Quantum oscillations revealing topological band in kagome metal ScV6Sn6

Compounds with kagome lattice structure are known to exhibit Dirac cones, flat bands, and van Hove singularities, which host numerous versatile quantum phenomena. Inspired by these intriguing properties, we investigate the temperature and magnetic field dependent electrical transports along with the theoretical calculations of ScV6Sn6, a nonmagnetic charge density wave (CDW) compound. At low temperatures, the compound exhibits Shubnikov-de Haas quantum oscillations, which help to design the Fermi surface (FS) topology. This analysis reveals the existence of several small FSs in the Brillouin zone, combined with a large FS. Among them, the FS possessing Dirac band is a non-trivial and generates a non-zero Berry phase. In addition, the compound also shows the anomalous Hall-like behaviour up to the CDW with the CDW phase, ScV6Sn6 presents a unique material example of the versatile HfFe6Ge6 family and provides various promising opportunities to explore the series further.

The kagome lattice gives rise to unavoidable exotic topological electronic states namely Dirac point [1,2], Van Hove singularity [3,4] and flat band [1,2].These features have been extensively studied by different spectroscopy experiments in various kagome compounds.
Depending on band filling and interactions, they accommodate a number of unconventional quantum phases, topological band structure [5][6][7], Chern insulator [8], unconventional superconductivity [9,10] and charge density wave (CDW) [11][12][13][14].The profound impacts of the kagome lattice have recently been highlighted by the discovery of CDW below 90 K in nonmagnetic AV3Sb5 and antiferromagnetic hexagonal-FeGe [11].The charge orders in these systems are distinct.In AV3Sb5 [14][15][16][17][18], it is chiral, whereas in FeGe [11,19] magnetism is mediated.Similar to AV3Sb5 and FeGe, the CDW transition temperature (at 92 K) was recently discovered in the ScV6Sn6 compound [20].ScV6Sn6 is the only known compound from the vast hexagonal HfFe6Ge6 family to exhibit a CDW phase transition, which is a first-order-like transition with propagation vector (1/3, 1/3, 1/3) revealed by early X-ray and neutron experiments [20].The strongly coupled out-of-plane lattice dynamics suggests an unconventional nature of the CDW phase, which is different from the in-plane lattice dynamics in AV3Sb5 [12].Furthermore, CDW of ScV6Sn6 is easy to tune [21] and exhibits various microscopic features such as critical role of phonons [22][23][24][25], large spin Berry curvature [26], partial bandgap opening [26][27][28], hidden magnetism [29].As the first such member of the versatile kagome family of HfFe6Ge6-type compounds, it is worthwhile to study the temperature and magnetic field dependent electrical transport properties of high-quality single crystals of ScV6Sn6 in order to obtain information about the Fermi surface (FS) and lattice dynamics, reporting in this letter.ScV6Sn6 crystallizes at room temperature in a hexagonal centrosymmetric structure with a P6/mmm space group.The V atoms form a kagome lattice in the ab-plane (Fig. 1(a) left), while the Sn atoms sit above and below, separating the lattice [20].High quality hexagonal single crystals (Fig. 1(b), inset) were grown by the flux method, see supplemental information (SI) [30].To facilitate further measurements, the crystal orientations were marked in Cartesian coordinates  " ,  " , ̂, which correspond to ( 2 1 1 0 ), ( 0 1 1 0 ), ( 0 0 0 1 ) crystallographic directions in the hexagonal crystal structure, respectively.Laue X-ray diffraction patterns were recorded along ̂ and show a clear six-fold symmetry (Fig. 1(a), right).The patterns fit well with the lattice parameters a = b = 5.4733(5) Å and c = 9.1724(8) Å derived from the fitting of power X-ray patterns (Fig. S1) [30,31].The measured chemical compositions from the energy dispersive X-ray spectrum are close to the stoichiometric atomic ratio of Sc:V:Sn = 1:5.97:6.07.
