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Ubiquitous signatures of nematic quantum criticality in optimally doped Fe-based superconductors

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Science  20 May 2016:
Vol. 352, Issue 6288, pp. 958-962
DOI: 10.1126/science.aab0103

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(Fe1-xCox)2As2, 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 (26) and Fe-based (712) high-temperature superconductors. In the case of underdoped Fe-based systems, recent measurements of the elastoresistance (1315), Raman spectra (1618), and elastic moduli (19, 20) of the representative electron-doped system Ba(Fe1–xCox)2As2 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 (2124) 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 (2729) 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 Tc in all symmetry channels (3133). 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(Fe1–xCox)2As2 and Ba(Fe1–xNix)2As2 (electron-doped), Ba1-xKxFe2As2 (hole-doped), and BaFe2(As1–xPx)2 (isovalent substitution) and the 11-type iron chalcogenide FeTe1–xSex. 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.

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 dxy symmetry (the B2g irreducible representation of the D4h point group, corresponding to nematic order oriented along a nearest-neighbor Fe-Fe bond, i.e., along the [110] or [–110] crystal axes) or Embedded Image symmetry (B1g, corresponding to the [100] or [010] crystal axes). Anisotropic strains ε with appropriate symmetry are then ε6 = (εxy + εyx)/2 and ε1 – ε2 = εxx– εyy, respectively. The strain-induced changes in resistivity ρ can be described using the dimensionless elastoresistivity tensor, mij (14)Embedded Image (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 χN are given byEmbedded Image (2)Embedded Image (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 BaFe2As2. As has been previously demonstrated (14, 15), the data can be fit very well by a Curie-Weiss temperature dependence Embedded Image(4)

Fig. 1 Temperature dependence of the B2g elastoresistance 2m66 of BaFe2As2.

The data exhibit Curie-Weiss behavior, which is the anticipated mean-field temperature dependence of the nematic susceptibility of a material approaching a thermally driven nematic phase transition (1315). The upper panel shows –2m66, which is proportional to the nematic susceptibility Embedded Image, in the tetragonal phase. The black line shows the Curie-Weiss fit. The quality of fit can be better appreciated by considering the inverse susceptibility Embedded Image, which is perfectly linear (left axis of lower panel, as indicated by the blue arrow; the fit is shown by the red line), and the Curie constant Embedded Image (right axis of lower panel, as indicated by the black arrow), which is independent of temperature. The Weiss temperature T* obtained from the Curie-Weiss fit, which gives the bare mean-field nematic critical temperature, is 109 ± 1.4 K. Coupling to the lattice renormalizes the critical temperature, leading to a coupled nematic-structural phase transition at Ts = 134 K (vertical gray line). The inset shows a photograph of a square crystal glued on a PZT piezoelectric stack for differential elastoresistance measurements, using Montgomery’s geometry. Four electrical contacts were made at the corners, and a strain gauge was glued on the top surface (36).

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

In Fig. 2, we show B2g elastoresistance data for a range of optimally doped materials, including BaFe2(As1–xPx)2 (isovalently substituted), Ba(Fe1–xNix)2As2 (electron-doped), Ba1–xKxFe2As2 (hole-doped), and FeTe1–xSex. In all cases, 2m66 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 FeTe0.6Se0.4, suggests strongly that a divergence of the nematic susceptibility in the B2g channel is a generic feature of optimally doped Fe-based superconductors.

Fig. 2 Divergence of the B2g elastoresistance 2m66 of several different families of optimally doped iron pnictide and chalcogenide superconductors.

(A) Optimally doped BaFe2(As0.68P0.32)2 (isovalent substitution), (B) optimally doped Ba(Fe0.955Ni0.045)2As2 (electron-doped), (C) optimally doped Ba0.58K0.42Fe2As2 (hole-doped), and (D) optimally doped FeTe0.58Se0.42. Insets indicate the dopant site (red) in the respective unit cells of each material (Fe and Ni, yellow; As and P, purple; Ba and K; Te and Se, pink). Induced resistivity anisotropies of comparable magnitudes are observed for each case (36). Upper panels show |2m66|; lower panels show Embedded Image (left axes, blue squares) and Embedded Image (right axes, black curves). Black (upper panels) and red (lower panels) lines show Curie-Weiss fits of 2m66 and Embedded Image, respectively. Gray horizontal lines (lower panels) show the average values of Embedded Image in the fitting temperature range. Regions of deviation from Curie-Weiss behavior in (B) and (C) are indicated by gray shaded regions. For (A) and (B), 2m66 is negative. For (C) and (D), 2m66 is positive. Fit parameters, together with the temperature range over which a good fit is obtained, are listed in (36). The fits shown in the upper and lower panels are extrapolated beyond that range to emphasize the deviations from Curie-Weiss behavior.

