Abstract
The Weyl semimetal (WSM), which hosts pairs of Weyl points and accompanying Berry curvature in momentum space near Fermi level, is expected to exhibit novel electromagnetic phenomena. Although the large optical/electronic responses such as nonlinear optical effects and intrinsic anomalous Hall effect (AHE) have recently been demonstrated indeed, the conclusive evidence for their topological origins has remained elusive. Here, we report the gigantic magnetooptical (MO) response arising from the topological electronic structure with intense Berry curvature in magnetic WSM Co_{3}Sn_{2}S_{2}. The lowenergy MO spectroscopy and the firstprinciples calculation reveal that the interband transitions on the nodal rings connected to the Weyl points show the resonance of the optical Hall conductivity and give rise to the giant intrinsic AHE in dc limit. The terahertz Faraday and infrared Kerr rotations are found to be remarkably enhanced by these resonances with topological electronic structures, demonstrating the novel lowenergy optical response inherent to the magnetic WSM.
Introduction
The materials hosting the topological electronic band structure potentially exhibit exotic or enhanced electromagnetic responses^{1,2,3,4,5,6,7,8}. The intrinsic anomalous Hall effect (AHE), which essentially differs from the scatteringprocess mediated extrinsic AHE, is a representative example of the topological transport phenomena, where the Berry curvature arising from the electronic band topology plays a decisive role for the Hall conductivity^{9}. The intrinsic AH conductivity and AH angle are particularly enhanced by tuning the Fermi level to the (anti)crossing point in the electronic band structure with Berry curvature^{10}. Accordingly, the topological materials with nontrivial crossing points near their Fermi levels are attracting much attention as the ideal platforms for exploring the large AHE, accelerating the extensive material research^{11,12,13,14,15,16,17,18,19,20,21}. One important advance is the recent discovery of the Weyl semimetal (WSM) possessing pairs of nondegenerate crossing points, i.e., Weyl points^{1,14,15,16,17,20}. The pairs of Weyl points are characterized by the opposite chiralities and act as the monopole and antimonopole of the emergent magnetic field in momentum space. The large AH conductivity and AH angle exceeding even 1000 Ω^{−1} cm^{−1} and 10%, respectively, are indeed reported for prospective WSMs^{7,8,19,21}. However, because of the lack of the conclusive experimental proof, the role of the topological electronic structure for the large AHE remains elusive.
The intrinsic AHE has been considered to exhibit the prominent feature in the optical Hall conductivity spectra σ_{xy}(ω); the interband optical transition on the topological electronic structure with the Berry curvature gives rise to the resonant structure in σ_{xy}(ω)^{22,23,24,25,26}. We introduce an archetypal theoretical model for understanding the optical Hall conductivity spectra owing to the intrinsic AHE (Fig. 1a)^{23}; the twodimensional electronic structure has a single anticrossing point with the mass gap 2 m, and the chemical potential μ is defined as the energy distance from the Weyl point. The vertical transition near the Weyl point causes the resonance peak at the energy of 2 μ in σ_{xy}(ω), which corresponds to the lowest energy of the optical transition (Fig. 1a, inset; see also Supplementary Note 1). Im σ_{xy}(ω) and Re σ_{xx}(ω) (Supplementary Fig. 1) show the step functionlike spectra at 2 μ^{9,24}, indicating the dissipative response above the interband transition threshold. On the other hand, Re σ_{xy}(ω) and Im σ_{xx}(ω) represent the dissipationless responses, both of which show the sharp peaks and take finite value even below 2 μ. At ω = 0, the lowerlying tail of the resonance in Re σ_{xy}(ω) equals to the Hall conductivity obtained by the DC transport measurement, so that the resonance intensity is responsible for the magnitude of AH conductivity. This spectral feature of the intrinsic mechanism is clearly distinguished from that of the extrinsic ones dominated by the intraband scattering. Therefore, the optical Hall conductivity spectra can provide the fingerprint of the intrinsic AHE.
Another intriguing aspect is the possibly giant magnetooptical (MO) Faraday/Kerr effect, i.e., the rotation of the lightpolarization of the transmitted/reflected light for the magnetic media^{27,28}, resulting from the enhanced Hall conductivity on the optical transition near the Weyl point; the magnitude of these MO effects is scaled by the optical Hall conductivity. This novel optical property inherent to the magnetic WSM remains to be explored.
