Small Fermi Pockets Intertwined with Charge Stripes and Pair Density Wave Order in a Kagome Superconductor

The kagome superconductor family $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{Cs}$, K, Rb) emerged as an exciting platform to study exotic Fermi surface instabilities. Here, we use spectroscopic-imaging scanning tunneling microscopy (SI-STM) and angle-resolved photoemission spectroscopy (ARPES) to reveal how the surprising cascade of higher- and lower-dimensional density waves in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ is intimately tied to a set of small reconstructed Fermi pockets. ARPES measurements visualize the formation of these pockets generated by a 3D charge density wave transition. The pockets are connected by dispersive $q^{*}$ wave vectors observed in Fourier transforms of STM differential conductance maps. As the additional 1D charge order emerges at a lower temperature, $q^{*}$ wave vectors become substantially renormalized, signaling further reconstruction of the Fermi pockets. Remarkably, in the superconducting state, the superconducting gap modulations give rise to an in-plane Cooper pair density wave at the same $q^{*}$ wave vectors. Our work demonstrates the intrinsic origin of the charge stripes and the pair density wave in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ and their relationship to the Fermi pockets. These experiments uncover a unique scenario of how Fermi pockets generated by a parent charge density wave state can provide a favorable platform for the emergence of additional density waves.

Popular Summary

In condensed matter physics, the concept of the Fermi surface is a powerful tool for describing electronic behavior. This surface in momentum space delineates occupied electronic states from unoccupied ones at zero temperature. In correlated electron systems—where electrons strongly interact with one another—the Fermi surface develops “pockets” that result from coexisting electronic orders vying for the same electronic states. Recent theory has predicted that such pockets might be instrumental in understanding exotic electronic states in the $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ family of superconductors. Here, we experimentally demonstrate the emergence of small ellipsoidal Fermi pockets in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ and their intimate connection to unusual charge density waves.

The rich array of electronic phenomena in the $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ family includes an unprecedented number of charge density waves with varying morphology and dimensionality, which emerge at different temperatures upon cooling and coexist with superconductivity. In contrast to the well-established “parent” 3D charge density wave, the surprising emergence of additional density waves at lower temperatures has been difficult to capture by techniques lacking real- and momentum-space resolution.

Using a combination of spectroscopic techniques, we provide direct evidence that the Fermi pockets are caused by the parent charge density wave transition at high temperature. Second, we reveal that these pockets further change in shape and size by the emergence of a charge-stripe order. Last, we find that the vectors connecting the pockets are identical to those of superconducting gap modulations, which form a Cooper pair density wave in the superconducting state.

Our experiments reveal the common roots of translation symmetry-breaking phenomena and their relation to other exotic phenomena observed in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$.

Figure 1

Crystal structure and emergent charge orders in $A{V}_3{\mathrm{Sb}}_5$. (a) A 2D and a 3D ball model of the crystal structure of $A{V}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$, Cs, Rb) crystals. STM topographs of (b) ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ (Sb termination) and (c) ${\mathrm{KV}}_3{\mathrm{Sb}}_5$ (Sb termination) taken at 4.5 K, and (e,f) their associated Fourier transform (FT). The bottom right inset in panel (b) is a STM topograph encompassing a smaller region of the Sb termination taken at 60 K, the temperature around which the $4a_0$ charge stripe order disappears. Atomic Bragg peaks ${\mathbf{Q}}_{\text{Bragg}}$, $2a_0$ CDW peaks ${\mathbf{Q}}_{2\mathrm{a}0}$, and charge-stripe order ${\mathbf{Q}}_{4\mathrm{a}0}$ peaks in panel (e,f) are circled in black, blue, and red, respectively. (d) Schematic of the theoretically expected Fermi pockets and reciprocal space vectors connecting them from Ref. [15]. For simplicity of visualizing relevant pockets in panel (d), we omit plotting the six additional pockets within the first Brillouin zone, “folded-in” by the $2a_0$-by-$2a_0$ CDW. STM setup conditions: $I_{\mathrm{set}}=300\mathrm{pA}$, $V_{\text{sample}}=50\mathrm{mV}$. Inset: $I_{\mathrm{set}}=30\mathrm{pA}$, $V_{\text{sample}}=50\mathrm{mV}$ (b); $I_{\mathrm{set}}=400\mathrm{pA}$, $V_{\text{sample}}=20\mathrm{mV}$ (c).

