Vacancy-Assisted Diffusion in Silicon: A Three-Temperature-Regime Model

In this Letter we report kinetic lattice Monte Carlo simulations of vacancy-assisted diffusion in silicon. We show that the observed temperature dependence for vacancy migration energy is explained by the existence of three diffusion regimes for divacancies. This characteristic has been rationalized with an analytical model. In the intermediate temperature regime the divacancy dissociation plays a key role and an effective migration energy $E_v^m\sim 2\mathrm{eV}$ is predicted, computed from either full ab initio values or mixed with experimental ones. The exact position of this temperature regime strongly depends on vacancy concentration. Previous contradictory experimental results are revisited using this viewpoint.

Figure 1

Energies for states and barriers as used in the KLMC calculations. The represented path is a divacancy dissociation.

Figure 2

Silicon diffusivity $D_{\mathrm{Si}}$ versus reverse temperature for various vacancy concentrations as measured from KLMC simulations with $E_{1v}^m=0.45\mathrm{eV}$. Three temperature regions are evidenced and different states are activated: monovacancy in region I (high temperature), divacancy in region III (low temperature), and both in region II (intermediate temperature). The resulting effective migration energy in this region is connected to divacancy dissociation that rules the fraction of time between mono- and divacancy diffusion (see text).

Figure 3

Effective migration energies for silicon diffusion evaluated from the KLMC simulations as a function of temperature (a) at $C_v^{*}=1\times {10}^{-7}{\mathrm{at}}^{-1}$ (squares) and $4\times {10}^{-6}{\mathrm{at}}^{-1}$ (circles) with $E_{1v}^m=0.45\mathrm{eV}$; (b) at $C_v^{*}=5\times {10}^{-7}{\mathrm{at}}^{-1}$ with $E_{1v}^m=0.30\mathrm{eV}$ (circles) and $E_{1v}^m=0.45\mathrm{eV}$ (squares). The lines are computed from Eq. (3) using corresponding values.