Modelling the raman spectrum of the amorphous- crystal Si system

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Journal of Physics C: Solid State Physics

Modelling the Raman spectrum of the

amorphous-crystal Si system

To cite this article: A S Carrico et al 1986 J. Phys. C: Solid State Phys. 19 1113

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J . Phys. C: Solid State Phys. 19 (1986) 1113-1122. Printed in Great Britain

Modelling the Raman spectrum

of

the amorphous-crystal Si

system

A S Carrico, R J Elliott and R A Barrio?

Department of Theoretical Physics. 1 Keble Road. Oxford OX1 3NP. U K

Received 23 August 1985

Abstract. A model to calculate the Raman response of small Si crystallites immersed in an amorphous Si matrix is developed. The structure of the system is modelled by an infinite crystalline slab of varying width, in which the atoms on the (111) surfaces on both sides of the slab are attached to Bethe lattices. The Raman response is obtained from a simple polarisability model using the Green function technique. The results are compared with recent experiments. The model can be used for Si crystallites with free and hydrogenated (111) surfaces.

1. Introduction

The study of interfaces in semiconductors has been greatly developed in recent years responding to its important technological implications. In particular good planar interfaces on the atomic scale exist in the new superlattice materials (Esaki 1985) and between Si and its oxide, as in MOS devices (Poindexter and Caplan 1983). A further interesting system has been studied recently, in which two phases, amorphous and crystalline, of the same substance coexist in the same sample. Such a situation is found in Si samples that have been partially laser-annealed (Morhange et a1 1975) and in some microcrystalline Si (Iqbal and Vepiek 1982). The properties of these systems are believed to be dominated by the presence of the interface between the two phases.

For instance, the vibrational Raman spectra obtained in both cases consists of a broad band, rather similar to the amorphous Si spectrum, and a narrow peak that is broadened and shifted with respect to the single peak in a perfect Si crystal. The softening of the frequency of this peak, and its width, get more pronounced when the size of the crystallites decreases and therefore the number of sites in the interface increases (Iqbal and Vepiek 1982). This is confirmed by the laser-annealing experiments that report a larger shift when the Raman spectrum is taken from regions away from the centre of the laser beam (Morhange er a1 1975).

A theoretical treatment has been developed by Kanellis et a1 (1980) in order to explain the shift of the high-frequency narrow peak. This model considers a two- dimensional infinite slab of varying width, in which the interatomic forces have been represented by a nearest-neighbour potential that includes central and angle-restoring forces. The density of vibrational states (DOS) is calculated for a number of planes in the

+ Permanent address: Instituto de Investigaciones en Materiales. U N A M , Adpo Postal 70-360, Mexico, DF. 0022-3719/86/081113

+

10 $02.50 @ 1986 The Institute of Physics 1113

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1114 A S Carrico, R J Elliott and R A Barrio

slab. This theory predicts a shift of the upper band edge of the DOS, although it is somewhat smaller than the one observed. In our opinion this theory suffers from two severe restrictions: (i) it does not take the amorphousxrystal interface into account, since the effect is calculated for free surfaces, and (ii) it only looks at the DOS, not the Raman spectrum. This could differ substantially because of the coherence of the light waves involved.

It is the purpose of this paper to remove both restrictions by calculating the Raman response of the same infinite slabs of varying size but embedded in an amorphous medium represented by Bethe lattices (BL). For comparison we also investigate free and hydrogenated (1 11) samples.

In § 2 we describe the structure of the model and set up the equations of motion for the displacement-displacement Green function, for the Born model with central and non-central forces. In

0

3 we develop a simple model for the Raman intensity in terms of the Green functions found before. Then we discuss the results in

8

4 and compare

them with the experiment and previous theoretical results.

2. Dynamical model

2.1. Crystalline slab

We assume that the interactions between the atoms of the system are only between nearest neighbours and of the Born type:

VI,, = l(cu -

p)

1

[U(/) - U(/’)] * f i l l ,

l 2

+

i P ( U ( 0 -

U ( 1 ’ ) ) * (1) where (Y is the restoring force constant along the unit vector

A,,.

that joins the two

neighbouring sites 1, l ‘ , and pis a non-central force constant.

