Abstract
CsV_{3}Sb_{5} exhibits superconductivity at T_{c} = 3.2 K after undergoing intriguing two hightemperature transitions: charge density wave order at ~98 K and electronic nematic order at T_{nem} ~ 35 K. Here, we investigate nematic susceptibility in single crystals of Cs(V_{1x}Ti_{x})_{3}Sb_{5} (x = 0.000.06) where doubledomeshaped superconducting phase diagram is realized. The nematic susceptibility typically exhibits the Curie‒Weiss behaviour above T_{nem}, which is monotonically decreased with x. Moreover, the Curie‒Weiss temperature is systematically suppressed from ~30 K for x = 0 to ~4 K for x = 0.0075, resulting in a sign change at x = ~0.009. Furthermore, the Curie constant reaches a maximum at x = 0.01, suggesting drastically enhanced nematic susceptibility near a putative nematic quantum critical point (NQCP) at x = ~0.009. Strikingly, T_{c} is enhanced up to ~4.1 K with full Meissner shielding realized at x = ~0.00750.01, forming the first superconducting dome near the NQCP. Our findings directly point to a vital role of nematic fluctuations in enhancing the superconducting properties of Cs(V_{1x}Ti_{x})_{3}Sb_{5}.
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Introduction
Metals with kagome lattices have drawn enormous attention due to their unique electronic structures, which can host Dirac points, flat bands, and saddle points in their crystal momentum space. Many of these peculiar features can lead to electronic instabilities associated with divergence in the density of states near the Fermi level. As such, it has been theoretically predicted that various emergent orders such as charge bond order^{1,2}, charge density wave (CDW) order^{1}, nematic order^{3}, and superconductivity^{1,2,4} might appear. For instance, a functional renormalisation group study on the kagomeHubbard model predicted that CDW order and unconventional dwave superconductivity can arise near a van Hove singularity at an electron filling of 5/4 per band^{1}.
Recently discovered Vbased kagome metals AV_{3}Sb_{5} (A = K, Rb, and Cs) represent experimental realisations of these various theoretical predictions; each member indeed exhibits signatures of the structural transition due to the CDW order below its CDW transition temperature T_{CDW}, where T_{CDW} = 80104 K^{5,6,7}, followed by the coexistence of superconductivity^{8,9,10}. Within this material class, CsV_{3}Sb_{5} has been most extensively studied due to its highest superconducting transition temperature (T_{c} = ~3.2 K) and the presence of multiple structural and electronic instabilities. For example, the CDW order found at T_{CDW} = 98 K^{7} induces the firstorder structural transition with a 2 × 2 modulation, showing a translational symmetry breaking in each kagome layer. Moreover, it was later found that the 2 × 2 modulation is accompanied by two kinds of outofplane modulations (i.e., 2 × 2 × 2^{11} and 2 × 2 × 4^{12}). The structure in the CDW states with the caxis modulation is compatible with the C_{2} rotational symmetry below T_{CDW}^{13}. However, it is known that the electronic rotational symmetry in each kagome layer is still maintained as C_{6} just below T_{CDW}^{14}. With a further decrease in temperature, the system finally reaches the electronic nematic transition at T_{nem} ~ 35 K, at which a rotational symmetry of the electronic state is continuously varied from C_{6} to C_{2} in the kagome layer, as evidenced by nuclear magnetic resonance (NMR), scanning tunnelling microscopy and elastoresistance measurements^{14}.
Alongside the experimental progresses, recent theoretical studies based on the kagomeHubbard model have successfully explained the multiple thermal phase transitions related to the CDW and the nematic orders in CsV_{3}Sb_{5}^{3,15,16,17,18}. Namely, the charge ordering at T_{CDW} is understood as stemming from a tripleqcharge bond order, arising from the Fermi surface nesting at the van Hove singularity points in CsV_{3}Sb_{5}^{3,15}. Moreover, near the suppression of these charge bond orders, it has been predicted that bond order fluctuations can mediate sizable pairing glue for superconductivity, possibly resulting in the superconductivity of various symmetries including singlet swave^{17}, triplet pwave^{17}, or dwave superconductivity^{1}.
