  We develop a first-principles quantum scheme to calculate the phonon magnetic
moment in solids. As a showcase example, we apply our method to study gated
bilayer graphene, a material with strong covalent bonds. According to the
classical theory based on the Born effective charge, the phonon magnetic moment
in this system should vanish, yet our quantum mechanical calculations find
significant phonon magnetic moments. Furthermore, the magnetic moment is highly
tunable by changing the gate voltage. Our results firmly establish the
necessity of the quantum mechanical treatment, and identify small-gap covalent
materials as a promising platform for studying tunable phonon magnetic moment.
