We compare three-dimensional electrostatics of semiconductor structures with graphene-like lattices of quantum dots and antidots formed in the plane of the two dimensional electron gas. With lattice constant fixed, the shape of the potential may be tuned so that both lattices have minband spectrum where the second Dirac feature is pronounced and not overlaid by the other states. We show that the lattice of quantum dots is more sensitive to fabrication imperfections, because sources of the disorder are located directly above the electronic channels. Thus the lattices of antidots should be preferred semiconductor artificial graphene candidates.
We consider oscillations of the length and width in rectangular quantum billiards, a two "degree-of-vibration" configuration. We consider several superpositon states and discuss the effects of symmetry (in terms of the relative values of the quantum numbers of the superposed states) on the resulting evolution equations and derive necessary conditions for quantum chaos for both separable and inseparable potentials. We extend this analysis to n-dimensional rectangular parallelepipeds with two degrees-of-vibration. We produce several sets of Poincaré maps corresponding to different projections and potentials in the two-dimensional case. Several of these display chaotic behavior. We distinguish between four types of behavior in the present system corresponding to the separability of the potential and the symmetry of the superposition states. In particular, we contrast harmonic and anharmonic potentials. We note that vibrating rectangular quantum billiards may be used as a model for quantum-well nanostructures of the stated geometry, and we observe chaotic behavior without passing to the semiclassical (ℏ → 0) or high quantum-number limits.
A two-dimensional scattering potential represents the quantum extension of a
diffractive lattice: a Dirac delta function with a modulated permeability
along the y-axis. This model does not have an explicit
classical analogue and quantum effects such as tunneling and diffraction play
an important role. An analytical solution for the one-harmonic case is found.
For the general case of an arbitrary number of harmonics a simple criterion is
derived for the range of parameters where quantum chaos is permitted (but does
not necessarily occur). The statistical properties of the
S-matrix for the given model have been investigated. The
deviations from the usual predictions for irregular scattering in the random
matrix theory (RMT) framework have been found and are discussed.
Transport in two-dimensional modulation-doped semiconductor structures