Quantum eigenstates of curved nanowire structures

Eigenstates and associated eigenvalues of a quantum-mechanical particle confined to a three-dimensional arbitrarily curved nanowire structure are determined. Special emphasis is given to the influence of nanowire geometry and curvature effects which are expected to play important roles for the physical properties of several nanowire structures recently grown in laboratories. Use of differential-geometry arguments allows separation of the three-dimensional Schrödinger equation into either (i) two partial differential equations plus one ordinary differential equation in the general nanowire cross-section case or [for simple cross-sectional nanowire shapes] (ii) three ordinary differential equations (ODEs) in appropriate curved coordinates for the case where the cross-sectional area is constant along the nanowire axis. Problems corresponding to item (ii) with three ODEs can be solved either completely analytically or by use of a simple one-dimensional finite-difference scheme. Three case studies are finally analyzed in details: the rectangular cross-sectional-shaped nanowire with a (a) straight-line axis, (b) a circular-shaped axis, and (c) the sinusoidal-shaped nanowire axis including discussion of symmetry properties.