Large Zeeman splitting induced anomalous Hall effect in ZrTe5

Berry phase effects have significant influences on the electronic properties of condensed matter. In particular, the anomalous Hall conductivity has been recognized as an intrinsic property of the systems with non-zero Berry curvature. Here, we present the anomalous Hall effect observed in the non-magnetic material ZrTe5, which hosts a large Zeeman splitting with Landé g-factor of 26.49. The quantum oscillation analysis reveals non-linear band dispersion near the top of valence band in bulk band structure, and no Weyl node forms with applied magnetic field. The anomalous Hall conductivity reaches 129 Ω−1 cm−1 at 2 K, and shows weak temperature dependence. All these combined with theoretical analysis suggest that the anomalous Hall effect observed in ZrTe5 originates from the non-vanishing Berry curvature induced by combining large Zeeman splitting and strong spin–orbit coupling. Remarkably, the anomalous Hall resistivity reverses its sign from negative to positive at a hydrostatic pressure P = 1.3 GPa, which confirms that the anomalous Hall effect in ZrTe5 is highly related to the band structure-dependent Berry curvature. Our results have verified the anomalous Hall mechanism in ZrTe5 and offer a new platform to study the anomalous Hall effect.


INTRODUCTION
The ordinary Hall effect arises from the Lorentz force bending of charge carriers' path under a magnetic field. On the other hand, the anomalous Hall effect (AHE) occurs due to the transverse velocity of carriers, which can be engendered by two classes of mechanisms 1,2 : the extrinsic mechanism owing to spin-dependent impurity scattering effects [3][4][5] , including skew-scattering and side jump, and the intrinsic mechanism, which is related to the nonzero Berry curvature 1,2,6-9 . The intrinsic mechanism is only dependent on the band structure and it is proposed that the non-zero Berry curvature can generate an anomalous velocity in the direction transverse to the electric field and external magnetic field, which gives rise to an anomalous Hall current resulting in intrinsic contribution to the Hall conductivity [10][11][12][13][14] . Such intrinsic contribution dominates when the Fermi level is located around the band crossing point where the Berry curvature is significantly enhanced 8,9 . Magnetic Weyl semimetals with broken time-reversal symmetry (TRS) are expected to exhibit strong intrinsic AHE owing to their large Berry curvature around the Weyl points. Recently, large intrinsic AHE has been observed in magnetic kagome lattice [15][16][17] and Heusler 18-20 compounds with possible non-trivial band topology. In principle, broken TRS can give rise to non-zero anomalous Hall conductivity in the presence of strong spin-orbit coupling (SOC), whether Weyl points are present or not 1,6 .
Here, we study the AHE in flux-grown non-magnetic singlecrystal ZrTe 5 and we propose an alternative route to realize AHE in ZrTe 5 , where the non-zero Berry curvature originates from the combination of large Zeeman splitting and strong SOC in our ZrTe 5 samples. Our samples show good crystalline quality, which enables the observation of quantum oscillations. By the analysis of quantum oscillation, we obtained non-linear band dispersion near the top of valence band in the bulk band structure and no Weyl node forms in our ZrTe 5 samples. We observed clear AHE that shows weak temperature dependence, indicating the anomalous Hall conductivity arises from intrinsic Berry curvature. Supported by theoretical analysis, it suggests the AHE in ZrTe 5 arises from the large Zeeman splitting in the presence of strong SOC, without the existence of Weyl nodes. Furthermore, the anomalous Hall resistivity reverses its sign at a hydrostatic pressure P = 1.3 GPa, which also corroborates that the AHE in ZrTe 5 arises from the band structure-dependent Berry curvature. Our work not only indicates the AHE observed in ZrTe 5 arises from the Berry curvature generated from combining large Zeeman splitting and strong SOC, but also provides a new insight into searching and studying the anomalous Hall effect in real materials.

RESULTS AND DISCUSSION
As depicted in Fig. 1a, the ZrTe 3 chains run along the a-axis and form a quasi-two-dimensional layer via linking zigzag chains of Te atoms along the c-axis. The quasi-two-dimensional layers stack along the b-axis crystalizing an orthorhombic structure with space group Cmcm ðD 17 2h Þ. Single layer ZrTe 5 is predicted to be a candidate for quantum spin Hall (QSH) insulator 21 , whereas it is still under debate whether the bulk ZrTe 5 is weak topological insulator or three-dimensional Dirac semimetal [22][23][24][25][26][27][28][29][30] . Recently, unconventional physical phenomena that may be related to the Berry phase effects have been observed in this compound [30][31][32][33] .
