⇦Chinese Physics Letters, 2020, Vol. 37, No. 9, Article code 090301 Butterfly-Like Anisotropic Magnetoresistance and Angle-Dependent Berry Phase in a Type-II Weyl Semimetal WP$_{2}$ Kaixuan Zhang (张凯旋)1†, Yongping Du (杜永平)2†, Pengdong Wang (王鹏栋)3, Laiming Wei (魏来明)1, Lin Li (李林)1, Qiang Zhang (张强)1, Wei Qin (秦维)1, Zhiyong Lin (林志勇)1, Bin Cheng (程斌)1, Yifan Wang (汪逸凡)1, Han Xu (徐晗)1, Xiaodong Fan (范晓东)1, Zhe Sun (孙喆)3,5, Xiangang Wan (万贤纲)4,5*, and Changgan Zeng (曾长淦)1* Affiliations 1International Center for Quantum Design of Functional Materials, Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key Laboratory of Strongly Coupled Quantum Matter Physics, Department of Physics, and Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei 230026, China 2Department of Applied Physics and Institution of Energy and Microstructure, Nanjing University of Science and Technology, Nanjing 210094, China 3National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China 4National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 5Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Received 4 June 2020; accepted 13 July 2020; published online 1 September 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11974324, 11804326, U1832151, and 11674296), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDC07010000), the National Key Research and Development Program of China (Grant No. 2017YFA0403600), the Anhui Initiative in Quantum Information Technologies (Grant No. AHY170000), the Hefei Science Center CAS (Grant No. 2018HSC-UE014), the Jiangsu Provincial Science Foundation for Youth (Grant No. BK20170821), the National Natural Science Foundation of China for Youth (Grant No. 11804160), and the Anhui Provincial Natural Science Foundation (Grant No. 1708085MF136).
†Kaixuan Zhang and Yongping Du contributed equally to this work.
*Corresponding authors. Email: xgwan@nju.edu.cn; cgzeng@ustc.edu.cn Citation Text: Zhang K X, Du Y P, Wang P D, Wei L M and Li L et al. 2020 Chin. Phys. Lett. 37 090301    Abstract The Weyl semimetal has emerged as a new topologically nontrivial phase of matter, hosting low-energy excitations of massless Weyl fermions. Here, we present a comprehensive study of a type-II Weyl semimetal WP$_{2}$. Transport studies show a butterfly-like magnetoresistance at low temperature, reflecting the anisotropy of the electron Fermi surfaces. This four-lobed feature gradually evolves into a two-lobed variant with an increase in temperature, mainly due to the reduced relative contribution of electron Fermi surfaces compared to hole Fermi surfaces for magnetoresistance. Moreover, an angle-dependent Berry phase is also discovered, based on quantum oscillations, which is ascribed to the effective manipulation of extremal Fermi orbits by the magnetic field to feel nearby topological singularities in the momentum space. The revealed topological character and anisotropic Fermi surfaces of the WP$_{2}$ substantially enrich the physical properties of Weyl semimetals, and show great promises in terms of potential topological electronic and Fermitronic device applications. DOI:10.1088/0256-307X/37/9/090301 PACS:03.65.Vf, 72.15.-v, 75.47.-m © 2020 Chinese Physics Society Article Text Type-II Weyl semimetals possess highly anisotropic electronic structures, e.g., the significantly tilted Weyl cones, and anisotropic Fermi surfaces.[1,2] Such anisotropies can be exploited to produce a wide spectrum of exciting physical properties, including the anisotropic version of the chiral anomaly,[3,4] photocurrent response,[5,6] magnetoresistance (MR),[7] and plasma mirror behavior.[8,9] Recently, WP$_{2}$ was revealed to be a new, robust type-II Weyl semimetal, having both anisotropic hole and electron Fermi surfaces.[10–13] Since MR behavior is intimately related to the topology of the Fermi surface,[14] the anisotropies of both hole and electron Fermi surfaces may collectively lead to exotic anisotropic magnetoresistance (AMR) patterns. In general, degenerate points (such as band crossings) may exist in the momentum space of a material, which act as monopoles to create Berry fields.[15] If an electron is moving in a closed orbit near a degenerate point in the momentum space, it acquires an additional nonzero Berry phase ($\varphi_{_{\rm B}}$) as the integral of the Berry field flux, i.e., Berry curvature, from nearby monopoles.