⇦Chinese Physics Letters, 2020, Vol. 37, No. 10, Article code 107504 Quantum Oscillations and Electronic Structure in the Large-Chern-Number Topological Chiral Semimetal PtGa Sheng Xu (徐升)1,2†, Liqin Zhou (周丽琴)3,4†, Xiao-Yan Wang (王小艳)1,2†, Huan Wang (王欢)1,2, Jun-Fa Lin (林浚发)1,2, Xiang-Yu Zeng (曾祥雨)1,2, Peng Cheng (程鹏)1,2, Hongming Weng (翁红明)3,4,5, and Tian-Long Xia (夏天龙)1,2* Affiliations 1Department of Physics, Renmin University of China, Beijing 100872, China 2Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China, Beijing 100872, China 3Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China 5Songshan Lake Materials Laboratory, Dongguan 523808, China Received 10 August 2020; accepted 24 August 2020; published online 29 September 2020 Supported by the National Key Research and Development Program of China (Grant Nos. 2019YFA0308602, 2018YFA0305700 and 2016YFA0300600), the National Natural Science Foundation of China (Grant Nos. 11874422 and 11574391), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant Nos. 19XNLG18 and 18XNLG14), the Chinese Academy of Sciences (Grant No. XDB28000000), the Science Challenge Project (Grant No. TZ2016004), the K. C. Wong Education Foundation (Grant No. GJTD-2018-01), the Beijing Municipal Science & Technology Commission (Grant No. Z181100004218001), and the Beijing Natural Science Foundation (Grant No. Z180008).
†Sheng Xu, Liqin Zhou, and Xiao-Yan Wang contributed equally to work.
*Corresponding author. Email: tlxia@ruc.edu.cn Citation Text: Xu S, Zhou L Q, Wang X Y, Wang H and Lin J F et al. 2020 Chin. Phys. Lett. 37 107504    Abstract We report the magnetoresistance (MR), de Haas-van Alphen (dHvA) oscillations and the electronic structures of single-crystal PtGa. The large unsaturated MR is observed with the magnetic field $B \parallel [111]$. Evident dHvA oscillations with the $B \parallel [001]$ configuration are observed, from which twelve fundamental frequencies are extracted and the spin-orbit coupling (SOC) induced band splitting is revealed. The light cyclotron effective masses are extracted from the fitting by the thermal damping term of the Lifshitz–Kosevich formula. Combining with the calculated frequencies from the first-principles calculations, the dHvA frequencies $F_1/F_3$ and $F_{11}/F_{12}$ are confirmed to originate from the electron pockets at $\mit\Gamma$ and $R$, respectively. The first-principles calculations also reveal the existence of spin-3/2 Rarita–Schwinger–Weyl fermions and time-reversal doubling of the spin-1 excitation at $\mit\Gamma$ and $R$ with large Chern numbers of $\pm4$ when SOC is included. DOI:10.1088/0256-307X/37/10/107504 PACS:75.47.-m, 81.10.Fq, 71.20.-b © 2020 Chinese Physics Society Article Text Weyl semimetals have attracted tremendous attention in condensed matter physics[1,2] due to the novel and significant transport properties, such as the high mobility, negative longitudinal magnetoresistance (MR) and three-dimensional quantum Hall effect.[3–8] The materials of TaAs family[2–4,9–17] are well-known as Weyl semimetals with spin-1/2 twofold Weyl fermion. In addition, some new types of multifold Weyl fermions, named as spin-1 chiral fermion, double Weyl fermion and spin-3/2 Rarita–Schwinger–Weyl (RSW) fermion,[18–27] are found in CoSi, RhSi, RhSn, AlPt and PdGa.[24,25,27–36] These materials with the space group $P2_13$ (No. 198) hold the spin-1 excitation at $\mit\varGamma$, double Weyl fermion at $R$ in the first Brillouin zone with the Chern number $\pm 2$, accompanied by two Fermi arcs in their surface states (SSs). Once the spin-orbit coupling (SOC) is included, the band splitting is induced. Thus, the spin-1 excitation and double Weyl fermion evolve into spin-3/2 RSW fermion and time-reversal (TR) doubling of the spin-1 excitation with the Chern number $\pm 4$, respectively. Correspondingly, Fermi arcs should split and result in four expected Fermi arcs in the SSs. However, the SOC is relatively weak and the band splitting is not observed in the recent angle-resolved photomission spectroscopy (ARPES) or transport studies on CoSi, RhSi and AlPt.[25,28–33] Band splitting in RhSn with stronger SOC has been confirmed by our previous transport study.