Multiple metamagnetism, extreme magnetoresistance and nontrivial topological electronic structures in the magnetic semimetal candidate holmium monobismuthide

Single crystals of HoBi were grown from the Bi flux method. Highly purified crude materials of Ho and Bi with a molar ration of 1:19 were put into a tantalum crucible and then sealed in an evacuated quartz tube. All the operation was conducted in an argon-filled glove box, where the oxygen and humidity content were less than 0.1 ppm. The quartz tube was heated to 1273 K, held for 24 h, and followed by slow cooling to 773 K at a rate of 0.5 K h⁻¹ in furnace. At this temperature, the excess Bi flux was removed by centrifugation. Then, shiny and cubic single crystals (figure 1(b)) were obtained. To determine the crystal structure and phase purity, x-ray diffraction (XRD) with Cu Ka radiation was performed with 2θ ranging from 10° to 100° at room temperature. Chemical stoichiometry of the polished crystals was characterized by a Zeiss Sigma field emission scanning electronic microscopy equipped with an Oxford INCA energy-dispersive x-ray spectrometer (EDX).

The temperature- and field-dependent magnetization measurements were carried out using the VSM module of a physical property measurement system (PPMS-VSM, Quantum Design), where the temperature and magnetic field vary from 2 to 300 K and from 0 to 9 T, respectively. The electric resistivity and Hall effect measurements were performed in a temperature ranging from 2 to 300 K and in an applied magnetic field up to 9 T on a Quantum Design PPMS platform. For all the electric measurement, standard four-probe method was used.

Band structure and Fermi surface of HoBi were built by first-principles calculations implemented in the Vienna ab initio simulation package code, where the projected augmented wave method and Perdew–Burke–Ernzerhof (PBE) type exchange-correlation interactions were employed [41, 42]. The plan-wave cutoff energy of 600 eV and a 30 × 30 × 30 Gamma-centered K-mesh were adopted in the electronic structure calculation. Spin–orbit coupling and on-site Hubbard U for the correlated 4f-electrons were included in all calculations with U = 7 eV, J = 0.6 eV. The lattice constants a = b = c of HoBi were taken as 6.223 Å. The optimization procedure was truncated until the residual forces for each relaxed atoms were less than 0.02 eV Å⁻¹. Considering that the band gap determined by the generalized gradient approximations (GGA) was possibly overestimated, a hybrid functional approximation for the exchange-correlation with Heyd–Scuseria–Ernzerhof (HSE06) was supplemented to further examine the relativistic effect on band structure [43].

Figure 1(c) shows the XRD pattern for HoBi single crystal along the (001) plane. Sharp peaks (00h) in the diffraction spectrum indicate high crystalline nature of the grown crystals. The composition of well-polished single crystal is determined by EDX spectroscopy, which is carried out at different positions on the crystal surfaces with an instrument error of 1%–2%. Typical results plotted in figure 1(d) confirm almost the perfect stoichiometry (Ho:Bi = 49.8:50.2), and no evidence of Bi thin film or nanocluster in the as-grown crystals. As shown in figures 1(e) and (f), Ho and Bi are uniformly distributed across the crystal surface, suggesting an absence of any observable impurity as well.

