Theory of Pressure Effects on Silicon Nanocrystallites

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ues obtained when the bulk conduction band is well described [23]. However, we have used this model because the s* orbital interaction with the p states at the bottom of the conduction band gives a negative pressure coefficient for the bulk band gap. We expect that for a cluster, it will realize a good interpolation between the bulk states and the corresponding bulk pressure coefficients. We have assumed that the interatomic nearest-neighbor Slater-Koster tight-binding parameters used in the Vogl model vary as d-"' where a is for ss(, spa, ppa, ppir and s*ppa. The n%'s

have been fitted to the four bulk pressure coefficients listed above. We get n,,,- = npp I= 1.5, npp I = npp Oand

3.25, np

nsp , = 3.

= 2.5, a2E,

We have verified using the calculated total energy E,from the LDA calculation that

T-2

is

very close to the bulk value and the pressure coefficient a = E (where E is the crystallite band gap) can be calculated as: VaE BVE

a-=

(1)

where B is taken equal to the bulk value (978.8 kbar [22]). The variation with the cluster size is given on fig. 1. The results are slightly scattered for small clusters and one can see from the tightbinding calculation that it smoothly tends towards the bulk value when the diameter increases. For small clusters, the difference is comparable to the one between tight-binding and LDA bulk values. Recent measurements [9] give a=-0.4 and -0.6 mevfkbar for cluster size - 3 nm. This is in very good agreement with the tight-binding model which is in that case more reliable than the LDA calculation as it is fitted to bulk values. Due to the underestimated cluster gap obtained with the Vogl model, comparison with experiment which measures the photoluminescence energy and not the cluster size can only be made using for example the relation between the gap EG and the cluster diameter we have obtained using a non orthogonal tight-binding model[26] and which is in agreement with empirical pseudopotential or first-principles LDA results: EG =

2.29 V

d

eV

(2)

0.0



/

-1.0

S2.0

"

/

-

0.0

0.5 1.0 1/d (nm"1) Fig 1: Band gap pressure coefficient of spherical nanocrystallites as a function of their diameter: LDA calculation [14] for small clusters (a), sp3s* tight-binding model (.)[16]. The values for l/D=0. are the bulk values. The dotted line is a guide foir the eyes between the LDA bulk and small clusters values. 76

In a confinement model, we get as for Si quantum wires negative pressure coefficients. This would not be the case if the luminescence comes from a surface defect as the self-trapped exciton we have previously studied [23-24]. In this case, the luminescence energy is directly related to the coupling between silicon hybrid orbitals. When the pressure is increased, the interatomic distance decreases and the coupling between the orbitals increase, which gives rise to a blue shift and a positive pressure coefficient BC8 PHASE NANOCRYSTALLITES The BC8 structure is metastable and is obtained after release of the pressure when the highpressure P-tin phase (Si-II) is