The discovery of superconductivity in fluorine doped LaFeAsO has stimulated tremendous research efforts in iron-based superconductors (FeSCs) in recent years.^[1] Different families of FeSCs have been detected since then. Among them, the 11 family iron–chalcogen (FeCh) with , Se, Te have been extensively studied due to them having the simplest crystal structure but intriguing superconducting properties. A central issue in studying 11 family FeCh is to determine the pairing symmetry and gap structure. For example, in FeSe, the gap structure is still under debate. The presence of gap nodes is observed by scanning tunneling spectroscopic (STS) measurements,^[2,3] while some experiments such as specific heat and thermal conductivity suggest nodeless gaps.^[4,5] With Te substitution, for optimally doped FeSe_0.5Te_0.5, the superconducting gap is believed to be isotropically full-gapped.^[6–8] For FeS (Ref. [9]), the recent low-temperature heat transport measurements have revealed a nodal quasiparticle behavior.^[10,11] Furthermore, theoretical calculation suggests that the paring symmetry is a wave.^[12] Although the 11 family FeCh share the same crystal structure and similar electronic structure, the gap structure seems to be different. Therefore, it is interesting to discuss the similarity and difference within 11 family FeCh.

To elucidate the mechanism behind the evolution of the paring symmetry, one clue is the effect of the lattice structure on the electronic structure near the Fermi energy, which may change the spin fluctuation.^[13] However, this interplay picture has not yet been clarified in FeSCs. The 11 family FeCh may offer a good platform to study the mechanism behind the pairing symmetry transition owing to their very simple crystal structures. In this regard, a comparison among different members of this family is necessary.

In this work, we mainly compare the electronic structures and microscopic pairing mechanism between FeS and FeSe_0.5Te_0.5. FeS exhibits nodal behavior, while FeSe_0.5Te_0.5 is believed to be a nodeless full-gapped superconductor clarified by experiments. We present the electronic calculation based on density functional theory (DFT) and construct a theoretical model using a maximally-localized Wannier functions (MLWFs) method. The microscopic interplay between spin fluctuation and superconductivity is studied using both random-phase-approximation (RPA) and singular-mode functional renormalization group (SMFRG) methods.

Our main findings are listed as follows. We find that for FeSe_0.5Te_0.5 and FeS, the electronic structures are similar except some small difference in projected density of states (pDOS) and Fermi surfaces. With Ch height changing, the Fe orbital components on Fermi surfaces change remarkably, which contributes to electron and hole pockets in FeSe_0.5Te_0.5 but only to hole pockets in FeS. The SMFRG results show that for FeSe_0.5Te_0.5, a full-gapped -wave pairing is triggered by spin fluctuations connecting electron and hole pockets. Whereas for FeS, a nodal d_xy-wave pairing is induced by small spin wave vectors. The form factors of pairing functions show that intraorbital scattering of plays an important role in pairing of FeS, related with the changes of orbital components. Therefore, we propose the different crystal structures and Ch atoms will bring two tendencies: the enhanced intraorbital scattering and the change of orbital components on Fermi surfaces, especially for the orbital, which facilitate the pairing symmetry from a full-gapped -wave evolved into a nodal d-wave.

Since the details of DFT calculations about FeS have been reported in our former work,^[12] we focus on FeSe_0.5Te_0.5 now. For FeSe_0.5Te_0.5 composition, the unit cell in the tetragonal phase of the PbO-type is used in our following studies, shown in Fig. 1. This arrangement requires only a replacement of one Se by the Te atom in the single unit cell of FeSe. Furthermore, different arrangements of Te are tested within the supercell by DFT calculation, and the total energies are very similar. So we choose this convenient atomic configuration, following the same way as former DFT studies.^[14–16]

Fig. 1.

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	Fig. 1. (color online) The unit cell of FeSe0.5Te0.5 with the alternate arrangement of Se and Te atoms.

The crystallographic parameters are taken from neutron diffraction data in Ref. [14]. The Se and Te ions occupy different atomic sites, which leads to two inter-cell positions, and . For experimental values and . Compared with FeS, the Ch heights are enlarged from 1.269 Å to 1.453/1.697 Å owing to the bigger ionic radius of selenium and tellurium atoms. We use the preliminary experimental values and then relax the height of the chalcogen. The fractional coordinates and are determined after relaxation. We perform our calculation for both of the experimental values and optimized values, the results are almost the same and we mainly discuss the results for experimental values below.

