Observation of quasi-two-dimensional Dirac fermions in ZrTe5

Since the discovery of graphene, layered materials have attracted extensive interests owing to their unique electronic and optical characteristics. Among them, Dirac semimetal, one of the most appealing categories, has been a long-sought objective in layered systems beyond graphene. Recently, layered pentatelluride ZrTe5 was found to host signatures of Dirac semimetal. However, the low Fermi level in ZrTe5 strongly hinders a comprehensive understanding of the whole picture of electronic states through photoemission measurements, especially in the conduction band. Here, we report the observation of Dirac fermions in ZrTe5 through magneto-optics and magneto-transport. By applying magnetic field, we observe a square-root-B dependence of inter-Landau-level resonance and Shubnikov-de Haas (SdH) oscillations with non-trivial Berry phase, both of which are hallmarks of Dirac fermions. The angular-dependent SdH oscillations show a clear quasi-two-dimensional feature with highly anisotropic Fermi surface and band topology, in stark contrast to the 3D Dirac semimetal such as Cd3As2. This is further confirmed by the angle-dependent Berry phase measurements and the observation of bulk quantum Hall plateaus. The unique band dispersion is theoretically understood: the system is at the critical point between a 3D Dirac semimetal and a topological insulator phase. With the confined interlayer dispersion and reducible dimensionality, our work establishes ZrTe5 as an ideal platform for exploring exotic physical phenomena of Dirac fermions.

