Vortex phase diagram of kagome superconductor CsV₃Sb₅

Correlated metals in reduced dimensions often exhibit propensity to multiple ordered phases with similar energies, which leads to the phenomenon of “intertwined order” [1, 2], where multiple phases emerge out of a primary phase. Kagome metals, featuring correlated electron effects and a topologically nontrivial band structure, have recently emerged as a compelling family of compounds to realize such multiple ordered electronic states. A particularly interesting material system in that respect is the class of quasi-two-dimensional kagome metals AV₃Sb₅, which exhibit charge order transitions at $\sim$80 K, 103 K and 94 K for A = K, Rb and Cs respectively [3]. Focusing on CsV₃Sb₅, a CDW transition is revealed, which is associated with a substantial reconstruction of the Fermi surface pockets linked to the vanadium orbitals and the kagome lattice framework [4]. Nuclear magnetic resonance studies on the different vanadium sites is consistent with orbital ordering at T$\sim$94 K induced by a first order structural transition, accompanied by electronic charge density wave (CDW) that appears to grow gradually below $T_{CDW}$, with possible intermediate subtle stacking transitions perpendicular to the kagome planes [5].

The appearance of superconductivity with $T_{c}$ ranging from $\sim$2.5 K to 4 K prompted further examinations of the relation between the normal and the superconducting states [6, 7], with initial studies suggesting a pair-density-wave (PDW) [8]. While the possibility of a time-reversal symmetry breaking CDW state [9] is still under debate [10], the emerging superconducting state appears to exhibit a gapped conventional s-wave order parameter [9]. With in-plane $H_{c2}$ clearly dominated by orbital effects and a large Ginzburg-Landau parameter (here we find $\kappa\sim 35$), we might expect a rather ordinary vortex physics behavior. Considering the non-trivial topological band structure of CsV₃Sb₅, Majorana zero modes (MZM) excitations in vortex cores [11] may be expected. However, it is not yet clear whether there exists a range of magnetic fields where vortex-induced MZM do not fade-out due to overlap and thus affect the observed $H$-$T$ phase diagram of this material.

In this letter we examine the vortex phase diagram of CsV₃Sb₅ through measurements of the superfluid response using a high-resolution mutual-inductance probe, supplemented by resistivity and magnetization measurements. Whereas the low-temperature zero-field data confirms the absence of gapless quasiparticles, we find an unusually wide temperature-range of phase fluctuations, which at a finite magnetic field evolves into a wide vortex liquid phase. For a constant magnetic field, an irreversibility line is observed, separating unpinned dissipative vortex liquid from a weak pinning regime characterized by weak sensitivity to the measurement time scale. The vortex lattice melting transition, appearing at lower fields and temperatures, is identified as an abrupt increase in inductive response. Finally, a pronounced “peak effect” is observed below the melting transition, exhibiting a hysteresis that weakens with increasing temperature. We therefore observe that CsV₃Sb₅provides a unique example of a clean anisotropic superconductor where a hierarchical succession of transitions and crossovers in the vortex state do not cross each other (by contrast to e.g. Bi₂Sr₂CaCu₂O_8+y [12]) and where the melting point is clearly distinct from other effects (by contrast to e.g. YBa₂Cu₃O_7-y [13]). Taking into account the strong anisotropy of CsV₃Sb₅ [3, 14], we calculate a low temperature limit for vortex-entropy change as $\sim 0.04\ k_{B}$ per layer.

Peak effect is often observed in the inductive part of superfluid response, while exhibiting an enhancement in the depinning critical current density $j_{c}$ near $H_{c2}$ [15]. Typically accompanied by a hysteresis, it is closely linked to the strengthening of pinning in a vortex glass regime [16, 17]. Such effects have been under scrutiny in a broad range of type-II superconductors including Nb(O, Ti) [18], NbZr [19], YBa₂Cu₃O_7-δ [20, 21], 2H-NbSe₂ [22], (Ba, K)Fe₂As₂ [23], ternary stannide Ca₃Rh₄Sn₁₃ [24], as well as germanides Lu₃Os₄Ge₁₃ and Y₃Ru₄Ge₁₃ [25]. Different scenarios for the interplay between vortex pinning and lattice elasticity defines melting and irreversibility lines, which play important roles in determining the magnetic properties of high-$T_{c}$ cuprates such as YBa₂Cu₃O₇ and Bi₂Sr₂CaCu₂O₈ [26], often led to conflicting phase diagrams, particularly in discerning the thermodynamic melting transition out of the pinning-induced vortex state. The clear feature that we observe between the irreversibility line and the peak effect is unique in that respect, especially as it seems to mark the melting transition.