Electronic structure calculations for FeSe_0.5Te_0.5 have been performed via the Quantum ESPRESSO code in the framework of DFT.^[17] The PAW pseupotentials are adopted and the exchange correlation is considered by the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof.^[18] After the total energy convergence test, we choose 62-Ry (1 Ry = 13.6056923 eV) energy cut off to ensure accuracy. Self-consistent calculations were converged on a k-point grid and k-point grid was used for non-consistent calculations.

The density of states are shown in Fig. 2. It is obvious that the DOS are divided into two parts. The Se 3p and Te 4p states concentrate from −6.0 eV to −2.5 eV below Fermi energy . The Fe 3d states dominate from −2.5 eV to 2.5 eV around . The pDOS onto five Fe 3d orbitals are shown in Fig. 2(b). At , one finds that the and (the X, Y, Z refer to axes for unit cell with two Fe atoms) provide the main contributions. Compared with FeS, the value of is larger than that of FeS, which has a value of . The pDOS of contributes a lot to this difference, which becomes bigger in FeSe_0.5Te_0.5 than FeS. The other trend noted is that the pseudogap of DOS near ) is rather smaller than FeS. It is ascribed to the evolution of the Ch height. When the Ch height decreases, the interorbital hoppings are enhanced, thus leading to the splitting of the Fe 3d bands into lower and upper bands.^[19]

Fig. 2.

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	Fig. 2. (color online) (a) Total DOS and pDOS obtained by DFT. (b) The pDOS from all five orbitals of Fe 3d, the X, Y, Z refer to axes for a unit cell with two Fe atoms.

Figure 3 shows the band structure of FeSe_0.5Te_0.5. There are five bands across , forming five disconnected Fermi surface sheets, as shown in Fig. 4. Two cylinders are around the M–A line to construct two rather similar electron pockets. Three hole cylinders including an innermost closed pocket are around the –Z line. The results share great similarities with a former angle-resolved photoemission spectroscopy (ARPES) study of Fe_1.04(Te_0.66Se_0.34).^[20] Moreover, for an optimized Ch height, it gives qualitatively similar band structures and Fermi surface around , except the innermost hole pocket gets bigger. For FeS, the innermost hole pocket is absent and both the electron and hole pockets are reduced. The band filling is reduced from 12.00 for FeSe_0.5Te_0.5 to 11.92 for FeS.

Fig. 3.

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	Fig. 3. (color online) Band structure of FeSe0.5Te0.5 obtained by DFT calculations (solid black lines) and MLWF fitting (dashed red lines).

Fig. 4.

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	Fig. 4. (color online) Lateral view (a) and top view (b) of Fermi surface sheets for FeSe0.5Te0.5.

The band structures are then fitted by means of the MLWF method.^[21,22] There are ten Wannier functions centered at two Fe sites in the unit cell. As shown in Fig. 3, the MLWFs fit the band structure rather accurately in the plane. It is noted that unlike the case of FeSe or FeS, the local arrangement of each Fe atom in the unit cell is different due to the different surroundings of Se and Te atom. Therefore, we cannot further unfold the Brillouin zone with one Fe atom per unit cell, and thus we only consider the ten-orbital theoretical model.

It has been suggested that the orbital character plays an important role in determining the pairing symmetry for FeSCs. To address this issue, the Fermi surface of FeSe_0.5Te_0.5 and FeS encoded with the spectral weights in d_XZ, d_YZ, and orbital components are obtained and shown in Figs. 5(a)–5(f). Specifically, for FeSe_0.5Te_0.5, the contribute mainly to the α₁, α₂ pocket and β₁, β₂ pockets. Whereas the (refers to d_xy in an unfolded Brillouin zone) contribute mainly to the α₃ pocket and some spectral weights to the β₁, β₂ pockets. For FeS, the orbital composition changes substantially. One major difference is the α₂, α₃ hole pockets around have the few character, instead the β₁, β₂ electron pockets around M have mainly the character. Seen from the Fermi surface in Fig. 5(a)–5(f), there are some different types of spin fluctuation, including the interpocket scattering between hole and electron pockets labeled as and scattering between electron pockets labeled as . Intuitively, the leading scattering will be changed due to the change of orbital components. To check this, we should consider the correlation effects.