[16–20] The nonzero Berry phase has been observed in graphene,[21,22] topological insulators,[23,24] Rashba semiconductors,[20] Dirac semimetals,[25] and Weyl semimetals,[26] manifesting in relation to quantum transport effects, particularly in relation to Shubnikov-de Haas (SdH) quantum oscillations.[16,20–27] For a Weyl semimetal without inversion symmetry, in addition to the Weyl nodes (the band crossing points of valence and conductance bands), other degenerate points may also occur, owing to strong spin-orbit coupling. Such topological features further enrich the fascinating physics of Weyl semimetals, presenting additional possibilities for the future design of topological electronic devices. In this work, we investigate the type-II Weyl semimetal WP$_{2}$ in relation to electronic transport. A four-lobed butterfly-like AMR emerges at low temperatures, gradually developing into a two-lobed variant as the temperature increases, a behavior primarily ascribed to anisotropic electron Fermi surfaces. Moreover, an angle-dependent Berry phase is also observed in the quantum oscillations, which is ascribed to topological singularities in the momentum space. High quality WP$_{2}$ single crystals were grown via the chemical vapor transport method, using iodine as the transport agent.[28] The P, WO$_{3}$ and $I_{2}$ sources were mixed and sealed in an evacuated quartz tube, and the WP$_{2}$ crystals were grown in a two-zone furnace with a temperature gradient from 1000 ℃ (source) to 900 ℃ (sink) for ten days. The stoichiometric ratio for W:P was confirmed to be $1\!:\!2$ by means of energy-dispersive x-ray spectroscopy. Low-field transport measurements were performed on the WP$_{2}$ crystals, employing resistivity with a rotator option in a Quantum Design physical property measurement system, where the highest magnetic field measured up to 14 T. High-field transport measurements were carried out using standard ac lock-in techniques, with a He-3 cryostat and a dc-resistive magnet ($\sim $35 T) at the High Magnetic Field Laboratory of the Chinese Academy of Sciences (CAS). In order to improve the electrical contact, Au/Ti electrodes with a thickness of 75 nm/5 nm were first deposited onto the sample by the hard-mask method, after which gold wires were employed to make contacts between the chip carrier and the Au/Ti electrodes with silver paint. As depicted in Fig. 1(a), the WP$_{2}$ crystals, grown as described above, possess an orthorhombic structure ($\beta$-phase).[29,30] Figure 1(b) shows a high-resolution transmission electron microscopy (TEM) image and a corresponding selected area electron diffraction (SAED) pattern, which illustrate the high quality of the single crystal at the atomic scale. The crystalline long axis, i.e., the direction of growth, is the [100] direction. The biggest crystal face of the as-grown WP$_{2}$ single crystals is the (021) plane (up to $3{\,\rm mm} \times 2{\,\rm mm}$), confirmed by an x-ray diffraction (XRD) measurement [Fig. 1(c)] with a small full width at half maximum (FWHM) in terms of the rocking curve ($\sim$$0.05^\circ$). The powder XRD results after single crystals ground into powders are displayed in Fig. 1(d), where the perfect match of the experimental result with the calculated pattern for $\beta$-phase WP$_{2}$ further demonstrates its high crystalline quality and purity. Figure 1(e) shows a typical STM image of the cleaved (021) surface, with a terrace height of $\sim $0.37 nm, very close to the (021) layer spacing $\sim $0.371 nm.[30] The high-resolution STM image in Fig. 1(f) shows the atomic chain structure along the [100] direction, and the white rectangular region represents a unit cell of the (021) surface.
cpl-37-9-090301-fig1.png
Fig. 1. Structure characterizations. (a) Crystallographic structure of WP$_{2}$ with a (021) surface. The blue and purple balls represent W and P atoms, respectively. The black lines indicate a unit cell. (b) TEM and SAED of a single WP$_{2}$ crystal. The scale bar is 2 nm. (c) Single-crystal XRD pattern of a WP$_{2}$ single crystal. The insets show the FWHM of the (021) peak in the rocking curve ($\sim$$0.05^\circ$), and the optical image of a shiny as-grown crystal. The scale bar is 3 mm. (d) Powder XRD pattern of ground WP$_{2}$ crystals. The measured peaks (blue) perfectly coincide with the calculated results (red). (e) STM image of the cleaved (021) surface with well-resolved terraces. The inset shows the height profile along the green dashed line. The scale bar is 10 nm. (f) STM image of the cleaved (021) surface with atomic resolution. The white solid rectangle denotes a surface unit cell. The scale bar is 1 nm.