The mutual inductance (MI) technique that we are adopting for our studies offers an alternative approach of measuring the full complex ac superfluid response [27, 28]. In zero external magnetic field, screening response due to a small ac drive can be related to superfluid density and (effective) penetration depth, given exact geometries of the sample and MI coils [29]. In the present study, the complex ac response $V=V^{\prime}-iV^{\prime\prime}$ of superconducting CsV₃Sb₅  single crystals was measured using a gradiometer-type MI probe [28] as shown in Fig. 1. In this configuration, both the drive coil and the astatically-wound pair of receive coils (1.5-mm in diameter) are positioned above the sample. The drive coil is supplied an ac current of $20\ \mu$A at $100$ kHz, thus creating a minuscule ac magnetic field $H_{ac}\lesssim 3\ \text{mOe}\ll H_{c1}$ and inducing screening supercurrents (and potentially dissipations) in the superconductor; meanwhile, the receive coils pick up both quadratures $V^{\prime}$ and $V^{\prime\prime}$ of the induced emf using a lock-in amplifier phase-locked at the drive frequency.

CsV₃Sb₅ single crystals were prepared by self-flux method [3, 30]. An approximately $2\times 2\times 0.1$ mm³ hexagonal-shaped CsV₃Sb₅ crystal was silver pasted onto the cold finger of a dilution refrigerator, where the MI coils are mounted above and gently pressed down onto the sample. Care was taken to minimize the stress exerted by the coil assembly, which together with the measurement wiring is well-thermalized at the mixing chamber temperature. Details of the coil assembly can be found in the supplemental information.

We first focus on the superconducting transition in zero magnetic field as shown in Fig. 2. With careful tuning of the applied magnetic field such that any effect of remnant field is eliminated, we identify a zero-field superconducting transition temperature $T_{c}(H=0)\approx 4.0$ K, where the onset of inductive response $V^{\prime\prime}$ at 100 kHz is located. Following a gradual increase of $V^{\prime\prime}(T)$ over a transition region of $\Delta T/T_{c}\sim 1/4$, the inductive signal saturates at the maximum value below $T\lesssim 2$ K. The finite size of the sample with respect to our coils limited our ability to extract an effective penetration depth $\lambda_{\text{eff}}(T)\propto T^{-0.5}\exp(-\Delta(0)/T)$ or a superfluid density $n_{s}(T)\propto\lambda_{\text{eff}}^{-2}(T)$ from the measured $V^{\prime\prime}(T)$. Nevertheless, such temperature independence as $T\to 0$ is supported by the recent measurements of penetration depth that yielded similar results, i.e. nodeless superconductivity exhibiting no signature of gapless quasiparticles in zero magnetic field [31, 32].

Upon applying a dc magnetic field along the c-axis $H_{\text{dc}}>H_{c1}\sim 30$ Oe at $0.12$ K, the inductive response $V^{\prime\prime}(T)$ is increasingly suppressed and becomes temperature-dependent as $T\to 0$ in Fig. 2. From the measured $H_{c1}$ and an estimated $H_{c2,\parallel c}$ [14], we find an effective penetration depth $\lambda_{\text{eff}}=\sqrt{\Phi_{0}\ln(\lambda/\xi)/4\pi}\approx 400$ nm, consistent with the tunneling diode oscillator (TDO) method [31]. In intermediate fields $2\ \text{kOe}\lesssim H\lesssim 6$ kOe, the evolution of $V^{\prime\prime}(T)$ appears non-monotonic as a function of magnetic field. For each curve below $6$ kOe, there also appears to be a “knee” marking a change in slope and a “break” that separates a gradual superconducting transition at high temperatures and a more rapid linear form at intermediate temperatures. Above $6$ kOe, $V^{\prime\prime}(T)$ drops monotonically to zero as the inductive superfluid response is increasingly suppressed. We will examine such a non-monotonic evolution in details by exploring the magnetic-field-dependence of $V^{\prime\prime}$ isotherms.