However, in spite of extensive progress in both experiments and theories, the superconducting properties of CsV_{3}Sb_{5} are still poorly understood. Firstly, no consensus has been reached on the pairing symmetry of CsV_{3}Sb_{5}. While unconventional superconductivity with the nodal gap structure has been suggested by thermal conductivity^{19} and scanning tunnelling microscopy data^{20}, tunnel diode oscillator^{21} and muon spin resonance measurements^{22} support a nodeless swave gap with a signpreserving order parameter. Secondly, an unusual superconducting phase diagram with a doubledome shape has been found independently either as a function of Sn doping in CsV_{3}Sb_{5x}Sn_{x} polycrystals^{23} or that of pressure in a CsV_{3}Sb_{5} single crystal^{24,25}, of which physical mechanisms remain elusive.
Elastoresistance measurements, which is a direct probe of the evenparity nematic susceptibility, have been found to be quite useful in unravelling the pivotal role of nematic fluctuation in mediating Cooper pairing, particularly in several Febased superconductors^{26,27,28,29,30}. To better understand the role of nematic order in the superconductivity of CsV_{3}Sb_{5}, we employ elastoresistance measurements to systematically study the nematic susceptibility in highquality single crystals of Tidoped CsV_{3}Sb_{5,} Cs(V_{1x}Ti_{x})_{3}Sb_{5}. It should be emphasised that such studies have not been available thus far, as both controlling the doping ratios and maintaining high quality in the doped single crystals of CsV_{3}Sb_{5} have been quite challenging^{31,32}.
Results
Figure 1a depicts the crystal structure of CsV_{3}Sb_{5}, forming the hexagonal P6/mmm space group with lattice constants of a = b = 5.508 Å and c = 9.326 Å. One unit cell comprises a VSb layer sandwiched by Cs planes, each of which consists of four Cs atoms. Within each VSb sheet, a kagome network of V atoms is interlaced with a hexagonal lattice of Sb. As the temperature is lowered below T_{CDW} = 98 K, the V atoms rearrange themselves to form an inverse starofDavid pattern (2 × 2 charge order) in the kagome plane^{7,33}. As the temperature is lowered further, an additional ordering is known to appear at T_{nem} ~ 35 K, below which the system forms a distinct nematic phase^{14} (Fig. 1b). Upon Ti doping, the titanium atoms can occupy the vanadium sites, resulting in progressive distortion of the vanadium kagome network.
Figure 1c shows the inplane resistivity ρ_{ab} of Cs(V_{1x}Ti_{x})_{3}Sb_{5} normalised by the resistivity at 300 K (ρ_{ab}/ρ_{ab,300 K}). The resistivity of the undoped CsV_{3}Sb_{5}, evidencing metallic behaviour near room temperature, exhibits a slight shoulder near 98 K due to the formation of the CDW order. With increasing x, the residual resistivity ratio (RRR) of Cs(V_{1x}Ti_{x})_{3}Sb_{5} systematically decreases. This result implies increased impurity scattering within the kagome plane in proportion to x. Along with the decreased RRR value, Cs(V_{1x}Ti_{x})_{3}Sb_{5} exhibits increasingly broadened CDW transitions with increasing x, which are well visualised in the dρ_{ab}/dT curves (Fig. 1d). Furthermore, the anomalies in dρ_{ab}/dT shift to lower temperatures, indicating the development of a lower T_{CDW} at higher x, e.g., ~59 K at x = 0.04. The decreasing trend of T_{CDW} is also confirmed by the dc magnetic susceptibility (M/H) data presented in Fig. 1e. A drop in the M/H curve observed in pristine CsV_{3}Sb_{5}, a Pauli paramagnet, indicates that depletion of the density of states has occurred due to the CDW gap opening at T_{CDW} = 98 K. Consistent with the behaviour found in ρ_{ab}, increasing x results in a shift in T_{CDW} (the temperature of the M/H drop) to lower temperatures as well as a progressively decreased and broadened drop. This indicates that with progressive perturbation in the V atom arrangement by Ti doping, depletion of the electronic density of states at the longrange CDW transition is systematically reduced along with the decrease of the averaged T_{CDW} and the increase of the T_{CDW} distribution.
