The Geometrical Effects on Electronic Spectrum and Persistent Currents   in Mesoscopic Polygon
  In this paper, a new mesoscopic polygon which possesses smooth transition at its corners is proposed. Because of the particularity of structure, this kind of mesoscopic polygon can also be a geometrical supperlattice. The geometrical effects on the electron states and persistent current are investigated comprehensively in the presence of magnetic flux. We find that the particular geometric structure of the polygon induces an effective periodic potential which results in gaps in the energy spectrum. The changes of gaps show the consistency with the geometrical twoness of this new polygon. This electronic structure and the corresponding physical properties are found to be periodic with period $\phi_{0}$ in the magnetic flux $\phi $ and can be controlled by the geometric method. We also consider the Rahsba spin-orbit interaction which make the energy levels splitting newly to double and leads to an additional small zigzag in one period of the persistent current. These new phenomena may be useful for the applications in quantum device design in the future. 
Shengli Zhang, Qi Wang, Erhu Zhang