Fermi surface studies of a non-trivial topological compound YSi
Vikas Saini, Souvik Sasmal, Ruta Kulkarni, Bahadur Singh, A. Thamizhavel
FFermi surface studies of a non-trivial topological compound YSi
Vikas Saini, Souvik Sasmal, Ruta Kulkarni, Bahadur Singh, and A. Thamizhavel ∗ Department of Condensed Matter Physics and Materials Science,Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India. (Dated: February 11, 2021)The Fermi surface properties of a nontrivial system YSi is investigated by de Haas-van Alphen(dHvA) oscillation measurements combined with the first-principle calculations. Three main fre-quencies ( α , β , γ ) are probed up to 14 T magnetic field in dHvA oscillations. The α -branchcorresponding to 21 T frequency possesses non-trivial topological character with π Berry phase anda linear dispersion along Γ to Z direction with a small effective mass of 0 . m e with second-lowestLandau-level up to 14 T. For B (cid:107) [010] direction, the 295 T frequency exhibits non-trivial 2 D character with 1 . π Berry phase and a high Fermi velocity of 6 . × ms − . The band structurecalculations reveal multiple nodal crossings in the vicinity of Fermi energy E f without spin-orbitcoupling (SOC). Inclusion of SOC opens a small gap in the nodal crossings and results in nonsym-morphic symmetry enforced Dirac points at some high symmetry points, suggesting YSi to be asymmetry enforced topological metal. PACS numbers:Keywords:
INTRODUCTION
Topological semimetals have gained significant atten-tion both on theoretical and experimental fronts in re-cent years owing to their exotic properties. They ex-hibit symmetry protected bulk and surface states whichresults novel phenomena and transport properties withtremendous potential for applications in quantum tech-nologies and energy sciences [1–6]. Topological semimet-als can be classified based on the dimensionality and de-generacy of the crossing points. In particular, topologicalDirac semimetal has zero-dimensional four-fold degener-ate crossing points in the vicinity of Fermi energy ( E f ).The low energy excitations of the so-called Dirac par-ticles follow linear energy dispersion around the Diracpoint and are the reasons for the observed exotic phe-nomenon in the electrical transport measurements. Thefour-fold degeneracy of the Dirac point is ensured by thepresence of both the time-reversal T and inversion I sym-metries along with specific crystalline symmetries. Thisfour-fold degeneracy is lifted by breaking either inver-sion or time-reversal symmetries, thus Dirac semimet-als (DSM) transitions to Weyl semimetals (WSM) withtwo-fold Weyl nodal crossings [7–10]. Cd As and Na Biare known prototype topological Dirac semimetals whichwas predicted theoretically and realized in experiments.They exhibit ultra-high mobility, extremely high mag-netoresistance and show exotic phenomena such as chi-ral anomaly, quantum hall effect, among others owingto symmetry protected Dirac band crossings. The exten-sive studies of Shubnikov de-Haas(SdH) and de Haas-vanAlphen(dHvA) oscillations confirmed topological non-trivial nature of these compounds from the Berry phaseanalysis and other unusual properties [8, 9, 11–16].In view of finding new topological semimetals, we in-vestigate Fermi surface and topological properties of bi- nary compound YSi by de Haas-van Alphen (dHvA) os-cillations and first-principles calculations. We synthesizehigh-quality single crystals of YSi using the Czochral-ski method. From dHvA quantum oscillations studies,YSi is found to have nontrivial Fermi pockets. The first-principle calculations show that it has a rich nodal struc-ture at the Fermi level with symmetry protected bandcrossings. Three different branches ( α, β , and γ ) areprobed up to a magnetic field of 14 T. The nontrivial α pocket is observed along the three directions with a subtlevariation in the frequency implying a non-uniform cross-section of the Fermi surface suggesting an anisotropicnature of the α pocket. On the other hand, the othertwo pockets ( β and γ ) are observed only along B (cid:107) [010]direction suggesting a 2 D nature of the Fermi surface,where β branch has the nontrivial character of topology.The detailed band structure analysis shows that boththe valence and conduction bands participate in the for-mation of the Fermi surface with the hole and electronpockets. The type-I nodal anti-band crossing along theΓ − Z direction in the vicinity of Fermi energy leads tothe nontrivial α pocket. METHODS
From the binary phase diagram of Y and Si, it is ob-vious that YSi melts congruently at 1845 ◦ C [17] andhence can be grown directly from its melt. A tetra-arcfurnace has been used to grow the single crystal of YSiby the Czochralski method. High purity starting mate-rials of Y (3N pure, Alfa-Aesar) and Si (5N pure, Alfa-Aesar) were taken in the ratio of 1 : 1 .
