Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section A(i) / 14

The competition between crystallization, exposed surfaces, adhesion and elastic strain can be summarized byHere $\Delta g>0$ measures the bulk driving force from supersaturation, and the $\gamma$ values are interfacial free energies. For a free spherical nucleus, $\Delta G=4\pi r^2\gamma -4\pi r^3\Delta g/3$ gives $r_{\ast }=2\gamma /\Delta g$. A support changes the heterogeneous nucleation barrier through adhesion and geometry.

Precursor activities and supply rate, temperature, time, solvent and ligands control nucleation, attachment, detachment and surface diffusion. Many nuclei divide a fixed solute inventory into smaller particles; continued growth or Ostwald ripening can enlarge them. Substrate defects determine preferred nucleation sites; lattice mismatch and interface bonding constrain orientation and elastic relaxation. Facet-selective adsorption can change both surface energies and growth rates.

Near equilibrium, a free nanocrystal follows a Wulff construction; a supported particle can follow a Winterbottom construction. Under kinetic control, slowly advancing facets remain visible while rapidly growing ones disappear. Equilibrium facet energy and kinetic growth rate are different controls, so a crystal structure does not uniquely determine particle shape.

The upper island has lower contrast than the rod in HAADF imaging, consistent with a lighter Zn-containing region above a Ru-rich support. Thickness, orientation and detector conditions also affect contrast, so intensity alone is not a compositional assay. The island is a flattened supported cap, not an isolated regular bulk crystal.

Using the 2 nm scale bar as a ruler gives a projected lateral size of order 4–6 nm and height of order 1–2 nm, depending on placement of the diffuse edges. These are approximate image estimates, not a three-dimensional diameter or size distribution. The height/width ratio is roughly $0.2$–$0.5$, and only a few atomic rows span the height. Thus surface and interface atoms form a substantial fraction of the particle. The truncated, irregular cap and weak edge order depart from an infinite, fully coordinated, perfectly periodic wurtzite crystal.

Adhesion can favor spreading across the rod. epitaxy can orient ZnO against a compatible surface repeat of the rutile structure, despite the different bulk symmetries. Coherent growth initially introduces strain; free edges allow relaxation, while larger islands may develop misfit dislocations. Facet-dependent attachment, polar surface energies, ligand coverage and unequal precursor delivery can further alter the cap. A final image does not uniquely reveal which kinetic pathway operated.

To measure strain, first index the zone axis and projected lattices. For pixel spacing $s$ and scale-bar length $s_{\mathrm{bar}}$, $d_{\mathrm{image}}=2s/s_{\mathrm{bar}}\mathrm{nm}$. Compare an indexed spacing with its unstrained reference:The latter is the hexagonal wurtzite relation. An ideal tetrahedral geometry has $c/a=\sqrt{8/3}$ and internal coordinate $u=3/8$; real bulk ZnO itself need not attain those ideal values. Measure several periods and compare interface and free-edge regions to distinguish strain from tilt and calibration errors.

A unique numerical lattice strain, vacancy concentration or phase assignment is not established by this unindexed single projection. Oxygen columns are weak, so a missing bright spot does not prove an oxygen vacancy. Tilted images, electron diffraction, chemical mapping and thickness-aware simulations would test those claims. The full rutile structure is primitive tetragonal with a basis: its metal positions can appear body-centered in a conventional cell, but the oxygen basis prevents calling the entire structure a body-centered tetragonal Bravais lattice.

Two applications are ultraviolet photodetection and ultraviolet/near-band-edge light emission. The bulk edge corresponds approximately toAbove-gap absorption creates carriers that can be collected in a photodetector. Measure dark current and wavelength-dependent response to separate photocarriers from heating or leakage. An optically or electrically excited emitter instead uses electron-hole recombination. quantum confinement in sufficiently small dots can shift absorption and emission above the bulk band gap. Surface states can produce broad defect luminescence or nonradiative loss, so a direct bulk gap does not guarantee an efficient, narrow-band emitter.