In this work, we study the intrinsic AHE in terms of the MO response in the recently discovered magnetic WSM Co_{3}Sn_{2}S_{2} by using terahertz Faraday/infrared Kerr spectroscopy on the thin film/bulk single crystal and firstprinciples calculation. The thin film allows us to measure the optical response of the metallic compound around the terahertz region with high accuracy, where the lowenergy limit of conduction electron dynamics can be examined. The observed lowenergy resonance structure in the optical Hall conductivity is consistent with the DC AHE and the theoretical calculation, manifesting the intrinsic AHE arising from the topological electronic band structure. The MO Faraday and Kerr effects are largely enhanced by the interband transition on the topological electronic structures near the Fermi level, exemplifying the novel optical functionality of the topological materials.
Results
Magnetic Weyl semimetal
Co_{3}Sn_{2}S_{2}, a Cobased shandite material, is a ferromagnetic metal with the Curie temperature T_{C} of ~175 K^{7,8,20,21,29,30,31}. The crystal structure has a hexagonal lattice belonging to the space group R\(\bar 3\)m with forming a kagome network of Co atoms within the Co_{3}Sn layer (Fig. 1b). Figure 1c reproduces the band structure of Co_{3}Sn_{2}S_{2} calculated by using the density functional theory (DFT) without/with the spin–orbit interaction, in accord with the literature^{7,8}. In the case without the spin–orbit interaction, the spinup band possesses the linear band crossings along the ΓL and LU directions above and beneath the Fermi level. Those crossing points are connected in threedimensional reciprocal space, forming the nodal ring structure protected by a mirror symmetry of the crystal structure (Fig. 1d, left). When the spin–orbit interaction is taken into account, this nodal ring structure almost opens a gap with anticrossing lines except for the pairs of the Weyl points (Fig. 1d, right). These topological electronic states including the Fermi arc are observed in the recent ARPES study, demonstrating that Co_{3}Sn_{2}S_{2} is a novel example of the magnetic WSM^{20}.
The giant AH conductivity and AH angle are observed in Co_{3}Sn_{2}S_{2}, both of which are categorized in the largest class among the known compounds^{7}. Figure 1e, f shows the magneticfield and temperature dependence of the Hall conductivity, respectively, for the bulk single crystal in the magnetic field parallel to the c axis. Below T_{C}, which is discerned also as the anomaly in the temperature dependence of the resistivity (Fig. 1f), the AHE rapidly grows up and reaches 1300 Ω^{−1} cm^{−1} at the lowest temperature (Fig. 1e, f). The very similar transport properties are also confirmed for the thin film used in this study^{21} (Supplementary Fig. 2). Several recent studies suggested that the observed large AHE is attributed to the intense Berry curvature associated with the Weyl points and anticrossing line structures^{7,8}.
MO effects
To address the optical transition related to these topological electronic structures, we measured the terahertz Faraday rotation at 1–8 meV for the thin film and the infrared Kerr rotation at 0.08–1 eV for the bulk single crystal. Figure 2 shows the temperature dependence of the terahertz Faraday and infrared Kerr effects for the c plane at zero field after the fieldcooling procedure; the field cooling is performed with applying the magnetic field along the c axis from 200 K (>T_{C}), which stabilizes the single ferromagnetic domain state due to the easyaxis anisotropy along the c axis (Fig. 1e; see also Methods and Supplementary Fig. 3). The optical rotation owing to the terahertz Faraday and infrared Kerr effects grows up below T_{C}. With decreasing the temperature, as a whole, the terahertz Faraday rotation θ_{F} is once enlarged and then reduced (Fig. 2a), whose lowenergy limit (1.38 meV) shows the similar temperature dependence to the DC Hall angle (Supplementary Fig. 4). In the Faraday ellipticity η_{F} spectra, the negative slope as a function of the energy is observed, developing at low temperatures (Fig. 2b). Meanwhile, in the infrared region, where the interband optical transitions across the Weyl points are anticipated to occur, the magnitude of the Kerr rotation shows the monotonous increase while keeping the spectral characteristics unchanged. The Kerr rotation spectra θ_{K}(ω) show the pronounced negative peak structure at 0.1 eV (Fig. 2c) and the Kerr ellipticity spectra η_{K}(ω) show two negative peak structures at 0.15 and 0.3 eV (Fig. 2d). The rotation angles θ_{F}(ω) and θ_{K}(ω) reach 160 mrad (~9.4 deg) and 57 mrad (~3.2 deg) at maximum, respectively, both of which are remarkably large as discussed later.
