Abstract
Recently, Weyl semimetals have been experimentally discovered in both inversionsymmetrybreaking and timereversalsymmetrybreaking crystals. The nontrivial topology in Weyl semimetals can manifest itself with exotic phenomena, which have been extensively investigated by photoemission and transport measurements. Despite the numerous experimental efforts on Fermi arcs and chiral anomaly, the existence of unconventional zeroth Landau levels, as a unique hallmark of Weyl fermions, which is highly related to chiral anomaly, remains elusive owing to the stringent experimental requirements. Here, we report the magnetooptical study of Landau quantization in Weyl semimetal NbAs. High magnetic fields drive the system toward the quantum limit, which leads to the observation of zeroth chiral Landau levels in two inequivalent Weyl nodes. As compared to other Landau levels, the zeroth chiral Landau level exhibits a distinct linear dispersion in magneticÂ field direction and allows the optical transitions without the limitation of zero z momentum or \(\sqrt B\) magnetic field evolution. The magnetic field dependence of the zeroth Landau levels further verifies the predicted particlehole asymmetry of the Weyl cones. Meanwhile, the optical transitions from the normal Landau levels exhibit the coexistence of multiple carriers including an unexpected massive Dirac fermion, pointing to a more complex topological nature in inversionsymmetrybreaking Weyl semimetals. Our results provide insights into the Landau quantization of Weyl fermions and demonstrate an effective tool for studying complex topological systems.
Introduction
Dirac semimetals can be viewed as special Weyl semimetals with spin degeneracy^{1}. The lifting of the Weyl nodes degeneracy can be accessed by breaking inversion symmetry (IS) in Dirac semimetals. It was recently realized in noncentrosymmetric compounds NbAs family^{1,2,3,4,5,6,7,8,9,10,11,12,13}. Such a Weyl semimetal phase is featured by Fermi arcs and chiral anomaly, both of which were intensively investigated^{6,7,8,9,10,11,12,13,14,15,16}. The Fermi arcs with unique spin texture have been observed by photoemission on the surfaces of Weyl semimetals^{7,910,12,13,14}. The opencontour feature distinguishes themselves from traditional surface states and results in novel cyclotron orbits under magnetic fields^{15}. Meanwhile, the predicted chiral or Adlerâ€“Bellâ€“Jackiw anomaly, a chirality imbalance phenomenon in Weyl semimetals, has been widely observed^{16,17,18,19,20,21,22,23,24}, although the origin of some experiments is still under debate^{25,26}. Besides the Fermi arcs and chiral anomaly, one of the most important features of Weyl semimetals is the presence of the zeroth chiral Landau level, which is linearly dispersed in K_{ z } direction^{27,28,29}. It was theoretically predicted to be a hallmark of Weyl fermions that leads to the widely observed chiral anomaly in Weyl semimetals^{16,17,18,19,20,21,22,24}. However, directly detecting this distinct zeroth Landau level through conventional photoemission or transport experiments remains an unprecedented challenge since both magnetic field and Landau level spectroscopy are required^{27}. Also, experimentally approaching the quantum limit is strictly demanded to distinguish the chiral Landau level from other degenerated Landau levels.
Here, we investigate the quasiparticle dynamics in Weyl semimetal NbAs by means of magnetooptics. The ability to detect the Landau levels via the optical reflectance under high magnetic fields enables the observation of the zeroth chiral Landau levels and complex topological nature near the quantum limit. Two sets of inequivalent Weyl cones with different Fermi velocity were identified through \(\sqrt B\) and \(\sqrt n\) dependence of the optical transition frequency. The intraband transition edges were observed as a direct consequence of the chiral Landau levels in both of inequivalent Weyl nodes. The dramatic distinction from other Landau levels, however, is the linear dispersion in k_{ z } (magneticÂ field)Â direction, resulting in a unique field evolution of transition energy. Detailed study of the transitions between the chiral and normal Landau levels reveals a large particlehole asymmetry in Weyl cones with higher Fermi velocity which agrees well with the predictions from firstprinciple calculations^{30}. Finite scattering helps to enhance the feature from transition edge^{27}. Also, corroborated with our theory, five kinds of optical transitions involving the chiral and conventional Landau levels are unambiguously distinguished, which construct the whole band structure of NbAs under magnetic field as summarized in the end. Besides these two inequivalent Weyl nodes, we also reveal the coexistence of massive Dirac fermions, as well as trivial band insulating state, suggesting a more complex topological nature in the Weyl semimetal family. The comparison among different optical transitions in NbAs exhibits the uniqueness of the chiral Landau levels. In the following, we will describe these five optical transitions and distinguish their origins one by one, which show the clear contribution from the chiral Landau levels. The comprehensive study of Landau level spectroscopy in NbAs provides insights into multiple novel topological carriers and constitutes an ideal experimental tool for studying complex quasiparticles with their own topological nature.
