Formation, migration, and clustering of delocalized vacancies and interstitials at a solid-state semicoherent interface

Atomistic simulations are used to study the formation, migration, and clustering of delocalized vacancies and interstitials at a model fcc-bcc semicoherent interface formed by adjacent layers of Cu and Nb. These defects migrate between interfacial trapping sites through a multistep mechanism that may be described using dislocation mechanics. Similar mechanisms operate in the formation, migration, and dissociation of interfacial point defect clusters. Effective migration rates may be computed using the harmonic approximation of transition state theory with a temperature-dependent prefactor. Our results demonstrate that delocalized vacancies and interstitials at some interfaces may be viewed as genuine defects, albeit governed by mechanisms of higher complexity than conventional point defects in crystalline solids.

Figure 1

(a) Perspective view of the simulation cell showing Cu (gold) and Nb (gray) layers with free surfaces parallel to the interface. (b) Perspective view of the terminal Cu and Nb atomic planes. (c) Top-down view of the Cu and Nb terminal planes showing the locations of representative members of two parallel misfit dislocation arrays. Angles between the dislocations and $x$ and $y$ axes are labeled. (d) Cu terminal plane with representative misfit dislocations and corresponding Burgers vectors marked. Protrusions occur at intersections between misfit dislocations (MDIs).

Figure 2

(a) Variations in energy upon multiple quenches from $T=800$ K indicate that the Cu-Nb interface has a rough potential energy landscape. (b) Thermal kink pairs are one type of structural fluctuation that occurs at the Cu-Nb interface. Atoms colored yellow form five-membered rings in the terminal Cu plane and correspond to the locations of kinks on misfit dislocations. Arrows indicate atomic half-planes that terminate at these kinks. (c) A dislocation model for the thermal kink pair in (b).

Figure 3

Differences in atomic energy in the Cu terminal plane between an interface containing a thermal kink pair at the center and the reference configuration in Fig. 1d. (a) Direct comparison of an interface structure obtained from MD and (b) of one where atomic positions outside a test region containing the kink pair were restored to their positions in the reference configuration.

Figure 4

Clusters of vacancies form kink-jog combinations at MDIs. Atom coloring is the same as in Fig. 2b.

Figure 5

Similar to isolated self-interstitials [Fig. 7g], clusters of interstitials form kink-jog pairs at MDIs. Atom coloring is the same as in Fig. 2b.

Figure 6

Formation energies of point defect clusters at an MDI in the Cu-Nb interface. The $x$ axis shows the number of point defects created in a defect-free interface, such as that shown in Fig. 1d. The lines are best fits to the vacancy and interstitial formation energies relative to the ground-state interface ($\Delta N=-2$ or $-3$).

Figure 7

Migration of a delocalized vacancy (interstitial) from one MDI (a) [(g)] to another (c) [(i)] through the intermediate state (b) [(h)]. Atom coloring is the same as in Fig. 2b. Dislocation models for configurations (a)–(c) are depicted (d)–(f) and those for (g)–(i) are depicted in (j)–(l).

Figure 8

Thermal kink pairs nucleated at (a) an MDI adjacent to that of a vacancy and (b), (c) between the kink-jogs of a vacancy extended over two MDIs. Dislocation models for configurations (a)–(c) are depicted in (d)–(f), respectively. Red arrows in (c) mark the locations of kink-jogs in the indicated kink-jog pair, showing that they are separated by fewer $\langle 110\rangle$ planes than those in (a) and (b). Atom coloring is the same as in Fig. 2b.

Figure 9

Examples of different MEPs for isolated vacancy migration. A, B, and C correspond to the states shown in Figs. 7a, 7b, 7c, respectively. [In MEPs of interstitials, which have similar qualitative features (see Fig. 4 of Ref. 40 for an example), they correspond to Figs. 7g–7i, respectively.] I corresponds to thermal kink pairs that aid defect migration, as in Figs. 8a and 8b. I${}^{\prime }$ is a kink pair that does not aid migration, as in Fig. 8c. Examples of transition states t are shown in Fig. 10.

Figure 10

Examples of saddle-point configurations encountered during the migration of (a)–(c) a vacancy and (d)–(f) an interstitial. The activation barrier increases with the size of the region that has been sheared to create thermal kinks (represented by the four-member rings marked with white contours). Atom coloring is the same as in Fig. 2b.

Figure 11

Clustering of two vacancies delocalized at adjacent MDIs (a) into a divacancy (c). The clustering mechanism is similar to that for migration of isolated point defects. In an intermediate state (b), the divacancy resides on two MDIs. Atom coloring is the same as in Fig. 2b.

Figure 12

The migration mechanism of a divacancy is similar to that of a single vacancy. In the intermediate state (b), the divacancy extends to reside on two MDIs. Atom coloring is the same as in Fig. 2b.

Figure 13

Dissociation of a trivacancy (a) into a divacancy and a monovacancy (c) is aided by the formation, migration, and annihilation of kink-jogs. In the intermediate state, the trivacancy resides on two MDIs. Atom coloring is the same as in Fig. 2b.

Figure 14

Migration of interstitial clusters occurs through an extended intermediate state where the clusters reside on two MDIs. Atom coloring is the same as in Fig. 2b.

Figure 15

(a) Total energy change (filled squares), kink-jog core energy (filled triangles), and the energy from the dislocation model (continuous curve) for the direct migration mechanism. Filled circles show the kink-jog core volume. The arrow on the left shows the range of formation energies computed for a $\frac{a_{\mathrm{Cu}}}{4}\langle 112\rangle$ jog on a screw dislocation in fcc Cu and the arrow on the right shows the corresponding formation volumes. (b) Plan view of the interface Cu (gold) and Nb (gray) atoms with a point defect in extended state B. Arrows mark the location of kink-jogs, the numbers are values of $S$, and red lines mark the nominal locations of set 2 misfit dislocation cores.

Figure 16

(a) Comparison between effective defect migration rates computed using kMC (black dots), a standard Arrhenius-type equation using only the highest activation energy event ($E_{\mathrm{eff}}^{\mathrm{act}}=0.4$ eV, gray line), and the modified rate expression in Eq. (8) with a temperature-dependent prefactor (black line). (b) Fits of the modified rate expression to MD data (gray dots). Temperatures are plotted on an inverse scale.

Figure 17

Comparison between the MD data (histograms) to the fits obtained by assuming that point defect migration follows a Poisson process (continuous curves and data points).