Longitudinal and Hall conductivity spectra
We show the longitudinal optical conductivity σ_{xx}(ω) and the optical Hall conductivity σ_{xy}(ω) for the comparison with the firstprinciples calculation as well as with the DC AHE (Fig. 3a–d). The σ_{xx}(ω) spectra for the bulk single crystal were deduced through the Kramers–Kronig transformation of the reflectivity spectra at 20 K (Fig. 3a; see also Supplementary Fig. 5). The interband transitions including two peak structures at 0.2 (circle) and 0.6 eV (square) are observed in Re σ_{xx}(ω) in addition to the strong zerofrequency peak due to the Drude response of conduction electrons below 0.02 eV (Fig. 3a). In fact, the Drude response is dominant for σ_{xx}(ω) observed in the terahertz region (Supplementary Fig. 6). Apart from the Drude component, the overall spectral characteristics of the interband transitions are well reproduced by the theoretical calculation as shown in Fig. 3b; two broad peaks are identified at ~0.2 and 0.6 eV, as indicated by the open circle and square, respectively. The recent optical study argues that the lowerlying peak at 0.2 eV is mainly the transition among Co 3d orbital in nearly the same spin channel while the higherlying peak at 0.6 eV is associated with the transition from Co 3d t_{2g} orbital to e_{g} orbital^{31}. Some fine structures discerned in the calculated spectra (Fig. 3b) presumably result from the small damping of the optical transition assumed in the calculation. It is noted that, to consider the electron correlation effect, we introduced the renormalization factor of 1.52 and that the energy was divided by this factor for all theoretical spectra; this treatment is verified by the recent ARPES study, where the DFT calculation with the renormalization factor of 1.43 well reproduces the experimental band structure^{20}.
The spectral responses in the MO Faraday/Kerr rotation are argued in a unified manner by the complex optical Hall conductivity spectra σ_{xy}(ω). The σ_{xy}(ω) spectra (Fig. 3c), which were deduced from the MO Faraday/Kerr rotation and σ_{xx}(ω) spectra (see Method), are totally distinct from the longitudinal optical conductivity σ_{xx}(ω) (Fig. 3a). Re σ_{xy}(ω) increases below ~0.3 eV towards zero photon energy with the sign change at ~0.2 eV (red line, Fig. 3c) and approaches to the lowermostenergy value in the terahertz region (red line, inset of Fig. 3c) or equivalently to the DC value, indicating the monotonous rise of Re σ_{xy}(ω) below 0.1 eV (red dotted line, Fig. 3c). On the other hand, Im σ_{xy}(ω) gradually increases towards zero frequency below ~0.4 eV (blue line, Fig. 3c) and converges to zero in the terahertz region (blue line, inset of Fig. 3c) in accord with the causality constraint Im σ_{xy}(ω = 0) = 0; therefore, the peak structure is inferred to position below 0.1 eV (blue dotted line, Fig. 3c). These spectral characteristics are consistent with the theoretical spectra that take the interband transition into account (Fig. 3d). Re σ_{xy}(ω) gradually increases towards zero frequency below ~0.4 eV with forming the peak structure near zero frequency, while Im σ_{xy}(ω) shows the broad peak structure around ~0.1 eV. Accordingly, the broad interband transition below 0.4 eV observed in both the experiment and theory is responsible for the DC AHE, evidencing the intrinsic origin. The possible extrinsic AHE is excluded because the spectral weight is distributed in a broader energy range than that of the Drude response with a scattering rate of ~3 meV in σ_{xx}(ω) (Supplementary Fig. 6). The interband transition below 0.2 eV can be ascribed mainly to the optical transition across the Weyl points and anticrossing lines in Co_{3}Sn_{2}S_{2}; in fact, the lowenergy spectra of total σ_{xy}(ω) (red and blue) almost coincide with the bandresolved σ_{xy}(ω) (pink and light blue) arising from interband transition between the two bands composing those topological electronic structures (bold curves in Fig. 3e), as shown in Fig. 3d. Thus, the optical Hall conductivity spectra demonstrate their vital role in producing the large AHE as well as the gigantic MO effect.