3 Layered materials, formed by stacking strongly bonded layers with weak interlayer coupling, have drawn immense attention in fundamental studies and device applications owing to their tunability in band structures and Fermi energy. 3,4,[11][12][13] Unlike other layered materials such as MoS2 and BN, graphene stands out as an appealing candidate as it is featured with a linear energy dispersion and low-energy relativistic quasi-particles. 9,14,15 Many exotic phenomena, such as half-integer quantum Hall effect 1, 2 and Klein tunneling 16 , have been realized in graphene. Along this line, extensive efforts were also devoted to explore new Dirac semimetal states in other layered systems beyond graphene. 5,6 Pentatelluride ZrTe5 with layered orthorhombic structure has been widely studied since 1980s for the resistivity anomaly [17][18][19] and large thermo-power 20,21 . For a long time, ZrTe5 was considered to be a semimetal or degenerated semiconductor with a parabolic energy dispersion. 10,22 However, a recent study 7 revealed a linear dispersion in ZrTe5 bulk states along with a chiral magnetic effect, hosting the signatures of Dirac semimetal. Nevertheless, owing to the relatively low Femi level in ZrTe5, a complete understanding of the band structure, especially the conduction band, remains elusive from angle-resolved photoemission spectroscopy (ARPES) measurements, which makes it challenging to confirm the existence of Dirac fermions. Meanwhile, the layered structure of ZrTe5 gives rise to a weak interlayer coupling, which should release the confinement on the interlayer dispersion of Dirac fermions as in the case of cuprates. 23 The interplay between Dirac fermions and the interlayer confinement 4 may result in intriguing physical properties yet to be explored.
Here we report the observation of massless Dirac fermions in layered ZrTe5 based on two independent experiments: magneto-optics and magneto-transport. External magnetic field leads to the Landau quantization of Bloch electrons, which enables us to probe the band structure and carrier dynamics in ZrTe5. A B -dependence of inter-Landau-level resonance is observed, indicating a linear band dispersion. Owing to the high electron mobility and low Fermi level, we are able to detect the Shubnikov-de Haas (SdH) oscillations close to the quantum limit, from which a non-trivial Berry phase is obtained. Both of them are well-established signatures for ultra-relativistic quasi-particles in crystals. Furthermore, the angular-dependent magneto-transport reveals a quasi-two-dimensional Fermi surface. A striking anisotropy is witnessed in the band dispersion characteristics along different crystal orientations, and along the b-axis (the layer-stacking direction), a carrier mass heavier than that of the free electron suggests a non-linear dispersion. This conclusion is further supported by the critical evidence of the trivial Berry phase and the observed bulk quantum Hall effect (QHE). ZrTe5 is theoretically analyzed to be at a critical phase between a 3D Dirac semimetal and a weak topological insulator. Reducible dimension in ZrTe5 (2D & 1D) attained upon exfoliation also promises possible device applications.
Under external magnetic field B, the charged particles can occupy discrete orbits and 5 form Landau levels.
Here n E follows a square root relation with the magnetic field as depicted in Fig. 1a.
The Fermi velocity determines the evolution of Landau level energy in Dirac systems with magnetic field. The assumption of kz = 0 is valid for magnetic fields applied along the b-axis direction. Although the Landau levels have a finite dispersion in the kz direction, magneto-optical reflection occurs at kz=0 where the joint density of states of the Landau levels corresponding to the transition are optimal. 24 Landau level transitions give resonance peaks in the reflection spectra. To confirm the validity of Equation (1) in ZrTe5, we measured the reflection spectra of bulk ZrTe5 crystals under various magnetic fields at liquid helium temperature. Single crystal ZrTe5 used in this study was grown by iodine vapor transport method which is different from flux method [25][26][27] . The as-grown crystals crystallized in the 6 orthorhombic layered structured with space group 17 2h D . 17 The prismatic chains of ZrTe3 run along a-axis and are linked by the zigzag chains of Te along c-axis. These two dimensional layers then stack along b-axis forming the bulk crystal. In order to avoid oxidation and contamination, the samples were freshly cleaved prior to experiments. During the magneto-infrared measurements, the magnetic field is parallel to the b-axis (stacking axis) of the samples. The measured reflection spectra are normalized by the spectrum at zero magnetic field. The detailed experimental setup for the magneto-infrared measurements is provided in the supplementary Fig.   S1. Several reflection peaks can be well resolved in the normalized reflection spectra measured under different magnetic fields (Fig. 1b inset). The blocked grey area is originated from 60 Hz harmonics. Since these reflection maxima systematically shift towards higher energy with increasing magnetic field, we conclude the resonance coming from inter-Landau-level transitions, as determined by electric-dipole-selection rules. 28 The incident photons excite the electrons from the occupied valance band to the unoccupied conduction band (Fig. 1a) following 1 nn    . Thus, the resonance energy of inter-Landau-level transitions is given by The energy ratio between the second (right) and the first (left) resonance peak (Fig. 1b inset) is 1.3, almost the same value as ( 2+ 3) : ( 1 2)  . So the first peak is assigned to be the transition from L-1 to L2 or from L-2 to L1 (orange arrows in Fig. 1a), where Ln represents the n th Landau level. Peaks at higher energy should correspond to higher Landau level index. Keeping this in mind, we plot the peak positions with 1 nn  . As shown in Fig. 1b, for different magnetic fields, the data points are well aligned on a straight line with the y-axis offset close to zero, proving the validity of Equation (1)  To acquire in-depth understanding of the Dirac quasi-particles, we carried out magneto-transport measurements with rotatable field direction. As schematically illustrated in Fig. 3a, conventional six-terminal devices were prepared with a constant 9 current applied along the a-axis of the single crystal. Similar to the previous studies, 17, 18, 32 we observed several unique features of ZrTe5, such as "resistivity anomaly" and Hall sign reversal at 150K (refer to Fig. S4). The anomaly temperature is related to the Fermi level position relative to the Dirac point. 33 As determined from the Hall effect ( Fig. S5), ZrTe5 has an n-type conductivity at low temperatures. Generally, under a perpendicular magnetic field B, closed cyclotron orbits follow the Lifshitz-Onsager quantization rule, Important transport parameters can be obtained from the SdH oscillation analysis. 39 The temperature dependence of the oscillation amplitude   is shown in Fig. 3e, observed which possibly arises from the axial anomaly. 43 The small amplitude of the negative MR could be attributed to the relatively high Fermi level. 43 Fig. 4b and  44 . Figure  4d shows a Fast-Fourier-Transform (FFT) analysis of the SdH oscillations. The single peak feature in the FFT spectra corresponds to a single-band transport. For an ideal 2D system, the oscillation frequency F increases linearly with 1/ cos and becomes infinitely at 90   due to the infinite cross-section of Fermi surface. In Fig. 4e and We have revealed that the Berry phase extracted from the SdH oscillations in a-c plane is exactly , corresponding to a topological non-trivial state. However, the same experiments on other orthogonal axes suggest the unexpected trivial state with the 13 complete elimination of Berry phase (Fig. S12). The anomalous feature of Berry phase along different crystal orientations drives us to further analyze the SdH oscillations at all three orthogonal axes for a detailed understanding of band dispersion characteristics. Fig.5a is the oscillation amplitude plotted as a function of temperature. The best fit to those data yields the effective mass for each plane (kb-kc, ka-kc, and ka-kb). The Fermi velocity of each plane can be also calculated as discussed above. The a-c plane shows a much smaller * m and a larger F v compared with the other two (Fig. 5b). This value in a-c plane ( The ab initio calculations have predicted the interlayer binding energy for ZrTe5 to be 12.5 meV/Å, a comparable value to graphite (9.3 meV/Å), suggesting a weak interlayer coupling. 49 Using the conventional scotch tape method, we have also successfully exfoliated ZrTe5 into 2D thin flakes or 1D nano wires (Fig. S10). We found that, upon the exfoliation, not only the inter-plane van der Waals force is easy to overcome, but also that the in-plane Te zigzag chains are readily broken, thus developing a quasi-one-dimensional structure (Fig. S11). The relatively weak bond along c-axis may be the reason for the increase of m* and the decrease of vF, similar to the case of the b-axis. We also show that the natural 1D structure can be achieved upon exfoliation, which promises related studies such as density wave in ZrTe5 at low dimension.
Since both the magneto-optics and magneto-transport measurements were performed near liquid helium temperature, our conclusion of massless Dirac fermion is valid in the low temperature range. But the resistivity and Hall measurements near 140 K (Fig.   S4) reveal a metal-semiconductor transition (anomaly peak) along with Hall sign reversal (Fig. S4), indicating a multi-carrier transport and a complex band structure evolving with temperature. However, for higher temperatures, when thermal energy is larger than the energy level separation ( Bc kT   ), Landau quantization is no longer accessible, which calls for further experimental efforts to elucidate the detailed band structure. Both the previous 10 and recent 50 photoemission experiments report the observation of a gap opening near the anomaly temperature. It indicates that the observed Dirac point is unstable against the thermal perturbation. As we discussed in the supplementary section X, the Dirac states that we observed are not likely to originate from the surface states. Recently, several surface sensitive probes were employed to examine the surface/bulk states in ZrTe5. 25,51 While it was claimed to be gapped with 2D-like structure in the bulk through scanning tunneling microscopy (STM) technique, another study using the same technique realized a gapped topological insulator on the surface layer. 51,52 Interestingly, an ARPES measurement suggested that the gap opening decreased with reducing the temperature whereas another separate work concluded for massless Dirac fermions at low temperature. 50 The electronic structure was also discussed theoretically. The gapless bulk state was predicted to possibly coexist with the gapped surface state. 33 Importantly, the calculated band structure is highly sensitive to the lattice parameters, which can be tuned by using different growth methods or under specific growth conditions. 53 ZrTe5 is likely to be close to the band topology transition and thus the topology of the band 17 structure is highly sensitive to the chemical composition and the lattice constants.
Those theories could explain the controversial arguments surrounding the possible gap. Detailed study on the surface state and clear ARPES experiments along kb direction are required to comprehensively understand the overall electronic structure.
In the following, we theoretically outline a situation where one can obtain a linear dispersion along two directions and a quadratic one along the third, based on the symmetry analysis of Yang and Nagaosa. 54 (Fig. 5d). Therefore, an energy dispersion in a three-dimensional system, which is quadratic along one direction and linear along the other two, is possible in ZrTe5. The system is at a critical point separating an insulating (or topological insulator) and a 3D Dirac semimetal phase. 18 The phase transition is possibly driven by changes in the lattice constants.
Judging from the Landau fan diagram in Fig. 3d, the first Landau level can be accessed within 6 T in the magneto-transport which is in stark contrast to the representative Dirac semimetal Cd3As2 in which the quantum limit requires a magnetic field at least 43 T or even larger 55,56 . In general, the higher the Fermi level is, the larger field is demanded to access the quantum limit. interests to see whether it will sustain the Dirac fermions as in bulk form or transfer into a quantum spin Hall insulator as theory predicted when approaching the 2D limit.
Further experiments in the nanoscale samples or the ultra-quantum limit of ZrTe5 are currently being pursued.
In conclusion, we confirm the existence of Dirac fermions in bulk ZrTe5 by magneto-optics and SdH oscillations. Angular-dependent quantum oscillations reveal a quasi-2D nature of the Dirac fermions, which is further supported by the bulk QHE 19 and the angle-dependent Berry phase at low fields. Theoretical analysis implies that the system is at the critical point between the Dirac semimetal phase and the topological insulator phase. The unusual interlayer dispersion characteristics, low Fermi level, and van der Waals structure demonstrate ZrTe5 as an intriguing system for both the fundamental studies and device applications.

