Abstract
The quantum limit is quite easy to achieve once the band crossing exists exactly at the Fermi level (E_{F}) in topological semimetals. In multilayered Dirac fermion systems, the density of Dirac fermions on the zeroth Landau levels (LLs) increases in proportion to the magnetic field, resulting in intriguing angle and fielddependent interlayer tunneling conductivity near the quantum limit. BaGa_{2} is an example of a multilayered Dirac semimetal with its quasi2D Dirac cone located at E_{F}, providing a good platform to study its interlayer transport properties. In this paper, we report the negative interlayer magnetoresistance induced by the tunneling of Dirac fermions between the zeroth LLs of neighboring Ga layers in BaGa_{2}. When the field deviates from the caxis, the interlayer resistivity ρ_{zz}(θ) increases and finally results in a peak with the applied field perpendicular to the caxis. These unusual interlayer transport properties are observed together in the Dirac semimetal under ambient pressure and are well explained by the model of tunneling between Dirac fermions in the quantum limit.
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Introduction
Topological semimetals have attracted tremendous attention due to their novel properties, such as high carrier mobility, large transverse magnetoresistance (MR), nontrivial Berry phase, and chiral anomaly^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32}. Quantum oscillations, from which the Berry phase, quantum mobility, and effective mass can be extracted, is a routine method in the transport measurement to study topological materials. Obvious Shubnikov–de Haas (SdH) oscillation has been observed in graphene and 3D topological semimetals, e.g. Cd_{3}As_{2}^{7,8,9,10,11,12}, Na_{3}Bi^{3,4,5,6}, TaAs family^{13,14,15,16,17,18,19,20,21,22,23,30,31,32}, and MoTe_{2}^{24,25,26,27,28,29}. A particular interesting case is that only the zeroth LL is occupied and no more quantum oscillations should be observed once the quantum limit is achieved. In 2D Dirac/Weyl system, the LL energy of the 2D Dirac/Weyl band can be described as \(\varepsilon =\pm \!{v}_{{\rm{F}}}\sqrt{2e\hslash B n }\) (n = 0, ±1, ±2...)^{33}. Once the Dirac/Weyl point locates at E_{F}, the quantum limit is easily reached when a small field is applied and the zeroth LL is always locked at E_{F}. Thus, the zeroth LL’s degeneracy increases dramatically with the field and the Dirac/Weyl fermions on the zeroth LLs participate in the interlayer magnetotransport directly through tunneling mechanism^{34}, inducing the exotic phenomena of negative interlayer magnetoresistance (NIMR) and interlayer resistivity peak as shown in α(BEDTTTF)_{2}I_{3} and YbMnBi_{2}^{34,35,36}. However, the 2D Dirac cone in α(BEDTTTF)_{2}I_{3} only exists under pressure and the NIMR is absent in YbMnBi_{2}^{34,35,36}.
BaGa_{2} is recently predicted to be a candidate of Dirac semimetals. The crystal structure of BaGa_{2} consists of Ga honeycombnet layers and Ba layers. Its quasi2D Dirac cone originates from Ga p_{z} orbitals with the Dirac point located at E_{F}^{37}, while Ba layer between two Ga honeycombnet layers act as the interlayer transport barrier. Thus, BaGa_{2} is believed as an ideal candidate of quasi2D Dirac material and it provides a good platform to investigate the unusual interlayer transport properties induced by the tunneling of Dirac fermions on the zeroth LLs. Figure 1a presents the schematic illustration of the Dirac fermion interlayer tunneling on the zeroth LLs in BaGa_{2}.
In this work, the quasi2D Dirac cone at K with the Dirac point located at E_{F} is confirmed by the firstprinciples calculations, angledependent dHvA oscillations and angleresolved photoemission spectroscopy (ARPES) measurements. The abovementioned NIMR and peak in angledependent interlayer resistivity are both observed clearly in BaGa_{2} for the first time and well explained by the model of Dirac fermion tunneling between the zeroth LLs^{34}.
