Comparative Theoretical Study of Amorphous and Crystalline Silicon Clusters
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some theoretical informations on these questions, particularly on the relative blue shift in amorphous and crystalline silicon clusters. LOCALIZED VERSUS DELOCALIZED STATES To calculate the electronic structure of amorphous silicon (a-Si) clusters, one has first to determine their atomic structure. The starting structure is obtained by isolating the atoms in a sphere of a given diameter in a 4096 atoms periodic cluster determined using the Wooten-Winer-Weaire (WWW) method [13-14], known to produce a correct radial distribution function. There is no dangling bond in this initial network. To simulate amorphous clusters, the center of the sphere is randomized. Dangling bonds occurring on atoms close to the surface of the sphere are saturated by hydrogen atoms. Even if the initial WWW atomic model was relaxed, the atoms in the sphere are no more in equilibrium positions since the boundary conditions are modified. We have thus relaxed the atomic positions in the sphere using a Keating potential [15] identical to the one used to build the initial periodic cluster. Two initial networks (with and without four-membered rings) have been used and give very close results. All the results in this paper have been obtained with the network with four-membered rings. The electronic structure is calculated by the sp 3 s* empirical tight-binding method [16]. The Hamiltonian between silicon atoms includes interactions up to nearest neighbors. Associated with a R` dependence upon distance they produce reasonable deformation potentials. Tight-binding 57 Mat. Res. Soc. Symp. Proc. Vol. 452 01997 Materials Research Society
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Diameter (nm) Fig.2: Confinement energy for an amorphous cluster as a function of its diameter for a-Si and a-Si:H. The center of the clusters are kept constant. Each column of points represents the energy levels of a cluster.
Fig. 1: Radius of cluster eigenstate as a function of the energy level. The clusters respectively have about 420 silicon atoms. Their diameter is close to 2.5 nm. The vertical dotted line shows the top of the crystalline cluster valence band. The horizontal lines give two limits: deep in the valence band, the radius is equal to 1.25 R for a uniform eigenstate and close to the gap, it is equal to 0.53 R for an effective mass state sin(kr)/r. 58
parameters between the silicon and hydrogen atoms have been obtained from Harrison's expressions [17]. Figure 1 compares the average radii p (En) of the eigenstates of energy En for a crystalline cluster and an amorphous one. This is defined as the root mean square distance from the center of gravity of the distribution defined by the wave function. The diameters of the crystalline and amorphous clusters on Figure 1 are close to 2.5 nm (-420 silicon atoms in the clusters). One can see that the variation of the eigenstate
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