Sensitivity of the superconducting state and magnetic susceptibility to key aspects of electronic structure in ferropnictides

Experiments on the iron-pnictide superconductors appear to show some materials where the ground state is fully gapped, and others where low-energy excitations dominate, possibly indicative of gap nodes. Within the framework of a 5-orbital spin fluctuation theory for these systems, we discuss how changes in the doping, the electronic structure or interaction parameters can tune the system from a fully gapped to nodal sign-changing gap with s-wave ($A_{1g}$) symmetry ($s^\pm$). In particular we focus on the role of the hole pocket at the $(\pi,\pi)$ point of the unfolded Brillouin zone identified as crucial to the pairing by Kuroki {\it et al.}, and show that its presence leads to additional nesting of hole and electron pockets which stabilizes the isotropic $s^\pm$ state. The pocket's contribution to the pairing can be tuned by doping, surface effects, and by changes in interaction parameters, which we examine. Analytic expressions for orbital pairing vertices calculated within the RPA fluctuation exchange approximation allow us to draw connections between aspects of electronic structure, interaction parameters, and the form of the superconducting gap.


I. INTRODUCTION
In any new class of superconductors, the structure of the order parameter is an important clue to the nature of the pairing mechanism, but the determination of this structure is seldom immediate. In the cuprates, for example, many different types of experiments on a variety of samples had to be analyzed and compared before a consensus was achieved, and the experimental picture was not clarified until the effects of disorder were understood and clean samples were prepared. It is therefore not unexpected that the symmetry and momentum structure of the gap in the Fe-based superconductors are still controversial nearly two years after their discovery. 2 Nevertheless, the range of behaviour seen in different materials is striking. 3 Here we pose, from the point of view of a weak coupling fluctuation exchange theory, 4,5 the questions: is it possible that the superconducting state of the ferropnictides is intrinsically sensitive to aspects of the electronic structure which "tune" the pairing interaction? If so, which degrees of freedom are most important?
Based on density functional theory (DFT), 6-8 quantum oscillations and angle-resolved photoemission experiments (ARPES) [9][10][11][12][13][14] , the Fermi surface of the Fe-pnictides is believed to consist of a few small hole and electron pockets, as shown in Fig. 1, where we have also indicated the predominant Fe-orbital character of the various parts of the Fermi surface taken from the DFT calculations of Cao et al. 8 for the LaFeAsO material. We will follow the convention in Ref. 5 and elsewhere and refer to the hole pockets around the (0, 0) point as the α sheets and the electron pockets around the (π, 0) and (0, π) points of the unfolded (1-Fe) Brillouin zone as the β sheets. Early on, it was proposed by Mazin et al. 15 and Dong et al. 16 that the nested structure of this Fermi surface would lead to a peak in the magnetic susceptibility near (π, 0), and that this might drive a sign change in the superconducting order parameter between the α and β sheets 15 . Several experiments on the Fe-based superconductors are indeed consistent with a gap which is isotropic (independent of momentum on a given pocket), but possibly with overall sign change of this type. Angleresolved photoemission spectroscopy (ARPES) experiments, while not sensitive to the sign of the gap, are the most direct measure of its magnitude, and have consistently provided evidence taken to support an isotropic gap structure in momentum space. 9,13 The observation of a resonance in inelastic neutron scattering is strong evidence for a sign change of the superconducting gap [17][18][19][20][21][22] . most of the phase diagram 26 ; these power laws are in contrast to the activated temperature dependences expected for an isotropic gap. Similar power laws have been observed in NMR [27][28][29][30][31][32] , thermal conductivity 25,[33][34][35][36][37][38] , and Raman scattering 39 . One obvious way of interpreting these observations is to suppose that the superconducting gap has nodes on parts of the Fermi surface, allowing for the excitation of quasiparticles at arbitrarily low energies. However, one may also show that in an isotropic "sign-changing s-wave" (s ± ) superconductor, disorder can create subgap states 40 under certain conditions, depending on the ratio of inter-to intraband impurity scattering. An impurity band at the Fermi level in an s ± state will also lead to ∆λ ∼ T 2 . There is no known scenario for producing ∆λ ∼ T with impurity scattering in a gapped state, however. It is extremely important to establish whether low-energy excitations are intrinsic (nodal) or extrinsic (disorder-induced), and under what circumstances fully developed gaps, as opposed to highly anisotropic gaps, possibly with nodes, should be expected. From the standpoint of fluctuation exchange theories of pairing based on realistic Fermi surfaces in these materials, the most likely states indeed appear to be preferentially of "s-wave" symmetry, with quasi-isotropic gaps on the hole pockets but highly anisotropic states on the electron pockets 4,5,41 . All of these calculations indicate the proximity of other pairing channels, particularly one with d x 2 −y 2 symmetry, but transitions between an s-wave state and a d-wave state would give rise to thermodynamic anomalies which have not yet been convincingly observed. Attention has therefore focussed primarily on the possibility of s-wave (A 1g symmetry) states with "accidental" nodes, i.e. nodes whose existence is due to details of the pairing interaction rather than symmetry. When the leading instability was of s-wave (A 1g )type, 5-orbital calculations found highly anisotropic states for all values of parameter space explored 5 , in apparent contradiction to the existence of nearly isotropic states experimentally observed in some materials.