The magneto-transport properties of ZrTe 5 were measured with the current applied along a-axis and magnetic field applied along b-axis (see inset of Fig. 1c). Typical temperature-dependent 1 resistivity curves of two ZrTe 5 samples are shown in Fig. 1c, labeled as S1 and S2, respectively. The resistivity ρ xx first decreases with decreasing temperature and reaches a minimum at about 220 K, and then increases with further decreasing temperature down to 20 K, below which it tends to a saturation. We note that this resistivity behavior is different from that has been observed in chemical vapor transport (CVT)-grown ZrTe 5 showing a resistivity peak 25 , but is similar to the resistivity curve reported in refs. 31,34,35 . We ascribe this absence of resistivity peak to the much smaller density of impurities and defects in our samples, since Te deficiency can be largely reduced in our samples by using the Te-flux method and the modified growth parameters. This results in a semiconducting-like behavior in the resistivity.
Our first task is to clarify the band-topological properties of our ZrTe 5 samples. The transverse magnetoresistance (MR) of sample S1 at various temperatures from 2 to 15 K is shown in Fig. 2a. It exhibits pronounced Shubnikov-de Haas (SdH) oscillations and reaches a non-saturated value of~1200% at 9 T. By subtracting a smooth background, we obtained the oscillation patterns at different temperatures, as shown in Fig. 2b. Following the Lifshitz-Kosevich (LK) formula, the quantum oscillation of resistivity could be written as 36,37 Δρ / R T R D R s cos 2π where F is the oscillation frequency, R T , R D , and R s are damping factors corresponding to temperature, scattering and spin splitting, respectively. γ is the Onsager phase factor and δ is an additional phase shift ranging from −1/8 to 1/8 depending on the degree of the dimensionality of the Fermi surface. Since in our ZrTe 5 samples, the value ρ xx /ρ yx is~2. 8-8.7 in the oscillation (see Supplementary Fig. 1 Consequently, the oscillation of ρ xx is totally out-of-phase with σ xx , and we labeled the valleys of the oscillating pattern as the integer Landau indices. The Landau index n versus 1/μ 0 H at 2 K are shown in Fig. 2c. The oscillation frequency F, determined by the slope in the linear fit of the index plots, is 4.36 T. According to the Onsager quantization rule, F is proportional to the extremal cross-sectional area S F of the Fermi surface normal to the magnetic field, F = (ħ/ 2πe)S F . This yields a tiny cross-sectional area S F = 0.042 nm −2 , manifesting that there is a quite small Fermi pocket. The extrapolation intercept gives γδ = 0.44 ± 0.03. The Onsager phase γ is defined as γ = ½ − ϕ B /2π, where ϕ B is Berry phase. In an electron system that exhibits parabolic energy dispersion, ϕ B is zero and γ = 1/2. Nevertheless, in a massless Dirac electron system with linear energy dispersion, a non-trivial Berry phase ϕ B = π exists, which leads to a zero Onsager phase factor 38 . The extracted intercept of our ZrTe 5 sample indicates that the SdH effect comes from a non-linear dispersive band, which is in accordance with previous angle-resolved photoemission spectroscopy (ARPES) results 31 and our electronic band structure calculations (Fig. 1b), but in striking contrast to that previously found in CVT-grown c Temperature-dependent resistivity of ZeTe 5 with current along a-axis for samples S1, S2. The inset shows the schematic for electrical transport measurements.
ZrTe 5 (ref. 25 ). This also excludes the emergence of Weyl nodes induced by Zeeman splitting in our ZrTe 5 samples 39 .
By analyzing the temperature dependence of the oscillation amplitude based on the Lifshitz-Kosevich formula via R T ¼ 2π 2 kBT= hωc sinhð2π 2 kBT= hωcÞ (Fig. 2d), we got a tiny carrier cyclotron mass m * = (0.031 ± 0.001)m e . Prominent spin splitting exists between n = 2 and n = 1 Landau level (Fig. 2b). We extracted the spin-splitting parameter S = (1/2)g(m*/m e ) = FΔ(1/H) from the oscillation curves, where Δ(1/H) is the spacing between the split peaks corresponding to the up and down spins on one Landau level. We obtained a large Landé g-factor with the value of 26.49 in our samples, which is slightly larger than the result in previous report 25,40 . This large Landé g-factor indicates that there is strong SOC in ZrTe 5 and large energy splitting of electronic band can occur once external magnetic field is applied.
