Andreev Spectroscopy of EuCsFe4As4 Stoichiometric Superconducting Pnictide

EuCsFe4As4 pnictide (1144 family) single crystals with a critical temperature of Tc ≈ 36.8 K and optimal superconducting properties in the stoichiometric composition are grown. Multiple-gap superconductivity determined by bulk order parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{{\text{L}}}^{{{\text{out}}}}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{{\text{L}}}^{{{\text{in}}}}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\Delta }_{{\text{S}}}}$$\end{document} with characteristic ratios \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\Delta _{{\text{L}}}^{{{\text{out}}}}(0){\text{/}}{{k}_{{\text{B}}}}{{T}_{{\text{c}}}} \approx 5.3$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\Delta _{{\text{L}}}^{{{\text{in}}}}(0){\text{/}}{{k}_{{\text{B}}}}{{T}_{{\text{c}}}} \approx 3.2$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2{{\Delta }_{{\text{S}}}}(0){\text{/}}{{k}_{{\text{B}}}}{{T}_{{\text{c}}}} \approx 1.3$$\end{document}, respectively, is found for the first time below Tc using incoherent multiple Andreev reflection effect spectroscopy. It is shown that the gap structure of EuCsFe4As4 is similar to that of BaFe2 – xNixAs2 superconducting pnictides of the sister 122 family.


INTRODUCTION
Among iron-containing pnictides, the recently discovered 1144 family, being a full stoichiometric analog of the BaFe 2 As 2 system (the so-called 122 family), is of particular interest. A typical member of 1144-pnictides, EuCsFe 4 As 4 has a tetragonal structure consisting of two inequivalent Eu-FeAs-Cs-FeAs blocks with the 122 structure alternating along the c direction [1]. Rapid degradation of superconducting properties of EuCsFe 4 As 4 on the surface by oxygen or water vapor caused by the presence of an alkali metal in the structure strongly complicates experimental work with such crystals [2]. Therefore, both the preparation and the experiment itself should be carried out in a protective atmosphere.
Unlike related 122-pnictides, which require a certain level of doping for the appearance and optimization of superconducting properties, the 1144 family shows optimal superconducting properties directly in the stoichiometric composition, and their critical temperature T c ≈ 36 K is close to the maximum T c ≈ 38 K reached in the optimally doped Ba-122 pnictides. Among other 1144 family compounds, magnetic ordering in the Eu sublattice in the superconducting state is a unique feature of EuCsFe 4 As 4 [1,3].
The band structure of EuCsFe 4 As 4 has not yet been calculated. However, for some compounds, calculations show that the bands formed by Eu orbitals are located on average 1.5 eV below the Fermi level (see Fig. 3 in [4]), while the Fermi level is crossed by about ten bands formed by various iron d orbitals and forming nested Fermi surface cylinders around Γ and M points of the Brillouin zone, slightly corrugated along the momentum direction [4,5].
Although the gap structure of the 1144 family also has not yet been calculated, by analogy with Ba-122 pnictides, the development of several superconducting condensates below T c with order parameters anisotropic in the k space can be expected within both [6] and approaches [7]. In this work, the gap structure of EuCsFe 4 As 4 stoichiometric pnictide is studied experimentally for the first time. The superconducting energy gap parameters of a superconductor at and their characteristic ratios are directly determined using incoherent multiple Andreev reflection spectroscopy. It is shown that the structures of the superconducting order parameter of EuCsFe 4 As 4 pnictides and their stoichiometric analogs Ba(Fe, Ni) 2 As 2 , which we studied earlier, are similar.

EXPERIMENTAL DETAILS
EuCsFe 4 As 4 single crystals were grown from the melt of the CsAs component by the so-called "selfflux" method [2]. To synthesize EuAs, Fe 2 As, and CsAs precursors, we used high-purity elements Eu (99.95%),