What aspects of the physics of these materials are responsible for the nodes or near-nodes seen in these theories? Some observations on this question have already been made. Maier et al. 42 pointed out that, within a model with intra-and inter-orbital interactions, nodes were driven by the intra-orbital Coulomb repulsion, the scattering between the two β sheets neglected in simpler 2-band approaches, and a tendency (observed for the parameters considered in that work which were consistent with local spin rotational invariance) of electrons in like orbitals to pair. Kuroki et al. 1 made an important connection between the lattice structure, electronic structure, and pairing state of the Fe-based superconductors, observing that in DFT calculations the pnictide atom height above the Fe plane appeared to control the appearance of a third γ Fermi surface sheet centered on the Γ point in the folded zone corresponding to a (π, π) pocket in the unfolded zone. This new hole-type pocket, not considered in Ref. 5, stabilizes a more isotropic s ± state. When the γ pocket is present, intra-orbital q ∼ (π, 0) and (0, π) scattering of d xy pairs between the γ and β pockets favor a nodeless s ± state.
Within a model with band interactions, Vildosola et al. 43 and Calderon et al. 44 have also discussed the change in the electronic structure caused by the shift of the pnictogen. In particular, the latter authors have noted that a change in the angle α formed by the Fe-As bonds and the Fe-plane can modify the orbital content as well as the shape of the Fermi surface sheets. In a similar model, Chubukov et al. 45 deduced a phase diagram manifesting a transition between a nodal and fully gapped state with s symmetry as a function of a parameter controlling the relative importance of intraband repulsion, and Thomale et al. 46 reached similar conclusions within a 4-band model, exploring the stability of the nodeless state with respect to doping and other changes in electronic structure. Ikeda et al. 47 considered the doping dependence of spin fluctuations and electron correlations within the renormalized fluctuation exchange (FLEX) approximation using LDA dispersions; they also observe anisotropic behaviour of the gap function around the M point for the electron doped material, and fully-gapped behaviour in the hole-doped material. In a subsequent work, they explored the doping dependence of the pairing state within FLEX. 48 Wang et al. 49 have also discussed the important role played by the γ Fermi surface and emphasized the role of the orbital matrix elements in determining the momentum structure of the gap. From their functional renormalization group calculations, they find that the degree of gap anisotropy depends upon the relative weight of the orbital matrix elements. They find that by tuning some of the tight-binding parameters to alter the relative orbital weights, one can change the anisotropy of the gap. Thomale et al. 50 have also recently reported functional renormalization group results and argue that a nodal superconducting gap can appear on the β electron Fermi surfaces when the γ Fermi surface is absent.
Many authors have considered a Hamiltonian with all possible two-body on-site interactions between electrons in Fe orbitals as a good starting point for a microscopic description of this system, where n iℓ = n i,ℓ↑ + n iℓ↓ . The interaction parametersŪ ,Ū ′ ,J, andJ ′ correspond to the notation of Kuroki et al. 4 , and are related to those used by Graser et al. 5 byŪ = U ,J = J/2,Ū ′ = V + J/4, andJ ′ = J ′ . The kinetic energy H 0 is described by a tight-binding model 5 spanned by the 5 Fe d orbitals, which, depending on the tight binding parameters and the filling n give rise to the Fermi surfaces shown in Fig. 1. Here in the 1 Fe per unit cell Brillouin zone that we will use, there are the usual α 1 and α 2 hole Fermi surfaces around the Γ point, the two β 1 and β 2 Fermi surfaces around (π, 0) and (0, π) and for the hole doped case n = 5.95 shown in Fig. 1 there is an extra hole FS γ around the (π, π) point. We use a notation in which a given orbital index ℓ ∈ (1, 2, . . . , 5) denotes the Fe-orbitals (d xz , d yz , d xy , d x 2 −y 2 , d 3z 2 −r 2 ). An important role will be played by the orbital matrix elements a ℓ ν (k) = ℓ|νk which relate the orbital and band states. The dominant orbital weights |a ℓ ν (k)| 2 on the Fermi surfaces are illustrated in Fig. 1.
