Introduction

Topological insulators (TIs) with novel properties such as extreme magnetoresistance and high carrier mobility are currently attracting intense interests in condensed matter and material science communities. A TI supports the presence of gapless topological surface states (TSSs) with Dirac-cone-like energy dispersions within an inverted gap between the bulk conduction and valence bands1,2,3,4. The discovery of TIs has spurred the exploration of gapless nontrivial states beyond the insulators to include a great variety of topological semimetals such as the Dirac-, Weyl-, and nodal-line/loop semimetals in which the bulk bandgap opening is prohibited by crystalline symmetries5,6,7,8,9. These developments have opened up exciting new routes for finding exotic quantum phases and improved materials platforms for constructing low-power electronics/spintronics devices1,10. A distinct nontrivial \({\mathbb {Z}}_2\) state with TSSs has been recently reported in a three-dimensional (3D) material with a vanishing global bandgap11,12 where the TSSs overlap with the bulk states. The rare-earth monopnictide (REM) family is drawing special interests as a rich playground for investigating Dirac fermionic excitations. REMs have been predicted to host topological Dirac semimetal as well as 3D TI states13 and exhibit extremely large magnetoresistance (XMR)14,15,16,17,18,19,20. Although a good deal of work in the literature concerns the linearly-dispersing states and their role in driving XMR in the REMs, a robust conclusion in this regard remains a matter of debate. Although the conventional explanation for the XMR effects involves electron-hole compensation15,17,19,20,21,22, the observation of XMR in the REM family and numerous other topological materials such as \(\hbox {WTe}_2\)23, \(\hbox {Cd}_3 \hbox {As}_2\)24, \(\hbox {PtSn}_4\)25, and TaAs26 has ignited the possibility that XMR might have its origin in the forbidden backscattering channels of topological materials24,26. Notably, TSSs have been reported in LaBi, LaSb, CeSb, and CeBi by various groups17, 27,28,29,30,31,32,33,34,35,36 whereas LaAs, LuBi, YBi, YSb, and CeSb have been suggested to be topologically trivial materials14,15,16,31,34,37. The presence of a Dirac semimetal state is suggested in NdSb38 and DySb18. With this background in mind, further experimental and theoretical studies are needed to gain an understanding of the nature of Dirac-like states and how they are connected with XMR in the REM family.

Figure 1
figure 1

Sample characterizations and electronic structure of HoSb. (a) NaCl-type (rock salt) crystal structure of HoSb. Yellow and blue balls represent Ho and Sb atoms, respectively. (b) Primitive bulk Brillouin zone (BZ) and the projected (001) surface BZ. High-symmetry points are marked. (c) Core level spectrum of HoSb showing sharp peaks of Sb 4d and Ho 4f. Right inset shows the variation of the temperature derivative of resistivity with temperature, sharp peak at \(\sim\)5.7 K marks the magnetic transition. Left inset shows the picture of a HoSb single crystal. (d), (e) Calculated bulk band structure along the various high-symmetry directions without and with the inclusion of spin-orbit coupling (SOC), respectively.

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Rare-earth elements with their f-electrons provide strongly-correlated, tunable magnetic ground states in the REM family. Moving from La to Lu in the lanthanide series, one observes a nonmagnetic to ferromagnetic transition. HoSb that supports an antiferromagnetic (AFM) ground state39 is not a well-studied member of the REM family. It displays a magnetic transition from an MnO-type AFM arrangement to a HoP-type ferromagnetic arrangement under external magnetic field40. More recently, an unusual XMR has been reported in HoSb41,42. Here, we report the observation of a gapped state at the X point of the BZ in HoSb. Using angle-resolved photoemission spectroscopy (ARPES) along with first-principles calculations and magneto-transport measurements, we examine in-depth the electronic structures of this material. Our analysis reveals the presence of a highly anisotropic Dirac-like cone at the X point. Our experimental data and theoretical results identify a small gap around 470 meV below the Fermi level. More importantly, our calculations of the \({\mathbb {Z}}_2\) invariant show that HoSb assumes a trivial topological state (\({\mathbb {Z}}_2 = 0\)). We analyze our magnetotransport results to show that the presence of nearly perfect electron-hole compensation could drive the appearance of non-saturating XMR effect in HoSb.

