Sample and second harmonic generation

The chiral crystal structure of CoSi seen from the (111) direction is depicted in Fig. 1c. As a first step, we pick up large homogeneous (111) natural facets by scanning second-harmonic generation (SHG) measurement³¹. To stimulate SHG, we focused light pulses centered at 800 nm under near-normal incidence to a 10-μm diameter spot on the sample and the second harmonic signal centered at 400 nm is measured. As shown in Fig. 1d, polar patterns of SHG are found as a function of the direction of the linear polarization of the incident light in the co-rotating parallel-polarizer (orange) and crossed-polarizer (blue) configurations. These patterns agree well with a fit with only one non-zero parameter based on the point group symmetry (see “Methods” section).

Longitudinal CPGE in CoSi

Figure 2a shows schematically the measurement of the longitudinal CPGE. When circularly polarized light is incident on the sample, a current flows along the light propagation direction inside of the material. Under normal incidence, the current flows perpendicular to the surface, which prevents THz emission into free space from CPGE in the bulk^24,29. THz emission does originate from the surface current density under oblique incidence²⁹. See Supplementary Note 1 for more details. Therefore, in order to emit THz radiation into free space, we utilize off-normal incidence at 45 degrees as shown in Fig. 2a. An achromatic quarter-wave plate is used to control the polarization of the incident light. Terahertz wave components in xz and y direction are detected by using a THz polarizer before the ZnTe detector. Figure 2b, e shows the reversal of the polarity of the time trace of emitted THz electric field under left and right circularly-polarized light at the incident photon energy of 0.50 eV, which indicates the change of the direction of the photo-current under opposite helicity of circularly polarized incident light (i.e., the CPGE).

To confirm that the CPGE we observe is a longitudinal photocurrent, we studied the polarization dependence of the CPGE by rotating both the achromatic quarter-waveplate and the samples. The experimental geometry is shown in Fig. 2a, and we detect the emitted THz components in the incident plane, E_xz(t), and perpendicular to the plane, E_y(t). In Fig. 2b (orange) we show the peak value of the emitted THz field E_y(t) at the incident photon energy of 0.50 eV as a function of the angle of the quarter-wave plate. The THz field under left and right circularly polarized light has the same direction and magnitude, which indicates no CPGE. The almost identical time traces of E_y(t) with opposite circular polarizations are shown in Fig. 2c. The CPGE component (E_↺(t) − E_↻(t))/2 is zero within the detection sensitivity, as shown in Fig. 2d. (E_↺(t) + E_↻(t))/2 is the linear photo-galvanic effect (LPGE) component under circularly-polarized light (see Supplementary Note 1B for details).

In contrast, the in-plane THz field E_xz(t) shows completely different polarization dependence as shown in Fig. 2b (blue). When the helicity of the circularly polarized light is reversed, the direction of the peak THz field in E_xz(t) changes, and the waveform is shown in Fig. 2e. They are not simply the same curve with opposite signs because of a sizable LPGE contribution. Nevertheless, (E_↺ − E_↻)/2 is not zero in E_xz(t) and relatively large compared to (E_↺ + E_↻)/2, as shown in Fig. 2f. The observation of a non-zero CPGE only in the incident xz plane is consistent with the longitudinal CPGE, where the current flows along the wave vector direction inside CoSi. This longitudinal CPGE is unchanged as we rotate the sample due to the cubic symmetry constraints, as shown for 0.50 eV incident photon energy in Fig. 2g. We also observed similar angle dependence at other photon energies. See Supplementary Note 2 for more details.

To quantify the longitudinal CPGE, we performed a symmetry analysis by fitting the angle dependence of the quarter-wave plate on E_xz and E_y. The solid lines in Fig. 2b are the best fit to the functions determined by the crystal symmetry, \({E}_{y}(\theta )={B}_{1}\sin (4\theta )+{C}_{1}\cos (4\theta )+{D}_{1}\) and \({E}_{xz}(\theta )=A\sin (2\theta )+{B}_{2}\sin (4\theta )+{C}_{2}\cos (4\theta )+{D}_{2}\), where the coefficients A, B₁, B₂, C₁, C₂, D₁, D₂ are determined by the CPGE and LPGE conductivity (see Supplementary Note 1B for details). Both curves are fitted simultaneously with the same fit weights. The \(\sin (2\theta )\) term describes the CPGE while \(\sin (4\theta )\), \(\cos (4\theta )\) and the constant terms describe the LPGE. The symmetry analysis shows that the out-of-plane component E_y does not contribute to the CPGE, while the in-plane component E_xz dominates the CPGE.