In Eq. (2), we have separated the intra-and inter-orbital Coulomb repulsion, as well as the Hund's rule exchange J and "pair hopping" term J ′ for generality, but note that if they are generated from a single two-body term with spin rotational invariance they are related byŪ ′ =Ū − 2J andJ ′ =J. Below we also consider the case where spin rotational invariance (SRI) is explicitly broken by other interactions in the crystal, such thatJ ′ andŪ ′ can take on independent values.
The analysis of individual orbital contributions to pair scattering can be a powerful tool to understand the influence of electronic structure on pairing. The basic picture which emerges from the spin fluctuation theories 4,5 is straightforward. The intra-orbital scattering of d xz and d yz pairs between the α and β Fermi surfaces by (π, 0) and (0, π) spin fluctuations leads naturally to a gap which changes sign between the d xz /d yz parts of the α Fermi surfaces and the d xz /d yz parts of the β 1 and β 2 electron pockets. However, intra-orbital scattering between the d xy portions of the β 1 and β 2 Fermi surfaces suppresses the gap in the d xy regions of the β Fermi surface and competes with the formation of an isotropic s ± gap. In addition, this anisotropy can also be driven by the need to reduce the effect of the on-site Coulomb interaction. 42,45 There can also be inter-orbital pair scattering, such that for example a d xz pair on α 1 scatters to a d xy pair on β 1 . These inter-orbital scattering processes which depend uponŪ ′ andJ ′ are weaker than the intra-orbital processes for spin rotationally invariant (SRI) parameters whereŪ ′ +J ′ =Ū −J andJ ′ =J. However, as noted by Zhang et al. 51 , the interaction parameters in the solid will be renormalized and one may have non spin rotationally invariant (NSRI) interaction parameters leading to enhanced inter-orbital pairings.
Here we explore these issues using results obtained from a 5-orbital RPA calculation to further examine the effects of the electronic structure, the doping and the interaction parameters on the structure of the gap. Analytical results for orbital components of the pairing vertices have allowed us to understand the factors which influence the structure of the gap, and in particular the transition from nodal to nodeless behavior. In Section II we introduce the effective pair scattering vertex, its fluctuation-exchange RPA form and the pairing strength functional that determines the momentum dependence of the gap function. 5 Results for the gap function g(k) obtained for a typical set of spin rotationally invariant (SRI) interaction parameters are discussed in Section III. The strongest pairings occur in the A 1g channel and the discussion in this section focuses on the role of the doping and the presence of the γ pocket in determining whether the gap is nodeless. The nodeless character is found to be related to the additional resonant pairing provided by the γ sheet, and is associated with the strong tendency for pairing between like orbitals when the γ pocket is of d xy character. The origins of the tendency to pair in like orbitals are then explored in Sec. III. Non spin rotationally invariant (NSRI) interaction parameters are discussed in Section IV, where we examine the effect of the Hund's rule exchangeJ and the pair hoppingJ ′ on the pairing interaction and the nodal structure of the gap. Section V contains a further discussion of the effect of the orbital character of the γ hole pocket as well as the possible role of surface effects in causing nodeless behavior. Our conclusions are contained in Section VII and in the appendix we discuss some approximate forms for the pairing vertex.

II. SPIN FLUCTUATION PAIRING
As in Ref. 5, we analyze the effective pair scattering vertex Γ(k, k ′ ) in the singlet channel, where the momenta k and k ′ are restricted to the various Fermi surface sheets with k ∈ C i and k ′ ∈ C j . The orbital vertex functions Γ ℓ1ℓ2ℓ3ℓ4 represent the particle-particle scattering of electrons in orbitals ℓ 1 , ℓ 4 into ℓ 2 , ℓ 3 (see Fig. 2) and in the fluctuation exchange formulation 52,53 are given by where each of the quantitiesŪ s ,Ū c , χ 1 , etc. represent matrices in orbital space as specified in the appendix. Note that the χ RPA 1 term describes the spin-fluctuation contribution and the χ RPA 0 term the orbital (charge)-fluctuation contribution.

Intra
Mixed Inter For the parameter regions we will discuss, the dominant contribution to the pairing comes from the S = 1 particlehole exchange given by the first term in Eq. (2). The forms of the interaction matricesŪ s andŪ c are given in the appendix. As illustrated in Fig. 2 there are intra-orbital, inter-orbital, and mixed-orbital pair scattering processes. The contributions of each to the total pair scattering vertex Γ ij in Eq. (2) are quite different. In particular, as discussed below, the orbital matrix elements for k and −k states on the Fermi surface favor pairs which are formed from electrons in the same orbital state. We therefore find that, in spite of the fact that the mixed-orbital scattering can be significant, its contribution to the pairing interaction is negligible.