In the crystal there are two different interactions between (111) planes, since con- secutive planes are joined alternately by one bond per site perpendicular to the planes and three bonds per site, as depicted in figure 1. If the 2 axis is chosen to be perpendicular to the (111) planes, the interaction between atoms of the first kind will be

m=l m =O n = l m=-l n.2 n =3 n.4 n.5 n=N-1 n = N m=-I m= 0

(4)

Modelling the Raman spectrum in a-c-Si 1115 and between the atoms of the other kind will be

4 a + 5 p 0 0

4 a + 5 P 0

0

C Y + 8p

Because of the tetrahedral symmetry

4

c

=

--E

c,

= $(a+ 2P)Z i = 1

(3)

(4) where the summation is over the four bonds arriving to a given site. If the planes are infinite one can take advantage of the translational symmetry of the crystal in directions parallel to the planes. Defining the displacement Green functions in a 3 X 3 matrix form

( ( u u ( l ) , ~ I , ( l ’ ) ) ) = Guu,(l. 1 ‘ ; U )

the sum over atoms in the planes gives

G k ( n , n ’ ; U ) =

E

G ( l ( n ) ; I(n’); U ) exp(ik.R.(I)

-

R n J ( l ‘ ) ) _{( 5 )} I/’

where U is the frequency and R,(I) is the equilibrium position of a particular atom 1 in

plane n , and

k

is a vector in the two-dimensional Brillouin zone.

The Raman scattering is related to the Green function at the k = 0

(r)

point which can be obtained from the equations of motion in the same way as Falicov and Yndurain (1975) treated the electronic problem in Si using a set of plane-dependent transfer matrices:

G o ( n , n ’ ; U ) = t ( n ) G o ( n - 1, n ’ ; U ) . ₍₆₎

This quantity G o ( n , n ’ ; U ) contains all the correlations between atoms belonging to

planes n , n’. There are recurrence relations for the transfer matrices: ( M u 2 1

-

C ( n ) ) t ( n ) = C 1

+

C 3 t ( n

+

l ) t ( n )

(Mu21 - C ( n

-

l ) ) t ( n - 1) = C 3

+

C , t ( n ) t ( n - 1)

( 7 a )

(7b) and

where n is an odd-numbered plane in figure 1, M is the mass of the Si and C ( n ) = C (equation ( 4 ) ) for all the planes except the surface ones. If the total number of planes in the slab N is even, there is a symmetry in (7):

t ( n ) = t ( N - n ) (8)

and the finite set of equations (7) can be solved exactly by imposing suitable boundary conditions.

2.2. Amorphous region

This is modelled by attaching Bethe lattices to each atom in layers n = 1 and n =

N.

The atomic arrangement is shown in figure 1. We label each shell of the Bethe lattices as a layer numbered m = 0, 1 etc as we move away from the crystal surface (which can be labelled m = -1) in either direction. In this model the atomic density in each layer

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1116 A S Carrico, R J Elliott and R A Barrio

increases exponentially with m and gives an overwhelming weighting to the properties of the outside layers. This is countered by reducing the weighting given to calculated physical properties of layer m by 3-" so all layers have equal weighting.

The interatomic coupling is assumed to take the same form as (1) with slightly modified force constants CY' and

p'

to give the correct position to the top of the mode

spectrum. The Green functions within a Bethe lattice have been considered elsewhere within this model (Thorpe 1981, Barrio and Elliott 1982, Elliott et a1 1982), but it is useful to repeat the essential results appropriate to the particular geometry of this problem. We define a transfer matrix between site 1 in shell m and a neighbouring site 1' in shell m - 1:

g ( p ( r ) ;

4 m ) )

= q,(m)g(p(r), l'(m

-

I)), (9)

where g are the site-to-site Green functions in the Bethe lattice. If all the matrices in (9)

are written in a frame of reference in which the Z axis is along the bond between 1 and l ' , then q,(m) is diagonal and takes the form

From now on a transfer matrix without a subscript will be referred to the local frame of each bond. If we define

SI

as the rotation matrix that transforms the bond I'

- 1

into another bond j

-

1 attached to site 1 (j is in shell m

+

l ) , then

q,(m

+

1) = S;'q(m

+

l)S,. (11)

Therefore one can write a recurrence relation for the transfer matrices:

(Mw'l - C' - _S;'C;q(m

+

_l)Sj)q(m)₌_C;(m)

] # l '

where the force constant matrix C; ( m ) for the pair I , I' and C' are as in (2) and (4) with

CY' and

p'

.

