First-Principles Calculation of the Optical Properties of Nanocrystalline Silicon
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t. Present address: Central Research Laboratory, Hitachi, Ltd., Kokubunji, Tokyo 185, Japan 3 Mat. Res. Soc. Symp. Proc. Vol. 358 0 1995 Materials Research Society
30 20
EF
10
20Lo~
;LUMO
40 Si29H36
S20
40Si66H64 ca10
C
6
0
240 20
nanocrystalline Si.
0 -15
.10
-5
0
Energy (eV) METHOD OF CALCULATION We calculated the electronic structure of hydrogenated nanocrystalline Si; Sil 0 H 16 , Si 29H 36, Si66H64, and Si 123 H 100. The crystallite diameters (L) are 0.73 nm, 1.0 nm, 1.27 nm and 1.55 nm, respectively. The first-principles pseudopotential method was used with local density approximation (LDA) to determine exchange-correlation interaction. Norm-conserving nonlocal pseudopotentials of Si valence electrons were constructed based on the results of all-electron atomic structure calculations. A plane-wave basis set, with a cut-off energy of 90 eV (6.61 Ry, 2517 to 13613 plane waves) was employed and the super cell method was used to treat crystallites in the band calculation scheme. Atomic coordinates in the crystallites were determined by spherical quarrying from the Si diamond structure, and the surface dangling bonds (DB) were terminated by hydrogen atoms. The SiH 3 species were replaced by H atoms, as evidenced by the relatively weak intensity of the SiH 3 species in the FTIR spectrum [9,25]. The state-to-state spontaneous emission probability between the occupied and unoccupied states was calculated to investigate luminescent ability. We paid particular attention to the transition between the band-edge orbitals, the highest occupied molecular orbital (HOMO), and the lowest unoccupied molecular orbital (LUMO). Since the PL spectra weakly depend on the excitation energy, PL is thought to be dominated by the transition between the states near the HOMO and LUMO. RESULTS AND DISCUSSION Electronic Structure The density of states (DOS) of hydrogenated nanocrystalline Si are shown in Fig. 1. The
Cluster Size
5 2 4 D.. 4Si cc
.
1
0.5 Si,.H 16
,
5
4-
3
S2
(nm)
S""9H36Si""
29H.
Fig. 2. Dependence of the energy gap on crystallite size. The gaps are shifted by 0.52 eV from the values calculated by LDA. The dotted line represents the relation
------Violet Si 123 Hlo i
0CI /L 2 , which is predicted by effective mass approximation.
.AEg 0
0
0.5
I
I
1.0
1.5
2.0
2.5
1IL (1/nm) energy gap (Eg) increases with increases in crystallite size. The electronic structure of Si 123 H 100 is surprisingly similar to that of bulk Si. The size dependence of Eg is plotted in Fig. 2. It is well known that the Eg estimated by LDA is smaller than the actual value. Self-energy corrections, such as in GW approximation [26], will widen the energy gaps. However, we simply shifted Eg uniformly in Fig. 2 by 0.52 eV to fit the calculated Eg of bulk Si to the experimental value. The enepgy-gap upshift from the bulk-Si value (1.17 eV) (AEg) is proportional to l/L, in contrast to the IlL relation predicted by EMA, which is shown by a dotted line in Fig. 2. This unusual relation was reported by Proot et al. [17],
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