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Lattice dilatation of small silicon crystallites - implications for amorphous silicon
Solid State Communications, Vol. 39, pp. 509-512. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/270509-04502.00/0
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES - IMPLICATIONS FOR AMORPHOUS SILICON S. Vep~ek, Z. Iqbal, H.R. Oswald, F.-A. Sarott, J.J. Wagner and A.P. Webb Institute of Inorganic Chemistry, University of Ztirich, Winterthurerstrasse 190, 8057 Ziirich, Switzerland
(Received 30 January 1981 by P. Wachter) The lattice constant, d, of polycrystalline, hydrogenated silicon with a mean crystallite size less than 100 )~ shows an expansion which increases with decreasing crystallite size, reaching a limiting value of Ad/do ,~ 0.9 - 1.2% for a crystallite size of about 30 A. Implications of this result for some properties of amorphous silicon are briefly discussed. 1. INTRODUCTION THE PRESENT PAPER deals with the crystal lattice dilatation of microcrystalline silicon. The preparative method used [1,2] allows one to controllably deposit hydrogenated polycrystalline silicon of various crystallite sizes down to about 30 A. This enabled us to perform fairly accurate measurements of the lattice expansion and obtain an estimate of the nearest neighbour distances in X-ray amorphous silicon. It will be shown that the lattice expansion amounts to 1.1% for polycrystalline-Si with crystallite size of 30 A which, when extrapolated to a microcrystalline, X-ray amorphous Si, would indicate an increase of the nearest-neighbour distances of 1.2-1.5%. Such a change can hardly be reliably determined by radial distribution function (RDF) methods. It might contribute to the increase in the band gap of amorphous Si as compared to c-Si. It is generally known that amorphous Si films have stress but no systematic study of this has been done so far. With this in view, we have studied the effect of stress on the lattice expansion and on the Raman spectra of the polycrystalline Si f'dms. 2. EXPERIMENTAL Polycrystalline silicon films of thickness ranging between a few 1000 A and > 20/am were prepared by chemical transport in a d.c. hydrogen plasma at low pressures. This method provides an excellent control of the deposition conditions which allows one to deposit polycrystalline silicon of variable crystallite size as well as X-ray amorphous films. The preparation method was first applied to Si and Ge by Vep~ek and Maregek [1] in 1968 and discussed in some detail more recently [2]. The physical properties of the polyerystalline films can be found in references [ 3 - 5 ] and further details on the plasma parameters controlling the structural properties of the films (i.e. polycrystaUine vs X-ray amorphous) will be published later.
The films were deposited on various substrates such as silica and silicate glass, sapphire, molybdenum-, aluminium- and stainless steel sheets of thickness ranging between ~ 40 and 200/am, amorphous metals (foils of 40/am thickness) and on single crystal silicon chips in the (11 l) orientation. The X-ray diffraction measurements were done with a powder diffractometer Picker Type 3488 K using the 20-scan at a fixed 0 angle, i.e. at a constant angle between the incident X-rays and the sample plane. This procedure also allows one to obtain the X-ray diffraction pattern of a few thousand A thick polycrystalline film deposited on a single crystal Si chip [4]. It was verified that the results to be reported here were independent of the angle chosen within reasonable limits, as well as the nature of the substrate used. The crystallite size has been evaluated from the width of the (111) and (220) peaks using the Scherrer formula [6] with a shape factor of 0.94. The ratio of the apparent crystallite sizes calculated from the (111) and (220) peaks of x/3 : ~/2 is indicative of a regular shape of the crystallites. Thus, a cube edge, a = a(111)/~/3 = a(220)/x/2 is used in the following part of this paper. Here, a(111) and a(220) are the apparent crystallite sizes calculated from the widths of the (111) and (220) peaks respectively, i.e. the extension of the crystallites in the direction perpendicular to the scattering planes. We have also verified that the instrumental broadening was negligible and the diffraction peaks were symmetrical. For the exact determination of the lattice constants of the small crystallites we have used crystalline silicon powder (Ventron, 100 mesh, purity better than 99.99%) ground to a size of ~ 30/am and dispersed directly on the polycrystalline film to be measured. CVD deposited films usually possess either compressive or tensile stress [7] which has to be accounted for in order to obtain a correlation between the lattice expansion and the crystallite size. However, polycrystalline films deposited at a floating potential did not show
