The discovery of Weyl and Dirac fermions in solid systems is a recent major breakthrough in the field of condensed matter physics. These materials exhibit extraordinary properties in terms of carrier mobility and magnetoresistance (MR). These two quantities are highly dependent in the Weyl semimetal transition monopnictide family, i.e. NbP, TaP, NbAs, and TaAs. Furthermore, the gathered mobility and MR (or slope of MR) at 2 K in 9 T of other well-known Weyl and Dirac semimetals follow a relation similar to the right turn symbol, i.e. the MR increases rapidly with mobility; thereafter it begins to saturate after reaching a value of 10(3). This suggests a nonlinear dependency. Nevertheless, for materials possessing high carrier mobility, it is valid to expect high MR

Felser, C.

Schmidt, M.

Shekhar, C.

Singh, S.

Süß, V.

English

MPG.PuRe

PAPER • OPEN ACCESSStrong correlation between mobility andmagnetoresistance in Weyl and Dirac semimetalsTo cite this article: Sukriti Singh et al 2020 J. Phys. Mater. 3 024003 View the article online for updates and enhancements.You may also likeElectronic properties of candidate type-IIWeyl semimetal WTe2. A reviewperspectiveP K Das, D Di Sante, F Cilento et al.-Twisted Fermi surface of a thin-film WeylsemimetalN Bovenzi, M Breitkreiz, T E O’Brien et al.-Experimental progress on layeredtopological semimetalsJunchao Ma, Ke Deng, Lu Zheng et al.-This content was downloaded from IP address 141.5.13.137 on 22/06/2022 at 15:18J. Phys.:Mater. 3 (2020) 024003 https://doi.org/10.1088/2515-7639/ab6c34PAPERStrong correlation betweenmobility andmagnetoresistance inWeyland Dirac semimetalsSukriti Singh, Vicky Süβ,Marcus Schmidt, Claudia Felser andChandra ShekharMaxPlanck Institute for Chemical Physics of Solids, D-01187Dresden, GermanyE-mail: shekhar@cpfs.mpg.deKeywords:Weyl semimetal, highmobility,magnetoresistanceAbstractThe discovery ofWeyl andDirac fermions in solid systems is a recentmajor breakthrough in thefieldof condensedmatter physics. Thesematerials exhibit extraordinary properties in terms of carriermobility andmagnetoresistance (MR). These two quantities are highly dependent in theWeylsemimetal transitionmonopnictide family, i.e. NbP, TaP,NbAs, andTaAs. Furthermore, the gatheredmobility andMR (or slope ofMR) at 2 K in 9 T of otherwell-knownWeyl andDirac semimetals followa relation similar to the right turn symbol, i.e. theMR increases rapidly withmobility; thereafter itbegins to saturate after reaching a value of 103. This suggests a nonlinear dependency. Nevertheless,formaterials possessing high carriermobility, it is valid to expect highMR.1. IntroductionMobility refers to the velocity acquired by the charge carrier in a unit of electric field, whereasmagnetoresistance(MR) indicates the extent of the resistance change on applying amagnetic field. These two effects are highlypronounced in semimetals, because the valence bandmaximumand conduction bandminimum lie close to theFermi level, and the charge carriers do not require significant energy tomove. Thismeans that the changes aresensitive even if there are small perturbations and that these changes are easily visible in the physical properties.Recently, the topology ofmaterials has attracted tremendous research attention, and it is currently a leadingresearch area in thefields of condensedmatter physics andmaterial science [1–10]. Similar to topologicalinsulators, semimetals are also topologically protected, nontrivialmetallic phases ofmatter; this nontrivialtopological nature guarantees the existence of exotic Fermi surface as a Fermi arc [2–5]. Owing to thesemimetallic character ofmaterials, the conduction and valence bands generally cross each other near the Fermienergy. Depending on their degeneracy, topological semimetals are further distinguished and classified intoWeyl semimetals (WSM) andDirac semimetals. The discovery ofWeyl fermions in transitionmetalmonopnictides [1–6]has facilitated the discovery of additionalDirac andWeyl semimetals