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Transverse acoustic phonons in amorphous silicon
] O U R N A L OF
Journal of Non-Crystalline Solids 141 (1992) 265-270 North-Holland
NON-CRYSTALLINE SOLIDS
Transverse acoustic phonons in amorphous silicon Christian T h o m s e n and E t i e n n e B u s t a r r e t 1 Max-Planck-Institut fiir Festk6rperforschung, Heisenbergstr. 1, W-7000 Stuttgart 80, Germany
The evolution of the broad transverse-acoustic p h o n o n density-of-states peak at ~ 150 cm - I in amorphous silicon produced by introducing increasing damage to a crystal is investigated. By comparison to the phonon density of states as derived from second-order R a m a n scattering on the same samples, and to a theoretical calculation of these branches, it is concluded that the L-branch ([111]-direction) is most susceptible to the loss in periodicity. Further, it is found that at low damage levels p h o n o n s with longer wavelengths contribute more to the p h o n o n density of states than zone-boundary vibrations.
1. Introduction Optical studies have traditionally been important for the investigation of amorphous solids. A classical contribution to this field was made by Tauc, Grigorovici and Vancu and is the discovery that the absorption edge in amorphous Ge may be described by the relation o)2~-z ~ ( h o . ) - E g ) 2. This so-called Tauc edge is nowadays commonly used to describe the absorption behavior of amorphous semiconductors [1]. Later, Wihl et al. [2] employed Raman scattering to deduce structural properties of a-Ge shortly after the work of Smith et al. [3] on a-Si and established that the shortrange order remaining in these amorphous solids suffices to preserve the main features of the phonon density of states of the corresponding crystalline material. In this paper we study the gradual change in the Raman spectra of silicon when a crystal is damaged and transformed into an amorphous solid. Compared with crystalline Si (c-Si), which has only one first-order peak at 521 cm-1 originating from the zone-center Raman-active optical phonon, the Raman spectrum of a-Si consists of a series of four broad bands. Such bands were already observed by Raman in optical glasses [4]. Their origin is well understood; the loss of trans1 P e r m a n e n t address: CNRS-LEPES, Grenoble, France.
lational symmetry in a-Si as compared with c-Si allows all vibrational modes to contribute to the Raman signal, which thus becomes roughly proportional to the phonon density of states (PDOS) [5,6]. The four bands correspond approximately to transverse and longitudinal acoustic (TA and LA) and optical (TO and LO) phonons, as can be seen from a comparison with theoretical calculations of the PDOS (see fig. 1). The T O peak in the amorphous spectra occurs slightly below the frequency of the degenerate zone-center optical phonon of the crystal; it has been used extensively to characterize the amorphous or crystalline character of a sample and shall not be our concern here [7-9]. Rather, we present a study of how the coupling to the acoutic-phonon density of states - as seen in Raman scattering of a-Si develops from that in a crystal under successively higher doses of ion bombardment. We base our discussion of the results on a comparison with the theoretical phonon dispersion relations of Weber [10] and on the two-phonon density of states as obtained in second-order Raman scattering for crystalline silicon. 2. Experimental procedure and sample preparation The Raman spectra were obtained on a Dilor XY triple spectrometer with a multichannel
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
266
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous sificon TO
LO
LO
LA
TA
÷
g TO{ ~
,
I ~ \
~N.''[' \
k 'W)! LA~
/
E
I I
'~(X) I II ~ '
,~-
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', t
t
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I I
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; .,-^ /#a,
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I
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, H/l~i I
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i
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ill
I --
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0
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I
I
~L
''
r
I
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>
iO0
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300
200
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larization of incident and scattered light were parallel in all spectra shown. N o further treatm e n t of spectra was performed; the Bose factor, for instance, was not removed. As we see below in the formulas for the R a m a n efficiency, at high t e m p e r a t u r e s the Bose and 1/o) factor partly c o m p e n s a t e the o92 d e p e n d e n c e of the coupling constant. Five samples were studied, of which one was a (100) crystalline reference sample (p-type, 5 0 0 1000 f~ cm resistivity, float zone), from the same wafer used for the p r e p a r a t i o n of ion-implanted sample. Ion implantation with 28Si was p e r f o r m e d at multiple energies to obtain an approximately h o m o g e n e o u s d a m a g e profile up to a depth of 0.5 ~xm (8% at 15 keV, 26% at 70 keV and 66% at 180 keV). C o m p l e t e a m o r p h i z a t i o n as d e t e r m i n e d from the absence of the first-order R a m a n p e a k at 521 cm -1 was only obtained for the highest implantation level. T h e integrated doses were 5 × 1012, 5 × 1013, 5 X 1014 and 5 × 1015 cm -2.
