Diffusion mechanisms for silicon di-interstitials

Tight-binding molecular dynamics and density-functional simulations on silicon seeded with a di-interstitial reveal its detailed diffusion mechanisms. The lowest-energy di-interstitial performs a translation/rotation diffusion-step with a barrier of $0.3\mathrm{eV}$ and a prefactor of $11\mathrm{THz}$ followed by a reorientation diffusion step with a $90\mathrm{meV}$ barrier and a $2300\mathrm{THz}$ prefactor. The intermediate reorientation steps allow di-interstitials to diffuse isotropically along all possible ⟨111⟩ bond directions in the diamond lattice. The dominating diffusion barrier of $0.3\mathrm{eV}$ is not inconsistent with the experimental value of $0.6\pm 0.2\mathrm{eV}$. In addition, this lowest energy di-interstitial may diffuse to neighboring sites through an intermediate structure which is the bound state of two single interstitials. The process in which migrating single interstitials combine into a di-interstitial is exothermic with almost zero energy barrier.

Figure 1

The density of states (DOS) for the ground state di-interstitial $I_2^a$ and excited state $I_2^b$ and $I_2^c$. DOS are computed within $216+2$ atom supercells. $I_2^a$ exhibits a clear energy gap of $0.58\mathrm{eV}$ with no localized state. $I_2^b$ has an occupied localized state in the energy gap while $I_2^c$ has an unoccupied localized state in the energy gap.

Figure 2

(Color online) Two unrelaxed end points for the ground state di-interstitial, $I_2^a$, diffusion path constructed from two end points for the compact tri-interstitial, $I_3^b$, diffusion-path (Ref. 15). (a) The initial state for the $I_3^b$ diffusion path. (b) The unrelaxed initial state for the $I_2^a$ diffusion path constructed by removing atom A from (a). (c) The final state for the $I_3^b$ diffusion path. (d) The unrelaxed final state for the $I_2^a$ diffusion path constructed by removing atom A from (c). The structures shown in (a), (b), (c), and (d) are viewed along the [111] direction.

Figure 3

(Color online) The diffusion pathway of the ground state di-interstitial $I_2^a$. The path consists of a combination of two steps, a translation/rotation and a reorientation. The translation/rotation step has an activation energy of $0.3\mathrm{eV}$ and a prefactor of $11\mathrm{THz}$ and the reorientation step has an activation energy of $90\mathrm{meV}$ and a prefactor of $2300\mathrm{THz}$. The initial, intermediate and final structures are shown with the saddle point structures between them. During the translation/rotation, atoms C and D define an axis of rotation along the ⟨111⟩ direction about which atoms A and B are rotating; meanwhile, atoms A, B, C, and D translate along the ⟨111⟩ axis. Following the reorientation, atoms E and A define a new possible ⟨111⟩ axis of rotation and translation. The atoms B, C, E, and A in the final state are now equivalent to the atoms A, B, C, and D in the initial structure and can now perform another translation/rotation step. The intermediate $I_2^a$ can either perform a translation/rotation step back along $\left[\overline{1}1\overline{1}\right]$, or perform a translation/rotation step along $\left[1\overline{1}\overline{1}\right]$ and $\left[\overline{1}\overline{1}1\right]$, respectively, after a corresponding reorientation step.

Figure 4

(Color online) Translation/rotation path with seven images projected onto two-dimensional reduced coordinates. Inset: showing initial, saddle, final images along the transition path. The dumbbell AB rotates around $\left[1\overline{1}1\right]$ while translating along the $\left[1\overline{1}1\right]$ direction.

Figure 5

(Color online) The transition path connecting the $I_2^b$, $I_2^a$, and $I_2^c$ di-interstitial structures. The excited state di-interstitial $I_2^b$ and $I_2^c$ readily decay to the ground state di-interstitial $I_2^a$ with almost zero barrier. The $I_2^b$ di-interstitial is comprised of two dumbbell single interstitials, atoms A and B and atoms C and D, respectively, which are related by $C_2$ symmetry. The separation between the center of mass of each dumbbell estimates a $3.2\mathrm{\AA }$ capture radius of two single interstitials.