CPGE spectrum in CoSi

After confirming the longitudinal direction of the CPGE, we now study the amplitude of the CPGE current inside the sample at different incident photon energies. We use circularly-polarized laser pulses with a duration 50–100 fs and a tunable incident photon energy from 0.2 eV to 1.1 eV to generate a CPGE inside of the sample. For the first time, we detected THz emission with incident photon energy below 0.5 eV, comparing with previous measurements^8,9,24,26,27. In order to convert the detected THz electric field into the CPGE current inside the sample, we measured a benchmarking ZnTe sample at the same position at each wavelength immediately after measuring CoSi. ZnTe is useful as a benchmark due to its relatively flat frequency dependence on the electric-optical sampling coefficient for photon energy below the gap³². See Supplementary Note 1F and Supplementary Fig. 3 for the raw data. We first convert the collected THz electric fields on CoSi and ZnTe from the time domain to the frequency domain by a Fourier transformation. By taking the ratio of the two Fourier transforms of the electric fields and considering the Fresnel coefficient, refractive indices and penetration depth, we obtain the ratio between the CPGE response of CoSi and the optical rectification of ZnTe. The use of ZnTe circumvents assumptions regarding the incident pulse length, the wavelength dependent focus spot size on the sample, and the calculation of collection efficiency of the off-axis parabolic mirrors (see Supplementary Note 1C for details).

The CPGE follows \(j(\Omega )=\frac{{\beta }_{xx}}{i\Omega +1/\tau }{E}_{0}^{2}(\Omega )\), where Ω is the THz frequency and τ is the hot-carrier lifetime. When the hot-carrier lifetime satisfies τ ≪ 1/Ω, \(j\approx {\beta }_{xx}\tau {E}_{0}^{2}\). This is the case for the current experiment as j(Ω) depends weakly on Ω. (See Supplementary Fig. 6.) The second-order photo-conductivity plotted in Fig. 3a was an average value of the CPGE over the frequency range of 0.5–2.0 THz in Supplementary Fig. 6, which should also be the DC limit. When τ is much longer than the pulse width, which is in the quantization regime, \({\rm{d}}j/{\rm{d}}t={\beta }_{xx}{E}_{0}^{2}\). When 1/τ is comparable to Ω, the CPGE conductivity, \(\frac{{\beta }_{xx}}{i\Omega +1/\tau }\), will have strong frequency dependence in the THz regime, which will enable the extraction of β_xx and τ separately. Anticipating our theory analysis, we note that the CPGE spectrum is determined by the only symmetry-independent non-zero CPGE response tensor β_xx, which is a photocurrent rate multiplied by the hot-carrier lifetime τ. The measured CPGE photocurrent per incident field squared as a function of frequency, which we will denote as the CPGE spectrum, is shown in Fig. 3a for room temperature, showing a peak value of  ~550 μ A/V² at 0.4 eV. The CPGE spectrum peak value is much larger than the photo-galvanic effect in any chiral crystals reported in the literature³³, BaTiO₃³⁴, single-layer monochalcogenides^34,35, the colossal bulk-photovoltaic response in TaAs³⁶ and RhSi in the same space group^24,25.

a Measured second-order CPGE photo-conductivity (β_xxτ) as a function of incident photon energy, and ab-intio calculations of the CPGE current with and without spin-orbit coupling at room temperature. b The band structure of CoSi without spin-orbit coupling. We define zero energy at the threefold node at the Γ point. The double Weyl node at the R point is at −185 meV. The dashed horizontal line indicates the chemical potential E_f = −37 meV in our sample, moderately lower from that obtained by DFT (E_f = −20 meV). The band structure of the k ⋅ p model is shown in green (band 1), blue (band 2), and orange (band 3) obtained by fitting the ab-intio band structure (black) up to quadratic corrections. For the Γ−X direction we define \({\omega }_{1}^{X}\), \({\omega }_{2}^{X}\), and \({\omega }_{3}^{X}\) as the minimum energy that allows transitions from band 1 to band 2, the maximum energy that allows transition from band 1 to band 2 and the minimum energy that allows transitions from band 2 to band 3, respectively. We define in the same way \({\omega }_{1}^{R}\) in the R direction (\({\omega }_{2,3}^{R}\) fall outside the applicability of the quadratic model). c Total (gold) and interband (blue) optical conductivity of CoSi at 300 K. d Momentum resolved contributions to the CPGE peak at 0.4 eV in the red curve in a.

