Fermi surface topology and magnetotransport in semimetallic LuSb

Several rare-earth monopnictides were shown to exhibit extreme magnetoresistance and field-induced low-temperature plateau of electrical resistivity. These features are also hallmarks of topological semimetals, thus the family is intensively explored with respect to magneto-transport properties and possible hosting Dirac fermion states. We report a comprehensive investigation of Fermi surface and electrical transport properties of LuSb, another representative of this family. At low temperatures, the magnetoresistance of LuSb was found to exceed 3000% without saturation in fields up to 9 T. Analysis of the Hall effect and the Shubnikov–de Haas oscillations revealed that the Fermi surface of this compound consists of several pockets originating from fairly compensated multi-band electronic structure, in full accordance with our first-principles calculations. Observed magnetotransport properties of LuSb can be attributed to the topology of three-dimensional Fermi surface and a compensation of electron and hole contributions.

specifying temperatures at which these quantities were obtained. In the very recent ARPES study no Dirac-like features were found 30 . In this paper we describe the magnetotransport properties and the Fermi surface of LuSb, and compare the experimental findings with the results of our electronic band structure calculations.

Results and Discussion
Electronic structure calculations. Figure 1 presents the electronic structure of LuSb calculated using LAPW method with the modified Becke-Johnson (mB-J) potential (details are described in Methods). Overall, it is very similar to those reported for other rare-earth monopnictides 5,12,13,27,31 , with four bands crossing the Fermi level: three hole bands in vicinity of Γ-point and one electron band close to the high-symmetry X-point of the Brillouin zone.
The d − p mixed orbital texture in LaSb and LaBi is due to the crossing of the lanthanum d-band with the pnictogen p-band and subsequent opening of a small gap at the point of crossing by the spin-orbit interaction. This was suggested to be responsible for topological nature of these compounds 5,31 . We reported very similar texture in YSb 27 .
In LuSb, however, the d-band of lanthanum and the p-band of antimony do not cross between Γ and X points, but are well separated by a direct gap of at least 0.38 eV (cf Fig. 1). The gaps opened by the spin-orbit coupling in lanthanum monopnictides 5,31 and YSb 27 were two or three orders of magnitude narrower than that in LuSb. These features exclude any significant d − p orbital mixing, which could give rise to topologically non-trivial electronic states in this compound.
The calculations showed that the FS of LuSb consists of four sheets (see Fig. 2). Three of them, centered at the Γ-point (labeled β, δ, ζ in Fig. 2(a)), are hole-like, whereas a triplicate one, centered at the X-point (labeled α in Fig. 2(a)), is electron-like. The smallest ζ pocket is a sphere placed inside the almost spherical β pocket, which in turn, is nested in the biggest hole-pocket δ, resembling an octahedron.
FS resulting from our calculations differs slightly from those reported before, which did not show the smallest hole-like pocket 24,25 . The Fermi surfaces calculated for lanthanum monopnictides are also different, having two hole-like Fermi sheets 5,13 . However, the data derived for LuSb are almost identical to those of YSb 27 . Comparing total volume of three hole pockets (0.06967 Å −3 ) with that of the electron pocket (0.07065 Å −3 ) leads to a conclusion that LuSb is fairly well compensated semimetal. Electrical resistivity and magnetoresistance. As shown in Fig. 3, the electrical resistivity of single-crystalline LuSb decreases with decreasing temperature in a metallic-like manner from ρ = 201.9 μΩ cm at T = 300 K down to ρ = 0.67μΩ cm at T = 2 K. The RRR of our sample is 30, slightly larger than 22, we reported for YSb 27 , but about tenfold smaller than those of several other rare-earth monopnictides 1,[11][12][13] . In the temperature interval from 0.7 K to 300 K, ρ(T) follows the Bloch-Grüneisen law: with the residual resistivity ρ 0 = 0.65 μΩ cm, Debye temperature Θ = 408 D K, factor A = 33.7 μΩ cm and exponent k = 2.19 (note red solid line in Fig. 3). The obtained value of k is similar to those observed for several La-or Lu-intermetallics 32 , and another lutetium monopnictide, LuAs, which has also a very similar Θ = 420 D K 33 . In magnetic fields applied perpendicular to the electric current, the electrical transport in LuSb has a completely different character than that in zero magnetic field (see the inset to Fig. 3). In B = 3 T, a shallow minimum in ρ(T) can be discerned at T m ≈ 34 K, and in stronger fields this feature becomes more pronounced and shifts to higher temperatures (T m ≈ 88 K in 9 T field). Below T m , the resistivity increases with decreasing T, and saturates below ≈10 K, forming a low-temperature plateau. Such a behavior of ρ(T) was described as a characteristic feature of topological semimetals 3 , or alternatively, attributed to band compensation effect occurring e.g. in WTe 2 34 or several rare-earth monopnictides 12,13,27 .