The Hall resistivity ρ yx (H) of sample S1 at 2 K is displayed in Fig. 3a. A typical anomalous Hall feature is observed, which is similar to that in previous report 31 . In the presence of AHE, the total Hall resistivity is composed of two components 1,2 where ρ O yx is the ordinary Hall resistivity that equals to R H B, and ρ A yx is the anomalous Hall resistivity. From the linear fitting of the highfield ordinary Hall resistivity, we obtained a hole carrier density n h = 2.68 × 10 18 cm −3 . Such low hole density is consistent with the small hole Fermi pocket revealed by our quantum oscillation analysis and band structure calculation. The anomalous Hall component ρ A yx (inset of Fig. 3a) was extracted by subtracting ρ O yx from the raw data. The Hall resistivity of ZrTe 5 at different temperatures is displayed in Fig. 3b. Clear AHE signal sticks to 50 K. When T > 50 K, a Hall resistivity peak appears at low field in the ρ yx (H) curves (see Supplementary Fig. 2), indicating the coexistence of both hole carriers and electron carriers above 50 K 34,35 . This peak broadens with increasing temperature, and the ρ yx (H) curves becomes almost linear when T > 160 K, implying that the hole carriers dominate at high temperature. Since ZrTe 5 is a non-magnetic material, the AHE we observed is not associated with spontaneous magnetization, or the topological Hall effect contributed to the non-zero spin chirality 1,2,41 . Further, the anomalous Hall conductivity jσ A H j at 2 K is 129 Ω −1 cm −1 under 5 T, which is much larger than the anomalous Hall conductivity originated from the extrinsic mechanism in the bad metal hopping regime (typically 0.01~1 Ω −1 cm −1 ) 1,2,42 . As displayed in Fig. 3c, the anomalous Hall conductivity jσ A H j shows quite weak temperature dependence, as expected for the intrinsic AHE. All these observations suggest that the AHE in ZrTe 5 arises from the intrinsic Berry curvature. Usually, the large Berry curvature is related to the Weyl nodes and intrinsic AHE has been observed in systems with magnetic Weyl fermions, where the non-zero Berry curvature is generated by the Weyl nodes [15][16][17][18][19][20] . However, there is no Weyl node appearing in our ZrTe 5 sample at zero or non-zero magnetic field. First, as ZrTe 5 is non-magnetic centrosymmetric material, Weyl nodes cannot exist at zero magnetic field. Second, quantum oscillation analysis and band structure calculation reveal non-linear band dispersion near the top of valence band and there is no Dirac node in our ZrTe 5 samples. This is also supported by previous ARPES data, which shows a hole Fermi pocket and non-linear band  dispersion in similar ZrTe 5 samples 31 . Thus, the Weyl nodes arising from splitting Dirac nodes under magnetic field cannot exist either. Third, the topologically trivial Berry phase revealed by the quantum oscillation analysis also suggests that no Zeeman splitting-induced Weyl nodes form under external magnetic field in our samples. Moreover, we can estimate the Zeeman energy to be 7.7 meV at 5 T using g-factor~26.49, which is much smaller than the band gap of 46 meV from the band structure calculation. Previous ARPES and scanning tunneling microscopy (STM) results [22][23][24]43 reported a 20-100 meV band gap in ZrTe 5 , which is in accordance with our theoretical calculation and larger than the estimated Zeeman energy~7.7 meV. Magneto-spectroscopy measurement also reported a Dirac mass term, corresponding to a 10 meV band gap in the ZrTe 5 thin flake 40 , which is also larger than our estimated Zeeman energy. These results indicate that the Zeeman splitting cannot induce band crossing and generate Weyl nodes in our ZrTe 5 samples. Given all above, there is no Weyl nodes formed at zero or non-zero magnetic field and the non-zero Berry curvature does not generate from Weyl nodes in our ZrTe 5 samples.
Theoretically, non-zero Berry curvature can be generated by combining large Zeeman splitting and strong spin-orbit coupling (SOC), regardless of whether Weyl node exists or not 1,6 . In nonmagnetic centrosymmetric systems, such as ZrTe 5 , the Berry curvature vanishes identically throughout the Brillouin zone from symmetry consideration when TRS preserves. Once magnetic field is applied, TRS is broken and Zeeman effect splits the doubly degenerate energy band into two separate ones. The spin-orbit interaction couples the spin-up and spin-down bands together, and transfers the time-reversal violation from spin degree of freedom to the orbital motion, giving rise to a non-zero Berry curvature 6 . This Berry curvature could be very large in materials with large spin-orbital coupling, such as ZrTe 5 . In general, the anomalous Hall conductivity can be expressed by the integral of Berry curvature over the Brillouin zone as 1 : where σ ac represents the in-plane anomalous Hall conductivity with magnetic field applied along b-axis, f(ε n (k)) is the Fermi-Dirac distribution and Ω n,b (k) is the Berry curvature of the n-th band along b-axis. Therefore, when external magnetic field set in, TRS is broken and the combination of large Zeeman splitting and strong SOC gives rise to a non-zero Berry curvature, which leads to the anomalous Hall conductivity in ZrTe 5 .
