Eliashberg theory of a multiband non-phononic spin glass superconductor

I solved the Eliashberg equations for multiband non-phononic $s\pm$ wave spin glass superconductor and I calculated the temperature dependence of the gaps and superfluid density that reveals unusual behavior as non monotonic temperature dependence and reentrant superconductivity. For particular values of input parameters the phase diagram is still more complex with two different ranges of temperature where the superconductivity appears.


Introduction
The delivery of new iron-based superconductors as EuF e 2 As 2 [1,2,3,4,5,6,7] allowed to investigate more deeply the interplay of magnetism and superconductivity. Compared to the past there is now a new aspect to be consider: not just that the magnetism competes with superconductivity, but also that it could be involved in the mechanism of superconductivity itself, as in the case of cuprates, heavy fermions and iron-based superconductors.
The case of iron-based superconductors EuF e 2 As 2 [1,2,3,4,5,6,7] is particularly interesting because the ferromagnetic and superconducting transition temperatures are near, where the first is connected to the Eu2+ local magnetic moments. Can also happen that the superconducting critical temperature is even higher than that of the magnetic ordering [1,2,3,4,5,6,7]. In these systems a complex phenomenology of magnetic phases is observed and below the critical superconductive temperature two distinct magnetic transitions are observed and the magnetic ordering at a higher temperature is associated with the antiferromagnetic interlayer coupling, whereas the behaviour at lower temperature is identified as the change over to a spin glass state, where the moments between the layers are decoupled [2,7]. Usually the spin glass state [8] occurs in substitutionally disordered alloys [9,10,11], where, by means the long-range Rudermann-Kittel-Kasuya-Yosida interaction, mediated by conduction electrons, the localized magnetic moments, randomly distributed, interact. Because not all magnetic moments can be simultaneously satisfied in their spin orientation with respect to the others happen that born frustration in the magnetic ordering. This fact produces an infinite number of random configurations degenerated in energy but separated by large energy barriers. In this way the ground state cannot evolve into another on the experimetal time scale. A typical freezing temperature T SG is associated with the spinglass state below which the spins freeze into one of these random configurations. The magnetic susceptibility in the spin glasses shows a cusp at T SG , while nothing happen to specific heat other than a broad maximum around T SG , and no Bragg peaks, which usually are a signal of long-range magnetic order, in neutron scattering experiments. The correct order parameter for these systems has to be related to probability that a spin with a given direction at a finite time. will have the same direction in the infinitetime limit. The frozen nature of the spin-glass state is reflected in this order parameter but no spatial correlations are present as happen in other magnetic order parameters. Is it possible to reproduce this phenomenology connected with the superconductive state inside a theory? In this paper I will discuss as reproduce the experimental data of a multiband spin glass non phononic s±-wave superconductor in the framework of Eliashberg theory and I will take as example the particular case of EuF e 2 (As 1−x P x ) 2 [5]. The starting point will be the theoretical work of M.J. Nass [12,13,14] and J.P Carbotte [15,16,17,18]. that describe a single band spin glass s-wave phononic superconductor always in the framework of Eliashberg theory.

