Linear and nonlinear optical responses in the chiral multifold semimetal RhSi

The study investigates how multifold fermions in the chiral topological semimetal RhSi manifest in linear (optical conductivity) and nonlinear (CPGE) optical responses. Chiral topological metals lack inversion and mirror symmetries, enabling topological band crossings at different energies and momenta and potentially allowing direct probes of topological charge via nonlinear optics. Prior work in Dirac and Weyl semimetals faced symmetry-imposed cancellations, hindering direct detection of topological charge. RhSi (space group 198) hosts multifold fermions (threefold at Γ, double Weyl-related at R) and was predicted to exhibit quantized CPGE when only one chirality contributes below ~0.7 eV. The research addresses whether and how quantized CPGE can be observed in practice, disentangling roles of band structure details, chemical potential position, and carrier lifetimes, and establishing how multifold fermions shape optical conductivity features.

- Dirac semimetals (e.g., Cd3As2, Na3Bi) possess symmetry-protected Dirac points where topological charges cancel, limiting topological response detection. - Noncentrosymmetric Weyl semimetals (e.g., TaAs family) exhibit large nonlinear optics (giant SHG, photogalvanic effects), but mirror symmetries typically force equal-energy nodes of opposite charge, obscuring direct topological signatures. - Type-II Weyl semimetals show remarkable photogalvanic effects linked to tilted cones and open Fermi surfaces, not directly to topological charge. - Chiral crystals (no inversion or mirror symmetries) can separate nodes in both momentum and energy; CPGE was predicted to be quantized in such systems. - CoSi and RhSi (SG 198) host multifold fermions (threefold at Γ, double/multifold at R), offering accessible Lifshitz energies (~1 eV) for optical probes. Prior experiments observed CPGE in RhSi between 0.5–1.1 eV but without clear quantization. Theoretical work highlighted potential spoiling of quantization by quadratic corrections and short hot-carrier lifetimes.

- Samples: High-quality single-crystal RhSi (2 mm × 5 mm) grown by Bridgman, (110) facet. - Linear optics: In-plane reflectivity R(ω) measured by FTIR (Bruker VERTEX 70v) from 30–12,000 cm^-1 (4 meV–1.5 eV) at 10–300 K; extended to 50,000 cm^-1 by ellipsometry (Woollam VASE). Kramers–Kronig analysis yielded ε1(ω) and σ1(ω). Drude–Lorentz modeling extracted plasma frequencies and scattering rates; alternative subtraction method removed Drude and phonon terms to isolate interband response. - Transport context: Hall resistivity measured at 10 and 300 K to determine dominant carrier sign and assess multiband behavior. - Nonlinear optics (CPGE): Broadband THz emission spectroscopy with tunable incident photon energies 0.2–1.1 eV at 300 K. Geometry: 45° incidence, circular polarization; THz detected by electro-optic sampling in ZnTe, with polarization selection to isolate longitudinal CPGE component. SHG on (110) surface determined crystal axes and symmetry properties. Absolute CPGE quantified by referencing a ZnTe “standard candle” at each photon energy to normalize experimental factors. - Modeling: (i) Low-energy linearized model of threefold fermion at Γ; (ii) four-band tight-binding model for SG 198 including orbital embedding; (iii) ab initio DFT (with SOC) to compute optical conductivity and CPGE. Disorder/hot-carrier lifetime incorporated via Lorentzian broadening L_η with τ = ħ/η. Dense k-point meshes used (up to 480^3 for CPGE). Chemical potential µ varied to match experiments. Momentum-resolved CPGE contributions analyzed to identify band/point origins (Γ, R, M).

