*3.1. Double Bubble Clusters*

With each layer (single bubble) composed of either ZnO or GaN, there are four possible *n* = 60 double bubble structures that can be constructed using the procedure discussed in Section 2. The cluster structures are relaxed so as to minimise the energy, which is initially defined using a semi-empirical potentials and then, during a final refinement stage, using density functional theory (DFT); see Sections 4.1 and 4.2 for details. The four double bubble clusters consist of: (a) only zinc oxide, denoted (ZnO)12@(ZnO)48; (b) only gallium nitride, denoted (GaN)12@(GaN)48; (c) a gallium nitride sodalite cage inside a zinc oxide bubble, denoted (GaN)12@(ZnO)48; and the inverse (d) a zinc oxide sodalite cage inside a gallium nitride bubble, denoted (ZnO)12@(GaN)48. During geometry optimization, although the high *Th* and *T* symmetry that is maintained when semi-empirical calculations are employed, there is a reduction of symmetry for all four systems to C*n*, where *n* = 1 or 2. As reported in Table 1, double bubble clusters with internal (ZnO)12 sodalite cages adopt C2 symmetry, whereas those that had gallium nitride sodalite cages adopt *C*1 symmetry—*i.e.*, there is no symmetry in those structures. The average relaxed bond lengths, separated into inner-bubble bonds, outer-bubble bonds, and inter-layer bubble bonds (M–X bonds connecting the inner to the outer bubbles) are also reported in Table 1. The average bond lengths of zinc oxide and gallium nitride are similar; although the average bond length for zinc oxide inner bubbles are slightly greater than the average bond lengths of gallium nitride inner bubbles.

Now, consider the distribution of bond lengths, *G*(*x*), using a Gaussian broadening function for each bond length, which is normalised to the number of linkages between the inner and outer bubble (*N* = 48 for our *n* = 60 double bubbles):

$$G(\mathbf{x}) = \mathcal{C} \sum\_{l=1}^{N} \exp(b\_l - \mathbf{x})^2 / 2\sigma^2 \tag{1}$$

*C* is a normalising constant, *bi* is the length of bond *i*, and ı is the dispersion (width) of the Gaussian function. This function is plotted in Figure 5 for two values of ı: 0.02 Å (red line) and 0.10 Å (blue line). The greater value of ı allows the resolution of two distinct peaks for the systems of interest. These two peaks are reported in Table 1, labelled as *A* and *B* inter-bubble bond distances.

**Table 1.** Structural parameters of double bubble clusters, where *D*outer is the mean distance between M–X atoms in the outer bubble, *D*inner is the mean distance between M–X atoms in the inner bubble, and *D*inter is the distance between the inner and outer bubbles. (Number in parentheses indicates standard error.)


**Figure 5.** Bond distribution plots for the double bubble cluster systems. Red line: Dispersion of Gaussian = 0.02, Blue line: Dispersion of Gaussian = 0.1.

We observe that the pure double-bubble clusters have similar bond distributions, and notice only a difference of a small peak at 2 Å for the pure GaN system, which appears as a shoulder on the 2.3 Å peak in the pure ZnO system. We mark this shoulder (at approximately 2.25 Å) as the split of the distribution into bonded and non-bonded linkages. The number of bonded linkages, in fact, is constant for all the systems except for (ZnO)12@(GaN)48 and has a value of twenty-four, which is related to the ideal T symmetry octahedral shape. In this type of linking, two extremes can be possible: four of the eight hexagonal rings form drums with the outer bubble, or only half of the possible bonds are formed in such drums—see Figure 3b. The (ZnO)12@(GaN)48 double bubble, in contrast, has only twenty-two bonded linkages, which is not due to an inner bubble displacement from the centre of the outer bubble but is caused by a distortion in the outer bubble. To relate these observations to macroscopic properties of the systems, we considered the deformation as seen from the displacement of the centre of mass (COM) of the inner bubbles with respect to the outer bubbles, and their normalized second moments of atom distribution, as given in Table 2.

**Table 2.** Centre of mass (COM) differences and normalised second moments of atom distributions for the double bubble clusters (*x*, *y*, *z* coordinates).


The largest COM displacement is seen in the (GaN)12@(ZnO)48 system and smallest in the inverse (ZnO)12@(GaN)48 system. The deformation is also lowest in the latter system, but has the largest values in pure ZnO. We explain this behaviour by considering the relative sizes of the inner and outer bubbles: the larger ZnO inner bubble fills in the space offered by the smaller GaN outer bubble better than the GaN counterpart. An additional point to take into account is the greater flexibility of the ZnO bubbles as compared with GaN: the size mismatch between the inner and outer bubble is accommodated easier by ZnO, the bubbles of which show the greater deformations. This flexibility is also seen in the bulk framework systems as discussed in Section 3.2 below. We show in Table 3 the energy of association, *E*Assoc, calculated as the difference in total energy of the double bubble cluster from their moieties, *i.e.*, the *n*<sup>1</sup> *=* 12 and *n*2 = 48 bubbles, and formation enthalpy, *H*f,:

$$H\_f = \frac{E\_{DB} - [n\_1(E\_a) + n\_2(E\_b)]}{n\_1 + n\_2},\tag{l}$$

where *EDB* is the total energy of the double bubble cluster, *Ea* and *Eb* are the total energies of the pure bulk wurtzite structures, where *a* and *b* can be ZnO or GaN. We find that the formation of the double bubble systems is most favourable for the (GaN)12@(ZnO)48 system and least favourable for the inverse system, and that the pure double bubbles have equal formation energies.

