⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 067201 Contactless Microwave Detection of Shubnikov–De Haas Oscillations in Three-Dimensional Dirac Semimetal ZrTe$_{5}$ * Min Wu (武敏)1,2†, Hongwei Zhang (张红伟)1,3†, Xiangde Zhu (朱相德)1, Jianwei Lu (陆建伟)1,2, Guolin Zheng (郑国林)1,4, Wenshuai Gao (高文帅)5, Yuyan Han (韩玉岩)1, Jianhui Zhou (周建辉)1, Wei Ning (宁伟)1**, Mingliang Tian (田明亮)1,5 Affiliations 1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031 2Department of Physics, University of Science and Technology of China, Hefei 230026 3Department of Physics, Shaanxi University of Science and Technology, Xi'an 710021 4School of Science, RMIT University, Melbourne, VIC 3001, Australia 5Department of Physics, School of Physics and Materials Science, Anhui University, Hefei 230601 Received 24 January 2019, online 18 May 2019 *Supported by the National Key Research and Development Program of China under Grant No 2016YFA0401003, the National Natural Science Foundation of China under Grant Nos 11774353, 11574320, 11374302, 11804340, U1432251 and U1732274, the Innovative Program of Development Foundation of Hefei Center for Physical Science and Technology under Grant No 2018CXFX002, and the China Postdoctoral Science Foundation under Grant No 2018M630718.
†Min Wu and Hongwei Zhang contributed to this work equally.
**Corresponding author. Email: ningwei@hmfl.ac.cn Citation Text: Wu M, Zhang H W, Zhu X D, Lu J W and Zheng G L et al 2019 Chin. Phys. Lett. 36 067201    Abstract We report Shubnikov–de Haas (SdH) oscillations of a three-dimensional (3D) Dirac semimetal candidate of layered material ZrTe$_{5}$ single crystals through contactless electron spin resonance (ESR) measurements with the magnetic field up to 1.4 T. The ESR signals manifest remarkably anisotropic characteristics with respect to the direction of the magnetic field, indicating an anisotropic Fermi surface in ZrTe$_{5}$. Further experiments demonstrate that the ZrTe$_{5}$ single crystals have the signature of massless Dirac fermions with nontrivial $\pi$ Berry phase, key evidence for 3D Dirac/Weyl fermions. Moreover, the onset of quantum oscillation of our ZrTe$_{5}$ crystals revealed by the ESR can be derived down to 0.2 T, much smaller than the onset of SdH oscillation determined by conventional magnetoresistance measurements. Therefore, ESR measurement is a powerful tool to study the topologically nontrivial electronic structure in Dirac/Weyl semimetals and other topological materials with low bulk carrier density. DOI:10.1088/0256-307X/36/6/067201 PACS:72.15.Gd, 71.55.Ak, 03.65.Vf © 2019 Chinese Physics Society Article Text Weyl semimetal, a three-dimensional (3D) gapless topological phase of matter, has discretely crossing points (called Weyl nodes) with linear dispersion in momentum space.[1–3] In the vicinity of each Weyl node, the electrons behave as massless Weyl fermions with definite chiralities. Due to the nontrivial topology of Weyl nodes, many exotic transport properties have been observed in Weyl semimetals, such as ultrahigh carrier mobility,[4–8] negative longitudinal magnetoresistance and planar Hall effect induced by the chiral anomaly,[6,9–16] anomalous Hall effect and anomalous Nernst effect,[17–20] as well as surface Fermi-arc state-mediated quantum oscillations.[21–23] A fundamental characteristic of Weyl fermions is the nontrivial $\pi$ Berry phase, which usually manifests itself in many physical phenomena,[24] such as de Haas–van Alphen oscillations and Shubnikov–de Haas (SdH) oscillations, and both quantum oscillations have been widely employed to investigate the topologically nontrivial features of topological materials.