PHYSICAL REVIEW B 84, 014521 (2011)

Stripes, spin resonance, and nodeless d-wave pairing symmetry in Fe2Se2-basedlayered superconductors

Tanmoy Das1,* and A. V. Balatsky1,2

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA2Center for Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 3 June 2011; published 27 July 2011)

We calculate RPA-BCS-based spin resonance spectra of the newly discovered iron-selenide superconductor byusing the two-orbital tight-binding model in a 1-Fe-unit cell. The slightly squarish electron pocket Fermi surfacesat (π,0)/(0,π ) momenta produce leading interpocket nesting instability at incommensurate vector q ∼ (π,0.5π )in the normal-state static susceptibility, pinning a strong stripe-like spin-density wave or antiferromagnetic orderat some critical value of U . The same nesting also induces dx2−y2 pairing in a 1-Fe-unit cell. The superconductinggap is nodeless and isotropic on the Fermi surfaces, as they lie concentric to the fourfold symmetric pointof the d-wave gap maxima, in agreement with various experiments. This produces a slightly incommensuratespin resonance with upward dispersion, in close agreement with neutron data on chalcogenides. Finally, wedemonstrate the conversion procedure from a 1-Fe-unit cell to a 2-Fe-unit cell in which the gap symmetrytransformed simultaneously into a dxy pairing and the resulting resonance spectrum moves from the q � (π,π )to the q � (2π,0)/(2π,0) region.

DOI: 10.1103/PhysRevB.84.014521 PACS number(s): 74.70.−b, 74.20.Rp, 74.20.Pq

I. INTRODUCTION

The newly discovered high-Tc superconductor in iron-selenide (Fe2Se2-) based compounds1–3 not only increases thenumber in the unconventional superconducting (SC) familybut also bears on several unique magnetic and SC proper-ties coming from its unusual Fermi surface (FS) topology.Both angle-resolved photoemission spectroscopy (ARPES)4–6

and local density approximation (LDA) calculations7,8 havedemonstrated that in Tl0.63K0.37Fe2−xSe2, at a critical valueof Fe vacancy x = 0.22 to tune the superconductivity, onecan eliminate the hole pocket at the � point and the FSconsists of only two electron pockets at (π,0)/(0,π ). Specificheat9 and ARPES4–6 studies find that these FSs host nodelessand isotropic SC gaps, whereas NMR measurements predictthat the corresponding pairing is spin-singlet.10 Evidence ismounting from Raman,11 μSR,12 optical,13 NMR,14,15 andother experiments,16 and supported by LDA calculations7,8 thatthese materials possess a magnetic-order ground state whichcoexists with superconductivity.

In cuprates, the FS is a hole-like pocket centering at thek = (π,π ) point and the nesting between inter-FSs is strongestalong Q = (π,π ), which is responsible for a spin-density wave(SDW), dx2−y2−wave pairing, and a spin resonance peak atQ with an “hourglass” dispersion in the SC state.17,18 Ina Ce-based heavy fermion, the hole pockets at k = (π,π )lead to Q = (π,π,π ) resonance with dx2−y2 -wave symmetry.19

In iron-pnictide20 and iron chalcogenides,21–24 the FS hastwo hole pockets at � and two electron pockets at k =(π,0)/(0,π ) in the unfolded 1-Fe-unit cell and the leadingnesting between the hole and the electron pockets constituteq = (π,0) SDW, s± pairing, and Q = (π,π ) spin resonancewith an hourglass or upward dispersion (only observed inchalcogenides). In KFe2As2, only the hole pocket is presentat � and is predicted to have nodal d-wave pairing.25 Onthe contrary, in iron-selenide only one (or two concentric)electron pocket(s) appeared at k = (π,0)/(0,π ) at a particularvalue of Fe vacancy.4–8 A similar FS topology appears in the

particular case of electron-doped cuprate near underdopingwhere the antiferromagnetic (AFM) gap eliminates the holepocket in the nodal region, resulting in a nodeless d-wavegap.26

