Extremely large magnetoresistance and chiral anomaly in the nodal-line semimetal ZrAs₂

Abstract

We performed the detailed magnetotransport measurements and first principle calculations to study the electronic properties of the transition metal dipnictides ZrAs₂, which is a topological nodal-line semimetal. Extremely large unsaturated magnetoresistance (MR) which is up to 1.9 × 10⁴ % at 2 K and 14 T was observed with magnetic field along the c-axis. The nonlinear magnetic field dependence of Hall resistivity indicates the multi-band features, and the electron and hole are nearly compensated according to the analysis of the two-band model, which may account for the extremely large unsaturated MR at low temperatures. The evident Shubnikov-de Haas (SdH) oscillations at low temperatures are observed and four distinct oscillation frequencies are extracted. The first principle calculations and angle-dependent SdH oscillations reveal that the Fermi surface consists of three pockets with different anisotropy. The observed twofold symmetry MR with electric field along the b-axis direction is consistent with our calculated Fermi surface structures. Furthermore, the negative magnetoresistance (NMR) with magnetic field in parallel with electric field is observed, which is an evident feature of the chiral anomaly.

Graphical abstract

Keywords

nodal-line semimetal / SdH oscillations / chiral anomaly

1 Introduction

The study of topological semimetals have attracted tremendous interests due to their novel electronic structure and unconventional transport properties in condensed-matter physics in recent years [1-3], such as large longitudinal magnetoresistance, high mobility, nontrivial Berry phase and chiral-anomaly induced negative longitudinal MR [4-10]. In these materials, the topological semimetallic states are characterized by band touching points or lines between conduction and valance bands in momentum space. Dirac and Weyl semimetals are characterized by band crossings at a single point in k-space, exhibiting linear dispersion and featuring fourfold and twofold degeneracies, respectively. Dirac points could be divided into two Weyl points by breaking time-reversal symmetry or inversion symmetry. Dirac semimetals have been confirmed in Cd{}_3As{}_2 and Na{}_3Bi [11-14] and Weyl semimetals have been theoretically proposed and experimentally observed in MX (M = Ta or Nb; X = P or As), WTe{}_2 and (Nb,Ta)IrTe{}_4 [15-21]. In contrast to Dirac or Weyl semimetals, topological nodal line semimetals showcase a unique characteristic: one-dimensional trajectory within the k-space, forming either an open path or a closed loop. Several candidate materials have been identified within the realm of nodal line semimetals, including PbTaSe{}_2, PtSn{}_4, ZrSiX (X = S, Se, Te), and the AB{}_3 family (A = Ca, Ba, Sr; B = P, As). Each of these materials exhibits intriguing topological properties [22-27].

The transition metal dipnictides (TMDPs) with the formula MX{}_2 (where M = Nb or Ta; X = As or Sb) exhibit a crystalline structure characterized by the symmetry of C2/m (No. 12), and they have attracted considerable attention primarily owing to their possession of compelling properties, including extremely large magnetoresistance (XMR) and negative magnetoresistance (NMR) [9, 10, 28]. Recently, TMDPs ZrX{}_2 (where X = P or As) with the space group Pnma (No. 62) were theoretically predicted to be topological nodal-line semimetals, garnering increasing attention. An extremely large magnetoresistance in ZrP{}_2 has been detected, which might be induced by the compensation of electrons and holes. The compensation of electrons pockets and hole pockets were further confirmed by angle-resolved photoemission spectroscopy (ARPES) as well as the first-principle calculations [29, 30]. The band structure of ZrP{}_2 is characterized by a nodal loop in the k_x = 0 plane without considering spin-orbit coupling (SOC) and a small gap along the nodal loop will open when considering SOC according to the first-principle calculations [29]. Compared with ZrP{}_2, the gap of ZrAs{}_2 along the node line should be larger due to the enhanced spin-orbit coupling [6].

Quantum oscillations, Hall resistivity, and anisotropic MR in ZrAs{}_2 have recently been reported [31]. However, the topological properties have yet to be confirmed and the transport properties are worth revisiting with enhanced sample quality. Only one oscillation frequency (i.e., single pocket in Fermi surface) was observed from the SdH oscillations with the current applying along the b-axis and the magnetic field in the ac plane of ZrAs{}_2. In contrast, multiple pockets were detected in de Haas-van Alphen (dHvA) oscillations. Furthermore, recent ARPES experiments have reported the surface bands and bulk states of ZrAs{}_2, suggesting multi-band feature [32]. Such a discrepancy between transport and magnetic measurements could result from the low sample quality indicated by the lower residual resistivity ratio (RRR) of the previous samples grown by the flux method.