[4–6,12,14,25–29] Recently, spectroscopy experiments, such as angle-resolved photoemission spectroscopy,[13] magnetoinfrared spectroscopy,[30] and optical spectroscopy,[31] have demonstrated that layered material ZrTe$_{5}$ is a Dirac semimetal. Furthermore, the nontrivial $\pi$ Berry phase was revealed by magnetotransport studies of SdH oscillations, which presented clear evidence of the existence of relativistic Dirac fermions in the bulk states of ZrTe$_{5}$.[14] However, these conventional magnetotransport measurements of conductivities require effort to prepare ohmic contacts, which may contaminate the samples and degrade the carrier mobility. Therefore, contactless measurement techniques with the observable signatures of quantum oscillations in topological semimetals are highly desirable. Here we investigate the SdH oscillations in topological semimetals by measuring the derivative of the microwave-cavity quality factor $Q$ with respect to the magnetic field in the electron spin resonance (ESR) spectrometer.[32–37] More importantly, since the ESR signals are directly proportional to the derivative of the conductivity,[33,35] $dQ/dB=A\ast d\sigma/dB$, where $\sigma$ is the electrical conductivity and $A$ is a material-dependent parameter, the ESR is more sensitive to the change of the density of states at the Fermi level. In this work, we present the SdH oscillations in 3D Dirac semimetal of layered ZrTe$_{5}$ single crystals by means of ESR measurements with the magnetic field up to 1.4 T. The significant anisotropy of angular dependence of SdH oscillations strongly indicates that the oscillations stem from the 3D bulk states with an anisotropic Fermi surface. Furthermore, the nontrivial $\pi$ Berry phase revealed by analyzing the Landau level (LL) fan diagram clearly demonstrates the existence of Dirac fermions in the bulk states of ZrTe$_{5}$. Our results provide an alternative and effective route to investigate the topologically nontrivial electronic structures in Dirac/Weyl semimetals with low bulk carrier density. ZrTe$_{5}$ single crystals were grown by the iodine vapor transport method with elements Zr (99.99%) and Te (99.99%) as reported in Ref.  [38]. The crystal structure of ZrTe$_{5}$ has an orthorhombic layered structure with $a=0.40$ nm, $b=1.44$ nm, $c=1.38$ nm and space group $Cmcm$ ($D_{2h}^{17}$),[39] as shown in Fig. 1. In this layered structure, the trigonal prismatic ZrTe$_{3}$ chains run along the $a$-axis and are linked along the $c$-axis by zigzag chains of Te atoms to form a 2D $a$–$c$ plane, and are then stacked along the $b$-axis into a crystal.[40] Both high-resolution transmission electron microscopy and selected area electron diffraction studies of ZrTe$_{5}$ nanoflakes exfoliated from single crystals indicate a high crystalline quality of the grown ZrTe$_{5} $.[41] The ESR measurements were carried out in the $X$ band (9.4 GHz) with a TE$_{102}$ resonance cavity with magnetic field up to 1.4 T. The temperature in the resonance cavity can be lowered down to 2 K using an Oxford continuous-flow cryostat.
cpl-36-6-067201-fig1.png
Fig. 1. The crystal structure of ZrTe$_{5}$ is an orthorhombic layered structure with space group $Cmcm$ ($D_{2h}^{17}$). The corresponding lattice constants are $a=0.40$ nm, $b=1.44$ nm, and $c=1.38$ nm.
