Nonlinear uniaxial pressure dependence of $T_c$ in iron-based superconductors

We have systematically studied the effects of in-plane uniaxial pressure $p$ on the superconducting transition temperature $T_c$ in many iron-based superconductors. The change of $T_c$ with $p$ is composed of linear and nonlinear components. The latter can be described as a quadratic term plus a much smaller fourth-order term. In contrast to the linear component, the nonlinear $p$ dependence of $T_c$ displays a pronounced in-plane anisotropy, which is similar to the anisotropic response of the resistivity to $p$. As a result, it can be attributed to the coupling between the superconducting and nematic orders, in accordance with the expectations of a phenomenological Landau theory. Our results provide direct evidences for the interplay between nematic fluctuations and superconductivity, which may be a common behavior in iron-based superconductors.


I. INTRODUCTION
Nematicity has been found in both cuprates and ironbased superconductors, consisting of an electronic state that breaks the in-plane C 4 rotational symmetry of the underlying lattice 1 . In the former, nematic order seems associated with the pseudogap state 2,3 , where many other types of orders are also found, such as stripes and charge density waves 4 . The scenario is much simpler in ironbased superconductors, where nematicity typically appears together with antiferromagnetism and superconductivity in the phase diagram 5 . In several materials, a putative nematic quantum critical point (QCP) has been proposed to exist around the optimal doping level 6-10 , suggesting a close interplay between nematicity and superconductivity. Interestingly, two-fold anisotropy in the magnetoresistivity has been reported in the vicinity of the superconducting transition of slightly overdoped Ba 1−x K x Fe 2 As 2 , which suggests the formation of a nematic superconductor 11 . Theoretically, it has been proposed that nematic fluctuations can induce attractive pairing interaction that may enhance or even lead to superconductivity [12][13][14][15][16][17][18] . However, direct experimental evidence for the interplay between superconductivity and nematic fluctuations in iron-based superconductors is scarce.
To shed light on this issue, here we study how the superconducting transition temperature T c changes with the in-plane uniaxial pressure p. It has already been shown that the values of dT c /dp for p applied within the ab plane is very different compared to p applied along the c axis [19][20][21][22][23] . This type of anisotropic behavior is expected as observed in many other superconductors with layered structures [24][25][26][27] . Since the nematic order breaks the inplane rotational symmetry, it is possible to observe inplane anisotropic superconducting properties if the cou-pling between the nematic order and superconductivity is significant. Even in optimally and overdoped regimes where neither the antiferromagnetic (AF) order nor the nematic order exists, nematic fluctuations can still be strong, giving rise to strongly anisotropic responses in the presence of uniaxial strain p. This is indeed observed in the resistivity of the normal state 9 , and thus may also affect the uniaxial pressure dependence of T c . One of the difficulties to single out the nematic contribution to the observed behavior of T c (p) arises from the fact that p induces not only the shear lattice distortion that couples to nematicity, but also lattice distortions associated with other symmetries that do not couple linearly to the nematic order parameter 28 . To disentangle these contributions, our strategy here is to compare T c (p) for pressures applied along the Fe-Fe direction, and along the Fe-As-Fe or Fe-Se-Fe direction. This is because the former is along the nematic direction and should exhibit more significant effects than the latter.
Following this idea, we systematically studied the uniaxial pressure dependence of T c in many iron-based superconductors for p applied within the ab plane. It is found that T c can be described as a fourth-order polynomial function of p. While the linear term is essentially unaffected by the direction of p, the second-order nonlinear term varies significantly along different directions and samples. Comparing the results with a phenomenological Landau model that includes the biquadratic coupling between the nematic and superconducting order parameters, we conclude that this quadratic term is associated with nematic fluctuations. Since these effects can also be found in heavily overdoped samples, our results provide key insights for the impact of nematic fluctuations on superconductivity in iron-based superconductors. arXiv:1803.00717v2 [cond-mat.supr-con] 16 Dec 2019

II. EXPERIMENTS
Single crystals of BaFe 2−x Ni x As 2 (BFNA), Ba 0.67 K 0.33 Fe 2 As 2 (BKFA), KFe 2 As 2 (KFA) and LaFeAsO 0.74 F 0.26 (LFAOF) were grown by the self-flux methods as reported previously 29,30 . The samples were cut into thin rectangular plates by a diamond saw along the desired directions determined by an x-ray Laue diffractometer. The tetragonal notation is used hereafter, i.e., the (110) and (100) directions correspond to the Fe-Fe and Fe-As-Fe directions, respectively. The uniaxial pressure dependence of the resistance was measured by a home-made uniaxial pressure device based on the piezo-bender as described previously 9,10 . The piezobender of the uniaxial pressure device results in a slight hysteresis behavior between the processes of increasing and decreasing pressure due to its intrinsic properties, which is removed by averaging the pressures with the same resistance. The positive and negative values of pressure correspond to compressing and tensiling the samples, respectively.

