Magnetoresistance in YBi and LuBi semimetals due to nearly perfect carrier compensation
Orest Pavlosiuk, Przemysław Swatek, Dariusz Kaczorowski, Piotr Wiśniewski
MMagnetoresistance in LuBi and YBi semimetalsdue to nearly perfect carrier compensation
Orest Pavlosiuk , Przemysław Swatek , , , Dariusz Kaczorowski , and Piotr Wiśniewski ,* Institute of Low Temperature and Structure Research,Polish Academy of Sciences, P.Nr 1410, 50-950 Wrocław, Poland, Division of Materials Science and Engineering, Ames Laboratory, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Dated: August 27, 2018)Monobismuthides of lutetium and yttrium are shown as new representatives of materialswhich exhibit extreme magnetoresistance and magnetic-field-induced resistivity plateaus. At lowtemperatures and in magnetic fields of 9 T the magnetoresistance attains orders of magnitudeof % and % , on YBi and LuBi, respectively. Our thorough examination of electrontransport properties of both compounds show that observed features are the consequence of nearlyperfect carrier compensation rather than of possible nontrivial topology of electronic states. Thefield-induced plateau of electrical resistivity can be explained with Kohler scaling. An anisotropicmultiband model of electronic transport describes very well the magnetic field dependence ofelectrical resistivity and Hall resistivity. Data obtained from the Shubnikov–de Haas oscillationsanalysis also confirm that the Fermi surface of each compound contains almost equal amounts ofholes and electrons. First-principle calculations of electronic band structure are in a very goodagreement with the experimental data.* Corresponding author: [email protected] PACS numbers: 71.20.Eh, 72.15.Gd, 74.25.F-, 74.70.Dd
Materials with extremely magnetic-field-dependent re-sistivity attract massive attention because of their possi-ble applications in sensors and spintronic devices. Rare-earth-metal monopnictides with the NaCl-type crystalstructure form a group of materials that possess rele-vant extraordinary properties. The very first observa-tion of extreme magnetoresistance (XMR) in lanthanummonopnictides has been reported by Kasuya et al. in1996 . Two decades later it has been proposed that lan-thanum monopnictides could be topologically nontrivialmaterials and magnetotransport properties of LaSb andLaBi have been found to resemble those of topologicalsemimetals . It was the starting point of an intensiverevival of interest in rare-earth-metal monopnictides. Upto date the question of the nontrivial topology of theirelectronic structures has remained open. Reports on theangle-resolved photoemission spectroscopy (ARPES) in-vestigations of rare-earth-metal monopnictides differ intheir conclusions. Some describe these materials as hav-ing Dirac-like features in their electronic structure ,others show that nontrivial topology is absent .Non-saturating (in magnetic field) XMR has earlierbeen reported for Dirac semimetals Cd As and ZrSiS,and Weyl semimetals NbP and TaAs . How-ever, their XMR could be often understood withoutinvoking nontrivial topology. In non-magnetic materi-als, charge carrier compensation , field-induced metal-insulator transition (all unrelated to nontrivial topol-ogy), or field-induced lifting of topological protectionfrom backscattering could be responsible for XMR.This work on YBi and LuBi is a continuation of ourprevious investigations of NaCl-type monoantimonideswith high magnetoresistance . These two compounds have been barely studied previously. The first report onYBi crystal structure appeared in Ref. 22, and then bi-nary phase diagrams Y-Bi and Lu-Bi, including YBi andLuBi, have been determined . Several theoretical pa-pers on lutetium monopnictides and YBi also appearedin the past few years . There was hitherto no in-formation about magnetotransport properties of yttriumand lutetium monobismuthides. Here we report on elec-tronic transport properties of high-quality single crystalsof YBi and LuBi studied in magnetic fields up to 9 T. Ex-perimental data are compared with results of electronicstructure calculations.We grew high-quality single crystals from Bi flux withthe starting atomic composition RE :Bi of 1:19 ( RE = Yor Lu). They had shapes of cubes with the dimensionsup to × × mm . Microanalysis of the crystals witha scanning electron microscope equipped with energy-dispersive X-ray spectrometer (FEI SEM with an EDAXGenesis XM4 spectrometer) yielded equiatomic chemicalcomposition of both compounds. Electrical resistivityand Hall effect measurements were carried out in a tem-perature range from 2 to 300 K and in applied magneticfields up to 9 T on a Quantum Design PPMS platform.Standard four-probe method was used for all measure-ments. Bar-shaped specimens with all edges along (cid:104) (cid:105) crystallographic directions were cut from single crystalsand then polished. Electrical contacts were made from µ m-thick silver wires attached to the samples by spotwelding and strengthened with silver epoxy. The elec-tric current was always flowing along the [100] crystal-lographic direction and the magnetic field was appliedalong the [001] crystallographic direction. a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug FIG. 1. Magnetoresistance isotherms of YBi (a) and LuBi (b) measured in magnetic field applied along [001] direction, transverseto electrical current.
