⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 077102 Single Crystal Growth and Magnetoresistivity of Topological Semimetal CoSi * D. S. Wu (吴德胜)1,2, Z. Y. Mi (米振宇)1,2, Y. J. Li (李延杰)1,2, W. Wu (吴伟)1,2, P. L. Li (李沛岭)1,2, Y. T. Song (宋友庭)1,2, G. T. Liu (刘广同)1,2,3, G. Li (李岗)1,2,3**, J. L. Luo (雒建林)1,2,3** Affiliations 1Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100190 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190 3Songshan Lake Materials Laboratory, Dongguan 523808 Received 7 May 2019, online 20 June 2019 *Supported by the National Natural Science Foundation of China under Grant No 11874399, and the National Key Research and Development Program of China under Grant No 2016YFA0300602.
**Corresponding author. Email: gli@iphy.ac.cn; jlluo@iphy.ac.cn Citation Text: Wu D S, Mi Z Y, Li Y J, Wu W and Li P L et al 2019 Chin. Phys. Lett. 36 077102    Abstract We report single crystal growth of CoSi, which has recently been recognized as a new type of topological semimetal hosting fourfold and sixfold degenerate nodes. The Shubnikov–de Haas quantum oscillation (QO) is observed on our crystals. There are two frequencies originating from almost isotropic bulk electron Fermi surfaces, in accordance with band structure calculations. The effective mass, scattering rate, and QO phase difference of the two frequencies are extracted and discussed. DOI:10.1088/0256-307X/36/7/077102 PACS:71.20.Be, 71.18.+y, 72.15.-v © 2019 Chinese Physics Society Article Text CoSi is a long known material with FeSi-type cubic structure (B20, space group $P2_{1}3$), of which the thermoelectric property and application have been the focus of study,[1,2] and it is generally regarded as a semimetal. With the development of characterizing materials by the topology of their electronic band structure, it has recently been suggested that in crystalline systems, in addition to Dirac, Weyl, and nodal line semimetals, band crossing points (nodes) with three-, six-, or eight-fold degeneracies can be stabilized,[3] and the low energy fermionic excitations close to the nodes are called 'new fermions'. For CoSi and isostructural transition metal silicides RhSi, RhGe, and CoGe, the crystal structure is asymmorphic with threefold rotation symmetry and twofold screw symmetry. Theoretical studies[4,5] found that with the inclusion of spin-orbit-coupling (SOC), close to the Fermi level there only exist bands containing a fourfold degenerate node at the Brillouin zone (BZ) center ${\it \Gamma}$, and a sixfold degenerate node at BZ corner $R$, both are chiral, thus could serve as a model system in the study of unconventional chiral fermions. Moreover, the phonon spectra of these monosilicides host double Weyl points.[6] Subsequent angle resolved photoemission spectroscopy (ARPES) measurements[7–9] have confirmed the overall bulk band structure and the existence of surface Fermi arcs. However, it has been noted that for CoSi preparation of a flat surface for ARPES is at least challenging. Therefore, other probes of its bulk electronic structure are needed. At the same time, in the context of searching for features characteristic of chiral fermions, the physical properties of CoSi are also worth revisiting on high quality single crystalline samples. In this work, we report single crystal growth by the flux method, different from the Czochralski, Bridgman, or chemical vapor transport techniques previously used. The resulting crystals are of decent quality. To our knowledge, it is the first time that the Shubnikov–de Haas (SdH) quantum oscillation (QO) is observed in CoSi. The angular dependence of the QO frequencies reveals that they are from the bulk electron Fermi surface centered at $R$ in BZ, agreeing with band calculation and ARPES data. Thus, it is a good starting point for further investigations. The single crystal growth starts with Co powder (99.999%, Alfa), Si powder (99.999%, Alfa), and Sn grains (99.99%), which were mixed in a molar ratio of 1:1:20, loaded into an alumina crucible, then sealed with a partial Ar pressure inside a silica capsule. The reactant was quickly heated up to 