A Supplemental Material

RPA Analysis of a Two-orbital Model for the -based Superconductors

Abstract

The random-phase approximation (RPA) is here applied to a two-orbital model for the -based superconductors that was recently proposed by Usui et al., arXiv:1207.3888. Varying the density of doped electrons per Bi site, , in the range , the spin fluctuations promote competing and superconducting states with similar pairing strengths, in analogy with the - near degeneracy found also within RPA in models for pnictides. At these band fillings, two hole-pockets centered at and display nearly parallel Fermi Surface segments close to wavevector , whose distance increases with . After introducing electronic interactions treated in the RPA, the inter-pocket nesting of these segments leads to pair scattering with a rather “local” character in k-space. The similarity between the and channels observed here should manifest in experiments on -based superconductors if the pairing is caused by spin fluctuations.

pacs:
74.20.Mn,74.20.Rp,74.70.-b

Introduction.—The recently discovered family of layered bismuth oxy-sulfide superconductorsMizuguchi et al. (2012a, b); Usui et al. (2012); Li et al. (2012); Demura et al. (2012); Tan et al. (2012a); Singh et al. (2012); Awana et al. (2012); Kotegawa et al. (2012); Zhou and Wang (2012); Wan et al. (2012); Takatsu et al. (2012); Sathish and Yamaura (2012); Jha et al. (2012a); Xing et al. (2012); Tan et al. (2012b); Jha et al. (2012b); Deguchi et al. (2012); Li and Xing (2012); Liu (2012); Zhang and Zhang (2012); Lei et al. (2012) has immediately attracted considerable attention from the Condensed Matter community due to its close similarities with the famous iron-pnictide superconductors.Kamihara et al. (2008); Stewart (2011); Johnston (2010); Dai et al. (2012); Dag () As in the case of other layered unconventional superconductors, such as the cuprates and the aforementioned iron pnictides/chalcogenides, this new family displays a layered structure involving planes where the observed superconductivity is believed to reside. The first report of superconductivity originated in , with K.Mizuguchi et al. (2012a) Superconductivity has also been reported in , where Re = La, Nd, Ce, and Pr, with corresponding ,Mizuguchi et al. (2012b) ,Demura et al. (2012) ,Xing et al. (2012) and K.Jha et al. (2012b) These compounds are metallic in the normal state and Density Functional Theory calculations indicate that the relevant bands crossing the Fermi surface (FS) originate mainly from the Bi 6 orbitals, as shown, e.g., for .Usui et al. (2012) However, contrary to the majority of the Cu- and Fe-based unconventional superconductors, no magnetically ordered phase has been detected thus far in the compounds. This apparent absence of magnetism in the compounds may still locate them in the same category as , , and possibly ,Stewart (2011) that are also non magnetic but their pairing properties are widely believed to still originate in short-range magnetic fluctuations. For these reasons, and despite the absence of observed long-range magnetism in , it is important to study the potential role of spin fluctuations in these novel materials and the pairing channels that those fluctuations tend to favor, to help in the analysis of experimental data.

In this manuscript, the two-orbital (2-orbital) model recently introduced by Usui et al. is adopted.Usui et al. (2012) The fact that the relevant orbitals in compounds are -type, where Coulomb interactions should be smaller than in orbitals, turns RPA into a suitable technique, whose results deserve a careful analysis if electron correlations are found to be important for superconductivity in these materials. Similar calculations for a related four-orbital modelUsui et al. (2012) are underway. Note that in Ref. Usui et al., 2012 a brief discussion of RPA calculations has already been presented. The results discussed by Usui et al. consisted of a single set of couplings (equivalent to our calculations below) at . Their early weak-coupling RPA analysis is here expanded via a systematic study of the influence of the band filling and the identification of the dominant channels for superconductivity under the assumption of a spin fluctuations mechanism. The main novel contribution of our present effort is the identification of closely competing and gap functions as the dominant pairing channels, particularly for band fillings around . At quarter filling (), another pair of almost degenerate gap functions (with symmetries and ) is found to closely compete with the previously mentioned dominant pair, especially at .

