Magnetoresistance and Kondo effect in the nodal-line semimetal VAs_2
Shuijin Chen, Zhefeng Lou, Yuxing Zhou, Qin Chen, Binjie Xu, Jianhua Du, Jinhu Yang, Haangdong Wang, Minghu Fang
MMagnetoresistance and Kondo effect in the nodal-line semimetal VAs Shuijin Chen, Zhefeng Lou, Qin Chen, Binjie Xu, Chunxiang Wu, Jianhua Du, Jinhu Yang, Hangdong Wang, and Minghu Fang
1, 4, ∗ Department of Physics, Zhejiang University, Hangzhou , China Department of Applied Physics, China Jiliang University, Hangzhou , China Department of Physics, Hangzhou Normal University, Hangzhou , China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing , China (Dated: December 23, 2020)We performed calculations of the electronic band structure and the Fermi surface as well as mea-sured the longitudinal resistivity ρ xx ( T, H ), Hall resistivity ρ xy ( T, H ), and magnetic susceptibilityas a function of temperature and various magnetic fields for VAs with a monoclinic crystal struc-ture. The band structure calculations show that VAs is a nodal-line semimetal when spin-orbitcoupling is ignored. The emergence of a minimum at around 11 K in ρ xx ( T ) measured at H = 0demonstrates that an additional magnetic impurity (V , S = 1/2) occurs in VAs single crystals,evidenced by both the fitting of ρ xx ( T ) data and the susceptibility measurements. It was foundthat a large positive magnetoresistance (MR) reaching 649% at 10 K and 9 T, its nearly quadraticfield dependence, and a field-induced up-turn behavior of ρ xx ( T ) emerge also in VAs , althoughMR is not so large due to the existence of additional scattering compared with other topologi-cal nontrival/trival semimetals. The observed properties are attributed to a perfect charge-carriercompensation, which is evidenced by both calculations relying on the Fermi surface and the Hallresistivity measurements. These results indicate that the compounds containing V (3 d s ) elementas a platform for studying the influence of magnetic impurities to the topological properties. Since the discovery of quantum Hall effect in the two-dimensional electron gas in the 1980s [1], it has been rec-ognized that the topology of band structure in the solidmaterials plays an important role in the classificationof maters and the understanding of physical properties.Then, many topological materials, such as, topologicalinsulators [2–4], Dirac semimetals [5, 6], Weyl semimet-als [7–9] and nodal-line semimetals [10–13] have been pro-posed theoretically and confirmed experimentally. Mostof them are free of strong correlation effects. In the pres-ence of strong electron interactions, very fruitful topolog-ical phases can be expected, such as the topological Mott[14] or Kondo insulators [15–17], topological supercon-ductors [18], and fractional topological insulators [19–21].In order to pursue these exotic phases, searching for suit-able compounds, which are strongly correlated (usuallyin d and f orbital systems) and topologically nontrivial,attracts much attention. For instance, SmB , as a typ-ical rare-earth mixed valence compound, in which, hasbeen proposed theoretically as a topological Kondo insu-lator [15–17] and recently has been confirmed by trans-port [22–25], photoemission [26–28] and scanning tunnel-ing microscope (STM) [29] experiments. At sufficientlylow temperature, the hybridization between 4 f orbitalsand highly dispersive 5 d bands results in the formationof “heavy fermion” bands, thus, SmB is a correlated Z topological insulator. Recently, a number of groups [30–33] predicted thousands of candidate topological materi-als by performing systematic high-throughput computa-tional screening across the databases of known materials.Following these predictions, it is possible to