Kagome metals have emerged as a new platform to investigate the interplay between topology and correlation owing to their unique lattice structures [1]. The frustrated corner-sharing triangular lattice naturally gives rise to electronic structures with flat bands, van Hove singularities, and Dirac crossings [2]. Several interesting phenomena have been discovered in kagome metals, including the giant anomalous Hall effect (AHE) in Weyl semimetal Co 3 Sn 2 S 2 [3], massive Dirac fermions in Fe 3 Sn 2 [4], and charge density wave (CDW) order in FeGe, ScV 6 Sn 6 , and the AV 3 Sb 5 (A ¼ K, Rb, Cs) family [5][6][7]. Among the kagome metals, the AV 3 Sb 5 (A ¼ K, Rb, Cs) family has attracted significant attention due to the exotic behavior of its CDW phase (T CDW ¼ 78-104 K) and superconducting phase (T c ¼ 1-3 K) [7][8][9][10][11][12][13][14][15][16][17][18]. In particular, the nature of the symmetry breaking in the CDW phase is not yet settled despite extensive investigations. Early studies including the measurements of AHE, optical Kerr effect, and change of muon relaxation rate all suggested that time-reversal symmetry is broken in the CDW phase [19][20][21][22][23][24][25][26], raising the intriguing possibility that the CDW is a form of loop current order [27,28]. Nevertheless, these observations are challenged by the most recent measurement of the highresolution polar Kerr effect, which found no observable Kerr response in zero field [29,30]. Therefore, whether time-reversal symmetry is truly broken remains an open question.
In addition to time-reversal symmetry, the rotational symmetry in the CDW phase has also been intensely studied. A twofold anisotropy in the CDW phase that breaks the C 6 rotational symmetry was first reported by angle-dependent magnetoresistance measurements [31]. Further studies including scanning tunneling spectroscopy (STM) [32,33], Raman spectroscopy [34,35], optical birefringence measurements [22,36], and angle-resolved photoemission spectroscopy [37][38][39], all confirmed the observed twofold anisotropy. However, the temperature at which rotational symmetry is broken remains a key question. While most studies agree that the breaking of rotational symmetry happens at T CDW , a diverging elastoresistivity response was reported within the CDW phase [40,41], which was argued as evidence for an electronic nematic instability well below T CDW .
The electronic nematic phase refers to a spontaneous rotational-symmetry-breaking phase while preserving translational symmetry, which has been extensively studied in strongly correlated electronic systems [42,43]. In the context of an Fe-based superconductor [44][45][46][47], nematicity is understood as a vestigial order of the underlying spin density wave or CDW phases [43,[48][49][50]. The partial melting of the density waves destroys long-range periodicity while continuing to break rotational symmetry; hence, the nematic transition is always above the density wave transition in these materials. A salient feature of the nematicity is a diverging nematic susceptibility at temperatures above the phase transition, which can be probed by elastoresistivity measurements, where resistivity anisotropy serves as a proxy for the nematic order parameter and the anisotropic strain serves as its conjugate field. For instance, a diverging elastoresistivity coefficient in the anisotropic strain channel (m B 2g ) with a Curie-Weiss temperature dependence was observed above the nematic transition in BaðFe 1-x Co x Þ 2 As 2 [44].
In this context, a diverging elastoresistivity in CsV 3 Sb 5 that peaks at T Ã ≈ 35 K, a temperature well below T CDW , is rather unusual [40]. It would imply the rotational symmetry is not broken at T CDW , which is inconsistent with the majority of previous experimental observations [22,33,36]. Another possible explanation is that the rotational symmetry is only weakly broken at T CDW by the π phase shift of the CDW order between different kagome layers [51], while the CDW within the two-dimensional kagome plane remains isotropic. At a lower temperature T Ã , the rotational symmetry within the kagome plane is broken, leading to diverging elastoresistivity [52]. In either scenario, the existence of a nematic instability below T CDW provides strong constraints to distinguish between various proposals for the CDW order parameter [17,28,[51][52][53][54][55]. In addition, a very recent elastoresistivity study proposed a nematic quantum critical point residing within the first superconducting dome in the phase diagram of Ti-doped CsV 3 Sb 5 [41], which was argued as evidence for nematic fluctuation-enhanced superconducting pairing. On the other hand, it has been shown that nonideal experiment configurations, such as what is incorporated in the differential elastoresistivity technique used in Refs. [40,41], may lead to the mixing of elastoresistivity signals in different symmetry channels, resulting in a false diverging nematic response even though the leading instability is in other symmetry channels [56][57][58][59]. The purpose of this work is to perform a comprehensive study of elastoresistivity and the elastocaloric effect in CsV 3 Sb 5 to elucidate the symmetry associated with the proposed transition at T Ã .
