Band-gap renormalization and excitonic binding in T-shaped quantum wires

We calculate the electronic structure for a modulation doped and gated T-shaped quantum wire using density functional theory. We calculate the band-gap renormalization as a function of the density of conduction band electrons, induced by the donor layer and/or the gate, for the translationally invariant wire, incorporating all growth and geometric properties of the structure completely. We show that most of the band-gap renormalization arises from exchange-correlation effects, but that a small shift also results from the difference of wave function evolution between electrons and holes. We calculate the binding energy of excitons in a finite length wire using a simpler, cylindrical geometry. For a single hole and a one-dimensional electron gas of density $n_e,$ screening of the exciton binding energy is shown to approximately compensate for band-gap renormalization, suggesting that the recombination energy remains approximately constant with $n_e,$ in agreement with experiment. We find that the nature of screening, as treated within our nonlinear model, is significantly different from that of the various linear screening treatments, and the orthogonality of free carrier states with the bound electron states has a profound effect on the screening charge. In particular, we find no Mott transition. Rather, the electron and hole remain bound for all densities up to $\sim 3\times {10}^6{\mathrm{cm}}^{-1}$ and, as $n_e$ increases from zero, trion and even “quadron” formation becomes allowed.