search for thetopological materials in the known compounds containing3 d elements based on the calculations of band structure, and the measurements of physical properties.VAs crystalizes in a monoclinic structure with spacegroup C /m (No. 12), as shown in Fig. 1(a), has anisostructure to the transition metal dipnictides XPn (X= Nb, Ta; Pn = P, As, Sb) [34–40], which have beenwidely studied theoretically and experimentally as a classof topological materials. For example, Baokai Wang etal. [41] identified the presence of a rotational-symmetry-protected topological crystalline insulator (TCI) statesin these compounds based on first-principles calculationscombined with a symmetry analysis. It was found that allthese compounds exhibit high mobilities and extremelylarge positive MR [35–40]. Interestingly, a negative lon-gitudinal MR when the applied field is parallel to thecurrent direction was observed in both TaSb [38] andTaAs [40], similar to that observed in the known weylsemimetals TaAs family [7–9, 42–47]. Compared to the4 d /5 d electrons in NbAs /TaAs , the more localizationof the 3 d electrons in VAs might lead to a stronger elec-tronic correlation, or introduce an additional magneticscattering to the carriers (Kondo effect).In this paper, we grew successfully VAs crystals witha monoclinic structure and measured its longitudinal re-sistivity ρ xx ( T, H ), Hall resistivity ρ xy ( T, H ), and mag-netic susceptibility as a function of temperature at var-ious magnetic fields, as well as calculated its electronicband structure and Fermi surface (FS). The band struc-ture calculations show that VAs is also a nodal-linesemimetal when spin-orbit coupling (SOC) is ignored. Itwas found that the ρ xx ( T ) measured at H = 0 exhibits aminimum at around 11 K, which is considered of originatefrom the magnetic impurities V ( S = 1/2) scattering( i.e. Kondo effect) , evidenced by both the susceptibil- a r X i v : . [ c ond - m a t . s t r- e l ] D ec ity measurements and the fitting of ρ xx ( T ) data at lowtemperatures. We further reveal a nearly quadratic fielddependence of MR, reaching 649% at 10 K and 9 T, afield-induced up-turn behavior of ρ xx ( T ) in this material,which are attributed to a perfect charge-carrier compen-sation, evidenced by both the calculations relying on theFS topology and the Hall resistivity measurements.VAs single crystals were grown by a chemical vaportransport method. High purity V and As powder weremixed in a mole ratio 1 : 2, then sealed in an evacuatedsilica tube containing I as a transport agent with 10mg/cm . The quartz tube was placed in a tube furnacewith a temperature gradient of 950 ◦ C - 750 ◦ C and heatedfor two weeks. Polyhedral crystals were obtained at thecold end of the tube. A single crystal with a dimensionof 2 . × . × .
29 mm [see Fig. 1(b)] and a cleavagesurface (001) was selected for the transport and magneticproperty measurements. The composition was detectedto be V : As = 32.7 : 67, using the Energy DispersiveX-ray Spectrometer (EDXS), the crystal structure wasdetermined by the single-crystal X-Ray diffraction, asshown in Fig. 1(c), from which, the lattice parameter, c = 7.481(7) ˚A, consistent with the result in Ref. [48].The longitudinal resistivity and Hall resistivity measure-ments were carried out on a physical property measure-ment system (Quantum Design, PPMS-9 T) with a stan-dard four-probe method [see Fig. 1(b)]. The magnetiza-tion measurements were carried out on a magnetic prop-erty measurement system (Quantum Design, MPMS-7T). The band structure was calculated by using the Vi-enna ab initio simulation package (VASP) [49, 50] witha generalized gradient approximation (GGA) of Perdew,Burke and Ernzerhof (PBE) [51] for the exchange