Transport through a monolayer-tube junction: Sheet-to-tube spin current

We develop a method to calculate the electron flow between an arbitrary atomic monolayer sheet and an arbitrary tube by expressing the corresponding sheet-tube tunneling matrix elements with those between sheets. We use this method to calculate the spin current from a monolayer silicene sheet with sublattice-staggered current-induced spin polarization to a silicene tube. The calculated sheet-to-tube spin current exhibits an oscillation as a function of the tube circumferential length because the Fermi points in the tube cross the Fermi circle in the sheet. Furthermore, the spin current with spin in the out-of-plane direction, which is absent in the sheet-sheet junction (including twisted sheets) with $C_3$ rotational symmetry, appears in an oscillating form in the tube-sheet junction due to the broken $C_3$ rotational symmetry. This is an example of the symmetry manipulation which realizes switching a particular component of the spin current.