Symmetry-enforced Fermi degeneracy in topological semimetal $\mathrm{Rh}{\mathrm{Sb}}_3$

Predictions of a topological electronic structure in the skutterudite ${\mathrm{TPn}}_3$ family ($T=\text{transition metal}$, $Pn=\text{pnictogen}$) are investigated via magnetoresistance, quantum oscillation, and angle-resolved photoemission experiments on $\mathrm{Rh}{\mathrm{Sb}}_3$, a semimetal with low carrier density. Electronic band structure calculations and symmetry analysis of $\mathrm{Rh}{\mathrm{Sb}}_3$ indicate this material to be a zero-gap semimetal protected by symmetry with inverted valence and conduction bands that touch at the $\mathrm{\Gamma }$ point close to the Fermi level. Transport experiments reveal an unsaturated linear magnetoresistance that approaches a factor of 200 at 60 T magnetic fields and quantum oscillations observable up to 150 K that are consistent with a large Fermi velocity ($\sim 1.3\times {10}^6$ m/s), high carrier mobility [$\sim 14$ ${\mathrm{m}}^2/\left(\mathrm{V}\mathrm{s}\right)$], and the existence of a small three-dimensional hole pocket. A very small, sample-dependent effective mass falls to values as low as 0.018(2) of the bare electron mass and scales with the Fermi wave vector. This, together with a nonzero Berry's phase and the location of the Fermi level in the linear region of the valence band, suggests $\mathrm{Rh}{\mathrm{Sb}}_3$ as representative of a material family of topological semimetals with symmetry-enforced Fermi degeneracy at the high-symmetry points.

Figure 1

Band inversion and symmetry-enforced Fermi degeneracy of unfilled skutterudite compound $\mathrm{Rh}{\mathrm{Sb}}_3$. (a) Schematic of band ordering in $\mathrm{Co}{\mathrm{Sb}}_3$ close to the Fermi level, showing the valence band with an irreducible representation ${\mathrm{\Gamma }}_5^{-}$ and the conduction bands with irreducible representations ${\mathrm{\Gamma }}_5^{+}$ and ${\mathrm{\Gamma }}_6^{+}+{\mathrm{\Gamma }}_7^{+}$ touching at the $\mathrm{\Gamma }$ point. (b) The stronger SOC in Rh atoms of $\mathrm{Rh}{\mathrm{Sb}}_3$ pushes the band ${\mathrm{\Gamma }}_5^{-}$ up. The energy difference $\eta$ between ${\mathrm{\Gamma }}_6^{+}+{\mathrm{\Gamma }}_7^{+}$ and ${\mathrm{\Gamma }}_5^{-}$, indicating the strength of the band inversion, becomes negative. (c) The SOC in $\mathrm{Rh}{\mathrm{Sb}}_3$ also lifts the degeneracy between ${\mathrm{\Gamma }}_5^{+}$ and ${\mathrm{\Gamma }}_6^{+}+{\mathrm{\Gamma }}_7^{+}$. (d) Finally, the overlap of the valence band ${\mathrm{\Gamma }}_5^{-}$ and the conduction bands ${\mathrm{\Gamma }}_5^{+}$ and ${\mathrm{\Gamma }}_6^{+}+{\mathrm{\Gamma }}_7^{+}$ induces a strong hybridization between the $p$ orbitals of Sb atoms and the $d$ orbitals of Rh atoms which is evidenced by the calculated negative energy difference $\xi$ between ${\mathrm{\Gamma }}_6^{+}+{\mathrm{\Gamma }}_7^{+}$ and ${\mathrm{\Gamma }}_5^{+}$, giving the final band order of $\mathrm{Rh}{\mathrm{Sb}}_3$.

Figure 2

Calculated electronic structure of $\mathrm{Rh}{\mathrm{Sb}}_3$. (a) and (b) The electronic structure of $\mathrm{Rh}{\mathrm{Sb}}_3$ was calculated from first principles using two methods: Without spin-orbit coupling using PBE potential (a) and with spin-orbit coupling and modified Becke-Johnson potential (b) (see text). The radii of circles denoting band lines are drawn proportional to the Rh-$4d$ orbital weight. The inset in (b) is an enlarged part of the band structure around the Fermi level. The green, red, blue, and pink dashed lines indicate the positions of the Fermi levels for samples D, E, C, and A. (c) The calculated Fermi surface of $\mathrm{Rh}{\mathrm{Sb}}_3$ is represented in the first Brillouin zone and is extended below and enlarged by a factor of 50 for clarity.

