or from the pillar. Eqs. (3)–(7) do not, however, properly take quantum confinement into account. We have therefore refined the assessment of the charge state of the QDs with the tight-binding model: we have tentatively assumed that the QDs were empty (setting N$_{v}$ = N$_{c}$ = 0 in the dots so that they are free of carriers), and checked the position of the tight-binding band edges with respect to the Fermi energy. We find that all the QDs of Fig. 7 are actually empty, except those with t$_{sup}$ = 4 nm and t$_{inf}$ ≥ 8 nm, which are filled with holes. As expected from Eq. (19b), the total charge density in the AlN barriers increases with t$_{het}$ , from n$_{s}$ = 2 . 39 × 10 13 cm − 2 for t$_{inf}$ = t$_{sup}$ = 4 nm, to n$_{s}$ = 3 . 54 × 10 13 cm − 2 for t$_{inf}$ = t$_{sup}$ = 8 nm, and n$_{s}$ = 4 . 25 × 10 13 cm − 2 for t$_{inf}$ = t$_{sup}$ = 16 nm. The Fermi energy is therefore pinned in the valence band of AlN at the top surface in all heterostructures considered here ( n$_{s}$ > N + tot = 2 × 10 13 cm − $^{2}$). 

We would finally like to discuss the role of the lateral surface states. The top surface states play a key role by releasing electrons in the GaN pillar, thereby screening the pyro- and piezo-electric field. The occupied lateral surface states of the upper and lower AlN barrier also act as a (secondary) source of electrons. Most of these extra electrons (as well as the donor electrons) are, however, trapped by the empty lateral surface states of the GaN pillar. As a consequence, the GaN pillar is effectively depleted far away from the heterostructure, and the Fermi level is pinned ≃ 1 . 25 eV below the conduction band edge (see Fig. 2). 41 This does not, however, have significant influence on the physics of the heterostructure. 

## IV. ELECTRONIC PROPERTIES OF GAN/ALN NANOWIRE HETEROSTRUC- TURES 

We now discuss the electronic and optical properties of the GaN/AlN nanowire het- erostructures, and compare our results with experimental data. 

The tight-binding band gap energy E$_{g}$ of empty 4 nm thick GaN QDs is plotted as a function of the electric field E$_{QD}$ in Fig. 8. The corresponding values of t$_{inf}$ and t$_{sup}$ are reported between parenthesis. The excitonic correction is not included in this calculation and should further decrease the optical band gap by at most ≃ 25 meV. The band gap energy is strongly red-shifted (below the bulk value) by the electric field (Stark effect). It depends almost linearly on E$_{QD}$ and spans around 1 eV in the investigated range of t$_{inf}$ and t$_{sup}$ . The ground-state electron and hole wave functions of a particular QD ( t$_{inf}$ = t$_{sup}$ = 8 