in the variousfamilies of compounds [11–27].Unlike their family and class, topological semimetals exhibit significantly highmobility andMR,accompanied by a low effectivemass of the charge carrier. This low effectivemass results in a highmobility; thus,thematerial reaches the quantum limit in amoderatemagnetic field. In this condition, the resistivity ofmaterialsvaries linearly with themagnetic field, resulting in a linearMR,which is known as quantumMR [28]. Amongtopologicalmaterials, the number of compounds that exhibit linearMR is comparatively low [14–16, 19].Generally,MRdeviates from linearity in amajority of the compounds and exhibits parabolic behaviour due tothe charge carrier compensation [11–13, 17, 18, 20–27]. An excellent example of parabolicMR is exhibited bytype-IIWeyl semimetals, inwhich unavoidably Fermi level possesses through the tiltedWeyl cone andsimultaneously creates electron and hole bands [12, 13, 23]. Irrespective of theMRbehaviour, the fluctuations inmobility arising due to disorder or inhomogeneity is another phenomenon that occurs in awide range ofmaterials [29]. Here, we investigate the relation betweenMR andmobility by considering the example ofWeylsemimetal transitionmetalmonopnictides. Based on the data of the best-knownWeyl andDirac semimetalsOPEN ACCESSRECEIVED8October 2019REVISED11December 2019ACCEPTED FOR PUBLICATION15 January 2020PUBLISHED25 February 2020Original content from thisworkmay be used underthe terms of the CreativeCommonsAttribution 3.0licence.Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.© 2020TheAuthor(s). Published by IOPPublishing Ltd(or slope ofMR) at∼2K in 9Tpresented in literature, themobility andMR are correlated, and they follow the‘right turn’ symbol, i.e.MR increases rapidly and thereafter begins to saturate after reaching the value of 103.2. Experimental detailsSingle crystals ofNbP, TaP,NbAs, andTaAswere grown via chemical vapour transport [30, 31]. As afirst step forpolycrystallinematerial, stoichiometric quantities ofNb, Ta (Alfa Aesar, 99.99%), P (Alfa Aesar, 99.999%), andAs (Chempur, 99.9999%)were accurately weighed in a quartz ampoule. Thereafter, theywereflushedwithAr,sealed under vacuum, and the sealed quart tubewas heated in two consecutive temperatures, i.e. 600 °C and800 °C, for 24 h. In the next step to grow the crystals, we used themicrocrystalline powders from step one andadded iodine (7–8 mgml−1) before sealing the powders in a quartz tube. The crystal was grown in a two-zonefurnace at a temperature range of 900 °C–1050 °C, for 2–4weeks. Here, I2 acts as a transport agent. To obtainhigh quality crystals, the temperature gradient is one of themost important parameters, and it varies based onthematerial.We optimized the temperature gradient for eachmaterialmentioned in the present investigation. Itis 900 °C (source)–1000 °C (sink) forNbP andTaP and 900 °C (source)–1050 °C (sink) forNbAs andTaAs.Additional details regarding crystal growth and their characterization can be found in our previous studies [6,32–34]. The experimental lattice parameters of the four compounds are a=3.3314(8)Åand c=11.3649(3)ÅforNbP, a=3.3166(9)Å and c=11.3304(6)Å for TaP, a=3.4492(6)Åand c=11.6647(3)Å forNbAs, anda=3.4310(4)Å and c=11.6252(6)Å for TaAs. These values are similar to those in previous reports [35, 36].The resistivitymeasurements were performed inQuantumDesign-physical propertymeasurement system inthe AC-Transport options. Ourwell characterized crystals were cut into bar-shapes by using thewire saw andkeeping the longest length along the crystallographic a-axis, inwhich direction the current was applied inresistivitymeasurements. The physical dimensions are 3.1mm×1.6mm×0.56mm forNbP, 1.1mm×0.42mm×0.16mm for TaP, 0.93mm×0.83mm×0.17mm forNbAs, and 1.5mm×0.42mm×0.28mmfor TaAs. Themeasurements for resistivity andHall resistivity were performed in four and five-probegeometries, respectively, for a constant current source of 3–4mA.3. Results and discussionThe resistivity, ρxx