ar X iv : c on d - m at / 9 80 31 66 v 1 1 3 M ar 1 99 8 Electron trapping by a current vortex

We investigate an electron in the plane interacting with the magnetic field due to an electric current forming a localized rotationally symmetric vortex. We show that independently of the vortex profile an electron with spin antiparallel to the magnetic field can be trapped if the vortex current is strong enough. In addition, the electron scattering on the vortex exhibits resonances for any spin orientation. On the other hand, in distinction to models with a localized flux tube the present situation exhibits no bound states for weak vortices. Interaction of charged particles with a localized magnetic field has been a subject of interest for a long time, both from the theoretical and experimental point of view — see, e.g., [GBG, ˇ St] and references therein. Such a field can have different sources, for instance, it may be induced by an electric current having one or more vortices. A lot of attention was payed in the last decade to vortex bound states in superconductors whose dynamics is governed by the Bogoliubov–de Gennes equation — cf. [HRD, SHDS, GS] and a bibliography given in [HIM]. Another, much simpler example involves a Pauli electron interacting with a flux tube modelling a vortex magnetic field — it is is appealing, in particular, since it has been observed that vortices appear often in the probability current associated with mesoscopic transport — see [E ˇ SSF] and the literature there. The system of a tube and an electron has been investigated in a fresh paper by Cavalcanti et al. [CFC] who, however, seem to be unaware of another recent studies of the problem — cf. [Mo] and references therein. In these papers, the field is assumed to be constant within a circle and zero otherwise — the conclusion is then that in one spin state the electron can be always trapped by the vortex, independently of the magnetic flux value, as long as the effective gyromagnetic factor g * > 2. This covers the physically important case of a free electron with g * = 2.0023. However, this claim depends substantially on the used magnetic field Ansatz. To illustrate this point we analyze in this letter the situation where the vortex current distribution represents the input. On one hand we are able to generalize the result 1