The basic transport properties, including the temperature and magnetic field dependences of the WP$_{2}$'s resistivity, are displayed in Fig. S1 in the Supplementary Material. Here, we focus on the interesting AMR behavior of WP$_{2}$. As illustrated in Fig. 2(a), the magnetic field $B$ was rotated in the plane consisting of the [100] direction and the normal direction of the (021) surface, and the current $I$ was injected along the [100] direction. The angle between $B$ and the [100] direction is denoted as $\theta$. As shown in Figs. 2(b) and 2(c), when $B$ is rotated, the resistivity $\rho$ peaks at $\theta = 60^\circ\!$, 120$^{\circ}$, 240$^{\circ}$, and 300$^{\circ}$, forming a butterfly-like pattern with four lobes. This is in sharp contrast to the two-lobed AMR reported previously for WP$_{2}$, with $B$ rotating in the [100]–[010] or [010]–[001] planes.[11,31,32] Since the electronic transport is dominated by the carriers in the vicinity of the Fermi level,[33] the MR behavior is intimately related to the topology of the Fermi surface.[14] Theoretical calculations reveal that the Fermi surfaces of WP$_{2}$ consist of two pairs of electron and hole pockets, split by spin-orbit coupling,[9–11,31,32] as depicted in Fig. 2(d). The electron Fermi surfaces are closed, with a bow-tie-like shape, whereas the hole Fermi surfaces are open, with a spaghetti-like shape extending along the [010] direction.[11,31,32] Note that the projected electron Fermi surfaces on the rotation plane of $B$ are also bow-tie-like, as shown in Fig. 2(e), similarly to the AMR shape. As detailed in the Supplementary Material, the butterfly-like AMR is largely determined by the intrinsic anisotropy of the electron Fermi surfaces, while both anisotropic hole and electron Fermi surfaces collectively contribute to the AMR.
cpl-37-9-090301-fig2.png
Fig. 2. Butterfly-like AMR. (a) Schematic of the measurement configuration. The current $I$ is injected along the [100] direction. The magnetic field $B$ is rotated in the plane consisting of the [100] direction and the normal direction of the (021) surface. The angle between the magnetic field and the [100] direction is denoted as $\theta$. (b) Resistivity $\rho$ as a function of $\theta$ at 2 K under various magnetic fields. (c) Polar plot of $\rho$ as a function of $\theta$, illustrating a butterfly-like AMR with four lobes. (d) Calculated Fermi surfaces of WP$_{2}$. The bow-tie-like closed pockets are electron Fermi surfaces, while the spaghetti-like open pockets are hole Fermi surfaces. (e) Projected electron Fermi surfaces in the rotation plane of $B$.
Angle-dependent magnetoresistance measurements were systemically performed under various magnetic fields and temperatures, and the results are displayed in Figs. 3(a) and S2. It is evident that when the temperature rises, the butterfly-like AMR with four lobes gradually evolves into a two-lobed AMR. To quantitatively characterize this feature, the angle where the resistivity peaks ($\theta_{\max}$) and the AMR ratio defined as ${[\rho (\theta_{\max})-\rho (90^{\circ})]} / {\rho (\theta_{\max})}$ are plotted versus temperature, as shown in Fig. 3(b). We observe that $\theta_{\max}$ varies from 60$^{\circ}$ to 90$^{\circ}$ with increasing temperature, whereas the AMR ratio vanishes. Such AMR evolution upon increasing temperature is primarily due to the reduced relative contribution of electron Fermi surfaces compared to hole Fermi surfaces in relation to magnetoresistance.
cpl-37-9-090301-fig3.png
Fig. 3. Temperature-dependent evolution of the AMR. (a) Characteristics of $\rho$ as a function of $\theta$ with $B = 12$ T at various temperatures. (b) The angle of the resistivity peak ($\theta_{\max}$) and the AMR ratio ${[\rho \left(\theta_{\max} \right)-\rho (90^{\circ})]} / {\rho (\theta_{\max})}$ versus $T$.