CsV 3 Sb 5 single crystals have a hexagonal shape with the a axis along the natural growth edges, consistent with the sixfold rotational symmetry of the kagome lattice [Fig. 1(a)]. The CDW transition occurs at T CDW ¼ 94 K, which can be seen from a resistivity anomaly and a sharp peak in dρ=dT as shown in Fig. 1(b). The superconducting transition is at T c ¼ 3 K. The residual resistivity ratio of ρð300 KÞ=ρð5 KÞ ¼ 143.5 indicates the high quality of the single crystals. The AHE is observed in the in-plane Hall resistivity below T CDW [Fig. 1(c)], consistent with previous experiments [20]. The Shubnikov-de Haas quantum oscillations can be observed in longitudinal resistivity below 30 K [Fig. 1(d)]. The oscillatory component Δρ xx as a function of inverse field and its fast Fourier transform are shown in Figs. 1(e) and 1(f), respectively. Four principal frequencies at 11, 28, 73, and 90 T were observed in the low-frequency regime (with some high-frequency peaks not shown here) in high field measurements up to 35 T. These observations are all in agreement with previous reports, providing a solid foundation to further study the elastoresponse in these single crystals [20,60,61].
Elastoresistivity is a fourth-rank symmetric tensor that relates the change of resistivity of a system to the externally induced strains. When using Voigt notation, the elastoresistivity tensor can be expressed as a second-rank 6 × 6 tensor:
where the indices i, j ¼ 1-6 represent 1 ¼ xx, 2 ¼ yy, etc. [45]. These elastoresistivity coefficients can be further grouped into different symmetry channels based on the irreducible representations of the point group of the crystal lattice. For example, in D 6h , the elastoresistivity coefficients associated with the isotropic A 1g irrep and the anisotropic
Here, the ν ab;ac are the in-plane and out-of-plane Poisson ratios (see Sec. V) [40,62]. In a material with an electronic nematic instability, the diverging susceptibility associated with the nematic transition will manifest in a diverging temperature dependence of the elastoresistivity coefficient in the anisotropic symmetry channel. When measuring elastoresistivity, single crystals are glued on the sidewall of a piezostack, which induces a combination of purely anisotropic strain 1 2 ðε xx -ε yy Þ and isotropic strain 1 2 ðε xx þ ε yy Þ when an external voltage is applied to the stack. In order to separate the isotropic and anisotropic symmetry channels, both ρ xx and ρ yy need to be measured. Three experimental techniques have been developed to measure elastoresistivity coefficients. The first technique that was developed is the differential elastoresistivity technique [44], which measures ρ xx and ρ yy using two separate bar-shaped samples cut along the same crystal axes but oriented perpendicularly on the stack, as shown in the top configuration in Fig. 2(a). However, it was soon realized that this technique inevitably introduces cross-contamination between different symmetry channels [56,57]. Exact symmetry decomposition requires identical strain transmission in both samples, which is never the case in any practical experiment. In fact, the bar-shaped sample results in more effective strain transmission for the uniaxial strain along the bar direction. Therefore, a barshaped sample glued along the x (y) direction experiences dominantly uniaxial strain ε xx ðε yy Þ even though nominally the same anisotropic strain ε xx -ε yy was applied. To address this issue, the modified Montgomery technique [45] and the transverse method [63] were subsequently developed. The modified Montgomery technique allows for obtaining ρ xx and ρ yy using a single square sample [middle configuration in Fig. 2(a)], and the transverse technique enables the direct measurement of the resistivity anisotropy ρ x 0 y 0 in a five-contact bar-shaped sample (bottom configuration; see details in Sec. V). Both methods measure the full resistivity tensors from the same single crystalline samples; hence, they do not suffer cross-contamination issues, and the symmetry decomposition is exact.
