Giant magnetoresistance, three-dimensional Fermi surface and origin of resistivity plateau in YSb semimetal

Very strong magnetoresistance and a resistivity plateau impeding low temperature divergence due to insulating bulk are hallmarks of topological insulators and are also present in topological semimetals where the plateau is induced by magnetic field, when time-reversal symmetry (protecting surface states in topological insulators) is broken. Similar features were observed in a simple rock-salt-structure LaSb, leading to a suggestion of the possible non-trivial topology of 2D states in this compound. We show that its sister compound YSb is also characterized by giant magnetoresistance exceeding one thousand percent and low-temperature plateau of resistivity. We thus performed in-depth analysis of YSb Fermi surface by band calculations, magnetoresistance, and Shubnikov–de Haas effect measurements, which reveals only three-dimensional Fermi sheets. Kohler scaling applied to magnetoresistance data accounts very well for its low-temperature upturn behavior. The field-angle-dependent magnetoresistance demonstrates a 3D-scaling yielding effective mass anisotropy perfectly agreeing with electronic structure and quantum oscillations analysis, thus providing further support for 3D-Fermi surface scenario of magnetotransport, without necessity of invoking topologically non-trivial 2D states. We discuss data implying that analogous field-induced properties of LaSb can also be well understood in the framework of 3D multiband model.

Scientific RepoRts | 6:38691 | DOI: 10.1038/srep38691 Tafti et al. also invoked the opening of insulating gap (i.e. metal-insulator transition) as a source of the field-induced resistivity plateau in LaSb 10 but it should be noted that in the case of WTe 2 the existence of a magnetic-field-driven metal-insulator transition has been excluded by means of Kohler scaling analysis of magnetoresistance 13 .
Motivated by these ambiguities in the interpretation of LaSb properties we decided to carry out a comprehensive characterization of magnetotransport properties of a sister compound YSb. We found that YSb displays physical properties in many aspects very similar to those of LaSb. Our results are in accord with those of other groups that appeared during preparation of our article 14,15 . The interpretation proposed by Ghimire et al. and Yu et al. follows that presented by Tafti et al. for LaSb 10 , implying the role of field-induced metal-insulator transition in YSb.
However, our analysis of magnetoresistance and Shubnikov-de Haas (SdH) effect provides strong support for 3D-Fermi surface scenario of magnetotransport, without necessity of invoking topologically non-trivial 2D states or metal-insulator transition in YSb.

Results
Magnetoresistance and the origin of its plateau. Electrical resistivity (ρ) was measured on two samples (denoted as #1 and #2) and its dependence on temperature in zero field is plotted in Supplementary Figure 1(a). Shape of ρ(T) curves is typical for a metal. When measured in different applied fields ρ(T) exhibits universal plateau at temperatures 2-15 K, as shown for sample #1 in Supplementary Figure 1(b). Temperature range of this plateau is very similar to that reported for LaSb 10 .
Magnetoresistance, MR ≡ [ρ(B) − ρ(B = 0)]/ρ(B = 0), is plotted versus magnetic field, B, in Fig. 1(a) and (b), for samples #1 and #2, respectively. Following the approach of Wang et al. 13 we performed Kohler scaling analysis of magnetoresistance to test for the existence of a magnetic-field-driven metal-insulator transition in YSb. Figure 1(c) and (d) show very good Kohler scaling of our data, MR ∝ (B/ρ 0 ) m , yielding at 2.5 K exponents m = 1.64 and 1.74, respectively, very close for both samples, despite significant difference of their MR values. Efficiency of this scaling indicates that resistivity plateau is due purely to the magnetoresistance, but not to a field-induced metal-insulator transition.
In order to elucidate the dimensionality of FS we measured magnetoresistance of the sample #2 in fields applied at different angles to its surface, Fig. 2(a). Here θ = 0° denotes the field perpendicular to the sample surface and the current direction, whereas θ = 90° means that the field is parallel to the current. As shown in the inset to Fig. 2(a), ρ at strongest field of 9 T follows θ ∝cos . This is typical behavior for materials without magnetic anisotropy, but the change of ρ expressed as anisotropic magnetoresistance, AMR ≡ [ρ(90°) − ρ(0°)]/ρ(0°), has an outstanding − 80% value.
