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arXiv:1105.1135v1 [cond-mat.supr-con] 5 May 2011

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

Chen Fang1, Yang-Le Wu2, Ronny Thomale2, B. Andrei Bernevig2, and Jiangping Hu1,3

1Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA

2Department of Physics, Princeton University, Princeton, NJ 08544 and

3Beijing 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 between

next-nearest-neighbor (NNN) sites while the nearest neighbor (NN) magnetic exchange coupling is

ferromagnetic (FM). This is different from the iron pnictides, where the NN magnetic exchange

coupling 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 from

the d-wave pairing result obtained by other theories where non-local bare interaction terms and

the NNN J2 term are underestimated. Detecting the pairing symmetry in (K, Cs)Fe2−xSe2 may

hence provide important insights regarding the mechanism of superconducting pairing in iron based

superconductors.

I.INTRODUCTION

The recent discovery of a new family of iron-based su-

perconductors A(K,Cs,Rb)yFe2−xSe21–3has initiated a

new round of research in this field.

new family shows distinctly different properties from

other 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.

angle-resolved photoemission spectroscopy (ARPES)5–7

and LDA calculations8–10show the presence of only elec-

tron Fermi pockets located at the M point of the folded

Brillouin zone (BZ). (Some signature of possible density

of states at the Γ point is still under current debate; in

any case this pocket, if present, is assumed to be very flat

and shallow). ARPES experiments have also reported

large isotropic superconducting gaps at these pockets5–7.

The absence of hole pockets around the Γ point of the

BZ provides a new arena of Fermi surface topology to in-

vestigate the pairing symmetries and mechanisms of su-

perconductivity proposed for iron-based superconductors

from a variety of different approaches11–35.

So far, the majority of theories for the pairing sym-

metry of iron-based superconductors are based on weak

coupling approaches13–18,27–32,36,37. Although there are

discrepancies, the theories based on these approaches

have reached a broad consensus regarding the pairing

symmetries in iron-based superconductors: for optimally

hole doped iron-pnictides, for example, Ba0.6K0.4Fe2As2,

an extended s-wave pairing symmetry, called s±, is

favored14(the sign of the order parameter changes be-

tween hole and electron pockets as potentially detectable

through neutron scattering38), as a result of repulsive in-

terband interactions and nesting between the hole and

electron pockets. For extremely hole-doped materials,

such as KFe2As2, the absence of electron pockets can

lead to a d-wave pairing symmetry18with a low transi-

Remarkably, this

Importantly, both

tion temperature; for electron doped materials such as

Ba2Fe2−xCoxAs2, the anisotropy of the superconducting

gap around the electron pockets in the s±state grows for

larger 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 dxy

orbital weight at the electron pocket Fermi surfaces39,40.

Finally, in the limit of the heavily electron doped case

when the hole pockets vanish and only the electron pock-

ets are left, the d-wave pairing symmetry may be favored

again18,41,42. The iron chalcogenide AyFe2−xSe2belongs

to the latter category and many theories based on weak

coupling approaches have suggested that the pairing sym-

metry should be d-wave as possibly detectable through

characteristic impurity scattering43–45.

A complementary approach based on strong coupling

likewise predicts an s-wave pairing symmetry in the iron

pnictides. Two of us showed that the pairing symme-

try is determined mainly by the next-nearest-neighbor

(NNN) AFM exchange coupling J2together with a renor-

malized narrow band width11,46. The superconducting

gap is close to a coskxcosky form in momentum space

(higher harmonic contributions are neglected in this ap-

proach). This result is model independent as long as

the dominating interaction is J2and the Fermi surfaces

are located close to the Γ and M points in the folded

BZ. The coskxcosky form factor changes sign between

the electron and hole pockets in the BZ. It resembles

the order parameters of s±proposed from weak-coupling

arguments14.

The J2 coupling will be of particular importance in

the following. We point out two key points on J2-related

physics 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 exchange

coupling originates mostly from superexchange processes

through As (P) or Se (Te).