1273 K and maintained there for 40 h, followed by cooling down in 3 d to 1073 K, at which the flux was separated by centrifuge. The harvested crystals with well-defined surfaces were further washed by dilute HCl to remove possible tin coating. There were mainly two types of shapes, one was a thin, prismatic rod, and the other was a polyhedron close to a cube. To check for chemical composition, we employed energy dispersive x-ray (EDX) spectroscopy equipped on a Hitachi S-4800 scanning electron microscope (SEM). The rod type crystals showed good consistency among different growth batches with the Co:Si atomic ratio close to unity, and the Co percentage ranging from 49.5% to 50.7%. However, the polyhedron type crystals showed a large variation in composition, with the Co% expanding from 39% to 61%, indicating the formation of other Co-Si binary compounds.[10] The quality of the rod type crystals was then checked by single crystal x-ray diffraction at room temperature in a Bruker D8 Venture diffractometer using Mo K$_\alpha$ radiation, $\lambda=0.71073$ Å. Structure refinement was performed by the program SHELXL-2014/7[11] embedded in the program suite Apex3. Employing the known crystal structure of CoSi, the best refinement had $R$-indices of $R=0.0334(74)$ and $wR2=0.0767(74)$, respectively. The lattice constant is determined to be $a=4.4376(6)$ Å, in accordance with the data reported.[12] In addition, the orientation along the rod is determined in the $\langle011\rangle$ direction of the crystal, and an optical image of a typical crystal is shown in the inset of Fig. 1. Based on initial composition and crystallographic screening, all the electrical resistivity measurements were performed on rod-type single crystals by the standard four-probe method with electrodes along the rod made by silver paste (DuPont 4922N) and platinum wires. The environment of cryogenic temperature and magnetic field was provided by either a 14 Tesla Quantum Design physical property measurement system (PPMS-14), or an Oxford 14 Tesla magnet with top-loading He-3 insert, both equipped with single-axis rotator. An external ac current source and lock-in amplifier were utilized. We note here that there are only a few inconsistent reports[13,14] on the temperature and magnetic field dependent magnetization of CoSi. Due to our current crystal size and limitation of instrumental resolution, it is planned in our future work.
cpl-36-7-077102-fig1.png
Fig. 1. Normalized temperature-dependent resistivity of several crystals in log-log scale. The corresponding resistivity at 2 K for each sample is labeled in the legend. The solid line represents a simulation by the Bloch–Grüneisen formula (see text) for S4. The inset is an optical image of a prismatic rod-shaped single crystal on millimeter scale.
The temperature dependence of resistivity $\rho(T)$ with $I\parallel\langle011\rangle$ for several samples is shown in Fig. 1. It is overall metallic while there is no up-turn at low temperature, against that of sample having nonstoichiometric cobalt.[15] The residual resistivity $\rho$ (2 K) for three out of four samples is around 20 $µ\Omega$$\cdot$cm, on the low side of the reported data.[12,16–19] The largest residual resistivity ratio (RRR=$\rho$ (300 K)/$\rho$ (2 K)) appears in sample S4, RRR$\sim$7.6, on the high side of the reported values. Since the residual resistivity and the RRR are commonly used as indicators of crystal quality of metallic systems, we judge that our flux grown crystals have reached at least the same level of quality as crystals grown by other methods. Compared to other known Dirac and Weyl semimetals, e.g., Cd$_{3}$As$_{2}$, for which residual resistivity $\sim$20 $n\Omega\cdot$cm, and RRR$\sim$4000 has been reported,[20] the corresponding values for CoSi are moderate. Simply considering that the residual resistivity is proportional to the product of carrier density and carrier mobility, while the carrier density of Cd$_{3}$As$_{2}$ is lower than that of CoSi ($\sim$10$^{19}$ versus $\sim$10$^{20}$ cm$^{-3}$), the carrier mobility of CoSi should be lower by orders of