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# Spectroscopic evidence for bulk-band inversion and three-dimensional massive Dirac fermions in ZrTe_{5}

Edited by J. C. Séamus Davis, Cornell University, Ithaca, NY, and approved December 13, 2016 (received for review August 7, 2016)

## Significance

Experimental verifications of the theoretically predicted topological insulators (TIs) are essential steps toward the applications of the topological quantum phenomena. In the past, theoretically predicted TIs were mostly verified by the measurements of the topological surface states. However, as another key feature of the nontrivial topology in TIs, an inversion between the bulk bands has rarely been observed by experiments. Here, by studying the optical transitions between the bulk LLs of ZrTe_{5}, we not only offer spectroscopic evidence for the bulk-band inversion—the crossing of the two zeroth LLs in a magnetic field, but also quantitatively demonstrate three-dimensional massive Dirac fermions with nearly linear band dispersions in ZrTe_{5}. Our investigation provides a paradigm for identifying TI states in candidate materials.

## Abstract

Three-dimensional topological insulators (3D TIs) represent states of quantum matters in which surface states are protected by time-reversal symmetry and an inversion occurs between bulk conduction and valence bands. However, the bulk-band inversion, which is intimately tied to the topologically nontrivial nature of 3D Tis, has rarely been investigated by experiments. Besides, 3D massive Dirac fermions with nearly linear band dispersions were seldom observed in TIs. Recently, a van der Waals crystal, ZrTe_{5}, was theoretically predicted to be a TI. Here, we report an infrared transmission study of a high-mobility [∼33,000 cm^{2}/(V ⋅ s)] multilayer ZrTe_{5} flake at magnetic fields (*B*) up to 35 T. Our observation of a linear relationship between the zero-magnetic-field optical absorption and the photon energy, a bandgap of ∼10 meV and a _{5} but also open up a new avenue for fundamental studies of Dirac fermions in van der Waals materials.

Topologically nontrivial quantum matters, such as topological insulators (1⇓⇓⇓⇓⇓⇓–8), Dirac semimetals (9⇓⇓⇓⇓⇓⇓⇓⇓⇓–19), and Weyl semimetals (20⇓⇓⇓⇓⇓⇓–27), have sparked enormous interest owing both to their exotic electronic properties and potential applications in spintronic devices and quantum computing. Therein, intrinsic topological insulators have insulating bulk states with odd *Z*_{2} topological invariants and metallic surface or edge states protected by time-reversal symmetry (4⇓–6, 28). Most of the experimental evidence to date for TIs is provided by the measurements of the spin texture of the metallic surface states. As a hallmark of the nontrivial *Z*_{2} topology of TIs (4⇓–6, 28), an inversion between the characteristics of the bulk conduction and valence bands occurring at an odd number of time-reversal invariant momenta has seldom been probed by experiments. An effective approach for identifying the bulk-band inversion in TIs is to follow the evolution of two zeroth Landau levels (LLs) that arise from the bulk conduction and valence bands, respectively. As shown in Fig. 1*A*, for TIs, due to the bulk-band inversion and Zeeman effects, the two zeroth bulk Landau levels are expected to intersect in a critical magnetic field and then separate (3, 29); and for trivial insulators, the energy difference between their two zeroth Landau levels would become larger with increasing magnetic field. Therefore, an intersection between the two zeroth bulk LLs is a significant signature of the bulk-band inversion in TIs. However, a spectroscopic study of the intersection between the two zeroth bulk LLs in 3D TIs is still missing. In addition, many typical 3D TIs, such as Bi_{2}Se_{3}, show massive bulk Dirac fermions with parabolic band dispersions, which are effectively described by massive Dirac models (6, 28, 29). By contrast, 3D massive Dirac fermions with nearly linear bulk band dispersions (7), which are interesting topics following 2D massive Dirac fermions in gapped graphene (30, 31), were rarely observed in 3D TIs.