Fig. 5.

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Fig. 5. (color online) Fermi surface sheets of FeSe0.5Te0.5 (top) and FeS (bottom) in the 2Fe unit cell. The width of each Fermi surfaces line is proportional to its spectral weights in dXZ, dYZ, and components, respectively. The hole pockets around are denoted by α1, α2, and α3 from the inside out. The electron pockets are denoted by β1 and β2, respectively. The α1 is absent for FeS. The arrows in panels (a) and (f) denote the possible dominant scattering vectors for FeSe0.5Te0.5 and for FeS, respectively.

We consider the following local interactions as,

We first calculate the bare susceptibilities and random-phase-approximation (RPA) spin susceptibilities for both systems based on the theoretical models discussed above. The main results are summarized in Fig. 6. The bare susceptibilities for both systems share great similarities with each other, owing to the similar Fermi surface topology. They both have dominant peaks at the zone corner and some broad peaks around the zone center. However, as the interaction is introduced, the spin fluctuations of these two systems diverge obviously. For FeSe_0.5Te_0.5, the predominant peaks around are enhanced rapidly. While for FeS, the peaks around the zone center become stronger, reflecting that the spin fluctuation is different from FeSe_0.5Te_0.5. We also check pairing interactions due to spin fluctuations from the RPA framework. For both FeSe_0.5Te_0.5 and FeS, we see different pairing channels compete with the value of U changes, the same as former RPA studies for other FeSCs.^[13,23,24] Thus, we should consider the competing collective fluctuation in density-wave and pairing channels in these systems, which we resort to the unbiased SMFRG developed recently. Meanwhile, it has an advantage in getting more reliable pairing functions in orbital basis, from which we can obtain the main difference of the pairing mechanism between FeSe_0.5Te_0.5 and FeS.

Fig. 6.

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	Fig. 6. (color online) Left panels: The bare suceptibilities for FeSe0.5Te0.5 (a) and FeS (c), respectively. Right panels: The RPA spin suceptibilities for FeSe0.5Te0.5 (b) and FeS (d), respectively. Here, U = 0.9 eV and .

Starting from the bare interactions in , SMFRG provides the flow versus a decreasing infrared energy cutoff of the effective one-particle-irreducible four-point interaction vertex function. In a nutshell, the latter can also be understood as a generalized pseudo-potential running with . The divergence of the vertex function versus decreasing signals an instability of the normal state. To see the instability channel, from the vertex function we extract the scattering matrices between fermion bilinears in the general charge-density-wave (CDW), SDW and SC channels. For a given channel, and for a given collective momentum ,^[26] the scattering matrix can be decomposed into eigen modes as

We first discuss the results for FeSe_0.5Te_0.5. Figure 7(a) shows the FRG flow versus (the infrared cutoff of the Matsubara frequency) for U = 1.40 eV and . Since the CDW channel remains weak during the flow, we shall not discuss it henceforth. We find the interaction in the SDW channel is enhanced in the intermediate stage but levels off at low-energy scales. The associated collective momentum evolves (as indicted by the arrows) from and eventually settles down at . The small hump in the flow of suggests the leading spin fluctuation undergoes a change. The inset shows versus at the final stage. There are predominant peaks around , corresponding to scattering between the hole pockets and electron pockets in Fig. 5(a). As spin fluctuations are enhanced, the attractive pairing interaction is enhanced significantly and eventually diverges. The details of the pairing form factors for can be found in Appendix A. The form factor turns out to be dominated by the nearest bond from mainly (d_XZ, d_YZ) orbitals, which is closely related with the orbital components discussed above. The pairing symmetry is dictated by the leading form factor. To see this straightforwardly, we further project the form factor into the band basis and plot it along the Fermi surfaces in Fig. 7(b). Obviously, the gap is isotropic on each pocket and describes a full-gapped -wave, consistent with former experiments.^[6–8]

Fig. 7.

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	Fig. 7. (color online) (a) FRG flow of versus for eV and for FeSe0.5Te0.5. The arrows indicate snapshots of the leading momentum (divided by π) in the SDW channel. The inset shows in the momentum space at the final energy scale. (b) The gap funtion on the k points of Fermi surfaces for FeSe0.5Te0.5.

