Electronic evidence of temperature-induced Lifshitz transition and topological nature in ZrTe5

The topological materials have attracted much attention for their unique electronic structure and peculiar physical properties. ZrTe5 has host a long-standing puzzle on its anomalous transport properties manifested by its unusual resistivity peak and the reversal of the charge carrier type. It is also predicted that single-layer ZrTe5 is a two-dimensional topological insulator and there is possibly a topological phase transition in bulk ZrTe5. Here we report high-resolution laser-based angle-resolved photoemission measurements on the electronic structure and its detailed temperature evolution of ZrTe5. Our results provide direct electronic evidence on the temperature-induced Lifshitz transition, which gives a natural understanding on underlying origin of the resistivity anomaly in ZrTe5. In addition, we observe one-dimensional-like electronic features from the edges of the cracked ZrTe5 samples. Our observations indicate that ZrTe5 is a weak topological insulator and it exhibits a tendency to become a strong topological insulator when the layer distance is reduced.

geometry, the electric field vector of the incident laser is perpendicular to the plane formed by the incident light and the lens axis of the electron energy analyzer, while in the ppolarization geometry, the electric field vector of the incident light lies within such a plane.
Supplementary Fig. 1 shows results of the constant energy contours and band structures measured under these two distinct polarization geometries. It is clear that the topology of the constant energy contours and the band dispersions are similar for these two cases. But the spectral weight distribution shows obvious dependence on the polarization geometry, as seen more clearly by comparing high binding energy contours (Supplementary Fig. 1d versus Supplementary Fig. 1j).  2c and Supplementary Fig. 2h) indicates that the hole-like energy bands consist of neither one single MDC or EDC peaks, nor two sharp MDC or EDC peaks. This can be seen more clearly in Supplementary Fig. 2e and Supplementary Fig. 2j where the EDCs at the Γ point are plotted. It is clear that, in this case, the seemingly two sub-bands in the secondderivative images (Supplementary Fig. 2e and Supplementary Fig. 2j) do not correspond to two sharp peaks in its corresponding EDCs. Rather, the EDC of the lower-branch hole-like bands represents a broad continuum that is encompassed by two relatively sharp edges. The position of the two "sub-bands" in the second-derivative images actually corresponds to the location of the two edges as marked in Supplementary Fig. 2d-e and Supplementary Fig.   1 2i-j by pink arrowed lines. These results clearly indicate that the hole-like bands near Γ are not composed of two sub-bands, but represent an envelope of spectral weight encompassed by two edges. The bandwidth of the hole-like band, i.e., the energy difference between the two edges, is maximal at the Γ point, and decreases with the momentum moving away from the Γ point. It is also clear that the bandwidth gets wider with decreasing temperature, as seen from comparing Supplementary Fig. 2(d-e) and Supplementary Fig. 2(i-j).