The angle-dependent quantum oscillations can be used to determine the geometry of the Fermi surface in ZrTe$_{5}$. Figure 2 shows pronounced SdH oscillations extracted from the ESR spectra at $T=2$ K. The angular dependence of the ESR spectra of S1 by rotating the magnetic field $B$ in the $a$–$b$ plane is presented in Fig. 2(a), where $\theta$ is the titled angle between the direction of $B$ and the $b$-axis. For $\theta =0^{\circ}$, the SdH oscillations can be tracked to a field as low as $B=0.2$ T, which is much lower than that required in the conventional magnetotransport measurement.[14] With increasing the titled angle $\theta$, the amplitude of the oscillations decreases gradually and ultimately vanishes at $\theta =70^{\circ}$. It is worth pointing out that the amplitude of the oscillation signatures of the ESR spectra increases initially and then decreases with the increase of the magnetic field $B$ at a fixed $\theta$, as shown in Fig. 2(a). It dramatically differs from the monotonous increase of the amplitude of SdH oscillations before the magnetic field reaches the quantum limit in the conventional magnetoresistance measurements.[4–7,12,14,25,26] This may be due to the different responses of the longitudinal magnetoresistivity $\rho_{xx}$ or the longitudinal magnetoconductivity $\sigma_{xx}$ to the externally applied magnetic field.[34] Similar behavior is also observed in S2, where the magnetic field is rotated in the $b$–$c$ plane, as shown in Fig. 2(c). The out-of-plane oscillation components of ESR spectra at different angles in S1 and S2 are plotted in Figs. 2(b) and 2(d), respectively. It is clearly seen that the positions of oscillation peaks shift continuously with the increase of the tilted angle $\theta$, as shown by the dashed lines, indicating that the SdH oscillations possess a 3D nature. Since the surface Fermi arc states hardly form a closed orbit in single crystal with a thickness of about hundreds of micrometers,[21] the SdH oscillations should be ascribed to the 3D bulk states.
cpl-36-6-067201-fig2.png
Fig. 2. The angular dependence of SdH oscillations measured at $T=2$ K by ESR in $X$ band (9.4 GHz) with a TE$_{102}$ resonance cavity. (a) The observed ESR spectra by rotating the magnetic field $B$ in $a$–$b$ plane of S1. Tilting the angle $\theta$ weakens the SdH oscillations. The amplitude of oscillations increases initially and then decreases with the magnetic field at a given angle $\theta$. Inset: the schematic structure for the angle-dependent ESR measurements. (b) The out-of-plane components of SdH oscillations at different angles. The shift of peak positions of the oscillations with $\theta$, shown as the black dashed line, indicates a 3D nature of the SdH oscillations. (c), (d) Similar behaviors are also observed in S2.
This 3D feature of SdH oscillations in these two samples is further confirmed by the angular dependence of the oscillation frequency derived from fast Fourier transform (FFT) analysis, as shown in Figs. 3(a) and 3(b). It is clear that the oscillation frequencies at different angles do not exactly follow the $1/\cos \theta$ or $1/\cos \varphi$ rule. Therefore, the significantly anisotropic oscillations measured by ESR with respect to the direction of the magnetic field strongly indicate an anisotropic Fermi surface in ZrTe$_{5}$, which is consistent with previous magnetotransport measurements.[14] Note that a single oscillation frequency is identified to be $F=4.86$ T and $F=5.36$ T with the magnetic field $B$ parallel to the $b$-axis at $T=2$ K, as shown in the inset of Figs. 3(a) and 3(b) for S1 and S2, respectively. According to the Onsager relation, $F=\frac{\hslash }{2\pi e}A_{\rm F}$, the cross-section of the Fermi surface is about $A_{\rm F}=4.91\times {10}^{-4}$ Å$^{-2}$ and $A_{\rm F}=5.42\times {10}^{-4}$ Å$^{-2}$ with the Fermi wave vector estimated to be $k_{\rm F}=0.0116$ Å$^{-1}$ and $k_{\rm F} =0.0131$ Å$^{-1} $ for S1 and S2, respectively. Such a small Fermi wave vector implies that the Fermi energy is close to the Dirac points.
cpl-36-6-067201-fig3.png
Fig. 3. The oscillation frequency $F$(T) as a function of $\theta$ ($\varphi $) in S1 (S2) at 2 K. The oscillation frequency deviated from the $1/\cos(\theta)$ and $1/\cos(\varphi)$ relation with $B$ rotating in $a$–$b$ and $b$–$c$ planes, respectively. Inset: the FFT spectra of SdH oscillations with the magnetic field along the $b$-axis.