Results
Singlecrystal NbAs
NbAs crystalizes in bodycentered tetragonal unit cell with space group I4_{1}md and point group C_{4V}^{31}, as shown in Fig.Â 1a. Since this type of structure lacks a horizontal mirror plane, the IS in NbAs is naturally broken, which leads to the lift of the spin degeneracy, necessary for developing the Weyl semimetal phase. Ab initio calculations predict 24 Weyl nodes in the first Brillouin zone of NbAs^{1,7}, where they always come in pairs with opposite chiralities or monopole charges. The studied single crystals were grown by transport method^{32} and their crystalline orientation and chemical ratio were examined by Xray diffraction (XRD) and electrondispersive Xray spectroscopy (EDX), respectively (Fig.Â 1b and Supplementary FigureÂ 1). Three prominent peaks can be clearly identified as (004), (008), and (0012) crystal planes^{31,33}. The lattice constant is calculated to be aâ€‰=â€‰3.45â€‰Ã… and câ€‰=â€‰11.68â€‰Ã… which is consistent with the previous studies^{31}. The stoichiometry of the sample confirmed by the EDX ensures the comparatively low Fermi level and less undesired defects^{34}.
Magnetooptical spectrum
Magnetoinfrared measurements have been proved to be an effective tool for study Landau quantization for topological materials that can provide rich information, such as band parameters and field response^{35}. The measurements of magnetoinfrared spectroscopy are combined with magnetotransport, where the important parameters are extracted that help to understand Landau levels. Here we take magnetooptical spectrum on the (001) crystal surface of NbAs as shown in Fig.Â 1b inset. The schematic experimental setup of the magnetooptical measurements is displayed in Fig.Â 1c. Faraday geometry were used in a superconducting magnet with the magnetic field up to 17.5â€‰T. More experimental details are available in the method section. As shown in Fig.Â 1d, the reflectivity spectra were normalized by the zerofield data. A series of reflection features can be resolved. The reflection maxima and minima systematically shift towards the higher energy with increasing magnetic field. Given the high mobility of NbAs at low temperatures, the fielddependent peaks can be assigned to the Landau level transitions. Although all of the peak energies increase with field, certain differences are discernible. In FigsÂ 1d and 2a, some of the peak energies increase linearly with the magnetic field (denoted as Mtype in dark green) while the others do not. The observed difference indicates the distinct origins of lowenergy excitations in the system. For a classical system with a quadratic energy dispersion Îµâ€‰=â€‰Ä§^{2}k^{2}/2m*, the Landau quantization under magnetic field reads
where n,e,Ä§ represent the Landau index, the elementary charge and the reduced Plankâ€™s constant, respectively. Optical selection rules require \(\Delta n = n_i  n_f = \pm 1\), where n_{ i },n_{ f } denote the Landau level index of the initial and final states. The Landau level transition with the Ttype and Ctype features are in stark contrast to the classical picture, while the Mtype peaks fit well to the linearinB rule (Fig.Â 2a). The notations C/T/M are named after several terms: Chiral Landau level, normal Transition, and Multiple bands, which will be discussed in the following sections. All these salient features are reproducible as demonstrated in Supplementary NoteÂ II from another NbAs sample of the same batch with an identical growth condition.
