In-plane uniaxial-strain tuning of superconductivity and charge-density wave in CsV3Sb5

Xiaoran Yang(杨晓冉), Qi Tang(唐绮),†, Qiuyun Zhou(周秋韵), Huaiping Wang(王怀平),Yi Li(李意), Xue Fu(付雪), Jiawen Zhang(张加文), Yu Song(宋宇),Huiqiu Yuan(袁辉球), Pengcheng Dai(戴鹏程), and Xingye Lu(鲁兴业),‡

1Center for Advanced Quantum Studies and Department of Physics,Beijing Normal University,Beijing 100875,China

2Center for Correlated Matter and School of Physics,Zhejiang University,Hangzhou 310058,China

3State Key Laboratory of Silicon and Advanced Semiconductor Materials,Zhejiang University,Hangzhou 310058,China

4Department of Physics and Astronomy,Rice Center for Quantum Materials,Rice University,Houston,TX 77005,USA

Keywords: kagome metal,superconductivity,charge-density wave,uniaxial-strain

Materials possessing kagome lattice structures have attracted intense attention due to their unique electronic properties, allowing for the exploration of new and exotic quantum phenomena.[1,2]Among the newly discovered kagome metals,AV3Sb5(A=K, Rb, Cs) exhibits rich quantum phenomena such as non-trivial topological bands, van Hove singularities near the Fermi energy, highly unusual superconductivity, and charge-density waves(CDWs).[3-9]These findings have stimulated a wave of research in this field.Our research focuses on CsV3Sb5, a specific member of theAV3Sb5class that has attracted substantial attention for its novel electronic properties.

The structure of CsV3Sb5(space groupP6/mmm) consists of V-Sb layers intercalated by cesium layers.Within the V-Sb layer, the vanadium cations are coordinated by Sb octahedra, forming a two-dimensional kagome lattice(Fig.1(a)).[3]CsV3Sb5undergoes a CDW transition atTCDW≈94 K,and enters into a superconducting ground state atTc≈3 K.[4]Various experimental studies revealed longrange CDW order[10-12]and suggested that the unconventional CDW may be related to van Hove filling, in addition to electron-phonon coupling.In addition, electron nematicity has been reported in this system and suggests the CDW to be highly unusual.[13]Despite the relatively lowTc, the superconducting state in CsV3Sb5could be highly unusual.For example,theoretical and transport measurements indicate that charge-4e and charge-6e superconductivity could exist in CsV3Sb5.[7,8]

It is well established that the interplay between superconductivity and CDW in CsV3Sb5, both of which are related to Fermi surface instability, is essential to understanding the microscopic nature of the electronic ground state.Many experimental techniques, such as elemental substitution (Sn, Nb, Ta),[14-16]mechanical exfoliation,[17,18]hydrostatic pressure,[19-22]uniaxial stress/strain[23,24]have been employed to study the complex interplay between superconductivity and CDW in CsV3Sb5.Among these methods,uniaxial strain warrants detailed studies because it is sensitive to symmetry-breaking orders and fluctuations, as well as tuning the physical properties of the system with high precision.

Previously, in-plane uniaxial strain (ε) along theaaxis(εa) was used to tune theTcandTCDWin CsV3Sb5.The measurements reveal a competition between superconductivity and the CDW.Note that a uniaxial strain applied along theaaxis (εa=ε[110]asa‖[110]) will induce opposite strains along the other two perpendicular directions (εcalong thecaxis andε[¯110]along the [¯110] direction, which is equivalent to the [100] direction).Through comparing the results with the tuning ofTcandTCDWby hydrostatic pressure which preserves theD6hsymmetry, the authors in Ref.[24] found that the strain-induced ∆Tcand ∆TCDWare driven by thec-axis uniaxial strain(εc),while the effect of the symmetry-breaking inplane uniaxial strain is negligible.This is surprising as the inplane kagome lattice is thought to be essential to the tuning of electronic properties including the superconductivity and the CDW.In addition to thea-axis that is parallel with the [110]direction,the[100]direction(30◦away from[110])is another high-symmetry direction, and the response to uniaxial strain along these directions is integral for a comprehensive understanding of the uniaxial-strain-tuning in CsV3Sb5.

Fig.1.(a) Top (c axis) view of the structure for CsV3Sb5.The vanadium ions form an ideal kagome lattice.(b) Temperature-dependent resistivity of CsV3Sb5.The left-upper inset shows a zoomed-in view of the superconducting transition at Tc ≈3 K.The vertical arrow marks the TCDW ≈94 K.The right-lower inset shows the zoomed-in view of the first derivative of resistivity dR/dT,in which the peak corresponds to the TCDW.(c)-(d)Schematics of the uniaxial strain application and the measurements of(c)resistivity and(d)AC magnetic susceptibility.Thin(∼20µm in thickness)CsV3Sb5 single crystal is glued onto a titanium platform(0.1 mm in thickness),which is fixed between the two ends of the sample gap(∼1 mm).The black thin layers represent the Stycast 2850FT epoxy to help fix the titanium platform.Four silver-paste electrodes are attached to the surface of the crystal in (c) for resistivity measurements.In (d), a commercial MTD100 coil for AC magnetic susceptibility measurement is put over the top of the crystal.The inner coil provides an excitation signal and the outer one collects the signal from the sample.The uniaxial strain is applied by the FC100 strain cell, and the AC susceptibility is measured through an SR830 lock-in amplifier.All the measurements are performed on PPMS.

