Abstract
Topological semimetals hosting bulk Weyl points and surface Fermiarc states are expected to realize unconventional Weyl orbits, which interconnect two surface Fermiarc states on opposite sample surfaces under magnetic fields. While the presence of Weyl orbits has been proposed to play a vital role in recent observations of the quantum Hall effect even in threedimensional topological semimetals, actual spatial distribution of the quantized surface transport has been experimentally elusive. Here, we demonstrate intrinsic coupling between two spatiallyseparated surface states in the Weyl orbits by measuring a dualgate device of a Dirac semimetal film. Independent scans of top and backgate voltages reveal concomitant modulation of doublydegenerate quantum Hall states, which is not possible in conventional surface orbits as in topological insulators. Our results evidencing the unique spatial distribution of Weyl orbits provide new opportunities for controlling the novel quantized transport by various means such as external fields and interface engineering.
Introduction
Topological phases of matter hosting nontrivial surface states have enriched our understanding of transport phenomena^{1,2,3}. In contrast to topological insulators (TIs), where the surface transport can be isolated from the bulk one by tuning the Fermi level (E_{F}) within the bulk energy gap^{1,2}, topological Dirac and Weyl semimetals (DSM/WSM) have a characteristic feature that the gapless bulk and surface states merge at Weyl nodes^{3}, leading to unique interplay between them^{4,5,6,7}. One of the consequences is the formation of Weyl orbit under magnetic fields, which weaves together two spatially separated surface Fermiarc states via the bulk nodes^{4,5}. Owing to the completion of a closed cyclotron motion by the two orbital segments on the opposite surfaces, the Weyl orbit has been predicted to form Landau levels, leading to twodimensional (2D) quantum transport^{4,5,8}.
Experimentally, transport signatures of the 2D surface states have been reported by several groups^{9,10,11,12,13,14}, mainly for the typical DSM material Cd_{3}As_{2}. Distinct from the 2D bulk transport triggered by subband splitting under confinement^{15,16}, the 2D surface transport is typically accompanied by coexisting 3D bulk state, and is characterized by a larger Fermi surface and band mass than their bulk counterparts^{10,11,12,13,14}. It exhibits not only quantum oscillations, but also quantum Hall (QH) effect in high magnetic fields or at low carrier concentrations^{12,13,14}, attracting growing attention as novel quantized transport in a 3D system. On the other hand, an important question still under debate is whether the two opposite surface states are coupled through the bulk nodes in the quantized state. In particular, DSM with two Fermiarcs on the same surface may also host TIlike surface orbits independently localized on the opposite surfaces^{17,18} (Fig. 1a), in contrast to the Weyl orbits (Fig. 1b).
A Weyl orbit has two features distinct from the TIlike conventional orbit. One is the bulk tunneling process via the 1D bulk chiral mode formed parallel to the magnetic field. As proposed based on a semiclassical picture, the bulk tunneling process may be experimentally detectable as phase shift of QH plateaus, when the bulk thickness is modulated^{4,5,12}. It should be noted, though, that a similar plateau shift can be induced also by inhomogeniety of sheet carrier density, which varies proportionally to the thickness^{13,14}. Therefore, it is challenging to verify the existence of Weyl orbits, excluding extrinsic origins.
Here, we take a different approach by focusing on another feature of the Weyl orbit, namely its unique spatial distribution extending over two distanced sample surfaces. The key observation to distinguish the Weyl orbit from the TIlike orbit is the response of surface transport to electric fields applied in a dualgate device configuration. In the TIlike conventional case (Fig. 1a), each of the two surfaces independently hosts a 2D electronic state^{19,20,21}. The total QH filling factor ν can be expressed by simple summation of those of the two surfaces (ν_{1} and ν_{2}),
The field effect applied on each surface modulates ν_{1} or ν_{2} only independently, thus giving a checkerboard QH plateau pattern of ν as depicted in Fig. 1c^{21}. In the Weyl orbit case (Fig. 1b), on the other hand, the two Weyl orbits with opposite chirality are formed across the top and bottom surfaces, and the electron density of both orbits can be modulated simultaneously by top and/or backgating. If the two Weyl orbits are perfectly degenerate in the DSM phase, the total ν is given by
resulting in a stripe pattern extending along the constant sheet density line as shown in Fig. 1d. Here, we assume the halfinteger type QH effect for the Weyl orbits. This is because the surface dispersion of the two Fermiarcs of DSM is expected to possess a single Dirac point at the Brillouin zone center, owing to the nontrivial Z_{2} invariant as in TIs^{17,18}.
