Giant topological longitudinal circular photo-galvanic effect in the chiral multifold semimetal CoSi

The absence of mirror symmetry, or chirality, is behind striking natural phenomena found in systems as diverse as DNA and crystalline solids. A remarkable example occurs when chiral semimetals with topologically protected band degeneracies are illuminated with circularly polarized light. Under the right conditions, the part of the generated photocurrent that switches sign upon reversal of the light's polarization, known as the circular photogalvanic effect (CPGE), is predicted to depend only on fundamental constants. The conditions to observe quantization are non-universal, and depend on material parameters and the incident frequency. In this work, we perform terahertz emission spectroscopy with tunable photon energy from 0.2 eV - 1.1 eV in the chiral topological semimetal CoSi. We identify a large longitudinal photocurrent peaked at 0.4 eV reaching $\sim$ 550 $\mu A/V^{2}$, which is much larger than the photocurrent in any chiral crystal reported in the literature. Using first-principles calculations we establish that the peak originates from topological band crossings, reaching 3.3$\pm$0.3 in units of the quantization constant. Our calculations indicate that the quantized CPGE is within reach in CoSi upon doping and increase the hot-carrier lifetime. The large photo-conductivity suggests that topological semimetals can be used as novel mid-infrared detectors.

The absence of mirror symmetry, or chirality, is behind striking natural phenomena found in systems as diverse as DNA and crystalline solids. A remarkable example occurs when chiral semimetals with topologically protected band degeneracies are illuminated with circularly polarized light. Under the right conditions, the part of the generated photocurrent that switches sign upon reversal of the light's polarization, known as the circular photogalvanic effect (CPGE), is predicted to depend only on fundamental constants. The conditions to observe quantization are non-universal, and depend on material parameters and the incident frequency. In this work, we perform terahertz emission spectroscopy with tunable photon energy from 0.2 eV -1.1 eV in the chiral topological semimetal CoSi. We identify a large longitudinal photocurrent peaked at 0.4 eV reaching ∼ 550 µA/V 2 , which is much larger than the photocurrent in any chiral crystal reported in the literature. Using first-principles calculations we establish that the peak originates from topological band crossings, reaching 3.3±0.3 in units of the quantization constant. Our calculations indicate that the quantized CPGE is within reach in CoSi upon doping and increase the hot-carrier lifetime. The large photo-conductivity suggests that topological semimetals can be used as novel mid-infrared detectors.
Several challenges remain to observe the quantized CPGE in chiral semimetals. In this family of materials, the presence of spin-orbit coupling leads to a splitting of the nodes, which can still display quantization but in a reduced frequency range determined by the strength of the spin-orbit coupling, for example as happens in RhSi [13][14][15]17]. In this work, we measure the CPGE in CoSi. Its spin-orbit coupling is much smaller than in RhSi, and its transport lifetime can reach around 1 ps at low temperature [18], two orders of magnitude longer than in RhSi [16]. We measure the CPGE by detecting radiated terahertz (THz) pulses emitted from the illuminated regions, a method with several advantages compared to DC current measurements. Firstly, the emitted THz pulse originates from a local illuminated region, and therefore thermal current and the non-local diffusion of photo-excited carriers to the contacts, typical of a DC current measurement, are not concerns [29]. Secondly, the time-resolved nature of the experiment can be used to directly detect the quantized CPGE so long as one uses pulses much shorter than the lifetime of the photo-  The chiral crystal structure of CoSi seen from the (111) direction is depicted in Fig. 1b. As a first step, we pick up large homogeneous (111) natural facets by scanning second-harmonic generation (SHG) measurement [30]. To stimulate SHG, we focused light pulses centered at 800 nm under near-normal incidence to a 10-µm diameter spot on the sample. As shown in Fig. 1c, polar patterns of SHG are found as a function of the direction of the linear polarization of the incident light in the corotating 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). Fig. 1d 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. THz emission originates from current oscillation parallel to the surface [31]. Under normal incidence, the current flows perpendicular to the surface, which prevents THz emission into free space from CPGE in the bulk [16,31]. Therefore, in order to emit THz radiation into free space, we utilize off-normal incidence at 45 degrees as shown in Fig. 1d and Fig. 2a. Fig. 1e 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 0.375 eV, which indicates the change of the direction of the photo-current under opposite helicity of circularly polarized incident light (i.e. the CPGE). For the first time, we detected THz emission with incident photon energy below 0.5 eV, comparing with previous measurements [10,11,16,32,33].
To confirm that the CPGE we observe is a longitudinal photocurrent, we studied the polarization dependence of the CPGE by rotating both the achromatic quarterwaveplate 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 photogalvanic effect (LPGE) component under circularly-polarized light.
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. This contrasts with our result at 0.375 eV in Fig.1d where the LPGE is negligible. 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 0.35 eV incident photon energy in Fig.2g.