509
510
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES
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Vol. 39, No. 3
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Fig. 1. Correlation between the relative lattice expansion Ad/do, and the crystallite size, a. do = 5.425 A is the lattice constant of silicon and Ad is the absolute value of the expansion. Filled circles represent samples deposited between ~ 60 and 260°C at a floating potential and open circles represent samples deposited between ~ 80 and 260°C under varying bias. Broken line shows the data of Hamasaki et al. [8] for Si crystalrites in a SiOx matrix. any observable stress. In addition, we compared in several cases the lattice expansion of an as-deposited film with the same one after dissolving the substrate and found identical results. Annealing of the samples under ultra-high vacuum at temperatures between "~ 300 and 800°C results in some enlargement of the crystallite size for films with a <~ 80 A and a relatively very small increase for films with a >~ 100 ~&, and a partial or complete evolution of hydrogen [4], but the correlation between the lattice expansion and the crystallite size remains within the experimental error the same as for the as-deposited samples (see below). Thus, the results presented here are believed to be representative of pure, hydrogenated silicon films which are free of stress (Fig. 1, solid line). Some samples were deposited also under a bias which was applied to the sample holder by means of an auxiliary power supply. The actual plasma potential at the position of the sample holder as well as the chosen value of the bias were determined from the Langmuir characteristic of the sample holder used as a probe. Samples deposited under a negative bias display a compressive stress and, as one would expect, a smaller lattice expansion as compared to the stress-free samples of the same crystallite size deposited at a floating potential (see below). 3. RESULTS AND DISCUSSION Figure 1 shows the correlation between the relative lattice expansion, Ad/do, and the crystallite size, a. For trims deposited at a floating potential (full circles,
s12
si4
si6
5i~
"
~2o
Fig. 2. Correlation between the relative lattice expansion, Ad/do, and the peak position of the k ~ 0 phonon frequency of the polycrystalline films. (See captions to Fig. I for further details.) Samples deposited under a negative bias o f up to ~ 200 V correlate w i t h the stressfree ones (open circles on the full line), those deposited under a negative bias exceeding ~ 300 V show a peak frequency o f ~ 519 cm -1 .
Fig. 1) the correlation extrapolates to (Ad/do) 1.2-1.5% for microcrystalline material with crystalrite size ~ 10 A. Films deposited under negative bias (i.e. under bombardment by positive ions) show somewhat smaller lattice expansion as compared with those deposited at a floating potential (see Fig. 1, open circles). The relatively large scattering of the data for these samples is due to the fact that the samples with different crystallite sizes were deposited at different values of the bias, showing for a given crystallite size, a decrease of the lattice expansion with increasing absolute value of the negative bias. Because of the lack of knowledge of the energy of the impinging ions we did not try to obtain a more complete set of data in the present study. The few examples shown in Fig. 1 are presented here only to illustrate the effect of the compressive stress in the films on the lattice expansion of small crystallites in order to warn against an overinterpretation of our results and conclusions. This example also illustrates that care has to be taken when comparing data on amorphous silicon since samples deposited under a high bias (i.e. the "cathode" of an r.f. discharge) may possess a larger stress than those deposited at a low bias (e.g. the "anode" of an r.f. discharge). Hamasaki et al. [8] have also found lattice expansion of silicon crystallites embedded in a matrix of silicon suboxide. Their experimental results can be fitted by an empirical formula
Ad/do = 0.17/(a -- 8.8) and are indicated by the broken line in Fig. 1.
