Band renormalization and Fermi surface reconstruction in iron-based superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Band renormalization and Fermi surface reconstruction iniron-based superconductors
Shun-Li Yu, Jing Kang, and Jian-Xin Li
National Laboratory of Solid State Microstructures and Department of Physics,Nanjing University, Nanjing 210093, China (Dated: October 31, 2018)
Abstract
Using the fluctuation exchange approximation and a three-orbital model, we study the bandrenormalization, Fermi surface reconstruction and the superconducting pairing symmetry in thenewly-discovered iron-based superconductors. We find that the inter-orbital spin fluctuations leadto the strong anisotropic band renormalization and the renormalization is orbital dependent. As aresult, the topology of Fermi surface displays distinct variation with doping from the electron typeto the hole type, which is consistent with the recent experiments. This shows that the Coulombinteractions will have a strong effect on the band renormalization and the topology of the electronFermi pocket. In addition, the pairing state mediated by the inter-orbital spin fluctuation is of anextended s -wave symmetry. PACS numbers: 74.70.-b, 74.25.Jb, 71.18.+y, 74.20.Mn . INTRODUCTION Recently, the discovery of superconductivity in the iron-based compounds has generatedenormous interest, because these materials are the first non-copper superconductors withhigh superconducting (SC) critical temperature. These compounds share the same FeAs(orFeP) layers that are believed to be responsible for the superconductivity. Two classes ofsuch compounds have been extensively investigated: (1) The LaOFeAs classes(denoted asFeAs-1111), space group P /mmm , with Tc ≈
26 K through electron doping with replacingO − by F − [1] and T c can be up to about 41 to 56 K by replacing lanthanum by otherrare earth ions [2, 3, 4]. (2) The BaFe As classes(denoted as FeAs-122), space group I /mmm , with Tc ≈
38 K through hole doping with substituting Ba for K + [5]. Allparent compounds show a spin-density-wave(SDW) abnormality below a temperature ∼ π, π ) consisting of disconnectingpatches [18, 25], which exhibits a significant difference from the result obtained in theband structure calculation. Therefore, the understanding of the role played by the electroncorrelation and the origin of the different electron FS topology between the hole-doped andelectron-doped system is of importance.As the band structure calculations having shown, the FS and band structures of thesecompounds are qualitatively similar [23, 24, 26] and the Fe-3d orbitals represent the maincontribution to the density of states, together with a contribution from As-p orbitals, withinseveral eV of the Fermi level. In the FeAs layers, the Fe atoms form a square lattice and a Featom is coordinated by four As atoms in a tetrahedron. Due to the direct Fe-Fe bonds andthe hybridization with the As-4p orbitals, the Fe-3d orbitals form a complex band structure.However, the main contribution to the bands near the Fermi level comes from the d xz , d yz d xy orbitals(the direct Fe-Fe bonds along the x and y axes) of the Fe atoms [26, 27,28]. In this paper, we employ a three-orbital (the d xz , d yz and d xy orbitals) model [29] toinvestigate the band renormalization, FS reconstruction and superconducting gap symmetryin the iron-based compounds with the fluctuation exchange(FLEX) approximation. We findthat a strong anisotropic band renormalization is resulted from the Coulomb interactionwith the strongest effect occurring around the ˜ X = (0 , π ) point, which is defined in theunfolded Brillouin zone(BZ), and this renormalization increases rapidly with the increase ofthe Hund’s coupling J when J > . U ( U is the intra-orbital Coulomb interaction). Dueto the band renormalization, the Fermi level for the undoped case is slightly below theflat band centered around the ˜ X point which is the bottom of the band along the ˜ X to˜ M = ( π, π ) direction. As a result, for the hole doped case, the