Basic modeling of open wave billiards using imaginary potentials for source and sink
K.-F. Berggren, I. I. Yakimenko and J. Hakanen
Due to advances in semiconductor technology it is now possible to fabricate high-quality modulation doped layered materials in which electrons form a two-dimensional high-mobilty electron gas at the interface of the two materials, typically AlGaAs/GaAs. By lithographic patterning together with suitably located gates and application of external voltages it is possible to "electrostatically squeeze" the electron gas into a varity of nano-sized closed structures. If they are sufficiently small the electrons inside such confinements occupy discrete quantized states. Elementary examples of "quantum dots" or "billiards" of this kind are squares, rectangles and circular discs. Quantum dots may be closed or open. Openness implies that there are attached leads by which a current may be passed through a dot. If the nominal mean free path exceeds the dimensions of a dot the transport is ballistic.
Because of the current scientific/technological focus on quantum dots and cavities it is of interest to model electron states, transport etc. There are many ways to do this but here we will present a basic heuristic model that extends ideas originally developed for inelastic scattering in nuclear physics. Thus complex potentials are introduced to mimic source and drain which turns the transport problem into quite an elementary one. In the article we have tested the method for a square quantum dots with leads. Wave functions and currents have been evaluated using the finite difference method. It turns out that quantum dots may be emulated by planar microwave resonators. "Macroscopic wave functions", currents etc
may be thus obtained from measurements.
The figure shows a typical case of an “open” quantum dot with “imaginary source and drain” at the very ends of the two stubs. Probability current flow (left panel) case) and probability density (right panel) as obtained from simulations. Dark and light areas in the density plot correspond to low and high densities, respectively. The state is reminiscent of a "particle-in-a box" state distorted by the leads and rounded corners. The accumulation of density in the leads tells that the cavity is quite open. As discussed in the article the results compare well with microwave measurements.
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Last updated: 10/21/10