  We perform self-consistent Schr\"odinger-Poisson calculations with exchange
and correlation corrections to determine the electron/hole gas in a radial
hetero-junction formed in a modulation doped GaAs/AlGaAs core-multi-shell
nanowire (CSNW) which is n-/p-doped. Realistic composition and geometry are
mapped on an symmetry compliant two-dimensional grid, and the
inversion/accumulation layers are obtained assuming mid-gap Fermi energy
pinning at the surface. We show that the electron and hole gases can be tuned
to different localizations and symmetries inside the core as a function of the
doping level. Contrary to planar hetero-junctions, conduction electrons do not
form a uniform 2D electron gas (2DEG) localized at the GaAs/AlGaAs interface,
but rather show a transition between i) an isotropic, cylindrical distribution
deep in the GaAs core (low doping), and ii) a set of six tunnel-coupled
quasi-1D channels at the edges of the interface (high doping). Holes, on the
other hand, are much more localized at the GaAs/AlGaAs interface and form
either i) six separated planar 2DEGs at the GaAs/AlGaAs interfaces (low
doping), ii) a quasi uniform six-fold bent 2DEG (intermediate doping), or iii)
six tunnel-coupled quasi-1D channels at the edges (high doping). We also
simulate the electron/hole gas in a CSNW-based field effect transistor. The
field generated by a back-gate may easily deform the electron or hole gas,
breaking the six-fold symmetry. Single 2DEGs at one interface or multiple
quasi-1D channels are shown to form as a function of voltage intensity,
polarity, and carrier type.