3. Results
Wove Number (cm -1) Fig. 1. Raman spectra of a-Si (lst order) and c-Si (2nd order) (upper and lower trace). The c-Si abscissa has been divided by two to provide a quantity proportional to the phonon density of states. The vertical dashed lines indicate critical points in the phonon dispersion relations of the crystal and are labelled accordingly. The labelling above the trace of a-Si refers to the predominant contribution to PDOS between pairs of dashed lines as derived from c-Si. The first-order peak in c-Si has been suppressed; its maximum is indicated by an arrow. The phonon dispersion relations have been reproduced after Go [13]. The notation of critical points is as conventional.
charge coupled device ( C C D ) as detector. T h e g o o d sensitivity and stray-light suppression of this system allowed us to m e a s u r e in the range 10-600 cm -1 (the absolute accuracy in the frequency reading was ~ 2 c m - 1 ) ; one spectrum typically accumulated for 1 0 - 2 0 min with an incident laser p o w e r of ~ 40 m W at a wavelength of 530.9 nm. T h e samples were held at r o o m t e m p e r a t u r e in v a c u u m to avoid the air rotational lines below about 150 c m - 1 in the spectra. T h e relative po-
In fig. 1 we c o m p a r e the p h o n o n density of states obtained for silicon from R a m a n data under two different assumptions. T h e u p p e r trace is a first-order spectrum of a-Si and supposes that the R a m a n intensity is proportional to [11]
/am(o)) ~ C ( o ) ) o ) - l ( n
-}- 1)Nd(o)),
(1)
where C is proportional to an average R a m a n susceptibility tensor (OX/O~). T h e other factors are the trivial Bose factor n = [exp(hw/kBT)1] -1 and the desired P D O S Nd(W). C(w) describes thus the coupling of light to the p h o n o n density of states and has to vanish for o)-+ 0 because of translational invariance. Often, in a m o r p h o u s solids, C(o)) ~ o)2 such that [12]
/am(o)) --o)(H q'- 1)Nd(o) ).
(2)
In the case of crystalline silicon, C(o)) is a delta function at the R a m a n frequency. For the damaged, not-fully-amorphous samples of this study, the precise form of C(o)) is not known, and eq. (1) must be used to describe the data. N o t e that
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
in general C(w) may be different for each phonon branch or band and may even be frequency-dependent for a given band. The lower trace is a second-order Raman spectrum of c-Si; while in principle the situation here is quite complicated through the involvement of all combinations of two phonons which yield ql + q2 = 0, it has been shown that overtone scattering, the scattering by two of the same phonons (o) 1 = o92), usually gives a good description of polarized scattering data (i.e. ~s]l ~L) which we will assume for the following. Then, for a q-independent second-order Raman susceptibility, the simplified expression holds for branch i [6]:
/i2-Ph(w) ~ C 2 ~
1 ^/a2x\
2
es/0~/2/ eL ( h i -~- 1) 2
XN(w/2).
(3)
The second-order spectrum of c-Si is hence also proportional to the density of phonon states. The coupling constant C 2 is also often ~ co2. There is a factor of 51 in the frequency of Nd, which we have reflected in the trace of fig. 1 by compressing its abscissa for the crystalline trace to one half of the experimental value. The overall features of Nd(~O) in the two traces are similar, the sharpness of the structures in the lower trace arising from the periodic nature of the crystal. The labelling of the critical points in the figure follows the results of the theoretical dispersion relation calculated from Weber's bond charge model [10] and agrees essentially with previous such assignments [6]. It is clear that a label of a phonon at a critical point in the figure implies a two-phonon process of that phonon for the c-Si trace, whereas for a-Si a single phonon process takes place. Let us focus now on the energy region of the transverse acoustic phonons in Si, i.e., ~o < 230 cm -1. The lowest band in amorphous silicon is centered at ~ 150 cm -x and is almost 100 cm 1 wide (FWHM). It has predominantly transverse acoustic character, because the dispersion of the longitudinal phonons is too high in this energy region for them to contribute substantially to the density of states. The distinct cutoff at 205 cm-x
267
corresponds in the theoretical model to a transverse phonon at the W-point [TA(W)]. In fig. 1 this is seen to be slightly lower than the singularity in the second-order spectrum, which occurs at the K-point and along ~2. On the low-energy side, the T A peak in a-Si falls off monotonically and shows no sign of the L-point in c-Si, where we can detect an abrupt drop in N a below 114 cm -1. Finally, the sharp peak in the c-Si trace at 152 cm-1 corresponds to a two-phonon process at the X-point in the dispersion relations, hence the peak is labelled TA(X). In the a-Si derived spectrum, no strong feature is discernible at that energy. We now turn to the results for (100) samples exposed to different amounts of Si ion bombardment and investigate how the T A peak develops as the structural damage (i.e., defects and disorder) increases. In fig. 2 we show the traces of the four increasingly damaged samples. The spectrum of the lowest-dose sample proves to be quite similar to that of the reference sample in fig. 1, provided that we plot the signal at the energy where it actually occurs in the experiment instead of dividing it by a factor of two. One slight difference, a monotonic increase in intensity below ~ 100 cm-1 in the reference sample, is not believed to be related to the vibrational aspects discussed here, given that it fails to appear in any of the implanted or in other doped samples which we studied [14]. The overtone part of the spectrum is essentially the same as in the reference sample. In the region of the amorphous T A peak (a-TA), a small bump appears with a maximum at about 100 c m - i. At the second level of implantation, a significant a-TA feature has developed, which is about equal in amplitude to the 2TA(X) peak arising from the still-periodic nature of the sample (curve B of fig. 2). The second-order structures in the energy region 200-500 cm 1 have begun to merge with the optical modes. While the 2TA(X) singularity is still present, the other points, 2TA(K) and 2TA(L) are less pronouced than in t h e undamaged reference sample. Further, a new, weak structure appeared at ~ 370 cm-1. For the most damaged sample with still some crystalline character remaining (the 521 cm -1
268
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
4. D i s c u s s i o n
"G I-
>, In
r"
o
E o n-
500
400 300 200 100 R o m a n S h i f t ( c m -1))
0
Fig. 2. Raman spectra of c-Si with increased damage done by ion implantation (curves A-C) and a-Si (curve D). Note the asymmetric increase of the peak associated with transverse acoustic phonons in a-Si and the blurring of the critical-point structure in the second-order part of the spectra.
