Modelling of interfacial structures and energies

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lustrates a triple junction formed by semipolar (1212) AlN, nonpolar (0110) AlN, and r-plane sapphire. This GB comprises the [1210]λ|| [1120]µand [0002]λ||[1100]µsets of low (or zero) misfit directions. However, since the [0002]λ|| [1100]µdirection is in- clined relative to the projection direction of Figure 5.7, the GB dislocations are not visi- ble edge-on.

Figure 5.7: HRTEM image of the triple junction region between sapphire, semipolar (1212) AlN, and nonpolar m-plane (0110) AlN. The projection direction is [121 3]λ || [21 10]µ ||

[0001]Al2O3. The GB between the two AlN orientations is (1010)λ|| (0001)µ.⁹

In this Section representative cases of GBs that comprise low misfit directions were presented. In order to appreciate if such GBs are energetically stable, their interfa- cial energies were investigated through atomistic optimized and relaxed models.

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lar the III-species environment approach was appropriated for the defects description.¹⁹ For the supercell construction all the possible CNIDs were considered.⁶ For each one the atomic configurations were constructed in the form of rectangular parallelepiped supercell volumes. Each supercell contained enough number of layers parallel to the in- terface to avoid size effects. The atomistic models contained 50000-180000 atoms with periodic boundary conditions imposed along all directions. The size of the supercells was appropriate in order to follow the periodicity of the coincident site lattice along the interfaces of each GB. The distance between the bicrystal components at the interfaces was optimized. The minimum energy configurations calculations were performed by using the quench molecular dynamics method. After the atomic configurations were re- laxed, HRTEM image simulations were generated by using the EMS software.²⁰

Figure 5.8(a) shows a comparison between the interfacial energies of the ener- getically favorable atomic configurations of three GBs observed by HRTEM in the previ- ous Section, that is the (1010)λ|| (0001)µ, (1210)λ|| (1120)µ, and (1212)λ|| (1010)µ GBs. The interfaces were assumed to be flat without disconnections. The (1010)λ||

(0001)µ GB was found to have the smallest energy compared to the rest. The atomic structure and HRTEM image simulation of this interface, in [1213]λ|| [2110]µprojec- tion, are illustrated in Figure 5.9(a). The low energy of this GB is justified by the zero misfit along the [1210]λ|| [1120]µdirection, meaning that only one array of interfacial dislocations is geometrically necessary. Along the [0001]λ|| [1100]µ direction, the GB dislocation spacings are 4.2 nm for GaN, 3.3 nm for AlN, and 4.1 nm for InN. This GB is frequently observed experimentally consistent with its characterization as a low-energy one. Moreover, since this interface is parallel to the growth direction its stability might promote the growth of the crystallites along this direction.

Figure 5.9(b) illustrates the low energy atomic structure of the (1210)λ||

(1120)µ GB. This is a GB that exhibits high mismatch along both the [1010]λ|| [0001]µ , and [0001]λ|| [1100]µdirections, as well as a change in the sign of the misfit between these two directions. As shown in Figure 5.8(a), this boundary exhibits the largest ener- gy of the three studied GBs for all three binary compounds.

The (1212)λ|| (1010)µ GB is the second largest in energy among the examined GBs. It comprises the same families of low-misfit planes as the (1210)λ|| (1120)µ GB, except that the (0002)λ|| (1100)µplanes are now inclined at θ=60^o relative to the GB plane. Hence, the spacing of the GB dislocations is d/sinθ, where d is the spacing be- tween the extra half planes. The resulting dislocation spacings are 4.8 nm for GaN, 3.8 nm for AlN, and 4.7 nm for InN. Hence, this GB exhibits the second lowest dislocation density of the three considered GBs.

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Figure 5.8: (a) Chart of the interfacial energies of principal GBs in the 90^o<1210>system. (b) Comparative results on interfacial energies of the (12 12)λ|| (10 10)µboundary after defect introduction. A: with steps along [0001]µ only. B: with steps along both [0001]µ and [1010]µ . C: with combined steps/SFs along [0001]µ only. D: with combined steps/SFs along [0001]µ, and steps along [1010]µ.⁹

Overall, the calculated energies of the studied GBs are in agreement with the en- ergies of planar defects, such as IDBs (0.03–0.29 eV/Ų), or tilt GBs (0.1–0.15 eV/Ų) that have been calculated previously for GaN using the same interatomic potentials.^16,19

,21 The GBs of the present study exhibit, in general, higher self-energies due to the more

complex bonding environment at the interface.