The thermomagnetic curves of HoBi single crystal in zero-field-cooled (ZFC) and field-cooled (FC) modes are measured under an external field of 0.1 T perpendicular (figure 2(a)) and parallel (figure 2(b)) to ab plane. As shown in figure 2(a), the magnetization along c-axis shows an antiferromagnetic transition at 5.9 K (Néel temperature T_N), which is consistent with the previous works [40, 41]. The ZFC and FC magnetization curves almost superimpose in the whole measured temperature range (2–300 K), indicating the formation of long-range antiferromagnetic orders below Néel temperature. On the other hand, when the magnetic field is applied in ab plane, an analogous antiferromagnetic transition is observed at 5.9 K, as displayed in figure 2(b), matching exactly with the Néel temperature as well. The out- and in-plane thermomagnetic curves suggest that antiferromagnetic configurations generate at low temperature, which coincides with the spin structure in figure 1(a). Since the magnetic moments of HoBi mainly originate from Ho ions, as shown in the insets of figures 2(a) and (b), the effective magnetic moments μ_eff are estimated to be 10.3 ${\mu }_{{\rm{B}}}$ and 10.2 ${\mu }_{{\rm{B}}}$ by fitting the linear part of inverse magnetic susceptibility 1/χ(T) between 100 and 300 K to Curie–Weiss law χ(T) = M/H = C/(T + Θ) with C = N_Aμ_eff²/3k_B (Here, k_B, Θ, C, and ${\mu }_{{\rm{B}}}$ are the Boltzmann constant, Curie–Weiss temperature, Curie constants and Bohr magneton respectively) [44]. Thereby, the Weiss temperatures are calculated to be Θ_⊥ = 10.1 K and Θ_// = 12.3 K. The difference in two susceptibilities indicates the remarkable magnetic anisotropy in HoBi, which is confirmed by the magnetic hysteresis loops shown below as well. Figures 2(c) and (d) show the ZFC and FC magnetization as a function of temperature under 1 and 3 T. It is clear that, other than the obviously enhanced magnetization, the antiferromagnetic orders remain persistent at 1 T, exhibiting a transition from paramagnetism to antiferromagnetism at low temperature. When a higher field (3 T) is applied, slightly different magnetization responses to the applied magnetic field can be observed in the two directions. As plotted in figure 2(d), the antiferromagnetic order is suppressed and a remarkable phase transition from paramagnetic to ferromagnetic-like state takes shape. In figure 2(c), the ferromagnetic-like phase has not completely formed, and a cusp, which is a generic feature of antiferromagnetic transition, still exists around the magnetic transition temperature. The insets of figures 2(c) and (d) illustrate the magnetic hysteresis loops of HoBi at 2 K. As shown, obvious metamagnetic behaviors are observed, where the onset fields for those transitions in the two directions are disparate. Furthermore, a significant hysteresis, which doesn't develop in the inset of figure 2(c), is evidenced in the inset of figure 2(d), showing an anisotropic behavior of magnetization. It is noticed that the magnetization of HoBi is saturated above 5 and 7 T, i.e. ${\mu }_{{\rm{sat}}}$ = $9.9\,{\mu }_{{\rm{B}}}$ and $10.1\,{\mu }_{{\rm{B}}}$ for the applied field along c axis and in ab plane, which significantly accord with the isolated Hund's rule ground state of Ho³⁺ ion (10 ${\mu }_{{\rm{B}}}$) [40, 41]. As is reported, RPn (R = Ce, Dy, Ho; Pn=Sb, Bi) shows metamagnetic transitions with diverse behaviors below the antiferromagnetic temperature [24, 36, 40, 45]. For instance, DySb shows a plateau at one half of the saturated moment in an intermediate phase between 2.5 and 4.2 T, which is ascribed to the competition between Zeeman effect and magnetocrystalline anisotropy [24]. More narrowly, DySb is a type-II antiferromagnet and its ground state can be driven into ferromagnetic state via a HoP-type intermediate phase, where the moment in one of each two nearest ferromagnetic plane are rotated 90° and the measured magnetization is one half of the saturated values, by applying a magnetic field along the easy axis direction [24]. Analogous metamagnetic transitions are observed in HoSb, in which the MnO-type antiferromagnetic ground state also goes through a HoP-type spin structure to a ferromagnetic spin structure with the increasing magnetic field [36]. Relatively speaking, the metamagnetic transition of HoBi shown in the inset of figures 2(c) and (d) are more complicated, indicating that richer magnetic phase diagram could be expected.

