Tunable Berry curvature and transport crossover in topological Dirac semimetal KZnBi

Topological Dirac semimetals have emerged as a platform to engineer Berry curvature with time-reversal symmetry breaking, which allows to access diverse quantum states in a single material system. It is of interest to realize such diversity in Dirac semimetals that provides insight on correlation between Berry curvature and quantum transport phenomena. Here, we report the transition between anomalous Hall and chiral fermion states in three-dimensional topological Dirac semimetal KZnBi, which is demonstrated by tuning the direction and flux of Berry curvature. Angle-dependent magneto-transport measurements show that both anomalous Hall resistance and positive magnetoresistance are maximized at 0° between net Berry curvature and rotational axis. We find that the unexpected crossover of anomalous Hall resistance and negative magnetoresistance suddenly occurs when the angle reaches to ~70°, indicating that Berry curvature strongly correlates with quantum transports of Dirac and chiral fermions. It would be interesting to tune Berry curvature within other quantum phases such as topological superconductivity.


INTRODUCTION
Berry's phase (Φ B ) is a geometric quantum phase, which is manifested in the magnetic field (B)-induced cyclotron motion of fermions along closed loop in momentum (k) space around a Dirac point 1,2 . In transport experiments, the Φ B has been verified from the analysis of Shubnikov-de Haas (SdH) oscillation, which gives a nonzero intercept of zeroth Landau level (LL) 3,4 . The Φ B with a quantized π value is defined by an integral of the Berry curvature (Ω k ) over the closed Fermi surface (FS) in k space, bearing a resemblance to an electric monopole of Gauss law 1,2 . This effective magnetic field of Ω k is strongly correlated with a diversity of quantum transport phenomena, such as anomalous and quantum Hall effects 5,6 , chiral magnetic effect 7,8 , and topological superconductivity 9 . The nonzero Ω k can be generated through either time-reversal and inversion symmetry breakings and the resulting Ω k is locally distributed in the Brillouin zone 1,2 . Regardless of the magnetization of a material, the modulation of Ω k , which can be achieved by manipulating symmetries and band structures, will dominate quantum transport properties of topological materials.
Three-dimensional topological Dirac semimetal (3D TDS) state can be a central hub to access diverse topological states with a distinct quantum transport phenomenon by lifting a Dirac degeneracy with external stimuli 10 . For instance, a topological phase transition (TPT) from the 3D TDS to Weyl semimetal occurs by time-reversal symmetry breaking under B, which splits a degenerate Dirac cone into two Weyl cones with opposite Ω k hotspots 10,11 (Fig. 1a). Interestingly, the axial splitting of 3D Dirac cones can be adjusted according to the direction of external B, allowing the tuning of the Ω k with the incidence angle (θ) of B. This angular dependence of the Ω k has been of interest in the TDSs such as Na 3 Bi (refs. 8,11 ), Cd 3 As 2 (refs. 12,13 ), and ZrTe 5 (refs. 5,6,14,15 ) due to the possibility to show a variety of exotic quantum transport phenomena, such as the negative magnetoresistance (NMR) induced by chiral anomaly 7,8,13,14 and anomalous Hall effect (AHE) 5,15 originate from the Ω k . Beyond that, tuning the Ω k would be important for controlling the mixed and separated states of such quantum transport phenomena to understand the correlation between Dirac and Weyl fermions in the topological semimetals, as we found that the NMR and AHE occurred with a reversal behavior on the angle-dependent Ω k (Fig. 1b). For the precise tuning of the Ω k , it is necessary to exploit a 3D TDS material with a simple band structure having only two Dirac cones robustly protected by a crystal symmetry and a FS topology accommodating only a single monopole without disturbance from other bands.
Such a 3D TDS can be found in ABC compounds (A: alkali metal, B: transition metal, C: chalcogen element) with a P6 3 / mmc group symmetry, whose crystal structure is constructed by an alternative stacking of honeycomb BC layers and A ion layers, guaranteeing the TDS state from the threefold rotational and inversion symmetries 16 (Fig. 1c). Among them, KZnBi has been found as the 3D TDS with only two Dirac cones that are exclusively composed from Zn-4s and Bi-6p orbitals of the ZnBi honeycomb lattice 17 (Fig. 1d), providing a platform to study the transport phenomena of Dirac quasiparticles. In contrast to Weyl semimetals that usually have a complex band structure with multiple Ω k hotspots 2 , the simple band structure of the 3D TDS KZnBi allows to resolve the Ω k distribution under B explicitly and to modulate the Ω k with a weak B efficiently ( Supplementary Fig. 1). Here, from the θ-dependent magnetotransport measurements, we find that the tunable Ω k determines the quantum transport characters of Dirac fermions (DFs) and chiral fermions (CFs), demonstrating an abrupt transition between anomalous Hall and CF states. 1

