  One-dimensional (1D) van der Waals (vdW) heterostructures, formed between
coaxial nanotubes of transition metal dichalcogenides (TMDCs), have emerged as
a new area of endeavor in nanoscience. A key to designing and engineering the
properties of such 1D vdW heterostructures lies on understanding the band
alignment of coaxial nanotubes in the heterostructures. However, how curvature,
tube diameters, and intertube coupling affect the band-edge levels and band
alignment of TMDC nanotubes in 1D vdW heterostructures remains unknown. Here,
through comprehensive first-principles calculations and analyses, we establish
a complete framework of band alignment in 1D vdW heterostructures of TMDC
nanotubes. We reveal that, as the diameter of a TMDC nanotube decreases, the
combined effects of curvature-induced flexoelectricity and intrinsic
circumferential tensile strain cause a rapid and continuous lowering of the
conduction band minimum (CBM), whereas the valence band maximum (VBM) exhibits
an initial lowering before rising, which originates from a change in the
orbital character of the VBM. The transition in the orbital character of VBM
also leads to direct-to-indirect bandgap transition in small-diameter armchair
and chiral nanotubes, as well as photoluminescence quenching in zigzag
nanotubes. As individual TMDC nanotubes form coaxial 1D vdW heterostructures,
the effect of intertube coupling via flexovoltage effect can result in a
transition of intertube band alignment from Type II to Type I in multiple
heterostructural systems, including large-diameter MoSe$_2$@WS$_2$,
MoTe$_2$@MoSe$_2$, and MoTe$_2$@WS$_2$ heterostructures. These results lay down
a foundation for the rational design of 1D vdW heterostructures.