magnitude. Whether it is an intrinsic property of CoSi or that samples at current stage still have a high level of impurity scattering requires further investigation. A simulation of $\rho(T)$ is also tested by the Bloch–Grüneisen (BG) formula[21] $$ \rho(T)=\rho_{0}+C\Big(\frac{T}{{\it \Theta}_{\rm R}}\Big)^{n}\int^{\frac{{\it \Theta}_{\rm R}}{T}}_{0}\frac{t^{n}}{(e^{t}-1)(1-e^{-t})}dt,~~ \tag {1} $$ in which ${\it \Theta}_{\rm R}$ is a characteristic temperature that is usually equal to the Debye temperature ${\it \Theta}_{\rm D}$, and $n$ is an integer depending on the scattering mechanism, $n=5$ for electron-phonon scattering. As shown in Fig. 1, for sample S4 BG simulation with $n$ fixed at 5 could only closely follow the temperature dependence below 170 K, while the derived ${\it \Theta}_{\rm R}$ is 300 K, only about 60% of ${\it \Theta}_{\rm D}$ obtained by heat capacity measurement.[13] Therefore, it is clear that a one-channel electron-phonon scattering mechanism could not explain the $\rho(T)$ behavior of CoSi. Other aspects including a non-constant ${\it \Theta}_{\rm D}$, multiple conducting channels, and electron-electron scattering may also need to be considered. With the aim of probing the bulk Fermi surface properties of CoSi, angular and temperature-dependent magnetoresistivity (MR) measurements were carried out on sample S4. Thanks to the thin rod shape of the crystal, the mixing of MR and Hall signals is minimal. Figure 2(a) shows the isothermal MR curves covering over 90$^{\circ}$, and the measurement configuration is illustrated in the inset. Both $B$ and $I$ are parallel to the $\langle011\rangle$ direction at $\theta=0^{\circ}$. The MR reaches 120% at 14 T, 350 mK, and $\theta\sim90^{\circ}$, about 2 times larger than reported values for CoSi,[17] although it is at least one order of magnitude smaller than the huge transverse MR reported for other Dirac and Weyl semimetals.[22] The MR versus tilt angle at selected magnetic fields is plotted in Fig. 2(b), which is symmetric with respect to 0$^{\circ}$ and 90$^{\circ}$ as expected. The data could be tracked by a $\sin^{2}(\theta)$ function, suggesting that the MR is proportional to a simple power law of $H$, with the power close to 2, and is mainly orbital. Moreover, the mobility of associated charge carriers should not have a significant anisotropy. The SdH is already apparent as small wiggles on those MR curves in Fig. 2(a). Since the amplitude of the oscillation is less than 2% of the total resistivity, it is feasible to keep the analysis of SdH on MR only. Figure 2(c) shows the oscillating part of MR (MR$_{\rm OSC}$) versus the reciprocal of magnetic field after a smooth background subtraction by a forth order polynomial. A beating pattern is evident for most of the curves, while the position of nodes of the envelope line has a weak evolution with angle. By fast Fourier transform (FFT) from 7 to 14 T, two prominent fundamental frequencies are identified on the spectra for all the angles stacked in Fig. 2(d). Their 1st harmonics are also visible when the oscillation amplitude is strong around $\theta=90^{\circ}$. By Gaussian peak fit, $F_1=563.9$ T and $F_2=664.6$ T at $\theta\sim87.7^{\circ}$, and $F_1=564.5$ T and $F_2=664.1$ T at $\theta\sim 3.1^{\circ}$, respectively. The standard errors of fitting are all less than 1 T, while the full width at half maximum (FWHM) is less than 30 T. If the FWHM is taken as the upper limit of frequency shift, then for both $F_1$ and $F_2$ the changes will be less than 5% when $B$ rotates from parallel to perpendicular to $I$.
cpl-36-7-077102-fig2.png
Fig. 2. (a) Angular dependent MR of CoSi at 350 mK, the inset shows the measurement configuration. (b) The MR at given fields extracted from (a), where points are connected by lines as a guide to the eyes. Fittings for 3 T, 7 T, and 14 T by $\sin^{2}(\theta)$ are superimposed as dark grey lines. The oscillating parts of MR are in (c), and the corresponding FFT spectra in (d), all stacked for clarity. In (d), those two vertical grey bars are centered at 564 T and 665 T, respectively. The width of the bars is 30 T.