peak is still present in curve C), the a-TA peak dominates, whereas the sharp structure at 2TA(X) has almost disappeared. The a-TA maximum has shifted up in energy to ~ 120 cm-1 and shows a high-energy shoulder, now resembling much of the peak in the truly amorphous sample. A slight dip separating the shoulder and the maximum occurs at 153 cm -1. At higher energies, on the tail of the optical phonons we observe also the feature at 370 cm -1, in addition to the formerly strong 2TA(X) contribution. The lowest trace is for the case of complete amorphization and similar to that in fig. 1; the a-TA peak has its fully developed shape. The high-energy part of the peak, in the less damaged sample only a shoulder, is now as strong as the low-energy part. Nothing of the second-order component seen in the other samples remains in the amorphous sample.
We first focus our discussion on the experimental results on the transverse acoustic phonons as derived from the peak characteristic of the amorphous phase (a-TA). Then we shall have a brief look at the disappearance of the singularities in the second-order spectrum of the TAphonons. Before comparing the various traces in fig. 2, we have to comment on the Raman intensity scale. The spectra are not normalized relative to each other. This may be done by integrating the T O peak and correcting for the change in absorption depth. Since we are interested here in the structure of the spectra and not their absolute strength, we have abstained from this procedure. The first-order peak at 521 cm - t is ~ 75 times stronger than the maximum at 2TA(X). However, we deduce from the continuous changes in the shape of the a-TA peak for the different samples that the Raman scattering process always occurs in a sample volume which has been 'treated' by the bombardment; i.e., the crystalline component of the traces originates also from a bombarded volume. This agrees with the stopping range of ~ 0.5 ~ m derived from the ion energies, which has to be compared with the absorption coefficient of light in c-Si of ~ (530.9 nm) = 2 × 104 cm -1. The absorption coefficient in a-Si is larger (aa_si = 2 × 10 s cm -1) and those for the bombarded samples fall in between [15]. The zone-center first-order peak of c-Si is present in all spectra except for that with the highest bombarded dose; this one is thus the only fully amorphous sample. From fig. 2 we find that the a-TA peak does not grow simply continuously in intensity with increasing damage as one might naively expect. Rather, for little damage, a coupling to the density of states with a maximum near 100 cm -1 appears (curve A); this is the energy of a phonon in the direction of the L-point in c-Si, which has the lowest frequency of all zone-boundary points in crystalline Si. Experimentally we determine the L-point T A in c-Si as being one half of the Raman shift frequency of the singularity in the second-order spectrum, i.e., 115 cm-1. This agrees
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
well with the value of 115 cm -1 from the theoretical model. Higher-energy acoustic phonons contribute much less; in particular, phonons above the X-point (152 cm -a) do not appear significantly in the samples with the two lowest implantation doses. For more damaged samples, the weight of the a-TA peak shifts towards higher frequencies: the maximum at 120 cm -1 is above the L-point and the center of the a-TA peak lies at ~ 140 cm-1. In particular, there is significant coupling to the density of states above 152 cm -1, the TA(X) phonon frequency. Also, there is a clear cut off of the transverse acoustic phonon strength at 200 cm -1, i.e., some of the highest-energy T A phonons contribute to N a in curve C. Finally, in a-Si, the center of the a-TA peak is at ~ 150 cm -1. The most striking difference in the a-TA peak to that in the damaged, crystalline ones is the strong contribution to the spetrum of phonons with energies between those of the Xand W-points, i.e., between 150 and 205 cm -1. These phonons apparently become disorderactivated predominantly in the amorphous state. Any remaining periodicity in the highly damaged samples suffices to suppress these phonons. It also suffices to allow coherently the zone-center optical phonon at 521 cm -~ in the first-order spectrum (although it is significantly broadened). We thus find an increase in coupling of phonons with an energy between those of the Xand W-point phonons with increasing damage. How can this be? We believe that the reason for such an increase is as follows. The lowerfrequency zone-boundary phonons have more bond-bending character (TA(L)) and the transverse bond-bending vibrations are apparently more sensitive to the loss of long-range order. The vibrations with more stretching character and a higher frequency appear to be less perceptive of the damage as seen for instance in the persistance of the first-order peak. Martin and Galeener have considered a related problem in a-SiO 2 [16]. Further, it depends on the wavelength of a phonon (or its q-vector) compared with the density of defects whether or not it feels a distortion in the lattice. The zone-boundary phonons are thus the last to be affected by in-