Vol. 39, No. 3
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES
Qualitatively, the results of Hamasaki et al. agree with ours in showing increasing lattice expansion with decreasing crystallite size but disagree in some details, the divergence of Ad/do for crystallite size about 8.8 A being the most pronounced one. It is surely not unreasonable to seek the origin of the latter difference in the different environments of the Si crystallites in their and in our case. We have shown in [4] and [5] that our polycrystalline Si samples are remarkably pure, that the only "amorphous-like" matrix in which the crystallites are embedded are their grain boundaries [5], and, last but not least, that their hydrogen content is very much the same as in hydrogenated a-Si. In our previous paper [5] we have shown that the crystalline component of the Raman spectra of the polycrystalline silicon displays a shift towards smaller frequencies with decreasing crystallite size. It is seen from Fig. 2 that this shift correlates with the lattice expansion determined from the X-ray diffraction. In fact this correlation is even better than that of Fig. 1 since the data from samples 4eposited under negative bias not exceeding -- 200 V also fit on the same curve with those deposited at a floating potential. Samples deposited at a negative bias greater than ~ 300 V however show a jump in frequency to ~ 519 cm -1 . More detailed study of this effect is in progress. Obviously, these results show that the decreasing frequency of the crystalline component in the Raman spectrum with decreasing crystallite size is due largely to an increase in the Si-Si bond distance rather than to a progressive relaxation of the k ~ 0 selection rule as suggested earlier [5]. For the purpose of the present work we conclude that the results presented by Figs. 1 and 2 are mutually consistent and convincingly illustrate that the measured lattice expansion reflects the increase of the Si-Si distances which should reach some limiting value of about 1.2-1.5% for a stress-free, X-ray amorphous silicon film of a microcrystalline nature and somewhat less or more for films under compressive or tensile stress respectively. This value can be somewhat modified if the topology of the amorphous network is different from that of the diamond lattice. Nevertheless, even if some cluster model is assumed* some expansion of the Si-Si distances is to be expected (see below).
* For example, the interesting topological theory of J.C. Phillips showing that a continuous random network o f four-fold coordinated elemental solid cannot exist in three-dimensional space [9], the EXAFS study of Evangelisti et al. on G e [ I0] indicating a change of the third peak upon the transition from crystalline to amorphous material and our Raman spectroscopic measurements on polycrystalline- and amorphous-Si [5, 11].
511
We have to point out that the data of Fig. 1. but not those of Fig. 2 have to be used for the extrapolation to a < 30 A since the frequency of the crystalline component of the Raman spectra does not scale down to any well-defined limiting value with decreasing crystalline size. As indicated earlier [5] and will be discussed in some detail in a paper to follow [11] the feature at 480 cm- 1 in the Raman spectrum from a-Si should be associated with a totally symmetric vibrational mode of small tetrahedral Si clusters. A shift of the first peak in the radial distribution function (RDF) ofa-Si and a-Ge indicating a comparable lattice expansion has been occasionally reported in the literature (e.g. [12]), the actual value being a function of the preparation conditions. Indications for expansion of up to several percent were reported also for some other amorphous materials (e.g. [6, 12, 23]). In general, the accuracy of the position of the first peak in the RDF is limited to several percent due to experimental and computational limitations [6, 24] and a shift of the same amount cannot be reliably established by this method [6, 12, 24, 26]. In some cases, nearest-neighbour distances in a-Si were reported to equal to those of c-Si, although the first peak in the corresponding RDF was broad (FWHM ~ 0.4 ~,) and had a relatively large curvature on the top [25]. The increase of the Si-Si bond distance reported in this paper is also supported by the theoretical studies of the structure and stability of small covalently bonded clusters [13, 14]. For a cluster size of ~ 10 A the calculation by Allpress and Sanders gives an expansion of nearest-neighbour distance between 3 and 8% depending on the choice of the cluster structure and of the pair interaction potential [13]. Possibly, the mutual interaction on grain boundaries reduces this expansion in the polycrystalline film. In films with a high hydrogen content (e.g. > 12 at.