On the Geometry of Nanowires and the Role of Torsion

A detailed analysis of the Schrodinger equation in curved coordinates, exact to all orders in the cross sectional dimension is presented, and we discuss the implications of the frame rotation for energies of both open and closed structures. For a circular cross-section, the energy spectrum is independent of the frame orientation for an open structure. For a closed curve, the energies depend on the holonomy angle of a minimal rotating frame (MR) which is equal to the area enclosed by the tangent image on the unit sphere. In the case of a curve with a well-defined torsion at all points this is up to a multiple of 2 pi equal to the total torsion, a result first found in 1992 by Takagi and Tanzawa. In both cases we find that the effect on the eigenstates is a phase shift. We validate our findings by accurate numerical solution of both the exact 3D equations and the approximate 1D equations for a helix structure and find that the error is proportional to the square of the diameter of the cross section. We discuss Dirichlet versus Neumann boundary conditions and show that care has to be taken in the latter case.