Fermi level will situate belowthe flat band along the ˜ X to ˜ M direction, though it still crosses the renormalized bandalong the ˜Γ to ˜ X direction. In this case, the FS around ˜ X consists of disconnecting patches.However, in the electron-doped case, the Fermi level is lifted to be above the flat band andthe circular-like FS is formed. This result provides a possible explanation for the differentFS topology observed in the electron doped and hole doped materials. We also carry out thesame calculation based on the two-orbital model [30], no similar FS reconstruction has beenfound. This difference is ascribed to be due to the orbital dependent renormalization. Onthe other hand, the most favored pairing state mediated by the inter-orbital spin fluctuationsis found to be the extended s -wave with a sign change between the electron and hole Fermipockets, which is consistent with the result obtained in the two-orbital model [31, 32, 33].This indicates that the two models share the similar physics as far as the pairing symmetryis concerned, but exhibits difference in the band renormalization.The paper is organized as following. In Sec.II, we present the three-orbital model andintroduce the FLEX method. In Sec.III, the numerical results for the band renormalizationis presented and discussed. We also give a brief discussion on the pairing symmetry in thissection. In Sec.IV, we give a summary of the results. II. MODEL AND FLEX METHOD
The model Hamiltonian consists of two parts, H = H + H int , (1)3here the bare Hamiltonian H is given by the three-orbital model as introduced in Ref. [29].In the unfolded(extended) BZ for the reduced unit cell(only one Fe atom in the unit cell asin Ref. [27]), it can be written as H = P k Ψ † k M k Ψ k with M k = ε xz ( k + Q ) ε xz,yz ( k + Q ) ε xz,xy ( k ) ε xz,yz ( k + Q ) ε yz ( k + Q ) ε yz,xy ( k ) ε ∗ xz,xy ( k ) ε ∗ yz,xy ( k ) ε xy ( k ) , (2)and Ψ k = ( c xzk + Q , c yzk + Q , c xyk ) T . Here the diagonal elements of M k denote the dispersion ofFe-3d orbitals d xz , d yz and d xy , while the others denote the hybridization among them.Keeping up to the next nearest neighbor hopping terms, we have ε xz ( k ) = − P k [2 t cos k x +2 t cos k y + 4 t cos k x cos k y ], ε yz ( k ) = − P k [2 t cos k y + 2 t cos k x + 4 t cos k x cos k y ], ε xy ( k ) = − P k [2 t (cos k x + cos k y ) + 4 t cos k x cos k y ], ε xz,yz ( k ) = ε yz,xz ( k ) = − P k t sin k x sin k y , ε xz,xy ( k ) = ε ∗ xy,xz ( k ) = − P k t sin k x [29], ε yz,xy ( k ) = ε ∗ xy,yz ( k ) = − P k t sin k y [29],In order to reproduce the FS and band structure feature, we set the parameters as t = − . ≈ . t = 0 . t = − . t = − . t = 0 . t = 0 . t = − .
35. Asthe three orbitals belong to the t g manifold, we set the same on-site energy to the threeorbitals. In Fig.1, we show the band structure and FS with µ = 1 . n = 4 .
0) corresponding to the parent compound. We find that this three-orbital modelcan basically reproduce the main features of the FS and band structure obtained in the LDAcalculation [23, 24]. The interaction between electrons is included in H int as following, H int = 12 U X i,l,σ = σ ′ c † ilσ c † ilσ ′ c ilσ ′ c ilσ + 12 U ′ X i,l = l ′ ,σ,σ ′ c † ilσ c † il ′ σ ′ c il ′ σ ′ c ilσ + 12 J X i,l = l ′ ,σ,σ ′ c † ilσ c † il ′ σ ′ c ilσ ′ c il ′ σ + 12 J ′ X i,l = l ′ ,σ = σ ′ c † ilσ c † ilσ ′ c il ′ σ ′ c il ′ σ , (3)where U ( U ′ ) is the intra-orbital(inter-orbital) Coulomb interaction, J the Hund’s couplingand J ′ the inter-orbital pair hopping.We carry out the investigation using the FLEX approximation [34], in which the Green’sfunction and spin/charge fluctuations are determined self-consistently. For the three-orbitalmodel, the Green’s function ˆ G and the self-energy ˆΣ are expressed in a 3 × B (b) k y / k x / ( ~ X~ (M)(X) M ~ d yz d xz (c) -8-404 E ne r g y X M (a) E F d yz d xy (d) FIG. 1: (color online) Band structure and FS of the three-orbital model. (a) The band structurein the folded BZ with t = − . t = 0 . t = − . t = − . t = 0 . t = 0 . t = 0 .