To thoroughly examine the elastoresistivity coefficients of CsV 3 Sb 5 , all three techniques mentioned above were employed. We found that resistivity as a function of strain is linear in both E 2g and A 1g symmetry channels at all temperatures for all measurements [Figs. 2(b)-2(e)], with only weak hysteresis near T CDW likely due to structural domains. This suggests that all the measured elastoresistivity coefficients are in the near-zero strain linear response regime. Figure 2(f) shows the temperature dependence of E 2g and A 1g elastoresistivity coefficients measured by the modified Montgomery technique. The m E 2g value jumps from 3 to 8 at T CDW , but it is essentially temperature independent both below and above T CDW . The m A 1g is also temperature independent above T CDW , albeit with a larger value (approximately 10). At T CDW , it exhibits a sharp peak and grows continuously as temperature decreases until reaching a maximum value of 90 at T Ã ¼ 20 K. The m E 2g measured by the modified Montgomery technique is very different from those reported in Refs. [40,41], which were measured by the differential elastoresistivity technique. The m E 2g reported in Refs. [40,41] is considerably larger and shows a temperature dependence that resembles m A 1g measured by the modified Montgomery technique. To gain more insight, we present the elastoresistivity coefficients measured by all three techniques in Figs. 2(g) and 2(h). It can be seen that m E 2g and m A 1g measured by the modified Montgomery technique and the transverse technique are consistent with each other, whereas the differential elastoresistivity technique yields a larger m E 2g and smaller m A 1g in comparison to the other two techniques. The temperature dependence of m E 2g
measured by the differential elastoresistivity technique is also more similar to that of m A 1g . All of these are consistent with the admixture of m A 1g into m E 2g in the differential elastoresistivity measurement due to unequal strain transmission in the two samples. Hence, we conclude that the divergent m E 2g in previous reports is not intrinsic. We note that two groups have also reported elastoresistivity measurements during the preparation of this manuscript, which are in broad agreement with our observations [64,65]. A detailed comparison of the results among different groups is discussed in Supplemental Material [66].
In addition to the elastoresistivity measurements, the elastocaloric effect (ECE) is another sensitive probe to measure the diverging susceptibility associated with a nematic phase transition [67,68]. The elastocaloric measurement detects the temperature change of a system due to an adiabatically induced strain. Experimentally, a small ac strain is induced in the sample simultaneously with a tunable dc bias strain, and the ac temperature variation is measured at the frequency of the induced strain by a thermocouple (see details in Sec. V). The elastocaloric coefficient (∂T=∂ε) can be related to the isothermal entropy change caused by strain via the following equation:
where C ε is the heat capacity at a given strain and S is entropy. The elastocaloric effect is sensitive to both the susceptibility of the nematic order parameter as well as its spontaneous onset. Depending on the symmetry of the induced strain, two ECE phenomena are expected near a phase transition [68,69], the first of which is associated with inducing strain that breaks the same symmetry as the order parameter [68]. When using such a strain to perturb the system, an enhancement of ECE above the transition temperature is expected. This enhanced ECE is described by the following equation:
where i labels the anisotropic symmetry channels. Thus, this enhancement of the ECE is proportional to both the temperature derivative of the susceptibility (dχ N =dT) and the dc bias anisotropic strain ε i ; hence, it switches sign from tensile to compressive dc strain. This effect was observed in the iron-based superconductors where ε B 2g couples linearly to electronic nematicity and showed excellent agreement with the nematic susceptibility obtained from previous elastoresistivity measurements [68,70].
The second effect is associated with the temperature shift of the phase transition induced by strains with symmetry that do not couple linearly to the order parameter. For example, ε A 1g is expected to linearly tune T CDW . In this case, we expect the ECE near the phase transition to be proportional to the critical contribution of heat capacity (C ðcÞ ε E 2g ) times the strain derivative of transition temperature:
We measured the ECE of a CsV 3 Sb 5 sample by applying a uniaxial stress, which induces both ε A 1g and ε E 2g . Figure 3(a) summarizes the ECE measured under different dc bias strains. A pronounced peak that mimics the heat capacity anomaly is observed near T CDW , and the peak is systematically shifted as a function of dc bias strain. This peak is consistent with the second effect mentioned above, where ε A 1g linearly shifts the transition temperature. The linear dependence of T CDW as a function of A 1g strain is also consistent with previous studies of CsV 3 Sb 5 [71].