The exploration of the exotic effects originating from particularities, such as strong anisotropy, of the Fermi surfaces, i.e., Fermitronics, is an emerging research area in the field of electronics.[34] Here, we demonstrate that the AMR effect in WP$_{2}$ can be effectively engineered by selectively tailoring the anisotropy of the carrier trajectories at the three-dimensional Fermi surfaces by, for example, varying the direction of the applied magnetic field and temperature. The nontrivial topological nature of Weyl semimetals is their most attractive property, and the Berry phase can be experimentally determined using phase information derived from SdH quantum oscillations.[18,22–30] Next, we exploit the Berry phase of WP$_{2}$, using quantum oscillation as an effective probe. Figure 4(a) shows the resistivity, $\rho$, as a function of $B$ at various $\theta$'s, with a $B$ value of up to 33 T. The magnetoresistance oscillations appear at high fields, and are attributed to SdH quantum oscillations.[20–27] The SdH oscillations originate from the quantization of the closed electronic orbits under an applied magnetic field, following the Lifshitz–Onsager quantization rule,[35–37] and incorporating the Berry phase.[15,20] To better illustrate the SdH quantum oscillations, the classical nonoscillatory background is subtracted from $\rho$[11,31,32,34] (see more details in the Supplementary Material), and the oscillatory component $\Delta \rho$ is depicted in Fig. 4(b).
cpl-37-9-090301-fig4.png
Fig. 4. Quantum oscillations. (a) Resistivity $\rho$ as a function of $B$ at 2 K for various $\theta$'s. (b) The oscillatory component $\Delta \rho$ as a function of $1/B$ after the subtraction of a smooth MR background from $\rho$. (c) FFT spectra of $\Delta \rho$ at various $\theta$'s. The dashed curves denote the calculated angular dependences of the SdH frequencies associated with the individual extremal cross-sectional orbits of the hole Fermi surface pockets ($\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, and $\beta_{2}$). (d) Magnitude of the total phase $|\gamma - \delta|$ as a function of $\theta$ for the $\alpha_{1}$ branch of the hole Fermi surfaces. The red dashed curve serves as a guide to the eyes for the evolution trend of the total phase. (e) Calculated bulk band structures of WP$_{2}$ along the $\varGamma$–$Y$–$X_{1}$–$A_{1}$ direction. The pink circles mark the band crossing points of the two valence subbands. (f) Theoretically calculated magnitude of the Berry curvature $|\varOmega|$ of the valence bands along the $\varGamma$–$Y$–$X_{1}$–$A_{1}$ direction. $|\varOmega|$ peaks at the band crossing points. (g) Schematic of the angle-dependent Berry phase in the presence of a monopole when rotating the magnetic field.
Figure 4(c) displays the fast Fourier transform (FFT) spectra of $\Delta \rho$ at different $\theta$'s. The multiple peaks in the FFT spectra can be ascribed to different extremal cross-sectional orbits of individual hole Fermi surface pockets by comparing their frequencies with the theoretically calculated values,[31] as confirmed by the Supplementary Material. The frequencies of these peaks vary with $\theta$, indicating the quasi-two-dimensional nature of the Fermi surface pockets.[31,32,34] Note that the FFT peak associated with the $\alpha_{1}$ branch of the hole Fermi surfaces splits into double peaks, which may originate from fine structures beyond the calculation resolution.