If one writes the superconducting gap ∆(k) as ∆g(k), with g(k) a dimensionless function describing the momentum dependence of the gaps on the Fermi surfaces, then g(k) is determined as the stationary solution of the dimensionless pairing strength functional 5 with the largest coupling strength λ. Here the momenta k and k ′ are restricted to the various Fermi surfaces k ∈ C i and k ′ ∈ C j and v F,ν (k) = |∇ k E ν (k)| is the Fermi velocity on a given Fermi surface. The eigenvalue λ provides a dimensionless measure of the pairing strength.

III. SPIN ROTATIONAL INVARIANT CASE
Below we discuss the essential physics of the gapped-nodal transition within the constrained interaction parameter subspace where spin rotational invariance is assumed. It is worthwhile recalling the main points of the argument for an isotropic s ± state: 1) that a repulsive effective interaction peaked near (π, 0) would drive such a state provided 2) this interaction did not vary significantly over the small Fermi surface pockets. 15 Our main points are as follows: • The largest pair scattering processes tending to stabilize an isotropic s ± state are the intra-orbital scattering pairing vertices Γ aaaa (c.f. Fig. 2), which are peaked near (π, 0).
• The intra-orbital processes primarily affect the gap on the sections of the Fermi surface with the corresponding orbital character. The relative signs between various orbital sections are determined by subdominant intra-and inter-orbital processes.
• The isotropic s ± state can be frustrated by intraband Coulomb scattering and by pair scattering processes between the two electron sheets with q ∼ (π, π), both of which favor nodes. 42 • The γ pocket of d xy character can overcome this frustration and stabilize the nodeless state. 1 • If the Hund's rule couplingJ is weak, the processes induced by the γ pocket are not sufficient to eliminate the nodes. The Hund's rule exchange is necessary to overcome an attractive (π, 0) interaction between d xz (d yz ) pairs and d xy pairs and drive a strong intra-orbital d xz and d yz repulsion.
To illustrate these points, we begin by plotting gap functions g(k) for two typical sets of SRI interaction parameters U = 1.3,J = 0.0, 0.2 and two different fillings n = 6.01 and 5.95, as shown in Fig. 3. In the electron-doped case (n = 6.01) the γ Fermi surface around (π, π) is absent and one finds an anisotropic gap with nodes on the β 1 and β 2 Fermi surfaces for zero and for finiteJ. 5 The doping dependence at fixedJ of the gap g(k) versus angle around the β 1 Fermi surface is shown in Fig. 4. The nodes arise in part because of the β 1 − β 2 pair scattering and other aspects of the interactions which frustrate the isotropic s ± state. 22 In particular, the sign change of the gaps on the end regions of the β Fermi surfaces reduces some of the frustration which arises from the β 1 − β 2 scattering. 1 In addition, this sign change acts to suppress the effect of the intra-band Coulomb repulsion.
For the hole doped case (n = 5.95) and finiteJ, the gap function, as seen in Figs. 3 and 4, is anisotropic but nodeless. As discussed in Ref. 1, the scattering between the γ (π, π) Fermi surface and the β 1 and β 2 Fermi surfaces stabilizes the nodeless gap. While there is still frustration associated with the β 1 − β 2 pair scattering, the additional β 1 − γ and β 2 − γ pair scattering processes overcome it. The presence of the pocket also increases the overall pairing strength, which can be seen by comparing the pairing eigenvalues in the Figure; this would correspond to an increase in the critical temperature. To see how these effects arise in more detail, in Fig. 5 we have plotted several of the orbital pairing vertices Γ ℓ1ℓ2ℓ3ℓ4 (q) for q along high symmetry directions in the Brillouin zone for both n = 5.95 and 6.01. The peak in Γ 2222 (q) near the X point arises from q = (π, 0) scattering processes in which d yz pairs are scattered between the α and β 1 Fermi surfaces. In the presence of the γ sheet, both Γ 3333 (q) intra-orbital d xy pair scattering between β 1,2 and γ as well as Γ 1331 (q) and Γ 2332 (q) inter-orbital d xz , d yz pair to d xy pair scattering between β 1,2 and γ are also possible. The pair scattering from the β sheets to the γ hole pocket, represented by Γ 3333 , provides a strong resonant stabilization of the nodeless state; we have verified that the nodal behaviour is recovered when it is neglected. As discussed previously, although mixed-orbital vertices such as Γ 2233 are present, their contribution to Γ ij (k, k ′ ) is suppressed by the orbital matrix elements in Eq. (2), and the pairing is dominated by intra-and inter-orbital scattering.