The difference in the appearance pattern of the QH states can be also described by the difference in the capacitive coupling of the surface carrier densities to the two gate voltages as presented in Fig. 1e. Denoting the carrier densities of the two surface orbits as n_{1} and n_{2}, and top and backgate voltages as V_{TG} and V_{BG}, their coupling relation is simply expressed by a 2 × 2 capacitance C matrix as follows,
Here, e is the elementary charge. Since the carrier densities of the two Weyl orbits can be modulated by both V_{TG} and V_{BG}, the offdiagonal term ideally equals to the diagonal term (C_{12}/C_{11} = 1) independent of the film thickness (Fig. 1e). This is different from the TI case, where the two surface orbits are perfectly isolated (C_{12}/C_{11} = 0) by the insulating bulk state in the threedimensionally thick limit^{21}. We note that reducing the film thickness in the TI case leads to finite coupling between the two surfaces (C_{12}/C_{11} ≠ 0). Even in such a case, the checkerboard pattern of the QH states (Fig. 1c) only deforms from a tetragonal to a diamond pattern^{21}, and is still distinct from the stripe pattern in the Weyl orbit case (Fig. 1d).
In this report, we demonstrate that the surface QH states in a 75 nm thick DSM (Cd_{1−x}Zn_{x})_{3}As_{2} film exhibit a clear stripe pattern in dualgate V_{TG}–V_{BG} scans. Our results indicate the intrinsic coupling of the two surfaces states even in a thick enough region, conclusively evidencing the unique distribution of the Weyl orbits extending across the entire film.
Results
Sample and quantum transport properties
Zndoped Cd_{3}As_{2} thin film was fabricated on a singlecrystalline SrTiO_{3} (100) substrate by the combination of pulsed laser deposition and solidphase epitaxy, following the same procedure described in previous papers^{15,22,23}. The film is oriented in the [112] crystallographic direction of Cd_{3}As_{2}, and the Fermiarc states on the film surfaces have finite length connecting between the projections of the bulk nodes residing on the [001] axis. Zndoping is for reducing the residual electron density of Cd_{3}As_{2}, and the doping concentration (x = 0.07) is designed to be low enough to maintain the nontrivial DSM phase (x < 0.2)^{14}. Figure 2a summarizes the magnetotransport of a 75 nm thick film measured at V_{TG}, V_{BG} = 0 V and temperature T = 3 K. Quantum oscillations in the resistance R_{xx} are clearly observed in both transverse (I ⊥ B) and longitudinal (I ∥ B) field configurations, which ensures that the film is thick enough to keep the 3D bulk state with gapless bulk nodes required for the formation of the Weyl orbit. At the electron density of 6 × 10^{17} cm^{−3}, the conduction is dominated mainly by the bulk transport, and the signature of the surface transport is barely observed without gating.
Once depleting the electron density by gating, on the other hand, the surface transport and its evolution into QH states become more evident. Figure 2b, c presents V_{TG} scans of R_{xx} and Hall resistance R_{yx} at 9 T with V_{BG} set to 0 and −20 V (see also Supplementary Fig. 1). In the lower panels, we also show R_{xx} curves measured with I ∥ B configuration and their second derivative with respect to V_{TG}. Plateau structures in R_{yx} accompanied by sharp drops of R_{xx} start to appear as the occupation of the bulk state is depleted to the quantum limit (see also Supplementary Fig. 3). When depleted by both V_{TG} and V_{BG}, the system crosses the charge neutrality point, changing the majority carrier type from n to ptype (Supplementary Fig. 2).
It is worth mentioning that even with bulk occupation, the QH plateaus with integer filling factors appear at the fields determined by B = n_{2D}h/eν with the total sheet carrier density n_{2D}, the elementary charge e, and the Planck constant h, as in a conventional 2D system. This feature indicates the unconventional involvement of bulk carriers in the formation of QH states, which may be behind the realization of quantized conduction even in the 3D system. It has been also discussed recently that the flatness of the Landau levels formed by the interacting bulk and surface states leads to a significant probability of the electron transport taking place at the sample boundary, enabling the quantization similar to a 2D system^{24}. In this sense, the conventional picture of parallel surface and bulk conduction established for TIs is not applicable here, because in such a case, the presence of bulk occupation would make the QH plateaus deviate from integer values and suppress the quantization^{20}.
