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iv:1

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1135

v1 [

cond

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201

1

Robustness of s-wave Pairing in Electron-Overdoped A1−yFe2−xSe2

Chen Fang1, Yang-Le Wu2, Ronny Thomale2, B. Andrei Bernevig2, and Jiangping Hu1,31Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA

2Department of Physics, Princeton University, Princeton, NJ 08544 and3Beijing National Laboratory for Condensed Matter Physics,

Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

Using self consistent mean field and functional renormalization group approaches we show that s-wave pairing symmetry is robust in the heavily electron-doped iron chalcogenides (K, Cs)Fe2−xSe2.This is because in these materials the leading antiferromagnetic (AFM) exchange coupling is betweennext-nearest-neighbor (NNN) sites while the nearest neighbor (NN) magnetic exchange coupling isferromagnetic (FM). This is different from the iron pnictides, where the NN magnetic exchangecoupling is AFM and leads to strong competition between s-wave and d-wave pairing in the elec-tron overdoped region. Our finding of a robust s-wave pairing in (K, Cs)Fe2−xSe2 differs fromthe d-wave pairing result obtained by other theories where non-local bare interaction terms andthe NNN J2 term are underestimated. Detecting the pairing symmetry in (K, Cs)Fe2−xSe2 mayhence provide important insights regarding the mechanism of superconducting pairing in iron basedsuperconductors.

I. INTRODUCTION

The recent discovery of a new family of iron-based su-perconductors A(K,Cs,Rb)yFe2−xSe2

1–3 has initiated anew round of research in this field. Remarkably, thisnew family shows distinctly different properties fromother pnictide families: the compounds are heavily elec-tron doped, but their superconducting transition tem-peratures are high, at more than 40 K. For compar-ison, such large Tc’s can only be reached in the op-timally doped 122 iron pnictides4. Importantly, bothangle-resolved photoemission spectroscopy (ARPES)5–7

and LDA calculations8–10 show the presence of only elec-tron Fermi pockets located at the M point of the foldedBrillouin zone (BZ). (Some signature of possible densityof states at the Γ point is still under current debate; inany case this pocket, if present, is assumed to be very flatand shallow). ARPES experiments have also reportedlarge isotropic superconducting gaps at these pockets5–7.The absence of hole pockets around the Γ point of theBZ provides a new arena of Fermi surface topology to in-vestigate the pairing symmetries and mechanisms of su-perconductivity proposed for iron-based superconductorsfrom a variety of different approaches11–35.So far, the majority of theories for the pairing sym-

metry of iron-based superconductors are based on weakcoupling approaches13–18,27–32,36,37. Although there arediscrepancies, the theories based on these approacheshave reached a broad consensus regarding the pairingsymmetries in iron-based superconductors: for optimallyhole doped iron-pnictides, for example, Ba0.6K0.4Fe2As2,an extended s-wave pairing symmetry, called s±, isfavored14 (the sign of the order parameter changes be-tween hole and electron pockets as potentially detectablethrough neutron scattering38), as a result of repulsive in-terband interactions and nesting between the hole andelectron pockets. For extremely hole-doped materials,such as KFe2As2, the absence of electron pockets canlead to a d-wave pairing symmetry18 with a low transi-

tion temperature; for electron doped materials such asBa2Fe2−xCoxAs2, the anisotropy of the superconductinggap around the electron pockets in the s± state grows forlarger electron doping and eventually the SC gap devel-ops nodes around the electron pockets due to the weak-ening of the nesting condition and the increase of dxyorbital weight at the electron pocket Fermi surfaces39,40.Finally, in the limit of the heavily electron doped casewhen the hole pockets vanish and only the electron pock-ets are left, the d-wave pairing symmetry may be favoredagain18,41,42. The iron chalcogenide AyFe2−xSe2 belongsto the latter category and many theories based on weakcoupling approaches have suggested that the pairing sym-metry should be d-wave as possibly detectable throughcharacteristic impurity scattering43–45.

A complementary approach based on strong couplinglikewise predicts an s-wave pairing symmetry in the ironpnictides. Two of us showed that the pairing symme-try is determined mainly by the next-nearest-neighbor(NNN) AFM exchange coupling J2 together with a renor-malized narrow band width11,46. The superconductinggap is close to a cos kx cos ky form in momentum space(higher harmonic contributions are neglected in this ap-proach). This result is model independent as long asthe dominating interaction is J2 and the Fermi surfacesare located close to the Γ and M points in the foldedBZ. The cos kx cos ky form factor changes sign betweenthe electron and hole pockets in the BZ. It resemblesthe order parameters of s± proposed from weak-couplingarguments14.

The J2 coupling will be of particular importance inthe following. We point out two key points on J2-relatedphysics as it has appeared in the literature up to now.First, the effect of J2 is underestimated in most an-alytic models constructed based on the pure iron lat-tice with only onsite interactions since the J2 exchangecoupling originates mostly from superexchange processesthrough As (P) or Se (Te). Second, in the effective{t} − J1 − J2 model studied before11, the superconduct-

2

ing state is obtained only when the magnetic exchangecoupling strength is of the same order as the hoppingparameters (or the bandwidth) of the model. Therefore,as t > J , it requires the effective bands given by {t} berenormalized. However, the absence of double occupan-cies in the standard t − J model is not strictly imposedin such an intermediately coupled effective model wherethe bandwidth is assumed to be of similar order as theinteraction scale.

