Temperature-dependent Fermi surface probed by Shubnikov–de Haas oscillations in topological semimetal candidates DyBi and HoBi

Rare earth-based monopnictides are among the most intensively studied groups of materials in which extremely large magnetoresistance has been observed. This study explores magnetotransport properties of two representatives of this group, DyBi and HoBi. The extreme magnetoresistance is discovered in DyBi and confirmed in HoBi. At \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=2$$\end{document}T=2 K and in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B=14$$\end{document}B=14 T for both compounds, magnetoresistance reaches the order of magnitude of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^4\%$$\end{document}104%. For both materials, standard Kohler’s rule is obeyed only in the temperature range from 50 to 300 K. At lower temperatures, extended Kohler’s rule has to be invoked because carrier concentrations and mobilities strongly change with temperature and magnetic field. This is further proven by the observation of a quite rare temperature-dependence of oscillation frequencies in Shubnikov–de Haas effect. Rate of this dependence clearly changes at Néel temperature, reminiscent of a novel magnetic band splitting. Multi-frequency character of the observed Shubnikov–de Haas oscillations points to the coexistence of electron- and hole-type Fermi pockets in both studied materials. Overall, our results highlight correlation of temperature dependence of the Fermi surface with the magnetotransport properties of DyBi and HoBi.


I. INTRODUCTION
Rare-earth monopnictides (REP n, where RE denotes one of the rare-earth elements and P n = As, Sb or Bi) have attracted broad scientific and technological interest due to their exceptional physical properties and potential applications in solar cells, [1] thermoelectrics, [2] plasmonics, [3] and spintronics.[4] In addition, the preparation of RESb thin films has been developed, [5,6] which may be important from the point of view of future applications.
Recently, interest in REP n materials has surged due to the possible topologically nontrivial nature of their electronic bands.[7] In addition, unsaturated and extreme magnetoresistance (XMR), accompanied by magnetic field-induced low-temperature resistivity plateau, has been reported in many REP n materials.[7][8][9][10][11][12] At first, these phenomena in LaSb have been attributed to the existence of topologically non-trivial electronic states.[7] However, alternative explanation that has been postulated for many REP n compounds is perfect or nearly-perfect compensation of charge carriers.[8,[12][13][14][15][16][17] In order to experimentally verify the presence of topologically non-trivial states in REP n compounds, their electronic structure has been studied directly by angle-resolved photoemission spectroscopy (ARPES).In LaBi, for example, several groups have found the presence of topologically non-trivial states of different natures: nodal-lines, [18] surface states, [10,19,20] and multiple Dirac cones.[21] In addition, pressure-induced superconductivity has been reported for LaBi and YBi, [22,23] which established them as possible topological superconductors, potentially hosting Majorana fermions.
The presence of a rare-earth element with partially filled f -shells in REP n compounds provides even more fascinating physical properties.PrBi has been reported to exhibit nontrivial Berry phase and quadrupolar moment, [24] whereas topological quantum phase transition has been proposed in HoSb.[25] Several antiferromagnets among REP n family have been reported to demonstrate XMR.[12,15,26,27] Interestingly, it has been proposed that within RESb and REBi series topological phase transitions could occur close to SmSb and DyBi, respectively.[28] Intriguing new magnetic splitting effect has recently been discovered in NdBi [4] and other REBi compounds.[29] This splitting takes place below the Néel temperature (T N ) and leads to the appearance of new sheets of the Fermi surface, hole-and electron-like "magnetic" Fermi arcs, and it can lead to a change in the shape of the bulk Fermi pockets.Interestingly, these arcs can be topologically non-trivial in nature.[4] We hypothesized that this behavior could be reflected in the properties of quantum oscillations (the appearance of new frequencies and the change in frequency with changing temperature), since the oscillation frequency is directly proportional to the cross-sectional area of the Fermi pocket.[30] In NdBi, T N = 24 K [4], a temperature too high for quantum oscillations to be observed in this material.Therefore, in order to examine our conjecture, we chose two other REBi compounds with significantly lower T N : DyBi (T N = 11.2K [31]) and HoBi (T N = 5.7 K [32]), and investigated their magnetotransport properties.HoBi has previously been reported to exhibit XMR, magnetic-field induced resistivity plateau and complex magnetic structure [33][34][35], which we also confirmed in our study.In contrast to HoBi, magnetotransport properties of DyBi were not studied.We found that, similar to HoBi, DyBi exhibits XMR and magnetic field-induced resistivity plateau.In addition, we observed that oscillation frequencies change with temperature variation in both compounds, which corresponds to a change in the topology of Fermi surface.