{˜t} − J1− J2model studied before11, the superconduct-

Second, in the effective

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2

ing state is obtained only when the magnetic exchange

coupling strength is of the same order as the hopping

parameters (or the bandwidth) of the model. Therefore,

as t > J, it requires the effective bands given by {˜t} be

renormalized. However, the absence of double occupan-

cies in the standard t − J model is not strictly imposed

in such an intermediately coupled effective model where

the bandwidth is assumed to be of similar order as the

interaction scale.

Comparing the predictions from weak coupling and

strong coupling, the 122 iron chalcogenides provide an

interesting opportunity to address the difference between

the two perspectives. In this paper, we predict that

the s-wave pairing symmetry is robust even in extremely

electron-overdoped iron chalcogenides because the AFM

J2is the main factor for pairing and the J1is ferromag-

netic (FM), a conclusion drawn from both neutron scat-

tering experiments47–49and the magnetic structure asso-

ciated with 245 vacancy ordering50,51. As we will show,

the FM J1 significantly reduces the competitiveness of

d-wave pairing symmetry. We substantiate this claim by

two different methods. First, we solve the three orbital

{˜t}−J1−J2model on the mean field level to show that

the s-wave pairing is the leading instability regardless of

the change of doping given that J2is large. We calculate

a full phase diagram as J1 varies from FM to AFM. If

J1 is AFM, we obtain a SC state with a mixed s-wave

and d-wave pairing. Second, we use the functional renor-

malization group (FRG) to analyze this trend obtained

by mean field analysis for a 5-band model of the chalco-

genides. We confirm that a dominant AFM J2generally

leads to robust s-wave pairing while an AFM J1 tends

to favor d-wave pairing in the electron overdoped region.

The competition between s-wave and d-wave weakens the

superconducting instability scale. In addition, it drives

the anisotropy feature of the superconducting form fac-

tor as consistently obtained for various weak coupling

approaches. Together, our analysis provides an expla-

nation for the different behavior of superconductivity in

the iron pnictides and iron chalcogenides in the electron

overdoped region since J1 has opposite signs for these

two classes of materials, i. e. J1 is AFM in the iron

pnictides58,59and FM in the iron chalcogenides. Our

study suggests that determining the pairing symmetry of

the 122 iron chalcogenides can provide important insight

regarding whether the local AFM exchange couplings are

responsible for the high superconducting transition tem-

peratures.

The paper is organized as follows. In Section II, we

present the mean field analysis of the˜t − J1− J2model

to show the differences between the iron pnictide setup

J1> 0 and the chalcogenide setup J1< 0 in the electron-

overdoped regime. This is followed by FRG studies in

Section III where we mainly investigate the competition

between s-wave and d-wave in the effective model, and

also analyze the possible effect of an additional pocket

at 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 J1

is important to explain the robustness of s-wave pair-

ing symmetry in these compounds. Furthermore, we set

our work into context of other approaches to the prob-

lem. We conclude in Section V that electron-overdoped

chalcogenides exhibit a robust s-wave pairing phase when

the NNN interactions are correctly taken into considera-

tion.

II. MEAN FIELD ANALYSIS

We calculate the mean-field diagram of an effective

model for the AFe2Se2compounds. As the main relevant

orbital weight is given by the dxz, dyz, and dxyorbital, we

employ a three-orbital kinetic model with J1and J2in-

teractions. For the case of strong electron doping we are

interested in, we do not find qualitative differences when

four or five orbital models are used. For a more thorough

discussion of these aspects, refer to Section III. The spe-

cific kinetic theory we use for the mean-field analysis is

a modified three-band model52, given by

ˆT(k) =

T11(k) − µ

T21(k)

T31(k)

T12(k)

T22(k) − µ

T32(k)

T13(k)

T23(k)

T33(k) − µ

, (1)

where

T11(k) = 2t2cos(kx) + 2t1cos(ky) + 4t3cos(kx)cos(ky),

T22(k) = 2t1cos(kx) + 2t2cos(ky) + 4t3cos(kx)cos(ky),

T33(k) = 2t5(cos(kx) + cos(ky)) + 4t6cos(kx)cos(ky) + δ,

T12(k) = 4t4sin(kx)sin(ky),T21(k) = T⋆

T13(k) = 2it7sin(kx) + 4it8sin(kx)cos(ky),

T23(k) = 2it7sin(ky) + 4it8sin(ky)cos(kx).

