Competition between charge-density-wave and superconductivity in the kagome metal RbV3Sb5

The interplay between charge-density-wave (CDW) order and superconductivity (SC) in the Kagome metal RbV3Sb5 is studied by tracking the evolutions of their transition temperatures, T* and Tc, as a function of pressure (P) via measurements of resistivity and magnetic susceptibility under various hydrostatic pressures up to ~ 5 GPa. It is found that the CDW order at T* experiences a subtle modification at Pc1 ~ 1.5 GPa before it is completely suppressed around Pc2 ~ 2.4 GPa. Accordingly, the superconducting transition Tc(P) exhibits a shallow M-shaped double superconducting dome with two extrema of Tconset ~ 4.4 K and 3.9 K around Pc1 and Pc2, respectively, leading to a fourfold enhancement of Tc with respect to that at ambient pressure. The constructed T-P phase diagram of RbV3Sb5 resembles that of CsV3Sb5, and shares similar features as many other unconventional superconducting systems with intertwined competing electronic orders. The strong competition between CDW and SC is also evidenced by the broad superconducting transition width in the coexistent region. Our results shed more light on the intriguing physics involving intertwined electronic orders in this novel topological kagome metal family.

So far, several high-pressure studies have been performed on KV3Sb5 and CsV3Sb5 to unveil the intimated correlations between the CDW order and SC [34][35][36][37][38][39]. For KV3Sb5, the application of high pressure was found to enhance the superconducting transition temperature Tc in concomitant with the suppression of the CDW order, suggesting a strong competition between CDW and SC [36]. However, for CsV3Sb5 with much larger interlayer distance, detailed high-pressure transport measurements reveal a more complex relationship between CDW and SC, displaying an unusual M-shaped double superconducting dome accompanying a monotonic suppression of CDW order [34,39]. Such an unusual phase diagram of CsV3Sb5 should arise from a subtle modification of the CDW order associated with the large compression of interlayer distance as indicated by the DFT calculations [34]. The more dispersive band structure along the c axis for KV3Sb5 with much reduced interlayer distance (or c-axis) [11,40] can explain the rapid suppression of CDW order at a lower critical pressure of Pc  0.4 GPa [36]. Moreover, resistance measurements on CsV3Sb5 by using diamond anvil cells over a much extended pressure range have uncovered the emergence of a second superconducting phase (SC-II) above 15 GPa with a maximum Tc  5 K [35,37,38,41]. Since the highpressure XRD rules out the occurrence of structural phase transition around this pressure [37], the observed SC-II phase at P > 15 GPa has been attributed to a Lifshitz transition as supported by the transport measurements and band structure calculations [35,37]. In comparison with K and Cs, Rb has an intermediate atomic radius and consequently the interlayer c-axis distance of RbV3Sb5 lies between that of KV3Sb5 and CsV3Sb5 [14]. It is thus interesting to investigate the evolutions of CDW and SC in RbV3Sb5 under hydrostatic pressures in order to gain a comprehensive understanding on the relationship between CDW order and SC in this class of Kagome superconductors.
In this work, we have performed detailed resistivity, direct-current field (dc) and alternate-current field (ac) magnetic susceptibility measurements on RbV3Sb5 single crystal by using the piston-cylinder cell (PCC) and cubic anvil cell (CAC) under various hydrostatic pressures up to 5.2 GPa. Our results reveal a shallow M-shaped double superconducting dome in RbV3Sb5, which should correlate with the cryptic modification of the CDW order at Pc1  1.5 GPa before it is completely suppressed around Pc2  2.4 GPa. Moreover, the maximum Tc can be enhanced to ~ 4.4 K at about 1.5 GPa, a fourfold enhancement compared with Tc at ambient pressure. The constructed T-P phase diagram, similar with that of CsV3Sb5 [34], clearly reveals the competition between CDW and SC, providing more insights into the high-pressure properties of this topological kagome superconducting family.