To obtain more information about the Fermi surface, we carried out extra ESR measurements at different temperatures of S3 with the magnetic field $B$ aligned along the $b$-axis, as shown in Fig. 4(a). After subtracting the smooth background, the oscillation components as a function of $1/B$ at various temperatures are depicted in Fig. 4(b). The temperature dependence of the oscillation amplitudes can be described by[32] $$ \frac{A(T)}{A(T_{0})}=\frac{T\sinh (\alpha T_{0}m^{\ast }/{Bm_{\rm e}})}{T_{0}\sinh ( {\alpha Tm^{\ast }}/{Bm_{\rm e}})}, $$ where $\alpha ={2\pi^{2}k_{\rm B}m_{\rm e}}/{\hslash e}$, with $k_{\rm B}$ the Boltzmann constant, $m_{\rm e}$ the electron rest mass, $\hslash$ the reduced Planck's constant, $T_{0}=2$ K the relative temperature for normalization, and $m^{\ast}$ the effective cyclotron mass of the electron. As shown in the inset of Fig. 4(b), the solid red line is the best fit to the oscillation amplitudes, which yields $m^{\ast }=0.011 m_{\rm e}$. The Fermi wave vector in S3, derived from the oscillation frequency $F=5.15$ T, as shown in the inset of Fig. 4(a), is about $k_{\rm F}=0.0129$ Å$^{-1}$. Consequently, the Fermi velocity is estimated to be $v_{\rm F}={\hslash k_{\rm F}}/m^{\ast}=1.35\times {10}^{6}$ m/s, which is consistent with the spectroscopy experiments.[13,30] Correspondingly, the Fermi energy is determined to be $E_{\rm F}=\hslash v_{\rm F}k_{\rm F}=110$ meV. It should be noted that the effective mass $m^{\ast}$ is smaller than that from magnetotransport measurement ($m^{\ast }=0.026 m_{\rm e}$),[14] while the oscillation frequency $F$(T) is slightly larger ($F$(T)=3.76 T in Ref.  [14]). These differences in the effective mass and oscillation frequency may originate from different carrier densities.
cpl-36-6-067201-fig4.png
Fig. 4. The SdH oscillations at different temperatures with the magnetic field along the $b$-axis in S3. (a) With increasing the temperature, the SdH oscillations decrease gradually. Inset: the FFT spectra of SdH oscillations at $T=2$ K. (b) The oscillation component obtained after subtracting a smooth background, as a function of $1/B$ at various temperatures. Inset: the temperature-dependent relative amplitude of the SdH oscillations for the sixth LL. The red curve is the theoretical fitting, leading to an effective mass $m^{\ast }=0.011m_{\rm e}$.
cpl-36-6-067201-fig5.png
Fig. 5. The Landau index $n$ versus $1/B$ at 2 K with the magnetic field parallel to the $b$-axis. The peak positions of SdH oscillations are defined as integer indices. Non-zero intercepts with the $n$-axis obtained by the best linear fit (the red curve(s)) indicate a nontrivial $\pi$ Berry phase in ZrTe$_{5}$ single crystals.