cor-relation potential. A cutoff energy of 520 eV and a10 × × k -point mesh were used to perform the bulkcalculations. The nodal-line search and FS calculationswere performed by using the open-source software Wan-nierTools [52] that is based on the Wannier tight-bindingmodel (WTBM) constructed using Wannier90 [53].First at all, we discuss the results of our electronicstructure calculations that extend the initial predictionof the high symmetry line semimetal for VAs [32]. Inorder to address the topological character of VAs , wecalculate its band structure and the FS. As shown inFig. 1(d), there are six different FS sheets: two hole-likesurfaces (green) at the L and Y points, and four electron-like (red) surfaces near L and Z points, the volume ofthe electron and hole pockets is roughly the same, indi-cating that VAs is nearly an electron-hole compensatedsemimetal, also evidenced by the Hall resistivity mea-surements discussed as follows. The bulk band structureof VAs without and with SOC is presented in Fig. 1(f)and 1(g), respectively. It can be seen clearly that thebands near the Fermi level mainly arise from the d orbitsof V atoms, and the valence and conduction bands crossalong the Y-X , Z-I and L-I high symmetry directions. Without SOC, in the Brillouin zones (BZ), the nodal linescan be found by using the open-source software Wannier-Tool [52], see in Fig. 1(h). There are two type of nodallines, one is two nonclosed spiral lines extending acrossthe BZ through point Z, another is two nodal loops nearthe L point. When SOC is included, these nodal linesare gapped [see Fig. 1(g)] and lead to a band inversionalong the Y-X , Z-I and L-I high symmetry directions,driving into a topological crystalline insulator (TCI), asdiscussed by Baokai Wang et al. [41] for the identicalstructure transition metal dipnitides RX (R = Nb orTa; X = P, As or Sb). Although the opening of a localband gap between the valence and conduction bands oc-curs when SOC is included, VAs preserves its semimetalcharacter with the presence of electron and hole pockets,similar to that in NbAs reported in Ref. [41].Next, we focus on the Kondo effect emerging in thelongitudinal resistivity for VAs . Figure 2(a) shows thetemperature dependence of resistivity, ρ xx ( T ). Withdecreasing temperature, the resistivity, ρ xx , decreasesmonotonously from ρ xx (300 K) = 78 µ Ω cm, reaches aminimum at around 11 K, then increases a little to 6.4 µ Ω cm at 2 K, thus the residual resistivity ratio (RRR) ρ xx (300 K)/ ρ xx (2 K) ∼
12, much smaller than thatobserved in the other transition metal dipnictide crys-tals, such as NbAs ( ∼
75) [37], TaAs ( ∼ s − d exchange interactions, exhibits a minimumin resistivity at lower temperatures, i.e. termed as theKondo effect [54, 55]. As discussed by S. Barua et al. forVSe [56], considering the correction to the resistivity dueto the Ruderman-Kittel-Kasuya-Yosida (RKKY) interac-tions between the paramagnetic V ions, the Kondo re-sistivity described by the Hamann expression is modifiedto: ρ sd = ρ − ln( T eff /T K ) (cid:2) ln ( T eff /T K ) + S ( S + 1) π (cid:3) / (1)where ρ is the unitarity limit, T K is the Kondo tempera-ture and S is the spin of magnetic impurity, the effectivetemperature T eff = ( T + T W ) / , in which k B T W is theeffective RKKY interaction strength [54], replaces to T in the original expression [56]. The temperature depen-dence of resistivity, ρ xx ( T ), is expressed as : ρ ( T ) = ρ sd + bT + cT + ρ b (2)where bT n term is the electron-phonon scattering contri-bution, ρ b is an independent resistivity. We used Eq. (2)to fit the resistivity data measured at low temperatures(2 - 40 K). The results are shown in the inset in Fig. 2(a),it is clear that Eq. (2) can