Dualgate mapping of surface quantum Hall effect
Next, we present V_{TG}–V_{BG} mappings of the surface QH states measured at 9 T in Fig. 3. To explicitly show the oscillations in R_{xx} and plateau transitions in R_{yx}, we plot the second derivative of R_{xx} in Fig. 3a and the first derivative of R_{yx} in Fig. 3b. The most striking feature of the mappings is that the R_{xx} minima and R_{yx} plateaus of the QH states (corresponding to darker regions in Fig. 3a, b) extend over a wide range of V_{TG} and V_{BG} in a stripe pattern as shown in Fig. 1d. Moreover, the filling factor ν does not take every integer, but shows double degeneracy such as in the ν = 1, 3, 5 states. These results clearly indicate the presence of a pair of Weyl orbits. Each Weyl orbit has orbital segments on the top and bottom film surfaces so that both V_{TG} and V_{BG} can concomitantly modulate the surface QH states. Importantly, the V_{TG} and V_{BG} scans cover a sufficiently wide carrier density range (>10 eB/h at 9 T), allowing to distinguish the checkerboard (Fig. 1c) and the stripe (Fig. 1d) patterns. We also note that the finite curvature of the stripe patterns observed in Fig. 3a, b are caused by the saturating behavior of the gatemodulated carrier density around higher gate voltages.
In addition to the unique coupling between the opposite surface states, the appearance of QH filling factors further reveal sensitive responses of the Weyl orbits to the external electric and magnetic fields, reflecting their degeneracy lifting induced by the topological phase transition from DSM to WSM. In DSM, the application of external electric fields E (magnetic fields B) results in splitting of the bulk Dirac node into a pair of Weyl nodes with opposite chirality by breaking the inversion (timereversal) symmetry^{25,26,27}. In both the Edriven and Bdriven cases, a pair of Weyl nodes shifts oppositely both in the energy and momentum directions, depending on the underlying band structure^{25,27}. The direct consequence of these bulk node splitting effects is the degeneracy lifting of the two Weyl orbits with opposite chirality. The Landau levels of the two Weyl orbits split and cross with each other, depending on the magnetic and electric fields, leading to deviations from the doubly degenerate pattern shown in Fig. 1d.
Degeneracy lifting of Weyl orbits in Dirac semimetal
Experimentally, the effects of Edriven and Bdriven splitting can be confirmed separately in some scans. For example, in the region between the V_{TG} scans at V_{BG} = −5 and −20 V in Fig. 3a, b, one can notice that there are several level crossings (denoted by the white dashed lines in Fig. 3b), which result in additional plateau transitions. Figure 3c compares the typical V_{TG} scans of R_{yx} and its first derivative at V_{BG} = −5 and −20 V as a function of sheet density n_{2D}. Not only a series of odd integer ν, but also even integer ν such as ν = 6 and 8 can be observed, indicating the lifting of the double degeneracy. These results obtained by scanning the gate voltages at a fixed magnetic field can be interpreted as the manifestation of the Edriven splitting effect.
On the other hand, the Bdriven splitting of the Weyl orbits is observed by scanning the magnetic field. We present in Fig. 4a the mapping of the QH states as a function of magnetic field B and filling factor ν. It is evident that complex level crossings take place, resulting in the alternate appearance of odd and even integer ν, depending on the magnetic field. The externalfieldinduced bulk node splitting in the energy and momentum directions is reflected in the asymmetric modulation of the surface band dispersion of opposite chirality, including its energy position, Fermi surface area, and Fermi velocity (Fig. 4b). Taking these effects into account, we simulate the situation observed in Fig. 4a by assuming the Bdriven splitting effect appearing as an effective Zeeman term and the Edriven splitting effect as relative energy offset between the two Weyl orbits (see Supplementary Note 5 for details of the model). Choosing suitable parameters, the simulated fan diagram shown in Fig. 4c exhibits multiple Landau level crossings, and well accounts for the complex plateau transitions seen in Fig. 4a. We have also confirmed that the level crossing pattern in Fig. 4a changes, when measured with different V_{BG}, indicating the Edriven splitting of the two Weyl orbits in the gate configuration (see Supplementary Fig. 6).
Discussion
In previous studies of the surface QH states^{10,12,13,14}, observed QH filling factors and degeneracy greatly differ by experimental setups. This variation has been hindering the consistent understanding of the quantized transport. At this point, our mapping results in Figs. 3 and 4 indicate that the fieldinduced splitting of the paired Weyl orbits in DSM plays a critical role in determining the detailed appearance of the QH filling factors. We therefore suggest that the Weyl orbit degeneracy lifting sensitive to external fields may also explain the variation in the previous observations^{10,12,13,14}.
