Superconductivity at the border of electron localization and itinerancy

Abstract

The superconducting state of iron pnictides and chalcogenides exists at the border of anti-ferromagnetic order. Consequently, these materials could provide clues about the relationship between magnetism and unconventional superconductivity. One explanation, motivated by the so-called bad metal behaviour of these materials proposes that magnetism and superconductivity develop out of quasi-localized magnetic moments that are generated by strong electron-electron correlations. Another suggests that these phenomena are the result of weakly interacting electron states that lie on nested Fermi surfaces. Here we address the issue by comparing the newly discovered alkaline iron selenide superconductors, which exhibit no Fermi-surface nesting, to their iron pnictide counterparts. We show that the strong-coupling approach leads to similar pairing amplitudes in these materials, despite their different Fermi surfaces. We also find that the pairing amplitudes are largest at the boundary between electronic localization and itinerancy, suggesting that new superconductors might be found in materials with similar characteristics.

ARTICLE

Received 1 Dec 2012 | Accepted 16 Oct 2013 | Published 15 Nov 2013

DOI: 10.1038/ncomms3783

Superconductivity at the border of electron localization and itinerancy

Rong Yu1,2,*, Pallab Goswami2,w,*, Qimiao Si2, Predrag Nikolic3 & Jian-Xin Zhu4

The superconducting state of iron pnictides and chalcogenides exists at the border of anti-ferromagnetic order. Consequently, these materials could provide clues about the relationship between magnetism and unconventional superconductivity. One explanation, motivated by the so-called bad metal behaviour of these materials proposes that magnetism and super-conductivity develop out of quasi-localized magnetic moments that are generated by strong electronelectron correlations. Another suggests that these phenomena are the result of weakly interacting electron states that lie on nested Fermi surfaces. Here we address the issue by comparing the newly discovered alkaline iron selenide superconductors, which exhibit no Fermi-surface nesting, to their iron pnictide counterparts. We show that the strong-coupling approach leads to similar pairing amplitudes in these materials, despite their different Fermi surfaces. We also nd that the pairing amplitudes are largest at the boundary between electronic localization and itinerancy, suggesting that new superconductors might be found in materials with similar characteristics.

1 Department of Physics, Renmin University of China, Beijing 100872, China. 2 Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA. 3 School of Physics, Astronomy and Computational Sciences, George Mason University, Fairfax, Virginia 22030, USA. 4 Theoretical Division and Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. * These authors contributed equally to this work.

w Present address: National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA. Correspondence and requests for materials should be addressed to Q.S. (email: mailto:[email protected]

Web End [email protected] ).

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Superconductivity often occurs near a magnetic order. This is the case not only in the iron-based compounds13 but also in heavy fermion intermetallics, organic charge-transfer

salts and copper oxides. An important question that is central to these diverse classes of unconventional superconductors is whether the mechanism of their superconductivity is in a way analogous to that of conventional superconductors, with spin uctuations replacing phonons as the glue for electron pairing, or it instead involves novel electronic states that are generated by strong electron correlations4.

Iron-based superconductors represent a unique setting to elucidate this basic issue: their large materials parameter space provides the opportunity to understand the microscopic physics by comparing the properties across their material families. A major recent development suitable for this important characteristics is the discovery of high-temperature super-conductivity in a new family of iron chalcogenides, the alkaline iron selenides K1 xFe2 ySe2 (ref. 5). Other related iron selenides,

with K replaced in part or in entirety by Rb, Cs or Tl behave similarly6,7. The key property is that the maximum of the superconducting transition temperature (Tc) observed in the alkaline iron selenides, above 30 K, is similar to that of their 122 iron pnictides counterpart, suggesting a commonality in the underlying mechanism for superconductivity across these systems.

Compared with those of the iron pnictides, the Fermi surfaces in the alkaline iron selenides are very different. Although the former contain both electron and hole pockets, respectively, at the boundary (M) and centre (G) of the one-Fe Brillouin zone, only electron pockets are present in the alkaline iron selenides810. The weak-coupling Fermi-surface-based mechanism will operate very differently in the alkaline iron selenides compared with the iron pnictides1113.

Here we demonstrate that, by incorporating the bad metal nature of the normal state, the strong-coupling approach provides the understanding for the comparable pairing strength in the alkaline iron selenides and iron pnictides.

