Determination of the Superconducting Order Parameter of BaFe1.92Ni0.08As2 Weakly Underdoped Pnictides by Two Complementary Techniques

The structure of the superconducting order parameter of weakly underdoped BaFe1.92Ni0.08As2 pnictides with the critical temperature Tc ≈ 18.2 K has been determined. Two microscopic superconducting order parameters—the small superconducting gap ΔS(0) and, supposedly, the edges of the large gap with anisotropy in the ab plane \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{{\text{L}}}^{{{\text{out}}}}(0)$$\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}$$\Delta _{{\text{L}}}^{{{\text{in}}}}(0)$$\end{document}—have been determined at T ≪ Tc, together with their temperature dependences, using incoherent multiple Andreev reflection spectroscopy. It has been shown that the determined temperature dependence of the lower critical field Hc1(T) can be described in the two-band approximation both within the so-called alpha model (using experimental ΔS(0), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{{\text{L}}}^{{{\text{out}}}}(0)$$\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}$$\Delta _{{\text{L}}}^{{{\text{in}}}}(0)$$\end{document} values) and using the temperature dependences ΔS(T) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{{\text{L}}}^{{{\text{out}}{\text{,in}}}}(T)$$\end{document} obtained by multiple Andreev reflection spectroscopy under the assumption of a fairly strong anisotropy of the small superconducting gap, where the experimental ΔS(0) value is taken as the outer edge of its angular distribution.


INTRODUCTION
Superconducting BaFe 2 As 2 pnictides of the Ba-122 family have a layered crystal structure and critical temperatures up to T c ≈ 21 K at partial substitution Fe 2 -x Ni x at the optimal doping level x = 0.1 [1]. Angle-resolved photoemission spectroscopy (ARPES) reveals embedded hole cylinders on the Fermi surface around the Γ point of the Brillouin zone and electron cylinders around the M point corrugated along the k z direction [2].
Although ARPES studies of BaFe 2 -x Ni x As 2 (BFNA) pnictides in the superconducting state have not yet been performed, the general feature of pnictides of the Ba-122 family is the anisotropy of superconducting gaps in the momentum space, which was observed in ARPES experiments [3,4] and considered theoretically [5,6]. The calculations of the gap structure for the Ba-122 family [5,6] show the existence of anisotropic superconducting gaps in both spin-fluctuation ( ) and orbital-fluctuation ( ) approaches. The combination of these interaction types [6] makes it possible to obtain various gap structures, including those with the "nodal" order parameter (having nodes in the k space distribution). A new comprehensive method proposed in this work allowed us to obtain reliable experimental data on the anisotropy of superconducting gaps in BFNA.
Experimental studies of the gap structure of BFNA (with the partial substitution of Ni) are scarce and were carried out primarily on optimally doped single crystals with x = 0.1. Data on the number and values of superconducting order parameters obtained by local, bulk, and surface methods [7][8][9][10][11][12][13] are poorly consistent: in particular, the characteristic ratio for the large superconducting gap is 2Δ L (0)/k B T c ≈ 3.7-13, i.e., varies by a factor of 3.5. The inconsistency of existing data [7][8][9][10][11][12][13] also raises a question of the applicability of some experimental methods to the study of properties of unconventional multiband superconductors with anisotropic order parameters.
It can be expected that the gap structure in the underdoped region of the phase diagram is different from those in other regions because of the possible

CONDENSED MATTER
effect of the antiferromagnetic and nematic phases. However, superconducting order parameters in underdoped BFNA with x < 0. 09 have not yet been studied. In this work, we propose a comprehensive multiband approach based on incoherent multiple Andreev reflection (IMARE) spectroscopy and the measurement of the lower critical field H c1 (T), which provides more reliable data on the gap structure of anisotropic superconductors. Using this approach, we determine for the first time the structure of the superconducting order parameter of weakly underdoped BaFe 1.92 Ni 0.08 As 2 pnictides, including the values, temperature dependences, and characteristic ratios of two superconducting gaps; estimate a moderate anisotropy of A L ≈ 30% of the large superconducting gap; and conclude that the small gap is highly anisotropic: A S ≥ 50-100%. The similarities and differences of the gap structures of BFNA pnictides with different degrees of doping are discussed.
2. DETAILS OF THE EXPERIMENT Large BaFe 1.92 Ni 0.08 As 2 single crystals up to 1 cm were grown by the "self-flux" technique. The characterization by X-ray diffraction, element analysis, electron microscopy, and resistive and magnetic measurements showed the existence of a single superconducting phase with T c ≈ 18.5 K and a high degree of homogeneity of crystals. Details of the synthesis and characterization are presented in [14][15][16] and in Section II of the supplementary material.
