## Discerning the nematic connection

The phase diagram of any given family of iron-based superconductors is complicated: Superconductivity competes with antiferromagnetism, with a structural transition often thrown in for good measure. Transport experiments have shown that in one of these families, Ba(Fe_{1-x}Co_{x})2As_{2}, a rotational electronic asymmetry, dubbed nematicity, drives the structural transition. Kuo *et al.* detected nematic fluctuations in five Fe-based superconductor families in the vicinity of optimal chemical doping: the doping that maximizes the superconducting transition temperature. Thus, nematicity may play a role in the mechanism of superconductivity in these compounds.

*Science*, this issue p. 958

## Abstract

A key actor in the conventional theory of superconductivity is the induced interaction between electrons mediated by the exchange of virtual collective fluctuations (phonons in the case of conventional *s*-wave superconductors). Other collective modes that can play the same role, especially spin fluctuations, have been widely discussed in the context of high-temperature and heavy Fermion superconductors. The strength of such collective fluctuations is measured by the associated susceptibility. Here we use differential elastoresistance measurements from five optimally doped iron-based superconductors to show that divergent nematic susceptibility appears to be a generic feature in the optimal doping regime of these materials. This observation motivates consideration of the effects of nematic fluctuations on the superconducting pairing interaction in this family of compounds and possibly beyond.

A growing body of evidence suggests the possibility of an intimate connection between electronic nematic phases (*1*) and high-temperature superconductivity. However, it is currently unclear to what extent there is any causal relationship between nematic fluctuations and superconductivity. Strongly anisotropic electronic phases have been found in the underdoped regime of both cuprate (*2*–*6*) and Fe-based (*7*–*12*) high-temperature superconductors. In the case of underdoped Fe-based systems, recent measurements of the elastoresistance (*13*–*15*), Raman spectra (*16*–*18*), and elastic moduli (*19*, *20*) of the representative electron-doped system Ba(Fe_{1–}* _{x}*Co

*)*

_{x}_{2}As

_{2}reveal a divergence of the electronic nematic susceptibility upon approaching the tetragonal-to-orthorhombic structural phase transition; this definitively establishes that the phase transition is driven by electronic correlation. In the case of the cuprates, recent x-ray diffraction (

*21*–

*24*) and nuclear magnetic resonance (

*25*) measurements have shown evidence for short-range charge density wave order in underdoped crystals. Although details of the charge-ordered states are still being established, these initial observations have at least motivated discussion of a possible “vestigial” nematic order (

*26*). Perhaps importantly, in the phase diagrams of both families of compounds, optimal doping is located close to putative quantum critical points (

*27*–

*29*) that potentially have a nematic character (

*13*,

*26*,

*30*).

From a theoretical perspective, recent treatments indicate that nematic quantum criticality (i.e., quantum critical fluctuations caused by proximity to a nematic quantum critical point) can enhance an existing pairing interaction. In particular, a pure nematic phase does not break the translational symmetry of the original crystal lattice; consequently, the nematic fluctuations have a wave-vector *q* = 0 so that pairing through the exchange of nematic fluctuations enhances the critical temperature *T*_{c} in all symmetry channels (*31*–*33*). It is therefore of considerable interest to empirically establish whether nematic fluctuations are a characteristic feature of optimally doped high-temperature superconductors, as well as to determine the extent to which these nematic fluctuations show intrinsically quantum behavior. Here we show that this is the case for iron pnictide and chalcogenide superconductors by considering the representative 122-type pnictide materials Ba(Fe_{1–}* _{x}*Co

*)*

_{x}_{2}As

_{2}and Ba(Fe

_{1–}

*Ni*

_{x}*)*

_{x}_{2}As

_{2}(electron-doped), Ba

_{1-x}K

_{x}Fe

_{2}As

_{2}(hole-doped), and BaFe

_{2}(As

_{1–}

*P*

_{x}*)*

_{x}_{2}(isovalent substitution) and the 11-type iron chalcogenide FeTe

_{1–}

*Se*

_{x}*. Furthermore, we find that the nematic susceptibility obeys a simple Curie-Weiss power law for all five optimally doped Fe-based superconductors over a wide temperature range. For the electron- and hole-doped 122-type pnictides, a sub–Curie-Weiss deviation was observed at low temperatures, which we tentatively attribute to an enhanced sensitivity to disorder in a quantum critical regime.*