Experimental details
Single crystals of RbV3Sb5 were synthesized by Rb ingot (purity 99.9%), V powder (purity 99.9%) and Sb grains (purity 99.999%) using the self-flux method [14]. Temperature dependences of resistivity and ac magnetic susceptibility for RbV3Sb5 samples were measured simultaneously by using a self-clamped PCC under various hydrostatic pressures up to 2.2 GPa [42]. Here, we use the Daphne 7373 as the pressure transmitting medium (PTM) in PCC. The resistivity was measured with standard fourprobe method with the electrical current applied within the ab-plane. The magnetic field was applied along the c-axis. The ac susceptibility of RbV3Sb5 together with a piece of Pb placed in the same coil was measured with the mutual induction method. An excitation current of ~ 1 mA with a frequency of 1117 Hz was applied to the primary coil and the output signal was picked up with a Standford Research SR830 lock-in amplifier. The measured superconducting transition of Pb was used to determine the pressure value in PCC and it was also used as a reference to estimate the superconducting shielding volume of the RbV3Sb5. We have also employed a palmtype CAC to measure the resistivity of RbV3Sb5 up to 5.2 GPa [43]. Glycerol was employed as the liquid PTM for CAC. Finally, we used a miniature BeCu PCC to measure the dc magnetization under various pressures up to 0.84 GPa in the commercial magnetic property measurement system (MPMS-3, Quantum Design). The RbV3Sb5 crystals together with a piece of lead (Pb) were loaded into a Teflon capsule filled with Daphne 7373 as the PTM, and the pressure value was determined from the relative shift of the Tc of Pb. All the measurements were performed in the zero-field-cooled mode.

Results and Discussions
Figure 1(a) shows the temperature dependence of in-plane resistivity ρ(T) of RbV3Sb5 single crystal at ambient pressure. The normal-state ρ(T) exhibits a typical metallic behavior with the residual resistivity ratio RRR = ρ(290 K)/ρ(1.5 K) = 39, indicating a high quality of our samples. As can be seen, a kink-like anomaly appears in (T) at T*  103 K as indicated by the downward arrow, and this feature is correlated to the formation of the CDW order [14]. An enlarged view of ρ(T) below 1.5 K is depicted in the inset of Fig. 1(e), which shows that the superconducting transition starts at around 1.1 K and reaches zero resistance at about 0.78 K. Here, the Tc onset is determined as the interception between two straight lines below and above the superconducting transition and Tc zreo is defined as the zero-resistivity temperature. These results are consistent with the previous report [14]. Then, we measure the field dependence of resistivity ρ(H) of RbV3Sb5 at various temperatures up to 0.93 K with the field applied along the ab plane and the c-axis, respectively, as shown in Figs. 1(b) and 1(c). We can see that the superconducting upper critical field 0Hc2 is continuously shifted to lower fields with increasing temperature gradually. Here, we determined the upper critical field, 0Hc2, from the 90% drops of ρ(H) curves and plotted the temperature dependence of 0Hc2(T) in Fig. 1(d). As can be seen, the 0Hc2(T) can be well fitted by using the Ginzburg- (0) is defined as zero-temperature upper critical field and t represents the reduced temperature T/Tc. The calculated μ0H //ab c2(0) and μ0H //c c2(0) are 0.3 T and 0.11 T, respectively. Moreover, the corresponding G-L coherent lengths are estimated to be  ab GL = 547.0 Å and  c GL = 200.6 Å based on the formula: μ0H //c c2(0) = 0/2πξ ab GL 2 and μ0H //ab c2(0) = 0/2πξ ab GLξ c GL, where 0 = hc/2e is the magnetic flux quantum [44]. Therefore, the obtained anisotropy parameter is γ = μ0H //ab c2(0)/μ0H //c c2(0)  2.73, which is about one third of that in CsV3Sb5 [45]. The reduced anisotropy is consistent with the reduced ionic radius of alkali metal from Cs to Rb. For the anisotropic superconductors, we can use the ratio γ = * / * to express the band-structure anisotropy, where the * and * are the effective mass of the quasiparticles along the c axis and within the ab plane, respectively. The estimated * / * ~ 7.5 indicates a relatively strong anisotropy of the band structure in RbV3Sb5.