Figure 2

Momentum-space mapping of Fermi pockets in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. Fermi surfaces of ${\mathrm{KV}}_3{\mathrm{Sb}}_5$ across the first and second Brillouin zone at 6 K (a) and 140 K (b). Black and blue dashed hexagons represent the Brillouin zone in normal and charge density wave phases, respectively. Red solid and dashed ellipses indicate the Fermi pockets with strong and weak spectral weights, respectively. The three double-sided arrows with gray, brown, and orange show wave vectors ${\mathbf{q}}_a^{*}$, ${\mathbf{q}}_b^{*}$, and ${\mathbf{q}}_c^{*}$, respectively. We show Fermi surfaces in the green dashed squares at 6 K (c) and 140 K (d). Green, blue, and red arrows indicate K1, K2, and $\mathrm{K}{2}^{\prime }$ bands, respectively. Red solid curves represent the momentum distribution curves (MDCs) along the red dashed lines. We show Fermi surfaces of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ (e,g) and ${\mathrm{RbV}}_3{\mathrm{Sb}}_5$ (f,h) variants. Green arrows in panels (g) and (h) are visual guides showing the Fermi pockets.

Figure 3

Reconstruction of the electronic structure across the CDW transition in ${\mathrm{KV}}_3{\mathrm{Sb}}_5$. (a) Comparison between experimental and calculated Fermi surfaces at 97 eV ($k_z=0.67\pi /\mathrm{c}$). The calculated Fermi surface assumes a staggered tri-hexagonal charge order and is threefold symmetrized to average over three symmetry-equivalent domains. (b) Calculated dispersion along high-symmetry points for $k_z=0.67\pi /\mathrm{c}$. The inset shows the magnified band dispersion inside the yellow box. We show the energy-momentum dispersion along $\overline{\mathbf{K}}\text{−}\overline{\mathbf{M}}\text{−}\overline{\mathbf{K}}$ [red dashed line in Fig. 2] at 140 K (c) and 6 K (e), $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathbf{M}}\text{−}\overline{\mathrm{\Gamma }}$ [navy dashed line in Fig. 2] at 140 K (d) and 6 K (f), and $\overline{\mathrm{\Gamma }}\text{−}{\overline{\mathbf{M}}}_2\text{−}\overline{\mathbf{K}}\text{−}\overline{\mathbf{M}}$ [purple dashed line in Fig. 2] at 140 K (g) and 6 K (h). The K1 and K2 bands (marked with green and blue arrows, respectively) exhibit a holelike dispersion along the $\overline{\mathbf{K}}\text{−}\overline{\mathbf{M}}\text{−}\overline{\mathbf{K}}$ direction (c) and electronlike dispersion along $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathbf{M}}\text{−}\overline{\mathrm{\Gamma }}$ (d).

