Anisotropic magnetotransport and extremely large magnetoresistance in NbAs2 single crystals

We report the extremely large magnetoresistance and anisotropic magnetoresistance in a non-magnetic semimetallic NbAs2 single crystal. Unsaturated transverse XMR with quadratic field dependence has been observed to be ~3 × 105 % at 2 K and 15 T. Up to 12.5 K, clear Shubnikov de Haas (SdH) quantum oscillations were observed from which two distinct Fermi pockets were identified. The corresponding quantum electronic parameters such as effective cyclotron mass and Dingle temperature were obtained using Lifshitz-Kosevich formula. From the field dependent Hall resistivity at 2 K, carrier concentrations ne(nh) = 6.7691 (6.4352) × 1025 m−3 and mobilities μe (μh) = 5.6676 (7.6947) m2 V−1 s−1 for electrons (e) and holes (h) were extracted using semiclassical two-band model fitting. We observed large anisotropic magnetoresistance about 84%, 75%, and 12% at 0.75 T and 6 K for three different orientations γ, θ and ϕ, respectively, similar to that in several topological semimetallic systems. Magnetic properties of NbAs2 are similar to the case of graphite, without any phase transition in the temperature range from 5 K to 300 K.


Results and Discussion
NbAs 2 crystallizes in a monoclinic system with the centrosymmetric space group of C12/m1. It belongs to a larger family of transition metal dipnictides MPn 2 (M = V, Nb, Ta, Cr, Mo, and W, Pn = P, As and Sb), which is found to crystallize in OsGe 2 structure type. In the NbAs 2 crystal structure (as shown in Fig. 1(a) and its inset), each Nb (Nb1) atom is bounded by six As (As1, As2) atoms and two As atoms lie outside the rectangular faces. Figure 1(b) shows the Rietveld refinement of the X-ray powder diffraction results (Bruker D8) using Cu-K α radiation for the pulverized crystalline sample of NbAs 2 . The inset of Fig. 2(a) shows the as-grown single crystals of NbAs 2 . The refined lattice parameters, a = 9.3560 (2) Å, b = 3.3828 (1) Å, c = 7.7966(2) Å, and β = 119.440 (15)°, are in good agreement with those reported in the literature 10 . Figure 2(a) shows the plot of resistivity as a function of magnetic field at various temperatures measured in the I⊥H configuration. The data obtained by sweeping the magnetic field from 15 T to −15T were then symmetrized using ρ(H) = [ρ(H) + ρ(−H)]/2. NbAs 2 exhibits quite large magnetoresistance (MR) at low temperatures with a strong Shubnikov de Haas (SdH) quantum oscillation and quadratic field dependence as shown in Fig. 2(b). The MR percentage calculated from [(ρ(H) − ρ(0))/ρ(0)] × 100%, reaches 303,200% at 2 K without any signature of saturation in a field of 15 T. It is observed that the MR of NbAs 2 is very sensitive with respect to sample quality, since sample-1 shows MR about 115,200% at 2 K and 9 T with a residual resistivity ratio (RRR = ρ 300K /ρ 3K ) of 107.04 ( Fig. 1(c)), whereas the MR of sample 2 shows 170,800% with a RRR of about 110.49 as shown in Fig. S1. The RRR of NbAs 2 crystals attest to the good metallicity and quality of the grown crystals, which is comparable to that reported for the Weyl semimetal NbP (RRR = 115) 9 , higher than those of TaAs (RRR = 49) 8 and NbAs (RRR = 72) 14 , but lower than less than that previously reported for NbAs 2 (RRR = 222, and 317) 11,12 crystals. The unsaturated MR behavior of NbAs 2 is similar to the semimetals WTe 2 15 and NbSb 2 4 . The large and unsaturated MR of NbAs 2 is higher than that for the semimetals NbSb 2 (MR = 1.3 × 10 5% at 2 K and 9 T) 4 , LaBi (MR = 0.38 × 10 5 % at 2 K and 14 T) 6 , the Dirac semimetals ZrSiS 7 (MR = 1.4 × 10 5 % at 2 K and 9 T) and Cd 3 As 2 (MR = 1.6 × 10 5 % at 2.5 K and 15 T) 16 . It is comparable with that recently reported for NbAs 2 (8 × 10 5% at 9 T at 1.8 K 11 , 1 × 10 5% at 14 T at 2.5 K 13 ), the topological semimetal LaSb (9 × 10 5 % at 9 T and 2 K) 17 and the Weyl semimetal candidates NbP (8.5 × 10 5% at 9 T at 1.85 K) 9 and NbAs (MR = 2.3 × 10 5 % at 9 T and 2 K) 14 . In order to analyse the SdH quantum oscillations, the second order polynomial smoothed background was subtracted from the field dependent resistivity, ρ(H). Figure 2(c) shows the total oscillatory pattern (Δρ) with obvious quantum oscillations