269
creasing disorder: longer-wavelength phonons in a particular branch of the Brillouin zone will contribute sooner to the spectra (albeit with much less density of states). Closer than about 71 of a reciprocal lattice vector to the F-point, the density of states of all branches becomes too weak to find significant Raman intensity in the spectra. We conclude this from the contribution to the signal in curves A and B with a maximum below the L-point energy. We also note that the highest-energy in the T A peak, i.e., the cutoff at 205 cm -~, is ~ 10-15 cm -~ lower in energy than the highest energies deduced from the second-order spectrum [2TA(F.) and 2TA(K)]. This is presumably due to a low density of these regions in the Brillouin zone. Alternatively, it could result from an overall decrease in energy of the phonon density of states by about 5 - 7 % in a-Si as compared with c-Si. The longitudinal acoustic phonons in a-Si are expected from the theoretical model in the relatively narrow region between 330 c m - a [LA(L W)] and 400 cm -1 [LA(L)]. We associate the small bump at 370 cm-1, which we find reproducibly in curve B, to the L A at the K-point in c-Si. Finally, we briefly discuss the second-order part of the Raman spectra in fig. 2. The 2TA(X) singularity is maintained up to the highest damage level (curve C), while the E-point minimum (the highest T A frequency in the crystal), the 2TA(K) and 2TA(L) features have been smeared out in curve C. The behavior of the 2TA(L) phonon is consistent with the proposed explanation for the asymmetric increase in the a-TA peak. The phonons at the L-point contribute most to it for low damage levels and, correspondingly, they have been blurred from the second-order spectrum due to their stronger sensitivity to the loss of translational symmetry.
5. Summary We have studied systematically the effect of the transition from crystalline to amorphous silicon on the density of state for transverse acoustic phonons as seen in Raman scattering. The peak
270
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
in the Raman spectra at 150 cm-1 was found to grow asymmetrically with increased damage done to the sample. We attribute this to a combination of two effects: one is the difference in sensitivity to damage for vibrations with different q-directions; the second effect is the length scale of the disorder; phonons in the middle of the Brillouin zone appear to become activated before the zone-boundary ones. We have given a qualitative reason for a microscopic picture why this may be so and found it to be consistent with the observed blurring in the second-order Raman signal in the same samples. According to this explanation, TA phonons near the L-point of the crystal feel stronger the loss of translational symmetry in a damaged sample than other zone-boundary TA phonons. Calculations for 'stick and balls' modes of the damaged samples would be of interest. We owe special thanks to M. Stutzmann for discussions on these issues and to him and M. Cardona for a critical reading of this manuscript. We are grateful to C. Summonte at the CNR LAMEL, Bologna for providing the ion-implanted samples.
References [1] J. Tauc, R. Grigorovici and A. Vancu, Phys. Status Solidi 15 (1966) 627.
[2] M. Wihl, M. Cardona and J. Tauc, J. Non-Cryst. Solids 8-10 (1972) 172. [3] J.E. Smith Jr., M.H. Brodsky, B.L. Crowder, M.I. Nathan and A. Pinczuk, Phys. Rev. Lett. 26 (1971) 642. [4] C.V. Raman, Inaugural Address, delivered to the South India Science Association, Bangalore, 1928, in: The Scattering of Light, ed. C.V. Raman (Indian Academy of Sciences, Bangalore, 1978) p. 467. [5] M.H. Brodsky, in: Light Scattering in Solids I, ed. M. Cardona, Topics in Applied Physics, 2nd Ed., Vol. 8 (Springer, Heidelberg, 1983) p. 205. [6] M. Cardona, in: Light Scattering in Solids II, eds. M. Cardona and G. Giintherodt, Topics in Applied Physics, Vol. 50 (Springer, Heidelberg, 1982) p. 76 ft. [7] L.J. Pillone, N. Maley, N. Lustig and J.S. Lannin, J. Vac. Sci. Technol. A1 (1983) 388. [8] V.Kh. Kudoyarova, O.I. Konkov, E.I. Terukov, A.P. Sokolov and A.P. Shebanin, J. Non-Cryst. Solids 114 (1989) 205. [9] H.S. Mavi, K.P. Jain, A.K. Shukla, S.C. Abbi and R. Beserman, J. Appl. Phys. 69 (1991) 3696. [10] W. Weber, Phys. Rev. B15 (1977) 4789. [11] R. Shuker and R.W. Gammon, Phys. Rev. Lett. 25 (1970) 222. [12] J. Lannin, Solid State Commun. 12 (1973) 947. [13] S. Go, PhD thesis, University of Stuttgart (1975). [14] C. Thomsen, unpublished results. [15] M. Stutzmann, private communication. [16] R.M. Martin and F.L Galeener, Phys. Rev. B23 (1981) 3071.
