  In monolayer hexagonal lattices, two inequivalent valleys appear in the
Brillouin zone. With inversion symmetry breaking, we find chiral phonons with
valley contrasting circular polarization and ionic magnetic moment. At valley
centers, there is a three-fold rotational symmetry endowing phonons with a
quantized pseudo angular momentum, which includes spin and orbital parts. From
conservation of the pseudo angular momentum, crystal momentum and energy,
selection rules in intervalley scattering of electrons by phonons are obtained.
The chiral valley phonons are verified and the selection rules are predicted in
monolayer Molybdenum disulfide. Due to valley contrasting phonon Berry
curvature, one can also detect a valley phonon Hall effect. The
valley-contrasting chiral phonon, together with phonon circular polarization,
ionic magnetic moment, phonon pseudo angular momentum, valley phonon Hall
effect, will form the basis for valley-based electronics and phononics
applications in the future.
