Introduction

Materials containing kagome lattices have emerged as a promising platform for studying the interplay of electronic correlations and topology1,2,3,4. Among these, kagome metals hosting charge density waves have gained significant attention due to their symmetry-breaking phases and rich phase diagrams5,6,7. The A V3Sb5 (A = K, Rb, Cs) family hosts a charge density wave (CDW) with TCDW at ≈80 K. Although controversies remain8, the CDW potentially breaks time-reversal symmetry9 and rotational symmetry10,11,12, leading to speculation about an orbital current loop state13,14,15 and electronic nematicity16. At lower temperatures (<3 K), a superconducting state coexists and competes with the CDW6,7,17, displaying signatures of a pair density wave18. 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 diagrams19.

\({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) is the latest addition to the set of kagome metals exhibiting CDWs, with a CDW transition temperature near 90 K20. Since this compound contains kagome layers comprised solely of V ions, it is natural to compare it to the A V3Sb5 family. However, early studies have found several distinct differences between \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) and these compounds. In \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) the CDW is associated with a \(\sqrt{3}\) × \(\sqrt{3}\) in-plane ordering20, which is different from the 2 × 2 ordering in A V3Sb5 where the wave vectors nest the van Hove singularities of the kagome-derived energy bands. In \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) the lattice distortion associated with the CDW is mostly along the c-axis20 whereas the distortion in A V3Sb5 is mostly in the ab-plane21. Unlike the A V3Sb5 family, no superconductivity has been found in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) down to 40 mK even under high pressures22. Nevertheless, similar to the A V3Sb5 family, signatures of time-reversal symmetry breaking have been suggested by muon spin relaxation rate measurements and a potential anomalous Hall effect23,24,25. Recent measurements, including scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and Raman spectroscopy suggest the CDW is primarily structurally driven26,27,28, resulting from the softening of a multitude of phonon modes29,30,31, 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 TCDW in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\)29,30,32. 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 \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\). 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 behavior33. Our conclusion of pseudogap formation in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) 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 \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) which strongly influences the electronic transport properties above the transition temperature.

Results

Zero-field resistivity

Figure 1 presents in-plane resistivity, ρxx, and interlayer resistivity, ρzz (divided by 5), of a typical \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) sample as a function of temperature while cooling. ρxx is consistent with previous reports20,22, with residual resistivity ratios of various samples ranging from 3 to 10. Drops in resistivity are present in both curves near 90 K, indicating the charge density wave transition (TCDW)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 noted22, 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.

Fig. 1: Resistivity vs. temperature for \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\).
figure 1

ρxx and ρzz (divided by 5) of \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) 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. Here “SC" and “SE" refer to Sample C and Sample E, respectively. TCDW is marked in the same way as the main figure.

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The drop of resistivity at TCDW is reminiscent of the spin density wave transition in the parent compounds of the iron pnictide superconductors, such as BaFe2As2, which is also characterized by a similar feature34. This unusual behavior in BaFe2As2 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 BaFe2As2, optical measurements of \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\)35 have also revealed a similar decrease in both the carrier density and the scattering rate below TCDW, which could explain the increase in conductivity. The enhanced electron scattering above TCDW can be explained by the competing CDW fluctuations above TCDW32.

The interlayer resistivity measurements reveal a moderate resistivity anisotropy in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) with ρzz roughly 5 times larger than ρxx at 2 K. This is considerably smaller than in CsV3Sb5 where ρzz is ≈20 times larger than ρxx at low temperatures10, implying \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) is more 3-dimensional than CsV3Sb5. Nevertheless, unlike CsV3Sb5, the temperature dependence of ρzz is dramatically different from ρxx, showing a broad maximum of roughly 15 K above TCDW at T*. As shown in the inset of Fig. 1, for temperatures above T* multiple \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) samples display insulating behavior (here “SC" and “SE" refer to Sample C and Sample E, respectively. A summary of the samples measured can be found in Supplementary Table 2). 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 in-plane resistivity have been observed in highly anisotropic layered materials such as Sr2RuO4, in which the much weaker interlayer tunneling results in incoherent c-axis transport36. However, such a phenomenon is usually seen in materials with resistivity anisotropy ρzz/ρxx ≫ 10, which is not applicable to \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) where ρzz/ρxx ≈ 5. Interestingly, a similar insulating temperature dependence with a broad maximum in ρzz above a density wave transition has also been observed in BaFe2As234,37,38,39,40, which also has a moderate resistivity anisotropy (ρzz/ρxx ≈ 7). The insulating ρzz in BaFe2As2 was interpreted as a 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 ρzz in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\), which is further supported by the Hall effect and magnetoresistance analysis presented in the following sections.