In this work,we have explored the in-plane uniaxial strain effects on the superconductivity and CDW along the two highsymmetry directions [110] and [100] using the FC100 stress cell(Razorbill Instruments Ltd)which can apply a force up toF=90 N atT=4 K.Our resistivity and AC magnetic susceptibility measurements under uniaxial strains show that theε-induced ∆Tcand ∆TCDWalong the [110] and [100] directions are almost identical.Through decomposing the uniaxial strains into three symmetry channels (εA1g,εE1g, andεE2g)under theD6hpoint group,we conclude that the in-plane uniaxial strainε[110]andε[100]show similar and small tuning effects onTcandTCDW,consistent with the conclusion reported in Ref.[24].Our results confirmed the dominant role ofc-axis uniaxial strain in tuning the competingTcandTCDW,and provide an experimental basis concerning the in-plane uniaxial strain effects on the intertwined orders in CsV3Sb5.

The CsV3Sb5single crystals used in this study were grown with the flux method, which was described elsewhere.[3]Our crystals exhibit a superconducting transition atTc≈3 K (Tc, offset≈2.6 K) and a CDW transition atTCDW≈94 K, as shown in Fig.1(b).The [100] and [110]directions are determined using Laue diffraction,along which the crystals are cut into rectangular bars to facilitate the application of uniaxial strain.

In Ref.[24], the uniaxial strain was applied through a home-built uniaxial-strain apparatus based on piezoelectric stacks.The two ends of a bar-shaped CsV3Sb5single crystal were attached to the two blocks of the apparatus with Stycast 2850FT epoxy.[24]However, given the CsV3Sb5crystals cleave or break easily under uniaxial stress,we use an alternative method developed in the study of the uniaxial-strain effect in FeSe (Ref.[25]).As shown in Figs.1(c) and 1(d), a thin(∼20µm in thickness)CsV3Sb5single crystal is glued onto a∼0.1-mm thick titanium platform,which is∼10-mm wide at its two ends and has a neck-like part (∼0.5 mm in width) in the center.The neck of the platform bridges the gap between the two moving blocks of the stress cell.The uniaxial stress is applied to the titanium platform through piezoelectric stacks which drive one of the moving blocks holding one end of the titanium platform.Here, the stress cell has a capacitance to monitor the force applied to the platform/sample,from which the strain can be determined.Further, the uniaxial strain on the titanium platform can be transferred to the CsV3Sb5thin crystal via a thin layer of epoxy(Stycast 2850FT).

For resistivity measurements,four silver-paste electrodes were made on the surface of the thin CsV3Sb5crystal(Fig.1(c)),and the longitudinal resistance can be measured by slowly sweeping the temperature under strain and magnetic field.For the measurements of AC magnetic susceptibility,two concentrically nested coils (Razorbill MTD100) placed directly above the sample were used to measure the real part(χ′) of the AC magnetic susceptibility (Fig.1(d)).This signal is provided by a Stanford SR830 lock-in amplifier,which also serves as a reference to extract the signal from the pickup coil.Due to the received signal’s pronounced sensitivity to environmental shifts, the magnetic fluctuations in the sample and its vicinity during the superconductive phase transition,prompted by the Meissner effect,result in a marked variation of the received signal, which allows us to measure the phase transition curve accurately.

Fig.2.(a)Photos of the CsV3Sb5 single crystals attached on titanium platforms for the measurements of resistivity(left panel)and AC χ′(right panel)under uniaxial strains.(b)Resistivity measurements under the uniaxial strain along the[110]direction,with the strain ε ranging from-0.29%to 0.29%.(c)The measurements of AC χ′ under uniaxial strains along the[110]direction with ε =[-0.22%,0.44%].The horizontal dashed lines mark the values used to track the relative change of Tc.(d)ε[110] dependence of ∆Tc extracted from the data in panels(b)and(c).The data points labeled by black squares are from Ref.[24].The solid lines are the fittings of the data with Eq.(1).

