Temperature-dependent $s_\pm \leftrightarrow s_{++}$ transitions in the multiband model for Fe-based superconductors with impurities

We study the dependence of the superconducting gaps on both the disorder and the temperature within the two-band model for iron-based materials. In the clean limit, the system is in the $s_\pm$ state with the sign-changing gaps. Scattering by nonmagnetic impurities leads to the change of sign of the smaller gap thus resulting in a transition from the $s_{\pm}$ to the $s_{++}$ state with the sign-preserving gaps. We show here that the transition is temperature-dependent, thus, there is a line of $s_\pm \to s_{++}$ transition in the temperature-disorder phase diagram. There exists a narrow range of impurity scattering rates, where the disorder-induced $s_\pm \to s_{++}$ transition occurs at low temperatures, but then the low-temperature $s_{++}$ state transforms back to the $s_\pm$ state at higher temperatures. With increasing impurity scattering rate, temperature of such $s_{++} \to s_{\pm}$ transition shifts to the critical temperature $T_c$ and only the $s_{++}$ state is left for higher amount of disorder.


Introduction
Iron-based pnictides and chalcogenides attract much attention due to the presence of unconventional superconducting state and peculiar normal state properties [1][2][3][4][5][6][7][8]. Electronic structure of these Fe-based high-T c superconductors (FeBS) demonstrate several features putting them apart from other conventional and unconventional superconductors. First of all, most of FeBS have similar topology of the Fermi surface that consists of two or three hole pockets centered at (0, 0) point and two electron pockets centered at (π, π) point of the Brillouin zone corresponding to one Fe per unit cell. Exceptions are the cases of the extreme electron or hole doping, where only one group of sheets remains. Second important feature is that all five iron 3d orbitals contribute to the formation of the Fermi surface and even a single Fermi pocket is formed by several d-orbitals. Thus the system can only be described within a multiorbital and, respectively, multiband model. Minimal model would be a two-band model, such as suggested in Refs. [9][10][11]. The third interesting feature is connected with the unconventional superconductivity. While, in general, there are some controversy in discussion of the gap symmetry in unconventional superconductors, growing evidences in iron-based materials support sign-changing gap. The structure of the Fermi surface provides practically ideal nesting at the antiferromagnetic wave vector Q = (π, π). Long-range spin-density wave antiferromagnetic state is destroyed by doping, however, the enhanced spin fluctuations remain. These fluctuations are claimed to be the main source of the Cooper pairing thus contributing to the formation of the superconducting ground state [8,12,13]. Since spin fluctuations with the large wave vector ∼ Q lead to the repulsive interband Cooper interaction, the gap function have to possess the momentum dependence and