Electronic structure calculations were carried out us-ing both the wien2k code with the full-potentiallinearized augmented plane wave (FLAPW) method,and the full-potential Korringa-Kohn-Rostoker (KKR)Green’s function method . The exchange and cor-relation effects were treated using the generalized gra-dient approximation (GGA) . Spin-orbit couplingwas included as a second variational step, using scalar-relativistic eigenfunctions as the basis, after the ini-tial calculation was converged to self-consistency. TheMonkhorst-Pack special k -point scheme with × × mesh was used in the first Brillouin zone sampling, andthe cutoff parameter ( R mt K max ) was set to 8. For theFermi surface, the irreducible Brillouin zone was sam-pled by 20225 k -points to ensure accurate determinationof the Fermi level . Shubnikov–de Haas (SdH) frequen-cies were calculated using the Supercell k -space ExtremalArea Finder tool . Magnetoresistance, electrical resistivity and Hallresistivity
Figure 1 shows magnetoresistance, MR = 100 % × [ ρ ( B ) − ρ ( B = 0)] /ρ ( B = 0) , of YBi and LuBi as a functionof magnetic field, B , measured at several temperatures, T , in the range from 2 to 300 K. For both compounds, MR has extreme values at low temperatures (for YBi, MR = 6 . × % and for LuBi, MR = 7 . × % at T = 2 K in B = 9 T). Up to T = 10 K, magnitudes of MR change only slightly, a pattern which corresponds tothe resistivity plateaus in ρ ( T ) (see Fig. 3). We supposethat such big MR of our samples could be due to nearlyperfect carrier compensation, as it has been reported forother rare-earth-metal monopnictides .The difference between MR values of LuBi and YBiseems to reflect the difference in sample quality, ratherthan difference in electronic structures (see the next sub-section). On the example of compensated semimetalWTe and lanthanum monopnictides, it has been shown that magnitude of MR strongly depends on sample qual-ity . On heating above 10 K, MR of both compoundsdecreases strongly, and at 300 K reaches 6 % and 7 % (in9 T) for LuBi and YBi, respectively.Figure 2 shows the results of Kohler scaling of MR forboth compounds. All MR isotherms measured at differ-ent temperatures collapse on a single curve. According FIG. 2. Kohler scaling of transverse magnetoresistance, with MR ∝ ( B/ρ ) m fitted to the data from temperature range2–300 K yielding m = 1 . for YBi (a) and m = 1 . for LuBi(b). to the Kohler rule, MR ∝ ( B/ρ ) m , (1)where m is a sample-dependent constant that dependson the level of compensation (for perfectly carrier com-pensated systems m = 2 ). From the fitting of Eq. 1 (redsolid lines in Fig. 2) to experimental data we obtained m = 1 . and m = 1 . for YBi and LuBi, respectively.These values of m are larger than previously reportedfor rare-earth-metal monoantimonides . Our m values are close to that determined for LaBi , but stillsmaller than 1.92 reported for WTe in Ref. 40. Theyshow that the carrier compensation in LuBi is slightlybetter than in YBi.Figure 3 presents the results of electrical resistivity, ρ , measurements for YBi and LuBi in varying tempera-ture in zero and in finite magnetic fields. When B = 0 ,both compounds demonstrate metallic behavior of ρ ( T ) , ρ gradually decreases with T lowering, from the values20.0 and . µ Ω cm at T = 300 K to the values 0.1 and . µ Ω cm at T = 2 K for YBi and LuBi, respectively.It means that residual resistivity ratios [ ρ (300 K) /ρ (2 K) ]are quite large and equal to 180 and 55 for YBi and LuBi,respectively.Applying a magnetic field drastically changes the ρ ( T ) behavior. Already in 3 T, ρ of each compound decreasesupon cooling only to certain temperature where it hasa minimum. Further decreasing of temperature leads toincrease of ρ and its saturation below T ≈ K. Higherfields increase the values of resistivity in plateau regionin accordance with MR ∝ B m behavior depicted inFig. 