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Anisotropic in-plane optical conductivity in detwinned Ba(Fe1-xCox)2As2

We study the anisotropic in-plane optical conductivity of detwinned Ba(Fe1-xCox)2As2 single crystals for x=0, 2.5% and 4.5% in a broad energy range (3 meV-5 eV) across their structural and magnetic transitions. For temperatures below the Neel transition, the topology of the reconstructed Fermi surface, combined with the distinct behavior of the scattering rates, determines the anisotropy of the low frequency optical response. For the itinerant charge carriers, we are able to disentangle the evolution of the Drude weights and scattering rates and to observe their enhancement along the orthorhombic antiferromagnetic a-axis with respect to the ferromagnetic b-axis. For temperatures above Ts, uniaxial stress leads to a finite in-plane anisotropy. The anisotropy of the optical conductivity, leading to a significant dichroism, extends to high frequencies in the mid- and near-infrared regions. The temperature dependence of the dichroism at all dopings scales with the anisotropy ratio of the dc conductivity, suggesting the electronic nature of the structural transition. Our findings bear testimony to a large nematic susceptibility that couples very effectively to the uniaxial lattice strain. In order to clarify the subtle interplay of magnetism and Fermi surface topology we compare our results with theoretical calculations obtained from density functional theory within the full-potential linear augmented plane-wave method.


I. INTRODUCTION
In many unconventional superconductors including cuprates and Fe-based pnictides, superconductivity emerges from a complicated soup of competing phases in the normal state when magnetism is suppressed by doping, pressure or other external parameters [1,2]. This the longer a-axis [4]. The two-fold anisotropy is also evident in quasiparticle interference observed via scanning tunnelling microscopy measurements [5] and further confirmed by angle-resolved-photoemission-spectroscopy (ARPES) data, collected on crystals for which the incident beam size was comparable to the size of a single structural domain [6]. Furthermore, quantum oscillations in the parent compound reveal that the reconstructed Fermi-surface (FS) comprises several small pockets [7].
The smallest of these pockets is essentially isotropic in the ab-plane, but the other, larger pockets are much more anisotropic.
The first ARPES observations of an anisotropic electronic dispersion [6] motivated an intensive research activity also with probes for which any impact of the electronic anisotropy would be obscured by the formation of dense twin domains. These correspond to adjacent microscopic domains as small as a few microns with alternating orthorhombic a and b axes [8]. Two distinct methods have been employed so far to detwin the specimen: application of uniaxial stress [8,9] and of an in-plane magnetic field [10]. The former method, employed here, is superior in order to achieve an almost complete detwinning. Recent ARPES measurements [11,12] of detwinned single crystals of Ba(Fe 1−x Co x ) 2 As 2 reveal an increase (decrease) in the binding energy of bands with dominant d yz (d xz ) character on cooling through T s [11], leading to a difference in orbital occupancy. The splitting of the d xz and d yz bands is progressively diminished with Co substitution in Ba(Fe 1−x Co x ) 2 As 2 , reflecting the monotonic decrease in the lattice orthorhombicity [2(a − b)/(a + b)].
For temperatures above T s , the band-splitting can be induced up to rather high temperatures by uniaxial stress.
Mechanically detwinned crystals also provide a suitable playground in order to explore the intrinsic in-plane anisotropy of the transport properties. Measurements of the dc resistivity as a function of temperature of the single domain parent compounds BaFe 2 As 2 , SrFe 2 As 2 and CaFe 2 As 2 (i.e., so called 122 iron pnictides) reveal a modest in-plane dc anisotropy for temperatures below T s , with the resistivity in the ferromagnetic direction larger than along the antiferromagnetic direction [9,13,14].
Substitution of Co, Ni or Cu suppresses the lattice orthorhombicity [15], but in contrast the in-plane resistivity anisotropy is found to initially increase with the concentration of the substituent, before reverting to an isotropic in-plane conductivity once the structural transition is completely suppressed [9,16]. Perhaps coincidentally, the onset of the large in-plane anisotropy for the cases of Co and Ni substitution occurs rather abruptly at a composition close to the start of the superconducting dome.
