# From Stable ZnO and GaN Clusters to Novel Double Bubbles and Frameworks

^{1}

^{2}

^{*}

## Abstract

**:**

_{h}, 24 and 96 atom, single bubbles of ZnO and GaN. These are used to construct bulk frameworks. Upon geometry optimization—minimisation of energies and forces computed using density functional theory—the symmetry of the double bubble clusters is reduced to either C

_{1}or C

_{2}, and the average bond lengths for the outer bubbles are 1.9 Å, whereas the average bonds for the inner bubble are larger for ZnO than for GaN; 2.0 Å and 1.9 Å, respectively. A careful analysis of the bond distributions reveals that the inter-bubble bonds are bi-modal, and that there is a greater distortion for ZnO. Similar bond distributions are found for the corresponding frameworks. The distortion of the ZnO double bubble is found to be related to the increased flexibility of the outer bubble when composed of ZnO rather than GaN, which is reflected in their bulk moduli. The energetics suggest that (ZnO)

_{12}@(GaN)

_{48}is more stable both in gas phase and bulk frameworks than (ZnO)

_{12}@(ZnO)

_{48}and (GaN)

_{12}@(GaN)

_{48}. Formation enthalpies are similar to those found for carbon fullerenes.

## 1. Introduction

_{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.

_{2}[14,15,16]. 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.

_{h}point group, the double bubble can relax into a lower symmetry form depending on the composition. The T

_{h}symmetry unit, however, can be stabilised when this unit is used in frameworks that were constructed previously from the individual single layered components.

## 2. Construction of Double Bubbles

#### High Symmetry Double Bubble Clusters as Secondary Building Units

_{h}, T

_{d}and T. These (MX)

_{n}structures are found if n = 4 (T

_{d}), 12 (T

_{h}), 16 (T

_{d}), 28 (T), 36 (T

_{d}), 48 (T

_{h}), 64 (T

_{d}), … Larger T

_{h}and T

_{d}clusters include n = 108 and 192 and n =100, 144 and 196, respectively; examples are shown in Figure 1. These clusters can be visualised as truncated octahedra, where there is one tetragonal ring of the bubble at each of the six truncated corners, and the hexagonal patchworks form the octahedron’s faces. In this morphology, the distance between all tetragonal rings is maximised for a given size n, and the separation increases monotonically with n.

**Figure 1.**Models of high symmetry (MX)

_{n}bubbles with: (

**a**) n = 12 with symmetry T

_{h}; (

**b**) n = 48 with symmetry T

_{h}; (

**c**) n = 28, 36 and 48 with octahedra superimposed.

_{d}bubble links a side of two tetragonal rings via a “ladder” of m hexagonal rings (one hexagon wide)—see the highlighted ladder in Figure 2a—with the remaining hexagonal rings completing the faces of the octahedron. Note that the line of the octahedron edge bisects the rings of this ladder and that the tetragonal ring is out of phase with the tetragonal face created by truncating the octahedron. Constructions with m = 0, 1, 2, 3, 4 and 5 corresponds to perfect bubbles at n = 4, 16, 36, 64, 100, 144 and 196, respectively. In contrast, each octahedron edge in a T

_{h}bubble links the corner of two tetragonal rings via m + 1 M–X sticks that are separated by m hexagonal rings, forming an alternating linear pattern. Each stick is actually a shared side of two hexagonal rings, with each ring part of a hexagonal patchwork that covers a face of the octahedron. These sticks form a line segment of the octahedron edge (see highlighting in Figure 2b), and this line bisects opposite angles of the hexagonal rings, rather than opposite sides, and the tetragonal rings are in-phase with the tetragonal face created by truncating the octahedron. The smallest T

_{h}bubble, m = 0, or n = 12, has just one stick between two neighbouring tetragonal rings. The next smallest size T

_{h}bubble, m = 1, or n = 48, is constructed using two sticks and one hexagonal ring; then, for m = 2, or n = 108, there are three sticks and two hexagonal rings. Comparing the growth of the octahedron edges, it is evident why there are more bubbles with T

_{d}rather than T

_{h}symmetry.