To further characterize the superconducting transition of RbV3Sb5, we measured the low-temperature magnetization M(T) at a magnetic field of 5 Oe under zero-fieldcooled (ZFC) and field-cooled (FC) conditions. As shown in Fig. 1(e), the obvious diamagnetic signal can be seen in the ZFC and FC curves, and it reveals the bulk SC after correcting the demagnetization factor. The onset of superconducting transition appears at Tc  0.78 K, consistent with the Tc zreo determined from the (T) data shown in the inset of Fig. 1(e).  2(a)). From d/dT, we can actually define two characteristic temperatures, i.e. T peak and T dip corresponding to the peak and dip temperatures, and determine the T* = (T peak + T dip )/2. With increasing pressure gradually, the anomaly in (T) at T* and the corresponding T peak and T dip in d/dT continuously move to lower temperatures. Interestingly, the peak feature is diminished gradually with pressure, and the T peak and T dip are merged to about 53 K at 1.5 GPa, above which the dip feature in d/dT becomes less obvious. The weakening of the anomaly in resistivity has a profound influence on the superconducting transition as we will discuss in detail below. It is noteworthy that the kink anomaly in (T) at T* changes to a hump-like feature with increasing pressure ( Fig. 2(a)). Similar feature has also been observed in CsV3Sb5 under pressure [34,39]. Due to the quasi-2D nature of the AV3Sb5 family, it is inevitable that the interlayer interactions will be enhanced upon reducing the ionic radius of A cations. As shown in our previous work, the CDW order involves a non-vanishing order wave-vector along the c-axis [34], which can explain the hump-like feature in the c(T) of RbV3Sb5 at ambient pressure [14].
An enlarged view of the low-temperature (T) data under various pressures are present in Fig. 3(a). At ambient pressure, the superconducting transition cannot be detected down to T = 1.5 K, the lowest temperature in our high-pressure measurements. When increasing pressure gradually, we start to see the weak drop of (T) at 0.43 GPa, and then an obvious superconducting transition at Tc onset  2.5 K at 0.76 GPa. But zeroresistivity cannot be achieved down to 1.5 K. Tc onset and Tc zero rapidly rise to ~ 3.5 K and ~ 1.6 K at 1.02 GPa, and then further increase to ~ 4 K and ~ 2.7 K at 1.29 GPa, respectively. In this pressure range, the superconducting transition is broad with a transition width Tc over 1.5 K; such a broad transition is consistent with the fact that SC coexists with the CDW in this regime. With increasing pressure to 1.5 GPa, the superconducting transition temperature reaches a maximum with Tc onset  4.4 K and Tc zero  4 K; accordingly, the superconducting transition width quickly shrinks to Tc  0.4 K. Interestingly, when the pressure is further increased from 1.5 to 2.19 GPa, Tc onset and Tc zero are reduced slightly to ~ 4.1 K and ~ 3.55 K, respectively. In the pressure range 1.76 -2.19 GPa, Tc increases slightly to ~ 0.6 K. The broadened Tc highlights a complex and intrinsic phenomenon that may originate from the cryptic modification of the CDW as discussed below.
To further track the evolution of Tc(P) under higher pressures, we measured the (T) of RbV3Sb5 up to 5.2 GPa with CAC, and display the low-temperature data in Fig. 3(b).
The (T) in the whole temperature range are given in Fig. S1. The (T) at 1.   Fig. 3 Based on the above resistivity and magnetic susceptibility measurements under high pressures, we construct the T-P phase diagram of RbV3Sb5 as shown in Figs. 4(a) and 4(b). From the phase diagram, we can easily visualize the evolution and intimated correlations between the CDW and SC as a function of pressure. With increasing pressure gradually, the CDW order is monotonically suppressed, accompanied by the initial enhancement of Tc with a broad superconducting transition width Tc ~ 2 K below 1.5 GPa as displayed in Fig. 5(b) and 5(c), showing a strong competition between CDW and SC. At Pc1  1.5 GPa, the highest Tc zero ≈ 4 K is achieved and it is over four times higher than that at ambient pressure. Above 1.5 GPa, the resistivity anomaly associated with CDW order becomes much weakened, while the superconducting transition temperature shows a shallow valley between 1.5 and 2.4 GPa. It seems that the long-range CDW order has been replaced by short-ranged one that coexists with SC in this pressure range and thus leads to a broadening of the superconducting transition, Fig. 5(b, c). The complete suppression of the short-ranged CDW order gives the second maximum of Tc zero around Pc2  2.4 GPa. Above 2.4 GPa, the superconducting transition monotonically moves to lower temperatures and becomes very sharp with Tc  0.1 K, Fig. 5(c).