35 and µ = 1 .
15. (b) The FS in the unfolded BZ. The dashed lines denote the boundary of the folded BZ.The line with an arrow denotes the nesting vector between the hole and electron Fermi pockets.Panels (c) and (d) replot the hole and the electron Fermi pockets shown in Fig.(b), respectively,with the different colors representing the weight of the different orbitals. χ and the effective interaction ˆ V have a 9 × G ( k ) − = ˆ G ( k ) − − ˆΣ( k ), where the self-energy is given by Σ mn ( k ) = TN P q P µν V nµ,mν ( q ) G µν ( k − q ) and the bare Green’s functionreads ˆ G ( k ) = (i ω n − ˆ M k + µ ) − . The fluctuation exchange interaction is given by: V µm,nν ( q ) = 12 [3 ˆ U s ˆ χ s ( q ) ˆ U s + ˆ U c ˆ χ c ( q ) ˆ U c −
12 ( ˆ U s + ˆ U c ) ˆ χ ( q )( ˆ U s + ˆ U c )+3 ˆ U s − ˆ U c ] µm,nν , (4)with spin susceptibility ˆ χ s ( q ) = [ ˆ I − ˆ χ ( q ) ˆ U s ] − ˆ χ ( q ) and charge susceptibility ˆ χ c ( q ) =[ ˆ I + ˆ χ ( q ) ˆ U c ] − ˆ χ ( q ). The irreducible susceptibility is given by χ µm,nν ( q ) = − TN P k G nµ ( k + q ) G mν ( k ). In the above, T is temperature, k ≡ ( k , i ω n ) with ω n = πT (2 n + 1), and ˆ I theidentity matrix. The interaction matrix for the spin(charge) fluctuation ˆ U s ( ˆ U c ) is given by:For i = j = k = l , U sij,kl = U ( U cij,kl = U ); For i = j = k = l , U sij,kl = J ( U cij,kl = 2 U ′ − J );For i = k, j = l and i = j , U sij,kl = U ′ ( U cij,kl = − U ′ + 2 J ); For i = l, j = k and i = j , U sij,kl = J ( U cij,kl = J ); For other cases, U sij,kl = 0( U cij,kl = 0).After obtaining the renormalized Green’s function ˆ G , we can solve the ”Eliashberg”equation, λφ mn ( k ) = − TN X q X αβ X µν V s,tαm,nβ ( q ) G αµ ( k − q ) × G βν ( q − k ) φ µν ( k − q ) , (5)where the spin-singlet and spin-triplet pairing interactions ˆ V s and ˆ V t are given by,ˆ V s ( q ) = 32 ˆ U s ˆ χ s ( q ) ˆ U s −
12 ˆ U c ˆ χ c ( q ) ˆ U c + 12 ( ˆ U s + ˆ U c ) , (6)ˆ V t ( q ) = −
12 ˆ U s ˆ χ s ( q ) ˆ U s −
12 ˆ U c ˆ χ c ( q ) ˆ U c + 12 ( ˆ U s + ˆ U c ) . (7)The most favorable SC pairing symmetry corresponds to the eigenvector φ mn ( k ) with thelargest eigenvalue λ .The Dyson equation, the self-energy and the interaction matrix Eq.(4) form a closedset of equations and will be solved numerically on 64 × k meshes with 1024 Matsubarafrequencies. By symmetry, we set J ′ = J and use the relation U = U ′ + 2 J . In the followingcalculation, the intra-orbital Coulomb interaction U = 3 . J .6 E F ~ M ~X~~ E ne r g y ~ FIG. 2: (color online) Renormalization of the energy band for J = 0 . U , the red dotted(blacksolid) lines are the renormalized(bare) bands. The green, red and blue dashed lines indicate theFermi levels for undoping, 10% electron doping and 20% hole doping, respectively. To show therenormalization more clearly, the energy bands presented here are plotted in the unfolded BZ. III. RESULT AND DISCUSSIONA. Renormalization of band and Fermi surface
We show the renormalized bands for J = 0 . U (correspondingly U ′ = 0 . U according tothe relation U = U ′ + 2 J ) together with the bare bands in Fig.2, in which the renormalizedbands are calculated from the spectral function A ( k , ω ) = − π Im G ( k , ω ) with G ( k , ω ) the an-alytic continuation of the Matsubara Green’s function G ( k , i ω n ) by the Pad´ e approximation.The Fermi levels depicted in Fig.2 are determined by calculating the number of electrons viathe renormalized Green’s function, and the undoped case is determined from the electrondensity per site n = 4 .