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 (θ) = B 1 sin(4θ) + C 1 cos(4θ) + D 1 and E xz (θ) = A sin(2θ) + B 2 sin(4θ) + C 2 cos(4θ) + D 2 , where the coefficients A,B 1 ,B 2 ,C 1 ,C 2 ,D 1 ,D 2 are determined by the CPGE and LPGE conductivity. Both curves are fitted simultaneously with the same fit weights. The sin(2θ) term describes the CPGE while sin(4θ), cos(4θ) 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 inplane component E xz dominates the CPGE.
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. 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 gap [34]. Its use 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.
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 (±45) µA/V 2 at 0.4 eV. 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 hotelectron lifetime τ . The CPGE spectrum peak value is much larger than the photo-galvanic effect in any chiral crystals reported in the literature [3], BaTiO 3 [35], single-layer monochalcogenides [35,36], the colossal bulkphotovoltaic response in TaAs [37] and RhSi in the same space group [16,17].
Next we address the relationship between large photoconductivity 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. 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 response [18].
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 [2,13]. It is plotted in Fig. 3a times τ , which was determined by matching the calculated peak width and magnitude to the CPGE data. This self-consistent constraint is satisfied with a broadening /τ ≈ 38 meV at 300 K. As shown from the right y axis of Fig. 3(a) Energy (eV) FIG. 3. CPGE spectrum and band structure for CoSi. a, Measured second-order CPGE photo-conductivity (βxxτ ) 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 ω X 1 , ω X 2 and ω X 3 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 ω R 1 in the R direction (ω R 2,3 fall outside the applicability of the quadratic model). c, CPGE current obtained by ab-intio calculations corresponding to the band structure (black in b) at different chemical potentials at 0 K and 300 K with SOC. Note that the broadening used at 300 K in a and c is 38 meV while it is 5 meV at 0 K. d, CPGE current calculation from the k · p model, with parameters (v, a, b, c) = (1.79, 1.07, −1.72, 3.26). The contributions to the CPGE current from transitions near the Γ point are shown by purple dots for transitions from band 1 to 2 and as orange dots 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. 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 ω R 1 and ω X 2 , 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.
terband excitations occur in the vicinity of the multifold bands at Γ and R, consistent with a recent linear optical conductivity measurement and calculations [18]. Therefore, the large CPGE peak is determined by interband excitations near multifold fermions.
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 when E f is below the node (E f < 0) in our sample, which is consistent with recent quasi-particle interference [38] and linear optical conductivity experiments [18], the CPGE shows a dip-peak structure (see Fig.3c). Note that the dip-peak structure with E f below the node was recently clearly observed 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 RhSi [17]. The dip-peak structure for E f < 0 is also produced by using a four-band tight binding model for CoSi [14]. The dip reaches the quantized value of 4β 0 at low temperatures in the clean limit, and remains quantized for hot electron lifetime broadening up to 5 meV at 100 meV photon energy. 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 eV [18]. The peak after the dip appears non-universal in general. However, if E f is decreased further to lie close to the R point, this peak can reach 4β 0 at room temperature even with a broadening of 38 meV (see E f = −67 meV curve in Fig.3c). 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.
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 = 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. 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 4β 0 when excitations at R are allowed in Fig. 3d [13].
The possible optical transitions in the band structure near the Gamma 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) [14]. As shown in Fig. 3 b,d, for E f below the node, however, two types of transitions contribute: 1 to 2 and 2 to 3. The first transition (1 to 2) is active for a small range of energies, and then decays to zero. The second transition (2 to 3) only starts picking up at larger frequencies, leaving a dip in the JDOS and, therefore, a dip in the CPGE. The different frequencies where the transitions become active or inactive are labeled in Fig.3d. Fig.3d 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. However, as shown in Fig. 3c, 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.
By studying the CPGE in the chiral topological semimetal CoSi we found a large longitudinal photoconductivity in the mid-infrared regime, which has a topological origin linked to the existence of multifold fermions in this material. CoSi could potentially be used as a new mid-infrared detector based on a topological mechanism. Moreover, our theory suggests that a quantized CPGE is within reach in CoSi by several means. The narrow quantized plateau corresponding to the Γ node could be accessible at low temperatures if the hotelectron lifetime increases at lower photon energy, as occurs in other semimetals [39]. Also, electron doping the E f above the threefold node will result in a wider quantized plateau from the Γ node corresponding to the dip we calculate at low temperatures and photon energies. A quantized plateau corresponding to the R node can be revealed at T =300 K by hole-doping, even with a similar hot electron lifetime as that of our current sample. We expect that these possibilities, opened by our work, motivate further effort on crystal growth with longer hot electron lifetime and different doping, as well as infrared time-resolved measurements below 0.2 eV to probe the hot carrier dynamics. (Alfa Aesar 99.99%) with the molar ratio of 1:1:15-20 were set in an alumina crucible and then sealed in a fused silica ampule in around 0.8 atm argon environment. The ampule was heated to 1100 • C with a speed of about 150 • C/hour. After soaking at 1100 • C for 10 hours, the ampule was cooled down to 700 • C at the rate of 2 • C/hour, and the excess flux was centrifuged out at that temperature to get several single crystals with large (111) facet [40]. Crystals larger than 2 mm × 2 mm were picked for THz emission experiments. We also performed a spatial SHG scanning of the sample and found an homogeneous signal. Second harmonic generation fit For CoSi (111), the fits are: (2) xyz is the only non-zero SHG tensor element in CoSi. θ is the angle between the incident polarization and the [1,1,-2] axis.