Comparing the predictions from weak coupling andstrong coupling, the 122 iron chalcogenides provide aninteresting opportunity to address the difference betweenthe two perspectives. In this paper, we predict thatthe s-wave pairing symmetry is robust even in extremelyelectron-overdoped iron chalcogenides because the AFMJ2 is the main factor for pairing and the J1 is ferromag-

netic (FM), a conclusion drawn from both neutron scat-tering experiments47–49 and the magnetic structure asso-ciated with 245 vacancy ordering50,51. As we will show,the FM J1 significantly reduces the competitiveness ofd-wave pairing symmetry. We substantiate this claim bytwo different methods. First, we solve the three orbital{t} − J1 − J2 model on the mean field level to show thatthe s-wave pairing is the leading instability regardless ofthe change of doping given that J2 is large. We calculatea full phase diagram as J1 varies from FM to AFM. IfJ1 is AFM, we obtain a SC state with a mixed s-waveand d-wave pairing. Second, we use the functional renor-malization group (FRG) to analyze this trend obtainedby mean field analysis for a 5-band model of the chalco-genides. We confirm that a dominant AFM J2 generallyleads to robust s-wave pairing while an AFM J1 tendsto favor d-wave pairing in the electron overdoped region.The competition between s-wave and d-wave weakens thesuperconducting instability scale. In addition, it drivesthe anisotropy feature of the superconducting form fac-tor as consistently obtained for various weak couplingapproaches. Together, our analysis provides an expla-nation for the different behavior of superconductivity inthe iron pnictides and iron chalcogenides in the electronoverdoped region since J1 has opposite signs for thesetwo classes of materials, i. e. J1 is AFM in the ironpnictides58,59 and FM in the iron chalcogenides. Ourstudy suggests that determining the pairing symmetry ofthe 122 iron chalcogenides can provide important insightregarding whether the local AFM exchange couplings areresponsible for the high superconducting transition tem-peratures.

The paper is organized as follows. In Section II, wepresent the mean field analysis of the t− J1 − J2 modelto show the differences between the iron pnictide setupJ1 > 0 and the chalcogenide setup J1 < 0 in the electron-overdoped regime. This is followed by FRG studies inSection III where we mainly investigate the competitionbetween s-wave and d-wave in the effective model, andalso analyze the possible effect of an additional pocketat the Γ point of the unfolded BZ which we find to fur-ther increase the robustness of the s-wave pairing. The

qualitative trends confirm the results obtained in Sec. II.In Section IV we provide a combined view on the chalco-genides and point out that the ferromagnetic sign of J1is important to explain the robustness of s-wave pair-ing symmetry in these compounds. Furthermore, we setour work into context of other approaches to the prob-lem. We conclude in Section V that electron-overdopedchalcogenides exhibit a robust s-wave pairing phase whenthe NNN interactions are correctly taken into considera-tion.

II. MEAN FIELD ANALYSIS

We calculate the mean-field diagram of an effectivemodel for the AFe2Se2 compounds. As the main relevantorbital weight is given by the dxz, dyz, and dxy orbital, weemploy a three-orbital kinetic model with J1 and J2 in-teractions. For the case of strong electron doping we areinterested in, we do not find qualitative differences whenfour or five orbital models are used. For a more thoroughdiscussion of these aspects, refer to Section III. The spe-cific kinetic theory we use for the mean-field analysis isa modified three-band model52, given by

T (k) =

T11(k)− µ T12(k) T13(k)T21(k) T22(k)− µ T23(k)T31(k) T32(k) T33(k)− µ

, (1)

where

T11(k) = 2t2 cos(kx) + 2t1 cos(ky) + 4t3 cos(kx) cos(ky),

T22(k) = 2t1 cos(kx) + 2t2 cos(ky) + 4t3 cos(kx) cos(ky),

T33(k) = 2t5(cos(kx) + cos(ky)) + 4t6 cos(kx) cos(ky) + δ,

T12(k) = 4t4 sin(kx) sin(ky), T21(k) = T ⋆12(k),

T13(k) = 2it7 sin(kx) + 4it8 sin(kx) cos(ky),

T23(k) = 2it7 sin(ky) + 4it8 sin(ky) cos(kx). (2)