[35] However, the splitting in the bulk states and SSs is still not resolved in ARPES.[34] Motivated by the above-mentioned results and discussions, we further grew PtGa single crystals with the strongest SOC among the recently studied materials of this family and investigated its transport properties and electronic structures. The topological properties and band structure characteristics of PtGa are similar with CoSi, RhSi, RhSn, AlPt and PdGa. According to our first-principles calculations, there is spin-1 excitation at $\mit\varGamma$ and double Weyl fermion at $R$ in the first Brillouin zone with the Chern number $\pm 2$ without considering SOC. When SOC is considered, the spin-3/2 RSW fermion and TR doubling of the spin-1 excitation are required to exist at $\mit\varGamma$ and $R$ with the Chern number $\pm 4$, respectively. PtGa displays a metallic behavior in the temperature-dependent resistivity measurement and large unsaturated longitudinal MR at 1.8 K and 14 T. Evident de Haas-van Alphen (dHvA) oscillations are observed and twelve fundamental frequencies are extracted from the fast Fourier-transform (FFT) analysis. Combining with the frequencies calculated from the first-principles calculations with SOC considered, the two frequencies at 11.2 T and 66.2 T are confirmed to originate from the electron-like pockets at $\mit\varGamma$ and the two frequencies at 1723.9 T and 1937.3 T are from the electron-like pockets at $R$. However, according to the first-principles calculations without SOC included, there should exist only one frequency at $\mit\varGamma$ or $R$. Thus, the SOC-induced band splitting is revealed by the dHvA oscillations, indicating the existence of spin-3/2 RSW fermion and TR doubling of the spin-1 excitation. The light cyclotron effective masses are extracted from the fitting of thermal damping term in the Lifshitz–Kosevich (LK) formula, indicating the possible existence of massless quasiparticles. PtGa provides a better platform to study the SOC-induced bulk bands and Fermi arcs splitting in the materials of this family. Methods. The single crystals of PtGa were grown by the Bi flux method. The platinum powder, gallium ingot and bismuth granules were put into the crucible and sealed into a quartz tube with the ratio of Pt:Ga:Bi=1:1:50. The quartz tube was heated to 1150℃ at 60℃/h and held for 20 h, then cooled to 550℃ at 1℃/h. The rod-like PtGa single crystals were obtained by centrifugation. The atomic composition of PtGa single crystal was checked to be Pt:Ga=1:1 by energy dispersive x-ray spectroscopy (EDS, Oxford X-Max 50). The measurements of resistivity and magnetic properties were performed on a Quantum Design physical property measurement system (QD PPMS-14T). The standard four-probe method was applied on the resistivity and magnetoresistance measurements on a long flake crystal (S1), which was cut and polished from the grown crystals. The electrode was made by a platinum wire with silver epoxy. Magnetization measurements were conducted on another smaller sample (S2) with higher quality. The first-principles calculations of PtGa were performed by the Vienna ab initio Simulation Package (VASP),[37] which was based on the density functional theory (DFT). The generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) type was selected to describe the exchange-correlation functional.[38] The cutoff energy was set to 450 eV, and the Brillouin zone (BZ) integral for self-consistent calculation was sampled by $10 \times 10 \times 10 k$ mesh. We constructed the tight-binding model of PtGa using the Wannier90 package with Ga $4p$ orbital and Pt $5d$ orbital, based on the maximally localized Wannier functions method (MLWF),[39] and we further calculated the bulk Fermi surfaces (FSs) using the WannierTools software package.[40]
cpl-37-10-107504-fig1.png
Fig. 1. (a) Temperature dependence of the resistivity $\rho_{xx}$. Inset: the crystal structure of PtGa. (b) Magnetic field-dependent MR at 1.8 K with magnetic field titled from $B\,\bot\, I$ ($\theta=0^\circ $) to $B \parallel I$ ($\theta=90^\circ $). Here $\theta$ is the angle between $B$ and $I$, and $\theta=0^\circ $ when $B \parallel [111]$.
