Evidence for topological type-II Weyl semimetal WTe2

Recently, a type-II Weyl fermion was theoretically predicted to appear at the contact of electron and hole Fermi surface pockets. A distinguishing feature of the surfaces of type-II Weyl semimetals is the existence of topological surface states, so-called Fermi arcs. Although WTe2 was the first material suggested as a type-II Weyl semimetal, the direct observation of its tilting Weyl cone and Fermi arc has not yet been successful. Here, we show strong evidence that WTe2 is a type-II Weyl semimetal by observing two unique transport properties simultaneously in one WTe2 nanoribbon. The negative magnetoresistance induced by a chiral anomaly is quite anisotropic in WTe2 nanoribbons, which is present in b-axis ribbon, but is absent in a-axis ribbon. An extra-quantum oscillation, arising from a Weyl orbit formed by the Fermi arc and bulk Landau levels, displays a two dimensional feature and decays as the thickness increases in WTe2 nanoribbon.

As the thickness decreases, the metallic behavior of WTe 2 slab becomes semiconductive, indicating a band-gap opening in ultrathin samples. 6 To obtain the quantum oscillation c and d, respectively. We add the instrument error 10% into the data of b-d.
The fast decay of the FFT amplitude with temperature follows the Lifshitz-Kosevich Figure 5Fermi surface mapping of bulk WTe 2 . a, SdH oscillation with magnetic field rotation in ac plane; θ = 0° corresponds to the direction of the magnetic field parallel to c-axis. b, FFT spectra of SdH oscillations in ac plane. c, SdH oscillation with magnetic field rotated in bc plane; θ = 0° corresponds to the direction of the magnetic field parallel to c-axis. d, FFT spectra of SdH oscillations in bc plane. The extracted oscillation frequency of four pockets (two electrons,  and ; and two holes,  and ) is given in e, ( This information about the Fermi surface comes from the magneto-transport data measured in the bulk WTe 2 , which was prepared by focused ion beams (FIB) with a dimension of 10 m  nm). The chiral-anomaly-induced positive conductivity can be fitted by 9,10 (2)

Supplementary Note 1. Modified Weyl orbit model in type-II Weyl semimetals.
A number of theoretical and numerical works on the Landau levels in type-II Weyl semimetals [11][12][13] showed that the chiral Landau levels are still visible when the magnetic field is applied along the tilting direction of Weyl cone. Motivated by these work, we calculated the Landau levels in type-II Weyl semimetals for the exact same configurations as that in WTe 2 : Weyl cone tilting along Y direction, 14 Figure 12a), which is consistent with recent calculations. [11][12][13] This is why we can still observe the chiral anomaly induced negative longitudinal magnetoresistance in b-axis WTe 2 nanoribbons (Fig. 3b&3c). Whereas, when the applied magnetic field is normal to the tilting direction of Weyl cones the chiral Landau levels are missing (Supplementary Figure 12b), which agree well with the recent calculations as well. [11][12][13] In Dirac and type-I Weyl semimetals, the chiral Landau levels ( 0  n ) act as the only one-way "conveyor-belt", because the bulk Landau levels near the Fermi level in Dirac and type-I Weyl semimetals are gapped. 16 However, in type-II Weyl semimetals, when the magnetic field is  Figure 8) is considered. This mean free path is even longer than the thickness of the thickest sample (~35 nm) in which we could observe the Weyl orbit as shown in Fig. 4. Therefore, the electrons will finish the transport between two , L is the thickness of nanoribbon). Therefore, The total time completing a Weyl orbit is . Given the average velocity along c is Therefore, we found that the key results in the Weyl orbit in type-II Weyl semimetals remain the same as that in type-I Weyl semimetals that was proposed by A.C. Potter, including the quantum oscillation frequency In the case of current jetting, dips, humps and even the negative voltage will appear in the angular dependence of longitudinal resistance R(). 17,18 To confirm the validity of our data, we measured R() of a typical b-axis nanoribbon, as displayed in Supplementary Figure 13. Neither dips nor humps were observed in the R() curve. This observation confirms that current jetting is not dominated in our nanoribbon samples.
Another evidence to exclude the current jetting is enough large aspect ratio l/w of the sample length l and width w. According to the requirement of homogenous current distribution, the aspect ratio l/w should be larger than A , where A is the resistance anisotropy. 18 The typical resistance anisotropy A in Supplementary Figure 13of  , the aspect ratio l/w is, therefore, required to be higher than 6.7 2.6  . In our nanoribbon devices, the width is 0.6-1 m and the length between two voltage probes (V + , V -) is 8-12 m with a typical aspect ratio l/w ~12. Even though we take A as 57 from the bulk value ( Supplementary   Figure 13b), the required aspect ratio is 57 7.5  , which is still smaller than the real aspect ratio value ~12. Hence, the large aspect ratio l/w in our nanoribbons ensures a homogenous current distribution between two voltage probes.
The resistance maximum related to Knudsen effect normally peaks at maximal boundary scattering of the bent electron trajectories, c 2r L  , in which the cyclotron radius is c F / r k eB   . [19][20][21] Therefore, the peak position of magnetic field in the MR B  curves will shift towards higher magnetic field with the decrease of nanoribbon thickness (L). Supplementary