A transition-metal pentatelluride, ZrTe_{5}, embodies both 1D chain and 2D layer features (32), shown in Fig. 1*B*. One Zr atom together with three Te (1) atoms forms a quasi-1D prismatic chain ZrTe_{3} along the *a* axis (*x* axis). These prismatic ZrTe_{3} chains are connected through zig-zag chains of Te (2) atoms along the *c* axis (*y* axis) and then construct quasi-2D ZrTe_{5} layers. The bonding between ZrTe_{5} layers is van der Waals type (33, 34). Thus, as displayed in Fig. 1*C*, bulk ZrTe_{5} crystals can be easily cleaved down to a few layers. Recently, the ab initio calculations indicate that monolayer ZrTe_{5} sheets are great contenders for quantum spin Hall insulators—2D TI and that 3D ZrTe_{5} crystals are quite close to the phase boundary between strong and weak TIs (33). Scanning tunneling microscopy measurements have shown that edge states exist at the step edges of the ZrTe_{5} surfaces (35, 36). Nonetheless, further investigations are needed to check whether the observed edge states in ZrTe_{5} are topologically nontrivial or not. Studying the bulk-band inversion or the intersection between the two zeroth bulk LLs can provide a crucial clue to clarifying the nature of the edge states in ZrTe_{5}. Except the edge states within the energy gap of the bulk bands around the Brillouin zone center (i.e., Γ point) of ZrTe_{5} (36, 37), 3D massless Dirac fermions with the linearly dispersing conduction and valence band degenerate at the Γ point were suggested to exist in this material by previous angle-resolved photon emission spectroscopy, transport, and optical experiments (38⇓⇓–41). Considering that (*i*) our ZrTe_{5} thick crystals were experimentally shown to be Dirac semimetals hosting 3D massless Dirac fermions, (*ii*) ZrTe_{5} monolayers were theoretically predicted to be quantum spin Hall insulators, and (*iii*) the bulk state of ZrTe_{5} is very sensitive to its interlayer distance, which might be a discrepancy in different samples (33, 40); it is significant to quantitatively verify whether 3D massive Dirac fermions with a bandgap and nearly linear bulk-band dispersions can be realized in dramatically thinned flakes of our ZrTe_{5} crystals.

Infrared spectroscopy is a bulk-sensitive experimental technique for studying low-energy excitations of a material. Here, to investigate the bulk-band inversion and the nature of the bulk fermions in ZrTe_{5}, we measured the infrared transmission spectra *T*(*B*) of its multilayer flake with thickness *d* ∼180 nm at magnetic fields applied along the wave vector of the incident light (*Materials and Methods* and *Supporting Information*). A series of intra- and inter-LL transitions are present in the relative transmission spectra *T*(*B*)/*T*(*B*_{0} = 0 T) of the ZrTe_{5} flake. The linear *B* ≤ 4 T and the nonzero intercept of the LL transitions at *B* = 0 T, combined with the linear relationship between the zero-magnetic-field optical absorption and the photon energy, indicates 3D massive Dirac fermions with nearly linear band dispersions in the ZrTe_{5} flake. Moreover, a 3D massive Dirac model with a bandgap of ∼10 meV can quantitatively explain the magnetic-field dependence of the measured LL transition energies very well. At high magnetic fields, we observed fourfold splittings of the LL transitions. In addition, our analysis of the split LL transitions shows that the intra-LL transitions, which are associated to the two zeroth LLs and disappear at *B* ∼ 2.5 T, reemerge at *B* > 17 T. Considering that the zeroth LL crossing in a Zeeman field would make the two zeroth bulk LLs intersect with the chemical potential here and then alter the carrier occupation on the zeroth LLs, we attribute the reemergence of the intra-LL transitions in the ZrTe_{5} flake to the energy crossing of its two zeroth bulk LLs, which originates from the bulk-band inversion. These results strongly support the theoretically predicted 3D TI states in 3D ZrTe_{5} crystals.

## Results

### Three-Dimensional Massive Dirac Fermions.

At zero magnetic field, the measured absolute transmission *T*(*A*(*T*(*d*, where *d* is the thickness of the sample and *D*(*A*(*D*(*D*(^{2}, and for 2D linear dispersions, *D*(*A*(*D* indicates 3D linear band dispersions in ZrTe_{5}. Moreover, at low energies, the absorption coefficient apparently deviates from the linear relationship with _{5} flake.