CONDENSED MATTER
Fe (99.98%), Cs (99.99%), and As (99.9999%). The resulting precursors were placed in corundum crucibles in the required stoichiometric ratio and brewed into tantalum containers, after which a long-term multistage heat treatment was carried out in an argon atmosphere at temperatures 1200 and 900°С, respectively. As a result, we obtained 2.5 × 2-mm EuCsFe 4 As 4 single crystals (Fig. 1a, inset).
Since the 1144 phase is unstable in air, the selection of single crystals, their preparation and assembly, and experiments were carried out in a glove box with a dry argon atmosphere to prevent the degradation of the superconducting properties. Four-point resistance measurements showed a fairly sharp superconducting transition at T c ≈ 36.8 K with a width of K (Fig. 1b). Above T c , the increase in and no structural transitions were observed. The temperature dependence of the magnetic susceptibility with the field orientation parallel to the crystallographic ab plane was measured on a PPMS-9 Quantum Design setup. At T m ≈ 15 K < T c , a magnetic transition of the EuCsFe 4 As 4 phase is observed (marked by an arrow in Fig. 1a). As a rule, the 1144 phase competes with the related 122 phase during crystal growth, and it is rather difficult to obtain a pure 1144 phase. Nevertheless, having worked out the growth temperature regimes, we managed to obtain a EuCsFe 4 As 4 single crystal with trace inclusions of the nonsuperconducting EuFe 2 As 2 phase (see a minor feature in at T m ≈ 19 K).
Using the mechanically controlled planar breakjunction technique [8], we formed superconductorthin normal metal-insulator-thin normal metalsuperconductor (SnInS) junctions. The design of the measuring setup and the details of the experiment with samples of layered compounds and the advantages and disadvantages of the mechanically controlled planar break-junction technique are described in detail in [9]. The cracking of a layered single crystal at 4.2 K gives rise to two cryogenic cleavages with steps and terraces on their surfaces. By fine mechanical adjustment, it is possible to ensure that the cryogenic clefts slide with respect to each other along the terraces in the crystallographic plane, while the current flows through the junction always along the c direction. The physical model of the junction is schematically shown in Fig. 1 in [10].
The multiple Andreev reflection effect takes place in the SnInS junction below T c . In the case of the high-transparency insulating layer I (the barrier parameter ) and the absence of phase coherence between the superconducting banks ("long junction"), the Andreev transport induces an excess current in the entire range of bias voltages (with respect to the normal current above T c ). In the spectrum of such a junction at , the dynamic conductance increases by several times with respect to that in the normal state (i.e., a foot region is formed), while the supercurrent branch is absent [11][12][13]. A series of subharmonic gap structure dips also appear, the positions of which are directly determined by  (1) The subharmonic gap structure dips from each gap will be present in the spectrum for a multigap superconductor.
The numerical calculation within the formalism proposed in [16] showed that sharp intense subharmonic gap structure dips are observed for an isotropic superconducting gap, while small amplitude subharmonic gap structure dips are expected in the case of an order parameter with zero points in its k distribution ("nodes"). In the case of extended s wave symmetry in the (k x , k y ) plane (without nodes), doublet subharmonic gap structure dips are expected. The width of the doublets is determined by the maximum Δ out and minimum Δ in coupling energies of Cooper pairs in the k space (see Fig. 4 in [9]). Hereinafter, the degree of anisotropy of the superconducting gap Δ is determined as . Since the normal junction resistance (ρ is the resistivity, l is the mean free path, and d is the contact dimension; for a planar junction, all these values are taken in the plane) in the ballistic regime increases together with the ratio , we can expect the ballistic transport in high-resistance SnS junctions (having a relatively small diameter). Indeed, the normal resistances R N of the studied planar high-transparency SnS junctions are up to 122 Ω (Fig. 2), which exceed typical values Ω obtained in pnictides of other families in our previous works [17,18] and allow us to conclude that the transport in the plane is ballistic. Thus, the x and y momentum components of charge carriers can be conserved in the multiple Andreev reflection process, which makes it possible to obtain some information about the Δ anisotropy in the (k x , k y ) plane. At the same time, in the c direction, inelastic scattering and "mixing" of carriers along the current direction can occur.
Summing up the advantages of the experimental technique used, incoherent multiple Andreev reflection spectroscopy of the mechanically controlled planar break-junction Andreev junctions makes it possible to directly determine the superconducting energy gap parameters at temperatures and indirectly estimate their possible anisotropy in the (k x , k y ) plane.