However, the enhancement of the ECE due to a diverging nematic susceptibility was not observed either above or below T CDW . The signal quickly converges both above and below T CDW [Fig. 3(b)], and there is no indication of concavity or slope change in the peak from tensile to compressive strains. This result is consistent with no diverging nematic susceptibility, further corroborating the conclusion established by the elastoresistivity measurements. A very recent study suggested an odd-parity nematic transition above T CDW which turns into a first-order transition under a c-axis magnetic field [65]. We also performed ECE measurements under a c-axis magnetic The magnetic field is along the c axis, and the dc-biased strain is near zero. There is no change of ECE signal as a function of the magnetic field except for a slight shift of the background due to the weak field dependence of the voltage response of a type-E thermocouple above 100 K [72].
field up to 9 T and observed no anomaly above T CDW as illustrated in Fig. 3(d). This result is consistent with the absence of such a transition measured by thermal expansion experiments [64].
Both elastoresistivity and elastocaloric effect measurements suggest the absence of the nematic instability below T CDW . The diverging m E 2g reported in Refs. [40,41] is most likely due to the mixing of the diverging m A 1g , a consequence of unequal strain transmissions in the differential elastoresistivity technique. We note that, in Ref. [41], the Montgomery elastoresistivity measurement was performed on one Ti-doped CsV 3 Sb 5 sample, which showed agreement with the differential elastoresistivity measurement. Given that the unequal strain transmission is a generic issue in all differential elastoresistivity measurements, a reinvestigation of the elastoresistivity of Ti-doped CsV 3 Sb 5 is highly warranted.
We also note that our result is not inconsistent with the rotational symmetry breaking at T CDW . The increase of m E 2g could be a result of aligning symmetry-breaking CDW domains by the anisotropic strain [22,32,73], but the small value of m E 2g (approximately 8) indicates that the electronic anisotropy is only moderately larger than the structural anisotropy [74], which could be consistent with the weak anisotropy introduced by the interlayer coupling. In this scenario, given that both rotational symmetry and translational symmetry are simultaneously broken at T CDW , this transition cannot be characterized as a nematic transition. The absence of diverging E 2g elastoresistivity and elastocaloric effect above T CDW also confirms that there is no fluctuating vestigial nematicity associated with the rotational symmetry-breaking CDW. The temperature-independent E 2g elastoresistivity also suggests that the previously reported 4a 0 order observed by STM is likely a surface phenomenon, which does not contribute to bulk electronic anisotropy [10,12]. Instead of diverging m E 2g , the dominant response is in the isotropic A 1g channel, manifested by the diverging m A 1g . Below, we discuss the possible origins of the large and strongly temperature-dependent m A 1g . There are two features in the temperature dependence of m A 1g . The first feature is a sharp peak at T CDW , which can be understood by the generalized Fisher-Langer relation [69]. The relation states that the temperature derivative of resistivity, dρ=dT, the ECE, and elastoresistivity coefficient all scale like the critical component of the heat capacity near a phase transition, if the strain used in the ECE and elastoresistivity is associated with the same symmetry channel that is orthogonal to the order parameter. Indeed, as shown in Fig. 4(a), all three quantities follow the same temperature dependence near T CDW . As a side note, this phenomenon can appear in the m E 2g channel as well, if the sample is at a nonzero E 2g strain, since T CDW is potentially a quadratic function of ϵ E 2g .
Below T CDW , the m A 1g increases rapidly as temperature decreases, showing a diverging behavior that peaks at T Ã . In contrast to the sharp peak at T CDW , we do not see similar temperature dependence in the ECE, and dρ=dT shows only a broad hump centered at T Ã that barely resembles the feature in m A 1g [Fig. 4(b)]. The lack of any feature in the ECE and the strong divergence in elastoresistivity is rather striking. If we attribute the divergence of m A 1g to a phase transition at T Ã , it implies that the order parameter has a very weak coupling to strain, such that it causes a minimum entropy change induced by strain, yet it has a very strong coupling to the conducting quasiparticles, leading to the strong divergence in elastoresistivity. This phenomenon has been observed in iron-based superconductors, where the magnitude of ECE decreases by factor of 35 as the doping approaches the nematic quantum critical point, while the elastoresistivity is enhanced by fivefold [68]. It was understood as a consequence of simultaneous reduction of nematoelastic coupling and enhancement of nematotransport coupling.