It should be emphasized that the broad spectral width of resonances in σ_{xy}(ω) is naturally expected from the dispersive anticrossing lines traversing the Fermi level in Co_{3}Sn_{2}S_{2} (Fig. 1c). As shown in Fig. 1a, a single anticrossing point gives rise to the sharp resonance peak in the optical Hall conductivity. On the other hand, the crosssections of the dispersive anticrossing lines at different k points can be viewed as the twodimensional twoband model (Fig. 1a) with different optical transition energies (Fig. 3f, inset). The integration of the optical response in k space thus results in the continuum resonance band in optical Hall conductivity σ_{xy}(ω) (Fig. 3f). The anticrossing point near the Fermi level has a larger resonance peak if assuming the constant mass gap. Therefore, the anticrossing lines traversing the Fermi level results in the lowenergy peak of σ_{xy}(ω) with the large spectral weight, producing the large AHE at zero photon energy, i.e., DC AHE.
Discussions
As for the observed MO effect in this WSM compound, it is worth noting that both of the Faraday and Kerr rotations are proportional to the Hall angle, σ_{xy}/σ_{xx} = tanΘ_{H}(ω); θ_{F} + iη_{F} = Z_{0}σ_{xy}d/(1 + n_{s} + Z_{0}σ_{xx}d) ~ tanΘ_{H}(ω) for thin film and θ_{K} + iη_{K} = −tanΘ_{H}(ω)/ε_{xx}^{1/2} for bulk surface, where n_{s}, Z_{0}, and ε_{xx} are the refractive index of the substrate, vacuum impedance, and the dielectric constant, respectively (See also Methods). In the present case, the real part of tanΘ_{H}(ω) shows the maximum as large as 0.5 at ~0.1 eV (Fig. 4a) due to the optical transitions on the anticrossing lines and Weyl points near the Fermi level.
The Faraday rotation observed at zero field, which exceeds 160 mrad (3.8 mrad/nm), is larger than that of the conventional ferromagnets represented by the archetypal MO material Bi:YIG with 0.77 mrad/nm^{25,26,32,33,34,35} (Fig. 4b). We note that the difference in the refractive indices for right and left circularly polarized light Δn and the figure of merit defined by ωΔnd_{p}/2c, where d_{p} is the penetration depth, are more appropriate for the quantitative discussion of the Faraday rotation, because the magnitude of the Faraday rotation for thin films depends on the configuration of film; the Δn and ωΔnd_{p}/2c of the present material take large values ~10.6 and 451 mrad, respectively at 7.5 meV (Supplementary Note 2 and Supplementary Table 1). Moreover, the large infrared Kerr rotation has never been reported for the (ferro)magnetic metals because it should be necessarily suppressed by the divergence of ε_{xx}^{1/2} due to the Drude response of the conduction electron towards zero frequency (Fig. 4c) as suggested by the relation that θ_{K} + iη_{K} = −tanΘ_{H}(ω)/ε_{xx}^{1/2}. In fact, the peak magnitudes of the Kerr rotation angle for several ferromagnetic metals tend to be small in lowenergy regions^{23,25,26,36,37,38,39,40} (Fig. 4d). In Co_{3}Sn_{2}S_{2}, by contrast, the Kerr rotation is enhanced even in the farinfrared region beyond this tendency owing to the large Hall angle originating from the topological electronic structure. These strong MO responses exemplify the novel optical property inherent to the magnetic WSM. It should be noted that these topological MO responses are in contrast with the usual plasmaedge enhancement, in which the reduction of ε_{xx} around the plasma edge increases the MO signal from nearinfrared to visible regions^{39}.
