Introduction

Berry’s phase (ΦB) 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 point1,2. In transport experiments, the ΦB 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 ΦB with a quantized π value is defined by an integral of the Berry curvature (Ωk) over the closed Fermi surface (FS) in k space, bearing a resemblance to an electric monopole of Gauss law1,2. This effective magnetic field of Ωk is strongly correlated with a diversity of quantum transport phenomena, such as anomalous and quantum Hall effects5,6, chiral magnetic effect7,8, and topological superconductivity9. The nonzero Ωk can be generated through either time-reversal and inversion symmetry breakings and the resulting Ωk is locally distributed in the Brillouin zone1,2. Regardless of the magnetization of a material, the modulation of Ωk, 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 stimuli10. 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 Ωk hotspots10,11 (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 Ωk with the incidence angle (θ) of B. This angular dependence of the Ωk has been of interest in the TDSs such as Na3Bi (refs. 8,11), Cd3As2 (refs. 12,13), and ZrTe5 (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 anomaly7,8,13,14 and anomalous Hall effect (AHE)5,15 originate from the Ωk. Beyond that, tuning the Ωk 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 Ωk (Fig. 1b). For the precise tuning of the Ωk, 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.

Fig. 1: Tunable Berry curvature and quantum transport phenomena in 3D TDS KZnBi.
figure 1

a B-dependent Weyl cone splitting and net Ωk, which determines quantum transport phenomena of the 3D TDS. The source (red arrows) and sink (green arrows) of the Ωk are located at two Weyl points. The net Ωk (purple arrows) is tunable by varying the angle (θ) of external B. TPT is an abbreviation of topological phase transition. b θ-dependent anomalous Hall resistivity (\(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\), red dots) at saturation and negative magnetoresistance (NMR, blue dots). Dashed lines are guides to the eyes. c Crystal structure of the KZnBi constructed by planar ZnBi honeycomb and K ion layers (left panel), which has C3z rotational and inversion symmetries (right panel). d Band structure of the KZnBi that contains two Dirac cones at ±kD (left panel) which is constructed by Zn-s (red lines) and Bi-p (blue lines) orbitals (right panel).

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Such a 3D TDS can be found in ABC compounds (A: alkali metal, B: transition metal, C: chalcogen element) with a P63/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 symmetries16 (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 lattice17 (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 Ωk hotspots2, the simple band structure of the 3D TDS KZnBi allows to resolve the Ωk distribution under B explicitly and to modulate the Ωk with a weak B efficiently (Supplementary Fig. 1). Here, from the θ-dependent magneto-transport measurements, we find that the tunable Ωk determines the quantum transport characters of Dirac fermions (DFs) and chiral fermions (CFs), demonstrating an abrupt transition between anomalous Hall and CF states.

Results

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 (ρxx (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 materials8,12,13. 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/ΔFTDO where the inverse of frequency variation, ΔFTDO, in the TDO data is proportional to the resistance in the transport data (Fig. 2c). A linear increase of the 1/ΔFTDO 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 (EF) (refs. 3,18,19).

Fig. 2: Evidence of nontrivial Berry phase in 3D TDS KZnBi.
figure 2

a T-dependent ρxx, which shows a metallic behavior of DFs. Illustration for transport measurements is shown in the inset. b MR data in the range of −14 T ≤ B ≤ 14 T at 4 K ≤ T ≤ 200 K. c TDO data in the range of 0 T ≤ B ≤ 49 T at 1.6 K. The linear MR (red dashed line) shows quantum limit behavior. d SdH oscillations in dRxx/dB at 4 K ≤ T ≤ 200 K (colored lines in the main panel) and T-dependent oscillation amplitudes (colored circles in the inset) at B = 4.03 T. m* ≅ 0.012 m0 is obtained from Lifshitz–Kosevich analysis (black line in the inset). e Landau fan diagram for three KZnBi samples (red, blue, and green for samples #1, #2 and #3, respectively), which has integer indices (filled circles) and half-integer indices (open circles). Landau quantization with integer indices is described in the left top inset. The x-intercepts (right bottom inset) that are lying between 0 and ±1/8 imply a nontrivial ΦB.

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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 (Rxx) component is given in the form of dRxx/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)SF, where ħ is reduced Plank’s constant and e is elementary charge3, a small cross-sectional area (SF = 0.00046 Å−2) of the FS is obtained. For a circular FS, a Fermi wavevector also shows a small value (kF = 0.012 Å−1) is extracted from the relation20 of SF = πkF2, implying a simple band structure with sharp 3D Dirac cones in the KZnBi. The extremely small value of the effective mass (m* = 0.012 m0 where m0 is electron mass) is obtained at B = 4.03 T (inset of Fig. 2d), by fitting the T-dependent oscillation amplitude to Lifshitz–Kosevich formula3, \({{{\mathrm{{\Delta}}}}}\rho _{xx}\left( {T,{{B}}} \right)/\rho _{xx}\left( {T = 4\;{{{\mathrm{K}}}},{{B}}} \right) = {{{\mathrm{e}}}}^{ - {\Lambda}_D}\left( {{\Lambda}/\sinh \left( {\Lambda} \right)} \right)\), where Λ = 2π2kBT/β, ΛD = 2π2kBTD/β, and β = ehB/2πm*. Here, kB is a Boltzmann constant and TD is a Dingle temperature. From the Fermi velocity (vF = ħkF/m* ≅ 1.19 × 106 m · s−1), the EF = ħvFkF ≅ 93.6 meV below the Dirac node is calculated20. All values we extracted from the SdH oscillations are well matched with the previous report17. 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 ΦB, which can be obtained from the analysis of the SdH oscillations. According to the Lifshitz–Onsager rule for the SdH analysis3, the Rxx is expressed by Rxx = R0 [1 + A (B, T)cos2π(F/B + γ – δ)], where R0 is the background resistance, A (B, T) is the amplitude of the SdH oscillation, and γ is the values of 1/2 – ΦB/2π, which are determined by the value of ΦB, where it is close to 0 for nontrivial ΦB 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 ΦB, the value of |γ – δ| = |1/2 – ΦB/2π – δ | located between 0 and 1/8. For the identification of nontrivial ΦB, we displayed the LL fan diagram (Fig. 2e) by plotting the maxima (solid balls) and minima (open balls) of the SdH oscillations as integer and half-integer indices, respectively. A linear extrapolation of the relation between 1/B and LL index gives the intercept value of |γ – δ| at zeroth LL. For the measured samples, the intercept values were in the range of the nontrivial ΦB (inset of Fig. 2e, see Supplementary Figs. 2–4 for MR and SdH oscillations), implying the presence of the topologically nontrivial state in the KZnBi. Later, combined with AHE and NMR data, the state of the nontrivial ΦB is revealed as the time-reversal-symmetry broken 3D TDS state.