The nontrivial $\pi$ Berry phase is remarkable evidence of Dirac fermions with linear dispersion and has been widely revealed through quantum oscillation spectra in topological materials, such as topological insulators[25,26] and topological semimetals.[4–6,12,14,27–29] According to the Lifshitz–Kosevich formula,[42,43] the oscillation components are proportional to $\cos[2\pi(F/B+\gamma +\delta)]$ with $\gamma =\frac{1}{2}+\beta$, where $F$ is the oscillation frequency, $B$ is the magnetic field, $\beta 2\pi$ is the $\pi$ Berry phase and $\delta 2\pi$ is the additional phase shift with $\delta$ determined by the dimensionality of the Fermi surface and changed from 0 for 2D to $\pm 1/8$ for 3D. The LL fan diagram is plotted in Fig. 5, and here the integer LL index $n$ is assigned to the peak positions of the oscillation.[32,33] The Lifshitz–Onsager quantization rule, $A_{\rm F}\frac{\hslash }{eB}=2\pi (n+\gamma +\delta)$, implies that the LL index $n$ is linearly dependent on $1/B$. As clearly shown in Fig. 5, the best linear fit of the data generates a non-zero intercept on the $n$-axis for these three samples, which unambiguously demonstrates that the nontrivial $\pi$ Berry phase exists in our ZrTe$_{5}$ single crystals and provides strong evidence of the 3D Dirac phase in ZrTe$_{5}$. Note that, for a clear observation of the quantum oscillation through ESR, one usually needs materials with low carrier density. We did not observe any quantum oscillation in high-mobility Cd$_{3}$As$_{2}$ single crystal through ESR measurement, probably due to the relative high carrier density (${10}^{18}$ cm$^{-3}$) in the bulk Cd$_{3}$As$_{2}$,[5] which screens the microwave penetrating into the bulk. In conclusion, we have investigated SdH oscillations of layered material ZrTe$_{5}$ single crystals by contactless ESR measurements in the microwave cavity. By analyzing the SdH oscillations of the ESR signals at 2 K, a nontrivial $\pi$ Berry phase together with an anisotropic 3D Fermi surface is revealed in ZrTe$_{5}$. Therefore, ESR measurement is an effective tool to study topologically nontrivial electronic structures in topological materials at a low magnetic field. References Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridatesTopological semimetals predicted from first-principles calculationsTopological semimetalsQuantum Transport Evidence for the Three-Dimensional Dirac Semimetal Phase in Cd 3 As 2 Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2Observation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 3D Weyl Semimetal TaAsExtremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbPLarge magnetoresistance in compensated semimetals TaAs 2 and NbAs 2 Evidence for the chiral anomaly in the Dirac semimetal Na3BiGiant negative magnetoresistance induced by the chiral anomaly in individual Cd3As2 nanowiresNegative magnetoresistance in Dirac semimetal Cd3As2Signatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetalChiral magnetic effect in ZrTe5Transport evidence for the three-dimensional Dirac semimetal phase in ZrT e 5 Probing the chiral anomaly by planar Hall effect in Dirac semimetal Cd 3 As 2 nanoplatesGiant anisotropic magnetoresistance and planar Hall effect in the Dirac semimetal Cd 3 As 2 Weyl Semimetal in a Topological Insulator MultilayerChern Semimetal and the Quantized Anomalous Hall Effect in HgCr 2 Se 4 Anomalous Nernst Effect in the Dirac Semimetal