In conclusion, we have studied the MO responses in magnetic WSM Co_{3}Sn_{2}S_{2}. Our work reveals two intriguing aspects of the WSM, the optical Hall conductivity associated with the AHE and the enhanced MO effects, both of which originate from the Berry curvature of the topological electronic structures. The comprehensive study based on the MO spectroscopy and firstprinciples calculation clearly establishes the lowenergy electron dynamics arising from the topological electronic structures, which is the source of the large intrinsic AHE. One other important aspect is that the optical transition related to the Weyl point/anticrossing line produces the enhanced MO effects. This mechanism is essentially distinct from the conventional plasmaedge enhancement and applies to many other topological materials including even nonmagnetic Dirac or WSMs in general^{11,12,13,14,15,16,17,18}, promising future optical/electronic applications.
Methods
Single crystal growth
Single crystalline samples of Co_{3}Sn_{2}S_{2} were prepared by Bridgman methods. Co, Sn, and S were mixed in a stoichiometric ratio and then sealed in a quartz tube. Afterwards, it was heated to 1323 K and cooled down to 973 K with a rate of 4 mm per hour. The single phase with a shanditetype structure was confirmed by using the powder xray analysis.
Thin film fabrication
The 42nmthick caxisoriented Co_{3}Sn_{2}S_{2} film capped with a 50nmthick insulating SiO_{2} layer was grown by radiofrequency magnetron sputtering on Al_{2}O_{3} (0001) substrates. The crystal structure and composition were confirmed by xray diffraction and energydispersive Xray spectroscopy, respectively.
Transport measurement
The magnetoresistivity and Hall resistivity were measured by using Physical Property Measurement System (Quantum Design).
MO Kerr effect measurement in infrared regions
MO Kerr rotation spectra were measured with the use of a photoelastic modulator^{41}. The detection of the synchronous signal of the reflected light with the fundamental and second harmonic of the modulation frequency enables us to measure the Kerr rotation θ_{K} and ellipticity η_{K}, respectively. For the measurement, we performed the field cooling in the magnetic field ~70 mT using a permanent magnet from 200 K, resulting in the single magnetic domain state. During the measurement, the permanent magnet was removed. To deduce the Kerr rotation spectra, we antisymmetrized the spectra for the positive and negative magnetizations.
Optical (longitudinal) conductivity σ_{xx}(ω) and Hall conductivity σ_{xy}(ω) spectra in infrared regions
The optical conductivity spectra were deduced through the Kramers–Kronig transformation of the reflectivity spectra from 0.01 to 5 eV. For the extrapolation of the reflectivity data, we used the Hagen–Rubens relation below the lowest energy measured and assumed that the reflectivity is proportional to ω^{−4} above the highest energy. The optical Hall conductivity spectra were calculated by using the following formula; σ_{xy}(ω) = −σ_{xx}(ω)ε_{xx}^{1/2}(ω)(θ_{K}(ω) + iη_{K}(ω)).
Terahertz timedomain spectroscopy (THzTDS)
In the THzTDS, laser pulses with a duration of 100 fs from a modelocked Ti:sapphire laser were split into two paths to generate and detect THz pulses using the photoconductive antenna. Transmittance spectra were obtained by measuring the transmission of both the sample and substrate. We used the following standard formula to obtain the complex conductivity σ_{xx}(ω) = σ_{1}(ω) + iσ_{2}(ω) of the thin film;
where t(ω), d, Z_{0}, and n_{s} are the complex transmittance, the thickness of the film, the vacuum impedance (377 Ω), and the refractive index of the sapphire substrate, respectively. As the reference, we used the bare sapphire substrate to deduce the optical conductivity of the Co_{3}Sn_{2}S_{2} thin film. We note that the terahertz conductivity of SiO_{2} cap layer is negligibly small compared with that of the Co_{3}Sn_{2}S_{2} thin film and therefore the SiO_{2} layer hardly contributes to the terahertz spectra (Supplementary Fig. 7).
Terahertz Faraday rotation measurements
The rotatory component of the transmitted THz pulses, E_{y}(t), which is the electric field perpendicular to the incident electric field, E_{x}(t), was measured in the crossedNicole configuration by using wiregrid polarizers. To eliminate the background signal, we calculated the polarization rotation E_{y}(t) by antisymmetrizing the rotatory components under positive and negative magnetic fields. The Fourier transformation of the THz pulses E_{x}(t) and E_{y}(t) gives the complex Faraday rotation spectra θ_{F}(ω) + iη_{F}(ω) = tan^{−1}(E_{y}(ω)/E_{x}(ω)). For the antisymmetrization, we performed the field cooling in the magnetic field of ±1 T from 200 K, resulting in the single magnetic domain state^{21}. During the terahertz MO measurement, the magnetic field was absent.
