Impurity-induced in-gap state and Tc in sign-reversing s-wave superconductors: analysis of iron oxypnictide superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Impurity-induced in-gap state and T c in sign-reversing s -wave superconductors:analysis of iron oxypnictide superconductors Yuko
Senga , and Hiroshi Kontani , Department of Physics, Nagoya University,Furo-cho, Nagoya 464-8602, Japan. JST, Transformative Research-Project on Iron Pnictides (TRIP),Chiyoda, Tokyo 102-0075, Japan. (Dated: October 30, 2018)
Abstract
The sign-reversing fully gapped superconducting state, which is expected to be realized in oxyp-nictide superconductors, can be prominently affected by nonmagnetic impurities due to the inter-band scattering of Cooper pairs. We study this problem based on the isotropic two-band BCSmodel: In oxypnictide superconductors, the interband impurity scattering I ′ is not equal to theintraband one I . In the Born scattering regime, the reduction in T c is sizable and the impurity-induced density of states (DOS) is prominent if I ∼ I ′ , due to the interband scattering. Althoughimpurity-induced DOS can yield a power-law temperature dependence in 1 /T , a sizable suppres-sion in T c is inevitably accompanied. In the unitary scattering regime, in contrast, impurity effectis very small for both T c and DOS except at I = I ′ . By comparing theory and experiments, weexpect that the degree of anisotropy in the s ± -wave gap function strongly depends on compounds. . INTRODUCTION Recently, the mechanism of superconductivity in high- T c superconductors with FeAs lay-ers [1–5] has been attracting considerable attentions. The superconducting state is realizedby introducing carrier into the parent compound, which shows the spin density wave (SDW)state at T N ∼ ∼ . µ B andthe ordering vector is Q ≈ ( π,
0) [6, 8–10]. NMR studies had clearly shown that the singletsuperconducting state is realized in iron oxypnictides [11–13]. A fully gapped superconduct-ing state has been determined by the penetration depth measurement [14], angle-resolvedphotoemission spectroscopy (ARPES) [15–18], specific heat measurement [19], and so on.In the first-principle band calculations [20–22], the Fermi surfaces in iron oxypnictides arecomposed of two hole-like Fermi pockets around the Γ = (0 ,
0) point and two electron-likeFermi pockets around M= ( π, , (0 , π ) points. The nesting between the hole and electronpockets is expected to give rise to the SDW state in undoped compounds. In doped com-pounds without SDW order, the antiferromagnetic (AF) fluctuations with Q ≈ ( π,
0) isexpected to induce a fully gapped s -wave state with sign reversal, which is called the s ± -wave state [23–30]. Moreover, near the SDW boundary, the Hall coefficient and Nernst signalshow prominent anomalous behaviors [7, 12, 31], which are similar to those observed in high- T c cuprates and in Ce M In ( M =Co,Rh,Ir) [32]. Theoretically, these anomalous transportphenomena indicate the existence of strong AF fluctuations [33]. At the same time, hugeresidual resistivity far beyond the s -wave unitary scattering is expected to appear near theSDW boundary theoretically [34].To investigate the pairing symmetry of superconductivity, impurity effects on the su-perconducting state offer us decisive informations. In iron oxypnictide superconductors,impurity effect on T c due to Co, Ni, or Zn substitution for Fe sites is very small or absent[11, 12, 35–37]. This result clearly rules out the possibility of line-node superconductivity.One may also expect that the s -wave state with sign reversal is also eliminated, since theCooper pair is destroyed by the interband scattering induced by impurities. However, wehave recently shown that T c is almost unchanged by strong (unitary) impurities, since theinterband impurity scattering potential I ′ is different from the intraband one I [38]. Thereason for this unexpected result is that the effective interband scattering is renormalized tozero in the unitary limit except at I = I ′ in the T -matrix approximation. Therefore, the ex-2erimental absence of impurity effect on T c in iron oxypnictides is well understood in termsof the s ± -wave state. On the other hand, T c will be prominently reduced by short-rangeweak (Born) impurities [38].Recently, several authors had revealed that in-gap density of states (DOS) is induced byimpurities 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 demonstratedthat the relation 1 /T ∝ T under T c , which had been reported by several groups [13, 42–44],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 unitaryscattering regime, as suggested in ref. [38]. Therefore, we have to study the impurity effectson 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 usingthe T -matrix approximation for general x . We stress that x is not unity in iron oxypnictidessince hole and electron pockets are not composed of the same d -orbitals. In the Born orintermediate scattering regime, a sizable impurity-induced DOS appears for x & .
