Nonlocal perimeter magnetoplasmons in a planar array of narrow quantum rings

The collective excitations (perimeter magnetoplasmons) for a square-lattice array of quantum rings are calculated in the presence of a perpendicular magnetic field B. These calculations are done for thin rings whose width W is taken to be much smaller than the radius R and also satisfying W≪${\mathit{L}}_{\mathit{H}}$ where ${\mathit{L}}_{\mathit{H}}$≡(ħc/eB${)}^{1/2}$ is the magnetic length. The Coulomb interaction between electrons produces a depolarization shift and also couples those transitions for which Δm is different, where Δm is the difference between the angular momentum quantum numbers for the initial and final states. This coupling induces a small gap between the Δm and -Δm modes as well as a large gap between the states with different values of ‖Δm‖. This gap decreases as ‖Δm‖ increases. The collective excitation energies are also a periodic function of the magnetic flux Φ=π${\mathit{BR}}^2$ within a ring, with period equal to one flux quantum ${\mathrm{\phi }}_0$=hc/e. Only those excitations having the smallest difference ±ħ in angular momentum have appreciable dispersion due to strong Coulomb interaction effects on them. There is a peak in the excitation energy spectrum for some value of the lattice constant a due to a competition between screening and the modification in the electron density. Moreover, there is an abrupt change in the slope of the dispersion curve as a function of 1/${\mathit{R}}^2$ when the magnetic flux Φ is either an integer or half-odd integer multiple of the flux quantum ${\mathrm{\phi }}_0$.