0. It is clearly seen that different bands exhibit different renormaliza-tion, and the strongest renormalization occurs around the ˜ X point. Compared to the bareFermi level, we find that the renormailzed Fermi level is shifted up by 0.3( ≈ . (d) -1 0 1-101 (c) -1 0 1-101 k y / k x / 3711 (b) R eno r m a li z a t i on F a c t o r J /U(a)
FIG. 3: (color online) (a) Renormalization factor of total bandwidth for different values of J .(b),(c) and (d) are the spin susceptibilities for J = 0 . U , J = 0 . U and J = 0 . U respectively. undoped case. These features are consistent with the ARPES data [19, 20]. We define atotal bandwidth renormalization factor as W B /W R , where W B ( W R ) denotes the bandwidthof bare(renormalized) band. The renormalization factors for different values of J are shownin Fig.3(a). It increases from 1.4 to 2.3 when J is increased to be around 0 . U . In particular,a rapid rise is observed when J is larger than 0 . U . This shows that the Hund’s couplingplays an important role to enhance the renormalization effects.For U > U ′ , the spin fluctuation will dominate over the charge fluctuation. In Fig.3(b),wepresent the static spin susceptibility χ s ( q , ω = 0) = P µν χ sµν,µν ( q , ω = 0) for J = 0 . U . Itshows four peaks around (0 , ± π ) and ( ± π,
0) points, which is in agreement with the neutronscattering experiments [6]. This arises from the nesting between the hole and the electronpockets connected with the vectors (0 , ± π ) and ( ± π, IG. 4: (color online) Renormalized FS for 10% electron doping ((a) and (b)) and for 20% holedoping ((c) and (d)). (a) and (c) are the results shown in the unfolded BZ, while (b) and (d) inthe folded BZ. representing different weights of the three orbitals in Fig.1(c) and (d), respectively. For thenesting part of the FS, it is found that the d yz + d xy orbitals contribute the main weight to theelectron pockets and the d xz orbital mainly contributes to the hole pockets. This shows thatthe spin fluctuation is mainly due to the inter-orbital particle-hole excitations. As shownin Eq.(3), the Hund’s coupling favors the inter-orbital excitations, so the spin fluctuationaround (0 , ± π ) and ( ± π,
0) will be enhanced by the increase of J . This is evidenced fromthe results shown in Fig.2(c) and 2(d), where the spin susceptibilities for J = 0 . U and J = 0 . U are presented, respectively. For a small J (0 . U ), the (0 , ± π ) spin fluctuationdisappears, while for a large J ( J = 0 . U ) it is enhanced. Because the band renormalizationis increased with the rise of J as discussed above, it suggests that the strong renormalizationaround the e X point mainly comes from the coupling to the spin fluctuation around (0 , ± π ).Inspecting the details of the renormalized bands around the e X point in Fig.2, one will9nd that the energy band near the Fermi level is renormalized strongly along the e Γ − e X direction, but less along the e X − f M direction. Another feature is that the band crossing theFermi level becomes flat around the e X point, and the flat portion situates at the bottom ofthe band along the e Γ − e X direction. To identify the FS after renormalization, we calculatethe distribution of the spectral weight A ( k , ω ) integrated over a narrow energy window0 . ≈ e Γ − e X and the e X − f M directions. This gives rise to a large complete electron Fermi pocket around the e X point as shown in Fig.2(a) in the unfolded BZ and in Fig.2(b) in the folded one, which issimilar to that predicted by the LDA calculation [23, 24, 26] and observed in the ARPESexperiments [19, 21]. However, for the hole doped case, the Fermi level(the blue dashed