05 and repeatedlymelted to make a homogenous polycrystalline ingot ofabout 8 to 10 g. A seed crystal was cut from this poly-crystalline ingot to grow the single crystal in a vacuum a r X i v : . [ c ond - m a t . s t r- e l ] F e b chamber filled with Ar gas. The polycrystalline ingotwas melted and the seed crystal was slowly inserted intothe melt and pulled very rapidly to start with. Once thesteady-state condition is achieved the pulling rate wasmaintained at 10 mm/h. As grown pulled ingot had adiameter ≈ ≈
70 mm and is shown inFig. 1(a). The composition analysis was performed us-ing energy dispersive analysis by x-rays (EDAX). A smallportion of the crystal was crushed to fine powders andsubjected to room temperature powder x-ray diffraction(XRD) measurement in PANalytical x-ray diffractome-ter equipped with a monochromatic Cu- K α source withthe wavelength λ = 1.5406 ˚A. To confirm the single-crystalline nature of the sample and to orient and cutthe crystal along the three principal crystallographic di-rections we have performed Laue diffraction in the backreflection geometry. The oriented crystal was cut intoa rectangular bar shape using a spark erosion cuttingmachine. Magnetic measurements were performed in avibrating sample magnetometer (VSM) (PPMS, Quan-tum Design, USA), down to 2 K and in a magnetic fieldof 14 T.Band structure calculations were carried out with theprojector augmented wave (PAW) method[18] withinthe density functional theory (DFT)[19] framework asimplemented in the Vienna ab initio simulation pack-age (VASP)[20, 21]. The exchange-correlation ef-fects were considered with the generalized gradient ap-proximation (GGA) with the Perdew–Burke–Ernzerhofparameterization[22]. The spin-orbit coupling (SOC) wasincluded self-consistently. An energy cut-off of 310 eVwas used for the plane-wave basis set and a 11 × × k mesh was used for the bulk Brillouin zonesampling. The Xcrysden program was used to visualizethe Fermi surface[23]. The robustness of results is fur-ther verified by calculating electronic properties using theWIEN2K code which considers a full-potential linearizedaugmented plane-wave formalism [24]. The quantum os-cillations calculation were performed using the SKEAFcode[25]. RESULTS AND DISCUSSIONCrystal Structure
We start discussing the crystal structure of our grownsingle crystals of YSi. The phase purity of the growncrystal was analysed by powder XRD measurement. Ithas been reported that YSi crystallizes in the centrosym-metric orthorhombic crystal structure with space group
Cmcm ( θ rage from 10to 90 ◦ . From the Rietveld analysis using FULLPROF I n t e n s it y ( c oun t s ) Observed
Calculated
Difference
Bragg Position(100) (010) (001)YSi(a)(b)(c) (d) (e)
FIG. 1: (a) As grown single crystalline pulled ingot. (b) Roomtemperature powder x-ray diffraction and structural refine-ment by Rietveld method. The inset shows crystal structureof YSi with b as the longest axis. The large (yellow) andsmall (red) balls identify Y and Si atoms, respectively. (c),(d), and (e) show the observed Laue diffraction pattern alongthe three principal crystallographic planes (100), (010), and(001), respectively. software [27] we confirmed that this compound crystal-lizes in the Cmcm space group. The obtained latticeparameter from the Rietveld analysis are a = 4 .