Supplementary Note 3. Thickness dependence of Weyl orbit oscillation phase
According to the theoretical prediction of Weyl orbit quantum oscillation, the dependence of peak position of Weyl orbit on thickness (L) is decided by 16 From Supplementary Eq. (4), the oscillation phase offset of the peaks in the curve ) is expected to be thickness (L) dependent although the oscillation period 1/ B  is independent on thickness. However, it is difficult to identify and extract the precise position (B -1 ) of Weyl orbit due to the superposition of Weyl orbit and bulk Fermi surfaces in SdH oscillations, because the peaks are close to each other in FFT spectra. Even though we obtain the position of nth Landau levels from the inverse FFT spectrum, the large uncertainty of the position will further hinder us from analyzing the oscillation phase on the thickness dependence. This is similar to the case in Moll's work. 21 Nevertheless, we can expect another kind of oscillation phase shift in the asymmetric Weyl orbit peak in FFT spectrum, arising from the non-adiabatic correction related to the field-induced tunneling between Fermi-arc states and bulk states. 16,21 We analyzed the Weyl orbit peak in FFT spectrum in details. As Supplementary Figure 15 shown, we found that Weyl orbit peak in several curves are quite asymmetric, compared to the peaks of bulk Fermi surfaces in the same spectrum. The asymmetric feature of Weyl orbit is a strong indication of subtle deviation from rigorous periodicity. The shift of peak position B n towards higher fields will lead to an asymmetric broadening towards low frequency in the FFT spectrum, which is expected for the non-adiabatic correction. 21 The asymmetric feature of Weyl orbit peak in our curves and nonadiabatic corrections are consistent with the observation by Moll et al. 21

Supplementary Note 4. Examination of Klein tunneling in WTe 2
Recently, O'Brien et al theoretically predict Klein tunneling through a single type-II Weyl point. 23 It is proposed to generate an additional quantum oscillation frequency F + -F - of two touching Fermi pockets when the magnetic field is perpendicular to the extremal area of Fermi pockets and Fermi level is near the position of potential Weyl points. As discussed in the literatures, this effect is very sensitive to the position of the chemical potential. In single crystal samples of WTe 2 , both the first principle calculations and ARPES data indicate that the type II Weyl points are located about 50~80 meV above the chemical potential. Therefore we are not very optimistic to see the Klein tunneling in this particular sample. As shown in Supplementary   Figure 16a, in bulk WTe 2 two hole pockets (, , light blue) and two electron pockets (, , purple) are present in the Brillouin zone 8 . We should note that one smaller electron pocket  (and one smaller hole pocket ) locates inside another larger electron pocket  (and another 29 larger hole pocket ), which is similar to a set of Russia dolls. 8 The potential Klein tunneling should occur between the larger electron () and larger hole () pockets. According to our experimental 3D mapping of Fermi pockets of WTe 2 (Supplementary Figure 5), the extremal area of Fermi surface in WTe 2 is the bc plane. We apply the magnetic field again, therefore, along a-axis to check the QO frequency very carefully. As shown in Supplementary Figure 16b, we found that location of four Fermi pockets is consistent with previous result in Supplementary . However, the data at low frequencies (0~100 T) is too noisy to identify the quantum oscillation due to Klein tunneling. This potential QO frequency F + -F - is neither evident nor can be excluded in our experiments. Hence we can't make a clear conclusion based on our observations. Therefore the fact that there is no obvious quantum oscillation at frequency F + -F - in our sample is consistent with the previous results from the ARPES measurements and DFT calculations.