7, andtherefore 1 /T may deviate from a simple exponential behavior. Although impurity-inducedDOS can yield a power-law temperature dependence in 1 /T [39–41], we find that a sizablesuppression in T c is inevitably accompanied. The anisotropy in the s ± -wave superconductinggap, which had been predicted theoretically [23, 25], might be responsible for the power-lawtemperature dependence of 1 /T under T c as discussed in ref. [45]. In contrast, unitaryimpurities 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 BCSmodel if we introduce the interband repulsive interaction, which represents the AF fluctua-tions due to the interband nesting in iron oxypnictides. In the present paper, we study theimpurity effect using the T -matrix approximation for general I ′ /I . In the presence of massenhancement due to many-body effect, m ∗ /m >
1, both the superconducting gap and theimpurity effect (or impurity concentration n imp ) are renormalized by the factor ( m ∗ /m ) − .In the present analysis, we neglect the mass-enhancement for simplicity.3n the Nambu representation, the two-band BCS model is given by [46, 47]ˆ H = X k ˆ c † k ˆ H k ˆ c k , (1)where ˆ c † k = ( c α † k ↑ , c β † k ↑ , c − k ↓ α, c β − k ↓ ), andˆ H k = ǫ α k α ǫ β k β ∆ α − ǫ α k
00 ∆ β − ǫ β k . (2)In eq. (2), ǫ α k , ǫ β k are the band dispersions measured from the Fermi level. Since we considerthe 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 thesuperconducting gap. When only the inter-band repulsive interaction ( g αβ = g βα >
0) istaken into consideration, the gap equation without impurities is given as [38, 40, 41],∆ α ( β ) = − g αβ N β ( α ) T X n f β ( α ) ( iǫ n ) θ ( ǫ n − | ω c | ) , (3)where ǫ n = πT (2 n +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 thelocal 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 > | ∆ α / ∆ β | ∼ ( N β /N α ) / since f α /f β ∼ ∆ α / ∆ β .The Nambu matrix representation for the impurity potential is given asˆ I = I I ′ I ′ I − I − I ′ − I ′ − I . (4)We can assume that I, I ′ ≥ G k (˜ ω ) = (˜ ω ˆ1 − ˆ H k − ˆΣ(˜ ω )) − , (5)where ˜ ω ≡ ω + iδ ( δ = +0), and ˆΣ(˜ ω ) is the self-energy due to impurities.4ereafter, we derive ˆΣ(˜ ω ) in the T -matrix approximation, which gives the exact resultfor n imp ≪ I, I ′ . The T -matrix in the Nambu representation is givenby ˆ T (˜ ω ) = (ˆ1 − ˆ I · ˆ g (˜ ω )) − ˆ I, (6)where ˆ g (˜ ω ) ≡ N P k ˆ G k (˜ ω ) is the local Green function, which is given by [47]ˆ g (˜ ω ) = g α (˜ ω ) 0 f α (˜ ω ) 00 g β (˜ ω ) 0 f β (˜ ω ) f α (˜ ω ) 0 g α (˜ ω ) 00 f β (˜ ω ) 0 g β (˜ ω ) . (7)In the above expression, g i and f i ( i = α, β ) are given by g i (˜ ω ) = − πN i ˜ ωZ i (˜ ω ) p − (˜ ωZ i (˜ ω )) + (∆ i + Σ ai (˜ ω )) , (8) f i (˜ ω ) = − πN i ∆ i + Σ ai (˜ ω ) p − (˜ ωZ i (˜ ω )) + (∆ i + Σ ai (˜ ω )) , (9) Z i (˜ ω ) = 1 − ω (Σ ni (˜ ω ) − Σ ni ( − ˜ ω )) , (10)where Σ ni and Σ ai ( 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) asΣ nα (˜ ω ) = n imp T (˜ ω ) , Σ nβ (˜ ω ) = n imp T (˜ ω ) , (11)Σ nα ( − ˜ ω ) = − n imp T (˜ ω ) , Σ nβ ( − ˜ ω ) = − n imp T (˜ ω ) , (12)Σ aα (˜ ω ) = n imp T (˜ ω ) , Σ aβ (˜ ω ) = n imp T (˜ ω ) . (13)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 justifiedwhen 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 superconductingstate, based on the numerical results given by the T -matrix approximation.5 a) (b) I’/I=10.90.80.70.60.5 N a =1 N b =1 − ∆ T/n imp
I 1/8N
I’/I=10.90.80.70.60.5 I − ∆ T/n imp N a =1 N b =0.5 FIG. 1: − ∆ T c /n imp as a function of I given by the T -matrix approximation in the case of (a) N α = N β = 1 and (b) N α = 1 , N β = 0 .
