Tuning charge density wave of kagome metal ScV6Sn6

Compounds with a kagome lattice exhibit intriguing properties and the charge density wave (CDW) adds an additional layer of interest to research on them. In this study, we investigate the temperature and magnetic field dependent electrical properties under a chemical substitution and hydrostatic pressure of ScV6Sn6, a non-magnetic CDW compound. Substituting 5% Cr at the V site or applying 1.5 GPa of pressure shifts the CDW from 92 K to ∼ 50 K. This shift is attributed to the movement of the imaginary phonon band, as revealed by the phonon dispersion relation. The longitudinal and Hall resistivities respond differently under these stimuli. The magnetoresistance (MR) retains its quasilinear behavior under pressure, but it becomes quadratic after Cr substitution. The anomalous Hall-like behavior of the parent compound persists up to the respective CDW transition under pressure, after which it decreases sharply. In contrast, the longitudinal and Hall resistivities of Cr substituted compounds follow a two-band model and originate from the multi carrier effect. These results clearly highlight the role of phonon contributions in the CDW transition and call for further investigation into the origin of the anomalous Hall-like behavior in the parent compound.


Introduction
Kagome-lattice compounds are known to exhibit unavoidable exotic topological electronic states [1][2][3][4][5][6][7].The discovery of the charge density wave (CDW) in some of these compounds has introduced a captivating dimension to the research [8][9][10].In AV3Sb5 (A= K, Rb, Cs), the CDW coexists with a superconducting ground state [5,[11][12][13][14][15][16][17].In antiferromagnetic hexagonal-FeGe, the CDW amplifies the ordered moment, justifying a deeper correlation between CDW and magnetism [9,18,19].Recently, the CDW in ScV6Sn6 was discovered at 92 K, making it the only known compound in the vast HfFe6Ge6 kagome family with the CDW state.The outof-plane lattice dynamics of the CDW suggests an unconventional nature [10].Additionally, the CDW exhibits various microscopic features, such as the critical role of phonons [20][21][22][23], substantial spin Berry curvature [24], partial bandgap opening [25,26], and hidden magnetism [27].It is worth noting that the CDW phase of ScV6Sn6 is very fragile without any additional features of superconductivity and magnetism.The application of either internal chemical pressure through doping or external physical pressure can completely suppress the CDW due to minor modifications in the Sc-Sn and Sn-Sn bonds [26,28,29].The phase remains robust at lower doping levels [26].The Fermi surface (FS) comprises of electron-type Fermi pockets except for a tiny hole-type pocket, which remain almost unchanged after the CDW transition [30].Furthermore, ScV6Sn6 exhibits an anomalous Hall-like behavior [30,31], which cannot be explained by multiple band model, but it rather indicates a hidden magnetism origin [27].Interestingly, this anomalous Hall-like feature coexists with the CDW phase and dies down upon approaching TCDW.This prompts a speculation of a correlation between CDW and anomalous Hall-like feature.This speculation can be tested by performing pressuredependent magneto-transport measurements since CDW phase seems to rely on loose Sc-Sn-Sn chains which are highly impacted by pressure [26,28,29].Owing to several exotic microscopic features associated with the Sc-Sn and Sn-Sn bonds, as well as hidden magnetism related to anomalous Hall-like behavior, it is compelling to investigate that how the anomalous Hall-like behavior is coupled with the CDW phase.In this study, we have performed the temperature-dependent electrical transport of ScV6Sn6 under pressure and Cr doping compounds at the V site.This comprehensive study highlights how physical and chemical pressure affect the CDW, emphasizing the changes in electronic properties, such as conductivity, band structure and FS.