Hall effect

Figure 2a presents ρxy as a function of the magnetic field at a variety of temperatures. These data are taken from Sample A which has a RRR of 6.0 and TCDW of 93 K (a summary of TCDW and RRR of the samples measured can be found in Supplementary Table 2). Across the entire measured temperature range ρxy is non-linear, but while it evolves smoothly as a function of temperature above TCDW, it stays relatively unchanged below TCDW. There are two possible sources for the non-linearity in ρxy: the multi-band effect and the anomalous Hall effect. We first present an analysis of fitting of ρxy 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 ρxy even without any two-band fitting.

Fig. 2: Hall effect and two-band analysis.
figure 2

a ρxy as a function of magnetic field at various temperatures. b ρxy as a function of magnetic field at 3 and 150 K with fits to the two-band model. c ne and nh as a function of temperature extracted from fits to the two-band model. d μe and μh as a function of temperature extracted from fits to the two-band model.

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In order to analyze the two-band Hall effect, the standard two-band model is used to simultaneously fit ρxx(μ0H) and ρxy(μ0H) at each temperature41. The carrier densities and mobilities were determined by a non-linear least squares minimization of the error \({({\rho }_{{{{xy}}}}-{\rho }_{{{{xy}}}}^{{\rm {fit}}})}^{2}+C{({\rho }_{{{{xx}}}}-{\rho }_{{{{xx}}}}^{{\rm {fit}}})}^{2}\) where C provided a weighting such that ρxy was prioritized (as ρxx has potentially more scattering contributions than those arising from the two-band model). Figure 2b shows the fits to ρxy at 150 and 3 K. Above TCDW the two-band model fits ρxy well, but below TCDW the two-band fit loses quality, particularly in the low field regime. This disagreement may be due to the contribution from an anomalous Hall effect24,25. Figure 2c and d present the carrier densities and mobilities, respectively, that are extracted from the two-band fitting. It should be noted that the quantitative values below TCDW should not be taken as exact due to the decrease in fit quality in ρxy. At all temperatures, nh is greater than ne by several orders of magnitude. Notably, above TCDW nh decreases significantly, although ne is roughly constant, and below TCDW nh is roughly constant while ne grows by several times. Both μe and μh increase as temperature decreases which is typical for a metal.

Due to the substantial deviation from the two-band model of the Hall data below TCDW, we estimate the carrier density using the Hall data in the high field limit in two more ways to ensure our conclusions are robust. First, the Hall coefficient, \({R}_{{{{\rm{H}}}}}=\frac{{\rho }_{{{{xy}}}}}{{\mu }_{0}H}\), should saturate to \(\frac{1}{({n}_{{{{\rm{h}}}}}-{n}_{{{{\rm{e}}}}})e}\) at the high field limit where e is the charge of an electron42. These data are shown in Fig. 3a. The blue data in Fig. 3c present the extracted nh−ne from the high field Hall coefficient. Second, the red data in Fig. 3c show nh using \(\frac{{\rm {d}}{\rho }_{{{{xy}}}}}{{\rm {d}}{\mu }_{0}H}\) (μ0H = 14 T) \(\approx \frac{1}{{n}_{{{{\rm{h}}}}}e}\). \(\frac{{\rm {d}}{\rho }_{{{{xy}}}}}{{\rm {d}}{\mu }_{0}H}\) as a function of magnetic field is shown in Fig. 3b. This method provides an estimate of carrier density even when the low-field data can not be well represented by the two-band model. 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 TCDW the number of holes decreases as a function of temperature, and below TCDW the number of holes is roughly constant.