The role of FS topology and the interplay between electronand hole pockets and the exact shape of the neutron scatteringintensity as a consequence of the change in the FS shape acrossthese compounds remain a major challenge in the field. In thispaper we investigate the unusually located electron pocketsin iron-selenide superconductors in a two-band tight-binding(TB) model and their role in magnetic susceptibility. Wefind that the strong instability in the static susceptibilityevolves around an incommensurate vector q = (π,0.5π ) atsome critical value of U , leading to a stripe-like SDW or AFMorder in these systems, in agreement with experiments.11–16

Unlike the stripe order in cuprates or other iron-basedsuperconductors, here the stripe SDW order has the same q

modulation at which a dx2−y2 pairing symmetry possesses asign change at the hot spots, which results in a spin resonancepeak at Q ∼ (π ± δ,π ± δ) in the SC state. Focusing mainlyon Tl0.63K0.37Fe1.78Se2, and at the experimental SC gap valueof 8.5 meV,5 we predict the resonance peak to be present in therange of 12.4 meV at Q � 0.78(π,π ). The resonance profilealso yields an hourglass or upward dispersion and 45◦ rotationof the resonance profile, in close agreement with observationsin chalcogenides20–22 and cuprates.17

An important aspect of identifying the d-wave pairing andthe resonance is that all these results have to be convertedsimultaneously to the actual 2-Fe-unit cell, in which theywill look as follows: (1) stripe-order nesting (π,0.5π ) →(0.5π,0.5π ), (2) dx2−y2 − wave pairing → dxy wave, and (3)spin resonance at (π ± δ,π ± δ) → (2π ± δ,0). Furthermore,all these results are solely governed by the FS topology andthus they are reproducible with the inclusion of a highernumber of bands away from the FS.

The paper is organized as follows. In Sec. II, we presentour two-orbital TB formalism and the fitting of the dispersion

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TANMOY DAS AND A. V. BALATSKY PHYSICAL REVIEW B 84, 014521 (2011)

and FSs to first-principle calculation and ARPES data,respectively. The evolution of the “stripe” competing orderthrough the calculation of static susceptibility within therandom phase approximation (RPA) calculations is given inSec. III. The spin-resonance spectra and the role of d-wavepairing are studied in Sec. IV. Section V gives a detaileddiscussion of how all the results of the 1-Fe-unit-cell Brillouinzone (BZ) were transformed to the 2-Fe-unit-cell BZ. Thetemperature dependence of the spin-resonance spectra atQ = (π,π ) in the 1-Fe-unit cell is studied in Sec. VI. Usinga five-band model, we show in Sec. VII that all the resultsobtained with the two-band model remain unchanged as thehigher orbitals lie away from the Fermi level. Finally, weconclude in Sec. VIII.

II. TIGHT-BINDING FORMALISM

First-principle calculations show that the main contributionto the density of states (DOS) near EF comes from t2g

orbitals of Fe 3d states which disperse only weakly in the kz

direction.7,8 Similarly to pnictide,27,28 the role of the dxy orbitcan be approximated by a next-nearest-neighbor hybridizationbetween dxz, dzy orbitals, and we consider a two-dimensionalsquare lattice with two degenerate dxz, dzy orbitals persite. Based on these observations, our model Hamiltonian28

is

H0 =∑

k,σ

ψ†k,σ

[ξ+

k 1 + ξ−k τ3 + ξ

xy

k τ1]ψk,σ , (1)

where σ is the spin index, τ values are Pauli matrices, andψ

†k,σ = [d†

xzσ ,d†yzσ ] is the two-component field operator. We

consider up to third-nearest-neighbor hopping for the presentcase as depicted in Fig. 1(b), which gives

ξ±k = −(

t1x ± t1

y

)(cx ± cy) − 2(txx ± tyy)cxcy

−(t2x ± t2

y

)(c2x ± c2y) − μ±, (2)

ξxy

k = −4txysxsy.

