Figure 1

Calculated surface energy of (a) ${\mathrm{Al}}_2{\mathrm{O}}_3(0001)$ and (b) ${\mathrm{Al}}_2{\mathrm{O}}_3(1\overline{1}02)$ surfaces as a function of total number of adsorption and desorption atoms $N_{\mathrm{at}}=\sum _{}^{}|n_i|$. Only data for $-1.5\le E_{\mathrm{surf}}\le 4.0\mathrm{eV}$ are displayed. Triangles correspond to the surfaces without N atoms, and squares, diamonds, circles, hexagons, and pentagons to the surfaces with one, two, three, four, and five nitrogen atoms in the unit cell, respectively. The surfaces with Al desorption are represented by inverted triangles. Patterns of symbols are defined by the difference between the number of incorporated N atoms and that of desorbing O atoms. Insets represent top and side views of stable surfaces with N atoms ($3/\mathrm{\text{−}}5a$ and $2/\mathrm{\text{−}}3a$ on (0001) and ($1\overline{1}20$) surfaces, respectively) and without N atom ($0/0a$), along with the in-plane directions. Large and filled (empty) small circles represent Al and O (N) atoms, respectively.Reuse & Permissions

Figure 2

Top and side views of charge density difference between the surface with diffusing atom and a simple sum of the surface without diffusing atom and $\mathrm{O}/\mathrm{N}$ atom for (a) N atom at the topmost layer $3/\mathrm{\text{−}}5b$, (b) transition state structure for oxygen outdiffusion, (c) N and O atoms at topmost layer $3/\mathrm{\text{−}}5c$, and (d) transition state structure for nitrogen indiffusion on ${\mathrm{Al}}_2{\mathrm{O}}_3(0001)$ surface. Positive (accumulated) and negative (depleted) values are represented by green and yellow regions, respectively. Isosurfaces are $\pm 0.01\mathrm{electron}/{\AA }^3$. Interatomic distances (in Å) are also shown.Reuse & Permissions

Figure 3

Geometries and energy profiles (in eV) of (a) ${\mathrm{Al}}_2{\mathrm{O}}_3(0001)$ and (b) ${\mathrm{Al}}_2{\mathrm{O}}_3(1\overline{1}02)$ surfaces during nitridation processes obtained by KMC simulations under typical nitridation conditions ($T=1400\mathrm{K}$, $p_{\mathrm{N}}=1\times {10}^{-4}\mathrm{Torr}$, and $p_{{\mathrm{O}}_2}=1\times {10}^{-8}\mathrm{Torr}$). The notation of atoms is the same as in Fig. 1. We add $\wedge$ to each configuration for transition state structures in $\mathrm{O}/\mathrm{N}$ diffusion (for instance, $n_{\mathrm{N}}/n_{\mathrm{O}}{x}^{\wedge }$). Arrowheads indicate diffusing N and O atoms. Note that the atomic coordinates in the figure are obtained by DFT calculations.Reuse & Permissions

Figure 4

Calculated nitridation rate on ${\mathrm{Al}}_2{\mathrm{O}}_3$ surfaces as a function of reciprocal temperature obtained by KMC simulations. Circles and squares represent the growth rate on ${\mathrm{Al}}_2{\mathrm{O}}_3(0001)$ and ($1\overline{1}02$) surfaces, respectively. Filled and empty symbols denote the results with nitrogen pressures at $p_{\mathrm{N}}=1\times {10}^{-4}$ and $5\times {10}^{-4}\mathrm{Torr}$, respectively. Experimental data for ${\mathrm{Al}}_2{\mathrm{O}}_3(0001)$ [13] and ($1\overline{1}02$) [9] surfaces are shown as filled and empty diamonds, respectively.Reuse & Permissions