260 ˚A, b = 10 .
530 ˚A and c = 3 .
830 ˚A, which is in closeagreement with the available data [26]. The Y and Siatoms occupy 4 c Wyckoff position with coordinates (0,0.3598, 0.25) and (0, 0.0764, 0.25), respectively. TheLaue diffraction pattern attests good quality of the growncrystal as shown in Fig. 1(c), (d), and (e) respectively for(100), (010), and (001) planes.
Electronic structure
Figure 2(a) shows the bulk Brillouin zone for the primi-tive crystal structure of YSi where high-symmetry pointsare marked explicitly. The calculated bulk band struc-ture of YSi along various high-symmetry directions with- Si p Y d (a) (b) (c) (d) (e) (f) (g) (h) (i) GGA GGA+SOCGGA GGA+SOCGGA γ α
FIG. 2: (a) Bulk Brillouin zone of orthorhombic YSi. Thehigh-symmetry points are marked. Bulk band structure ofYSi (b) without and (c) with spin-orbit coupling (SOC). (d)Orbital resolved band structure without SOC. Y d and Si p states are shown with gray and blue markers, respectively.(e)-(f) Closeup of the bands along Γ − Z direction in thearea highlighted by yellow rectangles in (b)-(c). The brokencircles highlight the three crossing bands. The crossing bandsare gapped in the presence of SOC. (g)-(h) The calculatedindividual Fermi pockets and (i) the Fermi surface of YSi. out SOC is shown in Fig. 2(b). It is seen to be metal-lic where various bands cross the Fermi level. Impor-tantly, many symmetry protected spinless band crossingsare found along the high-symmetry lines such as R − Z and T − Z at the Brillouin zone boundaries as well at thegeneric k − points. Along the Γ − Z direction three bandscross, forming both the type-I and type-II nodal cross-ings as shown in Fig. 2(e). The orbital resolved bandstructure shows that these band crossings are composedby Y d and Si p orbitals (Fig. 2(d)). The structure withSOC is shown in Figs. 2(c) and (f). The various nodalcrossings at the generic k points are gapped. Howeverowing to the presence of screw rotations { C z | } and { C y | } and glide mirror { M y | } , the band cross-ings at R , Z , and T points remain protected, referringhigh symmetry point Dirac states.The calculated bulk Fermi surface of YSi is shown inFig. 2(i) whereas its constituents individual pockets areillustrated in Figs. 2(g) and (h). Owing to the multi-band crossings at the Fermi level, the Fermi surface iscomposed of both the electron and hole bands. We marka small pocket along the Γ − Z direction as α whichis formed by bands highlighted in the broken red circlein Fig. 2(f). On the other hand, the pocket along the T − A γ . Additionally, giant Fermi pockets are enclosing Γ and Z points. The cal-culated quantum oscillations frequencies are summarizedin Table I. It is found that a shift of ∼ +11 meV in theFermi level is essential to reproduce the experimentallyobserved frequencies. The observed oscillations of the α pocket are observed in all three directions and it carriesthe lowest frequency of 21 T for B (cid:107) [100]. β pocket isobserved only for B || [010] and has a frequency of 61 T.Also, the γ pocket was observed when B || [010] with ahigh frequency of 295 T. These oscillations are well cap-tured in our first-principles results (see below for moredetails). de Haas-van Alphen quantum oscillations studies The field dependence of the magnetization measure-ment measured in a VSM at 2 K, up to a field of 14 T,along the three principal crystallographic directions isshown in Fig. 3(a), (b) and (c). Robust dHvA oscillationsare observed along all three directions. Long-period os-cillations are observed for B (cid:107) [100] and [001] directionswhile for [010] direction strong oscillations with multiplefrequencies are observed. At 2 K the oscillations beginto appear from 2 T field and this corresponds to themagnetic length l B = (cid:113) (cid:126) eB ≈