The other matrix elements are given by hermiticity.The parameters in the model are taken to be t =(0.02, 0.06, 0.03,−0.01, 0.35, 0.3,−0.2, 0.1), δ = 0.4, andµ = 0.412. (Throughout the article, energies are givenin units of eV unless stated otherwise). The parameterset chosen gives the Fermi surface shown in Fig. 1 witha filling factor of 4.41 electrons per site. Aside from anegligibly small electron pocket at the M point in theunfolded Brillouin zone, the main features are the largeelectron pockets at X which dictate the physics of themean-field phase diagram at this electron doping regime(see Fig.1). In the three band model, the small electronpocket appears around the M-point in the unfolded Bril-louin zone which may be related to the resonance featureexperimentally discussed for the Γ point in the foldedzone. In contrast, the 5-band fit to the chalcogenides weemploy in Section III suggests small electron pocket fea-tures around the Γ-point of the unfolded Brillouin zone.Despite this discrepancy, later we will see that both ap-pearances have a similar effect and can hence be discussed

3

B

A

FIG. 1. (color online) The Fermi surface used to represent thechalcogenides. Colors indicate the orbital components: Reddxz, Green dyz, and Blue=dxy. The A and B are auxiliarylabels used in Fig. 6.

on the same footing. The interaction part in our meanfield analysis is the pairing energy obtained by decou-pling the magnetic exchange couplings11, which can bewritten as

V = −∑

α,r

(J1b†α,r,r+xbα,r,r+x + J1b

†α,r,r+ybα,r,r+y (3)

+J2b†α,r,r+x+ybα,r,r+x+y + J2b

†α,r,r+x−ybα,r,r+x−y)

where bα,r,r′ = cα,r,↑cα,r′,↓ − cα,r,↓cα,r′,↑ represent sin-glet pairing operators between the r, r′ sites.Before we perform the self-consistent mean field calcu-

lation, we define the pairing order parameters as follows:in real space, the pairings on two NN bonds and twoNNN bonds are represented by ∆α

x = J1 < bα,r,r+x >,∆α

y = J1 < bα,r,r+y >, ∆αx+y = J2 < bα,r,r+x+y >

and ∆αx−y = J2 < bα,r,r+x+y >, where α denotes the

orbital index and x, y are the two unit lattice vectors.We only consider intra-orbital pairing and ignore inter-orbital pairing which is very small as shown in previouscalculations11. Considering the C4 symmetry of the lat-

tice, we can classify the pairing symmetries accordingto the one-dimensional irreducible representations of theC4 symmetry. Since the pairing is a spin singlet, we canclassify them as follows: an order parameter is of A-type(B-type) if it is even (odd) under a 90-degree rotation.This classification leads to six candidate pairings with A-symmetry and another six candidates with B-symmetryas the SC pairings include NN from J1 and NNN fromJ2 bonds, which manifests as the A-type symmetry

∆ANN,s = (∆xz

x +∆xzy +∆yz

x +∆yzy )/4,

∆ANN,d = (∆xz

x −∆xzy −∆yz

x +∆yzy )/4,

∆ANNN,s = (∆xz

x+y +∆xzx−y +∆yz

x+y +∆yzx−y)/4,

∆ANNN,d = (∆xz

x−y −∆xzx+y +∆yz

x+y −∆yzx−y)/4,

∆xyNN,s = (∆xy

x +∆xyy )/2,

∆xyNNN,s = (∆xy

x+y +∆xyx−y)/2. (4)

and the B-type symmetry

∆BNN,s = (∆xz

x +∆xzy −∆yz

x −∆yzy )/4,

∆BNN,d = (∆xz

x −∆xzy +∆yz

x −∆yzy )/4,

∆BNNN,s = (∆xz

x+y +∆xzx−y −∆yz

x+y −∆yzx−y)/4,

∆BNNN,d = (∆xz

x−y −∆xzx+y −∆yz

x+y +∆yzx−y)/4,

∆xyNN,d = (∆xy

x −∆xyy )/2,

∆xyNNN,d = (∆xy

x−y −∆xyx+y)/2. (5)

In reciprocal lattice space, the mean-field Hamiltonian isgiven by

H =∑

k

(

T (k) ∆(k)

∆†(k) −T ⋆(−k)

)

, (6)

where

∆(k) =

∆11(k) 0 00 ∆22(k) 00 0 ∆33(k)

, (7)

and

∆11(k) = (∆ANN,s +∆B

NN,s)(cos(kx) + cos(ky)) + (∆ANN,d +∆B

NN,d)(cos(kx)− cos(ky))

+2(∆ANNN,s +∆B

NNN,s) cos(kx) cos(ky) + 2(∆ANNN,d +∆B

NNN,d) sin(kx) sin(ky),

∆22(k) = (∆ANN,s −∆B

NN,s)(cos(kx) + cos(ky)) + (∆BNN,d −∆A

NN,d)(cos(kx)− cos(ky))

+2(∆ANNN,s −∆B

NNN,s) cos(kx) cos(ky) + 2(∆BNNN,d −∆A

NNN,d) sin(kx) sin(ky),

∆33(k) = ∆xyNN,s(cos(kx) + cos(ky)) + ∆xy

NN,d(cos(kx)− cos(ky)) + 2∆xyNNN,s cos(kx) cos(ky)

+2∆xyNNN,d sin(kx) sin(ky). (8)