Origin of the interLandaulevel transitions
To quantitatively study the spectrum and understand the origin of each transition, we extracted the frequency of interLandaulevel transitions and plotted it against the magnetic field in Fig.Â 2a. Generally, Landau level transition induces a peak in the optical conductivity. In order to extrapolate the transition energy, an absolute reflection spectrum with gold overcoatingÂ technique is required (Supplementary NoteÂ III). Then the experimental data were fitted to the Drudeâ€“Lorentz model (Supplementary NoteÂ X) in the presence of magnetic field^{36,37,38,39}. The fitted curve in Fig.Â 2b shows a good consistency with the original experimental data for all magnetic fields (Supplementary FigureÂ 8 in Supplementary NoteÂ X). The extracted transition energy typically locates nearby but with a lower energy than where the apparent peak is positioned in the original data (especially for Ttype and Mtype transitions). The interLandaulevel transition energy in Fig.Â 2a is obtained from the aforementioned fitting scheme.
Fan diagram reveals rich features that can be used to determine the electronic structure of NbAs under magnetic fields. A series of transition energies follow a straight line pointing to the origin of the plot on a scale of \(\sqrt B\), which provides key information for the Ttype transitions as shown in Fig.Â 2a. Massless Weyl fermions are described by the Hamiltonian \(H({\mathbf{k}}) = v_{\mathrm{F}}{\mathbf{k}} \cdot \sigma\) (Ä§â€‰=â€‰1 for notation simplicity) where Ïƒâ€‰=â€‰(Ïƒ_{ x },Ïƒ_{ y },Ïƒ_{ z }) and v_{F} represent the Pauli matrices and the Fermi velocity, respectively. With the introduction of a magnetic field along z direction, the formed Landau levels read^{18},
The lowest Landau level is of chiral nature which makes it quite different from the higher Landau levels. Based on Eq. (2), it is readily found that the density of states is given by
Î£ is a sum symbol. From Eq. (3) and Fig.Â 2c, the singularity occurs at k_{ z }â€‰=â€‰0 where the density of state becomes infinite. Therefore, the optical transition energy should follow \(\sqrt B\) and \(\sqrt n\) law for Weyl semimetals with nonzero index. Experimentally, all the Ttype transitions clearly adopt theÂ \(\sqrt B\) dependence, indicating the Landau level transition from nonzero index (illustrated by red arrows in Fig.Â 2c), and thus serving as a direct evidence of ultrarelativistic fermions^{29,40,41,42,43}. Note that all the schematic plots of Landau level dispersion are also applicable to the holedoped Weyl semimetals because the optical transition is schematically symmetric, i.e., it has the same absorption energy for the electron/hole doping with the same carrier density.
We now assign the specific Landau level index n and verify the \(\sqrt n\) dependence. Deduced from Eq. (2) and the optical selection rules, the resonance frequency^{42} or the slope of T1â€“T7 in Fig.Â 2a, designated as S1S7, should be proportional to \(\sqrt {n + 1} \pm \sqrt n\), where the positive (negative) sign is for interband (intraband) transition. Apparently, T1â€“T7 do not simply follow this rule and the observed optical transition cannot be explained by the Landau quantization of a single ideal Weyl cone. In fact, we have noticed that the ratio of the adjacent slopes keeps nearly a constant, i.e.,Â \(S7/S6 \approx S6/S5 \approx S5/S4 \approx S3/S2\). This clearly shows signatures of two inequivalent Weyl nodes. By further calculatingÂ \(S7:S6:S5:S3 \approx (\sqrt 4 + \sqrt 5 ):(\sqrt 3 + \sqrt 4 ) :(\sqrt 2 + \sqrt 3 ):\) \((\sqrt 1 + \sqrt 2 )\), these four transitions can be identified to originate from one Weyl node, denoted as Weyl node 1 (W1). Then the peak assignments can be made as follows: T3 corresponds to \(L_{  1} \to L_2\) or \(L_{  2} \to L_1\) and T5 \(L_{  2} \to L_3\) or \(L_{  3} \to L_2\), and so on so forth for T6 and T7. Here L_{ n } represents the n^{th} Landau level. Note that only \(L_{  n} \to L_{n + 1}\)Â is presented with labels for simplicity. These assignments can be further verified by the fan diagram in Fig.Â 2d, where \(\sqrt {n + 1} + \sqrt n\) dependence can be clearly seen. The assignments for the second Weyl node (W2, Fig.Â 2e) involving T6, T5, T4, and T2 can be made in the same manner as that for W1. Therefore, we elucidate the existence of two inequivalent Weyl nodes and the Landau level energy is indeed proportional to \(\sqrt B\) and \(\sqrt n\). By performing a linear fit, the Fermi velocity is derived to be v_{F1}â€‰=â€‰2.0â€‰Ã—â€‰10^{5}â€‰m/s and v_{F2}â€‰=â€‰1.7â€‰Ã—â€‰10^{5}â€‰m/s, which matches well with the previous firstprinciple calculations^{30}. Weyl nodes 1 and 2 are illustrated in red and blue in Fig.Â 2, respectively. It is worth mentioning that T5 and T6 include the transitions from both Weyl nodes and, therefore, are labeled in purple. This overlapping is a natural consequence of two inequivalent Weyl cones as shown in Supplementary NoteÂ X. Usually for the system close to quantum limit, the interband transition with higher index should show smaller amplitude. Here the comparatively higher amplitude of T5, T6 can be explained by the overlapping of these close transitions. The larger width of T5 and T6 originates from same mechanism. To distinguish more clearly how those different electronic bands influence the optical transitions, Fig.Â 2a is replotted and divided into several panels for further clarification in Supplementary NoteÂ IV and Supplementary FigureÂ 4. All the observed features originated from different bands are also summarized in the end.
