Elasto-Dynamical Induced Spin and Charge Pumping in Bulk Heavy Metals

Analogous to the Spin-Hall Effect (SHE), {\it ab initio} electronic structure calculations reveal that acoustic phonons can induce charge (spin) current flowing along (normal to) its propagation direction. Using Floquet approach we have calculated the elastodynamical-induced charge and spin pumping in bulk Pt and demonstrate that: (i) While the longitudinal charge pumping is an intrinsic observable, the transverse pumped spin-current has an extrinsic origin that depends strongly on the electronic relaxation time; (ii) The longitudinal charge current is of nonrelativstic origin, while the transverse spin current is a relativistic effect that to lowest order scales linearly with the spin-orbit coupling strength; (iii) both charge and spin pumped currents have parabolic dependence on the amplitude of the elastic wave.