We emphasize that in above definitions the symbols s andd merely represent the geometric factor of pairing in k-

space, and do not correspond to whether pairing is evenor odd under a 90-degree rotation in a multi-orbital sys-

4

0 0.375 0.75 1.125 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

J2

∆

A s−waves−wave on d

xy

B d−waved−wave on d

xy

FIG. 2. (color online) The mean field phase diagram alongJ1 = 0 in the parameter space (up); the quasiparticle spec-trum at J2 = 1 (middle) and J2 = 0.75 (bottom). At J2 = 1,one can see that the quasiparticle spectrum explicitly breaksC4 symmetry because of the mixing of A- and B-type pairingsymmetries.

tem. In general, there are more than one self-consistentset of {∆}’s as self-consistent meanfield solutions. Thefree energies in each solution hence have to be comparedto find the solution with the lowest free energy.

First consider pure NNN-pairings stemming from J2(this is a reasonable limit to start with since J1 inFeTe(Se) has been shown to be ferromagnetic, thus notcontributing to pairing in the singlet pairing channel).J2 is increased from zero to J2 = 1.5 while the bandwidth is W ∼ 4. The robust superconductivity solutionwith purely A-type s-wave pairing is obtained when J2is larger than 0.4. This is to say the pairing remainsthe same as in iron-pnictides with the geometric factorcos(kx) cos(ky)

11. The Bogoliubov particle spectrum iscompletely gapped in this state. When J2 becomes larger

0 0.375 0.75 1.125 1.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

J1

∆

A s−waves−wave on d

xy

d−wave on dxy

B d−wave

FIG. 3. (color online) The mean field phase diagram alongJ2 = 0 in the parameter space (up); the quasiparticle spec-trum at J1 = 0.75 (bottom). At J1 = 0.75 one can see thatthe quasiparticle spectrum explicitly breaks the C4 symmetrybecause of the mixing of A- and B-type pairing symmetries.

than 1, the ground state is a mixture of A- and B-typepairings. The nonzero B-type pairings all have the geo-metric factor sin(kx) sin(ky) (see the phase diagram showin Fig. 2). In the coexistence phase, the quasiparticlespectrum shows nearly gapless features at several points,and moreover, the dispersion explicitly breaks C4 rota-tion symmetry (see Fig. 2 displaying the quasiparticlespectrum of the lowest branch).Second, we study the phase diagram when only (anti-

ferromagnetic) J1 is present. In this case, only NN pair-ings are nonzero and there are six SC gaps. We increaseJ1 from J1 = 0 to J1 = 1.5 where the band width isW ∼ 4. The SC order becomes non-zero from J1 = 0.4on. However, in this case, the B-type SC order arisesslightly earlier than A-type SC order. The ground stateis always a mixture of A- and B-type pairings. The twoleading orders are A-type s-wave and B-type d-wave inxz, yz orbitals while the sub-leading ones are s- and d-waves in the xy orbitals (Fig. 3). Due to strong mixingof A- and B-type pairings, the quasiparticle spectrum isvery anisotropic. It is, however, still nodeless, in contrastto a pure s-wave pairing cos(kx) + cos(ky) where thereare nodes39 on the electron pockets (see Fig. 3).Finally, for J1 and J2 antiferromagnetic, we fix J1 +

J2 = 1 and change J1 − J2 as a parameter. We observethat NNN pairings dominate for J1 − J2 < −0.1 andNN pairings dominate for J1 − J2 > 0.2 (Fig. 4). In theintermediate range, there is only weak B-type pairing. A

5

−1 −0.5 0 0.5 1−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

J1−J

2

∆

A NN sA NN dxy NN sA NNN sA NNN dxy NNN sB NN sB NN dxy NN dB NNN sB NNN dxy NNN d

FIG. 4. (color online) The mean field phase diagram withJ1 + J2 = 1 in the parameter space.

A NNN s-wave

A NN s-wave +

B NN d-wave

B NN s-wave +

B NNN s-wave

J1

J2

A NN s-wave

FIG. 5. (color online) A schematic phase diagram for themodel (6) within 0 < J1, J2 < 1.

schematic phase diagram within the range 0 < J1, J2 < 1is shown in Fig. 5.

In the whole parameter region of (J1, J2), the SC or-der parameters always have the same sign for all threeorbitals. This can be seen in Fig. 6 where the orbitalresolved pairing amplitude is shown along electron pock-ets around X . This result is essentially consistent withthe FRG result shown in Sec. III (Fig.12). It is, how-ever, different from what one would expect from the verystrong coupling limit: There, the strong inter-orbital re-pulsion favors different signs of pairing for the dxy orbitaland the dxz/yz orbitals60. Some quantitative differencesbetween Fig. 6 and Fig. 12 may be explained as the in-completeness of a three-orbital model and the fact thatthe meanfield pairing is not constrained to the FS. InFig. 6 we also see that the orbital resolved pairing am-plitude is highly anisotropic: This is a natural reflectionof different orbital composition on different parts of the

FIG. 6. (color online) The orbital resolved pairing amplitudeon the FS for a typical s-wave (d-wave) pairing state in theupper (lower) panel, calculated within meanfield approxima-tion. The interaction parameters are J1 = 0, J2 = 0.8 forthe upper panel and J1 = 0.5, J2 = 0 for the lower panel.In the left half of these figures, the k-point traces the elec-tron pocket around X-point counterclockwise from point Ain Fig. 1 and in the right half, it traces the electron pocketaround the Y -point counterclockwise from point B in Fig. 1.