%) the latter is preferentially bonded on the grain boundaries [4, 22] and it can probably reduce their mutual interaction. Both crystalline [15] and amorphous [16] silicon display a decrease of the optical band gap with increasing pressure. One could speculate that the observed lattice expansion may cause certain contribution to the observed increase of the optical gap in a-Si as compared with the crystalline material. However, it is difficult to give some quantitative estimate at this stage. At this point a remark on the electronic properties of small clusters is appropriate. In general, one finds a more or less continuous transition from the properties typical o f a bulk towards that of molecules and atoms with decreasing cluster size. The photoionization potential of isolated Na, K and mixed clusters formed in a molecular beam [ 17] display a similar shift as the position of the valence band of Pd clusters supported on
512
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES
silica. In the latter case, the Pd d-band narrows and both its threshold and centre-of-gravity shift further from the Fermi level with decreasing cluster size [ 18-21 ]. Such shifts, which in isolated or supported clusters can amount to a few eV might have some relevance to the increase in the band gap for an amorphous material, particularly if the latter were composed of clusters with a relatively weak mutual interaction on their boundaries (see, e.g. [27]). However, it is difficult to give some quantitative estimate without detailed calculations. Another question arises as to the role of the lattice expansion of the microcrystalline silicon in the transition to the amorphous state. Such an expansion reflects an increase of the free energy of the microcrystalline material with decreasing crystallite size and it probably reflects the increase of the specific surface energy of the microcrystallites. Preliminary estimates indicate that this could set up a lower limit for the crystallite size of the microcrystaUine material making it unstable with respect to an amorphous phase having a different structural topology. This suggestion should be regarded as very preliminary and it is presently under more detailed study [28]. 4. CONCLUSIONS Fine grain, polycrystalline silicon shows lattice dilatation increasing with decreasing crystallite size to a limiting value of Ad/do ~ 0.9 -- 1.2% for a crystallite size o f ~ 30A (Fig. 1). A compressive or tensile stress in the films reduces or enhances this dilatation (Fig. 1). The lattice dilatation correlates with the position of the peak frequency of the Raman spectra regardless of a medium stress in the films deposited under a negative bias of up to ~ 200 V (Fig. 2). The increase of the Si-Si distance might contribute to the increase of optical gap in a-Si. It can also set up a lower limit for the crystallite size of the microcrystalline material making it unstable with respect to an amorphous phase. Both these points require further investigations.
Acknowledgements - We would like to thank Prof. P. Wachter for permission to use the Raman instrumentation in his laboratory at the ETH-Ziirich and to Dr G. Harbeke for valuable comments. This work has been supported in part by the Nationale Energie Forschungsfonds and the Schweizerischer Nationalfonds zur F6rderung Wissenschaftlicher Forschung. REFERENCES 1.
S. Vep~ek & V. Mare~ek, Solid State Electron. 11, 683 (1968).
2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. l 9. 20. 21. 22. 23. 24. 25. 26. 27. 28.