To further illustrate this, in Fig. 6 we have plotted the pairing interaction Γ ij (k, k ′ ) which determines the gap via Eq. (3). In these plots, one member of a (k ′ , −k ′ ) pair is located at the wave vector denoted by a black circle in the figure. The plot shows the strength of the pairing interaction Γ ij (k, k ′ ) associated with the scattering of this (k ′ , −k ′ ) pair to a (k, −k) pair on the various Fermi surfaces. One sees that if the initial pair is located in a region which has predominantly d xz -orbital character (top), the strongest scattering is to a pair (k, −k) on other d xz regions. 22 Similarly, there are strong d yz intra-orbital scattering processes which are obtained by rotating the figures by π/2. While inter-orbital scattering processes are also present, they are weaker for the parameters we have considered here, as seen, e.g. from the plot of Γ 2332 in Fig. 5. In addition to the d yz intra-orbital scattering, there is strong d xy intra-orbital scattering between β 2 and γ as well as between β 1 and γ. Note that in Fig. 6 the pairing strength is anisotropic along the Fermi sheets, violating one of the key assumptions underlying the argument for an isotropic s ± state.
We return to the role of the Hund's rule exchangeJ in stabilizing a nodeless state. The main effect, i.e. the lifting of the nodes asJ is turned on, is illustrated in Fig. 7. There are two reasons this occurs. First, as discussed in appendix A, the intra-orbital Γ's are controlled byŪ andJ (see Eqs.A7 and A8). For a fixedŪ , increasingJ drives the system closer to the instability and thus enhances the leading peak in the RPA susceptibility. However, Eq. (2) also contains the term 1 2 (Ū s +Ū c ), which involves only the bare interactions rather than the RPA-enhanced susceptibilities. This term contributes to the intra-band Coulomb interaction (which favors anistropic pairing on the electron sheets) and scales, in the dominant orbital channels, withŪ rather thanJ. Thus increasingJ increases the importance of the χ RP A 1 term in Eq. (2) relative to the intraband Coulomb interaction favoring the isotropic state. Secondly, we find that inter-orbital d xy to d xz pair scattering, represented by Γ 1331 , plays an important role in stabilizing the isotropic state when a Hund's rule exchange is present. IfJ = 0, Fig. 8 shows that the pair scattering has an attractive peak at (π, 0) which tends to induce nodes on the β Fermi surface. IfJ > 0, on the other hand, this attractive peak changes sign, as discussed in Sec. IV and in the appendix, and a nodeless state is stabilized by the Finally, we note that in Graser et al. 5 , upon electron doping a d-wave solution overtakes the anisotropic s-wave one. Here we find that upon strong hole doping (∼ 8%), the spin-fluctuation approximation also leads to a d-wave solution.

IV. BROKEN SPIN ROTATIONAL INVARIANCE
In leading order, the strength of the inter-orbital pair scattering is determined by U ′ and J ′ . For SRI parameters U ′ =Ū −J −J ′ the intra-orbital pair scattering tends to dominate the pairing interaction as we have seen in the previous section. However, as noted by Zhang et al. 51 , the actual interaction parameters appropriate for the Fesuperconducting materials need not be SRI. In this case, inter-orbital pair scattering may play a more important role in determining the momentum dependence of the gap function g(k). In addition, using non-SRI parameters we can separately explore the roles of the exchange couplingJ and the pair hoppingJ ′ interactions on the structure of the pairing interactions and the gap. Here, for a filling n = 5.95, we holdŪ andŪ ′ fixed and examine the roles ofJ and J ′ on the nodal-gapped transition on the β 1 pockets near (π, 0). The discussion below applies also to the β 2 pocket near (0, π) with orbital states rotated by 90 • .
