Competitive and Cooperative electronic states in Ba(Fe$_{1-x}$T$_x$)$_2$As$_2$ with T=Co, Ni, Cr

The electronic structure inhomogeneities in Co, Ni, and Cr doped BaFe2As2 122 single crystals are compared using scanning tunneling microscopy/spectroscopy (STM/S) at the nanoscale within three bulk property regions in the phase diagram: a pure superconducting (SC) dome region (Co-122), a coexisting SC and antiferromagnetic (AFM) region (Ni-122), and a non-SC region (Cr-122). Machine learning is utilized to categorize the various nanometer scale inhomogeneous electronic states, described here as in-gap, L-shape and S-shape states immersed into the SC matrix for Ni-and Co-doped 122, and L-shape and S-shape states into the metallic matrix for Cr-doped 122. Although the relative percentages of in-gap, L-shape and S-shape states vary in the three samples, the total volume fraction of the three electronic states is quite similar. This is coincident with the number of electrons (Ni0.04 and Co0.08) and holes (Cr0.04) doped into the 122 compound. By combining the volume fractions of the three states, the local density of states (LDOS), magnetic field dependent behavior and global properties in these three samples, the in-gap state is confirmed as a magnetic impurity state from the Co or Ni dopants. In addition, the L-shape state is identified as a spin density wave (SDW) which competes with the SC phase, and the S-shape state is found to be another form of magnetic order which constructively cooperates with the SC phase, rather than competing with it. The comparison of the vortex structures indicates that the inhomogeneous electronic states serve as pinning centers for stabilizing the vortex lattice.

inhomogeneous electronic states, described here as in-gap, L-shape and S-shape states immersed 23 into the SC matrix for Ni-and Co-doped 122, and L-shape and S-shape states into the metallic 24 matrix for Cr-doped 122. Although the relative percentages of in-gap, L-shape and S-shape states 25 vary in the three samples, the total volume fraction of the three electronic states is quite similar. 26 This is coincident with the number of electrons (Ni0.04 and Co0.08) and holes (Cr0.04) doped into the 27 122 compound. By combining the volume fractions of the three states, the local density of states 28 (LDOS), magnetic field dependent behavior and global properties in these three samples, the in-29 gap state is confirmed as a magnetic impurity state from the Co or Ni dopants. In addition, the L-30 shape state is identified as a spin density wave (SDW) which competes with the SC phase, and the 31 S-shape state is found to be another form of magnetic order which constructively cooperates with 32 the SC phase, rather than competing with it. The comparison of the vortex structures indicates that Introduction 37 The interplay between magnetism and superconductivity (SC) is one of the fundamental 38 topics for understanding the mechanism of superconductivity of unconventional superconductors 39 in the cuprates and iron-based superconductors (FeSCs) 1,2 , since SC appears near 40 antiferromagnetic (AF) order. In cuprates, experiments demonstrated that the superconducting 41 pairing state is the unconventional dx2-y2, which is believed induced by spin fluctuations [3][4][5] . 42 Similarly to the cuprates, magnetism in the FeSCs plays an important role in the electron pairing 43 mechanism 2,4 . However, in the FeSC it is more complicated and controversial since Fe 3d orbitals 44 form multiple Fermi surfaces (FS), while in cuprates only a single Cu d-band crosses the FS 6-9 . 45 For example, in the most studied FeSC family of pnictides BaFe2As2 (122), SC can be realized by 46 atomic substitution at any element site, for example by hole-doping onto Ba site 10 , electron-doping 47 on Fe site 11 12 and isovalent doping on As site 13 . However, it is intriguing that crystals with 48 electron-doping to Fe site (with Ni and Co) are superconducting while crystals with hole-dopants 49 (with Cr) are not 14 . 50 On the other hand, for the electron-doped 122 crystals, it is well established that upon 51 electron doping via Co/Ni substitution for Fe, the collinear long-range AF order is suppressed, and 52 SC then appears. However, there are ongoing debates concerning the relationship between the AF 53 order and SC. One perspective arises from a presumed itinerant nature of magnetism. In this view 54 the static AF order arises from the formation of a spin-density-wave induced by itinerant electrons 55 and Fermi surface nesting of the electron and hole pockets. Upon doping the pair scattering 56 between the electron and hole like FS pockets then leads to SC 15,16 . A different perspective arises 57 from a presumed localized nature of magnetism: the short-range incommensurate AF order is a 58 cluster spin glass phase arising from disordered localized moments 12,17 . There is an additional 59 perspective that local moments and itinerant electrons coexist. In this scenario part of the Fe d 60 bands are delocalized and contribute to the itinerancy, whereas others are localized due to a strong 61 correlation effect and provide the source for local moments 18,19 . This model was used to explain 62 the scanning tunneling microscopic observation of the coexistence as well as the co-disappearance 63 of a pseudogap-like feature and superconductor in the NaFe1-xCoxAs (Co-111) system where the 64 pure itinerant picture apparently fails 20 . A muon spin rotation study also observed a spatially 65 inhomogeneous magnetic state developing below Tc which, surprisingly, has a constructive 66 relationship with SC in near optimal or overdoped Co-122 samples 21,22 . This is intrinsically 67 different from the reported SDW phase in NaFe1-xCoxAs (Co-111) which competes with, and is 68 anti-correlated, with the SC phase 23 . 69 Here, we investigate the local electronic structure of 122 crystalline matrix BaFe2As2 that 70 is doped with magnetic elements (Ni, Co, or Cr) at the same amount of electron versus hole doping 71 levels per Fe atom. We systematically explore the electronic structure signatures of the magnetism 72 and SC phases using low temperature high magnetic-field scanning tunneling 73 microscopy/spectroscopy (STM/S). The machine learning technique of K-means clustering 74 method is employed to categorize the various nanometer-size inhomogeneous electronic states. In 75 addition to the SC state, an in-gap state, a competitive L-shape and a cooperative S-shape state are 76 found to coexist in the samples. The in-gap state in the SC crystals is confirmed to be a magnetic 77 impurity state from Co or Ni dopants, while the L-shape state is identified as an SDW that 78 competes with the SC phase. In addition, the S-shape state originates from a local magnetic order 79 that surprisingly, constructively cooperates with the SC phase rather than competing with it. A 80 comparison of the vortex structures indicates that those inhomogeneous electronic states serve as 81 pinning centers for stabilizing the hexagonal vortex lattice for the bulk superconductors.