**Figure 2.**Models of high symmetry (MX)

_{n}bubbles with: (

**a**) n = 64, symmetry T

_{d}and the ladder of hexagonal rings, highlighted in yellow, that corresponds to one of the twelve edges of an octahedron; (

**b**) n = 48, symmetry T

_{h}with a fragment that corresponds to one of twelve edges highlighted in yellow; (

**c**) n = 48, symmetry T

_{h}with one of twelve patchworks that correspond to the octahedron side highlighted in purple.

_{h}and T

_{d}symmetry this rotation is 45° out-of-phase, and, if one of the bubbles has T symmetry, between 0° and 45°. For stability, the best match is obtained when the inner and outer bubbles are taken from the set of T

_{h}bubbles and the highest density obtained by combining the smallest two of these: n = 12 (a sodalite cage) and n = 48 [10].

_{2}) [21,22] have been reported to be constructed from four and six (hexagonal) membered ring building units. CdSe cage structures have been experimentally observed to be stable and formed from truncated-octahedra [23]. DFT calculations on cage structures of CdSe that are similar to our structures have also been reported [24].

**Figure 3.**Models of the n = 60 T

_{h}double bubble, with inter-layer links between the inner n = 12 sodalite cage and the eight hexagonal rings that are in the centres of the octahedron faces of the outer n = 48 bubble, highlighted using ball-and-sticks rather than line representation for: (

**a**) no bridging links; (

**b**) four bridging links; and (

**c**) all eight bridging links.

_{h}bubble (sodalite cage) secondary building units (SBUs) of (ZnO)

_{12}and (GaN)

_{12}, see Figure 1a. As the typical Zn–O and Ga–N bond lengths are similar (1.98 Å and 1.95 Å in their ground state wurtzite form), their respective SBUs are also similar in size. Consider each SBU as an octahedron. By corner sharing the octahedra, and assuming an equal number of SBUs for each compound, we construct an fcc, rock-salt like lattice, as shown in Figure 4c. The second framework is constructed from the n = 60 T

_{h}double bubbles; see Figure 1c and Figure 4b. Again, imagining each SBU as an octahedron, but rather than corner sharing they are now stacked so that they share edges, each double bubble is surrounded by twelve others (see Figure 4d), and each edge of the outer bubble is one bond length from an edge of a neighbouring bubble forming an n = 6 double ring (a drum) and two n = 2 rings. Each tetragonal ring of an outer n = 48 bubble combines with five others to form an n = 12 T

_{h}bubble, i.e., the void is a sodalite cage. The inner sodalite cage of each double bubble is formed from (i) the same compound and (ii) two compounds, which we alternate.

_{h}symmetry. Whether there is an ideal match depends on the composition: if the two layers are of the same composition and there is not an ideal match then the inner bubble is too small. The outer eight planes of hexagonal patchworks, or octahedron faces, have more flexibility than the corners. During a geometry relaxation of the double bubble, the central hexagonal ring of each outer patchwork can move inward, maintaining the T

_{h}symmetry or, due to the repulsion between neighbouring patchworks, only the central hexagons from alternate patchworks, i.e., four of the eight, move inwards reducing the symmetry to T; see Figure 3.

_{h}and T) were constructed and then geometry optimised. Low symmetry structural distortions were allowed in the optimisation process in order to find the lowest energy double bubble configuration.

**Figure 4.**Ball and stick models of two framework structures. (

**a**) Constructed from T

_{h}bubbles of (GaN)

_{12}and (ZnO)

_{12}; (

**b**) Constructed from T

_{h}double bubbles of (ZnO)

_{48}and (GaN)

_{12}; (

**c**) the same structure as (a) but with each (GaN)

_{12}coloured red and each (ZnO)

_{12}coloured blue (lighter/darker shades used in the front/back row); (

**d**) the same structure as (b) but with each (GaN)

_{12}hidden and each (ZnO)

_{48}uniquely coloured.