18 nm thus indicating agood quality of the grown single crystal. As the tempera-ture is increased, the amplitudes of the dHvA oscillationstend to decrease and are not discernible for a tempera-ture greater than 21 K. The non-oscillatory backgrounddata was subtracted to extract the dHvA oscillation fre-quency by fast Fourier transform (FFT). The FFT spec-tra at T = 2 K along the three principal crystallographicdirections are shown in Fig. 3(d),(e) and (f). When themagnetic field is along the [100] direction, a single natu-ral frequency at 21 T has been observed which is namedas α . For B (cid:107) [010] three natural frequencies have beenobserved at 36 T ( α ), 61 T ( β ) and 295 T ( γ ) with thesecond harmonics at 590 T (2 γ ), while for B (cid:107) [001] asingle natural frequency at 34 T ( α ) and its correspond-ing second harmonics was observed at 68 T (2 α ). It is tobe mentioned here that the α -branch is observed along allthe three directions with a subtle anisotropy in the fre-quency thus representing a small anisotropic 3 D Fermisurface.We have estimated the quantum parameters by ana-lyzing the frequency spectrum of the observed oscilla-tions. The temperature dependence of the FFT ampli-tude of the three natural frequencies for B (cid:107) [010] isshown in Fig. 4(a). The amplitude of the frequencies de-creases as the temperature increases. The T -dependenceof the amplitude of the oscillation frequencies are shownFig. 4(a) and (b) and are fitted to thermal damping fac-tor of Lifshitz-Kosevich expression: R T = ( X/sinhX ),where X = ( λT m ∗ /H ), λ = (2 π k B m e /e (cid:126) )(= 14 .
69 T)
TABLE I: Quantum parameters estimated from dHvA oscillationsB F exp F cal m ∗ exp m ∗ cal τ q µ q k F v F l q n (T) (T) ( m e ) ( m e ) (10 − sec ) ( cm V − s − ) (˚A − ) (10 m/s) (nm)[100] 21 ( α ) 21.8 0.069(1) 0.082 0.025 4.26 5.56 *10 cm − [010] 295 ( γ ) 294.7 0.162(2) 0.147 2.2 2384 0.095 6.75 148.65 7.14 * 10 cm −
61 ( β ) 71.5 0.097(2) 0.162 1.14 2053 0.043 5.1 58.39 1.48*10 cm − [001] 34 ( α ) 28.1 0.096(4) 0.095 2.15 3927 0.032 3.85 82.77 1.109*10 cm − M ( - e m u ) H (T) B // [100] T = 2 K 630-3 M ( - e m u ) H (T) B // [010] T = 2 KYSi 50 M ( - e m u ) H (T) B // [001] T = 2 K0.30.20.10 FF T A m p lit ud e ( a . u ) B // [100] T = 2 K α FF T A m p lit ud e ( a . u ) B // [010] α YSi T = 2 K β γ γ FF T A m p lit ud e ( a . u ) B // [001] T = 2 K α α (a) (b) (c)(d) (e) (f) FIG. 3: (a), (b), and (c) dHvA quantum oscillations observedalong the three principal crystallographic directions in YSi at T = 2 K . (d), (e), and (f) The obtained FFT frequency spec-trum. The α branch is observed along all the three directionswhile β and γ are observed only along the [010] direction. and m ∗ is the cyclotron effective mass of the charge car-riers which is expressed in units of free electron mass m e [5, 28, 29] . The calculated effective masses for thevarious frequencies are listed in Table I. It is evident fromthe table that the effective masses of charge carriers arevery small for all the observed main frequencies suggest-ing Dirac-like dispersion of bands and comparable