Cd 3 As 2 Anomalous Hall effect in ZrTe5Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetalsTransport evidence for Fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2Recognition of Fermi-arc states through the magnetoresistance quantum oscillations in Dirac semimetal C d 3 A s 2 nanoplatesManifestation of Berry's Phase in Metal PhysicsQuantum Oscillations and Hall Anomaly of Surface States in the Topological Insulator Bi2Te3Two-dimensional surface state in the quantum limit of a topological insulatorProbing the Fermi surface of three-dimensional Dirac semimetal Cd 3 As 2 through the de Haas–van Alphen techniqueEvidence of Topological Nodal-Line Fermions in ZrSiSe and ZrSiTeA magnetic topological semimetal Sr1−yMn1−zSb2 (y, z < 0.1)Magnetoinfrared Spectroscopy of Landau Levels and Zeeman Splitting of Three-Dimensional Massless Dirac Fermions in ZrTe 5 Optical spectroscopy study of the three-dimensional Dirac semimetal ZrTe 5 Microwave detection of Shubnikov–de Haas oscillations in In 0.53 Ga 0.47 As/InP single quantum wellsApplication of microwave detection of the Shubnikov–de Haas effect in two‐dimensional systemsCarrier‐modulated, microwave‐detected Shubnikov–de Haas oscillations in two‐dimensional systemsContactless microwave studies of weak localization in epitaxial grapheneLandau-Level Spectroscopy of Relativistic Fermions with Low Fermi Velocity in the Bi 2 Te 3 Three-Dimensional Topological InsulatorThe oscillations in ESR spectra of Hg 0.76 Cd 0.24 Te implanted by Ag + at the X and Q-bandsFermi surface, effective masses, and Dingle temperatures of ZrTe 5 as derived from the Shubnikov – de Haas effectTransition-Metal Pentatelluride ZrTe 5 and HfTe 5 : A Paradigm for Large-Gap Quantum Spin Hall InsulatorsObservation of a semimetal–semiconductor phase transition in the intermetallic ZrTe 5Field-induced topological phase transition from a three-dimensional Weyl semimetal to a two-dimensional massive Dirac metal in ZrT e 5 Detection of Berry's Phase in a Bulk Rashba Semiconductor
[1] Wan X G, Turner A M, Vishwanath A and Savrasov S Y 2011 Phys. Rev. B 83 205101
[2] Weng H M, Dai X and Fang Z 2016 J. Phys.: Condens. Matter 28 303001
[3] Burkov A A 2016 Nat. Mater. 15 1145
[4] 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
[5] Liang T, Gibson Q, Ali M N, Liu M, Cava R J and Ong N P 2015 Nat. Mater. 14 280
[6] Huang X C, Zhao L X, Long Y J, Wang P P, Chen D, Yang Z H, Liang H, Xue M Q, Weng H M, Fang Z, Dai X and Chen G F 2015 Phys. Rev. X 5 031023
[7] Shekhar C, Nayak A K, Sun Y, Schmidt M, Nicklas M, Leermakers I, Zeitler U, Skourski Y, Wosnitza J, Liu Z, Chen Y, Schnelle W, Borrmann H, Grin Y, Felser C and Yan B 2015 Nat. Phys. 11 645
[8] Yuan Z J, Lu H, Liu Y J, Wang J F and Jia S 2016 Phys. Rev. B 93 184405
[9] Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J and Ong N P 2015 Science 350 413
[10] Li C Z, Wang L X, Liu H, Wang J, Liao Z M and Yu D P 2015 Nat. Commun. 6 10137
[11] Li H, He H, Lu H Z, Zhang H, Liu H, Ma R, Fan Z, Shen S Q and Wang J 2016 Nat. Commun. 7 10301
[12] Zhang C L, Xu S Y, Belopolski I, Yuan Z, Lin Z, Tong B, Bian G, Alidoust N, Lee C C, Huang S M, Chang T R, Chang G, Hsu C H, Jeng H T, Neupane M, Sanchez D S, Zheng H, Wang J, Lin H, Zhang C, Lu H Z, Shen S Q, Neupert T, Hasan M Z and Jia S 2016 Nat. Commun. 7 10735
[13] Li Q, Kharzeev D E, Zhang C, Huang Y, Pletikosić I, Fedorov A V, Zhong R D, Schneeloch J A, Gu G D and Valla T 2016 Nat. Phys. 12 550
