Pseudogap behavior in charge density wave kagome material ScV$_6$Sn$_6$ revealed by magnetotransport measurements

Over the last few years, significant attention has been devoted to studying the kagome materials AV$_3$Sb$_5$ (A = K, Rb, Cs) due to their unconventional superconductivity and charge density wave (CDW) ordering. Recently ScV$_6$Sn$_6$ was found to host a CDW below $\approx$90K, and, like AV$_3$Sb$_5$, it contains a kagome lattice comprised only of V ions. Here we present a comprehensive magnetotransport study on ScV$_6$Sn$_6$. We discovered several anomalous transport phenomena above the CDW ordering temperature, including insulating behavior in interlayer resistivity, a strongly temperature-dependent Hall coefficient, and violation of Kohler's rule. All these anomalies can be consistently explained by a progressive decrease in carrier densities with decreasing temperature, suggesting the formation of a pseudogap. Our findings suggest that high-temperature CDW fluctuations play a significant role in determining the normal state electronic properties of ScV$_6$Sn$_6$.


I. INTRODUCTION
Materials containing kagome lattices have emerged as a promising platform for studying the interplay of electronic correlations and topology [1][2][3][4].Among these, kagome metals hosting charge density waves have gained significant attention due to their novel symmetry breaking phases and rich phase diagrams [5][6][7].The AV 3 Sb 5 (A = K, Rb, Cs) family hosts a charge density wave (CDW) with T CDW at ≈80 K, which potentially breaks time reversal symmetry [8,9] and rotational symmetry [10][11][12], leading to speculation about an orbital current loop state [13][14][15] and electronic nematicity [16].At lower temperatures (<3 K), a superconducting state coexists and competes with the CDW [6,7,17], displaying signatures of a pair density wave [18].Many of these phenomena resemble the characteristics of other strongly correlated systems, such as high temperature superconductors, where the extended fluctuation regime gives rise to intertwined orders and complex phase diagrams [19].
ScV 6 Sn 6 is the latest addition to the set of kagome metals exhibiting CDWs, with a CDW transition temperature near 90 K [20].Since this compound contains kagome layers comprised solely of V ions, it is natural to compare it to the AV 3 Sb 5 family.However, early studies have found several distinct differences between ScV 6 Sn 6 and these compounds.In ScV 6 Sn 6 the CDW is associated with a √ 3 x √ 3 in-plane ordering [20], which is different from the 2 x 2 ordering in AV 3 Sb 5 where the wave vectors nest the van Hove singularities of the kagomederived energy bands.In ScV 6 Sn 6 the lattice distortion associated with the CDW is mostly along the c-axis [20] whereas the distortion in AV 3 Sb 5 is mostly in the abplane [21].Unlike the AV 3 Sb 5 family, no superconductivity has been found in ScV 6 Sn 6 down to 40 mK even under high pressures [22].Nevertheless, similar to the AV 3 Sb 5 family, signatures of time reversal symmetry breaking have been suggested by muon spin relaxation rate mea-surements and an anomalous Hall effect [23][24][25].Recent measurements, including scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and Raman spectroscopy suggest the CDW is primarily structurally driven [26][27][28], indicating a minor role of the electronic degrees of freedom in the CDW formation.However, despite the first-order nature of the CDW transition, recent studies have revealed short-range CDW fluctuations persisting well above T CDW in ScV 6 Sn 6 [29].Hence, it is crucial to examine whether these fluctuations impact the electronic properties, as observed in other strongly correlated systems.
In this paper, we present evidence of a pseudogap above the CDW transition in ScV 6 Sn 6 .Pseudogap formation was first observed in the cuprate superconducting family, and it refers to the suppression of the density of states which was revealed by various spectroscopy measurements and anomalous transport behavior [30].Our conclusion of pseudogap formation in ScV 6 Sn 6 is established from a comprehensive magnetotransport study, including measurements of the interlayer resistivity, magnetoresistance, and the Hall effect, all consistent with an abnormal decrease of carrier density with decreasing temperature above the CDW transition.In addition, we found several striking similarities to the proposed pseudogap phase in the Fe-based superconductors, in which strong spin density wave fluctuations persist well above the transition temperature.Our results suggest that there is an extended fluctuation regime in ScV 6 Sn 6 which strongly influences the electronic transport properties above the transition temperature.