With these equations we can find a quantity comparable with ( 5 ) in the slab, with the reduced weighting factor,

G o ( m , m ' ;

w )

= 3-("+")

2

g ( l ( m ) , l ' ( m ' ) ; 0) ₍₁₃₎ I/'

for the k = 0 case which is required in Raman scattering. A method of calculating the appropriate combinations of Go is given in the Appendix.

2.3. Boundary conditions

Fixing attention on the crystalline slab. we examine three cases with varying boundary conditions.

(i) Free surfaces, where t(0) = t(N) = 0

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Modelling the Raman spectrum in a-c-Si 1117 and we define Cy as in (2), then

t(0) = t(N) = C;/Moo2

C(1) = C ( N - 1) = -c3

-

cy.

( 1 5 a ) where M O is the hydrogen mass and

(15b) (iii) Embedded slabs. Here we must match the two solutions. The slab equations (7)

have

t(0) = t(N) = q(-1)

C(1) = C(N

-

1) = -c3

-

c;.

From these equations we can, in principle, calculate the Green functions and hence the vibrational response locally at any plane in the system. Carrico (1985) has for example shown that the density of states varies considerably between the surface and the bulk of the slab.

3. Raman response

The fundamental electronic processes involved in the Raman effect are extremely complicated to describe theoretically. In fact, a fair description of the Raman matrix elements is possible only in cases where the many-body problem can be reduced by symmetry arguments, as in the crystals. In general, the Raman matrix elements depend on the local polarisability of the bonds due to the atomic motions and on the changes of the dipole moments on every atomic site. Because the wave-vector of the light is very small compared with the relevant reciprocal distances in the crystal, the momentum conservation rule allows participation of phonons around the

r

point only. In fact, for silicon only the high-frequency optic mode at 520 cm-', in which all atoms vibrate against each other coherently, is Raman-active (the acoustic modes near k = 0 contribute to Brillouin scattering).

In the amorphous material the k = 0 selection rule breaks down because of the lack of translational symmetry. The Raman response in the BL can be modelled by summing all the correlated displacement motions with alternating phases (Elliott et a1 1982). In the simple model the overwhelming weight of the surface atoms causes pathological behaviour (Yndurain et a1 1983) but this is removed if the weighting factor is included to keep the atomic density uniform.

Considering the model of the polarisability tensors developed by Alben et a1 (1975)

we make the simple approximation that the depolarised Raman response is proportional to

up' 11'

For atoms in the slab the atomic polarisability alternates in sign from layer to layer. We assume that this property continues into the amorphous material. The sum over sites can now be amalgamated into a sum over layers, taking the

k

= 0 component within the plane. In this case the matrix G o defined in (5) and (12) becomes diagonal in the index ,u and differs for p parallel and perpendicular to the slab axis 2 . The overall Raman

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1118 A S Carrico, R J Elliott and R A Barrio response is then

I

-

wcz Im

x(-l)r-r’

GO,,(r, r ’ ; w ) .

p rr’

This comprises three components, I,, for the slab when ( r , r ’ ) are ( n , n ’ ) ; I,, from the amorphous part when ( r , r ’ ) = ( m , m ’ ) and a cross-term 21,, where ( r , r ’ ) = ( n , m’) = ( m , n ’ ) . The relevant Green functions are calculated using the recurrence relations discussed in § 2 .

The above approximation is clearly more satisfactory for the crystal slab than for the amorphous material. It is therefore correct, within the model, for the slab with free and hydrogenated boundaries, and becomes progressively less satisfactory for thin embedded slabs. The pure crystal has a single peak response at 520 cm-’ and the model gives this correctly in the limit. For thin slabs two peaks, arising respectively from the parallel and perpendicularly polarised motions, still dominate the spectrum. The peaks arise almost exclusively from I,,

+

21ac. The amorphous response, a broad peak at lower frequencies. arises from I,,

+

21ac. In this paper we concentrate on the behaviour of the peaks in small crystallites, and leave further detailed discussion of the broad amorphous peak to a subsequent paper. We shall therefore only calculate I,, and I,, in the slab geometry using the results of the last section. I,, is somewhat more tedious to calculate and will be approximated here by taking the result for a large purely amorphous system with appropriate normalisation. The expressions used to calculate I,, and I,, are given in the Appendix. I,, is obtained from (6) and (7).

