  Relativistic electronic properties of a nanospring under a static magnetic
field are theoretically investigated in the present study. The wave equation
accounting for the spin-orbit interaction is derived for the nanospring as a
special case of the Pauli equation for a spin-1/2 particle confined to a curved
surface under an electromagnetic field. We define the helical momentum operator
and show that it commutes with the Hamiltonian owing to the helical geometry of
the nanospring. The energy eigenstates are hence also the eigenstates of the
helical momentum. We solve the equation numerically to obtain the surface wave
functions and the energy spectra. The electronic properties are systematically
examined by varying the parameters that characterize the system. It is
demonstrated that either the nonzero spin-orbit interaction or the nonzero
external magnetic field suffices for the occurrence of the persistent spin
current on the nanospring. Two different mechanisms are shown to generate the
persistent spin current. One employs the spin-orbit interaction coming from the
local inversion asymmetry on the surface, while the other employs the curvature
coupling with the external magnetic field.