In this work, we reported the detailed magnetotransport properties and electronic structure of the topological nodal-line semimetal ZrAs{}_2. We used chemical vapor transport method to grow high quality crystals of ZrAs{}_2 to investigate topologital properties. The Hall resitivity and longitudinal MR measurements reveal the concentration of electron (n_e=3.58\times {10}^{20}{\mathrm{c}\mathrm{m}}^{-3}) with a mobility of {\mu }_e ≈ 5600 {\mathrm{c}\mathrm{m}}^2\cdot {\mathrm{V}}^{-1}\cdot {\mathrm{s}}^{-1} and the concentration of hole (n_h\approx 3.47\times {10}^{20}{\mathrm{c}\mathrm{m}}^{-3}) with a mobility of {\mu }_h = 5164 {\mathrm{c}\mathrm{m}}^2\cdot {\mathrm{V}}^{-1}\cdot {\mathrm{s}}^{-1} at 2 K, according to the analysis of the two-band model. The XMR up to 1.9\times {10}^4 % is oberved at 2 K and 14 T, which is considered to stem from the compensation mechanism [29]. The SdH oscillations at low temperature and high magnetic field are observed and four frequencies F_1= 155 T, F_2= 178 T, F_3= 443 T, F_4= 579 T are extracted. The nontrivial Berry phases are obtained from the plot of the Landau level index fan diagram. Moreover, we observed anistropic MR with two-fold symmetry when the current is applied along the b axis and the magnetic field lies within the ac plane, consistent with the two-fold Fermi surfaces (FSs) structure predicted by the DFT calculations. Furthermore, we observed the evident NMR when the electric field is in parallel with magnetic field applying along the b-axis. According to our analysis and previous theoretical works [33], we argued that the NMR should be induced by the chiral anomaly. Our study has provided compelling evidence for the topological properties of nodal-line semimetals in ZrAs{}_2.

2 Experimental methods

Single crystals of ZrAs{}_2 were synthesized by the chemical vapor transport method. The Zr power and As power were weighted with a molar ratio of 1:2 in sealed fused-silica tubes with 4 mg/mL grain of I{}_2 added to promote crystal growth. The quartz tubes were heated to 850 °C over 2 days, kept at 850 °C for 1 week and cooled to room temperature over 1 day, as described in Ref. [34]. Then needle-like crystals were obtained. The atomic composition of the ZrAs{}_2 single crystal was determined using energy dispersive X-ray spectroscopy (EDS), revealing a Zr to As ratio of 1:1.92. This observed ratio aligns closely with the nominal ratio of 1:2, taking into account experimental errors. A suitable crystal was selected and mounted on a Bruker D8 Venture diffractometer to measure the single crystal X-ray diffraction (XRD). The crystal was kept at 170.0 K during data collection. Using Olex2 [35], the structure was solved with the SHELXT [36] structure solution program using Intrinsic Phasing and refined with the SHELXL [37] refinement package using Least Squares minimisation.

The measurements of resistivity and magnetic transport properties were performed on a Quantum Design physical property measurement system (PPMS-DynaCool 14 T). The DFT calculations of electronic structure of bulk ZrAs{}_2 were performed by using the Vienna ab initio simulation package (VASP) [38], with the generalized gradient approximation (GGA) in the Perdew−Burke−Ernzerhof type as the exchange-correlation energy [39]. The cutoff energy was set to 350 eV and \mathrm{\Gamma }-centered 8 \times 15 \times 6 mesh were sampled over the Brillouin zone integration. The lattice constant a = 6.7964 Å, b = 3.6848 Å, c = 9.0214 Å was used for all the calculations. The tight-binding model of ZrAs{}_2 was constructed by the Wannier90 with 4d orbitals of Zr and 4p orbitals of As, based on the maximally-localized Wannier functions [40, 41].

3 Results and discussion

The crystal structure of ZrAs{}_2 is presented in Fig.1(a) with the space group of Pnma (No. 62). Fig.1(b) shows the temperature-dependent {\rho }_{xx} exhibiting a typical metallic behavior with the electric current along the b-axis. The residual resistivity {\rho }_{xx}(2K) is 0.53 μ\mathrm{\Omega }\cdotcm at 2 K and residual resistance ratio (RRR = {\rho }_{xx}(300 K)/{\rho }_{xx}(2 K)) is 56 which is higher than previously reported value in Ref. [31]. A higher RRR indicates a higher quality of single crystals. The inset shows a picture of the needle shaped single crystal ZrAs{}_2 with the easy growth direction along the b-axis. Fig.1(c−e) presents the diffracted reflections in (0kl) plane, (h0l) plane and (hk0) plane, respectively. The lattice constants of ZrAs{}_2 were obtained as a= 6.7964(3) Å, b= 3.6848(2) Å, c= 9.0214(4) Å, which are consistent with the previous reported results [34].