Results and discussion
The parent compound ScV6Sn6 crystallizes in a hexagonal centrosymmetric structure with a P6/mmm space group at room temperature.The V atoms form a double kagome lattice and are separated by ScSn2 and Sn2 layers along the c-axis (figure 1a).High quality hexagon shaped single crystals were grown using the flux method, see supplemental material (SM) [32].
To investigate the effects of internal chemical and external physical pressure on the CDW phase and related transport properties in ScV6Sn6, we performed systematic measurements on pristine single crystal under hydrostatic pressures and Cr substituted single crystals.The electric current and magnetic field were applied in the ab-plane and c-axis, respectively, as listed in table S1 in SM [32].For this study, we have selected two concentrations Sc(V1-xCrx)6Sn6 (nominal x = 0.05, 0.1) among the various levels of Cr substitution, along with the pristine compound.The radius of Cr atoms is similar to that of V atoms, which means that the lattice parameters would not change significantly.Table S2 in SM [32] summarizes the estimated lattice parameters of Cr substituted samples and the pristine crystal, obtained from the refinement of powder x-ray diffraction (PXRD) pattern (figure S1 in SM) [32].In figure 1a, the Cr occupies the V atomic position.The real substituting components x is approximately 0.086 for Sc(V0.95Cr0.05)6Sn6and ~0.157 for Sc(V0.9Cr0.1)6Sn6,which were averaged from several points measured by Energydispersive x-ray spectroscopy (EDXS) (see table S3 in SM) [32].The observed concentrations dependent TCDW of our samples are consistent with the previous report, where the CDW TCDW is approximately 60 K for x = 0.06 [26], but approximately 48 K for x = 0.086 in our work.
Figures 1b and 1c represent the clear Laue diffraction patterns with six-fold symmetry along the c-axis for both crystals, indicating their high quality.Table S1 in SM [32] lists the directions of external magnetic field and applied current used during the electronic transport measurements.The temperature-dependent longitudinal resistivity ( !! ) for both compositions shows metallic behavior like the parent compound, as displayed in figure 1e.However, the apparent change in  !! , which indicates the CDW phase transition, systematically varies.The CDW phase transition shifts from 92 K to 50 K for x = 0.086, and vanishes completely for x = 0.157, which is consistent with the previous report [26].The resistivity values at 2 K are typically 3.1 ´ 10 -5 Wcm and 4.5 ´ 10 -5 Wcm, and the residual resistivity ratio ( =  "##$  %$ ⁄ ) are 3.5 and 3.1 for x = 0.086 and 0.157, respectively.It is noteworthy that the substitutions of Y and Lu elements at the Sc position also affect the CDW in a similar manner due to the limitation of rattling in the Sc-Sn-Sn chains in substituted samples [29].The displacement of Sc and Sn atoms is responsible for strong electron-phonon coupling, the ultimate origin of the CDW [10,20,33,34].The substitution of Cr atoms for V has two effects: it weakens the bonding and destroys the kagome lattice, and it shifts the Fermi energy (EF) away from the von Hove singularity (vHS) (shown in figure 1d) [26].Therefore, the CDW phase transition is unlikely to be induced by the FS nesting mechanism, where the vHS is significant, but rather by the strong electron-phonon coupling [20,33,34].To investigate the effect of a magnetic field on  !! , we measured field-dependent  !! of the doped crystals and pressurized pristine ScV6Sn6 crystal at various temperatures.The estimated magnetoresistance is shown in figures 2a and 2b, where MR is defines as and  !! (0) are the resistivities with and without a field, respectively.The MR is significantly reduced to 10% for x = 0.086 and 6 % for x = 0.157, which is almost 20 and 30 times lower, respectively, than that of the pristine compound [30].This reduction can be attributed to the shift of the EF away from the Dirac band and vHS after substituting the Cr [26], resulting in the domination of the trivial charge carriers.