As the systematic suppression of T_{CDW} is established with Ti doping, we now turn to the evolution of superconductivity to understand its interplay with the preexisting CDW order. Figure 2 shows lowtemperature ρ_{ab}(T) and 4πχ(T) data of Cs(V_{1x}Ti_{x})_{3}Sb_{5}. T_{c} is clearly identified by either zero resistivity or the onset of diamagnetic behaviour in the χ(T) curves. Here, we define T_{c} from the transport data by the criterion of 0.5 ρ_{N} (ρ_{N}: normalstate resistivity)^{34}. It is noted that T_{c} exhibits nonmonotonic behaviour with x. At x = 0.0075, T_{c} is maximised to 4.1 K from T_{c} = 3.2 K for the pristine compound. Moreover, in the doping range of x = 0.0–0.01, a full Meissner shielding of −4πχ ≅ 1 is realised. However, with a further increase of x towards x = 0.02, T_{c} is progressively suppressed to ~1.9 K, accompanied by decreased Meissner shielding of −4πχ ~ 0.3 at x = 0.015 and −4πχ ~ 0 around x = 0.02. At x > 0.02, T_{c} increases again to 3.8 K at x = 0.05 and recovers the full Meissner shielding. Thus, the evolution of the superconducting properties, i.e., T_{c} and Meissner shielding, reveals a doubledome feature.
Finding a superconducting region with a doubledome feature among CDW materials is unusual because the competition between CDW and superconductivity often leads to either a monotonically extended region of superconductivity after the collapse of the CDW order^{35}, or a single superconducting dome stabilised at the putative CDW quantum critical point (QCP)^{36}. While an increase in T_{c} at x = ~0.05 may be associated with the putative QCP of the CDW order, identifying another superconducting dome within the CDW order is truly uncommon. This finding strongly suggests that additional fluctuating orders could be present, enhancing the pairing interaction.
To investigate the origin of the unusual trend in T_{c}, the elastoresistance in Cs(V_{1x}Ti_{x})_{3}Sb_{5} was investigated from 6 to 250 K. Figure 3a illustrates the experimental configuration for the elastoresistance measurements. It is known that nematic susceptibility \(\widetilde{n}\) can be obtained by measuring the electronic anisotropy induced by anisotropic strain. In other words, \(\widetilde{n}\) becomes linearly proportional to the anisotropic change in the resistance \(N\equiv {(\Delta R/R)}_{{xx}}{(\Delta R/R)}_{{yy}}\) in response to anisotropic strain \(\left({\varepsilon }_{{xx}}{\varepsilon }_{{yy}}\right)\), which results in \(\widetilde{n}=\alpha {{{{{\boldsymbol{\times }}}}}}\partial N/\partial ({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\). Here, α is a proportionality constant depending on microscopic details of the electronic structure^{37} (See, Supplementary Note 8, for details). Therefore, \(\partial N/\partial ({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\) reveals the essential temperature dependence of \(\widetilde{n}\) upon approaching a thermally driven nematic phase transition. According to the general definition of elastoresistance coefficients \({(\Delta R/R)}_{i}\equiv {\sum }_{j=1}^{6}{m}_{{i\, j}}{\varepsilon }_{j}\)^{37}, \(\partial N/\partial ({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\) can be expressed in terms of \({m}_{{ij}}\), where \({\varepsilon }_{j}\) represents the engineering strain, \({m}_{{ij}}\) are elastoresistance tensor components, and the subscripts i and j represent the Voigt notation (1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = zx, 6 = xy). For a crystal in the D_{6h} point group with x measured along the [100] axis^{14}, \(\partial N/\partial ({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\) directly corresponds to the elastoresistance coefficient \(({m}_{11}{m}_{12})\), representing the nematic susceptibility \(\widetilde{n}\) along the evenparity \({E}_{2g}\) symmetry channel (See, Supplementary Note 4).