4. Results and discussion

Calculations of the Raman response were carried out using values of the force constants

LY = 120.3 N m-’,

p

= 23.5 N m-’ to fit the density of states and Raman-active mode of the crystal at 520 cm-I, and choosing LY’ = 104.5 N m-’, /3’ = 19.0 N m-’ to give a

reasonable form to the high-frequency component of the density of states in amorphous Si. The Si-H force constants LY” = 227.4 N m-l and p” = 23.0 N m-’, were chosen to give correct frequencies for the infrared-active modes in hydrogenated Si (Barrio et a1 1983). For slabs less than 20 planes wide two well defined crystal peaks appear which soften with decreasing thickness. The Z mode polarised perpendicular to the surface is always softer than the X modes with parallel polarisation, since the surface effects are stronger for displacements in the 2 direction. In figure 2 we show the position of these peaks for hydrogenated and embedded slabs as a function of N , the number of planes. The results for free slabs lie between the two. As expected the hydrogenated case lies lower since at these frequencies the H atom follows the Si motion. (The Si-H stretching frequency is at 2000 cm-I.) The Si-H unit then behaves roughly as a single heavier atom creating a sort of isotope shift. In very thin slabs this would give a reduction of some 1.74% (-8 cm-I) aver the free slabs-which is of the order observed. The modes in embedded slabs have a smaller shift and occur at higher frequencies than in free slabs, due to the restoring force of the amorphous material. These results are calculated using an amorphous component with M = 21 layers. This value was chosen as the smallest that gave frequencies independent of M .

In figure 3 we plot the shift in frequency on a logarithmic scale. For a free slab we expect the mode of smallest wave-vector magnitude k - l / N a n d hence the peak to shift

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Modelling the R a m a n spectrum i n a-c-Si 1119 520 515

-

I

-

E, 3 510 505 0 - - I / - ' I I I I " H 1 J 0 1 2 3 L IniAl

Figure 3. The shift of peak positions of the Raman response. plotted logarithmically as a function of slab width for free ( F ) . embedded ( E ) and hydrogenated (H) slabs in X polarisation. The results of Kanellis er a/ (1980) are plotted-circles: X : crosses: Z .

by A w

-

k 2

-

1 / N 2 . This is in fact observed. For a hydrogenated slab the exponent is somewhat larger while for embedded slabs it is smaller.

We have also compared our results with those of Kanellis et a1 (1980) in figure 3.

They give the position of the peak in the density of states. For the higher frequency

X-

polarised modes the transverse optic modes are flat and the peak position is close to that

of

the coherent modes. In fact these authors used a Hamiltonian that differs from ours in its treatment of the non-central forces. However these are relatively unimportant for the high-frequency stretching modes and we get good agreement with their results. (Their curves are given in unit cells which corresponds to two layers in our model.)

A comparison with experimental measurements in laser-annealed Si is not possible since no detailed results have been published, but we have compared our calculations with measurements on hydrogenated microcrystalline Si by Iqbal and Vepiek (1982).

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1120 A S Carrico, R J Elliott and R A Barrio

The comparison is not straightforward since their material presumably had crystallites with a geometry more nearly spherical than slab-like. They report the Raman shift as a function of crystallite size C,, defined as the dimension of the cube that gives the correct widths of the x-ray peaks in their samples. If we simply equate our slab thickness 1Nd (where d is the lattice parameter d = 5.43A) to C, we obtain the result shown in figure 2.

Our calculations have demonstrated that the Raman response in the slab depends on the polarisation of the modes.and it will certainly also depend on the shape of the crystallites. However the shift is a surface effect and it seems reasonable to expect that the most significant parameter affecting the frequency is the surface-to-volume ratio. For a slab of

N

planes this is

R = N,,”, = 2/(N - 2). ₍₁₉₎

For other crystallites it depends on the shape. For a sphere of diameter C,

R = 6a/C, (20)

where a is a convenient normalising length giving the volume per atom as a 3 . A more

likely shape for crystallites in the diamond structure is octahedral, with (111) faces. For an octahedron contained in a cube of edge C, we find

N s = 2(2X2

+

4X

-

1)

where X = C,/d. Using R = N s / ( N T

-

N s ) and comparing with (19) to define an effective value of N we obtain the crosses in figure 2 which show the general trend expected for embedded and hydrogenated crystallites.

