Fermi Surface Mapping and the Nature of Charge-Density-Wave Order in the Kagome Superconductor ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$

The recently discovered family of $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A$: K, Rb Cs) kagome metals possess a unique combination of nontrivial band topology, superconducting ground states, and signatures of electron correlations manifest via competing charge density wave order. Little is understood regarding the nature of the charge density wave (CDW) instability inherent to these compounds and the potential correlation with the onset of a large anomalous Hall response. To understand the impact of the CDW order on the electronic structure in these systems, we present quantum oscillation measurements on single crystals of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. Our data provide direct evidence that the CDW invokes a substantial reconstruction of the Fermi surface pockets associated with the vanadium orbitals and the kagome lattice framework. In conjunction with density functional theory modeling, we are able to identify split oscillation frequencies originating from reconstructed pockets built from vanadium orbitals and Dirac-like bands. Complementary diffraction measurements are further able to demonstrate that the CDW instability has a correlated phasing of distortions between neighboring ${\mathrm{V}}_3{\mathrm{Sb}}_5$ planes, and the average structure in the CDW state is proposed. These results provide critical insights into the underlying CDW instability in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ kagome metals and support minimal models of CDW order arising from within the vanadium-based kagome lattice.

Popular Summary

The recent discovery of a new class of kagome compounds (an atomic network connected via corner-sharing triangles) of the form $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ (where $A$ is cesium, rubidium, or potassium) have brought with them several exciting ideas. Many of their intriguing properties, including a high-temperature charge-density-wave state reported to break time-reversal symmetry, currently puzzle the scientific community. Here, we study the nature of this charge-density-wave state in an exemplar of these new compounds, ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$, and establish that the charge-density wave arises solely because of reconstruction of the vanadium orbitals comprising its kagome lattice.

The charge-density-wave state is intimately linked to two other intriguing properties in these compounds: a potentially topologically nontrivial superconducting state, where widely sought quasiparticles for quantum computing may appear, and an unusually large anomalous Hall effect, where properties such as time-reversal-symmetry breaking and topologically nontrivial electronic phenomena manifest.

To understand how the charge-density wave alters the electronic structure of these systems, we perform quantum oscillation measurements on single crystals of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. Quantum oscillations are fingerprints of a material’s electronic band structure, and they appear in multiple properties of metals once a magnetic field is applied, and electrons are driven into orbits normal to the field direction. We use magnetoresistance observations to detect the oscillation frequencies. This allows us to map the Fermi surface—a visualization of occupied electron states—while in the charge-density-wave state, which we then interpret with a first-principles theory.

These results provide a crucial step forward for understanding the genesis of the numerous exotic properties linked to the charge-density-wave phase in this new class of materials.

Figure 1

${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ is a layered, exfoliatable, kagome metal consisting of a structurally perfect lattice of vanadium at room temperature. Upon cooling below $T^{*}=93\mathrm{K}$, ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ exhibits charge density wave order. A concurrent structural distortion emerges as well, which is suspected to be related to the kagome “breathing mode.” Upon distortion and relaxation in both the positive and negative displacements, the breathing mode gives rise to the Star of David (SoD) and trihexagonal (TrH) candidate structures.

Figure 2

(a)–(d) Two-dimensional slices through reciprocal space on half-integer Bragg planes at 15 K, highlighting superlattice peaks in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. (e)–(h) Line cuts through the 2D data, highlighting the periodicity of the superlattice peaks. Cuts performed at the base temperature of 15 K (blue) contrast data collected at 130 K (red), confirming that the superlattice peaks emerge simultaneous the charge density wave order at $T^{*}\sim 94\mathrm{K}$. Our data indicate that the superlattice in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ is described by a wave vector of (0.5, 0.5, 0.25).

Figure 3

Single-crystal diffraction data imply that ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ distorts into a $2\times 2\times 4$ supercell, where the kagome layers exhibit both TrH- and SoD-like distortions. Our current model indexes the cell in the $P\overline{3}$ space group. The SoD-like distortions are substantially weaker than the TrH-like distortion, and two unique SoD layers are noted.

