Abstract
The interplay among frustrated lattice geometry, non-trivial band topology and correlation yields rich quantum states of matter in kagome systems1,2. A series of recent members in this family, AV3Sb5 (A = K, Rb or Cs), exhibit a cascade of symmetry-breaking transitions3, involving the 3Q chiral charge ordering4,5,6,7,8, electronic nematicity9,10, roton pair density wave11 and superconductivity12. The nature of the superconducting order is yet to be resolved. Here we report an indication of dynamic superconducting domains with boundary supercurrents in intrinsic CsV3Sb5 flakes. The magnetic field-free superconducting diode effect is observed with polarity modulated by thermal histories, suggesting that there are dynamic superconducting order domains in a spontaneous time-reversal symmetry-breaking background. Strikingly, the critical current exhibits double-slit superconductivity interference patterns when subjected to an external magnetic field. The characteristics of the patterns are modulated by thermal cycling. These phenomena are proposed as a consequence of periodically modulated supercurrents flowing along certain domain boundaries constrained by fluxoid quantization. Our results imply a time-reversal symmetry-breaking superconducting order, opening a potential for exploring exotic physics, for example, Majorana zero modes, in this intriguing topological kagome system.
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Acknowledgements
We are grateful to L. Jiao and C. Guo for helpful discussions. This research is supported by Zhejiang Provincial Natural Science Foundation of China for Distinguished Young Scholars (Grant No. LR23A040001). C.W. is supported by the National Natural Science Foundation of China (Grant Nos. 12234016 and 12174317). Z.W. and Y.Y. are supported by the National Key R&D Program of China (Grants Nos. 2020YFA0308800 and 2022YFA1403400) and the Beijing Natural Science Foundation (Grants No. Z210006). T.L. acknowledges support from the China Postdoctoral Science Foundation (Grant Nos. 2022M722845 and 2023T160586). This work has been supported by the New Cornerstone Science Foundation. X.L. acknowledges support from the Research Center for Industries of the Future at Westlake University (Award No. WU2023C009). We are thankful for the support provided by C. Zhang from the Instrumentation and Service Center for Physical Sciences at Westlake University. Z.W. thanks the Analysis & Testing Center at BIT for assistance in facility support.
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T.L. fabricated the devices and did the transport measurements assisted by Z.X., J.W., Z.L. and X.Y. J.L. prepared the samples supervised by Z.W. and Y.Y. Z.P. did the theoretical calculations supervised by C.W. T.L. and X.L. prepared the figures. X.L. wrote the paper with input from T.L., Z.P. and C.W. X.L. led the project. All authors contributed to the discussions.
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Extended data figures and tables
Extended Data Fig. 1 Extended data of resistance and I-V curves.
a, T-dependence of ρ for the bulk single crystal (B1) and R for a mechanically exfoliated specimen (D1) in the full-T range. The residual-resistance-ratio (RRR) amounts to 250 in B1, among the highest value of the literature5,12,50,51, highlighting the ultra-high quality of our crystals. The transition to chiral charge order phase in B1 appears at TCDW ≈ 92 K12, accompanied by a SC phase transition at Tc ≈ 3 K (determined at zero-resistance), consistent with previous reports12. In D1, TCDW is reduced to 80 K with the enhancement of Tc to 3.5 K, as reported in the literature49. The inset presents the normalized resistance R/Rn around Tc, where Rn is the normal state resistance. b, T-dependence of R around Tc for D1-D3. For D2, we present the data collected at two sets of terminals: V3-4 and V6-7. The onset temperature of the SC transition (\({T}_{{\rm{c}}}^{{\rm{onset}}}\)) for D1-D3 is similar, about 4.3 K. Tc for D3 amounts to 4.1 K, which is higher than that of D1 and D2 (about 3.5 K). Note that Tc of D2 measured at V3-4 and V6-7 is slightly different, reflecting different domain characteristics in-between the terminals. c, Optical image of D2 and D3. All the terminals are numbered. The thickness of D1-D3 is about 40 nm. Scale bar, 6 μm. d-h, Multi-step SC phase transitions on dV/dI versus I for D2 measured at various terminals. Characteristics of dV/dI exhibit notable distinctions across different terminals, including the variation of Ic. Only V3−4 displays an apparent SDE (e). Note that the SDE at V6−7 can be excited by thermal cycling, as shown in Extended Data Fig. 2g-i. Given the ultra-high quality of CsV3Sb5 single crystals, mild device fabrication processes and the dynamic features on dV/dI, the multi-step transitions and the variation of Ic cannot be simply attributed to significant sample inhomogeneity, but rather implies the formation of SC domain structure (See more discussion in Methods). Ic is influenced by the strength of inter-domain connections.