Results and Discussions. The crystal structure of PtGa is illustrated in the inset of Fig. 1(a), which belongs to a simple cubic structure with the space group $P2_13$ (No. 198). Figure 1(a) displays the temperature-dependent resistivity, which exhibits the metallic behavior. The field-dependent longitudinal MR [MR=[$\rho_{xx}(B)-\rho_{xx}(0)]/\rho_{xx}(0)$] with the field tilted from $B \,\bot\, I$ to $B \parallel$ $I$ is exhibited in Fig. 1(b) ($B$ represents the magnetic field and $I$ the current). The large unsaturated longitudinal MR reaches 230% at 1.8 K and 14 T with the $B \parallel [111]$ configuration, which decreases gradually with $\theta$ increasing from $\theta=0^\circ $ to $\theta=90^\circ $ and tends to saturate at $\theta=90^\circ $ under high field. Because of the existence of trivial pockets, the negative longitudinal MR with the $B \parallel I$ configuration in PtGa has not been observed. The quantized Landau levels will cross the Fermi energy ($E_{\rm F}$) with the increasing field, which leads to the oscillations of density of state (DOS) at $E_{\rm F}$ and finally induces the quantum oscillations of magnetization. According to the Onsager relation $F=(\phi{_0}/2\pi^2)A_{\rm F}=(\hbar/2\pi e)A_{\rm F}$, the frequency $F$ is proportional to the extreme cross section ($A_{\rm F}$) of the FS normal to the magnetic field. Thus, dHvA oscillations is an effective method to study the electronic structures of topological materials, which can reveal the detailed information of the FS. Figure 2(a) presents the isothermal magnetization of a typical single crystal with $B \parallel [001]$ configuration, which exhibits evident dHvA oscillations. The oscillatory components of magnetization are obtained after subtracting a smooth background as displayed in Fig. 2(b). Twelve fundamental frequencies are extrcted from the FFT analysis. The low frequency at about 2.1 T comes from the data processing and is not the intrinsic signal of dHvA oscillations. The oscillatory components versus 1/$B$ of dHvA oscillations can be described by the LK formula:[41] $$ \Delta M \propto -B^{1/2}\frac{\lambda T}{{\rm \sinh}(\lambda T)}e^{-\lambda T_{\rm D}}{\rm \sin}\Big[2\pi\Big(\frac{F}{B}-\frac{1}{2}+\beta+\delta\Big)\Big],~~ \tag {1} $$ where $\lambda=(2\pi^2 k_{_{\rm B}} m^*)/(\hbar eB)$. $T_{\rm D}$ is the Dingle temperature. For the 2D system, $\delta=0$, and for the 3D system, $\delta=\pm1/8$. Here $\beta=\phi_{_{\rm B}}/2\pi$ and $\phi_{_{\rm B}}$ is the Berry phase. The inset of Fig. 2(c) shows the normalized temperature-dependent FFT amplitudes and the fitting by the thermal factor in LK formula. The effective masses are estimated to be $m^*_{\rm F_2}=0.05m_{\rm e}$, $m^*_{\rm F_3}=0.12m_{\rm e}$, $m^*_{\rm F_4}=0.10m_{\rm e}$, $m^*_{\rm F_5}=0.11m_{\rm e}$, $m^*_{\rm F_7}=0.08m_{\rm e}$ and $m^*_{\rm F_{11}}=0.15m_{\rm e}$.
cpl-37-10-107504-fig2.png
Fig. 2. (a) The dHvA oscillations at various temperatures with the $B \parallel [001]$ configuration. (b) The amplitudes of dHvA oscillations as a function of $1/B$. (c) The FFT spectra of the oscillations. Inset: the temperature dependence of relative normalized FFT amplitude of the frequencies.
cpl-37-10-107504-fig3.png
Fig. 3. [(a),(b)] Calculated bulk band structure of PtGa along high-symmetry lines and FSs in the bulk Brillouin zone without SOC. [(c),(d)] Calculated FSs in $k_z=0$ plane and $k_z=\pi$ plane without SOC.