Moreover, when field strength is scaled by a factor ε θ dependent on mass anisotropy and θ-angle, all ρ(T) data of Fig. 2(a) collapse on single curve, as shown in Fig. 2(b). Inset of Fig. 2(b) shows that values of ε θ plotted against field angle θ can be perfectly fitted with ε θ = (cos 2 θ + γ −2 sin 2 θ) 1/2 function, shown with red line. Such scaling has initially been proposed for anisotropic superconductors 16 , and recently used to interpret MR behavior of WTe 2 based on its 3D electronic nature 17 . Parameter γ represents effective mass anisotropy of carriers mostly contributing to the magnetoresistance. We ascribe this behavior to a strongly anisotropic sheet of 3D-FS revealed by SdH effect data, as shown below.
Hall effect. Hall resistivity of sample #1h (cut from the same single crystal as #1) measured at several temperatures between 2.2 K and 300 K is shown in Fig. 3(a). Nonlinear ρ xy (B) indicates that at least two types of charge carriers are responsible for the Hall effect observed in YSb. The ρ xy (B) curves for temperatures from 2.2-15 K range are almost identical, which points to nearly constant carrier concentrations and mobilities, and coincides with the plateau of ρ xx (T) observed in the same range of T. Changes of sign of ρ xy (B) observed at T ≤ 100 K, indicate conducting bands of both electrons and holes, at higher temperatures ρ xy (B) is positive in the whole range of magnetic field (0 < B ≤ 9T). Thus, the Hall contributions of holes and electrons nearly compensate, but both depend on temperature in different manner. Clear quantum oscillations are observed in ρ xy in temperature range 2.2-15 K (cf. Fig. 3(a) and (b)).
Since ρ xy ≪ ρ xx , the off-diagonal component of conductivity tensor σ ρ ρ ρ = − + /( ) xy xy xx xy 2 2 should be used for multiple-band analysis of Hall data. In this case simple Drude model can be used: summing up conductivities of individual bands, with n i and μ i denoting respectively concentration and mobility of carriers from the i-th band 18 . As Fig. 3(b) shows, for data collected at 300 K accounting for two bands yielded a good fit. On the other hand, fitting with two bands was insufficient for 2.2 K data, but addition of a contribution of another band with small concentration of more mobile holes brought a very satisfactory fit. Inset: resistivity at 2.5 K and in 9 T versus field rotation angle; blue line represents ρ θ ∝ cos dependence. (b) Data of (a) replotted with B scaled by angle-dependent factor ε θ . Inset: angle dependence of ε θ ; red line represents fit with ε θ = (cos 2 θ + γ −2 sin 2 θ) 1/2 function yielding mass anisotropy γ = 3.4. The fit to data collected at 300 K yielded: the concentrations: n e = 1.34 × 10 18 cm −3 , n h = 1.94 × 10 19 cm −3 , and mobilities μ e = 1.8 × 10 3 cm 2 /(Vs), μ h = 8.2 × 10 2 cm 2 /(Vs), so YSb has similar concentration of carriers but with significantly lower mobility than LaSb. This seems to be the main reason for its significantly smaller magnetoresistance, as it was well demonstrated for WTe 2 9 . Overall, Hall effect results are in perfect agreement with characteristics of Fermi surface presented in the next section.
Fermi surface analysis: electronic structure calculations and Shubnikov-de Haas effect. We performed the electronic structure calculations for YSb using a full potential all-electron local orbital code (FPLO) within GGA approximation. Figure 4 shows obtained energy band structure for YSb, with a few bands crossing Fermi level. Near X-point there is an anti-crossing present with a gap of ≈ 0.8 meV, similar to those reported for lanthanum monopnictides, which led to the proposal of 2D topologically non-trivial states at the origin of extraordinary behavior of their magnetoresistance 10,19 . Fermi surface resulting from our calculations is presented in Fig. 5: two electron sheets centered at X-points and three nested hole sheets centered at Γ -point. Our calculations are in good agreement with those of ref. 14.
Shubnikov-de Haas oscillations of resistivity are discernible for YSb at temperatures up to 15 K and in fields above 6 T, as seen in Figs 1(a), 1(b) and 2(a). Since ρ xy ≪ ρ xx , we may safely assume that the conductivity σ ρ −  xx xx 1 and analyze directly oscillations of ρ xx . Points in Fig. 6 represent resistivity of sample #2 measured at 2.5 K, after subtraction of smooth background, plotted versus inverse field 1/B (for 7 < B < 9T). Complex shape of Δ ρ xx (1/B) dependence indicates that observed SdH oscillation has several components. Indeed fast Fourier transform (FFT) analysis reveals clearly six well separated frequencies ( Fig. 7(a)). We chose to fit Δ ρ xx (1/B) with the multi-frequency Lifshitz-Kosevich function [20][21][22] (Eq. 2) because, as the maxima of total Δ ρ xx do not correspond to the maxima of particular components with different frequencies, it is inadequate to determine phases in a multicomponent SdH oscillation using the so-called Landau-level fan diagram (plot of the values of 1/B N corresponding to the N-th maximum in Δ ρ xx versus N). The fit including six components was of very good quality, as shown by blue line in Fig. 6. Obtained parameters are collected in Table 1. All frequencies converged almost exactly to f i FFT values obtained from FFT analysis. We ascribe oscillations with frequencies of 720 and 1072 T to second and third harmonic of the strongest one with f i = 360 T. The phases ϕ i of fundamental oscillations resulting from the fit are close to Onsager phase factor of 1/2 expected for free electrons. Thus all components of SdH oscillation can be assigned to 3D-FS sheets predicted by band calculations (shown above in Fig. 5) and no Berry phase of π was observed, which could reveal topologically non-trivial charge carriers.