Terahertz Hall conductivity
By using the longitudinal conductivity spectra σ_{xx}(ω) and Faraday rotation spectra θ_{F}(ω) + iη_{F}(ω), the Hall conductivity spectra were calculated from the following formula; \(\sigma _{xy}\left( \omega \right) = (\theta _{\mathrm{F}} + i\eta _{\mathrm{F}})\frac{{1 + n_s + Z_0d\sigma _{xx}\left( \omega \right)}}{{Z_0d}}\).
DFT calculation
The electronic structure of Co_{3}Sn_{2}S_{2} was calculated by using the OpenMX code^{42}, where the exchangecorrelation functional within the local spin density approximation^{43} and normconserving pseudopotentials^{44} were employed. The spin–orbit coupling was included by using total angular momentum dependent peudopotentials^{45}. The wave functions were expanded by a linear combination of multiple pseudoatomic orbitals^{46}. A set of pseudoatomic orbital basis was specified as Co6.0s3p3d3, Sn7.0s4p3d1, where the number after each element stands for the radial cutoff in the unit of Bohr and the integer after s, p, and d indicates the radial multiplicity of each angular momentum component. The cutoffenergy for charge density of 350.0 Ry and a kpoint mesh of 31 × 31 × 31 were used. The lattice constant of Co_{3}Sn_{2}S_{2} was set to a = 5.36 and c = 13.17 Å^{7}. The magnetic moment is evaluated as 0.9 μ_{B}/f.u. in this calculation.
Wannier representation and optical conductivities
From the Bloch states obtained in the DFT calculation described above, a Wannier basis set was constructed by using the Wannier90 code^{47}. The basis was composed of (s, p, d)character orbitals localized at each Co and Sn site and (s, p)character ones at S site, which are 106 orbitals/f.u. of Co_{3}Sn_{2}S_{2} in total if including the spin multiplicity. These sets were extracted from 194 bands in the space spanned by the original Bloch bands at the energy range from −20 to +50 eV.
The optical conductivity and optical Hall conductivity were calculated by using KuboGreenwood formula given by,
where e, \(\hbar\), Ω_{c}, N_{k}, η, \(f_{n,{\boldsymbol{k}}}\) are the elementary charge with negative sign, reduced Planck constant, cell volume, number of kpoint, smearing parameter, and the Fermi–Dirac distribution function with the band index n and the wave vector k, respectively. The \(\sigma _{\alpha \beta }\left( {\hbar \omega } \right)\) was calculated using the Wannierinterpolated band structure with a 100 × 100 × 100 kpoint grid and η = 20 meV. To consider the electron correlation effect, we introduced the renormalization factor of 1.52 phenomenologically and the energy was divided by this factor.
Data availability
The data that support the plots of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank N. Nagaosa for valuable comments and N. Ogawa and M. Sotome for experimental help. This work was partially supported by JSPS KAKENHI (Grant Nos. 19K14653, 16H06345) and JST CREST (JPMJCR16F1, JPMJCR1874 and JPMJCR18T2).
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Y.O. and Y. Takahashi conceived the project. Y.O. performed the optical measurement and analyzed the data with the assistance of J.M., Y. Kato, R.K., V.K., K.U., and Y. Taguchi. Y.F. and Y. Kaneko grew the single crystal and measured the transport properties with the assistance of N.K. J.I., K.F., and A.T. fabricated the thin film and measured the transport properties. S.M., T.K., and R.A. calculated the band structure and σ_{xx} and σ_{xy} spectra. Y.O., Y. Tokura, and Y. Takahashi discussed and interpreted the results with inputs from other authors. Y.O. and Y. Takahashi wrote the manuscript with the assistance of other authors.
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Okamura, Y., Minami, S., Kato, Y. et al. Giant magnetooptical responses in magnetic Weyl semimetal Co_{3}Sn_{2}S_{2}. Nat Commun 11, 4619 (2020). https://doi.org/10.1038/s41467020184700
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DOI: https://doi.org/10.1038/s41467020184700
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