The Model
By introducing the order parameter for the spin-glass state as q = lim t→+∞ < S i (t) · S i (0) > it is possible to describe mathematically this spin freezing [8]. This order parameter is proportional to the probability that a given spin that has a particular direction at t = 0 will still orientated in that direction an infinite time later. This situation is quite different from to have a order parameter in a ferromagnetic or antiferromagnetic system which reflect space as well as time correlations. Although each spin is essentially fixed in direction, in the absence of a magnetic field, upon averaging over all spins the total spin is zero at all temperatures. By introducing a probability distribution it is possible to reproduce the randomness of the exchange interaction, and then averaging over this distribution. It is necessary to use the replica approach in order to carry out the averaging of the free energy over this distribution of exchange interactions and succeeded in finding a new order parameter defined as the configuration average of the equal time spin operators at a given site in different replicas of the system [8].
In the past papers [12,13,14,15,16,17,18] the theory developed concerned phononic superconductors where it was also added in contribution of antiferromagnetic spin fluctuactions (dynamic part) and spin glasses (static part). In our case it is not necessary to introduce the dynamic part which is already being responsible for the mechanism of superconductivity but only the static part which is formally equal to the contribution of magnetic impurities with in addition a dependence on temperature. The contribution of the spin glass phase can be represented, in an approximate way, in Eliashberg equations by a term (Γ M (T )) similar to that associated with the presence of magnetic impurities with in addition a dependence on temperature. Precisely the magnetic impurities scattering rate [15,16,17,18] that mimics the spin glass state is the total density of states at the Fermi level, J is a exchange constant, S is the spin of the magnetic element, T SG is the spin-glass critical temperature and β is a number [15,16,17,18] that can be 1 or 2 depending from the physical characteristic of the magnetic element (Eu in this case) and of the host material (particular iron compound). At this moment I am not enough data to understand if β is 1 or 2 so I solve the Eliashberg equations in the two cases. For solving the Eliashberg equations are necessary a lot of input parameters connected with the characteristic of the physical system. In the following I will refer to EuF e 2 (As 0.835 P 1.65 ) 2 a material [5] of the family of iron compound. The electronic structure of the compound EuF e 2 (As 0.835 P 1.65 ) 2 can be approximately described, in principle, as almost all doped iron-based materials, by a three-band model with one hole band (indicated in the following as band 3)and two electron bands (indicated in the following as bands 1 and 2) and. In this way the gap of the hole band, ∆ 3 , has opposite sign to the gaps residing on the electrons bands ∆ 1 and ∆ 2 . The phonons are responsible for the intraband coupling (ph) [20] and usually are neglected while the antiferromagnetic spin fluctuations (sf ) are connected to interband coupling between holes and electrons bands (s± wave model [19,20]). In the optics to reduce the number of free parameters I use an effective two-band model (band 1 electrons, band 2 holes) where it is not possible to set to zero the intraband coupling and where the physical coupling constant are not an immediate interpretation [23] because this model simulates the true physical situation (three bands) with effective values of the electron boson coupling constant. Now I investigate what happens in a multiband system and for simplicity I study a two band system that simulates a real three band system. In the following the s± wave two-band Eliashberg equations [21,22] are written and in the way that, to calculate the critical temperature and the gap, it is necessary to solve 4 coupled equations: 2 for the renormalization functions Z i (iω n ) and 2 for the gaps ∆ i (iω n ), where i is a band index (that ranges between 1 and 2) and ω n are the Matsubara frequencies.
The imaginary-axis equations [24,25,26] read: where Γ N ij and Γ M ij (T ) are the scattering rates from non-magnetic and magnetic impurities that, in this model, represent the term connected with the spin glass phase. For spin-glasses superconductors the magnetic impurities scattering rates are where c ij are weight connected with the bands and I put the non magnetic scattering rates Γ N ij = 0 because I suppose to have good single crystals (no disorder). In the previous equations I have Λ Z ij (iω n , iω m ) = Λ ph ij (iω n , iω m ) + Λ sf ij (iω n , iω m ) and Θ is the Heaviside function and ω c is a cutoff energy. The quantities µ * ij (ω c ) are the elements of the 2 × 2 Coulomb pseudopotential matrix and finally, . In order to have the smallest number of free parameter and the simplest model that still grasps the physics of this system, I make further assumptions that have been shown to be valid for iron pnictides [26,24,25]. I assume, following ref. [20] that the total electron-phonon coupling constant is small (the upper limit of the phonon coupling in the usual iron-arsenide compounds is ≈ 0.35 [28])so I put, in first approximation, the phonon contribution equal to zero (λ ph ij = 0) and as, following Mazin [29], the Coulomb pseudopotential matrix: [26,24,25,29]. After all these approximations, I write the electron-boson coupling-constant matrix λ ij in this way: [26,24,25,30]: where ν 12 = N 1 (0)/N 2 (0), and N i (0) is the normal density of states at the Fermi level for the i-th band. Based on experimental data and theoretical calculations [26,24,25] I choose for the electron-antiferromagnetic spin fluctuation spectral functionsα 2 ij F sf ij (Ω) a Lorentzian shape, i.e.: where and C ij are normalization constants, necessary to obtain the proper values of λ sf ij , while Ω ij and Y ij are the peak energies and the half-widths of the Lorentzian functions, respectively [26]. Following the experimental data [31] I put Ω ij = Ω 0 , i.e. I assume that the characteristic energy of spin fluctuations is a single quantity for all the coupling channels, and Y ij = Ω 0 /2. The spectral function used here, normalized to one, is shown in the inset of Fig 1. The factors ν ij that enter the definition of λ ij (eq. 3) are unknown so I assume that they are equal, for example, to the Eu(F e 1−x Rh x ) 2 As 2 electron doped case [32] so ν 12 = 0.8333 as well as the coupling constant [32] and we change lightly just a value (λ 22 ) for obtaining the correct critical temperature. At the end the values are λ 11 = 1.00, λ 11 = −0.17 and λ 22 = 2.65 for a total coupling constant λ t = 1.75. For iron pnictides it was experimentally found [33,34] that the empirical law Ω 0 = 2T c /5 works so value of energy peak Ω 0 of the Eliashberg spectral functions α 2 ij F sf ij (Ω) is not more a free parameter. To finish, in the numerical calculations I used a cut-off energy ω c = 180 meV and a maximum quasiparticle energy ω max = 200 meV.