We performed FFT analysis for all data sets presented in Fig. 2(a), which allowed us to observe angular behavior of frequencies corresponding to all extreme cross-sections of Fermi sheets and compare them to those derived from our band structure calculations, as shown in Fig. 7(a). Three of observed frequencies were clearly changing Green lines indicate cyclotron orbits (extreme cross-sections of FS-sheets α, β, δ and α 1 ), for which we observed SdH oscillations in fields applied at θ = 0° (cf. Fig. 7 and Table 1). T at θ = 0°), and its harmonics, 2α and 3α. These frequencies are plotted versus θ in Fig. 7(b).
It became apparent from the shape of FS obtained from band calculations (cf. Fig. 5) that angular behavior of α f FFT follows a cross-section area of a prolate ellipsoid (which well approximates the shape of electron sheet centered at X-point shown in 1/2 . This holds for a two-axial ellipsoid described by the equation: (x/k x ) 2 + (y/k x ) 2 + (z/k z ) 2 = 1, with r = k z /k x . Band structure shown in Fig. 4 yields r ≈ 3.6 (with k x estimated as average size of α-sheet of FS along X-U and X-W lines, and k z as its size along Γ -X line).
After rotation by θ = 90° the α frequency meets the one denoted as α 1 , initially (i.e. at θ = 0°) corresponding to the largest cross-section of the same ellipsoidal FS sheet. Two other observed frequencies, β and δ do not change notably with θ, as expected for almost isotropic hole Fermi sheets centered at Γ -point.
We plotted S(θ) (for r = 3.6) in Fig. 7(b) as solid lines. The θ ∝ − cos 1 dependence, expected for two-dimensional FS, behaves similarly and is shown for comparison with dashed lines. The θ ∝ − cos 1 dependence was used by Tafti et al. 10 as a hint of possible topologically-nontrivial states in LaSb. However, Fermi surface of that compound has already been well characterized by band calculations and angle-dependent de Haas-van Alphen measurements 11,12 , revealing FS very similar to the one we found in YSb, namely consisting of one ellipsoidal electron sheet centered at X-point and two isotropic hole pockets centered at Γ -point. Hasegawa 11 and Settai et al. 12 assigned angular dependence of the principal de Haas-van Alphen frequency (identical to the SdH frequency in ref. 10) to the cross-section of the ellipsoidal sheet S(θ) in accord with our interpretation of S(θ) behavior for YSb. This underscores the similarities between these two compounds and implies that there is no need to invoke topologically non-trivial states to explain exotic magnetotransport properties neither in YSb nor in LaSb (contrary to ref. 10).

Discussion and Conclusions
YSb is another material displaying giant magnetoresistance (1100% in 9 T), three orders of magnitude smaller than that of sister compound LaSb 10 , thus it cannot be termed 'extreme magnetoresistance' (XMR). This is due mainly to its lower carrier mobility and weaker electron-hole compensation revealed by our Hall effect measurement.
Kohler scaling analogous to that shown in Fig. 1(c) and (d) has recently been used to explain the remarkable up-turn behavior of MR in WTe 2 without the field-induced metal-insulator transition or significant contribution of an electronic structure change 13 . The same authors have shown that perfect carrier compensation leads to  Fig. 2(a): α (circles) and 2α (squares), as well as 3α (triangles), versus sample rotation angle θ. Dashed lines represent θ ∝ − f cos 1 , solid line corresponds to f = f 0 (sin 2 θ + r 2 cos 2 θ) −1/2 with r = 3.6. This line is redrawn for θ + π/2, reflecting the symmetry of cubic YSb lattice. exponent m = 2 in this scaling. Kohler scaling for YSb yielded for our samples the exponents m = 1.64 and 1.74, which seems related to weaker carrier compensation than nearly perfect one in WTe 2 , where m = 1.92 13 . Thus, analogously to WTe 2 , Kohler scaling indicates that the field-induced metal-insulator transition is unnecessary to explain up-turn and low-temperature plateau of resistivity in YSb. The origin of the up-turn is a combination of magnetoresistance with the low-temperature resistivity plateau present already at zero field. Given the similarity of YSb and LaSb the same may also be true for the latter compound.