Hamiltonian. The 2-orbital model described by Usui et al.Usui et al. (2012) contains hopping parameters up to fourth neighbors, and in k-space is given by

(1)

where

(2)
(3)
(4)

The operator () in Eq. (1) creates (annihilates) an electron in band , with spin , and wavevector . The values for the hopping parameters are those from Ref. Usui et al., 2012, and are reproduced in Table I for completeness (in eV units, as used throughout this paper). Figure 1(a) shows the FS hole-pockets for four different band fillings , , , and , with corresponding chemical potentials , , , and (in principle, in ).Usui et al. (2012) Panel (b) shows the corresponding non-interacting magnetic susceptibilities . The leftmost peaks in , located at , with as the filling varies from to , can be associated to FS nesting once it is noticed that their position matches the horizontal separation between the two adjacent FS segments from the pockets centered at () and (), as highlighted by the dashed box in panel (a) and sketched in the inset to panel (b). Note that the horizontal separation is well defined if the two FS segments are parallel, which is the limiting case as increases, as shown in the inset, to (for details, see Fig. 5 and the associated discussion). It is also important to remark that once interactions are introduced, the leftmost peak in is the one that diverges in the RPA calculation of the spin susceptibility for almost all the fillings and various values of interaction parameters. This divergence indicates a tendency to magnetic order, or at least strong spin fluctuations (paramagnons), with characteristic wavelength determined by . Our analysis is not extended into the region since there the topology of the FS changes (see Ref. Usui et al., 2012 for details of the FS at lower fillings not ()).

Table 1: Tight-binding parameters (eV) for 2-orbital model.
Figure 1: (Color online) (a) Hole-pockets for four different electronic fillings: (solid red), (dashed green), (dotted blue), and (dot-dashed magenta). Note that close to the wavevector, where the pockets almost touch, the increase of decreases the radius of the hole-pockets and, more importantly, the adjacent FS segments (inside the dashed box) become more and more parallel. (b) Lindhard function for the same fillings as in panel (a). Note that the position in k-space of the leftmost peak is clearly associated to FS nesting through a vector, as indicated in the inset, which zooms-in the dashed box in panel (a). Indeed, the position of the leftmost peaks in agree (within a few percent) with the vectors indicated in the inset (see text for details, especially Fig. 5). Obviously, there are additional nesting vectors that become evident in a 2-d plot of [Fig. 5(b)].

The Coulomb interaction in the Hamiltonian is given by

(5)

where the notation is standard and the many terms have been described elsewhere.Luo et al. (2010) Here, the usual relation is assumed, and is a parameter. Calculations were done for , in steps of , for the four fillings , , , and . The multi-orbital RPA calculations performed here follow closely those described in Ref. Graser et al., 2009, and previous works by the authors.Luo et al. (2010); Nicholson et al. (2011) All results were obtained at temperature and an imaginary part was used to regularize the Green’s functions.

Figure 2: (Color online) RPA spin susceptibility (solid red curves in the main panels) and dominant gap function (red and blue dots in the insets) for (a) and (b) . In the inset to each panel, the dominant gap function with symmetry is shown. The subdominant gap function (not shown) has symmetry and its eigenvalue is almost degenerate with the dominant one (see text).

Our RPA results for spin-singlet pairing link the dominant superconducting gap functions to spin fluctuations, which originate in FS nesting and are enhanced by electronic interactions. The particular relative topology of the two adjacent hole-pockets (see Fig. 1) promotes pairing whose strength is independent of the global symmetry of the pairing functions [see Fig. 4(b)]. Indeed, the and symmetries have essentially the same pairing strength, which is determined by pair scattering between these two adjacent FS segments (see Fig. 5) close to in the Brillouin Zone (BZ). In addition, our results show that both dominant gap functions change sign between these two segments (Figs. 2 to 4), and the pairing is through the intraorbital scattering channel [Fig. 3(b)]. The near degeneracy - is the analog of the near degeneracy - found also in RPA calculations for the pnictides,Graser et al. (2009) since the pocket structures in both cases can be related by a 45 rotation. Results for spin-triplet pairing are presented in the supplemental material at the end of the manuscript.

Figure 3: (Color online) (a) RPA spin susceptibility and dominant gap function for . Orbital composition for the and FS pockets (), (c) and (d), respectively. The winding angle is counter-clockwise, starting from the direction. Assuming the nesting described in the inset to Fig. 1(b) as producing the spin fluctuations that provide pairing, the pair coupling is then intraorbital.
Figure 4: (Color online) (a) Dominant gap function with symmetry at . (b) Main panel: normalized pairing strengths for the dominant (, solid red curve) and subdominant (, dashed green curve) gap functions. Although the two curves are very close, the eigenvalues are not degenerate. In the inset, the structure of the subdominant gap function () is shown. When compared to that of the dominant one [ in panel (a)], it is clear that the structure around is very similar for both of them, explaining why the pairing strengths (eigenvalues) are the same. The region inside the dashed box, in panel (a), is analyzed in detail in Fig. 5(a).