Results
Materials characteristic and inplane transport properties
BaGa_{2} single crystals are grown by selfflux method (see the “Methods” section). The crystal structure consists of Ba planes and the Ga honeycombnet planes which shows 2D characteristic. The (00l) lattice plane and lattice parameters are determined by the Xray diffraction (XRD) measurements on BaGa_{2} single crystal and powder as shown in Fig. 1b and c. Both the inplane resistivity (ρ_{xx}) and the interlayer resistivity (ρ_{zz}) show metallic behavior (Fig. 1d). The ratio of ρ_{zz}/ρ_{xx} is about 26 at 2.5 K, indicating a moderate electronic structure anisotropy. The measurements of Hall resistivity ρ_{xy} reveal that the inplane dominant carrier is holelike with the carrier concentration and mobility estimated to be n_{h} = 4.28 × 10^{21} cm^{−3} and μ_{h} = 3277 cm^{2} V^{−1} s^{−1} at 2 K. The MR of BaGa_{2} reaches 400% and shows no sign of SdH oscillation (Fig. 1e).
dHvA quantum oscillations
dHvA oscillations at various temperatures with the magnetic field along [00l] direction is shown in Fig. 2a. Four fundamental frequencies F_{α} = 36.9 T, F_{β} = 56.6 T, F_{γ} = 356.0 T, F_{η} = 1862.0 T are confirmed to exist after fast Fourier transform (FFT) analysis of the oscillatory component ΔM as shown in Fig. 2b and c.
The dHvA oscillations can be described by the Lifshitz–Kosevich (LK) formula^{38}
where R_{T} = (λTμ/B)/sinh(λTμ/B), R_{D} = exp(−λT_{D}μ/B) and λ = (2π^{2}k_{B}m_{0})/(ℏe). μ is the ratio of effective mass m^{*} to free electron mass m_{0}, and T_{D} is the Dingle temperature. γ = 0, δ = 0 for a 2D system, and γ = 1/2, δ = ±/8 for a 3D system. β = ϕ_{B}/2π and ϕ_{B} is the Berry phase. The inset of Fig. 2c displays the temperaturedependent FFT amplitudes and fittings using the thermal factor R_{T} in LK formula. The obtained effective cyclotron masses are quite small, comparable with those in topological semimetals, due to the almost linear dispersion though they all originate from trivial bands lately determined by the magnetotransport analysis and firstprinciples calculations. In order to extract the Berry phase conveniently, we applied fourband LK formula fitting directly. The fitting result is displayed in Fig. 2d with the violet dots being experimental results and the red line being the fitting curve. All of the Berry phase of these four pockets greatly deviate from the nontrivial value π implying that these bands may be trivial. Detailed data is exhibited in Table 1. The angledependent dHvA oscillation measurements are applied to further reveal the Fermi surface of BaGa_{2} as exhibited in Fig. 3a. Figure 3b and c shows the FFT spectra of the corresponding dHvA oscillations with the B rotating from B//c to B//ab. The fundamental frequencies increase with the angle θ and vanish at θ = 90°, indicating the moderate electronic structure anisotropy of these trivial bands. If these trivial bands are highly 2Dlike, these frequencies should vanish once the magnetic field rotates by a small angle, which is inconsistent with the experimental measurements that F_{α}, F_{β}, and F_{γ} are still observable at θ = 56°. Besides, the twofold symmetry of polar plot MR (Fig. 3d) with B always vertical to I at 2.5 K and the ratio of ρ_{xx}(14 T, 0°)/ρ_{xx}(14 T, 90°) ≈ 5 also indicate a moderate electronic structure anisotropy in BaGa_{2}.