Fig. 3: High field limit Hall analysis.
figure 3

a \(\frac{{\rho }_{{{{\rm{xy}}}}}}{{\mu }_{0}H}\) as a function of μ0H to evaluate the high field limit of the two-band model. Note that the curves are not fully saturated at 14 T. b \(\frac{{\rm {d}}{\rho }_{{{{xy}}}}}{{\rm {d}}{\mu }_{0}H}\) as a function of μ0H to approximate the carrier concentration using a one-band model. c nh−ne extracted from the high field limit of the two-band model and nh from the high field regime of a one-band model. While quantitatively different than the results presented in Fig. 2 for reasons described in the main text, all three of these analyses are qualitatively consistent.

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Magnetoresistance (MR) and Kohler’s rule analysis

The ab-plane MR (MRxx\(=\frac{{{\Delta }}{\rho }_{{{{xx}}}}}{{\rho }_{{{{xx}}}}({\mu }_{0}H=0)}* 100 \%\)) with a c-axis magnetic field at various temperatures is shown in Fig. 4a. These data are from Sample B which has a RRR of 4.3 and two drops in the resistivity at 80 and 88 K. This apparent double transition at TCDW in ρxx has been noted before24, and the meaning of this feature is not yet understood. This MR looks qualitatively similar to that observed in the A V3Sb5 family in that the low field behavior shows a cusp at low temperatures and evolves to a more standard quadratic behavior at high temperatures43. At low temperatures quantum oscillations can be resolved once a background subtraction is performed. The c-axis magnetoresistance (MRzz) with a magnetic field in the ab-plane at several temperatures is presented in Fig. 4b. These data are from Sample C which has a TCDW of 86 K. The MRzz of another sample was measured and was similar to the data presented here. While MRzz looks qualitatively similar to MRxx, MRzz is about double the size at 2 K. Also, at low temperatures quantum oscillations are observed much more prominently in MRzz. These quantum oscillations, as well as accompanying electronic structure calculations, are discussed in Supplemental Note 1 and are in agreement with a very recent study25. Overall, the quantum oscillations reveal three-dimensional Fermi pockets occupying less than a percent of the Brillouin zone with effective masses between 0.1 and 0.2m0 (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.

Fig. 4: Magnetoresistance of \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\).
figure 4

a MRxx as a function of the magnetic field at various temperatures. b Magnetic field dependence of MRzz at several temperatures.

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Kohler’s rule of magnetoresistance44 is violated in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) as shown in Fig. 5a as MRxx is not simply a function of μ0H/ρxx,0 where ρxx,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 \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) was also reported in a recent study23. Similar violations of Kohler’s rule have been used as evidence of phase transitions45 or non-Fermi liquid behavior46,47. Recently, an extended Kohler’s rule has been developed48 in which the MR is expressed as a function of μ0H/(nTρxx,0) where nT 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 kBT48. Here we apply this formula with nT fixed to be 1 at 255 K to collapse the MR curves onto the linear part of the 255 K MR (Fig. 5b). Through this method, nT can be extracted as a function of temperature, and is presented in the inset of Fig. 5b. Intriguingly, Kohler’s rule is nearly followed below TCDW as evidenced by the nearly constant value of nT, but above this temperature Kohler’s rule is clearly violated. These results are consistent with those presented in the section “Hall effect”—nh extracted from two-band fitting is also plotted in the inset of Fig. 5b to show the similar behavior between the concentration of the dominant carriers (holes) and nT, as both decrease above TCDW and become relatively constant below TCDW. Analysis of MRzz data using the extended Kohler’s rule yields similar results to that of MRxx: above TCDW nT decreases quickly with decreasing temperature, and below TCDW nT changes far less drastically. These data are presented in Supplementary Note 2.

Fig. 5: Kohler’s rule analysis of \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\).
figure 5

a MRxx as a function of μ0H/ρxx,0 using the data from Fig. 4a 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 MRxx as a function of μ0H/(nTρxx,0) on a log–log scale. Inset: extracted nT as a function of temperature. nh extracted from the two-band Hall fitting as discussed in the section “Hall effect” is also plotted in the inset to highlight the similar temperature dependencies between nh and nT.