to thatof the gap-less Dirac system Cd As [30].To understand the topological character of the chargecarriers in YSi, we have performed the Berry phase anal-ysis of the dHvA oscillations using the Lifshitz-Kosevich(L-K) formula [31, 32] as given below,∆ M ∝ − B k R T R D R s sin (cid:20) π (cid:18) FB + ψ (cid:19)(cid:21) , (1)where R T , R D , and R S are the thermal, magnetic fieldand spin damping factors and ψ is the phase factor. Theexpression for R T has already been defined while that for R D is R D = exp ( − . m ∗ T D /B ) where T D is the Dingletemperature and R S = cos ( πgrm ∗ / m ), where g is theLand´e g -factor and r is the harmonic number. In Eqn. 1 I n t e n s it y ( a . u ) B // [010]
11 K
13 K
15 K
17 K
19 K
21 K YSi FF T A m p lit ud e ( a . u ) B // [100]YSi α Fit B // [001] α FF T A m p lit ud e ( a . u ) B // [010]YSi γ β Fit (a)(b) (c)
FIG. 4: (a) Temperature dependence of the FFT amplitudeof the oscillation. The FFT amplitude decreases with increas-ing temperature. (b), (c) Mass plot of the frequencies men-tioned along the three principal crystallographic directions.The solid lines are the fits to the thermal damping factor ofthe Lifshitz-Kosevich expression (see text for details). k = 1 / ψ is given by ψ = [( − Φ B π ) − δ ], where δ is the additional phase factor which dependson the dimensionality of the Fermi surface. δ = 0 for2 D Fermi surface and δ = ± / D Fermi surfacewhere the + sign corresponds to hole pocket and − signcorresponds to electron pocket.For B (cid:107) [010] two additional frequencies are observedat 61 T and 295 T while these two frequencies are absentin the other two directions. This indicates the anisotropicnature of the Fermi surface and these two frequencies rep-resents a 2 D Fermi surface. Using the Onsager’s relation F = (cid:126) πe A F , we have estimated the cross-sectional areafor γ frequency as 0.028 ˚A − . In order to estimate theDingle temperature T D of the γ -branch, we have used theband-pass filter to isolate the oscillations correspondingto the frequency 295 T and used the 2 D L-K expression inthe high magnetic field region as shown in Fig. 5(a). Thereasonably good fit resulted in a Dingle temperature T D as 5.52 K. A similar estimate of the Dingle temperaturewas made for the β which resulted in T D = 10.62 K. Fromthe obtained values of m ∗ , T D , we have estimated theFermi wave vector k F , Fermi velocity v F , quantum scat-tering τ q , quantum mobility µ q and the surface carrierdensity n as listed in Table I. The information aboutthe Berry phase in YSi has been extracted by plottingthe Landau level (LL) fan diagram. We have assignedthe LL-index n (= F/B + ψ ) to the minima of quantumoscillations. For B (cid:107) [010] there are multiple frequen-cies in the quantum oscillation hence we have used theband pass filtered oscillation to construct the LL-fan di-agram which is plotted as n as a function of the inversemagnetic field as shown in Fig. 5(b) and (d). The plotsare straight line and the slope corresponds to the oscilla-tion frequency while the intercept gives the phase factor ψ mentioned in Eqn. 1. For the γ -branch, the interceptis − .
12, this amounts to a Berry phase (Φ B ) of 1 . π ,which reveals the non-trivial character of γ pocket. Simi-larly, for the β -branch (61 T) the intercept was estimatedto be − .
3, this amounts to Φ B of 1 . π suggesting a triv-ial nature of this band and for the α -branch the interceptwas 0 .
114 and − .