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S. Vep~ek, Chimia 34,489 (1980). Z. Iqbal, A.P. Webb & S. Vep~ek, Appl. Phys. Lett. 36,163 (1980). S. Vep~ek, Z. Iqbal, H.R. Oswald & A.P. Webb, J. Phys. C: SolM State Phys. 14,295 (1980). Z. Iqbal, S. Vep~ek, A.P. Webb & P. Capezzuto, Solid State Commun. 39,991 (1981). H.P. Klug & L.E. Alexander, X-ray Diffraction Procedures. Wiley, New York (1974). (a) R.F. Bunshah & D.M. Mattox, Physics Today 33, 50 (1980). (b) K.K. Yee, Int. Metals Rev. No. 1,19, (1978). (c) N.J. Acher, Physics in Technology 10, 152 (1979). M. Hamasaki, T. Adachi, S. Wakayama & M. Kikuchi, J. Appl. Phys. 49, 3987 (1978). J.C. Phillips, J. Non-Cyst. Solids 34, 153 (1979), and in press. F. Evangelisti, M.G. Prioetti, A. Balzarotti, F. Comin, L. Inoccia & S. Mobilio, Solid State Commun. 37,413 (1981). Z. Iqbal & S. Vep~ek, J. Phys. C: Solid State Phys. (in press). R. Grigorvici, Amorphous and Liquid Semiconductors (Edited by J. Tauc). Plenum Press, New York (1974). J.G. Allpress & J.V. Sanders, Aust. J. Phys. 23, 23 (1970). M.R. Horae & P. Pal, Adv. Phys. 24,645 (1975). H.F. Wolf, Silicon Semiconductor Data. Pergamon, New York (1969). B. Welber & M.H. Brodsky, Phys. Rev. B16, 3660 (1977). A. Herrmann, E. Schumacher & L. W6ste,J. Chem. Phys. 68, 2327 (1978). M.G. Mason, L.J. Gerenser & S.-T. Lee, Phys. Rev. Lett. 39,288 (1977). Y. Takasu & A.M. Bradshaw, Chem. Phys. o f Solids and their Surfaces, Vol. 7, pp. 59-88. Chemical Society, London (1978). Y. Takasu, R. Unwin, B. Tesche & A.M. Bradshaw, Surf. Sci. 77,219 (1978). R. Unwin & A.M. Bradshaw, Chem. Phys. Lett. 58, 58 (1978). (a) J.C. Knights, J. Non-Cryst. Solids 35 & 36, 159 (1980), (b) J.A. Reimer, R.W. Vaughan & J.C. Knights, Phys. Rev. Lett. 44, 193 (1980). H.U. Beyeler & S. Vep~ek, Phil. Mag. B41,327 (1980). B.E. Warren, X-ray Diffraction. Addison-Wesley, Reading (1969). S. Minomura, K. Tsuji, H. Oyanagi & Y. Fujii, J. Non-Cryst. Solids 35 & 36, 513 (1980). P.H. Gaskell, J. Phys. C." Solid State Phys. 12, 4337 (1979). M.H. Brodsky, Solid State Commun. 36, 55 (1980). S. Vep[ek, Z. Iqbal and F.-A. Sarott, Phil. Mag. (in press).
0038-1098/81/270509-04502.00/0
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES - IMPLICATIONS FOR AMORPHOUS SILICON S. Vep~ek, Z. Iqbal, H.R. Oswald, F.-A. Sarott, J.J. Wagner and A.P. Webb Institute of Inorganic Chemistry, University of Ztirich, Winterthurerstrasse 190, 8057 Ziirich, Switzerland
(Received 30 January 1981 by P. Wachter) The lattice constant, d, of polycrystalline, hydrogenated silicon with a mean crystallite size less than 100 )~ shows an expansion which increases with decreasing crystallite size, reaching a limiting value of Ad/do ,~ 0.9 - 1.2% for a crystallite size of about 30 A. Implications of this result for some properties of amorphous silicon are briefly discussed. 1. INTRODUCTION THE PRESENT PAPER deals with the crystal lattice dilatation of microcrystalline silicon. The preparative method used [1,2] allows one to controllably deposit hydrogenated polycrystalline silicon of various crystallite sizes down to about 30 A. This enabled us to perform fairly accurate measurements of the lattice expansion and obtain an estimate of the nearest neighbour distances in X-ray amorphous silicon. It will be shown that the lattice expansion amounts to 1.1% for polycrystalline-Si with crystallite size of 30 A which, when extrapolated to a microcrystalline, X-ray amorphous Si, would indicate an increase of the nearest-neighbour distances of 1.2-1.5%. Such a change can hardly be reliably determined by radial distribution function (RDF) methods. It might contribute to the increase in the band gap of amorphous Si as compared to c-Si. It is generally known that amorphous Si films have stress but no systematic study of this has been done so far. With this in view, we have studied the effect of stress on the lattice expansion and on the Raman spectra of the polycrystalline Si f'dms. 2. EXPERIMENTAL Polycrystalline silicon films of thickness ranging between a few 1000 A and > 20/am were prepared by chemical transport in a d.c. hydrogen plasma at low pressures. This method provides an excellent control of the deposition conditions which allows one to deposit polycrystalline silicon of variable crystallite size as well as X-ray amorphous films. The preparation method was first applied to Si and Ge by Vep~ek and Maregek [1] in 1968 and discussed in some detail more recently [2]. The physical properties of the polyerystalline films can be found in references [ 3 - 5 ] and further details on the plasma parameters controlling the structural properties of the films (i.e. polycrystaUine vs X-ray amorphous) will be published later.
