Two-Dimensional Quantum Hall Effect and Zero Energy State in Few-Layer ZrTe5

Topological matter plays a central role in today’s condensed matter research. Zirconium pentatelluride (ZrTe5) has attracted attention as a Dirac semimetal at the boundary of weak and strong topological insulators (TI). Few-layer ZrTe5 is anticipated to exhibit the quantum spin Hall effect due to topological states inside the band gap, but sample degradation inflicted by ambient conditions and processing has so far hampered the fabrication of high quality devices. The quantum Hall effect (QHE), serving as the litmus test for 2D systems to be considered of high quality, has not been observed so far. Only a 3D variant on bulk was reported. Here, we succeeded in preserving the intrinsic properties of thin films lifting the carrier mobility to ∼3500 cm2 V–1 s–1, sufficient to observe the integer QHE and a bulk band gap related zero-energy state. The magneto-transport results offer evidence for the gapless topological states within this gap.


Sample thickness determined by AFM
In order to determine the thickness of the ZrTe 5 layer, atomic force microscopy (AFM) was deployed. Panel a of Fig. S1 shows an AFM image recorded on device D1, discussed in detail in the main text. Panel b is a line trace of the height across the active device area. We extract a thickness of 6.5 nm on this device. The surface of thin ZrTe 5 is rather rough. This ± 1.1 observation is in agreement with what has been reported previously 1,2 . Increased surface roughness is accompanied by a drop in the sample mobility. The more homogeneous the sample thickness, the more pronounced quantum oscillations are.  is a linear fit to the n Hall-data in order to extract the gate capacitance.

Zero field resistance map in the (V g , T) plane for samples of different thickness
Two examples of the resistance behavior as a function of back-gate voltage and temperature in the absence of an applied magnetic field are shown in Fig. S3. The left panel is for the 9.5 nm thick ZrTe 5 layer referred to as device D3, while the right panel plots the data for ± 1.5 device D5 with a ZrTe 5 layer thickness of 13 nm. Similar behavior is observed for all other ± 3 devices that exhibit a clear maximum in the resistance as a function of temperature for the case of electron doping. for device D5 with a 13 nm thick ZrTe 5 layer. ± 3

Transition from electron doping to hole doping
In order to determine the transition from electron doping to hole doping, three distinct procedures can be used, each of which is illustrated in the panels of Fig. S4 using data acquired on device D1. The first approach is based on the anomalous maximum that forms in the temperature dependence of the longitudinal resistance at a fixed back-gate voltage (panel a).
Alternatively, the transition point at a given temperature can be obtained from the zero crossing of the Hall resistance plotted for a fixed magnetic field (here 1 T and -1 T) as a function of the back-gate voltage (panel b), i.e.
. In order to remove the undesirable contribution of the longitudinal resistance to the measured resistance, the Hall resistance trace is symmetrized using the expression Finally, also field effect traces recorded at fixed temperature allow the extraction of the transition point by determining the peak position, i.e. .

Field effect mobility
Field effect curves recorded at zero magnetic field for different sample temperatures are plotted in Fig. S5a. For temperatures below 120 K pronounced peaks are visible. They remind of the so-called Dirac peak in graphene samples. At temperatures beyond 200 K, the maximum has disappeared indicating that the chemical potential has moved deep into the valence band and it is no longer possible to enter the electron doped regime. In panel b some of these traces have been replotted using conductivity rather than resistivity as the ordinate. From the slope of a linear fit to these traces in the regime of low hole density the field effect mobility is

Emergence of the QHE and SdH oscillations with decreasing ZrTe 5 layer thickness
Shubnikov-de Haas oscillations and the quantum Hall effect gradually appear when successively reducing the ZrTe 5 thickness, provided the material quality can be preserved.  layer thickness is reduced. All data are recorded at 1.7K and 13 T. a Longitudinal resistance R xx and Hall resistance R xy for device D4 with a ZrTe 5 layer thickness of 11.6 1.8 nm, b Same ± but for device D3 with a 9.5 1.5 nm thick ZrTe 5 layer, c Same but for device D2 with a 7.6 ± ± 1.2 nm thick ZrTe 5 layer. d Same but for device D1 with a 6.5 1.1 nm thick ZrTe 5 layer. ± Dotted lines highlight Shubnikov-de Haas oscillation minima. Numbers refer to the corresponding filling factor of the hole system.