Fig.2(a) shows the field-dependent longitudinal MR of ZrAs{}_2 at various temperatures, where MR is defined as MR = [{\rho }_{xx} (B)-{\rho }_{xx} (0)]/{\rho }_{xx} (0)\times100%. To eliminate the Hall contribution, the {\rho }_{xx}(B) is calculated from {\rho }_{xx}(B)=[ {\rho }_{xx}(-B)+{\rho }_{xx}(B)]/2. An extremely large unsaturated MR of up to 2.5\times {10}^3% is observed at 2 K and 8 T. The MR decreases as temperature increases, reaching 1.6\times {10}^2 % at 80 K and 8 T. To explore the origin of this large and unsaturated MR, we also measured the Hall resistivity. The measurements of {\rho }_{xy}(B) at different temperatures provide further transport features, as shown in Fig.2(b). The Hall resistivity exhibits nonlinear magnetic field dependence at low temperatures, implying its multi-band properties. Furthermore, the mobility and concentration of electron and hole can be obtained by a two-band model fitting. Firstly, we calculated the {\sigma }_{xy} according to the formula {\sigma }_{xy}=- {\rho }_{xy}/[({\rho }_{xy}{)}^2+({\rho }_{xx} {)}^2], then we fit the {\sigma }_{xy} using the two-band model formula as the following.

(1){\sigma }_{xy}( B)=[{\displaystyle \frac{n_h{\mu }_h^2}{1+({\mu }_{h}B{)}^2}}-{\displaystyle \frac{n_e{\mu }_e^2}{1+({\mu }_{e}B{)}^2}}]eB,

where n_{e,h} and {\mu }_{e,h} represent the concentration and mobility of electrons and holes, respectively. As shown in Fig.2(c), the fitted red curves are consistent with the experimental data, from which the temperature-dependent concentrations and mobilities are extracted and plotted in Fig.2(d). At 2 K, the concentration of electron n_e\approx 3.58\times {10}^{20} {\mathrm{c}\mathrm{m}}^{-3} which is almost equal to the concentration of hole n_h\approx 3.47\times {10}^{20} {\mathrm{c}\mathrm{m}}^{-3}. The mobilities increase with decreasing temperature. {\mu }_e and {\mu }_h reach up to 5600 {\mathrm{c}\mathrm{m}}^2\cdot {\mathrm{V}}^{-1}\cdot {\mathrm{s}}^{-1} and 5164 {\mathrm{c}\mathrm{m}}^2\cdot {\mathrm{V}}^{-1}\cdot {\mathrm{s}}^{-1} at 2 K, respectively. The high mobilities further illustrate the good quality of single crystals. According to these results and analysis, the compensation mechanism of electrons and holes can account for the origin of the XMR at low temperature. Similarly, the compensation mechanism of electrons and holes induced XMR has been reported in other topological semimetals such as TaAs{}_2, NbAs{}_2, and ZrP{}_2 [9, 10, 29].

Quantum oscillation experiments serve as a potent approach to investigate the properties of Fermi surfaces [42, 43]. Fig.3(a) shows the resistivity as a function of magnetic field at low temperatures with the electric current along the b-axis and the magnetic field along the c-axis. The SdH oscillations can be clearly observed at low temperatures and high magnetic fields. The inset shows obvious oscillations around 14 T. The oscillations gradually decreases as temperature increases and almost disappears at 12 K. The oscillatory components of the SdH oscillations versus 1/B can be described by the Lifshitz−Kosevich (LK) formula [44-47]:

(2)\mathrm{\Delta }\rho \propto {\displaystyle \frac{\lambda T}{\cosh (\lambda T)}} {\mathrm{e}}^{-\lambda T_D}\sin [2 \pi ({\displaystyle \frac{F}{B}}-{\displaystyle \frac{1}{2}}+\beta +\delta )],

