Phonon Hall effect (PHE) has attracted a lot of attention in recent years with many theoretical and experimental explorations published. While experiments work on complicated materials, theoretical studies are still hovering around the phenomenon-based models. Moreover, previous microscopic theory was found unable to explain large thermal Hall conductivity obtained by experiments in strontium titanate (STO). Therefore, as a first attempt to bridge this gap, we implement first-principles calculations to explore the PHE in real materials. Our work provides a new benchmark of the PHE in sodium chloride (NaCl) under a large external magnetic field. Moreover, we demonstrate our results in barium titanate (BTO), and discuss the results in STO in detail about their deviation from experiments. As a possible future direction, we further propose that the inner electronic Berry curvature plays an important role in the PHE in STO.
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