A. Zero-field Resistivity
Fig. 1 presents in-plane resistivity, ρ xx , and interlayer resistivity, ρ zz (divided by 5), of a typical ScV 6 Sn 6 sample as a function of temperature while cooling.ρ xx is consistent with previous reports [20,22], with residual resistivity ratios of samples ranging from 3-10.Drops in resistivity are present in both curves near 90 K, indicating the charge density wave transition (T CDW ) [20].It should be noted that this is a first order transition, but only the cooling curve is shown due to the small temperature hysteresis of this transition (≈1-2 K).As previously noted [22], charge density wave phase transitions in layered materials typically appear as hump-like increases in resistivity as a function of temperature, as parts of the Fermi surface get gapped out by the transition.
The drop of resistivity at T CDW is reminiscent of the spin density wave transition in the parent compounds of the iron pnictide superconductors, such as BaFe 2 As 2 , which is also characterized by a similar feature [31].This unusual behavior in BaFe 2 As 2 is understood as a more rapid decrease of scattering rates relative to the decrease of carrier density below the spin density wave transition.Like observed in BaFe 2 As 2 , optical measurements of ScV 6 Sn 6 [32] have also revealed a similar decrease in both the carrier density and the scattering rate below T CDW , which could explain the increase in conductivity.The enhanced electron scattering above T CDW can be explained by the competing CDW fluctuations above T CDW [29].
The interlayer resistivity measurements reveal a moderate resistivity anisotropy in ScV 6 Sn 6 with ρ zz roughly 5 times larger than ρ xx at 2 K.This is considerably smaller than in CsV 3 Sb 5 where ρ zz is ≈20 times larger than ρ xx at low temperatures [10], implying ScV 6 Sn 6 is more 3-dimensional than CsV 3 Sb 5 .Nevertheless, unlike CsV 3 Sb 5 , the temperature dependence of ρ zz is dramatically different from ρ xx , showing a broad maximum roughly 15 K above T CDW at T * .As shown in the inset of Fig. 1, for temperatures above T * multiple ScV 6 Sn 6 samples display insulating behavior.The difference in high temperature resistivity between samples may be explained by contamination from lower resistivity in-plane components.
The insulating interlayer resistivity and metallic inplane resistivity have been observed in highly anisotropic layered materials such as Sr 2 RuO 4 , in which the much weaker interlayer tunneling results in incoherent c-axis transport [33].However, such a phenomenon is usually seen in materials with resistivity anisotropy ρ zz /ρ xx ≫ 10, which is not applicable to ScV 6 Sn 6 where ρ zz /ρ xx ≈ 5. Interestingly, a similar insulating temperature dependence with a broad maximum in ρ zz above a density wave transition has also been observed in BaFe 2 As 2 [31,[34][35][36][37], which also has a moderate resistivity anisotropy (ρ zz /ρ xx ≈ 7).The insulating ρ zz in BaFe 2 As 2 was interpreted as signature of a pseudogap, resulting from the spin density wave fluctuations partially gapping the section of the Fermi surface where the Fermi velocity has a large z-component.We propose that a similar mechanism could be responsible for the insulating ρ zz in ScV 6 Sn 6 , which is further supported by the Hall effect and magnetoresistance analysis presented in the following sections.

