Weak-coupling superconductivity in a strongly correlated iron pnictide

Iron-based superconductors have been found to exhibit an intimate interplay of orbital, spin, and lattice degrees of freedom, dramatically affecting their low-energy electronic properties, including superconductivity. Albeit the precise pairing mechanism remains unidentified, several candidate interactions have been suggested to mediate the superconducting pairing, both in the orbital and in the spin channel. Here, we employ optical spectroscopy (OS), angle-resolved photoemission spectroscopy (ARPES), ab initio band-structure, and Eliashberg calculations to show that nearly optimally doped NaFe0.978Co0.022As exhibits some of the strongest orbitally selective electronic correlations in the family of iron pnictides. Unexpectedly, we find that the mass enhancement of itinerant charge carriers in the strongly correlated band is dramatically reduced near the Γ point and attribute this effect to orbital mixing induced by pronounced spin-orbit coupling. Embracing the true band structure allows us to describe all low-energy electronic properties obtained in our experiments with remarkable consistency and demonstrate that superconductivity in this material is rather weak and mediated by spin fluctuations.

In order to confirm that the slight anisotropy of the photoemission intensity in the energy-momentum cut shown in Fig.1a of the main text does not affect the conclusions of our study we have fitted the left and right branches of the outer band's dispersion independently and compared these fits to the results of the simultaneous fit of both branches presented in the text. Only the data points at high enough binding energies (unaffected by the reduction in the effective mass due to the spin-orbit coupling) were used for all fits (red and green filled circles); the entire dispersion of the band is shown for completeness (grey empty circles).
The results of this comparison are shown in Fig. S1a. The red and green solid lines are the parabolic dispersions obtained from the fit of the data points of the corresponding color (only the left and right branches of the parabola were used in the fit of the red and green data points, respectively). It is clear that the independent fit of the right (green solid line) branch gives results very similar to the simultaneous fit of both branches (blue solid line). The independent fit of the left branch deviates somewhat from the simultaneous fit, albeit it provides only a marginal improvement in the fit's figure of merit. To quantify the effect of this uncertainty on the extracted renormalization of the effective mass we have calculated the latter using the formula given in the main text, m(E) =h 2 k(E)dk/dE, independently for the left and right branches. The resulting energy-dependent effective mass is shown in  the independent fit of the left branch is still significantly larger than the renormalization of all the other bands and the largest in the family of the high-temperature iron-pnictide superconductors. In view of the independent fits shown in Fig. S1 it is more likely that the true value of the renormalization is closer to the average than to the lowest value. Similar results can be obtained through the analysis of quasiparticle dispersion in the symmetrized (with respect to zero wave vector) data. Figures S1c,d demonstrate the flattening of the middle hole band's dispersion (white arrows) near its top due to the orbital-mixing effect induced by the spin-orbit coupling.

II. INHERENT CHARACTER OF THE ORBITALLY SELECTIVE BAND SHIFTS IN THE ELECTRONIC STRUCTURE OF
NaFe 0.978 Co 0.022 As In this section we would like to briefly substantiate the bulk-related character of the orbitally selective band shifts or, equivalently, disprove their surface-related origin. Orbitally selective band width renormalization and band shifts are generic to iron-based superconductors, have been observed in essentially all materials of this family [S 1,1-9], and suggested to result from orbital blocking in the presence of strong Hund's coupling [S 10] and a pronounced particle-hole asymmetry [S 11], respectively. This renormalized and band-shifted electronic structure, as well as the superconducting energy gap that develops on it, observed via ARPES has been found to agree remarkably well with many bulk-sensitive techniques, strongly suggesting that the electronic band structure of the surface layer in these compounds is reflective of that in the bulk. The only exception to this rule is the 1111 type of compounds, in which the presence of both bulk-and surface-related states in the photoemission signal has been found [S 12-21], making the extraction of the inherently bulk electronic structure challenging. Recently, a detailed ARPES work succeeded in disentangling the bulk and surface contributions to the photoemission signal and demonstrated that the bulk electronic structure features the same orbitally selective band width renormalization and band shifts as observed in other classes of iron-based superconductors [S 11], proving its generic and inherent character. Finally, it is well-known that 111-type materials cleave between the two layers of the intercalant, which results in a non-polar surface. The equivalence of the bulk and surface electronic states has been further demonstrated in an explicit density-functional calculation [S 22].

