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Unusual magneto-transport from Si-square nets in topological semimetal HfSiS

The class of topological semimetals comprises a large pool of compounds. Together they provide a wide platform to realize exotic quasiparticles for example Dirac, nodal line Dirac and Weyl fermions. In this letter, we report the Berry phase, Fermi surface topology and anisotropic magnetoresistance of HfSiS which has recently been predicted to be a nodal line semimetal. This compound contains large carrier density, higher than most of the known semimetals. Massive amplitudes of de Haas-van Alphen and Shubnikov-de Haas oscillations up to 20 K in 7 T assist us in witnessing nontrivial pi-Berry phase which is a consequence of topological Dirac-type dispersion of bands originating from the hybridization of px + py and dx2-y2 orbitals of square-net plane of Si and Hf atoms, respectively. Furthermore, we establish the 3D Fermi-surface which consists of very asymmetric water caltrop-like electron and barley seed-like hole pockets which account for the anisotropic magnetoresistance in HfSiS.


Introduction:
In graphene, carbon atoms are arranged in honeycomb lattice that facilitates a linear relation between energy and momentum, Dirac-type dispersion [1] near the Fermi energy.
Additionally, a negligible spin orbit coupling (SOC) in carbon atom maintains the gapless Dirac crossing [2]. It has recently been realized that many other families of compounds also have such Dirac-type crossing in bulk and they, however, possess the square-net arrangement of atoms namely Bi square-net in AMnBi2, where A  Sr, Ba, Ca, and Eu [3][4][5], R2O2Bi [6,7] and RAgBi2 [8,9], where R = rare earth or Y, Se and Te square-net in selenene, tellurene and RTe3 [10,11], arsenic square-net in RCuAs2 [12] etc. Evolution of such a Diractype bands crossing upon the inclusion of SOC normally leads to a negligible small gap in graphene [2] while it is large in compounds containing heavy atoms like Bi. Furthermore, depending on the crystal structure and symmetries present in the compound some Dirac crossings are protected against the opening of the gap [13,14]. In nodal line semimetals, the valence and conduction bands, however, cross along in a particular direction in momentum space forming a drumhead like surface states [15][16][17][18]. Such compounds show extraordinary transport properties e.g. low effective mass of carrier, extremely high magnetoresistance (MR), extremely high mobility etc. [13,14,[19][20][21][22][23]. Very recently, a group of ternary materials with general formula WHM (W= Zr, Hf, or La; H = Si, Ge, Sn, or Sb; M = O, S, Se and Te) have been predicted as 2D topological insulators [24]. All these compounds have PbFCl-type crystal structure (space group P4/nmm) as shown in the Fig. S1(a) for HfSiS wherein Si atoms (in general H atoms) arrange in squarenet manner. This series of compounds exhibit two interesting features. First, a nontrivial nodal line states are protected by glide-mirror symmetry below the Fermi energy (-0.5 eV in HfSiS) [15][16][17][18]. Second, several other trivial Dirac-type crossings at Fermi level originate from px  py orbitals of square-net Si atoms and dx2-y2 orbitals of Hf atoms in HfSiS. These Dirac crossings are not protected by symmetries, for example a gap opening of 30 meV and 20 meV is observed in ZrSiO [24] and ZrSiS [25], respectively because of SOC.
The square-net arrangement of Si atoms in the ab-plane imparts layered structure in these compounds which results in the plate-like crystals growth [ Fig. S1(b)]. Among the predicted WHM series, only some Zr compounds have been studied ZrSi(S,Se,Te) [19][20][21][22][23][25][26][27] and all the compounds show very similar transport properties. Surprisingly, the noteworthy experimental application of quantum oscillations to construct 3D Fermi surface (FS) is still missing. It is more interesting to investigate HfSiS, a sister compound of ZrSiS, which has a larger spin-orbit coupling and hence shows a clear unexpected Dirac-node arc [28]. In this study, we report Berry phase, MR and the Fermiology of HfSiS. We find that HfSiS shows Berry phase, low effective mass of carriers, high MR and mobility that reveal the Diraclike dispersion of bands which indeed control the transport properties. Highly anisotropic MR is due to the asymmetric electron and hole pockets and large mass anisotropy. We further believe that the transport characteristics in HfSiS and related compounds significantly depend on the Dirac-like dispersion originated from the hybridized px  py orbitals of square-net Si atoms and dx2-y2 orbitals of Hf atoms near the Fermi level rather than on the nodal line states which lie deeper in the valence band [29].
Results and discussion: Thin platelike crystals of HfSiS were grown by chemical vapor transport method. The details about the synthesis route and characterization were described in the supplementary information (SI). From Fig. S1(d), the resistivity, xx (T) of HfSiS without magnetic field initially increases with T 2 up to 100 K after that it shows linear dependency with T like in a metal. The xx changes an order of magnitude from 2 K (3.1  cm) to 300 K (33  cm) and we found similar values in different batches of crystals. A dominating p-type charge carrier density of 6×10 20 cm -3 at 2 K was obtained from Hall resistivity which does not show a strong temperature dependence like in other low carrier semimetals [30].
To derive the physical parameters of the charge carriers from quantum oscillations, we measured The phase shift  is determined by the dimensionality of the Femi surface and it is 0 and 1/8 ("" for hole pocket and "" for electron pocket) for the 2D and 3D cases, respectively. From the above LK formula, the value of * can be determined through the fit of the temperature dependence of the oscillation amplitude with the thermal damping factor RT, as shown in Fig. 1(c). We find the values of * = 0.08 and  * = 0.17, which are very low and are likely to originate from the Diractype dispersion of bands. Further, the Dingle temperature TD is determined to be  3.5 K and  6.5 K for  and  pockets, respectively and corresponding quantum relaxation times, τq (= ħ/(2πkBTD) = 1.22  10 -12 /TD) are 3.5  10 -13 s and 1.9  10 -13 s. The quantum mobility, q (= eτq/ * ) is estimated to be 7.7  10 3 cm 2 /Vs for  pocket and 2.0  10 3 for  pocket.
We also measured the magnetization isotherms up to 20 K for B  [001] and the data of the whole temperature range can be found in Fig. S3. In this geometry also, HfSiS shows strong dHvA and no beating patterns are observed. Therefore, LK fit [ Fig. 2 [24]. Furthermore, the nodal line states in HfSiS which is protected by non-symmorphic crystal symmetry hardly affect the electronic properties because they are situated far below EF.
In the above, we discussed features of the electronic structure considering the maximal area of the  In this case, FFT shows only two frequencies  and  at 31 T and 264 T, respectively.