The films were deposited on various substrates such as silica and silicate glass, sapphire, molybdenum-, aluminium- and stainless steel sheets of thickness ranging between ~ 40 and 200/am, amorphous metals (foils of 40/am thickness) and on single crystal silicon chips in the (11 l) orientation. The X-ray diffraction measurements were done with a powder diffractometer Picker Type 3488 K using the 20-scan at a fixed 0 angle, i.e. at a constant angle between the incident X-rays and the sample plane. This procedure also allows one to obtain the X-ray diffraction pattern of a few thousand A thick polycrystalline film deposited on a single crystal Si chip [4]. It was verified that the results to be reported here were independent of the angle chosen within reasonable limits, as well as the nature of the substrate used. The crystallite size has been evaluated from the width of the (111) and (220) peaks using the Scherrer formula [6] with a shape factor of 0.94. The ratio of the apparent crystallite sizes calculated from the (111) and (220) peaks of x/3 : ~/2 is indicative of a regular shape of the crystallites. Thus, a cube edge, a = a(111)/~/3 = a(220)/x/2 is used in the following part of this paper. Here, a(111) and a(220) are the apparent crystallite sizes calculated from the widths of the (111) and (220) peaks respectively, i.e. the extension of the crystallites in the direction perpendicular to the scattering planes. We have also verified that the instrumental broadening was negligible and the diffraction peaks were symmetrical. For the exact determination of the lattice constants of the small crystallites we have used crystalline silicon powder (Ventron, 100 mesh, purity better than 99.99%) ground to a size of ~ 30/am and dispersed directly on the polycrystalline film to be measured. CVD deposited films usually possess either compressive or tensile stress [7] which has to be accounted for in order to obtain a correlation between the lattice expansion and the crystallite size. However, polycrystalline films deposited at a floating potential did not show
509
510
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES
~d/.¢ [o/J
Vol. 39, No. 3
d°
1,4
\
Io2
1.0 1.0
.8 .8
.6
o6 .4
.4
,2
.2 '
~0
40
60
80-
qoo
12-0
Fig. 1. Correlation between the relative lattice expansion Ad/do, and the crystallite size, a. do = 5.425 A is the lattice constant of silicon and Ad is the absolute value of the expansion. Filled circles represent samples deposited between ~ 60 and 260°C at a floating potential and open circles represent samples deposited between ~ 80 and 260°C under varying bias. Broken line shows the data of Hamasaki et al. [8] for Si crystalrites in a SiOx matrix. any observable stress. In addition, we compared in several cases the lattice expansion of an as-deposited film with the same one after dissolving the substrate and found identical results. Annealing of the samples under ultra-high vacuum at temperatures between "~ 300 and 800°C results in some enlargement of the crystallite size for films with a <~ 80 A and a relatively very small increase for films with a >~ 100 ~&, and a partial or complete evolution of hydrogen [4], but the correlation between the lattice expansion and the crystallite size remains within the experimental error the same as for the as-deposited samples (see below). Thus, the results presented here are believed to be representative of pure, hydrogenated silicon films which are free of stress (Fig. 1, solid line). Some samples were deposited also under a bias which was applied to the sample holder by means of an auxiliary power supply. The actual plasma potential at the position of the sample holder as well as the chosen value of the bias were determined from the Langmuir characteristic of the sample holder used as a probe. Samples deposited under a negative bias display a compressive stress and, as one would expect, a smaller lattice expansion as compared to the stress-free samples of the same crystallite size deposited at a floating potential (see below). 3. RESULTS AND DISCUSSION Figure 1 shows the correlation between the relative lattice expansion, Ad/do, and the crystallite size, a. For trims deposited at a floating potential (full circles,
s12
si4
si6
5i~
"
~2o
Fig. 2. Correlation between the relative lattice expansion, Ad/do, and the peak position of the k ~ 0 phonon frequency of the polycrystalline films. (See captions to Fig. I for further details.) Samples deposited under a negative bias o f up to ~ 200 V correlate w i t h the stressfree ones (open circles on the full line), those deposited under a negative bias exceeding ~ 300 V show a peak frequency o f ~ 519 cm -1 .