Universal T − B phase diagram was constructed for several materials showing XMR (LaSb, LaBi, NbSb 2 , PtSn 4 and WTe 2 ) 5,13 . It consisted of linear-with-field T m and field-independent T i (marking the inflection point of ρ(T)). The T m and T i temperatures discerned from the inset to Fig. 3 also form a similar phase diagram for LuSb.
The transverse magnetoresistance of LuSb, MR = ρ ρ ρ ]/ ( 0) (magnetic field applied perpendicular to electric current), measured at several temperatures in the range from 0.7 K up to 300 K, is displayed in Fig. 4. MR reaches 3025% at T = 0.7 K, and MR(B) does not saturate in a field of 9 T. Upon heating the sample up to 15 K, MR decreases very little, but between 15 and 50 K it drops drastically. The  magnetoresistance of LuSb is one or two orders of magnitude smaller that MR reported for other rare-earth monopnictides 5,12,13,29 . This seems due to rather small RRR of our crystals, since it was shown that MR depends on RRR in a quadratic manner 5,35 . In previous works, XMR of rare-earth monopnictides was attributed to: almost perfect electron-hole compensation 12,13,29 , metal-insulator transition 1 or d−p orbital mixing combined with carrier compensation 5 . The last scenario was also proposed for some other materials with XMR effect, like WTe 2 , NbSb 2 , or PtSn 4 5 . We suggest that electron-hole compensation leads to very large MR in LuSb. The Kohler scaling, which is presented in Fig. 5, allows to draw conclusions about the very large MR effect. For LuSb, all the MR curves measured at different temperatures collapse onto a single curve that can be described by the Kohler equation MR α ρ = = B B ( / ( 0)) m with the exponent m = 1.73 (note a red solid line in Fig. 5). The obtained value of m is nearly the same as that derived for YSb 27 but somewhat smaller than m = 1.866 and m = 1.92 reported for LaBi 13 , and WTe 2 34 , respectively. It is worth recalling that m = 2 is expected for a material with perfect carrier compensation.
In magnetic fields stronger than ~5 T, the transverse magnetoresistance of LuSb shows clear Shubnikov-de Haas (SdH) oscillations at temperatures up to 15 K (cf Fig. 4). The result of subtraction of a 2 nd -order-polynomial background from the resistivity data, Δρ, is plotted in Fig. 6(a) against inverse magnetic field for several temperatures from the range 2-15 K. Fast Fourier transform (FFT) analysis (see Fig. 6(b)), performed on the Δρ(1/B) data, revealed that the observed oscillations contain four basic frequencies and two harmonics (the values of f i FFT are listed in Table 1). Multifrequency quantum oscillations clearly indicate complex Fermi surface structure of LuSb, in accord with the results of the electronic structure calculations presented above.
According to Onsager relation 36 , oscillation frequencies are proportional to areas of extremal cross-sections of Fermi surface pockets. Thus, one can compare the experimental values of f i FFT with those estimated from the first-principle calculations, and ascribe each of them to a Fermi pocket. In the FFT spectrum of LuSb two different frequencies, α f FFT and α f FFT 1 correspond to two different extreme cross-sections of the Fermi sheet α (marked as green loops in Fig. 2(a,b)). In turn, the frequencies β f FFT and δ f FFT correspond to Fermi pockets β and δ, respectively (note blue and yellow loops in Fig. 2  the second harmonic of β f FFT . We could not observe the contribution due to the smallest ζ Fermi sheet because the range of magnetic field in which we recorded oscillations was too narrow to detect such a small frequency as ≈ ζ f 50 T predicted by calculations. Making rough approximations that extreme cross-sections of α, β and δ are circles, and extreme cut of α 1 is an ellipse (see Fig. 2), we calculated the Fermi wave-vectors, k F,i , gathered in Table 1. Subsequently, we calculated volumes, V F,i ), and corresponding carrier concentrations, 3 , for each Fermi pocket. The so-obtained values of n i are given in Table 1 (the value n α takes into account the triplicity of the α Fermi pocket).
In the inset to Fig. 6, the amplitudes of peaks corresponding to the α and β Fermi pockets are plotted as a function of temperature. From the least-square fitting with the equation describing thermal damping of SdH oscillation β ⁎ m m 0 27 0 . These values of m * are similar to those reported in the literature for other rare-earth monopnictides 6,12,13,27 .