Figure 2
figure 2

Fermi surface map and constant-energy contour plots. (a) Experimentally observed Fermi surface and constant-energy contour plots measured at a photon energy of 100 eV in HoSb. Binding energies are noted on the plots. Experiments were performed at the ALS beamline 4.0.3 at a temperature of 18 K. (b) Theoretical Fermi surface of HoSb.

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Results

Sample characterizations and electronic structure

HoSb crystalizes in a rock-salt-type crystal structure with space group Fm-3m like other members of the REM family (see Fig. 1a). Our X-ray diffraction measurements confirm the cubic crystal structure with the refined lattice parameter a = 6.13(1) Å. Here, yellow and blue balls correspond to the Ho and Sb atoms, respectively (Fig. 1a). Figure 1b shows the bulk BZ and its projection on the (001) surface. The core level spectrum is shown in Fig. 1c. Peaks of Sb 4d (\(\sim 32\) eV), Ho \(4\textit{f}_{5/2}\) (\(\sim 8.6\) eV) and Ho \(4\textit{f}_{3/2}\) (\(\sim 5.2\) eV) are clearly resolved. The observation of sharp peaks in the spectrum indicates that our HoSb samples are of good quality. The left inset of Fig. 1c shows a picture of our cubic crystal. In order to determine the magnetic transition temperature, we consider the temperature derivative of resistivity, \(d\rho /dT\), in the low-temperature regime (see Fig. 1c-right inset). One can clearly see the magnetic transition at around 5.7 K, which is in accord with previous report40.

We present the bulk band structure of HoSb without including spin-orbit coupling (SOC) effects and treating f-electrons as core orbitals in Fig. 1d. There are three hole-like bands at the \(\Gamma\) point and one electron-like band at the X point that crosses the Fermi level. On including SOC in the computations, salient features of this band structure picture are preserved (Fig. 1e), although the third hole-band at \(\Gamma\) moves closer to the Fermi level so that its top cross the Fermi level. At the X point, the gap between the Ho d and Sb p states is seen to nearly close. An inspection of the valence and conduction bands shows the presence of gaps of \(\sim 110\) meV and \(\sim 90\) meV at the \(\Gamma\) and X points, respectively.

Figure 3
figure 3

Dispersion maps along the various high-symmetry directions. (a), (b) Measured dispersion maps along the X-\(\Gamma\)-X and the M-\(\Gamma\)-M directions. In both plots two hole-like bands can be seen to cross the Fermi level. (c), (d) Measured dispersion map and its second derivative along the \(\Gamma\)-X-\(\Gamma\) symmetry lines, respectively. (e) Calculated energy dispersion along \(\Gamma\)-X-\(\Gamma\). (f), (g) Experimentally measured dispersion map and its second derivative along X-M-X. Photon energies are noted on the various panels. (h) Calculated energy dispersion along the \({{X}}\)-\({{X}}\) direction. Experiments were performed at the ALS beamline 4.0.3 at a temperature of around 18 K.

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Fermi surface and constant-energy contour plots

We now discuss the bulk band structure and the Fermi surface using our measured constant energy contours for the (001) surface, see Fig. 2a. \(\Gamma\) denotes the center, X the corner and M the midpoint between the two corners of the BZ. At the \(\Gamma\) point, we observe a diamond-like outer Fermi pocket and a circular inner pocket. Moving towards higher binding energies, we observe a third band and that the size of the pockets increases, confirming the hole-like nature of the bands around \(\Gamma\). At the X point, we clearly see two concentric elliptical-shaped pockets. However, moving to higher binding energies, the elliptical-pockets evolve into point-like features around 460 meV, which indicates the electron-like nature of the bands around the X point. Importantly, our calculations show the presence of a local band gap near this binding energy at the X point. Around the binding energy of 600 meV, size of the elliptical pocket increases, suggesting that the possible bulk bandgap lies around this binding energy. We will return below to present additional photon energy-dependent dispersion maps around the \(\Gamma\) and X points in order to ascertain the nature of the bands from our experimental measurements. Figure 2b shows the calculated Fermi surface contour, which is in substantial agreement with our experimental results.

