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Solid State Communications, Vol. 106, No.3, pp. 121-126,1998 @ 1998 Published by Elaevier Scicncc Ltd Printed in Great Britain. All righr. merved 0038-1098198 $19.00 + .00

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QUANTUM CONFINEMENT AND DIRECT-BANDGAP CHARACTER OF SilSi02 SUPERLATTICES Nacir Tita,b and M.W.C. Dharma-wardanaa,1 aInstitute for Microstructural Sciences, National Research Council, Ottawa, Canada KIA OR6 bDepartment of Physics, UAE University, P.O. Box 17551, AI-Ain, United Arab Emirates (Received 7 January 1998; accepted 20 January 1998 by D.l. Lockwood)

We present tight-binding bandstructure calculations for {Silm{Si02}1I crystalline superlattices (SLs) grown along the [001]direction. A striking new feature of the results is the essentially direct band-gap structure along the zr symmetry line of the SL-Brillouin zone which has a blue shifted energy gap due to quantum confinement. This feature is very attractivefor obtaining high radiative efficiencies. These results suggest the possibility of novel optical devices which exploitthe direct bandstructure, and have implications for our understanding of the luminescence in porous-Si and other Si-based nanostructures. @ 1998 Published by Elsevier Science Ltd. Keywords: A. semiconductors, A. surfaces and interfaces, D. electronic bandstructure. Information and communication technologies are increasingly turning to photonic devices which promise ever-faster processing speeds. However, silicon-based microelectronics cannot as yet use silicon itselffor its photonic needs because the indirect bandgap of silicon prevents the direct conversion of electronic energy into photons. Thus the discovery of intensevisible luminescence in porous silicon generated a world wide effortin fabricating Si-based light emitting crystallites, dots, wires, and other structures[1-3]. A recent studyof strongluminescence in amorphousSilSi02 superlattices (SLs) presented experimental evidence for quantum confinement (QC) [2]. The high luminescence efficiency has been attributed to the break down of momentum conservation in amorphous quantumconfined systems. Hereweshowtheoretically that crystalline SilSi02 SLs should show QC and also have a nested direct-band gap character. Thus, if such SLs could be fabricated, Si-based lasers and other novel photonic devices become real possibilities. The electronic structure of thin isolated films or clusters of Si have been studied by various authors in the past [1,3]. However, here we explicitly study I

Corresponding author. E-mail: [email protected]

the SilSi02 SLs to examine the possibility of direct bandgaps and in the context of recentexperiments [2] on SilSi02 [00I}SLs. In this study weuse well tested tight-binding methods to determine the electronic structure of crystalline SilSi02 [ool]-SLs and show that there is QC, as expectedfromthe largebarrier potential of the Si02 layers. More strikingly, the confined bands along the zr symmetry direction (see Fig. lea»~ are essentially dispersionless and showa strongly nesteddirect-bandgap character. They also contain the minimum energy gap of the system. That is, the minimum bandgap holds for a significant finite extent of k-space for which the valence and conduction bands remain"nested". Thus, although optical transitions in bulk Si is limited by its indirect energy gap along the r X direction, these Si-SLs can show strong luminescence along the fult zr symmetry line due to the direct-gap nature of the bands. The SilSi02 SLs of Ref. [2] contained amorphous layers, i.e., a-Si layers interposed with a-Si0 2 layers. New fabrication methods are now available where crystalline-Si layers interposed with a-Si02 layers can be realized [4]. The SL studied here have the structure ..AlabB1ba.., where A is a c-Si layercontaining m unit

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Fig. 1. (a) Thick lines:- the tetragonal Brilloiun zone (t.BZ) of the superlattice and its symmetry points. The familiar diamond B.Z (d.BZ) is shown in thin lines. (b) The valance- and conduction bands of bulk silicon (d.BZ) are compared with bands from the Vogl- and GP tight-binding methods. (c) The GP bands for bulk silicon in a t.BZ with 4-Si unit cells along [001]. cells of Si, while lab is the atomically thin interface layer from A-B. The B layer is made up of n unit cells of c-SiOz with the P-crystobalite structure. The detailed structure of the large-band gap Si02-barrier is not very critical. Also, the results become rapidly insensitive to the thickness n of the Si02-barrier layer as quantum confinement is quite strong. While the ease of fabrication dictated the use of a-Si layers instead of c-Si layers in the experimental samples, another motivation was the belief that the breakdown in k-conservation in the amorphous system favoured the radiative efficiency. However, the present study suggests that quantum-confining structures containing c-Si should be of urgent interest for realizing direct band gap Si-lasers, light emitting diodes etc., if fabrication problems could be overcome. The electronic properties of structures containing many atoms, can be conveniently studied via the tight binding method [5]. Here a minimal basis of atomic functions (e.g., s, p3 functions on each Si atom) which are orthogonalized using L6wdin's method is used to form linear combinations which diagonalize an effective Hamiltonian, The parameters of the effective Hamiltonian are the "tight binding"(fB) parameters of the problem. Modern tight binding methods have