Supplementary Note 3. Calculated Electronic Structures of ZrTe 5
Extensive band structure calculations have been performed on ZrTe 5 by considering the spin-orbit-coupling effect, surface and bulk states, k z effect, and the effect of layer distance along b-axis on the topological property. Supplementary Fig. 3 shows bulk electronic structure under different layer distance along b-axis. It shows a clear topological phase transition from strong topological insulator to weak topological insulator at about 1.04b 0 (b 0 = 14.502 A is the lattice constant b of ZrTe 5 at room temperature[1])). Considering the existance of an energy gap and its size at Γ point and the topological nature of ZrTe 5 samples, we beileve that our experimental results are more consistent with the calculated results with a layer distance of 1.06b 0 . Supplementary Fig. 4a shows bulk electronic structure of ZrTe 5 in the weak topological insulator case. The corresponding constant energy contours at different energies are shown in Supplementary Fig. 5. The primary features of the calculated bulk electronic structure are consistent with ARPES data (Fig. 1) in terms of the evolution of constant energy contours with energy, nearly-linear band dispersions and full gap formation near the Brillouin zone center. However, one apparent discrepancy exists as to the lineshape of the photoemission spectra. The EDC for the lower branch hole-like band is not a single-peak, nor a combination of two sharp peaks, but a spectral continuum encompassed by two edges ( Supplementary Fig. 2). In order to understand such a difference, we performed band structure calculations using a slab method. Supplementary Fig. 4b shows the calculated band structure of ZrTe 5 using the slab method, by taking the lattice constant along the b-axis as 1.06b 0 to reduce the interlayer interaction that gives rise to a weak three-dimensional topological insulator. Supplementary Fig. 4c shows calculated band structure of ZrTe 5 using the slab method by taking the lattice constant along the b-axis as 1.0b 0 which results in a strong three-dimensional topological insulator. In the calculated results using the slab method, the bands are composed of a spectral continuum that are consistent with our measured results.
It is also clear that with the decreasing of the lattice constant b, the bandwidth of the LB hole-like band near Γ gets broader; this is also consistent with the measured results (Supplementary Fig. 2) that the bands get broader with decreasing temperature, accompanied by the shrinking of the lattice constant b. The broad photoemission spectra in the hole-like bands near Γ can be explained as due to finite k z resolution in the photoemission process, in conjunction with k z -dependence of the band structure for a three-dimensional electronic structure. In the case of strong topological insulator ( Supplementary Fig. 4c), topological surface state appears, as marked by the red lines in Supplementary Fig. 4c and Supplemen-  At low temperature, the observation of four small β electron pockets, together with the α electron pocket at Γ, is consistent with the quantum oscillation measurements where two electron pockets are detected [3,4]. The appearance of these electron pockets will also enhance electrical conductivity at low temperature.

Supplementary Note 6. Electronic Structure of ZrTe 5 at 2K
To check whether the gap between the LB and UB bands is closed and possible emergence of the in-gap topological surface state at low temperature, we carried out ARPES measurement on ZrTe 5 to a very low temperature, ∼2 K ( Supplementary Fig. 9). At such an extremely low temperature, we detected no surface state in the bulk gap. It indicates that ZrTe 5 is still in weak topological state at 2 K.

Photon Energies
We have carried out ARPES measurements on ZrTe 5 samples using various laser photon energies, as shown in Supplementary Fig. 10a. It shows that the gap at Γ point decreases with the photon energy changing from 6.13 to 6.99 eV. Our results indicate that the electronic structure measured at 6.99 eV is close to Γ point. By taking the inner potential V 0 at 7.5eV for ZrTe 5 from literature[5]) , we also estimated the k z for various photon energies, as shown in Supplementary Fig. 10b. The measured data are consistent with the calculation ( Supplementary Fig. 4a), indicating that our 6.99eV data is close to the Γ point. According to the discussion in the main manuscript, because the relative energy position change of the upper conduction band and lower valence band near Γ with temperature, we conclude that the gap exists at Γ in the three-dimensional Brillouin zone. The nice agreement between our calculations and measurements shown here is a good indication on the reliability of our band structure calculations.

Supplementary Note 8. Orientation of Quasi-One-Dimensional Features in ZrTe 5
We measured two samples to check the orientation of the quasi-one-dimensional feature in ZrTe 5 , as shown in Supplementary Fig. 11. For the sample a, when the cracked lines on the surface are horizontal ( Supplementary Fig. 11a), the observed quasi-one-dimensional features run vertically (Supplementary Fig. 11b and Supplementary Fig. 11c). For another sample b, where we put the cracked lines on the surface vertically ( Supplementary Fig.   11d), i.e., it is 90 degree rotated with respect to sample a during the measurement, the observed quasi-one-dimensional features run horizontally. This confirms that the quasi-onedimensional feature is related to the intrinsic property of ZrTe 5 , rather than an artifact or due to photoemission matrix element effects. It also shows that the observed quasi-onedimensional feature is closely related to the one-dimensional lines and edges on the sample surface.