Results and Discussion. As mentioned above, the most important feature of the FS for fillings between and is that the hole-pockets centered at the and points present almost parallel segments close to the wavevector, becoming more and more parallel as the pockets shrink, with increasing filling [see Fig. 1(a) and inset in Fig. 1(b)]. In Figs. 2 and 3 it will be shown that this has important consequences for the spin fluctuations and the superconducting pairing associated to this 2-orbital model. Indeed, as displayed in the main panel of Fig. 2(a) (solid (red) curve), there is a divergence in the RPA spin susceptibility for very small values: for , and for [panel (b)]. A divergence in the spin susceptibility may point to magnetic order, or at least to strong spin fluctuations with wave vector . Figure 3(a) shows the same calculations, but now for . Note that although displays a broad-peak structure around [see Fig. 1(b)], does not present a divergence in this region. In the insets to Figs. 2(a) and (b), and Fig. 3(a), it is shown that the dominant gap function at the FS has symmetry for the three cases, showing that despite the changes in the size of the hole-pockets the results are qualitatively the same. Figures 3(b) and (c) contain the orbital contribution (, red solid curve; , green dashed curve) of the BZ states at the FS for the and pockets, respectively. It is interesting to note that the modifications in the position of the peak in correlates well with the “separation” between the and hole-pockets in the region around . For the purposes of describing our results, this separation will be defined as the horizontal distance between two parallel lines tangent to the hole-pockets at the points where each intercepts the () line. As described in more detail in Fig.5(a) [and already mentioned in connection with Fig.1(a)], as the filling increases these segments of FS approach more and more the parallel lines just defined, justifying the definition just given.

Figure 5: (Color online) (a) Region around point of the BZ [dashed box in Fig. 4(a)], showing the dominant () gap-function for and . (b) Two dimensional contour plot of also for . The horizontal (blue) vector in panel (a) connects the maximum amplitude of the gap-function in both pockets. Note also the horizontal (blue) vector in panel (b) along the direction, indicating the position of the maximum value of . These two vectors agree up to a difference smaller than the width of this maximum peak in . Therefore, it can be shown (see text) that the line describing the position of the points in the pocket in relation to the points in the pocket, as indicated by the two additional vectors (black and red) in panel (a), satisfies , where and are the positions of the maxima in (with ). This equation also describes the line of local maxima of , as seen in panel (b), originating from FS nesting.

The RPA results for the gap functions also point to an interesting effect, namely, the small value of for fillings results in the pairing strength depending on very “local” properties of the gap function at the adjacent segments of the hole-pockets. This implies that the pairing strength of gap functions with different symmetries is very similar, as long as they have the same “local” properties. To demonstrate that, in Fig. 4(a) the dominant gap function (with symmetry) is shown for and . It is clear that this is very similar in structure to the subdominant one shown in the previous figures. In the inset to Fig. 4(b) the subdominant gap function with symmetry is displayed for the same parameters. Comparing it with the dominant gap function in panel (a) note that, despite having different symmetries, the two gap functions are identical in the two adjacent hole-pocket segments that cross the line. For this reason, their pairing strengths as measured by (the eigenvalues of the Eliashberg Equation), and shown in the main panel of Fig. 4(b), are the same to the third decimal place. Note that the two eigenvalues for symmetries and are not degenerate. This seems a strong indication that the “local” aspect of the pair scattering, as mentioned above, seems to be determinant to establish the pairing properties of this model, at least in our RPA weak-coupling approach. It should be noted that the eigenvalue results shown in Fig. 4 are basically identical to those for lower fillings, shown in previous figures, with the only difference being the order of the dominant and subdominant symmetries. Since their eigenvalues are almost identical for all fillings studied, this does not have a special significance. Note that for and (not shown) follows the same trends as described in Figs. 2 and 3. From the orbital composition in Fig. 3(b) and the gap structure in Fig. 4 it appears that the symmetry of the B and A pairing operators is determined by the orbitals, while the spatial form in both cases is characterized by symmetric nearest-neighbor pairing with rotational invariance. Thus, the pairing operators have the form ) where the () sign corresponds to A (B) symmetry with , plus higher harmonics with A symmetry.

Figure 6: (Color online) Two-dimensional plot of the RPA spin susceptibility for . The parameter values are and . The similarity to the results in Fig. 5(b) is clear, showing also that there are relevant nesting features along the line. Note that a smaller value of than in Fig. 5(a) was used to avoid having a peak at that would wash out the features in the rest of the BZ.