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Discussion

Both Kohler’s rule analysis and the Hall effect demonstrate a pronounced temperature dependence of the carrier density in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\), decreasing by almost a factor of 2 from the value at 200 K to the value just above the CDW transition temperature TCDW = 90 K. This decrease in carrier density is consistent with the insulating temperature dependence observed in interlayer resistivity. Interestingly, similar characteristics have also been observed in Fe-based superconductors, such as BaFe2As2. The temperature dependence of in-plane and interlayer resistivity, as well as the resistivity anisotropy ratio, exhibit a remarkable resemblance between BaFe2As2 and \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\)34. Additionally, in BaFe2As2 the Hall coefficient also shows a substantial increase with decreasing temperature above the spin density wave transition49,50,51. This is also seen in the pseudogap regime of the cuprate superconductors52. Another striking similarity can be observed in the magnetic susceptibilities of BaFe2As2 and \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) (presented in ref. 20). In both materials there is a drop in the susceptibility at the phase transition, but these potentially stem from different mechanisms, for in BaFe2As2 this is likely due to enhanced magnetic correlations. Also in both materials, 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 susceptibility53,54.

In BaFe2As2, 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 \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\), the CDW transition is first-order, which could explain the nearly temperature-independent carrier density below TCDW, but contradicts with the existence of an extended fluctuation regime above TCDW. Nevertheless, theoretical studies have suggested that, in addition to the long-range \(\sqrt{3}\) × \(\sqrt{3}\) × 3 CDW that develops below TCDW, there are several other nearly degenerate CDW instabilities associated with different ordering wave vectors29. Furthermore, experimental evidence has shown the presence of a short-range \(\sqrt{3}\) × \(\sqrt{3}\) × 2 CDW well above TCDW, which is suppressed by the \(\sqrt{3}\) × \(\sqrt{3}\) × 3 CDW through a first-order transition at TCDW32. It is possible that the short-range CDW fluctuations are responsible for the anomalous decrease of carrier density and insulating interlayer resistivity observed in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\).

It should be noted that a pseudogap in the BaFe2As2 family has been identified using a variety of techniques including angle-resolved photoemission spectroscopy (ARPES)55 and optical conductivity56. While several works have presented ARPES26,28,57,58 and scanning tunneling microscopy26,58 data on \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\), to our knowledge no measurements were performed above 120 K. Thus, it is possible that future studies will reveal a pseudogap in this material using these experimental probes.

In conclusion, the transport behavior in the normal state of \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) is consistent with the formation of a pseudogap, which is likely arising from high-temperature CDW fluctuations. We have also highlighted several similarities between \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) and Fe-based superconductors with pseudogaps above their ordering temperatures. Due to the high degree of tunability in the RT6X6 family (R = rare earth, T = transition metal, X = Si, Ge, Sn), \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) 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 that reported the two-band behavior of the Hall effect in \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\)24. They discovered high carrier density and low mobility holes and low carrier density and high mobility electrons, which broadly corroborates our findings.

Methods

Sample preparation

Single crystals of \({{{{\rm{ScV}}}}}_{6}{{{{\rm{Sn}}}}}_{6}\) were grown using a flux method similar to the one previously reported20. Mixtures of Sc pieces (99.9%), V pieces (99.9%), and Sn shot (99.999%) were loaded into Canfield crucible sets59 with atomic ratios 1:3:30, then vacuum-sealed in quartz tubes. These were heated to 1150 °C in 12 h, held at this temperature for 15 h, then cooled to 780 °C in 200 h 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

Transport measurements were performed on samples that were polished and cut by a wire saw to be bar-shaped with dimensions roughly 1 mm × 0.4 mm × 0.05 mm (in-plane current) or 0.2 mm × 0.15 mm × 0.05 mm (out-of-plane 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 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 anti-symmetrized, respectively. For some of the measurements, the samples were rotated in situ using a Quantum Design in-plane rotator.

Density functional theory calculations

Density functional theory (DFT) calculations were performed using a full-potential linear augmented plane wave (FP-LAPW) method, as implemented in WIEN2K60. The primitive cell contains one formula unit, and experimental lattice parameters20 were adopted. The generalized gradient approximation of Perdew et al.61 was used for the correlation and exchange potentials. To generate the self-consistent potential and charge, we employed RMT⋅Kmax = 8.0 with Muffin–Tin (MT) radii RMT = 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 were <1 × 10−3e and the total energy difference lower than 0.01 mRy. After obtaining a self-consistent charge, band energies were calculated with a 64 × 64 × 33 fine k-mesh for the full Brillouin Zone (FBZ).

We employed FermiSurfer62 and SKEAF63 for visualizing FS and calculating de Haas–van Alphen (dHvA) frequencies, respectively.