Tuning of Ωk and θ-dependent AHE

The Ωk 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 (vA = E × Ωk) in the electrical field (E) (refs. 5,15). To verify the AHE in the nonmagnetic 3D TDS KZnBi, we measured the Hall resistivity (ρyx) in the range of −2 T ≤ B ≤ 2 T at 4 K (see Supplementary Fig. 5 for the ρyx in the whole measured range of −14 T ≤ B ≤ 14 T) as plotted in Fig. 3a. It is clear that the ρyx shows the kink around zero B, indicating the occurrence of AHE due to the Ωk. 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 (\(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\)) is separated from the total \(\rho _{{{{\mathrm{yx}}}}} = \rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{OHE}}}}} + \rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\), where \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{OHE}}}}}\) is the contribution of ordinary Hall effect (blue dotted line in the high B region) extracted by a linear fitting5,15. The ordinary Hall coefficient (RH), which is obtained from the linear slope, indicates that the DFs of the present KZnBi are hole carriers with a density of ~3.3 × 1018 cm−3. This p-type carrier is in a good agreement with the recent results of angle-resolved photoemission spectroscopy and DFT calculation17. Although both Dirac band and small pocket are lying at the EF ~ 100 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 \(\rho _{yx}^{{{{\mathrm{AHE}}}}}\) for all measured samples shows the EF independent value of ~80 μΩ · cm, clarifying that the AHE was induced by Ωk (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 DFs15.

Fig. 3: Unusual angle-dependent anomalous Hall effect.
figure 3

a The ρyx in the range of −2 T ≤ B ≤ 2 T at 4 K. A linear background (green dashed lines) is attributed to the ordinary Hall coefficient (RH) and the anomaly near B = 0 (red two-headed arrow) is responsible for the \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\). b θ-dependent \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) in the range of 0° ≤ θ ≤ 90°. Illustration of θ-dependent Hall measurements is given in the inset. c Unusual θ-dependent saturation of \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) (red dots and line) in contrast to conventional θ-dependent RH (blue dots) with a cosine profile (blue line).

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Figure 3b shows the separated \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\), which has strong dependences on both the strength of B and θ between B and z axis of sample. At the θ = 0°, the \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) increases with B and saturates to the value of ~78 μΩ · cm at 4 K under 2 T. The corresponding \(\sigma _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) ≈ 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, ZrTe5, with the AHE induced by nontrivial ΦB (Supplementary Fig. 5e). This phenomenon strongly indicates that the AHE of nonmagnetic KZnBi is intrinsically driven by the nonzero Ωk. As illustrated in the Supplementary Fig. 1, the θ-dependent Ωk can affect the transport properties of the AHE and reveal the correlation between B and Ωk. Indeed, a peculiar angular variation of the Ωk is uncovered from the θ-dependent \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) curves at 4 K (Fig. 3b, c and Supplementary Fig. 5 for raw data). For each θ-dependent \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) curve, the saturated \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) value appears at the different magnitude of B. The θ-dependent curve of the \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) saturation values (red dots in Fig. 3c) shows a sudden decrease from ~70° while the \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{OHE}}}}}\) (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 \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) 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 Cd3As2 (ref. 12) and ZrTe5 (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 ΦB 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 Ωk. At θ = 0°, two Weyl cones, which are split along the kz direction, have a monopole in their own FSs, giving the nonzero net Ωk. In contrast, over θ = 70°, both the source and sink monopoles can be simultaneously enclosed by each FS22 due to the anisotropy of the FS of the KZnBi (Supplementary Fig. 4b). This θ-dependent variation of the AHE verifies the modulation of the Ωk, 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 Ωk. The NMR of chiral Weyl fermions in nonmagnetic KZnBi originates from a backscattering-free ballistic current of Weyl fermions driven by a chiral charge pumping8,14,25,26. 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 curvature13 and the linear band dispersion of zeroth LL3,18,19. 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 Ωk. As already described in Fig. 1b, this 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.

Fig. 4: Unusual angle-dependent chiral magnetic effect.
figure 4

a θ-dependent MR for θ = 0°, 30°, 60°, and 90° at T = 4 K, showing the maximal positive value of ~700% (θ = 0°, B = ±14 T) with SdH oscillations. Illustration of the θ-dependent MR measurements is given in the inset. 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.

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Discussion

In summary, we succeeded in identifying and modulating the Ωk in the 3D TDS KZnBi by varying the incidence θ of B. The nontrivial ΦB 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 ΦB 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 Ωk of the 2D honeycomb structured graphene.

Methods

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 evaporation17. 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 T19,27,28,29,30.