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The de Hass-van Alphen quantum oscillations in BaSn3 superconductor with multiple Dirac fermions

By measuring the de Hass-van Alphen effect and calculating the electronic band structure, we have investigated the bulk Fermi surface of the BaSn3 superconductor with a transition temperature of ~ 4.4 K. Striking de Haas-van Alphen (dHvA) quantum oscillations are observed when the magnetic field B is perpendicular to both (100) and (001) planes. Our analysis unveiled nontrivial Berry phase imposed in the quantum oscillations when B is perpendicular to (100), with two fundamental frequencies at 31.5 T and 306.7 T, which likely arise from two corresponding hole pockets of the bands forming a type-II Dirac point. The results are supported by the ab initio calculations indicating a type-II Dirac point setting and tilting along the high symmetric K-H line of the Brillouin zone, about 0.13 eV above the Fermi level. Moreover, the calculations also revealed other two type-I Dirac points on the high symmetric {\Gamma}-A direction, but slightly far below the Fermi level. The results demonstrate BaSn3 as an excellent platform for the study of not only exotic properties of different types of Dirac fermions in a single material, but also the interplay between nontrivial topological states and superconductivity.


I. INTRODUCTION
The nontrivial topological states in solids have brushed up our knowledge on band theory, which provide a platform to conveniently study the intriguing physics of some elementary particles with properties akin to those theoretically predicted in high-energy physics, such as Dirac fermions [1][2][3], Weyl fermions [4][5][6][7], Majorana fermions [8][9][10][11][12][13][14], and even other exotic new fermions that have no counterparts in high-energy physics [15][16][17][18][19]. The realization of these particles is owing to the numerous symmetries of solids and their versatile operations, which whereas is somewhat difficult in high-energy physics and most of the particles consequently have not been discovered yet. With the protection by both time-reversal and space-inversion symmetries, the massless Dirac fermions can appear as low-energy quasiparticles in the Dirac semimetals (DSMs), such as Na 3 Bi [2,3] and Cd 3 As 2 [20,21], around the Dirac points (DPs) which are formed by the four-fold degenerate linear bands crossing. Once the time-reversal symmetry or space-inversion symmetry is broken, a DP will split into two Weyl points (WPs) that are characterized by a pair of Weyl fermions with opposite chiralities, such as in the Weyl semimetals (WSMs) TaAs series [4][5][6][7]. In the momentum space, both Dirac and Weyl cones can be tilted via modulating the tilting term in the Hamiltonian. Unlike the case in high-energy physics, where the Lorentz invariance is strictly required but not necessary in condensed matter physics, which could be broken by additional momentum dependent term, thus forming the type-II DSMs and WSMs, such as the PtTe 2 [22][23][24][25][26] and the MoTe 2 families [27][28][29][30][31]. The type-II DPs and WPs are still symmetry-protected while the spectra of the Dirac and Weyl cones are strongly tilted. Different from the type-I Dirac or Weyl fermions, the type-II Dirac or Weyl fermions are generally located at the boundary between the electron-like and hole-like pockets of the Brillouin zone (BZ), which are expected to display unique properties, such as Klein tunneling in momentum space [32], angle-dependent chiral anomaly [33], modified anomalous Hall conductivity [34], and topological Lifshitz transition [35], etc. In light of the different physical properties of type-I and type-II Dirac or Weyl fermions, the combination of them in a single material is rather attractive for the realization of multiple functionalities. However, it is definitely challenging because of the different symmetry requirements for them. The coexistence of different nontrivial topological states in a single material is expected to produce exotic physical properties, which has been reported in some compounds [15][16][17][18][19]. Regarding the combination of different types of Dirac fermions in a single material, the ternary wurtzite CaAgBi family of materials was predicted to be excellent targets, which however is still waiting for experimental verification [36].
Recently, the exploration of nontrivial topological states in superconductors has garnered increasing attentions [8][9][10][11][12][13][14], because superconductors hosting nontrivial topological states are thought to be a good place to discovery Majorana Fermion. The interplay of nontrivial band topology and superconductivity might give rise to topological superconductivity with spin-momentum locking of topological surface states hosting the long-pursuing Majorana Fermion, which is potential for the use in low decoherence topological quantum computation [37][38][39][40][41][42]. LnM 3 (Ln = Y and rare-earth elements, M = Pb, TI, In, Ga, and Sn) compounds with the AuCu 3 -type structure cover an array of interesting physical properties, such as superconductivity with relatively high superconducting transition temperature T c , heavy fermion behavior, quantum critical points [43,44], etc. Due to ionic radius mismatch between the alkaline earth metal (A) and Sn ions, ASn 3 generally crystallize in the variant structures of AuCu 3 . For example, CaSn 3 is in the AuCu 3 structure [45], while BaSn 3 (T c = 4.3 K) has a hexagonal structure and SrSn 3 (T c = 5.4 K) crystallizes in a rhombohedral structure [46]. Up to now, the physical properties of ASn 3 have been less studied as compared with LnM 3 . The recent studies on CaSn 3 have revealed it as a weakly coupled type-II superconductor hosting eight pairs of WPs in the BZ with closed loops of surface Fermi arcs [47][48][49]. In light of the intriguing properties of Quantum Design. Due to sensitivity of the crystal to air, a layer of paraffin was coated on the crystal to isolate the air for magnetization measurements.
A kinetic energy cutoff of 500 eV and a Γ-centred k mesh of 6×6×9 were utilized in all calculations. The energy and force difference criterion were defined as 10 -6 eV and 0.01 eV/Å for self-consistent convergence and structural relaxation, respectively. The WANNIER90 package [57][58][59] was adopted to construct Wannier functions from the first-principles results without an iterative maximal-localization procedure. The WANNIERTOOLS [60] code was used to investigate the topological features of surface state spectra.

III. RESULTS AND DISCUSSION
The magnetic susceptibility of BaSn 3 crystal is presented in Fig. 1 Here * = 4 is the harmonic value of . The angle dependent frequencies ,  1 , and  2 are comparable to those calculated for the Band 1 and Band 2, respectively, as shown in Fig. 3(b). The area of FS can be determined by the Onsager relation F = A F (/2 2 ), where  =h/e is the magnetic flux and A F is the FS area. The derived parameters are summarized in Table I.  Table I.
To gain more insights into the quantum oscillations which are imitatively related to the FS of the BaSn 3 superconductor, we performed first-principle calculations on the electronic structure of BaSn 3 by the density functional theory (DFT). The band structure of BaSn 3 in the absence of spin-orbit coupling (SOC) is shown in Fig. 4(a), revealing a metallic behavior with several bands crossing the E F and the multi-orbital nature of the superconductivity. When SOC is taken into account, we focus our special attention on the bands near the E F along the Γ-A and K-H high symmetry lines.
Along the Γ-A direction, seen in Fig. 4 This holds possibility for making BaSn 3 more attractive for topological applications.

ACKNOWLEDEMENTS
The authors acknowledge the support by the Natural Science Foundation of Shanghai