Trivial electronic structures of HoSb

Figure 3 presents energy dispersion maps along the high-symmetry directions in the BZ. Figure 3a shows the energy dispersion along the \(X-\Gamma -X\) direction measured at a photon energy of 70 eV. Two hole-bands are observed to cross the Fermi level while the top of a third hole-band can be seen at around 200 meV below the Fermi level. Figure 3b shows the measured dispersion map around the \(\Gamma\) point along the high-symmetry direction \(M-\Gamma -M\). Similar to Fig. 3a, three hole-bands are observed with two bands crossing the Fermi level. Importantly, two bands crossing the Fermi level along the \(\Gamma -M\) direction are sharply dispersive in comparison to the \(\Gamma -X\) direction, which indicates the highly anisotropic nature of these bands and also explains the distortion of the diamond-shaped pocket as we move towards the higher binding energies in Fig. 2. Figure 3c,d show the dispersion map and the second derivative plots of the spectra along the \(\Gamma -X-\Gamma\) direction at a photon energy of 91 eV. One can observe the gapped Dirac-like state at the X points of the BZ. The bulk band gap at the X point is better resolved in the second derivative plot (see also Supplementary Fig. 3b). Notably, the preceding experimental results are in substantial accord with our theoretically predicted dispersions in Fig. 3e. We further confirm this by presenting photon energy dependent dispersion nature of the bands along the X–M–X direction and its second derivative in Fig. 3f,g (see also Supplementary Fig. 5), respectively. The experimentally observed bulk band gap is approximately 120 meV. Figure 3h shows that the calculated dispersion along the X-X line is in excellent accord with the correponding experimental results. The theoretical bulk band gap at the X point is about 90 meV. There is a continuous band gap between the valence and conduction bands so that the \({\mathbb {Z}}_2\) invariant is well defined. Using the evolution of Wannier charge centers44, we find a trivial topological invariant \({\mathbb {Z}}_2\)=0. In this way, we conclude that HoSb is a trivial semimetal and not a topological semimetal.

Figure 4
figure 4

Large and non-saturating magnetoresistance in HoSb. (a) Temperature dependence of electrical resistivity of a PFIB-prepared micrometer-sized single crystal of HoSb. The Inset shows a picture of the PFIB prepared sample (\(33 \,\upmu \hbox {m} \times 0.8\, \upmu \hbox {m} \times 10 \,\upmu \hbox {m}\)). (b) The magnetic field dependence of magnetoresistance of HoSb for applied fields up to 30 T. The dashed line represents the relation \(\hbox {MR}\sim \hbox {B}^{1.75}\). Inset shows the data in the low-field region.

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Transport signature of electron-hole compensation

Our ARPES measurements indicate that HoSb is a trivial semimetal. In order to study the transport properties of HoSb, we used a micron-sized sample prepared by plasma focused ion beam (PFIB) microscope. Magneto-transport measurements were performed on a FIB\('\)ed single crystal, where the electrical current (i) was applied along the \([100]\) crystallographic direction and the magnetic field was applied perpendicular to i and \([100]\) (see inset in Figure 4a). Temperature dependence of electrical resistivity of the HoSb single crystal is shown in Fig. 4a. The overall behavior of \(\rho\)(T) and the presence of an antiferromagnetic ordering at \(\textit{T}_N = 5.7\) K agrees well with previous studies41. Application of magnetic field strongly increases electrical resistivity and \(\rho\)(T) curve saturates at lower temperatures (the resistivity plateau)41. The field dependent resistivity saturation at low temperatures has not been observed in the non-magnetic members of the REM family28. The origin of such a plateau has been recently attributed to the presence of a nearly perfect electron-hole carrier compensation and the high mobility of the compensated semimetals45. Figure 4b and its inset show the magnetic field dependence of transverse magnetoresistance of HoSb at various temperatures. Magnetoresistance denotes the change of the electrical resistance under applied magnetic field defined as, \(\hbox {MR} = [R(B)-R(0)]/R(0)\), where R(B) and R(0) are resistances with and without the magnetic field, respectively. As seen from the inset of Fig. 4b, at temperatures below and above \(T_N\), MR increases with the increasing magnetic field. The kinks in the MR(B) curves mark a metamagnetic transition that is related to the change in the magnetic structure with the increase of field46. At \(T = 1.8\) K, magnetoresistance is large, reaching 350 \(\%\) at 9 T without any sign of saturation, in agreement with previous reports41. In this connection, we measured MR in pulsed magnetic fields up to 30 T (Fig. 4b), and found MR measured at several temperatures below and above the Neel temperature increases with magnetic field. MR reaches a large values of about 1500 \(\%\) at 30 T (\(T = 4\) K) which is comparable to the values observed in topological semimetals LaSb47, NbP48, and \(\hbox {WTe}_2\)23. The MR(B) curve at 4 K can be described by the form MR \(\propto\) \(B^n\) (see dashed line in Fig. 4b), where \(n = 1.75\). The close proximity to quadratic field dependence indicates nearly complete electron-hole compensation in HoSb, as expected from a semiclassical two-band model45. In addition, for \(T = 4\) K and 16 K, obvious Shubnikov-de Haas (SdH) oscillations are observed at high fields. The presence of the quantum oscillations, even at 16 K, points to the high quality of our HoSb single crystals used in the present studies. Analysis of SdH data in fields up to 30 T will be shown elsewhere. It is also the characteristic of low effective mass and large mobility of carriers that could exist in HoSb. The large, non-saturating magnetoresistance, the resistivity saturation in the turn-on temperature behavior at very low temperatures together with metallic conductivity observed in HoSb are all comparable to the characteristic of topological electron-hole-compensated semimetals45,49.