To confirm the 3D massive Dirac fermions in ZrTe_{5}, we further performed infrared transmission experiments at magnetic fields applied perpendicular to the *ac* plane (*xy* plane) of the crystal (Faraday geometry). The low-field relative transmission *T*(*B*)/*T*(*B*_{0} = 0 T) spectra in Fig. 2*A* show seven dip features T_{n} (1 *n* *T*(*B*)/*T*(*B*_{0}) as the absorption energies, which is a usual definition in thin film systems, such as Bi_{2}Se_{3} films and graphene. Then, we plotted the square of the T_{n} energies (*B*. The linear *B* dependence of *C*, the nonzero intercept of the linear fit to *B*, together with the nonzero intercept, provides further evidence for 3D massive Dirac fermions in the ZrTe_{5} flake.

To quantitatively check the 3D massive Dirac fermions in the ZrTe_{5} flake, we use a 3D massive Dirac Hamiltonian, which was derived from the low-energy effective *k* ⋅ *p* Hamiltonian based on the spin–orbital coupling, the point group and time-reversal symmetries in ZrTe_{5}, and includes the spin degree of freedom (40). According to the 3D massive Dirac Hamiltonian expanded to the linear order of momenta, we can obtained the nearly linear band dispersions of ZrTe_{5} at zero magnetic field: *ac* plane, the LL spectrum of ZrTe_{5} without considering Zeeman effects has the form:*N* is Landau index, *N*) is the sign function, and *e* is the elementary charge. The **1**, the magnetic field makes the 3D linear band dispersions evolve into a series of 1D non-equally-spaced Landau levels (or bands), which disperse with the momentum component along the field direction. Specifically, because the 3D massive Dirac Hamiltonian of ZrTe_{5} involves the spin degree of freedom of this system, two zeroth LLs indexed by _{5} only allows the LL transitions from LL_{N} to LL_{N’}: Δ*N* = *B* scale as 1: 5.7: 9.3: 13.1: 16.7: 20.5: 24.1, respectively, which is close to the approximate ratio of the theoretical inter-LL transition energies based on Eq. **2**, 1: (^{2} (∼ 5.8): (^{2} (∼ 9.9): (^{2} (∼ 13.9): (^{2} (∼ 17.9): (^{2} (∼ 21.9): (^{2} (∼ 25.9). Therefore, the absorption features T_{n} are assigned as the inter-LL transitions: *B*) and we have *n* = *n* **2** from a least square fit yields a bandgap ^{5} m/s and ^{5} m/s (*Supporting Information*).

As another signature of the bandgap or the nonzero Dirac mass, the absorption feature T_{1}* is present at energies lower than the lowest-energy inter-LL transition T_{1} in Fig. 2*D*. The feature T_{1}* is attributed to the intra-LL transition *E* (*Supporting Information*). According to Eqs. **2** and **3**, the energy difference (*E*_{T1} *E*_{T1*}) between the transitions T_{1} and T_{1}* in the inset of Fig. 2*C* directly gives the bandgap value _{1}* energy in Fig. 2*C* can be well fitted by Eq. **3** with ^{5} m/s.

The carrier-charge mobility _{5} flake can be calculated using the general equation (47): *e**m**), where *m** is the carrier effective mass on the anisotropic Fermi surface (48, 49). Here, the transport scattering rate *ac* plane can be roughly estimated from the width of the T_{1} feature at low fields: *B* = 0.5 T. Moreover, considering the absence of Pauli blocking of the T_{1} transition at *B* = 0.5 T, we get the Fermi energy in ZrTe_{5}, *Supporting Information*), which means the Fermi level in our sample is quite close to the band extrema. In this case, the average effective mass *m** of the carriers within the *ac* plane can be described by (30): *m*_{ac}* ^{2}] ^{−33} kg *m*_{0}, where *m*_{0} is the free electron mass and the average Fermi velocity within the *ac* plane ^{5} m/s. Finally, we can estimate the mobility of the carriers within the *ac* plane of our ZrTe_{5} sample: ^{2}/(V ⋅ s), which is comparable to those in graphene/h-BN heterostructures (50, 51).