Fermi surface.

Following the Fermi surface topology in Fig. 1, thesemean-field results have been obtained in the case wherea small electron pocket was still present at the M -point.For completeness of the analysis, we also adapted theparameters such that the electron pocket around the Mpoint vanishes, leaving two pockets around X . WithouttheM pocket, we see that s-wave pairings are less favoredthan before, as its geometric factor is cos(kx) cos(ky)or cos(kx) + cos(ky), both being maximized around M .With the two-pocket FS, taking J1 = 0 and increasingJ2 > 0, B-type pairings blend in at smaller J2 than shownin Fig.2; taking J2 = 0 and increasing J1 > 0, A-typepairings appear at slightly larger J1 than shown in Fig. 3.The main features still remain unchanged. These trendsare in accordance with the FRG studies in the followingsection.

6

III. FRG ANALYSIS

To substantiate the mean field results above, we em-ploy functional renormalization group (FRG)53–55 to fur-ther investigate the pairing symmetry of the t-J1-J2model. As an unbiased resummation scheme of all chan-nels, the FRG has been extended and amply employed tothe multi-band case of iron pnictides. More details canbe found in Refs. 13, 17, 39, and 56. The conventionalstarting point for the FRG are bare Hubbard-type inter-actions which develop different Fermi surface instabilitiesas higher momenta are integrated out when the cutoff ofthe theory flows to the Fermi surface. To address the spe-cial situation found in the chalcogenides where the Fe-Secoupling is strong, not only local, but also further neigh-bor interaction terms would have to be taken into ac-count: in our FRG setup, the onsite Hubbard-type inter-actions of the same type as used in the study of pnictidestriggers no instability at reasonable critical scales. Thissuggests already at this stage that the chalcogenides maynecessitate a perspective beyond pure weak coupling. Inaddition, the total parameter space of bare interactions islarge and constrained RPA parameters are not yet avail-able for this class of materials. For the purpose of ourstudy, we hence constrain ourselves to the t-J1-J2 modelfrom the outset. This implies that the pairing interactionis already attractive on the bare level, and a developmentof an SC instability is expected as the physics is domi-nated by the pairing channel. Still, we can employ FRGto investigate the properties and competition of differentSC pairing symmetries for the chalcogenides for different(J1, J2) regimes.

Within FRG, we consider general J1-J2 interactionswhich are not limited to the spins in the same orbital:

H = J1∑

〈i,j〉

∑

a,b

(Sia · Sjb −1

4nianjb)

+ J2∑

〈〈i,j〉〉

∑

a,b

(Sia · Sjb −1

4nianjb).

The kinetic theory will differ in the various cases studiedbelow. For all cases, we will study the full 5-band modelincorporating all Fe d orbitals. Concerning the discretiza-tion of the BZ, the RG calculations were performed with8 patches per pocket, and a 10radius × 3angle mesh oneach patch. (This moderate resolution is convenient toscan wide ranges of the interaction parameter space;we checked that increasing the BZ resolution did notqualitatively change our findings.) The output of the RGcalculation is the four-point vertex on the Fermi surfaces:

VΛ(k1, n1;k2, n2;k3, n3;k4, n4)c†k4n4s

c†k3n3 sck2n2s

ck1n1s,

where the flow parameter is the IR cutoff Λ approachingthe Fermi surface, and with k1 to k4 the incomingand outgoing momenta. We only find singlet pair-ing to be relevant for the scenarios studied by us:∑

k,p VΛ(k,p)[O†kOp], where OSC

k = ck,↑c−k,↓. We

Γ M X Γ

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

ǫ−µ /

eV

−π −π/2 0 π/2 π−π

−π/2

0

π/2

π

1

23

4

5

6 7

8

9

10 11

12

13

1415

16

FIG. 7. (color online) The spectrum and the Fermi surfaces ofthe band structure proposed by Maier et al.44, colored accord-ing to the dominant orbital content. The color code is red dxz,green dyz, blue dxy, orange dx2

−y2 and magenta d3z2−r2 . Thenumbered crosses show the center of Fermi surface patchesused in the FRG calculations.

decompose the pairing channel into eigenmodes,

V SCΛ (k,−k,p) =

∑

i

cSCi (Λ)fSC,i(k)∗fSC,i(p), (9)

and obtain the band-resolved form factors of the leadingand subleading SC eigenmode (i.e. largest two negativeeigenvalues). This way we are able to discuss the inter-play of d-wave and s-wave as well as the degree of formfactor anisotropy for a given setting of (J1, J2). Com-paring divergence scales Λc gives us the possibility toinvestigate the relative change of Tc as a function of(J1, J2). Furthermore, we also investigate the orbital-resolved pairing modes39 by decomposing the orbital fourpoint vertex

V orbc,d→a,b =

5∑

n1,...,n4=1

{

VΛ(k1, n1;k2, n2;k3, n3;k4, n4)

×u∗an1

(k1)u∗bn2

(k2)ucn3(k3)udn4

(k4)}

, (10)

where the u’s denote the different orbital components ofthe band vectors. By investigating the intraorbital SCpairing modes in (10), we make contact to the findingsfrom the previous mean field analysis.