of all crystals at the zero field decreases with decreasing temperature, which is an expectedcharacteristic ofmetallic compounds. Their low residual resistivity and high residual resistivity ratio valuesreflect the high quality of the crystals [6]. Apart from thewell-establishedWeyl property, these compounds arealsowell known for exhibiting ultrahighmobility andMR.Wenow focus on themeasurements forMR.Generally,MR is calculated as the ratio of the change in resistivity due to the appliedmagnetic field and the initialresistivity, using the relation ( ) ( ) ( )/r r r= -MR B 0 0 ,xx xx xx whereB is appliedmagnetic field perpendicularto current. On the contrary,mobility is estimated based on the slope ofHall resistivity at the highmagnetic fieldregion, where it is almost linear with the field at different temperatures. ThemeasuredMRofNbP, TaP,NbAs,andTaAs at the selected temperatures are shown infigure 1. Themost noteworthy features are(i) Positive and unsaturatingMR that exhibits systematic variations with the temperature and field.(ii) TheMR is almost constant up to 50K,whereas it decreases sharply above 50K.(iii) More importantly, all the compounds exhibit Shubnikov de-Haas (SdH) oscillations below 20 K, therebyreflecting the high quality of the crystals.A detailed analysis of the SdHoscillations and their fermiology can be found in previous literature [6, 32].Among the four compounds, NbP exhibits the highestMRof 8.5×105% at 1.85K, in afield of 9 T.Additionally, it does not exhibit any sign of saturation under ultrahighfields of 62T, at 1.5 K [6]. This value isfivetimes higher than the value that was reported forWTe2, which is anotherWSM, in the samefield [12]. TheMRofNbP is as high as 250% even at 300K and 9T (inset of figure 1(a)).Moreover, the othermembers of this seriesalso exhibit similarMRorders as that ofNbP at all temperature and field ranges [37, 38]. In the presence ofmagnetic fields, a nonzero transverse current experiences a Lorentz force in the inverse-longitudinal direction.Such a backflowof carriers eventually increases the apparent longitudinal resistance, resulting in an extremelyhighMR. From the abovementionedmeasurements, we can conclude that thesematerials have a uniqueproperty of high and never saturatingMR.Wenow focus onmobility, which can be derived from themeasurements ofHall resistivity, ρxy, which isfurther defined as /r = V t I. ,yx y x whereVy is the voltage developed normal to the appliedmagnetic field andcurrent Ix and t is the thickness of the crystal. The carriermobility and concentration usually derived2J. Phys.:Mater. 3 (2020) 024003 S Singh et alfromryx are two important parameters of amaterial.We have performedHallmeasurements in the positive andnegative field directions to improve the accuracy of data. For the sake of simplicity, we used the single-carrierDrudemodel, ( ) ( ) ( )/m r=T R T T ,avg H xx where ( )m Tavg is the averagemobility, and ( )R TH is theHallcoefficient calculated from the linear slope of ( )r Tyx at the high field.However, the sign of theHall coefficientchanges fromnegative to positive between 125 and 170K, depending on thematerial, e.g. this temperature isapproximately 125K forNb-compounds [6] and 170K for Ta-compounds [32]. The charge carrier density liesbetween 1018 and 1020 cm−3 in the temperature range of 2–300K. The observed small carrier density at lowtemperatures and its significant change on increasing the temperature are typical characteristics of semimetals. AlargeMR is usually associatedwith highmobility, and it plays amajor role in the charge transport in amaterial.Consequently, it determines the efficiency of devices.We calculated the averagemobility and plotted it againsttemperature, as shown in figure 2. It is evident thatNbP exhibits the highestmobility of 5×106 cm2 V−1 s−1and the lowest residual resistivity of 0.63μΩ cmat 2K. These values are similar to those of Cd3As2 [15]. Themobilities of the othermembers are 3×105 cm2 V−1 s−1 for TaP, 5×105 cm2 V−1 s−1 forNbAs, and 