[38] On the other hand, no pronounced FFT peak related to the electron Fermi surfaces is observed (for further details, see the Supplementary Material). We now focus on the Berry phase for the $\alpha_{1}$ branch of the hole Fermi surfaces, since its FFT peaks are well resolved at all measured angles. Following the methods described in previous studies,[26,39,40] the SdH oscillation components associated with the double FFT peaks of the $\alpha_{1}$ branch are extracted, and the corresponding total phase magnitude $\left| \gamma -\delta \right|$ is derived from the two-band Lifshitz–Kosevich fitting [see Figs. 4(d) and S3, and additional details in the Supplementary Material]. Here $\gamma =1/2-\varphi_{_{\rm B}}/2\pi$ is the Onsager phase, and the phase shift $\delta$ equals zero for two-dimensional (2D) or $\pm$1/8 for three-dimensional (3D) cases.[26,39,40] We observe that $\left| \gamma -\delta \right|$ drops from $\sim $0.8 at 60$^{\circ}$ to $\sim $0.07 at 91.5$^{\circ}$ before increasing once more to $\sim $0.6 near 120$^{\circ}$, indicating a highly tunable Berry phase. In particular, $\left| \gamma -\delta \right|$ is almost zero at 91.5$^{\circ}$, suggesting a Berry phase close to $\pi$ [a $\pi$ Berry phase leads to $\left| \gamma -\delta \right| = 0$ (2D) or $\left| \gamma -\delta \right| = 1/8$ (3D)].[26,39,40] A similar angle-dependent Berry phase has also been recently reported in ZrSiS,[34] although the underlying mechanism is unclear. In relation to this work, such behavior can be attributed to the shape of the extremal Fermi orbit, perpendicular to the applied magnetic field $B$ and its relative position with respect to the Berry curvature monopoles. Based on our theoretical calculations, there are quite a few crossing points for the two valence subbands of WP$_{2}$, some of which, relating to the high-symmetry line, are depicted in Fig. 4(e). As shown in Fig. 4(f), the Berry curvature sharply peaks at these band crossing points, which can be theoretically regarded as monopoles, serving as the source or sink of the Berry field or Berry curvature in the momentum space.[15,17] Although the Berry curvature is generated by these monopoles, located at several hundreds of meV below the Fermi level, as marked in Fig. 4(e), the corresponding Berry connection tends to extend through entire energy bands.[27] In the presence of an external magnetic field, carriers near the Fermi surface move along the extremal cross-sectional orbit, accumulating a Berry phase equal to the path integral of the Berry connection.[16–20] When the magnetic field is rotated, the shape of the extremal cross-sectional orbit and its relative position with respect to these monopoles vary accordingly, thereby altering the path integral of the Berry connection along the orbit, i.e., the Berry phase. Fig. 4(g) depicts a simplified case with only a single monopole: when the extremal cross-sectional orbit of carriers perpendicular to $B$ happens to enclose the monopole, the corresponding Berry phase equals $\pi$.[16,17] While $B$ is rotated to generate an extremal cross-sectional orbit away from the monopole, the Berry phase gradually decreases. It is well established that the nonzero Berry phase of Fermi surfaces may induce exotic effects, e.g., the half-integer quantum Hall effect[21,22] and the valley Hall effect[41] in 2D graphene. In contrast to 2D systems, the Berry phase for carriers moving in varying closed orbits in 3D systems can effectively be tuned, as demonstrated in this work. This high tunability of the Berry phase in 3D systems paves the way to controlling interesting topological quantum effects. In summary, electronic transport reveals a butterfly-like AMR at low temperatures, and its temperature-dependent evolution, mainly arising from anisotropic electron Fermi surfaces. Moreover, an angle-dependent Berry phase is demonstrated, based on SdH quantum oscillations, which may be attributed to topological singularities in the momentum space. Our findings not only provide a deeper insight into the