- Optical conductivity: - σ1(ω) exhibits two quasi-linear interband regimes with a slope change near 0.4 eV. Below ~0.4 eV, transitions near Γ dominate; above ~0.4 eV, transitions involving R contribute, altering slope. - After subtracting Drude and phonon contributions, a low-energy quasi-linear σ1 with a subtle shoulder ~0.2 eV is observed, consistent with a flat middle band at Γ when µ lies below the threefold node. - High-energy features include a broad maximum near 0.85–1.1 eV (two Lorentzian components), attributed primarily to transitions near the M-point saddle and other high-symmetry saddle points (DFT). - Drude analysis: Two Drude components (narrow and broad) indicate multiband carriers (electron pocket at R, heavy hole pocket at Γ). Transport lifetimes estimated as ≤13 fs at 300 K and ≤23 fs at 10 K (from narrow Drude). - DFT/tight-binding with disorder broadening η ~100–150 meV and chemical potential below Γ node (e.g., µ ≈ −30 to −100 meV) reproduce the low-energy conductivity shape, including the ~0.2 eV shoulder and the upturn at ~0.4 eV. - CPGE: - CPGE magnitude rises with decreasing photon energy from 1.1 eV, peaking at 163 ± 19 µA/V^2 near 0.7 eV, then drops and undergoes a sign change at ~0.4 eV. - DFT CPGE (µ ≈ −30 meV, η = 100 meV, τ ≈ 6.6 fs) quantitatively captures peak position, width, magnitude, and the sign change. Momentum-resolved analysis shows opposite-sign contributions from Γ and R below ~0.6 eV and activation of M near ~0.75 eV; the sign reversal is linked to R-point contributions turning on around 0.4 eV under large broadening. - Tight-binding CPGE reproduces peak-dip and sign change trends but underestimates peak position/magnitude due to limitations capturing M-point physics. - Lifetimes and chemical potential: - Hot-carrier lifetime inferred as τ ≈ ħ/η ≈ 4.4–6.6 fs (η ≈ 100–150 meV), consistent with broad low-energy features and lack of sharp SOC-split structures. - Both optical conductivity and CPGE indicate the chemical potential lies below the Γ threefold node, crossing a relatively flat hole-like band at Γ. - Quantized CPGE prospect: - Quantization is absent due to finite quadratic corrections, chemical potential below the node, and short τ. DFT suggests that electron-doping by ~+100 meV and reducing broadening to η ~10 meV (i.e., increasing τ) could yield a near-quantized CPGE in a narrow window around ~0.6 eV without sign change below 0.7 eV.

The combined linear and nonlinear optical measurements, together with modeling, show that multifold fermions in RhSi control distinct spectral regimes. Below ~0.4 eV, interband transitions near the Γ threefold dominate, yielding a quasi-linear σ1 with features (shoulder ~0.2 eV) arising from curvature of the flat middle band when µ is below the node. Above ~0.4 eV, broadened transitions near R contribute significantly, changing the slope of σ1 and introducing an opposite-sign CPGE contribution that causes the observed sign reversal at ~0.4 eV. The absence of a quantized CPGE plateau reflects realistic material factors: the chemical potential sits below the Γ node (activating flat-band transitions), quadratic band curvature spoils ideal quantization beyond the strictly linear regime, and strong disorder/short hot-carrier lifetimes smear fine features and allow R contributions at lower energies. DFT with SOC and realistic broadening reproduces the magnitude and energy dependence of both σ1 and CPGE, validating the interpretation that band curvature at Γ and the M-point saddle crucially shape the responses. The analysis clarifies practical conditions needed for observing quantization: positioning µ above the nodes and significantly extending τ to reduce broadening and isolate single-chirality contributions over an energy window.

This work presents a comprehensive experimental-theoretical study of RhSi’s linear (optical conductivity) and nonlinear (CPGE) optical responses, establishing how multifold fermions at Γ and R determine key spectral features. Interband σ1 displays two quasi-linear regimes with a slope change near 0.4 eV, attributable to Γ-dominated transitions at low energies and R contributions at higher energies. The CPGE exhibits a prominent non-quantized peak (~163 µA/V^2 at ~0.7 eV) and a sign change at ~0.4 eV. Tight-binding and DFT calculations, including SOC and realistic disorder, are consistent with experiments if the chemical potential lies below the Γ threefold node and the hot-carrier lifetime is short (~4.4–6.6 fs). The study outlines a route to near-quantized CPGE: electron-doping by ~100 meV to place µ above the nodes and improving sample quality to increase τ (reduce η), potentially yielding a narrow quantized window near ~0.6 eV. The broadband THz emission approach demonstrated here is broadly applicable to probing photogalvanic effects and topological responses in noncentrosymmetric materials and may aid development of efficient infrared detectors based on topological semimetals. Future work should explore materials with smaller spin–orbit coupling and energy/momentum-dependent scattering to optimize conditions for CPGE quantization.

- Two-Drude decomposition introduces uncertainty, particularly in the broad Drude component; subtraction choices affect the inferred interband onset though the quasi-linear behavior is robust. - Tight-binding models without SOC and limited orbital detail cannot fully capture band curvature and saddle-point (M) contributions, leading to deviations >0.5 eV and underestimation of CPGE magnitude/peak position. - Large disorder broadening (η ~100–150 meV) washes out fine SOC-induced features and complicates isolation of single-node contributions; modeling assumes an energy-independent hot-carrier lifetime τ, which may be oversimplified. - DFT optical conductivity underestimates spectral weight in 0.2–0.4 eV, possibly due to neglect of surface Fermi arc contributions. - CPGE measurements were performed at room temperature; temperature dependence of τ and µ may influence quantization prospects and was not systematically explored.