M Texier

A Boselli

D Forget

J Colson

Bobroff

D Leboeuf

Y Texier

M Boselli

A Forget

D Colson

J Bobroff

NMR study of electronic correlations in Mn-doped Ba(Fe 1-x Co x ) 2 As 2 and BaFe 2 (As 1-x P x ) 2

Keywords:

NMR study of electronic correlations in

Mn-doped Ba(Fe 1 -x Co x ) 2 As 2 and BaFe 2 (As 1 -x P x ) 2 D. Leboeuf, Y.

I. INTRODUCTION

Superconductivity in Fe-based materials can be induced in a remarkable amount of fashions 1 : carrier doping 2,3 (heterovalent substitution), chemical pressure 4,5 (isovalent substitution) and hydrostatic pressure 6 . In each case the experimental phase diagrams are very similar: magnetism becomes unstable as a function of the tuning parameter and superconductivity arises around the magnetic instability. There is nonetheless a rich palette of variations from one material to the other, and physical properties such as superconducting gap structure, superconducting critical temperature T c , Fermi surface, magnetic ordered moments may vary significantly 7 . Such variations appear as well in the strength of electronic correlations present in those materials. Different experimental signatures of those electronic correlations have been reported, such as reduced kinetic energies 8 , enhanced masses 9 , deviations from conventionnal (T 1 T ) -1 Korringa law due to the presence of AF spin fluctutations 10,11 . The strength of electronic correlations assessed from those experimental findings vary significantly from one material to another 12 , and whereas mass enhancement (m /m band ) can reach 24 in KFe 2 As 2 13 , it can be as low as 2 in LaFePO 14 . The experimental observation of those variations raises a central question in the field : can those systems fall under a universal description? The answer is not clear so far and basic issues about how strong are correlations within each family or iron-based superconductors, and how correlation strength may vary from one family to another are still not completly resolved. In order to determine the importance of electronic correlations in the description of those systems, it is necessary to perform experiments sensitive to correlation strength on different families of iron-based superconductors, and to compare those experiments with well established examples of materials where correlations are significant.

Here we use NMR to study the strength of magnetic correlations in two particular BaFe 2 As 2 (122) materials at optimal doping: heterovalent doped Ba(Fe 1-x Co x ) 2 As 2 (Co-122) with x Co 6.5% and isovalent doped BaFe 2 (As 1-x P x ) 2 (P-122) with x P = 35%. To do this we introduce diluted Mn impurities that substitute to Fe. It has been shown that in Mn-doped BaFe 2 As 2 , that the hole carried by Mn does not delocalise but rather acts as a local moment 15 . This impurity local moment induces a staggered spin polarization in its vicinity. This phenomenon has been studied extensively in metals (RKKY interaction) and correlated materials 16 . The impurity induced staggered polarization can be probed using the NMR of nuclei neighboring these local moments. The polarisation results in the appearance of satellites on the sides of the central line and in a temperature dependent broadening ∆ν(T ) of the NMR spectrum which in the diluted regime is: ∆ν(T ) ∝ S imp (T ) • y imp • f (χ syst (r, T )) (1) where S imp (T ) is the thermodynamic average of the impurity spin, y imp the impurity concentration and f (χ syst (r, T )) a function of the real part of the real space susceptibility of the system, which determines the amplitude, the extension and the shape of the staggered spin polarization induced around the impurity. Knowing the temperature dependence of the susceptibility of the impurity local moments S imp (T ) , one can access the Tdependence of χ syst (r). In a metal, it has been shown that RKKY interaction leads to a Lorentzian broadening ∆ν(T ) that follows the T -dependence of the local moment susceptibility, because χ syst (r) is mainly a constant as a function of T 17 . However in correlated systems, ∆ν(T ) can have a different T -dependence, due to the existence of magnetic correlations that imprint a T -dependence to χ syst (r) 18 . Assuming that the presence of the impurity does not locally affect the nature of the magnetic interactions, the NMR response in the vicinity of a local moment is consequently a good probe of magnetic correlations in a system.

Here we show that in the case of Mn-doped Co-122 and P-122, the broadening of the central line follows the susceptibility of Mn local moments. This demonstrates that the susceptibility of FeAs layers is only weakly temperature dependent in both materials. This behaviour is in contrast with that of the underdoped cuprate superconductor YBa 2 Cu 2 O y , where the CuO 2 layer susceptibility has a strong 1/T dependence 18 .

The Article is organized as follows. In sec. II, we describe the samples studied here and the measurement technique used. In sec. III, we present the evolution of the NMR spectra as a function of Mn doping in both Co-122 and P-122 materials. In sec. IV, we determine the susceptibility of Mn impurities local moments. In sec. V, we show that the broadening of the central line follows the susceptibility of Mn local moments, and from that we conclude that the susceptibiltiy of FeAs plane in Co-122 and P-122 is only weakly temperature dependent. In sec. VI, we finally discuss the implications of this result and compare it with other probes of correlation strength in BaFe 2 As 2 .

II. SAMPLES AND METHODS

We studied 4 single crystals of Ba(Fe 1-x-y Co x Mn y ) 2 As 2 (Co-Mn-122) with Co content x Co ∼ 6.5% and Mn doping ranging from y Mn = 0.65% to y Mn = 1.5% (see table I). The crystals were grown using a self-flux method from a mixture of Ba and FeAs, CoAs and MnAs powders in the molar ratio 1:4-x-y:x:y. The mixtures were heated in evacuated quartz tubes at

Material

x (%) yMn (%) Tc (K) Ba(Fe1-x-yCoxMny)2As2

6.5 0.65 18 7.5 0.9 13.5 6.0 1 12.5 6.5

1. 5 8 Ba(Fe2-yMny) I), while staying well defined for all dopings. For the sake of clarity, most of the data presented here focus on samples with Mn doping y Mn = 0.65% and y Mn = 1.5%, though nearly identical results were obtained with the other samples. Ba(Fe 2-y Mn y ) 2 (As 1-x P x ) 2 (P-Mn-122) powders with x P = 35% and with Mn doping y Mn = 0%, y Mn = 1.5% and y Mn = 3.0% were synthesized using a solid state technique at 950 • C during 36 hours from a stoichiometric mixture of Ba lumps, Fe, MnAs, P and As powders. The MnAs powder was prepared from a stoichiometric mixture of Mn and As powders heated in an evacuated quartz tube at 520 • C for 10 hours and then 870 • C for 10 hours. X-ray powder diffraction revealed a small amount of Fe 2 P and a minority phase of an unidentified iron based compound. Thanks to the ability of NMR to adress and resolve nuclei originating from different phases, the presence of parasitic phases in the powders does not constitute an obstacle for the present study and can be ignored. The powders were then mixed with a standard two component epoxy. Using the anisotropy of the susceptibility of iron pnictides and a rotating sample holder placed in a static 7.5 T magnetic field applied perpendicularly to the rotating axis of the sample holder, the crystals constituting the powder can be aligned while the epoxy hardens. As a result, the components of the powder are oriented such that their c-axis is perpendicular to the rotation axis. Success of the orientation procedure has been checked by measuring the angle dependence of the 75 As and 31 P NMR frequency when rotating the field with respect to the c-axis. T c in P-Mn-122 decreases at a rate of ∼ -10 K per % of Mn, such that sample with y Mn = 3.0% shows no sign of superconducting transition down to 2 K (see table I). 75 As NMR in Ba(Fe 1-x-y Co x Mn y ) 2 As 2 and Ba(Fe 2-y Mn y ) 2 (As 1-x P x ) 2 was performed with an applied magnetic field of 13.3 T. 31 P NMR was measured at 7.5 T. All the spectra were acquired for field applied along the c-axis and using a standard Fourier transform recombination after a π/2 -τ -π pulse sequence.

III. DESCRIPTION OF THE NMR SPECTRUM IN DOPED BaFe2As2

Figure 1 shows the temperature dependence of 75 As NMR central line in Co-Mn-122 and 31 P NMR central line in P-Mn-122, for different Mn concentrations. 75 As NMR central line in Co-Mn-122 is splitted such that two additional peaks, associated with different local atomic environment, appear on each side of the main line (pannels a and b of figure 1). The low frequency peak is due to a modification of the second order quadrupole effect in the vicinity of Co atoms and the high frequency peak (indicated by the arrows in fig. 1a,b) comes from a modification of the shift in the vicinity of Mn impurities, as will be discussed below. 31 P NMR central line in P-Mn-122 with y Mn = 0 is not splitted (fig. 1c).The introduction of Mn induces a splitting of the central line with a satellite appearing on the right hand side of the main line, as indicated by the arrows in fig. 1d,e.