Figure 3b depicts the response of \(N\) to \(({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\) for CsV_{3}Sb_{5} at selected temperatures. \(N\) shows a linear relationship with \(({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\), enabling a precise measurement of \(\widetilde{n}\) by obtaining \(N/({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\) in a small strain limit of \(({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\to 0\). This parameter is redefined as \(\widetilde{n}=\partial N/\partial ({\varepsilon }_{{xx}}{\varepsilon }_{{yy}})\) (α is set to be 1), and the resulting \(\widetilde{n}\)(T) curve for CsV_{3}Sb_{5} is presented in the top panel of Fig. 3c. A sharp jump in \(\widetilde{n}\) is found at T_{CDW} = 98 K, implying that the firstorder structural transition^{38} due to the charge bond order results in an abrupt offset change in the elastoresistance anisotropy. It is found that at temperatures above 36 K and below the sharp jump near T_{CDW}, \(\widetilde{n}\) is well fitted by the Curie‒Weisstype temperature dependence,
Here, \({\widetilde{n}}_{0}\) describes the intrinsic anisotropy in the piezoresistivity effect, unrelated to electronic nematicity, θ_{nem} is the meanfield nematic transition temperature, and C is the Curie constant of the corresponding nematic susceptibility. A good agreement of the experimental data to Eq. (1) can be confirmed by a fitted red solid line with \({\theta }_{{{{{{\rm{nem}}}}}}}\) = 30 K, \({\widetilde{n}}_{0}\) = 14.7, and C = 39 K. A good fit to Eq. (1) can also be verified by the plots of \({(\widetilde{n}{\widetilde{n}}_{0})}^{1}\) and \(\left(\widetilde{n}{\widetilde{n}}_{0}\right)\left(T{\theta }_{{{{{{\rm{nem}}}}}}}\right)\) at the bottom panel of Fig. 3c, which exhibit linear (pink open circles) and constant (green open circles) behaviours with temperature, respectively. The nearly constant value of \(\left(\widetilde{n}{\widetilde{n}}_{0}\right)\left(T{\theta }_{{{{{{\rm{nem}}}}}}}\right)\) should directly correspond to the C value.
It should be further noted in Fig. 3c that below 36 K as represented by a black dotted line, the experimental data clearly deviate from the trace predicted by Eq. (1). This deviation indicates that the nematic correlation goes beyond the meanfield description below this temperature located near the longrange ordering temperature T_{nem}. The temperature of 36 K is indeed close to T_{nem} ~ 35 K, as found by the previous work^{14}. Therefore, our observation shows that a significant nematic correlation of evenparity type exists above T_{nem} up to T_{CDW} and even above T_{CDW} (vide infra).