Furthermore, the MR behavior can be described by the Kohler's law, where  ∝ , and m is a temperature independent constant.When single scattering dominates at all temperatures, the Kohler's law allows the MR curves of all temperatures to collapse onto a single line.However, the role of temperature dependent charge carrier was recognized later, and then the extended Kohler's law is used [35].This law describes the  ∝ , where the temperature dependent term  ' = e (∑ n ( µ ( ( included as a temperature correlation.The expression of  ' depends on the carrier concentration ( ( ) and mobility ( ( ) extracted from the Hall resistivity.Figures 2c and 2d show the extended Kohler's plots for x = 0.086 and 0.157.The plot for different temperatures at x = 0.086 almost coincides in the log-log plot, yielding m = 1.72.However, the curves slightly deviate from the line below 50 K from the line due to the CDW transition, which is consistent with the pristine ScV6Sn6 [36].The deviation in the plot is due to the evolution of electronic band structure at CDW transition.The plot for x = 0.157, as displayed in figure 2d, also shows linear dependence in  n ' ρ # ⁄ and yields m = 1.71 without any deviation, indicating the complete suppression of CDW.Figures 2e and 2f display the Hall resistivity  +! of x = 0.086 and 0.157 at several temperatures, respectively.The  +! exhibits a nonlinear dependence on the magnetic field at low temperatures.After substituting Cr, the  +! changes remarkably compared to the parent compound [30,31].We estimated the conductivity model in the pristine compound, where a kink appears at CDW transition [31].One type of electrons, named electron-1, exhibit weak temperature dependence with the density of 10 19 cm - 3 and a mobility of 10 3 cm 2 V -1 s -1 .However, the density of electron-2 decreases from 10 22 cm - 3 to 10 20 cm -3 as the temperature is decreased from 250 K to 2 K. On increasing Cr concentration up to x = 0.157, the carrier concentration and mobility show similar temperature dependence as x = 0.086 with closely related values.To understand the origin of the Hall resistivity in the Cr substituted sample, we compare it with the pristine ScV6Sn6 sample.Previous reports on pristine crystal [30,31] has shown anomalous Hall effect (AHE)-like behavior in the Hall resistivity, which may be due to the formation of a loop current in the V kagome lattice, breaking the time reversal symmetry (TRS) [17,27].However, substitution of the Cr atoms in V site in the kagome lattice destroys the V-V bonds by inducing disorder.The presence of an additional valence electron in Cr causes the EF to shift upwards by 120 meV [26].This shift also moves the vHS further away from the EF, which is the main cause of the loop current order in AV3Sb5 systems [40][41][42][43].After studying the chemical substitution effect in the ScV6Sn6 crystals, we investigated the impact of external pressure effect on the electromagnetic transport properties and phonon band structures of pristine ScV6Sn6 crystal.Our external hydrostatic pressure measurements on the parent compound, as shown in figure 4a, demonstrate a systematic suppression of the CDW with pressure, as similar to the chemical substitution.At 1.5 GPa, the CDW appears at 54 K, which eventually disappears at 2.5 GPa, similar to the findings of a previous report [28].The pressure-temperature phase diagram, shown in the inset of figure 4a, illustrates the pressuredependent CDW.The rate of change of TCDW with pressure is slower than that of AV3Sb5 systems [44][45][46].The vanishing of CDW due to external pressure may suggest the similar physics, as observed with the chemical substitution.Additionally, first-principles calculations using density functional theory (DFT (see method section) were carried out to examine the dynamic instability of ScV6Sn6 under pressure.The implementation of generalized gradient approximation (GGA) method typically overestimates both pressures and relaxed lattice constants in numerical calculations [47,48].Figure 4b shows the phonon spectrum at pressures of 1.5 and 5 GPa.It is evident that the entire spectrum shifts upward with compressed volume due to hydrostatic pressure.Remarkably, the nearly flat imaginary modes at the kz = p plane at lower pressure, such as 1.5 GPa, are renormalized at higher pressure 5 GPa, indicating the stabilization of the pristine structure under higher pressure.The frequencies at A, L, and H points were tracked under various pressures, as shown in figure 4c, suggesting a critical pressure of ~5 GPa and indicating a stabilized state under this pressure.The completely relaxed lattice constants are also summarized in figures 4d and 4e.Taking into account the temperature effect and overestimation of the lattice constants, the calculated result of approximately 5 GPa is in good agreement with the experimental value of 2.5 GPa.
