Quantum Transport Evidence of Topological Band Structures of Kagome Superconductor ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$

We report the transport properties of kagome superconductor ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ single crystals at magnetic field up to 32 T. The Shubnikov–de Haas oscillations emerge at low temperature and four frequencies of $F_{\alpha }=27\mathrm{T}$, $F_{\beta }=73\mathrm{T}$, $F_{\epsilon }=727\mathrm{T}$, and $F_{\eta }=786\mathrm{T}$ with relatively small cyclotron masses are observed. For $F_{\beta }$ and $F_{\epsilon }$, the Berry phases are close to $\pi$, providing clear evidence of nontrivial topological band structures of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. Furthermore, the consistence between theoretical calculations and experimental results implies that these frequencies can be assigned to the Fermi surfaces locating near the boundary of Brillouin zone and confirms that the structure with an inverse Star of David distortion could be the most stable structure at charge density wave state. These results will shed light on the nature of correlated topological physics in kagome material ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$.

Figure 1

(a) Temperature dependence of ${\rho }_{xx}(T)$. Inset: enlarged ${\rho }_{xx}(T)$ curve at low temperature. (b) Temperature dependence of $\chi (T)$ at ${\mu }_{0}H=1\mathrm{T}$ for $H\Vert c$ with ZFC and FC modes. Inset: ZFC and FC $4\pi \chi (T)$ curves at ${\mu }_{0}H=1\mathrm{mT}$ for $H\Vert ab$ below 3.5 K. (c) Field dependence of ${\rho }_{xx}({\mu }_{0}H)$ for $H\Vert c$ at various temperatures. The ${\rho }_{xx}({\mu }_{0}H)$ curves are shifted vertically for clarity. (d) The oscillatory component $\mathrm{\Delta }{\rho }_{xx}$ as the function of $1/{\mu }_{0}H$ at various temperatures. The $\mathrm{\Delta }{\rho }_{xx}$ is extracted from ${\rho }_{xx}$ by subtracting a smooth background.

Figure 2

(a) FFT amplitudes of the SdH oscillations at low-frequency region. Inset: FFT amplitudes of the SdH oscillations at high-frequency region. (b) Temperature dependence of FFT amplitudes of $F_{\alpha }$, $F_{\beta }$, $F_{\epsilon }$, and $F_{\eta }$ peaks. The solid lines represent the LK formula fits for $m^{*}$s. (c) Landau index $n$ vs $1/{\mu }_{0}H$ for the frequency of $F_{\beta }$, derived from the $\mathrm{\Delta }{\rho }_{xx}$ curve at 15 K. The closed squares denote the integer index ($\mathrm{\Delta }{\rho }_{xx}$ peak), and the open squares indicate the half integer index ($\mathrm{\Delta }{\rho }_{xx}$ valley). Inset: magnified view around the intercept. (d) The high-frequency oscillatory components at 1.8 K. The red line represents the fit using the two-frequency LK formula.

Figure 3

(a) Field dependence of FFT amplitudes of SdH oscillations at various field directions ($\theta$s) when $T=1.8\mathrm{K}$. The data at different field directions have been shifted for clarity. (b) Angular dependence of oscillation frequencies $F_{\alpha }$, $F_{\beta }$, $F_{\epsilon }$, and $F_{\eta }$. Inset in (b) shows the definition of $\theta$. Error bars are defined as the half width at the half-height of FFT peaks and the angle uncertainty of measurement ($\sim 3°$). The blue dashed line is calculated from the formula $F_{\beta }(\theta )=F_{\beta }(0°)/\cos \theta$. The solid symbols at $\theta =0$ represent the frequencies calculated from the cross sections of unfolded FSs in pristine BZ shown in Figs. 4 and 4. The error bars originate from the uncertainties when evaluating the areas of cross sections of unfolded FSs. The red solid line is obtained from theoretical calculations using the FS in the reconstructed BZ.

Figure 4

(a) Calculated unfolded band structure of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ along high-symmetry directions across the pristine BZ using the structure with an ISD distortion. Inset: schematic of the 3D BZ in the pristine phase. High-symmetry points and momentum lines are marked. (b) and (c) The cross sections of unfolded FSs at $k_z=0$ and $\pi /c$ planes in the pristine BZ, respectively. The extremal orbits related to $F_1–F_4$ are highlighted.