Quantum limit for inequivalent Weyl nodes
The existence of resonant peaks corresponding to \(L_{  1} \to L_2\) or \(L_{  2} \to L_1\) suggests that the system is close to the quantum limit. The Fermi velocity and the energy offset between two inequivalent Weyl nodes are also suggested by angleresolved photoemission spectroscopy^{9}. Different from interband transitions, which always come in pairs (blue and red for two Weyl nodes), as shown in Fig.Â 2a in the low frequency regime, there exists only one linearin\(\sqrt B\) transition (T1), which leads us to check the position of Fermi level and the quantum limit. Therefore, magnetotransport measurements were carried out with a Hallbar device on the same sample with the same geometry as the magnetooptical measurements. Figures.Â 3a, b present the longitudinal MR in different field regimes. Clear Shubnikovde Haas (SdH) oscillations can be resolved on a large upward background. The ultimate quantum limit is reached near a field of 18â€‰T similar to the previous magnetic torque measurements^{15}. A large MR reaches 60,000%, which features the compensated material and topological massless fermion with the suppressed backscattering^{44}. We extracted the peak positions of the resistivity and assigned the integer index. Two sets of SdH oscillations were found as shown in Fig.Â 3c (also, arrows in different colors in Fig.Â 3a, b). It is noted that the closed cyclotron orbit under magnetic field follows the Lifshitzâ€“Onsager quantization rule \(S_{\mathrm F}\frac{\hbar }{{eB}} = 2\pi \left( {n + {\textstyle{{1  \phi _{\mathrm B}} \over 2}}} \right)\), where S_{F},Ï•_{B} are Fermi surface area and Berry phase, respectively. Performing the linear fit in the fan diagram yields the intercepts, which are nearly zero, indicative of the topological nontrivial Berry phase. The nontrivial topology of the oscillation components evidences two gapless Weyl nodes. Consistent with the previous reports^{45,46}, our Hall effect measurements suggest a typical twocarrier transport behavior (Supplementary NoteÂ V and Supplementary FigureÂ 5) and the quantum limit is reached with magnetic field 19â€‰T and 12â€‰T for the inequivalent Weyl nodes. The fact that one of the intraband transitions is missing can now be explained by the entrance of the quantum limit.