been applied to very diverse elements. There is a well established knowledge base for dealing with the two elements Si, and 0 which occur in our system. Some of the older TB parametrizations for Si use only the s, p3 atomic basis limited to nearest-neighbour (NN) interactions [6,7]. They do not provide a good band structure or a correctcrystal structure for bulkSL These issues have been reviewed by Lenosky et al. [8]. The s, p3, s* basis of Vogi et al. provides a considerable improvement, but the energy dispersion in the Si-conduction band is still underestimated [9]. Hence we included NN- as well as next-nearest neighbour (NNN) interactions within the s, p3 basis, using the TB parameters of Grosso and Piermarochhi(GP) [1O}. In Fig. l(b) we compare the bulk-Si band structure from GP and Vogi et al., with a reference bandstructure from empirical pseudopotentials [II}. The GPTB calculation reproduces the indirect energy minimum along the f-X direction of'bulk-Si diamond Brillouin zone (d.BZ, cr. Fig. I(a» . This is important here since the imposed-SL periodicity along [00I} leads to a zone-folding and energy splittings along the r -X line, which maps to the zr line in the tetragonal BZ (t.BZ.) . The Si-bands for the t.BZ, with k x = ky = rr/asi and k, = rr/4as;, where as; is the Si-lattice parameter, are

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QUANTUM CONFINEMENT OF SilSi02 SUPERLATIICES

displayed in Fig. 1(c). TheTB-parametrization of LaFemina and Dukeare used for tJ-crystobalite (tJ-c), where NN- and NNNinteractions have been included [12]. In relating the TB parameters of Si-atoms in the Si-layers with the Si-atoms in the Si02-1ayers, band offsets have to be considered. We usea valence band offset (VBO) of 3.75 eV [13]. If e-Si were replaced by a-Si, the VBO shifts by about ~ 0.25- O. 3 eV [14,15]. The results presented here are unaffected by such shifts of the VBO. The SilSi02 SLs grown in the [001] direction contain three ingredients, viz., a e-Si layer, interfaces lab and lbo, and a Si02 layer. The tJ-c structure is adopted for the Si02 layer because it is cubic and leads to a simple model for the interface. The SilSi02 interface has been the object of decades of studies due to its importancein MOSFET structures[13,16-18]. A recent review is due to Lu [17]. Experiments show that the interface (containingsuboxide-charge states) is rather abrupt, and contains very few interface states. Such an abrupt interface implies a density bulgewhich has been observed [19]. It is also known that the interface between the bulk-Si and bulk-oxide layers containsall the suboxide-charge states Si1+, Si2+, Si3+. Further, the model must accommodate the nearly 30% lattice mismatch between aSi=5.43 A and aSioz=7.l4 A. Here, following Batra et al., [18] the tJ-c structure is applied along the diagonal of the (OOI)-surface unit cell of the e-Si layer. Thus we take asioz=aSi.J2, i.e., 7.65 A. This is onlyslightly larger than the usual tJ-c latticeparameter and implies a modest mismatch of about 5%. The atomic arrangement across the interface is shown in Fig. 2 and involves one extra oxygen atom which saturates the interface bonds. This model produces a density bulge at the interface and also contains the suboxide charge states Sil+ and 8i2+ , but lacks the state 8i3+ as is also the case in the energy optimized model due to Pasquarello et al. [16]. Actual SilSi02 interfaces are kinetically-prepared metastable structures and need not be energy optimal except in someconstrainedsense. In fact, the complete energy optimization of a superlattice structure would lead to a random alloy with no periodic structure. So we have not attempted any energy minimization but selected an interfacestructure withchemically acceptablebond lengths, chargestates, densityenhancements etc as already stated. The TB parameters for the Si atom in the interface are the average of the value in c-Si and in B«, The near-neighbour Si-atoms on both sides of the interfacefeel the presence of the interface. Thus by including, or by excluding, the NNN-effects of the interfaceatoms in the tight-binding calculation, we can effectively obtain two interface models (NN-