Journal of Non-Crystalline Solids 141 (1992) 265-270 North-Holland
NON-CRYSTALLINE SOLIDS
Transverse acoustic phonons in amorphous silicon Christian T h o m s e n and E t i e n n e B u s t a r r e t 1 Max-Planck-Institut fiir Festk6rperforschung, Heisenbergstr. 1, W-7000 Stuttgart 80, Germany
The evolution of the broad transverse-acoustic p h o n o n density-of-states peak at ~ 150 cm - I in amorphous silicon produced by introducing increasing damage to a crystal is investigated. By comparison to the phonon density of states as derived from second-order R a m a n scattering on the same samples, and to a theoretical calculation of these branches, it is concluded that the L-branch ([111]-direction) is most susceptible to the loss in periodicity. Further, it is found that at low damage levels p h o n o n s with longer wavelengths contribute more to the p h o n o n density of states than zone-boundary vibrations.
1. Introduction Optical studies have traditionally been important for the investigation of amorphous solids. A classical contribution to this field was made by Tauc, Grigorovici and Vancu and is the discovery that the absorption edge in amorphous Ge may be described by the relation o)2~-z ~ ( h o . ) - E g ) 2. This so-called Tauc edge is nowadays commonly used to describe the absorption behavior of amorphous semiconductors [1]. Later, Wihl et al. [2] employed Raman scattering to deduce structural properties of a-Ge shortly after the work of Smith et al. [3] on a-Si and established that the shortrange order remaining in these amorphous solids suffices to preserve the main features of the phonon density of states of the corresponding crystalline material. In this paper we study the gradual change in the Raman spectra of silicon when a crystal is damaged and transformed into an amorphous solid. Compared with crystalline Si (c-Si), which has only one first-order peak at 521 cm-1 originating from the zone-center Raman-active optical phonon, the Raman spectrum of a-Si consists of a series of four broad bands. Such bands were already observed by Raman in optical glasses [4]. Their origin is well understood; the loss of trans1 P e r m a n e n t address: CNRS-LEPES, Grenoble, France.
lational symmetry in a-Si as compared with c-Si allows all vibrational modes to contribute to the Raman signal, which thus becomes roughly proportional to the phonon density of states (PDOS) [5,6]. The four bands correspond approximately to transverse and longitudinal acoustic (TA and LA) and optical (TO and LO) phonons, as can be seen from a comparison with theoretical calculations of the PDOS (see fig. 1). The T O peak in the amorphous spectra occurs slightly below the frequency of the degenerate zone-center optical phonon of the crystal; it has been used extensively to characterize the amorphous or crystalline character of a sample and shall not be our concern here [7-9]. Rather, we present a study of how the coupling to the acoutic-phonon density of states - as seen in Raman scattering of a-Si develops from that in a crystal under successively higher doses of ion bombardment. We base our discussion of the results on a comparison with the theoretical phonon dispersion relations of Weber [10] and on the two-phonon density of states as obtained in second-order Raman scattering for crystalline silicon. 2. Experimental procedure and sample preparation The Raman spectra were obtained on a Dilor XY triple spectrometer with a multichannel
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
266
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous sificon TO
LO
LO
LA
TA
÷
g TO{ ~
,
I ~ \
~N.''[' \
k 'W)! LA~
/
E
I I
'~(X) I II ~ '
,~-
J[ ', (F)
I
~', I
i
I
I
', t
t
150,0 I 400 I ii
I I
TA (W)~
t
I
I
L ~ TA X t '
I
TA(Z -~
; .,-^ /#a,
',1I
I
~I
1300
12,00
I
III iii
' /--W" I
, H/l~i I
I
i
/
I
ill
I --
A iI
f~ ~ T A ( L )
100 ,
0
×
I I
[-
I
I
~L
''
r
I
'1
L
>
iO0
400
300
200
100
larization of incident and scattered light were parallel in all spectra shown. N o further treatm e n t of spectra was performed; the Bose factor, for instance, was not removed. As we see below in the formulas for the R a m a n efficiency, at high t e m p e r a t u r e s the Bose and 1/o) factor partly c o m p e n s a t e the o92 d e p e n d e n c e of the coupling constant. Five samples were studied, of which one was a (100) crystalline reference sample (p-type, 5 0 0 1000 f~ cm resistivity, float zone), from the same wafer used for the p r e p a r a t i o n of ion-implanted sample. Ion implantation with 28Si was p e r f o r m e d at multiple energies to obtain an approximately h o m o g e n e o u s d a m a g e profile up to a depth of 0.5 ~xm (8% at 15 keV, 26% at 70 keV and 66% at 180 keV). C o m p l e t e a m o r p h i z a t i o n as d e t e r m i n e d from the absence of the first-order R a m a n p e a k at 521 cm -1 was only obtained for the highest implantation level. T h e integrated doses were 5 × 1012, 5 × 1013, 5 X 1014 and 5 × 1015 cm -2.