Figure 3(a) exhibits the calculated band structures of PtGa without SOC. There is spin-1 excitation at $\mit\varGamma$ and double Weyl fermion at $R$ in the first Brillouin zone with the Chern number $\pm 2$, respectively. A spherical electronic FS exists at $\mit\varGamma$ with one FS's extreme cross section cutting at $k_z=0$, as shown in Fig. 3(c). The number of FS's extreme cross section at $R$ cutting at $k_z=\pi$ is also one, as shown in Fig. 3(d). Thus, there should be only one frequency originating from the FS at $\mit\varGamma$ or $R$ according to the Onsager relation. When the SOC is included, the bands split. In such a case, spin-1 excitation and double Weyl fermion evolve into spin-3/2 RSW fermion and TR doubling of the spin-1 excitation with the Chern number $\pm 4$, respectively, as shown in Fig. 4(a). Figure 4(b) displays the calculated FSs of PtGa with SOC in the first Brillouin zone. Figure 4(c) exhibits eight fundamental FS's extreme cross sections cutting at $k_z=0$. $S_5$, $S_6$, $S_7$ and $S_8$ at the $M$ point show a quarter of the FS's extreme cross sections. Four fundamental FS's extreme cross sections cutting at $k_z=\pi$ are exhibited in Fig. 4(d). $S_9$ and $S_{10}$ show half of the FS's cross sections at the $M$ point, $S_{11}$ and $S_{12}$ show a quarter of the FS's extreme cross sections at the $R$ point. Meanwhile, there are two FS's extreme cross sections at $\mit\varGamma$ cutting at $k_z=0$ and two FS's extreme cross sections at $R$ cutting at $k_z=\pi$, indicating that two frequencies will be induced by the FS at $\mit\varGamma$ or $R$, respectively. These characteristics are different from the FSs without SOC and are regarded as the evidences of SOC induced band splitting, which can be resolved by the quantum oscillations. The calculated frequencies originating from $S_1$ and $S_2$ are 13.0 T and 61.7 T, corresponding to the observed $F_1$ (11.2 T) and $F_3$ (66.2 T), respectively. The calculated frequencies originating from $S'_{11}$ $(4\times S_{11}$) and $S'_{12}$ $(4\times S_{12}$) are 1699.0 T and 1987.6 T, corresponding to the observed $F_{11}$ (1723.9 T) and $F_{12}$ (1937.3 T), respectively. Thus, the SOC induced band splitting is revealed by our dHvA oscillations. As shown in Fig. 4(c), the FSs of $S'_7$ and $S'_8$ touch along $M$–$X$, which enables the carriers to move between the two FSs. The observed frequency $F_9$ (333.7 T) may originate from the difference between $S'_{8}$ $(4\times S_{8})$ and $S'_{7}$ $(4\times S_{7})$ (346.8 T). The match between the calculated FS's extreme cross sections and the observed frequencies is shown in Fig. 4(e). The gaps between $S_3$ and $S_4$, the $S'_7$ and $S'_8$ along $M$–$\mit\varGamma$, $S'_9$ and $S'_{10}$, $S'_{11}$ and $S'_{12}$ along $M$–$R$ are very small. Thus, the magnetic breakdown is likely to exist under high magnetic field. The rest of observed frequencies may originate from the FS's extreme cross sections cutting between $k_z=0$ and $k_z=\pi$ and/or the complicated magnetic breakdown.
cpl-37-10-107504-fig4.png
Fig. 4. [(a),(b)] Calculated bulk band structure of PtGa along high-symmetry lines and FSs in the bulk Brillouin zone with SOC. [(c),(d)] Calculated FSs in $k_z=0$ plane and $k_z=\pi$ plane with SOC. (e) Matching between the FSs' extreme cross sections and the observed frequencies. Inset: the Brillouin zone of PtGa with $k_z=0$ and $k_z=\pi$ planes highlighted.
Table 1. The parameters extracted from dHvA oscillations in PtGa with $B \parallel [001]$ configuration. $F$ is the frequency of dHvA oscillations; $m^*/m_{\rm e}$ is the ratio of the effective mass to the electron mass. $A_{\rm F}$ is the FS's extreme cross section.