B. Hall Effect
Fig. 2(a) presents ρ xy as a function of magnetic field at a variety of temperatures.Across the entire measured temperature range ρ xy is non-linear, but while it evolves smoothly as a function of temperature above T CDW , it stays relatively unchanged below T CDW .There are two possible sources for the non-linearity in ρ xy : the multiband effect and the anomalous Hall effect.We first present an analysis of fitting of ρ xy using a two-band model, which reveals a strong temperature dependence of carrier density above the CDW transition.We will also argue that this conclusion can be made from the high field Hall coefficient of ρ xy even without any two-band fitting.
In order to analyze the two-band Hall effect, the standard two-band model is used to simultaneously fit ρ xx (µ 0 H) and ρ xy (µ 0 H) at each temperature [38].The carrier densities and mobilities were determined by a nonlinear least squares minimization of the error where C provided a weighting such that ρ xy was prioritized (as ρ xx has potentially more scattering contributions than those arising from the two-band (n h −ne)e at the high field limit where e is the charge of an electron [39].These data are shown in Fig. 3(a).
The blue data in Fig. 3(c) present the extracted n h − n e from the high field Hall coefficient.Second, the red data in Fig. 3(c) show n h using dρxy dµ0H (µ 0 H = 14 T) ≈ 1 n h e .dρxy dµ0H as a function of magnetic field is shown in Fig. 3(b).This method provides an estimate of carrier density in the presence of an anomalous Hall effect.While the three methodologies used to estimate the carrier densities vary quantitatively, they are all of the same order of magnitude and qualitatively consistent -above T CDW the number of holes decreases as a function of temperature, and below T CDW the number of holes is roughly constant.

C. Magnetoresistance (MR) and Kohler's Rule Analysis
The ab-plane MR (M R xx = ∆ρxx ρxx(µ0H=0) * 100%) with c-axis magnetic field at various temperatures is shown in Fig. 4(a).This MR looks qualitatively similar to that observed in the AV 3 Sb 5 family in that the low field behavior shows a cusp at low temperatures and evolves to a more standard quadratic behavior at high tempera-   tures [40].At low temperatures quantum oscillations can be resolved once a background subtraction is performed.The c-axis magnetoresistance (M R zz ) with a magnetic field in the ab-plane at several temperatures is presented in Fig. 4(b).The M R zz of another sample was measured and was similar to the data presented here.While M R zz looks qualitatively similar to M R xx , M R zz is about double the size at 2 K. Also, at low temperatures quantum oscillations are observed much more prominently in M R zz .These quantum oscillations, as well as accompanying electronic structure calculations, are discussed in the Supplemental Materials and are in agreement with a very recent study [25].Overall, the quantum oscillations reveal three-dimensional Fermi pockets occupying less than a percent of the Brillouin zone with effective masses between 0.1 -0.2 of the free electron mass.These pockets are consistent with the high mobility and low density of electron carriers extracted from the two-band Hall fitting.Kohler's rule of magnetoresistance [41] is violated in ScV 6 Sn 6 as shown in Fig. 5(a) as M R xx is not simply a function of µ 0 H/ρ xx,0 where ρ xx,0 is the zero-field resistivity.It should be noted that the data point spread at low fields and high temperatures is due to the small MR in this regime.The violation of Kohler's rule in ScV 6 Sn 6 was also reported in a recent study [23].Similar violations of Kohler's rule have been used as evidence of phase transitions [42] or non-Fermi liquid behavior [43,44].Recently, an extended Kohler's rule has been developed [45] in which the MR is expressed as a function of µ 0 H/(n T ρ xx,0 ) where n T describes the relative change in the carrier density.The extended Kohler's rule successfully explains the violation of conventional Kohler's rule by incorporating a temperature dependent carrier density, which could arise from a phase transition that partially gaps the Fermi surfaces or thermal excitations in topological semimetals where the Fermi energy is comparable to k B T [45].Here we apply this formula with n T fixed to be 1 at 255 K to collapse the MR curves onto the linear part of the 255 K MR (Fig. 5).Through this