[14] Zheng G L, Lu J W, Zhu X D, Ning W, Han Y Y, Zhang H W, Zhang J L, Xi C Y, Yang J Y, Du H F, Yang K, Zhang Y H and Tian M L 2016 Phys. Rev. B 93 115414
[15] Wu M, Zheng G L, Chu W W, Liu Y Q, Gao W S, Zhang H W, Lu J W, Han Y Y, Zhou J H, Ning W and Tian M L 2018 Phys. Rev. B 98 161110(R)
[16] Li H, Wang H W, He H, Wang J and Shen S Q 2018 Phys. Rev. B 97 201110(R)
[17] Burkov A A and Balents L 2011 Phys. Rev. Lett. 107 127205
[18] Xu G, Weng H M, Wang Z J, Dai X and Fang Z 2011 Phys. Rev. Lett. 107 186806
[19] Liang T, Lin J, Gibson Q, Gao T, Hirschberger M, Liu M, Cava R J and Ong N P 2017 Phys. Rev. Lett. 118 136601
[20] Liang T, Lin J, Gibson Q, Kushwaha S, Liu M, Wang W, Xiong H, Sobota J A, Hashimoto M, Kirchmann P S, Shen Z X, Cava R J and Ong N P 2018 Nat. Phys. 14 451
[21] Potter A C, Kimchi I and Vishwanath A 2014 Nat. Commun. 5 5161
[22] Moll P J W, Nair N L, Helm T, Potter A C, Kimchi I, Vishwanath A and Analytis J G 2016 Nature 535 266
[23] Zheng G L, Wu M, Zhang H W, Chu W W, Gao W S, Lu J W, Han Y Y, Yang J Y, Du H F, Ning W, Zhang Y H and Tian M L 2017 Phys. Rev. B 96 121407(R)
[24] Mikitik G P and Sharlai Y V 1999 Phys. Rev. Lett. 82 2147
[25] Qu D X, Hor Y S, Xiong J, Cava R J and Ong N P 2010 Science 329 821
[26] Analytis J G, McDonald R D, Riggs S C, Chu J H, Boebinger G S and Fisher I R 2010 Nat. Phys. 6 960
[27] Pariari A, Dutta P and Mandal P 2015 Phys. Rev. B 91 155139
[28] Hu J, Tang Z J, Liu J Y, Liu X, Zhu Y L, Graf D, Myhro K, Tran S, Lau C N, Wei J and Mao Z Q 2016 Phys. Rev. Lett. 117 016602
[29] Liu J Y, Hu J, Zhang Q, Graf D, Cao H B, Radmanesh S M A, Adams D J, Zhu Y L, Cheng G F, Liu X, Phelan W A, Wei J, Jaime M, Balakirev F, Tennant D A, DiTusa J F, Chiorescu I, Spinu L and Mao Z Q 2017 Nat. Mater. 16 905
[30] Chen R Y, Chen Z G, Song X Y, Schneeloch J A, Gu G D, Wang F and Wang N L 2015 Phys. Rev. Lett. 115 176404
[31] Chen R Y, Zhang S J, Schneeloch J A, Zhang C, Li Q, Gu G D and Wang N L 2015 Phys. Rev. B 92 075107
[32] Omling P, Meyer B and Emanuelsson P 1991 Appl. Phys. Lett. 58 931
[33] Linke H, Omling P, Ramvall P, Meyer B K, Drechsler M, Wetzel C, Rudeloff R and Scholz F 1993 J. Appl. Phys. 73 7533
[34] Linke H, Kowalski B, Ramvall P, Emanuelsson P, Omling P, Oettinger K, Drechsler M and Meyer B K 1993 Appl. Phys. Lett. 62 2725
[35] Drabinska A, Wołos A, Kaminska M, Strupinski W and Baranowski J M 2012 Phys. Rev. B 86 045421
[36] Wolos A, Szyszko S, Drabinska A, Kaminska M, Strzelecka S G, Hruban A, Materna A and Piersa M 2012 Phys. Rev. Lett. 109 247604
[37] Shestakov A V, Fazlizhanov I I, Yatsyk I V, Gilmutdinov I F, Ibragimova M I, Shustov V A and Eremina R M 2018 J. Semicond. 39 052001
[38] Kamm G N, Gillespie D J, Ehrlich A C, Wieting T J and Levy F 1985 Phys. Rev. B 31 7617
[39] Weng H M, Dai X and Fang Z 2014 Phys. Rev. X 4 011002
[40] McIlroy D N, Moore S, Zhang D, Wharton J, Kempton B, Littleton R, Wilson M, Tritt T M and Olson C G 2004 J. Phys.: Condens. Matter 16 L359
[41] Zheng G L, Zhu X D, Liu Y Q, Lu J W, Ning W, Zhang H W, Gao W S, Han Y Y, Yang J Y, Du H F, Yang K, Zhang Y H and Tian M L 2017 Phys. Rev. B 96 121401(R)
[42]Shoenberg D 1984 Magnetic Oscillations in Metals (Cambridge: Cambridge University Press)
[43] 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
Chin. Phys. Lett. (2019) 36(6) 066501 - Negative Thermal Expansion of the Dy$_{2}$Fe$_{16}$Cr Compound