Firstprinciples calculations
The band structure and Fermi surfaces of BaGa_{2} calculated with spin–orbit coupling (SOC) effect included are shown in Fig. 4. As exhibited in Fig. 4a, the Dirac cone locates exactly at K point (E_{F}) in the Brillouin zone (BZ), which is consistent with previous work^{37}. In addition, there exist three trivial bands crossing the Fermi level with the corresponding FSs demonstrated in Fig. 4b–d. The holetype FSs (Fig. 4b and c) are openorbit FSs along the k_{z} direction exhibiting quasi2D characteristic. Notably, different from the ideal cylindrical Fermi surface the lotusroot shape holetype FS in Fig. 4b has two evident cross sections indicating the existence of k_{z} dispersion, which is consistent with the angledependent dHvA oscillations (Fig. 3). According to the Onsager relation F = (ϕ_{0}/2π^{2}) = (ℏ/2πe)A, the frequency F is proportional to the extreme cross section A of FS normal to the magnetic field. The calculated dHvA frequencies from the holetype Fermi surface Fig. 4b are 61.4 and 430.1 T corresponding to the observed frequencies F_{β} and F_{γ}. The calculated frequency from the other holetype Fermi surface Fig. 4c is 2273.4 T corresponding to the observed frequency F_{η}. The calculated largest frequency from the electrontype Fermi surface (Fig. 4d) is 4608.3 T which is too high to be observed at 2.5 K and 14 T. The observed frequency F_{α} may originate from the electrontype Fermi surface (Fig. 4d) as there is a branch along K–H according to the calculations. Thus, the trivial Berry phases obtained from the fitting are acceptable since they are all from the trivial bands. As for the Dirac band, no frequencies corresponding to the oscillation should be observed since it is located exactly at E_{F}. In addition to the Dirac cone, our firstprinciples calculations reveal that there is an electrontype pocket along L–H and two holetype pockets at Γ point in the BZ, which is slightly different from previous results in which there seems to be an extra analogous holetype pocket at H point and only one holetype pocket at Γ point^{37}. The difference is acceptable considering that different calculation methods are employed. Besides, in real materials the Fermi surface is affected by many factors, such as temperature, magnetic field, defects, etc. According to the calculations, the Dirac point locates at the highsymmetry point K and E_{F}. Although the Dirac point is not protected by the crystal symmetry and opens a small gap of ~10 meV by considering the SOC effect, the lowenergy excitation can still be described by the Dirac equation and possesses the characteristic of Dirac fermions.
ARPES measurement
The electronic structure of BaGa_{2} is measured by ARPES and the quasi2D Dirac cone at the K point is revealed. Figure 5a shows a schematic drawing of the 3D BZ and its projected surface BZ onto the (00l) surface measured. The FS is shown in Fig. 5b. The most obvious feature is a nearly circular pocket around the \(\overline{{\rm{K}}}\) point. Besides this pocket, there are several much weaker features around the \(\overline{\Gamma }\) and \(\overline{{\rm{K}}}\) points where several bands cross the Fermi energy. This can be seen more clearly in the ARPES spectra along the highsymmetry directions as shown in Fig. 5c. The experimental spectra show an overall agreement with the calculated dispersions at different k_{z} values. There also exist a few extra band dispersions near the Fermi energy (indicated by red circles in Fig. 5c) originating from surface states which is consistent with the firstprinciples calculations, as shown in Fig. 5d. Besides, the experimental dispersions which are away from Fermi energy show a mixture of calculated dispersions ranging from k_{z} = 0 and k_{z} = 0.5c^{*} (the dispersions at the k_{z} boundaries are colored black in Fig. 5d), suggesting a very strong k_{z}broadening^{39}.
We zoom in the dispersion near E_{F} to further reveal the Dirac cone at the K point. The dispersion at k_{y} = 0 is shown in Fig. 5f. Due to the matrix element effect^{40}, the left branch is much weaker than the right branch. To enhance the visualization of the dispersion on both sides, we show in Fig. 5g the normalized spectra by integrating each energy distribution curve (EDC). In addition to the Dirac cone at the K point, there is an extra dispersion originating from the surface state, which is consistent with the calculations. The evolution of Dirac cone along the k_{y} direction is shown in Fig. 5f. Moving away from the K point, the dispersion shifts down in energy and becomes more parabolic, in agreement with a Dirac cone at the K point as shown in Fig. 5a. The k_{z}dependence of the Dirac cone is shown in Fig. 5g. Similar behavior is also observed, with the dispersion shifting down in energy when moving away from K point (k_{z} = 2c^{*}). However, the k_{z} dispersion of the Dirac cone is much weaker than k_{y} dispersion and the Fermi velocity along k_{z} is almost one order of magnitude less than k_{y}, suggesting that the Dirac cone is quasi2D like centered at K point.
