Impurity-induced in-gap state and Tc in sign-reversing s-wave superconductors: analysis of iron oxypnictide superconductors

The sign-reversing fully gapped superconducting state, which is expected to be realized in oxypnictide superconductors, can be prominently affected by nonmagnetic impurities due to the interband scattering of Cooper pairs. We study this problem based on the isotropic two-band BCS model: In oxypnictide superconductors, the interband impurity scattering $I'$ is not equal to the intraband one $I$. In the Born scattering regime, the reduction in Tc is sizable and the impurity-induced density of states (DOS) is prominent if $I\sim I'$, due to the interband scattering. Although impurity-induced DOS can yield a power-law temperature dependence in $1/T_1$, a sizable suppression in Tc is inevitably accompanied. In the unitary scattering regime, in contrast, impurity effect is very small for both Tc and DOS except at $I=I'$. By comparing theory and experiments, we expect that the degree of anisotropy in the $s_\pm$-wave gap function strongly depends on compounds.


I. INTRODUCTION
perimental absence of impurity effect on T c in iron oxypnictides is well understood in terms of the s ± -wave state. On the other hand, T c will be prominently reduced by short-range weak (Born) impurities [38].
Recently, several authors had revealed that in-gap density of states (DOS) is induced by impurities in the s ± -wave state using the Born approximation for general value of x ≡ |I ′ /I| [39], or using the T -matrix approximation only for x = 1 [40,41]. They also demonstrated that the relation 1/T 1 ∝ T 3 under T c , which had been reported by several groups [13,[42][43][44], can be reproduced by the impurity-induced DOS. However, the assumed impurity parameters (n imp , I and I ′ ) also yields a sizable suppression in T c according to the analysis in Ref. [38].
Furthermore, impurity-induced DOS should be sensitive to the value of x in the unitary scattering regime, as suggested in ref. [38]. Therefore, we have to study the impurity effects on the DOS and T c for general x, and compare their relationships in detail.
In this paper, we investigate the impurity-induced DOS and T c in the s ± -wave state using the T -matrix approximation for general x. We stress that x is not unity in iron oxypnictides since hole and electron pockets are not composed of the same d-orbitals. In the Born or intermediate scattering regime, a sizable impurity-induced DOS appears for x 0.7, and therefore 1/T 1 may deviate from a simple exponential behavior. Although impurity-induced DOS can yield a power-law temperature dependence in 1/T 1 [39][40][41], we find that a sizable suppression in T c is inevitably accompanied. The anisotropy in the s ± -wave superconducting gap, which had been predicted theoretically [23,25], might be responsible for the power-law temperature dependence of 1/T 1 under T c as discussed in ref. [45]. In contrast, unitary impurities affect both the superconducting DOS and T c only slightly, except at I = I ′ .

II. T -MATRIX APPROXIMATION IN THE TWO-BAND BCS MODEL
As studied in refs. [29,38,39,41], the s ± -wave state is realized in the two-band BCS model if we introduce the interband repulsive interaction, which represents the AF fluctuations due to the interband nesting in iron oxypnictides. In the present paper, we study the impurity effect using the T -matrix approximation for general I ′ /I. In the presence of mass enhancement due to many-body effect, m * /m 0 > 1, both the superconducting gap and the impurity effect (or impurity concentration n imp ) are renormalized by the factor (m * /m 0 ) −1 .
In the present analysis, we neglect the mass-enhancement for simplicity.
In the Nambu representation, the two-band BCS model is given by [46,47] In eq. (2), ǫ α k , ǫ β k are the band dispersions measured from the Fermi level. Since we consider the isotropic s ± superconducting state, only the DOSs for both bands at the Fermi level (N α , N β ) are taken into consideration in the present BCS study. ∆ α , ∆ β in eq. (2) are the superconducting gap. When only the inter-band repulsive interaction (g αβ = g βα > 0) is taken into consideration, the gap equation without impurities is given as [38,40,41], where ǫ n = πT (2n+1) is the fermion Matsubara frequency, and ω c is the cutoff energy. N β(α) is the DOS for β(α)-band at the Fermi energy in the normal state per spin. f β(α) (ǫ) is the local anomalous Green function for β(α) band, which will be given later. Since f β(α) ∝ ∆ β(α) , the s ± -state ∆ α = −∆ β is realized for g αβ > 0 [38,40,41]. Moreover, The Nambu matrix representation for the impurity potential is given aŝ We can assume that I, I ′ ≥ 0 without losing generality. In the presence of impurities, the Green function in the Nambu representation is given by [47] whereω ≡ ω + iδ (δ = +0), andΣ(ω) is the self-energy due to impurities.
Hereafter, we deriveΣ(ω) in the T -matrix approximation, which gives the exact result for n imp ≪ 1 for any strength of I, I ′ . The T -matrix in the Nambu representation is given byT whereĝ(ω) ≡ 1 N kĜ k (ω) is the local Green function, which is given by [47] In the above expression, g i and f i (i = α, β) are given by where Σ n i and Σ a i (i = α, β) are the normal and anomalous self-energies, respectively. In the T -matrix approximation, the self-energies are given by using eq. (6) as In the fully self-consistent T -matrix approximation, we have to solve eqs. (3) and (7)-(13) self-consistently. In this paper, however, we solve only eqs. (7)-(13) self-consistently, by neglecting the impurity effect on ∆ α and ∆ β in eq. (3). This approximation is justified when the reduction in ∆ α(β) due to impurity pair-breaking is small.