Extended Data Fig. 2 Thermal modulation of zero-field SDE for D1-D3 measured at T = 1.4 K.
a-b, dV/dI versus I for terminals V3-4 of D2 before (a) and after recooling from 4.5 K, slightly above Tc (b). c-d, dV/dI versus I for D1 before (c) and after thermal cycling (d). e-f, dV/dI versus I for terminal V3-6 of D3 before (e) and after thermal cycling (f). In e, the measurement includes four branches: sweeping I from zero to positive (I+), from positive back to zero (Ir+), from zero to negative (I−) and from negative back to zero (Ir−). The hysteresis between I+ (I−) and Ir+ (Ir−) is negligible, indicating the absence of thermal heating or current re-trapping effect. Below, several observations are made: 1. All the devices exhibit remarkable non-reciprocity. 2. Not only the polarity, but also the magnitude of \(\Delta {I}_{{\rm{c}}}^{{\rm{SDE}}}\) and \(\bar{{I}_{{\rm{c}}}}\) could be changed by thermal cycling. 3. The curves in c and d show multiple transition-like features with non-reciprocity (marked by arrows), probably related to the difference in Ic across different SC domain boundaries. 4. In c and d, the SDE polarity at A1 is reversed after thermal cycling, while the polarity at A2 remains unchanged. As discussed in Methods, the dynamic nature of SDEs with multiple transition peaks is unlikely to be fully explained by scenarios involving the combination of chiral charge order and certain sources of IRS-breaking such as geometric asymmetry and significant sample inhomogeneity. While, all of these could be reconciled with the existence of dynamic SC domains with broken TRS. Characteristics of the domains, such as domain asymmetry and inter-domain interaction, are randomly altered by thermal cycling. g-i, dV/dI versus I for terminals V6-7 of D2 before (g) and after thermal cycling (h, i). V6−7 shows negligible non-reciprocity in the initial state. After thermal cycling, a finite SDE with either positive and negative polarity is induced. j-k, Demonstration of half-wave rectification. Direction-selective supercurrent transmission is demonstrated at V6−7 of D2 with positive (j) and negative polarity (k). The measurements were performed by alternating the current polarity every 15.5 seconds. SDE remains stable after 100 cycles.
Extended Data Fig. 3 Zero-field SDE at various orientations.
a, Residual environmental field detected by a fluxgate magnetometer (CTM-6W, 0-1,000,000 nT, U≤ 1 nT). b, Measured field value after the application of a compensating field. c, Image of the setup encompassed by SC Nb films. d-g, Zero-field SDE measured at four setup configurations: 0o (d), 90o (e), 180o (f) and 270o (g).
Extended Data Fig. 4 SIPs for D1, D2 and D3 in a broader range of B.
