  The helicity of 3D topological insulator surface states has drawn significant
attention in spintronics owing to spin-momentum locking where the carriers'
spin is oriented perpendicular to their momentum. This property can provide an
efficient method to convert charge currents into spin currents, and vice-versa,
through the Rashba-Edelstein effect. However, experimental signatures of these
surface states to the spin-charge conversion are extremely difficult to
disentangle from bulk state contributions. Here, we combine spin- and
angle-resolved photo-emission spectroscopy, and time-resolved THz emission
spectroscopy to categorically demonstrate that spin-charge conversion arises
mainly from the surface state in Bi$_{1-x}$Sb$_x$ ultrathin films, down to few
nanometers where confinement effects emerge. We correlate this large conversion
efficiency, typically at the level of the bulk spin Hall effect from heavy
metals, to the complex Fermi surface obtained from theoretical calculations of
the inverse Rashba-Edelstein response. %We demonstrate this for film thickness
down to a few nanometers, Both surface state robustness and sizeable conversion
efficiency in epitaxial Bi$_{1-x}$Sb$_x$ thin films bring new perspectives for
ultra-low power magnetic random-access memories and broadband THz generation.