$F_1$ $F_2$ $F_3$ $F_4$ $F_5$ $F_6$ $F_7$ $F_8$ $F_9$ $F_{10}$ $F_{11}$ $F_{12}$
$F$(T) 11.2 33.2 66.2 100.4 132.3 254.7 284.9 324.6 333.7 380.4 1723.9 1937.3
$m^*/m_{\rm e}$ 0.05 0.12 0.10 0.11 0.08 0.15
$A_{\rm F}$ (10$^{-3}$ Å$^{-2}$) 1.07 3.17 6.32 9.58 12.63 24.31 27.20 30.99 31.85 36.31 164.56 184.93
In conclusion, we have successfully synthesized the single crystals of PtGa by the Bi flux method. It shows a metallic behavior at zero field. The large unsaturated MR reaching 230% at 1.8 K and 14 T with $B \parallel [111]$ has been observed, which decreases gradually with $\theta$ increasing from $\theta=0^\circ $ to $\theta=90^\circ $ and tends to saturate at $\theta=90^\circ $ under high field. The dHvA oscillations have been observed. The light cyclotron effective masses are extracted from the fitting of the thermal damping term in the LK formula, indicating the possible existence of massless Weyl fermions. The dHvA oscillations also reveal two concentric ball-like electronic FSs with two FS's extreme cross sections at $\mit\varGamma$ and two concentric circle-like FS's extreme cross sections at $R$, indicating the possible existence of spin-3/2 RSW fermion and TR doubling of spin-1 excitation, which is in agreement with our first-principles calculations with SOC. ARPES is required for further study of its band structures and Fermi arcs in the SSs. Note Added. When the paper is being finalized, we note one related work, in which similar magneto-transport measurements are reported with ARPES confirming the topological characteristics in PtGa crystals grown with a different method.[42] References Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridatesWeyl Semimetal Phase in Noncentrosymmetric Transition-Metal MonophosphidesObservation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 3D Weyl Semimetal TaAsSignatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetal3D Quantum Hall Effect of Fermi Arcs in Topological SemimetalsQuantum Hall effect based on Weyl orbits in Cd3As2Three-dimensional quantum Hall effect and metal–insulator transition in ZrTe5Theory for the charge-density-wave mechanism of 3D quantum Hall effectDiscovery of a Weyl fermion semimetal and topological Fermi arcsA Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs classExperimental Discovery of Weyl Semimetal TaAsObservation of Weyl nodes in TaAsDiscovery of a Weyl fermion state with Fermi arcs in niobium arsenideExperimental discovery of a topological Weyl semimetal state in TaPObservation of Weyl nodes and Fermi arcs in tantalum phosphideEvolution of the Fermi surface of Weyl semimetals in the transition metal pnictide familyNegative magnetoresistance without well-defined chirality in the Weyl semimetal TaPOn a Theory of Particles with Half-Integral SpinExistence of bulk chiral fermions and crystal symmetryBeyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystalsType-II Weyl points in three-dimensional cold-atom optical latticesSemimetal with both Rarita-Schwinger-Weyl and Weyl excitationsPseudospin- 3 2 fermions, type-II Weyl semimetals, and critical Weyl semimetals in tricolor cubic latticesMultiple Types of Topological Fermions in Transition Metal SilicidesUnconventional Chiral Fermions and Large Topological Fermi Arcs in RhSiDouble-Weyl Phonons in Transition-Metal MonosilicidesBand structure and unconventional electronic topology of CoSiTopological chiral crystals with helicoid-arc quantum statesObservation of unconventional chiral fermions with long Fermi arcs in CoSiObservation of Chiral Fermions with a Large Topological Charge and Associated Fermi-Arc Surface States in CoSiChiral topological semimetal with multifold band crossings and long Fermi arcsCrystal growth and quantum oscillations in the topological chiral semimetal CoSiSingle Crystal Growth and Magnetoresistivity of Topological Semimetal CoSiChiral fermion reversal in chiral crystalsQuantum oscillations and electronic structure in the large–Chern number semimetal RhSnObservation and control of maximal Chern numbers in a chiral topological semimetalEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setGeneralized Gradient Approximation Made SimpleAn updated version of wannier90: A tool for obtaining maximally-localised Wannier functionsWannierTools: An open-source software package for novel topological materialsU1 snRNP regulates cancer cell migration and invasion in vitro
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