Here c/siα = cos / sin (ikαa) for α = x,y, μ+ = μ, andμ− = 0. The Hamiltonian in Eq. (1) can be diagonal-ized straightforwardly, obtaining the eigenstates E±

0k =ξ+

k ± √(ξ−

k )2 + (ξxyk )2.28 The TB parameters are obtained

after fitting the dispersion to the LDA calculations7,8

[Fig. 1(c)]: (t1x ,t1

y ,txx,tyy,t2x ,t2

y ,txy) = (−0.12, − 0.108, −0.12, − 0.12,0.048,0.108,0.06) eV and μ = −0.36 eV. Toobtain the experimental FS, we use a rigid band shift toμ = −0.56 eV and get two concentric squarish electronpockets centering (π,0), in good agreement with ARPES5 [seeFig. 1(d)].

As shown below, for such an FS, nesting along Q = (π,π )is most pronounced, leading to a dx2−y2 pairing according toconventional views. We include an SC gap with the sameamplitude on each band to obtain the SC quasiparticle dis-persion as E±

k = √(E±

0k)2 + 2k, where k = 0(cos kxa −

cos kya)/2. 0 is taken from ARPES to be 8.5 meV5 forboth eigenstates throughout this paper, except in Fig. 1(e),where an artificially large value of 0 is chosen to explicatethe behavior of unconventional gap opening on the DOS.Despite the presence of nodes in the underlying gap symmetry,

FIG. 1. (Color online) (a) Unit cell of a KFe2Se2-layered com-pound. (b) x-y plane of the unfolded zone with Fe 3dxz and 3dyz

orbitals. (c) The two resulting bands are plotted along high-symmetrymomentum cuts and compared with the LDA ones [dotted (black)lines].7,8 We mainly concentrate on fitting the low-energy part of theband to get the FS shape and their orbital characters in agreement withARPES. (d) Computed electron pocket FSs overlain on ARPES data.5

(e) The normal-state (NS) DOS is compared with its SC counterpart.An artificially large value of gap = 100 meV is chosen here tohighlight the nodeless and isotropic nature of the gap opening atEF , despite d-wave pairing.

the dispersion at all momenta and the resulting DOS exhibitnodeless behavior in Fig. 1(e), as the FS pieces in thesecompounds are small in size and centering resides at thefourfold symmetric momenta of the gap maxima, similarlyto pnictide but unlike in cuprate. The nodeless and isotropicgap is observed in ARPES4–6 and specific heat measurements.9

III. STATIC SUSCEPTIBILITY AND STRIPEGROUND STATE

We now proceed to the spin susceptibility χ calculationof our model Hamiltonian. The calculation of the BCS χ fora multiband system including matrix-element effects of all

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FIG. 2. (Color online) (a) Bare static χ ′0 (q,ω = 0) and (b), (c)

its RPA part χ ′(q) for two representative values of U . (d) χ ′ (q) fordifferent values of U is drawn along the high-symmetry lines. Blackand green arrows separate the peaks in χ ′(q) coming from the innerand outer FS pockets (see text). (e) Schematic top view of the FeSelayer, with the Fe spins in the stripe AFM order expected in ourcalculation shown by the arrows.

the orbitals is standard and can be found, for example, inRefs. 27–30. We consider the electron-electron correlationon the same Fe atom within the RPA formalism.27 Theterms in the RPA interaction vertex Us that are includedin the present calculation are the intraorbital interactionU , an interorbital interaction U ′ = U − J/2, the Hund’srule coupling J = U/4, and the pair hopping strengthJ ′ = J .29

Due to the specific nature of the FS shown in Fig. 1(d),the interpocket nesting is the dominant and incommensuratecentering at q = (π,π ); see the static bare χ ′

0(q) in Fig. 2(a).An additional nesting occurs centered at q = 0 due to theintrapocket nesting of the semisquarish portion of the FSs.While the shape of χ ′

0(q) is governed by the shape of the FS,the corresponding intensity is related to the Fermi velocity ofthe corresponding hot spots.31 Each of the χ ′

0(q) pattern is splitinto two due to the splitting of the electron pockets into twoconcentric pockets within our model Hamiltonian.