a-b, SIPs for D1, covering the B range of 20 Gs and 240 Gs, respectively, as the Extended Data of Fig. 2a. In a, three SIPs (\({I}_{{\rm{c}}},{I}_{{\rm{c}}}^{{\prime} }\) and \({I}_{{\rm{c}}}^{{\prime\prime} }\)) are clearly resolved, corresponding to those in Fig. 2a. b displays more complex, periodic-like structures, alongside \({I}_{{\rm{c}}},{I}_{{\rm{c}}}^{{\prime} }\) and \({I}_{{\rm{c}}}^{{\prime\prime} }\). Notably, we observe periodic oscillations on Ic. The magnitude of Ic remains nearly unchanged in B up to 240 Gs, as expected from the LP effect. In contrast, \({I}_{{\rm{c}}}^{{\prime} }\) displays a broad Fraunhofer-like pattern, on top of which is a rapid double-slit periodic oscillation. The broad feature is likely associated with a local Josephson junction between neighboring domains (domain bulk contribution) and the rapid mode arises from the LP effect from the domain edge. It implies a composite contribution from the bulk Josephson supercurrent and the domain edge supercurrent. Relevant discussion is also presented in the main text. As discussed in Methods, such sharp double-slit SIPs are difficult to explain by alternative interpretations involving chiral charge orders or significant inhomogeneity with the absence of Ie. c and d, SIPs for D2, covering the B range of 40 Gs and 240 Gs, respectively. An explicit periodic oscillation appears on Ic, along with some vague patterns. At B > 10 Gs, distinct spikes (marked by white arrows) emerge on Ic, disrupting the periodic patterns, which is the result of the penetration of magnetic vortices into the domain bulk. e-f, SIPs for D3, covering the B range of 40 Gs and 240 Gs, respectively.
Extended Data Fig. 5 SIPs for D2 and D3 measured by varying the current or voltage terminals.
a, SIPs for D2 measured at V6-7 with current injected into I5-8, which is compared with the data in Fig. 2c (D2, V6-7 and I4-8). b-f, SIPs for D3 measured at V3-6, but by varing current terminals. The oscillation patterns on Ic is nearly unchanged when the current terminals are varied, indicating that the SIPs are associated with the domain structure between the voltage electrodes. g-l, SIPs for D3 collected by varying the voltage terminals while keeping the current terminals (I1-14) unchanged. Eplicit periodic oscillation patterns are observed in V3-6 (g) and V3-4 (h). However, the oscillation patterns are vague in V4-6 (i), V2-3 (j), V6-7 (k) and V7-8 (l). In close inspection of g, h and i, we find that the patterns in V3-6 appear to be the superposition of V3-4 and V4-6. And the dominant contribution to the SIP (Ic) comes from V3-4. These observations strongly suggest that the SIP arises from a proper domain structure between terminals 3 and 4.
Extended Data Fig. 6 SIPs at V6-7 of D2 measured at T = 1.9 K at 0° and 180° setup configurations.
SIPs measured after multiple zero-field cooling (ZFC) from 5 K (a), 7 K (b), 10 K (c). The relevant temperatures are much lower than TCDW ≈ 80 K. The measurements were conducted at low B to prevent vortex trapping during field sweeping, as referring to Extended Data Fig. 7. The data are displayed with the coordinate frame fixed to the device. We define the axis pointing outward from the top surface as the positive direction, as labeled by the arrow in Extended Data Fig. 3. Notably, we observe a remarkable phase shift, e.g. ϕT ≈ 0.8 Gs in panel a, as indicated by the intersection of two orange dashed lines. When the setup is reversed, the phase shift shows no discernible changes, providing strong support for the prevailing influence of internal B in determining the phase. This is because external B will lead to a phase reversal following the setup inversion. When comparing panels a, b and c, several observations can be made: First, the patterns beyond the primary oscillatory patterns exhibit remarkable modulation after undergoing thermal cycling. Second, the phase (ϕT) is initially about 0.8 Gs after ZFC from 5 K, becomes approximately 0.4 Gs after ZFC from 7 K, and becomes about 0.5 Gs after ZFC from 10 K. Third, the period (Bp), corresponding to the domain size, displays a slight but discernible variation after undergoing multiple thermal cycling from three different temperatures. All these collectively reflect the dynamic nature of SC domains tuned by temperature.
