Two types of charge order with distinct interplay with superconductivity in the kagome material CsV3Sb5

CsV3Sb5 is a kagome superconductor exhibiting multigap superconductivity and chiral charge order that breaks time-reversal symmetry below T_CO ≈ 94–95 K. Under hydrostatic pressure, the system shows a pronounced double superconducting dome in the temperature–pressure phase diagram while charge order persists, with Tc enhanced from ≈2.5 K at ambient pressure up to ≈8 K near 2 GPa. The research question is to elucidate the microscopic origin of this unusual double-dome behavior and the interplay between charge order and superconductivity. The study combines high-pressure muon spin relaxation/rotation (μSR) and first-principles calculations to probe how superfluid density and charge-order patterns evolve with pressure and how they correlate with superconductivity and time-reversal symmetry breaking.

Prior studies established superconductivity and time-reversal-symmetry-breaking charge order in AV3Sb5 (A = K, Rb, Cs), with CsV3Sb5 having the highest Tc (~2.5 K) and multigap superconductivity at ambient pressure. Transport and magnetization under pressure reported a double superconducting dome and enhanced Tc near ~2 GPa in CsV3Sb5, more prominent than in Rb and K analogs, which generally show a single dome or less pronounced features. STM, Kerr, and μSR indicated TRSB in the charge-ordered state. Theoretical works predicted multiple nearby charge-order instabilities associated with M and L point phonons and possible chiral flux phases, with structural periodicities 2×2×2 commonly observed. Phonon calculations suggested suppression of lattice instabilities at several GPa. These motivate a pressure-tuned exploration of CO–SC interplay and potential changes in CO topology.

- Samples: Single crystals of CsV3Sb5 grown via self-flux (Cs ingots 99.9%, V 3-N 99.9%, Sb grains 99.999%). Details reported previously. - μSR under pressure: Conducted at PSI (μE1 beam line, GPD spectrometer). 100% spin-polarized μ+ implanted; muon spin precesses around local Bμ. Temperatures down to 0.25 K using a He-3 Heliox cryostat. Pressures up to ~1.9 GPa using a low-background MP35N/CuBe double-wall piston-cylinder pressure cell. Daphne 7373 oil ensured hydrostatic conditions. Pressure calibrated at low T by ac susceptibility of indium. Approximately 40% muons stopped in the sample. Data acquired with four detector geometry; ~10^6 positrons per point; analysis with MUSRFIT. - TF-μSR: Field-cooled measurements at 10 mT to form a homogeneous vortex lattice. Extracted superconducting Gaussian relaxation rate σ_s(T), related to the inverse squared magnetic penetration depth λ^-2(T) via standard Brandt relation. Fourier transforms used to visualize internal field distributions. - ZF-μSR: Sensitive probe of internal magnetic fields (~0.1 G) to detect TRSB. Spectra modeled by Gaussian Kubo-Toyabe function multiplied by an exponential exp(-Λ t), separating nuclear and electronic relaxation contributions. Temperature dependence of electronic relaxation rate Λ(T) parameterized by Λ(T) = A + ΔA[1 - (T/T*)^n] below T*. - Penetration depth analysis: λ^-2(T) analyzed within local London model using a two-gap s+s-wave phenomenological α-model: λ^-2(T) = x λ^-2(0,Δ_01) + (1 - x) λ^-2(0,Δ_02), with BCS-like temperature dependence of gaps; x taken as a global parameter within each pressure region (I, II, III). The 1.68 GPa dataset modeled as a linear combination of neighboring SC+CO and SC states. - First-principles calculations: DFT with PAW in VASP 5.4.4 using PBEsol. Plane-wave cutoff 450 eV; Monkhorst-Pack k-mesh 20×20×10; Methfessel-Paxton smearing 10 meV. Non-spin-polarized calculations; enthalpy used for phase stability at finite pressure. Considered candidate CO structures arising from M and L point phonon instabilities: (MMM) planar tri-hexagonal (2×2×1), (MMM)+(LLL) superimposed tri-hexagonal Star-of-David (2×2×2), and (M00)+(OLL) staggered tri-hexagonal (2×2×2). Calculated pressure-dependent enthalpy differences and densities of states (DOS) at EF.