A. Electrical resistivity
In zero magnetic field, the temperature dependence of electrical resistivity, ρ(T ), of both compounds (see Figure 1 a,b) displays metallic-like behavior, with ρ decreasing as the temperature decreases, with values of ρ = 25.6 µΩ cm and ρ = 40.2µΩ cm at T = 300 K for HoBi and DyBi, respectively.At T = 2 K electrical resistivity falls to ρ = 0.46 µΩ cm and ρ = 0.63 µΩ cm, for HoBi and DyBi, respectively (and the residual resistivity ratios [≡ ρ(300 K)/ρ(2 K)] are 56 and 64).A significant drop in electrical resistivity is observed as the temperature decreases below T = 5.9 K for HoBi and T = 12 K for DyBi.These drops can be attributed to the elimination of spin-disorder scattering by the antiferromagnetic ordering, which according to the literature takes place at T N = 5.7 K and T N = 11 K for HoBi and DyBi, respectively.[31,35,36] Magnetic field of 14 T, applied transverse to the current, dramatically changes the ρ(T ) dependence: resistivity is enhanced in whole temperature range, and displays metallic-like behavior down to 60 K and 40 K, for HoBi and DyBi, respectively, followed by a sharp increase of ρ with further decreasing temperature.
At low temperatures magnetic field-induced resistivity plateaus are observed (see insets to  in other rare earth bismuthides and antimonides: LuSb, LuBi, YBi, LaBi, LaSb, YSb, PrBi, GdSb, TbSb [8-12, 16, 27] and in several topological semimetals: ZrSiS, NbP, TaAs.[37][38][39] Appearance of the magnetic field-induced resistivity plateau has been attributed to the metal-insulator transition, [40,41] perfect or nearly perfect carrier compensation [42,43] or Lifshitz transition.[44] In most of RESb and REBi compounds, the overall ρ(T ) dependence in applied magnetic field (including resistivity plateau and change of behaviour from metallic-to semiconducting-like) has been described in terms of carrier compensation.[8-10, 13, 14, 16, 43] For HoBi, the shape of ρ(T ) obtained in B = 14 T is similar to that of temperature dependent Hall coefficient (R H (T )) obtained for B = 14 T, shown in the inset to Figure 4.  Black solid lines in the insets correspond to the fits with Kohler's formula.