Quantum transport of DFs
First, we examined the basic character of quantum transport of the DFs in the 3D TDS KZnBi sample. Figure 2a shows the metallic behavior in the T-dependent electrical resistivity (ρ xx (T)) at 4 K ≤ T ≤ 300 K. In the MR measurements, we observed the unsaturated behavior and obtained the high value of~700% under 14 T at 4 K (Fig. 2b). It is noticeable that multiple SdH oscillations appear in the MR data in the range of 0 ≤ B ≤ 14 T, implying a high mobility of the DFs as a characteristic transport feature of TDS materials 8,12,13 . These clear Dirac transports are ascribed to the planar ZnBi honeycomb layers responsible for the formation of 3D Dirac cones and to the high-quality crystallinity of single crystal KZnBi with a very low impurity level. Moreover, a simple Dirac band benefited from the planar ZnBi honeycomb layer (Fig. 1c, d) is also featured in the remarkably low magnitude of B for the quantum limit. To investigate the transport behavior beyond the quantum limit, we performed tunnel diode oscillator (TDO) measurements. We presented the graph of B versus 1/ΔF TDO where the inverse of frequency variation, ΔF TDO , in the TDO data is proportional to the resistance in the transport data (Fig. 2c). A linear increase of the 1/ΔF TDO beyond quantum limit of~16 T (red dashed line in Fig. 2c) implies a crossing of the zeroth LL (n = 0) with the Fermi level (E F ) (refs. 3,18,19 ).
Next, we characterized the 3D Dirac state of the KZnBi from the analysis on the SdH oscillations, which reflects the Landau quantization under B. The longitudinal resistance (R xx ) component is given in the form of dR xx /dB to show the SdH oscillations, revealing the oscillation frequency, F = 1/d(1/B) = 4.69 T (Fig. 2d). Based on the Onsager relationship F = (ħ/2πe)S F , where ħ is reduced Plank's constant and e is elementary charge 3 , a small cross-sectional area (S F = 0.00046 Å −2 ) of the FS is obtained. For a circular FS, a Fermi wavevector also shows a small value (k F = 0.012 Å −1 ) is extracted from the relation 20 of S F = πk F 2 , implying a simple band structure with sharp 3D Dirac cones in the KZnBi. The extremely small value of the effective mass (m * = 0.012 m 0 where m 0 is electron mass) is obtained at B = 4.03 T (inset of Fig. 2d), by fitting the T-dependent oscillation amplitude to Lifshitz-Kosevich formula 3 6 meV below the Dirac node is calculated 20 . All values we extracted from the SdH oscillations are well matched with the previous report 17 . These physical parameters of the DFs in the KZnBi well describe the characteristic feature of the DFs (Supplementary Table 1 for all physical parameters).

