**1. Introduction**

A new class of materials is sought that will support the separation of electrons and holes typically generated during photo-excitation by solar radiation. In this context, heterostructures of ZnO/GaN attract particular interest, as such materials have great potential in a wide range of applications from semiconductor optoelectronics to photo-catalysis [1–5].

Previous computational simulations [6–8] have predicted that both ZnO and GaN, at the nanoscale, form clusters with a bubble architecture that are dramatically different from models cut from their wurtzite bulk structures. Using ZnO and SiC as two simple examples, we have shown [9,10] how individual bubbles can combine to form extended framework materials; alternative constructions and the viability (or stability) of similar frameworks using bubbles as building blocks have also been reported [6,9–11]. For framework structures, an increase in density is typically correlated with an enhanced stability, which can be achieved by connecting appropriate building units. In our approach to framework construction, we choose a new type of unit, the so-called "double bubble", that are themselves denser than single-shell bubbles and which are a preferred motif for larger sized clusters [12].

For binary oxide and semiconductor II-VI and III-V materials with a 1-1 stoichiometry, fullerene type structures have been the focus of materials modelling at the nanoscale in the last decade. This interest has partly been spurred by reports of synthesis of (MX)*n* clusters of these materials, where M denotes metals, or cations, and X represents anions, with the mass spectra of such systems showing unexpected preference for certain sizes *n*. The preferred values of *n* are widely known as "magic numbers" [8]. The stability of such clusters has been explained on thermodynamic grounds: the binding energy per formula unit as a function of size having a minimum (*i.e.*, the energy released on cluster formation has a maximum). Alternative explanations have been proposed using: (i) a kinetic argument based on whether the cluster growth or shrinkage is an energetically favourable process; and (ii) a statistical argument: a particular cluster size may be realized in a greater number of atomic configurations compared to others, and therefore is favoured entropically. For any particular experiment, one or a combination of these factors may in fact be relevant.

Considering the atomic structure of stable clusters as a function of increasing size, we recognise an evolution of basic structural units with increasing dimensionality: from 1D—sticks; to 2D—rings and patchworks of rings; and then to 3D units that are initially composed of one layer—a tube or bubble—and then multiple layers, and finally bulk like phases (those that could be stable or metastable on the macroscopic scale). The bubble structures are also denoted in the literature as cages, spheroids, or fullerenes. The above classification is also based on the atomic connectivity or bonding, which increases with increased dimensionality. Using additional definitions given in reference [13], perfect closed bubbles are an important subclass of single walled fullerene-like clusters, in which each atom has three nearest neighbours, and the surface of such fullerenes is composed of a patchwork of hexagonal faces that is wrapped in three dimensions by the introduction of six "defects", or tetragonal faces. The existence and stability of the fullerene-like inorganic structures have been known from both theory and experiment for BN, ZnO, and MoS2 [14–16]. Perfect bubbles can also include larger patches with an even number of sides complemented by an appropriate number of tetragonal faces. In contrast to carbon fullerenes, pentagonal faces are not realised for the heterogeneous semiconductor class of compound, as they would require formation of M–M and/or X–X bonds that are electrostatically unfavourable. Due to the ionic nature of bonding in these materials, the charge disproportionation is not compatible with electron localisation, required for metal-metal bond formation, or hole localisation, which stabilises di-oxygen or di-nitrogen species.

On further size increase (cluster growth), the appearance of layered structures becomes a possibility, in which a smaller sized cluster unit is contained within a larger bubble structure. Indeed, such structures have been discovered in molecular dynamical studies of ZnS, where the smallest is found for *n* = 60: an *n* = 48 bubble forms a concentric shell around an *n* = 12 sodalite cage [12]. Although both single bubbles have the same high symmetry *Th* point group, the double bubble can relax into a lower symmetry form depending on the composition. The *Th* symmetry unit, however, can be stabilised when this unit is used in frameworks that were constructed previously from the individual single layered components.

In this paper, we investigate the different possible atomic structural relaxations of the double bubble and the effect of mixing components of different compositions, for both the individual clusters and the constructed frameworks.