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Figure 5.9: (a) Simulated atomic model of the (1010)λ|| (0001)µ boundary in AlN and super- imposed simulated HRTEM image along the [121 3]λ|| [21 10]µ (thickness 4.6 nm and defocus

~62 nm; atomic columns are white). Large and small circles denote the III-type and N atoms, respectively. (b) Simulated atomic model of the (1210)λ|| (11 20)µboundary in GaN, and su- perimposed HRTEM image simulation along [1010]λ|| [0001]µ (thickness 5.0 nm, defocus ~39 nm).⁹

So far, interfacial structures were simulated by considering flat interfaces be- tween the bicrystal components. Interestingly, when relaxed in this manner, the

(1212)λ|| (1010)µ GB supercell was found to exhibit long range strain and it is not clear whether this is attributed to the inclined family of low misfit planes. Since the ex- perimental observation of this GB was only at nanofacets of curved boundaries, further study of this interface, with defect arrays that were observed experimentally, was con- ducted.

Initially, the dislocations of Figure 5.2(a) were introduced along the [1213]λ||

[2110]µ direction. The relaxed atomic structure comprising such defects and the HRTEM image simulation are illustrated in Figure 5.10(a). Next, emanating I1 SFs were introduced at the dislocation core sites, consistent with Figure 5.2(b). In this manner the interfacial structures on either side of the SFs to become energetically degenerate and the GB energy is lowered, as shown in Figure 5.8(b). The obtained relaxed atomic configuration is shown in Figure 5.10(b). The addition of I1 SFs does not contribute to increase the energy since the associated…ABC… stacking is indistinguishable energeti-

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cally from the wurtzite …ABAB… one with our interatomic potentials. Therefore a re- duction of the configurational energy is obtained due to the elimination of the surface tension force induced by the coexistence of distinct GB structures.

Figure 5.10: (a) Relaxed atomic models of the (12 12)λ|| (10 10)µGB in GaN along [121 3]λ||

[21 10]µ. In (a) the HRTEM simulation of the supercell comprising bij(λµ)=1/6 [022 3]µinterfa- cial dislocations is illustrated (thickness 7.0 nm, defocus ~62 nm). In (b) the relaxed supercell with the combination of I1 SFs and the interfacial dislocations is shown. Symbols are the same as in Figure 5.9.⁹

We next introduced the dislocations along the [1010]λ|| [0001]µdirection. The- se defects were not structurally resolvable in our TEM along the [1010]λzone axis, ex- cept from the basal planes. On the other hand, we have observed from Figure 5.5 that these dislocations are disconnections. We postulate their topological character taking the minimum Burgers vectors that result in the introduction of an extra half plane. This a priori characterization with a step sense toward the semipolar crystal λ is illustrated in Figure 5.11(a). The step vectors are t(λ)i=1/2[1211] and t(µ)j=1/3[1120] , resulting in bij(λµ)=1/6[1213] if expressed in the λ coordinate frame, and 1/3[1210] if expressed in the µ frame. In order to obtain periodic boundary conditions for the relaxation, we also introduced another step of opposite sense, consistent with the periodicity of the

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interface. This step is shown in Figure 5.11(b), and its Burgers vector is the same as in Figure 5.11(a). This Burgers vector is exhibited in the case of the 4 ML step observed in Figure 5.5(c), since it can be decomposed as 3 + 1 MLs, and the 3 ML step possesses al- most zero Burgers vector, as shown in Figure 5.11(c).

Figure 5.11: (a) Volterra-like schematic of the (12 12)λ|| (10 10)µ boundary showing the min- imum Burgers vector with introduction of an extra half-plane at the single step. The step vectors t(λ)i and t(µ)j and the step heights h(λ)i and h(µ)j are indicated, as well as the Burgers vector bij

and the line direction ξ of the defect. (b) Similar disconnection with a step sense toward crystal µ. The disconnection has the same Burgers vector as in (a). (c) Volterra-like schematic of a dis- connection with a step height equal to 3 MLs. (The projection direction is [1010]λ|| [0001]µ.

Symbols are as in Figure 5.9.⁹

Figure 5.12 shows the result of the simulation with a periodic array of ML-height steps of alternating senses. Through the disconnections, all interfacial regions are ener- getically degenerate. The employed supercell comprised kinks between the interesting orthogonal line defect arrays. In this manner, further reduction of interfacial energy was observed (Figure 5.8(b)). This was consistent with the experimental observations, i.e.

the observed disconnections yielded a stable interface with low energy. The nanomech- anism of energy reduction through emanation of SFs could increase further the SF den- sity in nonpolar and semipolar materials, and also lead to cubic pockets.¹ The energy reduction by the introduction of the defects was smaller in the case of InN. This is at- tributed to the lower energy of the In-N bond, leading to a relatively smaller effect when non-degenerate interfacial regions coexist. Furthermore, the defects of InN generally exhibit lower energy than corresponding defects in GaN.^19,22

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Figure 5.12: (a) Relaxed atomic model of the (12 12)λ|| (10 10)µboundary in GaN projected along [1010]λ|| [0001]µ. The minimum periodicity along [121 3]λ|| [1210]µ is presented. The steps given in Figure 5.11 have been introduced. (b) Corresponding HRTEM image simulation (thickness 4.3 nm, defocus ~39 nm). Symbols are as in Figure 5.9.⁹