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To study the metamagnetic properties of HoBi in detail, the magnetization as a function of magnetic field along different directions are plotted in figures 2(e) and (f). Clearly, the anisotropy of HoBi is reinforced again, where the isothermal magnetization curves show different shapes and critical fields for those magnetic transitions. As shown, several field-induced magnetically ordered phases can be readily observed at low temperature. Actually, to precisely determine the magnetic transitions, neutron diffraction on this compound is necessary, which is unavailable to us at the moment. Here, the critical fields for metamagnetic transitions in HoBi are roughly determined from the peaks in derivative dM/d(μ₀H). There are approximately five maximums at low temperature [40]. For clarity, the insets of figures 2(e) and (f) plot the field dependence of magnetization and dM/d(μ₀H) at 2 K. The magnetization jumps rapidly at 1.9, 2.5, 3.7, 4.2, and 4.8 T in the inset of figure 2(e), indicating that the metamagnetic transition is completed under the field less than 5.0 T along c axis. In the inset of figure 2(f), the complicated magnetic transitions at 2 K are mainly distributed in two field regions with a broad valley from 2.5 to 4.6 T in dM/d(μ₀H) curve. Intriguingly, the applied field can not terminate the metamagnetic transition until 6.5 T, which is slightly larger than that along c axis, reconfirming the remarkable magnetic anisotropy. Figures 2(g) and (h) show the field-dependent derivative dM/d(μ₀H) at various selected temperatures with the field perpendicular and parallel to ab plane. It can be observed that multiple metamagnetic transitions are induced at low temperature. With the gradual increment of temperature, the metamagnetic behaviors become more and more inconspicuous, and disappear above 6 K.

Figure 3(a) shows the variation of zero-field resistivity with temperature for HoBi from 2 to 300 K. As plotted, the resistivity is metallic in nature and decreases monotonically to 2 K, where the resistivity ${\rho }_{0}$ becomes as small as 2.6 μΩ cm, resulting in a large residual resistivity ratio RRR = ρ(300 K)/ρ(2 K) ∼ 193. In the inset of figure 3(a), the low-temperature resistivity obeys a power-law behavior, ρ(T) = ρ₀ + ATⁿ with n=3.9 similar in numerical value to that of LaSb (n = 4), which indicates the interband s–d e–ph scattering, rather than the intraband s–s e–ph scattering should be dominant [6, 27]. The low-temperature resistivity is also fitted by the above-mentioned relation with n = 3 (see the supplementary materials is available online at stacks.iop.org/NJP/21/093063/mmedia). It can be easily found that the fitted curve deviates from the measured one. Figure 3(b) shows the temperature-dependent resistivity of HoBi at different magnetic fields (0.5, 1, 3, 5, 7, and 9 T). As shown, the low-temperature resistivity is slightly enhanced in the applied magnetic fields of 0.5 T and 1 T. When a larger magnetic field (3, 5, 7, and 9 T) is applied, the low-temperature resistivity significantly increases and experiences a metal-insulator-like behavior along with a minimum at T_m, which is a generic feature of semimetals [16, 19].The inset of figure 3(b) shows the definition of two characteristic temperatures T_m and T_i, where T_m denotes the point of sign change in dρ/dT, and T_i corresponds to the minimum in dρ/dT [28]. Figure 3(c) exhibits the evolution of T_m and T_i as a function of magnetic field, in which the former one increases with increasing magnetic field and the latter one is almost field-independent, suggesting that the temperature window of insulator-like regime grows with the increment of applied magnetic field and the temperature of resistivity plateau is field-insensitive [46]. Figure 3(d) shows the plot of T⁻¹-dependent log(ρ), which can be used to estimate the insulating gap E_g in the insulator-like regime by employing the formula ρ(T) ∼ exp(E_g/k_BT) [16]. The derived values of insulating gap increase monotonically with the increasing magnetic field (not shown), from which sharper metal-insulator crossovers can be expected at higher magnetic field.