Fig. 1) the correlation extrapolates to (Ad/do) 1.2-1.5% for microcrystalline material with crystalrite size ~ 10 A. Films deposited under negative bias (i.e. under bombardment by positive ions) show somewhat smaller lattice expansion as compared with those deposited at a floating potential (see Fig. 1, open circles). The relatively large scattering of the data for these samples is due to the fact that the samples with different crystallite sizes were deposited at different values of the bias, showing for a given crystallite size, a decrease of the lattice expansion with increasing absolute value of the negative bias. Because of the lack of knowledge of the energy of the impinging ions we did not try to obtain a more complete set of data in the present study. The few examples shown in Fig. 1 are presented here only to illustrate the effect of the compressive stress in the films on the lattice expansion of small crystallites in order to warn against an overinterpretation of our results and conclusions. This example also illustrates that care has to be taken when comparing data on amorphous silicon since samples deposited under a high bias (i.e. the "cathode" of an r.f. discharge) may possess a larger stress than those deposited at a low bias (e.g. the "anode" of an r.f. discharge). Hamasaki et al. [8] have also found lattice expansion of silicon crystallites embedded in a matrix of silicon suboxide. Their experimental results can be fitted by an empirical formula
Ad/do = 0.17/(a -- 8.8) and are indicated by the broken line in Fig. 1.
Vol. 39, No. 3
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES
Qualitatively, the results of Hamasaki et al. agree with ours in showing increasing lattice expansion with decreasing crystallite size but disagree in some details, the divergence of Ad/do for crystallite size about 8.8 A being the most pronounced one. It is surely not unreasonable to seek the origin of the latter difference in the different environments of the Si crystallites in their and in our case. We have shown in [4] and [5] that our polycrystalline Si samples are remarkably pure, that the only "amorphous-like" matrix in which the crystallites are embedded are their grain boundaries [5], and, last but not least, that their hydrogen content is very much the same as in hydrogenated a-Si. In our previous paper [5] we have shown that the crystalline component of the Raman spectra of the polycrystalline silicon displays a shift towards smaller frequencies with decreasing crystallite size. It is seen from Fig. 2 that this shift correlates with the lattice expansion determined from the X-ray diffraction. In fact this correlation is even better than that of Fig. 1 since the data from samples 4eposited under negative bias not exceeding -- 200 V also fit on the same curve with those deposited at a floating potential. Samples deposited at a negative bias greater than ~ 300 V however show a jump in frequency to ~ 519 cm -1 . More detailed study of this effect is in progress. Obviously, these results show that the decreasing frequency of the crystalline component in the Raman spectrum with decreasing crystallite size is due largely to an increase in the Si-Si bond distance rather than to a progressive relaxation of the k ~ 0 selection rule as suggested earlier [5]. For the purpose of the present work we conclude that the results presented by Figs. 1 and 2 are mutually consistent and convincingly illustrate that the measured lattice expansion reflects the increase of the Si-Si distances which should reach some limiting value of about 1.2-1.5% for a stress-free, X-ray amorphous silicon film of a microcrystalline nature and somewhat less or more for films under compressive or tensile stress respectively. This value can be somewhat modified if the topology of the amorphous network is different from that of the diamond lattice. Nevertheless, even if some cluster model is assumed* some expansion of the Si-Si distances is to be expected (see below).
* For example, the interesting topological theory of J.C. Phillips showing that a continuous random network o f four-fold coordinated elemental solid cannot exist in three-dimensional space [9], the EXAFS study of Evangelisti et al. on G e [ I0] indicating a change of the third peak upon the transition from crystalline to amorphous material and our Raman spectroscopic measurements on polycrystalline- and amorphous-Si [5, 11].
