Introduction

The temperature-doping phase diagram of almost all iron-based superconductors (Fig. 1 for the hole doped materials) is characterized by an antiferromagnetic dome centered at the parent compound.1,2 The onset of the magnetically ordered state below TN is coincident with or follows a tetragonal-to-orthorhombic structural phase transition at Ts(≥TN), that is driven by electronic nematic order. The latter breaks the tetragonal symmetry of the underlying lattice, without altering the translational symmetry.2,3 Superconductivity emerges in proximity of such a broken symmetry state, at the bottom of the magnetic dome in both electron- and hole-doped systems (Fig. 1).

Fig. 1
figure 1

Schematic view of the experiment and flavour of the major findings within the generic phase diagram of Ba1−xKxFe2As2: The optimally doped (x = 0.4 and Tc = 38.5 K) sample is illuminated with polarized light while external uniaxial compressive stress (p) is applied on the lateral face of the specimen. We adopt an experimental protocol consisting in the measurement of the optical reflectivity signal (R) during a p-loop after a zero-p-cooling (ZPC) procedure (see Methods as well as Supplementary Information).24,25 We define the b-axis as the direction along which compressive p is applied and the a-axis as its orthogonal direction. Light is then polarized parallel to either axis (in the convention of our experimental set-up the polarization 0 is parallel to the b-axis and 90 to the a-axis). Below Tc (inset a) we clearly observe an anisotropy in the detected R-signal between the two axes when p is progressively applied, so that the ratio Ra/Rb deviates from 1 (i.e., isotropic limit). The ratios Ra/Rb are read from Fig. 2c at 0, at saturation (i.e., thick dotted line at 1.2 bar) and at 0 released p. Ra/Rb disappears upon releasing p back to zero (‘(r)’ denotes released p). At Tc (inset b) the onset of the optical anisotropy is only incipient and above Tc (inset c) there is no signature of the optical anisotropy upon sweeping p. We note that for compositions between \(x \simeq 0.25\) and 0.3, Ba1−xKxFe2As2 suffers a reentrant C4 antiferromagnetic phase transition at temperatures below the C2 antiferromagnetic/nematic transition.32 This aspect as well as the still on-going debate on the symmetry of the superconducting order parameter1 are neglected in this schematic representation of the phase diagram

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The in-plane dc resistivity anisotropy of the underdoped iron-based materials4,5,6,7 was initially of paramount importance in shedding light on the origin of the structural phase transition within the concept of the electronic nematic order. In this context, the most important fingerprint is provided by the diverging nematic susceptibility in the tetragonal phase, as evinced by elastoresistance measurements (i.e., measurements of the induced resistivity anisotropy due to anisotropic strain).8,9 Raman10,11,12 and elastic moduli13 investigations also point out the remarkable divergence of the nematic susceptibility. In a broader perspective, electronic nematicity is of relevance well beyond the iron-based superconductors since several cuprates and some heavy-fermion compounds, just to quote a few examples of other unconventional superconductors, do provide signatures for strongly anisotropic electronic phases.14

It has been empirically found,15 that the divergent nematic susceptibility is a generic property of several iron-based superconductors even up to optimally doped compositions. This leads to the question of whether nematic quantum criticality could perhaps enhance the pairing interaction;16,17,18 an intriguing possibility that has also been envisaged for the cuprates.19,20 While the relationship between superconductivity and nematicity has been debated theoretically,14 there remains a need to better experimentally scrutinize the influence of nematic fluctuations on the electronic properties over a large energy interval and also over a temperature (T) range extending under the superconducting dome, which is not accessible by elastoresistive technique.

Complementary to the dc transport data mostly affected by the energy scales close to the Fermi level, optical spectroscopy proves to be a powerful experimental tool in order to reveal the fingerprints of the nematic phase on the whole electronic structure.21,22,23,24,25,26,27,28,29 In our experiment, we use uniaxial and in-situ tunable applied stress (p), which acts as a conjugate field to the orthorhombic distortion and allows circumventing sample twinning below Ts (see Supplementary Information). Above Ts, stress as external symmetry-breaking field induces a finite value of the orthorhombic distortion.4 It is worth recalling that the microscopic mechanisms that result in the electronic anisotropy in the nematic phase are equivalent to those in the stressed tetragonal phase.9,26,30 Consequently, measurements of the anisotropic optical response in stressed samples in the tetragonal phase (i.e., in the T-window where nematic fluctuations are involved) must directly connect to the electronic nematicity.4,31 So far, optical investigations upon applying p have focused attention on the underdoped regime of the electron-doped 122 materials Ba(Fe1−xCox)2As2 as well as on FeSe. The main motivation of this work is the study of the optimally doped regime (Fig. 1). To this purpose, we choose the hole-doped Ba1−xKxFe2As2, which displays a nematic state up to x ~ 0.3, when the antiferromagnetic phase boundary is reached.32 Beyond this doping the nematic order disappears (i.e., Ts = 0), so that the optimally doped x = 0.4 compound is an ideal composition in order to address the impact of the nematic fluctuations, imaged in our experiment by the p-induced anisotropy of the electrodynamic response.

