Topological materials, a class of quantum materials, possess robust and intrinsic electronic properties that can potentially enhance electromagnetic responses. However, detecting these properties directly is challenging. Dirac semimetals, such as Cd3As2 and Na3Bi, have doubly degenerate bands crossing linearly at a Dirac point, but the topological contributions to their responses cancel due to symmetry. Weyl semimetals, with isolated twofold topological band crossings, offer an alternative but are often limited by mirror symmetry, which causes cancellation of opposite charges. Chiral topological metals, lacking inversion and mirror symmetries, provide a better avenue for studying topological charge. In these materials, CPGE, a photocurrent reversing with polarization, is predicted to be quantized. However, chiral Weyl semimetals with sufficient node separations are scarce. Recently, silicides like CoSi and RhSi emerged as promising candidates, hosting multifold nodes (where more than two bands meet) that generalize Weyl points. These materials, with Lifshitz energies around 1 eV (much larger than in previously studied materials), are ideal for studying topological excitations using optical conductivity measurements. This study aims to comprehensively understand the linear and nonlinear optical responses of RhSi in the energy range where multifold fermions dominate optical transitions and transport, addressing the challenge of observing quantization in practice.

Previous studies on Dirac and Weyl semimetals, such as Cd3As2, Na3Bi, and TaAs, focused on materials with Lifshitz energies below 100 meV. While giant second-harmonic generation (SHG) and interesting photogalvanic effects have been observed in Weyl semimetals, they are not directly linked to the topological charge of individual band crossings due to symmetry constraints. The prediction of a quantized CPGE in chiral Weyl semimetals, specifically RhSi, was based on the material's threefold fermion at the zone center and a double Weyl node at the zone boundary. Theory predicted a CPGE plateau below 0.7 eV and vanishing CPGE at higher frequencies. However, the practical observation of quantization remained a challenge due to factors like quadratic corrections and short hot-carrier lifetimes. Prior experimental signatures of multifold fermions were primarily limited to band structure measurements, lacking a comprehensive understanding of linear and nonlinear optical responses in the relevant energy range.

This study involved measuring both the linear and nonlinear optical responses of RhSi. Optical conductivity measurements were conducted from 0.004 to 6 eV and 10 to 300 K using Fourier transform infrared (FTIR) spectroscopy and ellipsometry. Terahertz (THz) emission spectroscopy was employed with incident photon energies from 0.2 to 1.1 eV at 300 K to measure the CPGE. The optical conductivity data were analyzed using a Drude-Lorentz model, separating contributions from itinerant carriers and interband transitions. Theoretical modeling included a low-energy linearized model around the Γ point, a four-band tight-binding model, and ab initio density functional theory (DFT) calculations. The CPGE measurements used SHG to determine the high-symmetry axes, and a THz emission spectroscopy setup at 45° incidence. The CPGE response was extracted by comparing THz emission under opposite circular polarizations. A ZnTe detector served as a benchmark for quantitative analysis, circumventing assumptions about pulse length and collection efficiency. First-principles calculations using FPLO (full-potential local-orbital minimum-basis) were used to calculate the CPGE response.

Optical conductivity measurements revealed two quasi-linear regimes below and above 0.4 eV, attributed to interband transitions around the Γ and R points. The analysis showed a relatively short transport lifetime (≤13 fs at 300 K, ≤23 fs at 10 K). The CPGE measurement displayed a sign change at 0.4 eV and a peak of approximately 160 µA/V² at 0.7 eV, considerably larger than in many other materials. The tight-binding and DFT calculations reproduced the experimental optical conductivity and CPGE reasonably well when the chemical potential was below the threefold node at the Γ point, crossing a relatively flat band, and with a short hot-carrier lifetime (4-7 fs). The absence of a quantized CPGE was attributed to the short hot-carrier lifetime. DFT calculations predicted that a quantized CPGE could be achievable by increasing electron doping by 100 meV and significantly improving sample quality to increase the hot-carrier lifetime. The four-band tight-binding model qualitatively reproduced the peak-dip structure and sign change in the CPGE, but it underestimates the peak position and magnitude, highlighting limitations in capturing the band structure curvature at high symmetry points like M. DFT calculations with a broadening factor of 100 meV (corresponding to a hot-carrier lifetime of 6.6 fs) showed good agreement with the experimental CPGE data, capturing the peak, width, and sign change. Momentum-resolved analysis indicated that the sign change is due to the contribution of excitations at R with opposite sign in βxx. By adjusting the chemical potential in DFT calculations and decreasing broadening to 10 meV, a close-to-quantized CPGE value was predicted around 0.6 eV.

The findings demonstrate a consistent picture of the optical transitions in RhSi, with the chemical potential likely crossing a large hole-like band at Γ. The short hot-carrier lifetime plays a crucial role in the observed optical and CPGE responses. The curvature of the flat band at the Γ point and the saddle point at M significantly impact the optical conductivity and CPGE. The two quasi-linear regions in the interband optical conductivity reflect the contributions of the threefold fermion at Γ and the R point transitions. The absence of a quantized CPGE highlights the importance of material quality and chemical potential tuning. The large CPGE response observed is significant for potential device applications.

This study provided a consistent picture of optical transitions in RhSi, explaining the observed linear and nonlinear optical responses. The findings underscore the impact of the chemical potential, hot-carrier lifetime, and band structure curvature. Achieving a quantized CPGE appears feasible with increased doping and improved sample quality. The methodology can be extended to other noncentrosymmetric materials to probe topological excitations, bulk photovoltaic effects, and spintronic responses.

The constant hot-carrier scattering time used in the theoretical models might be an oversimplification, as the scattering time is likely energy and momentum dependent. The tight-binding model has limitations in capturing the band structure accurately at all high-symmetry points. The effects of surface Fermi arcs on the CPGE were not fully considered, primarily because this study focused on the bulk contributions.