ResultsProximity to Mott transition. We seek for the commonality between the iron chalcogenides and pnictides based on the observation that the parent compound of the alkaline iron selenides are anti-ferromagnetic insulators6,14. These insulating selenides contain Fe vacancies that are ordered in the Fe-square lattice, so that the Fe valence is kept at 2. Because of their very

large ordered moment of about 3.3 mB/Fe (refs 15,16), they are naturally considered as Mott insulators, arising through a kinetic energy reduction induced by the ordered vacancies17,18. An important question is how U/W, the ratio of a combined onsite Coulomb and Hunds interaction to the electron bandwidth, compares with the Mott-transition threshold, Uc/W. Given that a modest reduction of the kinetic energy from the parent iron pnictides to the parent alkaline iron selenides turns the system from metallic to insulating, we can infer that U/W is larger than but close to Uc/W in the alkaline iron selenides, whereas smaller than but also close to Uc/W in the iron pnictides. Hence, both superconductors arise out of bad metals on the verge of a Mott localization.

This is illustrated in Fig. 1 showing the parent compounds of the alkaline iron selenides and iron pnictides in the vicinity of, albeit on the two sides of, the Mott transition point. Consequently, we use the Mott transition point as anchoring the regime of the phase diagram that has strong anti-ferromagnetic correlations, illustrated by the purple shading in Fig. 1. At the Mott transition point, all the electronic excitations are incoherent.

Integrating out the gapped electronic excitations gives rise to a model of localized spins with nearest-neighbour (n.n.) (J1) and

next-nearest-neighbour (n.n.n.) (J2) interactions on the Fe-square lattice. In the carrier doped regime, a ve-band tJ1J2 model

ensues1921. More generally, for the iron pnictides, the proximity to the Mott transition has also been supported by the bad metal behaviour of the normal state, as determined by the optical conductivity3 and other measurements, providing the basis for strong-coupling approaches to superconductivity1926.

Multiorbital tJ1J2 approach. We study spin singlet pairings in two such ve-band tJ1J2 models, respectively, for the alkaline iron selenides and iron pnictides. The different Fermi surfaces arise from different choices of the tight-binding parameters, which are specied by the kinetic energy part of the model. The ve 3d orbitals of iron are used to correctly describe the Fermi surfaces; they are denoted by a 1,...,5, which correspond to 3dxz,

3dyz, 3dx2 y

U/W

Alkaline

selenides

Mott trans.

Pnictides

0

Carrier doping x

Figure 1 | Schematic phase diagram near a Mott transition. In this zero-temperature phase diagram, the red point located on the U/W axis refers to the point of the Mott transition, whereas the purple shading illustrates the regime that has strong anti-ferromagnetic correlations. The parent compounds of alkaline iron selenides and iron pnictides are located inthe vicinity of, albeit on the two sides of, the Mott transition. Super-conductivity occurs at non-zero carrier doping, with the optimal doping located in the region indicated by the arrows.

2 (see Methods). The Fermi surface of K1 xFe2 ySe2 is shown in Fig. 2a. It comprises electron

pockets only, and corresponds to an electron doping of about 15% per Fe; both have been specied in accordance with the angle-resolved photoemission spectroscopy (ARPES) measurements810. The electron- and hole-like Fermi pockets for the iron pnictides27, also with d 15% electron doping, are displayed in

Fig. 2b. The xz/yz and xy 3d orbitals dominate the electronic states near the Fermi surfaces, as illustrated in Fig. 2c,d. We observe that, at and near zero doping (d 0), the ground state

will be anti-ferromagnetically ordered (see below). We also note that our study focuses on the couplings in the spin sector as driving the superconductivity, although our general analysis may also have implications for the considerations in the orbital sector26.

Figure 3a,b demonstrates how superconductivity in the ve-band tJ1J2 is magnetically driven by the short-range J1J2 exchange interactions. Shown here, respectively, for K1 xFe2 y

Se2 and iron pnictides, are the zero-temperature phase diagrams in the J1J2 plane, for 0rJ1 and J2r0.3D, where D is the

2 , 3dxy and 3d3z2 r

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a

b

1

1

k y/[afii9843]

0

0

ky /[afii9843]

1 1 0 1

1

1 0 1

kx/[afii9843]

kx/[afii9843]

c

d

1.0

1.0

xz yz x2y

2

xy z2r2

O.W.

0.5

0.5

O.W.