Interlayer transport properties
BaGa_{2} provides an ideal platform to study the unusual interlayer transport properties caused by the tunneling of Dirac fermions between the zeroth LLs since the Ga honeycombnet layer contributes the quasi2D Dirac cone and the Ba layer between adjacent two Ga honeycombnet layers provides the barriers.
Figure 6b presents the angledependent interlayer resistivity ρ_{zz}(B, θ) at 2.5 K under different fields. It shows the interlayer resistivity peak at θ = 90°. θ is the angle between B and I as defined in Fig. 6a. Besides, the interlayer resistivity decreases with the increasing field, resulting in the NIMR at θ = 0°, as shown in Fig. 6b, d and e. The NIMR decreases quickly when temperature changes from 2.5 to 70 K. Both of these anomalous interlayer transport properties can be understood based on the tunneling of Dirac fermions between the zeroth LLs. According to previous works^{34,36}, the interlayer Dirac fermion tunneling conductivity \({\sigma }_{{\rm{t}}}^{{\rm{LL}}0}({\bf{B}},\theta )\) can be described as
where A is a fieldindependent parameter, θ is the angle between B and I, and c is the distance of the adjacent two Ga layers. When the field is applied parallel to I along caxis (θ = 0°), the tunneling conductivity is simplified as \({\sigma }_{{\rm{t}}}^{{\rm{LL0}}}({\bf{B}},{0}^{\circ })\) = A\(\left{\bf{B}}\right\). In this case, the tunneling conductivity increases proportional to the magnetic field, leading to the NIMR as exhibited in Fig. 6d. When θ increases from 0° to 90°, the interlayer tunneling conductivity decreases gradually because of the suppression of 2D LL quantization, resulting in the interlayer resistivity peak as shown in Fig. 6b.
In an ideal 2D Dirac system, the interlayer transport is only contributed by the tunneling of Dirac fermions between the zeroth LLs (tunneling channel) and the resistivity exhibits the equivalent peak values at θ = 90° under different fields^{34,35}. However, as shown in Fig. 6b, the interlayer resistivity peak increases with the increasing field. The inconsistency is reasonable since there also exist trivial bands contributing the positive interlayer MR except for the Dirac band. As discussed in the angledependent dHvA oscillations, the results from firstprinciples calculations, the ratio of ρ_{xx}(14 T, 0°)/ρ_{xx}(14 T, 90°) ≈ 5, and ρ_{zz}/ρ_{xx} ≈ 26, the electronic anisotropy of these trivial bands are analyzed as moderate. Based on these discussion, we assume the trivial bands contribute to the interlayer transport through a momentum relaxation mechanism (i.e. coherent band transport) and describe it with Drude model. Considering both the tunneling mechanism and momentum relaxation mechanism, the total interlayer resistivity is described as
where σ_{c}(B, θ) is the conductivity from the coherent trivial bands. As shown in the Supplementary Table 1, all of the trivial bands do not reach the quantum limit, so the conductivity σ_{c}(B, θ) is taken as \({\sigma }_{0}/(1+k\cdot {B}_{xy}^{2})\) for low field approximation (σ_{0} is the Drude conductivity, k is a constant, B_{xy} = ∣B∣sin θ). B_{0} is a fitting parameter^{34}. At θ = 90°, the tunneling channel is suppressed [\({\sigma }_{{\rm{t}}}^{{\rm{LL0}}}({\bf{B}},9{0}^{\circ })\) = 0] and the interlayer resistivity mainly originates from the trivial bands. In such case, the interlayer resistivity is described as ρ_{zz}(B, 90°) = 1/σ_{zz}(B, 90°) = 1/(σ_{c}(B, 90°) + B_{0}). As shown in Fig. 6e the fitting curve matches well with the experimental data. Therefore, to a certain extent, the contribution from trivial bands to interlayer transport can be described by Drude model.