Since the oscillation frequency $F$ is directly proportional to the extremal cross-sectional area of a Fermi surface ($A_{\rm F}$) by the Onsager relation[23] $F=(\hbar/2\pi e)A_{\rm F}$, the non-dispersive angular dependence of oscillation frequencies $F_1$ and $F_2$ with respect to magnetic field direction indicates that the underlying bulk Fermi surfaces are close to spherical. From recent band structure calculations,[4,24] it is shown that those bands crossing the Fermi level close to the BZ center ${\it \Gamma}$ point have their node above the Fermi energy ($E_{\rm F}$), while the Fermi wavevector ($k_{\rm F}$) is different along ${\it \Gamma}$–$X$, ${\it \Gamma}$–$M$, and ${\it \Gamma}$–$R$, which are directions from ${\it \Gamma}$ to the face center, the edge center, and the corner of BZ, respectively. At the same time, for those bands crossing close to the point $R$, their node is $\sim$200 meV below $E_{\rm F}$. Without inclusion of SOC, there are two adjacent bands crossing the Fermi level along $R$–${\it \Gamma}$, one along $R$–$M$, and one along $R$–$X$, $k_{\rm F}$ is similar along the three aforementioned directions. Thus there exist two electron Fermi surfaces (FSs) close in size centered at $R$, from which we infer that the observed two oscillation frequencies are originated. For $F_1=564$ T and $F_2=665$ T, the corresponding $A_{\rm F1}=5.39\times10^{-2}$ Å$^{-2}$ and $A_{\rm F_2}=6.34\times10^{-2}$ Å$^{-2}$, respectively. The averaged value is compatible with that obtained from $E_{\rm F}$ intensity mapping at $R$ point in recent ARPES experiments.[7,8] Here quantum oscillation shows the advantage of resolving two frequencies. However, we have not yet detected any oscillation from the hole FSs at ${\it \Gamma}$.
cpl-36-7-077102-fig3.png
Fig. 3. (a) Temperature-dependent MRs at $\theta\sim 90^{\circ}$, a wide grey curve of MR $\sim H^{1.7}$ is set as background. [(b), (c)] The oscillating parts of MRs and the corresponding FFT spectra, respectively. (d) Fittings of FFT peak amplitude by the thermal damping factor $R_{\rm T}$, an average $B=9.3$ T of the FFT range is used.
For the observed oscillations, more information about charge carriers at the FS could be extracted utilizing the Lifshitz–Kosevitch (LK) formula,[23,25] considering summations over different oscillation frequencies and their harmonics $$\begin{alignat}{1} {\rm MR}_{\rm OSC}\sim\,&\sum_{i}\sum_{p}\frac{5}{2}\sqrt{\frac{B}{2pF_{i}}} R_{\rm T}^{p}R_{\rm D}^{p}R_{\rm s}^{p}\\ &\cdot \cos\Big\{2\pi p\Big[\frac{F_{i}}{B}+\frac{1}{2}\Big] +\delta_{i}+\phi_{i}\Big\},~~ \tag {2} \end{alignat} $$ where $R_{\rm T}^{p}=\alpha p m^{*} (T/B)/\sinh[\alpha p m^{*} (T/B)]$, $R_{\rm D}^{p}=\exp[-\alpha p m^{*}(T_{\rm D}/B)]$, and $R_{\rm s}^{p}=\cos(p\pi g m^{*}/2)$ are the thermal damping factor, the Dingle damping factor, and the spin factor, respectively, and $\alpha=2\pi^{2}k_{_{\rm B}}m_{\rm e}/e\hbar$ is a prefactor constructed by the Boltzmann constant $k_{_{\rm B}}$, the bare electron mass $m_{\rm e}$, the electron charge $e$, and the reduced Plank constant $\hbar$. For the additional phase shift of the cosine function, $\delta$ is the dimensional correction, and $\phi$ is often denoted as the Berry phase. The effective mass $m^{*}$ at a given field direction can be obtained by the temperature dependence of oscillation amplitude through $R_{\rm T}$. Figure 3 shows the analysis of MR data at $\theta\sim 90^{\circ}$ up to $\sim$9 K. The effective masses for previously determined $F_1$ and $F_2$ are $m_{1}^{*}=0.82$ and $m_{2}^{*}=0.88$, respectively. While in early experimental work, the effective mass of electrons was estimated to be 2.