B. Magnetoresistance
HoBi are slightly smaller than those reported in literature for this material, but the order of magnitude of MR is the same.[34] In case of DyBi, the magnitude of MR is almost the same as that reported for DySb in Ref. [26].Values of MR in both compounds are related to the residual resistivity ratios in a similar way as in other REP n. [16] XMR exhibited by REP n series has been attributed to: carrier compensation, [8,9,12,13,16,45] metal-insulator transition, [7] or d-p orbital mixing in the presence of carrier compensation.[46] In numerous, both magnetic and non-magnetic REP n demonstrating XMR, nearly quadratic behavior of MR∝ B m (m < 2) was observed.[12,[47][48][49] However, for HoBi and DyBi, MR(B) isotherms for temperatures ≤ 15 K display strikingly different character.
Those for T < T N exhibit two distinct regions, for example, at T = 2 K, in the low field region, MR(B) increases with fluctuating slope up to about 5 T for HoBi, and up to about 8 T for DyBi.Such behavior can be related to the series of metamagnetic transitions reported for both materials.[31,[33][34][35]50] Magnetic moments of Ho and Dy become fully aligned with the stronger magnetic field, and in such conditions the onset of typical MR (∝ B m ) occurs in both compounds.Interestingly, our magnetotransport results for HoBi are in a full agreement with those reported by Wu et al. [34], although they differ from those described by Yang et al. [33] That difference comes from the magnetic anisotropy in HoBi, since in those measurements direction of the applied magnetic field was different ([001] and [110], respectively).In Ref. [33] it has been concluded that HoBi is the first material where XMR appears in the fully field-polarized magnetic state.Our results indicate that the mechanism of XMR in DyBi is the same as in HoBi.
It has recently been proposed that the magnetotransport properties of semimetals displaying XMR can be understood in the scope of Kohler's scaling, which reflects the degree of charge carrier compensation.[51] According to standard Kohler's rule M R = (B/ρ 0 ) m , where m is a sample characteristic parameter, indicating the level of charge balance in the system.For a perfectly electron-hole compensated material m = 2.In order to test this approach for our materials, we performed the Kohler's scaling, as shown in Figure 3. Kohler's rule is violated for both compounds from 2 to 15 K (see main panels of Figure 3a,b).i.e. the temperature range covering antiferromagnetic order and noticeable short-range order.
In several compounds exhibiting extreme magnetoresistance, e.g.LaBi, YBi, WP 2 , WTe 2 or TaAs, Kohler's rule is violated as well, [39,44,[52][53][54] but in these cases violation is due to the change of concentration and mobility of carriers with temperature, not to the magnetic interactions.On the other hand, in the temperature interval from 50 to 300 K MR isotherms of HoBi and DyBi collapse onto a single curve (see insets to Figure 3a,b).For the MR data obtained in the temperature range 50-300 K, the least-squares fitting (black solid lines in the insets to Figure 3a,b) yielded m = 1.6 and m = 1.54, for HoBi and DyBi, respectively.
Similarly, it has recently been shown that Kohler's scaling is effective at T > 50 K in HoSb and ErBi.[25,45] For other REP n containing magnetic RE, Kohler's rule was disobeyed in the entire covered range of temperature.[12,27,34] The magnitude of m that we obtained for both compounds is one of the smallest among REP n. [8,9,14,16,45,52] To the best of our knowledge, the only material among REP n compounds with m smaller that we observed in this work is LaAs with m = 1.35.[55] The Kohler's rule is obeyed in simple non-magnetic metals, with one type of carriers and single scattering time [56] and in semimetals, only if mobilities of both types of carriers are small, and their ratio, as well as carrier concentrations are all temperatureindependent.[57] Therefore, violation of the Kohler's rule for HoBi and DyBi seems most likely due to the temperature-dependent carrier concentrations and/or mobilities.Results of the Hall effect data presented in Figure 4 support such assumption.The Hall coefficient (R H = ρ xy /B = 1/(en H )), which is inversely proportional to carrier concentration (n H ) varies strongly with increasing temperature (see inset to Figure 4).Moreover, the frequency of observed Shubnikov-de Haas (SdH) oscillations also depend on temperature (see below), which is consistent with the assumption of temperature-dependent carrier concentrations in HoBi and DyBi.
Recently, an extended approach to Kohler's rule has been reported, M R = (B/(n T ρ 0 )) m in which temperature-dependent carrier concentration is implemented through introducing a factor n T .[58] We tested that approach to our data for temperatures where standard Kohler's scaling failed.We demonstrate the outcome in Figure 5.For HoBi extended scaling is working, for T ≤ 15 K all data above MR∼ 5000% collapse on single curve with m = 1.7.This exponent is different from 1.6 obtained for T ≥ 50 K, which indicates small but clear change of electron-hole compensation between these two temperature ranges.In the case of DyBi extended scaling works, but the curve on which data collapse (for 4 < T < 6 K, and MR∼ 9000%) corresponds to m = 1.85, suggesting a higher degree of charge carriers compensation if compared with T ≥ 50 K, for which m = 1.54.Parameter n T strongly evolves with temperature, for both compounds, as shown in insets to Figure 5a,b, implying a significant change of carrier concentration (we will discuss this further below when analyzing the SdH oscillations).In this range of T , strong interactions of magnetic moments, and series of metamagnetic transitions strongly influence scattering rate of charge carriers, and magnetoresistance regains its typical (power function) behavior only in strong magnetic fields, when antiferromagnetic order is destroyed and critical fluctuations damped.