1. Such magnetic field-induced resistivity plateau isa characteristic feature of topological semimetals and has also been observed in several rare-earth-metalmonopnictides .The authors of Ref. 40 argued that analogous turn-onbehavior of ρ ( T ) in WTe could be understood in thescope of Kohler scaling. We used this approach to de-scribe electrical resistivity of both studied monopnictides(see Fig. 4). Previously, it has also been used by Han etal. to explain magnetotransport properties of LaSb .According to Wang et al. , ρ ( T ) measured in magneticfield can be described by the following equation: ρ ( T, B ) = ρ ( T,
0) + ∆ ρ ( T, B ) , (2)where the first term corresponds to the temperature de-pendence of resistivity in zero magnetic field and thesecond term describes magnetic-field-induced resistivity.Assuming that ρ ( T, can be well approximated withthe Bloch-Grüneisen law, ρ ( T ) = ρ + A (cid:32) T Θ D (cid:33) k (cid:90) Θ DT x k ( e x − − e − x ) dx, (3)and ∆ ρ ( T, B ) = γB m / [ ρ ( T, m − , (4) FIG. 3. Temperature variations of electrical resistivity of YBi(a) and LuBi (b) in various magnetic fields applied perpen-dicular to the current direction. we simultaneously fitted ρ ( T ) in zero field with Eq. 3and ρ ( T ) measured in B = 9 T with Eq. 2 using sharedparameters. Fits to these model with ρ , A , k , Θ D , and γ as free parameters, and parameter m fixed at its valueobtained from Kohler scaling are shown as red and purplesolid lines in Fig. 4. The obtained parameters for bothcompounds are rather similar and listed in Table I. Valuesof k are close to that previously determined for LuSb and several Lu- and La-containing intermetallics .The Debye temperatures are smaller than Θ D = 408 and420 K reported for LuSb and LuAs, respectively .Additionally, we show in the Fig. 4 magnetic field-induced resistivity versus temperature as a green cir- ρ A Θ D k m γ ( µ Ω cm) ( µ Ω cm) (K) ( Ω cm) m YBi 0.12 42.3 295 3.08 1.81 . × − LuBi 0.5 34 307 2.55 1.89 . × − TABLE I. Parameters obtained from the fitting of Eqs. 2 and3 to the ρ ( T ) data, as shown in Fig. 4. FIG. 4. Temperature variations of electrical resistivity mea-sured in magnetic fields of 9 and 0 T and their difference forYBi (a) and LuBi (b). The solid lines correspond to fits ofEqs. 2 and 3. cles. This data were obtained by subtraction of datameasured in zero magnetic field from those measured in9 T. Cyan solid lines in Fig. 4 represent Eq. 4 with pa-rameters yielded by the fitting of Eq. 2. In order to getmore insight in carrier concentration we measured Hallresistivity ( ρ xy ) at the temperature of 2 K, where MRattains its maximum. The ρ xy ( B ) plots for both com-pounds are shown in insets to Figs. 9a and 9b. Theircurved shapes indicate multiband character of conduc-tivity. Since ρ xy << ρ xx for both compounds, in furtheranalysis we use Hall conductivity σ xy calculated usingEq. 6. Electronic structure calculations and Shubnikov–deHaas effect
Figure 5 presents calculated bulk electronic band struc-tures of YBi and LuBi. The results of our calculations areconsistent with scalar-relativistic data obtained for YBi . Both compounds have very similar electronic struc-tures. Due to spin-orbit interaction three-fold degeneracyof Bi- p states is modified at the Γ point, i.e. one of the p -bands dips deeply below E F , whereas two other p -bands FIG. 5. Electronic band structure of YBi (a) and LuBi (b).Horizontal line marks the Fermi level. Red and blue colors de-note contributions from d -electrons of Y or Lu, and p -electronsof Bi, respectively. remains degenerated and stay above E F . Furthermore,the two-fold degeneracy of these two bands is graduallylifted along Γ − L and Γ − X lines, and they become wellseparated at points L and X. The corresponding shiftsof p -bands are noticeably smaller in analogous