Mechanically controlled planar superconductorthin normal metal-superconductor (SnS) nanojunctions with the current flow direction along the c axis were created in BFNA single crystals using the breakjunction technique [17]. Typical current-voltage characteristics of such junctions are shown in Fig. 1а, and the temperature dependence of the resistance near T c for a 4 × 2 × 0.2-mm platelet single crystal is presented in Fig. 2 (gray circles, right axis). The width of the resistive transition is about 1.1 K. The design of the sample holder, details of the experiment with layered compound samples, and advantages and disadvantages of the method are described in detail in review [18]. The physical model of the junction is schematically shown in [19, Fig. 1]. The controllable parameters of the tunnel junction in the planar break-junction technique experiments are the area of the junction (in the plane) and, correspondingly, its normal resistance R N rather than the thickness and transparency of the constriction (gap, see [18,Figs. 12 and 16] and Section II in the supplementary material).
Multiple Andreev reflections occur in the SnS junction below T c . Incoherent Andreev transport at a high-transparency NS interface (barrier parameter ) induces an excess current (compared to the normal-state current-voltage characteristic above T c ) in the current-voltage characteristics of the SnS junc-ab 0.3 Z tion in the entire bias range . The dynamic conductance in the spectrum in the limit is much higher than the conductance G N of the SnS junction in the normal state (so-called foot region), whereas the supercurrent branch is absent in this case [20,21]. A series of features called the subharmonic gap structure) also appear at the positions directly determined by the superconducting gap Δ at any temperatures up to T c [20][21][22][23][24]: (1) The number n* of observed subharmonic gap structure minima in the dI(V)/dV spectrum of the planar SnS junction is determined by the ratio of the junction dimension d c and the inelastic scattering length in . The violet solid line is the approximation of the shape of the n L = 1 doublet within the approach proposed in [25]. Arrows mark Andreev minima (n S = 1, 2) from the small gap Δ S ≈ 1.6 meV. the c direction: [19,21]. The spectrum of the multigap superconductor will contain the subharmonic gap structure from each gap.
The effect of the angular distribution in the k space on the shape of Andreev minima within the approach proposed in [25] is demonstrated in [18,Fig. 4]. Supposedly, sharp intense subharmonic gap structure dips are expected in the dI(V)/dV spectrum for the isotropic gap, whereas the amplitude of the subharmonic gap structure is suppressed in the case of the nodal order parameter. The subharmonic gap structure dips forming a doublet whose width is determined by the maximum, Δ out , and maximum, Δ in , binding energies of Cooper pairs in the k space are expected for the gap with the extended s-wave symmetry in the plane without nodes. The degree of anisotropy is defined below as . Within the formalism proposed in [25] and under the assumption of is the angle in the plane and H(eV) = G(eV, 4.2 K) -is the amplitude of details of the extended (doublet) Andreev feature with . The x and y momentum components of charge carriers in the planar SnS junction can be conserved in the case of ballistic transport in the plane, i.e., if the mean free path exceeds . At the same time, the component can be unconserved because of the "mixing" of charge carriers along the current direction. The estimate of the Sharvin resistance [26] similar to [7,8] for the SnS junction with the normal resistance R N ≈ 65 Ω makes it possible to obtain nm and . Thus, the feature of the planar junctions under study is the ballistic transport in the a and b momentum components, whereas inelastic scattering occurs in the c direction. This allows one to obtain some information on the anisotropy of the superconducting gap in the plane. Briefly speaking, the mechanically controlled planar break-junction technique allows one to locally (within the junction region with a size of about tens of nanometers) and directly determine the magnitudes, temperature dependences, and characteristic ratios of the superconducting order parameters in high resolution and to estimate their anisotropy in the plane [18].