A. Two-pocket scenario

We start by studying the 5-band model suggested be-fore by Maier et al.44. There are only two electron pock-ets at the X point of the unfolded Brillouin zone closelyresembling the Fermi surface topology and orbital con-tent employed for our mean-field analysis (Fig. 7).The RG flow and the form factors of the leading di-

verging channels are shown in Fig. 8 for dominant J2and in Fig. 9 for dominant J1. As stated before, the pair-ing interaction is already present at the bare level in themodel so that we achieve comparably fast instabilities asthe cutoff is flowing towards the Fermi surface. As foundin Ref. 11, the dominant J2 scenario exhibits a leading s-wave cos kx cos ky form factor which causes the same sign

7

10-2 10-1 100

Cutoff

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

Inte

ract

ion s

trength

1 5 9 13

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Form

fact

ors

FIG. 8. (color online) Typical RG flows and the supercon-ducting gaps associated with the Fermi surfaces for the two-pocket scenario with (J1, J2) = (0.1, 0.5) eV. Leading formfactor is denoted in blue, sub-leading form factor in green.

on both electron pockets (blue dots in Fig. 8). The sub-leading form factor is found to be of d-wave cos kx−coskytype, changing sign from one electron pocket to the other.The inverse situation is found for dominant J1. As shownin Fig. 9, the d-wave cos kx−cosky form factor establishesthe leading instability. As before, the form factor doesnot cross zero related to nodeless SC for this parametersetting.

With only pairing information available on the limitednumber of sampling points along FS, it is impossible toobtain as in the mean-field analysis the superconductinggap in the whole BZ. For illustration, a mixture of asmall A-type NNN d-wave pairing and large A-type NNNs-wave pairing is indistinguishable from a pure A-typeNNN s-wave pairing; a mixed state of a small B-type NNs-wave pairing plus a large B-type NN d-wave pairing,and a state with pure B-type NN d-wave pairing showlittle difference if one compares the gap on a few pointsalong the Fermi surfaces. For this reason, the symbolsx2y2 used in this section refers to a pairing consisting ofa large A-type NNN s-wave pairing and possible smallcomponents of A-type NNN d-wave pairing or A-typeNN s/d-wave pairing. In turn, the symbol dx2−y2 refersto a pairing made up with a large B-type NN d-wavepairing and possible small components of B-type NN s-wave pairing or B-type NNN s/d-wave pairing.

We have scanned a large range of (J1, J2). For eachsetup we have obtained Λc as well as the ratio of the in-stability eigenvalues between s-wave and d-wave in thepairing channel (encoded by the two-color circles shownin Fig. 10). The FRG result is qualitatively consistentwith the mean field analysis. In the antiferromagneticsector, the s-wave wins for dominant J2 while the d-wavewins wins for dominant J1. For ferromagnetic J1 cor-responding to the situation in chalcogenides, we find arobust preference of s-wave pairing. The anisotropy ofthe s-wave gap around the pockets in the FRG calcu-lation is also qualitatively consistent with the meanfield

10-1 100

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0

Inte

ract

ion s

trength

1 5 9 13

-0.3-0.2-0.10.00.10.20.3

Form

fact

ors

FIG. 9. (color online) Typical RG flows and the supercon-ducting gaps associated with the Fermi surfaces for the two-pocket scenario with (J1, J2) = (0.5, 0.1) eV. Leading formfactor is denoted in blue, sub-leading form factor in green.

result. The gap on the Fermi surfaces with dxy orbitalcharacter is smaller than the gap on those with dxz,yzorbital character.The predictions from the mean field analysis are fur-

ther confirmed for the mixed phase regime where s-waveand d-wave coexist in the mean field solution. In FRG,one of these instabilities will always be slightly preferred;still, when both instabilities diverge in very close prox-imity to each other, this regime behaves similarly to thecoexistence phase. For illustration, in Fig. 11 we haveplotted the dependence of Λc on J1 − J2, with J1 + J2fixed to 0.7 eV; there is a clear reduction of the criticalscale (and thus the transition temperature) when thereis a strong competition between s- and d-wave channels.Following (10) we also analyze the orbital decomposi-

tion of the SC pairing from FRG (Fig. 12). We constrainourselves to the most relevant three orbitals dxy, dxz, anddyz. In particular, we observe that the SC orbital pair-ing induces the same sign for all three dominant orbitalmodes, in correspondence with the mean field analysispresented before.