C. Shubnikov-de Haas oscillations
Magnetoresistance isotherms, MR(B), for both compounds show oscillatory behavior above B = 9 T, which we ascribe to the SdH effect.For both materials, the SdH oscillations survive at quite high temperatures, up to at least 10 K. Figures 6a, and 7a depict the oscillating part of the electrical resistivity (∆ρ) as a function of the inverse magnetic field (1/B), obtained by subtraction of a smooth background from the resistivity data.Fast Fourier transform (FFT) analysis of these oscillations revealed three fundamental frequencies (F α , F β and F γ listed in Table I), which points at possible complex form of Fermi pockets or/and at multi-band type of conductivity.F i values we obtained for HoBi are very close to those reported in Ref. [33].In contrast to that report, we did not observe the high-frequency oscillations (F > 900 T), most probably due to the narrower range of magnetic field in which our experiments were carried out.For DyBi, our observation of the SdH oscillations is the first.and the frequencies we obtained (see Table I) are quite close to those found for HoBi, which implies that relevant sheets of Fermi surfaces in both compounds are rather similar.According to the results of several theoretical calculations, Fermi surfaces of REP n compounds, which contain heavy rare earth elements, are qualitatively similar, all of them possess two or three nearly isotropic hole-like pockets in the center of the Brillouin zone and strongly anisotropic triplicate electron-like pocket.[8, 16, 25-27, 34, 59-61] In the magnetic-field induced polarized state (with all spins aligned with strong enough magnetic field) the electronic structure of magnetic REP n changes with each Fermi pocket splitting into two pockets due to the magnetic-field induced time-reversal symmetry breaking.[25,27,33,34] Such splitting has already been reported for HoBi.[33,34] Comparing the frequencies we obtained to those reported in Ref. [33], we may assume that F α originates from the smallest hole-like Fermi pocket (denoted as h 1 in Ref. [33]); F β and F γ come from two extreme crosssections of the electron-like pocket (denoted as e 2 in Ref. [33]).For another REP n, DySb, quantum oscillations were also reported [26] and their frequencies are listed in Table I along those theoretically calculated.The authors of Ref. [26] ascribed the frequencies F α and F γ to the electron-like Fermi pocket and frequency F β to the hole-like pocket.This assignment is different from that made for HoBi in Ref. [33] To find which option is valid for HoBi and DyBi, it might be necessary to perform angle-dependent studies of quantum oscillations and to calculate electronic structures for both materials.According to the Onsager relation, the oscillation frequencies are directly proportional to the area of the extreme cross-sections (S i ) of Fermi surface pockets: F i = (hS i )/e, where h stands for the Planck constant, and e is the elementary charge [30].Assuming, that the shapes of hole-like and electron-like Fermi pockets of both compounds can be approximated by a sphere and an ellipsoid, respectively, we calculated the corresponding Fermi wave vectors (k F ) and charge carrier concentrations (n = V F /(4π 3 ), where V F is volume of the Fermi pocket).The calculated values (listed in Table I) are smaller than those obtained from our Hall effect results, as well as reported in literature.[34] This difference is most likely due to the fact that not all Fermi pockets were detected with the SdH oscillations.Interestingly, in the FFT spectra for both compounds, clear shifts of the peaks related to the F α were observed.In the case of DyBi, the pronounced shift was also noticed for F β .To see it more clearly, the temperature dependence of F i are shown in the left insets to Figure 6b and Figure 7b, where one can see that F i increases with increasing temperature.Similar behavior has previously been noticed for several REP n, such as DySb, [26] SmBi, [62] HoSb, [63] and NdSb.[64] In the first three reports, the authors did not comment on these observations, while in the fourth one concluded that it indicated the change of topology of the Fermi surface.Interestingly, for REP n containing non-magnetic RE, there were no reports on the temperature dependence of oscillations frequencies, which may point that magnetism is responsible for this behavior.From another point of view, it is also known that in semimetals with small Fermi pockets and small effective masses, the Fermi level can be shifted with increasing temperature, leading to changes in the Fermi surface.[30] For our materials, the Fermi temperature (T F = E F /k B ) is much larger than temperatures at which SdH oscillations were observed, therefore we can exclude the above scenario.The thermal expansion of the material is another possible source of the Fermi level shift.