In Fig. 9 we plot both the various relevant orbital pairing vertices along high symmetry directions in momentum space, as well as the leading gap function g(k) on the β 1 sheet. Beginning with the nodal caseJ =J ′ = 0, we can see that increasingJ only weakly affects the momentum space structure of the gap. However, note that the g(k) plotted in Fig. 9 is normalized and should be scaled by the corresponding eigenvalue λ to obtain the true gap amplitude. Turning onJ indeed increases the pairing eigenvalue λ by causing an overall increase in the pairing vertices. The largest scattering is provided by Γ 2222 near (π, 0), which is driven up by increasingJ; this can be understood by observing that the RPA-renormalized susceptibility is enhanced by increasingJ (see Eq. A7). This clearly enhances the strength of the gap on the parts of the β 1 Fermi surface characterized by a strong d yz orbital content. And, as discussed above, the repulsive nature of the interaction forces the gap on the d yz sections of the β 1 sheet to have the opposite sign of the gap on the d yz sections of the α 1 and α 2 sheets. This also applies to the Γ 1111 vertex and corresponding d xz sections of the Fermi surface. The gap on the remaining portions of the Fermi surface, which are of d xy character, are left to be determined by the largest other pair scattering with a d xy component, which for these parameters is Γ 1331 . As discussed in appendix A, the Hund's rule couplingJ contributes to an increase in the intra-orbital repulsive scattering which favors a nodeless gap, while also (see Eq. A10) leading to a more negative Γ 1331 scattering, which favors a nodal gap. The net result, as seen in Fig. 9b), is that the nodes remain forJ = 0.1 andJ ′ = 0. Alternatively, for the case in whichJ ′ = 0.1 andJ = 0, as shown in Fig. 9c), the nodes are "lifted." In this case,J ′ is sufficient to overcome the negative contribution ofŪ ′2 χ 0 1331 . Here, Γ 1331 has changed sign due to the contribution of Eq. A9, which is stronger than the contribution ofŪ ′ (Eq. A10) due to its resonant structure. The change in sign now favors the same sign across the whole β sheet, which causes the nodes to lift.

V. ORBITAL CHARACTER OF HOLE POCKET
We have discussed the appearance of the γ hole pocket with doping in the context of doping by a rigid band shift of the 1111 Fermi surface of Cao et al. 8 . There appear to be various other scenarios in which electronic structure distortions might occur. For example, as noted by Kuroki et al. 1 , variations in the As height, which are known to occur in the 1111 family, can tune the size of the γ pocket and thereby the pairing itself, independent of doping. Another effect of the As height noted by Calderón et al. 44 is a switching of two bands near Γ such that the γ pocket which occurs upon hole doping is of primarily d 3z 2 −r 2 character rather than d xy . Within our framework we can imitate this effect simply by adjusting the size of certain tight-binding coefficients associated with the Fe-As hopping in order to create such a d 3z 2 −r 2 pocket at γ, and ask what effect this has on the pairing. As also found by Calderón et al., the new γ pocket which appears is the only Fermi surface sheet with d 3z 2 −r 2 character, so one may expect the pairing to be substantially altered relative to the situation with a γ pocket of d xy character. In Fig. 10, we see that when the electronic structure is adjusted to create a d 3z 2 −r 2 pocket instead of a d xy , the pairing eigenfunction reverts to the nodal s-type found in the electron doped case, as expected since the additional γ − β condensation energy which stabilized the isotropic state has been lost. In addition, the pairing strength λ is substantially reduced.

VI. EFFECT OF SURFACE ON PAIR STATE
As noted above, the presence of a hole pocket of mainly d xy character around (π,π) in the unfolded zone causes a nodeless state to be favored over a nodal one (c.f. Fig. 3). As pointed out by Kuroki et al. 4 , this can be accomplished by an increase of the pnictogen height h Pn . To provide some additional insight on the emergence of nodeless behavior, in particular in ARPES experiments 9-14 , we have performed first-principles calculations on a slab of BaFe 2 As 2 containing 6 FeAs layers.
The calculations were performed using the Quantum-ESPRESSO package 54 , which uses a plane wave basis. We used the Perdew-Burke-Ernzerhof 55 exchange-correlation functionals and ultrasoft pseudopotentials. The use of ultrasoft pseudopotentials enabled us to utilize an energy cut-off of 40 Ry for the plane wave basis, while the density cut-off was taken to be 400 Ry. To determine the positions of the surface ions, two layers in the middle were kept fixed, and the rest of the atoms were allowed to relax. We find that the pnictogen height near the surface and the c-axis lattice constant contract, by about 13% and 5%, respectively. Fig. 11 shows the band structure obtained for the BaFe 2 As 2 slab. To determine the origin of the bands, we projected the band structure on the atomic wavefunctions of the ions. A particular ǫ k is considered to belong to a certain ion if the projection onto the atom is larger than 50%. We have verified that the results below do not change appreciably if the projection threshold is varied.
As can be seen from the figure, the presence of the surface causes the band just below the Fermi energy in the bulk to rise slightly, causing the appearance of an additional Fermi surface. Note that these results are in the folded zone; zone folding causes the pocket at (π,π) to appear at Γ, and it is this pocket that is seen due to the surface. For this pocket to cause a nodeless gap it is necessary that it is of d xy character (or d x 2 −y 2 in the folded zone), which we have confirmed (not shown). Thus, it is possible that due to the sensitive nature of the band structure, the presence of a surface can cause surface probes to detect a nodeless gap, even when the bulk gap has nodes.