_{x}Nematic order couples linearly to anisotropic strain of the same symmetry. Consequently, the nematic susceptibility of a material can be measured by considering the electronic anisotropy that is induced by anisotropic in-plane strain. In the regime of infinitesimal strains, all forms of electronic anisotropy are linearly proportional. Hence, the rate of change of resistivity anisotropy with respect to anisotropic strain, defined in the limit of vanishing strain, is linearly proportional to the nematic susceptibility (*34*). The proportionality constant depends on microscopic physics, but away from any quantum critical point, it does not exhibit any singular behavior. Thus, the induced resistivity anisotropy reveals the essential divergence of the nematic susceptibility upon approaching a thermally driven nematic phase transition (*14*, *15*). For a tetragonal material, nematic order has either d* _{xy}* symmetry (the B

_{2}

*irreducible representation of the D*

_{g}_{4}

*point group, corresponding to nematic order oriented along a nearest-neighbor Fe-Fe bond, i.e., along the [110] or [–110] crystal axes) or symmetry (B*

_{h}_{1}

*, corresponding to the [100] or [010] crystal axes). Anisotropic strains ε with appropriate symmetry are then ε*

_{g}_{6}= (ε

*+ ε*

_{xy }*)/2 and ε*

_{yx}_{1 }– ε

_{2}= ε

_{xx}_{}– ε

*, respectively. The strain-induced changes in resistivity ρ can be described using the dimensionless elastoresistivity tensor,*

_{yy}*m*(

_{ij}*14*) (1)where the indices are defined using standard Voigt notation (1 =

*xx*, 2 =

*yy*, and so forth). Consequently, the corresponding components of the nematic susceptibility tensor χ

_{N }are given by (2) (3)where

*c*and

*c*' are proportionality constants that depend on microscopic physics (

*34*).

We measured elastoresistivity by applying an in situ tunable anisotropic strain, using a piezoelectric lead-zirconate-titanate (PZT) stack. In this approach, a square plate sample (typical dimensions, 750 × 750 × 20 μm) is glued on the side wall of the PZT stack with a commercial two-part epoxy. The PZT stack deforms when a voltage is applied and hence strains the sample glued on top of it (*35*). The amount of strain can be measured by a strain gauge glued either on the backside of the PZT stack or on the top surface of a larger sample [the latter arrangement enables a full determination of the strain transmission (*36*, *37*)]. The in-plane resistivity tensor of the sample is measured via the Montgomery technique, with electrical contacts made at the four corners of the square sample (*36*). Representative data collected using this new technique are shown in Fig. 1 for the specific case of BaFe_{2}As_{2}. As has been previously demonstrated (*14*, *15*), the data can be fit very well by a Curie-Weiss temperature dependence
(4)

where λ/*a* is the Curie constant, and *T** is the Weiss temperature.

In Fig. 2, we show B_{2}* _{g}* elastoresistance data for a range of optimally doped materials, including BaFe

_{2}(As

_{1–}

*P*

_{x}*)*

_{x}_{2}(isovalently substituted), Ba(Fe

_{1–}

*Ni*

_{x}*)*

_{x}_{2}As

_{2}(electron-doped), Ba

_{1–}

*K*

_{x}*Fe*

_{x}_{2}As

_{2}(hole-doped), and FeTe

_{1–}

*Se*

_{x}*. In all cases, 2*

_{x}*m*

_{66}rises pronouncedly with decreasing temperature, with comparably large values for each compound. The observation of this effect across such a wide variety of doping methods, including in the case of the iron chalcogenide FeTe

_{0.6}Se

_{0.4}, suggests strongly that a divergence of the nematic susceptibility in the B

_{2}

*channel is a generic feature of optimally doped Fe-based superconductors.*

_{g}Regardless of microscopic models, from a purely empirical perspective, it is apparent that the optimally doped superconductor is born out of an electronic state that is characterized by strongly fluctuating orientational (nematic) order in this specific symmetry channel. This result is especially notable for FeTe_{0.6}Se_{0.4}, given that the orientation of both the magnetic ordering wave vector and the in-plane component of the structural distortion of undoped FeTe are 45° away from the orientation of those in the iron arsenide materials.

