resents a normal to a scattering plane, defined by the electron incidence direction on a gold foil and two detec- tors mounted on either side that measure the left-right asymmetry A$_{y}$ ′ ,z ′ = [( I y $^{′}$,z ′ L − I y $^{′}$,z ′ R ) / ( I y $^{′}$,z ′ L + I y $^{′}$,z ′ R )] of electrons backscattered off the gold foil [15]. Figure 3(d) shows the spin polarization for both components given by P = (1 /S$_{eff}$ ) × A y $^{′}$,z $^{′}$, where S$_{eff}$ = 0 . 085 is the Sherman function. Following the procedure described in [24], we take the spins to be fully polarized, assign a spin resolved spectra for each of the fitted peaks I i shown in Fig. 3(c), and fit the calculated polarization spectrum to measurement. The spin resolved spectra for the y com- ponent derived from the polarization fit is shown in Fig. 3(e), given by I ↑ , ↓ y = ∑ 4 i $_{=1}$I $^{i}$(1 ± P i $_{y}$) / 6 + B/ 6, where B is a background and P i y is the fitted y component of polarization. There is a clear difference in I ↑ y and I ↓ y at each of the four MDC peaks indicating that the surface state bands are spin polarized. Each of the pairs l 2 /l 1 and r 1 /r 2 have opposite spin, consistent with the behav- ior of a spin split Kramers pair, and the spin polarization of these bands are reversed on either side of ¯ Γ in accor- dance with time reversal symmetry [Fig. 3(f)]. Similar to Au(111) [21] and W(110)-(1 × 1)H [18], the spin polar- ization of each band is largely in-plane consistent with a predominantly out-of-plane electric field at the surface. However unlike the case in Au(111), where the surface band dispersion is free electron like and the magnitude of the Rashba coupling can be quantified by the momentum displacement between the spin up and spin down band minima [21], the surface band dispersion of Sb(111) is highly non-parabolic. A comparison of the k -separation between spin split band minima near ¯ Γ of Sb(111) [Fig. 3(b)] with those of Bi(111) [17], which are 0.03 ˚ A − 1 and 0.08 ˚ A − 1 respectively, nevertheless are consistent with the difference in their atomic p level splitting of Sb(0.6 eV) and Bi(1.5 eV) [25]. Therefore, despite Sb having a similar atomic spin-orbit coupling strength to Au(0.5 eV), the k splitting between spin polarized surface bands near E$_{F}$ is greater in Sb due to its unique dispersion. This could also be due to the nature of spin-polarized photoemission and detection methods. Figure 4(a) shows the full ARPES intensity map from ¯ Γ to ¯ M together with the calculated bulk bands of Sb projected onto the (111) surface. Although the six-fold rotational symmetry of the surface band dispersion is not known a priori due to the three-fold symmetry of the bulk, we measured an identical surface band disper- sion along ¯ Γ- ¯ M’. The spin-split Kramers pair near ¯ Γ lie completely within the gap of the projected bulk bands near E$_{F}$ attesting to their purely surface character. In contrast, the weak diffuse hole like band centered near k$_{x}$ = 0.3 ˚ A − 1 and electron like band centered near k$_{x}$ = 0.8 ˚ A − 1 lie completely within the projected bulk va- lence and conduction bands respectively, and thus their ARPES spectra exhibit the expected lifetime broadening due to coupling with the underlying bulk continuum [26]. Figure 4(b) shows the ARPES intensity plot at E$_{F}$ of Sb(111) taken at a photon energy of 20 eV, where the In general, the spin degeneracy of surface bands on spin-orbit coupled insulators can be lifted due to the breaking of space inversion symmetry. However Kramers theorem requires that they remain degenerate at four spe- cial time reversal invariant momenta (TRIM) on the 2D surface BZ, which for Sb(111) are located at ¯ Γ and three ¯ M points rotated by 60 ◦ from one another. According to recent theory, there are a total of four Z$_{2}$ topolog- ical numbers ν$_{0}$ ;( ν$_{1}$ν$_{2}$ν$_{3}$ ) that characterize a 3D spin- orbit coupled insulator’s bulk band structure [3, 9, 11]. One in particular ( ν$_{0}$ ) determines whether the spin po- larized surface bands cross E$_{F}$ an even or odd number of times between any pair of surface TRIM, and conse- quently whether the insulator is trivial ( ν$_{0}$ =0) or topo- logical ( ν$_{0}$ =1). An experimental signature of topologi- cally non-trivial surface states in insulating Bi$_{1}$$_{−}$$_{x}$Sb$_{x}$ is that the spin polarized surface bands traverse E$_{F}$ an odd number of times between ¯ Γ and ¯ M [2, 12, 19]. Although this method of counting cannot be applied to Sb because it is a semimetal, since there is a direct gap at every bulk bulk band near H is completely below E$_{F}$ [Fig. 2(b)]. Therefore this intensity map depicts the topology of the Fermi surface due solely to the surface states. By com- paring Figs 4(a) and (b), we see that the inner most spin polarized V-shaped band produces the circular electron Fermi surface enclosing ¯ Γ while the outer spin polarized V-shaped band produces the inner segment (0.1 ˚ A − 1 ≤ k$_{x}$ ≤ 0.15 ˚ A − $^{1}$) of the six hole Fermi surfaces away from ¯ Γ. Previous ARPES experiments along the ¯ Γ- ¯ K direction [13] show that this outer V-shaped band merges with the bulk valence band, however the exact value of k$_{x}$ where this occurs along the ¯ Γ- ¯ M direction is unclear since only occupied states are imaged by ARPES. The outer seg- ment of the six hole pockets is formed by the hole like surface resonance state for 0.15 ˚ A − 1 ≤ k$_{x}$ ≤ 0.4 ˚ A − $^{1}$. In addition, there are electron Fermi surfaces enclosing ¯ M and ¯ M’ produced by surface resonance states at the BZ boundaries. Altogether, these results show that in a sin- gle surface BZ, the bulk valence and conduction bands are connected by a lone Kramers pair of surface states [Fig. 4(c)]. FIG. 5: Spin split surface states survive alloying disorder in Sb 0 . 9 Bi 0 . 1 . (a) ARPES intensity map at E$_{F}$ of single crystal Sb 0 . 9 Bi 0 . 1 (111) in the k$_{x}$ - k$_{y}$ plane taken using 20 eV photons. (b) ARPES intensity map of Sb 0 . 9 Bi 0 . 1 (111) along the ¯ Γ- ¯ M direction taken with hν = 22 eV photons. SbGLYPH<.notdef>GLYPH<.notdef>GLYPH<.notdef>BiGLYPH<.notdef>GLYPH<.notdef>GLYPH<.notdef>GLYPH<.notdef>(111) 0.9 0.1 0.0 -0.2 kGLYPH<.notdef>GLYPH<.notdef>(ÅGLYPH<.notdef>GLYPH<.notdef>) y -1 0.0 -0.2 -0.4 EGLYPH<.notdef>GLYPH<.notdef>(eV) B (b) kGLYPH<.notdef>GLYPH<.notdef>(ÅGLYPH<.notdef>GLYPH<.notdef>) x -1 -0.6 kGLYPH<.notdef>GLYPH<.notdef>(ÅGLYPH<.notdef>GLYPH<.notdef>) x -1 -0.5 -0.3 -0.1 0.2 (a) 0.0 -0.4 0.4 0.0 0.4 0.8 GLYPH<c71> M GLYPH<c174> 4