Phonon-mediated spin Hall effect in quantum paraelectric metals

Recent years have seen discovery of ferroelectric phase transition in various metallic materials. Given that incipient ferroelectricity usually implies the crystal near continuous phase transition, its existence in metals with spin-orbit coupling typically gives rise to soft transverse optical phonons with Rashba-type coupling to itinerant electrons. We find through the Kubo formula calculation that such Rashba electron-phonon coupling has a profound impact on electron spin transport. While finite polarity is required for spin Hall current induced by uniform electric field, the Rashba electron-phonon coupling is sufficient for giving rise to spin Hall current proportional to a second derivative of electric field with distinct quadrupolar characteristics. Furthermore, this spin conductivity displays an unusual thermal activation behavior characterized by a scaling laws dependent on the phonon frequency to temperature ratio. These findings shed light on exotic electronic transport phenomena originating from ferroelectric quantum criticality, highlighting the intricate interplay of charge and spin degrees of freedom.