Effects of disorder induced by heavy-ion irradiation on (Ba1−xKx)Fe2As2 single crystals, within the three-band Eliashberg s± wave model

One of the open issues concerning iron-based superconductors is whether the s± wave model is able to account for the overall effects of impurity scattering, including the low rate of decrease of the critical temperature with the impurity concentration. Here we investigate Ba1−xKxFe2As2 crystals where disorder is introduced by Au-ion irradiation. Critical temperature, T c, and London penetration depth, λ L, were measured by a microwave resonator technique, for different values of the irradiation fluence. We compared experimental data with calculations made on the basis of the three-band Eliashberg equations, suitably accounting for the impurity scattering. We show that this approach is able to explain in a consistent way the effects of disorder both on T c and on λ L(T), within the s± wave model. In particular, a change of curvature in the low-temperature λ L(T) curves for the most irradiated crystals is fairly well reproduced.

• as we refer to experimental data taken from single crystals of high quality, we can safely assume a negligible disorder for the unirradiated samples. The scattering rate from non-magnetic and magnetic impurities Γ N ij and Γ M ij can thus be taken to be zero for the unirradiated crystals. The way to account for non-negligible scattering rates in irradiated (disordered) crystals is discussed in the article.
• Following Mazin et al. 1 , we can assume that: a) the total electron-phonon coupling constant is small (the upper limit of the phonon coupling in the usual ironarsenide compounds is ≈ 0.35) 2 ; b) phonons do not contribute significantly to interband coupling so that λ ph i j ≈ 0. Moreover, the phonon contribution to intraband coupling is negligible, so that λ ph ii ≈ 0, so as the Coulomb pseudopotential matrix: µ * ii (ω c ) = µ * i j (ω c ) = 0. 3−6 ; c) spin fluctuations mainly provide interband coupling between holes and electrons bands, so that λ s f ii ≈ 0; Within these approximations, the electron-boson coupling-constant matrix λ i j becomes: 3−5 where ν i j = N i (0)/N j (0), and N i (0) is the normal density of states at the Fermi level for the i-th band. The coupling constants λ s f i j are defined through the electron-antiferromagnetic spin fluctuation spectral functions (Eliashberg functions) α 2 i j F s f i j (Ω). We choose these functions to have a Lorentzian shape: 3−5 and C i j are normalization constants, necessary to obtain the proper values of λ i j , while Ω i j and Y i j are the peak energies and the half-widths of the Lorentzian functions, respectively (see fig.1). 6 In all the calculations we set Ω i j = Ω 0 , i.e. we assume that the characteristic energy of the spin fluctuations is a single quantity for all the coupling channels, and Y i j = Ω 0 /2, based on the results of inelastic neutron scattering measurements. 7 Figure 1. Antiferromagnetic spin fluctuactions spectral function normalized to λ =1.

•
The peak energy of the Eliashberg functions, Ω 0 , can be directly associated to the experimental critical temperature, T c , by using the empirical law Ω 0 = 2T c /5 = 15.47 meV that has been demonstrated to hold, at least approximately, for iron pnictides. 8 We use a cut-off energy ω c = 464 meV and a maximum quasiparticle energy ω max = 619 meV. The factors ν i j that enter the definition of λ i j (eq.1) can be extracted from the ARPES measurements 9 by assuming that the Fermi momentum in the i-th band is proportional to the normal density of states at the Fermi level in the same band, i.e. k Fi ∝ N i (0). In this way, the ARPES results 6 lead to ν 12 = 2, ν 13 = 1 and ν 23 = 0.5.

Calculation of the energy gaps and T c •
Now the model contains only two free parameters, λ s f 13 and λ s f 23 , and we want to reproduce the low-temperature gap amplitudes, which are actually obtained by analytical continuation of the imaginary solutions of the Eliashberg equations to the real axis by using the technique of the Padé approximants. Before irradiation, we find the values ∆ 1 =12.0 meV, ∆ 2 = 5.2 meV and ∆ 3 =−12.0 meV, in good agreement with earlier ARPES data. Actually, it turns out that in order to reproduce all the experimental gap values, the unique possibility is to set λ s f 13 = 3.41 and λ s f 23 = 0.75 for a total coupling λ s f tot = 3.05.

•
Once all the parameters of the model have been fixed, we can calculate the critical temperature, that turns out to be equal to T * c = 47.72 K, while the experimental T c is much lower (38.7 K for the unirradiated sample, if the temperature where the London penetration depth diverges is considered). However, we should still take into account the f eedback effect 6,10 of the electronic condensate on the antiferromagnetic spin fluctuations. To this aim, we consider the electron-boson spectral functions with an energy peak following the same temperature dependence of the superconductive gap (Ω(T ) = Ω 0 tanh(1.76 T * c /T − 1)). 6 Of course, at T = T * c the energy peak is equal to zero, while at T =0, the new spectral functions are equal to the old ones. This procedure leads to a critical temperature in agreement with the experimental one.

Microwave measurements
As mentioned in the manuscript, the geometrical factor (V s /V r ) in Eqs. 2 and 3 is determined in a self-consistent way from data above T c , where the crystals show a metallic behavior. In such conditions, Re(k) = Im(k) = 1/δ , where δ = 2/ω µσ is the classical skin depth. Moreover, the finite dimensions of the crystals has to be taken into account, resulting in the penetration of the field also from the lateral sides. This, in combination with Eqs. 2 and 3 gives: where where 2c is the crystal thickness and 2a and 2b are the lateral dimensions of the sample. For such small crystals, it can be assumed that above T c the temperature dependence of the shifts of both resonance frequency and quality factor is mainly due to the temperature dependence of the skin depth, and the small contribution given by the thermal expansion of the sample can be neglected. Figure 3 shows the experimental f and Q shifts fitted by the above equations, with the constraint to keep for both the same δ (T ). The inset shows examples of direct measurements of resonance curves, at different temperatures, below and above T c .