III. RESULTS AND DISCUSSIONS
The way of obtaining the pressure dependence of T c in this work is to measure the resistance change under the uniaxial pressure at various temperatures and then convert the data to the temperature dependence of resistance to calculate the T c at each pressure. Figures 1(a) and 1(b) show the results of the optimally doped BaFe 1.9 Ni 0.1 As 2 for uniaxial pressure along the (100) and (110) directions, respectively. For p (100), the resistance R is nearly linear with p for the whole temperature range. Taking the values of R at the same p, the temperature dependence of R is shown in Fig. 1(c), which clearly demonstrates the change of T c under pressure. When the pressure is along the (110) direction, deviations from the linear behavior of R(p) is observed around the superconducting transition. Again, the converted temperature dependence of R is shown in Fig. 1(d). Accordingly, the uniaxial pressure dependence ∆T c can be derived as shown in Fig. 1(e), where ∆T c = T c (p) − T c (0), i.e., the relative change of T c to that at zero pressure. An almost linear relationship between the pressure and T c is observed for p (100). When the pressure is along the (110) direction, ∆T c shows clear nonlinear relationship with T c . Here T c is determined as where R becomes zero by the linear extrapolation of R(T ) during the transition. We have also tried to determine the value of T c by the onset and the middle temperature of the transition. The results are similar but with less certainty since the normal-state resistance along the (110) direction is affected by the uniaxial pressure 9 .
To quantitatively analyze the pressure dependence of T c , we fit ∆T c as Bp + ∆T nl c , where B is constant and ∆T nl c is the nonlinear component of T c (p). It is found that the following function is good enough to describe the data, where C and D are all constants. Figure 1(f) shows the pressure dependence of ∆T nl c , which clear shows the anisotropic behavior.
After having established the nonlinear pressure dependence of T c in optimally-doped BaFe 1.9 Ni 0.1 As 2 , we fur- The nonlinear pressure dependence of T c is also observed in other iron-based superconductors. Figure 3(a) shows the results of optimally doped Ba 0.67 K 0.33 Fe 2 As 2 , whose anisotropic behavior of ∆T nl c is similar to that in BaFe 2−x Ni x As 2 ; i.e., it is more significant along the (110) direction. For KFe 2 As 2 , the nonlinear behavior becomes weaker and isotropic, as shown in Fig. 3(c). The nonlinear behavior of T c in LaFeAsO 0.74 F 0.26 is similar to that in Ba 0.67 K 0.33 Fe 2 As 2 , as shown in Fig. 3(e), although the isotropic contribution is larger. Figure 4(a) shows the fitting parameter B in different samples. Since B corresponds to the linear dependence of T c on the uniaxial pressure p, it is similar to dT c /dp in previous reports 19 . In BaFe 2−x Ni x As 2 , B increases with increasing doping level, which is consistent with previous reports on Ba(Fe 1−x Co x ) 2 As 2 19 . Overall, the values of B in BaFe 2−x Ni x As 2 , KFe 2 As 2 and Ba 0.67 K 0.33 Fe 2 As 2 are very close for pressure applied along the (110) and (100) directions, suggesting a nearly isotropic response of superconductivity to p. In optimally doped Ba 0.67 K 0.33 Fe 2 As 2 , B also shows a slightly anisotropic behavior, probably because the exact doping levels of the samples are slightly different due to the inhomogeneity in the growing process and dT c /dp changes sign around optimal doping 22 . The values of the quadratic C coefficient for different compounds are plotted in Fig. 4(b). Except for KFe 2 As 2 , the values of C all show large anisotropic behavior, and are very similar for different samples when strain is applied along the (110) direction. Note that C becomes isotropic and rather small in KFe 2 As 2 .
Our results demonstrate that the response of superconductivity to the uniaxial pressure within the ab plane is composed of linear and nonlinear components. The latter can be described as an even function of the pressure, where the quadratic term dominates. Moreover, the quadratic coefficient along the (110) direction is usually larger than that along the (100) direction. The nonlinear behavior seems to be unique for iron-based superconductors since it is not found in a cuprate sample, as shown in the appendix. As the (110) direction is associated with the nematic order direction and the uniaxial pressure acts as an external field to the nematic order 9 , we propose that the nonlinear response of T c to the uniaxial pressure is due to the coupling between superconductivity and nematic fluctuations. Indeed, the change of resistivity under p along the (110) direction is usually much larger than that along the (100) direction 7,9 . This also explains why the quadratic coefficient C is small and nearly isotropic in the case of KFe 2 As 2 , since this compound seems to be far from a nematic instability and, as shown by the inset of Fig. 3(d), its resistance response to uniaxial pressure is nearly isotropic already in the normal state.
A phenomenological symmetry analysis sheds important light on the behaviors observed here 31,32 . Pressure along the (110) direction induces not only shear strain in the B 2g channel, ε B2g = ∂ x u y + ∂ y u x , where u is the displacement vector, but also strain in the other symmetry channels due to the finite Poisson ratio 28 -including the isotropic strain ε A1g = ∂ x u x + ∂ y u y . The latter couples to the square of the superconducting order parameter ∆ in a Landau free energy expansion, resulting in the linear dependence of T c on p. On the other hand, ε B2g acts as a conjugate field to the nematic order parameter ϕ, inducing a finite value ϕ ∝ ε B2g χ n that can be sizable if the nematic susceptibility χ n is large -as expected near a (quantum) nematic phase transition. Now, because ∆ and ϕ have a biquadratic coupling in the Landau free energy, T c acquires a quadratic dependence on p. The fact that the quadratic coefficient C in Eq. (1) is negative implies that this biquadratic coefficient is positive, i.e., nematicity and superconductivity compete with each other. The additional quartic coefficient D in Eq. (1) is likely a consequence of the relatively large pressures applied experimentally.
On the other hand, pressure along the (100) direction induces both ε A1g = ∂ x u x +∂ y u y and ε B1g = ∂ x u x −∂ y u y . The fact that the linear coefficient B in T c (p) is essentially the same for both uniaxial pressure directions implies that the induced ε A1g strain is nearly the same in both cases. In contrast, the very small quadratic coefficient C in the case of p (100) can be attributed to the absence of nematicity in the B 1g channel, i.e., the B 1g nematic order parameter induced by ε B1g is small. Our results suggest that the interplay between nematic fluctuations and superconductivity is ubiquitous in ironbased superconductors, which is consistent with previous results that nematic fluctuations are present above T c in various systems [6][7][8][9][10] . Thus, elucidating the superconducting state in these systems likely requires understanding the effects of nematic fluctuations. Within the framework of the above analysis, the negative values of C in Fig. 4 reveal the competition between the nematic and superconducting order parameters. It is surprising that C changes little with doping in BaFe 2−x Ni x As 2 and is still observable in KFe 2 As 2 , since one would expect the nematic fluctuations to be weak in these samples as they are far away from optimally doping levels. It should be pointed out that we have no reliable way to obtain the amplitude of nematic fluctuations in overdoped samples, but previous works indicate that it increases with increasing doping in the underdoped regime 10 . Whether the doping-dependence of C is compatible with the existence of nematic quantum critical fluctuations, which have been shown to exist in many systems 6-10 and thought to be important to superconductivity [12][13][14][15][16][17][18] , remains to be established. Importantly, the interplay between nematic fluctuations and superconductivity is by no means limited to the form we have discussed here. Future experiments and theories are desired to further elucidate these issues.

