Observation of ubiquitous charge correlations and hidden quantum critical point in hole-doped kagome superconductors

Abstract

The interplay between superconductivity and charge-density wave (CDW) order, and its evolution with carrier density, is central to the physics of many quantum materials, notably high-$T_c$ cuprates and kagome metals. Hole-doped kagome compounds exhibit puzzling double-dome superconductivity and, as chemical substitution inevitably introduces quenched disorder, their properties remain poorly understood. Here, by leveraging the sensitivity of nuclear quadrupole resonance to local and static orderings, we uncover new features, primarily the incipient and fragmented CDW phases, in the charge landscape of CsV$_3$Sb$_{5-x}$Sn$_x$. Static CDW puddles are observed well above the transition temperature, a hallmark of pinning by defects. Their doping and temperature evolution indicate that, in the absence of disorder, the inverse Star-of-David $π$-shifted (ISD-$π$) CDW order would vanish near $x=0.12$, between the two superconducting domes. This critical doping represents a hidden quantum critical point. Nevertheless, the ISD-$π$ pattern persists well beyond previous reports, although its volume fraction is progressively reduced up to the critical doping at which it saturates. We establish that carrier doping promotes fragmentation of the ISD-$π$ order, whereas randomness preserves the ISD-$π$ patches.

Figures (4)

• Figure 1: Commensurate inverse Star-of-David CDW. (a) Crystallographic structure of CsV$_3$Sb$_5$ in the pristine phase. The kagome pattern is realized by the V atom in red. We distinguish two Sb sites: the basal Sb1 site, blue, and the apical Sb2 site, green. (b) Crystallographic structure of CsV$_3$Sb$_5$ below $T_\text{CDW}$. The hexagonal lattice becomes orthorhombic with an inverse Star-of-David $\pi$-shifted modulation (ISD-$\pi$) (c) NQR $^{121}$Sb spectra for the 5/2 $\rightarrow$ 3/2 transition as a function of temperature and Sn-doping $x$ in CsV$_3$Sb$_{5-x}$Sn$_x$. The solid curves represent Lorentzian-based models, while the data is the shaded color area. Even at high Sn-doping, charge ordering $T_\text{CDW}$ is unambiguously determined by the presence of the Sb2-a peak (brown curve) at approximately 156 MHz. For the undoped sample, below $T_\text{CDW}$, the whole sample is in the ISD-$\pi$ phase. The Sb1-a and Sb1-b = Sb1-b' & Sb1-b" peaks are omitted for $x\ge 0.04$. The extra room temperature peaks belong to the Sb2 site based on site-occupancy analysis and dopant-induced neighboring-site effects.
• Figure 2: CDW order parameter. (a) The frequency of the Sb2-a site is proportional to the amplitude of the CDW modulation. The minimal change in the low temperature Sb2-a frequency shows that the CDW amplitude is almost unaltered by the dopant. (b) The ISD-$\pi$ volume fraction as a function of temperature. (c) The low-temperature volume-fraction ($f_\text{CDW}$) as a function of Sn-doping shows that the ISD-$\pi$ survives in at least half of the sample. Open symbols are for data calculated by two techniques (Supplementary Note II). Error bars represent two standard deviations and are not shown when smaller than the symbol.
• Figure 3: High-temperature charge correlations. (a) Full-width-at-half-maximum ($\Sigma$) of the dominant Sb2 peak in the pristine phase $T>T_\text{CDW}$. The blue-shaded area represents the $T_\text{CDW}$ range for different dopings. (b) Linear fit to 1/$\Sigma$ demonstrates the Curie-Weiss (CW) behavior of the pristine-phase Sb2 linewidth and directly shows the intersection with the $x$-axis. The solid lines are CW fits. The inset is the undoped data. The dashed line is the CW fit of the undoped compound. Note that the undoped compound is a pure crystal as opposed to doped samples which are powders. (c) The doping dependence of the CW parameter $A$ shows that Sn-doping is linearly proportional to the intrinsic CDW features. (d) The CW temperature extracted from the linear fit shows a sign change in the vicinity of $x = 0.12$. The dashed green line is a guide to the eye. Error bars represent two standard deviations and are not shown when they are smaller than the symbol size.
• Figure 4: Temperature phase diagram of CsV$_3$Sb$_{5-x}$Sn$_x$ as a function of Sn-doping. (a) Incipient CDW exists in the high temperature region, $T > T_\text{CDW}$. The ISD-$\pi$ onset temperature is denoted by $T_\text{CDW}$, where LR denotes long-range order. The gradient represents crossover between the LR and fragmented order. Open diamond and square symbols are magnetization $\chi$ measurements of the bulk CDW, dark blue, and SC, burned orange, transition temperatures, from Oey2022. (b) Volume fraction of ISD-$\pi$. Shown error bars reflect two standard deviations. (c) Doping-dependence of the CW temperature $\theta_\text{CW}$. The value $\theta_\text{CW} = 0$ indicates critical doping at which a putative hidden quantum critical point (QCP) lies. Error bars not shown for clarity. Solid lines serve as a guide to the eye. (d-f) Schematic of the $T$-dependent formation of CDW in the presence of quenched disorder for low and high doping. Black dots represent Sn-dopants and disorder, while purple lines represent charge modulations. (d) For $T>T_\text{CDW}$, Friedel oscillations occur around impurities to form incipient CDW. (e) On approaching $T_\text{CDW}$, the incipient CDW spreads. (f) For $T<T_\text{CDW}$, long-range CDW develops at low doping, while increased doping induces domain walls (white ribbon) that fragment the parent CDW.