= Solution
Two suitable epitaxial techniques are \b[<metalorganic vapour-phase epitaxy> and <molecular beam epitaxy>].
<MOVPE> supplies metalorganic group-III precursors and a reactive nitrogen source in a flowing gas. Surface reactions, mass transport and selective nucleation control growth; it is well suited to conformal radial growth and relatively high throughput. <molecular beam epitaxy> supplies molecular/atomic beams in high vacuum, with beam fluxes and shutters controlling delivery. It offers precise flux switching and in-situ surface monitoring, but flux, geometry and nitrogen activation can limit growth rate and conformality. Neither method automatically gives identical growth on all wire <crystal facets>.
For an overall target near \b[$40\ {\rm nm\,min^{-1}}=2.4\ {\rm \mu m\,h^{-1}}$], <MOVPE> is a natural choice. This is a practical selection, not a theorem that <molecular beam epitaxy> cannot attain that rate.
Prepare the substrate/buffer and patterned SiN growth mask; grow donor-doped <gallium nitride> cores with controlled axial morphology; change conditions to favor radial deposition and incorporate indium for the thin <indium gallium nitride> active shell; then overgrow acceptor-doped p-GaN without degrading the active layer. Finally expose/contact the core and shell appropriately. Monitor facet-dependent thickness and indium content, and characterize the interfaces. If the active-layer normal rate itself were $40\ {\rm nm\,min^{-1}}$, 5 nm would take 7.5 s; the stated overall rate does not establish that the alloy shell actually grows at that same rate.
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