0.0 0.0 0.5 1.0 1.5 2.0

0.0

0.0 0.5 1.0 1.5 2.0

[afii9835]/[afii9843][afii9835]/[afii9843]

Figure 2 | The contrasting Fermi surfaces of K1 xFe2 ySe2 and iron pnictides. a and b, respectively, show the Fermi surfaces of K1 xFe2 ySe2 and iron

pnictides in the extended Brillouin zone corresponding to one iron per unit cell, as obtained by using a ve-orbital tight-binding model described in the Methods section, for electron doping d 0.15. There are only electron pockets at the M points (p,0) and (0,p) for K1

xFe2 ySe2. For iron

pnictides, there are in addition two hole pockets near the G point (0,0). (c,d) The corresponding orbital weights (O.W.) on the electron pockets centred at (p,0) in a and b. y is the winding angle of the pocket with respect to its centre. The weights on the electron pocket centred at (0,p) can be obtained by interchanging the xz and yz components and shifting y by p/2.

0.20.10.0

0.3

0.2

J 0.1

2

D

a

b

0.3

0.2

0.1

0.0

J 2/D

0.3

0.2

0.1

0.0

I

II

J 2/D

I

II

III

IV

III

0.0 0.1

0.2

0.3 0.0

0.1

0.2

0.3

J1/D

/

D

c d

J1

A1g sx2y2A1g d x2y2

B1g dx2y2

B1g

A1g sx2+y2A1g dxyB2g dxyB2g

0.20.10.0

P.A.

s x2y2

s x2+y2

P. A.

0.3

0.2

0.2

0.3

0.1

J1

D

J2

D

0.1 0.1

0.2

0.3

J1

D

Figure 3 | Phase diagram and pairing amplitudes of K1 xFe2 ySe2 and iron pnictides. The results presented here are for electron doping d 0.15.

a and b, respectively, show the zero-temperature phase diagrams of K1 xFe2 ySe2 and iron pnictides. Region I corresponds to an A1g state with dominant

sx y channel. Regions II and III mark an A1g iB1g state with dominant sx y and dx y channels (II), and dominant sx y and dx y channels (III);

the phase locking occurs only at low temperatures. Region IV for K1 xFe2 ySe2 is a pure B1g state with dominant dx y channel. (c,d)The corresponding

pairing amplitudes for the xy 3d orbital of K1 xFe2 ySe2 and iron pnictides.

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renormalized bandwidth. For K1 xFe2 ySe2, the J2-dominated

region I has A1g symmetry, with sx2y2 (cos kxcos ky) wave being the dominant pairing component. Regions II, III and IV are primarily of the B1g dx2 y

2 wave pairing. For similar ratios of J1/D and J2/D, the corresponding pairing amplitude is naturally comparable to that of the iron pnictides, in which both the electron pockets near M and the hole pockets near G promote these two pairing components.

Enhancement of pairing amplitudes near Mott transition. The results for both classes of materials (Figs 3c,d and 4) also show that the pairing amplitudes are larger when J1/D and J2/D are increased. This conclusion, in turn, leads to a general principle. To see this, we note that the exchange interactions increase as the Mott transition is approached from the insulating side (cf., Fig. 1) because of the reduction of the charge gap. At the same time, the renormalized bandwidth D is reduced as the Mott transition is approached from the metallic side. Correspondingly, at the boundary between electronic localization and delocalization, the ratios J1/D and J2/D will be maximized and so will the super-conducting pairing amplitudes.

Our results provide the understanding for some key experimental observations. Inelastic neutron-scattering experiments have shown that the exchange interactions in the alkaline iron selenides and iron pnictides have the same order of magnitude16. Further, as Fig. 1 illustrates, the alkaline iron selenides and iron pnictides have approximately the same doping concentration and similar degree of proximity to the Mott transition. Therefore, J1/D and J2/D are similar in the two materials. This leads to our key conclusion, namely, the two classes of iron-based superconductors have comparable pairing amplitudes and, by extension, comparable Tc.

It is instructive to contrast the situation here with KFe2As2. The latter system, with the Fermi surface containing only hole pockets28, represents another material lacking Fermi-surface nesting. However, it is strongly doped, with a hole doping of0.5 per Fe. This extreme overdoping in KFe2As2 means that the system is far away from the Mott transition-anchoring point discussed here and should, therefore, have considerably reduced

2 (cos kx cos ky) wave character. The

distinction among the three regions reects the difference in the admixture of a small A1g component at zero (and low)

temperatures (see Methods). This is similar to the phase diagram of the pnictides case, where the J2-dominated region I also is primarily A1g sx2y2 wave, and regions II and III are primarily B1g dx2 y

2 wave.