Chiral Landau levels
It has a profound influence on intraband transitions when the system reaches the quantum limit. As shown in Fig.Â 3d, different from the discussed Ttype or Mtype transitions, there remains three other peaks denoted as C1 to C3, which follow neither linearinB nor linearin\(\sqrt B\). Since all the allowed transitions with nonzero index have been exclusively identified at high energy, the remaining C1 to C3 are anticipated to come from the intraband transitions between chiral L_{0} and L_{Â±1}. The linearly dispersed zeroth Landau level in Weyl semimetal is unique compared with traditional systems. It has been used to explain the fourfold splitting in magnetooptical spectrum of ZrTe_{5} and was further studied by circular polarization resolved experiments^{42,47}. It also differs from other topological materials such as Kane semimetal HgCdTe^{48} or Dirac semimetal graphene^{49}, where a flat zeroth Landau level along k_{ z } locates at zero energy regardless of various magnetic fields, resulting in an energy gap between the zeroth and the first Landau level. A previous magnetooptical study also suggests Cd_{3}As_{2} to be Kane semimetal partly due to the lack of chiral Landau level^{29}. The transitions involving the zeroth Landau level should occur at k_{ z }â€‰=â€‰0 in Kane or 2D Dirac semimetals while a fixed zeroth Landau level should result in a linearin\(\sqrt B\) relation for \(L_0 \to L_1\) ^{43,42,4849,}. As to Weyl semimetals, the 3D structure with Weyl cones ensures a different form of zeroth Landau level, as described by Eq. (2), with unique optical features that agree well with the observed Ctype transitions in the following four aspects. Firstly, for the lowest energy intraband transition, it does not necessarily occur at k_{ z }â€‰=â€‰0 due to the lack of density of state maxima. Instead, the allowed transition has a certain frequency edge where the density of states experiences a sudden increase from zero when the Fermi level crosses the zeroth Landau level as shown in Fig.Â 3e, f (colored arrows). Another fact that needs toÂ be pointed out is that in the presence of impurity scattering, the edge for the transition involving chiral Landau level will be enhanced and the peaks for interband transitions will be suppressed^{27} (see details in Supplementary NoteÂ XV). Thus, it is reasonable to expect that the peak due to the transition edge has a similar feature with comparable amplitude as the feature from interband transitions. Secondly, the energy of the normal Landau level transitions, for example T1â€“T7, is determinedly by the distance between Landau levels at k_{ z }â€‰=â€‰0, which is proportional to \(\sqrt B\). Varying the Fermi level can only allow or abandon certain optical transitions, but it cannot affect the transition energy. For the linearly dispersed chiral Landau level, however, the transition edge energy is determined by the Fermi level as well, resulting in non\(\sqrt B\) field evolution. Thirdly, to observe the optical feature of the zeroth Landau level transition, it requires the system near the quantum limit (i.e., lowcarrierdensity sample under high magnetic fields). For the system far from that, the transition energy rapidly decreases to a vanishing value beyond our experimental resolution (Supplementary NoteÂ VI and Supplementary FigureÂ 6, the experimental challenge of probing chiral Landau level). Fourthly, the optical behavior involving the zeroth Landau levels can be significantly influenced as the system approaches the quantum limit. Here, in our field regime, one of the Weyl nodes does not approach the quantum limit. Therefore, the Landau level intersects with the first Fermi level resulting in three intraband transition edges as displayed in Fig.Â 3e. The upper two have the exactly same energy (Supplementary NoteÂ VIII). For those two transitions at same k_{ z }, the lower one is the transition between chiral Landau level and 1^{st} Landau level while the upper one is the transition between 1^{st} and 2^{nd} Landau level. Further increasing the magnetic field enlarges the transition energy (Fig.Â 3d, red line). As long as the system reaches the quantum limit (Fig.Â 3f), the right edges will disappear since the intersection between Fermi level and L_{1} no longer exists. The behavior of C3 transition fits well to the discussed scenario and the peak disappears near 12T.