Model and Method
Here we use the same two-band model as in Refs. [10,11,43] with the following Hamiltonian, where operator c † kασ (c kασ ) creates (annihilates) electron with momentum k, spin σ, and band index α = a, b, ξ k,α = v Fα (k − k Fα ) is the quasiparticles' dispersion linearized near the Fermi level with v Fα and k Fα being the Fermi velocity and the Fermi momentum of the band α, respectively. Hamiltonian contains the term with the impurity potential U at site R i and the term responsible for superconductivity, H SC . The exact form of the latter is not important for the present discussion but we assume that it provides superconducting pairing due to the exchange of spin fluctuations (repulsive interaction) and electron-phonon coupling (attractive interaction), see details in Ref. [11].
To describe the effect of disorder, we use the Eliashberg approach for multiband superconductors [44]. The connection between the full Green's functionĜ(k, ω n ), the self-energy matrixΣ(k, ω n ), and the 'bare' Green's function, is established via the Dyson equation,Ĝ(k, , where ω n = (2n + 1)πT is Matsubara frequency. Green's function is the matrix in the band space (denoted by bold face) and combined Nambu and spin spaces (denoted by hat). The Pauli matricesτ i andσ i refers to Nambu and spin spaces, respectively.
Assuming the isotropic superconducting gaps at the Fermi surface sheets, we set the impurity self-energy to be independent on the momentum k but preserving the dependence on the frequency and the band indices,Σ Therefore the task is simplified by averaging over k. As a result, all equations are written in terms of the quasiclassical ξ-integrated Green's functions, whereĝ αn = g 0αnτ0 ⊗σ 0 + g 2αnτ2 ⊗σ 2 , Here g 0αn and g 2αn are the normal and anomalous (Gor'kov) ξ-integrated Green's functions in the Nambu representation, They depend on the density of states per spin at the Fermi level of the corresponding band (N a,b ), and on the order parameterφ αn and frequencyω αn . Both the order parameter and the frequency are renormalized by the self-energy, where Σ 0(2)α and Σ imp 0(2)α are the parts of the self-energy coming from the pairing interaction (spin fluctuations, electron-phonon coupling, etc.) and impurity scattering, respectively. Impurity scattering will be considered in the s-wave channel only. Treating a more complicated impurity potential is a separate cumbersome task, although this goal was pursued by some authors [45][46][47]. The order parameterφ αn is connected with the gap function ∆ αn via the renormalization factor Z αn =ω αn /ω n , i.e.
The part of the self-energy due to the pairing interaction can be written as where λ that is defined by coupling constants λ φ,Z αβ and includes density of states N β and the normalized bosonic spectral function B(Ω). The matrix elements λ φ αβ may be positive (attractive) or negative (repulsive) due to Coulomb repulsion, spin fluctuations and electron-phonon interaction, while matrix elements To calculate the impurity part of the self-energyΣ imp , we use noncrossing diagrammatic approximation that is equivalent to the T -matrix approximation, . Without loss of generality we set R i = 0. Intra-and interband parts of the impurity potential are given as v and u respectively, The relation between intra-and interband impurity scattering is set by a parameter η = v/u.
It is convenient to introduce the generalized cross-section parameter σ = π 2 N a N b u 2 1 + π 2 N a N b u 2 → 0, Born limit, 1, unitary limit (14) and the impurity scattering rate Both are shown to have special values in the limiting cases: (i) the Born limit corresponding to the weak impurity potential (πuN a(b) 1), and (ii) the unitary limit corresponding to the strong impurity potential (πuN a(b) 1). Equations presented above are general and can be applied to a wide range of multiband systems. Below we concentrate on iron-based superconductors within the minimal two-band model. Bands are labelled as a and b. Spin fluctuation spectrum enters through the bosonic spectral function B(Ω) [48][49][50]; the superconducting interaction is also determined by the choice of the coupling constants λ αβ .