In Fig. 3, isothermal field sweeps in CsV₃Sb₅ demonstrate a broad array of qualitative features related to various vortex states. Below a lower critical field $H_{c1}\sim 30$ Oe at low temperatures, the inductive ac response $V^{\prime\prime}(H)$ is independent of magnetic field as in the Meissner state. Above $H_{c1}$, $V^{\prime\prime}(H)$ drops sharply, corresponding to a rapidly weakening screening response, until a peak develops in each of the inductive response isotherms below $\sim$2 K. These peaks span a wide portion of the intermediate field range and correspond to the non-monotonic behavior in the temperature-dependence, reminiscent of the widely-known peak effect [19, 18, 15, 26, 22, 24, 23, 25].

The peaks in CsV₃Sb₅ appear more prominent at lower temperatures, approaching $30$% of the maximal zero-field signal, while a peak was also found in critical current measurements (see supplementary). The peak’s onset field ($H_{\text{onset}}$), peak field ($H_{\text{peak}}$), and width ($|H_{\text{base}}-H_{\text{onset}}|$) all increase with lower temperature, consistent with the canonical peak effect behaviors. Nevertheless, the peaks in CsV₃Sb₅ are located at an intermediate range of magnetic field, rather than close to the onset of screening response at higher fields.

Hysteresis in $V^{\prime\prime}(H)$ is largely seen at the low-field edges of the peaks with the largest split of $\sim$7 nV between upward and downward sweeps, and can be detected until at least $0.8$ K where the split drops below measurement sensitivity. There also exists a much weaker hysteresis at lower fields (see supplemental information.) Such a hysteresis has been identified as a signature of vortex glass phase [33, 17]. In addition, the low-field edges of the peaks in all isotherms together form an envelop, corresponding to a weak or an absence of temperature-dependence in the field windows where the isotherms overlap. These windows are located accordingly near the “knees” ($T_{\text{knee}}$) in the finite-field superconducting transition in Fig. 2.

The rich features in the temperature- and field-dependence of superfluid screening response $V^{\prime\prime}$ allow us to identify various vortex phases in an $H$-$T$ phase diagram Fig. 4. Close to zero magnetic field, the Meissner state with a constant diamagnetic susceptibility is bounded by the low $H_{c1}\sim 30$ Oe and is barely visible. From the field-dependence $V^{\prime\prime}(H)$, we extract the peak’s onset $H_{\text{onset}}(T)$ (violet), peak $H_{\text{peak}}(T)$ (blue), and base $H_{\text{base}}(T)$ (green) as a function of temperature. Independently, from the temperature-dependence $V^{\prime\prime}(T)$ we extract $T_{\text{knee}}(H)$ (light blue) and $T_{\text{break}}(H)$ (light green) as a function of field as explained previously. The resulting curves are surprisingly consistent with each other, where $T_{\text{knee}}(H)$ and $H_{\text{peak}}(T)$ as well as $T_{\text{break}}(H)$ and $H_{\text{base}}(T)$ are perfectly aligned, respectively. The peaks become indistinguishable above $\sim 0.5T_{c}\approx 2$ K.

We begin with the low-field and low-temperature regimes of the $H$-$T$ phase diagram. The strong hysteretic behavior (shadow) is bounded between $H_{\text{onset}}(T)$ (violet) and $H_{\text{peak}}(T)$ (blue), with its magnitude dropping below our sensitivity above $\sim 1$ K. Consequently, $H_{\text{onset}}$ can be designated as the onset of a strongly hysteretic vortex glass phase, replacing a less-disordered dislocation-free “Bragg glass” phase at lower fields [17]. The vortex glass region is highly correlated with the peaks in the screening response $V^{\prime\prime}(H)$ which signifies an enhancement in flux pinning [24]. Such an enhancement is commonly achieved through a crossover from the Larkin-Ovchinnikov collective pinning to individual pinning of the flux lines, following the softening of vortex lattice elastic moduli [16] in the presence of a significant anisotropy [14].