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 FeTe0.6Se0.4, 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 m66 coefficient of optimally doped BaFe2(As0.68P0.32)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 Tc (below which the resistance is zero), with a T* close to zero. For FeTe0.6Se0.4, the m66 coefficient can be perfectly fitted with the Curie-Weiss dependence down to Tc 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 BaFe2(As0.68P0.32)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 Embedded Image 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 BaFe2(As0.64P0.36)2 sample; the elastoresistivity coefficient m66 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 BaFe2(As1-xPx)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 B2g elastoresistance of optimally doped BaFe2(As1–xPx)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 B2g elastoresistance as a function of composition for the specific case of Co-doped BaFe2As2.

The progression of 2m66 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), 2m66 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, Ts. It has been previously established that this behavior is essentially independent of disorder, at least when comparing undoped BaFe2As2 materials with different residual resistance ratios and Co- and Ni-substituted crystals with the same Neel temperature TN (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(Fe0.553Co0.047)2As2, a downward deviation from mean-field behavior at low temperatures is noticeable, and this pattern is more pronounced for optimally doped Ba(Fe0.93Co0.07)2As2; similar deviations are apparent for optimally Ni- and K-doped BaFe2As2. 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 2m66 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.

Fig. 3 Variation of the B2g elastoresistance of Ba(Fe1–xCox)2As2 for four representative compositions.

(A and B) Underdoped compositions Ba(Fe0.975Co0.025)2As2 and Ba(Fe0.953Co0.047)2As2; (C) optimally doped Ba(Fe0.93Co0.07)2As2; and (D) overdoped Ba(Fe0.92Co0.08)2As2. The very underdoped composition (A) is well described by a Curie-Weiss temperature dependence over the entire temperature range (black lines in upper panels), with only subtle deviations at the highest temperatures that are sensitive to the value of m660 in the fit. For the compositions near optimal doping, 2m66 can be well fit by a Curie-Weiss T dependence at intermediate temperatures. At temperatures below a characteristic value (different for each composition; indicated by the gray shaded regions), a strong downward deviation from Curie-Weiss behavior is observed; this deviation is also evident in the inverse susceptibility [upward deviation of the data (blue symbols) from an inverse Curie-Weiss fit (red line)] and in Embedded Image (strong downturn, black curve), which are shown in the lower panels. Gray vertical lines indicate Ts; gray horizontal lines indicate average values of Embedded Image. The deviation from Curie-Weiss behavior is the strongest at optimal doping and diminishes on either side of the phase diagram. Fit parameters, together with the temperature range over which a good fit is obtained, are given in (36). As in Fig. 2, the fits shown in the upper and lower panels are extrapolated beyond that range to emphasize the deviations from Curie-Weiss behavior..

Fig. 4 Phase diagram of Ba(Fe1–xCox)2As2, showing the variation of 2m66 in the x-T plane (color scale).

Squares, circles, and triangles indicate Ts, TN, and Tc respectively. Stars indicate the bare mean-field nematic critical temperatures (T*) extracted from the Curie-Weiss fits of 2m66 above the temperature at which disorder effects suppress the nematic susceptibility (light gray stars are used for cases in which deviations from Curie-Weiss behavior are observed at low temperatures). As has been previously determined by longitudinal elastoresistance measurements (13), and as established in this study by the full B2g differential elastoresistance, the Weiss temperature T* goes through zero as a function of x close to optimal doping. The color scale shows the magnitude of –2m66, which peaks between slightly underdoped to optimally doped compositions.

P-substituted BaFe2As2 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 BaFe2As2 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 χN for underdoped compositions of Ba(Fe1–xCox)2As2, 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 Embedded Image 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 Embedded Image initially increases with x, peaking for slightly underdoped compositions where x ~ 0.05. The apparent enhancement of the elastoresistance 2m66 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 BaFe2As2 (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 (4663)

References and Notes

  1. 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.
  2. Typical strains using this method are on the order of 10−3 to 10−4.
  3. Materials and methods are available as supplementary materials on Science Online.
  4. 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).
  5. 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).
  6. Part no. PSt 150/557, from Piezomechanik, Munich, Germany.
  7. Part no. WK-XX-062TT-350, General Purpose Strain Gages - Tee Rosette, from Vishay Precision Group.
  8. 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.
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