[30] In antiferromagnets, at temperatures close to the T N thermal expansion very often demonstrates drastic change, [65,66] which was observed for HoBi as well.[32] However, thermal expansion as the only source of change of the Fermi surface can also be ruled out for the following reasons: (i) for HoBi above T N thermal expansion does not change significantly, but F α continues to increase; (ii) for DyBi both F α and F β increase with increasing temperature, but if a simple Fermi level shift would be the reason of this, then one of these frequencies should decrease and the other increase with increasing temperature, since we ascribe them to Fermi pockets of different type, electron-and hole-like.Very recently, it has been proposed that change of oscillation frequency can be a signature of Dirac bands [67], however, in this case magnitude of frequency change is two orders of magnitude smaller than we observed for DyBi and HoBi.It has recently been shown that in several REP n compounds a new type of magnetic splitting leads to the appearance of additional Fermi arcs below T N , [4,29] moreover, such splitting can change the shape of the 3D Fermi surface.Our analysis of the SdH oscillations did not reveal new oscillation frequencies due the Fermi arcs, but we can speculate that the strong temperature dependence of oscillation frequencies comes from a new type of magnetic splitting, especially because the rate of change of F with temperature is very different below and above T N (cf.left inset to Figure 6b).That splitting is also likely to underlay dramatic change in Kohler's scaling between paramagnetic and antiferromagnetic state of HoBi (cf.inset to Figure 5a).
Recently, temperature variations of the oscillation frequency have been found in MnBi 2−x Sb x Te 4 , PrAlNi and CePtBi.[68][69][70] For MnBi 2−x Sb x Te 4 this variation was attributed to the change in the field-induced magnetization [68], and for PrAlNi it has been proposed that spin polarization is responsible for this effect.[69] We suppose, that the former explanation may also be reasonable for HoBi and DyBi.In turn, for CePtBi, this rare behavior was attributed to the temperature variation of 4f electrons hybridization.[70] This explanation can be excluded in our case, as the electronic bands related to the 4f electrons are located much above the Fermi level in REP n. [27] In Ref. [70], the authors explained the possible role of the so-called background magneti- zation and its temperature dependence on the oscillation frequency.For CeBiPt described in that paper, this effect has been rejected (because of opposite than expected change of F vs. T ), but for our results it could be applicable.However, such mechanism would affect all observed frequencies in a similar way, whereas for HoBi the value of F β does not change at all, in contrast to strong change of F α .Moreover, in high fields, between 9 and 14 T, the magnetization is very close to saturation and weakly changes with temperature.These allow us to rule out the role of background magnetization.
Despite the fact that the Fermi surfaces in both compounds vary with temperature, we made a working assumption that the effective masses are almost temperature independent.
This allowed us to use the Lifshitz-Kosevich theory for determining the effective masses.[30] FFT peaks' amplitudes as a function of temperature are shown in the right insets to Figure 6b and Figure 7b for HoBi and DyBi, respectively.Effective masses (m * i ) were obtained through least-square fitting with the equation representing thermal damping of SdH oscillations: R i (T ) = (λm * i T /B eff )/ sinh(λm * i T /B eff ), (i = α, β) where B eff = 2/(1/B 1 + 1/B 2 ) = 10.96T and λ=14.7 T/K.The value of B eff was obtained for B 1 = 14 T and B 2 = 9 T, limiting magnetic field range of the analysis.Obtained effective masses for HoBi and DyBi are listed in Table I, these values are similar to those reported for other REP n. [8,33,63] III.CONCLUSIONS We have performed a comprehensive study of magnetotransport properties of two XMR materials, DyBi and HoBi.While the magnetotransport properties that we observed for HoBi are in agreement with previous reports, [33,34] no systematic studies of magnetotransport of DyBi have previously been reported.For both materials, we observed XMR of the order of 10 4 % and magnetic field-induced resistivity plateau at low temperatures.We found that standard Kohler's scaling is obeyed at temperature range from 50 to 300 K. On Fermi pocket.This suggests that electron-hole compensation can be responsible for XMR in these compounds.
FIG. 1. Temperature dependence of the electrical resistivity of HoBi (a) and DyBi (b), measured in zero magnetic field (yellow line) and in magnetic field of 14 T (green line), applied transverse to the electrical current (j) direction.Insets show the low-temperature data, with arrows indicating Néel temperatures.