The total interlayer transport properties can be considered as the result of the competition between the tunneling mechanism and momentum relaxation mechanism. When θ is close to 0°, the trivial bandinduced positive MR is relatively small because the fieldinduced scattering is very weak (absence of Lorentz force). However, the density of states (DOS) at Dirac point increases dramatically with the increasing field since the Dirac cone is located exactly at E_{F}. In this case, the tunneling mechanism dominants the interlayer transport, resulting in the NIMR as shown in Fig. 6b, d and e. The angledependent and fielddependent unusual interlayer resistivity can be well fitted by Eq. (3), as shown in Fig. 6c and e. When θ increases from 0° to 90°, the tunneling of Dirac fermions between the zeroth LLs is suppressed while the trivial band induced positive MR increases due to the enhancement of the Lorentz forceinduced scattering. It is consistent with the observed interlayer MR, where the NIMR vanishes gradually and positive MR increases with θ varying from 0^{∘} to 90^{∘}. These resistivity curves are well fitted by Eq. (3) as shown in Fig. 6c and e. These anomalous interlayer transport properties induced by the tunneling of Dirac fermions between the zeroth LLs have also been observed in (BEDTTTF)_{2}I_{3}, where the Dirac cone only exists under pressure^{34,35}. YbMnBi_{2} is another quasi2D topological material that exhibits the same anomalous interlayer transport properties^{36}, in which the NIMR was not observed possibly due to the large positive MR induced by other bands overwhelming the signature of NIMR. To our knowledge, BaGa_{2} is the only material that exhibits both NIMR and interlayer resistivity peak at ambient pressure.
Discussions
To further understand the observed NIMR in BaGa_{2}, we discuss on the possibly different origins of the negative magnetoresistance (NMR) below. BaGa_{2} is a nonmagnetic material, spin flip may not be a suitable reason for the NIMR in BaGa_{2}^{41}. Secondly, the NMR has also been observed in topologically trivial materials at high field such as graphite, where the NMR is induced by the ellipsoidal Fermi surfaces approaching the quantum limit^{42}. As displayed in the Supplementary Table 1, all of the trivial bands in BaGa_{2} are not under the quantum limit with the field of 14 T. In addition, the NIMR in BaGa_{2} appears once a quite small field is applied. Thus, the scenario of trivial bands under quantum limit does not apply in BaGa_{2}. Thirdly, the NMR induced by chiral anomaly has been widely observed in 3D Dirac and Weyl semimetals^{5,9,12,21,22}, which usually has the form of Δσ_{xx} ∝ B^{2} or B depending on the Fermi energy^{43}. However, the Dirac cone in BaGa_{2} opens a small energy gap (~10 meV) when SOC is considered and it will not evolve into Weyl cones when time reversal symmetry is broken (applying the magnetic field). So the chiral anomaly is absent in BaGa_{2} when I//B. Besides, the inplane NMR is not observed when the field is parallel to the current in BaGa_{2} (Supplementary Fig. 1). Thus the scenario of chiral anomaly induced NMR does not apply either. Fourthly, current jetting can often cause the NMR^{44}, which has been explained in detail in ref. ^{45}. Specifically, it is caused by the inhomogeneous spatial distribution of the current in the sample^{22,45}. In order to eliminate the current jetting effect, the fourprobe AC transport method is applied in the interlayer transport measurement and the current probe with silver paste almost contact fully on the samples’ upper and lower surfaces (Supplementary Fig. 2) to make the current flow cross the sample homogeneously. Besides, the NIMR in BaGa_{2} decreases quickly with the increasing temperature, inconsistent with the current jetting effect^{44}. Finally, the NIMR can arise in highpurity layered metal PdCoO_{2} with the residual resistivity ρ_{xx} ranging from about 10 to 40 nΩ cm in most single crystals^{46}. As for BaGa_{2}, the residual resistivity ρ_{xx} and ρ_{zz} are about 440 nΩ cm and 11.44 µΩ cmwhich are larger than that of PdCoO_{2} at least one order of magnitude. So the condition 4t_{c} > ℏω_{c} (ε_{n} = (n + 1/2)ℏω_{c} + 2t_{c}cos(k_{z}c), t_{c} is interlayer transfer integral) describing the Landau levels for quasi2D Fermi surface under the magnetic field in the quasi2D metal material PdCoO_{2} may not be fulfilled here. Besides, the interlayer MR in PdCoO_{2} increases quickly at low field and decreases at high field when field deviates from caxis which is quite different from that of BaGa_{2}. From the above, we conclude that the interlayer NMR and the interlayer resistivity peak result from the tunneling of Dirac fermions between the zeroth LLs.
