Multiple Dirac nodes and symmetry protected Dirac nodal line in orthorhombic $\alpha$-RhSi

Exotic multifold topological excitations have been predicted and were recently observed in transition metal silicides like $\beta$-RhSi. Herein, we report that interesting topological features of RhSi are also observed in its orthorhombic $\alpha$ phase, which displays multiple types of Dirac nodes very close to the Fermi level ${\varepsilon }_F$. We discuss the symmetry analysis, band connectivity along high-symmetry lines using group representations, band structure, the nature of the Dirac points and of a nodal line occurring near ${\varepsilon }_F$ which is protected by the crystalline symmetry. The de Haas–van Alphen effect indicates a Fermi surface in agreement with the calculations. We find an elliptically shaped nodal line very close to ${\varepsilon }_F$ around and near the $S$ point on the $k_y\text{−}{k}_z$ plane that results from the intersection of two upside-down Dirac cones. Both Dirac points of the participating Kramers degenerate bands are only 5 meV apart; hence, an accessible magnetic field might induce a crossing between the spin-up partner of the upper Dirac cone and the spin-down partner of the lower Dirac cone, possibly explaining the anomalies observed in the magnetic torque. $\alpha$-RhSi is a unique system since all bands crossing ${\varepsilon }_F$ emerge from Dirac nodes.

Figure 1

(a) Crystal structure of $\alpha$-RhSi. (b) Magnetic torque $\tau$ as a function of the magnetic field ${\mu }_{0}H$ for several angles $\theta$ at $T=350$ mK. $\theta$ is the angle between ${\mu }_{0}H$ and the (101) direction of the crystal. Red, orange, and purple lines indicate different polynomial fittings.

Figure 2

(a) Electrical resistivity ${\rho }_{xx}$ for an $\alpha$-RhSi single crystal as a function of the temperature $T$. $\alpha$-RhSi displays a low residual resistivity ${\rho }_0=0.78\mu \mathrm{\Omega }$ cm and a residual resistivity ratio (RRR) of 160. Inset: same plot but in a log-log scale. (b) and (c) Longitudinal resistivity ${\rho }_{xx}$ and Hall resistance ${\rho }_{xy}$ as a function of the magnetic field and for several temperatures. Red lines are fits of ${\rho }_{xx}$ and ${\rho }_{xy}$ to the two-band model [see Eq. (1)]. (d) Charge carrier densities $n_e$ and $n_h$ for electron and holes, respectively, as a function of $T$. (e) Charge carrier mobilities ${\mu }_e$ and ${\mu }_h$ for electron and holes, respectively, as a function of $T$. Charge carrier densities and mobilities were extracted from fits to the two-band model. Inset: picture of typical crystals of $\alpha$-RhSi grown through the use of Te as the flux. (f) Temperature dependence of the magnetic susceptibility measured under fixed magnetic fields after zero-field cooling (ZF) and field cooling (FC). Inset: magnetization as a function of the field at $T=1.8$ K.

Figure 3

Electrical conductivity ${\sigma }_{xx}$ and Hall conductivity ${\sigma }_{xy}$ for an $\alpha$-RhSi single crystal as a function of the magnetic field ${\mu }_{0}H$. These parameters were obtained by inverting the components ${\rho }_{ij}$ of the resistivity tensor as a function of ${\mu }_{0}H$. ${\sigma }_{xy}$ displays the typical “dispersive-resonance” profile response discussed in detail in Ref. [23], displaying a sharp peak at ${\mu }_0{H}_{\text{max}}$ that reflects the elliptical cyclotron orbits executed by the charge carriers at low fields. From the position of the peak, i.e., 1.5 T, a mean transport mobility ${\overline{\mu }}_{tr}\simeq {\left({\mu }_0{H}_{\text{max}}\right)}^{-1}\simeq 6667{\mathrm{cm}}^2$/Vs is extracted.

Figure 4

(a) High-symmetry points on the BZ of orthorhombic $\alpha$-RhSi. (b) Electronic band structure along various high-symmetry directions where the fourfold high-symmetry BZ boundary lines are indicated. (c)–(f) Band structure along the various high-symmetry directions starting from the $S$ point, i.e., along the $Y\mathrm{\Gamma }$, $SR$ and $S\mathrm{\Gamma }$, $SX$, and $ZS$ and $SY$ directions, respectively. Our group theory analysis regarding the connectivity of the $S$ point indicates that there are band crossings along the $SR$ and $SX$ directions.

Figure 5

Band connectivity along the high-symmetry lines (a) SR and (b) SX.

Figure 6

(a) The band structure at the $S$ point as a function of $(k_y,k_z)$. (b) Tilted band structure exposing the nodal line on the $k_y\text{−}{k}_z$ plane around the HSP $S$. Circles and arrows indicate the Dirac nodes and the Dirac nodal line, respectively. (c) Constant-energy contour for the lower band forming the nodal line. (d) Energy difference between the bands forming a nodal line.

Figure 7

(a) Fast Fourier transform spectra of the oscillatory component superimposed onto the magnetic torque $\tau$ for several angles between ${\mu }_{0}H$ and the (101) direction. These traces are vertically displaced for clarity. The oscillatory data collected at $T=350$ mK are displayed in Fig. S3(a). Superimposed clear blue, magenta, and orange traces depict the angular dependence of the cyclotron orbits according to the DFT calculations. (b) Same as in (a), but for ${\varepsilon }_F$ displaced by $-5$ meV. Blue dotted rectangles enclose peaks that correspond to harmonics and hence were not included in our calculations. The top panels display the FS sheets according to the DFT calculations for ${\varepsilon }_F=0$ meV. Their colors match those of the markers depicting the associated dHvA frequencies.