Another notable observation is that the temperature dependence of m A 1g shows a striking resemblance to that of the AHE discovered in the CDW phase of the AV 3 Sb 5 family [3,19,20], since the temperature dependence of m A 1g can be decomposed into two components: a peak near T CDW that resembles dρ=dT and a diverging behavior toward T Ã . To isolate diverging behavior toward T Ã , we normalized both m A 1g and dρ=dT [Fig. 5(a)] and subtract the latter from the former. The anomalous Hall resistivity was extracted by removing the ordinary Hall from the ρ xy in Fig. 1(c) by linearly fitting the data for fields between 1.5 and 2 T, shown in Fig. 5(b). The extracted ρ AHE xy as a function of the temperature is plotted against the background-subtracted Δm Ã
A 1g , shown in Fig. 5(c). The remarkable resemblance of these transport observables highlights the impact of the T Ã instability on the quasiparticles at the Fermi level. We notice that the recently observed chiral transport effect in CsV 3 Sb 5 also shows a similar temperature dependence to AHE [25]. Intriguingly, while the AHE and the chiral transport effect are sensitive to time-reversal symmetry and inversion symmetry breaking, respectively, the A 1g elastoresistivity is not a direct probe of either symmetry-breaking order parameters, because both resistivity and strain are even-parity operators. Future studies on the strain dependence of AHE and nonlinear transport effect may elucidate the origin of the common diverging behavior of all three transport observables. There has also been a suggestion that the small Fermi pockets observed in quantum oscillations may be relevant to AHE or field-tunable chirality [11,[24][25][26]. This may also be resolved by performing strain-dependent quantum oscillation measurements.
In summary, we investigated the isotropic and anisotropic elastoresistivity coefficients m A 1g and m E 2g , as well as the elastocaloric effect in CsV 3 Sb 5 . The lack of enhancement in elastocaloric effect and the temperature-independent m E 2g below T CDW are both consistent with the absence of nematic instability in this system. The previously reported diverging elastoresistivity m E 2g is likely due to the cross-contamination from m A 1g . Both the elastoresistivity coefficient m A 1g and the ECE show a peak at T CDW , which is a consequence of the tuning of T CDW by A 1g strain, and it is consistent with the extended Fisher-Langer relation. In addition to the peak at T CDW , m A 1g also shows a diverging temperature dependence below T CDW that reaches a maximum at T Ã , but there is no corresponding elastocaloric anomaly. Our results show that there is no nematic phase transition within the CDW phase in CsV 3 Sb 5 , and the nature of T Ã requires further investigation.
Single crystals of CsV 3 Sb 5 were synthesized using a self-flux method described elsewhere [7,71,75]. The electrical transport measurements were performed in DynaCool (Quantum Design, Inc.). The high magnetic field experiments were carried out in a 35 T resistive magnet at the National High Magnetic Field Laboratory in Tallahassee, Florida.
The symmetry decomposition of elastoresistivity tensor in the D 6h point group is different from that in D 4h . There are two irreducible representations in D 6h that are relevant to this study: isotropic A 1g and anisotropic
The same procedure was applied to dρ=dT to obtain ðdρ=dTÞ Ã . We shifted m à A 1g by 3 K because of the change of T CDW due to thermal built-in strain. (b) Δρ xy extracted by subtracting the linear ordinary Hall background at various temperatures (see details in Supplemental Material [66]). The black dashed line is located at -1.5
T, where we extracted the ρ AHE xy values. (c) ρ AHE xy and Δm à A 1g as a function of the temperature, where Δm à A 1g ¼ m à A 1g -ðdρ=dTÞ Ã . This presents the temperature dependence of m A 1g without the peak feature. E 2g symmetry. The in-plane biaxial strain employed in this study can be decomposed into these two symmetry channels: ε A 1g;1 ¼ ðε xx þ ε yy Þ=2, ε A 1g;2 ¼ ε zz , and ε E 2g ¼ ½ðε xx -ε yy Þ=2; ε xy . We note that E 2g is a two-dimensional irreducible representation and ðε xx -ε yy Þ=2 and ε xy are the two bases. The in-plane resistivity tensors can also be decomposed into these two irreducible representations: ðΔρ=ρÞ A 1g;1 ¼ 1 2 ½ðΔρ=ρÞ xx þ ðΔρ=ρÞ yy ; ðΔρ=ρÞ A 1g;2 ¼ ðΔρ=ρÞ zz ; ðΔρ=ρÞ E 2g ¼ 1 2 ½ðΔρ=ρÞ xx -ðΔρ=ρÞ yy ; ðΔρ=ρÞ xy :