III. DETAILS OF THE FAR-INFRARED DATA AND ANALYSIS
Here we would like to demonstrate the degree of uncertainty in the extracted parameters of the free-charge-carrier response shown in Fig.1g that the finite signal-to-noise ratio of and the uncovered lowest-frequency range in the experimental data allow. We do so by comparing the optical response corresponding to the best fit shown in Figs.1f,g with that derived from the Drude-term parameters slightly detuned from those optimal values. In order to eliminate the integral transformation via Kramers-Kronig relations and to emphasize the functional shape of the reflectance assumed in the extrapolation region at lowest frequencies for Kramers-Kronig analysis, we demonstrate these trends in the raw far-infrared reflectance data. The latter are shown for different temperatures between 10 K and 300 K in Fig. S2a. Figures S2b-e show the reflectance obtained in our experiment as a function of photon energy at 30 K overlaid with that produced by assuming two Drude terms (ω pl,1 , γ 1 ; ω pl,2 , γ 2 ) with the parameters specified in the figure legends. The high-frequency dielectric function is taken to be ε ∞ = 60, determined by the contribution of the interband transitions at higher energies. The default (best-fit) parameters of the free-charge-carrier response referred to in these panels are those specified in Fig.1g of the main text.

IV. DETERMINATION OF BAND-SPECIFIC PLASMA FREQUENCIES BASED ON ARPES DATA
In optimally doped NaFe 0.978 Co 0.022 As the Fermi surface consists of three sheets: two electonlike in the corners of the Brillouin zone and one holelike at the center [S 23]. It is well-known and a generic feature of iron-based superconductors that the bands generating the electron sheets are only weakly dispersive in the out-of-plane direction, giving rise to two quasi-cylindrical sheets of the Fermi surface with relatively small corrugations [S 24]. Therefore, we can approximate these two sheets by two identical ideal cylinders with their radii equal to the average of the long and the short half-axis of the ellipsoids giving rise to the corrugation. The electronic band generating the hole sheet of the Fermi surface also disperses rather weakly along the Γ-Z direction of the Brillouin zone, as shown in Fig. S3: the Fermi wave vector of this band remains within 0.07-0.09Å −1 between Γ and Z. Therefore, the Fermi-surface sheet generated by this band can likewise be approximated by a cylinder with a radius of about 0.08Å −1 . Thus the task of calculating the plasma frequency has been reduced to a two-dimensional problem. In general, the plasma frequency of free charge carriers can be obtained from the semi-classical expression for the dielectric function [S 25] assuming the intraband case and low temperatures and is given by the following expression (in SI units): where σ = x, y, z enumerates the coordinate axes, n is the band index, δ (x) is Dirac's delta function, E nk is the energy dispersion of the nth band, E F is the Fermi energy, e is the electron charge,h is the reduced Planck constant, and ε 0 is the vacuum permittivity. Assuming our two-dimensional approach and the tetragonal symmetry of NaFe 0.978 Co 0.022 As, ω pl,z = 0 and ω pl,x = ω pl,y = ω pl . Thus only one quantity ω pl needs to be calculated for each band. In the two-dimensional case and for an isotropic parabolic band with an effective band mass m eff [E k =h 2 k 2 /(2m eff )] Eq. 1 can be rewritten as a simple and familiar expression: where N 2D (E F ) = k 2 F (2π) 2 2π c is the density of states at the Fermi level for an isotropic parabolic band in two dimensions, k F is the Fermi wave vector, and c is the out-of-plane lattice constant. Thus the calculation of the band-specific plasma frequencies is reduced to the determination of the Fermi wave vector and the quasiparticle effective mass for each band, which can easily be done based on the photoemission intensity distributions near the Fermi level presented in Figs.1a,b. As already mentioned above and shown in Fig. S3, the Fermi wave vector of the hole band is about 0.08Å −1 , and its effective mass is found to be 3.83 m e (where m e is the bare electron mass), which results in a plasma frequency of ω hole,1 pl = 0.23 eV. The Fermi wave vectors and effective masses of the electron bands at the M-point are 0.086Å −1 , 0.92 m e and 0.153Å −1 , 2.34 m e for the inner and the outer band, respectively. These values result in almost identical plasma frequencies of ω el,in pl = 0.5 eV and ω el,out pl = 0.56 eV, respectively.