82 for B (cid:107) [100] and B (cid:107) [001] direc-tions (Fig. 5(e) and (f)), respectively with Φ B = π and2 . π suggesting a non-trivial nature of the Fermi sur-face. For α -branch the second-lowest Landau level hasbeen achieved in a field of 14 T that exhibits a smallFermi surface which is smaller than the reported Fermisurfaces for Cd As and ZrSiS systems [30, 33]. CONCLUSION
We have grown the single crystal of YSi using theCzochralski method. The band structure shows mul-tiple nodal band crossings near the Fermi level. TheFermi surface is formed by both the electron and holebands, resulting in multi Fermi pockets with both smalland large areas. Importantly, our calculated quantumoscillation frequencies match well with the observed fre-quencies. The dHvA oscillation measurements performedalong the three principal crystallographic directions haverevealed three Fermi pockets, namely, α , β , and γ , whichare well seen in our first-principles calculations. The α band obeys linear dispersion around the Fermi energywhich results in a very small effective mass 0 . m e for B (cid:107) [100]-direction and in fact, it is lighter than otherpopular Dirac and Weyl semimetals [34–40]. The esti-mated Berry phase for this branch is π which refers toa non-trivial topological character for this pocket and is -4-2024 Δ M ( a . u ) H (T -1 ) YSi B // [010]295 T ( γ ) LL i nd e x n H (T -1 ) YSi B // [010]
295 T ( γ ) Fit -202 Δ M ( a . u ) H (T -1 ) YSi B // [010]61 T ( β ) LL i nd e x n H (T -1 ) YSi B // [010]
61 T ( β ) Fit (a) (b)(c) (d) LL i nd e x n H (T -1 ) YSi H // [100]
21 T ( α ) Fit LL i nd e x n H (T -1 ) YSi H // [001]
34 T ( α ) Fit (e) (f)
FIG. 5: (a) The band pass filtered dHvA oscillations of the γ -branch measured at T = 2 K for B (cid:107) [010], the solid linecorresponds to the L-K formula fitting. (b) LL-fan diagramcorresponding to the γ -branch. (c) L-K formula fitting ofthe filtered β -branch oscillations, (d) the LL-fan diagram cor-responding to β -branch. (e) and (f) LL-fan diagram of the α -branch along the [100] and [001] directions. observed along all the three principal directions with sub-tle anisotropy. Similarly, γ -branch also possess very lightmass because of linear band dispersion. The observednon-trivial Fermi pocket corresponding to the γ -branchshows a 2 D character with a quantum scattering time2 . × − s, which is of the same order as observedin ZrSiS and PtBi systems [33, 35]. The large quantummean free path and high quantum mobility signifies thesuppression of backscattering in YSi. These results makeYSi is an interesting topological material where nontriv-ial band crossings can be explored in angle-resolved pho-toemission spectroscopy (ARPES) and scanning tunnel-ing microscopy (STM). ∗ Electronic address: [email protected][1] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. ,021004 (2016).[2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[3] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057 (2011).[4] M. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A.Bernevig, and Z. Wang, Nature , 480 (2019).[5] J. Hu, S.-Y. Xu, N. Ni, and Z. Mao, Annual Review ofMaterials Research , 207 (2019).[6] L. Fu and C. L. Kane, Physical Review B , 045302(2007).[7] B. Singh, A. Sharma, H. Lin, M. Z. Hasan, R. Prasad,and A. Bansil, Phys. Rev. B , 115208 (2012).[8] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, PhysicalReview B , 125427 (2013).