12(k),

(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, and

µ = 0.412. (Throughout the article, energies are given

in units of eV unless stated otherwise). The parameter

set chosen gives the Fermi surface shown in Fig. 1 with

a filling factor of 4.41 electrons per site. Aside from a

negligibly small electron pocket at the M point in the

unfolded Brillouin zone, the main features are the large

electron pockets at X which dictate the physics of the

mean-field phase diagram at this electron doping regime

(see Fig.1). In the three band model, the small electron

pocket appears around the M-point in the unfolded Bril-

louin zone which may be related to the resonance feature

experimentally discussed for the Γ point in the folded

zone. In contrast, the 5-band fit to the chalcogenides we

employ 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

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3

B

A

FIG. 1. (color online) The Fermi surface used to represent the

chalcogenides. Colors indicate the orbital components: Red

dxz, Green dyz, and Blue=dxy. The A and B are auxiliary

labels used in Fig. 6.

on the same footing. The interaction part in our mean

field analysis is the pairing energy obtained by decou-

pling the magnetic exchange couplings11, which can be

written as

ˆV = −

?

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 two

NNN bonds are represented by ∆α

∆α

and ∆α

x−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 previous

calculations11. Considering the C4symmetry of the lat-

x= J1 < bα,r,r+x >,

x+y= J2 < bα,r,r+x+y >

y= J1 < bα,r,r+y >, ∆α

tice, we can classify the pairing symmetries according

to the one-dimensional irreducible representations of the

C4symmetry. Since the pairing is a spin singlet, we can

classify 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-symmetry

as the SC pairings include NN from J1 and NNN from

J2bonds, which manifests as the A-type symmetry

∆A

∆A

∆A

∆A

∆xy

∆xy

NN,s= (∆xz

NN,d= (∆xz

NNN,s= (∆xz

NNN,d= (∆xz

NN,s= (∆xy

NNN,s= (∆xy

x+ ∆xz

x− ∆xz

x+y+ ∆xz

x−y− ∆xz

x+ ∆xy

x+y+ ∆xy

y+ ∆yz

y− ∆yz

x−y+ ∆yz

x+y+ ∆yz

x+ ∆yz

x+ ∆yz

x+y+ ∆yz

x+y− ∆yz

y)/4,

y)/4,

x−y)/4,

x−y)/4,

y)/2,

x−y)/2.(4)

and the B-type symmetry

∆B

∆B

∆B

∆B

∆xy

∆xy

NN,s= (∆xz

NN,d= (∆xz

NNN,s= (∆xz

NNN,d= (∆xz

NN,d= (∆xy

NNN,d= (∆xy

x+ ∆xz

x− ∆xz

x+y+ ∆xz

x−y− ∆xz

x− ∆xy

x−y− ∆xy

y− ∆yz

y+ ∆yz

x−y− ∆yz

x+y− ∆yz

x− ∆yz

x− ∆yz

x+y− ∆yz

x+y+ ∆yz

y)/4,

y)/4,

x−y)/4,

x−y)/4,

y)/2,

x+y)/2. (5)

In reciprocal lattice space, the mean-field Hamiltonian is

given by

ˆH =

?

k

?

ˆT(k)

ˆ∆†(k) −ˆT⋆(−k)

ˆ∆(k)

?

,(6)

where

ˆ∆(k) =

∆11(k)

0

0

00

0∆22(k)

0∆33(k)

,(7)

and

∆11(k) = (∆A

NN,s+ ∆B

+2(∆A

NN,s)(cos(kx) + cos(ky)) + (∆A

NNN,s+ ∆B

NN,s− ∆B

+2(∆A

∆33(k) = ∆xy

+2∆xy

NNN,dsin(kx)sin(ky).