For temperatures above T s , there is a remarkably large sensitivity to uniaxial pressure, leading to a large induced in-plane resistivity anisotropy that is not observed for overdoped compositions [9]. There is no evidence in thermodynamic or transport measurements for an additional phase transition above T s for unstressed crystals, implying that the induced anisotropy is the result of a large nematic susceptibility, rather than the presence of static nematic order. The observation of a large in-plane resistivity anisotropy, at least for the electrondoped 122 Fe arsenides, bears witness to the orthorhombicity of the material, but does not distinguish between anisotropy in the electronic structure and anisotropy in the scattering rate. To this end, reflectivity measurements of detwinned single crystals using polarized light can provide important insight to the effects of the magnetic and structural transitions on the anisotropic charge dynamics and the electronic band structure. Indeed, the counterintuitive anisotropic behavior of ρ(T ) is also reflected in the finite frequency response of the charge carriers as observed by the optical measurements reported in our previous Letter work [17]. Optical measurements of detwinned single crystals of Ba(Fe 1−x Co x ) 2 As 2 in the underdoped regime reveal large changes in the low-frequency metallic response on cooling through T s and T N together with a pronounced optical anisotropy (i.e., ∆σ 1   the full-potential linear augmented plane-wave method (LAPW) [18].

A. Samples
Single crystals of Ba(Fe 1−x Co x ) 2 As 2 with x = 0, 2.5 and 4.5% were grown using a self-flux method [9]. The crystals have a plate-like morphology of 0.1÷0.3 mm thickness, with the c-axis perpendicular to the plane of the plates. Crystals were cut in to a square shape, approximately 2 mm on the side, oriented such that below T s the orthorhombic a/b axes are parallel to the sides of the square [9]. Detailed thermodynamic, transport and neutron scattering measurements for the studied dopings of Ba(Fe 1−x Co x ) 2 As 2 give evidence for structural, mag-netic and superconducting phase transitions occurring at different temperatures [19,20]: for x = 0, the coincident structural (tetragonal-orthorhombic) and magnetic transitions where the system forms antiferromagnetically ordered stripes occur at T s = T N = 135 K, whereas for x = 0.025 they develop at T s = 98 K and T N = 92 K, respectively. The compound with x = 0.045 undergoes first a structural transition at T s = 66 K then a magnetic transition at T N = 58 K and finally a superconducting one at T c = 15 K.

B. Technique
It has recently been shown that almost single-domain specimens can be achieved by application of uniaxial pressure in situ [9]. This is crucial in order to reveal the intrinsic anisotropy of the orthorhombic phase. To this goal we have extended the basic cantilever concept originally developed for transport measurements to allow optical measurements under constant uniaxial pressure.
The device consists of a mechanical clamp ( Fig. 1(a)) and an optical mask ( Fig. 1(b) and 1(c)) attached on top of it in tight contact. The pressure-device was designed according to the following specific criteria: i) the uniaxial stress is applied to the sample (S) by tightening a screw and drawing the clamp against the side of the crystal ( Fig. 1(a) and Fig. 1(c)). Even though our clamp set-up still lacks of a precisely tunable pressure, the uniaxial stress was gradually increased, so to observe optical anisotropy. The applied pressure is modest, such that T N is unaffected, and can be adjusted over a limited range Hagen-Rubens (HR) formula (R(ω) = 1−2 ω σ dc ), inserting the dc conductivity values (σ dc ) from Ref. [9], while above the upper frequency limit R(ω) ∼ ω −s (2 s 4) [21].