Next, we performed a quantitative study for these intraband transitions. Previously, the firstprinciple calculations reveal that NbAs is particlehole asymmetric^{30}. To capture the effect of particlehole asymmetry in the massless system, we can add a diagonal term to the lowenergy Hamiltonian that describes the Weyl fermions with different Fermi velocity on conduction and valence band^{30} by
where Î±â€‰=â€‰1,2 corresponds to the Weyl node reaching or not reaching the quantum limit, and Î»_{ Î± } denotes the effect of asymmetry. For Î»_{ Î± }â‰ 0, the conduction and valence bands are asymmetric. For simplicity, we assume Î·_{1}â€‰=â€‰Î·_{2}â€‰=â€‰1 and Î·_{3}â€‰=â€‰0. Similar to the previous derivation of Landau level, we consider that a magnetic field is exerted in z direction, then the Landau level reads
As shown in Supplementary NoteÂ VII, the frequency influenced by the asymmetry is always orders of magnitude lower for the interband transition frequency. However, the effect on the intraband transition cannot be neglected. We define the energy of the low (the left arrow in Fig. 3e and the arrow in Fig.Â 3f) and high (the bottomright arrow in Fig.Â 3e) transition edges as Ï‰_{0}â€‰â†’_{â†’}â€‰_{1,1} and Ï‰_{0}â€‰_{â†’}â€‰_{1,2}. For Ï‰_{0}â€‰_{â†’}â€‰_{1,1}, the Fermi level intersects with chiral Landau level, therefore \(\omega _{0 \to 1,1} = \left. {E_{1,\alpha }  \left {E_{\mathrm{F}}} \right} \right_{v_\alpha k_z = \lambda _\alpha \sqrt {eB}  \left {E_{\mathrm{F}}} \right}\) can be calculated. For the high energy edge Ï‰_{0}â€‰_{â†’}â€‰_{1,2} from the Weyl node not in the quantum limit, the Fermi level intersects with the first Landau level. In the same manner, k_{ z } is firstly determined and then \(\omega _{0 \to 1,2} = \left {E_{\mathrm{F}}} \right  E_{0,\alpha }\) can be calculated. Using the similar approach for the first and second Landau levels, we can calculate Ï‰_{12} for T1 as well. The energy of the transition edge becomes a function of E_{F,}v,Î»,B. All details including the derivation of equations and data fitting are available in Supplementary NoteÂ VII. For a given band parameter, the position of Fermi level determines the quantum limit. Therefore, E_{F} is no longer a free parameter for the data fitting. C2 transition can be best fitted only by higher energy edge Ï‰_{0}_{â€‰â†’}â€‰_{1,2} as shown in Figs.Â 2a and 3d (fitting curves plotted as red solid line), which means that C2 is originated from the chiral Landau level in W1 (not in quantum limit). The fitting parameters are \(v\prime1 = 2.2 \times 10^5\)â€‰m/s and Î»_{1}â€‰=â€‰1.1â€‰Ã—â€‰10^{5}â€‰m/s which agree well with the previouslydiscussed origin of W1. When putting \(v\prime _1,\lambda _1\) into Ï‰_{0}_{â€‰â†’}â€‰_{1,1} and attempting to extract the position of the lower edge from W1, the calculated curve is almost overlapping with T1 as shown in Figs.Â 2a and 3d (red solid line).
For C1 from W2, the best fit for Ï‰_{0}â€‰_{â†’}â€‰_{1,1} gives \(v\prime_2 = 1.7 \times 10^5{\kern 1pt} {\mathrm{m/s}}\) and Î»_{2}â€‰=â€‰0.26Ã—10^{5}â€‰m/s. The fitted curve is plotted as lower blue solid line in Fig.Â 3d. By examining the slope of T1 and Ï‰_{12}, the particlehole asymmetry parameter is determined to be Î»_{1}â€²â€‰=â€‰1.0Ã—10^{5}â€‰m/s. Note that the equations in tables and figures use the Weyl model without particlehole asymmetry for simplicity (see more details in Supplementary NoteÂ VIII). Using the fitted parameters from W2, one can predict the positions of higher frequency edge Ï‰_{0â€‰â†’â€‰1,2} from W2 as shown by the upper blue lines in Fig.Â 3d. The experimentally extracted C3 peaks fit well to the predicted curve. Therefore, the C3 peak comes from the higher frequency edge transition Ï‰_{0â€‰â†’â€‰1,2} from W2. In both Weyl nodes, the frequency of higher edge increases faster with magnetic field than the lower edge. As also determined by the transport measurements, W2 reaches the quantum limit near 12â€‰T. Since the intersect between the Fermi level and the first Landau level is gone, it is reasonable to find that C3 peak disappears near 12â€‰T.
The fitting parameters v,Î» further verify the anticipated origin of the intraband transitions accompanied by the following independent evidences: Firstly, experimental data points from all chiral Landau level transitions fit well with the magnetic field evolution as predicted by theory^{27,29} that clearly differs from the contribution of other bands; Secondly,Â the extracted velocity v from C1 and C2 is very close to the value given by interband transitions where v is acquired by \(\sqrt B\) and \(\sqrt n\) fitting. The edge behavior agrees well with our analysis for different situations (reaching the quantum limit or not); Thirdly, the extracted particlehole asymmetry generally agrees with the features predicted by previous density functional theory (DFT) calculations^{30}, where the Weyl node with higher Fermi velocity possesses a much stronger particlehole asymmetry. Our conclusion of Î»_{1}>Î»_{2} accords well with this prediction. The difference of Fermi velocity between particle and hole bands in previous DFT is 92% and 6% for W1 and W2^{30}, respectively, while as a comparison our Landau level spectroscopy gives 53% and 15%. With such a quantitative analysis, we indeed verify the existence of the chiral Landau level and particlehole asymmetry in NbAs. To quantitatively describe all 10 features (C1â€“C3 and T1â€“T7) in the spectra, it requires only two parameters of v and Î» with a simple Weyl particle picture as described by Eq. (4).
