Bicrystallography

In the topological theory of interfaces and interfacial defects, the dichromatic complex (DCC) is employed for the a priori determination of admissible interfacial defects and the pertinent changes of interfacial structure and plane.^4,5 Based on the DCC, GB super- cells can be constructed for energetic relaxation.^6,7

The DCC of the 90^o <1210> orientation relationship (OR) was constructed us- ing a disymmetrization procedure.⁸ Initially the dichromatic pattern (DCP) was ob- tained by allowing the two hexagonal lattices [designated white (λ) and black (µ)] to in- terpenetrate (Figure 5.1(a)). The DCP spacegroup is P4’/mmm’ where the four-fold an- tisymmetry operation is the 90^o [1210] rotation. The DCC was then obtained by allow- ing the crystal structures to interpenetrate as shown in Figure 5.1(b). Due to the non- holosymmetric wurtzite spacegroup ( 6P ₃mc), the DCC symmetry is reduced to Pmm′ ′2 . The point symmetry and antisymmetry operations of the DCP and DCC are listed in Ta- ble 5.I.

Figure 5.1: (a) Projection of the DCP along [1210]λ|| [1120]µ. Shading indicates levels 0 and a/2 along the projection direction. (b) Schematic of the DCC viewed along [1210]λ|| [1120]µ. The common origin is taken at 0,0,0. Shading is as in (a). (c) Schematic of the correspondence between lattice vectors ti=1/3<1210> and structural vectors vk=1/6<2203>, with residual Burgers vectors b(λµ) indicated.⁹

In Table 5.II the lattice mismatch between lattice planes that are aligned (or al- most aligned) in the abutting crystals for three binary compounds (GaN, AlN and InN) is given. The 90^o [1210] OR yields multiple parallel families of low-index crystal planes.

For the case of zero misfit, aside from the {1210} planes normal to the 90^o axis, a small angular deviation must be accommodated for planes of {1011} and {1012} type.

123

Table 5.I: Point symmetry and antisymmetry operations in the DCP.

Operation Orientation in crystal λ Orientation in crystal µ Operations in the DCC

1 -

m (1210) (1120)

2’ 2

[101 ]{ [1011]}≅

Λ * 2

[110 ]{ [1101]}≅

Λ *

m’ (101(Λ 2 )){ (10 12)}≅ * (110(Λ 2 )){ (1102)}≅ * Operations suppressed in DCC

1 - -

2 [1210] [1120]

2 [0001] [1100]

2 [1010] [0001]

M [0001] [1010]

M [1010] [0001]

4’ [1210] [1120]

4’ [1210] [1120]

2’ 2

[101(− )]{ [1011]}≅

Λ * 2

[110(− )]{ [1101]}≅

Λ *

m’ (101(−Λ 2 )){ (1012)}≅ * (110(−Λ 2 )){ (1102)}≅ *

*Off by 1.8^o for GaN, 2.3^o for AlN and 2.1^o for InN.

Table 5.II: Misfit between the principal parallel or almost parallel lattice planes for the 90 [1210]^o GBs.

Misfit* (%)

Parallel crystal planes GaN AlN InN

(1210) ||(1120)_{λ µ} 0 0 0

(0002) ||(1100)_{λ µ} 6.53 8.2 7.44

(1010) ||(0002)_{λ µ} -6.13 -7.58 -6.92

(0111) ||(1011)_{λ µ} , (1101) ||(1011)_{λ µ} ,

(0111) ||(0111)_{λ µ} , (1101) ||(0111)_{λ µ} 0 (2.34^o) 0 (2.64^o) 0 (2.5^o) (1012) ||(1102)_{λ µ} , (1012) ||(1102)_{λ µ} 0 (3.6^o) 0 (4.1^o) 0(3.9^o) (1212) ||(1010)_{λ µ} , (1212) ||(0110)_{λ µ} 1.67 (1.6^o) 2.11 (2.0^o) 1.91 (1.8^o) (0110) ||(1122)_{λ µ} , (1100) ||(1122)_{λ µ} -1.64 (1.6^o) -2.07 (2.0^o) -1.87 (1.8^o)

*The values are expressed with respect to crystal λ. The numbers in parenthesis show rotational misalignments between the lattice plains.

Using the DCC, we considered possible low energy GB planes based on the crite- ria of structural continuity and misfit minimization. Based on Table 5.II, we may postu- late that GB planes that comprise at least two low misfit directions should exhibit low

124

energy. Random GBs could comprise alternating terraces of low energy GBs. In addition to this simple criterion, one should also consider the atomic density at the GB plane, bonding constraints, and energy of interfacial defects. Thus energetic simulations are needed in order to obtain a clear picture.