In summary, BaGa_{2} is a multilayered Dirac semimetal with quasi2D Dirac cone located at E_{F}, which is confirmed by the firstprinciples calculations, angledependent dHvA oscillations and ARPES measurements. The unusual interlayer transport properties (NIMR and peak in angledependent interlayer resistivity) originating from the tunneling of Dirac fermions on the zeroth LLs are both observed in BaGa_{2} under ambient pressure for the first time. Our findings highlight the unusual role of the zeroth LLs in interlayer magnetotransport and enrich the novel transport properties of topological materials at quantum limit.
Methods
Crystal growth and magnetotransport measurements
The single crystal BaGa_{2} was grown by selfflux method. The ingredient ratio with Ba:Ga = 41.7:58.3 were put into the alumina crucible and sealed into a quartz ampoule. Then the ampoule was heated to 950 °C and kept for 50 h to make sure the materials have melted and mixed thoroughly. The process of crystal growth was carried up in cooling at 3 °C/h to 700 °C. The BaGa_{2} single crystal was obtained after centrifuging the excess flux at that temperature. The atomic composition of BaGa_{2} was checked by energydispersive xray spectroscopy (EDS, Oxford XMax 50). The crystal structure was determined by XRD in Bruker D8 Advance xray diffractometer. TOPAS4.2 was employed for refinement. The electric transport and magnetic properties were collected in Quantum Design PPMS14 T and MPMS3 SQUID VSM system.
Angleresolved photoemission spectroscopy
ARPES measurements were taken at BL13U of Hefei National Synchrotron Radiation Laboratory and our home laboratory. The crystals were cleaved in situ and measured at a temperature of T ≈ 15 K in vacuum with a base pressure better than 1 × 10^{−10} torr.
Band structure calculations
The electronic structures of BaGa_{2} were studied by using firstprinciples calculations. The projector augmented wave (PAW) method^{47,48} as implemented in the VASP package^{49,50,51} was used to describe the core electrons. The SCAN type exchangecorrelation potential was adopted^{52}. The kinetic energy cutoff of the planewave basis was set to be 400 eV. A 20 × 20 × 18 kpoint mesh was utilized for the BZ sampling and the Fermi surface was broadened by the Gaussian smearing method with a width of 0.05 eV. Both cell parameters and internal atomic positions were fully relaxed until the forces on all atoms were smaller than 0.01 eV/A. Once the equilibrium structures were obtained, the electronic structures were calculated with the SOC effect. The Fermi surfaces were studied by using the maximally localized Wannier functions (MLWF) method^{53,54}. To study the surface states, a 1 × 1 twodimensional supercell with an BaGa_{2} slab containing 59 atoms and a 20 Å vacuum was employed to simulate the BaGa_{2} (001) surface.
Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information. Extra data are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank T. Qian, G.F. Chen, and H.C. Lei for helpful discussions. This work is supported by the National Natural Science Foundation of China (Nos.11874422, 11574391, 11725418, 11774422, 11774424), the Ministry of Science and Technology of China (Nos. 2019YFA0308602, 2019YFA0308603, 2017YFA0302903) and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Nos. 18XNLG14, 19XNLG18, 19XNLG13). Computational resources were provided by the Physical Laboratory of High Performance Computing at Renmin University of China. The Fermi surfaces were prepared with the XCRYSDEN program^{55}.
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T.L.X. coordinated the project and designed the experiments. S.X. and Y.Y.W. synthesized the single crystals of BaGa_{2} with the assistance of Q.H.Y., L.L.S. and Y.S. C.B. and S.Z. performed the ARPES measurements. P.J.G., K.L. and Z.Y.L. performed ab initio calculations. S.X. performed the magnetotransport measurements with the assistance of Y.Y.W. S.X., C.B. and P.J.G. plotted the figures and analyzed the experimental data. S.X., Y.Y.W., and T.L.X. wrote the paper. All authors discussed the results and commented on the manuscript.
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Xu, S., Bao, C., Guo, PJ. et al. Interlayer quantum transport in Dirac semimetal BaGa_{2}. Nat Commun 11, 2370 (2020). https://doi.org/10.1038/s41467020158540
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DOI: https://doi.org/10.1038/s41467020158540
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