[9] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu,H. Weng, X. Dai, and Z. Fang, Physical Review B ,195320 (2012).[10] B. Lv, N. Xu, H. Weng, J. Ma, P. Richard, X. Huang,L. Zhao, G. Chen, C. Matt, F. Bisti, et al. , NaturePhysics , 724 (2015).[11] S.-Y. Xu, C. Liu, S. Kushwaha, T.-R. Chang, J. Krizan,R. Sankar, C. Polley, J. Adell, T. Balasubramanian,K. Miyamoto, et al. , arXiv preprint arXiv:1312.7624(2013).[12] C. Zhang, Y. Zhang, X. Yuan, S. Lu, J. Zhang,A. Narayan, Y. Liu, H. Zhang, Z. Ni, R. Liu, et al. , Na-ture , 331 (2019).[13] C. Zhang, E. Zhang, W. Wang, Y. Liu, Z.-G. Chen, S. Lu,S. Liang, J. Cao, X. Yuan, L. Tang, et al. , Nature com-munications , 1 (2017).[14] S. K. Kushwaha, J. W. Krizan, B. E. Feldman, A. Gye-nis, M. T. Randeria, J. Xiong, S.-Y. Xu, N. Alidoust,I. Belopolski, T. Liang, et al. , APL materials , 041504(2015).[15] C.-K. Chiu and A. P. Schnyder, in Journal of Physics:Conference Series , Vol. 603 (IOP Publishing, 2015) p.012002.[16] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, andN. Ong, Nature materials , 280 (2015).[17] T. Button, I. McColm, and J. Ward, Journal of the LessCommon Metals , 205 (1990).[18] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[19] P. Hohenberg and W. Kohn, Phys. Rev. , B864(1964).[20] G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999).[21] G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996).[22] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). [23] A. Kokalj, Journal of Molecular Graphics and Modelling , 176 (1999).[24] P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka,J. Luitz, et al. , An augmented plane wave+ local orbitalsprogram for calculating crystal properties (2001).[25] P. Rourke and S. Julian, Computer Physics Communica-tions , 324 (2012).[26] E. Parthe, Acta Crystallographica , 559 (1959).[27] J. Rodr´ıguez-Carvajal, physica B , 55 (1993).[28] M. Matin, R. Mondal, N. Barman, A. Thamizhavel, andS. Dhar, Physical Review B , 205130 (2018).[29] R. Mondal, S. Sasmal, R. Kulkarni, A. Maurya, A. Naka-mura, D. Aoki, H. Harima, and A. Thamizhavel, Physi-cal Review B , 115158 (2020).[30] L. He, X. Hong, J. Dong, J. Pan, Z. Zhang, J. Zhang,and S. Li, Physical review letters , 246402 (2014).[31] N. Kumar, K. Manna, Y. Qi, S.-C. Wu, L. Wang, B. Yan,C. Felser, and C. Shekhar, Physical Review B , 121109(2017).[32] J. Hu, Z. Tang, J. Liu, X. Liu, Y. Zhu, D. Graf, K. Myhro,S. Tran, C. N. Lau, J. Wei, et al. , Physical review letters , 016602 (2016).[33] J. Hu, Z. Tang, J. Liu, Y. Zhu, J. Wei, and Z. Mao,Physical Review B , 045127 (2017).[34] J. Du, H. Wang, Q. Chen, Q. Mao, R. Khan, B. Xu,Y. Zhou, Y. Zhang, J. Yang, B. Chen, et al. , Sci-ence China Physics, Mechanics & Astronomy , 657406(2016).[35] W. Gao, N. Hao, F.-W. Zheng, W. Ning, M. Wu, X. Zhu,G. Zheng, J. Zhang, J. Lu, H. Zhang, et al. , Physicalreview letters , 256601 (2017).[36] T. Butcher, J. Hornung, T. F¨orster, M. Uhlarz, J. Klotz,I. Sheikin, J. Wosnitza, and D. Kaczorowski, PhysicalReview B , 245112 (2019).[37] F. Wu, C. Guo, M. Smidman, J. Zhang, Y. Chen, J. Sin-gleton, and H. Yuan, npj Quantum Materials , 1 (2019).[38] R. Singha, B. Satpati, and P. Mandal, Scientific reports , 1 (2017).[39] M. Inamdar, M. Kriegisch, L. Shafeek, A. Sidorenko,H. M¨uller, A. Prokofiev, P. Blaha, and S. Paschen, in Solid State Phenomena , Vol. 194 (Trans Tech Publ, 2013)pp. 88–91.[40] B. Chen, X. Duan, H. Wang, J. Du, Y. Zhou, C. Xu,Y. Zhang, L. Zhang, M. Wei, Z. Xia, et al. , npj QuantumMaterials3