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For the purpose of comparison, we have studied three sets of samples of the transition-  red and yellow balls in the temperature-composition (T-x) phase diagram in Fig. 1(a); the crystal 96 structure of the undoped parent 122-unit cell is in inset.

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After low-temperature cleavage, the three sets of samples show similar topography, which 98 is the coexistence of 2×1 and √2 × √2 reconstructions, as reported for Co-122 before 25,26 . Fig. 1 shows a typical large-scale topographic image of Ni-122, with terraces, steps in 0.7 nm (half unit 100 cell in c axis) and defects. The atomic resolution image collected from the red box of Fig. 1  Dramatic local electronic inhomogeneity is found on the Ni-122 surface. Fig. 1(d) presents 109 the STS spectra along the black arrow in Fig. 1(c). Although the morphology of the areas around   Fig. 4(d). At higher fields, the vortex lattice gradually changes to a rectangular 233 shape at 4 T ( Fig. 4(e)) and becomes a more isotropic ring-like structure at 6 T ( Fig. 4(f)). The six-

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The magnetic field influence on all states is summarized in Fig. 5, which shows the result 259 from Co-122 under magnetic field up to 6 Tesla. In this plot, each set of field dependent spectra is Tesla, but the intensity of the peak increases slightly with the magnetic field as shown in Fig. 5(b).

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A similar robust zero-energy bound state was also observed on Fe(Te,Se) 32 which in that material 265 is caused by interstitial Fe impurities.

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For both the S-shape and L-shape states, there is no significant change detected up to 6 T 267 as shown in Fig. 5(c) and (d). These observations exclude observation of the Kondo effect of 268 magnetic moment in metal/superconductor since no splitting was found in the magnetic field 269 dependent spectra. Although the S-shape and L-shape spectra look similar to the Fe-vacancy-270 induced bound state found in the K-doped iron selenide 33 , their extremely weak field dependence indicates that either the g-factor in our case is far smaller than 2.1 or that these spectra are from a 272 different origin.

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The L-shape like state has been identified as a spin density wave spectrum with the The S shape state was not reported previously. Although the present data set cannot firmly 287 prove the origin of this phase, we believe it is another form of magnetic state, based on its spectral 288 features. The main differences between the S and L shape states are: 1) the asymmetric spectrum 289 is tilted to the negative bias; 2) the Fermi surface residual DOS of the L shape is low, very close 290 to the bottom of SC gap, but for S shape the Fermi surface residual is much higher (Fig. 3 (c)  The data sets that support the findings in this study are available from the corresponding 373 author upon request.

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Code availability 375 The codes that support the findings in this study are available from the corresponding 376 author upon request.     Fig. 1(a), V Bias = -20 mV and I t = 100 pA, the scale bar represents 6 nm. (b) STS map at Fermi level 497 (0 meV) and (c) superconducting gap map from the same area (blue box in (a)), the scale bar represents 3 nm. (d) and 498 (e) The line STS spectra along the brown and green arrows in (b). Brown and green circles in (b) and spectra in (d)

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and (e) outline the non-SC areas. (f) Three K-means principal responses and (g) cluster map which shows the spatial 500 distribution of each type of spectra from the same area as (b).