Let us first describe and explain in more detail the different features of the spectra shown in fig. 1. We do so with the help of figure 2 where we compare the 75 As NMR central line in Mn doped materials: Mn-122 ( Ba(Fe 1-y Mn y ) 2 As 2 ) (green curve), Co-Mn-122 (red curve) and P-Mn-122 (blue curve). Since 75 As nuclear spin is I = 3/2, its NMR frequency depends on both local magnetic field and electric field gradients (EFG):

75 ν = 75 γ 2π H 0 + 75 γ 2π H 0 75 K + α ν 2 Q 75 γH 0 (2)
where ν Q is proportionnal to the main axis EFG, α depends on the orientation of the crystal with respect to H 0 and on the asymmetry in the EFG and 75 γ/2π = 7.2919 MHz/T is the gyromagnetic factor of 75 As nucleus.

75 K is the shift induced by the hyperfine interaction between the As nuclei and the Fe electrons orbitals and spin susceptibilities. The frequency of the main central line of the 75 As NMR spectra originates from As nuclei located in regions far from any doping atom, a configuration sketched in fig. 2, inset b. In 122 materials, where one As nucleus is surrounded by 4 Fe atoms, 75 K is given by:

75 K central = 4A Fe • χ Fe + K orb (3)
with χ Fe the Fe layer susceptibility probed far from any doping atom, A Fe the hyperfine coupling between one Fe and the As nucleus, and K orb a temperature independent shift due to orbital effects.

In Mn-doped 122, the NMR frequency of As nuclei located as nearest neighbor of Mn atoms (a situation depicted in fig. 2,inset c) is modified. This results in the splitting of the NMR central line such that a satellite on the right-hand side of the main line appears (green curve in fig. 2). Eq.( 2) indicates that this satellite may arise from a modification of 75 K or of the local EFG. A previous NMR study of Mn-122 has shown that the satellite shift is proportionnal to H 0 and that it is therefore due to a modification of the shift 75 K (ref. 15 ). By substituting Fe atoms with x Co = 6.5% Co atoms, we get Co-Mn-122 which typical 75 As NMR spectrum is shown in red in figure 2. Co substitution has three main consequences that we will discuss now. Eq.( 3) indicates that 75 K central depends on carrier population in Fe orbitals. The first consequence of Co subsitution is thus a shift of the 75 As NMR spectrum as observed in previous work 11 . The second consequence of Co substitution is the appearance of an additional satellite on the left-hand side of the central line, as shown in figures 1a, 1b and 2. This satellite originates from the modification of the second order quadrupolar contribution to the NMR shift (the last term in eq. ( 2)) in the vicinity of Co atoms, as shown earlier 11 . For symmetry reasons, α = 0 in the specific case of BaFe 2 As 2 , on the As atomic site, for H 0 c, so that the position of the central line does not depend on the EFG. However Co substitution locally breaks this symmetry and shifts the frequency of As nuclei located as nearest neighbor of Co atoms (inset a in fig. 2), giving rise to this low frequency satellite (see figure 2). This identification is confirmed by the fact that the T -dependence of the shift of this satellite follows the T -dependence of the sift of the central line (see fig. 1a,b). The third conse-quence of Co doping, is a broadening of the NMR lines, due to increased structural and electronic disorder induced by Co-doping. This broadening is however small enough so that both the position of the satellite line and the full width at half maximum of the central line can be followed as a function of temperature (see fig. 1a andb). Those quantities will be central for the discussion that follows in the next sections.

Finally, the blue curve in fig. 2 is a typical 75 As NMR spectrum for P-Mn-122, with x P = 35% and y Mn = 3%. Disorder is significantly increased in comparison to Co-Mn-122 with x Co = 6.5%. Now the high frequency Mnrelated satellite is merged with the central line. Signature of this satellite is still slightly visible though in the asymmetry of the central line, with a higher weight on the right-hand side, where the satellite associated with As nuclei nearest neighbor to Mn atoms is expected. Note that there is no visible splitting due to second order quadrupolar effect. This can be a result of higher disorder than in Co-Mn-122, and/or of the fact that the As atomic sites are closer to Co atoms than to P atoms, meaning that this quadrupolar satellite could merge with the central line in the case of P-Mn-122. The fact that the 75 As NMR frequency is sensitive to EFG disorder results in a very broad line in the case P-Mn-122 sample, due to high P-doping level. This can be a major drawback in the study of the magnetic broadening of this line. To circumvent this problem we use 31 P NMR in P-Mn-122. 31 P has a nuclear spin I = 1/2 and is hence insensitive to EFG :

31 ν = 31 γ 2π H 0 + 31 γ 2π H 0 31 K (4)
with 31 γ/2π = 17.2356 MHz/T the gyromanetic factor of P nucleus. Figures 1c,1d and 1e show the temperature dependence of 31 P NMR in P-Mn-122 for three different Mn dopings. In the case of y Mn = 0 and x P = 35.0%, the 31 P NMR line width is 31 ∆ν ∼ 20 kHz whereas the 75 As NMR line width in the same sample is 75 ∆ν ∼ 60 kHz. 31 P NMR is thus a cleaner probe of magnetic braodening. The 31 P NMR line width observed here is a factor ∼ 4.5 smaller than that observed previously in mosaic of single crystals 10 , demonstrating the quality of the powders and the efficiency of our orientation procedure.

In the following we will study the Mn-induced broadening of the 75 As NMR central line in Co-Mn-122 and that of the 75 As and 31 P NMR central line in P-Mn-122. But in order to understand these broadenings we first need to determine the susceptibility of the local moments carried by Mn impurities.

Texier et al. 15 used the temperature dependence of the Mn satellite shift 75 K sat (T ) in order to determine the susceptibility of the local moments carried by Mn impurties χ Mn (T ) in Mn-122. We apply the same method to our data on Co-Mn-122. In the diluted limit, each As site participating to the high-frequency satellite intensity is hyperfine coupled to 1 adjacent Mn atom and 3 Fe atoms (see inset c in fig. 2) such that eq.( 3) is modified in the following way:

75 K sat = 3A Fe • (χ Fe + ∆χ ) + A Mn • χ Mn + K orb (5)
where χ Fe is the Fe susceptibility far from Mn, and ∆χ accounts for possible modification of this susceptibility in the vicinity of Mn. Combining this with eq.( 3), assuming that A Fe is unaffected and that ∆χ ∝ χ Mn leads to:

χ Mn (T ) ∝ 75 K sat (T ) - 3 4 75 K central (T ) + K orb (6)
with K orb = 3/4K orb -K orb , a T -independent orbital term. We first determine 75 K sat by fitting the spectra of fig. 1a,b. Fig. 3 shows a zoom of the satellite and the fits to the spectra used to determine 75 K sat (T ). χ Mn (T ) is then computed and plotted in fig. 4a. The errorbars increase considerably when the temperature is lowered because both the satellite and the main line broaden faster than they shift apart. For T 50 K the satellite line merges with the main line such that the shift of the satellite line cannot be reliably determined anymore. The fact that curves for y Mn = 0.65% and y Mn = 1.5% overlap shows that the Mn induced polarisation does not depend on Mn concentration, and is evidence that As nuclei are coupled to at most 1 Mn atom. Additionnal confirmation that the Mn content is within the diluted regime comes from the Mn-doping dependence of the intensity ratio between the satellite and the central line, which follows the probability law of finding As atoms with exactly one Mn nearest neighbor 15 . χ Mn in Co-Mn-122 shows a similar temperature dependence to what was observed earlier in Mn-122 15 . A Curie-Weiss fit to χ Mn (T ) gives θ = 20 ± 20 K. Such a low value of θ, and independent of Mn doping, indicates that Mn-Mn interactions are weak. This is further confirmed by an analysis of the macroscopic susceptibility of Co-Mn-122 samples measured by SQUID. By subtracting the Fe contribution to the macroscopic homogeneous susceptibility, namely χ Fe (T ) ∝ 75 K central (T ), one can derive the Mn contribution to this macroscopic susceptibility (see Appendix A for details). The resulting macroscopic χ Mn (T ) is shown in fig. 4b. The data fits nicely to a Curie model (black lines in fig. 4b). A Curie-Weiss fit to those data would yield θ = 0 ± 10 K. In a Curie model χ (T ) = y 0 + C/T , C is proportionnal to µ 2 eff , with µ eff = g S(S + 1) the magnetic moment carried by the Mn impurity. Fits to the data of fig. 4b give µ eff = 1.63 ± 0.3µ B for all samples. This value is in agreement with µ eff = 1.73 calculated for a spin S = 1/2 and g = 2, as expected in a case where each Mn atom carries one localized hole.

In summary, this study of the Mn local moment susceptibility is consistent with a scenario where each Mn atom carries a local S = 1/2 moment, and where those local moments are only weakly coupled to each other, and behave as isolated Curie moments.