5. In both figures, the unit of energy is 1 /N α , whichcorresponds to 18000 K for (a) and 14000 K for (b), since N α + N β = 1 .
31 eV − in iron oxypnictides. − ∆ T c /n imp for ( x = 1, I = ∞ ) is 1 / N α ∼ / . N α ∼ A. Impurity effect on T c As derived in Ref. [38], the expression for the reduction in T c per impurity concentrationbased on the two-band BCS model is given as − ∆ T c n imp = π (cid:2) N α + N β ) − p N α N β (cid:3) I ′ A , (14)For n imp ≪
1, the transition temperature is given by T c = T − ( − ∆ T c /n imp ) · n imp , where T is the transition temperature without impurities. In eq. (14), ¯ A = 1 + π I ( N α + N β ) +2 N α N β π I ′ + N α N β π ( I − I ′ ) . In the case of x ≡ I ′ /I = 1, the right hand side of eq. (14)is [3( N α + N β ) − p N α N β ] / [8( N α + N β ) ]+ O ( I − ) in the unitary regime. In the case of x = 1,in contrast, eq. (14) is given by x [3( N α + N β ) − p N α N β ] / [8 π N α N β (1 − x ) I ] + O ( I − ).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 ironoxypnictides, the total DOS per Fe atom ( N α + N β ) is 1.31 eV − per spin [20]. Then,1 /N = 1 corresponds to 18000 K. When x = 1, − ∆ T c /n imp approaches 1 / N ∼ IN ≫ n imp ≈ N · T = 0 . ∼ .
02 [1 ∼ x = 1, in high contrast, − ∆ T c /n imp decreases and approaches zero as I increases in the unitary regime, since theeffective interband scattering is renormalized as I ′ eff ∼ I ′ · ( IN ) − ≪ I ′ [38].According to the first principle calculations, N β /N α & . N β /N α = 0 . T c for the theparticle-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 / . N α ∼ N α + N β = 1 .