3. Results
Wove Number (cm -1) Fig. 1. Raman spectra of a-Si (lst order) and c-Si (2nd order) (upper and lower trace). The c-Si abscissa has been divided by two to provide a quantity proportional to the phonon density of states. The vertical dashed lines indicate critical points in the phonon dispersion relations of the crystal and are labelled accordingly. The labelling above the trace of a-Si refers to the predominant contribution to PDOS between pairs of dashed lines as derived from c-Si. The first-order peak in c-Si has been suppressed; its maximum is indicated by an arrow. The phonon dispersion relations have been reproduced after Go [13]. The notation of critical points is as conventional.
charge coupled device ( C C D ) as detector. T h e g o o d sensitivity and stray-light suppression of this system allowed us to m e a s u r e in the range 10-600 cm -1 (the absolute accuracy in the frequency reading was ~ 2 c m - 1 ) ; one spectrum typically accumulated for 1 0 - 2 0 min with an incident laser p o w e r of ~ 40 m W at a wavelength of 530.9 nm. T h e samples were held at r o o m t e m p e r a t u r e in v a c u u m to avoid the air rotational lines below about 150 c m - 1 in the spectra. T h e relative po-
In fig. 1 we c o m p a r e the p h o n o n density of states obtained for silicon from R a m a n data under two different assumptions. T h e u p p e r trace is a first-order spectrum of a-Si and supposes that the R a m a n intensity is proportional to [11]
/am(o)) ~ C ( o ) ) o ) - l ( n
-}- 1)Nd(o)),
(1)
where C is proportional to an average R a m a n susceptibility tensor (OX/O~). T h e other factors are the trivial Bose factor n = [exp(hw/kBT)1] -1 and the desired P D O S Nd(W). C(w) describes thus the coupling of light to the p h o n o n density of states and has to vanish for o)-+ 0 because of translational invariance. Often, in a m o r p h o u s solids, C(o)) ~ o)2 such that [12]
/am(o)) --o)(H q'- 1)Nd(o) ).
(2)
In the case of crystalline silicon, C(o)) is a delta function at the R a m a n frequency. For the damaged, not-fully-amorphous samples of this study, the precise form of C(o)) is not known, and eq. (1) must be used to describe the data. N o t e that
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
in general C(w) may be different for each phonon branch or band and may even be frequency-dependent for a given band. The lower trace is a second-order Raman spectrum of c-Si; while in principle the situation here is quite complicated through the involvement of all combinations of two phonons which yield ql + q2 = 0, it has been shown that overtone scattering, the scattering by two of the same phonons (o) 1 = o92), usually gives a good description of polarized scattering data (i.e. ~s]l ~L) which we will assume for the following. Then, for a q-independent second-order Raman susceptibility, the simplified expression holds for branch i [6]:
/i2-Ph(w) ~ C 2 ~
1 ^/a2x\
2
es/0~/2/ eL ( h i -~- 1) 2
XN(w/2).
(3)
The second-order spectrum of c-Si is hence also proportional to the density of phonon states. The coupling constant C 2 is also often ~ co2. There is a factor of 51 in the frequency of Nd, which we have reflected in the trace of fig. 1 by compressing its abscissa for the crystalline trace to one half of the experimental value. The overall features of Nd(~O) in the two traces are similar, the sharpness of the structures in the lower trace arising from the periodic nature of the crystal. The labelling of the critical points in the figure follows the results of the theoretical dispersion relation calculated from Weber's bond charge model [10] and agrees essentially with previous such assignments [6]. It is clear that a label of a phonon at a critical point in the figure implies a two-phonon process of that phonon for the c-Si trace, whereas for a-Si a single phonon process takes place. Let us focus now on the energy region of the transverse acoustic phonons in Si, i.e., ~o < 230 cm -1. The lowest band in amorphous silicon is centered at ~ 150 cm -x and is almost 100 cm 1 wide (FWHM). It has predominantly transverse acoustic character, because the dispersion of the longitudinal phonons is too high in this energy region for them to contribute substantially to the density of states. The distinct cutoff at 205 cm-x
267
corresponds in the theoretical model to a transverse phonon at the W-point [TA(W)]. In fig. 1 this is seen to be slightly lower than the singularity in the second-order spectrum, which occurs at the K-point and along ~2. On the low-energy side, the T A peak in a-Si falls off monotonically and shows no sign of the L-point in c-Si, where we can detect an abrupt drop in N a below 114 cm -1. Finally, the sharp peak in the c-Si trace at 152 cm-1 corresponds to a two-phonon process at the X-point in the dispersion relations, hence the peak is labelled TA(X). In the a-Si derived spectrum, no strong feature is discernible at that energy. We now turn to the results for (100) samples exposed to different amounts of Si ion bombardment and investigate how the T A peak develops as the structural damage (i.e., defects and disorder) increases. In fig. 2 we show the traces of the four increasingly damaged samples. The spectrum of the lowest-dose sample proves to be quite similar to that of the reference sample in fig. 1, provided that we plot the signal at the energy where it actually occurs in the experiment instead of dividing it by a factor of two. One slight difference, a monotonic increase in intensity below ~ 100 cm-1 in the reference sample, is not believed to be related to the vibrational aspects discussed here, given that it fails to appear in any of the implanted or in other doped samples which we studied [14]. The overtone part of the spectrum is essentially the same as in the reference sample. In the region of the amorphous T A peak (a-TA), a small bump appears with a maximum at about 100 c m - i. At the second level of implantation, a significant a-TA feature has developed, which is about equal in amplitude to the 2TA(X) peak arising from the still-periodic nature of the sample (curve B of fig. 2). The second-order structures in the energy region 200-500 cm 1 have begun to merge with the optical modes. While the 2TA(X) singularity is still present, the other points, 2TA(K) and 2TA(L) are less pronouced than in t h e undamaged reference sample. Further, a new, weak structure appeared at ~ 370 cm-1. For the most damaged sample with still some crystalline character remaining (the 521 cm -1
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C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
4. D i s c u s s i o n
"G I-
>, In
r"
o
E o n-
500
400 300 200 100 R o m a n S h i f t ( c m -1))
0
Fig. 2. Raman spectra of c-Si with increased damage done by ion implantation (curves A-C) and a-Si (curve D). Note the asymmetric increase of the peak associated with transverse acoustic phonons in a-Si and the blurring of the critical-point structure in the second-order part of the spectra.