Figure 5(a) shows in more detail the almost parallel FS segments of the two hole-pockets for . In this figure, the horizontal (blue) vector that was defined above as the separation between the two FS segments is displayed. A vector with the same length is reproduced in panel (b), where a 2d plot of in the first quadrant of the BZ is also shown. It clearly indicates that the position of the main peak in is exactly given by the horizontal separation. Not only that, the (red) vector along the line in panel (a) is also reproduced in panel (b) and it coincides also exactly with a local maximum of . In fact (see in both panels the black vectors located at angle ), the locus of the ridge of local maxima in in panel (b) exactly coincides with the BZ points defined by the vectors connecting the two FS segments for . Figure 6 shows the RPA spin susceptibility for . The similarity between these results and those in Fig. 5(a) is clear, indicating that the FS nesting for the interacting system is the one described by the vectors in Fig. 5. Finally, an important issue should be highlighted: the four points in the hole-pockets in Fig. 5(a) where the gap function has a very pronounced peak, are exactly the two pairs of points (one in each pocket) connected by and . This fact clearly links the pairing properties with the spin fluctuations. Note also that for and , the second pair of eigenvalues ( and ) corresponds to symmetries and , respectively (not shown). The same occurs for and , also for (but the eigenvalues are smaller). Yet, the same explanation as described in Fig. 5 applies. See the supplemental material for a connection between the emergence of a symmetry at with the one-dimensionality of the bands.

Conclusions. Summarizing, a weak-coupling RPA analysis of a minimal 2-orbital model was used to investigate the pairing properties of the BiS-based superconductors. Fillings between and were analyzed. The Hund’s coupling was varied in the range . Qualitatively, the results are similar for all values of and different fillings. In the RPA results described here, a clear relationship is found between quasi FS nesting, spin fluctuations, and superconductivity: the topology of the two hole-pockets is such that they present almost parallel segments close to the wavevector in the BZ. It is found that the horizontal distance between the tangents to these segments at the points where they cross the line is also where the non-interacting susceptibility has a pronounced peak at , for . Once interactions are introduced, this peak will diverge at a certain critical coupling for each filling, and all the values of studied (with exception of one: , ). In addition, a line of local maxima, connecting the BZ points and , is clearly observed in a 2-d plot of . As expected, this line can also be associated to FS nesting. This nesting structure gives origin to pairing functions with similar eigenvalues, i.e., similar pairing strengths, and symmetries and . This close competition originates in the FS quasi nesting properties, which determine the spin-fluctuation-mediated inter-pocket pair scattering. This pair scattering is overwhelmingly between two adjacent FS segments, therefore the properties of the pairing functions, including the pairing strength, are quite “local”, having almost no dependence on their global symmetry. One can then predict that pairing symmetry measurements may contain a mixture of both symmetries if the pairing mechanism is driven by spin fluctuations.

GBM acknowledges fruitful conversations with K. Kuroki, Q. Luo, and H. Usui. ED and AM were supported by the National Science Foundation Grant No. DMR-1104386. After finishing this manuscript, a related effort addressing the pairing symmetry of these materials using a spin model was published. Liang et al. (2012) There, it is found a dominant state analogous to ours, but no competing state.

Appendix A Supplemental Material

Spin-triplet pairing. We also tested the two-orbital model for the case of spin-triplet pairing. Using the same RPA all the four fillings studied in this work were investigated, but calculations were carried out only for and . All the critical values obtained for the Hubbard repulsion were slightly above those obtained for the singlet pairing channel. However, they were close enough to warrant a brief discussion in this supplemental material. Note that in Usui et al. Usui et al. (2012) the possibility of spin-triplet pairing was mentioned, in connection with the similarity of the bands with those of , in regards to their common one-dimensionality character. Figure S1 shows the gap functions [dominant in panel (a) and subdominant in (b)] for parameters , , and . This critical value of should be compared with that obtained for singlet pairing for the same parameters (i.e., , see Fig. 2(b) in the main text). The symmetries for both the dominant () and subdominant () gap functions in the spin-triplet channel are the same as for the spin-singlet channel. The main difference here is that they do not have as competing pairing strengths as in the spin-singlet channel. Indeed, the eigenvalues for Fig. S1 are and , while for the same parameters in the spin-singlet channel their values are and .

Figure S1: (Color online) (a) Dominant and (b) subdominant gap functions in the spin-triplet channel, for parameters , , and .