V. CENTRAL LINE BROADENING

The analysis of the shift of the NMR spectral weight originating from the population of 75 As nuclei located as nearest neighbors to Mn impurities, namely the high frequency satellite in fig. 1a,b and3, allowed to determine the T -dependence of the Mn local moments susceptibility. 75 As nuclei located further away from Mn impurities do not contribute to the satellite amplitude, but rather to the central line broadening ∆ν(T ). In combination with χ Mn (T ), the determination of ∆ν(T ) allow to reveal the FeAs layer susceptibility χ Fe (T, r) (see eq.( 1)), and the strength of magnetic correlations in Mn-doped Co-122 and P-122.

We measure the full width at half maximum ∆ν(T ) of 75 As NMR central line in Co-Mn-122 and of both 75 As and 31 P NMR central lines in P-Mn-122. In order to isolate the Mn-induced broadening from over sources of broadening, we subtract the line width measured at the highest T (where the Mn impurity contribution to the line width becomes negligeable) to the measured ∆ν(T ). The resulting quantity, normalized by the Mn content y Mn , is plotted in figure 5a in the case 75 As NMR for Co-Mn-122 and P-Mn-122 and in figure 5b in the case of 31 P NMR for P-Mn-122. In each case, the normalized ∆ν(T ) obtained for different Mn contents overlap, confirming the linear dependence of ∆ν(T ) on y Mn indicated in eq.( 1).

A Curie model nicely reproduces the temperature dependence of ∆ν(T ) in Co-Mn-122, as shown by the red line in fig. 5a. It indicates that ∆ν(T ) follows the same T -dependence as χ Mn (T ). Figure 6 shows the central line broadening ∆ν as a function of χ Mn determined earlier in fig. 4a. Within the errorbars, the two quantities track each other in a linear fashion. There is no strong deviation from proportionality between ∆ν(T ) and χ Mn (T ). This indicates that the magnetic susceptibility of FeAs layers χ Fe (T ) is only weakly temperature dependent in Co-Mn-122.

In P-Mn-122, we have not been able to determine χ Mn (T ) via K sat because the high frequency satellite is merged with the central line, even in 31 P NMR spectra (see fig. 1d,e). We assume that χ Mn (T ) in P-Mn-122 follows a Curie law T -dependence just as in Co-Mn-122. Two facts make this assumption reasonable. First, Mn acts as local moment in P-122, just as in Co-122 (no modification of the Knight shift as function of Mn doping, broadening of the central line, appearance of a satellite at the same position as in Co-Mn-122 in 75 As NMR spectra, see fig. 2, as a consequence of the polarization around the Mn local moments). Second, Mn contents FIG. 6: (Color online) 75 As NMRcentral line broadening plotted as a function of the Mn local moments susceptibility χ Mn (T ) derived from 75 As NMR satellite shift, for all Co-Mn-122 samples studied here. This plot shows that, within errorbars, ∆ν(T ) ∝ χ Mn (T ), such that the FeAs layer suscpetibility has a very weak T -dependence. There is remarkable agreement for all Co and Mn contents, indicating that Mn doping is within the diluted regime and that magnetic correlations are not dependent on carrier concentration within the errorbars.

studied here in P-Mn-122 are within the same range of concentration studied in Co-Mn-122, such that the effects of Mn on the spectrum are also proportional to Mn content in the case of P-Mn-122 (ratio of the satellite intensity with respect to central line intensity, and central line broadening proportionnal to y Mn as shown in fig. 5a andb).

Based on this assumption, the fact that ∆ν(T ) from 75 As NMR in P-Mn-122 is quantitatively identitical to ∆ν(T ) from 75 As NMR in Co-Mn-122, as plotted in fig. 5a, shows that χ Fe (T ) is quantitatively similar in P-122 and Co-122. χ Fe (T ) is thus weakly T -dependent in both heterovalent and isovalent-doped 122. It also suggests that magnetic correlations have comparable strength in both materials such that χ Fe (T ) appears to be insensitive to the number of carriers in the system. 31 P NMR can be used to confirm the temprature dependence of ∆ν(T ) in P-Mn-122. The absence of quadrupolar effect in 31 P NMR makes it a cleaner probe of magnetism. As plotted in fig. 5b, 31 P NMR central line broadening follows a Curie model almost perfectly. 31 P NMR shows more compelling evidence that χ Fe (T ) is weakly T -dependent and strengthens the results derived from 75 As NMR.

VI. DISCUSSION

Studies of magnetic correlations using the NMR of nuclei located in the vicinity of an impurity have been per-formed in high-T c cuprate superconductors 16 . Those systems are doped antiferromagnetic (AF) charge-transfer insulators, and strong magnetic correlations extend over a wide range of doping. In the underdoped regime, the broadening of the NMR spectra induced by impurity local moments, is found to diverge as 1/T 2 . If χ imp ∝ 1/T , then according to eq.( 1), it implies that the susceptibility of the CuO 2 plane, the basic common building block of cuprate superconductors, has a strong 1/T dependence. The staggered polarization induced by the impurity local moment grows faster than the impurity susceptibility itself. This behaviour is a consequence of the growth of significant AF correlations as a function of T , in underdoped cuprates. In the overdoped regime, the 1/T 2 dependence of the broadening is lost and a more conventionnal 1/T law is recovered 19 .

In figure 5 we compare ∆ν(T ) of both 75 As NMR and 31 P NMR with a fit of the form y 0 + C/T 2 (dashed red line). Departure from this 1/T 2 dependence is observed in both cases though more strikingly for 31 P NMR. While a 1/T dependence nicely reproduces the data, a 1/T 2 dependence is clearly incompatible with our data. We thus conclude that both isovalent and heterovalent doped 122 materials lie closer to a conventionnal metal or an overdoped cuprate than to an underdoped cuprate, in terms of correlation strength as inferred from this NMR technique.

Numerical simulations of the NMR spectrum of Co-Mn-122 were performed in order to determine the shape and the spatial extension of the staggered polarization in the vicinity of Mn impurities (see Appendix B). The best simulations are achieved when AF correlations are turned on, whether at Q stripe = ( 12 , 1 2 ) or Q Néel = (1, 0) (in r.l.u. with respect to the tetragonal structure). The staggered magnetization around the impurity rapidly decreases as a function of distance to the impurity, typically extending over λ ∼ 3-5 units of in-plane lattice parameter, or λ = 22 ± 5 Å. This characteristic length can be compared to some extent with the AF correlation length ξ found in inelastic neutron scattering (INS). In Co-122, with x Co = 15%, the 9.5meV INS AF peak at T = 40 K has a finite width that corresponds to ξ = 20 ± 4 Å20 , comparable to the spatial extension of the staggered polarization we measured here in real space by NMR. Moreover, we found that for y Mn = 0.65% we can simulate the temperature dependence of the spectrum by simply considering the 1/T dependence of χ Mn while keeping the spatial extension λ constant and equal to 4 units of inplane lattice parameter (see fig. 8b and8c). This is consistent with our previous conclusion drawn from fig. 5: magnetic correlations in the FeAs plane are only weakly T -dependent.

Short range, weakly temperature dependent AF correlations seem to be the origin of the 75 As NMR central line broadening in Co-Mn-122. In fig. 5a we see that ∆ν(T ) is almost identical in both Co-Mn-122 and P-Mn-122. This suggests that the very same short range, weakly T -dependent AF correlations exist in both heterovalent and isovalent doped BaFe 2 As 2 , around optimal doping. This similarity between the two systems adds up to an increasing list of others: similar maximum T c (25 K for Co-122, 30 K for P-122), similar thermodynamic phase diagrams and similar coupling of magnetism and superconductivity to the lattice 21 . (T 1 T ) -1 measured with 75 As NMR in Co-122 is also found to be very similar to (T 1 T ) -1 measured with 31 P NMR in P-122 10,11 . A remarkable resemblance between the two compounds and yet very different tuning parameters (chemical pressure vs carrier doping) and very different levels of disorder (as illustrated in fig. 2 by the width of the 75 As NMR central line). Authors of ref. 21 argue those similiraties might be related to the fact that Co and P doping act similarly on the Fe-As(P) bond length, which has been shown to be a central parameter to control T c and to suppress magnetism (ref. 21,22 and ref. therein). Our results are consistent with such an explanation: they indeed suggest that AF correlations, which are believed to be the underlying architects of the phase diagrams of those systems, do not depend on charge carrier doping. Nonetheless there is at least one dissimilarity between the two systems: they have different superconducting gap structures. Indeed while Co-122 is fully gapped around optimal doping 23 , line nodes were found in the superconducting gap of optimally doped P-122 24 . Our results suggest that this difference in gap structure cannot be accounted for by a difference in magnetic correlation strength.