### Buk-Band Inversion.

As shown in Fig. 3*A*, applying a higher magnetic field enables us to observe the splitting of the T_{1} transition, which indicates a nonnegligible Zeeman effect in ZrTe_{5} (40). For TIs, due to the Zeeman field, each LL except the two zeroth LLs splits into two sublevels with opposite spin states, while the LL_{−0} and LL_{+0} are spin-polarized and have spin-up and -down state, respectively (3, 29). The energy of the sublevel has the form (40):*g* factor of LL_{N}. The spin-orbit coupling (SOC) in ZrTe_{5} mixes the spin states of the two sublevels, so two extra optical transitions between the sublevels with different spin indices become possible. The inter- and intra-LL transition energies including the Zeeman effect can be written as (40):

Fig. 3*B* displays the false-color map of the –ln[*T*(*B*)/*T*(*B*_{0})] spectra of the ZrTe_{5} flake. Interestingly, a cusplike feature around 18 T, which is indicated by a white arrow, can be observed in Fig. 3*C* (i.e., the magnified image of a region in Fig. 3*B*). To quantitatively investigate the physical meaning of this cusplike feature, we plot the energies of the four split T_{1} transitions [i.e., 1*B*, which are defined by the onsets of the absorption features due to the Zeeman splitting (see figure 3 of the Supplemental Material of ref. 40 and *Supporting Information*). As displayed by the green dashed lines in Fig. 3*B*, fitting the energy traces of the inter-LL transitions, 1**5** with the obtained values of the Fermi velocity *g* factors of the two zeroth LLs and LL_{±1}: *g*_{eff}(LL_{+0}) = *g*_{eff}(LL_{−0}) ∼ 11.1, *g*_{eff}(LL_{−1}) ∼ 31.1 and *g*_{eff}(LL_{+1}) ∼ 9.7 [or *g*_{eff}(LL_{+1}) ∼ 31.1 and *g*_{eff}(LL_{−1}) ∼ 9.7] (*Supporting Information*).

It is known that as a hallmark of TIs, the band inversion causes the exchange of the characteristics between the valence- and conduction-band extrema (2, 6, 28), so as shown in Fig. 1*A* and 3*D*, the LL_{−0} and LL_{+0}, which come from the inverted band extrema, have reversed spin states and cross at a critical magnetic field (3, 29). According to Eq. **4** with the above values of *g*_{eff}(LL_{±0}) and *∆*, we estimated the critical magnetic field *B*_{c} ∼ 17 T. In Fig. 2 *A* and *D*, the disappearance of the intra-LL transition T_{1}* around *B* ∼ 2.5 T indicates that LL_{+0} (or LL_{−0}) becomes fully depleted (or occupied) with increasing magnetic field and that at *B* > 2.5 T, the chemical potential of ZrTe_{5} can be considered to be located at zero energy. In this case, the two zeroth LLs intersect with the chemical potential at the same magnetic field *B*_{c}. More importantly, this intersection means at *B* > *B*_{c} ∼ 17 T, LL_{−0} and LL_{+0} becomes empty and occupied, respectively, which leads to the gradual replacement of the inter-LL transitions T_{1}, 1_{1}*, 1*D*. Furthermore, in Fig. 3*B*, the energy traces of the four split transitions (gray dots) observed at *B* > 17 T are shown to follow the white theoretical curves for the intra-LL transitions T_{1}*, which are based on Eq. **6**. Therefore, the four split transitions observed at *B* *B* can be assigned as the intra-LL transitions T_{1}*, 1_{1}* deviate markedly from those of the inter-LL transitions T_{1}, the reemergence of the T_{1}* transitions at *B* _{5} flake.

In summary, using magnetoinfrared spectroscopy, we have investigated the Landau level spectrum of the multilayer ZrTe_{5} flake. The magnetic-field dependence of the LL transition energies here, together with the photon-energy dependence of the absorption coefficient at zero field, quantitatively demonstrates 3D massive Dirac fermions with nearly linear dispersions in the ZrTe_{5} flake. Due to the Zeeman splitting of the LLs, the energy splitting of the LL transitions was observed at *B* ≥ 6 T. Interestingly, the intra-LL transitions T_{1}* reemerge at *B* > 17 T. We propose that the reemergence of the T_{1}* transitions results from the band-inversion-induced crossing of the two zeroth LLs, LL_{+0} and LL_{−0}. Our results make ZrTe_{5} flakes good contenders for 3D TIs. Moreover, due to the 3D massive Dirac-like dispersions and the high bulk-carrier mobility [∼ 33,000 cm^{2}/(V ⋅ s)], the ZrTe_{5} flake can also be viewed as a 3D analog of gapped graphene, which enables us to deeply investigate exotic quantum phenomena.