IV. CONCLUSIONS
In conclusion, our results provide direct evidence for the coupling between nematicity and superconductivity by revealing the in-plane anisotropic behavior of T c (p). A quadratic p dependence of T c is found and can be explained by the biquadratic coupling between the superconducting and nematic order parameters. The fact that it is ubiquitous in iron-based superconductors indicates the importance of nematic fluctuations for superconductivity.
Experimental work is supported by the National Key   We performed the same resistance measurements on single crystals of optimally doped copper-oxide hightemperature superconductors Bi 2−x Pb x Sr 2 CaCu 2 O 8+δ (Bi-2212). High-quality single crystals of Bi-2212 were grown by the traveling solvent floating zone technique. Note that, for Bi-2212, the orthorhombic notation is used in the experiments. The (100) and (110) directions are for Cu-O-Cu and diagonal directions, respectively. The electronic contacts were made by standard a two-part silver paste with heating up at 350 • for 2 h. The typical contact resistance is less than 5 Ω. Other sample preparation and measuring methods were similar as those in iron-based superconductors depicted in main text.
Here, since the flake of Bi-2212 single crystals are very fragile under uniaxial pressure, we measured them with quite small pressure range, i.e., less than about ±3 Mpa. As shown in Fig. 6(a) and 6(b), the uniaxial pressure dependence of resistance in either (100)  range from normal state to superconducting state, which is quite different from the nonlinear behaviors in other iron-based superconductors near optimal dopings. To clearly show the linear dependence on uniaxial pressure of resistance, we display ∆R at middle transition temperatures for each samples in Fig. 6(d). The statistics R 2 's of linear fitting for both data exceed 0.9999. The temperature dependence of resistance at 0 Mpa for both samples (Fig. 6(c)) are converted from data in Figs. 6(a) and 6(b). The sharp superconducting transitions demonstrate high homogeneity in our cuprate samples. Although the two samples measured along different directions were cut from same crystal rod, the T c 's are slightly different (∼2 K) which were mainly caused by different current densities during the measurements due to different sample cross sections.
We obtained the uniaxial pressure dependence of ∆T c and ∆T nl c in Figs. 7(a) and 7(b), respectively. Both ∆T c show perfect linear relationship with the uniaxial pressure and ∆T nl c is extremely small compared with results in other iron-based superconductors in the main text. Therefore, there is no nonlinear behavior of T c under uniaxial pressure in Bi-2212, at least for small pressure.