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Figure 4(a) shows the magnetic field dependence of magnetoresistance for HoBi single crystal at various temperatures, where the external field is applied along c axis and the electric current flows in ab plane (the sketch map is plotted in the inset of figure 4(a)). Obviously, the external field can greatly change the resistivity of HoBi. As displayed in figure 4(a), the positive magnetoresistance reaches an approximate value of 3.9 × 10⁴% at 2 K without any sign of saturation under the magnetic field as large as 9 T. Upon slightly increasing the temperature, the magnetoresistance decreases gradually with its order of magnitude unchanging. At higher temperature, especially above the antiferromagnetic phase temperature about 6 K, the magnetoresistance drops drastically, indicating that the ordered spin structure benefits the generation of extreme magnetoresistance. It can be clearly observed that the magnetoresistance at room temperature (figure 4(a)) is only the order of 4%, which suggests that lattice scattering dominates the electron mobility of HoBi in high temperature regime. Usually, the evolution of resistivity with magnetic field in its sister compounds, like LaBi, YBi, and so on, behaves in a quadratic manner [27, 28]. While, a distinct transport scenario emerges in HoBi, where kinks generate in the magnetoresistance curves. To investigate the origin for occurrence of these features in magnetotransport, the field-dependent magnetoresistance and magnetization at 2 K are replotted in figure 4(b). As shown, the magnetoresistance increases gently under low magnetic field and dramatically changes after 4.6 T. Accordingly, after this critical field, all metamagnetic transitions are completed and HoBi enters its ferromagnetic state, which indicates that the kinks may originate from low-field-induced some magnetic transitions. Similar scenarios in the magnetization and magnetoresistance curves of HoSb have been observed during the evolution of magnetic structure from HoP-type orders to ferromagnetic orders [36]. The correlation effect between the magnetic orders and resistivity of HoBi can be easily reflected in the field-dependent derivative dMR/d(μ₀H). As shown in figure 4(c), the critical fields (H_C) for inflexions in the magnetoresistance curves increase with the increasing temperature, which also suggests that the ordered low-temperature magnetic states benefit the formation of extreme magnetoresistance. Figure 4(d) shows the results of Kohler scaling for magnetoresistance of HoBi. The semiclassical transport theory based on Boltzmann equation predicts that the relative change of resistivity Δρ/${\rho }_{0}$ in a magnetic field is a universal function of H/${\rho }_{0},$ Δρ/${\rho }_{0}$ = F[H/${{\rm{\rho }}}_{0}$], where ${\rho }_{0}$ is the zero-field resistivity at a certain temperature [28]. When there is a single type of charge carrier and relaxation rate τ in the transport process, this rule will be tenable and then all the magnetoresistance isotherms collapse on a single curve. If not, the Kohler's rule will be violated, which has been widely observed in the materials with extreme magnetoresistance, like LaBi, TaPn and TaPn₂ (Pn = As and Sb) [27, 47, 48]. As shown, the magnetoresistance of HoBi obviously deviates from Kohler's rule as well, which may originate from multiband carriers at low temperature, or multiple scattering mechanisms at high temperature. Evidently, as shown in figure 4(b), the field-dependent magnetoresistance at 2 K can be roughly fitted by MR ∝ (μ₀H)ⁿ, where the obtained n = 2.05 is closed to 2. Due to the existence of kinks, fitting of the magnetoresistance curve is not perfect. While, as shown in the inset of figure 4(b), the data at high magnetic field can be well fitted, yielding the obtained n = 2.14, which is slightly larger than 2 as expected for a perfectly compensated metal. Such deviations have been observed in other materials hosting extreme magnetoresistance as well, like PtSn₄, WTe₂, and some rare earth manopnictides, e.g. LaSb/Bi, NdSb, YSb/Bi, LuSb/Bi [21, 25, 26, 28, 46, 49, 50]. Traditionally, the extremely large magnetoresistance is discussed in the framework of two-band model, which can be expressed as ${\rho }_{xx}=\tfrac{({n}_{e}{u}_{e}+{n}_{h}{u}_{h})+({n}_{e}{u}_{e}{{u}_{h}}^{2}+{n}_{h}{u}_{h}{{u}_{e}}^{2}){B}^{2}}{e[{({n}_{e}{u}_{e}+{n}_{h}{u}_{h})}^{2}+{({u}_{h}-{u}_{e})}^{2}{{u}_{e}}^{2}{{u}_{h}}^{2}{B}^{2}]},$ where ${\rho }_{xx}$ is the longitudinal resistivity, e is the electron charge, n is the carrier density, u is the mobility, and the subscripts e and h denote the electron-type and hole-type carriers, respectively [28]. As predicted by this model, the longitudinal resistivity will increase as a function of quadratic in the magnetic field, when the carrier densities of electron and hole are compensated. A slight deviation from the perfect resonant condition can also result in a large quadratic magnetoresistance as long as the applied field is not too strong. In the case of perfect compensation between electrons and holes, another prediction from the formula manifests that the magnetoresistance ratio strongly depends on the average mobility u_ave, and can be rewritten as MR = (u_aveB)² (u_ave = $\sqrt{{u}_{e}{u}_{h}}$) [51]. Based on this formula, the obtained u_ave is evaluated to be 2.1 × 10⁴ cm² V⁻¹ s⁻¹ at 2 K. Although the average mobility of HoBi stands no superiority with respect to the carrier mobility of topological semimetals like Cd₃As₂ and WTe₂ [52, 53], it is still larger than those of the common magnetic metals, which could contribute to the extreme magnetoresistance of this compound.

