## Lattice Trailing the Electrons

Superconducting order recedes with decreased chemical doping in a typical iron-based superconductor family. Thus, the symmetry of the material is broken by almost simultaneous antiferromagnetic and structural phase transitions. However, some pnictides also exhibit an electronic nematic transition manifested by anisotropy of the electrical resistance. Because this anisotropy occurs at the same time as the structural transition, it is not clear whether it is a consequence of the broken crystal lattice symmetry or its cause. **Chu et al.** (p. 710) performed constant strain experiments on the series Ba(Fe

_{1−x}Co

_{x})

_{2}As

_{2}, which can distinguish between the two scenarios, and confirm that the electrons drive the lattice transition.

## Abstract

Within the Landau paradigm of continuous phase transitions, ordered states of matter are characterized by a broken symmetry. Although the broken symmetry is usually evident, determining the driving force behind the phase transition can be complicated by coupling between distinct order parameters. We show how measurement of the divergent nematic susceptibility of the iron pnictide superconductor Ba(Fe_{1−}* _{x}*Co

*)*

_{x}_{2}As

_{2}distinguishes an electronic nematic phase transition from a simple ferroelastic distortion. These measurements also indicate an electronic nematic quantum phase transition near the composition with optimal superconducting transition temperature.

In an electronic nematic phase transition, the electronic system breaks a discrete rotational symmetry of the crystal lattice without altering the existing translational symmetry (*1*). Examples include half-filling quantum Hall states (*2*) and the field-induced metamagnetic state in Sr_{3}Ru_{2}O_{7} (*3*). In the latter case, the electronic ground state exhibits a strong twofold anisotropy, which is only weakly reflected in a subtle structural anisotropy (*4*) indicative of an electronically driven phase transition. Recently, both cuprates (*5*–*8*) and iron pnictides (*9*–*12*) have been proposed as candidates for an electronic nematic phase, in which nematic order might coexist with high temperature superconductivity. However, the argument for electronic nematicity in these compounds is not straightforward because the crystal lattices suffer a relatively large deviation from fourfold symmetry. We report measurements of the resistivity anisotropy of Ba(Fe_{1−}* _{x}*Co

*)*

_{x}_{2}As

_{2}induced by an in situ tunable strain, which reveals the intrinsic electronic nematic response under a constant lattice distortion. On the basis of a phenomenological Ginzburg-Landau model, we show that the structural phase transition in this representative iron pnictide superconductor is purely driven by an instability in the electronic part of the free energy.

We apply a tuneable in-plane uniaxial strain to single crystal samples of Ba(Fe_{1−}* _{x}*Co

*)*

_{x}_{2}As

_{2}to probe the nematic response. By gluing the sample on the side wall of a piezo stack (Fig. 1A), strains can be applied by the deformation of the piezo, which is controlled by an applied voltage (

*V*

_{P}) (

*13*,

*14*). The strain (i.e., the fractional change of length along the current direction, ϵ

_{P}= ∆

*L*/

*L*) was monitored via a strain gauge glued on the back side of the piezo stack. Both ϵ

_{P}and the fractional change of resistivity (η = ∆ρ/ρ

_{0}, where ρ

_{0}is the resistivity of the free-standing sample before gluing on the piezo stack) were measured at constant temperature while the applied voltage was swept (Fig. 1B). The voltage dependence of η and ϵ

_{P}shows hysteretic behavior because of the ferroelectric nature of the piezo materials, yet the two quantities exhibit a linear relationship without any hysteresis (Fig. 1B). The negative slope of η(ϵ

_{P}) indicates that the resistivity is higher along the shorter bonding direction, consistent with previous results (

*10*,

*15*,

*16*).

The amount of strain transmitted to the sample (ϵ_{S}) can be assessed by gluing another strain gauge on the top surface of the crystal (Fig. 1A, lower panel). The comparison of ϵ_{S} and ϵ_{P} for a Ba(Fe_{0.955}Co_{0.045})_{2}As_{2} sample is summarized in Fig. 1, C and D. For applied voltages |*V*_{p}| < 150 V, the strain is fully transmitted to the sample for temperatures below about 100 K for typical thickness crystals (less than 100 μm). For lower voltages, |*V*_{p}| < 50 V, the strain is fully transmitted to even higher temperatures (Fig. 1D). The maximum strain that can be applied (|ϵ| < 5 × 10^{−4}) is substantially less than the lattice distortion developed below the phase transition (10^{−2} ∼ 10^{−3}), and, as we show below, the system is always in the regime of linear response.