monoan-timonides , and eventually become just-noticeable inarsenides (data not shown), reflecting decreasing spin-orbit coupling strength.Fermi level crosses two hole-like bands near the Γ pointof Brillouin zone and one electron-like band around theX point. Besides, at ≈ . eV below the Fermi level,there is a tiny gap between the bands and the band in-version occurs. This is where the Dirac cones potentiallymay form. Analogous gaps have previously been reportedin lanthanum monopnictides and YSb , calculatedusing the GGA with Perdew-Burke-Ernzerhof exchange-correlation potential. Our electronic structure calcula-tions reveals also some d − p − mixed orbital texture nearthe X point of the Brillouin zone (visualized with red andblue colors in Fig. 5). This finding resembles those forPtSn , NbSb , LaBi and WTe . The Fermi surface isvery similar in both compounds and thus schematicallydepicted, together with its projection on (001) plane, in acommon Fig. 6. It consists of a triplicate electron pocketcentered at the X points (denoted as α ) and two holepockets ( β and δ ) nested in the center of the Brillouinzone. The calculations of electronic structure brought FIG. 6. (a) Fermi surface of YBi and LuBi. It consists ofa triplicate electron pocket α and two hole pockets δ and β . (b) Projection of the Fermi surface on the (001) plane.Proportions between the Brillouin zone and Fermi pocketssizes were not preserved. also the carrier concentrations, n calcp , cyclotron frequen-cies for maximal cross-sections of Fermi pockets by planesperpendicular to [001] direction, f calcp , and correspond-ing cyclotron masses, m ∗ calcp . Their values are collectedin Table II. Comparing the ratios of the concentrationsof electrons and holes n calcα / ( n calcβ + n calcδ ) , being 1.003in YBi and 1.002 in LuBi, suggests that carrier compen-sation is nearly perfect in both compounds.Good quality of our samples allowed us to observe FIG. 7. Oscillating part of electrical resistivity as a functionof inverted magnetic field for YBi (a) and LuBi (b), measuredat several different temperatures. quantum oscillations of electrical resistivity in magneticfield, i.e., the Shubnikov–de Haas (SdH) effect. The sub-traction of the third-order polynomial background fromthe ρ (1 /B ) data resulted in experimental curves pre-sented in Fig. 7. Strong SdH oscillations were clearlyobserved at temperatures up to at least 10 and 15 K forYBi and LuBi, respectively. The shape of ∆ ρ (1 /B ) sug-gests multifrequency character of the oscillations. In-deed, their fast Fourier transform (FFT) analysis showsfor each of two compounds, six pronounced maxima (seeFig. 8). Corresponding SdH frequencies f F F Tp are listedin Table II. These, denoted with f F F Tα and f F F Tα (cid:48) , weascribe to the electrons on orbits being maximal crosssections of ellipsoid-like Fermi pocket α , perpendicularto its long and short axis, respectively. f F F T α and f F F T α are the second and the third harmonics of f F F Tα . Fre-quencies f F F Tβ and f F F Tδ are due to the hole pockets.We obtained very similar FFT spectra, matching verywell the results of our electron structure calculations,for both compounds. According to the Onsager rela-tion: f SdH = hS/e , where S is the area of Fermi sur-face cross section . Assuming perfect ellipsoidal shapeof the α sub-pockets and the spherical one of pocket β ,we calculated the Fermi wave vectors and than carrierconcentrations using the formula n p = V F,p / (4 π ) , where FIG. 8. Fast Fourier transform analysis of oscillating part of electrical resistivity of YBi (a) and LuBi (b). Insets: temperaturedependence of the amplitude of the highest peak in the FFT spectra. Red solid line represents fits of Eq. 5 to the experimentaldata. V F,p is the volume of Fermi pocket p . The n e /n h ratiosresulting from analysis of SdH oscillations are 