Calculation of the superconductive gaps
In the iron compound usually the impurities are almost all concentrated in one band: i.e. in the hole band for the electron doped materials as this case and in the electron band [35] for the hole doped materials [36]. This means that, in the electron doped materials, k 22 >> k 11 , k 12 . I choose k 11 = k 12 = 0.2k 22 as happen in the Ba(F e 1−x Co x ) 2 As 2 [35]. By using the typical parameters of iron compounds and spin glass systems I find that k 22 ≃ 3.1 meV (N(0) = 5.6 states/eV, S = 7/2, J = 0.12 meV and T SG = 15 K) [5,37]. Because the true values of the parameters in the last bracket are just approximative I solve the Eliasberg equations for k 22 = 0, 2, 3, 4, 4.15, 4.756 meV in the two cases: β = 1 and β = 2. In the ideal case it would be necessary to know the law that links T SG at the value of k 22 . Here T SG is a experimental input. In the Figs 1 and 2 the temperature dependence of the gaps ∆ 1,2 (iω n=0 ) are shown: it is possible to see that, for k 22 ≥ 4 meV reentrant superconductivity is obtained. I solved the Eliashberg equations, for completeness, also in the case k 12 = 0.2 with k 11 = 0.2k 22 always with β = 1 and β = 2. The results are shown in Fig 3. In all case, of course, for T > T SG the effect of "magnetic iinpurities" disappeared and the behaviour is the same of a standard twoband superconductor.

Calculation of the penetration depth
The penetration depth (or the superfluid density as it is possible to see in Figs. 4, 5 and 6) can be computed starting from the renormalization functions Z i (iω n ) and the gaps ∆ i (iω n ) by using the following formula: [38] where ω p,i is the plasma frequency of the i-th band and ω p is the total plasma frequency in order that the w i = (ω p,i /ω p ) 2 are the weights of the single bands. The low-temperature value of the penetration depth λ L (0) should, in principle, be related to the plasma frequency by ω p = c/λ L (0) and appears as a multiplicative factor before to symbol of summatory. Here w 1 = 0.72 and w 2 = 0.28 as in the Co doped iron compounds [35]. In the Figs 4, 5 the superfluid density in function of temperature is shown when k 11 = k 12 = 0.2k 22 , k 22 = 0, 2, 3, 4, 4.15, 4.75, 6 meV with β = 1 and β = 2. In the cases with β = 2 and k 22 = 4.15 meV (orange lines, Fig 5) three critical temperatures appear, the superconductive state is in two different temperature ranges. In the Fig. 6 I show the superfluid density when k 12 = 0.2k 11 = 0.2k 22 , k 22 = 0, 1, 3, 5 meV with β = 1 and β = 2. These results are a clear prediction of possible situations that can be easily identified. Unfortunately, there is still no experimental data to compare with these theoretical predictions. The behavior of the penetration length as a function of temperature shows how presence of a spin glass state in competition with superconductivity substantially changes the phase diagram of a superconductor making it extremely richer.

Conclusions
In conclusion, I have calculated the temperature dependence of the gaps and superfluid densities for a two-band s± wave spin-glass superconductor. In general, the temperature dependence of superconducting properties shows a lot of different behaviours that should be observable in experiment. In addition, reentrant behavior is a possible signature of a spin glass state. In the case corrisponding to β = 2 the phase diagram is still more complex with two different ranges of temperature where the superconductivity appears.