Terehertz emission spectroscopy A laser beam from a Ti:sapphire amplifier (center photon energy 1.55 eV, repetition rate 1 kHz, duration ∼ 35 fs) was split by a beam splitter into pump and probe beams. On the pump side, an optical parametric amplifier is used to convert the photon energy to 0.47-1.1 eV (pulse duration 40-70 fs), and a different frequency generation is used to further convert photon energy to 0.20-0.48 eV (pulse duration 70-110 fs). The laser beams were then focused by a 40-cm BaF 2 lens or a 40-cm germanium lens onto the sample with a diameter of 1 mm under 45 degree angle of incidence. A typical pump power of 15 µJ per pulse was used, which falls into the linear response range. The emitted THz wave was collected by an off-axis parabolic mirror (OAP) and focused by another OAP onto an electrooptic (EO) crystal, ZnTe (110). The probe beam was co-propagating with the THz wave into the EO crystal to detect the THz electric field using EO-sampling method [31]. All of the measurement were performed in a dry-air environment with relative humidity less than 3% to avoid water absorption. To control the polarization of pump pulses, a quartz-MgF 2 achromatic quarter-wave plate (600-2700 nm, retardance error ≤ λ/500) and a MgF 2 achromatic quarter-wave plate (2500-7000 nm, retardance error ≤ λ/100) were used. A THz wire-grid polarizer was used to extract out-of-plane (E y ) and inplane (E xz ) components of THz electric field. A benchmarking crystal ZnTe (110) was used as a standard candle to extract the photogalvanic response from CoSi. Both crystals were mounted on a computer-controlled motor to reliably change the position. For each incident photon energy, measurement of CoSi was immediately followed by ZnTe to avoid long-term fluctuation of laser power.
By comparing the THz electric field of CoSi and ZnTe in frequency domain, the photogalvanic response of the CoSi crystal could be precisely obtained.
First principle calculation To calculate the CPGE current, we obtain the density-functional theory (DFT) Bloch wave functions from the Full-Potential Local-Orbital program (FPLO) [41] within the generalized gradient approximation (GGA) [42]. By projecting the Bloch wave functions onto Wannier functions, we obtain a tight-binding Hamiltonian with 104 bands from 3d, 4s, 4p orbitals of Co and 3s, 3p orbitals of Si, which we use for efficient evaluation of the CPGE photocurrent.
To implement the CPGE integrals in Eq. (4), the Brillouin zone was sampled by k-grids from 200 × 200 × 200 to 960 × 960 × 960 [43]. Satisfactory convergence (less than 2% change) was achieved for a k-grid of size 400 × 400 × 400. The temperature dependence is implemented by the Fermi-Dirac distribution function and we also include a hot-electron lifetime broadening factor. CoSi is in space group P 2 1 3 (#198), with point group 23 (T ). Owing to the two-fold glide rotation symmetry s 2x , s 2y , s 2z , only diagonal tensor elements are nonzero, and the C 3 rotation symmetry further leads to a single independent component β xx = β yy = β zz . We carefully checked the symmetry of numerically calculated tensor elements with the tensor shape given by lattice symmetry and found the errors to be within 10 −6 . The full circular photogalvanic effect tensor is given by where E mn ≡ E m − E n , f mn ≡ f m − f n are the difference of band dispersion and Fermi-Dirac distribution respectively, ∆ a mn ≡ ∂ ka E mn / , r a mn ≡ i m|∂ ka n is the interband transition matrix element or off-diagonal Berry connection. The finite relaxation time τ is considered via the Lorentzian function L τ (E nm − ω).
The CPGE follows j = βxx iΩ+1/τ E 2 0 (Ω), where the THz frequency Ω is comparable to 1/τ . When the hot-electron lifetime τ 1/Ω, which is the case for the current experiment, j ≈ β xx τ E 2 0 . When τ is much longer than the pulse width, which is in the quantization regime, dj/dt = β xx E 2 0 .

II. ADDENDUM
We thank C.L. Kane for helpful discussions and N. P. Armitage and J. Stensberg for proof-reading the manuscript. N.Z, X.H. and L.W. are supported by Army Research Office under the Grant W911NF1910342. L.W. and E.M. also acknowledge support by a seed grant from NSF MRSEC at Penn under the Grant DMR-1720530 for part of the instrumentation development. Research at the University of Maryland was supported by the Gordon