  $A$B$_{2}$ ($A=$K, Rb, Cs) compounds crystallize in the cubic Laves phase
(symmetry Fd$\bar{3}$m). The geometry of the crystal structure allows the
realization of chiral phonons, which are associated with the circulation of
atoms around their equilibrium positions. Due to the inversion symmetry and
time reversal symmetry, total pseudo-angular momentum (PAM) of the system
vanishes. We show that the doping of these system can lead to a new phase with
symmetry F$\bar{4}3$m. New systems (KRbBi$_{4}$ and RbCsBi$_{4}$) do not
exhibit soft modes (are stable dynamically). Due to the inversion symmetry
breaking, realized chiral phonon modes posses a non-zero total PAM. In both
type of systems the chiral phonons are realized for the wavectors at the edge
of the Brillouine zone. This study explores the possibility of chiral phonon
engineering via doping, and predicts two new materials. Discussing the problem
opens a new way to study the phonon Hall effect.