NN,d+ ∆B

NNN,d+ ∆B

NN,d− ∆A

NNN,d− ∆A

NN,d)(cos(kx) − cos(ky))

NNN,s)cos(kx)cos(ky) + 2(∆A

NN,s)(cos(kx) + cos(ky)) + (∆B

NNN,s− ∆B

NN,s(cos(kx) + cos(ky)) + ∆xy

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

∆22(k) = (∆A

NN,d)(cos(kx) − cos(ky))

NNN,s)cos(kx)cos(ky) + 2(∆B

NN,d(cos(kx) − cos(ky)) + 2∆xy

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

NNN,scos(kx)cos(ky)

(8)

We emphasize that in above definitions the symbols s and

d merely represent the geometric factor of pairing in k-

space, and do not correspond to whether pairing is even

or odd under a 90-degree rotation in a multi-orbital sys-

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0 0.3750.75

J2

1.1251.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

∆

A s−wave

s−wave on dxy

B d−wave

d−wave on dxy

FIG. 2. (color online) The mean field phase diagram along

J1 = 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 breaks

C4 symmetry because of the mixing of A- and B-type pairing

symmetries.

tem. In general, there are more than one self-consistent

set of {∆}’s as self-consistent meanfield solutions. The

free energies in each solution hence have to be compared

to 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 in

FeTe(Se) has been shown to be ferromagnetic, thus not

contributing to pairing in the singlet pairing channel).

J2 is increased from zero to J2 = 1.5 while the band

width is W ∼ 4. The robust superconductivity solution

with purely A-type s-wave pairing is obtained when J2

is larger than 0.4. This is to say the pairing remains

the same as in iron-pnictides with the geometric factor

cos(kx)cos(ky)11. The Bogoliubov particle spectrum is

completely gapped in this state. When J2becomes larger

0 0.375 0.75

J1

1.125 1.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

∆

A s−wave

s−wave on dxy

d−wave on dxy

B d−wave

FIG. 3. (color online) The mean field phase diagram along

J2 = 0 in the parameter space (up); the quasiparticle spec-

trum at J1 = 0.75 (bottom). At J1 = 0.75 one can see that

the quasiparticle spectrum explicitly breaks the C4 symmetry

because of the mixing of A- and B-type pairing symmetries.

than 1, the ground state is a mixture of A- and B-type

pairings. The nonzero B-type pairings all have the geo-

metric factor sin(kx)sin(ky) (see the phase diagram show

in Fig. 2). In the coexistence phase, the quasiparticle

spectrum shows nearly gapless features at several points,

and moreover, the dispersion explicitly breaks C4 rota-

tion symmetry (see Fig. 2 displaying the quasiparticle

spectrum of the lowest branch).

Second, we study the phase diagram when only (anti-

ferromagnetic) J1is present. In this case, only NN pair-

ings are nonzero and there are six SC gaps. We increase

J1 from J1 = 0 to J1 = 1.5 where the band width is

W ∼ 4. The SC order becomes non-zero from J1= 0.4

on. However, in this case, the B-type SC order arises

slightly earlier than A-type SC order. The ground state

is always a mixture of A- and B-type pairings. The two

leading orders are A-type s-wave and B-type d-wave in

xz,yz orbitals while the sub-leading ones are s- and d-

waves in the xy orbitals (Fig. 3). Due to strong mixing

of A- and B-type pairings, the quasiparticle spectrum is

very anisotropic. It is, however, still nodeless, in contrast

to a pure s-wave pairing cos(kx) + cos(ky) where there

are nodes39on the electron pockets (see Fig. 3).

Finally, for J1 and J2 antiferromagnetic, we fix J1+

J2= 1 and change J1− J2as a parameter. We observe

that NNN pairings dominate for J1− J2 < −0.1 and

NN pairings dominate for J1− J2> 0.2 (Fig. 4). In the

intermediate range, there is only weak B-type pairing. A

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−1 −0.50 0.51

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

J1−J2

∆

A NN s

A NN d

xy NN s

A NNN s

A NNN d

xy NNN s

B NN s

B NN d

xy NN d

B NNN s

B NNN d

xy NNN d

FIG. 4. (color online) The mean field phase diagram with

J1+ 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 the

model (6) within 0 < J1,J2 < 1.

schematic phase diagram within the range 0 < J1,J2< 1

is shown in Fig. 5.

