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The effect of the surface curvature on the magnetic moment and persistent currents in twodimensional (2D) quantum rings and dots is investigated. It is shown that the surface curvature decreases the spacing between neighboring maxima of de Haas -van Alphen (dHvA) type oscillations of the magnetic moment of a ring and decreases the amplitude and period of AharonovBohm (AB) type oscillations. In the case of a quantum dot, the surface curvature reduces the level degeneracy at zero magnetic fields. This leads to a suppression of the magnetic moment at low magnetic fields. The relation between the persistent current and the magnetic moment is studied. We show that the surface curvature decreases the amplitude and the period of persistent current oscillations.

We propose a microscopic theory of the optical second-harmonic generation (SHG) from π electrons in two-dimensional (2D) honeycomb lattice structures with broken space inversion symmetry, such as graphene epitaxially grown on a SiC substrate and boronitrene (a single sheet of hexagonal boron nitride (h-BN)). The approach developed is based on a simple two-band π-electron tight-binding model combined with the original Genkin-Mednis formalism of the second-order nonlinear optical response theory, detailed in our recent paper (2010 Phys. Rev. B 82 235426). Within the framework of the approach, we derive an explicit expression for the SHG susceptibility χ2(SHG(ω), which involves two distinct contributions originating from a mixture of interband and intraband motion of π electrons. Both the contributions, and, hence, the χ2(SHG(ω) on the whole, are found to tend to zero when the π-electron energy bands involved are treated at the simplest level of approximation, neglecting the effect of their trigonal warping around the corners of the Brillouin zone of the 2D hexagonal lattice. Through numerical calculations, it is shown that this effect, though rather small, leads to a fairly large magnitude of the SHG susceptibility, reaching the order of 10(-4) esu for the graphene/SiC overlayer system and 10(-7) esu for monolayer h-BN, when the pump photon energy ħω approaches half the bandgap energy Eg of those structures. These theoretical findings suggest that SHG can be used as a sensitive optical probe of the electronic structure of the examined 2D hexagonal crystals and simultaneously demonstrate that those crystals may be an appropriate material for practical uses in future optoelectronic nano-devices.

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The Hall conductivity of an electron gas on the surface of constant negative curvature (the Lobachevsky plane) in the presence of an orthogonal magnetic field is investigated. It is shown that the effect of the surface curvature is to change the break locations and the plateau widths in the Hall conductivity. An increase of temperature results in smearing of the steps.

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