31 eV − in iron oxypnictides. By comparingwith the results for N α = N β = 1 in Fig. 1 (a), we find that − ∆ T c /n imp is insensitive to thevalue of N β /N α , under the condition that N α + N β =constant. = → (cid:58) + (cid:58) (cid:58) (cid:58) + ... ' 0 II N π = →+ (cid:58) + (cid:58) (cid:58) (cid:58) + ... ( ')1 ( ' ) i I N iI N π N π π + (cid:58) (cid:58) ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I T αβ = α β T αα = α αα α α αα βββ β β I = I = FIG. 2: Intraband and interband T -matrices in the normal state, T I =0 αα and T I =0 αβ . Previously, impurity effect on T c in two-band BCS models had been studied by manyauthors in various contexts [49–54], and it was found that T c is unchanged in the unitarylimit [50, 51]. However, eq. (14) for s ± -state had not been derived. Here, we present a clearexplanation 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 T -matrices 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 ′ ). Ifwe construct T αβ of ( T I =0 αα , T I =0 αβ , I ), it contains at least one ˆ T I =0 αβ . For this reason, interband T -matrix is expected to approach zero in the unitary regime. This expectation is correctunless x = 1, as shown in Ref. [38]. 7 a) (b) I=0.25
I’/I=1
DOS n imp =0.008N a =N b =1I’/I=0.9I’/I=0.8I’/I=0.7I’/I=0.6I’/I=0.5 ω I=0.5
DOS ω n imp =0.008N a =N b =1 n imp =0.001I’/I=1I’/I=0.9I’/I=0.8 (c) (d) I=2
DOS ω n imp =0.008N a =N b =1 n imp =0.001I’/I=1I’/I=0.9I’/I=0.8 I=8
DOS ω n imp =0.008N a =N b =1 n imp =0.001I’/I=1I’/I=0.9I’/I=0.8 FIG. 3: Obtained DOS in the superconducting state for N α = N β = 1 ( ∼ .
66 eV − ) and ∆ α = − ∆ β = 0 .
005 ( ∼
100 K), in the case of (a) I = 0 .
25, (b) I = 0 .
5, (c) I = 2, and (d) I = 8. Impurityconcentration n imp is 0 . n imp = 0 . N ( − ω ) = N ( ω ). 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 is8iven by the imaginary part of the local Green function, which is expressed in eq. (8), asfollows: N ( ω ) = − π Im { g α (˜ ω ) + g β (˜ ω ) } . (15)If n imp ≪
1, the obtained DOS will be reliable for any I and I ′ in the present T -matrixapproximation. 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 . | ∆ α,β | = 0 .
005 corresponds to 90 K.Experimentally, in Ba . K . Fe As , | ∆ | = 11 ∼
12 meV for α Fermi surface (hole-like) andfor γ and δ Fermi surfaces (electron-like), and | ∆ | = 5 . β Fermi surface (hole-like)[18]. As shown in Fig. 3 (a), the superconducting gap is almost filled by the impurity-induced DOS when I = I ′ = 0 .
25 (Born regime), which corresponds to ∼ /T shows a power-law temperature dependence since theimpurity-induced DOS is approximately linear in ω , like in line-node superconductors.Figure 4 shows 1 /T below T c for I = I ′ = 0 .
25, where σ ≡ ( πN I ) / (1 + ( πN I ) ) = 0 . | ∆ α ( β ) | = 0 . p − T /T c and T c = 0 . x = 1 . /T is inside of T - and T -lines for T c > T > . T c ,consistently with the analysis by Parker et al. for σ = 0 . 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 anincrease function of n imp for T c . T / /T for T ≪ T c due to the galpess superconducting state alwaysaccompanies a sizable suppression in T c , − ∆ T c &
10 K, for | ∆ | = 90 K. When | ∆ | = 40K, the gapless superconducting state can be realized when − ∆ T c ∼ /T ∝ T below T c [42–44] in clean sampleswith high T c to the impurity effect.In the intermediate ( I = 0 .
5) or unitary ( I ≥
2) regime, in Figs. 3 (b)-(d), a largeimpurity-induced DOS appears at the zero energy in the case of x = 1, which is consistentwith previous theoretical studies [40, 55]. In this case, however, − ∆ T c ∼
13 K for n imp =0 . x ≤ .
9, in contrast, − ∆ T c in the unitary regime is prominently reducedas shown in Fig. 1. At the same time, the impurity-induced DOS quickly moves to the gapedge and disappears, as demonstrated in Figs. 3 (c) and (d). The reason for these resultsis that the effective interband scattering is renormalized as I ′ eff ∼ I ′ · ( IN ) − ≪ I ′ in the9 T/T c −3 −2 −1 / T x=1x=0.9x=0.8x=0.7 I=0.25 T T n imp =0.008N α =N β =1 FIG. 4: Obtained 1 /T below T c for I = 0 .
25 and x = 0 . ∼ .