**Table 3.** Energy of association, *E*Assoc of single-shell cages and enthalpy of formation, *H*f per atom for double bubble clusters as defined in Equation (2).


We find that the formation of the homogeneous (ZnO)12@(ZnO)48 system is the most favourable closely followed by the heterogeneous (GaN)12@(ZnO)48 systemcompared to the homogeneous bulk wurtzite phases. Systems that have an outer-bubble of GaN are less likely to form when compared with bulk (at zero temperature). If we consider the mixing energies per atom:

$$E\_{ml\chi} = \frac{E\_{DB} - [0.8(E\_a) + 0.2(E\_b)]}{120} \tag{2}$$

where *Ea* and *Eb* are the energies of the pure double bubbles that make up the mixed system, we find that the energy of mixing for (GaN)12@(ZnO)48 and (ZnO)12@(GaN)48 are 0.07 kJ/mol and í0.96 kJ/mol respectively.

#### *3.2. Double Bubble Frameworks*

We took the double bubble frameworks that were constructed using the procedure discussed in Section 2, and also corresponding systems of pure ZnO and GaN, and optimised their geometry (see Section 4.2 for details). The structural analysis performed in Section 3.1 was repeated for these frameworks. The calculated average bond lengths are presented in Table 4, again separated into inner-bubble bonds, outer-bubble bonds and, inter-layer bubble bonds (bonds connecting the inner to the outer bubbles). The graphs of the corresponding bond-length distribution analysis can be seen in Figure 6. Table 4 also has two additional pieces of information—the lattice parameter and the bulk modulus, which are available for these extended crystalline frameworks. Similar to the double bubble clusters, we find that the bonds in the ZnO inner bubble are slightly larger than the equivalent GaN bonds. In the framework systems this has a noticeable effect on the bond distribution: when the inner bubble is composed of ZnO, the bond length distribution is no longer bi-modal but has a single peak at 2.3 Å (Figure 6), which, similar to the double bubble clusters, is due to the larger ZnO bubble occupying the space inside the outer bubble. In this case, however, as the outer bubble is in a framework, it is unable to deform to the same degree as the gas-phase cluster, and only a single peak forms in the bond length distribution.

We also see that the lattice parameters for the double bubble frameworks are, as expected, related to the composition of the outer bubble, and that when the outer bubble is composed of ZnO the lattice parameters are larger. Comparing the bulk moduli of the systems, we find that the pure GaN system is the least compressible, whereas the pure ZnO system is the most. This agrees with the double bubble cluster findings, where the ZnO systems exhibit the greatest deformations. Table 5 shows the corresponding structural parameters for the wurtzite systems used in the framework analysis, and the bulk modulus of the GaN system is much larger than that of the ZnO.



**Figure 6.** Bond distribution plots for the double bubble frameworks. Red line: Dispersion of Gaussian = 0.02, Blue line: Dispersion of Gaussian = 0.1.


**Table 5.** Structural parameters of wurtzite phases.

We observe that the framework system of (GaN)12@(ZnO)48 has a similar inter-bond length distribution to that found in the double bubble systems which is again due to the fact that the smaller GaN cage has more freedom to move inside the larger ZnO bubble. The (GaN)12@(ZnO)48 system has a more clearly defined bi-modal distribution for the framework systems than observed for the double bubble cluster systems, and is likely due to reduced degrees of structural freedom with the extended bulk framework. Table 6 gives the formation enthalpies for the framework systems, and although these energies are positive *i.e.*, unfavourable with respect to the pure bulk wurtzite phases, they are small enough to be accessible at experimental temperatures, and are comparable to the formation of fullerene (C60) with respect to bulk carbon (*ca.* 40 kJ/mol) [25,26]. The pure GaN double bubble framework was found to be the least likely to form, whereas the (GaN)12@(ZnO)48 framework was found to be most favourable—again agreeing with the formation energy double bubble cluster findings.

**Table 6.** Enthalpy of formation per atom of double bubble frameworks as defined in Equation (2).


When we compare the energies of mixing (using Equation (3)) we find that the energies per atom for (GaN)12@(ZnO)48 and (ZnO)12@(GaN)48 are 1.98 kJ/mol and í2.61 kJ/mol, respectively.