VII. CONCLUSIONS
One of the striking features of the Fe superconductors is the sensitivity of the momentum space structure of the superconducting gap to relatively small changes in the electronic structure. Indeed, the electronic structure of these materials is quite special. First, they are semi-metallic with multiple, nearly compensated electron and hole Fermi surfaces. Secondly, multiple Fe orbitals lie near the Fermi energy. This means that relatively small changes in the doping or atomic structure can alter the nesting, the orbital composition of the band states on the Fermi surface, and even the number of k z =0 Fermi surfaces. The most prominent example of this phenomenon is the appearance of the so-called (π, π) hole pocket, studied first by Kuroki et al., with hole doping or pnictogen height. 1 Within the framework of an RPA fluctuation exchange approximation, we can understand how these changes affect the structure of the superconducting gap. For the parameter regime that we have studied, the leading pairing state has A 1g symmetry. It is basically an s ± state, but the anisotropy of the gap, and particularly the presence or absence of gap nodes on the electron Fermi surface sheets is sensitive to the electronic structure. It is important to note that this Γ X M Γ type of a unconventional nodal state is much more sensitive than for example, the d x 2 −y 2 pair state of the cuprates, where nodes owe their existence to symmetry; instead, in the Fe-pnictides the nodes appear to be "accidental", i.e. determined by details of the pair interaction. As discussed in earlier works, the dominant pairing processes involve intra-orbital scattering. In this case, the intra-orbital (π, 0) and (0, π) scattering processes lead to a change of sign between the regions on the α and β Fermi surfaces where the d xz and d yz orbital weights are dominant. However, β 1 to β 2 intra-orbital d xy scattering tends to frustrate the isotropic s ± state. Furthermore, the effect of the intraband Coulomb interaction can be reduced in an anisotropic state. Thus, for light electron doping (n = 6.01), where there are only the α and β Fermi surfaces, we find that the gap has nodes on the β sheets. This anisotropy is further enhanced by the inter-orbital scattering of d xz (d yz ) pairs on α 1 (α 2 ) to d xy pairs on the β sheets.
In the hole doped case (n = 5.95), or if the band structure is adjusted, a hole pocket appears around the (π,π) point. For our band structure, this γ pocket has d xy character. Intra-orbital d xy scattering between the β and γ pockets favors a more isotropic gap, removing the nodes on the β Fermi surfaces. We note that if the band parameters were such that the orbital character of the γ Fermi surface were, for example, d 3z 2 −r 2 instead of d xy , the gap nodes would return to the β Fermi surfaces. In addition, the pairing strength λ is substantially reduced, reflecting the important role played by the orbital weights on the Fermi surface. 56 This illustrates the important role played by the orbital weight.
We also pointed out that various effects in addition to the pnictogen height identified as a key factor by Kuroki et al. can influence the γ pocket. In particular, the presence of a surface can create such a pocket in a nominally electrondoped system, stabilizing a nodeless state. This may explain why ARPES experiments have reported quasi-isotropic gaps in these systems.
We have investigated in some detail the way in which the various single-site interaction parametersŪ ,Ū ′ ,J and J ′ influence the different types of pair scattering processes. The RPA technique is crude, but it allows analytical insights into questions of this type. We have been therefore able to trace the effects of varying these parameters to the strength of the relevant orbital pair scattering vertices at appropriate nesting vectors, which in turn determines not only the overall magnitude of the pairing strength (i.e., T c ), but also the anisotropy of the gap on the Fermi surface. It is to be hoped that the influence of various changes in crystal structure, morphology, etc. on the pair state and transition temperature may now be better understood through this type of analysis. (c) pair scattering processes. Some second order contributions are shown in Fig. A.1 (e)-(g). The orbital matrix elements for k and −k states on the Fermi surface favor pairs which are formed from electrons in the same orbital state. Thus in spite of the fact that the mixed-orbital scattering can be significant, its contribution to the pairing interaction is negligible and the intra-and inter-orbital scattering processes give rise to the superconducting pairing 51 . The relative l-orbital contribution to the