linefor 20% hole doping) is shifted below the flat band, so that it does not cross the energyband along the e Γ − e X direction anymore. Consequently, the complete electron Fermi pocketin the bare case is broken and the spot-like portions are formed around the e X point, asclearly shown in Fig.4(c). After being folded, the electron FS shown in Fig.4(d) reproducesthe experimental results in ARPES measurements [18, 25]. We note that due to the bandrenormalization the crossing between the two energy band along the e X − f M directions isshifted up and approaches the Fermi level for 20% hole doping. This crossing gives rise toa large density of states. Therefore it adds an enhancement in the spectral weight aroundthe spot-like portions and makes it be easily observed in experiments.The anisotropic band renormalization along the e Γ − e X and the e X − f M direction can betraced to the different orbital weights composing the energy bands along these two directions.Specifically, the energy band near the e X point along the e X − f M direction is composed ofonly the d yz orbital, while that along the e Γ − e X direction is a mix of the d xy and d yz orbitals,as shown in Fig.1(c)and Fig.1(d) in which the different colors represent the weight of therespective orbitals. As noted above, the band renormalization is mainly due to the scatteringoff the inter-orbital spin fluctuations composed of the components χ xz,yz , χ xz,xy and χ yz,xy .10 ~~~ E ne r g y E F (b) ~ X M -1 0 1-101 d yz d xz (a) k y / k x / FIG. 5: (color online) (a) Renormalization of the energy band for the two-orbital model, the reddotted(black solid) lines are the renormalized(bare) bands. The green line indicate the Fermi levelfor undoping. (b) The FS for the two-orbital model, the weight of d xz and d yz orbitals are depictedby different colors. We find that the magnitude of the components χ xz,xy and χ yz,xy is the same, while it islarger than that of the χ xz,yz . Therefore, the d xy orbital is renormalized most strongly bythe spin fluctuation. This suggests that the anisotropic renormalization of the energy bandnear the e X point is orbital dependent.We note that in the two-orbital model [30] the energy band near the e X point alongboth directions is composed of one orbital d yz as shown in Fig.5(a), so the similar bandrenormalization is expected. To show this, we have carried out the same calculation for thetwo-orbital model and the result is presented in Fig.5(b). Though the strong renormalizationstill occurs in one of the bands around the e X point, no strong anisotropic renormalizationbetween the e Γ − e X and the e X − f M direction can be seen. Thus, no similar FS reconstructionas observed in the three-orbital model and in experiments [18, 25] can be obtained here. B. Symmetry of superconducting pairing
With the static spin susceptibility, we will further investigate the pairing symmetry me-diated by spin fluctuations. For J = 0 . U , which is the case with peaks around (0 , ± π ) and( ± π,
0) points in spin fluctuations, we find that the eigenvalue λ has the maximum value11 x / -1 0 1-101 -11 (b) -1 0 1-101 -11 (d) -1 0 1-101 k y / (c) -1 0 1-101 k y / (a) k x / FIG. 6: (color online) Pairing gap functions in the unfolded BZ for U = 3 . T = 0 .
02 for 10%electron doping and 20% hole doping. (a), (b), (c), (d) for J = 0 . U and (e), (f) for J = 0 . U .(a) ∆ hh for electron doping. (b) ∆ ee for electron doping. (c) ∆ hh for hole doping. (d) ∆ ee for holedoping. (e) ∆ hh for electron doping. (f) ∆ ee for electron doping. in the spin-singlet channel and it is nearly zero in the spin-triplet channel at temperature T = 0 .
