Spin Polarization of Gapped Dirac Surface States Near the Topological Phase Transition in $\mathrm{TlBi}({\mathbf{S}}_{1\mathrm{\text{−}}x}{\mathrm{Se}}_x{)}_2$

We performed systematic spin- and angle-resolved photoemission spectroscopy of $\mathrm{TlBi}({\mathrm{S}}_{1\mathrm{\text{−}}x}{\mathrm{Se}}_x{)}_2$ which undergoes a topological phase transition at $x\sim 0.5$. In ${\mathrm{TlBiSe}}_2$ ($x=1.0$), we revealed a helical spin texture of Dirac-cone surface states with an intrinsic in-plane spin polarization of $\sim 0.8$. The spin polarization still survives in the gapped surface states at $x>0.5$, although it gradually weakens upon approaching $x=0.5$ and vanishes in the nontopological phase. No evidence for the out-of-plane spin polarization was found, irrespective of $x$ and momentum. The present results unambiguously indicate the topological origin of the gapped Dirac surface states, and also impose a constraint on models to explain the origin of mass acquisition of Dirac fermions.

Figure 1

(a) ARPES intensity at $E_F$ around the $\overline{\Gamma }$ point for ${\mathrm{TlBiSe}}_2$ plotted as a function of two-dimensional wave vector $\mathbf{k}$ measured with the Xe I resonance line ($h\nu =8.437\mathrm{eV}$) at $T=30\mathrm{K}$. The ARPES intensity is integrated within $\pm 20\mathrm{meV}$ with respect to $E_F$. In this experiment, the incident photon beam lies in the $xz$ plane, and hence, the dipole matrix elements for photoelectrons for the forward side and the back side (i.e., $+k_x$ and $-k_x$) are different from each other, naturally leading to an asymmetric ARPES intensity distribution with respect to the $k_y$ axis as seen here. The definition of an in-plane up-spin vector is also indicated by an arrow. (b) Near-$E_F$ ARPES intensity along the $\overline{\Gamma }\overline{K}$ cut for ${\mathrm{TlBiSe}}_2$. (c, d) Spin-resolved EDCs for the $\mathbf{k}$ points ($A\mathrm{\text{−}}E$ and $F\mathrm{\text{−}}J$) whose positions in the 2D Brillouin zone are indicated by open circles in (a).

Figure 2

(a) $x$-dependence of near-$E_F$ ARPES intensity around the $\overline{\Gamma }$ point for $\mathrm{TlBi}({\mathrm{S}}_{1\mathrm{\text{−}}x}{\mathrm{Se}}_x{)}_2$. Dashed lines are guides to the eyes to trace the band dispersion. The vertical red (thick) line represents the $\mathbf{k}$ point where the spin-resolved EDCs were obtained. (b, c) Corresponding spin-resolved EDCs for the specified $\mathbf{k}$ point and their spin polarization $P_{\mathrm{IP}}$, respectively. In (c), the averaged $P_{\mathrm{IP}}$ value in the energy window of $\pm 0.1\mathrm{eV}$ with respect to the energy of the SS ($E_{\mathrm{SS}}$) is indicated by a number and also by a dashed line. The experimental uncertainty (error bar) in the averaged $P_{\mathrm{IP}}$ value is $\pm 0.05$.

Figure 3

(a) Representative results of the two-component fitting to the EDC obtained by the regular ARPES experiment to estimate the bulk and surface spectral weights as indicated by blue (thick) and red (thin) curves, respectively. The plotted EDCs are generated by summing the raw EDCs within the $\mathbf{k}$ window of the spin-resolved measurement, to take into account the $\mathbf{k}$ broadening effect. In the fitting, we assumed a Lorentzian peak for the surface, bulk VB, and bulk CB, multiplied by the Fermi–Dirac distribution function. A constant term was subtracted from the peaks for bulk VB and CB to reproduce the bulk band gap. (b) Ratio of the surface spectral weight to the total spectral weight, $N_{\mathrm{SS}}/N_{\mathrm{total}}$ (where $N_{\mathrm{total}}=N_{\mathrm{SS}}+N_{\mathrm{bulk}}$), at $E_{\mathrm{SS}}$ plotted against $x$ as estimated by the fitting of EDCs [see Fig. 3a]. (c) $P_{\mathrm{IP}}$ divided by $N_{\mathrm{SS}}/N_{\mathrm{total}}$, which represents the intrinsic spin polarization $P_{\mathrm{int}}$ of the SS (circles). $P_{\mathrm{IP}}$ is also plotted with triangles.

Figure 4

(a) Spin-resolved EDCs and corresponding spin polarization for the out-of-plane component in the topological phase ($x=1.0$, 0.8, and 0.6) measured at the $\mathbf{k}$ point along the $\overline{\Gamma }\overline{M}$ direction indicated by the red (thick) line in Fig. 2a. The EDCs along the $\overline{\Gamma }\overline{K}$ direction are also plotted for $x=1.0$. (b) Near-$E_F$ EDCs for $x=0.9$. Spectrum at the $\overline{\Gamma }$ point is highlighted by thick black curve. (c) Spin-resolved EDCs and corresponding spin polarization for the out-of-plane component measured at the $\overline{\Gamma }$ point for $x=0.9$. (d) Schematic picture of the evolution of the topological SS and bulk bands, shown by red (gray) and blue (light gray) colors, respectively, as a function of $x$. The spin polarization vector is indicated by black arrows.