To further probe the evolution of the superconducting electronic states of RbV3Sb5 under high pressures, we measured the upper critical field 0Hc2 at various pressures up to 5.2 GPa. All the (T) data under various magnetic fields and different pressures in PCC and CAC are shown in Fig. S2. Tc moves to lower temperatures gradually with increasing magnetic fields. In order to track the evolution of 0Hc2, we employed the criteria of middle-point temperature Tc mid as the superconducting transition temperature. As shown in Fig. 5, we plot all μ0Hc2(T) data measured in PCC and CAC, and then estimate the zero-temperature μ0Hc2(0) by employing the empirical G-L equation as discussed above to fit the μ0Hc2(T) data. The best fitting results are indicated by the broken lines in Fig. 5(a, b). Surprisingly, the μ0Hc2(0) as a function of pressure exhibits a pronounced peak around Pc1  1.5 GPa (Fig. 4(d)), but not at Pc2. This result is different from the double peak feature observed in CsV3Sb5 [34,39]. We also extract the initial slope of μ0Hc2(T) at each pressure, i.e., -dHc2/dT|Tc, and a similar peak feature shows around Pc1 (Fig. S3). As the slope -dHc2/dT|Tc is proportional to the effective mass of charge carriers [47], and the divergence of -dHc2/dT|Tc around Pc1  1.5 GPa indicates an enhancement of effective mass as shown in Fig. 4(d). In general, the divergence of effective mass is considered as a hallmark of quantum criticality due to a complete suppression of certain electronic order [9,48]. It should be noted that the optimal superconducting phase usually emerges at the QCP in many unconventional superconductors [49][50][51][52][53].

Discussions.
By combining resistivity and magnetic susceptibility measurements, we have tracked the evolutions of the CDW order and SC in RbV3Sb5, and unveiled a shallow M-shaped double superconducting dome under pressure as described above. With increasing pressure, T*(P) decreases monotonically and vanishes completely around Pc2  2.4 GPa, while Tc(P) exhibits two maxima at Pc1  1.5 GPa and Pc2, respectively. Above Pc2, the CDW order is completely eliminated and the superconducting transition shows a monotonic reduction. The highest Tc onset  4.4 K is achieved around Pc1 1.5 GPa, rather than the putative QCP of the CDW order located at Pc2  2.4 GPa. The optimal Tc onset  4.4 K around Pc1 is about fourfold enhanced in comparison with that at ambient pressure. All these characteristics in the T-P phase diagram of RbV3Sb5 are similar with those of the sister compound CsV3Sb5 [34], but having some quantitative differences between them. In addition, their double superconducting domes are also distinct from the observed single dome in KV3Sb5 under pressure [36].Thus, side-by-side comparisons among them are merited in order to have a better understanding on the unique properties of the AV3Sb5 family.
First of all, the character of the superconducting dome seems to correlate intimately with the A-cation size or the interlayer distance; the double-dome feature is weakened and changed to a single dome upon reducing the A-cation size from Cs though Rb to K. For CsV3Sb5, the larger Cs ion or interlayer distance should reduce the interlayer hopping and make the bands less dispersive along the c-axis. In principle, the highly two-dimensional character favors the formation of CDW order through the nesting scattering between van Hove points. Under high pressures, the bands become more dispersive along the c-axis with reducing the interlayer distance, and thus weaken the nesting scattering effect. The modification or vanishing of this out-of-plane CDW wavevector along the c axis under pressure would give rise to the first SC dome around Pc1. In comparison with CsV3Sb5, the interlayer distance has been compressed chemically in RbV3Sb5, and thus the modification of CDW component along c-axis is expected to be weakened that would lead to a shallow M-shaped superconducting phase. While for KV3Sb5 with much smaller interlayer distance, the bands along the c-axis become more dispersive and therefore the double-dome character becomes much more weakened or even vanished as observed.