## Materials and Methods

### Sample Preparation and Characterizations.

Bulk single crystals of ZrTe_{5} were grown by Te flux method. The elemental Zr and Te with high purity were sealed in an evacuated double-walled quartz ampule. The raw materials were heated at 900 °C and kept for 72 h. Then they were cooled slowly down to 445 °C and heated rapidly up to 505 °C. The thermal–cooling cycling between 445 and 505 °C lasts for 21 d to remelt the small size crystals. The multilayer ZrTe_{5} flake (*ac* plane) for magnetotransmission measurements were fabricated by mechanical exfoliation, and deposited onto double-side-polished SiO_{2}/Si substrates with 300 nm SiO_{2}. The flake thickness ∼180 nm and the chemical composition were characterized by atomic force microscopy (AFM) and energy dispersion spectroscopy (EDS), respectively (*Supporting Information*).

### Infrared Transmission Measurements.

The transmission spectra were measured at about 4.5 K in a resistive magnet in the Faraday geometry with magnetic field applied in parallel to the wave vector of incident light and the crystal *b* axis. Nonpolarized IR light (provided and analyzed by a Fourier transform spectrometer) was delivered to the sample using a copper light pipe. A composite Si bolometer was placed directly below the sample to detect the transmitted light. The diameter of IR focus on the sample is _{5} flake, an aluminum aperture was placed around the sample. The transmission spectra are shown at energies above 10 meV, corresponding to wavelengths shorter than 124 µm. The wavelength of infrared light here is smaller than the size of the measured sample, and thus the optical constants can be used for a macroscopic description of the data.

## Basic Characteristics of the ZrTe_{5} Sample

We show the thickness of the ZrTe_{5} flake characterized by atomic force microscopy and the energy dispersion X-ray spectra of the ZrTe_{5} flake in Fig. S1. The multilayer flake studied in the main text has the thickness *d* ∼ 180 nm. For our sample, the atom ratio, Zr: Te ∼ 1: 5.

## Bandgap and Effective Fermi Velocity Obtained by Least Squares Fit

Using the method of least squares fit, we fit the measured Landau level transition energies of the ZrTe_{5} flake present in Fig. 2 *B* and *C*, based on Eqs. **2** and **3** in the main text. The two parameters, bandgap ∆ and the effective Fermi velocity *A*, the fit to the T_{1} transition data with ∆ and ^{5} m/s. For the other transitions, T_{2}, T_{3}, T_{4}, T_{5}, T_{6}, and T_{7}, we fit the data using two approaches (*i*): setting both ∆ and *ii*) fixing the gap value ∆ = 10 meV and allowing *C*–*H*, for T_{2}, T_{3}, T_{4}, T_{5}, T_{6}, and T_{7}, these two approaches produce the similar fits within the error bars of the data and the similar values of the sum of square residuals _{2}, T_{3}, T_{4}, T_{5}, T_{6}, and T_{7} transition energies at low fields (*B* _{1} transition. From the gap value ∆ ∼ 10 ± 2 meV, we can obtain comparable Fermi velocities of the T_{2}, T_{3}, T_{4}, T_{5}, T_{6}, and T_{7} transitions, ranging from (4.95 ± 0.04) × 10^{5} m/s to (5.04 ± 0.04) × 10^{5} m/s, as shown in Table S1.

When _{1} transition based on Eq. **2** in the main text yields *i*) and (*ii*), respectively. As displayed in Fig. S2*A*, the first approach can describe the T_{1} transition [with ^{5} m/s] better than the second approach with ∆ = 10 meV. Given that all of the bandgap extracted from the LL transitions correspond to the zero-field gap of the ZrTe_{5} flake, we assume the same gap value of 10 ± 2 meV for the T_{1} transition and attribute the deviations of the T_{1} transition from the behaviors described by Eq. **2** to an effective Fermi velocity _{1} transition, ^{5} m/s, which is smaller than the fitted values of *n*

For T_{1}*, the first approach for fitting deduces the bandgap ^{5} m/s, and the second approach with fixed ∆ =10 meV has the Fermi velocity ^{5} m/s. A smaller gap value was obtained from the first method, which is consistent with the case in T_{1}, but these two approaches have similar values of the sum of square residuals _{1}* corresponds to the zero-field gap and should be the same as those from the other LL transitions, we also attribute the deviations of the T_{1}* transition from the behaviors described by Eq. **3** to an effective Fermi velocity. Thus, we obtained the effective Fermi velocity for the T_{1}* transition ^{5} m/s.

