams represent contributions of from 1000 to 2000 modes with ITc+ T I

=I Q I within+0. 22 k of the values indicated.

(T is a reciprocal-lattice vector. )

the appropriate factor converting the ratio ofweights to that of heights, this function is given inFig. 23 by the full line. The agreement is seen tobe excellent even for comparatively low values ofIQI. We expect that this oscillation should be

found in real materials. We might note, however,0

that for I QI ~ 8 A, the oscillation is sufficientlysmall that we would predict that the neutron energyloss does effectively reproduce the density of

states. For a more accurate measure of the den-sity of states, it appears that the loss spectrumshould be averaged over one or more complete os-cillations.

VII. CONCLUSIONS

As is the case with many electronic properties,the vibrational properties of Si and Ge are domi-nated by the effects of short-range order. For the

2293VIBRATIONAL PROPERTIES OF AMORPHOUS S' AND 6

10.0—

5.0—

Q'

2.0V)

I.o+3

CO

0.2

—-4

4 D444

~ ~

X- DIRECTION4 Y- DIRECTION

The availability in the forseeable future of neu-tron-scattering data of high resolution should be ofconsiderable assistance, since this will present apicture of the density of states which, while subjectto various uncertaintj. es, requires no subtractionof matrix-element effects.

ACKNOWLEDGMENTS

We wish to thank D. Henderson and D.. E. Polkfor prove. sng'd' the coordinates of their random-net-

ork modes and J. D. Axe, G. S. Cargill III, G.A. N. Connell, N. Shevchik, F. Stern, and M.Thorpe ior helpful conversations.

60 (A-I)

I

IO

APPENDIX A: INFRARED AND RAMAN MATRIX

ELEMENTS

'der the infrared absorption matrix elementCons& er e irofdff(5). This can be rearranged in a number o s er-

ent ways, of which the one given below would appearto be the most illuminating and useful. Wr&t~ng

C~(l ) for the compression of the 4th bond a.ssociat-ed with atom l, for a given vibrational eigenstate,M may be written

M=BQ(Q c (l))(Q r .())) . A)

This may be obtained simply by multiplying out theexpression (5), keeping the cross terms, and not-ing that the others cancel on each bond.

The second factor in the summand vanishes forexact tetrahedral symmetry, since then.

. 23. Ratio of calculated neutron scattering crossfor frequencies corresponding to the upp

lower densify--o -s a'fy-- f- t tes peaks for the 85-atom modelcompare o e cd to the calculated results for independently scat-

ives the ra-terin bond contributions. The dashed line gives e ra-terzng on contio of the density-of-states peak heights. us. Full line is theth t cal result described in Appendix C. eThe frequen-cies cu' and cu are, respectively, 0.97 and 0.39 tiTO frequency of the diamond cubic structure of Ge.

amorphous p ase,h the broad features of the densityof states and the variation with frequency of thematrix elements which weight the Raman and infra-red spectra can be rather well explained in termsof such short-range order and the complete ab-sence of any effects due to correlations of longerran e ("k-sejection" effects). Approaches whichstart from results for perfect crystals and attemptto smear out spectra to the point where they agreewith results for the amorphous phase are there-fore in some difficulty. Even were they to suc-ceed, we feel that they obscure the simplicity ofthe situation by including effects due to long-rangecorrelations and then trying to dispel them by un-certain procedures f

While it is an interesting intellectual exercise toshow that short-range order is, pew se sufficientto explain a grea ea, it deal it must be admitted that itwould be highly desirable to go one stage furtherand achieve an understanding of the finer detailsof the spectrum and their relation to structure.This is, however, a dif ficult task, ne cessitatingconsideration of forces of longer range as well asthe achievement of calculations for clusters oflarger size. In the latter regard, the outlook ishardly encouraging, since a significan'f icant reductionin the norseh " ' " due to the finite size of the clustersdemands a high price in computer time and stor-a e. For the present, we would caution againstoverinterpretation of the finer details of the nu-

merical calculations performed so far.

(A2)

Thus the chosen mechanism is such that it giveszero for any structure wi. th exact tetrahedral co-ordination of nearest neighbors, '3 and the total in-frared i.ntens ity for random- network structuresshould scale roughly as the square of the dsstortI. onfrom tetrahedral symmetry, as given by mean-square value of

~ g~ r~(l ) ~.

There is, however, a remaining factor in thesumma, nd of (Al) which depends on the characterof the vi.brational eigenstate. Since it involves thecompression of bonds, it might be expected to in-crease monotonically, roughly as was explained inSec. VIB in connection with the Raman tensor.However, for the case of exact tetrahedral symme-try and central forces only, we know, &n particular,that

QC (l)=0 (A3)

f the modes in the 5 function at the top of thespectrum. This follows from the development ofSec. VI, si.nce these modes are analogous to "pure-p- on sng" ed' "' lectronic wave functions; e contribu-tions to the compressi. on of each bond due to the

2294 ALBEN, KEAIRE, SMITH, JR. , AND BRODSKY

n, = g [r.(f)r.(f)--: l]C.(f),bon ds

(A4)

n~= Q TC~(f ),bon ds

(A6)

where each bond is counted only once in the sums.If the contributions from each bond add incoherent-ly, then the Raman cross section from these mech-anisms will be given by the mean-square value ofC~(l), i.e. , the mean-square compression of thebonds. The latter has a simple frequency depen-dence in the case of central forces. The equationof motion for this case is

m(g'u, = 3ng C~(l)r~(E) . (A6)

Taking the scalar product of both sides of (A6) withu, and summing over all l, we obtain

displacements of the two atoms at its end are equal.This, together with (A2), gives (A3).