SdH-oscillations in thicker samples with electron doping
In devices made out of thicker ZrTe 5 (layer thicknesses > 10 nm), oscillations appear for large electron doping (V g > 30 V). An example is shown in Fig   . c, as a function of magnetic field for V g = -12 V and for the temperatures listed in the Ω legend. d, Same as in panel c, but for the Hall resistance .

Attempt to extract the Berry phase and g-factor
By analyzing the location of successive Shubnikov-de Haas extrema for hole doping in device D1, discussed in the main text, we have attempted to extract the Berry phase for hole charge carriers. The procedure is identical to what was discussed previously in section S8 for electron doping on the sample with the thickest ZrTe 5 layer (device D5, 13 3 nm thick ZrTe 5 ). We

Arrhenius analysis for the determination of the activation energy gap of the = 0 state
In order to extract the energy gap for thermal activation of charge carriers when the electronic system condenses in the incompressible ground state at filling = 0, the longitudinal resistance was measured as a function of temperature at fixed magnetic fields ranging from 0 T to 10 T.
Such raw data sets obtained at small fields are shown in panel a of Fig. S13, whereas the data sets acquired at larger fields are shown in panel b. Above 2 T, curves nearly overlap with each other, indicating that the activation energy does not or is only weakly dependent on magnetic field. Activation energies are determined from an Arrhenius plot of the data with the inverse temperature as abscissae and ln(R xx ) as the ordinates. 8,9 An example of such an analysis is shown for B = 2 T in panel c. A plot of the activation energies can be found in panel d of Fig.   4 in the main text.

Conductivity tensor correction considering in-plane anisotropy
In ZrTe 5 , the large anisotropy of the Fermi surface is accompanied with a large difference of the effective mass and Fermi velocity among different crystal directions. This causes an anisotropy of the in-plane electrical transport properties 3,10,11 . In order to calculate the conductivity tensor elements ) and , the = /( resistivity in the y direction (c axis) is needed. However, it is not possible to measure at the same time as in our device geometry. Therefore, we introduce the anisotropy ratio to estimate its impact on the conductivities. Previous measurements on bulk α = / samples 3,5 reported anisotropy ratios larger than 10 using the expression . In ~~ * * these samples, the chemical potential was very close to the Dirac point. Typical carrier densities were 10 16 cm -3 and the mobilities were higher than 10 5 cm 2 V -1 s -1 indicating a low degree of impurity scattering. However, in an L-shaped ZrTe 5 device with a thickness of only 8 nm, comparable to the thicknesses of the samples discussed here, the anisotropy ratio was experimentally determined to be around 2 only. 10 Contrary to the previously mentioned bulk samples, the chemical potential in this L-shaped structure is pinned well inside the valence band as it exhibits a large carrier density (10 18

Methods
Sample preparation. High-quality single crystals of ZrTe 5 were synthesized from high-purity elemental materials (99.9999% zirconium and 99.9999% tellurium) using either the tellurium flux method or chemical vapor transport 3,4 . The crystals are needle like and are placed on atomically flat hBN in order to achieve the best sample quality. The starting hBN material was purchased from HQ graphene and mechanically exfoliated using sticky tape to obtain thin h- We note that just capping the active device layer with an inert hBN layer before exposig the sample to ambient air did not prevent sample degradation. It seems that air is still able to leak into the heterointerface and large hole doping is induced. Therefore in this work we have implemented the much more cumbersome approach described above. We have investigated samples that in addition were covered with hBN and the results were not necessarily better. In some cases, the results were worse (in terms of the measured mobility) than without encapsulation. This may be related to the significant number of bubbles that form at the interface between hBN and ZrTe 5 . We stress that commonly used techniques to reduce bubbles such as heating cannot be applied here, since ZrTe 5 degrades even during mild temperature treatments.
Measurements. The thickness of the exfoliated ZrTe 5 crystal was measured using non-contactmode atomic force microscopy (AFM) (NX-10 from Park System). The low-temperature magneto-transport measurements were performed in a dry cryostat system equipped with a superconducting magnet offering an axial field up to 14 T and a variable temperature insert that allows tuning the temperature between 1.7 K and 400 K. Low-frequency lock-in detection at 17.77 Hz was deployed for noise suppression to record the longitudinal and transverse resistance. Measurement currents ranged between 1 nA and 10 µA depending on the two-point resistance of the sample. A back-gate voltage was applied with the help of a Keithley 2400 Source Measure Unit.