The results of this paper show the effect of various boundary conditions on the vibrations and the Raman effect of finite crystallites of Si with slab geometry. We believe that the embedded and hydrogenated cases are more realistic than the free slabs that have been previously used as models for these systems. Comparison with experiment is very satisfactory in a qualitative way. No detailed results were presented for the Raman response of the amorphous component. The investigation of this within the model will be reported separately.

Acknowledgments

This work was supported by the US Army through its European Research Office. One of us (ASC) wants to acknowledge financial support from CAPES (Brazil).

Appendix

In a single Bethe lattice, the Green functions involving the atoms in shell m can be projected onto those in shell m

-

1 (equation ( 9 ) ) , and sequentially back to an initial point m = 0. The transfer matrices qr(m) depend on the directions of the bonds. Let us denote by 6, a unit vector along one of the three bonds between site l(m) and shell m

+

1; the bond between [’(m

-

1) and Z(m) is 6!. We know that along the direction of the bond

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Modelling the Raman spectrum in a-c-Si 1121

the transfer matrices are diagnonal; therefore

g(0,

W )

* 61 = ql,(m)

d o ,

l'(m

-

1 ) ) * 61

g(0,

+4)

x 61 = qi(m>g(07 1'(m

-

1 ) ) x 61. ( A I )

We define a Green function with components in arbitrary directions p , p' in terms of parallel and perpendicular projection on a bond 6[:

where the minus sign takes into account the alternating phase in ( 1 8 ) and the factorf =

B

will give the same weight to every shell in the Bethe lattice (see ( 1 3 ) ) . Then (A3) is

r(0, l ( m ) ) = A(m)r(O, l'(m - 1)) (A51

1

with

The recurrence relation (A5) holds for every shell; therefore the correlation between the central atom m = 0 and all the atoms in shell m emerging from a given branch in shell

m = l i s

m

m ' = 2

The sum of correlations between the central atom and all shells emerging from a branch in the lattice is

M m

We can use this result to obtain an expression for Zaa. As discussed in the text, we approximate further by assuming that the Bethe lattice is not attached to the slab but to

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1122 A S Carrico, R J Elliott and R A Barrio

other lattices. In this case four lattices grow from the central site to give

angg(0,O) is the isotropic scalar Green function for a site on a Bethe lattice (Barrio and Elliott 1982). The final result is then

I,, = --o Im pX,,, vg(0,O) ('410) where p = ( 1 , l ) is introduced to sum over components.

We can also use the result (A7) to find an expression for the interference term between the amorphous zone and the crystal. We notice that the total correlation between the atoms in the BL and one atom in the nth plane in the crystal is equal to the correlations between all atoms in plane n and those in a single BL stemming from a given surface site. This latter can be obtained exactly by substituting for r(0, 1) by

r(-l,

n ) , where

and Go( - 1, n ) is the k = 0 component of the Green function coupling of atoms in the first amorphous layer and the plane n (see figure l),

G o ( - l , n ) = q(-1) G(O,O)t(O)t(l).

. .

t ( n ) (A121 where G(0,O) is the Green function in the surface plane obtained via the equations in § 2. Then

.v

I,,

= -2w 1m

i

p ~ ,

2 r(-i,o))

n=O ('413)

where a factor of two has been introduced because there are two sides in the crystal.

References

Alben R , Weaire D, Smith J E and Brodsky M H 1975 Phys. Reu. B 11 2271 Barrio R A and Elliott R J 1982 J . Phys. C: Solid State Phys. 15 4493

Barrio R A . Elliott R J and Thorpe M F 1983 J . Phys. C: Solid State Phys. 16 3425 Carrico A da S 1985 DPhil Thesis Oxford University

Elliott R J, Barrio R A and Thorpe M F 1982 KINAM 42 55

Esaki L 1985 Proc. 17th Int. Conf. Physics of Semiconductors, San Francisco 1985 (Berlin: Springer) p 473 Falicov L M and Yndurain F 1975 J . Phys. C: Solid State Phys. 8 1563

Iqbal 2 and Vepiek S 1982 J . Phys. C: Solid State Phys. 15 377

Kanellis G , Morhange J F and Balkanski M 1980 Phys. Reu. B 21 1543

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Poindexter E H and Caplan P J 1983 Prog. Surf. Sci. 14 201

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