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To evaluate the carrier densities and mobilities of HoBi, the Hall resistivity is obtained by measuring the transverse resistivity under positive and negative fields, namely ${\rho }_{xy}=\left[{\rho }_{xy}\left(+B\right)-{\rho }_{xy}\left(-B\right)\right]/2,$ which is used to eliminate the longitudinal resistivity contribution due to voltage probe misalignment [27]. As shown in figure 5(a), the field-dependent Hall resistivity exhibits nonlinear behavior, indicating the involvement of more than one type of charge carrier in the transport process [28, 30]. The negative/positive slopes of high-field Hall resistivity at low/high temperature suggest that the dominant carriers change from electron to hole with the increment of temperature [27, 28, 30]. Figure 5(b) plots the Hall coefficient R_H as a function of temperature at 6 T and 9 T [19], which shows a strong temperature dependence and changes its sign around 6 K. The strong nonlinear behaviors and sign changed Hall coefficient in Hall resistivity have been widely observed in several semimetals, which are interpreted as the electron–hole compensation mechanism [19, 20, 28]. Obviously, sign change of the Hall coefficient together with the quasi-quadratic magnetoresistance at low temperature illustrate that this interpretation can also be applicable to HoBi. Here, it is assumed that the carrier concentration and mobilities are insensitive to magnetic field, and the carrier concentration are well compensated. Under this assumption, the magnetoresistance, zero-field resistivity, and Hall coefficient in two-band model can be written as: MR = [ρ(H,T) − ρ(0,T)]/ρ(0,T) = μ_e · μ_hH², ρ(T) = 1/[n_e(μ_e + μ_h)], and R_H = (μ_h − μ_e)/[n_e(μ_e + μ_h)] [19, 54]. Consequently, the obtained carrier mobilities at 2 K are μ_e = 1.9 × 10⁴ cm² V⁻¹ s⁻¹ and μ_h = 2.6 × 10⁴ cm² V⁻¹ s⁻¹ respectively, which are comparable to those of the Dirac semimetals HfTe₅, TlBiSSe, and so on [55, 56]. Besides, the extracted electron and hole densities are n_e ≈ n_h = 5.3 × 10²¹ cm⁻³. Such high mobility and nearly perfect carrier compensation in HoBi could be the major cause for generation of the unprecedentedly large and quasi-quadratic magnetoresistance. Figure 5(c) shows the extracted concentrations and mobilities for both types of carriers as a function of temperature, in which both the two parameters are strongly temperature-dependent. As plotted, the mobilities of electron and hole decrease with the temperature warming up, and are comparable to each other, which suggests that the two types of carriers contribute significantly to the transport properties. This behavior is reflected in the Kohler scaling analysis as well, where magnetoresistance curves deviate from the scaling at low temperature.