Finally we compare our results with other probes of correlation strength in 122 materials. Yang et al. 25 performed x-ray absorption (XAS) and resonant inelastic x-ray scattering (RIXS) measurements in BaFe 2 As 2 that showed remarkable similarity with Fe-based metals while showing only poor comparability with Fe-oxide insulators, highlitghing the itenerant character of ironpnictides. They then extracted from their spectra a value of the Coulomb repulsion U , and Hund's coulping J Hund : U ∼ W > J Hund , with W the bandwidth, indicating the existence of only moderated electronic correlations. Qazilbash and coworkers 8 used the ratio of kinetic energy of the electron extracted from optical conductivity data K exp to the kinetic energy computed from band theory K band . They found that the value of K exp /K band in BaFe 2 As 2 is similar to K exp /K band in ovderdoped high-T c cuprates. Our study of electronic correlation strength with NMR is in agreement whith those findings.

Probes of correlation strength in 122 materials indicate that they feature only moderated correlations, weaker than in underdoped cuprates superconductors. However, high value of Sommerfeld coefficient and mass enhancement of certain bands derived from quantum oscillations in KFe 2 As 2 suggest electronic correlations may be stronger in the case of hole-doped 122 materials 13,26,27 . Recent theoritical work proposed that hole-doped KFe 2 As 2 is more correlated than electrondoped Co-122 because it stands closer, in terms of band filling, to an orbital-selective Mott transition driven by Hund's coupling between the different orbitals at play 28 .

The significant variation of band renormalization from one band to another observed in KFe 2 As 2 is supporting evidence for proximity to this selective Mottness. This scenario calls for an experimental investigation of correlation strength in KFe 2 As 2 , and comparisons with the work achieved here in Co-122 and P-122 using NMR might help quantify and understand the evolution of electronic correlations in different iron-based superconductors.

VII. SUMMARY

We studied the strength of electronic correlations around optimal doping in both isovalent doped (P-122) and heterovalent doped (Co-122) BaFe 2 As 2 . To do so, we introduced local magnetic moments carried by diluted Mn impurities that polarise the FeAs plane in its vicinity. This polarization results in the appearence of a satellite and in a broadening of the NMR spectra that we measured with 75 As NMR in Co-122 and with both 75 As and 31 P NMR in P-122. We used the satellite shift to demonstrate that the Mn impurity local moments behave as nearly isolated free moments. We then studied the Tdependence of the NMR line width and showed that it follows the T -dependence of the Mn-impurity susceptibility in both Co-122 and P-122. This behaviour indicates that magnetic correlations are only weakly T -dependent in both materials. Moreover, the quantitative agreement between the T -dependent NMR line broadening found in P-122 and Co-122 suggests that those magnetic correlations are independent of carrier doping. Numerical simulations of the 75 As NMR spectrum of Co-Mn-122 show that AF correlations, extending over 3-5 units of in-plane lattice parameters, reproduce satisfactorily the experimental data.

The T -dependence of the NMR line broadening observed here is in contrast with that observed in underdoped cuprate, where the CuO 2 layer susceptibility measured with NMR in the vicinity of impurities shows a significant T -dependence, due the development of strong AF correlations as a function of temperature. We conclude that both isovalent and heterovalent doped 122 materials lie closer to a conventionnal metal or an overdoped cuprate than to an underdoped cuprate, in terms of correlation strength as inferred from this NMR technique. Here we explain in more details the method we use to extract the Mn local moments contribution to the macroscopic susceptibility of Co-Mn-122. Magnetization measurements were performed up to 7 T using a Magnetic Properties Measurement System (MPMS) produced by Quantum Design. Isothermal M (H) curves were measured and showed linear behaviour in the temperature range we focus on. Fits to M (H) allow to extract the macroscopic susceptibility of the sample as a function of T , that is plotted in fig. 7 (green squares) for Co-Mn-122 with x Co = 6.5%, y Mn = 1.5%. The sample susceptibility is mainly the sum of two contributions: the FeAs layer susceptibility and the Mn local moments susceptibility. The FeAs layer susceptibility is measured in the same sample using NMR: it is given by 75 K(T ), the shift of the central line of fig. 1a,b, shown with red triangles in fig. 7 for x Co = 6.5%, y Mn = 1.5%. By subtracting 75 K(T ) (suitably multiplied by a factor such that 75 K(T ) and the sample macroscopic susceptibility overlap for T > 200 K) to the sample macroscopic susceptibility (green squares in fig. 7), we obtain the Mn local moments contribution, shown with blue circles in fig. 7.

Appendix B: Simulations of the NMR spectrum

We can elaborate to some extent about the shape and the spatial extension of the staggered magnetization induced in the vicinity of the Mn local moment by simulating the NMR spectrum. We do so in the case of 75 As NMR in Co-Mn-122, following some of the methodology developed earlier in the case high-T c cuprate YBCO 19 . The idea is to compare the simulated spectra given by a set of models and to find which model best reproduce the important features of the experimental data.

We simulate the NMR spectra under the assumptions that Mn concentration is within the diluted regime such that one As nucleus is nearest neighbor to at most one Mn atom, and such that the effect of several Mn atoms on the Fe electronic susceptibility is additive. Such assumption has been checked experimentally (see figures 4, 5 and 6). In these simulations, we ignore the effect of Co substitution, since they mainly affect the local EFG environment of As nuclei. We assume that the hyperfine coupling between As nuclei and Fe atoms is isotrope. We compute the staggered magnetization in the FeAs layer and the resulting simulated spectra are convoluted with a Gaussian function to account for other sources of broadening that are already present in Co-122 without Mn doping.

We used two classes of model of the staggered magnetisation close to the impurity local moment, that we will discuss now.

RKKY model

The first class of model relies on a RKKY interaction. For a two-dimensionnal interaction, the staggered magnetization in the vicinity of the magnetic impurity takes the general form 29 : m Fe (k F , r) ∝ cos(2k F (r -1) + φ) r 2 (B1)

with k F the Fermi wavevector, r the distance to the impurity and φ a phase we choose to be zero. Eq.(B1) is however not appropriate in the case of a multi-band system.

The simulated spectra strongly depends on the value of k F but systematically feature more than one satellite line, in disagreement with the experiments.

Phenomenological models with AF correlations

In this second class of model, we assume that the natural tendency of iron-pnictides towards the formation of AF order should appear in the shape of the staggered magnetisation such that:

m Fe (Q, r, λ) ∝ cos(2Q AF • r) • f (r, λ) (B2)
with Q AF the wavevector at which AF correlations are maximum, r the vector coordinate of each atomic site in a coordinate system where the Mn impurity seats at the origin, λ a characteristic length controling the spatial extension of the staggered magnetization, and f (r, λ) a function for the spatial damping of the staggered magnetisation. We simulated NMR spectra with several f (r, λ):

f Bessel (r, λ) = Y k (0, r/λ) Y k (0, 1/λ) (B3)
with Y k a Bessel function of the second kind. In all the models mentionned above, we allow for a difference between the magnitude of the moment carried by Mn impurity and the moment carried by nearest neighbor Fe atoms.

Comparison to the data We now compare the different simulations we get from the different models of the staggered magnetisation we considered with the experimental spectra. There are two main constraints that allow to eliminate models which are not compatible with the data: first is the number of satellite lines, second is the shape and width of the central line. Fig. 8a shows the simulated spectra for the different models exposed above. Amongst those models, three can already be discarded because they show more than one satellite line. Moreover, a simulation based on f Gaussian do not yield a satisfactory shape for the central line. In the end, the model that best reproduces the experimental spectra is based on AF correlations at Q stripe , with a damping function given by eq.(B5) with γ = 1.5 and λ ∼ 3 -5 units of in-plane lattice parameter.

Using this particular model (f Stretched , γ = 1.5, Q = Q stripe and λ = 4), and taking into account the 1/T dependence of χ Mn (T ), we were able to simulate the temperature dependence of the 75 As NMR spectrum in Co-Mn-122 with y Mn = 0.65% and x Co = 6.5%, as shown in fig. 8b,c. Both the width of the central line and the satellite shift of the experimental spectra are well reproduced by this specific model. These simulations do not however constitute a demonstration that the model we favour here (f Stretched , γ = 1.5, Q = Q stripe and λ = 4) is the only model that could describe the data. Other models we did not consider here may work as well. Nonetheless, this approach allows to theorize the basic electronic correlations required to describe the NMR spectra, and since all models compatible with our data give a consistent value for λ, it also allows to constrain the spatial range of those correlations.

FIG. 1 :

 1 FIG. 1: (Color online) Pannels a and b: 75 As NMR measured with H0 = 13.3 T, H0 c, in Co-Mn-122 single crystals with xCo = 6.5% yMn = 0.65% (pannel a) and yMn = 1.5% (pannel b). The main line develops a structure at high temperature indicating that it is composed of several peaks with similar intensities. This structure might be a consequence of a modification of the hyperfine coupling or quadrupolar environment due to Co atoms. Pannel c, d and e: 31 P NMR measured with H0 = 7.5 T, H0 c, in P-Mn-122 powders with xP = 35%, yMn = 0 (pannel c), yMn = 1.5% (pannel d), yMn = 3.0% (pannel e). In pannels a,b,d and e, the arrow points to the satellite associated with As nuclei nearest neighbor to Mn atoms. In all pannels, curves are arbritarily shifted vertically for clarity.