We have now studied the evolution of MR of parent compound with pressure.The MR shows a quasilinear behavior with magnetic field without any remarkable change up to 2 GPa as shown in figure 5a.However, at the critical pressure of 2.5 GPa, the quasilinear behavior turns to linear behavior.Conversely, the temperature evolution of MR at a fixed pressure does not change (figure 5b) and it is similar to that without pressure [30].At a pressure of 0.5 GPa, the Shubnikov de Haas oscillations are observed at a high magnetic field range as shown in figure S3 in SM [32].After analyzing the quantum oscillation, two frequencies are observed, corresponding to 25 T and 47 T, which are consistent with the results obtained at ambient pressure [30,32].A study of the pressure dependence of these oscillations can provide further insight into changes of the FS.Interestingly, our measured nonlinear Hall resistivity (figure 5c) exhibits a rapid increase with magnetic field and then saturates at around 2 T, which is a striking feature similar to the AHE behavior in soft ferromagnets [49,50].At the particular pressure of 1.5 GPa, a sign change in Hall signal appears at TCDW ~ 54 K, as displayed in figure 5d (see figure S5 for the Hall resistivity of the other pressures in SM) [32].The Hall resistivity data at 5 K and 80 K, which are above and above TCDW ~ 54 K, respectively, were carefully fitted by using a two-band model (see figure S4b in SM) [32].The data roughly fit and give unreliable fitting parameters, as similar results of ambient pressure [30,32].Meanwhile, no sign change of the carrier is observed from two-band model as expected from the Hall resistivity, indicating that the nonlinear Hall effect is possibly an AHE. ++ is assumed for simplicity.The values of  !! and  !+ .are typically found to be 10 5 W -1 cm -1 and 10 4 W -1 cm -1 , respectively.To understand the origin of the AHE like behavior from the unified theory, three conductivity regimes are normally identified [51,52].(i) The high conductivity regime ( !! > 10 / Ω 0* cm 0* ) where the  !+ .follows  !+ .∝  !! , (ii) the intrinsic regime (10 1 Ω 0* cm 0* <  !! < 10 / Ω 0* cm 0* ) where  !+ . is a constant and it is contributed by the Berry curvature, (iii) the low conducting hopping regime ( !! < 10 / Ω 0* cm 0* ) where  !+ .follows a power law relation with σ !+ .∝ σ !! *./ .Our pressure-dependent  !+ .exhibits a power law relation with an exponent close to 2, as displayed in figure 5e.The observed exponent of ~2 goes beyond the above-mentioned empirical regimes.However, the range of  !! (= 10 5 W -1 cm -1 ) lies in the region (ii), indicating an intrinsic origin of the AHE.Meanwhile, figure 5f shows the temperature-dependent evolution of AHE-like behavior at different pressures.At each pressure, the AHC monotonically decreases up to its respective TCDW and declines rapidly above it.For example, the slope changes at 50 K in 1.5 GPa and at 35 K in 2 GPa, which are quite similar to the observation at ambient pressure at 92 K [30].This unambiguously indicates that the AHE-like behavior is closely related to the CDW phase and needs further exploration.
The absence of any magnetic order in ScV6Sn6 makes the AHE-like behavior intriguing and opens up further discussion.Both the chemical substitution and pressure tune the Sc-Sn and Sn-Sn bonds distance [26,28,29], leading to the renormalization of imaginary phonon modes and suppression of the CDW.However, the pressures below 2 GPa do not significantly affect the behavior of Hall resistivity, indicating a marginal effect on the electronic instability.
Once CDW disappears, the AHE-like Hall signal also disappears.The AV3Sb5 family has also shown AHE-like behavior, with various speculations made, including the concept of loop current order [40][41][42][53][54][55][56].A very recent study of muon-spin spectroscopy (µSR) measurements on RbV3Sb5 shows a clear distinction at the CDW transition.The µSR signal enhances in thin layer samples and diminishes after doping in the kagome lattice [57].In ScV6Sn6, µSR spectroscopy has uncovered the hidden magnetism, with the µSR signal increasing with field [27].The loss of AHE-like feature in Cr-substituted samples strengthens the claim of hidden magnetism in the form of the loop current.