0.97 and0.95 for YBi and LuBi, respectively. This shows thatthe electron-hole compensation is very close to perfect inboth compounds, as hinted above by Kohler scaling andDFT calculations.Effective masses ( m ∗ ) of the carriers of α Fermi pocketwere calculated from the temperature dependence of FFTamplitude, R α , at f F F Tα frequency, obtained from thefield window 7–9 T, using the following relation : R α ( T ) ∝ ( λm ∗ T /B eff ) / sinh( λm ∗ T /B eff ) , (5)with B eff = 7 . T being the the reciprocal of averageinverse field from the window where FFT was performed: B eff = 2 / (1 /B + 1 /B ) (with B = 7 T and B = 9 T),and the constant λ = 2 π k B m /e (cid:126) ( ≈ . T/K), we ob-tained m ∗ = 0 . m for both compounds. This valueof effective mass is close to those reported previously forother rare-earth-metal monopnictides andalso to effective masses m ∗ calcα = 0 . m and . m ,obtained from our electronic structure calculations forYBi and LuBi, respectively.Observing good agreement of SdH analysis, the calcu-lations and multiband fitting of magnetotransport (de-scribed in next section), all revealing or taking into ac-count strong anisotropy of electron pocket, we decidednot to pursue angle-dependent SdH measurements aswe expect that they would yield results very similar tothose presented in other papers on similar monopnictides . Multiband model of magnetotransport
After establishing the presence of three distinct Fermipockets, we proceeded to analyze how their form deter-mines the field dependence of transverse magnetoresis-tivity, ρ xx , and Hall resistivity, ρ xy . Cubic crystal symmetry of YBi and LuBi allows us todefine components of conductivity tensor as follows: σ xx = ρ xx / [( ρ xx ) + ( ρ xy ) ] σ xy = − ρ xy / [( ρ xx ) + ( ρ xy ) ] . (6)In semiclassical Drude model, conductivities of indi-vidual electron and hole pockets (indexed with p ) aresummed up to obtain total transverse and longitudinalcomponents of conductivity tensor as follows: σ xx = (cid:80) p e n p µ p / [1 + ( µ p B ) ] σ xy = (cid:80) p e n p µ p B/ [1 + ( µ p B ) ] . (7)Following the idea of Xu et al. and stressing the inad-equacy of an isotropic multiband model for the transportproperties of a system with anisotropic Fermi pockets,we used the same analysis as those authors, namely ananisotropic three-band model, taking into account pro-nounced anisotropy of the electron band α by using sep-arate conductivities for pockets elongated parallel andtransverse to the current direction, distinguished by twomobilities µ (cid:107) and µ ⊥ .Since in the case of LaBi several authors used the effec-tive two-band model, neglecting the anisotropy of elec-tron pocket , we also tested that model for YBi andLuBi. However, the fittings with the three-band modelwere clearly better (see the Supplemental Material ).We fitted simultaneously both σ xx and σ xy of Eq. 7to σ xx ( B ) and σ xy ( B ) data recorded at T = 2 K, withshared parameters [using as ρ xx ( B ) the data shown inFig. 1 plots of MR for 2 K]. Resulting n α , n β , and κ ( ≡ µ ⊥ /µ (cid:107) ), together with µ ⊥ , µ β , n δ , and µ δ obtained fromthe fitting of Eq. 7 are listed in Table III.These parameters allow us to estimate again the levelof compensation of electrons and holes, expressed by theratio n α / ( n β + n δ ) being equal to 0.95 for YBi and 0.97 FIG. 9. Electrical conductivity and Hall conductivity versus magnetic field measured at T = 2 K of (a) YBi and (b) LuBi. Redlines correspond to the fits with Eqs. 7 for LuBi. Comparing them to analogous values fromanalysis of SdH oscillations (0.97 for YBi and 0.95 forLuBi), we conclude that electron-hole compensation isnearly perfect in both compounds. Small discrepanciesbetween compensation values derived by different meth-ods are most likely due to the approximations of Fermipocket’s shapes we made in our analyzes.