Secondly, although T*(P) displays monotonic suppression under physical pressure for these three compounds, the evolution of T* does not exhibit a similar trend as a function of A-cation size; it is peaked out at RbV3Sb5 with T* = 104 K in comparison with that of 94 K for CsV3Sb5 and 78 K for KV3Sb5, respectively [12][13][14]. Accordingly, the superconducting Tc at ambient pressure exhibits exactly opposite trend, illustrating a competition nature between these two intertwined orders. These comparisons highlight that the physical and chemical pressures should play some distinct roles in modifying the crystal and electronic structures that may need further investigations. Nonetheless, the critical pressures for the suppression of CDW order has a positive correlation with T* at ambient pressure; i.e. the corresponding critical pressures are decreased gradually from Pc1  1.5 GPa and Pc2  2.4 GPa for RbV3Sb5, to Pc1  0.6 -0.9 GPa and Pc2  2 GPa for CsV3Sb5 [34], and finally to Pc1  0.4 -0.5 GPa for KV3Sb5 [36]. In addition, the optimal Tc achieved under pressure also follows the same trend of Tc at ambient.
Thirdly, the most interesting difference between CsV3Sb5 and RbV3Sb5 under pressure is the distinct behaviors of 0Hc2(P) and its connection with the optimal Tc. For the former, the μ0Hc2(0) shows two peaks at Pc1 and Pc2 and the maximum Tc emerges at Pc2 accompanying the complete suppression of CDW order; however, for the latter, both the μ0Hc2(0) and Tc are peaked out at Pc1 rather than Pc2. Then, the question naturally arises why the maximal μ0Hc2(0) and Finally, it is noteworthy that the unusual double-dome superconducting phase observed in CsV3Sb5 and RbV3Sb5 is reminiscent of the phase diagrams of high-temperature cuprates [51,54,55] and FeSe-based superconductors [52], showing the presence of competing intertwined CDW/SDW or nematic orders. As indicated from the theoretical calculations [28][29][30], multiple electronic orders can be achieved as a function of on-site repulsion U and nearest-neighbor Coulomb interaction V, such as ferromagnetism, intra-unit-cell antiferromagnetism, charge bond order or spin bond order. Thus, more experiments such as high-pressure nuclear magnetic resonance should be performed to further investigate the evolution of microscopic electronic orders in these V-based kagome metals.

Conclusion
In summary, we have performed a comprehensive high-pressure study on RbV3Sb5 single crystals by employing the electrical transport and magnetic susceptibility measurements. At ambient pressure, the kagome metal RbV3Sb5 shows a charge order or CDW-like order at T* = 103 K and SC at Tc zero = 0.78 K. Our results reveal a subtle modification of the CDW order around Pc  1.5 GPa, and the modified CDW is completely suppressed around 2.4 GPa. Correspondingly, the superconducting Tc(P) displays the unusual M-shaped double superconducting dome structure with the optimal Tc onset  4.4 K and Tc zero  4 K at 1.5 GPa, and another maxima Tc onset  3.93 K and Tc zero ~ 3.8 K occurring at 2.4 GPa. Therefore, our phase diagram reveals the intimate interplay and strong competition between the CDW and SC in the pressure range 0 GPa  P  1.5 GPa as evidenced by the broad superconducting transition width. Between 1.5 GPa and 2.4 GPa, the superconducting phase shows a valley character with possible underlying modification of the CDW. In addition, the 0Hc2(0) shows a prominent peak character around 1.5 GPa, showing the characteristics of quantum criticality associated with the suppression of CDW order. The constructed T-P phase diagram is similar to those of many unconventional superconductors with intertwined electronic orders and quantum criticality. Therefore, RbV3Sb5 together with CsV3Sb5 provide a new platform to study the correlations between the electronic instabilities and SC in this novel topological kagome metal family. In addition, the optimal Tc of RbV3Sb5 reaches about 4.4 K at 1.5 GPa, which gives the possibility to further enhance Tc of these V-based kagome superconductors. Further studies on RbV3Sb5 are need to address the open issues such as the character of CDW-like order in the intermediate pressure range.