First-principle calculation

Next, we address the relationship between the large photo-conductivity peak and the multifold fermions near the Fermi level shown in Fig. 3b. In Fig. 3a, we show our ab-initio calculations of the CPGE for CoSi with and without spin-orbit coupling (SOC) at room temperature and at a chemical potential E_f crossing the flat hole band, as indicated by the dashed line in Fig. 3b (see “Methods” section and Supplementary Note 4). They quantitatively reproduce the experimental data across a wide frequency range. The SOC splitting,  ≈20 meV at the Γ point node, determines the finer structure in the optical response³⁷.

To match the first-principles calculations with the CPGE spectrum, we considered β_xx, which is related to the CPGE trace by β = 3β_xx due to cubic symmetry^11,12. Note that β_xx is directly calculated from the band structure at certain chemical potential. It is plotted in Fig. 3a times τ, the only free parameter, which was determined by matching the calculated peak width and magnitude to the CPGE data. This self-consistent constraint is satisfied with a broadening ℏ/τ ≈ 38 meV at 300 K. As shown from the right y axis of Fig. 3a, the CPGE trace β reaches 3.3 (±0.3) in units of the quantization constant β₀.

For frequencies below 0.6 eV, all the interband excitations on CoSi occur in the vicinity of its multifold bands at the nodes Γ and R³⁷. We conclude this from the optical conductivity on CoSi at 300 K that is shown in Fig. 3c. With the subtraction of the Drude response and the four phonon peaks, the interband contribution has a kink at  ~0.2 eV that separates two quasi-linear regimes. The details of the theoretical and experimental studies could be found in ref. ³⁷. Below 0.2 eV, the interband excitations involve the threefold fermion at the Γ point, while the excitations near the double Weyl fermion at R become active only above 0.2 eV³⁷. The main contribution to the peak around 0.6 eV comes from the saddle point M³⁷. In addition to the optical conductivity, our momentum resolved calculation (Fig. 3d) for the CPGE peak at 0.4 eV reveals that it originates from these multifold fermions only, which contribute with opposite signs to the CPGE current. Therefore, this giant CPGE peak has a purely topological origin, although it is not quantized due to the sum of contributions of two kinds of multifold fermions and the quadratic contributions to the energy bands.

The position of the chemical potential is crucial to relate the CPGE to quantization. Our ab-initio calculations, supported by our low-energy analysis below, reveal that the CPGE shows a dip-peak structure as the one of Fig. 4a, b) when E_f is below the Γ node (E_f < 0) in our sample. Such Fermi energy is consistent with recent quasi-particle interference³⁸ and linear optical conductivity experiments³⁷. Note that this dip-peak structure was clearly observed recently in RhSi as the energy splitting between the nodes at the Γ and R in RhSi is around twice larger than CoSi so that the sign change in CPGE is pushed to around 0.4 eV in RhSi²⁵. The dip-peak structure for E_f < 0 is also produced by using a four-band tight binding model for CoSi¹³ (see Supplementary Note 5). As shown in Fig. 4a, the dip reaches the quantized value of 4β₀ at low temperatures in the clean limit, and remains quantized for hot carrier lifetime broadening up to 5 meV at 100 meV photon energy (see Supplementary Note 4 and Supplementary Fig. 7 for details.). The quantization of the dip is determined by the threefold fermion at the Γ point as the vertical excitations at the R point are Pauli blocked below 0.2 eV³⁷. Therefore, the CPGE will not be quantized in the current sample at low temperature in the photon energy range of 0.2–1.1 eV as the peak around 0.4 eV after the dip appears non-universal in general due to contributions from both nodes at Γ and R (see see E_f = −37 meV curve in Fig. 4a). However, if E_f is decreased further to lie close to the R point, this peak can reach 4β₀ at room temperature even with a broadening of 38 meV (see E_f= −67 meV curve in Fig. 4b). As discussed below, this peak originates from the double Weyl fermion at the R point, and it is enabled by an accidental window of vanishing CPGE contribution from the Γ point. Finally, we note that electron-electron interactions can also correct the quantized value, as occurs for chiral Weyl semimetals³⁹. While it is currently unknown how relevant these corrections are for multifold fermions, the large hole and electron pockets at Γ and R in CoSi suggest that screening should be strong and therefore interactions should have a small effect. The good agreement of our model calculations with the data, shown in Fig. 3a, is also consistent with this point of view. Experimentally, from the optical conductivity measurements we estimate a relative dielectric constant ϵ₁ of the order of −2500 at 300 K and −10,000 at 10 K, further supporting a normal metallic behavior with very large screening of interactions. Also, specific heat measurements on CoSi also showed that it is a weakly correlated semimetal, as evidenced by a normal metallic Sommerfeld constant⁴⁰. Because of these reasons interactions are neglected in this work.