Conclusions
We investigated electron transport properties of high-quality single crystals of two compounds YBi and LuBi.The electronic structure that emerges from our resultsis almost identical for both compounds and points totheir semimetallic character with nearly perfect compen-sation of electron and hole carriers. We found that low-temperature field-induced resistivity plateau could be in-terpreted in terms of Kohler scaling with the main pa-rameter confirming good compensation. This outcomeis strengthened by our electronic structure calculationsand analysis of Shubnikov-de Haas oscillations reveal-ing Fermi surfaces that consist of two hole pockets anda triplicate electron pocket. The multiband anisotropicmodel of electronic transport describes very well the ex-perimental results of σ xx ( B ) and σ xy ( B ) for both com-pounds. Therefore, our experimental results confirmedthat prominent magnetotransport properties of YBi andLuBi could be explained without invoking nontrivialtopology of electronic bands.Electronic structure calculations showed that band in-version exists in both compounds, but plausible Diracpoints could appear about 0.5 eV below the Fermi level(that is about twice as deep as in LaSb or LaBi ). Thereis also considerable d − p -orbital mixing of electron statesvisible in the same region. How such structures wouldinfluence magnetotransport of a semimetal remains anopen question.The mobilities, of both electrons and holes, are con- siderably larger in YBi than in LuBi (Table III), whichis reflected in almost four times smaller residual resistiv-ity of the former compound, and consequently leads toits three times larger magnetoresistance. But the bandstructure region where important orbital mixing occursdiffers very little between YBi and LuBi (cf Fig. 5). Thissuggests that d − p -orbital mixing is not the predomi-nant mechanism in magnetoresistance of these two com-pounds.A scenario of mobility mismatch between electron andhole bands, proposed recently to explain reduced MR inLaAs , does not seem appropriate for LuBi because itsmobilities of holes and electrons differ very little, and theHall coefficient is over two orders of magnitude smallerthan in LaAs (for which the large Hall coefficient re-flected strong mismatch of mobilities) . In the Supple-mental Material we show also how YBi and LuBi follow MR ∝ RRR behavior, in common with several othermonopnictide samples, but in contrast to LaAs .Future research with the ARPES technique would bevery helpful in making the final conclusion on the hy-pothetical presence of topologically nontrivial electronicstates in YBi and LuBi. Acknowledgements
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Magnetic Oscillations in Metals (Cam-bridge University Press, Cambridge, 1984). N. Wakeham, E. D. Bauer, M. Neupane, and F. Ronning,Phys. Rev. B , 205152 (2016). H.