One of the reliable methods of studying features of the macroscopic order parameter in superconductors is the analysis of the temperature dependence of the superfluid density ρ sc (T) ~ 1/λ 2 (T), where λ is the London penetration depth. This method makes it possible to indirectly determine the number of superconducting gaps and their magnitudes, as well as to estimate their angular structure in the momentum space. The temperature dependence of ρ sc can be obtained from the dependence of the lower critical field using the Ginzburg-Landau formula for type-II superconductors (2) according to the Ginzburg-Landau theory. The lower critical field H c1 can be determined from magnetization curves using various methods. As known, the magnetization curve at the external field below H c1 of the superconducting sample is linear with a slope of -1/(4π). When the external field reaches H c1 , Abrikosov vortices formed in a type-II superconductor, enter the crystal and effectively reduce the volume of the superconducting phase, and are responsible for the deviation of the magnetization curve from the lin- ear behavior. In this work, magnetization curves were measured using a Hall microsensor [27]. In this method, a 1 × 1 × 0.2-mm cylindrical sample is placed on the 50 × 50-μm Hall sensor parallel to the plane and the sensor signal is measured as a function of the static external magnetic field applied along the c axis. The anisotropy of the London penetration and, correspondingly, the density of Cooper pairs in the plane are integrated into the measured data. This allows the complementary usage of the two methods because IMARE spectroscopy also detects the anisotropy of the gap in the plane. We note that the effect of edge defects at the measurement of curves changes only the absolute value of H c1 (T), whereas the determined parameters of the superconducting state depend only on the shape of the H c1 (T)/H c1 (0) curve. The choice of the criterion of determining H c1 from magnetization curves also affects only the amplitude of H c1 : the resulting normalized temperature dependences H c1 (T)/H c1 (0) are similar to each other with an error of less than 2%.
To describe the results of the measurement of the lower critical field, the so-called alpha model is commonly used to calculate the macroscopic order parameter of an isotropic superconductor [28]; this model is easily extended for a two-gap superconductor [29]: (3) This model describes the normalized density of Cooper pairs of two noninteracting superconducting condensates with the corresponding weights (it is assumed that the temperature dependence of the energy of each gap has the Bardeen-Cooper-Schrieffer form). The normalized superfluid density is related to the critical field as (4) With allowance for anisotropy of both gaps, this model generally has seven free parameters: the lower critical field at zero temperature H c1 (0), critical temperature T c (common for both condensates), characteristic ratios and at T ≪ T c , the degrees of anisotropy of the gaps A L and A S (the angular distribution in the momentum space , where θ is the angle in the plane, is taken for the calculations), and the weight contribution ϕ of one of the effective bands. As a result, several solutions can exist. Using , , and Δ S determined directly by IMARE spectroscopy and the critical temperature, we can reduce the number of free parameters to three: H c1 (0), the weight contribution ϕ of bands with the small gap, and the small gap anisotropy A S .
The normalized macroscopic order parameter of the isotropic superconductor in the dirty limit can be estimated by the expression [30] (5) where . In the case of microscopic order parameter anisotropic in the k space, it is necessary to integrate Eq. (5) over the angle θ. Thus, the temperature dependences of the superconducting gaps Δ L (T) and Δ S (T) obtained by IMARE spectroscopy can be used for comparison with the normalized dependence H c1 (T)/H c1 (0). Unlike the alpha model, this approach allows one to take into account the interband interaction, which affects Δ L (T) and Δ S (T). Taking into account the angular distribution of microscopic order parameters Δ L (θ) and Δ S (θ) and their integral value determined above, we obtain in the two-band approximation (6) 3. EXPERIMENTAL DATA AND DISCUSSION Figure 1 presents the current-voltage characteristics I(V) and spectra of the SnS junction measured below and above T c . The current-voltage characteristics are symmetric with respect to and show no hysteresis or the supercurrent branch, which excludes the Josephson nature of features observed in . On the contrary, an excess current (compared to the normal curve at T > T c ) is observed in current-voltage characteristics at T < T c in the entire bias range. At the same time, spectra in the superconducting state clearly demonstrate a foot at and a series of Andreev minima. These features of current-voltage characteristics and curves correspond to the IMARE regime and a fairly high transparency of the SnS junction according to all the existing theoretical models of multiple Andreev reflections [20,21,23,24].
At a temperature of 4.2 K, a doublet is observed at bias voltages and 6.4 meV. The positions of the minima forming the doublet are inconsistent with Eq. (1) and cannot be interpreted as Andreev subharmonics of the orders or of the isotropic superconducting gap. Features of at higher bias voltages are absent; i.e., the linear segment of the spectrum corresponds to energies beyond the gap, where the electron density of states does not have intense peaks. Thus, both minima of the doublet are fundamental n = 1 Andreev features, and their positions are directly determined by the energy parameters meV and meV. The minimum at meV corresponds to the second subharmonic from . Minima observed at lower bias voltages and 1.6 meV (n S = 1 and 2 in Fig. 1b) are apparently the first and second Andreev features of the subharmonic gap structure of the small gap Δ S ≈ 1.6 meV. The position of the second subharmonic of expected at meV coincides with the position of the fundamental dip of Δ S .
According to ARPES data [3] obtained on Ba-122 family single crystals, the (anisotropic) large superconducting gap assumingly develops on the inner hole cylinder at the Γ point and in electron bands, whereas the small gap develops on the outer Γ cylinder.