peak is still present in curve C), the a-TA peak dominates, whereas the sharp structure at 2TA(X) has almost disappeared. The a-TA maximum has shifted up in energy to ~ 120 cm-1 and shows a high-energy shoulder, now resembling much of the peak in the truly amorphous sample. A slight dip separating the shoulder and the maximum occurs at 153 cm -1. At higher energies, on the tail of the optical phonons we observe also the feature at 370 cm -1, in addition to the formerly strong 2TA(X) contribution. The lowest trace is for the case of complete amorphization and similar to that in fig. 1; the a-TA peak has its fully developed shape. The high-energy part of the peak, in the less damaged sample only a shoulder, is now as strong as the low-energy part. Nothing of the second-order component seen in the other samples remains in the amorphous sample.
We first focus our discussion on the experimental results on the transverse acoustic phonons as derived from the peak characteristic of the amorphous phase (a-TA). Then we shall have a brief look at the disappearance of the singularities in the second-order spectrum of the TAphonons. Before comparing the various traces in fig. 2, we have to comment on the Raman intensity scale. The spectra are not normalized relative to each other. This may be done by integrating the T O peak and correcting for the change in absorption depth. Since we are interested here in the structure of the spectra and not their absolute strength, we have abstained from this procedure. The first-order peak at 521 cm - t is ~ 75 times stronger than the maximum at 2TA(X). However, we deduce from the continuous changes in the shape of the a-TA peak for the different samples that the Raman scattering process always occurs in a sample volume which has been 'treated' by the bombardment; i.e., the crystalline component of the traces originates also from a bombarded volume. This agrees with the stopping range of ~ 0.5 ~ m derived from the ion energies, which has to be compared with the absorption coefficient of light in c-Si of ~ (530.9 nm) = 2 × 104 cm -1. The absorption coefficient in a-Si is larger (aa_si = 2 × 10 s cm -1) and those for the bombarded samples fall in between [15]. The zone-center first-order peak of c-Si is present in all spectra except for that with the highest bombarded dose; this one is thus the only fully amorphous sample. From fig. 2 we find that the a-TA peak does not grow simply continuously in intensity with increasing damage as one might naively expect. Rather, for little damage, a coupling to the density of states with a maximum near 100 cm -1 appears (curve A); this is the energy of a phonon in the direction of the L-point in c-Si, which has the lowest frequency of all zone-boundary points in crystalline Si. Experimentally we determine the L-point T A in c-Si as being one half of the Raman shift frequency of the singularity in the second-order spectrum, i.e., 115 cm-1. This agrees
C. Thomsen, E. Bustarret / Transverse acoustic phonons in amorphous silicon
well with the value of 115 cm -1 from the theoretical model. Higher-energy acoustic phonons contribute much less; in particular, phonons above the X-point (152 cm -a) do not appear significantly in the samples with the two lowest implantation doses. For more damaged samples, the weight of the a-TA peak shifts towards higher frequencies: the maximum at 120 cm -1 is above the L-point and the center of the a-TA peak lies at ~ 140 cm-1. In particular, there is significant coupling to the density of states above 152 cm -1, the TA(X) phonon frequency. Also, there is a clear cut off of the transverse acoustic phonon strength at 200 cm -1, i.e., some of the highest-energy T A phonons contribute to N a in curve C. Finally, in a-Si, the center of the a-TA peak is at ~ 150 cm -1. The most striking difference in the a-TA peak to that in the damaged, crystalline ones is the strong contribution to the spetrum of phonons with energies between those of the Xand W-points, i.e., between 150 and 205 cm -1. These phonons apparently become disorderactivated predominantly in the amorphous state. Any remaining periodicity in the highly damaged samples suffices to suppress these phonons. It also suffices to allow coherently the zone-center optical phonon at 521 cm -~ in the first-order spectrum (although it is significantly broadened). We thus find an increase in coupling of phonons with an energy between those of the Xand W-point phonons with increasing damage. How can this be? We believe that the reason for such an increase is as follows. The lowerfrequency zone-boundary phonons have more bond-bending character (TA(L)) and the transverse bond-bending vibrations are apparently more sensitive to the loss of long-range order. The vibrations with more stretching character and a higher frequency appear to be less perceptive of the damage as seen for instance in the persistance of the first-order peak. Martin and Galeener have considered a related problem in a-SiO 2 [16]. Further, it depends on the wavelength of a phonon (or its q-vector) compared with the density of defects whether or not it feels a distortion in the lattice. The zone-boundary phonons are thus the last to be affected by in-
