Unconventional pairing originating from disconnected Fermi surfaces in the iron-based superconductor

For the iron-based high $T_c$ superconductor LaFeAsO$_{1-x}$F$_x$, we construct a minimal model, where all of the five Fe $d$ bands turn out to be involved. We then investigate the origin of superconductivity with a five-band random-phase approximation by solving the Eliashberg equation. We conclude that the spin fluctuation modes arising from the nesting between the disconnected Fermi pockets realise, basically, an extended s-wave pairing, where the gap changes sign across the nesting vector.


Introduction
While the physics of high-T c cuprate has matured after the two decades since the discovery, the superconductivity the iron-based pnictide LaFeAsO doped with fluorine discovered by Hosono's group [1] is more than welcome as a fresh challenge for yet another class of high-T c systems. Indeed, the iron-based material, along with various other ones in the same family of compounds with higher transition temperatures (T c ) [2], are remarkable as the first non-copper compound that has T c 's exceeding 50 K. This immediately stimulates renewed interests in the electronic mechanism of high T c superconductivity. In order to investigate the pairing mechanism, here we first construct an electronic model for LaFeAsO 1−x F x using maximally localised Wannier orbitals obtained from first principles calculation. The minimal model turns out to involve all the five Fe d orbitals. [3] Hence the iron-based material is contrasted with the cuprate, which is a one-band, doped Mott insulator. We then apply the random-phase approximation (RPA) to solve the Eliashberg equation. We conclude that a nesting between multiple Fermi surface (pockets) results in a development of a peculiar spin fluctuation mode, which in turn realises an unconventional pairing, which is basically an extended s-wave where the gap function changes sign across the nesting vector. [3,4] The result is intriguing as a realisation of the general idea that the way in which electron correlation effects appear is very sensitive to the underlying band structure and the shape of the Fermi surface. [5] LaFeAsO has a layered structure, where Fe atoms form a square lattice in each layer, which is sandwiched by As atoms. [6] Due to the tetrahedral coordination of As atoms, there are two Fe atoms per unit cell. The experimentally determined lattice constants are a = 4.03Å and c = 8.74Å, with two internal coordinates z La = 0.142 and z As = 0.651. [7] We have first obtained the band structure ( Fig.1) for these coordinates with the density-functional approximation with plane-wave basis [8], which is then used to construct the maximally localised Wannier functions (MLWFs) [9]. These MLWFs, centered at the two Fe sites in the unit cell, have five orbital symmetries where X, Y, Z refer to those for this unit cell with two Fe sites as shown in the bottom panel of Fig.1). The two Wannier orbitals in each unit cell are equivalent in that each Fe atom has the same local arrangement of other atoms. We can then take a unit cell that contains only one orbital per symmetry by unfolding the Brillouin zone, [10] and we end up with an effective five-band model on a square lattice, where x and y axes are rotated by 45 degrees from X-Y . We refer all the wave vectors in the unfolded Brillouin zone hereafter. We define the band filling n as the number of electrons/number of sites (e.g., n = 10 for full filling). The doping level x in LaFeAsO 1−x F x is related to the band filling as n = 6 + x.
The five bands are heavily entangled as shown in Fig.2 (left panel) reflecting strong hybridisation of the five 3d orbitals, which is physically due to the tetrahedral coordination of As atoms around Fe. Hence we conclude that the minimal electronic model requires all the five bands. In Fig.2(right), the Fermi surface for n = 6.1 (corresponding to x = 0.1) obtained by ignoring the inter-layer hoppings is shown in the two-dimensional unfolded Brillouin zone.
The Fermi surface consists of four pieces (pockets in 2D): two concentric hole pockets (denoted here as α 1 , α 2 ) centered around (k x , k y ) = (0, 0), two electron pockets around (π, 0) (β 1 ) or (0, π) (β 2 ), respectively. α i (β i ) corresponds to the Fermi surface around the ΓZ line (MA in the original Brillouin zone) in the first-principles band calculation. [11] Besides these pieces of the Fermi surface, there is a portion of the band near (π, π) that is flat and touches the E F at n = 6.1, so that the portion acts as a "quasi Fermi surface (γ)" around (π, π), which has in fact an important contribution to the spin susceptibility. As for the orbital character, α and portions of β near Brillouin zone edge have mainly d XZ and d Y Z character, while the portions of β away from the Brillouin zone edge and γ have mainly d X 2 −Y 2 orbital character (Fig.3, bottom panels).
An interesting feature in the band structure is the presence of Dirac cones, i.e., places where the upper and the lower bands make a conical contact. [12,13] The ones closest to the Fermi level lies at positions where the d X 2 −Y 2 and the d XZ /d Y Z bands cross, just below the β Fermi surface.

