Abstract
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 timereversal symmetry (protecting surface states in topological insulators) is broken. Similar features were observed in a simple rocksaltstructure LaSb, leading to a suggestion of the possible nontrivial 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 lowtemperature plateau of resistivity. We thus performed indepth analysis of YSb Fermi surface by band calculations, magnetoresistance, and Shubnikov–de Haas effect measurements, which reveals only threedimensional Fermi sheets. Kohler scaling applied to magnetoresistance data accounts very well for its lowtemperature upturn behavior. The fieldangledependent magnetoresistance demonstrates a 3Dscaling yielding effective mass anisotropy perfectly agreeing with electronic structure and quantum oscillations analysis, thus providing further support for 3DFermi surface scenario of magnetotransport, without necessity of invoking topologically nontrivial 2D states. We discuss data implying that analogous fieldinduced properties of LaSb can also be well understood in the framework of 3D multiband model.
Introduction
Yttrium monoantimonide has mainly been used as a nonmagnetic reference or as a ‘solvent’ in monoantimonides of felectronelements solid solutions with anomalous physical properties such as dense Kondo behavior and complex magnetic groundstates^{1}. Within that context it has been characterized as a metal by lowtemperature specific heat measurements^{2}. Later Hayashi et al. have shown that the firstorder phase transition from the NaCltype to a CsCltype crystal structure occurs in YSb at 26 GPa^{3}. That discovery induced numerous calculations of electronic structure of the compound, among them those by Tütüncü, Bagci and Srivastava, who directly compared electronic structure of YSb with that of LaSb^{4}. Results of those calculations were very similar for both compounds, revealing low densities of states at Fermi level and characteristic anticrossings leading to band inversion at Xpoints of the Brillouin zone. LaSb has a simple NaCltype structure without broken inversion symmetry, perfect linear band crossing or perfect electron–hole symmetry, yet it exhibits the exotic magnetotransport properties of complexstructure semimetals like TaAs, NbP (Weyl semimetals)^{5,6}, Cd_{3}As_{2} (Dirac semimetal)^{7} and WTe_{2} (resonant compensated semimetal)^{8,9}. Recently Tafti et al. discovered in LaSb fieldinduced resistivity plateau at low temperatures up to ≈15 K, ultrahigh mobility of carriers in the plateau region, quantum oscillations, and magnetoresistance (MR) of nearly one million percent at 9 T^{10}. Their calculations, including spinorbit coupling (SOC) effect, suggested that LaSb is a topological insulator with a 10 meV gap open near the Xpoint of the Brillouin zone. They also observed specific angular dependence of frequencies of quantum oscillations and ascribed them to twodimensional Fermi surface (FS) possibly formed of topologically nontrivial states, and thus proposed LaSb as a model system for understanding the consequences of breaking timereversal symmetry in topological semimetals^{10}.
However, such angular dependence has already been observed in LaSb by de HaasVan Alphen measurements and well explained by the presence of elongated pockets of 3DFermi surface^{11,12}.
Tafti et al. also invoked the opening of insulating gap (i.e. metalinsulator transition) as a source of the fieldinduced resistivity plateau in LaSb^{10} but it should be noted that in the case of WTe_{2} the existence of a magneticfielddriven metalinsulator 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 fieldinduced metalinsulator transition in YSb.
However, our analysis of magnetoresistance and Shubnikov–de Haas (SdH) effect provides strong support for 3DFermi surface scenario of magnetotransport, without necessity of invoking topologically nontrivial 2D states or metalinsulator 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 magneticfielddriven metalinsulator 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 fieldinduced metalinsulator 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 . 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 θangle, 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 3DFS 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 offdiagonal component of conductivity tensor should be used for multipleband 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 ith 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.
For LaSb Tafti et al.^{10} estimated uncompensated carrier concentration, n, and the Hall mobility m_{H}, using the relations n = 1/eR_{H}(0) (with R_{H}(0) being the zerotemperature limit of R_{H}(T)) and μ_{H} = R_{H}(0)/ρ_{0}. They obtained n ≈ 10^{20} cm^{−3} and μ_{H} ≈ 10^{5} cm^{2}/(Vs).
For our YSb sample the fit with with multipleband model (Eq. 1), shown in Fig. 3(b), yielded at T = 2.2 K the concentrations: n_{e} = 1.52 × 10^{20} cm^{−3}, n_{h} = 1.16 × 10^{20} cm^{−3}, and mobilities μ_{e} = 2.7 × 10^{3} cm^{2}/(Vs), μ_{h} = 1.9 × 10^{3} cm^{2}/(Vs). Third band necessary for that fit consists of more mobile holes [n = 3.4 × 10^{19} cm^{−3}, μ = 7.9 × 10^{3} cm^{2}/(Vs)]. These results are consistent with band calculations presented below.