III. DISCUSSION
Both Kohler's rule analysis and the Hall effect demonstrate a pronounced temperature dependence of the carrier density in ScV 6 Sn 6 , decreasing by almost a factor of 2 from the value at 200K to the value just above the CDW transition temperature T CDW = 90K.This decrease in carrier density is consistent with the insulating temperature dependence observed in interlayer resistivity.Interestingly, similar characteristics have also been observed in the Fe-based superconductors, such as BaFe 2 As 2 .The temperature dependence of in-plane and interlayer resistivity, as well as the resistivity anisotropy ratio, exhibit remarkable resemblance between BaFe 2 As 2 and ScV 6 Sn 6 [31].Additionally, in BaFe 2 As 2 the Hall coefficient also shows a substantial increase with decreasing temperature above the spin density wave transition [46][47][48].This is also seen in the pseudogap regime of the cuprate superconductors [49].Another striking similarity can be observed in the magnetic susceptibilities of BaFe 2 As 2 and ScV 6 Sn 6 (presented in [20]).In both materials, in addition to the drop below the phase transition due to gap formation, the susceptibility shows a linear increase with increasing temperature above the phase transition, which cannot be explained by either Pauli paramagnetic susceptibility or Curie Weiss susceptibility [50,51].
In BaFe 2 As 2 , the anomalous transport and magnetic properties observed above the transition temperature have been attributed to strong spin density wave fluctuations.However, in the case of ScV 6 Sn 6 , the CDW transition is first-order, which could explain the nearly temperature independent carrier density below T CDW , but contradicts with the existence of an extended fluctuation regime above T CDW .Nevertheless, theoretical studies have suggested that, in addition to the long range √ 3 x √ 3 x 3 CDW that develops below T CDW , there are several other nearly degenerate CDW instabilities associated with different ordering wave vectors [52].Furthermore, experimental evidence has shown the presence of a short-range √ 3 x √ 3 x 2 CDW well above T CDW , which is suppressed by the √ 3 x √ 3 x 3 CDW through a first-order transition at T CDW [29].It is possible that the short-range CDW fluctuations are responsible for the anomalous decrease of carrier density and insulating interlayer resistivity observed in ScV 6 Sn 6 .
In conclusion, the transport behavior in the normal state of ScV 6 Sn 6 is consistent with the formation of a pseudogap, which is likely arising from high temperature CDW fluctuations.We have also highlighted several similarities between ScV 6 Sn 6 and Fe-based superconductors with pseudogaps above their ordering temperatures.Due to the high degree of tunability in the RT 6 X 6 family (R = rare earth, T = transition metal, X = Si, Ge, Sn), ScV 6 Sn 6 offers an exciting platform to study exotic electronic ordering in a kagome material.
Note: During the preparation of this paper we became aware of a separate study which reported the two-band behavior of the Hall effect in ScV 6 Sn 6 [24].They discovered high carrier density and low mobility holes and low carrier density and high mobility electrons, which broadly corroborates our findings.