269
creasing disorder: longer-wavelength phonons in a particular branch of the Brillouin zone will contribute sooner to the spectra (albeit with much less density of states). Closer than about 71 of a reciprocal lattice vector to the F-point, the density of states of all branches becomes too weak to find significant Raman intensity in the spectra. We conclude this from the contribution to the signal in curves A and B with a maximum below the L-point energy. We also note that the highest-energy in the T A peak, i.e., the cutoff at 205 cm -~, is ~ 10-15 cm -~ lower in energy than the highest energies deduced from the second-order spectrum [2TA(F.) and 2TA(K)]. This is presumably due to a low density of these regions in the Brillouin zone. Alternatively, it could result from an overall decrease in energy of the phonon density of states by about 5 - 7 % in a-Si as compared with c-Si. The longitudinal acoustic phonons in a-Si are expected from the theoretical model in the relatively narrow region between 330 c m - a [LA(L W)] and 400 cm -1 [LA(L)]. We associate the small bump at 370 cm-1, which we find reproducibly in curve B, to the L A at the K-point in c-Si. Finally, we briefly discuss the second-order part of the Raman spectra in fig. 2. The 2TA(X) singularity is maintained up to the highest damage level (curve C), while the E-point minimum (the highest T A frequency in the crystal), the 2TA(K) and 2TA(L) features have been smeared out in curve C. The behavior of the 2TA(L) phonon is consistent with the proposed explanation for the asymmetric increase in the a-TA peak. The phonons at the L-point contribute most to it for low damage levels and, correspondingly, they have been blurred from the second-order spectrum due to their stronger sensitivity to the loss of translational symmetry.
5. Summary We have studied systematically the effect of the transition from crystalline to amorphous silicon on the density of state for transverse acoustic phonons as seen in Raman scattering. The peak
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in the Raman spectra at 150 cm-1 was found to grow asymmetrically with increased damage done to the sample. We attribute this to a combination of two effects: one is the difference in sensitivity to damage for vibrations with different q-directions; the second effect is the length scale of the disorder; phonons in the middle of the Brillouin zone appear to become activated before the zone-boundary ones. We have given a qualitative reason for a microscopic picture why this may be so and found it to be consistent with the observed blurring in the second-order Raman signal in the same samples. According to this explanation, TA phonons near the L-point of the crystal feel stronger the loss of translational symmetry in a damaged sample than other zone-boundary TA phonons. Calculations for 'stick and balls' modes of the damaged samples would be of interest. We owe special thanks to M. Stutzmann for discussions on these issues and to him and M. Cardona for a critical reading of this manuscript. We are grateful to C. Summonte at the CNR LAMEL, Bologna for providing the ion-implanted samples.
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[2] M. Wihl, M. Cardona and J. Tauc, J. Non-Cryst. Solids 8-10 (1972) 172. [3] J.E. Smith Jr., M.H. Brodsky, B.L. Crowder, M.I. Nathan and A. Pinczuk, Phys. Rev. Lett. 26 (1971) 642. [4] C.V. Raman, Inaugural Address, delivered to the South India Science Association, Bangalore, 1928, in: The Scattering of Light, ed. C.V. Raman (Indian Academy of Sciences, Bangalore, 1978) p. 467. [5] M.H. Brodsky, in: Light Scattering in Solids I, ed. M. Cardona, Topics in Applied Physics, 2nd Ed., Vol. 8 (Springer, Heidelberg, 1983) p. 205. [6] M. Cardona, in: Light Scattering in Solids II, eds. M. Cardona and G. Giintherodt, Topics in Applied Physics, Vol. 50 (Springer, Heidelberg, 1982) p. 76 ft. [7] L.J. Pillone, N. Maley, N. Lustig and J.S. Lannin, J. Vac. Sci. Technol. A1 (1983) 388. [8] V.Kh. Kudoyarova, O.I. Konkov, E.I. Terukov, A.P. Sokolov and A.P. Shebanin, J. Non-Cryst. Solids 114 (1989) 205. [9] H.S. Mavi, K.P. Jain, A.K. Shukla, S.C. Abbi and R. Beserman, J. Appl. Phys. 69 (1991) 3696. [10] W. Weber, Phys. Rev. B15 (1977) 4789. [11] R. Shuker and R.W. Gammon, Phys. Rev. Lett. 25 (1970) 222. [12] J. Lannin, Solid State Commun. 12 (1973) 947. [13] S. Go, PhD thesis, University of Stuttgart (1975). [14] C. Thomsen, unpublished results. [15] M. Stutzmann, private communication. [16] R.M. Martin and F.L Galeener, Phys. Rev. B23 (1981) 3071.