Within the RPA, at different critical values of U , one orthe other piece of χ ′(q) will dominate, leading to an SDWorder phase [see Figs. 2(b)–2(e)]. To clarify this, we studythe evolution of the RPA susceptibility at different values ofU , keeping the other interactions the same with respect tothe value of U (following the expressions given above). Forthe broad range of U studied in this case, the incommensuratepeak around q = (π,π ) is the winner, suggesting that a stripe

like the SDW order will be present in this class of materials.These observations are consistent with the LDA calculations7,8

as well as with a number of experiments.11–16 In a smallrange of U , the inner electron pocket gains more intensity(as in the case of χ0, indicated by black arrows), while theouter one is comparatively small (green arrows). Interestingly,when U > 2.2 eV, the situation is dramatically reversed andthe outer nesting develops remarkable instability propertiesaround both q = (π,π ) and q = 0, although the former isalways dominant in the range of U studied here. With carefulobservation, one can find that the strong intensity within theRPA is shifted toward q = (π,0.5π )/(0.5π,π ). This specificincommensurate vector is responsible for the possible 45◦rotation of the dynamical spin resonance spectra studiedbelow.

The small q = (0,0.46π )/(0.46π,0) nesting, which we findto be considerably large for our choice of parameters, is close tothe q = 0 ferromagnetic (FM) instability or may induce someform of charge-density wave (CDW) with finite q modulation.Note that the coexistence of a CDW and SDW due to multiplenestings is an emergent phenomenon predicted in pnictide32

and cuprates.31,33 The presence of vacancy order in thesesystems can promote the coexistence of several competetingorders.34,35 Scanning tunneling microscopy (STM) may beable to detect it.

Unlike the (π,0)-CDW order in cuprates or the (π,0)-SDWorder in pnictide, here the q = (π,0.5π )/(0.5π,π )-stripe SDWorder will facilitate d-wave pairing. This is due to the fact thatthe q = (π,0.5π )/(0.5π,π ) nesting comes from the interelec-tron pockets at which the d-wave pairing possesses a changein sign of the gap, which is the criterion for spin resonanceto occur as calculated below. But q = (0,0.46π )/(0.46π,0)instability will provide pair-breaking contributions to thed-wave symmetry.

IV. DYNAMICAL SUSCEPTIBILITY, d-WAVE PAIRING,AND SPIN RESONANCE

Based on these findings, we proceed to study the evolutionof the spin resonance spectra in the SC state, i.e., χ ′′(q,ω)(RPA-BCS), which is directly measured by an inelastic neutronscattering (INS) experiment. In the SC state, χ ′

0(q,ω) isstrongly enhanced for ω < 2 due to turning-on of thepair-scattering terms. This reduces the critical value of theinteraction U to satisfy the RPA denominator to be positiveat Uχ ′

0(q,ω) � 1. We show the resonance spectrum forU = 1.5 eV in Fig. 3(a) and that for U = 2.14 eV inFig. 3(b), which obtain a commensurate and an incommensu-rate resonance, respectively, in addition to their characteristicdispersion.