Figure 4

Dispersive scattering peaks ${\mathbf{q}}^{*}$ near the Fermi level in ${\mathrm{KV}}_3{\mathrm{Sb}}_5$. (a) Twofold symmetrized FT of $dI/dV(\mathbf{r},V=0\mathrm{mV})$ map acquired on the Sb surface of ${\mathrm{KV}}_3{\mathrm{Sb}}_5$. Atomic Bragg peaks ${\mathbf{Q}}_{\text{Bragg}}$ are enclosed in dark blue circles, while the new scattering peaks ${\mathbf{q}}_i^{*}$ ($i=a$,$b$, or $c$) near $3/4·{\mathbf{Q}}_{\text{Bragg}}$ are shown by dashed circles in gray (${\mathbf{q}}_a^{*}$), brown (${\mathbf{q}}_b^{*}$), and orange (${\mathbf{q}}_c^{*}$). (b) FT linecuts along the three ${\mathbf{Q}}_{\text{Bragg}}$ directions (circles) and the Gaussian fittings to each peak in the curve (solid line). The linecut is averaged by seven pixels perpendicular to the linecut. (c) Approximate schematic of the relevant portions of the Fermi surface of the ${\mathrm{KV}}_3{\mathrm{Sb}}_5$ band structure, with an example of a Fermi pocket shown by the green arrow. The three double-sided arrows in panel (c) denote scattering processes corresponding to different ${\mathbf{q}}^{*}$ wave vectors in panel (a). In principle, other scattering vectors between the pockets should be possible, in addition to ${\mathbf{q}}_i^{*}$, but we are not able to clearly resolve them in our data (Fig. 9 in the Supplemental Material [49]). (d)–(f) Position of the ${\mathbf{q}}_i^{*}$ peaks as a function of energy. All three peaks are dispersive with similar band velocities [the thick gray line in panels (d)–(f) is a visual guide]. The position of the peak is extracted from Gaussian fitting as shown for one energy in panel (b), and the error bars represent the standard error from Gaussian fitting. STM setup conditions: (a) $I_{\mathrm{set}}=400\mathrm{pA}$, $V_{\text{sample}}=20\mathrm{mV}$, $V_{\mathrm{exc}}=1\mathrm{mV}$, 4.5 K.

Figure 5

Renormalization of ${\mathbf{q}}^{*}$ scattering wave vectors in the presence of the charge-stripe order in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. (a) FT of the $dI/dV(\mathbf{r},V=-2\mathrm{mV})$ map acquired over a 96-nm-square Sb surface of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. Atomic Bragg peaks ${\mathbf{Q}}_{\text{Bragg}}$ are enclosed in blue circles, while the new scattering peaks ${\mathbf{q}}_i^{*}$ ($i=a$,$b$, or $c$) are shown by dashed circles in dark gray (${\mathbf{q}}_a^{*}$), brown (${\mathbf{q}}_b^{*}$), and orange (${\mathbf{q}}_c^{*}$). (b) Dispersion of the ${\mathbf{q}}_i^{*}$ peaks as a function of energy. In contrast to the dispersion of ${\mathbf{q}}_i^{*}$ in the absence of charge-stripe order in Fig. 3, only ${\mathbf{q}}_b^{*}$ and ${\mathbf{q}}_c^{*}$ are dispersive, while ${\mathbf{q}}_a^{*}$ is static and at a slightly different q-space position. (c) Approximate schematic of the Fermi surface of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ without (left panel) and with (right panel) charge-stripe order. (d) Comparison of ${\mathbf{q}}_i^{*}$ scattering wave vectors seen in our experiments, consistent with the schematics in panel (c). (e) Zoom on a region in FT space [denoted by a triangle in panel (a)], more clearly showing ${\mathbf{q}}_b^{*}$ and the Bragg reflection of ${\mathbf{q}}_c^{*}$ (labeled ${\mathbf{q}}_c^{**}$). The red dashed line connects ${\mathbf{Q}}_{\text{Bragg}}$ (blue circle) and ${\mathbf{Q}}_{2\mathrm{a}0}$ (purple circle). The bottom curve in panel (e) is the linecut (green open circles) taken along the green arrow direction in the top panel, and Gaussian fits (solid black line). The linecut is averaged by three pixels along the linecut and 15 pixels perpendicular to the linecut. (f) Positions of the ${\mathbf{q}}_b^{*}$ and ${\mathbf{q}}_c^{**}$ in panel (e) relative to the high-symmetry line at each energy for ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ (vertical red dashed line). The position of the peak at each energy is extracted from Gaussian fits as seen in the example in panel (e). The error bars represent the standard error from Gaussian fitting. STM setup conditions: (a) $I_{set}=40\mathrm{pA}$, $V_{\text{sample}}=-2\mathrm{mV}$, $V_{\mathrm{exc}}=1\mathrm{mV}$, 4.5 K.