Results and Discussions
For the calculations, we choose the ratio between densities of states of different bands as N b /N a = 2 and set the values of the coupling constants to be (λ aa ; λ ab ; λ ba ; λ bb ) = (3; −0.2; −0.1; 0.5), so the averaged coupling constant is positive, This set of parameters leads to the s ± superconducting state with unequal gaps -larger positive (band a) and smaller negative (band b) gap -and the critical temperature T c0 = 40 K in the clean limit.
We present the smaller gap ∆ b,n for the lowest Matsubara frequency, n = 0, as a function of the scattering rate Γ a and the temperature T in two different cases: (i) weak scattering with σ = 0 (Born limit), see Figure 1, and (ii) intermediate scattering limit with σ = 0.5, see Figure 2. The larger gap, ∆ a,n , is always positive and, therefore, there is a line of s ± → s ++ transition corresponding to the change of ∆ b,n sign from negative to positive. The transition goes through the gapless state with the finite larger gap and the vanishing smaller gap [10]. Note that the line of transition is not vertical in Figures 1 and 2. Therefore, there is no single critical scattering rate but a temperature-dependent Γ crit a (T). Moreover, if we stay at a fixed Γ a in the range 1.1T c0 < Γ a < 1.6T c0 for σ = 0 or 1.5T c0 < Γ a < 3.2T c0 for σ = 0.5 and increase the temperature, we observe an interesting behavior of the gap. Namely, while at low temperatures the transition to the s ++ state already took place (∆ b,n > 0), at higher temperatures the system goes back to the s ± state (∆ b,n < 0). Thus, there is a temperature-dependent s ++ → s ± transition. With the increasing Γ a , the temperature of this transition is shifted to T c and the system becomes s ++ for the whole temperature range.  To illustrate the mentioned points, in Figures 3 and 4 we present results for the temperature dependence of the gap function ∆ α,n=0 , order parameterφ α,n=0 , and the renormalization factor Z α,n=0 for several fixed values of Γ a for σ = 0. Temperature dependencies for σ = 0.5 are shown in Figures 5  and 6. The gap in band a has the same positive sign for all values of Γ a and vanishes at T c , see Figures 3(a) and 5(a). There is, however, a small range of Γ a values around Γ crit a , for which the gap is an increasing function of T at low temperatures. At the same time, the order parameter behaves quite conventionally and decreases towards T c , see Figures 4(a) and 6(a). The unusual temperature dependence of the gap at the lowest Matsubara frequency is due to the renormalization factor Z α,n=0 that is shown in Figures 4(b) and 6(b). That is, according to Equation (9), large values of Z α,n taking place at low temperatures lead to the decrease of ∆ α,n . For the band b the same effect is not seen in the Born limit, but becomes pronounced in the intermediate scattering limit; compare the gap function in Figure 5(b) and the order parameter in Figure 6(c), and notice also the low-temperature behavior of Z α,n in Figure 6(d). Similar behavior of gaps, in principle, may occur from the functional form of gaps at real frequencies, see Eq. (7) in Ref. [51].   T / T c0  In the clean limit and for the small Γ a , the sign of the smaller gap ∆ b,n is negative at all temperatures, see Figures 3(b) and 5(b). With the increase of the impurity scattering rate, the gap at low temperatures changes sign while at higher temperatures the sign is either reversed again (small Γ a ) or the gap vanishes (Γ a 2T c0 ). Therefore, the transition from the s ± state to the s ++ state is characterized by two parameters, namely, the critical scattering rate Γ crit a and the critical temperature T crit . The latter changes from zero to T c . Thus the s ++ state becomes dominant in the initially clean s ± system for Γ a > Γ crit a and T < T crit . This is true in both Born limit and the intermediate scattering limit. Also, we have checked that the similar behavior holds for the higher Matsubara frequencies, see Figure 7 for the illustration of the gaps behavior for n = 1 and n = 10.  T / T c0 Γ a = 0T c0 Γ a = 0.9T c0 Γ a = 1.7T c0 Γ a = 3.5T c0 Γ a = 10.4T c0  Previously we have found a steep change in the smaller gap as a function of the scattering rate in the weak scattering limit [43]. Here we observe that the same jump is present in the temperature dependence of the gap, see Figure 8, where ∆ b,n in the Born limit is shown. Discontinuous jump in the temperature dependence of the smaller gap appears at T < 0.1T c0 .  Figure 8. Temperature dependencies of lowest-frequency Matsubara gap ∆ b,n=0 normalized by T c0 in the Born limit for several values of Γ a . Solid curves correspond to a smooth evolution of the gap in the s ± state and across the s ++ → s ± transition while the temperature dependencies with the discontinuous jump of the gap are shown by symbols.

Conclusions
We considered the two-band model for FeBS that has the s ± superconducting ground state in the clean limit. Here we studied dependence of the superconducting gaps ∆ α,n on both the temperature and the nonmagnetic impurity scattering rate. We show that the disorder-induced transition from s ± to s ++ state is temperature-dependent. That is, in a narrow region of scattering rates, while the ground state is s ++ , it transforms back to the s ± state at higher temperatures up to T c . With the increasing impurity scattering rate, temperature of such a s ++ → s ± transition shifts to the critical temperature T c . The s ± → s ++ transition is characterized by two parameters: (i) the critical scattering rate Γ crit a and (ii) the critical temperature T crit ≤ T c . The s ++ state becomes dominant in the initially clean s ± system for Γ a > Γ crit a and T < T crit . The similar situation takes place for the case of the magnetic disorder where the s ± state at low temperatures occurring due to the s ++ → s ± transition [42], at higher temperatures may transform back to the s ++ state. Experimentally, one can observe the reentrant s ± state by increasing the temperature for the fixed amount of disorder that results in the low-temperature s ++ state. For example, the spin resonance peak in the inelastic neutron scattering should be absent in the low-temperature s ++ state, but have to appear in the s ± state at higher temperatures [8,13,17,18]. Temperature dependence of the penetration depth should also bear specific signatures of the gapless behavior accompanying the s ++ → s ± transition [11].
Author Contributions: All author contributed equally.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:

FeBS
Fe-based superconductors NMR Nuclear Magnetic Resonance