The dispersion of χ ′′(q,ω), which has a one-to-one cor-respondence with the dispersion of χ ′

0(q,ω), comes fromthree conditions27,29,30,36 in addition to the matrix-elementeffects: (1) ωres(q) = ∑

kνF ,kν′

F|kν

F| + |kν′

F|, (2) sgn(kν

F) �=

sgn(kν′F

), and (3) q = kνF + kν ′

F , where ν,ν ′ are the bandindices. Condition (1) comes from the energy conservationof inelastic scattering of Bogoliubov quasiparticles, while(2) constitutes the coherence factors of BCS susceptibility topossess a nonzero value. As both FS pockets have the same gap

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FIG. 3. (Color online) Computed BCS-RPA χ ′′(q,ω) for (a) U = 1.5 eV and (b) U = 2.14 eV. The constant-energy spectra at tworepresentative energies are shown in the lower panels [(e)–(h)] of the corresponding spectra. The right-hand column shows the neutron dataon FeTe1−xSex , demonstrating the observations of (c) an upward dispersion,21 (d) an “hourglass” or “camel-back” pattern,22 and (i), (j) 45◦

rotation of the spectra23 in this compound.

symmetries and amplitudes, and particularly for FeSe systems,where all the FS pockets have isotropic gaps at all kF values,conditions (1) and (2) are satisfied equally at all kF valuesfor inter-electron-pocket scatterings with dx2−y2 symmetry. Nointensity modulation in χ0(q) is governed by conditions (1) and(2), and the only source of intensity variation is the interplaybetween the orbital matrix-element effect and condition (3);the latter is simply related to the number of degenerate kF

values. This means that the upward dispersion of the resonancebehavior in FeSe cannot be understood by tracking how thegap varies on the FS as happens in cuprates,36 but only fromthe FS nesting and the matrix element in the tensor form of χ0

and, also, on the interaction vertex Us .27

Furthermore, unlike in cuprates, where the hourglassdispersion extends to ω = 0 due to the presence of nodalquasiparticles on the FS, in the nodeless FS of FeSe compoundsthe resonance behavior spans only from to 2. For U =1.5 eV, the maximum intensity lies at the bottom of thespectrum at the commensurate vector at ωres = 13.5 meVwith ωres/2 = 0.79. Again, for U = 2.14 eV, the maximumintensity shifts to the top of the spectra at an incommensuratevector q = 0.78(π,π ) at ωres = 12.4 meV with ωres/2 =0.73, slightly larger than the average value of 0.64 foundin all other superconductors.38 The upward dispersion ofthe magnetic spectrum is qualitatively analogous to theone observed by neutrons in FeTe0.6Se0.4 superconductors21

shown in Fig. 3(c), while pnictides show more commensurateresonance within the experimental resolution.20 By includingthe weak intensity of the dispersion below the (π,π ) resonance,one can draw an hourglass or camel-back feature, also seen inchalcogenides,22 in Fig. 3(d).

The constant-energy profile of the resonance spectra ismore interesting and bears important physical insights. Atω � ωres for both values of U , the resonance profile is squarish,with the maximum intensity lying along the diagonal and thefinite intensity along the (100) direction [see Figs. 3(f) and3(h)]. For U = 1.5 meV, the intensity remains concentrated at(π,π ) even when ω � ωres. Interestingly, for U = 2.14 eVat ω � ωres the peak along the diagonal sharply loses in-tensity, while the intensity peak along the bond directionremains almost the same. As a result, the resonance profileis rotated by 45◦ as one goes below the resonance. The45◦ rotation of the resonance spectra is observed earlier inhole-doped cuprates17 and can also be seen in chalcogenidesbetween two different dopings in the SC state23 in Figs. 3(i)and 3(j).

V. ONE-FE-UNIT CELL TO TWO-FE-UNITCELL CONVERSION

It has been argued that when FS pockets are translatedfrom a one-Fe unfolded zone to a two-Fe folded zone, itswitches location from k = (π,0)/(0,π ) to k = (π,π ). Thusin dx2−y2 symmetry, the SC gap on the FS is assumed tochange from nodeless to nodal behavior.39 We show that thisapparent inconsistency is merely a result of misinterpretationcoming from not performing the same unitary transformationin the gap symmetry as in the FS. To perform this transfor-mation, we denote a quantity with a tilde in the 2-Fe-unitcell.