We further cut the crystals in defined orientations of Hall-bar geometry and fabricated four-probe Pt-wire contacts with highly conductive Ag paint.The I-V characteristics at 2 K in different fields are linear (Fig. 1(b)), confirming the high quality of the ohmic contacts.The appearance of CDW in this compound is a key property, and we found it in the temperaturedependent magnetization and the electrical resistivity (Figs.1(c-e)).A sharp transition appears at 92 K, depicting the CDW transition temperature similar to the previous reports [19,20].The measured magnetic susceptibility in Fig. 1(c) describes a weak Pauli paramagnetic-like behavior, which is different from its structural sister compound YV6Sn6 [32].The zero-field electrical resistivity shows a metallic behavior with temperature, and both ryy and rzz drop suddenly at the CDW phase transition due to the large softening of the acoustic phonon modes (Figs.1(d, e)) [23,24].The typical values of ryy (rzz) at 2 K are found to be 1 ´ 10 -5 (4.58 ´ 10 - 5 ) W cm and the resulting residual resistivity ratio (RRR = r300K/r2K) is 8.9(4.3).Noticeably, the rzz changes significantly than the ryy at the CDW phase transition.
After the basic characterization of the crystals, we now focused on the magnetic field, B-dependent transverse resistivity, which were measured at different temperatures and angles in the field of ± 9 T. The angular dependence of the field is defined as q (= x " ® z ") and f (= x " ® y "), where 0 o = B ||  ", 90 o = B ||  " or .First, we describe the behavior of rzz when B ||  " (q and f = 0), and the measured data are shown in Fig. S6(a) [30].At low temperatures, rzz exhibits clear Shubnikov de-Haas (SdH) quantum oscillations which are a striking feature that helps to probe the low-energy bands.At 2 K, the quantum oscillation starts from the field around 1.5 T, as seen in Fig. 2(a) for the drzz/dB plot and it is easily visible up to 12 K (Fig. [30].The magnetoresistance MR (= !! ()  !! (0) − 1 ⁄ ) is estimated, and the highest value is found to be 170 % at the lowest temperature of 2 K (Fig. S6(a)) [30].Unlike the other typical semimetals, e.g., NbP, the present compound ScV6Sn6 does not show an extremely large MR [33].However, the SdH oscillation indicates a low effective mass of the electron charge carrier in the CDW phase [23,24].To obtain the amplitude of the SdH oscillations, a smooth polynomial background was subtracted from the measured rzz.The resultant D !" at several temperatures is plotted as a function of 1/B, as shown in Fig. 2(b).As expected, the oscillations are periodic in 1/B and they arise from quantization of the energy level further, forming Landau levels (LLs) [34].The analysis of the temperature and field-dependent periodic oscillations provides insight into the fermiology of the compound and the associated physical characteristics of the charge carriers.directly related to the Berry phase within δ, equals to 0 for the 2D system and ± ' ( for the 3D system ( ± corresponds to the contribution from the minimal/maximal cross section) [36,39,40].Among the various parameters of FS that can be estimated using L-K formula, the estimation of the Berry phase is very promising for proving the non-trivial topology of the band.Accordingly, we designed the Landau fan diagram by assigning LLs to the oscillatory extrema.The integer n-th LLs are assigned to maxima when  )) ≫  *) , while they are assigned to minima when  )) ≪  *) [36,39,41].For ScV6Sn6, where  !! ~10 *! which satisfies the condition of  )) ≫  *) , the integer n-th are assigned to the maxima and the half integer ( + ' ( for q ³ 60 o .While it is possible to estimate the Berry phase factor q > 20 o , since the Fb is absent, but the SdH amplitude declines rapidly.At all other angles of q and f , the Fb is consistently present as can be seen in Fig. 4(b), which intervenes the amplitude in terms of the beating patterns [44,45], as can be seen in Figs.S9(c) and (d) [30].One method to separate the amplitude corresponding each frequency is to use the band-passing filter as shown in Figs.S6(c) and (d) [30,46]), but may not be very reliable.We also tried to estimate to Berry phase related to Fa and Fb.As we can see, Fa shows very weak oscillatory amplitude with limited cycle up to 9 T. One the other hand, the presence of the beating patterns harms the amplitude of the Fb, which only appears for q £ 20 o and after that the Fb vanishes.These together make difficult to carry out the Berry phase analysis for Fa and Fb.