To clarify the admixture of the different pairing components, we show the evolution of the amplitudes of these components, projected onto the 3dxy orbital, in Fig. 3c,d. Comparing the two gures clearly shows that, for the alkaline iron selenides, the dominant pairing amplitudes in the sx2y2 and dx2 y

2 channels are comparable to their counterparts in the iron pnictides. This is further illustrated in Fig. 4a,b, which shows the pairing amplitudes, also projected to the 3dxy orbital, versus J1/D for a xed J2/D 0.1. The same conclusion applies to the pairing

amplitudes projected to the other 3d orbitals, as shown in Fig. 4c,d for the case of 3dxz/3dyz orbitals.

Pairing amplitudes in alkaline iron selenides. We therefore reach the conclusion that the pairing amplitudes in models, respectively, for the alkaline iron selenides and iron pnictides are similar for given dimensionless exchange interactions, J1/D and

J2/D. This is surprising because the alkaline iron selenides lack any hole Fermi pocket and, therefore, do not possess any Fermi-surface nesting. Our result is inherent to the strong-coupling approach; here, although the details of the Fermi surfaces are important, the superconducting pairing does not require coupling between hole and electron Fermi pockets. Instead, the driving force for the pairing lies in the close-neighbour exchange interactions, J1 and J2. The presence of electron pockets near the M points of the Brillouin zone is adequate to promote the cos kxcos ky A1g sx2y2 wave pairing, as well as the cos kx cos ky

B1g dx2 y

J1/D

J1/D

0.0 0.1 0.2 0.3

0.0 0.1 0.2 0.3

a

b

0.2

0.1

0.0

0.2

0.1

0.0

P.A.

xy orbital

P.A.

xy orbital

c

A1g (sx2y2)

B1g (dx2y2)

A1g (sx2+y2)

d

0.1

0.0

0.1

0.0

P.A.

P.A.

xz/yz orbital

xz/yz orbital

0.0 0.1 0.2 0.3

J1/D

0.0 0.1 0.2 0.3

J1/D

Figure 4 | Pairing amplitudes of K1 xFe2 ySe2 and iron pnictides for xy and xz/yz orbitals. (a,b) Comparison of the competing dominant pairing

amplitudes A1g sx y , A1g sx y and B1g dx y for the xy 3d orbital of K1 xFe2 ySe2 and iron pnictides, both for electron doping d 0.15 and J2/D 0.1.

(c,d) The same as a and b but for the xz/yz 3d orbitals. For K1 xFe2 ySe2 the amplitude for the A1g sx y channel is strongly suppressed compared

with the pnictides case. Correspondingly, a pure B1g dx y state is observed for K1 xFe2 ySe2 but is absent in iron pnictides.

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pairing amplitudes. Experimentally, it indeed has a much smaller Tc of about 3 K.

For the alkaline iron selenides, our results show dominating A1g sx2y2 and B1g dx2 y

2 wave states in competition. Both have nearly isotropic and nodeless gaps on the electron pockets, and this is consistent with all existing measurements of the super-conducting gap. Neutron-scattering experiments have identied a resonance associated with the superconducting state29. This is most readily understood in terms of a B1g dx2 y

2 wave pairing29, although it may also arise from an A1g sx2y2 channel once the effect of two iron ions per unit cell is taken into account30. On the other hand, ARPES experiments have suggested that the superconducting gap is nodeless on the faint electron pocket near the G point30,31. This appears to favour an A1g sx2y2 wave pairing30,31. Further experiments are needed to settle which of the two possible pairing channels operates in the alkaline iron selenides. For the iron pnictides, the dominance of A1g sx2y2 state

in a large portion of the phase diagram is consistent with the sign-changing s paired state arising in both weak and strong-coupling approaches. Compared with the alkaline iron selenides, an important difference is that the pairing amplitude in the A1g

sx2 y

2 channel is sizable (Figs 3,4). This arises because the corresponding form factor, (cos kx cos ky), while negligible at

the dominant electron Fermi surfaces located near the M points for the alkaline iron selenides, is large near the hole Fermi pockets around the G point in the case of the iron pnictides. The relevance of the A1g sx2 y

2 channel is important for understanding a possible strong momentum dependence or even the development of nodes in the superconducting gap; nodes arise on the electron Fermi pockets near the M points when the amplitude of the A1g sx2 y

2 component exceeds a threshold value compared with that of the coexisting A1g sx2y2 component. Indeed, experimentally, some iron pnictide superconductors show fully gapped behaviour, whereas others display properties that suggest the existence of nodes.