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Fig. 2. A 8i1Si02 superlattice with one unit of Si and one unit of 8i02, i.e, m = n= 1.The l3-crystobalite unit cell is matched to the Si cell by rotating the former about the [001] axis by 1T/4. An extra oxygen (shaded square) is added to saturate the interface bonds. The interfaces lSi-SiOz and lSiOz-Si differ by a reflection and a 1T/2 rotation. and NNN-models) to gaugethe robustness of the results to variants of the interface structure. It should be noted that the atoms entirely in the Si-layers and in the tJ-c layers are always treated within the NNN schemes of Refs [10] and [12] respectively. The energy gap Eg{m. n) as a function of the layer thicknesses m,n for a sequence of SLs {Si}m{Si02}" grown along [001] was determined. The layer thicknesses are L si= masi and L sioz= nasioz. As expected, the value of Eg is insensitive to the thickness of the Si02 layer but depends on the thickness of the Silayer, and this determines the quantum confinement. As m-increases the QC decreases and Eg tends to that of bulk-Si. Thus Eg{m. 1) for m=l to 5 are 2.l9±.61, 1.71 ±.34, 1.43±.l8, 1.29±.l3, 1.23±,.lO respectively. The variationis 1/ L~i for m > 4, as was also found experimentally [2]. Each Eg in the abovelist is accompa-

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the essentiallyisolated quantum-well nature ofthe system. However, the underlying Bloch functions define ....- ... a direct-gap structure not present in bulk silicon. u Figure 4 shows the density of states per atom for a " 2.1 bands (aJ ~ 0.8 number ofSLs. Scanning tunneling microscopy should a 3.3 bands 0.0 be able to probe this enhanced band-edge density of states. We envisage that the increased density of fully-0.8 nested direct-band gap states of the ideal crystalline system would, in an amorphous-Si/Sm, SL still pro2.0 duce a broad, strongly luminescent peak. In fact Lock1.0 wood et al. (Ref. [2D showed that the luminescence in0.0 tensity, while being small for systems with < loA Silayers, increased sharply for the regime of 15 to 20 A, and then again dropped rapidly for larger-layer thicknesses. One reason for this is the confinement of the 2.0~~~ electron and the hole in the same physical region, for ~ 1.0 layer thickness comparable to the exciton radius. How0.0 ever, here we are looking at a more microscopic effect arising from the nature of the Bloch functions of the X R Z r M A Z SL and the associated energy dispersions. For larger-Si layers, the band structure, density of states and the gap approach those of bulk Si. AlFig. 3. (a) The valence- and conduction bands along the Zf-line for (2,1), and (3,3) superlattices and for though there is still an apparent accumulation of bulk-Si (t.BZ.). The k-ranges along all the symmetry states along the zr direction, the amount of momenbranches have been normalized to unity. (b) The (1,1) tum space (Ak) covered by this branch decreases as superlattice bands in the t.BZ. Eight conduction bands the inverse of the SL period. The direct property of and four valence bands are shown. (c) Results for the the band gap is increasingly compromised. Hence the (3,3) superlattice bands. net effect is to return to the (poor) radiative efficiency of normal bulk silicon. But intermediate layer thicknied by a ±A. This defines the effect of using the NN- nesses (lo-30A) corresponds to m,n = 2-5 periods or NNN-interface models for which the gaps become in the m,n SLs and these provide excellent direct Eg + A and Eg - 6 respectively. As expected, the value band gaps (to better than 0.01 of an eV) as well as a of 6 decreases as the Si-Iayer thickness increases. The good spread of phase space (range of .1.k) along the dispersion (i.e, the change in the energy gap from r to zr direction. Thus a high radiative efficiency is to Z) of the energy gaps Eg(m, I, k) is only 44 meV for be expected from this regime of "critical layer thickm=l, drops to 0.5 meV for m=3, and rises to about nesses" where a favourable compromise between the 2.0 meV for m=5. Note that the electronic energy gaps competing demands of band nesting and phase space of nanostructures (e.g, as reported here) are usually are achieved. If porous Si contained microcrystalline smaller than the optical energy gaps (e.g, obtained by regions « 30 A) which incorporate Si02 interfaces experiment) unless thermal and quasi-particle effects into some type of self-organized local structure, a mechanism similar to what has been explored here have been included [20]. In Fig. 3(a) we show the SL-valence band and the could be operative in the enhanced luminescence of Sl.-conduction band together with the bands for bulk such Si nanostructures. The direct-gap properties of Si calculated in a t.BZ with k x = ky = 1T! as;, and the crystalline-Sl, bands shown here, if found to hold k z = rr!(4asi), i.e, with four units-cells along [001]. in fabricated samples, would pave the way for novel The k-values along the zr line (and other symmetry Si-based laser and optical components. directions appearing in panels (b) and (c) of the figure) have been normalized (by their respective magni- Acknowledgements-We thank Zheng-Hong Lu for tudes 6k in the different directions) to afford a valid many discussions, David Lockwood, Geof Aers and comparison among structures with differentsuperpe- Francois Perrot for comments on the manuscript. riods. Figure 3(b,c) shows the first eight low-lying, closely-spaced conduction bands for a (1,1) SL and a (3,3) SL respectively. The flat, nested nature .of the bands is basically a consequence of the coherence of 1.1