S6(b)
To determine the effective mass for the carriers corresponding to each FS, the temperature dependence of the amplitude of each peak in the FFT is plotted, and then it is fitted with RT term (Fig. 2(d)) from the L-K formula.From these fittings, we find the values of m * a, b,g = 0.105 m0, 0.128 m0, 0.184 m0.These masses are very low (even lower than CsV3Sb5 [35,36]).Furthermore, the Dingle temperature TD can be estimated from the semi-log plot described by ln(A/RT) ∝ -14.69m * TD/B (Fig. S8) [30], where A corresponds to the amplitude of the SdH oscillations.We found TD to be 2.5 K for γ-pocket and the corresponding  S1 [30].
To capture the topology in electronic band structure of ScV6Sn6, we performed detailed band structure calculations by using density functional theory (DFT).Due to the presence of CDW transition, we transformed the band structure of low temperature (LT) phase into the high-temperature (HT) phase, and the LT phase (CDW phase) has been characterized at a temperature of 50 K and the lattice parameters are taken from the previous report [20].From Utilizing the transformation matrix, we ascertain that the original M point at (1/2, 0, 0) and the Γ point at (0, 0, 0) have been mapped to the new M1 point at (1/2, 0, -1/2) and Γ point at (0, 0, 0), respectively.As a result, we discovered that the genuine Dirac point (labeled as DP in Fig. 3b) has been mapped to the position of (0.341666, 0.316666, -0.658327).From this magnified view of the band structure near this point, its Dirac nature is clearly confirmed.Similar to the Dirac points in graphene systems, this point develops a band gap when considering spin-orbit coupling as shown in Fig. when the q > 20 o , the b disappears.This helps further to improve the accuracy of Berry phase analysis related to g.In order to allocate the experimentally observed frequencies, we theoretically calculated 3D FSs and their corresponding frequency from the unfolded band structure of the LT phase.As we can see in Fig. 4(b), all observed frequencies are accurately replicated by the calculated 3D FSs, except for higher frequency (green line).In addition, the calculated Fb is present at all angles, but it disappears q > 20 o in the experiment, which needs a further investigation.Fig. 4(c) shows 3D FSs within the BZ, in which a large electron pocket (blue color) around G covers the whole BZ together with a small banana-shaped FS (red color) along the M-L direction, a small apple-shaped FS (blue color) at K, and a larger FS at M (yellow color).Another spindle-shaped FS at K which comes from the same band with the FS at M and is covered by the apple-shaped pocket.After comparing with DFT results, we found that the Fa belongs to the banana-shaped FS (Fig. 4(e)).The Fb belongs to the spindle-shaped FS (Fig. ( The field dependent rzy shows a non-linear behavior, which is primarily thought to be the presence of multi-carriers in a nonmagnetic system such as ScV6Sn6, since the multiple bands are around the EF.To clarify this apparent reason, we fitted both the Hall and longitudinal conductivities by using the standard two-band model (see Fig. S5 in SI) [30,[47][48][49][50].Both fits are seemingly good, but their fitted parameters are very different from each other.The fitting of the syz reflects both electron carriers, while the fitting of syy shows both electron and hole carriers.For the implementation of a reliable and accurate model, the parameters from the both fittings must be the same within the error bar as reported in the typical multiband systems [49,50].Therefore, given the type of carriers, the concentrations and mobilities are highly inconsistent, indicating that the two-band model may not be applicable to the nonlinear Hall effect for ScV6Sn6.It is also worthy to note here that the Hall resistivity from a typical multi-band system is nonlinear close to zero field, while it is become linear at high field [33,[51][52][53].In contrast, the present compound ScV6Sn6 shows the linear Hall close to zero field and it bends at high field (> 2 T) like a typical soft ferromagnetic system [54,55].
By considering the anomalous behavior naively, we subtracted the linear part from the high field and the anomalous part of the Hall resistivity at different temperatures is shown in Fig.

5(a)
. Like in soft ferromagnet, an anomaly in the rzy (a sharp increase at a lower field followed by saturation with a further increase in the field) is observed, which is attributed to anomalous Hall like behavior.Such an anomaly usually appears as a hallmark in magnetic systems and it resembles the magnetization curve, wherein magnetic spins play the role (Fig. 5(d, e)).
However, magnetization is linear with the field (see Fig. S2 in SI) [30] and does not follow the Hall resistivity in the present case.The as calculated value of anomalous Hall conductivity systems [14,15,38].The AHC remains almost constant up to 45 o and starts to decrease with increasing angle further (Fig. 5(c)), while it sustains up to the CDW transition temperature.In contrast to the involvement of magnetism (Figs.5(d), (e)), one of the possible explanations could be similar to the origin in AV3Sb5 (Fig. 5(f)), where the CDW forms a current loop and breaks the time reversal symmetry (TRS) [16,56,57], since ScV6Sn6 also breaks the TRS measured by the muon spectroscopy [29] and to speculate such loop current is a bit premature.
In the present scenario, it is difficult to understand the observed Hall behavior of ScV6Sn6, and needs to do further study.Moreover, the behavior of  *! is closer to ZrTe5, assuming the presence of a single Dirac band that splits into a pair of Weyl points in presence of the magnetic field [58].However, the several bands are present in ScV6Sn6, where the concept of Weyl points is not easily applicable.In conclusion, with the help of field-dependent electrical transports and theoretical calculations, we have revealed the fermiology of ScV6Sn6.Along with a large FS, the BZ is comprised of the several other small electron FSs, except for the one tiny hole FS.The small FSs exhibit SdH oscillations, resulting in the small effective mass, and the symmetric and asymmetric nature of the FSs for electron and hole, respectively.The electron Dirac band located at the K point gives rise to the non-zero Berry phase, proving that it is a non-trivial topological FS.Furthermore, the non-linear Hall, which resembles the anomalous Hall, does not fit well with the two-carrier model.Finding the microscopic origins for the anomalous Halllike signal remains an important issue that requires further theoretical and experimental investigations.
Note added.During the preparation of our work, a similar work including anomalous Hall-like behavior in ScV6Sn6 was found [59].

