  We have performed spin- and angle-resolved photoemission spectroscopy of the
topological insulator Pb(Bi,Sb)2Te4 (Pb124) and observed significant
out-of-plane spin polarization on the hexagonally warped Dirac-cone surface
state. To put this into context, we carried out quantitative analysis of the
warping strengths for various topological insulators (Pb124, Bi2Te3, Bi2Se3,
and TlBiSe2) and elucidated that the out-of-plane spin polarization Pz is
systematically correlated with the warping strength. However, the magnitude of
Pz is found to be only half of that expected from the kp theory when the
warping is strong, which points to the possible role of many-body effects.
Besides confirming a universal relationship between the spin polarization and
the surface state structure, our data provide an empirical guiding principle
for tuning the spin polarization in topological insulators.
