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The energy spectrum of a particle confined within a circle in the presence of an external magnetic field perpendicular to the plane of confinement is studied both exactly and approximately by the quasiclassical formalism (Wentzel–Kramers–Brillouin). For pure spatial confinement (without magnetic field) the energy spectrum for states other than the ground state is twofold degenerate, while in the case of pure magnetic confinement the spectrum shows the infinite-fold degeneration, typical of Landau states. For both types of confinement, the latter infinite-fold degeneration is lifted due to spatial confinement. Interestingly enough, for a given ratio between spatial and magnetic confining lengths, the magnetic flux is quantized. The conditions for the quantization of the flux are established; the nature and peculiarity of the energy spectrum are also discussed.

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A formalism to calculate the dipolar local field in homogeneously magnetized quasi-two-dimensional (Q2D) crystals is comprehensively presented. Two fundamental tests for this formalism are accomplished: the transition from the Q2D quantities to the corresponding 3D ones; and the recovering of the macroscopic quantities of the 3D continuum theory. The additive separation between lattice and shape contributions to the local field allows an unambiguous interpretation of the respective effects. Calculated demagnetization tensors for square and circular lateral geometries of dipole layers show that for a single crystal layer an extremely thin film, but still with a finite thickness, is a better physical representation than a strictly 2D plane. Distinct close-packed structures are simulated and calculations of the local field at the nodes of the stacked 2D lattices allow one to establish the number of significantly coupled dipole layers, depending on the ratio between the interlayer distance and the 2D lattice constant. The conclusions drawn are of interest for the study of the dipolar interaction in magnetic ultrathin films and other nanostructured materials, where magnetic nanoparticles are embedded in non-magnetic matrices.

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