Figure 4

(a) Unfolded electronic structure of the SoD and TrH distortions in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ with the undistorted electronic structure superimposed for comparison (black). The largest perturbation to the structure appears near the Dirac-like bands near M. (b) Close-up of the changes near the M point, highlighting the new bands that appear as a result of the CDW and the associated structural distortion.

Figure 5

(a) Oscillatory component of the magnetoresistance extracted from the temperature-dependent SdH data collected on single crystals of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ mounted with the c axis parallel to the magnetic field (0°). (b) A high-resolution scan at 1.8 K shows significant contributions from high-frequency modes, particularly at the peaks and troughs of the general oscillatory behavior. (c) Magnified view of the high-frequency features, providing visual confirmation for multiple high-frequency modes. (d,e) Fourier transformation of the quantum oscillation data, showing the low- and high-frequency components of the power spectrum. The nine unique frequencies have been assigned greek letters. (f) Temperature dependence of the Fourier coefficients is used in conjunction with the Lifshitz-Kosevich (LK) formula to extract the cyclotron “effective masses.” All low-frequency modes show very low effective masses, consistent with transport originating from the Dirac-like crossings at M.

Figure 6

(a) Raw electronic resistivity as a function of field and angle, where the angle is defined as being between the $c$ axis and the magnetic field. Oscillations are clearly visible at greater than 2 T. The oscillations appear to be dampened with angle, vanishing for $\theta >60°$. (b) Fourier transform of angle-resolved SdH data, showing primary low-frequency contributions to the power spectrum. The high-frequency contributions are suppressed rapidly by rotation. (c) Two of the most prominent frequencies ($\beta$, $\delta$) are shown as a function of rotation angle. Dashed lines serve as a guide to the eye. The frequencies were estimated from (b) using Gaussian functions to approximate both the broadening and shortening of peaks.

Figure 7

(a) Fermi surface for undistorted ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ calculated with $E_{\mathrm{F}}=E_{F,\exp }$, which shows a variety of potential orbits. Orbits calculated with the serendipity python package are shown highlighted in gray. For clarity, in panels (b) and (c), we show isoenergy contours at $k_z=0$ and $k_z=0.5$ for the $E_{\mathrm{F}}=E_{F,\exp }$ surface with all unique extremal orbits marked. Several high- and low-frequency orbits can be identified. The Fermi surface pocket of origin is indicated through the orbit color, which will be used for comparison throughout our discussion. While there are clearly multiple frequencies in the predicted spectrum, note that there is only one symmetry unique (39 T) low-frequency mode, which is at odds with our experimental observations. Further, for frequencies $400<f<2000\mathrm{T}$, there are only four predicted modes, as opposed to the five experimentally observed components.

Figure 8

(a) Fermi surfaces for the SoD and TrH distortions in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ generated from the unfolded supercell band structures. Undistorted Fermi surface (black lines) contours are shown for comparison. A significant reconstruction of the V $d$ states occurs. Most high-frequency modes are preserved, though the C-II orbit (dashed line) is gapped in the SoD structure. (b) Schematic description of Fermi surface reconstruction. Gray orbits are well above measurable frequencies. Band reconstruction in the TrH supercell introduces one additional triangular orbit around the M point while preserving higher-frequency native V $d$ and Sb $p$ orbits. Notably, the distortion also introduces three, smaller, Dirac-like orbits (purple) shown in Fig. 4, consistent with our measurements.

Figure 9

Frequencies extracted as a function of Fermi level in undistorted ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$, which allow us to directly compare the experimental SdH data (gray bars) with the DFT models. The undistorted oscillation frequencies are shown as a function of Fermi level (left). Colored frequencies have been grouped by the pocket and orbit of the origin. A conservative numerical estimate for the supercell orbits for the SoD and TrH structures is shown (left) at $E_{\mathrm{F}}=E_{F,\exp }$. While five mid-frequency orbits are indeed expected in the range $E_{\mathrm{F}}\pm 0.1\mathrm{eV}$, no singular choice of $E_{\mathrm{F}}$ produces complete agreement with experiment. However, the TrH distortion introduces a new high-frequency V $d$ orbit, similar to B-III, which provides a consistent hypothesis. The DFT resolution limit (pink shaded) indicates the frequency range below which we cannot accurately resolve closed orbits.