## Assigning T_{1}* to the Intra-LL transition from LL_{+0} (or LL_{−1}) to LL_{+1} (or LL_{−0})

To identify the nature of the T_{1}* feature, we plot the energy ratios between T_{1} and T_{1}* for the ZrTe_{5} flake in Fig. S3*A*. If there is no bandgap here, the T_{1}* feature, which locates at lower energies than the T_{1} feature, should correspond to the intra-LL transition T* from LL_{+1} (or LL_{−2}) to LL_{+2} (or LL_{−1}), as illustrated in Fig. S3*B*. In this case, according to Eqs. **2** and **3** in the main text without a bandgap (namely, massless Dirac fermions), the energy ratio between the inter-LL transitions from LL_{0} to LL_{+1} (or from LL_{−1} to LL_{0}) and the intra-LL transition T* should be ∼2.415, as displayed by the magenta dashed line in Fig. S3*A*. However, the energy ratios between the measured T_{1} and T_{1}* features, which are represented by the red stars, are much lower than 2.415. Instead, the red stars in Fig. S3*A* can be fitted by the red solid line corresponding to the energy ratio between the inter-LL transition from LL_{−0} to LL_{+1} (or from LL_{−1} to LL_{+0}) and the intra-LL transition from LL_{+0} to LL_{+1} (or from LL_{−1} to LL_{−0}), both of which are illustrated in Fig. 2*E* of the main text. Thus, the consistence between the red stars and the red solid line produced with a bandgap of 10 meV indicates that the observed T_{1}* feature originates from the intra-LL transition from LL_{+0} to LL_{+1} (or from LL_{−1} to LL_{−0}).

For the T_{2} and T_{1} features, if there is no bandgap, their theoretical energy ratio should correspond to the navy dashed line in Fig. S3*A*, which is produced by Eq. **2** of the main text. Definitely, this theoretical dashed line is much higher than the energy ratio between the measured T_{2} and T_{1} features, as represented by the blue stars. Instead, the blue stars can be fitted by the blue solid line based on Eq. **2** of the main text with a bandgap of 10 meV. So, the above two energy ratios between the observed LL transitions again reveal the existence of a bandgap in the ZrTe_{5} flake.

Because the bandgap is comparable to the energies of T_{1}* and T_{1} at magnetic fields *B* **2** and **3** of the main text without a bandgap (namely, for massless Dirac fermions). However, if the bandgap is much smaller than the energies of the LL transitions, such as T_{2} and T_{3}, it will be difficult to distinguish the difference in the energy ratios between the high-energy LL transitions for massive and massless Dirac fermions at high magnetic fields, which has been revealed by the purple stars, and the purple dashed and the solid lines for the energy ratio between T_{3} and T_{2}.

## Estimation of the Fermi Energy *E*_{F} from the Landau Level Transition

The Fermi energy in the ZrTe_{5} flake can be estimated from the magnetotransmission data. According to Pauli’s exclusion principle, the LL transitions T_{1} and T_{1}*, LL_{−0} (or LL_{−1}) _{+1} (or LL_{+0}) and LL_{−0} (or LL_{+0}) _{−1} (or LL_{+1}), should be blocked if the LL_{+1} (or LL_{−1}) is fully occupied (or depleted), as the magnetic field decreases below a critical value *B*_{T1} (shown in Fig. S4*A*). Fig. 2 of the main text shows T_{1} and T_{1}* can still be observed at 0.5 T, indicating *B*_{T1} < 0.5 T. Thus, as displayed in Fig. S4*B*, the Fermi level is between LL_{+0} and LL_{+1} (or between LL_{−1} and LL_{+0}) at *B* = 0.5 T and correspondingly, the Fermi energy *E*_{F} is not higher than 15 meV.