Identification of nontrivial Φ B from Landau fan diagram
The 3D Dirac cones of the KZnBi accommodate a nontrivial Φ B , which can be obtained from the analysis of the SdH oscillations. According to the Lifshitz-Onsager rule for the SdH analysis 3 where R 0 is the background resistance, A (B, T) is the amplitude of the SdH oscillation, and γ is the values of 1/2 -Φ B /2π, which are determined by the value of Φ B , where it is close to 0 for nontrivial Φ B of π , and δ is the degree of two dimensionality of FS (δ = 0 for 2D and ±1/8 for 3D). In case of a nontrivial Φ B , the value of |γ -δ| = |1/2 -Φ B /2π -δ | located between 0 and 1/8. For the identification of nontrivial Φ B , we displayed the LL fan Tuning of Ω k and θ-dependent AHE The Ω k emerges and acts like an effective B in the 3D TDS when time-reversal symmetry is broken, giving net anomalous Hall current induced by the transverse velocity (v A = E × Ω k ) in the electrical field (E) (refs. 5,15 ). To verify the AHE in the nonmagnetic 3D TDS KZnBi, we measured the Hall resistivity (ρ yx ) in the range of −2 T ≤ B ≤ 2 T at 4 K (see Supplementary Fig. 5 for the ρ yx in the whole measured range of −14 T ≤ B ≤ 14 T) as plotted in Fig. 3a. It is clear that the ρ yx shows the kink around zero B, indicating the occurrence of AHE due to the Ω k . We note that the easily  distinguishable contribution of AHE (red arrow) indicates that the KZnBi crystal is in the clean limit where the quasiparticle lifetime is sufficiently long, as evidenced by the high mobility in the previous report (ref. 17 ). The anomalous Hall resistivity (ρ AHE yx ) is separated from the total ρ yx ¼ ρ OHE yx þ ρ AHE yx , where ρ OHE yx is the contribution of ordinary Hall effect (blue dotted line in the high B region) extracted by a linear fitting 5,15 . The ordinary Hall coefficient (R H ), which is obtained from the linear slope, indicates that the DFs of the present KZnBi are hole carriers with a density of~3.3 × 10 18 cm −3 . This p-type carrier is in a good agreement with the recent results of angle-resolved photoemission spectroscopy and DFT calculation 17 . Although both Dirac band and small pocket are lying at the E F~1 00 meV, we showed that two-carrier model analysis cannot explain the nonlinearity of the Hall resistivity (Supplementary Fig. 6), ruling out the possibility of ordinary Hall effect. Also, the magnitude of ρ AHE yx for all measured samples shows the E F independent value of~80 μΩ · cm, clarifying that the AHE was induced by Ω k (Supplementary Fig. 7). We note that the absence of the Zeeman splitting-induced separation in the SdH oscillation indicates the AHE does not originate from the spin-polarized massive DFs 15 . Figure 3b shows the separated ρ AHE yx , which has strong dependences on both the strength of B and θ between B and z axis of sample. At the θ = 0°, the ρ AHE yx increases with B and saturates to the value of~78 μΩ · cm at 4 K under 2 T. The corresponding σ AHE yx ≈ 80 Ω −1 · cm −1 of the KZnBi is a high value compared to that (typically 0.01−1 Ω −1 · cm −1 , ref. 15 ) of the conventional materials with the AHE induced by extrinsic origins and it is comparable to that of topological semimetal, ZrTe 5 , with the AHE induced by nontrivial Φ B ( Supplementary  Fig. 5e). This phenomenon strongly indicates that the AHE of nonmagnetic KZnBi is intrinsically driven by the nonzero Ω k . As illustrated in the Supplementary Fig. 1, the θ-dependent Ω k can affect the transport properties of the AHE and reveal the correlation between B and Ω k . Indeed, a peculiar angular variation of the Ω k is uncovered from the θ-dependent ρ AHE yx curves at 4 K (Fig. 3b, c and Supplementary Fig. 5 for raw data). For each θ-dependent ρ AHE yx curve, the saturated ρ AHE yx value appears at the different magnitude of B. The θ-dependent curve of the ρ AHE yx saturation values (red dots in Fig. 3c) shows a sudden decrease from~70°while the ρ OHE yx (θ) (blue dots in Fig.  3c) smoothly follows a cosine profile of effective B magnitude along the z axis. This indicates that the critical crossover of ρ AHE yx is irrelevant to the θ-dependent change in the B magnitude. This θ-dependence of AHE resembles to that of Landau plot of other 3D TDS systems such as Cd 3 As 2 (ref. 12 ) and ZrTe 5 (ref. 21 ), where the origin was attributed to the TPT into a trivial phase. However, our study rules out such TPT scenario from the fact that the intercept value of zeroth LL is in the characteristic range of nontrivial Φ B over the whole θ range ( Supplementary  Fig. 8). Alternatively, we suggest that a change in the FS topology plays a key role in modulating the Ω k . At θ = 0°, two Weyl cones, which are split along the k z direction, have a monopole in their own FSs, giving the nonzero net Ω k . In contrast, over θ = 70°, both the source and sink monopoles can be simultaneously enclosed by each FS 22 due to the anisotropy of the FS of the KZnBi (Supplementary Fig. 4b). This θdependent variation of the AHE verifies the modulation of the Ω k , which can stimulate a utilization of the ultrastrong B (refs. 23,24 ) for exploring exotic quantum phenomena inside the honeycomb layered TDS.
θ-dependent evolution of NMR and its correlation with AHE Another representative quantum transport phenomenon in Dirac semimetals, the chiral anomaly-induced NMR, is also governed by the tuning of Ω k . The NMR of chiral Weyl fermions in nonmagnetic KZnBi originates from a backscattering-free ballistic current of Weyl fermions driven by a chiral charge pumping 8,14,25,26 . While the measured MR shows positive values with prominent SdH oscillations under the condition of B perpendicular to current I (θ = 0°), its magnitude significantly decreases as θ increases to 90° (Fig. 4a, see Supplementary Fig.  4 for FS analysis). We note that the sign of the MR remains positive up to θ = 70°but it abruptly turns to the negative value above θ = 70° (Fig. 4b). The NMR becomes larger when the θ is greater than 70°and maximized at 90° (Fig. 4c), which can be explained by the chiral magnetic effect determined by a product of B and I (inset of Fig. 4c) 8,14,25,26 . The upturn in MR values after~0.25 T may have originated from positive MR components of the weak antilocalization effect of nonzero Berry curvature 13 and the linear band dispersion of zeroth LL 3,18,19 . This competition between the positive and negative MR components inevitably changes with the applied B, giving different values of total MR at the different B (Supplementary  Table 2 for comparison with other Dirac and Weyl semimetals). Remarkably, the sudden NMR change occurs at~70°where the anomalous Hall resistance also shows a clear anomaly (Fig. 3c). It is surprising that the NMR is clearly observed although the chirality of Weyl fermions is ill defined over 70°due to the sudden shrink of the net Ω k . As already described in Fig. 1b, this b Magnified MR in the range of −1 T ≤ B ≤ 1 T. NMR starts to appear at 70°and its magnitude reaches the minimum of~0.25% at 90°. c Magnitudes of the NMR (blue dots and line) as a function of θ, showing a crossover at~70°. The θ dependence of NMR is described by chiral magnetic effect (inset) in 3D TDS, which occurs under the collinear B and I. χ = ±1 indicates chiralities of both Weyl cones. The chiral charge pumping induces a subsequent relaxation current, which is free from backscattering because of chirality protection and, therefore, is leading to a reduction of resistance.
observation of the critical crossover between the anomalous Hall state and CF state in the 3D TDS indicates the experimental demonstration of their mixed and separated states. Because this intriguing crossover is revealed to be irrelevant to the TPT ( Supplementary Fig. 8), its origin is needed to be studied further in the viewpoint of a hidden correlation between DFs and CFs.