The characteristic ratios of the energy parameters are , , and 2Δ S (0)/k B T c ≈ 2 < 3.53 (which is characteristic of a "weak" condensate in the multigap superconductor).
We reproducibly observed such doublets on the spectra of other SnS junctions obtained in BaFe 1.92 Ni 0.08 As 2 from the same batch, as well as in overdoped and optimally doped BFNA compounds [7,8], which could be caused by the anisotropy of the large gap Δ L in the plane ( and correspond to the maximum and minimum binding energy of Cooper pairs depending on the momentum direction). In favor of this assumption, the shape of the doublet can be approximated within the approach proposed in [25]: the result of the numerical calculation at 4.2 K is shown by the violet solid line in Fig. 1, and the corresponding angular distribution Δ L (θ) is presented in the upper inset of Fig. 2. At the same time, the existence of two distinct isotropic superconducting gaps and (developing below T c on different sheets of the Fermi surface) is not excluded. Since pronounced doublets are not reproducibly observed on spectra for the small gap, one may suppose the small gap Δ S either isotropic or having a degree of anisotropy higher than 50%, as well as even nodal in the (k x , k y ) plane. A small amplitude (~0.2G N ) of Andreev minima from the small gap Δ S can also indicate its strong anisotropy along the direction. In order to distinguish the above-listed cases and to reliably determine the symmetry of Δ L and Δ S , the existing theoretical models of multiple Andreev reflection [20,21,23,24] should be extended for the case of the anisotropic superconducting gap and the shape of doublets on spectra should be examined in more detail.
As the temperature increases, the amplitude of all Andreev minima decreases and their positions are shifted toward zero, reflecting the temperature dependence of the order parameter according to Eq. (1). At the same time, the excess current at Fig. 1a) and the zero bias conduction decrease (Fig. 1b). On the contrary, the normal conductance of the junction at remains approximately constant (cf. the black solid line at K and the dashed line at T > T c in the bottom of Fig. 1b at maximum eV values; for convenience, the other spectra are shifted vertically in the order of increasing temperature), which unambiguously indicates the ballistic transport through this junction. Features caused by Andreev transport are absent in the current-voltage characteristic and the spectrum (dashed line in Fig. 1b) at T ≈ 18.9 K > T c , which corresponds to the transition of the junction region in the normal state.
For clarity, the normal-state dynamic conductance, which is reproducibly nonlinear in BFNA SnS junctions and has the shape similar to that presented in [8, Fig. 1], was suppressed for all spectra in Fig. 1b. This nonlinearity is not due to the overheating of the junction (in this case, the maximum of at low values would observed in the normal state [31]) and can be interpreted as the manifestation of features of the electron density of states near the Fermi level (for more detail, see [8, Section III(a)]).
The temperature dependences of three observed energy gap parameters , (circles), and Δ S (T) (triangles) in Fig. 2 directly obtained from data in Fig. 1b are typical of the case of the moderate interband interaction. The degree of supposed anisotropy of the large gap is (squares, lower inset of Fig. 2). Although this estimate of anisotropy is indirect, nodes of Δ L in the k space are certainly absent. The small gap near T c is closed more rapidly (stars, the lower panel of Fig. 2). Since the ratio Δ L (T)/Δ S (T) is not constant and increases at T → T c , Andreev minima marked in Fig. 1b by n S = 1 and 2 cannot be interpreted as higher order subharmonics from the large gap Δ L , therefore, the small gap Δ S is a distinct superconducting order parameter. The dip indicated in Fig. 1b by the blue vertical straight line with the mark is the second subharmonic of because it has the bias voltage corresponding to the expected value and the temperature dependence resembling that of (blue circles in the lower panel of Fig. 2; see also Section II in the supplementary material).
The magnetization curves of the BaFe 1.92 Ni 0.08 As 2 single crystal are presented in Fig. 3. The criterion of the determination of H c1 (disregarding the demagnetization factor) from the dependence (inset of Fig. 3) is typical of measurements with the Hall sensor [32]. The initial horizontal segment in this dependence theoretically corresponds [33] to the Meissner state, whereas the linearly increasing segment corresponds to the mixed state. Experimental curves can demonstrate noises and nonlinearities, which are due in our opinion to small inhomogeneity inclusions of the superconducting phase, which can be located near the sensitive region of the Hall sensor. We reached a very low noise level of 0.02-0.03 Oe (at a sensor sensitivity of 0.5 μV/Oe), which is a factor of about 20 lower than that in [32]. The signal of the Hall sensor exceeding the level of noises and nonlinear parasitic contributions on the dependences was chosen as the criterion of the determination of the lower critical field H c1 (horizontal straight line in the inset of Fig. 3).