APPENDIX B: ELECTRONIC BAND STRUCTURE CALCULATIONS
Ab initio calculations were performed using density functional theory (DFT) implemented in the Vienna Ab initio Simulation Package (VASP) [60,61].The projector augmented wave method [62] and the generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [63] were used for the calculation.Here, we used refined lattice parameters for the band structure calculations, including the pristine structure and the CDW structure.A 15 × 15 × 7 k-mesh was used to sample the Brillouin zone of the pristine structure, with an energy cut-off of 500 eV.For the CDW structure, an 8 × 8 × 8 kmesh was used, with an energy cut-off of 450 eV.The CDW band structures were unfolded using the VASPKIT toolkit [64].To obtain the quantum oscillation frequencies, we constructed a Wannier tight-binding Hamiltonian using the Wannier90 code [65], including Sc-3d, V-3d, and Sn-5p orbitals.
current transport (ACT) option was used for measurement with an excitation current of 1.5 mA at a fixed frequency of 93 Hz.Data were collected in the temperature range of 2-300 K, and the magnetic field range of -9 to 9 T in sweep mode.
For the magnetic property measurements of ScV6Sn6, we selected flux-free crystals and a typical optical image as shown in the inset of Fig. S2 (a).For ease of further reference, we marked the crystal axes in the Cartesian coordinates  ",  ", , which are correspond to (2 This is also a distinct behavior from the well-known AV3Sb5 (A= K, Rb, Cs) family, which follows the Curie-Weiss law with a small effective moment (0.22 μB / V atom for KV3Sb5) [5].

Additional electronic transport measurements and analyses
Figure S3 shows the additional measurement of electronic transport properties when magnetic field is applied along  " and electric current is injected along  ".We have calculated the band structures of ScV6Sn6 in the high temperature (HT) phase as well as in the low temperature (LT) CDW phase.The LT phase band structure is unfolded to match the one of HT phase when considering spin orbital coupling (SOC) (Fig. S11a) and neglecting SOC (Fig. S11b), respectively.The band structures didn't change a lot with or without considering SOC.Specifically, for LT phase, energy bands along A-H-L-A (i.e.kz = π plane) are gapped by CDW phase transition, which is consist with the results of angle-resolved photoemission spectrum (ARPES) [13].The orbital projected band structure is also calculated and plotted in Fig. S11c and d

FIG. 1 .
FIG. 1. Kagome lattice, Laue pattern, I-V characteristic, CDW phase transition of ScV6Sn6.(a) Kagome lattice of ScV6Sn6 viewed along the c-axis (left) and recorded single crystal Laue patterns along the c-axis.(b) I-V characteristics at various fields at 2K.The inset is an optical image of the crystal with Cartesian coordinate.(c) Magnetic susceptibility along  ".Resistivity in zero-field (d) along  " and (c) along , where the sudden jumps show CDW phase transition.

FIG. 2 .
FIG. 2. Shubnikov-de Haas (SdH) oscillations and Berry phase.(a) Longitudinal rzz at 2 K and its first derivative with field showing quantum oscillations.(b) Background subtracted SdH oscillations amplitude at several temperatures.(c) Corresponding FFT amplitude exhibiting a, b and g frequencies.(d) FFT amplitude fit estimating effective mass.(e) SdH oscillations when B || ̂ and their FFT spectrum (inset) showing only the g frequency in the field window of 5-8 T. (f) Intercept of the Landau fan diagram revealing non-zero Berry phase.(g) Angular dependent of the Berry phase.