where \lambda =(2{\pi }^{2}k{}_B{m}^{\ast })/(\hslash eB). T{}_D is the Dingle temperature. The value of \delta depends on the dimensionality, \delta =0 for 2D systems and \delta =\pm 1/8 for 3D systems. \beta ={\varphi }_B/(2\pi ), in which {\varphi }_B is the Berry phase. After substracting a background, the oscillation component \mathrm{\Delta }\rho versus 1/B is shown in Fig.3(b). The oscillation intensity decreases with increasing temperature. Four oscillation frequencies can be identified from the fast Fourier transform (FFT) analysis with F_1=155 T, F_2=178 T, F_3=443 T, and F_4=579 T at low temperatures as shown in Fig.2(c). Furthermore, the harmonic frequency 2F_3=886 T and 2F_4=1158 T can also be observed. According to the Onsager relation F=[\varphi {}_0/(2{\pi }^2)]=[ \hslash /(2 \pi e)]A{}_F, the oscillation frequency is proportional to the extreme cross-section (A_F) of FS normal to the magnetic field. The calculated values of A_F associated with different frequencies were shown in Tab.1. The inset of Fig.3(c) shows the temperature-dependent FFT amplitudes and the red curves are fitted by the thermal factor R_T=(\lambda T)/\sinh (\lambda T). The fitting curves are consistent with experiment data and the effective masses are estimated to be m_1^{\ast }=0.10 m_e, m_2^{\ast }=0.11 m_e, m_3^{\ast }=0.13 m_e and m_4^{\ast }=0.12 m_e. The detailed fitting results are listed in Tab.1. Compared to previous reports [31], we obtained lighter effective masses, which are coresponding to the Dirac fermions with linear dispersion. This suggests that the electronic properties of this material are more akin to ideal Dirac fermions. Besides, according to the Lifshitz−Onsager quantization rule N=A_F[\hslash / (2\pi eB)]-1/2+\beta +\delta, the Berry phase can be extracted from the intercept of the linear extrapolation. The Landau indices N as a function of 1/B and the fitting curves are shown in Fig.3(d). The values of {\varphi }_B from the different \delta are listed in Tab.1. The Berry phase of F_3 with (\delta = 1/ 8) is 1.1 \pi while that of F_4 with (\delta =-1 /8) is 0.8 \pi, indicating the nontrivial \pi Berry phases. The nontrivial Berry phases indicate the topological properities of corresponding bands and FSs.

To further investigate the FS topology and compare it with results of the SdH experiments, we performed the DFT calculations. The FSs structures in the first Brillouim zone are displayed in Fig.4(a)−(c). ZrAs{}_2 has small \alpha hole pockets (blue) near U point and two big \beta hole pockets as shown in Fig.4(c) and (d) and three type \delta electron pockets (purple) near \mathrm{\Gamma } point can be devided as {\delta }_1 and {\delta }_2, which is similar to its sister compound ZrP{}_2 [6]. In addition, there is also a small electron-like FS inside {\delta }_2. There are two additional small Fermi surfaces located inside the {\delta }_2 Fermi surfaces. Moreover, both large Fermi surfaces, \beta and {\delta }_2, exhibit two-fold symmetry in the first Brillouin zone.

Angular-dependent magnetotransport measurements provide a valuable approach for investigating the detailed structure of the FS [43]. Fig.4(d) shows the angle-dependent MR as a function of magnetic field at 2 K. The inset of Fig.4(d) presents the definition of \phi where the magnetic field is always perpendicular to the current. The evident SdH oscillations are observed when H rotates from the a axis to c axis. The angle-dependent \mathrm{\Delta }\rho as a function of 1/B at different angles are shown in Fig.4(e). The itensities of oscillations increase with decreasing \phi which is consistent with the tendency of MR. Fig.4(f) shows the angle-dependent frequency of SdH oscillations as the rotation angle changes from the a axis (\phi = 0°) to c axis (\phi = 90°). As we mentioned hereinbefore, the four frequencies, F_1, F_2, F_3, and F_4, appear as B//c. Due to the complicated FSs, we draw a red dash line in Fig.4(f) to carefully tracke the position of these frequencies at different angles. The frequencies of F_1 and F_2 change slightly. However, the signals of F_1 and F_2 disappear at \phi = 0°, which possibly is obscured by the signal from a larger pocket. The frequencies of F_3 increased with decreasing \phi. However, the frequency of F_4 increases first and then decreases with decreasing \phi. The Fermi surfaces \beta and {\delta }_2 in ZrAs{}_2 are quite complex, which makes it difficult to distinguish the SdH oscillations of different FSs. The calculated frequencies of {\delta }_1, {\delta }_2 and \beta pockets are 135 T, 394 T, and 551 T respectively, with magnetic field along the c-axis. The frequency F_1 = 155 T may be associated with a small Fermi surface hidden within {\delta }_2. Combined with DFT calculations, we propose that F_2 is associated with electron pocket {\delta }_1, F_3 is associated with electron pocket {\delta }_2 and F_4 is associated with hole pocket \beta as shown in Fig.4(a)−(c). However, the oscillation frequency of small hole \alpha pocket is not observed in the SdH oscillations. The discrepancy between the SdH experiments and the DFT calculations may be induced by the slight variation in chemical potential of our sample. The chemical potential is often influenced by the small changes in the lattice parameters or impurity doping. In addition, the angle-dependent MR at 2 K is shown in Fig.4(g) with current along the b axis and magnetic field in the ac plane. Besides, the obvious two-fold symmetry of MR were obsrved and the ratio of {\rho }_{xx} (14\mathrm{T},\phi ={90}^{\circ } )/{\rho }_{xx} (14\mathrm{T},\phi =0^{\circ }) \approx 3 also indicates a moderate electronic structure anisotropy [43]. The twofold FSs can acount for the two-fold symmetry anisotropic MR [48]. At 2 K and 14 T, the extremely large unsaturated MR reaches up to 1.9 \times {10}^4% with magnetic field applied along the c axis, which is significantly higher than that of 1.01 \times {10}^3% in previous report [31], indicating higher quality of the single crystals.