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QUANTUM CONFINEMENT OF SilSi02 SUPERLATIICES

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energy(eV) Fig. 4. The density of states per atom for several SilSi02 superlattices compared with that of Bulk silicon. A 50 meV smoothing has been included and the curves are arbitrarily offset along the y-axis for clarity; the zero value of each DOS curve is implicit from the position of the flat region in the energy gap, e.g, between 0-1 eV., of each curve. REFERENCES

phys.

u«, 9, 1989,701.

8. Lenosky, TJ., Kress, 10., Kwon, I., Voter, A.F., 1. Iyer, S.S. and Xie, Y.H., Science, 260, 1993, Edwards, B., Richards, D.F., Yang, S. and Adams, 40; Coillins, R.T., Fauchet, P.M. and Tischler, ra, Phys. Rev. B,55, 1997, 1528. M.A., Physics Today, SO, 1997, 24; Lockwood, 9. Vogl,P., Hjalmarson, H.P. and Dow, 10., J. Phys. 0.1, (Ed.), Light Emission in Silicon, Academic, Chern. Solids, 44, 1983, 365. Boston, 1977. 10. Grosso, G. and Piermarochhi, C., Phys. Rev. B,51, 2. Lockwood, OJ., Lu, Z.H. and Baribeau, I-M., 1995, 16772. Phys. Rev. Lett., 76, 1996, 539. 11. Chelikowsky, IR. and Cohen, M.L., Phys. Rev. B, 3. Allan, G., Delerue, e. and Lannoo, M., Phys. Rev. 14, 1976,556. Lett., 78, 1997,3161. 12. LaFemina, IP. and Duke, c.a, J. Vac. Sci. Tech4. Private communication. nol., A9, 1991, 1847. 13. Williams, R., J. Vac. Sci. Technol., 14, 1977, 1106. 5. Bullet, D.W, Solid State Physics (Edited by H. Ehrenreich, F. Seitz and D. Turnbull), Vol. 35, p. 14. Lu, Z.H., Mitchell, D.F. and Graham, MJ., Appl. 129, Academic Press, New York, 1980; PapaconPhys. Lett., 65, 1994,552. stantopoulos, D.A., Handbook of the Bandstruc- 15. Van de Walle, c.o, and Yang, L.H., J. Vac. Sci. ture ofElementalSolids, Plenum, New York, 1986. Technol. B, 13, 1995, 1635. 6. Chadi, OJ. and Cohen, M.L., Phys. stat. sol. (b), 16. Pasquarello, A., Hybertsen, M.S. and Car, R., 68, 1975,405. Appl. Phys. Leu., 68, 1996,625; Phys. Rev. Lett., 74, 1995, 1024. 7. Godwin, L., Skinner, A.I and Pettifor, D., Euro-

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17. Lu, Z.H., Proceedings of the NATO ASI, St. Pe- 19. Kosowsky, S.D., Pershan,P.S., Kirsch, K.S., Bevk, tersburg, 1997, Kluwer, Dordrecht. J., Green, M.L., Brasen, D., Feldman, L.c. and Roy, P.K., Appl. Phys. Lett., (in press). 18. Batra, 1, The Physics of SiOz and its Interfaces (Edited by S.T. Pantelides), p. 333, Pergamon 20. Ogiit,S., Chelikowsky, J.R. and Louie, 8.0., Phys. Rev. Lett., 79, 1997, 1770. Press, ~ 1978.