Extended Data Fig. 7 SIPs of D1 measured using different field-sweep procedures in sequence.
a, SIPs obtained by sweeping B from -6 to 6 Gs. b, SIPs subsequently acquired by sweeping B from 6 to -6 Gs. c, SIPs obtained by sweeping B from 0 to -6 Gs and then 0 to 6 Gs. The vertical lines are guides to eyes. It’s obvious that the SIPs show no phase shift when comparing the three SIPs, strongly suggesting the absence of trapped vortices in either the device or the magnet during field sweeps.
Extended Data Fig. 8 Extended data for thermal modulation of SIPs in D2.
a, SIPs in the initial state. b, SIPs measured after thermal cycling. The SIPs show counter-shift in phase between the I+ and I− branches, indicating thermal modification of domain asymmetry. Note that the patterns in this figure and Extended Data Fig. 6 exhibit variations, because the measurements were conducted at different time.
Extended Data Fig. 9 Field modulation of SIPs for D2.
a-b, SIPs measured at 1.4 K: Initial SIPs (a) and SIPs measured after field cooling (FC) from T ≳ Tc at B = 120 Gs (b). After FC, the oscillation patterns underwent substantial modifications, including the alteration of period and the increase of ∣Ic∣. The patterns roughly returned to their initial state after subsequent multiple thermal cycling from T above Tc. c-e, Another measurement at 1.9 K for the same device but at different time: Initial SIPs (c), SIPs obtained after FC from T ≳ Tc at B = 1600 Gs (d), SIPs measured after FC at B = −1600 Gs (e). The period of the initial pattern is distinct from that in a-b, which is due to the effect of thermal cycling from 300 K, as referring to Fig. 4d. Remarkably, the oscillation pattern remains nearly unchanged after FC with B even up to 1600 Gs, a feature distinct from that observed in a-b. On closer inspection, we discern a peculiar ‘advanced’ nature of hysteresis in comparing the patterns among the initial, FC 1600 Gs, and FC -1600 Gs curves, i.e. the zero flux line (vertical green dashed line) shifts to positive (negative) B after FC at positive (negative) field. In the following, let’s discuss the two possible explanations for the distinct phenomena observed in a-b and c-e: vortex trapping and domain rearrangement in response to external B. First, let’s consider vortex trapping. It’s conceivable that vortex trapping could potentially alter the oscillation pattern. Following thermal cycling from T > Tc, the trapped vortices are expelled, thereby restoring the pattern to its initial state. This aligns with the observations in a-b. Nevertheless, the presence of vortices usually results in a reduction in ∣Ic∣, which contradicts the enhancement observed in a-b. Now, let’s look at c-e. Vortex trapping appears to cause a phase shift in the patterns. Typically, this would result in a retarded manner of hysteresis, i.e. the zero flux line shifts to negative (positive) B to compensate for the vortices introduced during positive (negative) FC. However, this contradicts the observations made in c-e. Then, let’s discuss the scenario of domain rearrangement. External B lifts the degeneracy among TRS-breaking domains, consequently, inducing the movement of domain walls. This scenario roughly explains the phenomena observed in a-b, including the modification of period and the enhancement of ∣Ic∣. The aligned domain structure induced by external B is energy costly and is pinned in place by local defects/strains within the material. This pinned state could in principle revert to its initial configuration after thermal cycling, thereby accounting for the restored manner mentioned in a-b. Regarding c-e, the peculiar ‘advanced’ hysteresis in oscillation patterns has been reported to identify the presence of dynamic TRS-breaking domains in potential chiral SCs42,52,53. This phenomenon could be explained by the motion of domain walls driven by external B53. However, this scenario appears insufficient to account for the nearly unchanged oscillation patterns observed after FC. It is possible that this device exhibits a soft manner with respect to domain reorientation, if the pinning strength is weak. In this context, the domain structure largely returns to its initial configuration as B decreases to zero.
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Le, T., Pan, Z., Xu, Z. et al. Superconducting diode effect and interference patterns in kagome CsV3Sb5. Nature (2024). https://doi.org/10.1038/s41586-024-07431-y
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DOI: https://doi.org/10.1038/s41586-024-07431-y
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