III. NUMERICAL RESULTS
Here, we discuss the impurity effect on the DOS and T c in the s ± -wave superconducting state, based on the numerical results given by the T -matrix approximation. A. Impurity effect on T c As derived in Ref. [38], the expression for the reduction in T c per impurity concentration based on the two-band BCS model is given as For n imp ≪ 1, the transition temperature is given by In the case of x ≡ I ′ /I = 1, the right hand side of eq. (14) . Therefore, eq. (14) approaches zero in the case of x = 1 in the unitary regime. Figure 1 (a) shows −∆T c /n imp given in eq. (14) in the case of N α = N β = 1. In iron oxypnictides, the total DOS per Fe atom (N α + N β ) is 1.31 eV −1 per spin [20]. Then, 1/N = 1 corresponds to 18000 K. When x = 1, −∆T c /n imp approaches 1/8N ∼ 2300 K in the unitary regime (IN ≫ 1). Therefore, the superconductivity in iron oxypnictides will vanish only at n imp ≈ 8N · T 0 c = 0.01 ∼ 0.02 [1 ∼ 2 %]. When x = 1, in high contrast, −∆T c /n imp decreases and approaches zero as I increases in the unitary regime, since the effective interband scattering is renormalized as . According to the first principle calculations, N β /N α 0.7 in iron oxypnictides [48]. Here, we study the case of N β /N α = 0.5 in order to clarify the the impurity effect on T c for the the particle-hole asymmetric case; N β /N α = 1. Figure 1 (b) shows −∆T c /n imp for N α = 1 and N β = 0.5. According to eq. (14), −∆T c /n imp for x = 1 and I = ∞ is 1/5.84N α ∼ 2400 K, by taking account of the relation N α + N β = 1.31 eV −1 in iron oxypnictides. By comparing with the results for N α = N β = 1 in Fig. 1 (a), we find that −∆T c /n imp is insensitive to the value of N β /N α , under the condition that N α + N β =constant. Previously, impurity effect on T c in two-band BCS models had been studied by many authors in various contexts [49][50][51][52][53][54], and it was found that T c is unchanged in the unitary limit [50,51]. However, eq. (14) for s ± -state had not been derived. Here, we present a clear explanation why the interband scattering (pair breaking) is absent in the unitary regime, which had not been discussed previously. Figure 2 shows the intraband and interband Tmatrices in the normal state, T I=0 αα and T I=0 αβ , in the case of I = 0 and N α = N β = N. Apparently, T I=0 αβ approaches zero for I ′ → ∞. Next, we consider T αβ for general (I, I ′ ). If we construct T αβ of (T I=0 αα , T I=0 αβ , I), it contains at least oneT I=0 αβ . For this reason, interband T -matrix is expected to approach zero in the unitary regime. This expectation is correct unless x = 1, as shown in Ref. [38].

