Prediction of Triple Point Fermions in Simple Half-Heusler Topological Insulators

We predict the existence of triple point fermions in the band structure of several half-Heusler topological insulators by ab initio calculations and the Kane model. We find that many half-Heusler compounds exhibit multiple triple points along four independent $C_3$ axes, through which the doubly degenerate conduction bands and the nondegenerate valence band cross each other linearly nearby the Fermi energy. When projected from the bulk to the (111) surface, most of these triple points are located far away from the surface $\overline{\mathrm{\Gamma }}$ point, as distinct from previously reported triple point fermion candidates. These isolated triple points give rise to Fermi arcs on the surface, that can be readily detected by photoemission spectroscopy or scanning tunneling spectroscopy.

Figure 1

Evolution of band structures with increasing numbers of triple points (TPs). (a) HgTe-type band structure along the line $L-\mathrm{\Gamma }-L$. The ${\mathrm{\Gamma }}_8$ bands (solids curves) lie above ${\mathrm{\Gamma }}_6$ (dotted curves). The ${\mathrm{\Gamma }}_8$ bands split into doubly degenerate ${\mathrm{\Lambda }}_6$ (thick solid blue curve) and nondegenerate ${\mathrm{\Lambda }}_4$ and ${\mathrm{\Lambda }}_5$ bands (thin solid black curves) along the $C_3$ axis ($\mathrm{\Gamma }-L$). Here ${\mathrm{\Lambda }}_6$ crosses ${\mathrm{\Lambda }}_{4,5}$, forming TPs (filled red circles) very close to the $\mathrm{\Gamma }$ point, where No. 1 and No. $1^{\prime }$ represent a TP and its time-reversal partner, respectively. (b) Heusler-type band structure. The ${\mathrm{\Lambda }}_6$ bands exhibit a double valley shape, pushing a pair of TPs out from the $\mathrm{\Gamma }$ point. (c),(d) Heusler-type band structure with two and three pairs of TPs along one $C_{3v}$ axis, respectively. (e) Corresponding to the band structures in (a) and (b), three nodal lines (blue curves) inside the $C_{3v}$ mirror planes connect a pair of TPs (1 and $1^{\prime }$) along the $C_3$ axis by passing the $\mathrm{\Gamma }$ point. (f) Corresponding to the band structure in (d), the nodal lines connects these three pairs of TPs. (g) Phase diagram of TPs with respect to the band inversion strength $\mathrm{\Delta }/{\epsilon }_0$ and the linear splitting term $C/C_0$ of ${\mathrm{\Lambda }}_{4,5}$. Within the phase diagram, (a)–(d) correspond to the band structures of (a)–(d), respectively. (h) Distribution of TPs in the bulk Brillouin zone and their projection onto the (111) surface. The surface Fermi arcs that connect different TPs are illustrated by red dashed lines.

Figure 2

Bulk band structures with triply degenerate band crossings. The long-TP materials (a) GdPtBi with 2 TPs along the $C_3$ axis and (b) YPtBi with 6 TPs. The band dispersion along the $C_3$ axis near the $\mathrm{\Gamma }$ point are magnified to show the TPs on the right panels. (c) The short-TP material LaPtBi are shown for comparison.

Figure 3

Surface band structures. (a),(b) Band structure calculated by the tight-binding model on a half-infinite surface with 1 and 3 TPs along $\overline{\mathrm{\Gamma }}-\overline{M}$, respectively. Red color stands for low surface intensity and white for strong surface intensity. The triple points are marked by blue points. (c) DFT band structure calculated on a slab model for GdPtBi (paramagnetic phase). (i) The size of the white circles represents the surface contribution and, thus, large circles show the surface states. Bulk bands are indicated by blue curves as a background, where the triple point is shown as the red point. $R1$ and $R2$ are a pair of Rashba-like surface states. (ii) and (iii), Fermi surfaces corresponding to energy $E1$ (crossing the triple point) and $E2$, respectively. The flowerlike Fermi surface in (iii) was measured in previous ARPES experiments. (iv) Magnified inner Fermi ring of (ii). Fermi arcs (red dotted lines) are artificially added as guides to the eye, for they are missing in the DFT band structure due to the finite slab thickness. (v) Expanded view near the triple point. Red dotted lines highlight $R1$ and $R2$ bands. The $R1$ band crosses $R2$ and later ends at the triple point.