DiscussionThe bad metal behaviour near the Mott transition also pertains to the relationship between magnetism and superconductivity in the alkaline iron selenides. For the vacancy-ordered parent insulating system (the so-called 245 phase), it causes the electronic excitations to have a large incoherent component; the latter gives rise to a large spin spectral weight even in the absence of any itinerant carriers, as have been observed by neutron-scattering experiments15,16.

In superconducting compounds, there is direct evidence for a phase separation32 between a superconducting component and the parent insulating anti-ferromagnetic part. The superconducting component is free of ordered vacancies32, suggesting a tetragonal 122 structure as we have used in our model. Our study here focuses on the pairing instabilities in the underlying metallic state of this component while taking advantage of its connection with the Mott-insulating phase in the overall phase diagram33 discussed in the Supplementary Note 3 (and Supplementary Fig. S5) and evidenced by recent experiments3436.

It is worth emphasizing that the physical pathway linking the superconducting phase and 245 Mott-insulating phase involves varying both vacancy order and carrier concentration. Indeed, the existence of the multiple phases of the alkaline iron selenides suggests an overall, extended, parameter space in which the different phases can be connected. This is described in some detail in the Supplementary Note 3 and is in particular illustrated in Supplementary Fig. S5. The details of this physical pathway is a distinct issue that is intriguing and important in its own right33,34, and other alkaline iron selenides such as KFe1.5Se2,

with one vacancy per four iron sites6,37 (and small carrier doping34,38), and K0.5Fe1.75Se2, with one vacancy per eight iron sites39, may also be placed in this extended phase diagram. For the purpose of the present work, what is particularly pertinent is the clue these multiple phases have provided for the strength of the electron correlations. As described in the introduction and illustrated in Fig. 1, the existence of the Mott-insulating phase in the 245 compound suggests not only that the underlying U/W is larger than Uc/W in the vacancy-ordered alkaline iron selenides but also that U/W is below but close to Uc/W in the iron

pnictides. This proximity to the Mott transition allows for the present study on the pairing amplitudes in both classes of iron-based superconductors.

Recently, superconductivity of around 50 K was reported in a single-layer FeSe lm grown on SrTiO3 substrate40. ARPES measurements41 indicate that the Fermi surface consists of only electron pockets, similar to that of the alkaline iron selenides. Thus, it is instructive to model the single-layer FeSe lm and compare its pairing properties with those of the other iron-based superconductors. We have thus studied the singlet superconductivity in a similar ve-orbital tJ1J2 model for the single-layer FeSe lm (see Methods), with an electron doping of0.1 per Fe; we consider the role of the substrate as providing the structure that dopes this amount of electron carriers into the single-layer FeSe42,43. The results are shown in Supplementary Figs S4d and S7. The pairing phase diagram of the single-layer FeSe is comparable to those for both alkaline iron selenides and iron pnictides, and so are the overall pairing amplitudes.

By showing that pairing amplitudes are similar for the iron chalcogenides and pnictides in spite of their drastically different Fermi surfaces, our results uncover a universality in the existing and emerging iron-based high-temperature superconductors with very diverse materials and Fermi-surface characteristics. Moreover, our demonstration, that the pairing amplitudes increase with the ratio of the short-range spin exchange energy to the renormalized kinetic energy, reveals an important principle. Namely, superconductivity is optimized at the border between itinerancy and electronic localization. This principle should apply beyond the context of iron pnictides and chalcogenides, and is expected to guide the search for superconductors with even higher transition temperatures.

Methods

Multi-orbital tJ1J2 model. We describe the methods used in our study of the phase diagram for the singlet superconducting pairing of ve-orbital tJ1J2 models, described as H Ht HJ. Here, Ht and HJ, respectively, correspond to the

fermion hopping and the J1J2 exchange Hamiltonians.