The admissible interfacial defects have been classified using a Volterra-like ap- proach.^4,5 Here, we confine to interfacial dislocations described by translation opera- tions. If these operations are lattice vectors, the dislocations conserve the GB structure and energy. Such dislocations have Burgers vectors given by the equation

b(λµ)ij = t(λ)i – Pt(µ)j (5.1)

expressed in the λ coordinate frame, where t(λ)i and t(µ)j are the white and black lattice vectors respectively, and P is the µ-to-λ coordinate transformation. Such dislocations can also induce an interfacial step. A step/dislocation combination is termed a discon- nection.¹⁰ Dimitrakopulos et al.⁷ have shown that, in addition to lattice vectors ti, other translation vectors, vi, that are dictated by the crystal structure may be important.

Hence, Equation (5.1) is generalized as

b(λµ)ij = v(λ)i – Pv(µ)j (5.2)

However, energetical equivalence of interfacial regions on either side of the defect is en- sured only if lattice vectors are employed.

Using the DCP, the smallest Burgers vectors given by Equation (5.1) correspond to t(µ) = 0 and t(λ)i = 1 / 3[2110]± or 1 / 3[1120]± , or we may also set t(λ)i = 0 and t(µ)j= 1 / 3[2110]± or 1 / 3[1210]± . Hence the b(λµ)ij = 1/3 2110< > Burgers vectors must be the most usual ones in this system. Given that these are lattice vectors, we can deduce that TD emanation from such GBs is possible, for example by bending of an in- terfacial dislocation at a facet junction.

If any of these vectors, say t(µ)j = 1 / 3[1210]± , is expressed in the coordinate frame of the other crystal, that is, in this case the λ frame, then obtain a vector close to v(λ)k = 1/6[2203] that describes a I1 intrinsic stacking fault (SF).¹¹ However, the two vectors, v(λ)k and t(µ)j, are misoriented by ~17^o. So Equation (5.2) yields a small resid- ual Burgers vector b(λµ)kj =

3( 21/2)

1 / 6 101  Λ −  1 / 6[1010]

  Λ ≈

  , expressed in the λ co-

ordinate frame {or

1/2

1/2 1/2 2 1/2

1 /2[(1 2 )( 1 2 )0 ] 1 /6[000(2 / )]

3

− −  

− Λ − + Λ  Λ  ≈ Λ ^{if ex-}

pressed in the µ frame}, as shown in Figure 5.1(c). Hence these GBs may also become sources of I1 SFs. This is illustrated schematically in Figure 5.2, whereby an interfacial dislocation bij(λµ)=-Pv(µ)j=-P1/6 [0223] was introduced to relax the misfit along [0001]. The step vector causes the introduction of a (0002) extra half plane but the GB structure changes because v(µ)j is not a lattice vector. However, if a I1 SF emanates into

125

crystal µ, as shown in Figure 5.2(b), the GB structure on either side of the defect is con- served. Figure5.2(c) illustrates schematically a nodal balance between a TD impinging on the GB, and a I1 SF emanating into crystal µ. The residual Burgers vector at the inter- face is b(λµ)ij = 1/3[1120] – P1/6[0223]. It is noted that these configurations are con- structed by reference to the DCC and, hence, they are general in concept and not neces- sarily associated to the particular GB plane.

Figure 5.2: (a) Volterra-like schematic of an interfacial dislocation introduced by step vector v(µ)j=1/6 [0223]. The interfacial regions on either side of the defect are energetically distinct (large and small circles denote III-type and nitrogen atoms, respectively). (b) Schematic illustra- tion of an interfacial dislocation as a result of v(µ)j combined with an emanating I1 SF. The defect now separates energetically degenerate interfacial regions. (c) Schematic of the interaction be- tween a 1/3 [1120] TD in crystal λ and a I1 SF emanating in crystal µ. The residual Burgers vec- tor is ~1/6 [000(-2^1/2/Λ)].⁹

126

5.2 HRTEM observations of 90

< < < < 1210 > > > > _GBs

In this section we present HRTEM observations of GBs in semipolar (1212) AlN epi- layers that were grown on m-plane sapphire, and contain nonpolar m-plane (0110) crystallites. The growth conditions and the TEM observations have been given in detail in Chapter 4, Section 4.1.

Figure 5.3 is a low magnification HRTEM image of the interfacial region of (1212) AlN grown without substrate nitridation. Nonpolar m-plane crystallites are pre- sent close to the substrate and the presence of Moiré fringes is due to the overlap with the semipolar matrix.¹² As indicated by dashed lines in Figure 5.3, these nanocrystals exhibit a facetted morphology. Such morphology has also been observed in the case where the matrix was nonpolar a-plane GaN that contained semipolar (1216) -oriented crystals from which TDs were often observed to emanate.¹³

Figure 5.3: Cross sectional HRTEM image of semipolar (1212) AlN on m-plane sapphire. AlN is viewed along the [1010]z.a. Misoriented grains are manifested by Moiré fringes. Two facetted grains are outlined. The nonpolar crystallites are projected along the [0001] z.a.⁹

The faceted morphology that is observed in Figure 5.3 is promoted by low- energy GBs between the crystallites and the matrix. In order to identify the facets it has to be noted that the presence of Moiré fringes and also the possibility that the facets might not be viewed edge-on in the specific cross-section of the sample lead to diffuse projected interfaces. Thus, the identification process is not always straightforward.