511
We have to point out that the data of Fig. 1. but not those of Fig. 2 have to be used for the extrapolation to a < 30 A since the frequency of the crystalline component of the Raman spectra does not scale down to any well-defined limiting value with decreasing crystalline size. As indicated earlier [5] and will be discussed in some detail in a paper to follow [11] the feature at 480 cm- 1 in the Raman spectrum from a-Si should be associated with a totally symmetric vibrational mode of small tetrahedral Si clusters. A shift of the first peak in the radial distribution function (RDF) ofa-Si and a-Ge indicating a comparable lattice expansion has been occasionally reported in the literature (e.g. [12]), the actual value being a function of the preparation conditions. Indications for expansion of up to several percent were reported also for some other amorphous materials (e.g. [6, 12, 23]). In general, the accuracy of the position of the first peak in the RDF is limited to several percent due to experimental and computational limitations [6, 24] and a shift of the same amount cannot be reliably established by this method [6, 12, 24, 26]. In some cases, nearest-neighbour distances in a-Si were reported to equal to those of c-Si, although the first peak in the corresponding RDF was broad (FWHM ~ 0.4 ~,) and had a relatively large curvature on the top [25]. The increase of the Si-Si bond distance reported in this paper is also supported by the theoretical studies of the structure and stability of small covalently bonded clusters [13, 14]. For a cluster size of ~ 10 A the calculation by Allpress and Sanders gives an expansion of nearest-neighbour distance between 3 and 8% depending on the choice of the cluster structure and of the pair interaction potential [13]. Possibly, the mutual interaction on grain boundaries reduces this expansion in the polycrystalline film. In films with a high hydrogen content (e.g. > 12 at.%) the latter is preferentially bonded on the grain boundaries [4, 22] and it can probably reduce their mutual interaction. Both crystalline [15] and amorphous [16] silicon display a decrease of the optical band gap with increasing pressure. One could speculate that the observed lattice expansion may cause certain contribution to the observed increase of the optical gap in a-Si as compared with the crystalline material. However, it is difficult to give some quantitative estimate at this stage. At this point a remark on the electronic properties of small clusters is appropriate. In general, one finds a more or less continuous transition from the properties typical o f a bulk towards that of molecules and atoms with decreasing cluster size. The photoionization potential of isolated Na, K and mixed clusters formed in a molecular beam [ 17] display a similar shift as the position of the valence band of Pd clusters supported on
512
LATTICE DILATATION OF SMALL SILICON CRYSTALLITES
silica. In the latter case, the Pd d-band narrows and both its threshold and centre-of-gravity shift further from the Fermi level with decreasing cluster size [ 18-21 ]. Such shifts, which in isolated or supported clusters can amount to a few eV might have some relevance to the increase in the band gap for an amorphous material, particularly if the latter were composed of clusters with a relatively weak mutual interaction on their boundaries (see, e.g. [27]). However, it is difficult to give some quantitative estimate without detailed calculations. Another question arises as to the role of the lattice expansion of the microcrystalline silicon in the transition to the amorphous state. Such an expansion reflects an increase of the free energy of the microcrystalline material with decreasing crystallite size and it probably reflects the increase of the specific surface energy of the microcrystallites. Preliminary estimates indicate that this could set up a lower limit for the crystallite size of the microcrystaUine material making it unstable with respect to an amorphous phase having a different structural topology. This suggestion should be regarded as very preliminary and it is presently under more detailed study [28]. 4. CONCLUSIONS Fine grain, polycrystalline silicon shows lattice dilatation increasing with decreasing crystallite size to a limiting value of Ad/do ~ 0.9 -- 1.2% for a crystallite size o f ~ 30A (Fig. 1). A compressive or tensile stress in the films reduces or enhances this dilatation (Fig. 1). The lattice dilatation correlates with the position of the peak frequency of the Raman spectra regardless of a medium stress in the films deposited under a negative bias of up to ~ 200 V (Fig. 2). The increase of the Si-Si distance might contribute to the increase of optical gap in a-Si. It can also set up a lower limit for the crystallite size of the microcrystalline material making it unstable with respect to an amorphous phase. Both these points require further investigations.
Acknowledgements - We would like to thank Prof. P. Wachter for permission to use the Raman instrumentation in his laboratory at the ETH-Ziirich and to Dr G. Harbeke for valuable comments. This work has been supported in part by the Nationale Energie Forschungsfonds and the Schweizerischer Nationalfonds zur F6rderung Wissenschaftlicher Forschung. REFERENCES 1.
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