Results and discussion

First of all, we offer a comprehensive view of the collected optical reflectivity (R(ω)) data within the p-loop experiment at 10 K after a ZPC protocol (Fig. 1, and Methods section). The R(ω) spectra along the a- and b-axis, shown in Fig. 2a, b, are obviously metallic and approach total reflection at finite frequencies below νg ~ 180 cm−1, as expected at T < Tc. Further details about this experiment and additional data can be found in the Supplementary Information. While the discovery of the unprecedented p-induced anisotropy is at the center of our discussion, it is worthwhile to confirm the consistency of our data (averaged between the two axes) at all T with those in refs 33,34,35,36,37,38, which comprehensively address the optical signature of both normal and superconducting state in un-stressed Ba1−xKxFe2As2 samples (Fig. S1 in Supplementary Information). The reversible anisotropy upon sweeping p is already evident in the raw data and can be further emphasized by the calculation of the reflectivity ratio RRatio = Ra/Rb, shown in Figs. 1a and 2c. RRatio drops below 1 (i.e., the isotropic situation) in the FIR range around 600 cm−1 upon reaching an applied p of 1.2 bar and reconverts to unity when p is released back to zero (see also Fig. 1a). The anisotropy is imaged in the real part σ1(ω) of the optical conductivity as well, as shown in Fig. 2d,e. Particularly the so-called dichroism, defined as \({\mathrm{\Delta }}\sigma _1(\omega ) = \sigma _1^a(\omega ) - \sigma _1^b(\omega )\) and depicted in Fig. 2f for the p-loop experiment at 10 K, allows the optical anisotropy to be resolved in the range between 300 and 1000 cm−1 for the stressed specimen, which further implies an anisotropy in the spectral weight distribution at FIR frequencies (Fig. S3 in Supplementary Information). Furthermore, we recognize the superconducting gap at 10 K for every p, identified by the drop of σ1(ω) to almost zero below νg, as illustrated in Fig. 2d, e) as well as Fig. 3b. Figures S1 and S2 in Supplementary Information for data at zero p and saturation, respectively, reinforce the gap opening at T < Tc. The sharp onset of absorption at about νg in σ1(ω) indicates a fully gapped superconducting material and the upturn of σ1(ω) at frequencies towards zero accounts for the residual ungapped charge carriers (i.e., thermally activated across the gap) at finite T.

Fig. 2
figure 2

Stress dependence of the optical anisotropy of Ba0.6K0.4Fe2As2 at T < Tc: a, b Representative data of the optical reflectivity (R(ω)) at 10 K for the p-loop experiment after a ZPC protocol along the a- and b-axis (Ra and Rb, respectively, Fig. 1) in the spectral range below 1400 cm−1. c The reflectivity ratio RRatio = Ra/Rb. d, e Real part σ1(ω) of the optical conductivity, achieved through KK transformation of the data in panels a and b. f The so-called dichroism \({\mathrm{\Delta }}\sigma _1(\omega ) = \sigma _1^a(\omega ) - \sigma _1^b(\omega )\) at 10 K within the p-loop experiment. c, f focus the attention to the FIR energy interval 200–1000 cm−1, where the reversible optical anisotropy upon sweeping p is discovered. A first-neighbor interpolation procedure is used in order to generate the color maps. Released pressures are denoted by ‘(r)’. The thick dotted line in all panels marks p = 1.2 bar (i.e., at saturation)

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Fig. 3
figure 3

Temperature dependence of the anisotropic optical response of Ba0.6K0.4Fe2As2 upon applying stress: a T dependence of the quantity ΔRRatio(ω0) (see text) within the p-loop experiment after a ZPC protocol, which emphasizes the T evolution of the p-induced optical anisotropy. We choose ω0 = 600 cm−1. b Measured σ1(ω) at 10 K and 60 K for p = 1.2 bar along both crystallographic axes. c–e T dependence of Δσ1(ω) at the selected p of 0, 1.2 and released 0 bar. These panels focus the attention to the FIR energy interval 200–1000 cm−1, thus emphasising the reversible optical anisotropy upon sweeping p. The thick dashed line in all panels indicates Tc. A first-neighbor interpolation procedure is used in order to generate the color maps. Released pressures are denoted by ‘(r)’