Several precautions were taken in order to avoid experimental artifacts: i) The polarizers chosen for each measured frequency range have an extinction ratio greater than 200, thus reducing leakages below our 1% error limit. ii) As control measurements for the detwinning setup we collected at different temperatures the optical reflectivity of a Cu sample of comparable surface dimensions and thickness with respect to the pnictide crystals and under equivalent uniaxial pressure. As expected, we could not observe any polarization dependence of the Cu reflectivity from room temperature down to 10 K (see e.g. the data at 250 K in Fig. 2). The Cu test measurements set to about 1-2% the higher limit of the polarization dependence due to any possible experimental artifacts (i.e. bended surfaces, leakage of the polarizers etc.), which is notably lower then the anisotropy ratio measured for the iron-pnictides ( Fig. 2 and 3). iii) Prior to performing optical experiments as a function of the polarization of light, the electrodynamic response of the twinned (i.e., unstressed) samples was first checked with unpolarized light, consistently recovering the same spectra previously presented in Ref. [22]. vi) We achieved the same alignment conditions of M and S ( Fig. 1(c)) by imaging on both spots a red laser point source.

A. Reflectivity
The three investigated compositions display overall similar features in their optical response but their polarization and temperature dependences show small but significant differences as we clarify in the presentation and discussion of our results below. Figure 3 presents the optical reflectivity R(ω) in the whole measured frequency range of detwinned Ba(Fe 1−x Co x ) 2 As 2 (x = 0, x = 0.025 and x = 0.045) at different temperatures and for the two polarization directions E a and E b. As already recognized in the twinned (i.e., unstressed) specimens [22], R(ω) gently increases from the UV to the MIR region displaying an overdamped-like behavior. Below the MIR energy range, R(ω) gets progressively steeper with a sharp upturn at frequencies lower than 200 cm −1 (Fig.   3). Close to zero frequency, R(ω) consistently merges in the HR extrapolations calculated with the σ dc values from Ref. [9]. For all measured dopings we observed a polarization and temperature dependence of R(ω) from the FIR up to the MIR-NIR range, while between 5000 and 6000 cm −1 the R(ω) spectra tend to merge together.
The optical anisotropy is rather pronounced at low temperatures in the FIR region, when approaching the zero frequency limit. Interestingly for x = 0, R(ω) increases with decreasing temperature along the a-axis, in agreement with the metallic character of the dc transport prop-  , which was ascribed to antiferromagnetically ordered stripes [18]. The optical anisotropy, as observed experimentally, was even shown to agree with the solution of a three dimensional five-orbital Hubbard model using the mean-field approximation in the presence of both orbital and magnetic order [25]. Moreover, it has been recently pointed out that interband transitions, whose relevance is manifested by first-principle calculations, give a non negligible contribution already in the infrared region, spanning the experimental energy interval of the MIR-band [26]. We will return later on to the comparison between experiment and theory. The complex dielectric functionε = ε 1 (ω) + iε 2 (ω) can be expressed as follows: where ε ∞ is the optical dielectric constant, ω 2 P N , ω 2

P B
and Γ N , Γ B are respectively the plasma frequencies, de-fined as ω 2 P = 4πe 2 n m * , and the widths of the narrow and broad Drude peaks. The latter parameters represent the scattering rates of the itinerant charge carriers, of which n, m * and e are then the density, the effective mass and the charge, respectively. The parameters of the jth Lorenz h.o. as well as those of the MIR-band are: the center-peak frequency (ω j and ω M IR ), the width (γ j and γ M IR ) and the mode strength (S 2 j and S 2 M IR ). The fit constraints are such that the measured reflectivity and the real part of the optical conductivity are simultaneously reproduced by the identical set of fit-parameters [22], which reduces the degree of freedom in the parameters choice. The upper boundary for the temperature dependence of the optical conductivity is found to be close to the NIR peak in σ 1 (ω) at about 5000 cm −1 . Thus, for all temperatures and dopings we fit R(ω) and σ 1 (ω) by varying the parameters for both Drude terms, the MIR-band and the I 1 h.o., while keeping constant