In addition to the non-trivial Berry phase identified through quantum oscillation analysis, a compelling piece of evidence for the topological characteristics is the observation of chiral anomaly induced negative MR (NMR), which have been observed in Weyl semimetal NbAs and Dirac semimetal TaAs{}_2 and NbAs{}_2 [9, 10]. Fig.5(a) shows the angle-dependent longitudinal MR at 2 K with magnetic field rotating from perpendicular to in parallel with current (I/ /b). In order to eliminate the influence of the Hall signal, the {\rho }_{xx}(B) were averaged by measuring the resitivity over positive and negative field directions. At \theta = 0°, the MR is the largest. As \theta decreases, the MR at 2 K decreases dramatically under magnetic field. When magnetic field is collinear with current (\theta = 0°), the chiral anomaly plays a significant role in the transport. As shown in Fig.5(b), when the magnetic field is parallel to current (B//I), a clear negative MR is observed. When the magnetic field increases, the MR becomes positive at low fields ( T) and negative at high fields below 50 K, at \theta = 90°. It should be noted that such a NMR was not in Ref. [31]. Similar NMR induced by the chiral anomaly has widely observed in topological semimetals like NbAs{}_2 in Ref. [9]. According to the theoretical works and related experiments [5, 26, 33], when magnetic field is collinear with electric field in the topological nodal line semimetal, a chiral anomaly will appear remarkably which is similar to the Weyl semimetal. The chiral anomaly can account for the NMR when the magnetic field is collinear with the electric field (B//I), which indicates again the topological characteristic of ZrAs{}_2.

Besides the chiral anomaly, there may be other mechanisms that can also give rise to negative MR. Firstly, our ZrAs{}_2 sample is high quality single crystalline which should not contain any magnetic atoms, thus the influence of magnetic moments can be excluded. Secondly, Anderson weak antilocalization (WAL) can also cause negative MR at low fields [49]. However, as shown in Fig.5(b), the negative MR in our case appears in high magnetic field (B>5 T), therefore the WAL effect could be excluded. Thirdly, the NMR can also be induced in the quantum limit, where the Fermi energy lies in the only the lowest Landau level [50, 51]. In our case, the quantum oscillations can be observed at high magnetic field as shown in Fig.2(a), the influence of quantum limit can be excluded. Finally, the current jetting effect can induce NMR when inhomogeneous currents are injected into the sample. To mitigate this, we positioned the current injection electrodes across the sample to ensure a uniform current distribution throughout the sample, thereby avoiding the influence of the current jetting effect [5]. To summarize, based on above discussions, and also considering theoretical calculations and analysis, we argued that the NMR in ZrAs{}_2 should be induced by the chiral anomaly.

4 Conclusions

In summary, the detailed magnetotransport properties and electron structures of topological nodal-line semimetal ZrAs{}_2 were measured. The extremely large unsaturated MR is observed which reaches up to 1.9\times {10}^4% at 2 K and 14 T. The electron- and hole-type charge carriers are almost compensated from the analysis based on the two-band model, which may account for the extremely large unsaturated MR at low temperature. The evident SdH oscillations are observed and four frequencies F_1=155 T, F_2=178 T, F_3=443 T and F_4=579 T are extracted from the SdH oscillations. The nontrivial \pi Berry phase and the chiral anomaly induced NMR suggested its nontrivial topological characteristic. Our study further confirmed the topological characteristics of nodal-line semimetal ZrAs{}_2.

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