- Superfluid density and Tc vs pressure: Both Tc and λ^-2(0) show a non-monotonic double-peak dependence on pressure, defining three regions: • Region I (SC + CO): Tc increases from 2.85(9) K at ambient to 6.9(3) K at pc1 ≈ 0.63 GPa; λ^-2(0) also increases. • Region II (SC + CO): Tc drops sharply to 2.87(7) K near 0.8 GPa, then increases slowly with pressure. • Region III (pure SC): At pc2 ≈ 1.74 GPa, Tc jumps to Tmax ≈ 8.0(1) K and nearly saturates; λ^-2(0) rises to 17.3(9) μm^-2 from 6.5(1) μm^-2 at ambient, then saturates. Relative variation δλ^-2(0)/λ^-2(0) = 63%. - Multigap superconductivity persists: λ^-2(T) across all pressures is well described by a two-gap s+s-wave model, indicating robust, nodeless multigap SC under pressure. At 1.68 GPa (border of SC+CO and SC), λ^-2(T) shows a two-step feature consistent with inhomogeneity or phase separation near a first-order transition. - Tc–superfluid density scaling: Linear relations Tc vs λ^-2(0) with distinct slopes in each region: • Region I slope ≈ 2.72 K·μm^2; Region II slope ≈ 0.78 K·μm^2; Region III shows λ^-2(0) nearly independent of Tc. The large Tc/λ^-2(0) ratios indicate unconventional superconductivity. - ZF-μSR evidence for TRSB: Electronic relaxation increases below Tc in all regions: • Ambient (Region I): ΔA0 = 0.024(8) μs^-1 with onset slightly above Tc, attributed to Meissner screening of CO-induced loop-current fields. • 1.18 GPa (Region II): ΔA1.18 = 0.011(2) μs^-1 with CO + SC coexistence. • 1.74 GPa (Region III): ΔA1.74 = 0.031(3) μs^-1 with smooth onset at Tc; corresponds to internal field width ΔB = ΔA/γμ ≈ 0.04 mT, comparable to known TRSB superconductors, indicating spontaneous TRSB in the pure SC state. - DFT results on CO evolution: Among candidate COs, the staggered tri-hexagonal (2×2×2) generally has the lowest Kohn-Sham enthalpy, with small enthalpy differences (few meV/f.u.) relative to the superimposed tri-hexagonal Star-of-David (2×2×2), allowing entropy to alter stability. CO is predicted to be fully suppressed above ~5 GPa. DOS analysis shows the undistorted lattice has the largest DOS at EF and van Hove peaks closest to EF; the staggered tri-hexagonal has the smallest DOS and more separated van Hove peaks than the Star-of-David phase. This supports a pressure-driven change from Star-of-David (Region I) to staggered tri-hexagonal (Region II), which competes more strongly with SC, and the highest Tc when CO is absent (Region III).

The double-dome superconductivity and the correlated threefold enhancement of superfluid density arise from competition between superconductivity and charge order that evolves with pressure. Regions I and II both host SC + CO, but with distinct Tc–λ^-2(0) scaling slopes and a sharp drop in both Tc and superfluid density across pc1, indicating a change in the nature of CO rather than SC. DFT suggests that pressure tunes the CO pattern from a superimposed tri-hexagonal Star-of-David at lower pressures to a staggered tri-hexagonal at intermediate pressures; the latter more strongly reduces the Fermi-level DOS and thus more strongly suppresses SC, consistent with the reduced Tc and superfluid density in Region II. At pc2, the abrupt increases in Tc and λ^-2(0) and the near temperature-independence of λ^-2 in Region III indicate the disappearance of competing CO and entry into a pure SC state. ZF-μSR shows TRSB-associated relaxation increases in Regions I and II likely due to CO loop-current fields. In Region III, where CO is suppressed, the emergence of spontaneous internal fields below Tc indicates that the superconducting state itself breaks time-reversal symmetry. Together, the results demonstrate that pressure drives a transformation of the CO landscape that modulates its competition with superconductivity, culminating in a TRSB superconducting state without CO.

The study reveals two distinct charge-ordered phases in CsV3Sb5 under pressure that compete differently with superconductivity: a superimposed tri-hexagonal Star-of-David phase at low pressures (Region I) and a staggered tri-hexagonal phase at intermediate pressures (Region II). Their evolution produces a double-peak in both Tc and superfluid density, while the superconducting state remains nodeless and multigap across all pressures studied. Once charge order is fully suppressed (Region III), superconductivity itself breaks time-reversal symmetry. These findings clarify the microscopic interplay of charge order and superconductivity in kagome metals and motivate future studies to directly resolve the proposed CO transitions under pressure and their coupling to superconductivity.

- First-principles calculations were non-spin-polarized and cannot directly capture time-reversal symmetry breaking; they focus on lattice energetics and therefore may miss TRSB-specific electronic effects. - DFT overestimates the pressure for CO suppression (predicts >5 GPa vs experimental pc2 ≈ 1.74 GPa), reflecting uncertainties from exchange-correlation approximations, large unit cells, and sensitivity of the electronic structure under pressure; entropic effects not included could change phase ordering. - The 1.68 GPa λ^-2(T) two-step feature suggests inhomogeneity that could arise from pressure inhomogeneity in the cell or intrinsic phase separation near a first-order boundary. - Samples were randomly oriented within the pressure cell, and μSR averages over muon stopping sites, which may obscure anisotropic details. - The Tc–λ^-2 scaling is empirical; without a complete microscopic theory for pairing in CsV3Sb5, causality between DOS changes and Tc enhancements is inferred but not directly proven.