Physics – Condensed Matter – Mesoscale and Nanoscale Physics
Scientific paper
2004-02-18
Phys. Rev. B 69, 245321 (2004)
Physics
Condensed Matter
Mesoscale and Nanoscale Physics
15 pages, 7 figures
Scientific paper
10.1103/PhysRevB.69.245321
We consider a set of electrostatic problems relevant for determining the real-space structure and the ground-state energy of a two-dimensional electron liquid subject to smooth external potentials. Three fundamental geometries are investigated: an elongated metallic island, an antidot, and a constriction. In the first two cases complete closed-form analytical solutions are obtained, despite the absence of rotational or translational symmetries. These solutions govern the shape and size of large quantum dots, and also the size of the depletion regions and the density profiles around isolated antidots. For the constriction, an exact asymptotical formula for boundary shape is derived and arguments are given in favor of its universality. For the cases where the full analytical solution cannot be obtained, an approximate method is proposed as an alternative. Its accuracy is verified against numerical simulations in a periodic (checkerboard) geometry.
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