well describe the ρ ( T ) databelow 40 K, and the obtained parameters in Eq. (1) andEq. (2) from the fitting are listed in TABLE I. The exis-tence of V ( S = 1/2) impurities in our VAs crystal was Figure 1. (a) Crystal structure of VAs . (b) Photograph of a VAs crystal with 4 wires for resistance measurements. (c) Singlecrystal XRD pattern of VAs . (d) The calculated Fermi surface of VAs . (e) The Brillouin zone. (f) and (g) Band structurescalculated without and with considering SOC. (h) and (i) Nodal lines in the first Brillouin zone from different perspective.TABLE I. The obtained parameters by the fitting to ρ ( T ) data using Eq. (2). ρ ( µ Ω cm) ρ b ( µ Ω cm) b ( µ Ω cm/K ) c ( µ Ω cm/K ) T K (K) T W (K) S × − × − confirmed by the magnetic susceptibility measurements.The temperature dependence of magnetic susceptibility, χ ( T ), measured at 1 T with a field-cooling (FC) processis presented in Fig. 2(b). With decreasing temperature,the susceptibility χ decreases a little, reaches a minimumat around 180 K, then increases strikingly below 100 K.No magnetic transition was observed in the whole tem-perature range (2 - 300 K), and the significant increase of χ in the lower temperatures was considered to originatefrom the contribution of V ( S = 1/2) impurities exist-ing in crystal as the interstitial ions, as observed in VSe crystals [56]. We used the Curie-Weiss law χ = CT − θ , tofit the data below 40 K, as shown in Fig. 2(a), the Curieconstant C = 1.27 ( ± × − emu K/mol, and thecurie temperature θ = -2.61 ( ± χ and T below40 K [see the inset in Fig. 2(b)] demonstrates the relia-bility of the fitting. The V ( S = 1/2) impurity molarfraction was estimated to be of 0.34 ( ± , with the pres-ence of Kondo scattering of V . Figure 3(a) presentsthe temperature dependence of longitudinal resistivity, ρ xx ( T ), measured at various magnetic fields H , with cur-rent I applied in the (001) plane, and H ⊥ (001) plane.Similar to many other nontrivial and trivial topologicalsemimetals [35, 36, 38], VAs also exhibits a large MR.As shown in Fig. 3(a), an up-turn in ρ xx (T) curves un-der applied magnetic field occurs at low temperatures: ρ xx increases with decreasing T and then saturates. Fig-ure 3(b) shows MR as a function of temperature mea-sured at various magnetic fields, with the conventionaldefinition M R = ∆ ρρ (0) =[ ρ ( H ) − ρ (0) ρ (0) ] × α -WP [59],for the nodal-line semimetal MoO [13], and the work ofThoutam et al. on the type-II weyl semimetal WTe [60].Figure 3(c) displays ρ xx ( T ), measured at 0 and 9 T, aswell as their difference ∆ ρ xx = ρ xx ( T , 9 T) - ρ xx ( T , 0T). It is clear that the resistivity in a applied magneticfield consists of two components, ρ ( T ) and ∆ ρ xx , withopposite temperature dependencies. As discussed by usfor α -WP [59], MoO [13] and by Thoutam et al . for (cid:1) xx (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:4) (cid:5) (cid:3) (cid:1) (cid:1) T ( K ) K o n d o E f f e c t (cid:1) xx (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:4) (cid:5) (cid:3) (cid:1) (cid:1) T ( K ) c (×10-4 emu/mol) T ( K ) / c (×105 mol/emu) T ( K ) Figure 2. (a) Temperature dependence of resistivity ρ ( T ),inset: the ρ ( T ) data below 40 K and the fitting by Eq. (2).