Figure 1 : Detection of spin-momentum locking of spin-helical Dirac electrons in Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$ using spin-resolved ARPES. (a) ARPES intensity map at E$_{F}$ of the (111) surface of tuned Bi$_{2}$$_{−}$$_{δ}$Ca$_{δ}$Se$_{3}$ (see text) and (b) the (111) surface of Bi$_{2}$Te$_{3}$. Red arrows denote the direction of spin around the Fermi surface. (c) ARPES dispersion of tuned Bi$_{2}$$_{−}$$_{δ}$Ca$_{δ}$Se$_{3}$ and (d) Bi$_{2}$Te$_{3}$ along the k$_{x}$ cut. The dotted red lines are guides to the eye. The shaded regions in (c) and (d) are our calculated projections of the bulk bands of pure Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$ onto the (111) surface respectively. (e) Measured y component of spin-polarization along the ¯ Γ- ¯ M direction at E$_{B}$ = -20 meV, which only cuts through the surface states. Inset shows a schematic of the cut direction. (f) Measured x (red triangles) and z (black circles) components of spin- polarization along the ¯ Γ- ¯ M direction at E$_{B}$ = -20 meV. Error bars in (e) and (f) denote the standard deviation of P$_{x,y,z}$ , where typical detector counts reach 5 × 10 $^{5}$; Solid lines are numerical fits [21]. (g) Spin-resolved spectra obtained from the y component spin polarization data. The non-Lorentzian lineshape of the I ↑ y and I ↓ y curves and their non-exact merger at large | k$_{x}$ | is due to the time evolution of the surface band dispersion, which is the dominant source of statistical uncertainty. a.u., arbitrary units. (h) Fitted values of the spin polarization vector P = ( S$_{x}$ , S$_{y}$ , S$_{z}$ ) are (sin(90 $^{◦}$)cos(-95$^{◦}$), sin(90$^{◦}$)sin(-95$^{◦}$), cos(90$^{◦}$)) for electrons with + k$_{x}$ and (sin(86 $^{◦}$)cos(85$^{◦}$), sin(86 $^{◦}$)sin(85$^{◦}$), cos(86$^{◦}$)) for electrons with - k$_{x}$ , which demonstrates the topological helicity of the spin-Dirac cone. The angular uncertainties are of order ± 10 ◦ and the magnitude uncertainty is of order ± 0.15. Figure 2 : Tuning the bulk Fermi level through systematic bulk charge compensa- tion monitored through systematic transport and ARPES measurements. (a) Resistivity at T = 4 K measured for samples of Bi$_{2}$Se$_{3}$ that are bulk electron doped due to varying concen- trations of Se vacancies [16] ( x ) or bulk hole doped through Ca/Bi substitution ( δ ). These are compared to analogous values for topological insulators Bi$_{0}$$_{.}$$_{9}$Sb$_{0}$$_{.}$$_{1}$ (black square, arrowed; [13]) and Bi$_{1}$$_{.}$$_{9933}$Sn$_{0}$$_{.}$$_{0067}$Te$_{3}$ (purple triangle, arrowed). The stoichiometric Bi$_{2}$Se$_{3}$ (Bi$_{1}$$_{.}$$_{9975}$Ca$_{0}$$_{.}$$_{0025}$Se$_{3}$) is found to be the most insulating of these topological insulators. In Bi$_{2}$$_{−}$$_{δ}$Ca$_{δ}$Se$_{3}$, bulk resistivity in excess of 75 mΩcm is possible, which will be shown elsewhere. The bulk insulating state in Bi$_{0}$$_{.}$$_{9}$Sb$_{0}$$_{.}$$_{1}$ [13] is intrinsic and not due to disorder which will also be shown elsewhere. Bi$_{1}$$_{.}$$_{9975}$Ca$_{0}$$_{.}$$_{0025}$Se$_{3}$ is known to be most metallic-like among the three classes studied so far. (b) Hall carrier density 13