Unified pressure and field response across distinct charge-order regimes in Ti-doped CsV$_3$Sb$_5$

Abstract

Understanding the phase diagram of kagome superconductors from a microscopic perspective is crucial for clarifying the interplay between charge order and superconductivity. Ti-doped CsV$_{3}$Sb$_{5}$ exhibits a nonmonotonic temperature-doping phase diagram in which both $T_{\rm c}$ and the charge-order temperature initially decrease with doping, followed by a crossover from long-range to short-range charge order and a subsequent increase in $T_{\rm c}$. Here, we report a muon spin rotation ($μ$SR) study of Ti-doped CsV$_{3}$Sb$_{5}$ at two representative compositions: underdoped (Ti$_{0.05}$-CVS) and optimally doped (Ti$_{0.22}$-CVS). Using zero-field, high-field, and high-pressure $μ$SR, we find spontaneous time-reversal-symmetry (TRS) breaking in the normal state of both compositions, strongly enhanced by an applied magnetic field and associated with long-range and short-range charge-order correlations, respectively. In the superconducting state, both samples exhibit anisotropic nodeless pairing with low superfluid density. Hydrostatic pressure substantially enhances both $T_{\rm c}$ and the superfluid density (by $\sim$2.5), revealing a linear correlation between them and pointing to unconventional pairing. Above $\sim$1 GPa, a crossover from anisotropic to isotropic nodeless pairing is observed. Despite the different nature of charge order in the two doping regimes, the superconducting responses are remarkably similar, suggesting that the competition between superconductivity and charge order occurs on a local scale, largely independent of the long-range coherence of the charge-ordered state.

Figures (4)

• Figure 1: Schematic phase diagram and superconducting scaling in Ti-doped CsV$_3$Sb$_5$. a Electronic phase diagram of Ti-doped CsV$_3$Sb$_5$, illustrating the Ti-concentration dependence of charge order and superconductivity. Color gradient indicates the transition between long-range (LRCO) and short-range (SRCO) charge order and corresponding superconducting regimes SC 1 and SC 2. Vertical gray line corresponds to the crossover doping level. Black lines represent the schematic Ti-concentration dependencies of charge-order transition temperature (solid for long-range ($T_{\rm{LRCO}}$) and dashed for short-range ($T_{\rm{SRCO}}$) regimes, respectively). The dashed-dotted schematic line for the superconducting transition temperature $T_{\rm{c}}$ is taken as an average through a large number of data points reported in the literature yang2022titaniumhou2023effectwu2023unidirectionalwu2025competitivepokharel2025evolution. Vertical white lines indicate two particular doping levels, ${x=0.05}$ and ${x=0.22}$ investigated in this study. The transition temperatures determined from our ${\mu}$SR experiments are marked by star (CO) and diamond (SC) symbols, in blue and red colors for Ti${_{0.05}}$-CVS and Ti${_{0.22}}$-CVS, respectively. b Plot of $T_\text{c}$ versus $\lambda_\text{eff}^{-2}(0)$ in logarithmic scale summarizing results of our ${\mu}$SR experiments as a function of pressure. Dependencies reported previously for hole- and electron-doped cupratesuemura1989universaluemura1991basicuemura2004condensationuemura2009energyshengelaya2005muon and various kagomemielke2022timeguguchia2023tunablegupta2022microscopicmielke2021nodelessmielke2022local and TMDguguchia2017signaturesvon2019unconventionalsazgari2025competing materials (gray shaded area) are shown for comparison, as well as the region of typical BCS values.
• Figure 2: Normal-state properties of Ti-doped CsV$_3$Sb$_5$ probed by ${\mu}$SR. a Zero-field and longitudinal-field ${\mu}$SR spectra recorded at 5 K for Ti${_{0.22}}$-CVS. The black solid lines represent fits using Eq. \ref{['eqGKT2']}. Error bars are the standard error of the mean (s.e.m.) in about ${10^6}$ events. b Temperature dependence of the relaxation rates ${\sigma}$ (light) and ${\Gamma}$ (dark) obtained for Ti${_{0.05}}$-CVS (blue) and Ti${_{0.22}}$-CVS (red), respectively. Vertical lines in corresponding colors mark the long-range and short-range charge-order transition temperatures $T_\text{LRCO}$wu2025competitive and $T_\text{SRCO}$xiao2026evolution, determined in previous works wu2025competitivexiao2026evolution. c,d Temperature dependence of the high transverse-field muon spin relaxation rates measured in selected external fields for Ti${_{0.05}}$-CVS (c) and Ti${_{0.22}}$-CVS (d), respectively.
• Figure 3: Superconducting-state properties of Ti-doped CsV$_3$Sb$_5$ probed by ${\mu}$SR. a,b Transverse field (TF) ${\mu}$SR spectra collected above (6 K) and below (30 mK) the superconducting transition after field-cooling from above $T_\text{c}$ in an applied field of 10 mT, for Ti${_{0.05}}$-CVS and Ti${_{0.22}}$-CVS, respectively. Error bars are the standard error of the mean (s.e.m.) in about ${10^6}$ events. c Temperature dependence of the superconducting muon spin depolarization rate, $\sigma_\text{SC}$ (left axis) and inverse squared effective penetration depth, $\lambda_\text{eff}^{-2}(T)$ (right axis) measured in 10 mT, together with the fits to theoretical models. The error bars represent the standard error extracted from the covariance matrix in the fitting. d Diamagnetic shift $\Delta B_\text{int}$ at 10 mT.
• Figure 4: Summary of the pressure impact on superconducting properties of Ti-doped CsV$_3$Sb$_5$ as revealed by ${\mu}$SR measurements. a,b Superconducting muon spin depolarization rates of Ti${_{0.05}}$-CVS and Ti${_{0.22}}$-CVS, respectively, as functions of pressure, together with the fits to theoretical models. Solid lines correspond to the anisotropic nodeless model, transitioning to the isotropic nodeless gap structure at elevated pressures (dashed-dotted lines). c Pressure phase diagram summarizing the superconducting properties of both systems, showing the pressure dependencies of the superconducting transition temperature $T_\text{c}$ (left axis) and the anisotropy parameter of the superconducting gap (right axis). Background gradient indicates the transition between anisotropic and isotropic nodeless gap structure.