(b) magnetic susceptibility, measured at 1 T, inset: 1/ χ ( T ). WTe [60], the resistivity can be written as : ρ xx ( T, H ) = ρ ( T )[1 + α ( H/ρ ) m ] (3)The second term is the magnetic-field-induced resistivity∆ ρ xx , which follows the Kohler rule with two constants α and m . ∆ ρ xx is proportional to 1/ ρ (when m = 2)and competes with the first term upon changing temper-ature, possibly giving rise to a minimum in ρ ( T ) curves.Figure 4(a) presents MR as a function of field at varioustemperatures. The measured MR is large at low tem-peratures, reaching 649 % at 10 K and 9 T, and doesnot show any of saturation up to the highest field (9 T)in our measurements. As discussed above, MR can bedescribed by the Kohler scaling law [61, 62]: MR = ∆ ρ xx ( T, H ) ρ ( T ) = α ( H /ρ ) m (4)As shown in Fig. 4(b), all MR data from 2 - 100 K col-lapse onto a single straight line in the plotted as M R ∼ H/ρ curve, and α = 0.038 ( µ Ω cm/T) . and m =1.76 were obtained by fitting. The nearly quadratic fielddependence of MR observed in this nodal-line semimetal
9 T 7 T 5 T 3 T 0 T (cid:3) xx (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:4) (cid:5) (cid:3) T ( K )( c ) (cid:1) xx (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:4) (cid:5) (cid:3) T ( K ) MR (×102 %) T ( K ) MR/MR (2K) T ( K ) (cid:3) xx , (cid:2) (cid:3) (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:4) (cid:5) (cid:3) (cid:1) (cid:1) T ( K ) (cid:1) (cid:1) = (cid:1) x x ( 9 T ) - (cid:1) (cid:1) (cid:1) x x ( 9 T ) Figure 3. (Color online) (a) Temperature dependence of re-sistivity measured at various magnetic fields. (b) The MRvs. temperature under various magnetic fields. The inset isnormalized MR. (c) Temperature dependence of resistivity at0 and 9 T and their difference. The red and blue lines are thefitting lines using the Kohler scaling law. ( b ) MR (×102 %) (cid:1) H ( T ) MR (%) (cid:1) H / (cid:1) x x (cid:2) (cid:4) (cid:3) (cid:1) (cid:2) (cid:5) (cid:1) (cid:6) - 1 c m - 1 )
2 K 5 K 1 0 K 2 0 K 3 0 K 4 0 K 5 0 K 6 0 K 1 0 0 K F i t
M R (cid:1) (cid:3) (cid:1) (cid:2) H / (cid:1) x x ( 0 ) ] ( a ) Figure 4. (a) Field dependence of MR at various tempera-ture. (b) MR plotted as a log scale as a function of H / ρ xx (0). VAs is attributed to the electron-hole compensation, ev-idenced by FS calculations mentioned above, as well asthe Hall resistivity measurements discussed below, whichis a common characteristics for the most topologicallynontrivial and trivial semimetals [35, 36, 38]. Figure5(a) displays the Hall resistivity, ρ xy ( H ), measured atvarious temperatures for a VAs crystal with H (cid:107) c axis.The nonlinear field dependence of ρ xy ( H ) below 100 Kdemonstrates its semimetal characteristics, in which bothelectron and hole carriers coexist. Following the analysisof Ref. [63] for γ − MoTe [64] and by us for MoO [13],we analyze the longitudinal and Hall resistivity by usingthe two-carrier model. In this model, the conductivitytensor, in its complex representation, is giving by [65]: σ = en e µ e iµ e µ H + en h µ h iµ h µ H (5)where n e ( n h ) and µ e ( µ h ) denote the carrier concentra-tions and mobilities of electrons (holes), respectively. To appropriately evaluate the carrier densities and mobili-ties, we calculated the Hall conductivity σ xy = - ρ xy ρ xx + ρ xy and the longitudinal conductivity σ xx = ρ xx ρ xx + ρ xy by us-ing the original experimental ρ xy ( H ) and ρ xx ( H ) data.Then, we fit both σ xy ( H ) and σ xx ( H ) data by using thesame fitting parameters and the field dependence givenby [64]: σ xy = e ( µ H ) n h µ h µ h ( µ H ) − e ( µ H ) n e µ e µ e ( µ H ) (6) σ xx = en h µ h µ h ( µ H ) + en e µ e µ e ( µ H ) (7)Figures 5(c) and 5(d) display the fitting of both the σ xy ( H ) and σ xx ( H ) measured at T = 2 - 60 K, respec-tively. The excellent agreement between our experimen-tal data and the two-carrier model over a broad range oftemperature, confirms the coexistence of electrons andholes in VAs . Figure 5(b) shows the obtained n e , n h , µ e and µ h values by fittings as a function of tempera-ture. The almost same values of n e and n h below 60 K[see the inset Fig. 5(b)], such as n e = 1.77 × cm − and n h = 1.69 × cm − at 2 K, indicate