To further investigate the AHE in ZrTe 5 , we have performed detailed measurements under hydrostatic pressure on sample S2. Previous works have reported pressure can largely tune the structure of ZrTe 5 and induce topological phase transition and superconductivity in ZrTe 5 samples with resistivity peak 44,45 . Since the Berry curvature is only related to the electronic structure, the AHE should respond dramatically to the external pressure in ZrTe 5 in which the band structure is very sensitive to the changes in lattice parameters 21 . Figure 4a shows the pressure-dependent Hall resistivity ρ yx of S2 at 2 K and the anomalous Hall resistivity component at various pressures is shown in Fig. 4b. Figure 4c displays the pressure dependence of anomalous Hall resistivity ρ A yx . The applied pressure reduces the negative anomalous Hall resistivity and the sign of the AHE changes from negative to positive at P = 1.3 GPa. Above this pressure, the positive anomalous Hall resistivity increases gradually.
The anomalous Hall conductivity arising from Berry curvature is sensitive to the Fermi level shift around the band crossing point 13 . The sign change behavior of AHE has been observed experimentally when the Fermi level passes though the band crossing point by controlling doping or temperature 9,[46][47][48] . Nevertheless, we have demonstrated that there is no band crossing induced by Zeeman splitting in ZrTe 5 at ambient pressure. The hole carrier density increases continuously with pressure and carrier-type does not change (Fig. 4d). These suggest that the sign reversal of AHE does not originate from the Fermi level scanning through the band crossing point. Instead, since bulk ZrTe 5 is theoretically proposed to locate in the vicinity of a transition from strong topological insulator to weak topological insulator 21 , the pressuretuned band structure may lead to a significant change of the total Berry curvature, which can result in the sign reversal of AHE. Moreover, the pressure dependence of carrier density changes its slope at P = 1.3 GPa, implying that the Berry curvature of the occupied band has a drastic change around the sign-reversal pressure. To elucidate the sign-reversal behavior of AHE, we calculated the band structure of ZrTe 5 under different pressures, as shown in Supplementary Fig. 3. With increasing pressure, the valence band at Γ point moves downward gradually and a hole Fermi pocket emerges near Ζ point at about 1.3 GPa. Consequently, it implies that the Fermi pocket near Ζ point makes an inverse contribution to the total Berry curvature and it dominates above 1.3 GPa, resulting in the sign reversal of AHE. In addition, our numerical calculated results indicate that ZrTe 5 is in the strong topological insulator phase between 0 and 9 GPa. However, the topological characteristic of bulk ZrTe 5 is still controversial [22][23][24][25][26][27][28][29][30] and we can only exclude Weyl nodes in our samples, but cannot confirm the topological phase of ZrTe 5 based on our experiments.
In summary, we have studied the AHE in ZrTe 5 with non-linear band dispersion near the top of valence band, and verified that it originates from the combination of large Zeeman splitting and strong SOC, without the emergence of Weyl nodes. Our results reveal that ZrTe 5 , owing to its simple electronic structure near the Fermi level, is a perfect candidate for investigating the AHE in nonmagnetic system. Moreover, the continuous control of AHE in ZrTe 5 under hydrostatic pressure provides an ideal platform for exploring how the Berry curvature is affected by the detailed electronic structures.

Single-crystal growth
The single crystals of ZrTe 5 in our studies were grown using the Te-flux method. Zr slug and Te shots in an atomic ratio of 1:49 were loaded into a canfield crucible set, then sealed into a silica ampoule under vacuum. The ampoule was heated to 1000°C and kept for 12 h to homogenize the melt. It was cooled down to 650°C rapidly, and then cooled to 460°C in 60 h. ZrTe 5 crystals were isolated from Te flux by centrifuging at 460°C.

Transport measurement
The samples are needlelike shape with typical dimensions of 1.8 × 0.08 × 0.03 mm. The electrical contacts were arranged in a conventional Hall-bar configuration with the current along a-axis and magnetic field along b-axis.
To have a good contact, we first deposited 100 nm Au on the electrode pattern and then contacted the electrical leads with platinum wires by silver paint. The magneto-transport measurements were performed in a Quantum Design PPMS-9T system. Hydrostatic pressure was established in Quantum Design high-pressure piston-cylinder cell with Daphne 7373 oil used as pressure transmitting medium. Pressure was calibrated by measuring the superconducting transition temperature of pure tin with an uncertainty of 0.02 GPa.

Density functional theory (DFT) calculations
The first-principles calculations based on the density functional theory (DFT) were implemented in Vienna ab-initio Simulation Package (VASP) code 49,50 . The Perdew-Burke-Ernzerhof (PBE) functional of generalized gradient approximation (GGA) 51,52 was employed as the exchangecorrelation potential, and the projector augmented wave (PAW) method 53 was chosen to treat core-valence electron interactions. The plane wave cutoff energy was 460 eV, with the Brillouin zone sampled by a 12 × 12 × 7 mesh. Spin-orbit coupled (SOC) effect was taken into account in all calculations.
Z. Sun et al.