Metallic surface electronic state in half-Heusler compounds $R$PtBi ($R$$=$ Lu, Dy, Gd)

Rare-earth platinum bismuth ($R$PtBi) has been proposed recently as a potential topological insulator. In this paper, we present measurements of the metallic surface electronic structure in three members of this family, using angle-resolved photoemission spectroscopy (ARPES). Our data show clear spin-orbit splitting of the surface bands and the Kramers’ degeneracy of spins at the $\mathrm{\Gamma ̄}$ and $\mathrm{M̄}$ points, which is reproduced nicely with our full-potential linearized augmented plane wave calculation for a surface electronic state. Topologically nontrivial behavior is signified by band inversion in the calculated bulk electronic structures, yet no direct indication of such behavior is detected by ARPES except for a weak Fermi crossing detected in close proximity to the $\mathrm{\Gamma ̄}$ point, making the total number of Fermi crossings odd. In the surface band calculation, however, this crossing is explained by a Kramers pair of bands that are very close to each other. The classification of this family of materials as topological insulators remains an open question.

Figure 1

Surface Fermi maps of half-Heusler compounds $R$PtBi ($R=\text{Lu}$,Dy,Gd). (a) $C{1}_b$ crystal structure of $R$PtBi. The crystallographic axes are rotated so that the (111) direction points along $z$. The red parallelogram marks the Bi(111) cleaving plane. (b) The surface and bulk Brillouin zone for the rotated crystal structure in (a). Here $k_z$ corresponds to the (111) direction of the fcc Brillouin zone. (c)–(e) Surface Fermi maps of $R$PtBi. All data are taken with 48 eV photons at $T=15$ K. Yellow lines denote the surface Brillouin zone.

Figure 2

Surface electronic structure of GdPtBi: Comparison between ARPES data and calculational result. (a) Fermi map of GdPtBi observed by ARPES, same as in Fig. 1e. (b) Calculational surface Fermi map of GdPtBi at the Bi(111) cleaving plane. See text for details. (c) ARPES band structure along the contour $\mathrm{\Gamma ̄}$-$\mathrm{M̄}$-$\mathrm{K̄}$-$\mathrm{\Gamma ̄}$. Inset of (c) shows enhanced ARPES intensity near $\mathrm{M̄}$ and $\mathrm{K̄}$ for better visibility of the bands. (d) Calculational band structure with respect to (c). Sizes of hollow circles represent the contribution of surface Pt atoms. (e),(f) Expanded figures for (b) and (d), respectively, showing six Fermi crossings. Panel (e) is rotated by 30${}^{ˆ}$ with respect to (b).

Figure 3

Absence of $k_z$ dispersion as proof for the observation of a surface electronic structure. (a),(b) $k_z$ ($\Gamma$-$A$) dispersion maps for LuPtBi. Data are obtained by scanning the incident photon energy $h\nu$ from 30 to 80 eV along (a) the $\mathrm{\Gamma ̄}$-$\mathrm{K̄}$ and (b) the $\mathrm{\Gamma ̄}$-$\mathrm{M̄}$ direction. (c) Calculational Fermi surface map for the bulk state in the $A$-$\Gamma$-$K$ plane. See panel (a) for comparison. (d)–(g) Band dispersion maps along the the $\mathrm{\Gamma ̄}$-$\mathrm{K̄}$ direction for selected $h\nu$’s. It is clear that all observed bands are independent of $h\nu$ ($k_z$). (h) Detailed peak analysis for the momentum distribution curves (MDC’s) at the chemical potential for four different photon energies. The $k$ range is indicated by an orange double arrow in (d). Bars in different colors indicate the Fermi crossings for different bands.

Figure 4

Band-structure analysis at the vicinity of $\mathrm{\Gamma ̄}$ [red arrows in Fig. 1e]. Data are taken on LuPtBi and GdPtBi samples at $T=15$ K. (a),(b) Band dispersion maps along the $\mathrm{\Gamma ̄}$-$\mathrm{M̄}$ direction. Green arrows point to the position of the inner hole band, which has lower intensity than the two other hole bands. (c),(d) Corresponding MDC’s for panels (a) and (b). (e),(f) Extraction of the band position for panels (a) and (b). By linearly extrapolating the bands above the chemical potential $\mu$, we show an approximate band crossing point (Dirac point) at $E~0.4$ eV for GdPtBi.

Figure 5

Band-structure analysis in the vicinity of $\mathrm{M̄}$ [red box in Fig. 1c]. Data are taken on LuPtBi samples. (a)–(d) Binding-energy dependence of band structure near $\mathrm{M̄}$. Map location in the surface Brillouin zone is shown in (e). (f) Theoretical band map at the chemical potential for GdPtBi. (g),(h) Band maps for two perpendicular directions marked by red lines in (g). There are in total two Fermi crossings along the $\mathrm{\Gamma ̄}$-$\mathrm{M̄}$ line segment at the vicinity of $\mathrm{M̄}$.