FIG. 2 :

 2 FIG. 2: (Color online) Typical 75 As NMR spectra at T = 260 K in Mn-122 compounds, measured with H0 = 13.3 T, H0 c. Pure Mn-122 (yMn = 2.5%) is shown in green (bottom axis) and features a satellite on the right wing of the central line associated with As nuclei nearest neighbor to Mn, as sketched in inset c (from ref. 15 ). Red curve (top axis) shows Co-Mn-122 with xCo = 6.5%, yMn = 0.65%. The satellite associated with Mn is still visible (right hand side of central line), and an additional satellite due to the modification of the second order quadrupole effect of As nuclei nearest neighbor to Co atoms (inset a) appears on the left wing of the central line. Blue curve (top axis) shows P-Mn-122 with xP = 35%, yMn = 3.0%. The central line width has significantly increased in comparison to Co-Mn-122 because of higher disorder for xP = 35% than for xCo = 6.5%. The satellite associated with Mn is still here, but is merged with the central line. Curves are shifted vertically.

FIG. 3 :

 3 FIG. 3: (Color online) Zoom on the 75 As NMR satellite associated with As nearest neighbor to Mn atoms in Co-Mn-122 with xCo = 6.5%, for 15 < T < 300 K, measured with H0 = 13.3 T, H0 c. Green lines are fits to the experimental data used to determine the shift of satellite line 75 Ksat. Top pannels : yMn = 0.65%. Bottom pannels : yMn = 1.5%. In all cases, the satellite line merges with the main line for T 50 K, preventing us to confidently determine the shift of the satellite line in this temperature range.

FIG. 4 :

 4 FIG. 4: (Color online) Determination of Mn impurity local moments susceptibility χ Mn in Co-Mn-122 (xCo = 6.5%) for yMn = 0.65% (red) and yMn = 1.5% (blue). a) χ Mn (T ) determined with the 75 As NMR shift. Continuous and dashed black lines are fits to the data with a Curie-Weiss model y0 + C/(T + θ) with θ respectively fixed to 0 and 40 K. b) χ Mn (T ) determined with macroscopic susceptibility measurements where the Fe layer contribution to the homogeneous susceptibility, namely χ Fe (T ) ∝ 75 K central (T ) has been subtracted. Black lines are fits to the data with a Curie law y0 + C/T .

FIG. 5 :

 5 FIG. 5: (Color online) Central line broadening plotted as (∆ν -∆ν high-T )/yMn. a) From 75 As NMR spectra in both Co-Mn-122 (dark and light blue markers) and P-Mn-122 (black and green squares). b) From 31 P NMR spectra in P-Mn-122. In both pannels, red dashed line is a least square y0 + C/T 2 fit to the data and red continuous line is a least square Curie law y0 + C/T fit to the data. While 75 As NMR ∆ν(T ) is best reproduced by a Curie model in both cases. Incompatibility with the 1/T 2 fit is more obvious in the case of ∆ν(T ) deduced from 31 P NMR in P-Mn-122.

  Appendix A: Determination of the macroscopic susceptibility of Mn local moments

FIG. 7 :

 7 FIG. 7: (Color online) Illustration of the method for the extraction of the macroscopic susceptibility of Mn local moments, using SQUID and NMR measurements. Green squares (left axis): sample (raw) macroscopic susceptibility obtained using the slope of M (H) isotherms measured with SQUID. Blue circles (left axis): Mn local moments susceptibility contribution to the sample macroscopic susceptibility. It is obtained by subtracting the FeAs layer susceptibility, measured here with the shift 75 K of the central NMR line in the same sample of Co-Mn-122 (red triangles, right axis), to the sample macroscopic susceptibility (green squares).

FIG. 8 :

 8 FIG. 8: Simulations of 75 As NMR in Co-Mn-122 with yMn = 0.65% and xCo = 6.5%. a) Simulated spectra obtained with different forms of polarisation (see text). Green circles highlight the satellite line which would be compatible with the experiment. Models giving rise to more than one satellite line (red circles) are not compatible with the experimental data. b) Experimental spectra (H0 = 13.3 T, H0 c) at different temperatures compared with simulations obtained with f Stretched , γ = 1.5, Q = Qstripe and λ = 4 (red lines). A zoom on the satellite line is shown in pannel c.

ff

  Gaussian (r, λ) = exp -Stretched (r, λ) = exp -r -1 2λ/(2ln2) 1/γ γ (B5) f Exponential (r, λ) = exp -

TABLE I :

 I Characteristics of the samples : Mn contents and superconducting transition temperatures Tc for both Co-Mn-122 and P-Mn-122. In Co-Mn-122, the growth technique described in the text yields homogeneous Mn content amongst different samples from the same batch, except for yMn = 0.65 where a variation yMn = 0.65 ± 0.15 was observed.

		35	0	30
	2(As1-xPx)2	35	1.5	16
		35	3.0	< 2

Acknowledgements

The authors would like to thank H. Alloul, F. Bert, V. Brouet, P. Mendels, F. Rullier-Albenque for constructive discussions, S. Poissonnet (SRMP/CEA) for chemical analysis on the crystals and A. Larcelet for help with the experimental setup. We acknowledge support from the French ANR PNICTIDES.


Y.-X Yang

Y Gallais
email: yann.gallais@univ-paris-diderot.fr

R M Fernandes

I Paul

L Chauvière

M.-A Méasson

M Cazayous

A Sacuto

D Colson

A Forget

Raman scattering as a probe of charge nematic fluctuations in iron based superconductors

We report Raman scattering measurement of charge nematic fluctuations in the tetragonal phase of BaFe 2 As 2 and Sr(Fe 1-x Co x ) 2 As 2 (x=0.04) single crystals. In both systems, the observed nematic fluctuations are found to exhibit divergent Curie-Weiss like behavior with very similar characteristic temperature scales, indicating a universal tendency towards charge nematic order in 122 iron-based superconductors.

Introduction

In a nematic phase, the rotational invariance is broken while the translational symmetry remains intact. Widely studied in classical soft condensed matter systems like liquid crystals [START_REF] Prost | The Physics of Liquid Crystals[END_REF], the existence of their quantum analogue, electron nematics, has also been postulated in various strongly correlated systems [START_REF] Fradkin | [END_REF]. Examples include quantum Hall systems, underdoped cuprates, bi-layer ruthenate and heavy fermions [3][4][5][6]. In these systems, rather than a breaking of the continuous rotational symmetry as in liquid crystals, electronic nematic order usually implies a breaking of the discrete C 4 tetragonal symmetry. More recently, the strongly anisotropic electronic properties of iron-based high temperature superconductors have been linked to the existence of a underlying electron nematic order [7,8]. In AFe 2 As 2 (A=Ba,Sr,Ca) (122 family) compounds, a structural transition in which the Fe-Fe bonds in the x and y direction become inequivalent occurs simultaneously, or precedes, a magnetic transition below which the system displays stripe-like magnetic ordering with antiferromagnetic (ferromagnetic) alignment along the longer (shorter) Fe-Fe bonds. The structural transition between a tetragonal and an orthorhombic phase, observed in close proximity to superconductivity, is believed to be a direct consequence of electron nematic ordering [9][10][11].

Evidences that the structural transition is a consequence of electron nematic ordering come mostly from measurements under uni-axial stress in the orthorhombic phase. Transport, ARPES and optical conductivity measurements on mechanically detwinned crystals have revealed strong inplane electronic anisotropies which cannot be ascribed to the small orthorhombic distortion alone [7,[12][13][START_REF] Yi | Proc. Natl Acad. Sci[END_REF][START_REF] Dusza | [END_REF][16]. However since these experiments were performed under a symmetry breaking field, uniaxial stress, which already breaks the C 4 symmetry, most of them could not unambiguously determine whether the structural transition is a true nematic phase transition, nor the exact nature of the primary nematic order parameter, spin or charge/orbital [17][18][19][20]. In the case of transport, the observed anisotropies seem to depend strongly on disorder, raising the question of the role played by scattering mechanisms [21].

Another route for studying nematicity is to probe the associated response functions in the lattice, spin or charge sector in the tetragonal phase in the absence of any symmetry breaking field. In the case of the lattice, orthorhombic softening has been observed for a wide range of doping via shear modulus measurements in Co doped BaFe 2 As 2 single crystals [11,22]. However corresponding measurements of the electronic nematic response function in the charge/orbital or spin degrees of freedom are currently lacking. Here we show that the dynamical charge nematic response can be probed directly using electronic Raman scattering. For both SrFe 2 As 2 (Sr122) and BaFe 2 As 2 (Ba122) systems, in the tetragonal phase of strain-free crystals, the low energy Raman response displays a strongly temperature dependent quasi-elastic peak whose intensity increases upon approaching the structural transition and collapses in the orthorhombic phase. By extracting the static charge nematic susceptibility from our Raman measurements, we are able to unravel the presence of an incipient charge nematic order with very similar characteristic temperature scales for Ba122 and Sr122 systems, indicating the universal character of charge nematic fluctuations in the 122 family of iron-based superconductors.

