Diffusion of point defects in crystalline silicon using the kinetic activation-relaxation technique method

We study point-defect diffusion in crystalline silicon using the kinetic activation-relaxation technique (k-ART), an off-lattice kinetic Monte Carlo method with on-the-fly catalog building capabilities based on the activation-relaxation technique (ART nouveau), coupled to the standard Stillinger-Weber potential. We focus more particularly on the evolution of crystalline cells with one to four vacancies and one to four interstitials in order to provide a detailed picture of both the atomistic diffusion mechanisms and overall kinetics. We show formation energies, activation barriers for the ground state of all eight systems, and migration barriers for those systems that diffuse. Additionally, we characterize diffusion paths and special configurations such as dumbbell complex, di-interstitial (IV-pair+2I) superdiffuser, tetrahedral vacancy complex, and more. This study points to an unsuspected dynamical richness even for this apparently simple system that can only be uncovered by exhaustive and systematic approaches such as the kinetic activation-relaxation technique.

Figure 1

Topological classification using nauty. (a) Atoms within a sphere of a given radius around a central atom are extracted from the system. (b) Edges are drawn between pairs of atoms below a cutoff distance. (c) The resulting graph is analyzed by nauty and (d) characterized by a unique identifier.

Figure 2

Representation of all 5000 k-ART accepted activation events for the divacancy system (inset: zoom-in between 0.5 to 0.9 eV). All initial and final energies are measured from 2Va, the ground state (set to 0.0 eV). While the $x$ and $y$ axes represent the initial and final energy for each event, the color defines the saddle energy. The energy of the various minima is indicated above the graph and the black arrows represent the trajectory of the first three accepted events.

Figure 3

Top panel: Energy histogram of the visited states during the 5000-event simulation of the divacancy with the ground state set at 0.0 eV. Middle panel: Histogram of the activation energy barrier for selected events during the 5000-event simulation. Bottom panel: Energy histogram final state for each of the 5000 accepted events with the ground state set to 0.0 eV. Colors are set according to the energy of initial configurations then used in the two others histograms to follow the possible transition path for each of them. These colors are associated to configurations 2Va (blue), 2Vc (green), 2Vd (red), 2Ve (yellow), 2Vf (cyan), and 2Vb (purple).

Figure 4

Squared displacement of a function of time (in blue) for the divacancy system. The green line shows a linear regression used to extract the self-diffusion coefficient at 500 K $(\mathrm{D}=1.7\times {10}^{-11}{\mathrm{cm}}^2{\mathrm{s}}^{-1})$.

Figure 5

Representation of all 2000 k-ART accepted activation events for the trivacancy system. All initial and final energies are measured from 3Va, the ground state (set to 0). The energy position of the three dominant configurations is indicated. Configurations 3Va and 3V(2,5) are shown above the panel; blue sphere represents empty crystalline silicon site.

Figure 6

Top panel: Energy histogram of the visited states during the 2000-event simulation of the trivacancy with the ground state set at 0.0 eV. Middle panel: Histogram of the activation energy barrier for selected events during the 2000-event simulation. Bottom panel: Energy histogram final state for each of the 2000 accepted events with the ground state set to 0.0 eV. For the trivacancy system, see Fig. 3 for more information. Configurations 3Va (blue), 3V(3,6) (green), 3V(2,5) (red).

Figure 7

Shortest diffusion mechanisms for the trivacancy (left panel) and the tetravacancy systems (right panel) as measured from the 3Va and 4Va configurations, respectively.

Figure 8

(a)–(d) Cartoon representation of three events connecting different 3V(2,5) configurations. Blue and green color spheres represent silicon atoms and vacancies, respectively. (a) Initial configuration 3V(2,5). (b), (c) Final configuration 3V(2,5) without break of the divacancy. (d) Final configuration 3V(2,5) with break of the divacancy. Red, green, and blue arrows represent movement of silicon atom (thin arrows) and energy barrier (thick arrows) associated to the transition from (a) to (b), (c), and (d), respectively.