The *m*_{66} coefficient of optimally doped BaFe_{2}(As_{0.68}P_{0.32})_{2} (Fig. 2A) follows a perfect Curie-Weiss temperature dependence from *T* = 250 K (the highest temperature at which strain can be effectively transmitted by the epoxy) down to *T*_{c} (below which the resistance is zero), with a *T** close to zero. For FeTe_{0.6}Se_{0.4}, the *m*_{66} coefficient can be perfectly fitted with the Curie-Weiss dependence down to *T*_{c} but substantially deviates at temperatures greater than 100 K, possibly related to the loss of quasiparticle coherence as a result of its extremely small Fermi energy, as has been observed in photoemission spectroscopy and transport (*38*, *39*). For the electron- and hole-doped 122-type pnictides, a downward deviation from Curie-Weiss behavior is apparent below a characteristic temperature that is different for each of the materials. The quality of fit to the Curie-Weiss functional form for BaFe_{2}(As_{0.68}P_{0.32})_{2}, in comparison with all of the other optimally doped compositions that we studied, can be readily appreciated when the data are plotted on log axes (fig. S11). Only the data for the P-substituted system can be fit by a single power law over the entire temperature range, yielding the power law exponent γ = 0.985 ± 0.005 and *T** = 11.7 ± 3.1 K.

We also measured a slightly overdoped BaFe_{2}(As_{0.64}P_{0.36})_{2} sample; the elastoresistivity coefficient *m*_{66} for this case can also be well fitted by a Curie-Weiss temperature dependence over the entire temperature range, with a negative *T** = –11.5 ± 2.3 K (fig. S14). The small value of the Weiss temperature observed in BaFe_{2}(As_{1-x}P_{x})_{2} near optimal doping, and the fact that it crosses zero as doping increases, motivates consideration of the nematic susceptibility in the context of a quantum critical point. An Ising nematic phase transition in an insulator would generally be expected to have a dynamical exponent *z* = 1, so the effective dimensionality of the system would be *d* + *z* = 3 + 1. Because the materials in question are metallic, the situation is more complicated (*40*); the Hertz-Millis paradigm is expected to break down in the case of electronic nematic order in metallic systems, for which Landau damping associated with the gapless Fermions leads to a larger value of the dynamical exponent, *z* = 3. Although the analysis of this problem is not straightforward, both analytic (*41*) and numerical studies (*42*, *43*) indicate that the nematic susceptibility at criticality should diverge in proportion to 1/*T* (up to possible logarithmic corrections), consistent with the observed behavior of the B_{2}* _{g}* elastoresistance of optimally doped BaFe

_{2}(As

_{1–}

*P*

_{x}*)*

_{x}_{2}.

To gain insight into the physical origin of the low-temperature downward deviation from Curie-Weiss behavior in the electron- and hole-doped pnictides, we consider the evolution of the B_{2}* _{g}* elastoresistance as a function of composition for the specific case of Co-doped BaFe

_{2}As

_{2}.

The progression of 2*m*_{66} as a function of Co doping for several representative compositions [more were measured (*36*)], from underdoped through optimal doping to the overdoped regime, is shown in Fig. 3, A to D. For very underdoped compositions (*x* = 0.025), 2*m*_{66} can be well described by the mean-field Curie-Weiss *T* dependence down to the critical temperature for the tetragonal-to-orthorhombic structural phase transition, *T*_{s}. It has been previously established that this behavior is essentially independent of disorder, at least when comparing undoped BaFe_{2}As_{2} materials with different residual resistance ratios and Co- and Ni-substituted crystals with the same Neel temperature *T*_{N} (*15*). The *T** extracted from high-temperature fits to Curie-Weiss behavior crosses zero close to optimal doping (Fig. 4). However, for slightly underdoped Ba(Fe_{0.553}Co_{0.047})_{2}As_{2}, a downward deviation from mean-field behavior at low temperatures is noticeable, and this pattern is more pronounced for optimally doped Ba(Fe_{0.93}Co_{0.07})_{2}As_{2}; similar deviations are apparent for optimally Ni- and K-doped BaFe_{2}As_{2}. A similar effect has been observed in recent measurements of the sheer modulus through three-point bending experiments (*20*). The deviation from Curie-Weiss behavior diminishes again as the doping is further increased: For the overdoped composition (Fig. 3D), the data can be fit to the Curie-Weiss functional form down to a lower temperature than for optimal doping, and the magnitude of the deviation below ~45 K is smaller than for optimal doping. The 2*m*_{66} of four additional Co dopings has also been measured, showing a similar nonmonotonic doping dependence of the deviation from Curie-Weiss behavior. The fact that the effect is maximal near optimal doping when *T** ~ 0 suggests that it is associated with proximity to the putative nematic quantum phase transition.