B. Three-pocket scenario

Recent ARPES data57 on the chalcogenides may sug-gest the existence of a shallow flat pocket around the Γpoint (the location, and especially the kz position of sucha pocket are still under debate). By tuning parameters,we have obtained in the previous section a three-orbitalmodel that has an additional electron-pocket around M -point in the unfolded BZ.In our FRG approach we can take a more profound mi-

croscopic perspective on this issue. From the true bandstructure calculations at hand for the chalcogenides, weconsider it unlikely that it will be a hole band regular-ized up towards the Fermi surface. Instead, we investi-gate the effect of a possible electron band at the Γ point

8

-1.0 -0.5 0.0 0.5 1.0

J1 / eV

0.0

0.2

0.4

0.6

0.8

1.0

J2 /

eV

FIG. 10. (color online) The phase diagram of the two-pocket model. Each pie shows the relative strengths of thetwo leading pairing channels, with the radius proportional to[8+ log10(Λc/eV)]2. The color code for pairing symmetries isgreen sx2y2 and red dx2

−y2 .

-1.5 -1.0 -0.5 0.0 0.5 1.0

J1−J2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Λc

FIG. 11. (color online) The variation of the critical scaleΛc along a line through parameter space which interpolatesbetween s and d wave. A minimum is visible for comparableordering tendency in s-wave and d-wave. See the caption ofFig. 10 for more details on the pie charts.

in the unfolded Brillouin zone. This is suggested fromthe folded 10-band calculations, where one electron-typeband closely approaches the Fermi level around the Γpoint44. This band should be very flat and shallow. Fromthe weak coupling perspective of particle-hole pairs cre-ated around the Fermi surfaces, this will probably have asmall effect: particle-hole pairs will only be created up toenergy scales of the depth of the electron band at the Xpoint below the Fermi surface, providing some hole-typephase space for the electron band at Γ. In a (J1, J2) pic-ture, however, this may still significantly promote scat-tering along Γ ↔ X , which may further stabilize the s-wave phase regime. We have hence developed a modifiedband structure designed for this scenario. There, we havebent down the band dominated by dxy in the two-pocketmodel band structure44 without changing its band vectorand created an electron pocket around Γ, accordingly ofmainly dxy orbital content (Fig. 15). The band bending

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Points on Fermi pockets

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Pair

ing f

orm

fact

ors

(π,0) (0,π)

(J1 ,J2 ) =(0.1, 0.5) eV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Points on Fermi pockets

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Pair

ing f

orm

fact

ors

(π,0) (0,π)

(J1 ,J2 ) =(0.5, 0.1) eV

FIG. 12. (color online) The orbital-resolved pairing form fac-tors of two typical RG flows. The upper resides in the dom-inant s-wave and the lower in the d-wave regime. The colorcode is the same as in Fig. 7, i. e. red dxz, green dyz, bluedxy, orange dx2

−y2 and magenta d3z2−r2 .

was achieved by

H → H +∑

k,a,b,s

ξ(k)c†kasua(k)u

∗b (k)ckbs,

where u(k) is the eigenvector of the band dominated bydxy. The shift of energy ξ(k) was intentionally chosensuch that the Γ pocket exhibits some nesting with the Xelectron pockets.The phase diagram is shown in Fig. 16. FRG results

for typical scenario for the s-wave and d-wave regime areshown in Fig. 13 and Fig. 14, respectively. As suspected,the additional pocket strengthens the tendency to forman s-wave in the system, aside from exhibiting an addi-tional constant s-wave instability in a small regime fordominant J1.

IV. DISCUSSION

The above calculations demonstrate that the s-wavepairing symmetry is always robust if the AFM NNN J2 isstrong while a d-wave pairing can be strong if J1 is AFMfor the electron overdoped region. Moreover, if both of

9

100

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ion s

trength

1 5 9 13 17 21

-0.3

-0.2

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0.0

0.1

0.2

0.3

Form

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FIG. 13. (color online) Typical RG flows and the super-conducting gaps associated with Fermi surface for the three-pocket scenario with (J1, J2) = (0.2, 0.8) eV. Leading formfactor is denoted in blue, sub-leading form factor in green.

100

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ion s

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1 5 9 13 17 21

-0.3

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FIG. 14. (color online) Typical RG flows and the super-conducting gaps associated with Fermi surface for the three-pocket scenario with (J1, J2) = (0.9, 0.3) eV. Leading formfactor is denoted in blue, sub-leading form factor in green.

them are AFM, there is strong competition between thes-wave and d-wave pairing. When there are hole pock-ets, as shown before11, even in a range of J1 ∼ J2, thecontribution to pairing from J1 is much weaker than theone from J2. In that case, an AFM J1 will not gener-ate strong d-wave pairing so that the s-wave wins easily.From neutron scattering experiments, it has been shownthat a major difference between iron-pnictides and iron-chalcogenides is that the NN coupling J1 change fromAFM in the former58,59 to FM in the latter47. In fact,J1 is rather strongly FM in the latter, which explainsthe high magnetic transition temperature (500 K) in the245 vacancy ordering state as shown in Ref. 51. Combin-ing these results, we can partially answer the questionregarding the different behaviors between iron-pnictidesand iron-chacogenides in the electron-overdoped region:why can the high SC transition temperature be achieved

