Fully screened two-dimensional magnetoplasmons and rotational gravity shallow water waves in a rectangle

We study plasmons in a rectangular two-dimensional (2D) electron system in\nthe vicinity of a planar metal electrode (gate) and in the presence of a\nperpendicular uniform magnetic field, using Maxwell's equations and neglecting\nretardation effects. The conductivity of the 2D system is characterized by the\ndynamical Drude model without taking collisional relaxation into account, which\nwell describes both high mobility graphene and other field effect transistor\nstructures, including quantum wells like Ga(Al)As, in the terahertz and in some\ncases sub-terahertz frequency ranges. Without a magnetic field, we analytically\nfind the current distribution and frequency of plasma eigenmodes when the\nplasmon wavelength is much larger than the distance to the gate, i.e. in the\nfully screened limit. To find an approximate solution in a magnetic field, we\nexpand current in the complete set of eigenmodes without magnetic fields.\nAnalytical asymptotic expressions in weak and strong magnetic fields were\nobtained for the lower modes. Unlike the disk and stripe, the frequencies of\nthese modes tend to zero as the magnetic field tends to infinity. We also\ndiscuss a direct analogy to rotational gravity shallow water waves, where\nsize-quantized Poincare waves correspond to size-quantized magnetoplasmons,\nwhile Kelvin waves correspond to edge magnetoplasmons.\n