P-substituted BaFe_{2}As_{2} is the least disordered of all of the known 122-type iron pnictide families, as evidenced by the fact that quantum oscillations can be observed across the phase diagram (*44**, **45*). The deviation from Curie-Weiss behavior for optimally doped compositions of all other dopants in BaFe_{2}As_{2} suggests that disorder plays an important role in the quantum critical regime. All forms of quenched disorder produce locally anisotropic effective strains, which thus couple to the orientation of the nematic order; this is known as random-field disorder (*26*). Analysis of the random-field Ising model yields several generically expected effects of random-field disorder, including suppression of the nematic susceptibility below mean-field expectations for a clean system, and, for the case of a quantum phase transition, the enhanced sensitivity of quantum critical phenomena to disorder (*36*).

The temperature dependence of the nematic susceptibility can also be extracted from measurements of the elastic moduli (*20*). The two measurements (elastoresistance and elastic moduli) are in broad agreement—for example, in terms of the Curie-Weiss *T* dependence of χ_{N} for underdoped compositions of Ba(Fe_{1–}* _{x}*Co

*)*

_{x}_{2}As

_{2}, and in terms of the deviation from Curie-Weiss behavior near optimal doping. However, there is an important distinction in the relative magnitude of the measured quantities as a function of doping. In particular, the normalized lattice softening extracted in (

*20*) monotonically decreases in magnitude and extends over a smaller window of temperature as the Co concentration

*x*is increased. In contrast, the quantity initially increases with

*x*, peaking for slightly underdoped compositions where

*x*~ 0.05. The apparent enhancement of the elastoresistance 2

*m*

_{66}over the softening of the elastic modulus for compositions near optimal doping is potentially related to renormalization of the quasiparticle effective mass in the quantum critical regime, as has been observed in P-substituted BaFe

_{2}As

_{2}(

*29*). Such a mass renormalization is an expected consequence of nematic quantum critical fluctuations (

*40*). It is precisely these low-energy quasiparticles, which are responsible for the large elastoresistivity, that are also involved in the eventual superconductivity.

## Supplementary Materials

www.sciencemag.org/content/352/6288/958/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S16

Table S1

## References and Notes

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- ↵There are several different ways to define a nematic order parameter in terms of equilibrium electronic correlation functions. Correspondingly, there is ambiguity in defining the nematic susceptibility. Thus, the constants
*c*and*c*' depend not only on microscopic details but also on the precise microscopic definition given to the nematic order parameter. - ↵Typical strains using this method are on the order of 10
^{−3}to 10^{−4}. - ↵Materials and methods are available as supplementary materials on
*Science*Online. - ↵As described in (
*36*), samples that have in-plane dimensions that are larger than ~700 × 700 μm do not exhibit a size dependence of the strain transmission for typical thicknesses. Although most of the samples used in this study are in this size range, the P-substituted iron arsenide samples have in-plane dimensions of ~300 × 300 μm, resulting in a smaller strain transmission. This does not affect the measured Weiss temperature (*36*). - ↵
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- ↵Preliminary confirmation that the nematic susceptibility at criticality diverges as 1/T has been obtained from determinantal Monte Carlo calculations of a two-dimensional metallic nematic quantum critical system (
*43*). - ↵
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- Part no. PSt 150/557, from Piezomechanik, Munich, Germany.
- Part no. WK-XX-062TT-350, General Purpose Strain Gages - Tee Rosette, from Vishay Precision Group.
- Part no. 14250, General Purpose Adhesive Epoxy, from Devcon, USA.
- ↵

**Acknowledgments:**The authors thank S. Raghu, S. Lederer, and E. M. Spanton for helpful discussions. J.C.P. is supported by a Gabilan Stanford Graduate Fellowship and a NSF Graduate Research Fellowship (grant DGE-114747). This work was supported by the Department of Energy, Office of Basic Energy Sciences, under contract no. DE-AC02-76SF00515.