DISCUSSION
In summary, we succeeded in identifying and modulating the Ω k in the 3D TDS KZnBi by varying the incidence θ of B. The nontrivial Φ B was deduced from the nonzero intercept of LL fan diagram. Moreover, we observed the AHE and NMR in nonmagnetic 3D TDS, which are the intrinsic signature of nontrivial Φ B in the KZnBi. It was suggested that the crossover between AHE and NMR can be explained with the FS topology without introducing the TPT. Our work provides a platform to study an interesting quantum transport entangled with the topological nature in the planar honeycomb layer structured 3D TDS, overcoming the difficulty in engineering the Ω k of the 2D honeycomb structured graphene.

Sample preparation
The KZnBi single crystals are synthesized via self-flux method with the excess amount of K metal to compensate the loss due to its evaporation 17 . The K:Zn:Bi mixture with a ratio of 1.3:1:1 is placed in alumina crucible and then sealed under an evacuated quartz tube to avoid an oxidization during the reaction. The alumina crucible containing the mixture is place in a furnace and is heated to 630°C for 12 h and is cooled down to 300°C for 100 h and then to room T by furnace quenching. High-quality copper ohmic contacts are formed on the sample by using silver conducting epoxy (inset of Fig. 2a) in a six-point probe configuration for measuring both longitudinal and transverse resistivities. This contact wiring process is carried out in glove box filled with a high-purity Ar (99.999%) gas to avoid the contact of the KZnBi from oxygen gas and moisture.

Magneto-transport measurements
To measure the transport properties of the KZnBi sample, physical property measurement system (PPMS Dynacool, Quantum Design) is used over the T range from 4 to 300 K. Transport behavior in extreme quantum limit is also investigated through TDO measurements on the KZnBi sample attached to the copper coil of the TDO circuit resonating at the frequency of~82 MHz. The sample with coil on the circuit is loaded to the chamber under a perpendicular B up to 49 T by using a nondestructive pulsed magnet at T = 1.6 K. In all the θ-dependent measurements, the rotator motor was calibrated with θ-dependent resistance of the samples at B of 14 T 19,27-30 .