starting from 4 T as a function of inverse magnetic field (1/μ 0 H) for (I⊥H) geometry. From the fast Fourier transformation spectrum (FFT) as shown in Fig. 2(d), two major peaks are observed at F α = 266 T and F β = 32 T as well as their harmonics, where α and β are denoted as high and low frequency peaks, respectively. Since the amplitude of total oscillation (Δρ xx ) seems to show complex periodic behavior, FFT frequency filtering was used to extract the respective oscillation patterns denoted as (δρ) for the observed frequency of 266 T to , where the A F is the extremal Fermi surface cross-sectional area perpendicular to the field, F is the frequency of the oscillation, and φ 0 is the magnetic flux quantum. The Fermi surface cross sections are calculated to be 25.3 × 10 −3 Å −2 and 3.04 × 10 −3 Å −2 for 266 T and 32 T, respectively. The total oscillatory pattern (∆ρ) can be expressed based on the Lifshitz-Kosevich (L-K) formalism 18 Here, m* refers to the effective cyclotron mass, and T D is the Dingle temperature. The temperature dependence of δρ is fitted well with the L-K formula as shown in Fig. 2(e). The fitting results yield the effective cyclotron mass = .
where m e is the electron rest mass. Figure 2(f) shows the fitting results of the respective δρ for various inverse fields (1/μ 0 H) at a fixed temperature of 3 K, which yields the Dingle temperature = .
From the Dingle temperature, the single particle scattering rate is calculated to be τ = π s k T 2 B D  = 4.35 × 10 −13 s. The obtained results of NbAs 2 are consistent with previous studies [11][12][13] .
The anisotropic magnetoresistance (AMR) is measured along three different field orientations of γ, θ and φ at different field strengths from 0.1 T to 0.75 T as shown in Fig. 3(a,c,e). The inset of Fig. 3(b) shows the AMR measurement geometry for the γ, θ and φ orientations. The AMR effect for the γ orientation is presented in Fig. 3(a) as a polar plot, which illustrates the two-fold symmetry with a variation of period π. In this configuration, the magnitude of AMR reaches a maximum about 75% at γ = 10° and minium about 32% at γ = 100° for the field of 0.75 T and temperature of 6 K, and a similar trend continues in the opposite way up to 180°. In the φ orientation, the magnitude of AMR shows a minimum about 12% at φ = 0° when the current is parallel to H, and maximum about 29.5% at φ = 90° when the current is perpendicular to H. The magnitudes of AMR are observed to be 75% and 12% for I⊥H (θ = 0°) and I∥H (θ = 90°), respectively, in the θ orientation. From the field dependent AMR measurements in three different orientations, it is clear that the AMR is positive, and its maximum always appears when the current is perpendicular to H. In order to analyse the power law dependence of MR, a double-logarathmic value between H and MR was taken for γ, θ and φ orientations as shown in Fig. 3(b,d,f), and the linear fitting of these plots yield the different slopes (m values) at various angles. The slope (m) varies from 1.233 at 0° to 1.632 at 90° for γ orientation. In the θ orientation, m values are found to be 1.260 at θ = 0° (I⊥H) and 1.201 θ = 90° (I∥H). For in-plane orientation (φ), m varies from 1.175 at φ = 0° to m = 1.743 at φ = 90°. It is noteworthy that the power law dependence of MR is close to 1 particularly at φ = 0° and θ = 90° with I∥H orientation. We also remark that the behavior of AMR with two-fold symmetry for γ and θ orientations remains the same regardless of the magnetic field strengths up to 0.75 T, whereas, for φ orientation, the two-fold symmetry in AMR gradually faded away in low field regime. The response of the charge carriers to the rotating magnetic field of magnitude about 0.75 T for three different orientations is studied as a function of temperature as shown in Fig. 4(a-c). The variation of AMR, ρ ρ ρ ρ ∆ = −

[ ]/
peak valley valley , with respect to temperature is presented in Fig. 4(d) for three different orientations. From the temperature dependence of magnetoresistance, AMR increases with decreasing temperature, and it is almost saturated at low temperatures. The two-fold symmetry is well pronounced at low temperature (6 K), and it is sustained up to a measured temperature of 150 K.
