  In crystals with time-reversal symmetry but without inversion symmetry, the
phonon angular momentum can be generated by the temperature gradient, and it is
called phonon thermal Edelstein effect. On the other hand, when both symmetries
are broken and their product is conserved, the phonon angular momentum for any
phonon modes at any wave vectors vanishes, and the phonon thermal Edelstein
effect does not occur. In this paper, we propose another mechanism of
generation of the phonon angular momentum. We show that in such crystals the
electric field generates the phonon angular momentum, via the lattice
distortion due to the electric field. This effect is in the same symmetry class
with the magnetoelectric effect, and we call this effect phonon rotoelectric
effect. We discuss the temperature dependence of the phonon angular momentum
generated by the temperature gradient and by the electric field in the high-
and the low-temperature limits.