Discussion
Other than C/Ttype transition, there exist several Mtype transitions at higher energy (also see the midinfrared spectra in Supplementary NoteÂ IX and Supplementary FigureÂ 7), which follow linearinB evolution and clearly possess positive intercepts as shown in Fig.Â 4a. These features usually indicate a gapped structure corresponding to massive fermions^{50,51} that could coexist with Weyl band. Distinct from M4â€“M5 transitions in Figs.Â 2a and 4a where the slope ratio between the adjacent ones is 2:1 (reasonable in classical quadratic band). However, a very large slope ratio is found between M1 and M2. To understand this, we have provided the main physics of an ISbreaking Weyl semimetal by a fourband minimal model whose form takes
where v_{x,y,z} represent the velocities, m and C are two constants, and Î” denotes the ISbreaking term. With the introduction of magnetic field in z direction, the Landau levels corresponding to the two gapped bands follow
where \(m(n,k_z) = m  Ck_z^2  2eBC(\left n \right + \frac{1}{2})\), and here we have assumed v_{ x }â€‰=â€‰v_{ y }â€‰=â€‰v_{ z } for simplicity. From topological insulators, we understand that the band inversion is the key to the realization of many exotic topological phases^{44} and the nearly zero slope feature of the M1 transition indicates the transition energy does not change much with magnetic field, which suggests the identical change of the initial and final Landau levels. Thus, it is of great significance to distinguish the inverted band structure from the normal band structure in experiments. The corresponding Landau fan diagrams are shown in Fig.Â 4b, c, where the zeroth Landau levels for the normal case and inverted case are quite different. For the normal case with quadratic dispersion (Fig.Â 4c), the energies of all Landau levels in particle (hole) branch increase (decrease) monotonically with the magnetic field and the slope ratio of transition \(L_{  1} \to L_0\) and \(L_{  2} \to L_1\) should be around 2. However, for the inverted case, the energy of the zeroth Landau level always shows an opposite monotonic behavior to other Landau levels belonging to the same branch. Consequently, the lowest spectrum corresponding to the optical transition \(L_{  1} \to L_{0^ + }\) or \(L_{0^{\mathrm{  }}} \to L_1\) can be very flat in a certain magnetic field regime leading to a large M2/M1 ratio. Based on the theoretical calculations, the spectra in Figs.Â 2a and 4a with finite zerofield intercepts can be understood. M1 and M2 are from a gapped band with band inversion. As shown by Fig.Â 4c, e and Supplementary NoteÂ XI, M4 and M5 are originated from normal (noninverted) bands. The overall understanding of the Landau quantization from multiple bands are summarized in Fig.Â 5 and TableÂ 1.
Here we provide a general understanding of zeroth Landau levels in topological materials. In twodimensional Dirac semimetals, such as graphene or the surface state of threedimensional topological insulators, the lower dimension makes Landau levels (including the zeroth one) nondispersed along k_{ z } directions. Therefore, a gap will be generated between adjacent Landau levels. This effect is most prominent between the zeroth and the first Landau level. The transition energy thus follows \(\sqrt B\). However, in conventional bulk materials, the zeroth Landau level is parabolically dispersed along k_{ z }. Although the gap vanishes, density of state diverges at k_{ z }â€‰=â€‰0, resulting in the resonant energy (between the zeroth and the first Landau level) following B. In the bulk state of topological insulator and the inverted gap state observed in NbAs, while the optical transition is still limited at k_{ z }â€‰=â€‰0, the energy of the zeroth Landau level decreases with magnetic field due to the band inversion. In Weyl semimetal, the zeroth chiral Landau level is linearly dispersed in k_{ z }, resulting in two distinct features: optical transition away from k_{ z }â€‰=â€‰0; and the transition frequency and momentum position highly dependent on the Fermi level. The transition energy is a function of both magnetic field and Fermi level.