02. We note that this is not the case considered in Ref. [29], where the spin-tripletstate is expected with a larger J (such as J > U/
3, see also the analysis in Ref. [33]). Theobtained gap functions of the hole band ∆ hh and the electron band ∆ ee for 10% electrondoping and 20% hole doping are shown in Fig.3(a)-(d). It can be seen that the gap isbasically an extended s -wave, which is nodeless around all the Fermi pockets and changesign between the hole pockets and the electron pockets. This result is consistent with theexperiment measurements [9, 10, 11, 12, 13, 18] and the calculation based on the two-orbitalmodel [31]. It is noted that the pairing symmetry is similar for both the electron doped andthe hole doped systems, although their FS topology is different as discussed above.12ecause the spin fluctuation is dominant over the charge fluctuation, the pairing inter-action in the spin-singlet channel is positive(repulsive) [See Eq.(6)] and strongest aroundthe wave vectors Q at which the spin fluctuation has a largest intensity. For a repulsivepairing symmetry, the SC gap will satisfy the condition ∆( k )∆( k + Q ) < λ of the ”Eliashberg” equation, as can be seen from Eq.(5). For J > . U ,the inter-band (inter-orbital) spin fluctuation is dominant and has peaks around ( ± π,
0) and(0 , ± π ) which is the nesting wave vectors connecting the hole and the electron Fermi pockets.Thus, the gap will have an opposite sign between these two Fermi pockets. This gives riseto the extended s -wave. Because the SC pairing is mediated by the spin fluctuation around( ± π,
0) and (0 , ± π ), which is the same for the three-orbital and the two-orbital models(seeRef. [31]), the same pairing symmetry will be obtained based on these two models. IV. CONCLUSION
In a summary, we have investigated the band renormalization, FS reconstruction and thesymmetry of the supercondcuting gap in iron-based superconductors using a three-orbitalmodel. A strong anisotropic band renormalization due to the scattering off the inter-orbitalspin fluctuations is found. The band renormalization is shown to be orbital dependent. Asa result, the electron Fermi pocket exhibits different topology between the electron dopedand the hole doped cases, which provides a natural explanation for the recent ARPESexperiments. In addition, we have found that the most favorable superconducting pairingsymmetry mediated by the inter-band (inter-orbital) spin fluctuations is the extended s -wave. Acknowledgments
We thank Z. Fang, X.J. Zhou, Q.H. Wang, Z.J. Yao and H.M. Jiang for many help-ful discussions. The work was supported by the NSFC (10525415), the 973 project(2006CB601002,2006CB921800). [1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. , 3296 (2008).
2] X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F. Fang, Nature(London) , 761(2008).[3] 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. , 247002 (2008).[4] C. Wang, L. Li, S. Chi, Z. Zhu, Zhi Ren, Y. Li, Y. Wang, X. Lin, Y. Luo, S. Jiang, X. Xu, G.Cao, and Z. Xu , Europhys. Lett. , 67006 (2008).[5] M. Rotter, M. Tegel, and D. Johrendt , Phys. Rev. Lett. , 107006 (2008).[6] C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff ll, J. L. Zarestky, H. A. Mook, G. F.Chen, J. L. Luo, N. L. Wang, and P. Dai, Nature(London) , 899 (2008).[7] J. Dong, H. J. Zhang, G. Xu, Z. Li, G. Li, W. Z. Hu, D. Wu, G. F. Chen, X. Dai, J. L. Luo,Z. Fang, and N. L. Wang, Europhys. Letters, , 27006 (2008).