Similar measurements and analyses were performed for Cs(V_{1x}Ti_{x})_{3}Sb_{5} up to x = 0.06 (Fig. 3d–i) (for plots with x ≥ 0.03, see Supplementary Fig. 6). It is noted that the jump in \(\widetilde{n}\) at T_{CDW} systematically decreases with increasing x, indicating weakened elastoresistance anisotropy at the CDW ordering. More importantly, we find that all the samples up to x = 0.03 exhibit the Curie‒Weiss temperature dependence of \(\widetilde{n}\) in broad temperature ranges above deviation temperatures represented by black dotted lines. For x = 0–0.0075, \(\widetilde{n}\) data clearly develop a peak at T_{ñ, peak}. In previous work on the pristine sample^{14}, T_{nem} identified by NMR and T_{ñ, peak} determined by the \(\widetilde{n}\) measurements were indeed nearly the same as ~35 K. Therefore, T_{ñ, peak} can be used as a good estimate of T_{nem} for each doping^{39}. In our case, the resultant T_{ñ, peak} = 34 K for x = 0 is indeed close to the known value of T_{nem} = ~35 K^{14}. For other x, T_{ñ, peak} shows a monotonous decrease; T_{ñ, peak} = 18 K (x = 0.005) and 14 K (x = 0.0075). For x = 0.01, however, we did not identify any peak feature at least down to 6 K, except finding the deviation temperature from the Curie–Weiss behaviour at ~12 K. This observation indicates that for x = 0.01, the true longrange nematic ordering is located well below 6 K or might not even exist at a finite temperature. For x = 0.0125 and 0.015, \(\widetilde{n}\)(T) doesn’t show any peak feature either, and only the deviation from the Curie–Weiss behaviour is identified around ~8 K. This indicates that in x = 0.0125 and 0.015, only nematic correlation exists without the development of true longrange order at a finite temperature.
In order to understand quantitatively the evolution of \(\widetilde{n}\) over the broad doping ranges, we have tried to fit the experimental \(\widetilde{n}\)(T) of all the samples by Eq. (1) below T_{CDW}. (see Supplementary Table 2 for detailed fit parameters). Firstly, we discuss the evolution of C for each doping. In contrast to the monotonic decrease in the jump of \(\widetilde{n}\)(T) and T_{ñ, peak} with x, C is found to exhibit nonmonotonic behaviour, i.e. increasing trend with x for x = 0.0–0.01 and decreasing trend for x ≥ 0.0125; C, as indicated by the slope of \({(\widetilde{n}{\widetilde{n}}_{0})}^{1}\) or the value of \(\left(\widetilde{n}{\widetilde{n}}_{0}\right)\left(T{\theta }_{{{{{{\rm{nem}}}}}}}\right)\), increases with x from C = 39 K (x = 0) to 124 K (x = 0.0075), and exhibits a maximum value of C = ~ 157 K at critical doping of x_{c} = 0.01. As a result, the highest value of \(\widetilde{n}\)(T = 6 K) ~ 23.8 can be found at x_{c}. For x ≥ 0.0125, C decreases with x to exhibit C = 2 K at x = 0.03, resulting in a flattening of the \(\widetilde{n}\)(T) curve at higher doping ratios. For x ≥ 0.04, Eq. (1) cannot be fitted very well to the \(\widetilde{n}\)(T) curves due to the almost temperatureindependent behaviour below and above the T_{CDW}.
The fit to Eq. (1) strikingly reveals that θ_{nem} is systematically suppressed from 30.0 K (x = 0) to 3.6 K (x = 0.0075), and to eventually exhibit a sign change (x = ~0.009). At higher x, θ_{nem} is suppressed further, resulting in θ_{nem} = −42 K at x = 0.03. In general, a nematic quantum critical point (NQCP) is often located at the phase space where θ_{nem} goes to zero temperature and strongly enhanced nematic susceptibility exists. The systematic suppression of θ_{nem} to zero temperature at x = ~0.009, combined with the sharp maximum of the C value and the disappearance of T_{ñ, peak} near x_{c}, strongly suggests the presence of an NQCP near the doping level close to x ~ 0.0090.01. Indeed, similar phenomena have been observed in numerous Febased systems having the NQCP, such as Ba(Fe_{1x}Co_{x})_{2}As_{2}^{27}, Fe(Se_{1x}S_{x})^{28}, LaFe_{1x}Co_{x}AsO^{29}, and Fe(Se_{1x}Te_{x})^{30}.