The induced fractional change of the resistivity η provides a direct measure of the electronic nematic order parameter. In general, the resistivity is determined by both the electronic structure and the scattering. In the case of iron pnictides, the proposed orbital ordering more likely results in an anisotropy of electronic structure, whereas the spin-nematic ordering leads to an anisotropy of electron scattering. Measurements of resistivity anisotropy ψ = (ρ_{b} − ρ_{a})/(ρ_{b} + ρ_{a}) pick up both effects and reveal the electronic nematic order (*17*–*19*). For a strained crystal in the tetragonal state, ρ_{b} and ρ_{a} refer to the resistivity in directions parallel and perpendicular to the applied compressive stress. It can be easily shown that η = ψ if the increase in ρ* _{b}* equals the decrease in ρ

_{a}and that the two quantities are directly proportional even if this is not the case. The same is also true for the derivatives of these quantities such that

*d*η/

*d*ϵ ∝

*d*ψ/

*d*ϵ (

*13*).

Representative data showing the electronic nematicity (η) as a function of strain (ϵ_{P}) for a BaFe_{2}As_{2} sample are shown in Fig. 2A at various temperatures above the structural transition temperature *T*_{s}. Data were fit by a straight line in a small range of strain near zero applied voltage. As shown in Fig. 2B, the quantity *d*η/*d*ϵ, which essentially measures the nematic response induced by a constant strain, diverges upon approaching *T*_{s} from above. This divergent behavior is reminiscent of the resistivity anisotropy observed above *T*_{s} for samples held in a mechanical clamp (*10*). However, as we explain below, there is an important distinction between measurements made under condition of constant stress (mechanical clamp) and constant strain (measurement of *d*η/*d*ϵ in the current set up).

From the thermodynamic point of view, the stress and strain are conjugate variables, and the stress (here denoted as *h*) is the externally controllable force, whereas strain is the response of a mechanical system. Intuitively it might be more reasonable to regard stress as a symmetry breaking field. However, from the electron nematic stand point, stress only couples indirectly to the nematic order parameter through strain. This relationship can be best understood from the following Ginzburg-Landau free energy:(1)Here, ψ represents the electronic nematic order parameter, measured by the resistivity as discussed above, ϵ is the elastic strain, and *h* is its conjugate stress. *a*, *b*, *c*, and *d* are the coefficients of the two order parameters in the usual power series expansion, and λ is the coupling constant. If there is a phase transition driven by the electronic degree of freedom, then the coefficient *a* becomes zero at some temperature, that is, *a* = *a*_{0}(*T* − *T**), whereas the other coefficients are temperature independent. On the other hand, if the phase transition is caused by a structural instability, then it is the coefficient *c* that becomes zero [*c* = *c*_{0}(*T* − *T**)] (*20*). Therefore, the driving force can be distinguished by determining the temperature dependence of the bare *a* and *c* coefficients.

With this in mind, we can now ask what the difference is between measuring the response of electronic nematicity ψ under constant strain ϵ rather than constant stress *h*. This can be answered explicitly by calculating the quantities *d*ψ/*dh* and *d*ψ/*d*ϵ under the constraint of minimizing the free energy (*13*):
(2)
(3)From these expressions, the nematic response under a constant stress (Eq. 2) will show a 1/*T* divergence whether the driving force is a structural or electronic phase transition. However, the nematic response under a constant strain will only diverge when it is a true electronic nematic phase transition (Eq. 3). In this sense, the divergence in *d*η/*d*ϵ shown in Fig. 2 is direct evidence that BaFe_{2}As_{2} suffers a true electronic nematic instability, and the structural transition merely passively follows the nematic order. Because strain is a field to the nematic order parameter, we refer to the quantity *d*ψ/*d*ϵ as the nematic susceptibility.