[8] Q. Huang, Y. Qiu, Wei Bao, M. A. Green, J. W. Lynn, Y. C. Gasparovic, T. Wu, G. Wu, andX. H. Chen , Phys. Rev. Lett. 101, 257003 (2008).[9] T. Y. Chen, Z. Tesanovic, R. H. Liu, X. H. Chen, and C. L. Chien, Nature(London) , 1224(2008).[10] H. Ding, P. Richard, K. Nakayama, K. 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).[11] P. Richard, T. Sato, K. Nakayama, S. Souma, T. Takahashi, Y.-M. Xu, G. F. Chen, J. L. Luo,N. L. Wang, and H. Ding, arXiv: 0808.1809 (2008).[12] L. Malone, J.D. Fletcher, A. Serafin, A. Carrington, N.D. Zhigadlo, Z. Bukowski, S. Katrych,and J. Karpinski, arXiv: 0806.3908 (2008).[13] G. Mu, H. Luo, Z. Wang, L. Shan, C. Ren, and H. -H. Wen, arXiv: 0808.2941 (2008).[14] K. Matano, Z.A. Ren, X.L. Dong, L.L. Sun, Z.X. Zhao, G.Q Zheng, Europhys. Lett. , 57001(2008).[15] H.-J. Grafe, D. Paar, G. Lang, N. J. Curro, G. Behr, J. Werner, J. Hamann-Borrero, C. Hess,N. Leps, R. Klingeler, and B. Buchner, Phys. Rev. Lett. 101, 047003 (2008).[16] R. T. Gordon, N. Ni, C. Martin, M. A. Tanatar, M. D. Vannette, H. Kim, G. Samolyuk, J.Schmalian, S. Nandi, A. Kreyssig, A. I. Goldman, J. Q. Yan, S. L. Bud’ko, P. C. Canfield, R.Prozorov, arXiv:0810.2295 (2008).[17] J.D. Fletcher, A. Serafin, L. Malone, J. Analytis, J-H. Chu, A.S. Erickson, I.R. Fisher, and . Carrington, arXiv:0812.3858 (2008).[18] L. Zhao, H. Liu, W. Zhang, J. Meng, X. Jia, G. Liu, X. Dong, G. F. Chen, J. L. Luo, N. L.Wang, G. Wang, Y. Zhou, Y. Zhu, X. Wang, Z. Zhao, Z. Xu, C. Chen, X. J. Zhou, Chin.Phys. Lett. 25, 4402(2008).[19] D. H. Lu, M. Yi, S. -K. Mo, A. S. Erickson, J. Analytis, J. -H. Chu, D. J. Singh, Z. Hussain,T. H. Geballe, I. R. Fisher, Z. -X. Shen, Nature(London) , 81 (2008).[20] H. Ding, K. Nakayama, P. Richard, S. Souma, T. Sato, T. Takahashi, M. Neupane, Y.-M. Xu,Z.-H. Pan, A.V. Federov, Z. Wang, X. Dai, Z. Fang, G.F. Chen, J.L. Luo, and N.L. Wang,arXiv:0812.0534 (2008).[21] C. Liu, T. Kondo, M. E. Tillman, R. Gordon, G. D. Samolyuk, Y. Lee, C. Martin, J. L.McChesney, S. Bu´dko, M. A. Tanatar, E. Rotenberg, P. C. Canfield, R. Prozorov, B. N.Harmon, A. Kaminski, arXiv: 0806.2147 (2008).[22] L. X. Yang, Y. Zhang, H.W. Ou, J. F. Zhao, D. W. Shen, B. Zhou, J. Wei, F. Chen, M. Xu,C. He, Y. Chen, M. Arita, K. Shimada, M. Taniguchi, Z. Y. Lu, T. Xiang, D. L. Feng, arXiv:0806.2627 (2008).[23] D. J. Singh and M. H. Du, Phys. Rev. Lett. , 237003 (2008).[24] I. A. Nekrasov, Z. V. Pchelkina, and M. V. Sadovskii, JETP Lett. , 144 (2008).[25] 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. Xu, C. Chen, X. J. Zhou , Phys. Rev.B 78, 184514 (2008).[26] L. Boeri, O. V. Dolgov, and A. A. Golubov, Phys. Rev. Lett. , 026403 (2008).[27] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev. Lett. , 057003 (2008).[28] F. J. Ma and Z. -Y. Lu, Phys. Rev. B , 033111 (2008).[29] P. A. Lee and X. -G. Wen, arXiv: 0804.1739 (2008).[30] S. Raghu, X. -L. Qi, C. -X. Liu, D. J. Scalapino, and S. -C. Zhang, Phys. Rev. B , 220503(R)(2008).[31] Z. -J. Yao, J. -X. Li, and Z. D. Wang, arXiv: 0804.4166 (2008).[32] Y. Ran, F. Wang, H. Zhai, A. Vishwanath, and D. -H. Lee, arXiv: 0805.3535 (2008).[33] W. Chen, K. -Y. Yang, Y. Zhou, and F. -C. Zhang, arXiv: 0808.3234 (2008).[34] N. E. Bickers, D. J. Scalapino and S. R. White, Phys. Rev. Lett. , 961 (1989); N. E. Bickersand D. J. Scalapino, Ann. Phys. (N.Y.) , 206 (1989); T. Takimoto, T. Hotta, and K. Ueda, hys. Rev. B , 104504 (2004)., 104504 (2004).