Another salient feature found in Fig. 3 is that the fit based on Eq. (1) below T_{CDW} for each doping level can be successfully extended to explain \(\widetilde{n}\)(T) above T_{CDW} with the same θ_{nem} and C but with a different \({\widetilde{n}}_{0}\) (orange solid line). This observation indicates that the evenparity nematic correlation might persist even above T_{CDW} for CsV_{3}Sb_{5} and Cs(V_{1x}Ti_{x})_{3}Sb_{5} (x ≤ 0.03). As a result, the Curie‒Weiss tail above T_{CDW} becomes clearly visible for 0 ≤ x ≤ 0.03, reaching at least up to the maximum investigated temperature of 250 K (see, Supplementary Fig. 6 for \(\widetilde{n}\)(T) plots up to 250 K and for x ≥ 0.03). Because of the most enhanced C value, the Curie–Weiss tail of \(\widetilde{n}\) becomes most conspicuous at x_{c} = 0.01. It is striking that the enhanced \(\widetilde{n}\) up to at least 250 K is observed at x_{c} = 0.01 where the actual nematic ordering temperature is close to zero. This directly suggests that large quantum fluctuation of evenparity nematic order near the NQCP could be responsible for the enhancement of \(\widetilde{n}\) at high temperatures.
Figure 4a, b summarises the phase diagram of Cs(V_{1x}Ti_{x})_{3}Sb_{5} plotted on top of the colour contour of \(\widetilde{n}\); T_{CDW} as derived from the data of ρ_{ab} (orange circles), M/H (yellow octagons), and \(\widetilde{n}\) (brown crosses) are plotted for each x. Moreover, T_{ñ, peak} (blue squares) and θ_{nem} (purple stars) obtained from \(\widetilde{n}\) in Fig. 3 are plotted with the T_{nem} (pink cross) of x = 0 determined in a previous work^{14}. At x = 0, a jump in \(\widetilde{n}\)(T) near T_{CDW} can be clearly identified by the abrupt change of colour in \(\widetilde{n}\) from blue at T > T_{CDW} to green at T < T_{CDW}. Near T_{nem}, the contour exhibits a yellow colour, indicating a local maximum of \(\widetilde{n}\) at T_{ñ, peak}. With an increase in doping, the T_{ñ, peak} shifts to low temperatures, resulting in the most intensified \(\widetilde{n}\) (6 K) indicated by the red colour near x = 0.00750.01. This behaviour is also confirmed by the maximum of the C value, indicated by the red diamonds.
The superconducting phase diagram of Cs(V_{1x}Ti_{x})_{3}Sb_{5} is also plotted in Fig. 4c. Here, the green triangles indicate the T_{c} obtained from transport measurements, while the Meissner volume fraction (−4πχ) is represented below the trace of T_{c} as a green colour contour. Surprisingly, it is found that T_{c} is optimised to 4.1 K (3.7 K) near this doping range of x = 0.0075 (0.01), when the nematic correlations indicated by the \(\widetilde{n}\) and C values are sharply enhanced near the NQCP. Our observation thus raises an intriguing possibility that fluctuation of the nematic order plays an important role in the pairing interaction to optimise superconductivity in the first superconducting dome of Cs(V_{1x}Ti_{x})_{3}Sb_{5}. At higher doping ratios of 0.01 ≤ x ≤ 0.02, T_{c} is monotonically suppressed with doping, which could be related to the reduced nematic fluctuations as indicated by the decreased \(\widetilde{n}\) and C values.