In summary, we have systematically investigated the surface QH states in a dualgated DSM film. Our results provide transport evidence of an intrinsic coupling between the two spatially separated surface states in the presence of Weyl orbits. The entire mappings of the QH states as functions of gate voltages and magnetic field further reveal the complex appearance of the filling factor, reflecting the degeneracy lifting of the paired Weyl orbits by external magnetic and electric fields. Considering that the unique spatial distribution of the Weyl orbit is implemented by electron tunneling via the bulk chiral mode parallel to the magnetic field, it would be interesting to further investigate how the connectivity between the two Fermiarcs on the opposite surfaces can be modulated, such as by applying tilted magnetic fields^{28} or by designing different sample geometries. Combined with conventional device fabrication techniques, such measurements are more feasible with film samples, and may provide an experimental answer also to the discussion, regarding the additional phase acquisition caused by bulk tunneling of the Weyl orbit^{4,5,8,12,13,14}. Fabrication of heterointerfaces for proximitizing the surface Fermiarcs with ferromagnets and superconductors, as has been in progress in different contexts for Cd_{3}As_{2} (refs. ^{29,30,31,32}), could also lead to novel transport physics and functionalities. In this respect, our work revealing the unique distribution of Weyl orbits in the quantized transport paves the way for further exploring the potential of exotic surface transport phenomena in topological semimetals.
Methods
Film growth and device fabrication
The 75 nm thick (Cd_{1−x}Zn_{x})_{3}As_{2} film was grown on a SrTiO_{3} (100) singlecrystalline substrate by the combination of pulsed laser deposition and solidphase epitaxy^{15,22,23}. By introducing a shadow mask during the deposition, the film was patterned into a Hall bar shape with a channel width of 60 μm. The deposited film was further capped by MgO (5 nm)/Si_{3}N_{4} (200 nm) to prevent its evaporation and oxidization in the subsequent thermal annealing at 600 °C in air. For the dualgate measurements, the topgate configuration was fabricated by first etching the topmost Si_{3}N_{4} capping layer down to 10 nm by ion milling, then depositing a 30 nm thick Al_{2}O_{3} dielectric layer by atomic layer deposition, and lastly depositing a 50 nm thick Au layer as topgate electrode. The backgate configuration, on the other hand, was fabricated by simply using the SrTiO_{3} substrate as backgate dielectric.
Lowtemperature transport measurement
The lowtemperature measurements up to 9 T were performed in a physical property measurement system (PPMS, Quantum Design). The electric current was kept along the [100] axis of the SrTiO_{3} substrate, corresponding to the [\(11\bar{1}\)] or [\(1\bar{1}0\)] axis of the (Cd_{1−x}Zn_{x})_{3}As_{2} film which has 90° inplane rotated domains formed by interfacial stacking of the (Cd_{1−x}Zn_{x})_{3}As_{2} hexagonal lattice on top of the SrTiO_{3} tetragonal lattice^{22}. All of the transport measurements with gate bias were performed using a conventional lockin technique. The excitation current was kept constant at 0.5 μA with the frequency set to 13 Hz. The V_{TG}–V_{BG} (V_{TG}–B) mapping was obtained by scanning V_{TG} with increasing V_{BG} (B) stepwise. The conversion of the V_{TG} axis to the sheet density n_{2D} axis is performed by referring to the lowfield Hall measurement at each V_{TG}.
Data availability
The data supporting the plots within the paper and its Supplementary Information file are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank N. Nagaosa, K. Kanoda, M. Tokunaga, K. Ishizaka, and K. Muraki for fruitful discussions. This work was supported by JST PRESTO Grant No. JPMJPR18L2 and CREST Grant No. JPMJCR16F1, Japan, and by GrantinAids for Scientific Research (B) No. JP18H01866 from MEXT, Japan.
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S.N., M.U., and M. Kawasaki designed the experiments. S.N., M.U., and Y.N. synthesized the bulk targets with M. Kriener and performed thin film growth. S.N. performed the dualgate measurements and analyzed the data with M.U. S.N. and M.U. wrote the manuscript with contributions from all authors. Y.T. and M. Kawasaki jointly discussed the results. M.U. and M. Kawasaki conceived the project. All authors have approved the final version of the manuscript.
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Nishihaya, S., Uchida, M., Nakazawa, Y. et al. Intrinsic coupling between spatiallyseparated surface Fermiarcs in Weyl orbit quantum Hall states. Nat Commun 12, 2572 (2021). https://doi.org/10.1038/s41467021229048
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DOI: https://doi.org/10.1038/s41467021229048
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