The dependence H c1 (T) obtained from data presented in Fig. 3 and its fits within the two-band alpha model for two cases are shown by red and blue lines in Fig. 4. In both cases, the large gap Δ L = 3.2-4.5 meV with the degree of anisotropy A L = 29% and the characteristic small gap Δ S = 1.6 meV, which were obtained from IMARE spectroscopy data, were taken. In the first case (red lines in Fig. 4), Δ S is assumed isotropic (A S = 0). A significant difference between data and model calculations is seen at K. In this variant, the best approximation is reached at a weight contribution of 82% from bands with the small gap. In the second case (blue lines in Fig. 4), we took an anisotropic small gap, which ensures better agreement between Δ ( ) M H Δ ( ) M H < 10 T experimental data and calculation at a degree of anisotropy of A S = 58% and a weight contribution of 62% from bands with the small gap. Under the variation of the criteria for the determination of the lower critical field H c1 from magnetization curves, the best approximation of H c1 (T) allows the spread of the weight contribution from bands with the small gap within ±5% and the spread of the anisotropy of Δ S within ±12% from the average A S = 55%. Approximations of H c1 (T) curves with other criteria are given in the supplementary material.
The approximation of the normalized temperature dependence H c1 (T)/H c1 (0) by IMARE spectroscopy data on the microscopic order parameters and is shown by green lines in Fig. 4 (right axis). We note that the critical temperature T c of the bulk crystal according to magnetization data coincides with the critical temperature T c of the Andreev junction according to IMARE spectroscopy data. For the case of the isotropic small gap Δ S ≈ 1.6 meV, the contribution of the corresponding bands to the superfluid density (short dashed line in Fig. 4) begins to sharply decrease at rather high temperatures K; thus, the low-temperature part of the temperature dependence H c1 (T)/H c1 (0) cannot be approximated. Experimental data can generally be described by integrating Eq. (6) over the angle with the weight coefficient for bands with the small gap Δ S under the assumption of its 100% anisotropy (which gives the effective value meV) (solid green line in Fig. 4).
Thus, the best approximation of H c1 (T) corresponds to the second considered case (blue line in Fig. 4 and the second row in Table 1).
The comparison of IMARE spectroscopy data with the lower critical field measurements shows that H c1 (T) data cannot be described with the isotropic small gap Δ S . This indirectly indicates a fairly strong anisotropy of the small gap and that a feature only from the its outer edge meV can be observed in the tunnel experiment.
Since the strong anisotropy of the superconducting gap in the space can hardly be obtained within the pure approach [6], spin fluctuations moderately contribute in the superconducting pairing mechanism in underdoped BFNA. Nevertheless, the gap structure of BFNA with x = 0.08 is quite similar to our data on optimally doped and overdoped crystals [7,8] (except for the strong anisotropy of the small gap Δ S , which was not observed in our previous works and requires further study). We did not observe a noticeable change in the anisotropy of the large gap Δ L in a weakly underdoped compound: the value A L ≈ 30% is in a range of 27-37% estimated for compounds with x = 0.1 and 0.14 [7,8].
The deviations of three approximations from the experimental data on H c1 (T) are shown in the corresponding colors in Fig. 4. It is seen that all used approaches hardly reproduce experimental data quantitatively in the region of the fast decrease in superfluid density in bands with the small gap Δ S at T ~ 4 K. It can be assumed that this inconsistency is due either to an increase in the effect of some unincluded channel of inelastic scattering on impurities at these temperatures

CONCLUSIONS
A comprehensive approach to the study of the anisotropic superconducting order parameter involving IMARE spectroscopy of planar SnS junctions and the measurement of the lower critical field has been proposed. We have found that the anisotropy of the large superconducting gap (without nodes in the k space) in weakly underdoped BaFe 1.92 Ni 0.08 As 2 pnictides with T c ≈ 18.2 K is A L ≈ 30%. No noticeable difference of the characteristic ratio 2Δ L (0)/k B T c ≈ 4.1-5.7 (range corresponds to A L ) and the degree of anisotropy from the respective parameters in other regions of the phase diagram of BFNA has been observed [7,8]. Nevertheless, a significant anisotropy of A S ≈ 50-100% has been found for the small gap by fitting the temperature dependence H c1 (T)/H c1 (0).
The directly measured temperature dependences and Δ S (T) are typical of the moderate interband interaction, and the degree of supposed anisotropy of the large gap hardly depends on the temperature.