Quantum oscillations revealing topological band in kagome metal ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$

Compounds with kagome lattice structure are known to exhibit Dirac cones, flatbands, 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 ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$, 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 nontrivial and generates a nonzero Berry phase. In addition, the compound also shows the anomalous Hall-like behavior up to the CDW phase transition, and they might be correlated. Combining these interesting physical properties with the CDW phase, ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$ presents a unique material example of the versatile $\mathrm{Hf}{\mathrm{Fe}}_6{\mathrm{Ge}}_6$ family and provides various promising opportunities to explore the series further.

Figure 1

Kagome lattice, Laue pattern, I-V characteristic, and CDW phase transition of ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$. (a) Kagome lattice of ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$ viewed along the $c$ axis (left) and recorded single-crystal Laue patterns along the $c$ axis. (b) I-V characteristics at various fields at 2 K. The inset is an optical image of the crystal with Cartesian coordinate. (c) Magnetic susceptibility along $\widehat{x}$. Resistivity in zero field (d) along $\widehat{y}$ and (c) along $\widehat{z}$, where the sudden jumps show CDW phase transition.

Figure 2

Shubnikov–de Haas (SdH) oscillations and Berry phase. (a) Longitudinal ${\rho }_{zz}$ at 2 K and its first derivative with field showing quantum oscillations. (b) Background-subtracted SdH oscillation amplitude at several temperatures. (c) Corresponding FFT amplitude exhibiting α, β, and γ frequencies. (d) FFT amplitude fit estimating effective mass. (e) SdH oscillations when $B\left|\right|\widehat{z}$ and their FFT spectrum (inset) showing only the γ frequency in the field window of 5–8 T. (f) Intercept of the Landau fan diagram revealing nonzero Berry phase. (g) Angular dependence of the Berry phase.

Figure 3

Band structure in charge-density wave (CDW) phase. (a) Band structure of ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$ CDW phase with the presence of spin-orbital coupling (SOC). The structural parameters are the same as in Ref. [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.

Figure 4

SdH oscillations and fermiology. (a) SdH oscillation frequency at different rotating angles $\theta (=\widehat{x}\to \widehat{z})$ and $\varphi (=\widehat{x}\to \widehat{y})$, where dotted lines represent their tracking with angle. The insets show a sketch of the field rotation. (b) Angular dependence of $F_{\alpha ,\beta ,\gamma }$, where $F_{\alpha ,\beta }$ do not change while $F_{\gamma }$ shifts slightly. Solid lines are corresponding calculated frequencies. 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) at the Fermi energy. (d)–(g) Enlarged FSs, including the FS at M point (d, yellow), the banana-shaped α pocket 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).

Figure 5

Anomalous Hall-like behavior in ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$. (a) Field-dependent measured anomalous behavior of Hall resistivity, ${\rho }_{zy}$ at various temperatures (inset) and extracted Hall resistivity ${\rho }_{zy}^A$. (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) Nonzero Berry phase from ferromagnetic spins. (e) Topological orbital moment due to spin chirality. (f) Formation of local loop currents.