Boundary condition for phonon distribution functions at a smooth crystal interface and interfacial angular momentum transfer

We theoretically elucidate the boundary conditions for phonon distribution functions of long-wavelength acoustic phonons at smooth crystal interfaces. We first derive boundary conditions that fully incorporate reflection, transmission, and mode conversion. We obtain these conditions for phonons from those for classical lattice vibrations, using the correspondence between the quantum and classical descriptions. This formulation provides a theoretical foundation for the acoustic mismatch model, widely used to analyze Kapitza resistance. We then refine the boundary conditions to include spatial dependence parallel to the interface. The refined form captures transverse shifts of elastic wave packets, analogous to the optical Imbert--Fedorov shift, and ensures conservation of total angular momentum. Consequently, circularly polarized phonons carrying spin angular momentum (SAM) generate phonon orbital angular momentum (OAM) at the interface. We analytically determine the spatial profile of this OAM and demonstrate that SAM and OAM are both involved in the interfacial diffusion of chiral phonons. Our theory provides concise boundary conditions for phonons, with applications ranging from heat transport to phonon angular momentum transport.