Discussion

We have carried out in-depth photoemission and transport measurements on HoSb single crystals along with parallel first-principles modeling of the electronic structure of this member of the REM family. Our ARPES measurements show the presence of Dirac-like cone with a small gap at around 470 meV below the chemical potential, but our analysis shows that HoSb is a trivial and not a topological Dirac semimetal. The observed experimental bulk band gap at X point is around 120 meV which is consistent with the theoretically predicted value. Magnetoresistance is found to be large and non-saturating, even at a magnetic field as large as 30 T. The characteristic behavior of electrical resistivity at low temperatures indicates that HoSb is likely a new electron-hole-compensated semimetal. Notably, the XMR effect has also been reported in other nontrivial members of the REM family suggesting that the presence of non-trivial state might not be directly responsible for the XMR. However, the high carrier mobility associated with the topological states might play a role in generating the XMR. Furthermore, our magnetoresitivity measurements up to 30 T show a similar unsaturated behavior below and above \({T}_N\) indicating similarity of the overall Fermi surface topology in the vicinity of Fermi energy above and below the Neel temperature. Keeping in mind that the rare-earth monopnictide family can host complex magnetic structures including the possibility of a Devil’s Staircase transition50, HoSb offers a unique platform for exploring the interplay between XMR, magnetism and topology in an antiferromagnetic matrix.

Methods

Crystal growth and characterization

Single crystals of HoSb were grown by the Sn flux technique as described elsewhere43. The crystal structure was determined by X-ray diffraction on a Kuma-Diffraction KM4 four-circle diffractometer equipped with a CCD camera using Mo K\(\alpha\) radiation, while chemical composition was checked by energy dispersive X-ray analysis using an FEI scanning electron microscope equipped with an EDAX Genesis XM4 spectrometer.

Spectroscopic characterization

Synchrotron-based ARPES measurements of the electronic structure were performed at the Advanced light Source (ALS) beamline 4.0.3 with a Scienta R8000 hemispherical electron analyzer. The samples were cleaved in situ in a ultra high vacuum conditions (\(5\times 10^{-11}\) Torr) at 18 K. The energy resolution was set to be better than 20 meV. The angular resolution was set to be better than \(0.2^\circ\) for the synchrotron measurements.

Transport measurements

For transport measurements, the HoSb sample was prepared by a Plasma Focused Ion Beam (PFIB) method. The electrical resistivity and magnetoresistivity were measured using a Quantum Design Dynacool-9 device in magnetic field up to 9 T. Magnetoresistance measurements at pulsed magnetic fields up to 30 T were performed at NHMFL, and LANL using the same HoSb sample as that used in DC fields. For both measurements, a standard four probe measurement technique was applied.

Theoretical calculations

Electronic structure calculations were performed within the framework of the density-functional theory (DFT) using the projector-augmented-wave (PAW) method51 as implemented in the VASP suite of codes52,53. The exchange-correlation functional was treated using SCAN meta-GGA54. An energy cutoff of 400 eV was used for the plane-wave basis set and a \(\Gamma\)-centered \(11\times 11\times 11\) k-mesh was used for BZ integrations. In order to examine the topological properties, we constructed a tight-binding model with atom-centered Wannier functions using the VASP2WANNIER90 interface55. The surface energy spectrum was obtained by using the iterative Green’s function method via the WannierTools package56,57,58.