Γ M X Γ

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

ǫ−µ /

eV

−π −π/2 0 π/2 π−π

−π/2

0

π/2

π

1

2 34

5

67

8

9

1011

12

1314 15

16

17

18 19

20

21

2223

24

FIG. 15. (color online) The band structure and the Fermisurfaces of the modified band structure, colored accordingto the dominant orbital content. The color code is red dxz,green dyz, blue dxy, orange dx2

−y2 and magenta d3z2−r2 . Thenumbered crosses show the center of Fermi surface patchesused in the FRG calculations.

-1.0 -0.5 0.0 0.5 1.0

J1 / eV

0.0

0.2

0.4

0.6

0.8

1.0

J2 /

eV

FIG. 16. (color online) The phase diagram of the three-pocketmodel. Here we are able to resolve the s-wave channel intoconstant s-wave (grey), extended sx2y2 -wave (green), and thenodal sx2+y2 -wave (purple). Parameter sets with J1 ∼ −1and J2 ∼ 0 have highly oscillating form factors which are dueto artifacts in the calculation; the triplet channel would haveto be considered in these cases.

in the latter, but not in the former? Since J1 in iron-pnictides is AFM while it is FM in iron-chacogenides, J1will weaken the SC pairing in the former but not in thelatter.

A few remarks regarding this work follow: (i) Ourmean-field result is qualitatively consistent with the re-sults from a similar model with five orbitals61–63. Thecritical difference is regarding J1 being FM, and has notbeen addressed previously; (ii) s-wave pairing symmetryhas also been obtained in Refs. 43, 64, and 65. How-ever, the s-wave pairing only shows up either in a narrowregion or with drastically different parameter settings.Therefore, the s-wave is not robust from a microscopicpoint of view. Instead, the d-wave is a robust resultin these studies. Still, even the d-wave pairing strengthbased on the scattering between two electron pockets isgenerally weak, as specifically discussed in61, which isanother difficulty for this type of mechanism. (iii) Ourresults suggest that there is no difference between iron-pnictides and iron-chalcogenides in terms of pairing sym-metry. Both of them are dominated by s-wave pairing.

10

If both hole and electron pockets are present, the signsof the SC order in hole and electron pockets are oppo-site, namely s±. However, the mechanism causing s±

is different in the weak and strong coupling approach.In the weak coupling approach, the sign change is dueto the scattering between the hole and electron pocketswhile in the strong coupling approach, the sign change isdue to the form factor of the SC order parameters whichis specified to be cos kx cos ky since the pairing mainlyoriginates from the AFM J2. Therefore, to obtain s±

pairing symmetry, the existence of both hole and elec-tron pockets is necessary in the weak coupling approach,but not in the strong coupling one. (iv) The reason thatthe superconductivity vanishes in the iron pnictides inthe electron overdoped region is not solely due to thecompetition between s-wave and d-wave pairing symme-try. It is also due to the weakening of local magneticexchange coupling themselves and the reduction of bandwidth renormalization. (v) The prospective experimentalconfirmation of s-wave pairing symmetry in KFe2Se2 willsupport that superconductivity in iron-based supercon-ductors might be explained by local AFM exchange cou-plings. (vi) Neutron scattering also suggests that thereis significant AFM exchange coupling between two thirdnearest neighbor sites, i.e. J3

47,49. The existence of J3will further enhance the s-wave pairing since it gener-ates pairing form factors as cos 2kx + cos 2ky in recipro-cal space which in turn can enhance the pairing at the

electron pockets.

V. CONCLUSION

In summary, we have shown that the pairing symmetryin electron-overdoped iron-chalcogenides is a robust s-wave. The fact that the NN magnetic exchange couplingis FM, which diminishes the possibility of d-wave pairingsymmetry in these materials. From a unified perspectiveof high-Tc cuprates and high-Tc chalcogenides, the NNAFM exchange coupling gives rise to the robust d-wavepairing in the cuprates while the NNN AFM exchangecoupling gives rise to the robust s-wave pairing in iron-based chalcogenide superconductors.

ACKNOWLEDGMENTS

We thank S. Borisenko, J. van den Brink, XianhuiChen, A. Chubukov, Hong Ding, Donglei Feng, S. Graser,W. Hanke, P. Hirschfeld, S. Kivelson, C. Platt, D.Scalapino, Qimiao Si, Xiang Tao, Fa Wang, and HaihuWen for useful discussions. JPH thanks the Institute ofPhysics, CAS for research support. RT is supported byDFG SPP 1458/1 and a Feodor Lynen Fellowship of theHumboldt Foundation and NSF DMR-095242. BAB wassupported by Princeton Startup Funds, Sloan Founda-tion, NSF DMR-095242, and NSF China 11050110420,and MRSEC grant at Princeton University, NSF DMR-0819860.

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