*3.1. Interaction between an 1/2[111] IDL to a Vacancy*

Two types of the 1/2[111] dislocation loop, IDL and VDL, can be experimentally observed [21]. To understand the interaction mechanism of the IDL and VDL to a vacancy, we calculate their binding energies by varying the relative position of the vacancy. Figure 2 depicts the binding energies of the IDL to a vacancy on the HP of the loop along the Direction I and Direction II.

**Figure 2.** The distribution of binding energies of an IDL to a vacancy when the vacancy is placed on the HP (**a**), along the Direction I (**b**), and along the Direction II (**c**), respectively. The inset figure shows the binding energies at a small energy scale.

The distribution of the binding energies of IDL (R = 40.0 A˚ ) to a vacancy placed on the HP (XY plane) is shown in Figure 2a. We choose the center of the loop as the origin of the coordinate. The binding energy is calculated by creating a vacancy at every lattice position on the HP. The color bar indicates the values of the binding energies of the IDL to the vacancy. It can be seen that the binding energy is relatively small when the vacancy is placed far away from the dislocation loop. The binding energy becomes large when the vacancy is placed close to the edge of the IDL, where the vacancy can be absorbed. We define the regime that the IDL can absorb the vacancy as their absorption area. This absorption area has a ring-like shape and its width is defined as the absorption distance. The calculated absorption distance is ~8.0 Å. The binding energies approach zero when the vacancy is not in the absorption area.

Due to the symmetrical distribution of the binding energies on the HP as shown in Figure 2a, we calculate the binding energies only along one particular direction to save computational resources. Figure 2b shows the calculated distribution of binding energies of the IDL to a vacancy as a function of their distance along the Direction I. With the increase of the distance, the binding energies gradually decrease for the same sized IDL. The binding energies are almost zero (<0.025 eV) when the distance is larger than 30.0 Å for all sized IDL. As the radius of the IDL is increased from 10.0 Å to 50.0 Å, the highest binding energy decreases from 0.25 eV to 0.01 eV. Figure 2c shows the distribution of the binding energies between the IDL and vacancy as a function of their distance along the Direction II. The binding energies remain constants within a certain distance for the same sized IDL. This indicates the vacancy is absorbed by IDL. Beyond this distance, the binding energies are nearly zero (<0.15 eV). This distance is defined as the absorption distance along the Direction II. With the radius of the IDL increasing from 10.0 Å to 50.0 Å, the largest binding energy decreases from 5.1 eV to 1.4 eV and the absorption distance is 14.0–17.0 Å. There is no clear evidence that the IDL size has an influence on the absorption distance based on our simulation results.

Comparing the binding energies calculated along the Direction I and Direction II, we can find that the binding energies along the Direction II are always larger than that along the Direction I for the same sized IDL with the same relative distance. This implies that the attraction of the IDL to the vacancy is stronger along the Direction II than that of the Direction I. The vacancy can be annihilated by IDL only when it is placed close to the edge of the loop.

We find that a stable VDL cannot be formed in the system if the radius of the VDL is smaller than 40.0 A˚ in our simulations. It will evolve into a void. Therefore, we construct a series of VDLs with a radius of 40.0 Å to 60.0 A˚ . Figure 3 shows the distribution of the binding energies of VDL to a vacancy on the HP of the loop along the Direction I and Direction II, respectively.

The distribution of the calculated binding energies of the VDL (40.0 A˚ ) to a vacancy placed on the HP is shown in Figure 3a. It shows the absorption area of the VDL to a vacancy also has a ring-like shape, which is very similar to the case of the IDL to a vacancy. The largest binding energy is around 2.2 eV, which is smaller than that of the IDL to a vacancy. The binding energies approach to zero when the vacancy is placed outside the absorption area.