4×105 cm2 V−1 s−1 for TaAs, at 2 K. Their residual resistivities are 3.2μΩ cm for TaP, 6.2μΩ cm forNbAs, and4.2μΩ cm for TaAs. It is noteworthy that the estimatedmobility from a single-carriermodel only differs slightlyfrom themobility obtained from the two-carriermodel; consequently, its overall temperature dependentbehaviour remains almost the same. From the abovementioned values ofmobility and residual resistivity, it isclear that an anomalously low residual resistivity can be achieved in a clean systemwhere themobility attainsultrahigh values.When themagnetic field is applied, the low temperature resistivity increases steeply, resultingin a significantly largeMR [6, 12, 32]. To examine the role of themobility onMR,we calculated the slope of theMRat approximately 9 T, and themobility at different temperatures forNbP, TaP,NbAs, andTaAs. The slope/¶ ¶MR B is plotted on the right axis, and the carriermobility is plotted on the left axis as a function oftemperature, as shown in figure 2. /¶ ¶MR B andμavg exhibit identical variationwith temperature, indicatingthatMR is governed by themobility, at least in these four compounds. A slight deviation of themobility graph inthe range 100–200K belongs to the regionwhere themajority of charge carriers switch from electrons to holes.We conclude that the averagemobility andMR arewell correlated for all the four compounds.Figure 1.Estimated transversemagnetoresistance at different temperatures from the field dependent resistivity using( ) ( ) ( )/r r r= -MR B 0 0xx xx xx for (a)NbP, (b)TaP, (c)NbAs, (d)TaAs. Inset show their respectivemagnetoresistance at hightemperature150K.3J. Phys.:Mater. 3 (2020) 024003 S Singh et alWegeneralize our findings for a broader range ofWeyl andDirac semimetals. It is essential to examine thevariability inMRwith respect to themobility in these compounds.We gathered these two quantities for variouscompounds at∼2K and 9T from literature [11–27, 39–42]. AsMR varies with thefield, it ismore valid toconsider the slope of theMRwith the field than considering the value ofMR in a fixfield.However, we plottedMRand its slope at approximately 9 T against themobility at 2K, as shown infigures 3(a) and (b), respectively.The behaviour of both the graphs is highly similar, and both show an intriguing relation betweenMR andmobility. TheMR initially increases sharply withmobility, and thereafter it begins to saturate after reaching thevalue of 103, forming a shape similar to the right turn symbol. Nevertheless, this relation broadly confirms thathighmobility compounds possess highMR. It seems that this right turn behavior is limited to charge carrierdensity. A group of compoundswith similar order ofmobility (for example,WP2,MoSi2 andNbAs2) but thehigh carrier density possess highMRwhile the low carrier density compounds (for example, Cd3As2, ZrTe5 andNbAs) show lowMR. This approximation can be understood in termof charge saturation in quantum limit inwhich the semimetals with lower carrier density get easily saturated at rather lowerfield, exhibiting lowerMRcompared to the semimetals with higher carrier density. Noticeably, in the presence ofmagnetic fields, a highmobile carrier experience a stronger Lorentz force, and exhibits highMR.However, it appears that apart fromthis, some other factors also contribute to the Lorentz force, otherwiseMRwould be linearly related to themobility.Figure 2.Temperature dependent averagemobility, mavg (left axis) from single carriermodel and slope ofmagnetoresistance (rightaxis) around 9T field for (a)NbP, (b)TaP, (c), NbAs and (d)TaAs. From the two-carriermodel, the obtainedmobility is similar assingle-carriermodel, and therefore its temperature dependent does not altermuch.Figure 3.Magnetoresistance (MR) andmobility of various well-knownWeyl andDirac semimetals at∼2K. (a)MR in 9T versusmobility. (b) Slope ofMR around 9T versusmobility. The compoundswith filled circle show linearMR. The data were taken from the[11–27, 39–43].4J. Phys.:Mater. 