Figure 3

Linear dispersion of $\mathrm{Rh}{\mathrm{Sb}}_3$ band structure from photon-energy-dependent ARPES at $T=80$ K. (a) Fermi surface at 140 eV, $p$ polarization (p-pol). The red square shows the first surface-projected Brillouin zone for the (100) surface, while the blue and green squares are the second and third zones (see more details in the Supplemental Material). (b) ARPES band mapping along the $k_x, k_y$, and $k_z$ directions. Left panel: band mapping along the $k_y=-0.68$ Å${}^{-1}$ line in the third BZ [yellow dashed line in (a), 170 eV ($k_z\sim 5\times 2\pi /c$), $p$ polarization]. White solid curves show the peak position of the momentum distribution curves from a double-Gaussian fit, while blue solid lines are the result of a linear fit of the band dispersion. Central panel: band mapping along the $k_x=0$ Å${}^{-1}$ line in the second BZ [white dashed line in (a), 140 eV ($k_z\sim 4.5\times 2\pi /c$), $p$ polarization]. Right panel: $k_z$ dependence, assuming 10 eV inner potential, of the hole pocket in the second BZ at $(k_x,k_y)=(-0.57$ Å${}^{-1},0$ Å${}^{-1})$ [orange circle in (a), $s$ polarization (s-pol)].

Figure 4

Large magnetoresistance, high carrier mobility, and quantum oscillations of $\mathrm{Rh}{\mathrm{Sb}}_3$ single crystals. (a) Magnetoresistance of $\mathrm{Rh}{\mathrm{Sb}}_3$ sample D for field orientation transverse to current direction, showing very large nonsaturating enhancement up to 65 T pulsed field and prominent Shubnikov–de Haas oscillations up to temperatures of 150 K. (b) Hall resistivity ${\rho }_{xy}$ of sample D [same temperature values as in (a)], with carrier mobility (red squares) and density (black circles) extracted from fitting of data shown in the inset. (c) Temperature dependence of the resistivity of sample D at different magnetic fields. (d) Amplitudes of Shubnikov–de Haas oscillations of the Fermi pocket as a function of temperature $T$, for four samples with varying carrier densities. The solid curves are fits to the Lifshitz-Kosevich formula for samples A, C, D, and E, which exhibit effective mass ratios $m^{*}/m_e$ of 0.102(8), 0.066(7), 0.018(2), and 0.033(4), respectively, corresponding to oscillation frequencies of 71, 60, 33, and 43 T, respectively. The inset shows the linear relation of $\frac{1}{m^{*}}$ vs $k_F^{-1}=\sqrt{\frac{\pi }{F}}$ for these samples (spheres), revealing the Dirac-like dispersion (solid line fit) for $\mathrm{Rh}{\mathrm{Sb}}_3$.

Figure 5

Field dependence of the transverse magnetoresistance at different temperatures for four different $\mathrm{Rh}{\mathrm{Sb}}_3$ single crystals: sample A (a), sample B (b), sample C (c), and sample E (d). The inset in (b) shows the magnetoresistance of sample B in fields up to 35 T and 1 K. Samples A, C, and E show clear quantum oscillation beginning at $\sim 10$ T.

Figure 6

The FFT spectra of Shubnikov–de Haas oscillations measured at different temperatures for sample A (a), sample C (b), sample D (c), and sample E (d).

Figure 7

Berry phase analysis of Shubnikov–de Haas oscillations in $\mathrm{Rh}{\mathrm{Sb}}_3$. (a) Quantum oscillations of background-subtracted conductivity $\mathrm{\Delta }{\sigma }_{xx}$ as a function of inverse fields up to 64 T for samples D and E. The discrete symbols are the experimental data, while the curves are fits using a global fitting routine as described in the text. (b) Landau level (LL) index plot of oscillations of three samples, with integer levels assigned to the minima of $\mathrm{\Delta }{\sigma }_{xx}$ and maxima assigned as half-integer indices. Solid lines are linear fits to the data, extrapolating to infinite-field intercepts that correspond to finite Berry's phase values of $0.92\left(1\right)\pi , 1.02\left(1\right)\pi$, and $0.98\left(1\right)\pi$ for samples C, D, and E, respectively (see text).