We now discuss the case for FeS. For comparison, a ten-orbital model with two Fe atoms is used. The FRG flow for U = 1.70 eV and is shown in Fig. 8(a). The interaction in the SDW channel, behaves qualitatively similar to FeSe_0.5Te_0.5 at the beginning but does not undergo transition to at intermediate stages. Eventually, levels off at low-energy scales, associated with small collective momentum . The inset of Fig. 8(a) shows versus at the final stage of the flow. Incommensurate peaks around the zone center are obvious. According to the orbital components in Figs. 5(d)–5(f), this small momentum corresponds mainly to the interpocket scattering between β₁ and β₂, which is labeled by in Fig. 5(f). The reason consists in two aspects. First, the interorbital hoppings are enhanced due to the reduction of Ch height in FeS, leading to the enhanced interorbital scattering. Second, α₁, α₂ pockets have few components whereas the β₁ and β₂ have mainly components. Therefore, the small momentum can benefit from different types of scattering and become the leading eigenvalue. Triggered by such spin fluctuations, the pairing interaction is enhanced and eventually diverges. The pairing form factors can be seen in Appendix A for details. It describes pairing on the nearest bond. Moreover, the most active orbital for this pairing state is , corresponding to the interpocket scattering between β₁ and β₂. We further project the form factor into the band basis. As is shown in Fig. 8(b), the pairing symmetry transforms as d_xy (refer to in an unfolded Broullin zone) along Fermi surfaces. It is fully gapped on the β₁, β₂ pockets but nodal on the α₁, α₂ pockets, which is consistent with nodal behavior in experiments.^[10,11]

Fig. 8.

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	Fig. 8. (color online) (a) FRG flow of versus Λ for U = 1.7 eV and for FeS. The arrows indicate snapshots of the leading momentum (divided by π) in the SDW channel. The inset shows in the momentum space at the final energy scale. (b) The gap function on the k points of Fermi surfaces for FeS.

Firstly, we emphasize that the pairing symmetry we obtained from FRG is consistent with our RPA results when U is big enough. Secondly, we have performed systematic calculations for other values of (). We find FeSe_0.5Te_0.5 is in close proximity to the antiferromagnetism phase. For , the quickly diverges at high-energy scales and the system develops an SDW instability at large wave vectors. On the other hand, for the case of FeS, the system develops SC instability until . For , the system develops SDW instability at small wave vectors. Given the experimental results, the wave vectors around connecting the and M pockets has been observed in FeSe_xTe and BaFe₂As₂-derived superconductors, which is seen as a common feature among most FeSCs.^[34–38] For FeS, the magnetic properties are sensitive to structural details and the previously reported sample with Å show bulk superconductors with no magnetism. The recent temperature neutron diffraction data for the Å sample has real spin wave vectors ,^[39] reflecting FeS have different magnetic ordering with Fe_0.5Te_0.5. Combined with our results, we propose that, for moderate U, the much stronger spin fluctuation in FeSe_0.5Te_0.5 will induce the superconductivity at a higher energy scale. It may explain why FeSe_0.5Te_0.5 () have higher superconducting transition temperature than FeS ().

We have compared the electronic structure and correlation effects of FeSe_0.5Te_0.5 with FeS. The electronic structures are similar except for some difference, the of FeSe_0.5Te_0.5 is larger than FeS owing to the increase of Fe orbital pDOS. An innermost hold pocket is present (absent) in FeSe_0.5Te_0.5 (FeS). Furthermore, for orbital components along the Fermi surfaces, we find the orbital character significantly changed, which mainly contribute to an electron pocket around M in FeS. To study the interplay of the electronic structure and pairing symmetry, we have performed RPA and SMFRG calculation based on ten Fe 3d orbital models. The results shown in FeSe_0.5Te_0.5, a full-gap is induced by wave vectors around connecting electron and hole pockets. Whereas in FeS, a nodal d_xy (refer to in the unfolded Brillouin zone) pairing is induced by a small wave vector coming from the interpocket scattering of electron pockets. This evolution of pairing symmetry is closely related to the change of orbital components on Fermi surfaces. As crystal structures change from FeSe_0.5Te_0.5 to FeS, the intraorbital scattering of is enhanced and the nesting becomes dominant. The leading form factor of the pairing function obtained by SMFRG also support this propose.