topological and Fermi surface properties of Weyl semimetals, but also promise potential applications in corresponding topological electronic and Fermitronic devices. A portion of this work was performed using the Steady High Magnetic Field Facilities, High Magnetic Field Laboratory, CAS. We thank Li Pi and Chuanying Xi for their experimental support and helpful discussions. References Type-II Weyl semimetalsExperimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe2Gate-tunable negative longitudinal magnetoresistance in the predicted type-II Weyl semimetal WTe2Experimental Observation of Anisotropic Adler-Bell-Jackiw Anomaly in Type-II Weyl Semimetal WTe 1.98 Crystals at the Quasiclassical RegimeNonlinear photoresponse of type-II Weyl semimetalsRobust edge photocurrent response on layered type II Weyl semimetal WTe2Magnetotransport properties of the type-II Weyl semimetal candidate Ta 3 S 2 Anisotropic electrodynamics of type-II Weyl semimetal candidate WTe 2 Room-Temperature Anisotropic Plasma Mirror and Polarization-Controlled Optical Switch Based on Type-II Weyl Semimetal WP 2 Robust Type-II Weyl Semimetal Phase in Transition Metal Diphosphides X P 2 ( X = Mo , W)Extremely high magnetoresistance and conductivity in the type-II Weyl semimetals WP2 and MoP2Stable Weyl points, trivial surface states, and particle-hole compensation in WP 2 Observation of Weyl Nodes in Robust Type-II Weyl Semimetal WP 2 Topology and solid-state physics (Review)Quantal Phase Factors Accompanying Adiabatic ChangesManifestation of Berry's Phase in Metal PhysicsThe Anomalous Hall Effect and Magnetic Monopoles in Momentum SpaceBerry Phase and de Haas–van Alphen Effect in L a R h I n 5 First Principles Calculation of Anomalous Hall Conductivity in Ferromagnetic bcc FeDetection of Berry's Phase in a Bulk Rashba SemiconductorTwo-dimensional gas of massless Dirac fermions in grapheneExperimental observation of the quantum Hall effect and Berry's phase in grapheneGate-tuned normal and superconducting transport at the surface of a topological insulatorQuantum Oscillations and Hall Anomaly of Surface States in the Topological Insulator Bi2Te3Anisotropic Fermi Surface and Quantum Limit Transport in High Mobility Three-Dimensional Dirac Semimetal Cd 3 As 2 π Berry phase and Zeeman splitting of Weyl semimetal TaPBerry phase effects on electronic propertiesReduction of WO3 by Phosphorus.X-Ray Studies of Molybdenum and Tungsten Phosphides.�ber Polyphosphide von Chrom, Molybd�n und WolframFermi surface of the Weyl type-II metallic candidate WP 2 Large magnetoresistance in the type-II Weyl semimetal WP 2 Butterfly magnetoresistance, quasi-2D Dirac Fermi surface and topological phase transition in ZrSiSInterpretation of the de Haas-van Alphen effectNegative magnetoresistance without well-defined chirality in the Weyl semimetal TaPPressure-induced topological phase transitions and strongly anisotropic magnetoresistance in bulk black phosphorusUnusual magnetotransport in holmium monoantimonideDetecting topological currents in graphene superlattices
[1] Soluyanov A A, Gresch D, Wang Z, Wu Q, Troyer M, Dai X and Bernevig B A 2015 Nature 527 495
[2] Deng K, Wan G, Deng P, Zhang K, Ding S, Wang E, Yan M, Huang H, Zhang H, Xu Z, Denlinger J, Fedorov A, Yang H, Duan W, Yao H, Wu Y, Fan S, Zhang H, Chen X and Zhou S 2016 Nat. Phys. 12 1105
[3] Wang Y, Liu E, Liu H, Pan Y, Zhang L, Zeng J, Fu Y, Wang M, Xu K, Huang Z, Wang Z, Lu H Z, Xing D, Wang B, Wan X and Miao F 2016 Nat. Commun. 7 13142
[4] Lv Y Y, Li X, Zhang B B, Deng W Y, Yao S H, Chen Y B, Zhou J, Zhang S T, Lu M H, Zhang L, Tian M, Sheng L and Chen Y F 2017 Phys. Rev. Lett. 118 096603
[5] Ma J, Gu Q, Liu Y, Lai J, Yu P, Zhuo X, Liu Z, Chen J H, Feng J and Sun D 2019 Nat. Mater. 18 476
