Abstract
Topological Dirac semimetals have emerged as a platform to engineer Berry curvature with timereversal 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 threedimensional topological Dirac semimetal KZnBi, which is demonstrated by tuning the direction and flux of Berry curvature. Angledependent magnetotransport 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.
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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 timereversal 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.
Threedimensional 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 timereversal 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 angledependent 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 Zn4s and Bi6p 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 θdependent 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.
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 Tdependent 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 highquality 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 crosssectional 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 Tdependent oscillation amplitude to Lifshitz–Kosevich formula^{3}, \({{{\mathrm{{\Delta}}}}}\rho _{xx}\left( {T,{{B}}} \right)/\rho _{xx}\left( {T = 4\;{{{\mathrm{K}}}},{{B}}} \right) = {{{\mathrm{e}}}}^{  {\Lambda}_D}\left( {{\Lambda}/\sinh \left( {\Lambda} \right)} \right)\), where Λ = 2π^{2}k_{B}T/β, Λ_{D} = 2π^{2}k_{B}T_{D}/β, and β = ehB/2πm^{*}. Here, k_{B} is a Boltzmann constant and T_{D} is a Dingle temperature. From the Fermi velocity (v_{F} = ħk_{F}/m* ≅ 1.19 × 10^{6} m · s^{−1}), the E_{F} = ħv_{F}k_{F} ≅ 93.6 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}, the R_{xx} is expressed by R_{xx} = R_{0} [1 + A (B, T)cos2π(F/B + γ – δ)], 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 diagram (Fig. 2e) by plotting the maxima (solid balls) and minima (open balls) of the SdH oscillations as integer and halfinteger 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 Φ_{B} (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 Φ_{B} is revealed as the timereversalsymmetry broken 3D TDS state.
Tuning of Ω_{k} and θdependent AHE
The Ω_{k} emerges and acts like an effective B in the 3D TDS when timereversal 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 (\(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\)) 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 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 ptype carrier is in a good agreement with the recent results of angleresolved photoemission spectroscopy and DFT calculation^{17}. Although both Dirac band and small pocket are lying at the E_{F} ~ 100 meV, we showed that twocarrier 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 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 splittinginduced separation in the SdH oscillation indicates the AHE does not originate from the spinpolarized massive DFs^{15}.
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, 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 θdependent 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 θdependent \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) curves at 4 K (Fig. 3b, c and Supplementary Fig. 5 for raw data). For each θdependent \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) curve, the saturated \(\rho _{{{{\mathrm{yx}}}}}^{{{{\mathrm{AHE}}}}}\) value appears at the different magnitude of B. The θdependent 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 θdependent change in the B magnitude. This θdependence 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.
θdependent evolution of NMR and its correlation with AHE
Another representative quantum transport phenomenon in Dirac semimetals, the chiral anomalyinduced NMR, is also governed by the tuning of Ω_{k}. The NMR of chiral Weyl fermions in nonmagnetic KZnBi originates from a backscatteringfree 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 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.
Methods
Sample preparation
The KZnBi single crystals are synthesized via selfflux 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. Highquality copper ohmic contacts are formed on the sample by using silver conducting epoxy (inset of Fig. 2a) in a sixpoint probe configuration for measuring both longitudinal and transverse resistivities. This contact wiring process is carried out in glove box filled with a highpurity Ar (99.999%) gas to avoid the contact of the KZnBi from oxygen gas and moisture.
Magnetotransport 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 θdependent measurements, the rotator motor was calibrated with θdependent resistance of the samples at B of 14 T^{19,27,28,29,30}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science, ICT & Future Planning) (No. 2015M3D1A1070639). B.C.P. acknowledges the support from the National Research Foundation of Korea (NRF2019R1A6A3A01096112).
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S.W.K. conceived the idea and organized the research. J.S. grew the single crystals and performed transport measurements. J.S., B.C.P., J.B., S.K., Y.Ki. and S.W.K. analyzed all experimental data. K.I.S. and B.C.P. performed theoretical calculations of Berry phase distribution. Z.Y. and Y.Ko. conducted highfield transport measurements with a tunnel diode oscillator. J.S., B.C.P. and S.W.K. wrote the manuscript with the input from all the authors.
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Song, J., Park, B.C., Sim, K.I. et al. Tunable Berry curvature and transport crossover in topological Dirac semimetal KZnBi. npj Quantum Mater. 6, 77 (2021). https://doi.org/10.1038/s41535021003787
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DOI: https://doi.org/10.1038/s41535021003787
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