0. For x ≤ .
8, 1 /T decreases muchfaster than T at low temperatures because of the absence of the in-gap state near ω = 0. Sincethe quasiparticle damping rate is γ = n imp πN ( I + I ′ ) / ¯ A [38], γ/ ∆ = 0 .
18 for x = 1. unitary regime [38].Next, we study the case of N α = 1 , N β = 0 .
5. We put ∆ α = 0 .
005 and ∆ β = − . | ∆ α / ∆ β | ∼ ( N β /N α ) / in the two-band BCS model withrepulsive interband interaction, as explained in § II. Figure 5 (a), (b) show the DOS in thesuperconducting state for n imp = 0 . I = I ′ = 0 .
25, in Fig. 5 (a), the impurity-induced DOS is reduced by changing N β from 1 to 0 .
5, by comparing with Fig. 3 (a). As x decreases from unity, the impurity-induced DOS moves to the gap edge. In the intermediate( I = 0 .
5) or unitary ( I ≥
2) regime, impurity-induced DOS covers the zero energy state inthe case of x = 1 and n imp = 0 . x ≤ .
9. In the case of I = 8, in Fig. 5 (d), impurity-inducedDOS is very large at x = 1, whereas it is strongly suppressed for x ≤ .
9. When the impurityconcentration is very low ( n imp ∼ . ω = 0 even if x = 1 asshown in insets in Figs. 5 (b)-(d), since f α + f β given in eq. (7) is non-zero in the case of N α = N β [41]. 10 a) (b) I=0.25
DOS ω n imp =0.008N a =1 N b =0.5I’/I=1I’/I=0.9I’/I=0.8I’/I=0.7I’/I=0.6I’/I=0.5 I=0.5
DOS ω n imp =0.008N a =1 N b =0.5 n imp =0.001I’/I=1I’/I=0.9I’/I=0.8 (c) (d) I=2
DOS ω n imp =0.008N a =1 N b =0.5 n imp =0.001I’/I=1I’/I=0.9I’/I=0.8 I=8
DOS ω n imp =0.008N a =1 N b =0.5 n imp =0.001I’/I=1I’/I=0.9I’/I=0.8 FIG. 5: Obtained DOS in the superconducting state for N α = 1 , N β = 0 . α = 0 . , ∆ β = − . I = 0 .
25, (b) I = 0 .
5, (c) I = 2, and (d) I = 8. Impurity concentration n imp is 0 . n imp = 0 . IV. DISCUSSION
In the present paper, we studied the impurity effects on the s ± -wave superconductingstate, which is expected to be realized in iron oxypnictide superconductors. There, nonmag-11etic impurities can induce both the in-gap bound state and the reduction in T c . Based onthe two-band BCS model, we have found that the zero-energy in-gap state emerge underthe conditions that (i) x ≡ | I ′ /I | = 1 and (ii) | I | N α , | I | N β ≫
1. Deviating from theseconditions, in-gap state shifts to a finite energy, and disappears eventually.Here, we discuss the case of unitary scattering: In iron oxypnictide superconductors, Fesubstitution by other elements (such as Co, Ni, and Zn) will cause the unitary scatteringpotential. 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], x ∼ h P d O d,α ( k ) O d,β ( k ′ ) i FS k ∈ α, k ′ ∈ β , where O d,α ( k ) = h d ; k | α ; k i represents the transformation matrix between the orbital representation (orbital d ) and theband-diagonal representation (band α ). In iron oxypnictide superconductors, x ∼ . d xz , d yz orbitals of Fe in the two-Fe unit cell, whereas halfof the electron-pockets are composed of d x - y orbitals [38]. Since the impurity effect is weakexcept when x = 1 in the unitary regime as shown in Figs. 3 (d) and 5 (d), Fe substitutionby other elements will affect the superconducting DOS and T c only slightly.We also discuss the case of Born scattering due to “in-plane” weak random potential ordisorder: 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 alsocaused 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 ′ | . /R ∼ /a . Since | k − k ′ | ≈ π/a in the interband scattering between electron-pockets and hole-pockets, I ′ should be muchsmaller than I . Therefore, the effect of off-plane impurities on the s ± -wave state will besmall since the relationship x ≪ s ± -wave state is violated by the interbandscattering. Only one percent Born impurities with x & . T c . For this reason, relation 1 /T ∝ T below T c observed in clean LaFeAsO − x F x [42, 44] and in LaFeAsO . [43] samples, which would bealmost absent from the impurity reduction in T c , cannot be explained by the present analysisbased on the isotropic BCS model. Thus, anisotropy in the s ± -wave superconducting gap12ight be responsible for the relation 1 /T ∝ T [45]. Recently, rapid suppression in 1 /T ( ∝ T α ; α >
5) below T c had been observed in a clean LaFeAsO . F . sample with T c = 28K (=intrinsic T c ) [57]. This result is consistent with the penetration depth [14] and ARPES[15–18], 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 sensitiveto model parameters such as the nesting condition [23, 25]. The wide variety of behaviorsin 1 /T would reflect the large sample dependence of the gap anisotropy in iron oxypnictidesuperconductors. Acknowledgments
We are grateful to M. Sato for enlightening discussions on the impurity effect and the gapanisotropy. We are also grateful to D.S. Hirashima, Y. Matsuda, T. Shibauchi, H. Aoki, K.Kuroki, R. Arita, Y. Tanaka, S. Onari and Y. ¯Ono for useful comments and discussions. Thisstudy has been supported by Grants-in-Aid for Scientific Research from MEXT of Japanand from the Japan Society for the Promotion of Science, and by JST, TRIP. [1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono: J. Am. Chem. Soc. (2008) 3296.[2] Z.-A. Ren, W. Lu, J. Yang, W. Yi, X.-L. Shen, Z.-C. Li, G.-C. Che, X.-L. Dong, L.-L. Sun,F. Zhou, and X.-X. Zhao: Chin. Phys. Lett. (2008) 2215.[3] X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F. Fang: Nature (2008) 761.[4] G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, P. Zheng, J. L. Luo, and N. L. Wang:Phys. Rev. Lett. (2008) 247002.[5] H. Kito, H. Eisaki, and A. Iyo: J. Phys. Soc. Jpn. (2008) 063707.[6] H. Luetkens, H.-H. Klauss, M. Kraken, F. J. Litterst, T. Dellmann, R. Klingeler, C. Hess,R. Khasanov, A. Amato, C. Baines, J. Hamann-Borrero, N. Leps, A. Kondrat, G. Behr, J.Werner, and B. Buechner: arXiv:0806.3533.