Tunability of the $k$-space location of the Dirac cones in the topological crystalline insulator Pb${}_{1-x}$Sn${}_x$Te

We have performed systematic angle-resolved photoemission spectroscopy of the topological crystalline insulator (TCI) Pb${}_{1-x}$Sn${}_x$Te to elucidate the evolution of its electronic states across the topological phase transition. As previously reported, the band structure of SnTe ($x=1.0$) measured on the (001) surface possesses a pair of Dirac-cone surface states located symmetrically across the $\overline{X}$ point in the (110) mirror plane. Upon approaching the topological phase transition into the trivial phase at $x_c\approx 0.25$, we discovered that Dirac cones gradually move toward the $\overline{X}$ point with its spectral weight gradually reduced with decreasing $x$. In samples with $x\le 0.2$, the Dirac-cone surface state is completely gone, confirming the occurrence of the topological phase transition. Also, the evolution of the valence band feature is found to be consistent with the bulk band inversion taking place at $x_c$. The tunability of the location of the Dirac cones in the Brillouin zone would be useful for applications requiring Fermi-surface matching with other materials, such as spin injection.

Figure 1

(a) Bulk fcc BZ and corresponding tetragonal (001) surface BZ of Pb${}_{1-x}$Sn${}_x$Te. The mirror plane is indicated by the green shaded area. The $k_{x^{\prime }}$ and $k_{y^{\prime }}$ axes are rotated by 45${}^{\circ }$ around the $k_z$ axis with respect to the standard fcc reciprocal lattice vector. (b) Mapping of ARPES intensity at $E_{\mathrm{F}}$ around the $\overline{X}$ point for $x=1.0$, 0.5, 0.3, and 0.0 plotted as a function of in-plane wave vector ($k_{x^{\prime }}$ and $k_{y^{\prime }}$) measured with the He I$\alpha$ line ($h\nu =21.218$ eV) at $T=30$ K. The intensity is obtained by integrating the spectra within $\pm$10 meV of $E_{\mathrm{F}}$. (c) Corresponding near-$E_{\mathrm{F}}$ ARPES intensity along the $\overline{\Gamma }\overline{X}$ cut plotted as a function of $k_{x^{\prime }}$ and binding energy ($E_{\mathrm{B}})$. (d) Band dispersions derived from the second derivatives of the MDCs along the $\overline{\Gamma }\overline{X}$ cut. The $\mathbf{k}$ location of the Dirac point (the $\overline{\Lambda }$ point) is indicated by white arrows in (d). (e) MDCs at $E_{\mathrm{F}}$ (open circles) for $x=1.0$, 0.5, and 0.3, together with numerical fittings with two Lorentzians (black solid curves). Triangles denote the $k$ positions of the maxima in MDCs determined from the fittings. (f) $x$ dependence of the Dirac momentum ($k_{\mathrm{D}}$) relative to the $\overline{X}$ point estimated from the MDCs at $E_{\mathrm{F}}$ (red circles) and the second derivative plots (blue circles). (g) Comparison of the band dispersions extracted from the peak position of the second derivatives of the MDCs for $x=1.0$, 0.5, and 0.3.

Figure 2

(a) and (b) Photon-energy dependence of near-$E_{\mathrm{F}}$ ARPES intensity map around the $\overline{X}$ point along a cut across the $\mathbf{k}$ point where the bulk VB is located closest to $E_{\mathrm{F}}$ at each $h\nu$, shown for $x=0.2$ and 0.8, respectively. (c) and (d) Comparison of the energy distribution curves (EDCs) at $k_{y^{\prime }}=0$ extracted from the intensity plots for $x=0.2$ and 0.8, respectively.

Figure 3

(a) and (b) $x$ dependence of near-$E_{\mathrm{F}}$ ARPES intensity and corresponding EDCs, respectively, measured along the $k_{y^{\prime }}$ cut across the bulk VB top that can be accessed with $h\nu =78.5$ eV. (c) Comparison of the EDCs at $k_{y^{\prime }}=0$ [thick curves in (b)]. Black and red arrows denote the features stemming from the bulk VB and the surface state, respectively. (d) Plots of the $\Delta E_{\mathrm{VB}}$ value (black circles) and the intensity at $E_{\mathrm{F}}$ (red circles) as a function of $x$. The intensity at $E_{\mathrm{F}}$ is normalized to that at the VB top.

Figure 4

Schematic picture of the energy band evolution around the $\overline{X}$ point in Pb${}_{1-x}$Sn${}_x$Te summarizing the present ARPES experiments. CB, VB, and SS denote the bulk conduction band, bulk valence band, and surface state, respectively.