Assuming fixed pyro- and piezoelectric polarizations, E$_{inf}$ and E$_{sup}$ are independent on t$_{inf}$ and t$_{sup}$ for given t$_{QD}$ and t$_{het}$ . Equations (14) then show that the QD can be empty only in a finite range of positions within the heterostructure. The QD is indeed filled with electrons if it is too far from the surface, and filled with holes if it is too far from the pillar. Note, however, that quantum confinement will practically hinder the charging of the QDs by rising the electron and hole energies. The above constraints thus provide safe bounds for the design of nanowire heterostructures. Equations (11) and (13) also give an estimate of the charge densities σ$_{s}$ and σ$_{p}$ accumu- lated at top surface and interface with the pillar, respectively. The continuity equation for the electric field indeed reads at this interface: where E$_{pil}$ is the electric field in the pillar. Since E$_{pil}$ decreases rapidly away from the interface, Assuming that the tip of the nanowire is charge neutral at equilibrium, we then get: We can further split σ$_{s}$ and σ$_{p}$ into polarization and induced charges: where Δ P$_{pil}$ = P$_{pil}$ − P$_{inf}$ , n$_{p}$ is the density of the electron gas at the interface with the pillar, and n$_{s}$ is the density of charges (ionized surface states+holes) at the top surface. The latter thus finally read: Note that t$_{het}$ must be large enough for the electron gas to form at the interface with the pillar ( n$_{p}$ > 0), 8 but this is usually not limiting the design of the heterostructure. n$_{s}$e ≃ − P$_{QD}$ + t$_{inf}$ + t$_{sup}$ t$_{het}$ Δ ¯ P − κ$_{0}$κ Δ V t$_{het}$ . (19b) n$_{p}$e ≃ P$_{pil}$ − P$_{QD}$ + t$_{inf}$ + t$_{sup}$ t$_{het}$ Δ ¯ P − κ$_{0}$κ Δ V t$_{het}$ (19a) σ$_{s}$ = P$_{sup}$ + n$_{s}$e , (18b) σ$_{p}$ = Δ P$_{pil}$ − n$_{p}$e (18a) σ$_{s}$ ≃ − ( σ$_{p}$ − Δ P$_{inf}$ + Δ P$_{sup}$ ) . (17) σ$_{p}$ ≃ κ$_{0}$κE$_{inf}$ . (16) κ$_{0}$κ ( E$_{inf}$ − E$_{pil}$ ) = σ$_{p}$ , (15) 13