PHONONS IN AMORPHOUS MATERIALS

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PHONONS IN AMORPHOUS MATERIALS

M. Thorpe

To cite this version:

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JOURNAL DE PHYSIQUE

CoZZoque C6, suppZ6ment a~ n012, Tome 42, d6cembre 1981 page C6-19

PHONONS I N AMORPHOUS M A T E R I A L S

M.F. Thorpe

P h y s i c s Department, Michigan S t a t e U n i v e r s i t y , E a s t k n s i n g , M I 48 824, U. S . A.

Abstract.-

A

brief review is given of phonons in amorphous mate-

rials as seen in Raman scattering, infrared absorption and inelastic neutron scattering. It is shown that phonons (i.e. quantised harmonic vibrations) are well defined in network

structures and a description is given of the standard theoretical technique which is to build a finite model and then use one of a number of numerical techniques to obtain the eigenvalue spectrum of the dynamical matrix. Particular emphasis is given to newer theoretical techniques that do not require the building

of a model. The results are i l l u s t r a t ~ w i t h reference to ex-

periments in elemental semiconductors like Si and Ge and two component glasses like Si02, GeS2, etc.

1. Introduction.- The vibrational s?ectra of amorphous solids and glasses have been extensively studied particularly in the last decade. The principle experimental techniques used are inelastic neutron scattering, Raaan scattering and infra-red absorption. Although there is no one theoretical approach that can explain all aspects of these vibrational spectra, the main features are now understood. However, many significant details renain to be put in place.

We will try to briefly review the most important theoretical techniques that are available to calculate the phonon spectra of glasses. The results are illustrated mainly with a. Si and a. Si02 on which most work has been done. i.lore exotic types of glasses have also been studied but less theoretical progress has been made on these.

2. Model Building and Numerical Methods.- The standard and most success ful calculations of phonon spectra have involved numerical methods on particular models. It is of course essential to know the structure of a glass before proceding to calculate the vibrational spectrum.

The earliest of these models were hand built ball and stick models

of a. sio2('). These were based on the known local bonding arrangements

and typically contained. s 300 atoms in a roughly spherical cluster.

Bell and ~ean(l) used these models to set up a dynamical matrix using the Born model which has nearest neighbor central and non-central forces. The eigenvalue spectrum was obtained as a histogram using the negative eigenvalue theorem.

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JOURNAL DE PHYSIQUE

In figure 1 we show a recent comparison between theory and an inelastic neutron scattering experiment(2) for Si02. The experiment is done at large wave vector transfer so that essentially the density

Fig: 1 : A comparison(3) between ex-

perlment and theory for inelastic neutron scatterin in a. Si02. [Adapted from reference 3.9

of states is measured. This has the disadvantage that the multiphonon background has to be subtracted. This is indicated in the figure by

the dashed line(2). The theory is the density of states weighted by

the appropriate neutron scattering lengths for Si and 0. The agreement overall is impressive.

Similar calculations(3) have been performed for the Raman and I

.R.

spectra. Again reasonablely good agreement is obtained. However, the matrix elements are difficult to put in as they are not simple as in

neutron scattering. Usually sone local bond ~olarisabilities are in-

troduced which have the correct synrmerry. This introduces a major theoretical uncertainty that is not present in inelastic neutron scattering.

Similar work has also been done on a. Si. Hand built models have been constructed by many groups. While the hand construction fixes the topology of the network, the positions of the atoms can be refined by relaxing according to a suitably chosen potential.(5) The equation of motion method has been used by Alben and collaborators to determine all quantities of interest [density of states, Xaman scatting, I.R.

absorption and the neutron scattering law

~(6,

w ) ]

.

The method tracks

the bahavior of the displacements with time and then Fourier trans- forms. It can be used in any harmonic system that is disordered. Good overall agreement is achieved between theory and experiment using this

method(4)

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.

The negative eigenvalue theorem was used to construct histograms for the density of states(l) for SiOZ type glasses. To calculate other quantities, sample eigenvectors must be obtained. This method could

also be ap~lied to a. Si as could the equation of motion method be

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has n o t happened.

I t i s p a r t i c u l a r l y d i f f i c u l t t o c a l c u l a t e t h e I . R . a b s o r p t i o n

i n elemental g l a s s e s because t h e r e i s no I . R . a b s o r p t i o n i n t h e corresponding c r y s t a l u s u a l l y . A new mechanism must be invoked. Two have been suggested; d i p o l e moments a s s o c i a t e d w i t h t h r e e n e a r neighbor atoms(4) o r f l u c t u a t i n g dynanic charges on a l l t h e atoms. (7)

D i r e c t d i a g o n a l i s a t i o n of t h e dynamical m a t r i x f o r a f i n i t e model

i s now r a r e l y used a s i t i s i n e f f i c i e n t .

These numerical techniques may be regarded a s providing a s o l u -

t i o n t o t h e problem

-

although some e f f e c t s such a s t h o s e produced

by t h e long range Colllomb f o r c e s have y e t t o be i n c l u d e d . However, t h e y do n o t o f t e n p r o v i d e a g r e a t d e a l of i n s i g h t and c a l c u l a t i o n s

must be r e p e a t e d every time a change i s made i n t h e model

-

such a s

i n c r e a s i n g t h e l o c a l d i s t o r t i o n s o r changing t h e masses. For t h i s r e a s o n , more a n a l y t i c approaches a r e now being p e r s u e d .