From Eq. 3, *d*ψ/*d*ϵ = λ/*a* = λ/[*a*_{0}(*T* − *T**)], it is natural to fit the data of *d*η/*d*ϵ in Fig. 2B with a Curie-Weiss temperature dependence. However, Eq. 3 is only valid in the limit of vanishing strain, at which one can disregard the higher-order nonlinear terms. In the realistic experiment situation, there is always some intrinsic built-in strain, even at zero applied voltage, because of the different thermal contraction of the sample and the piezo stack. To take into account this built-in strain, we perform a numerical fit based on the following expression:
(4)The effect of the next-order nonlinear term is included in the 3*b*η_{0}^{2} in the denominator, where η_{0} is the resistivity anisotropy induced by the built-in strain as a function of temperature, measured by the difference of resistivity of a sample before and after gluing on the piezo stack. In addition to *a*_{0} and *b* introduced before, χ_{0} is a fitting parameter to model the intrinsic piezoresistivity effect of the materials that is unrelated to the electron nematic phase transition.

The result of this fitting is plotted in Fig. 2B as a solid red curve, which is in excellent agreement with measured data *d*η/*d*ϵ. The mean field critical temperature *T** obtained from the fitting is 116 K, 22 K lower than the actual phase transition temperature (*T*_{s} = 138 K). This can also be understood from the Ginzburg-Landau free energy in Eq. 1. By minimizing the free energy, we can derive the result that the nonzero nematic and structural order parameters onset simultaneously at a temperature *T*_{S} = *T** + λ^{2}/(*a*_{0}*c*), higher than *T** (*13*). This is a consequence of the bilinear coupling between the electronic nematic system and the crystal lattice, which lifts the critical temperature of the electronic instability to a higher temperature. Physically, the lattice provides a polarizable medium, which enhances the nematic instability.

The divergence of *d*η/*d*ϵ not only reveals the tendency toward an electronic nematic phase transition but also measures the strength of nematic fluctuations, according to the fluctuation-dissipation theorem. We have measured *d*η/*d*ϵ of Ba(Fe_{1−}* _{x}*Co

*)*

_{x}_{2}As

_{2}samples for doping concentration ranging from the undoped parent compound to overdoped compositions (Fig. 3). The magnitude of

*d*η/

*d*ϵ is plotted as a color map in the composition versus temperature phase diagram in Fig. 4. For the underdoped part of the phase diagram,

*d*η/

*d*ϵ increases rapidly near the structural phase transition boundary. As the doping concentration increases, the intensity of fluctuations increases and reaches a maximum near optimal doping concentration, where structural and magnetic transitions are fully suppressed. The nematic fluctuations persist to the overdoped regime, eventually decreasing as the superconducting

*T*

_{c}decreases. The associated softening of the sheer modulus has been extensively studied by resonant ultrasound measurements (

*11*,

*21*).

To quantitatively track the evolution of nematic fluctuations across the phase diagram, we performed numerical fits to the data for each composition based on Eq. 4. The obtained *T** is also plotted as a function of composition in Fig. 4. The mean field nematic critical temperature *T** closely tracks the actual structural transition temperature *T*_{s} in the underdoped regime and is suppressed to zero at the optimal doping. *T** becomes negative as the doping further increases beyond optimal doping, indicating a “paranematic” state. Our experimental data and analysis indicate an electronic nematic quantum phase transition for a composition close to optimal doping (*22*). It remains to be seen whether fluctuations associated with this quantum phase transition play an important role in enhancing *T*_{c} in the superconducting phase. Nevertheless, the existence of nematic fluctuations across such a wide temperature and doping range suggests that they are a fundamental ingredient to describe the normal state of the system (*11*).

## Supplementary Materials

www.sciencemag.org/cgi/content/full/337/6095/710/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S5

## References and Notes

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**Acknowledgments:**The authors thank A. Cano, C.-C. Chen, P. Coleman, R. M. Fernandes, S. A. Kivelson, A. Mackenzie, I. Paul, and Q. Si for helpful discussions. This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract no. DE-AC02-76SF00515.