Based on our experimental findings and implications, the nematic fluctuations may be important in understanding the superconductivity in the AV_{3}Sb_{5} family. A very small T_{c} ~ 0.0008 K in CsV_{3}Sb_{5} has been indeed predicted based on the McMillan equation^{33}, suggesting that electronphonon coupling alone is not enough to explain the superconducting transition; nematic fluctuation should be considered as an essential ingredient to result in the relatively high T_{c} ~ 3.2 K in CsV_{3}Sb_{5}. In support of this scenario, it should be noted that the T_{c} values in KV_{3}Sb_{5} (T_{c} = 0.93 K^{8}) and in RbV_{3}Sb_{5} (T_{c} = 0.92 K^{9}) are lower than that in CsV_{3}Sb_{5} (T_{c} = 3.2 K). In addition, unlike the Cs variant, a recent study of Sn doping in polycrystalline KV_{3}Sb_{5} and RbV_{3}Sb_{5} revealed single superconducting domes near the suppression of the CDW orders^{40}. All these results, if interpreted correctly, potentially indicate that nematic order and its fluctuations might be absent in both compounds, motivating similar experiments for these materials. Furthermore, our scenario supports that nematic fluctuations should be also considered an important factor to understand the double superconducting domes reported in pressurised CsV_{3}Sb_{5}^{24,25} and in CsV_{3}Sb_{5x}Sn_{x}^{23} polycrystals.
It should be pointed out that the experimental results found here well resemble those found in the Febased superconductors, e.g., Ba(Fe_{1x}Co_{x})_{2}As_{2}^{27}, LaFe_{1x}Co_{x}AsO^{29} and Fe(Se_{1x}Te_{x})^{30}, where unconventional superconductivity is optimised near the NQCP. However, the microscopic origin for having the nematic order seems to be quite different; in contrast to the ironbased superconductors where the spin density wave order is closely coupled to the nematic order^{27,28,29,30,41}, the nematic order in CsV_{3}Sb_{5} is intertwined with the unconventional CDW order, possibly in a form of charge bond order^{3,14,18}. Recent theoretical studies^{3,18} considering the kagomeHubbard model have indeed shown that a tripleqcharge bond order is stabilised below T_{CDW}, described by three complex CDW order parameters. Furthermore, those theories suggest that these CDW order parameters can undergo a continuous variation of their phases at T_{nem} from a triplydegenerate, isotropic phase of π/2 into two different values, without the change of the isotropic amplitude, thereby resulting in onedimensional nematic modulation. Therefore, if the charge bond order developed in CsV_{3}Sb_{5} is assumed to be still maintained over the significant doping level x, the enhanced nematic correlation might be linked to the quantum phase transition involving a continuous variation of the phases of the tripleq CDW order parameters, at which the charge bond order with anisotropic phases, thus called nematic charge bond order, develops from the one with a homogenous phase.
Several other theoretical investigations considering the kagomeHubbard model have also predicted that CsV_{3}Sb_{5} might have unconventional superconductivity with a superconducting gap function of s or p or dwave type when the CDW state is melted^{1,17}, possibly via suppression of the amplitudes in the three qcharge bond orders. Investigations on the superconducting order parameters at the doping level x = ~0.00750.01 near the NQCP (first dome) and x = ~0.05 (second dome) may thus provide an opportunity for studying comparative characteristics of superconductivity instigated by anisotropic phase fluctuations and amplitudes of the three qcharge bond orders, respectively.
In conclusion, our experimental findings coherently suggest that an NQCP is located near x = ~0.0090.01. Moreover, a maximum T_{c} = ~4.1 K with full Meissner shielding is realised at x = ~0.00750.01, forming the first superconducting dome near the NQCP. This not only points out the vital role of nematic fluctuation in enhancing superconductivity but also provides important insights into understanding the link between the multiple orders and superconductivity in CsV_{3}Sb_{5} and related kagome superconductors.