In order to corroborate the FFT results, we evaluated the SdH oscillations in LuSb in terms of Lifshitz-Kosevich (LK) theory [36][37][38] . The multi-frequency LK function: i stands for the Dingle factor and ϕ i is the phase of oscillation. In order to avoid over-parametrization all frequencies were fixed at FFT-derived values (cf Table 1). The red fitting curve almost perfectly follows experimental ρ Δ B (1/ ) data, as shown in Fig. 7. This confirms that our FFT analysis revealed all significant contributions to SdH oscillations.
Hall effect. The results of Hall effect measurements of single-crystalline LuSb are summarized in Fig. 8

. The
Hall resistivity isotherms shown in Fig. 8(a) were obtained after removing the ρ xx contribution by subtraction of the data recorded in magnetic fields of opposite directions. The magnitude of ρ xy is very small and does not exceed 0.6 μΩ cm for any field or temperature values accessible in the experiment. The ρ xy (B) dependences are  curvilinear and nonmonotonous, which indicates that several bands are responsible for the transport properties of LuSb, in agreement with the results of the first-principle calculations and the SdH oscillations analysis presented above. Accordingly, the Hall-effect data of LuSb were evaluated within a multi-band Drude model of electrical conductivity. First, ρ xy was converted into the σ xy component of conductivity tensor using the simple relation where e is an elementary charge, n el and μ el , n i and μ i (for = i h h 1, 2) stand for the carrier concentrations and the carrier mobilities of one electron-and two hole-like bands, respectively. Such three bands give main contributions to the conductivity of LuSb, as concluded from the analysis of the SdH oscillations.
Since the initial fit using free n el , μ el , n i and μ i parameters was over-parameterized, in the next step we fixed n h1 at the carrier concentration value obtained from the SdH oscillations for the β pocket (the almost isotropic one) ≡ = . × β n n 1 31 10 h1 20 cm −3 . With this simplification, the experimental σ B ( ) xy data measured at T = 2 K were properly described with equation (4), as shown in Fig. 8(b) by a red solid line. The fit confirmed that two of three bands are hole-like and the third one is electron-like, and yielded the following parameters: μ = . × It is worth noting that this analysis gave the n h2 concentration almost identical to δ n and the n el value intermediate between α n and α n calc obtained from the SdH oscillations and band-structure calculations, respectively (cf Table 1). Small discrepancies may originate from the approximations of the α and δ Fermi sheets in LuSb with an ellipsoid and a sphere, respectively. On the other hand, our band-structure calculations were performed using lattice parameter obtained from X-ray diffraction at room temperature, slightly larger than at T = 2 K  20 cm −3 is almost equal to the carrier concentration n el of the electron-like band. This indicates again that LuSb is close to perfect carrier compensation. In turn, the carrier mobilities in LuSb are comparable to those found for YSb 27 , yet an order of magnitude smaller than those reported for LaBi 12,13 . The moderate values of μ i imply that the magnetoresistance of LuSb is notably smaller than MR measured for other rare-earth monopnictides 35 .
Angle-dependent magnetotransport. Figure 9(a) illustrates changes in the field-dependent electrical resistivity of LuSb with varying the direction of applied magnetic field with respect to the direction of electric current. In a transverse configuration (field direction is perpendicular to current direction, which corresponds to θ = 0°), MR reaches its maximum magnitude, whereas in longitudinal configuration (field and current directions are parallel to each other, i.e. θ = 90°) MR attains its minimum values. It should be noted that the anisotropy of the m a g n e tore s i s t an c e of Lu S b ( at T = 2 K an d i n B = 9 T ) i s g i ant an d e qu a l s A M R ρ ρ ρ ]/ (0 ) 87 %. This value is slightly larger than that observed for YSb 27 . In both compounds, AMR arises mainly due to carriers from the strongly anisotropic electron-like Fermi pockets.
In order to quantify the anisotropy of the Fermi surface in LuSb, the model developed in ref. 39 was applied. First, the magnetic field was scaled by a θ-dependent factor ε θ , so as all the ρ = B T ( 9 ) datapoints from all the MR curves of Fig. 9(a) moved on the ρ θ =  B ( , 0 ) curve. In result, all the ρ θ B ( , ) curves collapsed onto a single ρ ε 0 ) variation, as it is apparent from Fig. 9(b). The values ε θ which were used in the MR data  conversion are shown in the inset to Fig. 9(b). Interestingly, they can be almost perfectly approximated by the function which represents a proportionality of ε θ to the cross-section of ellipsoidal Fermi pocket with the parameter γ standing for the mass anisotropy 40 . From the fit of equation (5) to the data of LuSb (red solid line in the inset to Fig. 9(b)), we got γ = . 4 55. This value is larger than γ = 3.4 obtained for YSb in ref. 27 and γ = 4 found for MoTe 2 in ref. 41 , but smaller than γ = .