[26] By assuming a circular cross section of FS, the Fermi wave vectors are estimated to be $k_{\rm F_1}=1.31\times 10^{-1}$ $Å^{-1}$ and $k_{\rm F2}=1.42\times10^{-1}$ $Å^{-1}$, and the Fermi velocities ($v_{\rm F}=\hbar k_{\rm F}/m^{*}$) are $v_{\rm F_1}=1.85\times 10^{5}$ m$\cdot$s$^{-1}$ and $v_{\rm F2}=1.87\times 10^{5}$ m$\cdot$s$^{-1}$, respectively. As a comparison, $v_{\rm F}$ of Cd$_{3}$As$_{2}$ from quantum oscillation experiments is around $1\times 10^{6}$ m$\cdot$s$^{-1}$.[20,27]
cpl-36-7-077102-fig4.png
Fig. 4. (a) LK formula (see text) fit of one $MR_{\rm OSC}$ curve with large oscillation amplitude. Fitting parameters are listed in the legend. (b) Illustration of how the beating pattern is constructed by the product of two cosine functions with periods determined by the difference and summation of two oscillation frequencies, respectively. Note the phase shifts.
Since there are two nearby oscillation frequencies, the extraction of Dingle temperature $T_{\rm D}$ is performed by a direct fitting of ${\rm MR}_{\rm OSC}$ to the LK formula with known $F$ and $m^{*}$. In addition, the Landé g-factor in $R_{\rm s}$ simply takes the free electron value of 2.0023, and the dimensional phase factor $\delta$ for both $F_1$ and $F_2$ takes the value of $(-1/4)\pi$ for three-dimensional electron FS with maximum cross section. By only including fundamental harmonics, a good fit is achieved, as shown in Fig. 4(a) for $\theta=83^{\circ}$, while the residual mainly contains the 2nd harmonic. Within fitting error the obtained $T_{\rm D}$ for both $F_1$ and $F_2$ is 3.2 K. The quantum scattering time $\tau_{\rm q}=\hbar/(2\pi k_{_{\rm B}}T_{\rm D})$ for electrons is then computed to be $\sim$$3.8\times 10^{-13}$ s. By taking the values of electron mobility and effective mass from Ref.  [26] $\mu_{\rm e}=800$ cm$^{2}$V$^{-1}$s$^{-1}$ and $m^{*}=2$, the transport scattering time $\tau_{\rm t}=\mu m^{*}/e=9.1\times 10^{-13}$ s. The small difference between $\tau_{\rm t}$ and $\tau_{\rm q}$ could be one reason for the relatively small MR in CoSi when compared to other topological semimetals. As in Cd$_3$As$_2$, $\tau_{\rm t}$ is 4 orders larger than $\tau_{\rm q}$ because of the suppression of backscattering, which is lifted in magnetic field and leads to huge positive MR.[20] The importance of phase factor in quantum oscillation associated with the symmetry of FS has recently been further stressed.[28] In the above LK fitting, the Berry phase is a free parameter and non-zero values within [0, 2$\pi$] for both $F_1$ and $F_2$ are obtained. However, we treat them as nominal values and make our interpretation in another route without Landau level index plot. Since the two observed frequencies are close, with similar effective masses and Dingle temperatures, a clear beating pattern is present. Then without the field dependent damping factors, the envelope line is given by a 'slow' cosine function $\cos[\pi(F_1-F_2)/B+(\delta_{1}-\delta_{2})/2+(\phi_{1}-\phi_{2})/2]$. Given that the dimensional correction terms $\delta_{i}$ are the same for $F_1$ and $F_2$, the position of nodes