Addition spectra of arrays of interacting quantum dots: Dependence on array geometry and magnetic field

We use the extended Hubbard model to calculate the addition spectra of linear chains and ring-shaped arrays of three, four, and five quantum dots. We find that for linear arrays, the addition spectra obtained with the pure Hubbard model and the pure charging model show no qualitative differences. For rings of quantum dots, however, the Hubbard model always produces doubly degenerate states that are absent in the charging model. We attribute the double degeneracy to the presence of a periodic boundary condition. The degeneracies are lifted by a perpendicular magnetic field $\overrightarrow{B}.$ As $\overrightarrow{B}$ varies, the spectra are periodic with a period of one magnetic flux quantum ${\phi }_0=h/e$ and symmetric about one-half magnetic flux quantum enclosed by the ring of quantum dots.