4 762 reported for WTe 2 in ref. 39 . The so-obtained γ value slightly disagrees with ∼ , which also corresponds to the Fermi sheet anisotropy. The discrepancy may be due to a contribution from anisotropic features of the δ pocket, which was not taken into account in our analysis of the SdH effect.
In all the ρ(B) curves taken at different θ angles there were discernible SdH oscillations, which could be analyzed in the same way as it was described above. Figure 10(a) shows ρ Δ as a function of inverted magnetic field, and Fig. 10(b) demonstrates their FFT evaluation. The angle dependences of the particular FFT frequencies are displayed in Fig. 10(c). The experimental data are compared on this figure with the results of the first-principles calculations. In order to get good agreement between the measured and calculated values of the SdH frequencies, the latter ones had to be multiplied by 0.855 (which does not influence the carrier compensation), such adjustment may be due to slightly enhanced lattice parameter (determined at room temperature) used in band structure calculations.

Conclusions
Our comprehensive investigation of the magnetotransport behavior in single-crystalline LuSb revealed substantial MR effect, which is most likely a consequence of nearly perfect charge compensation in this material, as supported by the results of our first-principles electronic band structure calculations. The measured magnitude of MR slightly exceeded 3000%, but improving the sample quality would certainly allow to obtain significantly larger MR. The quality of our crystals permitted recording of clear Shubnikov-de Haas oscillations, which allowed for the determination of the Fermi surface topology in LuSb. The experimental SdH data were very well reproduced by our band structure calculations. Kohler scaling of magnetoresistance accounts very well for its temperature behavior, whereas the field-angle-dependent magnetoresistance could be scaled with the effective mass anisotropy perfectly agreeing with electronic structure and quantum oscillations analysis. The magnetotransport properties of LuSb could thus be well accounted for without invoking any topologically non-trivial electronic features.
This conclusion is in concert with the lack of band inversion and mixed d − p orbital texture in LuSb as well as findings from the most recent ARPES experiments 30 . Our results and previous reports on magnetotransport in lanthanum monopnictides show that studies of other rare-earth monopnictides might be crucial to elucidating the role of Dirac states in extreme magnetoresistance.

Methods
Electronic structure calculations were performed with the all-electron general potential linearized augmented plane-wave (LAPW) method using the WIEN2k code 42 . Spin-orbit coupling was included as a second variational step, using scalar-relativistic eigenfunctions as the basis, after the initial calculation was converged to self-consistency. The Monkhorst-Pack special k-point scheme with 46 × 46 × 46 mesh was used in the first Brillouin zone sampling, and the cutoff parameter (R K mt max ) was set to 8. The modified Becke-Johnson potential 43 was applied for improved estimates of the band gaps followed by regular self-consistent-field calculation using the GGA-PBE scheme 44 for the exchange-correlation potential.
For the Fermi surface, the irreducible Brillouin zone was sampled by 20225 k points to ensure accurate determination of the Fermi level 45 . SdH frequencies were calculated using the Supercell K-space Extremal Area Finder tool 46 .
Single crystals of LuSb were grown from Sn flux in temperature regime selected accordingly to the binary phase diagram Lu-Sb 47 . The crystals were oriented and their quality was verified by backscattering Laue method using a Proto LAUE-COS system. Powdered crystals were examined by powder X-ray diffraction (XRD) employing a PANanalytical X'pert Pro diffractometer with Cu-K α radiation. The NaCl-type crystal structure was confirmed, with the cubic lattice parameter of 6.0577(1) Å, which is close to the literature value 6.0555 Å 18 . No impurity phases were observed on the XRD pattern. In addition, phase purity of the crystals and their chemical composition was checked by energy-dispersive X-ray spectroscopy on FEI scanning electron microscope equipped with an EDAX Genesis XM4 spectrometer.
Measurements of electrical resistivity, magnetoresistance and Hall resistivity were carried out in the temperature range from 0.7 to 300 K and in magnetic fields up to 9 T employing a Quantum Design PPMS-9 platform. Electrical contacts were made of silver wires attached to the rectangular-prism-shaped samples with silver epoxy, and additionally strengthened by spot welding. Electrical current was always flowing along [100] crystallographic direction. Data availability. The datasets analysed during the current study are available from the corresponding author on reasonable request.