Figure 3b depicts the distribution of the binding energies as a function of their distance along the Direction I. The binding energies are negative for all sized VDL, implying the repulsive interaction between the VDL and the vacancy. The largest binding energy increases from −0.032 eV to −0.02 eV with the radius of the VDL increasing from 40.0 Å to 60.0 Å. As the distance increases, the value of the binding energy increases for the same sized VDLs. If the distance is larger than 30.0 Å, the value of the binding energy is less than 0.015 eV. The distribution of the binding energies as a function of their distance along the Direction II is shown in Figure 3c. The largest binding energy decreases from 2.4 eV to 0.9 eV with the increase of the radius of the loop from 40.0 Å to 60.0 Å. Within the absorption distance (~15.0 Å), the binding energy of a VDL to a vacancy decreases

gradually that is different from the case of the IDL and a vacancy. Beyond this absorption distance, the binding energies are nearly zero.

**Figure 3.** The distribution of binding energies of a VDL to a vacancy when the vacancy is placed on (**a**) HP, along (**b**) the Direction I and (**c**) Direction II, respectively. The inset figure shows the binding energies at a small energy scale.

Based on the above simulation results, the distribution of the binding energies of a VDL to a vacancy are similar to that of an IDL to a vacancy. However, we find that the binding energies of the IDL to a vacancy are positive while the binding energies of the VDL to a vacancy are negative. Thus, the IDL can attract the vacancy while the VDL can slightly repulse it along the Direction I. The reason can be explained by the distribution of the stress for different dislocation loops. As shown in Figure 4a, there is a compressive stress inside the IDL but a tensile stress outside it. While the signs of the stress are opposite for the VDL as shown in Figure 4b. A vacancy is very easily combined with the IDL, which is the main reason to have positive binding energies inside the IDL. The different distribution of the stress contributes to the different interaction behaviors of the IDL and VDL to a vacancy.

**Figure 4.** The stress distributions of the (**a**) interstitial dislocation loop and (**b**) vacancy dislocation loop with a radius of 40.0 Å.

## *3.2. Comparison of the Binding Energies of IDL-Vacancy by Using ET and MS*

To better understand the interaction behaviors between a vacancy and a 1/2[111] IDL, we compare their binding energies with different relative positions calculated by using both MS and ET as shown in Figure 5. The radius of the IDL is chosen as 40.0 A˚ as an example. The vacancy is placed at several selected positions along the Direction II and Direction III.

**Figure 5.** Comparison of the binding energies of the IDL (40.0 Å) to a vacancy calculated by MS and ET along (**a**) the Direction II and (**b**) the Direction III, respectively.

Figure 5a shows the comparison of the binding energies between an IDL and a vacancy calculated by ET and MS as a function of their distance along the Direction II. When the distance is less than 14.0 Å, the binding energies calculated by ET decrease dramatically from an infinity value to zero with increasing the distance, while the binding energies calculated by MS are nearly a constant. The difference can be attributed to the ET cannot accurately describe the stress field when two defects are close to each other, and also the vacancy is completely absorbed by IDL in MS simulations that is not the case in ET. Beyond this distance, the binding energies given by ET and MS are in good agreements. Figure 5b shows the comparison of the binding energies calculated by the ET and MS as a function of their distance along the Direction III. Based on the difference of the binding energies between the IDL and a vacancy, we can categorize the distribution of the binding energies into two different regimes, inside the absorption distance that is greater than 35.0 Å and less than 42.0 Å and beyond the absorption distance. Within the absorption distance, the binding energies predicted by ET tend to be infinity, while the binding energy given by MS is 5.93 eV. Beyond the absorption distance, the binding energies calculated by ET agree well with that given by MS. Furthermore, the stress inside the loop is larger than that outside the loop as shown in Figure 4a. Thus, it can be seen from Figure 5b that the binding energies given by ET are slightly larger for the vacancy placed inside the loop than that outside the loop along the Direction III. In conclusion, we basically validate our MS simulation results although the ET can not fully describe the interaction between an IDL and a vacancy.