Conclusion
In summary, we have presented a comprehensive study of temperature-dependent magnetoelectrical properties of the kagome metal ScV6Sn6 through chemical substitution and hydrostatic pressure.Our results show that the CDW phase transition decreases with either substituting Cr at the V site or applying pressure.For substituted samples, the quadratic MR follows the extended Kohle's scaling law, and the Hall resistivity can be well described by the two-band model.In contrast, the parent compound under pressure retains the quasilinear MR and AHE-like behavior, which declines rapidly above the corresponding TCDW for each pressure.The observed AHE-like behavior seems to result from hidden magnetism that breaks TRS.Our findings light a path for the study of unconventional electronic properties in kagome metallic materials under external stimuli.

Crystal growth
Single crystals of Sc(V1-xCrx)6Sn6 (x = 0, 0.05, 0.157) are grown by Sn-flux method following the previous report [ref PRL, PRB].High-purity Sc, V, Cr, and Sn elements are cut into small pieces, loaded into an alumina crucible, and then sealed into a high-evacuated silica tube.The tube then was heated to 1327 K, maintained for 20 hours and then slowly cooled down to 1073 K at a ratio of 2 K/h.The hexagon-shaped single crystals are yielded on the bottom of crucible after centrifuge.

Crystal characterization
The x-ray diffraction (XRD) data is collected in a home-made x-ray diffractometer equipped with Cu Ka1 radiation (l ~ 1.54 Å).Crystal structure is refined via FullProf package [58].The chemical components are confirmed by EDXS option on a commercial scanning electronic microscopy.The Laue backscattering diffraction pattern are collected in a home-made Laue diffractometer to confirm the crystal orientation.

Transport properties at ambient and hydrostatic pressure
The crystals are cut into cuboid along specific crystallographic orientation to facilitate the measurement of magnetoelectrical transport properties.The longitudinal and Hall measurements for doped crystals were utilized in Physical Properties Measurement System (PPMS, Quantum Design Inc.).For the measurement under pressure, a standard piston-cylinder hydrostatic pressure cell (Quantum Design Inc.) was used in PPMS.The pressure was simultaneously calibrated by measuring the superconducting critical temperature shift of Pb element.A six probes method for all measurements in ambient pressure for doped crystals and high pressure for pristine crystals.

Phonon band calculation
The first-principles calculations are performed using the density functional theory (DFT) and PHONOPY code was utilized to extract the force constants and phonon spectra [63].The spinorbital coupling (SOC) was not included in the phonon calculations.The discrepancy between the relaxed lattice constants and experimental values could be caused by the pseudopotentials and GGA.

Tuning charge density wave of kagome metal ScV6Sn6
Table S1.Contacts geometry for measurements of transport proper7es.

Figure 1 .
Figure 1.Crystal structure, Laue pattern, band structure and resistivity.(a) A representative crystal structure of Cr substituted ScV6Sn6.Laue diffraction patterns along the c-axis for (b) x = 0.086 and (c) x = 0.157 single crystals.(e) Electronic band structure of ScV6Sn6 at high temperature (HT) above TCDW and low temperature (LT) in the CDW phase.Green shadows are the possible shifting of Fermi surface.(d) Resistivity as a function of temperature for x = 0, 0.086, 0.157 single crystals.
as shown in figures 3a and 3b.The two-band model can simultaneously reproduce  !! and  !+ , yielding two electron-type carriers with different mobility and density[32,[37][38][39]. Figures3c-3dand 3e-3f show the temperature dependent carrier concentrations and mobilities of both type of electrons for x = 0.086 and 0.157.The carrier concentration and mobility display a monotonous relation with temperature without any significant change around TCDW = 50 K for x = 0.086.This result differs from the two-band

Figure 4 .
Figure 4. Hydrostatic pressure-tuned charge density wave in parent compound ScV6Sn6.(a) Temperature-dependent longitudinal resistivity at several hydrostatic pressures.The inset is the change in CDW transition with pressure.(b) Calculated phonon band dispersions of two selected pressure, 1.5 and 5.0 GPa.(c) Pressure dependent imaginary phonon frequency at high

Figure 5 .
Figure 5. Hydrostatic pressure tuned magnetoresistance (MR) and Hall resistivity of ScV6Sn6.(a) MR at T =5 K for several pressures.(b) MR at 1.5 GPa for several temperatures.(c) Hall resistivity at T =5 K for several pressures.Inset shows anomalous Hall resistivity (AHE) after subtracting linear part at high field.(d) Hall resistivity at 1.5 GPa for several temperatures.Inset shows AHE after subtracting linear part at high field.(e) Scaling relationship between anomalous Hall conductivity (AHC) and zero-field longitudinal conductivity at different pressures.(f) AHC as a function of temperature at selective pressures.

Figure S3 .
Figure S3.Shubnikov de Haas oscilla7on observed at P = 0.5 GPa and T = 5 K under high magne7c field range.Inset is the Fast Fourier transform of the oscilla7on, giving two frequencies.

Figure S4 .
Figure S4.Two-band model analysis of pressurized pris7ne samples.Simultaneously fiAng of (a) longitudinal conduc7vity and (b) Hall conduc7vity for P = 1.5 GPa at T = 5 K. Simultaneously fiAng of (c) longitudinal conduc7vity and (d) Hall conduc7vity for for P = 1.5 GPa at T = 80 K.

Figure S5 .
Figure S5.Hall resis7vity as func7on of magne7c field at various temperatures and pressures.