In the whole parameter region of (J1,J2), the SC or-

der parameters always have the same sign for all three

orbitals. This can be seen in Fig. 6 where the orbital

resolved pairing amplitude is shown along electron pock-

ets around X. This result is essentially consistent with

the FRG result shown in Sec. III (Fig.12). It is, how-

ever, different from what one would expect from the very

strong coupling limit: There, the strong inter-orbital re-

pulsion favors different signs of pairing for the dxyorbital

and the dxz/yzorbitals60. Some quantitative differences

between Fig. 6 and Fig. 12 may be explained as the in-

completeness of a three-orbital model and the fact that

the meanfield pairing is not constrained to the FS. In

Fig. 6 we also see that the orbital resolved pairing am-

plitude is highly anisotropic: This is a natural reflection

of different orbital composition on different parts of the

FIG. 6. (color online) The orbital resolved pairing amplitude

on the FS for a typical s-wave (d-wave) pairing state in the

upper (lower) panel, calculated within meanfield approxima-

tion. The interaction parameters are J1 = 0, J2 = 0.8 for

the 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 A

in Fig. 1 and in the right half, it traces the electron pocket

around the Y -point counterclockwise from point B in Fig. 1.

Fermi surface.

Following the Fermi surface topology in Fig. 1, these

mean-field results have been obtained in the case where

a small electron pocket was still present at the M-point.

For completeness of the analysis, we also adapted the

parameters such that the electron pocket around the M

point vanishes, leaving two pockets around X. Without

the M pocket, we see that s-wave pairings are less favored

than 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 increasing

J2> 0, B-type pairings blend in at smaller J2than shown

in Fig.2; taking J2 = 0 and increasing J1 > 0, A-type

pairings appear at slightly larger J1than shown in Fig. 3.

The main features still remain unchanged. These trends

are in accordance with the FRG studies in the following

section.

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6

III. FRG ANALYSIS

To substantiate the mean field results above, we em-

ploy functional renormalization group (FRG)53–55to fur-

ther investigate the pairing symmetry of the ˜t-J1-J2

model. As an unbiased resummation scheme of all chan-

nels, the FRG has been extended and amply employed to

the multi-band case of iron pnictides. More details can

be found in Refs. 13, 17, 39, and 56. The conventional

starting point for the FRG are bare Hubbard-type inter-

actions which develop different Fermi surface instabilities

as higher momenta are integrated out when the cutoff of

the theory flows to the Fermi surface. To address the spe-

cial situation found in the chalcogenides where the Fe-Se

coupling 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 pnictides

triggers no instability at reasonable critical scales. This

suggests already at this stage that the chalcogenides may

necessitate a perspective beyond pure weak coupling. In

addition, the total parameter space of bare interactions is

large and constrained RPA parameters are not yet avail-

able for this class of materials. For the purpose of our

study, we hence constrain ourselves to the˜t-J1-J2model

from the outset. This implies that the pairing interaction

is already attractive on the bare level, and a development

of an SC instability is expected as the physics is domi-

nated by the pairing channel. Still, we can employ FRG

to investigate the properties and competition of different

SC pairing symmetries for the chalcogenides for different

(J1,J2) regimes.

Within FRG, we consider general J1-J2 interactions

which 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 studied

below. For all cases, we will study the full 5-band model

incorporating all Fe d orbitals. Concerning the discretiza-

tion of the BZ, the RG calculations were performed with

8 patches per pocket, and a 10radius× 3angle mesh on

each patch. (This moderate resolution is convenient to

scan wide ranges of the interaction parameter space;

we checked that increasing the BZ resolution did not

qualitatively change our findings.) The output of the RG

calculation is the four-point vertex on the Fermi surfaces:

VΛ(k1,n1;k2,n2;k3,n3;k4,n4)c†

where the flow parameter is the IR cutoff Λ approaching

the Fermi surface, and with k1 to k4 the incoming

and outgoing momenta.We only find singlet pair-

ing to be relevant for the scenarios studied by us:

?

k4n4sc†

k3n3¯ sck2n2sck1n1¯ s,

k,pVΛ(k,p)[ˆO†

kˆOp], where

ˆOSC

k

= ck,↑c−k,↓.We

Γ

MX

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

3

4

5

6

7

8

9

10

11

12

13

14

15

16

FIG. 7. (color online) The spectrum and the Fermi surfaces of

the 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. The

numbered crosses show the center of Fermi surface patches

used in the FRG calculations.

decompose the pairing channel into eigenmodes,

VSC

Λ (k,−k,p) =

?

i

cSC

i (Λ)fSC,i(k)∗fSC,i(p),(9)

and obtain the band-resolved form factors of the leading

and subleading SC eigenmode (i.e. largest two negative

eigenvalues). This way we are able to discuss the inter-

play of d-wave and s-wave as well as the degree of form

factor anisotropy for a given setting of (J1,J2). Com-

paring divergence scales Λc gives us the possibility to

investigate the relative change of Tc as a function of

(J1,J2). Furthermore, we also investigate the orbital-

resolved pairing modes39by decomposing the orbital four

point vertex

Vorb

c,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 of

the band vectors. By investigating the intraorbital SC

pairing modes in (10), we make contact to the findings

from 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 closely

resembling 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 J2

and in Fig. 9 for dominant J1. As stated before, the pair-

ing interaction is already present at the bare level in the

model so that we achieve comparably fast instabilities as

the cutoff is flowing towards the Fermi surface. As found

in Ref. 11, the dominant J2scenario exhibits a leading s-

wave coskxcoskyform factor which causes the same sign

Page 7

7

10-2

10-1

100

Cutoff

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

Interaction strength

15913

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Form factors

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 form

factor 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 coskx−cosky

type, changing sign from one electron pocket to the other.

The inverse situation is found for dominant J1. As shown

in Fig. 9, the d-wave coskx−coskyform factor establishes

the leading instability. As before, the form factor does

not cross zero related to nodeless SC for this parameter

setting.

With only pairing information available on the limited

number of sampling points along FS, it is impossible to

obtain as in the mean-field analysis the superconducting

gap in the whole BZ. For illustration, a mixture of a

small A-type NNN d-wave pairing and large A-type NNN

s-wave pairing is indistinguishable from a pure A-type

NNN s-wave pairing; a mixed state of a small B-type NN

s-wave pairing plus a large B-type NN d-wave pairing,

and a state with pure B-type NN d-wave pairing show

little difference if one compares the gap on a few points

along the Fermi surfaces. For this reason, the symbol

sx2y2 used in this section refers to a pairing consisting of

a large A-type NNN s-wave pairing and possible small

components of A-type NNN d-wave pairing or A-type

NN s/d-wave pairing. In turn, the symbol dx2−y2 refers

to a pairing made up with a large B-type NN d-wave

pairing 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 each

setup we have obtained Λcas well as the ratio of the in-

stability eigenvalues between s-wave and d-wave in the

pairing channel (encoded by the two-color circles shown

in Fig. 10). The FRG result is qualitatively consistent

with the mean field analysis. In the antiferromagnetic

sector, the s-wave wins for dominant J2while the d-wave

wins wins for dominant J1. For ferromagnetic J1 cor-

responding to the situation in chalcogenides, we find a

robust preference of s-wave pairing. The anisotropy of

the s-wave gap around the pockets in the FRG calcu-

lation is also qualitatively consistent with the meanfield

10-1

100

Cutoff

-1200

-1000

-800

-600

-400

-200

0

Interaction strength

15913

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Form factors

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 form

factor is denoted in blue, sub-leading form factor in green.

result. The gap on the Fermi surfaces with dxy orbital

character is smaller than the gap on those with dxz,yz

orbital character.

The predictions from the mean field analysis are fur-

ther confirmed for the mixed phase regime where s-wave

and 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 the

coexistence phase. For illustration, in Fig. 11 we have

plotted the dependence of Λc on J1− J2, with J1+ J2

fixed to 0.7 eV; there is a clear reduction of the critical

scale (and thus the transition temperature) when there

is 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 constrain

ourselves to the most relevant three orbitals dxy, dxz, and

dyz. In particular, we observe that the SC orbital pair-

ing induces the same sign for all three dominant orbital

modes, in correspondence with the mean field analysis

presented before.