Methods
Single crystal growth
Cs(V_{1x}Ti_{x})_{3}Sb_{5} (0 ≤ x ≤ 0.06) single crystals were grown by the CsSb flux method. A mixture of Cs chunks (99.8%, Alfa Aesar), V powders (99.9%, Sigma Aldrich), Ti powders (99.99%, Alfa Aesar) and Sb shots (99.999%, Alfa Aesar) with a molar ratio of Cs: (V, Ti): Sb = 2: 1: 3 were put in an alumina crucible and were doublesealed in an evacuated quartz ampule with pressures less than 2 × 10^{−5} mbar. The sealed ampules were heated at a rate of 100 °C/h and kept at 1000 °C for 24 h to fully dissolve the V and Ti into the CsSb eutectic mixture. Later, the ampoules were cooled down to 600 °C at a rate of 2 °C/h. At 600 °C, the ampules were centrifuged to separate the crystals from the molten flux. To avoid oxidation, all the sample preparation was done inside a glove box, which was kept in an Ar atmosphere with oxygen and moisture concentrations less than 1 ppm. Shiny platelike crystals were obtained with a typical lateral area of 3 × 2 mm^{2}. The CsV_{3}Sb_{5} single crystals exhibited a residual resistivity ratio (RRR) value as high as 129.2.
Inplane resistivity and magnetisation measurements
Inplane resistivity measurements were performed in the PPMS^{TM} using a conventional fourprobe method. The electrical contacts were attached by silver paint (Dupont 4929 N). The M/H after zerofield cooling (ZFC) were obtained by MPMS^{TM} (Quantum Design) at temperatures between 5 and 300 K, while the magnetic susceptibility χ between 1.8 and 5 K were measured by a vibrating sample magnetometer in a PPMS^{TM} (Quantum Design). The demagnetisation factors in the M/H and χ measurements have been corrected after measuring the samples in a needlelike configuration.
Elastoresistance measurements
The elastoresistance was measured in a closedcycle cryostat (Sumitomo RDK101D) using a commercial piezoelectric actuator (Piezomechanik PSt 150) at various temperatures from 6 to 250 K. The samples were glued to the piezoelectric actuator by using an adhesive epoxy (Devcon 14250). Two samples were cut in rectangular shapes with a lateral size of ~1 × 0.2 mm^{2}; the longer direction was parallel (perpendicular) to the aaxis for the R_{xx} (R_{yy}) sample. All the samples were cleaved to the thickness of ~20 μm to ensure efficient strain transmission to the entire sample. The electrical contacts were attached directly by silver paint (Dupont 4929 N) to the freshly cleaved surface. Two strain gauges were glued on the other side of the actuator with perpendicular orientation to each other.
Xray diffraction and wavelength dispersive Xray spectroscopy measurements
Cs(V_{1x}Ti_{x})_{3}Sb_{5} single crystals were ground and inserted inside a quartz capillary tube with an inner diameter of 0.5 mm. The tube was measured by xray diffraction (XRD) θ−2θ scans using a highresolution xray diffractometer (PANalytical Empyrean). Wavelength dispersive xray spectroscopy (WDS) were performed in a field emission electron probe microanalyzer (JEOL Ltd., JXA8530F) by taking V (99.99%), Ti (99.9%) and Sb (99.99%) metals as standard specimens. The standard specimen data of Cs was taken from the JEOL database due to the high air sensitivity of the elemental Cs metal.
Data availability
Source data are provided with this paper. The data generated in this study have been deposited in the Figshare database^{42}.
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Acknowledgements
Y.S., K.–T.K., S.K. and K.H.K. were financially supported by the Ministry of Science and ICT through the National Research Foundation of Korea (2019R1A2C2090648, 2022H1D3A3A01077468) and by the Ministry of Education (2021R1A6C101B418).
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Y.S. and K.H.K. initiated the project. Y.S., K.–T.K. and S.K. prepared the single crystalline samples. Y.S. characterised the samples and performed the measurements. Y.S. and K.H.K. analysed the data and wrote the manuscript. K.H.K. devised the project and advised the research. All authors discussed the results and commented on the manuscript.
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Sur, Y., Kim, KT., Kim, S. et al. Optimized superconductivity in the vicinity of a nematic quantum critical point in the kagome superconductor Cs(V_{1x}Ti_{x})_{3}Sb_{5}. Nat Commun 14, 3899 (2023). https://doi.org/10.1038/s41467023394951
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DOI: https://doi.org/10.1038/s41467023394951
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