## 3. Results and Discussion

#### 3.1. Double Bubble Clusters

_{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 T

_{h}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 C

_{2}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.

_{i}is the length of bond i, and σ is the dispersion (width) of the Gaussian function. This function is plotted in Figure 5 for two values of σ: 0.02 Å (red line) and 0.10 Å (blue line). The greater value of σ 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.)

System | Symmetry | D_{outer} (Å) | D_{inner} (Å) | D_{inter} (Å) | |
---|---|---|---|---|---|

A | B | ||||

(GaN)_{12}@(ZnO)_{48} | C1 | 1.92 | 1.93 | 2.05 (0.1) | 3.08 (0.1) |

(ZnO)_{12}@(GaN)_{48} | C2 | 1.89 | 1.96 | 2.13 (0.1) | 2.94 (0.1) |

(ZnO)_{12}@(ZnO)_{48} | C2 | 1.93 | 1.95 | 2.10 (0.0) | 2.94 (0.2) |

(GaN)_{12}@(GaN)_{48} | C1 | 1.93 | 1.92 | 2.10 (0.1) | 2.94 (0.1) |

**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.

_{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).

System | COM difference (COM_{Outer}–COM_{Inner}) | Normalised second moments of atom distribution | |
---|---|---|---|

Inner | Outer | ||

(GaN)_{12}@(ZnO)_{48} | 0.00, 0.11, 0.05 | 1.05, 1.01, 0.94 | 1.05, 1.01, 0.94 |

(ZnO)_{12}@(GaN)_{48} | 0.00, 0.00, 0.01 | 1.02, 1.00, 0.98 | 1.01, 1.00, 0.99 |

(ZnO)_{12}@(ZnO)_{48} | 0.00, 0.00, 0.04 | 1.09, 1.00, 0.91 | 1.05, 1.00, 0.95 |

(GaN)_{12}@(GaN)_{48} | 0.00, 0.01, −0.06 | 1.04, 1.01, 0.96 | 1.04, 1.01, 0.95 |

_{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

_{1}= 12 and n

_{2}= 48 bubbles, and formation enthalpy, H

_{f},:

_{DB}is the total energy of the double bubble cluster, E

_{a}and E

_{b}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).

System | E_{Assoc} (kJ/mol) | H_{f} (kJ/mol) |
---|---|---|

(GaN)_{12}@(ZnO)_{48} | −11.27 | 78.32 |

(ZnO)_{12}@(GaN)_{48} | −8.17 | 104.55 |

(ZnO)_{12}@(ZnO)_{48} | −9.38 | 68.50 |

(GaN)_{12}@(GaN)_{48} | −9.16 | 116.18 |

_{12}@(ZnO)

_{48}system is the most favourable closely followed by the heterogeneous (GaN)

_{12}@(ZnO)

_{48}system compared 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:

_{a}and E

_{b}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 −0.96 kJ/mol respectively.

#### 3.2. Double Bubble Frameworks

**Table 4.**Structural parameters of double bubble frameworks. (Number in parentheses indicates standard error).

System | Lattice parameter (Å) | Bulk modulus (GPa) | D_{outer} (Å) | D_{inner} (Å) | D_{inter} (Å) | |
---|---|---|---|---|---|---|

A | B | |||||

(GaN)_{12}@(ZnO)_{48} | 19.26 | 78.84 | 1.96 | 1.94 | 2.08 (0.1) | 2.96 (0.2) |

(ZnO)_{12}@(GaN)_{48} | 18.84 | 77.88 | 1.90 | 2.01 | 2.27 (0.1) | - |

(ZnO)_{12}@(ZnO)_{48} | 19.26 | 69.78 | 1.94 | 2.00 | 2.26 (0.0) | - |

(GaN)_{12}@(GaN)_{48} | 18.94 | 101.92 | 1.93 | 1.94 | 2.18 (0.2) | 3.04 (0.1) |

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

System | Lattice parameter, a (Å) | Lattice parameter, c (Å) | Bulk modulus (GPa) | u |
---|---|---|---|---|

ZnO | 3.251 | 5.204 | 146.136 | 0.382 |

GaN | 3.187 | 2.760 | 188.367 | 0.378 |

_{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 (C

_{60}) 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.

