Atomistic approach to simulate kink migration and kink-pair formation in silicon: The kinetic activation-relaxation technique

The energy conversion efficiency of solar cells based on multicrystalline silicon is greatly deteriorated by dislocations. However, an in-depth understanding on the dislocation motion dynamics down to atomic scale is still lacking. In this paper, we propose a novel atomistic approach to simulate the kink migration and kink-pair formation which govern dislocation motion in silicon, namely the kinetic activation-relax technique (k-ART). With this method, long timescale events can be simulated and complex energy landscapes can be explored. Four mechanisms for kink migration are observed, with total activation energy of 0.16, 0.25, 0.32, and 0.25 eV. New nontrivial kink structures that participate in kink migration are identified due to the open-ended search algorithm for saddle points in k-ART. In addition, a new pathway for kink-pair formation, with a minimum activation energy of 1.11 eV is discovered. The effect of shear stress on kink migration is also investigated. It shows that shear stress shifts the energy barriers of available events to lower energies, resulting in a change of the preferred kink-migration mechanism and a reduction of kink-pair formation energy.

Figure 1

Flowchart of the k-ART structure.

Figure 2

Schematic illustration of the topological classification procedure where the red atom is in the center of the local graph. (a) is the initial configuration with the topology sphere, and (b) is the cluster of atoms within the sphere. A connectivity graph is extracted and analyzed by NAUTY, which returns (c) a unique label characteristic of the graph's topology.

Figure 3

Illustration of a complex energy landscape. The searches for saddle require random displacements followed by relaxation of the orthogonal forces in the hyperplane until the first-order saddle point is reached. The dashed circles represents the configuration on the energy surface before relaxation and the solid circles represent configuration after relaxation.

Figure 4

(a) Schematic illustration of simulation box with two dislocation segments of length 10b. The atoms are represented by the gray area, which is surrounded by a vacuum layer of 10 Å along $x$ and $y$ surfaces. Dislocation segment 1 lies in position $A$, while segment 2 lies in position $A^{\prime }$, resulting in a kink positioned in position $B$ (mixed shuffle/glide set). The periodic boundary condition along $z$ direction results in a kink pair (top and bottom kink). In (b), the atomic structure projected along $\left[10\overline{1}\right]$ is shown, while (c) shows the atomic structure in the bottom kink projected along $\left[1\overline{2}1\right]$. The red atom represents an overcoordinated atom, connecting the two opposing dimers.

Figure 5

Simulated time (solid line) and cumulative topologies (dashed line) as a function of KMC step for (a) 0.0, (b) 0.5, and (c) 1.0 GPa.

Figure 6

Atomistic representation of stable kink structures projected along $\left[1\overline{2}1\right]$. The solid, dashed-dot, and dashed lines correspond to mechanism ${\mathrm{M}}_1^{\mathrm{m}}, {\mathrm{M}}_2^{\mathrm{m}}$, and ${\mathrm{M}}_3^{\mathrm{m}}$, respectively. The color bar represents the bond length, and red atoms are overcoordinated atoms.

Figure 7

The minimum energy pathway for the different mechanisms is shown, where the circles and crosses represent minimum and saddle points, respectively. The saddle point energy relative to the ground state is indicated by the dashed horizontal line. The labels correspond to their respective kink configuration as shown in Fig. 6. The lines act as a guide for the eye.

Figure 8

Cumulative distribution of the energy barriers of the available events during kink migration. The associated energy barriers for kink migration is marked with the black arrows. A shift in energy barriers for mechanism ${\mathrm{M}}_2^{\mathrm{m}}$ is observed with application of shear stress, changing the preferred mechanism for kink migration to mechanism ${\mathrm{M}}_2^{\mathrm{m}}$.

Figure 9

Two mechanisms are encountered, ${\mathrm{M}}_1^{\mathrm{f}}$ (solid and dashed-dotted lines) and ${\mathrm{M}}_2^{\mathrm{f}}$ (dashed line), the former is observed in the nonstressed and stressed conditions, while the latter is observed only with 0.5-GPa shear stress. The ball-stick models illustrates the atomic configurations with same color scheme as Fig. 4. The numbers indicates the atoms that participate in kink-pair formation. The forward barriers for each mechanism is indicated between the configurations.

Figure 10

Minimum energy pathway for kink-pair formation with shear stress of 0.0 and 0.5 GPa. Two mechanisms are observed, named ${\mathrm{M}}_1^{\mathrm{f}}$ and ${\mathrm{M}}_2^{\mathrm{f}}$, where the latter is only observed for the 0.5 GPa simulation. The labels correspond to their respective kink configuration as shown in Fig. 9. The lines act as a guide for the eye.