Resonant, non-resonant, and anomalous states of Dirac electrons in a parabolic well in the presence of magnetic fields

摘要

We report on several new basic properties of a parabolic dot in the presenceof a magnetic field. The ratio between the potential strength and the Landaulevel (LL) energy spacing serves as the coupling constant of this problem. Inthe weak coupling limit the energy spectrum in each Hilbert subspace of anangular momentum consists of discrete LLs of graphene. In the intermediatecoupling regime non-resonant states form a closely spaced energy spectrum. Wefind, counter-intuitively, that resonant quasi-boundstates of both positive andnegative energies exist in the spectrum. The presence of resonantquasi-boundstates of negative energies are a unique property of massless Diracfermions. As the strong coupling limit is approached resonant and non-resonantstates transform into anomalous states, whose probability densities develop anarrow peak inside the well and another broad peak under the potential barrier.These properties may investigated experimentally by measuring opticaltransition energies that can be described by a scaling function of the couplingconstant.