Comparing results of SdH measurements with those of electronic structure calculations we obtained comprehensive description of the Fermi surface of YSb. Presence of both electron and hole sheets of similar volumes provides partial charge compensation responsible for its strong magnetoresistance. Band structures of YSb and LaSb are very similar. All Fermi sheets in YSb but the smallest one centered at Γ -point have their counterparts in LaSb 10-12 . Our analysis of angular behavior of SdH frequencies in YSb indicates it is related to the three-dimensional FS, in line with Hasegawa 11 and Settai et al. 12 findings for LaSb, but not connected to possible non-trivial topology of electronic structure analogous to that suggested by Tafti et al. 10 Angular behavior of MR can also be perfectly explained by anisotropy of 3D-FS. When field strength is scaled by the angle-dependent factor ε θ , all data of Fig. 2(a) collapse on single curve. The effective mass anisotropy factor γ = 3.4, obtained from the fit of ε θ (θ) with the expression θ γ θ + − (cos sin ) 2 2 2 1/2 , is in excellent agreement with k z /k x = 3.6 we estimated for α-sheet of FS. This is not surprising, since the mass anisotropy directly reflects the shape of FS, but it shows that angular behavior of MR in YSb is mainly governed by anisotropic form of α-sheet of FS. That sheet corresponds to the electron band, all other FS-sheets contain holes and are nearly isotropic. The effective mass and mobility of α-sheet electrons change significantly with field angle, which strongly modifies the magnetoresistance.
It has been proposed that the magnetic field induces the reconstruction of the FS in a Dirac semimetal by breaking the time reversal invariance [23][24][25][26] . Assisted by the high mobility of carriers such reconstruction has been suggested to induce very large MR observed in Cd 3 As 2 and NbSb 2 27,28 . We also observe features in the electronic structure of YSb, buried under the Fermi level, which may possibly allow the magnetic field to transform this compound into Dirac semimetal. A small gap between inverted bands near the X-point (cf. inset to Fig. 4) might result in topologically non-trivial states. The effect of FS reconstruction could be similar to temperature-induced Lifshitz transition in WTe 2 29 , whereas its mechanism might be related, for example, to that of Lifshitz transition driven by magnetic field in CeIrIn 5 30 . Very recently Dirac states have been observed by angle-resolved-photoemission spectroscopy in NbSb, a compound with bulk electronic structure very similar to that of YSb 31 , however topologically protected states were not detected in YSb by this method 32 .
Although a small contribution of topologically non-trivial 2D states cannot be completely excluded our analysis of magnetoresistance and Shubnikov-de Haas effect provides strong support for 3D-Fermi surface scenario of magnetotransport properties in YSb. Analogous field-induced properties of LaSb can most probably be also described in the framework of 3D multiband model.

Methods
Measurements were performed using a Physical Property Measurement System (Quantum Design) on two samples cut from one single crystal and labeled as #1 and #1 h, and a sample cut from another single crystal and labeled #2. All samples had shapes of rectangular cuboid with all edges along 〈 1 0 0〉 crystallographic directions. Their sizes were: 0.56 × 0.25 × 0.12 mm 3 , 0.4 × 0.47 × 0.13 mm 3 and 0.41 × 0.32 × 0.09 mm 3 , for samples #1, #1 h and #2, respectively. The electric current was always flowing along [1 0 0] crystallographic direction. Single crystals were grown from Sb flux and their NaCl-type crystal structure was confirmed by powder X-ray diffraction carried out using an X'pert Pro (PANanalytical) diffractometer with Cu-Kα radiation. No other phases were detected and lattice parameter of 6.163 Å was determined, reasonably close to literature value 6.155 Å 3 . Electronic structure calculations were carried out using FPLO-9.00-34 code within generalized gradient approximation (GGA) method 33 . The full-relativistic Dirac equation was solved self-consistently, treating exactly all relativistic effects, including the spin-orbit interaction without any approximations. The Perdew-Burke-Ernzerhof exchange-correlation potential 34 was applied and the energies were converged on a dense k mesh with 24 3 points. The convergence was set to both the density (10 −6 in code specific units) and the total energy (10 −8 Hartree). For the Fermi surface a 64 3 mesh was used to ensure accurate determination of the Fermi level.