Thus, in the absence of noncentral forces, thedipole assocj.ated with these particular modesshould be of second order in the distortions (i. e. ,the intensity is of fourth order) A. s argued in themain text, the inclusion of the small noncentralforces should not alter this conclusion qualitatively.This therefore would appear to be a sufficient ex-planatj. on of the relatively low infrared intensity as-sociated with these modes i.n the calculations ofFig. 8.

Note that in the case of III-V compounds thedominant contribution to the infrared intensity isnot of the form (Al), so the arguments given hereare not applicable.

The Raman polarizability tensors n, [Eq. (6)]and n3 [Eq. (8)] can also be written in terms ofthe C~(l )'s:

P~.(f) =0, (B2)

since P~r~(l) is 0 for each f. The potential energyassociated with nearest-neighbor central forces,given by Eq. (2), may be written

v= — 4~Ex;(I )'+4agx, (l )x,O&)) .lh l&

The kinetic energy may be written

(B3)

&=km+X~(f)', (B4)

since

Qr (f)r (f) =—I .

s24P —4Q Z p

4a Va,

x(- )-y, .(B6)

However, the spectrum of the tight-binding Hamil-tonian of Thorpe and Keaire is in turn related tothat of the simpler "one-band" Hamiltonian

Now, the eigenvalues of the dynamical matrix de-rived by taking second derivatives of V with re-spect to the y's would give the normal-mode fre-quencies (to be precise, m&u~) were it not for thelinear dependence (B2) of the variables. To allowfor this an extra term —,

'A. g, g~ y~(l )]2 can be added

to Vand, in the limit A. -~, that part of the spec-trum which remains at finite frequencies is the re-quired solution; i.e. , the extra term projects outvectors which obey (B2). With this addition theproblem is now identical to that studied by Thorpeand Keaireeo in the context of electronjc properties,provided we make the identifications

m~'Qu f =3n Q [C,(f)]'.bonds

(Av) ff'"= ~Elf&«~l . (BV)

%hen both sides are divided by the number of bonds,this equation shows that the mean-square bond com-pression is proportional to ~2 times the mean-square displacement. The reduced Raman intensi-ty per mode from mechanisms Nos. 1 and 3 is thusseen to vary as ~ over the whole spectrum in thecase of central forces when correlations amongdifferent bonds are neglected.

APPENDIX 8

In this appendix we give a proof of the theoremof Sec. VIA. As explained there, we introducevariables X according to

Here there is just one basis function i f) associat-ed with each site.

In the limit V&- ~, the eigenvalues of the fulltight-binding Hamiltonian are related to those of(B6), denoted by e, according to

(BS)

apart from two 5 functions at E =+ Vz. Thistranslates, using (B6), into the theorem stated inSec. VI.

APPENDIX C: COMPARISON OF TA AND TO MODES INNEUTRON SPECTRUM

y~(&) = (v'-')u( ~ r~(& ) (Bl)

Kith perfectly symmetric tetrahedral coordinationof nearest neighbors they are related by

TA and TO modes are such that, approximately,the projections of the displacements of neighboringatoms onto their common bond are equal i.n mag-nitude and, respectively, of the s;;."fe or opposite

VIBRATIONAL PROPERTIES OF AMORPHOUS Si AND Ge 2295

sign. Furthermore, the factor Q ~ u, which entersthe neutron-scattering transition probability is ap-proximately proportional to

I To~ Q (nTo)2 I TA~ Q ( TA)2 (c3)bon ds bon ds

%ith such an approximation, a simple sphericalaverage gives

4+LQ ~ r (l)][u, ~ (f)], (Cl)

since 7= pA rA(l )rA(l ) for tetrahedral symmetry(I is the unit dyadic). Using this we can write thescattering amplitude, i.e. , the expression whoseabsolute square gives the intensity in (13), as asum over bonds. The contribution of one bond con-necting two atoms A and B to the amplitude for aparticular TO or TA mode is

To si4 BQ 'Q Q RAB sjnl Q ~RABIIABAB & 2 - i (C2)

e4'~0 Q ' AR cos Q ' &R BUAB

where R0 is the center of the gB bond, 4H0 isthe vector from atom g to atom 8, and UA, is (U„+ UB) ~ r„B . We assume that the phase of the U"

~X,JABtogether with the factor e' '"0, varies sufficientlyrandomly from one bond to the next to justify the"incoherent approximation, "which gives for theintensities

~QR/2p, sin p, dp,

ITA 0

QR/2

p cos p, dJLL

—,' ——,(QR)

' sinQR —3(QR) cosQR+ 3(QR) sinQR—,'+ —, (QR) ' sinQR+3(QR) cosQR —3(QR) sinQR

(co)This has the asymptotic behavior

ITo/ITA-1 —6(QR) ' sinQR, as Q- ~ (cv)

in keeping with the assertion that the spectrumshould represent the density of states in this limit.

Note also that the sum rule

IT„+I» = (const)Q (cs)

since the system is assumed to be (macroscopical-ly) isotropic and there are no preferred directionsfor the bonds. R is the magnitude of the nearest-neighbor distance, taken to be constant. Equation(B4) may be rewritten

QR/2ITO ~ p sin j(L dp. )QR -oB i2

(c5)4 QR/2

gT cos JLL dp.QR oB(2

The ratio of the intensities of the two types ofmodes is

ITo ~ (QR cos8)2 sin2( —,'QR cos8) sin8 d8,0

I "~ (QRcos8) cos (-,'QR cos8) sin8d80

(c4)also follows from (C5); so the combined contribu-tion to the spectrum from the lowest and highestpeaks is constant in this approximation. There isalso a sum rule on the whole spectrum. '3

Supported in part by NSF Grant No. GH 3599 and U. S.AFOSR Contract No. F44620-71-C-0039.

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