Figure
Figure 1a,b) for both compounds.Similar behavior of ρ(T ) dependence has been reported

FIG. 3 .
FIG. 3. Standard Kohler's scaling for HoBi (a) and DyBi (b).Main panels show the data taken in the temperature range 2-15 K; insets show the data recorded in the temperature range 50-300 K.

FIG. 4 .
FIG. 4. Hall resistivity (ρ xy ) of HoBi as a function of magnetic field taken at several different temperatures.Inset: temperature dependence of Hall coefficient in B = 14 T. Vertical dashed line indicates the Néel temperature.

FIG. 5 .
FIG. 5. Extended Kohler's scaling, M R = (B/(n T ρ 0 )) m , for HoBi (a) DyBi (b); black dashed lines show power functions representing Kohler's rule.Insets show changes of the factor n T with temperature (lines are guides for the eye).Blue dashed lines indicate the Néel temperatures.

FIG. 6 .
FIG. 6.(a) The SdH oscillations extracted from electrical resistivity data in transverse magnetic field at different temperatures for HoBi.(b) Fast Fourier transform analysis of the data presented in (a).Left inset: Amplitude of F α principal frequencies in FFT spectra as a function of temperature (black solid line is guide for the eye).Blue dashed line indicates the Néel temperature.Right inset: Temperature dependence of the FFT peak height.Red solid lines represent the fits of Equation 1 to the experimental data.

FIG. 7 .
FIG. 7. The SdH oscillations extracted from electrical resistivity data in transverse magnetic field at different temperatures for DyBi.(b) Fast Fourier transform analysis of the data presented in (a).Left inset: Amplitudes of principal frequencies in FFT spectra as a function of temperature (black solid lines are guides for the eye).Right inset: Temperature dependence of the FFT peak height.Red solid lines represent the fits of Equation 1 to the experimental data.
the other hand, satisfying the extended Kohler's rule at T ≤ 15 K supports the variation of carrier concentration with temperature.The results of Hall effect measurements and analysis of the SdH oscillations also confirmed the temperature dependent carrier concentrations.The rare change of quantum oscillation frequencies with temperature increasing was observed for both materials and could be attributed to the change of the Fermi surface shape due to the emergence of the new type of Fermi arcs below T N , recently discovered for several REP n[4,29].This mechanism is strongly supported by the clear difference in both, n T (T ) and F (T ) behaviour at temperatures below and above T N .Similarly to other REP n, the SdH oscillations in both studied compounds show multi-frequency character, where one of the frequencies comes from the hole-type Fermi pocket and others from electron-type

TABLE I .
[26]meters obtained from the analysis of the Shubnikov-de Haas oscillations for HoBi and DyBi, compared to the data reported for HoBi in Ref.[33]and for DySb in Ref.[26].