System | H_{F}/atom (kJ/mol) |
---|---|

(GaN)_{12}@(ZnO)_{48} | 13.17 |

(ZnO)_{12}@(GaN)_{48} | 21.46 |

(ZnO)_{12}@(ZnO)_{48} | 18.54 |

(GaN)_{12}@(GaN)_{48} | 27.71 |

_{12}@(ZnO)

_{48}and (ZnO)

_{12}@(GaN)

_{48}are 1.98 kJ/mol and −2.61 kJ/mol, respectively.

## 4. Computational Detail

#### 4.1. Interatomic Potentials Calculations

#### 4.2. Density Functional Theory Calculations

_{12}@(B)

_{48}systems, where A and B stand for either ZnO or GaN, provide convergence in total energy up to 10

^{−5}eV for the framework systems, which is comparable with our double bubble cluster calculations.

## 5. Conclusions

_{12}@(ZnO)

_{48}, (ZnO)

_{12}@(GaN)

_{48}, (ZnO)

_{12}@(ZnO)

_{48}and (GaN)

_{12}@(GaN)

_{48}, were first geometry optimized using a semi-empirical potential within the GULP code and then refined using FHI-aims (for the double bubble clusters) or VASP (for the frameworks) at the DFT level of theory using the PBEsol exchange-correlation functional. We found that although the average bond lengths of both ZnO and GaN are similar, the average bond lengths for ZnO inner bubbles were larger than the GaN inner bubbles of both the double bubble cluster systems and the frameworks. This relative size difference, we believe, means that the larger ZnO inner bubble fills in the space offered by the smaller GaN outer bubble better than the GaN counterpart. In addition, we found that the greater flexibility of the ZnO bubbles from calculations of bulk moduli, as compared with that of GaN bubbles, means that the size mismatch between the inner bubble and outer bubble is more readily accommodated by ZnO. Furthermore, the structural analysis of the pure ZnO double bubbles also showed the greater deformations. The average M-X inter-bubble bonds were found to exhibit a bi-modal distribution for both clusters and frameworks, except for the pure ZnO and (ZnO)

_{12}@(GaN)

_{48}framework systems. These single-peak distributions were due to the larger ZnO inner bubble that has less freedom to move than in the inverse systems. The association energies of the double bubble clusters show that the systems investigated here are favourable when compared to individual bubbles, although when compared to bulk wurtzite phases, the clusters are less favourable.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Farrow, M.R.; Buckeridge, J.; Catlow, C.R.A.; Logsdail, A.J.; Scanlon, D.O.; Sokol, A.A.; Woodley, S.M.
From Stable ZnO and GaN Clusters to Novel Double Bubbles and Frameworks. *Inorganics* **2014**, *2*, 248-263.
https://doi.org/10.3390/inorganics2020248

**AMA Style**

Farrow MR, Buckeridge J, Catlow CRA, Logsdail AJ, Scanlon DO, Sokol AA, Woodley SM.
From Stable ZnO and GaN Clusters to Novel Double Bubbles and Frameworks. *Inorganics*. 2014; 2(2):248-263.
https://doi.org/10.3390/inorganics2020248

**Chicago/Turabian Style**

Farrow, Matthew R., John Buckeridge, C. Richard A. Catlow, Andrew J. Logsdail, David O. Scanlon, Alexey A. Sokol, and Scott M. Woodley.
2014. "From Stable ZnO and GaN Clusters to Novel Double Bubbles and Frameworks" *Inorganics* 2, no. 2: 248-263.
https://doi.org/10.3390/inorganics2020248