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After having demonstrated that an optical anisotropy can be p-induced in the tetragonal structure of Ba0.6K0.4Fe2As2 at T = 10 K < Tc, we move to its T dependence and its relationship to the overall phase diagram of iron-based superconductors (Fig. 1). We first focus our attention on the quantity \({\mathrm{\Delta }}R_{\mathrm{Ratio}}(\omega _0)\,=\, \frac{{R_{\mathrm{Ratio}}(p\,=\,0,T)\,-\, R_{\mathrm{Ratio}}(p,T)}}{{R_{\mathrm{Ratio}}(p\,=\, 0,T)}}\,{\times}\, 100\), which amplifies the relative change of the optical anisotropy during the p-loop experiment at each T with respect to the p = 0 initial situation (i.e., the isotropic limit since \(R_{\mathrm{Ratio}}(p\,=\,0,T)\sim 1\) at all T) and helps truly pointing out the variation of the optical anisotropy beyond the experimental data noise. This latter quantity, read at the fixed frequency ω0 = 600 cm−1, is shown in Fig. 3a. Our choice of ω0 is motivated by the largest anisotropy (Fig. 2c) at that frequency. Even though there is a substantial Curie-like nematic susceptibility in the normal state (i.e., at T > Tc) from elastoresistive investigation,15 our experiment does not have enough resolution in order to clearly recognize an optical anisotropy above Tc, which if any must be vanishingly small. It is however evident that the p-induced anisotropy develops upon entering the superconducting state at applied p large enough to cause a polarization dependence of the excitation spectrum. At saturation (thick horizontal dotted line at p = 1.2 bar) the color map in Fig. 3a reveals the pronounced increase of the optical anisotropy only at T < Tc. This is evidence for the growing of the p-induced nematicity even in the superconducting state, despite the established hardening occurring below Tc13 and the mild reduction at low T of the Raman susceptibilities11 for dopings across the whole phase diagram. Indeed, the discovered optical anisotropy is in contrast to expectations based solely on anticipated strain for a fixed stress; because of the hardening on cooling beneath Tc, a fixed stress would cause a smaller strain and thus an anisotropy of the optical response getting smaller upon cooling through Tc. On the contrary, we observe the opposite. The T evolution of the optical anisotropy is also mapped by Δσ1(ω), as shown in the energy interval between 200 and 1000 cm−1 at three selected p in Fig. 3c–e. These panels again highlight the p-induced as well as p-reversible optical anisotropy below Tc in the FIR range. This is the most important results of this work, signaling the presence of electronic nematicity at T deep into the superconducting dome and that the band structure is responding to nematic fluctuations.

Before going any further and in order to set the stage for our conclusions, it is instructive to briefly recall our previous optical findings with respect to the nematic phase in the underdoped regime of Ba(Fe1−xCox)2As2.24,25,26,27 At T < Ts ( = 135 K for the parent compound and decreasing upon Co-doping1,4), the optical anisotropy exhibits a remarkable hysteretic response to the applied p even at energies pertinent to the electronic structure, within 0.3 eV from the Fermi level (Fig. 3 in ref. 25). Referring to RRatio(ω0) as a representative quantity for the optical anisotropy, the largest one at saturation was identified at ω0 ~ 1500 cm−1 for the parent compound, being of the order of 5% at the lowest investigated T and decreasing to values of about 3% for T → Ts. The optical anisotropy at saturation for T < Ts decreases upon doping in the underdoped regime, showing a correlation with the Co-doping dependence of the orthorhombicity (Fig. 4 in ref. 25). The anisotropy at ω0 turns into a reversible linear p dependence at T ≥ Ts, further decreasing to values of about 1–3% at the highest applied stress (Fig. 3 in ref. 25). Our results indicate an important polarization of the electronic structure in the nematic phase below Ts and a significant p-induced one above Ts. The hysteretic behavior was also revealed in the optical anisotropy of FeSe at T < Ts(= 90 K) with ω0 ~ 1000 and 3000 cm−1,28,29 among which even a change of sign occurs in the polarization dependence of the measured spectra (Fig. 2 in ref. 28). The magnitude of the optical anisotropy itself is comparable to the values discovered in the underdoped regime of Co-doped 122 materials. However, the optical anisotropy at ω0 in FeSe displays a rather sudden drop on approaching Ts from low T. We conclude that the anticipated p-induced lattice distortion above Ts in FeSe may be less strongly or not obviously bound to the electronic structure as in other iron-based superconductors.28,29 Common to both families of Co-underdoped 122 and FeSe is nonetheless the persistence even at T < Tc of the optical anisotropy evinced in the nematic phase, so that superconductivity develops in an electronically polarized state.26,27,28,29 Overall, the anisotropy in the optical response at mid-infrared energy scales unfolded in Ba(Fe1−xCox)2As2 and FeSe signals the impact of the nematic phase to all iron d orbitals (squeezed in energy by correlation effects) and highlights the relevance of the orbital degree of freedom, as advanced by photoemission data.31,39,40,41,42,43,44 Such an anisotropic optical response at mid-infrared energy scales originates from the split of the relevant bands of the order of only a few tens of meV.39