Electronic Raman scattering and charge nematicity

Electronic Raman scattering in the symmetry channel µ is proportional to the weighted charge correlation function S µ (ω)= ρ µ (ω)ρ µ (-ω) . The correlation function S µ is in turn directly linked to the imaginary part of the Raman response (χ µ ) ′′ via the fluctuation dissipation theorem

S µ (ω) = -π (1 + n(ω, T ))(χ µ ) ′′ (ω) (1)
where n(ω, T ) is the Bose-Einstein distribution function. The electronic Raman response function χ µ is given by

χ µ (ω) = i ∞ 0 dte iωt [ρ µ (t), ρ µ (0)] (2)
where ρ µ = k γ µ k n k with n k the charge density operator and γ µ k (e i , e s ) a form factor, also called the Raman vertex. The symmetry index µ of the form factor depends on the polarizations of the incident and scattered lights e i and e s respectively. [23] Since the photon wave vector is several orders of magnitude smaller than the typical Brillouin zone size, there is negligible momentum transfer in the electron-photon scatterings, and therefore Raman spectroscopy probes the system uniformly.

The x 2 -y 2 (B 1g ) symmetry can be selected by choosing crossed incoming and outgoing photon polarizations at 45 • with respect to the Fe-Fe bonds (see figure 1a). Similarly the xy (B 2g ) symmetry can be selected by choosing incoming and outgoing photon polarizations along the Fe-Fe bonds. The momentum space structure of γ µ k is constrained by symmetry. In the case of x 2 -y 2 symmetry, γ µ k must change sign under mirror symmetry with respect to the direction at 45 • of the x and y axis. The k-space structure of these form factors are shown in figure 1a. In x 2 -y 2 symmetry, the effective charge ρ x 2 -y 2 is the product of the charge density n k and a form factor which changes sign under a 90 • rotation. This corresponds to the order parameter of a charge nematic φ = k γ x 2 -y 2 k n k with orientational order along the Fe-Fe bonds [START_REF] Fradkin | [END_REF]24]. Thus in this symmetry electronic Raman scattering is a direct probe of charge nematic order parameter fluctuations [25]. Besides, since χ µ is proportional to the square of the charge nematic order parameter φ 2 , it is possible to extract charge nematic fluctuations directly from the Raman response without applying any external symmetry breaking field such as uniaxial stress.

Sample and experiments

Single crystals of BaFe 2 As 2 (Ba122) and Sr(Fe 1-x Co x ) 2 As 2 (Co-Sr122) were grown using the self-flux method [26]. The as-grown BaFe 2 As 2 crystal was further annealed for three weeks at 700 • C and then slowly cooled down to 300 • C, resulting in an increase of T S by several degrees. Electronic Raman scattering results on two different single crystals, one for each family, are presented here: undoped BaFe 2 As 2 and Sr(Fe 1-x Co x ) 2 As 2 with x=0.04. Their structural transition temperatures, as determined by transport measurements, are identical: T S =138 K. Residual resistivity ratios (RRR=ρ 300K /ρ 0K ) for the two single crystals indicate different degrees of disorder: RRR ∼ 9 for annealed Ba122 and RRR ∼ 1 for Co-Sr122. The full Co doping dependence of the electronic Raman response for the Ba122 system can be found elsewhere [27]. In order to extract the imaginary part of the Raman response function, the raw spectra were corrected for the Bose factor and the instrumental spectral response. Data were taken in two different polarization configurations probing respectively x 2 -y 2 (B 1g ) and xy (B 2g ) symmetry. As explained above these two configurations are sensitive to charge nematic fluctuation along Fe-Fe bonds and at 45 • of the Fe-Fe bonds respectively. .'

Results and discussion

''(a )

! " ! # ! " ! # ! ! " " # γ ! (" # $% # & ' '() *+ &' γ ! ("%&'() #+ &' ! % ' ! " ' " # $% #' '() *+ &' " # $% #' '() *+ &' "% ' '() #+ &' "% ' '() #+ &'
,' -' .' [27]. In both symmetries, the spectra show a broad electronic continuum with sharp phonon lines superimposed to it. The structural transition induces a splitting of the in-plane Fe-As phonon mode at ∼ 130 cm -1 and an activation of the As out-of plane mode in the x 2 -y 2 symmetry at ∼ 180 cm -1 . These changes are particularly visible in the spectrum of Ba122 at 120 K and are discussed elsewhere [28,29]. Focusing on the electronic continuum, Fig. 1c shows that the spectra in xy symmetry display an essentially temperature independent and flat continuum in the tetragonal phase T ≥T S . By contrast the spectra in x 2 -y 2 symmetry are strongly temperature dependent: upon cooling the electronic Raman continuum shows a significant built-up of spectral weight at low energy, below ∼500 cm -1 . This quasi-elastic peak (QEP) emerges on top of a flat continuum similar to the one observed in xy symmetry, and extends up to much higher energies. Similarly to Ba122, the QEP intensity increases upon approaching T S before collapsing in the orthorhombic phase. The distinctive symmetry of this QEP and its temperature dependence clearly link it to growing dynamical charge nematic fluctuations of x 2 -y 2 symmetry in the tetragonal phase. A quantitative analysis of the charge nematic fluctuations can be obtained by extracting the static charge nematic susceptibility, χ x 2 -y 2 0 , from χ ′′ using Kramers-Kronig relation:

χ x 2 -y 2 0 = 2 π ∞ 0 dω(χ x 2 -y 2 ) ′′ (ω)/ω (3)
Because we only have access to a limited energy window, the integral was restricted to energies below 500 cm -1 since above this energy the spectra are temperature independent in the tetragonal phase. At low energy, below ∼10 cm -1 , the spectra were extrapolated using an overdamped Lorentzian form (see below). The resulting static susceptibilities for Ba122 and Co-Sr122 as a function of temperature are compared in Fig. 2. The enhancements of χ x 2 -y 2 0 are remarkably similar and display a maximum at T S =138 K. In the tetragonal phase, above T S , the nematic susceptibility can be fitted using:

χ x 2 -y 2 0 = a + b T -T 0 (4)
where a represents the temperature independent flat continuum and the Curie-Weiss term describe the diverging behavior of the QEP. The extracted T 0 are very similar, T 0 =105 K (± 5 K) for Co-Sr122 and T 0 =95 K (± 5 K) for Ba122. Comparison of the charge nematic susceptibilities of Ba122 and Co-Sr122 thus indicates an universal behavior among members of the 122 family, and a relative insensitivity of charge nematic fluctuations to static disorder. The divergent part of the charge nematic susceptibility can also be extracted by subtracting the flat temperature independent continuum from the electronic Raman response. This can be done by fitting the electronic Raman continuum at 300 K, far away from T S where it becomes almost temperature independent, and subtracting it from the data at temperatures close to T S . The resulting QEP contribution is shown in Fig. 3 (a.u.)

T(K) T 0 =105K χ !"# =AωΓ/(ω 2 +Γ 2 )
$% &% '% (% ity, χ x 2 -y 2 0,QEP , is given by the amplitude A of the QEP:

χ x 2 -y 2 0,QEP = 2 π ∞ 0 dωχ ′′ QEP (ω)/ω = A.
As shown in figure 3, this relaxational form provides a good description of the energy dependence of the QEP for temperatures up to at least ∼ 60 K above T S . In the case Ba122 the extracted A -1 , equivalent to the charge nematic stiffness, shows a linear temperature dependence, A -1 ∼ T -T 0 , in agreement with a Curie-Weiss-like behavior. We note that deviations from linearity above 180 K might be ascribed to the fact that the subtracted background is approximated by the 300 K spectrum. This approximation makes departure from Curie-Weiss behavior unavoidable at high temperatures. On the other hand no solid conclusions can be drawn from the behavior of A -1 for Co-Sr122 due to a more limited number of data points. Nevertheless for both Ba122 and Co-Sr122 the linear interpolation of A -1 yields similar values of T 0 as the ones found in the previous analysis: 100 K(± 5 K) and 105 K (± 10 K) respectively.

