  We systematically study the impact of various electron-acoustic-phonon
coupling mechanisms on valley physics in two-dimensional materials. In the
static strain limit, we find that Dirac cone tilt and deformation potential
have analogous valley Hall response since they fall into the same universality
class of pseudospin structure. However, such argument fails for the coupling
mechanism with position-dependent Fermi velocity. For the isotropic case, a
significant valley Hall effect occurs near charge neutrality similar to the
bond-length change, whereas for the anisotropic case, the geometric valley
transport is suppressed, akin to the deformation potential. Gap opening
mechanism by nonuniform strain is found to totally inhibit the valley Hall
transport, even if the dynamics of strains are introduced. By varying gate
voltage, a tunable phonon-assisted valley Hall response can be realized, which
paves a way toward rich phenomena and new functionalities of valley
acoustoelectronics.