Many-body Hamiltonian and 5-band RPA
We consider a two-dimensional model where the inter-layer hoppings are neglected. For the many body part of the Hamiltonian, we consider the standard interaction terms that comprise the intra-orbital Coulomb U, the inter-orbital Coulomb U ′ , the Hund's coupling J and the pair-hopping J ′ . The many body Hamiltonian then reads where i, j denote the sites and µ, ν the orbitals, and t µν ij is the transfer energy obtained in the previous section. The orbitals d XY are labeled as ν = 1, 2, 3, 4, and 5, respectively. As for the electronelectron interactions, there have been some theoretical studies that estimate the parameter values. Some give U ≫ J, [14,15] while others U ∼ J. [16] We assume here that U ≫ J, and take the values U = 1.2, U ′ = 0.9, J = J ′ = 0.15 throughout the study. These values are smaller than the values obtained in ref. [14,15] because the self energy correction is not taken into account in the present calculation, so that small values of interaction parameters are necesssary to avoid magnetic ordering at high temperatures.
Having constructed the model, we move on to the RPA calculation, where the modification of the band structure due to the self-energy correction is not taken into account. Multiorbital RPA is described in e.g. ref. [17,18]. In the present case, Green's function G lm (k) (k = (k, iω n )) is a 5 × 5 matrix. The irreducible susceptibility matrix (l i = 1, ..., 5) has 25 × 25 components, and the spin and the charge (orbital) susceptibility matrices are obtained from matrix equations, where We denote the largest eigenvalue of the spin susceptibility matrix for iω n = 0 as χ s (k). The Green's function and the effective singlet pairing interaction, are plugged into the linearised Eliashberg equation, λφ l 1 l 4 (k) = − T N q l 2 l 3 l 5 l 6 V l 1 l 2 l 3 l 4 (q)G l 2 l 5 (k − q)φ l 5 l 6 (k − q)G l 3 l 6 (q − k).  Fig.3. The susceptibility χ s has peaks around (k x , k y ) = (π, 0), (0, π). In addition we note that there is a ridge extending from (π, π/2) to (π/2, π) around (π, π). To explore the origin of these spin structures, we show χ s3333 and χ s4444 in the middle of Fig.3, which represent the spin correlation within d Y Z and d X 2 −Y 2 orbitals, respectively. χ s3333 peaks solely around (π, 0) and (0, π), which reflects the nesting between the XZ, Y Z-charactered portions of α and β Fermi pockets as shown in a bottom panel of Fig.3. On the other hand, χ s4444 has peaks around (π, 0), (0, π) and around (π, π/2)/(π/2, π) as well. The former is due to the nesting between the γ quasi Fermi surface and the d X 2 −Y 2 portion of the β Fermi surface as was first pointed out in ref. [4], while the latter originates from the nesting between the d X 2 −Y 2 portion of the β 1 and β 2 Fermi surfaces. [3] The (π, 0), (0, π) feature is consistent with the stripe (i.e., collinear) antiferromagnetic order for the undoped case, which was suggested by transport and optical reflectance, [19] and further confirmed by neutron scattering experiments. [7] The stabilization of such an antiferromagnetic ordering has also been pointed out in first principles calculations. [12,19,20] 5 Result: superconductivity The presence of multiple set of nesting vectors revealed above provides an interesting case of the gap function in a spin-fluctuation mediated superconductivity, since multiple nestings can not only cooperate but also compete with each other. Namely, the α-β and γ-β nestings tend to favour the sign reversing s-wave pairing, in which the gap changes sign between α and β with a fixed sign (i.e., full gap) on each of the Fermi surface. [4] On the other hand, β 1 − β 2 nesting tends to change the sign of the gap between these two pockets, which can result in either d-wave pairing or an s-wave pairing with nodes on the β Fermi surface [21,22]. For the band structure of LaFeAsO (obtained by using the experimentally determined lattice structure), the (π, 0) spin fluctuation dominates, and the sign-reversing s-wave with no nodes on the Fermi pockets dominates for the present set of parameter values. [22] In Fig.4, we show the gap function in the band representation for the third and the forth bands, which produce the α 2 and β Fermi surfaces, respectively. A number of theoretical studies that adopt effective two band models, [23,24,25] the present five band model, [26,27,28] or a 16 band dp model [29] have also found that this sign reversing s-wave is a good candidate of the gap function in this material.
The sign change in the s-wave gap is analogous to those in models studied by Bulut et al., [30] and also by the present author for the disconnected Fermi surfaces [31,32]. It is also reminiscent of the unconventional s-wave pairing mechanism for Na x CoO 2 · yH 2 O[33] proposed by Kuroki et al. [34] We have to realise, however, that the gap can vary significantly along the β Fermi surface, and also between different pieces of the Fermi surface due to the multiorbital character of the system. In the present case, the gap for the d X 2 −Y 2 -charactered portion of the β Fermi surface is about twice as large as that for the XZ, Y Z-charactered portions, namely near the Brillouin zone edge of the β Fermi surface and on the α 1 , α 2 Fermi surface. However, we find, in further calculations, that this gap variance is not universal, and depends strongly on the electron density and also on the details of the band structure.
This is because the β − γ nesting and thus the (π, 0) vs. (π, π/2) spin fluctuation competition is sensitively affected by the position of the d X 2 −Y 2 portion of the band near (π, π) with respect to the Fermi level. The details of this band filling and band structure dependences will be published elsewhere.

Conclusion
To summarise, we have constructed a five-band electronic model for which we consider to be the minimum microscopic model for the iron-based superconductor. Applying a five-band RPA to this model, we have found that spin-fluctuation modes around (π, 0), (0, π) develop due to the nesting between  So the general picture obtained here is that the iron compound is a multiband system having electron and hole pockets, as sharply opposed to the cuprate which is a one-band and nearly half-filled system with a simply connected Fermi surface. This poses a challenging future problem of elaborating respective pros and cons for the iron compound and the cuprate for superconductivity.