[6] Wang Q, Zheng J, He Y, Cao J, Liu X, Wang M, Ma J, Lai J, Lu H, Jia S, Yan D, Shi Y, Duan J, Han J, Xiao W, Chen J H, Sun K, Yao Y and Sun D 2019 Nat. Commun. 10 5736
[7] Chen D, Zhao L X, He J B, Liang H, Zhang S, Li C H, Shan L, Wang S C, Ren Z A, Ren C and Chen G F 2016 Phys. Rev. B 94 174411
[8] Frenzel A J, Homes C C, Gibson Q D, Shao Y M, Post K W, Charnukha A, Cava R J and Basov D N 2017 Phys. Rev. B 95 245140
[9] Zhang K, Du Y, Qi Z, Cheng B, Fan X, Wei L, Li L, Wang D, Yu G, Hu S, Sun C, Huang Z, Chu J, Wan X and Zeng C 2020 Phys. Rev. Appl. 13 014058
[10] Autes G, Gresch D, Troyer M, Soluyanov A A and Yazyev O V 2016 Phys. Rev. Lett. 117 066402
[11] Kumar N, Sun Y, Xu N, Manna K, Yao M, Suss V, Leermakers I, Young O, Forster T, Schmidt M, Borrmann H, Yan B, Zeitler U, Shi M, Felser C and Shekhar C 2017 Nat. Commun. 8 1642
[12] Razzoli E, Zwartsenberg B, Michiardi M, Boschini F, Day R P, Elfimov I S, Denlinger J D, Süss V, Felser C and Damascelli A 2018 Phys. Rev. B 97 201103(R)
[13] Yao M Y, Xu N, Wu Q S, Autes G, Kumar N, Strocov V N, Plumb N C, Radovic M, Yazyev O V, Felser C, Mesot J and Shi M 2019 Phys. Rev. Lett. 122 176402
[14] Kosevich A M 2004 Low Temp. Phys. 30 97
[15] Berry M V 1984 Proc. R. Soc. London A 392 45
[16] Mikitik G P and Sharlai Y V 1999 Phys. Rev. Lett. 82 2147
[17] Fang Z, Nagaosa N, Takahashi K S, Asamitsu A, Mathieu R, Ogasawara T, Yamada H, Kawasaki M, Tokura Y and Terakura K 2003 Science 302 92
[18] Mikitik G and Sharlai Y V 2004 Phys. Rev. Lett. 93 106403
[19] Yao Y, Kleinman L, MacDonald A, Sinova J, Jungwirth T, Wang D S, Wang E and Niu Q 2004 Phys. Rev. Lett. 92 037204
[20] 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
[21] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197
[22] Zhang Y, Tan Y W, Stormer H L and Kim P 2005 Nature 438 201
[23] Sacepe B, Oostinga J B, Li J, Ubaldini A, Couto N J, Giannini E and Morpurgo A F 2011 Nat. Commun. 2 575
[24] Qu D X, Hor Y S, Xiong J, Cava R J and Ong N P 2010 Science 329 821
[25] Zhao Y, Liu H, Zhang C, Wang H, Wang J, Lin Z, Xing Y, Lu H, Liu J, Wang Y, Brombosz S M, Xiao Z, Jia S, Xie X C and Wang J 2015 Phys. Rev. X 5 031037
[26] Hu J, Liu J Y, Graf D, Radmanesh S M, Adams D J, Chuang A, Wang Y, Chiorescu I, Wei J, Spinu L and Mao Z Q 2016 Sci. Rep. 6 18674
[27] Xiao D, Chang M C and Niu Q 2010 Rev. Mod. Phys. 82 1959
[28] Mathis H, Glaum R and Gruehn R 1991 Acta Chem. Scand. 45 781
[29] Rundqvist S and Lundstrom T 1963 Acta Chem. Scand. 17 37
[30] Rühl R and Jeitschko W 1983 Monatsh. Chem. - Chem. Mon. 114 817
[31] Schönemann R, Aryal N, Zhou Q, Chiu Y C, Chen K W, Martin T J, McCandless G T, Chan J Y, Manousakis E and Balicas L 2017 Phys. Rev. B 96 121108(R)
[32] Wang A, Graf D, Liu Y, Du Q, Zheng J, Lei H and Petrovic C 2017 Phys. Rev. B 96 121107(R)
[33]Ashcroft N and Mermin N 1976 Solid State Physics (San Diego: Harcourt College Publisher)
[34] Ali M N, Schoop L M, Garg C, Lippmann J M, Lara E, Lotsch B and Parkin S S 2016 Sci. Adv. 2 e1601742
[35] Onsager L 1952 Philos. Mag. 43 1006
[36]Lifshitz I M and Kosevich A M 1956 Sov. Phys.-JETP 2 636
[37]Lifshitz I M and Kosevich A M 1958 Sov. Phys.-JETP 6 67
[38] Arnold F, Shekhar C, Wu S C, Sun Y, Dos Reis R D, Kumar N, Naumann M, Ajeesh M O, Schmidt M, Grushin A G, Bardarson J H, Baenitz M, Sokolov D, Borrmann H, Nicklas M, Felser C, Hassinger E and Yan B 2016 Nat. Commun. 7 11615
[39] Li C H, Long Y J, Zhao L X, Shan L, Ren Z A, Zhao J Z, Weng H M, Dai X, Fang Z, Ren C and Chen G F 2017 Phys. Rev. B 95 125417
[40] Wang Y Y, Sun L L, Xu S, Su Y and Xia T L 2018 Phys. Rev. B 98 045137
[41] Gorbachev R V, Song J C W, Yu G L, Kretinin A V, Withers F, Cao Y, Mishchenko A, Grigorieva I V, Novoselov K S, Levitov L S and Geim A K 2014 Science 346 448