Two such models are considered. The difference in the fermiology of the alkaline iron selenides and iron pnictides are specied via the kinetic part Ht, and we have chosen the tight-binding parameters by tting the band dispersions obtained from density functional calculations. The short-range anti-ferromagnetic exchange interactions in HJ drive the singlet pairings. We have chosen the same exchange-coupling constants J1, J2 for each orbital, and have performed a mean-eld decoupling of the exchange part in the pairing channel. Each orbital contributes four pairing amplitudes Da,a, which are, respectively, given by sx y ,a,

dx y ,a, sx y ,a, dxy,a, where a 1,...,5 labels the orbitals; these amplitudes are

related to their real-space counterparts, De,a /ciamcieak ciakcieamS/2. We

have minimized the free energy to nd the self-consistent solution for the 20 pairing amplitudes. We now expound on these in some detail.

Our ve-orbital tJ1J2 model arises from an expansion of the ve-orbital Hubbard model with respect to the Mott transition point (the w-expansion)20.

The Hamiltonian for the model is given by

H X

ioj;a;b;s

tabijcyiascjbs h:c: m X

i;a

nia

X

hiji;a;b

Jab1 ~Sia

14 nianjb

X

hhijii;a;b

Jab2 ~Sia

~Sjb

1

~Sjb

14 nianjb

where cy

ias creates an electron at site i, with orbital index a and spin projection s; m is the chemical potential and tabij the hopping matrix. The orbital index a 1, 2, 3, 4

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(iii) A2g : dAxy gxy;ktz; and (iv) B2g : dBxy gxy;kt0. The eight pairing amplitudes sAx y

and so on are obtained as linear combinations of Da11 and Da22, which are in turn linear combinations of De,11 and De,22.

Mixed-symmetry-pairing states breaking time-reversal symmetry. The admixed states II and III of Fig. 3a,b break time-reversal symmetry and have the form A1g iB1g. In regions II and III, the A1g components are, respectively, sx y

cos kxcos ky and sx y (cos kx cos ky). In contrast to the pnictides, the pairing

phase diagram of 122 chalcogenides has a region IV, which is of purely B1g dx y

character even at zero temperature. The time-reversal symmetry breaking in region II is because of quasi-degeneracy between sx y and dx y pairings induced by

strong magnetic frustration. On the other hand, the quasi-degeneracy between sx y and dx y pairings in region III arises because of bandwith suppression.

These mixed-symmetry-pairing states are expected to be relevant only at sufciently low temperatures.

Generalizations of the models. The pairing phase diagrams, shown in Fig. 3a,b, have been determined by assuming orbitally diagonal exchange couplings J1 and J2.

The interorbital exchange couplings do not qualitatively modify the phase diagram. The interorbital couplings introduce some subdominant components to A1g and

B1g regions, while leaving the competition between A1g and B1g pairing symmetries intact. For example, an interorbital second neighbour coupling between xz and yz orbitals gives rise to interorbital dxy(sin kxsin ky) component, which is a part of A1g pairing. Similarly, we can also consider further neighbour intraorbital exchange couplings. For example, the third neighbour anti-ferromagnetic exchange-coupling J3 does not change the competing pairing channels; however, it introduces sub-dominant A1g (cos 2kx cos 2ky) and B1g (cos 2kx cos 2ky) components. There

fore, the phase diagram obtained from an orbitally diagonal J1J2 model is generic and robust.

We also note that an extended J1J2 model16,45, with ferromagnetic n.n. coupling, dened on a modulated square lattice, has been used to explain the

5

p 5

p block spin anti-ferromagnetic order. Although this is believed to reect the modulation to the exchange interactions that exists only in the presence of ordered vacancies45,46, it is interesting to consider the effect of a ferromagnetic J1.

The latter will suppress the B1g pairing and increase the A1g region. We stress that the A1gB1g competition described earlier is the feature of the paramagnetic state, which is devoid of vacancy order and should have an anti-ferromagnetic J1.

Finally, higher-spin interactions such as a biquadratic term can also be incorporated in the model. These interactions will generate contributions to the free energy that are of higher order in the pairing amplitudes. Therefore, they will not signicantly affect the competition among the different pairing channels.

