Nearly triple nodal point topological phase in half-metallic GdN

Recent developments in topological semimetals open a way to realize relativistic dispersions in condensed-matter systems. One recently studied type of topological feature is the “triple nodal point” where three bands become degenerate. In contrast to Weyl and Dirac nodes, triple nodal points, which are protected by a rotational symmetry, have nodal lines attached, so that a characterization in terms of a chirality is not possible. Previous studies of triple nodal points considered nonmagnetic systems, although an artificial Zeeman splitting was used to probe the topological nature. Here instead we treat a ferromagnetic material, half-metallic GdN, where the splitting of the triple nodal points comes from the spin-orbit coupling. The size of the splitting ranges from 15 to 150 meV depending on the magnetization orientation, enabling a transition between a Weyl-point phase and a “nearly triple nodal point” phase that exhibits very similar surface spectra and transport properties compared with a true triple-node system. The rich topological surface states, manipulable via the orientation of the magnetization, make half-metallic GdN a promising platform for future investigations and applications.

Figure 1

(a)–(e) Band structure of GdN calculated without SOC. (a), (b) The purple (green) colors represent spin majority (minority) bands, while in the blow-ups in panels (c)–(e) the colors are keyed to indicated band index numbers, where band 6 is the highest valence band. Triple nodal points are marked with red arrows. (f) Relation of the bulk Brillouin zone (BZ) to the two-dimensional (2D) projected BZ on the (001) surface. Six red dots denote triple nodal points; yellow rectangle defines the three variables ${\kappa }_1$, ${\kappa }_2$, and ${\kappa }_3$ used as the horizontal axes in panels (c)–(e). (g) Top view of the BZ, showing the overlap between bulk and surface BZs. Blue square is the area used for constant-energy scans in Figs. 2 and 4–6. (h) 2D band structure plotted on the $k_z=0$ plane near the $X_1$ point; the arrow at the base is directed toward the $\mathrm{\Gamma }$ point, and the shifted blue line terminating in two dots illustrates the nodal line connecting the two triple nodal points.

Figure 2

Surface states on the (001) surface calculated for a semi-infinite geometry without SOC. (a), (b) Surface band spectral function. (c), (d) Constant-energy scan at $E=E_W$. Bright (dim) yellow color represents surface (bulk) state. The origin has been shifted to the $\overline{M}$ point. Panels (b) and (d) are zoomed in around the $\overline{M}$ point; red dots indicate triple nodal points.

Figure 3

Electronic structure of GdN calculated with SOC. The magnetic moment is aligned along the [001] direction. (a) Band dispersion along the three Cartesian axes. The purple (green) colors represent spin majority (minority) bands. (b) Band structure on the $k_z$ axis, where band labels denote eigenvalues of the $C_4^{\widehat{z}}$ operator. (c)–(e) Band structure on ${\kappa }_1$, ${\kappa }_2$, and ${\kappa }_3$ lines. Blue and red arrows denote Weyl and nearly triple nodal points, respectively. (f), (g) 2D band structure on $k_x\text{–}{k}_y$ and $k_z\text{–}{k}_x$ planes at $X_1$ and $X_3$, respectively. The arrows at the base are directed toward the $\mathrm{\Gamma }$ point.

Figure 4

Electronic structure on the (001) surface calculated for a semi-infinite geometry with the magnetization orientation of [001] direction. (a) Band structure at the $\overline{M}$ point. (b) Constant-energy scan at $E=E^{\left[001\right]}$. Bright yellow color denotes intense spectral density. (c) Schematic view of the nearly triple-nodal-point surface state emerging from the $X_1$ and $X_2$ points. Red dots are nearly triple nodal points.

Figure 5

Electronic structure on the (001) surface calculated for a semi-infinite geometry with the magnetization along the [110] direction. (a)–(c) Band structure at the $\overline{M}$ point along $k_x$, $k_1$, and $k_2$ directions [see Fig. 1]. Constant-energy-scans at (d) $E=E_1^{\left[110\right]}$ and (e) $E=E_2^{\left[110\right]}$. Magenta (cyan) dots denote the position of Weyl points with positive (negative) chirality. In panels (d) and (e), large dots lie on the energy plane of the plot, while small dots lie off the plane.

Figure 6

Electronic structure of the (001) surface calculated for a semi-infinite geometry with the magnetization along the [100] direction. (a)–(d) Band structure at the $\overline{M}$ point along $k_x$, $k_y$, $k_1$, and $k_2$ directions. Constant-energy scans at (e) $E=E_0^{\left[100\right]}$, (f) $E=E_1^{\left[100\right]}$, (g) $E=E_2^{\left[100\right]}$, and (h) $E=E_3^{\left[100\right]}$. Red dots represent nearly triple nodal points. Magenta (cyan) dots denote Weyl points with positive (negative) chirality. In panels (e)–(h), large dots lie on the energy plane of the plot, while small dots lie off the plane. (i) Schematic view of the chiral surface states. Red and blue arrows indicate direction of magnetization and surface group velocity, respectively.

Figure 7

Anomalous Hall conductivity for magnetization along the [100] direction. (a) Electronic states projected on the $k_x$ axis from the tetragonal Brillouin zone (BZ) having the same volume as the conventional Wigner–Seitz BZ, as shown in panel (b). (c) Two-dimensional anomalous Hall conductivity vs $k_x$ calculated on constant-$k_x$ planes, one of which is shown as a colored square in panel (b), including all states below $E_F$ (blue) or $E_W$ (brown). (d) Bulk anomalous Hall conductivity as a function of chemical potential.

Figure 8

Schematic view of the GdN surface-state transition on the (001) surface for magnetization orientation along (a) [100], (b) [001], and (c) $\left[\overline{1}00\right]$. Orange and blue lines represent two bulk bands on the $k_x$ axis having $C_4^{\widehat{x}}$ eigenvalues of $\exp (i3\pi /4)$ and $\exp (i7\pi /4)$, respectively [see Fig. 3].

Figure 9

Topological phase diagram of GdN as a function of strain and exchange coupling with the magnetic moment along [001]. Plotted lines indicate closures of the direct band gap: without SOC (black), simultaneously at all three $X$ points; and with SOC, at $X_1$ (blue) and $X_3$ (red). Vertical dashed line indicates the equilibrium lattice constant. Acronyms are WSM for Weyl semimetal, NTNP for nearly triple nodal point, FMI for ferromagnetic insulator, TI for topological insulator, AI for axion insulator, and NI for normal insulator.

Figure 10

Landau-level spectra of GdN calculated with the $\mathit{k}·\mathit{p}$ model. (a), (e) Band structure calculated with the $\mathit{k}·\mathit{p}$ model along $k_z$ and $k_x$ directions, respectively. (b), (c) Landau-level spectra along $k_z$ axis for triple nodal point and nearly triple nodal point phases, respectively. (d) Landau-level spectra along the $k_z$ axis for the Weyl semimetal phase. (f), (g) Landau-level spectra along the $k_x$ direction for the triple nodal point and nearly triple nodal point phases, respectively. Dark blue color in panels (b)–(d) and (f), (g) indicates low-index Landau levels [see color bar in panel (d)].