In order to be consistent with the analysis given in Section 5.1, the semipolar ma- trix will be referred as white crystal (λ) and the nonpolar nanocrystals as the black crys- tal (µ). In Figure 5.4 the crystal on the right-hand side of Figure 5.3 is shown in higher magnification. The solid line indicates one facet which is identified as the (1210)λ||

127

(1120)µ GB. The axis which is normal to this facet corresponds to the 90^o rotation axis that relates the two orientations. This GB comprises two low misfit in-plane directions and in particular the [0001]λ|| [1100]µ and [1010]λ|| [0001]µ.

A second facet that is identified, and is indicated by dashed lines in Figure 5.4(a), is the (1216)λ|| (1210)µ GB. This GB has also been observed in the nonpolar a-plane GaN case.¹⁴ For the semipolar growth in this case the interface is oriented parallel to the growth orientation and the corresponding low misfit in-plane directions are the [1211]λ|| [1010]µand [1010]λ||[0001]µ.

Figure 5.4: (a) HRTEM image of a facetted nonpolar AlN nanocrystal inside semipolar (1212) AlN. The projection directions are[1120]_{Al O2 3}, [1010]_{s AlN−} , and [0001]_{n AlN−} . The dashed lines indicate GBs with orientation (1216)λ|| (1210)µ. The solid line indicates a GB facet with ori- entation (1210)λ|| (11 20)µ . (b) Enlarged part of (a) showing in detail the tip of the crystallite with associated facets. The solid lines indicate families of parallel low-misfit planes [(0002)λ ||

(1 100)µ].⁹

128

In addition to the abovementioned facets the nanocrystal is also bounded by a curved GB on the left-hand side. Such curved boundaries were often seen to comprise segments of low-index facets connected between them by disconnections. In Figure 5.5(a) the curved boundary is shown along with the corresponding strain map along the [0001]λ || [1100]µdirection, obtained by GPA.¹⁵ In the strain map the periodic intro- duction of interfacial dislocations is shown, and the (0002)λ extra half planes that are introduced in order to accommodate the misfit between the two orientations are out- lined in Figure 5.5(c) which is an enlargement of the indicated region in Figure 5.5(a).

The two lower dislocations are associated with an upward-oriented interfacial step equal to one monolayer (ML) in height. The third one is associated with a downward- oriented step equal to four MLs in height, meaning that all these defects are disconnec- tions. The interfacial terraces between the disconnections have the (1212)λ|| (1010)µ orientation which comprises the [1213]λ|| [1210]µand [1010]λ|| [0001]µ low misfit directions.

Figure 5.5: (a) HRTEM along [1010]λ|| [0001]µ showing the curved boundary of one of the nanocrystals. (b) Corresponding strain map along the [0002]λ || [1100]µdirection. (c) En- largement of the indicated part of (a) showing three disconnections between (12 12)λ||

(10 10)µterraces.⁹

The (0002)λ|| (1100)µ family of inclined low misfit planes is visible in Figure 5.5. Another variant of this boundary is the horizontal (1212)λ|| (0110)µboundary. A

129

large flat segment of this variant has not been observed as can be seen in the HRTEM image of Figure 5.6. While a nearly flat (1212)λ|| (0110)µ interface is observed on the left-hand side, the right-hand side exhibits a stepped structure. This stepped structure comprises alternating (1212)λ|| (0110)µand (0110)λ|| (0001)µsegments with visible interfacial dislocations. The Burgers circuit around one of these dislocations shows that they are of the type presented in Figure 5.2(a) and also have been observed in GaN ma- terial. In Figure 5.2(b) a semipolar GaN nanocrystals is observed in the interfacial region and it is connected with a TD.¹

Figure 5.6: (a) HRTEM image along [121 3]λ|| [21 10]µand superimposed strain map, showing an interface that consists of an almost flat (left-hand side) and a stepped (right-hand side) re- gion. Both regions comprise (1212)λ || (01 10)µboundary segments. In the stepped area dislo- cations are located at such segments and a Burgers circuit around one of these dislocations is indicated. (b) Cross-sectional HRTEM image that shows an interfacial semipolar GaN nanocrys- tals in the nonpolar matrix. The image is recorded by using only the 0002n-GaN and1104 Al2O3 re- flections. The image shows a TD connected to the semipolar island.^1,9

One of the most noticeable GBs in this system is the (1010)λ|| (0001)µGB, and it can be viewed along [1213]λ|| [2110]µdirection as shown in Figure 5.7. Figure 5.7 il-

130

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.