of the envelope line is dependent on $(\phi_{1}-\phi_{2})/2$. It is illustrated in Fig. 4(b) that a finite value of phase shift in $\cos[\pi(F_1-F_2)/B]$ is needed to correctly track the beating nodes, thus the difference of Berry phase terms for $F_1$ and $F_2$ is non-zero at this field direction. Here the absolute value $\sim$$0.32\pi$. As a deduction, at least one of the terms should have a non-zero value, possibly due to chirality of the associated Fermions. For our current angular dependent ${\rm MR}_{\rm OSC}$ of CoSi, the node positions of the envelope line with respect to field direction have a weak but observable evolution. Meanwhile the oscillation frequencies are almost constant, then following the above analysis, without invoking other trivial phase factors, the difference of the two Berry phase terms will show an angular dependence. This could serve as a constraint when analyzing how the individual Berry phase term associated with an FS changes with respect to $k$-space directions. In summary, decent quality CoSi single crystals have been grown with the aid of tin flux. QO originated from the bulk electron FSs at the $R$ point of BZ is observed in magnetoresistivity, and the additional phase in QOs arising from electron orbit geometry varies with field direction with non-zero values present. However, a full characterization of the bulk electronic structure of CoSi is still at the beginning stage. Further studies are needed to probe the electronic structure at ${\it \Gamma}$. Due to the heavier effective mass, we plan to carry out transport and magnetization measurements at lower temperature and up to higher magnetic field. During preparation of this manuscript, we become aware of a preprint by Xu et al.[29] The authors used Te flux for crystal growth and reached a better RRR for CoSi. QO was observed and yields similar frequencies as ours. We thank H. M. Weng, Y J. Sun and T. Qian for helpful discussions. References Thermoelectric Power in Transition-Metal MonosilicidesExploration of CoSi-based alloys promising for thermoelectricityTernare kohlenstofflegierungen von palladium und platin mit magnesium, aluminium, zink, gallium, germanium, kadmium, indium, zinn, quecksilber, thallium und bleiMultiple Types of Topological Fermions in Transition Metal SilicidesUnconventional Chiral Fermions and Large Topological Fermi Arcs in RhSiDouble-Weyl Phonons in Transition-Metal MonosilicidesObservation of Chiral Fermions with a Large Topological Charge and Associated Fermi-Arc Surface States in CoSiObservation of unconventional chiral fermions with long Fermi arcs in CoSiTopological chiral crystals with helicoid-arc quantum statesThe Co-Si (Cobalt-Silicon) systemSelf-doping effects in cobalt silicide CoSi: Electrical, magnetic, elastic, and thermodynamic propertiesMesures de Chaleur Specifique de Susceptibilite dt de R.M.N. dur CoSi Autour de la StoechiometrieStudying the magnetic properties of CoSi single crystalsTransport properties of cobalt monosilicide and its alloys at low temperaturesElastic, thermodynamic, and electronic properties of MnSi, FeSi, and CoSiMagnetoresistance measurements of FexCo1-xSiThermoelectric performance and electronic properties of transition metal monosilicidesSpin, charge, and bonding in transition metal mono-silicidesUltrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2The Bloch-Gruneisen function