In conclusion, we have studied the quasiparticle dynamics with contributions from multiple bands in NbAs through magnetooptical spectroscopy. All the optical features are summarized in Fig.Â 2a, Fig.Â 5, TableÂ 1, and Supplementary NoteÂ IV. Taking the advantage of probing Landau levels under high magnetic fields, unconventional optical transitions reveal the existence of the zeroth chiral Landau level. Different from other degenerated Landau levels with nonzero index, it unconventionally disperses linearly in k_{ z } direction and allows the optical transition without the limitation of k_{ z }â€‰=â€‰0 or \(\sqrt B\) evolution. Finite scattering helps to enhance the feature from chiral Landau level. Besides, two inequivalent Weyl nodes are found with different Fermi velocity. Fitting the magnetic field dependence of optical transitions unveils that the Weyl node in NbAs with higher Fermi velocity possesses much stronger particlehole asymmetry. Massive Dirac fermions and massive trivial fermions are also found by observing the linearinB cyclotron resonance. We summarize the overall Hamiltonian in Supplementary NoteÂ XIII and also provide a general picture of the zeroth Landau levels in topological materials. A comparison of the Landau quantization between Weyl fermions and massive fermions was made to further verify the unique feature of the zeroth chiral Landau levels.
Method
Sample growth
NbAs single crystals were grown by enhanced chemical vapor transport method using iodine as the agent with tilted ampules angle to maximize the growth rate and crystal quality. A temperature gradient is required to provide the necessary driving force for gaseous species in diffusion and connection between the cooler and hotter end. The stoichiometry and crystal structure were examined by XRD (Bruker D8 Discover) and EDX in a scanning electron microscopy.
Magnetooptical measurements
The farinfrared and midinfrared reflection were measured in a Faraday configuration with a superconducting magnet up to 17.5â€‰T. NbAs samples were exposed to the globar infrared light through light pipes with (001) surface perpendicular to both the incident light and magnetic field. Infrared light was focused with parabolic cone. The reflected light was detected by a bolometer and analyzed by a Fourier transform infrared spectrometer. All the light tube, samples and bolometer were kept at liquid helium temperature in a cryostat. The light path was pumped under vacuum to avoid the absorption of water and other gases.
Zerofield reflection measurements
Zerofield reflection spectrum was measured with E // (001) surface at liquid helium temperature on a combination of Bruke 113â€‰V and 80â€‰V. An insitu overcoating technique was used as reference.
Magnetotransport measurements
Magnetotransport measurements were carried out using physical property measurement system and pulsed magnetic field up to 60â€‰T at Wuhan National High Magnetic Field Center. A standard Hallbar geometry is made. Lockin amplifiers were used to measure the electrical signals.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11474058, 61674040, 11674189) and the National Key Research and Development Program of China (Grant No. 2017YFA0303302, 2016YFA0203900, 2017YFA0303504). We thank Ying Wang, Xue Jiang, Zhiqiang Li, and Nanlin Wang for helpful discussions. H.Y. is also grateful to the financial support from Oriental Scholar Program from Shanghai Municipal Education Commission. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR1157490, DMR1644779, and the State of Florida. Part of the sample fabrication was performed at Fudan Nanofabrication Laboratory.
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F.X. conceived the idea and supervised the experiments. Z.L. & X.C. carried out the growth. X.Y., C.Z., C.S., W.W., Y.L., J.L., D.S., T.X. and H.Y. performed the magnetooptical experiments. C.S. and H.Y. did magnetoDrudeâ€“Lorentz fitting. M.Z. carried out the zerofield reflection measurements. X.Y., C.Z., Y.L., X.Z. and Z.X. conducted magnetotransport experiments. Z.Y. and Z.W. performed theoretical calculations. X.Y., Z.Y., C.Z. and F.X. wrote the paper with help from all other authors.
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Yuan, X., Yan, Z., Song, C. et al. Chiral Landau levels in Weyl semimetal NbAs with multiple topological carriers. Nat Commun 9, 1854 (2018). https://doi.org/10.1038/s41467018040804
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DOI: https://doi.org/10.1038/s41467018040804
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