## Calculation of the *g* Factors of the Landau Levels

According to Eq. **5** of the main text (with ^{5} m/s), we fit the energy traces of the split-T_{1} transitions (i.e., 1*g* factors of LL_{±0} and LL_{±1}:**S1**–**S4**, we obtain *g*_{eff}(LL_{+0}) = *g*_{eff}(LL_{−0}) ∼ 11.1, *g*_{eff}(LL_{−1}) ∼ 31.1 and *g*_{eff}(LL_{+1}) ∼ 9.7. In another case with electron and hole LLs reversed, we have *g*_{eff}(LL_{−1}) ∼ 9.7 and *g*_{eff}(LL_{+1}) ∼ 31.1.

## Definitions of the Absorption Energies of T_{1} at Low and High Magnetic Fields

At high magnetic fields *B* *B* of the main text, the LL transition T_{1} is shown to split into four submodes due to the Zeeman splitting of LLs. According to the theoretical calculations displayed in figure 3 of the Supplemental Material of ref. 40, the energies of the split T_{1} submodes, are defined by the onsets of the absorption features. The definition of the submode energies in the ZrTe_{5} flake is consistent with that in the material with 3D massless fermions—HgCdTe (see figure 4 of ref. 42). However, when the magnetic field becomes low enough, the split submodes merge and cannot be well distinguished, so at *B* *B* _{1} of the ZrTe_{5} flake are defined by the minimal positions of the dip features, which is a standard method for film systems, such as graphene (44). In this supplementary section, we want to check whether the definition of the T_{1} energy at magnetic fields *B* _{5} flake.

The close-up image of Fig. S5*A* shows a royal blue dashed curve for fitting the low-field minimal positions of T_{1} in Fig. 2 of the main text (with the Fermi velocity *B* _{1} (royal blue dots) at *B* *B*. At *B* _{1} are shown to match well with the four olive dashed curves for fitting the energies of the four submodes at high magnetic fields and the green dashed curve. Thus, although the Landé *g* factors in thin ZrTe_{5} flake are large and the splitting of T_{1} transition can be well distinguished at high magnetic fields, the consistence between the minimal position of the dip features at *B* _{1} modes indicates that defining the minimal position as the absorption energy is a valid method for obtaining the T_{1} transition energies at low magnetic fields (*B*

## Acknowledgments

We thank X. C. Xie, F. Wang, Z. Fang, M. Orlita, M. Potemski, H. M. Weng, L. Wang, C. Fang, and X. Dai for very helpful discussions. We acknowledge support from the Hundred Talents Program of Chinese Academy of Sciences, the National Key Research and Development Program of China (Project 2016YFA0300600), the European Research Council (ERC ARG MOMB Grant 320590), the National Science Foundation of China (Grants 11120101003 and 11327806), and the 973 project of the Ministry of Science and Technology of China (Grant 2012CB821403). A portion of this work was performed in National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement DMR-1157490 and the State of Florida. Work at Brookhaven was supported by the Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, US Department of Energy, through Contract DE-SC00112704.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: zgchen{at}iphy.ac.cn or nlwang{at}pku.edu.cn.

Author contributions: Z.-G.C. designed the research; Z.-G.C. carried out the optical experiments; Z.-G.C. wrote the paper; Z.-G.C., R.Y.C., R.Y., and N.L.W. analyzed the data; R.D.Z., J.S., and G.D.G. grew the single crystals; and C.Z., Y.H., F.Q., and Q.L. performed the basic characterization.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1613110114/-/DCSupplemental.

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_{5}

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- Article
- Abstract
- Results
- Materials and Methods
- Basic Characteristics of the ZrTe
_{5}Sample - Bandgap and Effective Fermi Velocity Obtained by Least Squares Fit
- Assigning T
_{1}* to the Intra-LL transition from LL_{+0}(or LL_{−1}) to LL_{+1}(or LL_{−0}) - Estimation of the Fermi Energy
*E*_{F}from the Landau Level Transition - Calculation of the
*g*Factors of the Landau Levels - Definitions of the Absorption Energies of T
_{1}at Low and High Magnetic Fields - Acknowledgments
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