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As presented above, high magnetic field drives the emergence of a particular case instead of the generally plain quadratic magnetoresistance curves expected in many other semimetals, which has been scarcely reported [18, 29, 36]. Therefore, an in-depth study on electronic structure of the ferromagnetic phase of HoBi could help to understand the origin of extreme magnetoresistance in rare earth monopnictide family. The bulk band structure of ferromagnetic phase (figure 6) is calculated by the GGA method without and with SOC effect respectively. Figure 6(a) shows the partial density of states (PDOS) of the ferromagnetic state of HoBi, from which it can be observed that the f-electrons of Ho fully localize at non-occupied states near the Fermi level and the valence and conduction bands are mainly dominated by Bi-6p and Ho-5d. Figure 6(b) illustrates the obtained energy band structure without SOC for HoBi, in which there are a few bands crossing the Fermi level and the valence and conduction bands clearly dip into each other along the high-symmetry Γ-X line, suggesting the semimetal features without band crossing points. Similar to those reported for lanthanum monopnictides, double degenerate bands with a gap of 0.19 eV develop at the X-point, which leads to the proposal of two-dimensional topologically nontrivial states [57]. It is common knowledge that in Bi-based compounds the effect of SOC is very dramatic and shouldn't be ignored for band structure calculation with potential topological nature [28]. Indeed, as shown in figure 6(c), possibly due to the relativistic effect, more complicated electronic structures develop, where the sixfold degeneracy of Bi-6p states is modified at Γ point, namely, four of the p bands dip deeply around the Fermi level with the other two p bands still degenerated. Furthermore, twofold degeneracy of the two bands is gradually lifted along Γ–L and Γ–X direction, and they are well separated at L- and X-points. Noticeably, the band crossing points forming by Ho-5d and Bi-6p states (visualized with red and blue colors in the enlarged drawing of figure 6(c)) at X-point, exhibit inverted electronic structures and a very tiny gap, where the Dirac cones may potentially generate. For HoBi, there are eight time-reversal invariant points in the body-centered-cubic Brillouin zone, including (0,0,0), (0,0,0.5), (0,0.5,0), (0,0.5,0.5), (0.5,0,0), (0.5,0,0.5), (0.5,0.5,0) and (0.5,0.5,0.5). Due to the cubic crystal symmetry, the parity criteria proposed by Fu and Kane is employed to calculate the Z₂ index for determining the topological features [58]. The topological index is rigorously evaluated from the parity of valence band wave functions at each time-reversal invariant momentum and can be identified as (1; 000), which indicates that HoBi has nontrivial electronic structures and is a topological semimetal candidate.

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For better estimating the band gap as well as the band topologies of HoBi [28], the electronic structures are also checked by the HSE06 hybrid functional calculation without and with SOC interactions. Here, the discussion mainly focuses on the band structure around Fermi level. As shown in figure 7(a), the band topologies obtained through the hybrid calculation without SOC also show no band inversion along the Γ–X line, in which a larger gap (comparing to that in figure 6(b)) about 0.55 eV is observed. Agreeing with figure 6(c), the band gap will be closed when SOC effect is included, resulting in inverted electronic structures in a deeper position with two crossing points near the X-point (figure 7(b)). Besides, smaller gaps are obtained in the magnified region of band crossing points from HSE06 method. Through further comparison of the band structures in figure 7, it can be found that hole occupy states dominate the electronic structure around the Fermi level in the case without SOC, which deviates from the experimental observation of electron–hole compensation behavior, indicating that this relativistic effect plays an important role in determining the band structure of HoBi. Sincerely, the band structures calculated by HSE06 are slightly different from those obtained from GGA method, which seems to be inevitable. While, the compensated and inverted band structures still can be obtained, as plotted in figure 7(b), confirming again that HoBi is a semimetal candidate and hosts Z₂ topological nature.

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Based on the GGA and HSE06 calculations, a schematic illustration of the fcc Brillouin zone of HoBi can be extracted. The band calculations reveal that there are two electron pockets and four hole pockets (figure 8) in the first Brillouin zone. The electron pockets (α and α') are strongly anisotropic and locate at X points. While, the hole pockets (γ and γ') are identical in shape with the other two anisotropic hole pockets (β and β') being entirely wrapped at the Γ point. All pockets have only one extremal orbit supporting that the magnetic field is applied along the crystallographic axes. By rough assessment, it is found that the volumes of electron and the hole pockets are comparable, which verifies the semimetal nature of HoBi.

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