that VAs is indeed a electron-hole compensated semimetal, con-sistent with the above results from the calculation FS.Both electron and hole densities are estimated to be 10 cm − in our VAs crystal, significantly higher than thosein Dirac semimetals, such as, Cd As ( ∼ cm − [66])and Na Bi ( ∼ cm − [67]), but comparable to these ofanother nodal-line semimetals ZrSiS ( ∼ cm − [11]),and MoO ( ∼ cm − [13]), demonstrating further thenodal-line characteristics of VAs . As shown in Fig. 5(b),it is clear that the hole mobility µ h is higher than µ e inthe whole temperature range (2 - 300 K), such as, at 2K, µ h = 4.08 × cm /Vs, µ e = 1.54 × cm /Vs,but smaller one order of magnitude than these observedby us in the nodal-line semimetal MoO ( ∼ cm /Vs)[13], and both µ h and µ e decrease notably with increas-ing temperature due to the existence of phonon thermalscattering at higher temperatures. It is worth noting thatboth µ h and µ e have a little decrease below 11 K, cor-responding to the Kondo scattering from V magneticimpurities mentioned above.In summary, we calculated the electronic structure,and measured the longitudinal resistivity, Hall resistiv-ity and magnetic susceptibility for VAs . It was foundthat VAs exhibits many common characteristics of thetopological nontrivial/trivial semimetals, such as a largeMR reaching 649% at 10 K and 9 T, a nearly quadraticfield dependence of MR, and a field-induced up-turn be-haviour in ρ xx ( T ). Both the FS calculations and the Hallresistivity measurements verify these properties being at-tributed to a perfect carrier compensation. Interestingly,the Kondo scattering due to the existence of V ( S =1/2) magnetic impurities in our VAs crystals occurs, (cid:2) xy (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:4) (cid:5) (cid:3) (cid:1) H ( T )
2 K 5 K 1 0 K 2 0 K 3 0 K 4 0 K (cid:1) (cid:1) (cid:1) e (cid:1) h (cid:1) e, (cid:1) h (×103 cm2/V s) (cid:1) H ( T ) (cid:1) (cid:1) n e n h ne, nh (×1020 cm-3) (cid:1) H ( T ) ( b )( d ) (cid:1) (cid:1) (cid:1) (cid:3) yx (×103 (cid:1) (cid:6) -1 cm-1) (cid:1) H ( T ) ( c ) (cid:1) (cid:1)
2 K 5 K 1 0 K 2 0 K 3 0 K 4 0 K 5 0 K 6 0 K (cid:1) xx (×104 (cid:1) (cid:2) -1cm-1) (cid:1) H ( T ) ( a ) Figure 5. (a) Field dependence of Hall resistivity ρ xy measured at various temperatures for VAs crystal. (b) Charge-carriermobilities µ e and µ h , and (inset) the carrier concentrations, n e and n h , as a function of temperature extracted from the two-carrier model analysis of both σ xy and σ xx data. Components of the conductivity tenser, i.e. σ xy and σ xx in the panels (c) and(d), respectively, as a function of magnetic field for different temperatures ( <
60 K). Hollow dots represent experimental dataand solid lines are the fitting curves by using the two-carrier model. indicating that VAs crystal can be used to study theKondo effect in the nodal-line semimetal.ACKNOWLEDGEMENTS: This research is sup-ported by the Ministry of Science and Technology ofChina under Grant No. 2016YFA0300402 and the Na-tional Natural Science Foundation of China (NSFC)(NSFC-12074335 and No. 11974095), the Zhejiang Nat-ural Science Foundation (No. LY16A040012) and theFundamental Research Funds for the Central Universi-ties. ∗ Corresponding author: [email protected][1] R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983).[2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005).[3] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. ,106802 (2006). [4] L. Fu, Phys. Rev. Lett. , 266801 (2009).[5] Z. Liu, B. Zhou, Y. Zhang, Z. Wang, H. Weng, D. Prab-hakaran, S.-K. Mo, Z. Shen, Z. Fang, X. Dai, Z. Hussain,and Y. Chen, Science (New York, N.Y.) , 864 (2014).[6] Z. 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