Method
Single Crystal synthesis.
High-quality ZrTe5 single crystals were synthesized by iodine vapor transport method in a two-zone tube furnace. Stoichiometric amounts of high-purity Zr and Te elements were placed in a quartz tube and sealed under vacuum. ZrTe5 is crystallized during a chemical transport reaction process (14 days) with a temperature gradient from 500℃ to 450℃. We use a transport agent of iodine with a concentration of 10 mg cm -3 . The growth rate along a-axis is much faster than those in b-axis and c-axis, yielding needle-like samples. The chemical ratio is found to be 1:5 for Zr/Te, determined by energy dispersive x-ray analysis in scanning electron microscopy.

Magneto-optics measurements.
ZrTe5 was freshly cleaved, subject to an applied magnetic field parallel to the stacking axis (Faraday geometry) at liquid helium temperature. An infrared light from a broadband light source is modulated by interferometer and vibration mirrors in the Fourier-transform infrared spectrometer (FTIR). The light was guided through a light 20 pipe and focused on the sample with millimeter spot by a parabolic cone (Fig. S1). A bolometer was used to measure the intensity of reflection light simultaneously with FTIR.
Freshly cleaved ZrTe5 was measured using standard six-terminal Hall-bar geometry in a physical property measurement system. Stanford Research 830 Lock-in amplifiers were used to measure the electrical signals with magnetic field up to 9T that applied for various orientations of the applied magnetic field.

Sample exfoliation.
ZrTe5 was firstly exfoliated on scotch tape and then transferred onto 300nm/300μm SiO2/Si substrate. The flakes with different thickness can be identified by different colors. In order to enhance the productivity of one dimensional structure and avoid contamination, PDMS (polydimethylsiloxane) was used to transfer the exfoliated samples.