the parameters associated to the two high frequency oscillators I 2 and I 3 . We systematically adopted our fitting procedure (Fig. 5) [28]. Of interest is also for x = 0 the temperature independent total Drude weight, which is larger along the b-axis than along the a-axis for T > T s , thus inverting the polar- ization dependence observed below T s . This astonishing behavior for x = 0 might be compatible with a recent multi-orbital model [25]. Given the ARPES results [11], showing that the band splitting diminishes with increasing temperature, one would have eventually anticipated that the Drude weight gets indeed isotropic above T s . Nonetheless, the Drude weight anisotropy above T s is suppressed upon doping (Fig. 6). Experimentally, such a trend of SW Drude remains to be verified under controlled uniaxial pressure conditions, while theoretically it awaits confirmation within doping-dependent models. The overall temperature dependence of the scattering rates below T N as evinced from the analysis of the optical response is expected with respect to the well-established magnetic order [4]. Particularly for x = 0 and x = 0.025, the larger scattering rates along the elongated antiferromagnetic a-axis than along the shorter ferromagnetic b-axis for temperatures below T N may arise because of reduced hopping or of scattering from spin-fluctuations with large momentum transfer (i.e., by incoherent spin waves) [29,30]. The anisotropic scattering rate in the paramagnetic state, at least for x = 0 and x = 0.025, might also be in agreement with predictions based on interference between scattering by impurities and by critical spin fluctuations in the Ising nematic state [31]. Similarly to our previous discussion on the Drude weight, we shall caution the readership, that the trend in the scattering rates, particularly above T s , as well as their doping dependence should be also verified with tunable uniaxial pressures, in order to guarantee equal experimental conditions. A comprehensive theoretical framework, approaching different temperature regimes and considering the impact of doping, is also desired.
Having determined the two parameters governing the dc transport properties, it is worth pursuing at this point the compelling comparison between the temperature dependence of the optical anisotropy and the anisotropy ratio of the dc transport properties, defined as ∆ρ Fig. 7) [29]. From the Drude terms, fitting the effective metallic contribution of σ 1 (ω) over a finite energy interval, we can estimate the dc limit of the conductivity (σ opt 0 = (ω N p ) 2 /4πΓ N + (ω B p ) 2 /4πΓ B ) more precisely than simply extrapolating σ 1 (ω) to zero frequency. The anisotropy ratio ∆ρ opt ρ , reconstructed from the optical data, is thus compared in Fig. 7 to the equivalent quantity from the transport investigation. The agreement in terms of ∆ρ ρ between the optical and dc investigation is outstanding for x=0.025 and 0.045 at all temperatures.
For x= 0, ∆ρ opt ρ is nonetheless slightly larger than the dc transport anisotropy for T < T s . This disagreement might originate from a difference in the applied stress in the optical and dc transport measurements, or from differences in scattering rate of samples used for the two types of measurements.
Significantly, analysis of the optical properties for all compositions seems to indicate that anisotropy in the Fermi surface parameters, such as the enhancement(depletion) of the total Drude spectral weight occurring along the a(b)-axis, outweighs the (large at some compositions) anisotropy in the scattering rates (Fig. 6) that develops below T N in terms of the effect on the dc transport properties (Fig. 7). This is an important result from the optical investigation, which indeed enables to extract both pieces of information governing the behavior of the dc transport properties.
In order to emphasize the relevant polarization dependence at high frequencies, we calculate the linear dichroism ∆σ 1 (ω), as defined in the Introduction. For the purpose of further enhancing the optical anisotropy, we show in Fig. 8 ∆σ 1 (ω, T ), which is defined as ∆σ 1 (ω, T ) from the MIR to the UV for x = 0, 0.025 and 0.045 at various temperatures after having subtracted its corresponding room temperature values and being appropriately normalized. The dichroism persists above T N in the MIR range (Fig. 8), pairing our direct observations in terms of R(ω) and σ 1 (ω) (Fig. 3 and 4). This representation highlights once more that the MIR-feature moves towards lower frequencies upon increasing doping.