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 Shubnikovde Haas effect
We performed the electronic structure calculations for YSb using a full potential allelectron local orbital code (FPLO) within GGA approximation. Figure 4 shows obtained energy band structure for YSb, with a few bands crossing Fermi level. Near Xpoint there is an anticrossing present with a gap of ≈0.8 meV, similar to those reported for lanthanum monopnictides, which led to the proposal of 2D topologically nontrivial 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 Xpoints and three nested hole sheets centered at Γpoint. 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 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 Δρ_{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 Δρ_{xx}(1/B) with the multifrequency LifshitzKosevich function^{20,21,22} (Eq. 2) because, as the maxima of total Δρ_{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 socalled Landaulevel fan diagram (plot of the values of 1/B_{N} corresponding to the Nth maximum in Δρ_{xx} versus N).
where for each ith component: f_{i} is the frequency, represents the Dingle factor R_{D} and comprises the temperature reduction factor R_{T} and the spin factor R_{S}. Detailed form of Eq. 2, taking into account harmonic components, and description of R_{D}, R_{T} and R_{S} is presented in Supplementary Material.
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 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 3DFS sheets predicted by band calculations (shown above in Fig. 5) and no Berry phase of π was observed, which could reveal topologically nontrivial 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 crosssections 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 upon rotation of the magnetic field: the principal, labeled as α (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 follows a crosssection area of a prolate ellipsoid (which well approximates the shape of electron sheet centered at Xpoint shown in Fig. 5) by the ((1 − cos θ) 0 1) plane passing by Xpoint. When magnetic field is applied along [0 0 1] direction the plane is perpendicular to ΓX line. When field is tilted from [0 0 1] towards [1 0 0] by the angle θ the crosssection increases as . This holds for a twoaxial 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 αsheet of FS along XU and XW lines, and k_{z} as its size along ΓX 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 crosssection 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 Γpoint.
We plotted S(θ) (for r = 3.6) in Fig. 7(b) as solid lines. The dependence, expected for twodimensional FS, behaves similarly and is shown for comparison with dashed lines. The dependence was used by Tafti et al.^{10} as a hint of possible topologicallynontrivial states in LaSb. However, Fermi surface of that compound has already been well characterized by band calculations and angledependent 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 Xpoint and two isotropic hole pockets centered at Γpoint. 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 crosssection 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 nontrivial 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 electronhole 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 upturn behavior of MR in WTe_{2} without the fieldinduced metalinsulator transition or significant contribution of an electronic structure change^{13}. The same authors have shown that perfect carrier compensation leads to 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 fieldinduced metalinsulator transition is unnecessary to explain upturn and lowtemperature plateau of resistivity in YSb. The origin of the upturn is a combination of magnetoresistance with the lowtemperature 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 Γpoint have their counterparts in LaSb^{10,11,12}. Our analysis of angular behavior of SdH frequencies in YSb indicates it is related to the threedimensional FS, in line with Hasegawa^{11} and Settai et al.^{12} findings for LaSb, but not connected to possible nontrivial 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 3DFS. When field strength is scaled by the angledependent 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 , is in excellent agreement with k_{z}/k_{x} = 3.6 we estimated for αsheet 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 αsheet of FS. That sheet corresponds to the electron band, all other FSsheets contain holes and are nearly isotropic. The effective mass and mobility of αsheet 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 Xpoint (cf. inset to Fig. 4) might result in topologically nontrivial states. The effect of FS reconstruction could be similar to temperatureinduced 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 angleresolvedphotoemission 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 nontrivial 2D states cannot be completely excluded our analysis of magnetoresistance and Shubnikov–de Haas effect provides strong support for 3DFermi surface scenario of magnetotransport properties in YSb. Analogous fieldinduced 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 NaCltype crystal structure was confirmed by powder Xray diffraction carried out using an X’pert Pro (PANanalytical) diffractometer with CuKα 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 FPLO9.0034 code within generalized gradient approximation (GGA) method^{33}. The fullrelativistic Dirac equation was solved selfconsistently, treating exactly all relativistic effects, including the spinorbit interaction without any approximations. The Perdew–Burke–Ernzerhof exchangecorrelation 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.
Additional Information
How to cite this article: Pavlosiuk, O. et al. Giant magnetoresistance, threedimensional Fermi surface and origin of resistivity plateau in YSb semimetal. Sci. Rep. 6, 38691; doi: 10.1038/srep38691 (2016).
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Acknowledgements
We are very grateful to Dr. Zhili Xiao for discussion. This work was supported by the National Science Centre of Poland, grant no. 2015/18/A/ST3/00057. The band structure calculations were carried out at the Wrocław Centre for Networking and Supercomputing, grant no. 359.
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O.P. conducted all the experiments, O.P. and P.W. conceived the experiments and analyzed their results, P.S. carried out the electronic structure calculations and prepared Figures 5 and 6. All authors reviewed the manuscript.
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Pavlosiuk, O., Swatek, P. & Wiśniewski, P. Giant magnetoresistance, threedimensional Fermi surface and origin of resistivity plateau in YSb semimetal. Sci Rep 6, 38691 (2016). https://doi.org/10.1038/srep38691
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