of arbitrary order and its series representationsRecent observations of negative longitudinal magnetoresistance in semimetalBand structure and unconventional electronic topology of CoSiDetection of Berry's Phase in a Bulk Rashba SemiconductorSemimetallic Properties of Co 1 − x Fe x Si Solid SolutionsQuantum Transport Evidence for the Three-Dimensional Dirac Semimetal Phase in Cd 3 As 2 Revealing the Topology of Fermi-Surface Wave Functions from Magnetic Quantum OscillationsCrystal growth and quantum oscillations in the topological chiral semimetal CoSi
[1] Sakai A, Ishii F, Onose Y, Tomioka Y, Yotsuhashi S, Adachi H, Nagaosa N and Tokura Y 2007 J. Phys. Soc. Jpn. 76 093601
[2] Longhin M, Rizza M, Viennois R and Papet P 2017 Intermetallics 88 46
[3] Bradlyn B, Cano J, Wang Z, Vergniory M G, Felser C, Cava R J and Bernevig B A 2016 Science 353 aaf5037
[4] Tang P, Zhou Q and Zhang S C 2017 Phys. Rev. Lett. 119 206402 (and the supplementary material therein)
[5] Chang G, Xu S Y, Wieder B J, Sanchez D S, Huang S M, Belopolski I, Chang T R, Zhang S, Bansil A, Lin H and Hasan M Z 2017 Phys. Rev. Lett. 119 206401
[6] Zhang T, Song Z, Alexandradinata A, Weng H, Fang C, Lu L and Fang Z 2018 Phys. Rev. Lett. 120 016401
[7] Takane D, Wang Z, Souma S, Nakayama K, Nakamura T, Oinuma H, Nakata Y, Iwasawa H, Cacho C, Kim T, Horiba K, Kumigashira H, Takahashi T, Ando Y and Sato T 2019 Phys. Rev. Lett. 122 076402
[8] Rao Z C, Li H, Zhang T T, Tian S J, Li C H and Fu B B 2019 Nature 567 496
[9] Sanchez D S, Belopolski I, Cochran T A, Xu X, Yin J X, Chang G and Xie W 2019 Nature 567 500
[10] Ishida K, Nishizawa T and Schlesinger M E 1991 J. Phase Equilib. 12 578
[11]Http://shelx.uni-goettingen.de
[12] Stishov S M, Petrova A E, Sidorov V A and Menzel D 2012 Phys. Rev. B 86 064433
[13] Amamou A, Bach P, Gautier F, Robert C and Castaing J 1972 J. Phys. Chem. Solids 33 1697
[14] Narozhnyi V N and Krasnorussky V N 2013 J. Exp. Theor. Phys. 116 780 (and reference therein)
[15] Burkov A T, Novikov S V, Zaitsev V K and Reith H 2017 Semiconductors 51 689
[16] Petrova A E, Krasnorussky V N, Shikov A A, Yuhasz W M, Lograsso T A, Lashley J C and Stishov S M 2010 Phys. Rev. B 82 155124
[17] Ohta H, Arioka T, Kulatov E, Mitsudo S and Motokawa M 1998 J. Magn. Magn. Mater. 177–181 1371
[18] Ou-Yang T Y, Shu G J and Fuh H R 2017 Europhys. Lett. 120 17002
[19] D van der Marel, Damascelli A, Schulte K and Menovsky A 1998 Physica B 244 138
[20] Liang T, Gibson Q, Ali M N, Liu M, Cava R J and Ong N P 2015 Nat. Mater. 14 280
[21] Cvijović D 2011 Theor. Math. Phys. 166 37
[22] Xu X T and Jia S 2016 Chin. Phys. B 25 117204
[23]Shoenberg D 1984 Magnetic Oscillations in Metals (Cambridge: Cambridge University)
[24] Pshenay-Severin D A, Ivanov Y V, Burkov A A and Burkov A T 2018 J. Phys.: Condens. Matter 30 135501
[25] Murakawa H, Bahramy M S, Tokunaga M, Kohama Y, Bell C, Kaneko Y, Nagaosa N, Hwang H Y and Tokura Y 2013 Science 342 1490 (and the supplementary materials therein)
[26] Asanabe S, Shinoda D and Sasaki Y 1964 Phys. Rev. 134 A774
[27] He L P, Hong X C, Dong J K, Pan J, Zhang Z, Zhang J and Li S Y 2014 Phys. Rev. Lett. 113 246402
[28] Alexandradinata A, Wang C, Duan W and Glazman L 2018 Phys. Rev. X 8 011027
[29] Xu X, Wang X, Cochran T A, Sanchez D S, Belopolski I, Wang G, Liu Y, Tien H J, Gui X, Xie W, Hasan M Z, Chang T R and Jia S 2019 arXiv:1904.00630[cond-mat.mtrl-sci]