B. Impurity effect on the DOS
In the s ± -wave superconducting state, impurity interband scattering not only reduces T c , but also induces the in-gap state in the superconducting DOS [40,49,51,55]. The DOS is given by the imaginary part of the local Green function, which is expressed in eq. (8), as follows: If n imp ≪ 1, the obtained DOS will be reliable for any I and I ′ in the present T -matrix approximation. Figure 3 shows the DOS in the superconducting state in the case of N α = N β = 1 and ∆ α = −∆ β = 0.005 for n imp = 0.008. |∆ α,β | = 0.005 corresponds to 90 K.
Experimentally, in Ba 0.6 K 0.4 Fe 2 As 2 , |∆| = 11 ∼ 12 meV for α Fermi surface (hole-like) and for γ and δ Fermi surfaces (electron-like), and |∆| = 5.8 meV for β Fermi surface (hole-like) [18]. As shown in Fig. 3 (a), the superconducting gap is almost filled by the impurityinduced DOS when I = I ′ = 0.25 (Born regime), which corresponds to ∼ 5000 K. In this case, the nuclear relaxation ratio 1/T 1 shows a power-law temperature dependence since the impurity-induced DOS is approximately linear in ω, like in line-node superconductors.  [40]. In these cases, however, reduction in T c due to impurities, which is given by n imp times −∆T c /n imp in Fig. 1 (a), reaches 13 K. The estimated reduction in T c would be underestimated since −∆T c /n imp is an increase function of n imp for T c T 0 c /2 [41,47]. In all cases we have studied ( Fig. 3 (a)-(d)), power-law behavior in 1/T 1 for T ≪ T c due to the galpess superconducting state always accompanies a sizable suppression in T c , −∆T c 10 K, for |∆| = 90 K. When |∆| = 40 K, the gapless superconducting state can be realized when −∆T c ∼ 6 K. Thus, it will be difficult to ascribe the experimental relation 1/T 1 ∝ T 3 below T c [42][43][44] in clean samples with high T c to the impurity effect. unitary regime [38].

IV. DISCUSSION
In the present paper, we studied the impurity effects on the s ± -wave superconducting state, which is expected to be realized in iron oxypnictide superconductors. There, nonmag-netic impurities can induce both the in-gap bound state and the reduction in T c . Based on the two-band BCS model, we have found that the zero-energy in-gap state emerge under the conditions that (i) x ≡ |I ′ /I| = 1 and (ii) |I|N α , |I|N β ≫ 1. Deviating from these conditions, in-gap state shifts to a finite energy, and disappears eventually.
Here, we discuss the case of unitary scattering: In iron oxypnictide superconductors, Fe substitution by other elements (such as Co, Ni, and Zn) will cause the unitary scattering potential. In this case, the impurity potential is diagonal with respect to the d-orbital [38]. The impurity potential has off-diagonal elements in the band-diagonal representation.
As discussed in Ref. [38] We also discuss the case of Born scattering due to "in-plane" weak random potential or disorder: As shown in Figs. 3 (a) and (b), the impurity effect is rather insensitive to x.
Therefore, a broad impurity-induced in-gap state will emerge in the superconducting DOS, and a sizable reduction in T c occurs at the same time. Born impurity scattering will be also caused by "off-plane" impurities like the As substitution by other elements. In this case, the radius of impurity potential R for Fe sites will be about the unit cell length a. Then, the impurity scattering (k → k ′ ) is restricted to |k − k ′ | 1/R ∼ 1/a. Since |k − k ′ | ≈ π/a in the interband scattering between electron-pockets and hole-pockets, I ′ should be much smaller than I. Therefore, the effect of off-plane impurities on the s ± -wave state will be small since the relationship x ≪ 1 is expected to be realized.
In summary, in iron oxypnictide superconductors, Born or intermediate in-plane impurities cause prominent impurity effects since the s ± -wave state is violated by the interband scattering. Only one percent Born impurities with x 0.5 induce not only plenty of ingap DOS, but also sizable reduction in T c . For this reason, relation 1/T 1 ∝ T 3 below T c observed in clean LaFeAsO 1−x F x [42,44] and in LaFeAsO 0.7 [43] samples, which would be almost absent from the impurity reduction in T c , cannot be explained by the present analysis based on the isotropic BCS model. Thus, anisotropy in the s ± -wave superconducting gap might be responsible for the relation 1/T 1 ∝ T 3 [45]. Recently, rapid suppression in 1/T 1 (∝ T α ; α > 5) below T c had been observed in a clean LaFeAsO 0.9 F 0.1 sample with T c = 28 K (=intrinsic T c ) [57]. This result is consistent with the penetration depth [14] and ARPES [15][16][17][18], and it is naturally explained by the present analysis. Theoretically, in fully gapped s-wave superconductor, the gap function becomes anisotropic due to magnetic fluctuations, in a way that the two superconducting gap minima are connected by the nesting vector [56].
In iron oxypnictides, the degree of anisotropy in the s ± -wave gap function is rather sensitive to model parameters such as the nesting condition [23,25]. The wide variety of behaviors in 1/T 1 would reflect the large sample dependence of the gap anisotropy in iron oxypnictide superconductors.