Our present data of the optimally doped Ba0.6K0.4Fe2As2 bear testimony to a p-induced anisotropy of the excitation spectrum in the purely tetragonal phase, which is reversible upon applying and releasing p, respectively. Therefore, an anisotropy in the charge dynamics due to (p-induced) nematicity24,25,26,27 seems to be a generic feature in 122 iron-based superconductors even into the optimally-doped regime, similar to the dc transport properties.15 Somehow peculiar with respect to the previously investigated underdoped 122 materials and even FeSe,24,25,26,27,28,29 the optical anisotropy in the optimally K-doped 122 compound occurs at the much lower FIR energy scales, relevant to the superconducting gap(s) (Figs. 1a–c, 2f and 3c–e), thus excluding the involvement of bands deep into the electronic structure and possibly implying a less prominent impact of orbital ordering in the optimally doped than in the underdoped regime. While the different electronic structures between the various families of iron-based superconductors cannot be neglected a priori with respect to the anisotropic excitation spectrum, those (low) energy scales, at which the p-induced optical anisotropy occurs in Ba0.6K0.4Fe2As2, are possibly related to the response of the conduction bands to an external symmetry breaking field. The excitation spectrum of the itinerant charge carriers at the energy scales close to the Fermi level, ultimately of relevance for the transport properties as well as superconductivity, could be affected by p-induced anisotropic scattering. Precisely such (anisotropic) scattering may uncover some kind of spin-orbital interplay, where for instance the p-induced nematicity is caused by spin fluctuations and is vestigial to stripe magnetism.16 Alternatively, an external symmetry breaking field, like uniaxial stress, leads to a so-called differentiation of the orbital effective masses, which is even further enhanced by the presence of strong electron correlations.45 Indeed, the latter are supposed to be pronounced in hole-doped materials. Anisotropic effective masses affect the optical response of the conduction bands as well. This could be also reflected by an anisotropic reshuffling of spectral weight at T < Tc between the superconducting collective mode and the FIR energies (Fig. 3c–e), as elaborated in Fig. S3 in Supplementary Information.

In conclusion, the discovery of such an unprecedented p-induced optical anisotropy in (optimally doped) Ba0.6K0.4Fe2As2 only at T < Tc (Figs. 1a–c and 3a) suggests that its electronic structure is extremely susceptible to symmetry breaking stress below Tc. This may shed new light on the putative relationship between quantum critical nematic fluctuations and unconventional superconductivity.18,19 In this respect, establishing the exact extent to which uniaxial stress couple to nematic fluctuations (i.e., at T > Ts ~ 0) is of paramount importance and is a challenge left to the future.

Methods

The large single crystals of Ba0.6K0.4Fe2As2 (Tc = 38.5 K) for this study were grown using a self-flux method, as described previously46 (see Supplementary Information). The tunable stress (p) is applied with our custom-made mechanical device, thoroughly described in refs 24,25. It consists of a spring bellows, which can be extended or retracted by flushing He-gas into its volume or evacuating it, thus exerting or releasing uniaxial p on the lateral side of the specimen (Fig. 1), respectively. As in refs 24,25, we refer here to the He-gas pressure inside the volume of the bellows (pbellows): the effective stress felt by the sample (psample) depends on its size and thickness, so that pbellows = 0.1 bar corresponds to an effective uniaxial stress of about Psample ~ 2.5 MPa on our crystals. It has been widely established that an effective stress of at least 10 MPa is enough to reveal the underlying symmetry-breaking.4

The optical reflectivity R(ω) as a function of temperature (T) and p is measured at nearly normal incidence47 with the electromagnetic radiation polarized along the a- (Ra(ω)) and b- (Rb(ω)) axis (Fig. 1) from the far-infrared (FIR) up to the ultra-violet. We discuss here data collected after the zero-pressure-cooled (ZPC) protocol.24,25 At the target T achieved with p = 0, stress was first increased step-wise from 0 to 1.2 bar (above which a saturation of the (anisotropic) optical response for T < Tc has been found) and then decreased back to zero, thus completing the p-loop. We use a different stress grid upon increasing and decreasing p. This could lead to some tiny data asymmetry along the p-loop within the first-neighbor interpolation procedure used in order to generate the color maps in the figures. The real part σ1(ω) of the optical conductivity was obtained via the Kramers-Kronig (KK) transformation of R(ω) by applying suitable extrapolations at low and high frequencies (see Supplementary Information).47