  A generation, propagation, and transfer of phonon angular momenta are
examined on thermal transport in chiral insulative and diamagnetic crystals of
$\alpha$-quartz. We found that thermally-driven phonons carry
chirality-dependent angular momenta in the quartz crystals and they could be
extracted from the quartz as a spin signal. Namely, chirality-induced
selectivity of phonon angular momenta is realized in the chiral quartz. We
argue that chiral phonons available in chiral materials could be a key element
in triggering or enhancing chirality-induced spin selectivity with robust spin
polarization and long-range spin transport found in various chiral materials.