B.Three-pocket scenario

Recent ARPES data57on the chalcogenides may sug-

gest the existence of a shallow flat pocket around the Γ

point (the location, and especially the kzposition of such

a pocket are still under debate). By tuning parameters,

we have obtained in the previous section a three-orbital

model 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 band

structure calculations at hand for the chalcogenides, we

consider it unlikely that it will be a hole band regular-

ized up towards the Fermi surface.

gate the effect of a possible electron band at the Γ point

Instead, we investi-

Page 8

8

-1.0-0.50.00.51.0

J1 / eV

0.0

0.2

0.4

0.6

0.8

1.0

J2 / eV

FIG. 10.

pocket model. Each pie shows the relative strengths of the

two leading pairing channels, with the radius proportional to

[8+log10(Λc/eV)]2. The color code for pairing symmetries is

green sx2y2 and red dx2−y2.

(color online) The phase diagram of the two-

-1.5

-1.0-0.50.00.51.0

J1−J2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Λc

FIG. 11.

Λc along a line through parameter space which interpolates

between s and d wave. A minimum is visible for comparable

ordering tendency in s-wave and d-wave. See the caption of

Fig. 10 for more details on the pie charts.

(color online) The variation of the critical scale

in the unfolded Brillouin zone. This is suggested from

the folded 10-band calculations, where one electron-type

band closely approaches the Fermi level around the Γ

point44. This band should be very flat and shallow. From

the weak coupling perspective of particle-hole pairs cre-

ated around the Fermi surfaces, this will probably have a

small effect: particle-hole pairs will only be created up to

energy scales of the depth of the electron band at the X

point below the Fermi surface, providing some hole-type

phase 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 modified

band structure designed for this scenario. There, we have

bent down the band dominated by dxyin the two-pocket

model band structure44without changing its band vector

and created an electron pocket around Γ, accordingly of

mainly dxyorbital content (Fig. 15). The band bending

1234

Points on Fermi pockets

56

7

8910 11 12 13 14 15 16

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Pairing form factors

(π,0)(0,π)

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

1234

Points on Fermi pockets

56

7

8910 11 12 13 14 15 16

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Pairing form factors

(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 color

code is the same as in Fig. 7, i. e. red dxz, green dyz, blue

dxy, 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 by

dxy. The shift of energy ξ(k) was intentionally chosen

such that the Γ pocket exhibits some nesting with the X

electron pockets.

The phase diagram is shown in Fig. 16. FRG results

for typical scenario for the s-wave and d-wave regime are

shown in Fig. 13 and Fig. 14, respectively. As suspected,

the additional pocket strengthens the tendency to form

an s-wave in the system, aside from exhibiting an addi-

tional constant s-wave instability in a small regime for

dominant J1.

IV.DISCUSSION

The above calculations demonstrate that the s-wave

pairing symmetry is always robust if the AFM NNN J2is

strong while a d-wave pairing can be strong if J1is AFM

for the electron overdoped region. Moreover, if both of

Page 9

9

100

Cutoff

-1200

-1000

-800

-600

-400

-200

0

Interaction strength

159 13

17

21

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Form factors

FIG. 13.

conducting gaps associated with Fermi surface for the three-

pocket scenario with (J1,J2) = (0.2,0.8) eV. Leading form

factor is denoted in blue, sub-leading form factor in green.

(color online) Typical RG flows and the super-

100

Cutoff

-1200

-1000

-800

-600

-400

-200

0

Interaction strength

15913

17

21

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Form factors

FIG. 14.

conducting gaps associated with Fermi surface for the three-

pocket scenario with (J1,J2) = (0.9,0.3) eV. Leading form

factor is denoted in blue, sub-leading form factor in green.