EXPERIMENTAL DATA AND DISCUSSION
Typical current-voltage characteristics (CVCs) of the planar break-junctions obtained in EuCsFe 4 As 4 single crystals are given in Fig. 2. In the superconducting state, these CVCs reproducibly lack the supercurrent  Fig. 2, an Andreev excess current is observed K) in the superconducting state compared to the CVC above T c (dashed line) over the entire range of bias voltages shown at the bottom of the figure (right axis). While increasing sharply at , tends to a constant value at large bias voltages in accordance with all theoretical models describing incoherent multiple Andreev reflection spectroscopy in a highly transparent SnS junction [11][12][13]. Thus, the features clearly seen in the in-gap region of the CVC (gray region in Fig. 2) represent the Andreev subharmonic gap structure of the superconducting order parameters. The local critical temperatures of these junctions (corresponding to the transition temperature of the junction region with sizes of about tens of nanometers to the normal state) corresponding to the disappearance of I exc in CVCs (see the inset of Fig. 2) are approximately the same K.
Dynamic conductance spectra of these junctions measured at are shown in the corresponding colors in Fig. 3. Considering from high bias voltages along a smooth, almost horizontal region, the first sharp dynamic conductance feature occurs at about 16 mV. The next feature with a major (with respect to the fine structure) amplitude is reproducibly visible at 9.6 mV. We note that a significant amplitude of features (comparable in scale to the normal conductance of the junction) indicates the major contribution of the corresponding effective band to the Andreev conductance. The intense dips form a doublet at and 16 mV, which directly determine two energy gap parameters meV and meV according to Eq. (1). The second subharmonic of ( ), expected at mV, overlaps with the fundamental dip of , because of which, most probably, it is poorly resolved. However, the feature of is clearly visible at bias voltages mV. Near zero bias, the dips at 3.8 mV can be interpreted as the onset of the subharmonic gap structure of another superconducting order parameter, which has a smaller magnitude meV. The subharmonic of is also resolved in the bottom spectrum. The lower intensity out-of-gap fine structure of the spectrum requires a separate study.
Although the prepared junctions have different normal resistances Ω and areas, the CVC shape and position of gap features in the pre- spectra are well reproduced. Consequently, the observed features reflect the bulk superconducting properties of EuCsFe 4 As 4 and are not caused by surface effects or spatial resonances.
The found energy parameters and can be two separate isotropic (nodeless) superconducting gaps developing at different Fermi surface sheets. On the other hand, they can characterize a single anisotropic superconducting condensate with a large gap: in this case, and are extrema of the large superconducting gap with an extended s wave symmetry type being minimum and maximum coupling energies of Cooper pairs in the k space, respectively. Then, the anisotropy of can be estimated as . In any case, we can confidently assert the absence of nodes of the large superconducting gap in the (k x , k y ) plane.
We reproducibly observe similar doublets on spectra of SnS junctions in related Ba(Fe, Ni) 2 As 2 122-type single crystals with different degrees of substitution [18,19]. Moreover, the gap structure of EuCsFe 4 As 4 determined above is generally similar to that of Ba(Fe, Ni) 2 As 2 . In order to compare the data on the compounds with the critical temperature differing by a factor of almost 2, Fig. 4 shows spectra with the bias voltage axes normal-  Fig. 3) and in optimally doped BaFe 1.9 Ni 0.1 As 2 (bottom line, raw data taken from [19]). For each spectrum, the bias voltage axis is normalized to the corresponding value, and the normal monotonic background is suppressed for clarity. ized to the position of the gap feature. The bottom spectrum in Fig. 4 is obtained in optimally doped BaFe 1.9 Ni 0.1 As 2 with T c ≈ 19.7 K (taken from [19]). The positions of the dips of the doublet, the shape of the arc, and its relative width are almost exactly the same for both compounds (vertical lines in Fig. 4). The similarity of the doublet shape to numerical calculations (line 2 in [9, Fig. 4]) indirectly indicates the nodeless superconducting order parameter with a moderate anisotropy in the k space. Thus, the found and energies presumably characterize a single superconducting condensate with a large gap being its edges, i.e., the minimum and maximum coupling energies of Cooper pairs in the corresponding bands. The assumed anisotropy value up to 40% without nodes is also similar in pnictides of the Ba-122 and EuCsFe 4 As 4 families.
The positions of the Andreev features are also in agreement, although they are slightly shifted toward zero bias in the spectra of junctions obtained in EuCsFe 4 As 4 (arrows); i.e., the ratio increases in EuCsFe 4 As 4 . For the small superconducting gap, we do not observe any distinct doublets in spectra reproducibly, which can be due to either the isotropy of or, on the contrary, its strong anisotropy larger than 50% or even nodes in the k space.
The characteristic ratios for the edges of the anisotropic large superconducting gap in EuCsFe 4 As 4 are and . For the small gap, the characteristic ratio is typical of a "weak" condensate in a multiple-gap superconductor. All values obtained are close to each other, although slightly less than the corresponding average values we determined earlier for Ba(Fe, Ni) 2 As 2 of the optimal and overdoped compositions [18,19]. This can be caused by a change in the ratio of intraband to interband coupling strengths in EuCsFe 4 As 4 in comparison with Ba(Fe, Ni) 2 As 2 . Thus, in the first approximation, the scaling of with T c occurs in pnictides of EuCs-1144 and Ba-122 families.

CONCLUSIONS
We have grown EuCsFe 4 As 4 single crystals with the bulk critical temperature T c ≈ 36.8 K. The gap structure of EuCsFe 4 As 4 has been studied for the first time using incoherent multiple Andreev reflection spectroscopy of planar SnS contacts formed by the mechanically controlled planar break-junction technique. A small superconducting gap of 1.9 meV and the edges of the large gap meV and 8 meV (presumably having anisotropy in the k space without nodes), as well as their characteristic , and , have been directly determined at . It has been shown that the gap structure of EuCsFe 4 As 4 is similar to that for Ba(Fe, Ni) 2 As 2 pnictides of the isostructural 122 family; i.e., the scaling of all three energy gap parameters, , , and with T c has been observed in EuCs-1144 and Ba-122 family pnictides.