Quasi one-dimensionality. As mentioned in the main text, the two-orbital model has a quasi one-dimensional (1d) character, with the hopping between next-nearest-neighbor being dominant (, see the Table in the main text containing the hoppings). It is then interesting to verify how the results are modified in case all the other hoppings are removed from the two-orbital model Hamiltonian, except for . The energies of the orbitals were kept the same as in the original model. RPA calculations for the spin-singlet pairing channel were done for (with corresponding chemical potential ), , and . In addition, the spin-triplet pairing channel was investigated for , but, again, the critical value obtained for the Hubbard was higher than for the singlet channel, therefore, these results are not shown. Singlet pairing results for both values of the Hund’s coupling were similar, therefore, just the results for will be presented.

Figure S2: (Color online) Band structure for a quasi 1d Hamiltonian obtained from the two-orbital model discussed in the main text by keeping only two hopping terms: and (see text for details). The Fermi energy is at .

Figure S2 shows the band structure for high symmetry lines of the BZ. The Fermi energy is located at . The two hole-pockets obtained are identical and nearly square (see Fig. S4), their corners being slightly rounded due to the presence of the finite hopping. For the hole-pockets are perfectly square and the two bands are degenerate along the () line.

Figure S3: (Color online) Main panel: non-interacting magnetic susceptibility (Lindhard function) for the quasi 1d model. In the inset, for the truly 1d Hamiltonian (obtained when just the dominant hopping is taken in account, ). The arrows indicating peaks located at in the BZ are reproduced in Fig. S4. These peaks indicate different spin fluctuations which may lead to electronic pairing.
Figure S4: (Color online) (a) Dominant and (b) subdominant gap functions in the singlet channel for the quasi 1d model. The parameters used are , , and . The arrows indicate pair scattering processes associated to the peaks in Fig. S3, which are there indicated with the corresponding line style (see text for details).

Figure S3 shows, in the main panel (solid red curve), the non-interacting magnetic susceptibility (Lindhard function) obtained from the bands in Fig. S2. The inset shows, as a reference, for the truly 1d Hamiltonian (i.e., ). A comparison of these two curves in Fig. S3 with the one for the fully two-dimensional (2d) two-orbital model (Fig. 1(b) in the main text) shows that the introduction of a small brings the 1d model (green curve in the inset) qualitatively close to the 2d result. To see that, compare the solid (red) curve in the main pannel of Fig. S3 with the dashed (green) curve in Fig. 1(b) of the main text. As will be described next, the extra peaks introduced in the line (main panel of Fig. S3) have a marked influence in the singlet pairing gap functions. Indeed, Fig. S4(a) shows the dominant gap function (with symmetry ) for and . The four vectors connecting local maxima (with opposite signs) of the gap function are exactly the same that locate the four peaks in in the main panel of Fig. S3. This, once again, shows the strong connection between spin fluctuations and electron pairing. Panel (b) shows the subdominant gap function, with symmetry . Their eigenvalues are the same up to the third decimal place. It is easy to see that the vectors displayed in the gap function [panel (a)] apply identicaly to the subdominant in panel (b). It is also interesting to observe that a possible extra set of pair scattering processes, leading to change of sign in the gap function, are the ones connecting adjacent sides of the same hole-pocket. Two of them are indicated by double-headed arrows. However, these processes do not occur, as there are no peaks in that can provide spin fluctuations with these two wave vectors (see Fig. S3). This results in the pairing strengths of both gap functions being basically the same.

The presence of a relatively large number of different pairing spin fluctuations, as implied in Fig. S4, suggests that the RPA spin susceptibility should have competing peaks when is close to the critical value. Figure S5 indicates that this is indeed the case. There we show for the same parameters as Fig. S4, for three different values of Hubbard interaction (dotted blue curve), (dashed green curve), and (solid red curve). As a comparison, at the same filling and , for the full 2d model studied in the main text, the leftmost peak (Fig. 2(b), main text), at a comparable ratio as the ones in Fig. S5, is a few orders of magnitude above the other peaks. Similar results are seen for the other fillings and values, indicating that there is mainly a single dominant pairing process in the 2d model in the main text. In the 1d model it seems as if the different wave vector spin-fluctuations cooperate to produce pairing.

Figure S5: (Color online) RPA spin susceptibility for the quasi 1d two-orbital model Hamiltonian, and . Three curves are shown for values of Hubbard close to the critical value : , , and . A competition between two peaks can be clearly observed. They are indicated by the same type of arrows as the ones for the corresponding peaks in Fig.S3.

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