(color online) Typical RG flows and the super-

them are AFM, there is strong competition between the

s-wave and d-wave pairing. When there are hole pock-

ets, as shown before11, even in a range of J1∼ J2, the

contribution to pairing from J1is much weaker than the

one 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 shown

that a major difference between iron-pnictides and iron-

chalcogenides is that the NN coupling J1 change from

AFM in the former58,59to FM in the latter47. In fact,

J1 is rather strongly FM in the latter, which explains

the high magnetic transition temperature (500 K) in the

245 vacancy ordering state as shown in Ref. 51. Combin-

ing these results, we can partially answer the question

regarding the different behaviors between iron-pnictides

and iron-chacogenides in the electron-overdoped region:

why can the high SC transition temperature be achieved

Γ

MX

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

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18 19

20

21

22

23

24

FIG. 15. (color online) The band structure and the Fermi

surfaces of the modified band structure, colored according

to the dominant orbital content. The color code is red dxz,

green dyz, blue dxy, orange dx2−y2 and magenta d3z2−r2. The

numbered crosses show the center of Fermi surface patches

used in the FRG calculations.

-1.0-0.50.00.5 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-pocket

model. Here we are able to resolve the s-wave channel into

constant s-wave (grey), extended sx2y2-wave (green), and the

nodal sx2+y2-wave (purple). Parameter sets with J1 ∼ −1

and J2 ∼ 0 have highly oscillating form factors which are due

to artifacts in the calculation; the triplet channel would have

to 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, J1

will weaken the SC pairing in the former but not in the

latter.

A few remarks regarding this work follow: (i) Our

mean-field result is qualitatively consistent with the re-

sults from a similar model with five orbitals61–63. The

critical difference is regarding J1being FM, and has not

been addressed previously; (ii) s-wave pairing symmetry

has also been obtained in Refs. 43, 64, and 65. How-

ever, the s-wave pairing only shows up either in a narrow

region or with drastically different parameter settings.

Therefore, the s-wave is not robust from a microscopic

point of view.Instead, the d-wave is a robust result

in these studies. Still, even the d-wave pairing strength

based on the scattering between two electron pockets is

generally weak, as specifically discussed in61, which is

another difficulty for this type of mechanism. (iii) Our

results 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.

Page 10

10

If both hole and electron pockets are present, the signs

of 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 due

to the scattering between the hole and electron pockets

while in the strong coupling approach, the sign change is

due to the form factor of the SC order parameters which

is specified to be coskxcosky since the pairing mainly

originates 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 that

the superconductivity vanishes in the iron pnictides in

the electron overdoped region is not solely due to the

competition between s-wave and d-wave pairing symme-

try. It is also due to the weakening of local magnetic

exchange coupling themselves and the reduction of band

width renormalization. (v) The prospective experimental

confirmation of s-wave pairing symmetry in KFe2Se2will

support that superconductivity in iron-based supercon-

ductors might be explained by local AFM exchange cou-

plings. (vi) Neutron scattering also suggests that there

is significant AFM exchange coupling between two third

nearest neighbor sites, i.e. J347,49. The existence of J3

will further enhance the s-wave pairing since it gener-

ates pairing form factors as cos2kx+ cos2kyin recipro-

cal space which in turn can enhance the pairing at the

electron pockets.

V.CONCLUSION

In summary, we have shown that the pairing symmetry

in electron-overdoped iron-chalcogenides is a robust s-

wave. The fact that the NN magnetic exchange coupling

is FM, which diminishes the possibility of d-wave pairing

symmetry in these materials. From a unified perspective

of high-Tc cuprates and high-Tc chalcogenides, the NN

AFM exchange coupling gives rise to the robust d-wave

pairing in the cuprates while the NNN AFM exchange

coupling gives rise to the robust s-wave pairing in iron-

based chalcogenide superconductors.

ACKNOWLEDGMENTS

We thank S. Borisenko, J. van den Brink, Xianhui

Chen, 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 Haihu

Wen for useful discussions. JPH thanks the Institute of

Physics, CAS for research support. RT is supported by

DFG SPP 1458/1 and a Feodor Lynen Fellowship of the

Humboldt Foundation and NSF DMR-095242. BAB was

supported 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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