Estimate of the superconducting energy gaps. Our results allow an order-of-magnitude estimate of the pairing energy gaps for the alkaline iron selenides and optimally doped iron pnictides. For both alkaline iron selenides and iron pnictides, the energies of the zone-boundary magnetic excitations are on the order of200 meV, which imply that the effective exchange interactions are on the order of 2050 meV16,47. This species the order of magnitude of the parameters J1 and J2 in our model, although they are for individual orbitals. Our calculated valuesfor the pairing amplitudes D (Fig. 3c,d) then imply that the corresponding pairing gaps, 2JD, appropriately weighted over the different orbitals, will be on the

order of 10 meV. This is compatible with what have been inferred from the ARPES and tunnelling measurements810,32.

Pairing for the single-layer FeSe lm. We consider the spin singlet pairing of a ve-orbital tJ1J2 model for the newly discovered single-layer FeSe lm40. Recent experiments40,41 suggest that superconductivity arises from the FeSe layer but not from the FeSe/SrTiO3 interface. Hence, we study, using density functional theory, the electronic structure of a single-layer FeSe without including a substrate. We then use a ve-orbital tight-binding parametrization described earlier in the Methods section to t the density functional theory bandstructure. The best-tted tight-binding parameters are listed in Supplementary Table S3. Supplementary Figure S6a shows the bandstructure of the tight-binding model with these parameters. We have xed the Fermi energy so that the electron doping is 0.1 per Fe, which is close to the value of 0.09 per Fe determined by ARPES measurement41. As shown in Supplementary Fig. S6b, the calculated Fermi surface comprises electron pockets only, which is consistent with the ARPES results41. The overall bandwidth of this model is somewhat larger than that of K1 xFe2 ySe2, suggesting

the electrons are less correlated in the single-layer FeSe than in alkaline iron selenides. This is consistent with the weaker mass renormalization observed in the single-layer FeSe lm41.

Given these tight-binding parameters for the single-layer FeSe, the corresponding ve-orbital tJ1J2 model is specied as described earlier in the Methods section. We again consider spin singlet pairing. Supplementary Figure S7 shows the pairing phase diagram and pairing amplitudes in several channels. Both phase diagram and strength of the pairing amplitudes are similar to those of alkaline iron selenides K1 xFe2 ySe2 and Tl1 xFe2 ySe2.

and 5 respectively, correspond to ve 3d orbitals 3dxz, 3dyz, 3dx y , 3dxy and

3d3z r of iron. The n.n. (/ijS) and n.n.n. (//ijSS) exchange interactions are, respectively, denoted by Jab1 and Jab2. The spin operator ~Sia

1

2

Ps;s cyiasss cias and

the density operator nia P

s cy

iascias, with representing the Pauli matrices.

For the calculation in these ve-orbital models, we have considered the effects of the fermion no double-occupancy constraints as being implicitly accounted for by the reduction of the effective bandwidth D. We expect that this treatment does not affect the results qualitatively. We have veried this expectation for a two-orbital model by explicitly keeping track of the occupancy constraints. This is described in some detail in the Supplementary Note 1; Supplementary Methods and Supplementary Fig. S3. As shown there, the phase diagram and pairing amplitudes are insensitive to this when the exchange interactions are measured with respect to the renormalized bandwidth D.

In a previous study of a multiorbital tJ1J2 model for iron pnictides21, it has been demonstrated by three of the present authors that the dominant pairing symmetry is governed by the intraorbital exchange interactions, and the consideration of the orbitally off-diagonal exchange interaction only introduces quantitative modications of the phase diagram. Therefore, to keep the analysis simple, we will consider the orbitally diagonal exchange couplings Jab1J1dab and

Jab2J2dab. To consider a pure superconducting phase, we will study here the case

with the 122 tetragonal symmetry (see Supplementary Note 3 for further discussions).

To describe the fermiology, we use a tetragonal symmetry-preserving tight-binding model, involving all ve 3d orbitals of iron. In the momentum space, the 5 5 tight-binding matrix, and its eigenvalues will be, respectively, denoted by

^xk

and Ej,k. The total number of electrons is determined by the chemical potential and the dispersion relations, according to n2 P

j;k ym Ej;k, and carrier doping

d |n 6|. The tight-binding parameters are xed by tting the band structure

obtained from the local density approximation (LDA) calculation. The details of the tight-binding parametrization for iron chalcogenides and the associated band dispersions are discussed below, in the next subsection. For (K,Tl)1 xFe2 ySe2, the

band dispersions correctly produce the electron pockets near the M points, as illustrated in Fig. 2a for K1 xFe2 ySe2. The electron- and hole-like Fermi pockets

obtained from a similar ve-orbital tight-binding model of iron pnictides27 are shown in Fig. 2b to contrast the fermiology of the two materials. The fermiology of Tl1 xFe2 ySe2 is shown below, in Supplementary Note 2 and Supplementary Figs

S1 and S2, which again consists of only electron pockets. We note that ARPES experiments810 have suggested that very weak electron-like pockets may also exist near the G points. Unlike their hole counterpart in the iron pnictides, these electron pockets have very small spectral weight and are expected to have at most a secondary role in driving superconductivity.