'+
)-th are assigned minimum of D !" , as shown in Fig.2(e, f)).The Landau indices n and 1/B satisfy the Lifshitz-Onsager relationship, which is described by the equation = % & +  − .The slope of the plot between n and 1/B provides the oscillatory frequency, while the intercept offers the information on the Berry phase.To attend an accurate estimation of Berry phase of Fg, we selected a field window between 5 to 8 T to exclude the weak Fa oscillatory component (Fig.2(e) inset).From Fig.2(f), the intercept of the Landau fan diagram for Fg in terms of the phase factor γ − δ is estimated to be -0.101,indicating a nontrivial Berry phase[42,43].Similarly, we obtained the Berry phase further for the angular dependence of the field from  " to ̂ without interference from the other exiting frequencies, and the values are given in Fig.2(g).The figure indicates the Berry phase factor γ − δ falls within the range of 0 ±

Fig. S11 [ 30
Fig. S11[30], the unfolded band structures of LT and HT phases are in a good agreement with FIG. 3. Band structure in charge density wave (CDW) phase.(a) Band structure of ScV6Sn6 CDW phase with the presence of spin orbital coupling (SOC).The structural parameters are the same with the report [20] at 50 K.The magnified view of the band structure around K1 locating the Dirac point (DP) (b) without the SOC and (c) with the SOC.

FIG. 4 .
FIG. 4. SdH oscillations and fermiology.(a) SdH oscillation frequency at different rotating angles q (=  " ® ) and f (=  " ®  "), where dotted lines represent their tracking with angle.The insets show a sketch of the field rotation.(b) Angular dependence of Fa,b,g, where Fa,b do not change while Fg shifts slightly.Solid lines are calculated frequencies corresponding.The error bar is taken from the half width of the half maximum value of the frequency peaks.(c) Brillouin zone and three-dimensional (3D) Fermi surfaces (FSs) of the bands at the Fermi energy.(d-g), Enlarged FSs, including the FS at M point (d, yellow), the banana-shaped αpocket along M-L (e, red), the spindle-shaped inner β-pocket at K point (f, yellow), and the apple-shaped outer γ-pocket at K point (g, blue).

FIG. 5 .
FIG. 5. Anomalous Hall-like behavior in ScV6Sn6.(a) Field-dependent measured anomalous behavior of Hall resistivity, rzy at various temperatures (inset) and extracted Hall resistivity  !* -.(b) Temperature and (c) angular dependent corresponding anomalous Hall like conductivity.Electrons picking up anomalous velocity from the different sources and their typical example.(d) Non-zero Berry phase from ferromagnetic spins.(e) Topological orbital moment due to spins chirality.(f) Formation of local loop currents.
This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) under SFB1143 (project no.247310070), the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat(EXC 2147, project no.390858490) and the QUAST-FOR5249-449872909. S.R. thanks the Alexander von Humboldt Foundation for a fellowship.APPENDIX A: MAGNETIC AND ELECTRICAL MEASUREMENTS Temperature-dependent susceptibility (2-300 K) under various magnetic fields was measured in zero field cooled and field cooled configuration in a Magnetic Properties Measurement System (MPMS, Quantum Design Inc.) equipped with a Superconducting Quantum Interference Device -Vibrating Sample Magnetometer (SQUID-VSM) option.Temperaturedependent four-probe transverse and Hall resistivities were measured on a standard rotating sample holder in Physical Property Measurement System (PPMS, Quantum Design Inc.).The alternative current transport (ACT) option was used for measurement with an excitation current of 1.5 mA at a fixed frequency of 93 Hz.Data were collected in the temperature range of 2-300 K, and the magnetic field range of -9 to 9 T in sweep mode.
, (0001) crystallographic directions in the hexagonal crystal structure.The isomagnetic and isothermal magnetic properties were measured by applying a field along  " and ̂, the measured data are shown in Figs.S2 (a) (b) and S2 (c) (d), respectively.The magnetization decreases linearly with temperature and then suddenly drops at 92 K.The sudden decrease marks the charge density wave (CDW) transition temperature, TCDW.Noticeably, this magnetic above TCDW does not follow the Curie-Weiss paramagnetic characteristic as its sister compound YV6Sn6[3]  follows.After 50 K which is far below the TCDW, the magnetization starts to increase with decreasing temperature and behaves like a week Pauli paramagnetic behavior[4].
FIG. S3.Resistivity, MR, carrier density and mobility, Hall resistivity.(a) Temperaturedependent resistivity ρyy at various fields when I ||  ", B ||  ".(b) Magnetoresistance as functions of magnetic field at various temperatures.(c) Estimated carrier concentration and mobility as functions of temperature.The sudden change is related to CDW phase transition, which is also highlighted by the vertical black dashed line.(d) Hall resistivity as functions of magnetic field at different rotating angles.The field is rotated from the x " to the y " directions as shown in the inset.
. It is clear that the kagome V d orbitals dominate the bands close to the Fermi level.Only a few bands along M-K, G-A, and H-K are contributed by Sn p orbitals, while the d orbitals from Sc are negligible.