Tight-Binding Molecular Dynamics Simulations on Point Defects Diffusion and Interactions in Crystalline Silicon
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33 Mat. Res. Soc. Symp. Proc. Vol. 396 01996 Materials Research Society
COMPUTATIONAL METHODOLOGY In this work, we make use of the Kwon et al.[3] TB model for silicon (KBWHS). Here the short-ranged repulsive potential Urep as well as the scaling functions for the TB hopping integrals are improved compared to the previous Goodwin et al.[4] parameterization (GSP). More specifically, the form of U,,p is embedded-atom like, while in the GSP model only two-body interaction were considered. Furthermore, different scaling functions for the TB hoppings are introduced according to the orbital symmetry in a close agreement to first-principles calculations. The resulting TBMD scheme is more accurate, as demonstrated in Table I that will be discussed in the next section. Most of the results presented here have been obtained using large cubic periodicallyrepeated simulation supercells containing 216 atoms plus (minus) the number of interstitial (vacancy) defects involved. Typical simulations for annealing and relaxation are performed a few picoseconds. The diffusivities are currently calculated using the 64 atoms cell and with a simulation time up to 100 ps. The formation energy E t is defined as the energy difference between the defected system and the perfect system with same number of atoms of the defected system. The binding energy for a cluster of size N is defined as E -• _ + Ef - E/,, where E{ is the formation energy of a single defect, and E/f_ and E1N are the formation energies for clusters of size N - 1 and N, respectively. Defect
LDA
KBWHS
GSP
SW
Vacancy
3.65
3.69
3.96
2.64
T interstitial
3.5
4.39
4.40
4.84
H interstitial
3.3
4.93
5.90
6.58
(110) dumbbell
3.2
3.80
5.04
3.65
TABLE I - Formation energies (eV) of single point defects in crystalline Si obtained using TBMD of Kwon et al. (KBWHS) [3] and Goodwin et al. (GSP),[4] compared with local density approximation (LDA) [6] results and Stillinger-Weber potential (SW) [5]. RESULTS AND DISCUSSION Single Point Defects The formation energies of single vacancy and interstitial defects in silicon are calculated by TBMD at low temperature. The interstitial defects considered are the hexagonal site (H), the tetrahedral site (T) and the (110) dumbbell (two atoms share one common lattice site), which are believed to be the most important interstitial defects in crystalline silicon from previous studies.[5-8] The initial structures of each type mentioned above are set up in a perfect Si crystal, TBMD is then performed to relax the structures until they reach their energy minimum. The formation energies are finally calculated. The vacancy formation energy is computed by taking out one atom from its lattice site and relaxing the crystal to reach its energy minimum. The results of the formation energies are summarized in Table I, and compared to data obtained from first-principle local density approximation (LDA) [6] calculation, GSP [4] and classical (SW) [5] molecular dynamics. The comparisons show that, among the four methods, the formation energy of s
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