Figure 2(a) displays the photos of CsV3Sb5crystals on the FC100 stress cell for the measurements of resistivity (left panel) and ACχ′(right panel).We first measure temperature-dependent resistivity and ACχ′underε[110]at low-temperature range to determine the effect onTc, which had been reported in Ref.[24].Figures 2(b) and 2(c) show the results of resistivity and ACχ′measured underε[110]=[-0.29%,0.29%] andε[110]=[-0.22%,0.44%], respectively.To determine the strain-induced changes inTc,we use the temperatures corresponding to 10% of the resistance at 4 K, and 99.4%of the received signal in the pick-up coil at 4 K,in our resistivity and ACχ′measurements, respectively.Both resistivity and ACχ′reveal thatTcis monotonically tuned withε[110].Figure 2(d)plots the ∆Tcas a function ofε[110]extracted from the data in Figs.2(b)and 2(c).Theεa-dependent ∆Tcin Ref.[24] (black squares) is also plotted as a reference.The∆Tcdetermined by resistivity and ACχ′results are quantitatively consistent with each other,indicative of the consistency of our experimental methods.Solid lines are fittings of the∆Tc(ε[110])with[24]

The fitting parametersaandbare 0.56 and 0.12 in Ref.[24], and 1.27 (1.12) and 0.54 (0.69) in the resistivity(ACχ′) measurements, revealing that the linear dependence is dominant,consistent with the results from Ref.[24].However, the slope d∆Tc/dTof our data is much larger than that in Ref.[24].This could be caused by different ways of determining ∆Tc, a possible overestimate of the uniaxial strain applied on the crystals in Ref.[24],and a slight underestimate of the uniaxial stress applied on the titanium platform in our measurements.

Figures 3(a)-3(c) show the resistivity curves under different magnetic fields (H) withε[110]=-0.251%, 0%, and 0.285%, respectively.The field dependence ofTcextracted from Figs.3(a)-3(c)are summarized in Fig.3(d).By describing theHc2(T)data in Fig.3(d)using the empirical Ginzburg-Landau equationHc2(T)=Hc2(0)(1-t2)/(1+t2),theHc2(0)is estimated to be 0.23 T forε[110]=-0.251%, 0.30 T forε[110]=0%,and 0.46 T forε[110]=0.285%.Again,the results are consistent with those reported in Ref.[24].

Having presented the ∆Tc(ε[110]), we show in Fig.4(a)the ACχ′measured under uniaxial strains in the rangeε[100]=[-0.27%,0.27%],through which theε[100]dependence of ∆Tcis plotted in Fig.4(b).The ∆Tc(ε[110])data shown in Fig.2(d)are also plotted in Fig.4(b)for a comparison,revealing that the∆Tc(ε[110])and ∆Tc(ε[100])are almost identical.The fitting of∆Tc(ε[100]) with Eq.(1) givesa=0.97 andb=0.34, which are close to the fitting parameters of ∆Tc(ε[110]) (a=1.12,b=0.69).The results indicate thatε[110]andε[100](together with the induced strains along their perpendicular directions)have a similar tuning effect onTc.

Fig.3.(a)-(c),Magnetic field dependence of resistivity curves and Tc measured under ε =-0.251%(a),ε =0%(b),and ε =-0.285%(c).(d) Estimate of Hc2 for CsV3Sb5 under ε =-0.251%, 0%, and 0.285%.Solid lines are fittings to the Ginzburg-Landau model.The unit 1 Oe=79.5775 A·m-1.

Figures 4(c)and 4(e)are temperature-dependent resistivity curves nearTCDWmeasured under different tensile strains,whose first-order derivatives dR/dTshow systematic change with uniaxial strain (Figs.4(d) and 4(f)).We use the middle value of the slope (rather than the peak position as the peaks are too broad) of dR/dTto characterize the relative change ofTCDW.Figure 4(g) summarizes theε[100]dependence of∆TCDWand compares the ∆TCDW(ε[100]) to that reported in Ref.[24](black squares).The ∆TCDW(ε[100])is basically the same as ∆TCDW(ε[110]).Taking together the same ∆Tc(ε)along the [110] and the [100] directions, it seems indeed the uniaxial strain applied to these two directions have the same tuning effect on the intertwining orders.These results are consistent with the conclusion in Ref.[24]that a purely in-plane strain has little effect in tunning the competing intertwined orders,and the observed tuning effects of ∆Tcand ∆TCDWresult from changes in thecaxis induced by the applied in-plane strain,through a nonzero Poisson ratio(νac).

Our results clarify the effects of in-plane uniaxial strains in tuning the competing orders in CsV3Sb5and corroborate the central conclusion drawn in Ref.[24].The experimental strategies used in our work can be employed to study a wide class of layered quantum materials possessing intertwined orders.

Acknowledgements

The work at Beijing Normal University is supported by the National Key Projects for Research and Development of China (Grant No.2021YFA1400400) and the National Natural Science Foundation of China (Grant Nos.12174029 and 11922402).The work at Zhejiang University was supported by the National Key Research and Development Program of China (Grant No.2022YFA1402200), the Pioneer and Leading Goose Research and Development Program of Zhejiang Province, China (Grant No.2022SDX- HDX0005), the Key Research and Development Program of Zhejiang Province,China(Grant No.2021C01002),and the National Natural Science Foundation of China(Grant No.12274363).