Therefore, if we focus on the in-plane elastoresistivity, coefficients in isotropic A 1g and anisotropic E 2g symmetry channels are related to the elastoresistivity tensors in the Cartesian coordinate systems by the following expressions:
Since the sample is glued down to the side wall of the piezostack by the Stycast, its in-plane deformation will be constrained by the deformation of the piezostacks. Consequently, ν ab ¼ -ϵ yy =ϵ xx and ν ac ¼ -ϵ zz =ϵ xx are the Poisson ratios of the piezostacks and the sample, respectively. We note that, because
A tunable strain was applied to single crystal samples by gluing them on the sidewall of a piezostack (P-885.11, from PI-USA) using a thermally conductive epoxy (Stycast 2850FT with Catalyst 24LV, from Loctite). The uniaxial strain along the poling direction of the piezostack, ε yy [Fig. 2(a)], was measured by a foil strain gauge (CEA-06-062UWA-350, from Micro-Measurements), which was glued to the other side of the piezostack. The uniaxial strain along the orthogonal direction, ε xx , was determined by the known in-plane Poisson ratio of the piezostack. As discussed in the main text, the elastoresistivity coefficients can be measured by three techniques.
(i) Differential technique.-Two bar-shaped samples are glued to the piezostack. The bar directions are aligned with the transverse and poling directions of the piezostack. Using the standard four-terminal technique, the resistivity ρ xx and ρ yy are measured separately in each sample. (ii) Modified Montgomery technique.-Four electric contacts are made at the corners of a square-shaped sample, which is glued on the piezostack with the edges aligned to the poling direction. R xx and R yy are measured by sourcing current using the contacts on one side and measuring the voltage using the contacts on the other side. The measured resistance R xx and R yy can then be converted to resistivity ρ xx and ρ yy following the procedure in Supplemental Material [66]. (iii) Transverse technique.-The transverse technique measures the transverse resistivity ρ xy induced by shear strain ε xy , which is different from the other two techniques that measures the longitudinal resistivity ρ xx and ρ yy [63]. The shear strain ε xy is equivalent to an anisotropic strain ðε xx -ε yy Þ=2 under 45°rotation. In the transverse technique, a Hall bar with five electrical contacts is glued onto the piezostack and is aligned at a 45°angle with respect to the y axis. Note that there is no magnetic field; hence, the transverse resistivity ρ xy is not a result of the Hall effect. The main problem with the differential technique is the contamination of the isotropic strain channel to other anisotropic channels due to unequal strain transmission between two samples. This effect is inevitable as long as the samples are cut into bar shape and glued along orthogonal directions [45,63]. This effect becomes crucial when the elastoresistivity coefficient in the isotropic channel is much larger than the anisotropic channel. In this respect, both modified Montgomery and transverse techniques can perfectly eliminate this effect by measuring a single sample.
The elastocaloric effect was measured using an ac (dynamic) ECE technique with a homemade three-piezostack uniaxial strain cell, similar to the piezoelectric strain cell developed by Hicks et al. [76]. The sample was glued across the gap of the sample mounts of the strain cell. The temperature of the sample was measured at the center of the sample using a Chromel-AuFe0.07% (25 μm wire diameter) or type-E thermocouple for lower or higher temperature range, respectively. To apply a dynamical strain together with a static offset, ac and dc voltages were applied to the piezostacks using a TEGAM 2350 high-voltage amplifier sourced by a Stanford Research SR860 Lock-in amplifier and Keithley 2450 voltage supply, respectively. signals from the sample and thermocouple were amplified by the Stanford Research SR554 preamplifier and measured by SR860. We present the thermal transfer function and phase as a function of frequency in Fig. S5 in Supplemental Material [66]. Accordingly, we set the frequency 25.5 Hz to achieve the quasiadiabatic condition in our measurements. The details of the ac ECE technique can be found in Refs. [67,68,70].