a, b CPGE current obtained by ab-intio calculations corresponding to the CoSi band structure with spin-orbit coupling at different chemical potentials at a, 0 K with 5 meV broadening and b 300 K with 38 meV broadening. c, d CPGE current calculation from the k ⋅ p model, with parameters (v, a, b, c) = (1.79, 1.07, −1.72, 3.26). c The contributions to the CPGE current from transitions near the Γ point are shown by open purple circles for transitions from band 1 to 2 and as open gold squares for transitions form band 2 to 3. A solid blue line shows the sum of the contributions form transitions from 1 to 2 and from 2 to 3, which is the total CPGE contribution from the Γ point. d The contribution from the R point to the CPGE is shown by a step of 4 in red. The total contribution to the CPGE from the Γ and the R point is shown in green. A quantized dip/plateau is observed when the frequency is between \({\omega }_{1}^{R}\) and \({\omega }_{2}^{X}\), which allow transitions from band 1 to band 2 only. The dip is determined by transitions around Γ only, while the peak has contributions from excitations near the Γ and R points.

k ⋅ p model

To understand the origin of the dip-peak structure, it is necessary to describe the curvature of the middle band. To this end we derived a low-energy k ⋅ p type model keeping symmetry-allowed terms up to quadratic order in momentum k. The resulting Hamiltonian reads

where S is the vector of spin-1 matrices, and k = ∣k∣. We fixed its coefficients v, b, c, and \({c}_{1}=\frac{1}{3}(3a+2c)\) with a fit to the band structure shown in Fig. 3b around the Γ point. The second term includes three out of the four symmetry-allowed quadratic terms because the fourth has a negligible effect on the CPGE (see Supplementary Note 6 for details). The energies expanded to second order in momentum for the three bands are plotted as colored lines in Fig. 3b. The coefficients b and c determine the curvature in the Γ−X and Γ−R directions, respectively. For the R point bands, we use a spin-degenerate double Weyl model that has a step increase in the CPGE current by 4β₀ when excitations at R are allowed in Fig. 4d¹².

The possible optical transitions in the band structure near the Γ point are illustrated in Fig. 3b. We label the bands with increasing energies as 1, 2, 3. For E_f above the threefold node, the only possible transition is from bands 2 to 3. As the frequency increases, this transition becomes active and yields a monotonically increasing joint density of states (JDOS)¹³. As shown in Fig. 4c, for E_f below the node, however, two types of transitions contribute: 1 to 2 and 2 to 3. The first transition from band 1 to band 2 (open purple circles) is active for a small range of energies, and then decays to zero. The second transition from band 2 to band 3 (open gold squares) only starts picking up at larger frequencies, leaving a dip in the JDOS and, therefore, a dip in the CPGE (solid blue line). The different frequencies where the transitions become active or inactive are labeled in Figs. 3b, 4c, d. Figure. 4d show that when we add the contributions from the threefold fermions at Γ and double Weyl fermion at R, the existence of the dip from the threefold fermions leads to the dip-peak structure observed in the ab-initio calculations, only when E_f is below the threefold node. In the k ⋅ p model, we also show that while the dip is universally quantized, the peak is not because of the incomplete transitions from Γ. Note that the quantization of the peak not only depends on the Γ contribution but it also may be altered by the quadratic dispersion of the double Weyl fermion when it fully contributes to CPGE. However, as shown in Fig. 4b, decreasing E_f further could be used to diminish the contribution from the threefold fermions at around 0.4 eV and reveal the quantization due to the R point (see Supplementary Note 4 for details).