Spin Hall Conductivity in Bi_1–<i>x</i>Sb_<i>x</i> as an Experimental Test of Bulk-Boundary Correspondence

Bulk-boundary correspondence, a foundational principle underlying the electronic band structure and physical behavior of topological quantum materials, has been rigorously tested in topological systems that involve conserved charge currents. However, it remains unclear whether bulk-boundary correspondence should hold for nonconserved spin currents. We address this unresolved question by using spin-torque ferromagnetic resonance to accurately probe the charge-to-spin conversion efficiency in epitaxial thin films of a canonical topological insulator, Bi_1-<i>x</i>Sb_<i>x</i>. We find that the measured effective spin Hall conductivity (SHC) agrees well with tight-binding calculations for the intrinsic SHC of the bulk bands. These results indicate that the strong spin-orbit entanglement of bulk states well below the Fermi energy connects directly to the SHC of surface states in epitaxial Bi_1-<i>x</i>Sb_<i>x</i> films interfaced with a metallic ferromagnet. The excellent agreement between theory and experiment affirms the generic value of analyses focused entirely on bulk properties, even for nonconserved topological spin currents.