It is interesting to note that shear modulus measurements on Ba122 found a similar Curie-Weiss-like softening [11,22]. However the Curie-Weiss temperature extracted from shear modulus measurements was found to be very close to T S . This is in contrast to the one extracted here which is well below T S by at least ∼ 40 K, implying that the charge nematic fluctuations grow slower than the lattice ones [27]. It indicates that the observed charge nematic fluctuations are not sufficient to drive the structural transition, and that the charge nematic order which would otherwise occurs at a lower temperature T 0 , is pre-empted by the structural transition. It therefore seems that another degree of freedom is responsible for the softening of the shear modulus. A recent analysis of NMR spin-lattice relaxation rate data indicates that spin fluctuations can account for the observed lattice softening via magneto-elastic coupling [17,[30][31][32]. It remains to be seen if magnetic fluctuations can also account for the observed charge nematic softening via spin-charge coupling. In summary, electronic Raman scattering experiments unveil the presence of charge nematic fluctuations in the tetragonal phase of Ba122 and Sr122 systems. Their divergent behavior indicates the presence of an incipient charge nematic instability in the phase diagram of 122 iron superconductors. Our results also establish Raman scattering as a key probe of uniform charge order and fluctuations such as nematics, which might be relevant in other strongly correlated electron systems.

Fig. 1 .

 1 Fig. 1. (Color online) a: Polarization configurations showing the incoming, e i , and outgoing e s photon polarizations with respect to the Fe square plane for x 2 -y 2 (B 1g ) and xy (B 2g ) symmetries. The notation B 1g refers to the one Fe unit cell whose axes are along the Fe-Fe bonds. The k-space structure of the Raman form factors (or vertex) γ k is depicted for each symmetry. b: Temperature dependent Raman spectra for Sr(Fe 1-x Co x ) 2 As 2 (x=0.04) and c for BaFe 2 As 2 (data taken from [27]) in the two symmetries.

Figure 1b ,

 1b Figure 1b, c display low energy temperature dependent Raman spectra for Co-Sr122 in x 2 -y 2 (B 1g ) (b) and xy (B 2g ) (c) symmetries. The data for Ba122 are shown for comparison[27]. In both symmetries, the spectra show a broad electronic continuum with sharp phonon lines superimposed

Fig. 2 . 2 -y 2 0 for BaFe 2

 2202 Fig. 2. (Color online). Temperature dependence of the static charge nematic susceptibility χ x 2 -y 2 0 for BaFe 2 As 2 and Sr(Fe 0.96 Co 0.04 ) 2 As 2 . The lines are Curie-Weiss law fits of the susceptibilities in the tetragonal phase (T≥T S ).

Fig. 3 .

 3 Fig. 3. (Color online). a,c: Temperature dependence of the extracted quasi-elastic peak (QEP) contribution for BaFe 2 As 2 and Sr(Fe 0.96 Co 0.04 ) 2 As 2 along with fits using a relaxational Lorentzian form (see text). b, d: temperature dependences of the inverse of the divergent contribution to the static charge nematic susceptibility, A -1 , extracted from the Lorentzian fits. The lines are linear (Curie-Weiss) extrapolation of the data between 200 K and T S =138 K, allowing a determination of the characteristic temperature T 0 for both systems.

  as a function of temperature for Ba122 and Co-Sr122. The QEP can be modeled by a Lorentzian relaxational form with an amplitude A and a scattering rate Γ:χ ′′ QEP = A ωΓ Γ 2 +ω 2 .Within this framework the divergent part of the static charge nematic susceptibil-

		Sr(Fe 0.96 Co 0.04 ) 2 As 2	198K
		T S =138K		177K
				158K
	χ'' QEP (a.u.)			140K	A -1
	0	50	100	150	200	50	100 150 200 250
			Raman shift (cm -1 )
		T S =138K BaFe 2 As 2		192K 156K 178K
	χ'' QEP (a.u.)			142K		Α -1 (a.u.)
							T 0 =100K
	0	50	100	150	200	50	100 150 200 250
			Raman shift (cm -1 )				T(K)

We thank F. Rullier-Albenque for providing us with transport data. Y.X. Y., Y.G., L.C., M.C, M.-A.M., A.S., D.C. and A.F. acknowledge support from Agence Nationale de la Recherche through ANR Grant PNICTIDES.


Sylvie Boldo
email: sylvie.boldo@inria.fr

Computing a correct and tight rounding error bound using rounding-to-nearest

When a floating-point computation is done, it is most of the time incorrect. The rounding error can be bounded by folklore formulas, such as ε|x| or ε|•(x)|. This gets more complicated when underflow is taken into account as an absolute term must be considered. Now, let us compute this error bound in practice. A common method is to use a directed rounding in order to be sure to get an over-approximation of this error bound. This article describes an algorithm that computes a correct bound using only rounding to nearest, therefore without requiring a costly change of the rounding mode. This is formally proved using the Coq formal proof assistant to increase the trust in this algorithm.

Introduction

Floating-point (FP) arithmetic is the way computers deal with computations on real numbers. It is clearly defined [START_REF]IEEE Standard for Floating-Point Arithmetic. IEEE Std[END_REF][START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF], but it is not perfect. In particular, FP numbers have a finite precision, therefore even a single computation may be incorrect when the result does not fit in a FP number. This error is called a rounding error, and a part of the computer arithmetic literature tries to improve or bound these errors on some given algorithms.

The most common model for bounding this rounding error is the standard model [START_REF] Nicholas | Accuracy and stability of numerical algorithms[END_REF] where

•(x) = x(1 + δ)
with |δ| ≤ u with • being the rounding to nearest and u being the machine epsilon, therefore 2 -p when p is the number of bits of the FP mantissa. This model has two main drawbacks. The first one is that it does not take underflow into account. For example in binary64, if you round 3 × 2 -1075 , you get the FP number 2×2 -1074 , which gives a huge relative error of 33 % compared to the input, but a small absolute error.

The second drawback is when this error bound needs to be computed. That is the case for example when considering midpoint-radius interval arithmetic [START_REF] Rump | Fast and parallel interval arithmetic[END_REF]. Indeed, computing • (2 -p × •(x)) may not be an overestimation of the error bound. For example, let us consider x = 2 -1022 +2 -1074

2

. Then •(x) = 2 -1023 and the error is 2

-1075 . But • (2 -p × •(x)) = • 2 -53 × 2 -1023 = • 2 -1076 = 0, which is not an overestimation of 2 -1075 .
This article aims at providing a correct algorithm that computes a bound on the error of a FP operation. As we also want this algorithm to be fast, we wish to avoid changing the rounding mode, as it breaks the pipeline. We will therefore only consider rounding to nearest, both for the operation considered, and for our algorithm. Moreover, we want to prevent tests as they also break the pipeline.

More than a pen-and-paper proof, this work gives a high guarantee of its correctness and gives precise hypotheses on the needed precision and underflow threshold. We will rely on the Coq proof assistant. From the formal methods point of view, we will base our proof on the Flocq library [START_REF] Boldo | Flocq: A unified library for proving floatingpoint algorithms in Coq[END_REF]. Flocq is a formalization in Coq that offers a multi-radix and multi-precision formalization for various floating-and fixed-point formats (including FP with or without gradual underflow) with a comprehensive library of theorems. Its usability and practicality have been established against test-cases [START_REF] Boldo | Deductive Formal Verification: How To Make Your Floating-Point Programs Behave[END_REF]. The corresponding Coq file is named Error bound fp.v and is available in the example directory of Flocq1 , available in the current git version, and in the next released versions > 2.5.1.

Notations We denote by • the rounding to nearest, ties to even in radix 2. We denote by •[expr] the rounding of the expression into brackets, where all the operations are considered to be rounded operations. For example,

•[3 × x + y] denotes •(•(3 × x) + y).
The smallest subnormal number is denoted by 2 Ei and the number of bits of the mantissa is p > 0. For basic knowledge about FP arithmetic (roundings, subnormal, error bounds), we refer the reader to [START_REF] Goldberg | What every computer scientist should know about floating-point arithmetic[END_REF][START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF].

Theorem

Theorem 1. Let x be a real number. Assume that

E i ≤ -p. Then |•(x) -x| ≤ • 2 -p × |•(x)| + 2 Ei .
The assumption is very light: E i ≤ -p only means that 2 -p is in the format. It holds in all IEEE formats: for example in binary64, p = 53 and E i = -1074 and in binary32, p = 24 and E i = -149.

Proof The first step is to prove that 2 -p and 2 Ei are in the format, therefore not rounded. This is trivial as long as E i ≤ -p.

Then, the error bound we are used to is

|•(x) -x| ≤ 2 -p × |•(x)| + 2 Ei-1 or max 2 -p × |•(x)|, 2 Ei-1 .
The first formula looks like our the theorem, rounding excepted. The other difference is the 2 Ei instead of 2 Ei-1 . Let us prove that the roundings do not endanger the result. Ei . Then, we split the proof into 4 cases, depending on the value of |x|.

Let t = • 2 -p × |•(x)| + 2
1. Assume x = 0. Then •(x) = 0, and |•(x) -x| = 0, while t = • 0 + 2 Ei = 2 Ei . So the result holds. 2. Assume 0 < |x| < 2 Ei+p . Then x has exponent E i and |•(x)-x| ≤ 1 2 ulp(x) = 2 Ei-1 . Moreover, t = • 2 -p × |•(x)| + 2 Ei ≥ • 2 Ei = 2 Ei . So the result holds. 3. Assume 2 Ei+2p-1 ≤ |x|. Then 2 -p × |•(x)| ≥ 2
Ei+p-1 and is normal. Therefore, the multiplication by 2 -p is correct and does not create any rounding error.