[7] R. H. Liu, G. Wu, T. Wu, D. F. Fang, H. Chen, S. Y. Li, K. Liu, Y. L. Xie, X. F. Wang, R. . Yang, L. Ding, C. He, D. L. Feng, and X. H. Chen: Phys. Rev. Lett. (2008) 087001.[8] S. Kitao, Y. Kobayashi, S. Higashitaniguchi, M. Saito, Y. Kamihara, M. Hirano, T. Mitsui,H. Hosono, and M. Seto: J. Phys. Soc. Jpn. (2008) 103706.[9] J. Zhao, D.-X Yao, S. Li, T. Hong, Y. Chen, S. Chang, W. Ratcliff II, J. W. Lynn, H. A.Mook, G. F. Chen, J. L. Luo, N. L. Wang, E. W. Carlson, J. Hu, and P. Dai: Phys. Rev. Lett. , 167203 (2008).[10] Y. Qiu, M. Kofu, Wei Bao, S.-H. Lee, Q. Huang, T. Yildirim, J. R. D. Copley, J. W. Lynn,T. Wu, G. Wu, and X. H. Chen: Phys. Rev. B (2008)103704.[12] A. Kawabata, S.C. Lee, T. Moyoshi, Y. Kobayashi, and M. Sato: arXiv:0808.2912.[13] K. Matano, Z. A. Ren, X. L. Dong, L. L. Sun, Z. X. Zhao, and G.-Q. Zheng: Europhys. Lett.83 (2008) 57001.[14] K. Hashimoto, T. Shibauchi, T. Kato, K. Ikada, R. Okazaki, H. Shishido, M. Ishikado, H.Kito, A. Iyo, H. Eisaki, S. Shamoto, and Y. Matsuda: Phys. Rev. Lett. , 017002 (2009);K. Hashimoto, T. Shibauchi, S. Kasahara, K. Ikada, T. Kato, R. Okazaki, C. J. van der Beek,M. Konczykowski, H. Takeya, K. Hirata, T. Terashima, Y. Matsuda: arXiv:0810.3506.[15] H. Liu, W. Zhang, L. Zhao, X. Jia, J. Meng, G. Liu, X. Dong, G.F. Chen, J.L. Luo, N.L.Wang, W. Lu, G. Wang, Y. Zhou, Y. Zhu, X. Wang, Z. Zhao, Z. Xu, C. Chen, and X.J. Zhou:Phys. Rev. B , 184514 (2008).[16] H. Ding, P. Richard, K. Nakayama, T. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S.Souma, T. Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo, and N. L.Wang: Europhys. Lett. , 47001 (2008).[17] T. Kondo, A.F. Santander-Syro, O. Copie, C. Liu, M.E. Tillman, E.D. Mun, J. Schmalian,S.L. Bud’ko, M.A. Tanatar, P.C. Canfield, and A. Kaminski: Phys. Rev. Lett. (2008)147003.[18] K. Nakayama et al. arXiv:0812.0663.[19] G. Mu, H. Luo, Z. Wang, L. Shan, C. Ren, and H.-H. Wen: arXiv:0808.2941.[20] D.J. Singh, and M.H. Du: Phys. Rev. Lett. (2008) 237003.[21] S. Ishibashi, K. Terakura, and H. Hosono: J. Phys. Soc. Jpn. (2008) 053709.[22] K. Nakamura, R. Arita, and M. Imada: J. Phys. Soc. Jpn. (2008) 093711.
23] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki: Phys. Rev. Lett. (2008) 087004.[24] I. I. Mazin, D.J. Singh, M.D. Johannes, and M.H. Du: Phys. Rev. Lett. (2008) 057003.[25] F. Wang, H. Zhai, Y. Ran, A. Vishwanath, and D.-H. Lee: arXiv:0807.0498v4.[26] T. Nomura: J. Phys. Soc. Jpn. (2008) Suppl. C, 123. T. Nomura: arXiv:0811.2462.[27] Y. Yanagi, Y. Yamakawa, and Y. ¯Ono: arXiv:0808.1192; Y. Yanagi, Y. Yamakawa, and Y.¯Ono: J. Phys. Soc. Jpn. (2008) 123701.[28] H. Ikeda: J. Phys. Soc. Jpn (2008) No.123707.[29] V. Cvetkovic, and Z. Tesanovic: arXiv:0804.4678.[30] Y. Fuseya, T. Kariyado, and M. Ogata: arXiv:0811.3052.[31] Z. W. Zhu, Z. A. Xu, X. Lin, G. H. Cao, C. M. Feng, G. F. Chen, Z. Li, J. L. Luo, and N. L.Wang: New J. Phys. (2008) 063021.