Figure 9

Representation of all 2671 k-ART accepted activation events for the tetravacancy system. All initial and final energies are measured from 4Va, the ground state (set to 0). Inset: zoom-in between 0 and 2 eV. Configurations 4Va and 4V(chain) are shown above the figure.

Figure 10

Top panel: Energy histogram of the visited states during the 2671-event simulation of the tetravacancy with the ground state set at 0.0 eV. Middle panel: Histogram of the activation energy barrier for selected events during the 2671-event simulation. Bottom panel: Energy histogram final state for each of the 2671 accepted events with the ground state set to 0.0 eV. For the tetravacancy system (see Fig. 3 for more information). Configurations 4Va (blue), 4V(chain) (green), 4V(3,4,6) (yellow), 4V(6,7,11) (pink), 4V(2,3,5) (orange), 4V(2,3,5)* (purple).

Figure 11

Diffusion mechanisms for the monointerstitial. Lines represent events passing by the low saddle point (Sad1), dot lines represent events passing by the high saddle point (Sad2). Mechanisms denote A, B, and C are events, from and to the ground state 1Ia, from and to the metastable state 1Ib, and from 1Ia to 1Ia passing by 1Ib, respectively. C1, C2 correspond to transition from 1Ia to 1Ib, from 1Ib to 1Ia, respectively.

Figure 12

Representation of all 1048 k-ART accepted activation events for the di-interstitial system. All initial and final energies are measured from 2Ia, the ground state (set to 0). Inset: zoom-in of the region 0 to 0.5 eV. The ground state 2Ia and metastable state 2Ib are represented above the panel. Blue and beige spheres represent empty crystalline site and off-lattice Si atoms, respectively.

Figure 13

Top panel: Energy histogram of the visited states during the 1048-event simulation with the ground state set at 0.0 eV. Middle panel: Histogram of the activation energy barrier for selected events during the 1048-event simulation. Bottom panel: Energy histogram final state for each of the 1048 accepted events with the ground state set to 0.0 eV. For the di-interstitial system (see Fig. 3 for more information). Configurations 2Ia (blue), 2Ib (red).

Figure 14

Top panel: Energy histogram of the visited states during the $13623$-event simulation with the ground state set at 0.0 eV. Middle panel: Histogram of the activation energy barrier for selected events during the $13623$-event simulation. Bottom panel: Energy histogram final state for each of the $13623$ accepted events with the ground state set to 0.0 eV. For the tri-interstitial system (see Fig. 3 for more information). Configurations 3Ia (blue), 3Ib (green), 3Ic (red), 3Id (yellow), 3Ie (pink).

Figure 15

Dominant configurations and transitions for the tri-interstitial system. Si atoms are in beige color and empty crystalline sites are in blue. A cyan circle is used as a reference point for the configurational changes. Arrows show movement of atoms (thin arrow) over structural transitions (thick arrow).

Figure 16

Representation of all $13623$ k-ART accepted activation events for the tri-interstitial system. All initial and final energies are measured from 3Va, the ground state (set to 0). Relative energies for the various states are indicated.

Figure 17

Representation of all 1595 k-ART accepted activation events for the tetrainterstitial system. The ground state 4Ia and configuration 4Ib are shown above the panel. All initial and final energies are measured from 4Ia, the ground state (set to 0).

Figure 18

Top panel: Energy histogram of the visited states during the 1595-event simulation with the ground state set at 0.0 eV. Middle panel: Histogram of the activation energy barrier for selected events during the 1595-event simulation. Bottom panel: Energy histogram final state for each of the 1595 accepted events with the ground state set to 0.0 eV. For the tetrainterstitial system (see Fig. 3 for more information). Configurations 4Ia (blue), 4Ib (green), class 4Ic (red), configuration 4Id (yellow), class 4Ie (cyan).

Figure 19

Shortest diffusion mechanisms for the tetrainterstitial system. (The 0 eV is 4Ia configuration for the tetravacancy.)