Since the MR value of conventional metals is usually small in magnitude and saturated at high fields, and the consequences of unsaturated XMR and ultrahigh mobility in nonmagnetic topological semimetals such as Cd 3 As 2 16 , TaAs, etc, is related to Dirac and Weyl fermions (topological surface states and linear band dispersion), the fact that NbAs 2 exhibits unsaturated XMR is extremely important. In order to identify the intrinsic magnetic property of NbAs 2 , magnetization measurements as a function magnetic field and temperature were carried as shown in Fig. 5(a,b). The linear field dependence of magnetization in NbAs 2 is similar to that observed for graphite 19,20 . Even though a sudden rise of magnetization below 25 K due to small amount of magnetic impurities, there is no significant effect in the AMR behavior of NbAs 2. In ferromagnetic metals, AMR typically shows the maximum resistivity when the current is parallel to magnetic field due to spin-orbit scattering, and minimum resisitivity when the current is perpenducular to magnetic field 21,22 . Since the NbAs 2 belongs to nonmagnetic material category, the physical origin of the AMR effect in the present system is thus different from that in magnetic materials.
According to the semiclassical two-band model 23,24 , the total conductivity tensor is expressed in the complex form of  In equation (2), the Re and Im σ equal to σ xx and σ xy , respectively where σ = From equation (2) n , p, μ n and μ p can be obtained by fitting the σ xy (H) data. For a perfect compensated system (n = p), MR follows a quadratic field dependence which is shown in Fig. 2(b). Figure 6(a) shows the Hall resistivity as a function of magnetic field with nonlinear behavior at low temperatures. The expanded view of ρ xy below 1 T of H at 2 K is shown in the inset of Fig. 6(a). It is clear that the sign of ρ xy changes from positive (ρ xy > 0 in H < 0.4 T) to negative at high fields with nonlinear band, suggesting the multiband effect in the NbAs 2 system. At 200 K, ρ xy shows positive value below 8 T of magnetic field as shown in Fig. S1(c) which suggests that holes dominate over electrons in the transport properties. By fitting the σ xy data as shown in Fig. 6(b), carrier concentrations of n = 6.7691 × 10 25 m −3 and p = 6.4352 × 10 25 m −3 , and mobilities of μ p = 7.6947 m 2 V −1 s −1 and  Fig. 6(c). Figure 6(d) shows the ratio of n to p as the function of temperatures, suggesting that these two carriers are almost compensated in the NbAs 2 system.
The magnetic field and temperature dependent transport measurements revealed the highly compensated electron and hole pockets, which may be responsible for the observed XMR. Recently, the AMR effect is observed in several nonmagnetic materials such as ZrSiS 25    theoretical calculations and band structure characterizations are keenly required to reveal the possible nontrivial band topology in NbAs 2 .
In summary, the single crystals of NbAs 2 were grown using the chemical vapour transport method. We observed extremely large, unsaturated and anisotropic MR in NbAs 2 . Transverse magnetoresistance of NbAs 2 reaches a large value of about 303,200% at 2 K and 15 T, and MR follows a quadratic field dependence, which is in accord with the electron-hole compensation with the n/p ratio of about 1.05 determined from semiclassical two-band model fittings. From the SdH quantum oscillations, two distinct Fermi pockets were identified, and its effective electron mass and Dingle temperature were extracted from the L-K fitting. Interestingly, apparent two-fold symmetry and large magnitude in AMR are observed for three different field orientations, and power law dependence of MR is close to 1 for I∥H orientation. The origin of such large AMR effect in a non-magnetic semimetal NbAs 2 may be related to the presence of non-trivial Berry curvature in NbAs 2 , where the magnetic contribution to the AMR effect has been excluded based on magnetization measurements.

Experimental Section
Sample preparation. Two step chemical vapor transport processes were used to synthesize and grow single crystals of NbAs 2 . A quartz ampoule with a length of 30-40 cm was used for the synthesis and growth. At first, stoichiometric amounts of 5 N purity precursors of Nb and As in a molar ratio of 1:2 were sealed in an evacuated quartz ampoule. The vacuum-sealed quartz ampoule containing the binary mixture was treated at 950 °C for two days and then cooled to room temperature, yielding polycrystalline NbAs 2 . Secondly, the polycrystalline powder of NbAs 2 was mixed with I 2 in a weight ratio of 100:1 and vacuum-sealed in a two-zone tube furnace having a thermal gradient of about 950-850 °C within ~40 cm. The resulting NbAs 2 single crystals have shiny surfaces with well-defined crystal facets as shown in the inset of Fig. 1(b).