[32] Y. Nakajima, H. Shishido, H. Nakai, T. Shibauchi, K. Behnia, K. Izawa, M. Hedo, Y. Uwatoko,T. Matsumoto, R. Settai, Y. Onuki, H. Kontani, and Y. Matsuda: J. Phys. Soc. Jpn. ,024703 (2007); Y. Nakajima, H. Shishido, H. Nakai, T. Shibauchi, M. Hedo, Y. Uwatoko, T.Matsumoto, R. Settai, Y. Onuki, H. Kontani, and Y. Matsuda: Phys. Rev. B , 214504(2008).[33] H. Kontani, K. Kanki, and K. Ueda, Phys. Rev. B (1999) 14723; H. Kontani: Phys. Rev.Lett. , 237003 (2002); S. Onari, H. Kontani, and Y. Tanaka, Phys. Rev. B , 224434(2006); H. Kontani, Rep. Prog. Phys. (2008) 026501.[34] H. Kontani and M. Ohno: Phys. Rev. B , 014406 (2006); H. Kontani and M. Ohno: J. Mag.Mag. Mat. , 483 (2007).[35] A.S. Sefat, A. Huq, M.A. McGuire, R. Jin, B.C. Sales, and D. Mandrus: Phys. Rev. Lett. , 117004 (2008); A. Leithe-Jasper, W.Schnelle, C. Geibel, and H. Rosner: Phys. Rev. Lett. , 207004 (2008).[36] L. J. Li, Q. B. Wang, Y. K. Luo, H. Chen, Q. Tao, Y. K. Li, X. Lin, M. He, Z. W. Zhu, G. H.Cao, and Z. A. Xu: arXiv:0809.2009.[37] Y. K. Li, X. Lin, C. Wang, L. J. Li, Z. W. Zhu, Q. Tao, M. He, Q. B. Wang, G. H. Cao, andZ. A. Xu: arXiv:0808.0328.[38] Y. Senga and H. Kontani: J. Phys. Soc. Jpn. (2008) 113710.[39] A.V. Chubukov, D. Efremov, and I. Eremin: Phys. Rev. B (2008) 134512.
40] D. Parker, O.V. Dolgov, M.M. Korshunov, A.A. Golubov, and I.I. Mazin: Phys. Rev. B ,134524 (2008).[41] Y. Bang, H.-Y. Choi, and H. Won: arXiv:0808.3473.[42] Y. Nakai, K. Ishida, Y. Kamihara, M. Hirano, and H. Hosono: J. Phys. Soc. Jpn. (2008)073701.[43] H. Mukuda, N. Terasaki, H. Kinouchi, M. Yashima, Y. Kitaoka, S. Suzuki, S. Miyasaka, S.Tajima, K. Miyazawa, P. Shirage, H. Kito, H. Eisaki, and A. Iyo: J. Phys. Soc. Jpn. (2008)093704.[44] H.-J. Grafe, D. Paar, G. Lang, N. J. Curro, G. Behr, J. Werner, J. Hamann-Borrero, C. Hess,N. Leps, R. Klingeler, B. Buechner, Phys. Rev. Lett. , 047003 (2008).[45] Y. Nagai, N. Hayashi, N. Nakai, H. Nakamura, M. Okumura, and M. Machida: New J. Phys. (2008) 103026.[46] R.J. Schrieffer, Theory of Superconductivity , Benjamin, New York (1964).[47] P.B. Allen, and B. Mitrovic: Solid State Physics (1982) 1.[48] O.V. Dolgov, I.I. Mazin, D. Parker and A.A. Golubov: arXiv:0810.1476.[49] G. Preosti and P. Muzikar: Phy. Rev. B (1995) 3489; their one-band BCS model corre-sponds to the present two-band s ± wave model with I = I ′ and N α = N β .[50] M.L. Kulic and O.V. Dolgov: Phys. Rev.B (1999) 13062.[51] Y. Ohashi, J. Phys. Soc. Jpn. (2002) 1978; Y. Ohashi, Physica C (2004) 41.[52] A.Y. Liu, I.I. Mazin, and J. Kortus: Phys. Rev. Lett. , 087005 (2001).[53] B. Mitrovic, J. Phys.: Condens. Mat. (2004) 9013.[54] E.J. Nicol and J.P. Carbotte: Phys. Rev. B (2005) 014520.[55] Y. Bang and H.-Y. Choi: arXiv:0808.0302.[56] H. Kontani, Phys. Rev. B , 054507 (2004).[57] Y. Kobayashi, A. Kawabata, S. C. Lee, T. Moyoshi, and M. Sato: arXiv:0901.2830., 054507 (2004).[57] Y. Kobayashi, A. Kawabata, S. C. Lee, T. Moyoshi, and M. Sato: arXiv:0901.2830.