Optical conductivity of multifold fermions: The case of RhSi

We measured the reflectivity of the multifold semimetal RhSi in a frequency range from 80 to $20000{\mathrm{cm}}^{-1}$ (10 meV–2.5 eV) at temperatures down to 10 K. The optical conductivity, calculated from the reflectivity, is dominated by the free-carrier (Drude) contribution below 1000 ${\mathrm{cm}}^{-1}$ (120 meV) and by interband transitions at higher frequencies. The temperature-induced changes in the spectra are generally weak: only the Drude bands narrow upon cooling, with an unscreened plasma frequency that is constant with temperature at approximately 1.4 eV, in agreement with a weak temperature dependence of the free-carrier concentration determined by Hall measurements. The interband portion of conductivity exhibits two linear-in-frequency regions below 5000 ${\mathrm{cm}}^{-1}$ ($\sim 600$ meV), a broad flat maximum at around 6000 ${\mathrm{cm}}^{-1}$ (750 meV), and a further increase starting around 10 000 ${\mathrm{cm}}^{-1}$ ($\sim 1.2$ eV). We assign the linear behavior of the interband conductivity to transitions between the linear bands near the band crossing points. Our findings are in accord with the predictions for the low-energy conductivity behavior in multifold semimetals and with earlier computations based on band structure calculations for RhSi.

Figure 1

Optical reflectivity (a) and the real parts of the optical conductivity (b) and the dielectric permittivity (c) of RhSi at $T=10$, 200, and 295 K. Note the logarithmic $x$ scale. The insets show: the dc resistivity vs. $T$ (d), the Hall electron concentration vs. $T$ (e), and a zoom of the permittivity spectra near the zero crossings (f).

Figure 2

Drude-Lorentz fits (lines) of the measured optical spectra (symbols) of ${\sigma }_1$ and ${\varepsilon }_1$ of RhSi at 10 and 295 K, as indicated. Note the logarithmic $x$ scale.

Figure 3

Comparison of the experimental (this work and Ref. [11]) and calculated (Ref. [20]) optical spectra of RhSi: optical conductivity (a) and dielectric function (b).

Figure 4

Low-energy electronic structure of RhSi (a) and its optical conductivity (b). The electronic structure is calculated within the Topological Materials Database project [27]. Note that the band color code follows this reference and is not always related to the band position relative to the Fermi level. Both the as-measured conductivity spectra and the spectra after subtraction of the free-carrier response, ${\sigma }_1^{\mathrm{inter}}\left(\nu \right)$, are shown in (b). The optical transitions responsible for the characteristic features in ${\sigma }_1^{\mathrm{inter}}\left(\nu \right)$ are depicted as vertical arrows in (a). The corresponding frequency scales are indicated by the horizontal arrows of the same color in (b). The solid orange arrows represent the transitions between the approximately linear bands (including the almost flat bands) near the $\mathrm{\Gamma }$ point and the corresponding linear ${\sigma }_1^{\mathrm{inter}}\left(\nu \right)$. The solid grey arrows indicate two different processes coinciding in frequency: the onset of the downturn of the flat bands and the onset of interband transitions in the vicinity of the $R$ point. These processes lead to additional contributions to ${\sigma }_1\left(\nu \right)$. The dashed green arrows correspond to the transitions between the almost parallel bands along the $M\text{−}R$ line, leading to a maximum in ${\sigma }_1\left(\nu \right)$. Transitions with approximately, but not exactly, the same energy (e.g., along the $\mathrm{\Gamma }\text{−}R$ line) are responsible for smearing out this maximum and are indicated with the same arrow. The dashed blue lines are a guide to the eye. The solid purple line is an extrapolation of the effective-Hamiltonian computations from Ref. [19] to higher frequencies.

Figure 5

Optical conductivity of RhSi at the frequencies near the observed phonon modes for selected temperatures. The $y$ scale corresponds to the 10-K spectra. The spectra for other temperatures are shifted upwards for clarity. Results of Lorentzian fits of the phonon modes (on an electronic background) are shown for 10, 100, and 295 K as solid red lines. The black vertical lines are a guide to the eye.