IV. METHODS
Single crystals of ScV 6 Sn 6 were grown using a flux method similar to one previously reported [20].Mixtures of Sc pieces (99.9%),V pieces (99.9%), and Sn shot (99.999%) were loaded into Canfield crucible sets [53] with atomic ratios 1:3:30, then vacuum-sealed in quartz tubes.These were heated to 1150 • C in 12 hours, held at this temperature for 15 hours, then cooled to 780 • C in 200 hours where the growths were decanted in a centrifuge to separate the excess flux from the single crystals.Dilute HCl was used to etch the remaining flux from the surface of the crystals.The phase of the crystals was confirmed using energy-dispersive X-ray spectroscopy with a Sirion XL30 scanning electron microscope.The orientation of the crystallographic axes was determined using a Rigaku MiniFlex 600 system, with a Cu source and Hy-Pix 400MF 2D-detector.
Transport measurements were performed on samples that were polished and cut by a wire saw to be bar-shaped with dimensions roughly 1 mm x 0.4 mm x 0.05 mm (inplane current) or 0.2 mm x 0.15 mm x 0.05 mm (out-ofplane current).Silver paste or two-part silver epoxy (H20-E) and gold wires were used to make 4 point and 5 point (Hall pattern) measurements.These measurements were performed in a Quantum Design Dynacool Physical Property Measurement System with standard lock-in techniques in temperatures ranging from 1.7 K to 300 K and in magnetic fields up to 14 T. To eliminate any contributions from contact misalignment, the in-line and Hall resistivities were symmetrized and antisymmetrized, respectively.For some of the measurements the samples were rotated in situ using a Quantum Design in-plane rotator.
Density functional theory (DFT) calculations were performed using a full-potential linear augmented plane wave (FP-LAPW) method, as implemented in wien2k [54].The primitive cell contains one formula unit, and experimental lattice parameters [20] were adopted.The generalized gradient approximation of Perdew, Burke, and Ernzerhof [55] was used for the correlation and exchange potentials.To generate the self-consistent potential and charge, we employed R MT • K max = 8.0 with Muffin-Tin (MT) radii R MT = 2.4, 2.4, and 2.5 a.u., for Sc, V and Sn, respectively.The self-consistent calculations were performed with 490 k-points in the irreducible Brillouin zone (BZ).They were iterated until charge differences between consecutive iterations are smaller than 1 × 10 −3 e and the total energy difference lower than 0.01 mRy.
where µ 0 H avg is the average of the magnetic field used in the FFT and m * is the effective mass.The inset of Supplementary Fig. S2(b) and (d) shows fitting which reveals an effective mass of 0.15m 0 and 0.18m 0 (m 0 is the free electron mass) for the 14 T and 43 T peaks when field is along the c-axis, and 0.12m 0 and 0.15m 0 for the 9 and 17 T peaks for field in-the effective mass (Supplementary Fig. S5(b)) of these small orbitals.For the c-axis, the external magnetic field is applied along the hexagonal axis.For the a-axis, it is parallel to the Γ-M direction, and for the b-axis, it is along the Γ-K direction.Supplementary Table S1 displays dHvA frequencies (for frequency < 100 T) and effective masses with the magnetic field applied along c, a and b-directions.sponding orbitals.Since our focus is on smaller pockets at the zone-boundary, we are presenting data that are smaller than 100T.For the c-axis, the external magnetic field is applied along the hexagonal axis.For the a-axis, it is parallel to the Γ-M direction, and for the b-axis, it is along the Γ-K direction.Color scheme is consistent with Fig. S4.TABLE S1.Calculated dHvA frequencies (frequency < 100 T) and effective masses with external magnetic field (B-field) applied in three different directions.For the c-direction, B-field is applied along the hexagonal axis.For the a-direction, it is along the Γ-M direction and for the b-direction, it is parallel to the Γ-K direction.The effective masses are given in units of electron mass.

FIG. 1 .
FIG. 1. ρxx and ρzz (divided by 5) of ScV 6 Sn 6 as a function of temperature.TCDW is marked with a vertical line.Inset: ρzz of two different samples each normalized to be 1 at T * as discussed in the main text.TCDW is marked in the same way as the main figure.

FIG. 4 .
FIG. 4. (a) M Rxx as a function of magnetic field at various temperatures.(b) Magnetic field dependence of M Rzz at several temperatures.

FIG. 5 .
FIG. 5. (a) M Rxx as a function of µ0H/ρxx,0 using the data from Fig. 4(a) on a log-log scale.Kohler's rule is violated as the data from different temperatures do not collapse onto each other.(b) Extended Kohler's rule applied to the same data as presented in (a) by plotting M Rxx as a function of µ0H/(nT ρxx,0) on a log-log scale.Inset: extracted nT as a function of temperature.n h extracted from the two-band Hall fitting as discussed in Section II B is also plotted in the inset to highlight the similar temperature dependencies between n h and nT .
Fig. x FIG. S4. 3D Fermi surface (FS) plot along with cross sections for four chosen planes.(a) corresponds to the (0 0 1) plane with k z = 0.0, (b) corresponds to the (0 0 1) plane with k z = 0.25, (c) corresponds to the (1 1 0) plane passing through the Γ-point, and (d) corresponds to the (1 0 0) plane passing through the Γ-point.Different colors in the plot represent different bands.Blue, green, and red correspond to band 1, 2, and 3, respectively.
FIG. S5.We have calculated the dHvA frequencies (a) and the effective mass (b) of the corre-