Then |•(x)-x| ≤ 2 -p ×|•(x)| = • [2 -p × |•(x)|] ≤ • 2 -p × |•(x)| + 2
Ei by monotony of the rounding. 4. Assume 2 Ei+p ≤ |x| < 2 Ei+2p-1 . This is the most complex case, as all roundings may go wrong.

2

This pen-and-paper proof exactly corresponds to the Coq proof (as it was written from it). The itemized cases corresponding to the interval values of x become several lemmas in the Coq proof for the sake of readability.

Tightness of the bound

The next question is how tight is the proved bound. In particular, is it much coarser than the usual bound? The answer is no and a graphical explanation is given in Figure 3.

When |x| is small, then the optimal bound is 2 Ei-1 . As this bound is not a FP number, our algorithm returns the best possible bound, that is to say 2 Ei . When |x| is big enough, meaning greater than 2 Ei+2p (that is 2 -968 in binary64), we have the optimal bound, meaning 2 -p × |•(x)|. In this case, the 2 Ei is indeed negligible and is neglected as we use rounding to nearest (and not rounding towards +∞). In the middle, meaning between 2 Ei+p-1 and 2 Ei+2p , we have a slight overestimation compared to the usual bound. Note that this overestimation is bounded by 2 Ei .

|x| Error bounds max 2 -p × |•(x)| , 2 E i -1 • 2 -p × |•(x)| + 2 E i 2 E i 2 E i +1 2 E i +p-1 2 E i +p 2 E i +2p-1 2 E i +2p 2 E i -1 2 E i 2 E i +1 2 E i +p-1 2 E i +p

Conclusion

We have formally proven that the following algorithm using only the rounding-to nearest mode: fabs(x)*0x1.p-53+0x1.p-1074

gives a correct tight bound on the rounding error of x in rounding-to-nearest binary64. Note that overflow cannot happen here as all values involved are strictly smaller than the input x.

As for efficiency, random tests have shown it is quite efficient, as it involves neither tests, nor rounding change (2 flops plus the memory accesses). Nevertheless, on some architectures, subnormal numbers are trapped and handled in software, and are therefore much slower than normal FP operations. In this case, computing x*0x1.p-53+0x1.p-1022 might be a better idea. Indeed, the previ-ous theorem implies that • 2 -p × |•(x)| + 2 Ei+p-1 is also an overestimation of the rounding error. And as 2 Ei+p-1 is the smallest normal number, this algorithm does not involve any subnormal number if x is not one and will be faster in most cases, at the price of a worse bound. A use of the processor max function may also prevent the use of operation on subnormal numbers, which is known to be quite costly.

A perspective is to be able to compute an error bound on a given rounding, using only this given rounding. The formula will probably need to be modified, for example suppressing the addition when rounding towards +∞, negating twice the values when using rounding towards -∞. But this needs to be worked out in depth and formally proved to get correct algorithms.

A harder perspective is to deal with radix 10. Then the multiplication should not be by 2 -p , but by 5 × 10 -p , and this multiplication is not always exact with big numbers, as was the case here with radix 2. The provided algorithm does therefore not hold in radix 10.

  First |•(x)-x| ≤ 2 -p ×|•(x)|. Let y be 2 -p ×|•(x)|. Then y < 2 Ei+p-1 and is therefore in the subnormal range. As |•(x)-x| ≤ y, what is left to prove is that y ≤ • •(y) + 2 Ei . As y is small, •(y)is a positive FP number in the subnormal range, therefore •(y) + 2 Ei is also in the FP format and • •(y) + 2 Ei = •(y) + 2 Ei . What is left to prove is then y ≤ •(y) + 2 Ei . Finally, as y is in the subnormal range, |•(y) -y| ≤ 2 Ei-1 . So y ≤ •(y) + 2 Ei-1 and the result holds.

Fig. 1 .

 1 Fig. 1. Drawings comparing the error bounds of •(x)

http://flocq.gforge.inria.fr/.

This work was supported by the FastRelax (ANR-14-CE25-0018-01) project of the French National Agency for Research (ANR).


This paper is about the formal verification of a floating-point case study where the common iterators fold_left and fold_right have not the wanted behaviors. We then had to define other iterators, which are very similar in most cases, but that do behave well in our case study.

Our case study: floating-point expansions

This is about floating-point arithmetic, defined by the IEEE-754 standard [START_REF]IEEE Standard for Floating-Point Arithmetic[END_REF][START_REF] Goldberg | What every computer scientist should know about fl oating point arithmetic[END_REF]. It has been formalized in the Coq proof assistant by the Flocq library [START_REF] Boldo | Flocq: A unified library for proving floating-point algorithms in Coq[END_REF].

Our case study is a way to increase the precision of a computation, in order to get more accuracy on critical systems. To retain efficiency, we rely on the floating-point unit (FPU) of the processor. The idea of these floating-point expansions is to consider x as x = x i with the x i being floating-point numbers, see Figure 1. Using exact addition and multiplication [START_REF] Dekker | A Floating-Point Technique for Extending the Available Precision[END_REF][START_REF] Ervin | The Art of Computer Programming: Seminumerical Algorithms[END_REF], we can build correct operations on these expansions such as addition, multiplication, division, and so on [START_REF] Dekker | A Floating-Point Technique for Extending the Available Precision[END_REF][START_REF] Douglas | Algorithms for arbitrary precision floating point arithmetic[END_REF][START_REF] Richard | Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates[END_REF][START_REF] Bailey | A Portable High Performance Multiprecision Package[END_REF].

We are in the progress of proving operations on expansions. It is important as there may be intermediate zeroes in these expansions that may endanger the algorithms. Moreover, the condition of use of these algorithm: allowed overlap, underflow, overflow, and so on are hardly known, and not formally verified. In particular, it relies on the definition of a0:A that stands for the zero of A. For real numbers, we can consider fold_right Rplus R0 (1::2::3::nil) that will be simplified in 1 + (2 + (3 + 0)). This is perfect as Rplus is the mathematical + on R with R0 = 0 is the identity element. If you consider f as the floating-point addition, it is fine as 0 is also an identity element for this function.

Now consider the function errf (x, y) = •(x + y) -(x + y), with • being a rounding to nearest. Then f computes the error of the floating-point addition, which is also a floating-point number when x and y are floating-point numbers, and that may be computed using only floating-point operations [START_REF] Dekker | A Floating-Point Technique for Extending the Available Precision[END_REF][START_REF] Ervin | The Art of Computer Programming: Seminumerical Algorithms[END_REF]. Then fold_right errf R0 (a::b::c::nil) is (errf a (errf b (errf c 0))). As errf (x, 0) = 0, this value is 0 whatever a, b and c. And this is not what we want as we wished to have (errf a (errf b c)). This is not a toy example as such an iteration of errf on a sequence is the first part of a renormalization algorithm [START_REF] Joldes | Arithmetic algorithms for extended precision using floating-point expansions[END_REF].

Solution

Of course, this problem with fold_right also applies to fold_left. We have therefore defined two other iterators:

Definition my_iter_l f l := match l with | nil = > 0 | x :: l = > fold_left f l x end . Definition my_iter_r f l := match l with | nil = > 0 | _ = > fold_right f ( last l 0) ( removelast l ) end .
They have the expected behavior on the previous example. The correctness of my_iter_r is given by the following theorem: If there exists an identity element b, then the two iterators have the same result, providing either b = 0 or the list is not empty. A similar result for my_iter_l is useless as it simplifies immediately on fold_left as soon as the list is non empty. Several useful lemmas have also been proved, such as my_iter_l f l = my_iter_r (fun x y =>f y x) (rev l).

As a conclusion, we have defined other iterators that comply with our needs. In particular, they do not expect an identity element on the function, but are more difficult to handle in the proofs. They could have been made more generic, in the returned value when the list is empty (here 0), and in the input type (here R). It is nevertheless surprising that this problem has not arisen before on other applications.

  Iterators: where folds fail Sylvie Boldo * † Inria, Université Paris-Saclay, F-91893 Palaiseau LRI, CNRS & Univ. Paris-Sud, F-91405 Orsay

Figure 1 :

 1 Figure 1: Example of a floating-point expansion.

  Lemma m y _ i t e r _ r _ f o l d _ r i g h t : forall ( f : R ->R ->R ) b l , ( forall x , f x b = x ) -> ( l <> nil \/ b = 0) -> my_iter_r f l = fold_right f b l .

* This work was supported by the FastRelax (ANR-14-CE25-0018-01) project of the French National Agency for Research (ANR).