In previous studies of the vortex phase diagram in layered materials, particularly the cuprates, a gradual evolution in pinning strength is commonly expected between the boundary of the peak effect at $H_{\text{peak}}$ and $H_{\text{irr}}$. In these cases and with no otherwise observable features, $H_{\text{peak}}$ is identified with the melting transition (see e.g. [34, 35].) However, where a magnetization discontinuity is observed without a peak effect, a melting line was found to cross the irreversibility line in Bi₂Sr₂CaCu₂O₈ [12]. In the present case of CsV₃Sb₅, we find a pronounced break in the response of the vortex system between $H_{\text{peak}}$ and $H_{\text{irr}}$, which we denote as $H_{\text{base}}(T)$. This feature is also visible in the temperature scans of the inductive response, which we identify as $T_{\text{break}}(H)$. Such a break was previously found in surface impedance measurements of Bi₂Sr₂CaCu₂O₈ [36] and was identified with the melting line. Similar to our case, their technique, while performed at higher frequency, was sensitive to the superfluid density, which is expected to exhibit a sharp increase in superfluid density as the field is lowered through the melting transition. It is therefore natural to identify the line of ($T_{\text{break}}(H)$, $H_{\text{base}}(T)$) as the melting transition [37]. This now allow us to calculate the entropy change through the transition. Following Zeldov et al., [38]we can use Clausius-Clapeyron equation to calculate the latent heat of the transition: $L=T_{m}\Delta S=-(\Delta B/4\pi)(dH_{m}/dT)T_{m}$, where $(T_{m},H_{m})$ is the melting line and $\Delta B$ is the discontinuity in magnetic induction at the first order melting transition. Using a standard Lindeman criterion for melting [37] to estimate $\Delta B$, we find [38] $\Delta S\approx 0.1d\gamma\sqrt{B_{m}/\Phi_{0}}$, where $\gamma\approx\sqrt{600}$ is the anisotropy of CsV₃Sb₅ and $d\approx 9.3$ Å is the interlayer distance [3]. At low temperatures and high fields we find that $\Delta S\approx 0.04\ k_{B}$ per kagome layer, which is a factor of 10 smaller than the expected value from fully decoupled layers [39]. This implies correlations along the vortex line, which thus limit the number of degrees of freedom per vortex.

Returning to the peak effect, when compared with typical characteristics in both layered and non-layered superconductors [18, 19, 20, 21, 22, 23, 24, 25], the peak effect in CsV₃Sb₅ appears more prominent and spans a large range in parameter space. The details of pinning mechanisms are still unclear in CsV₃Sb₅, whether being randomly distributed point-like impurities, or correlated effects related to the underlying CDW and its associated distortions. However, at least down to $\sim T_{c}/2$ the temperature dependence of the irreversibility line and peak effect line follow the melting line, which could indicate that in that regime surface barrier effects are not important and bulk pinning dominate [12].

In the high-field and high-temperature regimes, we attribute the onset of $V^{\prime\prime}$ ($H_{\text{irr}}(T)$), for different frequencies, including dc, to the presence of pinned vortices, and thus an ac “irreversibility line” or “depinning line” $H_{\text{irr}}(T)\sim(T_{c}-T)^{\alpha}$, where the exponent typically ranges from $\frac{4}{3}$ to 2 [26]. As we are measuring the response of a coherent sum of screening currents over a select range of wave vectors determined by the receive coil geometry [28] and over a time scale determined by the drive frequency, our measurement is mostly sensitive to flux lines that appear pinned within the experiment’s parameters. In that respect, the region bounded by the irreversibility and peak effect lines indicate the increase in pinning efficiency, starting from thermally activated flux flow (TAFF) [40] at high temperatures, with its hallmark of frequency dependence (see Fig. 4) to the strong pinning in the peak effect regime with intermediate change in pinning behavior below the freezing of the vortex lattice.

Finally we note that debates persist for an unambiguous identification of the irreversibility line since it is highly sensitive to either geometrical barriers or the external drive current that causes depinning of vortices [12, 26]. It is important to note that the drive field in our experiment is several orders of magnitude lower compared to typical ac susceptibility measurements at $\sim 1$ Oe, whereas our MI experiment is complementary to transport measurements that typically require much higher drive current. Further clarifications on the irreversibility line may be achieved through torque magnetometry [41] or local magnetic induction experiments[12].

In summary, the screening response of vortices in kagome superconductor CsV₃Sb₅ has been measured using the ac mutual inductance technique. Besides confirming the absence of gapless quasiparticles in zero external magnetic field, we observe the peak effect, corresponding to enhanced pinning strength and critical current, in a broad intermediate range of magnetic field. The peaks vanishes at a melting transition from strong to weak pinning, unlike the usual peak effect that ends near $H_{\text{irr}}$ or $H_{c2}$. Hysteresis in $V^{\prime\prime}$ leads us to the identify a vortex glass phase, where its onset is highly correlated with that of the peaks. The various features in the temperature- and field-dependence of the screening response, corroborated by transport and dc magnetization measurements, have allowed us to construct an $H$-$T$ phase diagram of the vortex states and to infer the irreversibility line $H_{\text{irr}}(T)$.