3 . Beyond Numerical Methods.- Some p r o g r e s s has been made i n two

d i r e c t i o n s . The phonon spectrum of a B e t l ~ e l a t t i c e 8 can be obtained

u s i n g Born f o r c e s f o r a . S i . This shows many f e a t u r e s i n common w i t h t h e d e n s i t y of s t a t e s of c r y s t a l l i n e S i although t h e van Hone s i n g l - a r i t i e s and some o t h e r f e a t u r e s a r e a b s e n t . ( s e e f i g . 2)

1':

A

I

' U l L

P

:

i

z

3 - I U. I Showing t h e d e n s i t y of s t a t e s t a l l i n e S i i n t h e diamond e ( s o l i d l i n e ) and t h e Bethe (dashed l i n e ) . [From r e f e r e n c e W I W m x

This o r o t h e r Bethe l a t t i c e s can be t i e d o n t o s m a l l p i e c e s of

network and used a s a convenient boundary c o n d i t i o n i n t h e " C l u s t e r

-

Bethe

-

l a t t i c e " method. This avoids t h e l a r g e e f f e c t s t h a t t h e f r e e

s u r f a c e s of small c l u s t e r s can produce. A review of t h i s method con-

t a i n s many of t h e r e s u l t s o b t a i n e d t o d a t e ( 7 ) . Good agreement w i t h experiment i s obtained i n many c a s e s .

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C6-22 JOURNAL DE PHYSIQUE

soluble9 and very simple expressions are obtained for the positions

and widths of the main peaks in the density of states. These are

particularly valuable when the network is modified in sone sinple way.

For example 018 can,be substituted for 016 and the Raman sDectra of

the two glasses compared.10

It

may be possible eventually to include

non-central forces and Coulomb forces in this model using perturba-

t

ion theory.

4. Future Prospects. Although many Raman and

I.R.

experiments have

been performed on many glasses whose structure is known, progress has

been hampered

by

the absence of a good knowledge of the optical matrix

elements. Recent work by Plartin and Galeener

(I1)

has shown that these

matrix element effects are extremely important and can produce peaks

in the optical spectra where non exist in the density of states

(see figure

3).

I i 4 1 I I ' I ' I VITREOUS Ga02 lbl I I

-:

Y.

;

Ill Ill W,

-

I ~\;(11 - 2 7 Ill

!,:.a:

, W l ,

I I

, .

0 500 1000 I S W

The clearest measurements to interpret theoretically are from

inelastic neutron scattering. However, the measurements at large

Q

have problems with multiphonon scattering. It may be that measurements

at smaller Q will give the most information. These can be obtained

theoretically using the equation of motion technique.

L " " " " ' 1 " " - VITREOUS :

-

12- =a, - z kl

,

-

L

-

3 :

Showing the Raman scattering,

F-

enslty of states in the central force

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I s h o u l d l i k e t o t h a n k R . J . B e l l , F . L . G a l e e n e r a n d A. W r i g h t f o r many u s e f u l d i s c u s s i o n s . S u p p o r t from t h e O . N . R . i s g r a t e f u l l y acknowledge.

R e f e r e n c e s .

1. See P. Dean, Rev. Mod. P h y s . 4 4 , 127 (1972) a n d R . J . B e l l ,

R e p o r t s o f P r o g . i n Phys. 2 , 7 2 1 5 ( 1 9 7 2 ) .

2 . A, J . L e a d b e t t e r a n d M.

W.

S t r i n g f e l l o w i n "Neutron I n e l a s t i c

S c a t t e r i n g , P r o c . G r e n o b l e Conf. 1972", p . 5 0 1 , I.A.E.A. Vienna 1974.

3 . R. J . B e l l , A . C a r n e v a l e , C . R. K u r k j i a n a n d G . E . P e t e r s o n ,

J . Non C r y s t . S o l i d s 3 5 / 3 6 , 1185 ( 1 9 2 0 ) .

4 . R. A l b e n , D . W e a i r e , J . E . S m i t h a n d M . Brodsky

,

Phys

.

Rev.

e,

2271 ( 1 9 7 5 ) .

5 . See f o r example

-

D . Beeman a n d 3. L . Bobbs, P h y s . Rev.

812,

1399 ( 1 9 7 5 ) .

6. D . Beeman and R . A l b e n , Ad. i n Phys.

3,

339 (1977)

7. J . J o a n n o p o u l o u s , A . I . P . C o n f e r e n c e P r o c e d i n g s 3 1 , 103 ( 1 9 7 6 ) .

9 . P.I. F. T h o r p e , P,hys. Rev.

E ,

5352 ( 1 9 7 3 ) .

9 . P .

M.

Sen a n d M. F . T h o r p e , P h y s . Rev. B15, 4030 ( 1 9 7 7 ) ; If. F

Thorpe and F . L . G a l e e n e r , Phys. Rev.

-

BE

3078 ( 1 9 9 0 ) .

1 0 . F . L. G a l e e n e r a n d J . C . b l i k k e l s e n , Phys. Rev.

m,

5527 ( 1 9 8 1 ) .