3 (2020) 024003 S Singh et al4. ConclusionTo summarize, we demonstrate the dependence ofMRandmobility on temperature by considering theexamples of transitionmetal−monopnictidesWeyl semimetals.We also show that a similar dependencymayexist in other compounds. The plots between theMRor the slope ofMRversus themobility of themost popularWeyl andDirac semimetals exhibit a shape similar to the right turn symbol, i.e. theMR increases sharply andthereafter undergoes saturation after reaching 103.Our analyses reveal that theMRhighly depends onmobility,and a highMR can be expected in high carriermobility compounds.AcknowledgmentsThisworkwas supported by theDeutsche Forschungsgemein‐ schaft DFG (SFB 1143) and by the ERCAdvancedgrant 742068 ‘TOPMAT’.ORCID iDsChandra Shekhar https://orcid.org/0000-0002-3330-0400References[1] WengH, FangC, FangZ, Bernevig BA andDai X 2015Phys. Rev.X 5 011029[2] Xu S-Y et al 2015Nat. Phys. 11 748–54[3] Yang LX et al 2015Nat. Phys. 11 728–32[4] LvBQ et al 2015Phys. Rev.X 5 031013[5] Liu ZK et al 2016Nat.Mater. 15 27–31[6] ShekharC et al 2015Nat. Phys. 11 645–9[7] Bradlyn B, Cano J,WangZ, VergnioryMG, Felser C, CavaR J andBernevig BA 2016 Science 353 aaf5037[8] ArmitageNP,Mele E J andVishwanathA 2018Rev.Modern Phys. 90 015001[9] Hu J, Xu S-Y,NiN andMaoZ 2019Ann. Rev.Mater. Res. 49 207–52[10] GaoH,Venderbos JWF, KimY andRappeAM2019Ann. Rev.Mater. Res. 49 153–83[11] MunE,KoH,Miller G J, SamolyukGD, Bud’ko S L andCanfield PC 2012Phys. Rev.B 85 035135[12] AliMN et al 2014Nature 514 205–8[13] KeumDH et al 2015Nat. Phys. 11 482[14] Tafti F F,GibsonQD,Kushwaha SK,HaldolaarachchigeN andCavaR J 2015Nat. Phys. 12 272[15] LiangT,GibsonQ, AliMN, LiuM,Cava R J andOngNP2015Nat.Mater. 14 280–4[16] YuanZ, LuH, Liu Y,Wang J and Jia S 2016Phys. Rev.B 93 184405[17] KumarN, Shekhar C,Wu S-C, Leermakers I, YoungO, ZeitlerU, YanB and Felser C 2016Phys. Rev.B 93 241106[18] AliMN, Schoop LM,GargC, Lippmann JM, Lara E, Lotsch B and Parkin S 2016 Sci. Adv. 2 e1601742[19] Xiong J, Kushwaha S, Krizan J, Liang T, Cava R J andOngNP2016Europhys. Lett. 114 17002[20] PavlosiukO, Swatek P andWiśniewski P 2016 Sci. Rep. 6 18691[21] JoNH et al 2017Phys. Rev.B 96 165145[22] GaoW et al 2017Phys. Rev. Lett. 118 256601[23] KumarN et al 2017Nat. Commun. 8 1642[24] KumarN,MannaK,Qi Y,Wu S-C,Wang L, YanB, Felser C and ShekharC 2017Phys. Rev.B 95 121109[25] Leahy I A, Lin Y-P, Siegfried P E, Treglia AC, Song J CW,Nandkishore RMandLeeM2018Proc.Natl Acad. Sci. 115 10570–5[26] MatinM,Mondal R, BarmanN, Thamizhavel A andDhar SK 2018Phys. Rev.B 97 205130[27] PavlosiukO andKaczorowski D 2018 Sci. Rep. 8 11297[28] AbrikosovAA 1998Phys. Rev.B 58 2788–94[29] ParishMMand Littlewood PB 2003Nature 426 162–5[30] SchäferH and FuhrW1965 J. Less CommonMet. 8 375–87[31] Martin J andGruehnR 1988Z. Kristallogr. 182 180–2[32] Arnold F et al 2016NatCommun 7 11615[33] dos Reis RD,WuSC, SunY, AjeeshMO, Shekhar C, SchmidtM, Felser C, YanB andNicklasM2016Phys. Rev.B 93 205102[34] Klotz J et al 2016Phys. Rev.B 93 121105[35] BollerH andParthe E 1963Acta Crystallogr. 16 1095–101[36] Saini G S, Calvert LD andTaylor J B 1964Can. J. Chem 42 630–4[37] HuangX et al 2015Phys. Rev.X 5 031023[38] GhimireN J, LuoY,NeupaneM,WilliamsD J, Bauer ED andRonning F 2015 J. Phys.: Condens.Matter. 27 152201[39] Fauqué B, YangX, TabisW, ShenM, ZhuZ, Proust C, Fuseya Y andBehnia K 2018Phys. Rev.Mater. 2 114201[40] Zhao L, XuQ,WangX,He J, Li J, YangH, LongY, ChenD, LiangH and Li C 2017Phys. Rev.B 95 115119[41] SouleD 1958Phys. Rev. 112 698[42] Li P, ZhangQ,HeX, RenW,ChengH-M andZhangX-X 2016Phys. Rev.B 94 045402[43] PavlosiukO, Swatek P, Kaczorowski D andWiśniewski P 2018Phys. Rev.B 97 2351325J. Phys.:Mater. 3 (2020) 024003 S Singh et al

Strong correlation between mobility and magnetoresistance in Weyl and Dirac semimetals

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Strong correlation between mobility and magnetoresistance in Weyl and Dirac semimetals

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