Since the dichroism directly relates to a reshuffling of spectral weight in σ 1 (ω) in the MIR-NIR range ( Fig. 4a and 4b), ∆σ 1 (ω) at ω 1 is interrelated to that at ω 2 (i.e., the right y-axis for ∆σ 1 (ω i ) in Fig. 7 are inverted between ω 1 and ω 2 ), so that the behavior of ∆σ 1 (ω) is monotonic as a function of temperature and opposite in sign between ω 1 and ω 2 ( Fig. 7 and 8). This doping-dependence needs to be studied in a controlled pressure regime in order to exclude effects arising from different degrees of detwinning (T < T s ) and different magnitude of induced anisotropy (T > T s ). Even so, it is encouraging that, contrary to the dc resistivity, the changes in the electronic structure appear to follow a similar trend to doping as the lattice orthorhombicity [15]. Our data might thus reveal a pronounced sensitivity of the electronic properties to structural parameters, like the iron-pnictogen angle α [32], altered by external tunable variables like uniaxial pressure. Indeed, changes in α seem to induce relevant modifications in the shape of the Fermi surface and its nesting properties as well as in its orbital makeup, thus implying consequences in terms of the superconducting order parameter, critical temperature and magnetic properties [32].
The origin of the orthorhombic transition has been dis-cussed from the closely related perspectives of spin fluctuations (a so called spin-induced nematic picture [33][34][35][36]), and also in terms of a more direct electronic effect involving, for instance, the orbital degree of freedom [25,28,29,[37][38][39][40]. The present measurements alone cannot distinguish between these related scenarios, because in all cases some degree of electronic anisotropy is anticipated, and indeed it is likely that both orbital and spin degrees of freedom play a combined role in the real material. Nevertheless, it is instructive to compare the observed dichroism with specific predictions made within models based on orbital order. The two well-defined energy scales ω 1 and ω 2 (Fig. 4)  The resulting theoretical Co-dopings agree within 0.5% variation with the experimental ones. Figure 9 show the low temperature measured (top panels) and calculated (bottom panels) optical conductivity for both the a-and b-axis and for the three dopings.
For a direct comparison we have normalized all the measured and calculated σ 1 (ω) to their respective maxima, thus obtainingσ 1 (ω). We clearly see a fairly good agreement between theory and experiment in the general shape ofσ 1 (ω). In the MIR range the DFT calculatedσ 1 (ω) finely reproduces the observed polarization dependence, both as far as the MIR-band position and its polarization dependence (black arrows in Fig. 9) are concerned. magnetic stripe configuration which was shown to correspond to the energy-minimum configuration of these systems [18]. Therefore, the DFT calculation strongly supports a "magnetic origin" of the MIR-band which would originate from the Fermi topology reconstruction in the magnetically ordered state. In this scenario one would reasonably expect that the MIR-band disappears above the magnetic phase transition, contrary to our observations. However, a dynamic antiferromagnetic order due to spin fluctuations could persist in the paramagnetic phase well above the phase transition temperature [42]. The fingerprints of such underlying spin fluctuations would be frozen to fast enough probes as optics, thus explaining the persistence above the phase transition of the MIR-band in our spectra. At higher (NIR) frequencies the agreement result depleted because of the finite k-point sampling. The major interband peak ofσ 1 (ω) experimentally observed at about 5000 cm −1 is shifted to slightly higher frequencies in the theoretical calculations and shows a steeper rise. We notice that spin-polarized DFT calculations require a smaller renormalization fac-tor with respect to unpolarized ones in order to account for these frequency-shifts. The dc anisotropy below T N is principally determined by the anisotropy in the low frequency Drude weight (i.e., changes in the electronic structure close to the Fermi energy), outweighing the non-negligible anisotropy of the scattering rates between the a-and b-axis. Of equal or perhaps greater interest is the temperature regime above T N for which the Fermi surface is not reconstructed.
One would like to understand whether the resistivity anisotropy at high temperatures also originates from the Fermi surface, perhaps due to the difference in the orbital occupancy revealed by ARPES [25,29], or from anisotropic scattering, perhaps associated with incipient spin fluctuations [31]. peak, ascribed to antiferromagnetic ordered stripes.
The measured large in-plane anisotropy of the optical response and its doping dependence is consistently tracked by the LAPW calculations.