Details of tight-binding parametrization. For the ve orbital, tetragonal symmetry preserving tight-binding model, we adopt the parametrization method of Graser et al.27 We have tted the LDA band structure with the band dispersion found from the tight-binding model to determine the hopping parameters. For (K,Tl)1 xFe2 ySe2, we have performed calculations with two different sets of

hopping parameters (see Supplementary Tables S1 and S2). These two sets, respectively, were derived from tting the LDA results for KFe2Se2 and TlFe2Se2;

see Supplementary Fig. S1. For iron pnictides, we use the tight-binding parameters of ref. 27. In Figs 2a and 3a, we have shown the electron pockets and pairing phase diagram for the band structure corresponding to K1 xFe2 ySe2. In Supplementary

Fig. S1, we provide the band dispersions for both tight-binding models and also show the electron pockets derived from Tl1 xFe2 ySe2.

Spin singlet-pairing states. We rst consider degenerate pairing states in the absence of kinetic energy. The spin singlet, intraorbital-pairing operators are dened as De,a De,aa /ciamcieak ciakcieamS/2, where e ^x; ^y; ^x ^y. Two

types of pairing states D^x;a D^y;a, respectively, denoted as sx y and dx y

states with momentum space form factors gx y ;k cos kx cos ky; gx y ;k

cos kx cos ky arise because of the n.n. exchange interaction, and they are ener

getically degenerate in the absence of kinetic energy. Similarly, the n.n.n. exchange gives rise to two degenerate pairing states D^x^y;a D^x ^y;a, respectively, denoted

as sx y and dxy states, with momentum space form factors gx y ;k cos(kx

ky) cos(kx ky), gxy,k cos(kx ky) cos(kx ky). A strong magnetic frustration,

characterized by J1 J2, leads to an enhanced degeneracy among the four paired

states.

The kinetic energy term lifts most of these degeneracies. In the strong frustration regime (J1 J2), it leaves a quasi-degeneracy among a reduced set of

pairing states. To study the full problem, we perform a mean-eld decoupling21,44 of the exchange interaction terms. We introduce four complex singlet-pairing amplitudes for each orbital and write the following 5 5 pairing matrix

Dk P

a diagDak;11; Dak;22; Dak;33; Dak;44; Dak;55 , where Dak;aaDaaaga;k and the index a

corresponds to sx y , dx y , sx y and dxy symmetries. In the subspace of xz and

yz orbitals, which transform as a doublet under the tetragonal point group operations, there are the following four classes of intraorbital-pairing states for an orbitally diagonal J1J2 model: (i) A1g : sAx y gx

y ;k

sAx y gx y ;k t0

dAx y gx

y ;ktz; (ii) B1g : dBx y gx

y ;kt0 sBx y gx

y ;k

sBx y gx y ;k tz;

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Acknowledgements

This work was supported in part by the NSF Grant number DMR-1309531 and the

Robert A. Welch Foundation Grant number C-1411 (R.Y., P.G. and Q.S.), the National

Science Foundation of China Grant number 11374361 (R.Y.), the NSF Cooperative

Agreement No. DMR-0654118, the State of Florida, and the U. S. Department of Energy

through the National High Magnetic Field Laboratory (P.G.), the Ofce of Naval

Research Grant N00014-09-1-1025A and the National Institute of Standards and

Technology Grant 70NANB7H6138, Am001 (P.N.), and the U.S. DOE under Contract

number DE-AC52-06NA25396, the US DOE Ofce of Basic Energy Sciences and the

Center for Integrated Nanotechnologiesa US DOE user facility (J.-X.Z.).

Author contributions

All authors contributed substantially to this work.

Additional information

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How to cite this article: Yu, R. et al. Superconductivity at the border of electron

localization and itinerancy. Nat. Commun. 4:2783 doi: 10.1038/ncomms3783 (2013).

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