Bloch state k on the ν th Fermi surface is given by the square of the orbital matrix element |a l ν (k)| 2 . As shown in Fig. 1, the l = 1 (d xz ) and l = 2 (d yz ) orbitals give the main contributions on the α Fermi surfaces. Likewise the l = 1 and l = 3 (d xy ) orbitals contribute to the β 2 Fermi surface, while the l = 2 and l = 3 orbitals contribute to the β 1 Fermi surface. For our tight-binding fit of the Cao et al. bandstructure 8 , the γ Fermi surface (around M =(π, π)) has predominantly l = 3 (d xy ) weight. The orbital weight distribution favors (k, −k) pairs with similar orbitals so that both, intra-orbital (d xz , d xz ) α ↔ (d xz , d xz ) β2 , (d yz , d yz ) α ↔ (d yz , d yz ) β1 as well as inter-orbital (d xz , d xz ) α ↔ (d xy , d xy ) β2 and (d yz , d yz ) α ↔ (d xy , d xy ) β2 pair scattering processes contribute. In addition, when the γ Fermi surface around (π, π) is present there are important intra-orbital (d xy , d xy ) γ ↔ (d xy , d xy ) β1 and (d xy , d xy ) γ ↔ (d xy , d xy ) β2 pair scatterings. From Fig. A.1, one sees that the first order intra-orbital scattering processes involveŪ andJ while the inter-orbital processes depend uponŪ ′ andJ ′ . and Here µ and ν are summed over the band indices and f is the usual Fermi function. We have typically taken the temperature T = 0.02 eV. The orbital indexing convention for the susceptibility is illustrated in figure A.2. As seen in Fig. 1, three orbitals 1 (d xz ), 2 (d yz ), and 3 (d xy ) have significant weight on the Fermi surfaces. Therefore in this case, the interaction matrix U s reduces to the 9×9 matrix shown in Table I. Furthermore, as seen in Fig. A.3, The first order (a-d) and some second order (e-g) scattering vertices corresponding to intra-(a,e), inter-(d,f), and mixed-orbital (b,c,g) scattering processes. the bare off-diagonal elements of the susceptibility involve single-particle propagators projected on different orbitals which makes them smaller than the diagonal elements.
Keeping only the diagonal terms in χ 0 , it is straightforward to evaluate the RPA susceptibility matrix. For example,    with χ 0 a ≡ χ 0 aaaa . It is clear from these expressions that, for repulsive interactions and within the current approximation, while the intra-orbital spin susceptibities depend onŪ andJ, the inter-orbital susceptibilities depend on U ′ ±J ′ .
Although we have only kept the diagonal terms in the bare susceptibility, due to the structure of the interaction matrices there are off-diagonal terms in the RPA enhanced susceptibility. From these components, we focus on the intra-and inter-orbital pair scattering processes, as the mixed-orbital processes are suppressed by the external matrix elements.
Within this diagonal bare susceptibility approximation, the inter-orbital pair scattering elements of Γ are simple because they involve only a 2×2 interaction matrix. For example, the pairing strength for the spin-fluctuation scattering of a d xz pair to a d xy pair is where c † iℓσ creates a particle with spin σ in orbital ℓ at site i. Diagonalizing the non-interacting part of the Fouriertransformed Hamiltonian the orbital phases can be absorbed into the matrix elements in the form c ℓσ (k) = νã ℓ ν (k)ψ νσ (k) where ψ † νσ (k) creates now a particle with spin σ and momentum k in band ν and the matrix elementã ℓ ν (k) = e iφ ℓ a ℓ ν (k). For the bare multiorbital susceptibility as defined in Eq. A5 we thus find the following transformatioñ while the spin susceptibility as an observable is gauge invariant χ S (q) = 1 2 ℓ1ℓ2χ 0 ℓ1ℓ1ℓ2ℓ2 (q) = χ S (q).
If we now proceed to the interaction Hamiltonian Eq. 2 we note that the pair hopping term is not gauge invariant It is straightforward to show that this relation also yields the correct gauge transformation for the RPA enhanced multiorbital susceptibilityχ under this transformation, the orbital dependent pairing vertex (Eq. 2) transforms as Γ ℓ1ℓ2ℓ3ℓ4 (k, k ′ , ω) = Γ ℓ1ℓ2ℓ3ℓ4 (k, k ′ , ω)e −i(φ ℓ 1 −φ ℓ 2 −φ ℓ 3 +φ ℓ 4 ) and we see that the effective pairing vertex entering the linearized gap equation is gauge invariant.
In the above, we have presented the orbital pairing vertices (in Figs. 5, 8 and 9) and orbital susceptibilities (in Fig.  A.3). Both of these quantities depend on the choice of of orbital gauge. We have chosen to present them in the basis where (1) the d xz and d yz orbitals are aligned along the Fe-Fe directions, and (2) none of the orbitals have a purely imaginary phase with respect to any other. In the notation above, φ ℓ = 0, for all ℓ In this basis, a l ν (−k) = a l * ν (k), so that the sign of the intra-and inter-orbital pairing vertices accurately reflects their contribution to the gauge invariant pairing vertex Γ ij (k, k ′ ).