-Y. Yang, T. Nummy, H. Li, S. Jaszewski, M. Abram-chuk, D. S. Dessau, and F. Tafti, Phys. Rev. B , 235128(2017). See appended Supplemental Material (pp. 10–12) forderivation and detailed description of anisotropic three-band model and comparison of its fitting results with thoseof the effective two-band model, as well as for a comparisonof MR values for several monopnictide samples described indifferent papers (YSb, LuSb, LaSb, LaBi, YBi and LuBi),plotted against RRR . Compound p = α α (cid:48) α α β δ n e /n h YBi f FFTp (T) 490 1766 966 1478 900 2069 k F (Å − ) 0.122 0.439 - - 0.165 0.251 n p (10 cm − ) 6.63 - - 1.53 5.33 0.97 f calcp (T) 544 1853 - - 1018 2492 n calcp (10 cm − ) 7.52 - - 1.80 5.70 1.003 m ∗ calcp ( m ) 0.24 0.60 - - 0.20 0.61LuBi f FFTp (T) 477 1784 953 1535 884 2112 k F (Å − ) 0.120 0.451 - - 0.164 0.253 n p (10 cm − ) 6.61 - - 1.49 5.50 0.95 f calcp (T) 680 1868 - - 980 2738 n calcp (10 cm − ) 8.39 - - 1.73 6.64 1.002 m ∗ calcp ( m ) 0.29 0.56 - - 0.18 0.59TABLE II. Parameters obtained from analysis of SdH oscillations measured at T = 2 K and from electronic band structurecalculations.Compound n α µ ⊥ n β µ β n δ µ δ κ n e /n h (cm − ) (m V − s − ) (cm − ) (m V − s − ) (cm − ) (m V − s − ) YBi . × . × . × . × . × . × SUPPLEMENTAL MATERIAL
Multiband models of magnetotransport
Analyzing magnetotransport of YSb, Xu etal. stressed inadequacy of isotropic multi-band model to the properties of a system withanisotropic Fermi pockets . On the other handin three papers devoted to LaBi , their authorshave used effective two-band model, neglectingthe anisotropy of electron Fermi pocket centeredat X-point, and found it satisfactory. We de-cided to compare how these both models workfor YBi and LuBi. We used the same analysis asproposed by Xu et al. :magnetoconductivity due to a Fermi pocket, p ,when magnetic field B is applied along z -axisand electrical current is flowing along x -axis, isdescribed by a tensor: ˆ σ p = (cid:32) σ pxx σ pxy − σ pxy σ pyy (cid:33) , (8)with the components: σ pxx = e n p µ px / [1 + µ px µ py B ] σ pyy = e n p µ py / [1 + µ px µ py B ] σ pxy = e n p µ px µ py B/ [1 + µ px µ py B ] , (9)where n p stand for carrier concentrations, µ px and µ py are two first diagonal components of mobilitytensor for a Fermi pocket p , and e is elementarycharge.The hole pockets, centered at Γ point have thecubic symmetry ( m ¯3 m point group), and mobil-ities of the holes are isotropic, i.e. µ px = µ py = µ pz ( ≡ µ p ) for p = β, δ . Therefore: σ pxx = e n p µ p / [1 + ( µ p B ) ] σ pxy = e n p ( µ p ) B/ [1 + ( µ p B ) ] . for p = β, δ (10)Triplicate electron pocket consists of symmetry-equivalent parts of almost ellipsoidal shape, withlong axes along main crystallographic axes. Eachof them is centered at an X-point and there-fore has point symmetry /mmm . We distin-guish them with symbols: α x , α y and α z . There-fore mobility tensors for electrons of these sub- pockets can be written as: ˆ µ α x = µ (cid:107) µ ⊥
00 0 µ ⊥ , ˆ µ α y = µ ⊥ µ (cid:107)
00 0 µ ⊥ , (11) ˆ µ α z = µ ⊥ µ ⊥
00 0 µ (cid:107) . We can express the ratio of independent compo-nents of these mobilities as: µ ⊥ /µ (cid:107) = κ . Thisparameter is equivalent of ( k (cid:107) F /k ⊥ F ) , where k (cid:107) F and k ⊥ F are the Fermi wave vectors for each α i pocket, parallel and perpendicular to its 4-foldsymmetry axis, respectively.Thus, the symmetry reduces the number of pa-rameters, and using Eq. 9, the components of ˆ σ for the Fermi pockets α x , α y and α z can be writ-ten as: σ α x xx = σ α y yy = e ( n α / µ ⊥ / [ κ + ( µ ⊥ B ) ] ,σ α y xx = σ α x yy = σ α z xx = σ α z yy = e ( n α / µ ⊥ κ/ [ κ + ( µ ⊥ B ) ] ,σ α x xy = σ α y xy = e ( n α / µ ⊥ B/ [ κ + ( µ ⊥ B ) ] ,σ α z xy = e ( n α / µ ⊥ B/ [1 + ( µ ⊥ B ) ] . (12)Now total conductivity components are: σ ij = (cid:88) p = α x ,α y ,α z ,β,δ σ pij . (13)From Eqs. 10 and 12 it is obvious that σ xx = σ yy ,which reflects the cubic crystal symmetry.We fitted simultaneously both σ xx and σ xy of Eq. 13 to σ xx ( B ) and σ xy ( B ) datarecorded at T = 2 K, with shared parame-ters. Both fitted functions have 7 parameters: n α , µ ⊥ , n β , µ β , n δ , µ δ and κ , which are collectedin Table S1, together with n e /n h ratios, and ad-justed R parameter reflecting the quality of thefit .We tested also effective two-band model, rep-resented by Eq. 13 but after fixing values of κ FIG. S1
Magnetic field dependence of components ofconductivity tensor σ xx and σ xy , for YBi (a) and LuBi (b).Solid red lines represent fitted with the effective two-bandmodel. to 1 (assuming isotropic electron band) and n δ to zero (leaving only one effective hole band).Such fits are shown in Fig. S1 for data collectedfor both YBi and LuBi and can be compared tothose made with anisotropic-three-band modelshown in Fig. 9 of the paper. The effective two-band model yields worse fits than the anisotropicthree-band model as seen in Figures and indi- cated by R values given in Table S1. Also thevalues of carrier concentrations and n e /n h ra-tios from the anisotropic-three-band model muchbetter correspond to those obtained from ouranalysis of SdH oscillations (cf. Table I of the pa-per), than these yielded by the two-band model. MR dependence on RRR In Figure S2 we plotted magnetoresistance(MR) values versus square of residual-resistivityratios (RRR ) collected for several monopnic-tide samples described in different papers: YSb , LuSb , LaSb , LaBi , LaAs , as wellas for our samples of YBi and LuBi. Common FIG. S2
MR (at 2 K and 9 T) plotted versus RRR fordifferent monopnictide samples. MR ∝ RRR dependence is very closely followedfor all the samples except LaBi and LaAs ofRef. 8. For LaAs it has been argued that MRcan be significantly reduced by strong mismatchof electron and hole mobilities . Since for bothYBi and LuBi MR ∝ RRR and their mobilitiesof electrons and holes are not very different fromeach other, a scenario of mobility mismatch canbe dismissed. J. Xu, N. J. Ghimire, J. S. Jiang, Z. L. Xiao, A. S. Botana,Y. L. Wang, Y. Hao, J. E. Pearson, W. K. Kwok, Phys.Rev. B , 075159 (2017). F. F. Tafti, Q. Gibson, S. Kushwaha, J. W. Krizan, N. Hal-dolaarachchige, R. J. Cava, Proc. Natl. Acad. Sci. ,E3475 (2016). N. Kumar, C. Shekhar, S.-C. Wu, I. Leermakers, O. Young,U. Zeitler, B. Yan, C. Felser, Phys. Rev. B , 241106(2016). S. Sun, Q. Wang, P. J. Guo, K. Liu, H. Lei, New J. Phys. , 082002 (2016). O. Pavlosiuk, P. Swatek, P. Wiśniewski, Sci. Rep. , 38691(2016). O. Pavlosiuk, M. Kleinert, P. Swatek, D. Kaczorowski,P. Wiśniewski, Sci. Rep. , 12822 (2017). H.-Y. Yang, T. Nummy, H. Li, S. Jaszewski, M. Abram-chuk, D. S. Dessau, F. Tafti, Phys. Rev. B , 235128(2017). Compound n α µ ⊥ n β µ β n δ µ δ κ n e /n h R model (cm − ) (m /(Vs)) (cm − ) (m /(Vs)) (cm − ) (m /(Vs))YBi 3-band . × . × . × . × . × . × . × . × . × . × TABLE S1
Comparison of parameters obtained from the simultaneous fitting of magnetic field dependences of electricalconductivity tensor components σ xx and σ xy with the anisotropic three-band model and with the effective two-band model( κκ