The unitary transformation consists of 45o rotation ofthe lattice with a = √

2a which gives kx/y = (kx ± ky)/2.

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FIG. 4. (Color online) Correspondence between (a) the 1-Fe-unit cell and (b) the 2-Fe-unit cell in the FS topology and SC gap symmetry.The black-to-white color scale denotes the negative-to-positive value of the gap. (c) Transformation of the dxy-wave gap and the FS are plottedin the same 2-Fe-unit cell as in (b) but with one Fe vacancy. (d)–(e) Real-space view of the gap symmetry corresponding to momentum-spaceones given in (b) and (c). The dx2−y2 wave in the 1-Fe-unit cell (magenta box) automatically takes the form of a dxy = sin(kxa/2) sin(kya/2)wave in the 2-Fe-unit cell (blue box). (e) Vacancy order obtained from the experimental data in Ref. 37. Dashed yellow lines in each plot denotethe nodal lines for the corresponding pairing symmetry.

In doing so, we find that the pairing symmetry dx2−y2 =cos(kxa) − cos(kya) in the 1-Fe-unit cell transforms to an ex-tended dxy = sin(kxa/2) sin(kya/2) [see Figs. 4(a) and 4(b)].This transformation can be more clearly understood fromthe real-space view of the gap symmetry in the lattice inFig. 4(d). The 1-Fe-unit cell and 2-Fe-unit cell are highlightedby magenta and blue boxes, respectively. It is clear thatthe nodal line passes through the diagonal direction in theformer case, which automatically becomes aligned alongthe zone boundary in the latter unit cell. The rotation of theSC gap phases follows similarly. We note that such unitarytransformation preserves the nodeless and isotropic propertiesof the SC gap in any unit cell.

An important consequence of the dxy-wave SC gap is thatit breaks the crystal symmetry. As can be seen from themomentum-space view of the pairing phases in Fig. 4(b), forexample, as we translate from k = (−π/a, − π/a) to k =(π/a, − π/a) by the reciprocal vector 2π/a, the phase of theSC gap is reversed. One can recover the translational symmetryof the crystal by moving to k = (2π/a, − π/a) by a reciprocalvector of 4π/a. In the real-space approach, this would corre-spond to a case when one Fe atom from the center of the 2-Fe-unit cell is removed [see Fig. 4(e)]. Remarkably, such a crystalsymmetry can be achieved when the vacancy order and themagnetic order conspire with each other to favor the dxy-wave

SC gap. This vacancy arrangement was found indeed to occurin the real system.34 In this context, we note that superconduc-tivity occurs in this class of materials when a suitable amountof vacancy is achieved, which forms order, and the crystalsymmetry hops from the I4/mmm to the I4/mm group.35,37

A. Rotation of spin resonance in the 2-Fe-unit cell

The same unitary transformation leads to both the nestingvectors in the 1-Fe-unit cell shown in Fig. 2(a) coincidingwith each other around q = (π/2,π/2) in the 2-Fe-unit cell asshown by the blue arrows in Figs. 5(a) and 5(b). Subsequentlythe nesting along the (200) direction shown by the greenarrow serves as the hot spot to the dxy , where the SCgap changes signs. This leads to resonance spectra around(2π ± δ,0)/(0,2π ± δ) as shown in Fig. 5(d).

Note that the smaller intrapocket hot-spot vector (bluearrow) will give a sign change of the SC gap on the sameFS pocket, and thus a nodal SC gap (dx2−y2 in the 2-Fe-unitcell and dxy in the 1-Fe-unit cell] will occur. The resultingINS spectrum in a nodal SC state is confined near q ∼ 0 anddoes not show any true resonance peak, but a spin-excitationdispersion, as shown in Fig. 5(e). In this case the associatedweak magnetic excitation strength may not be sufficient toproduce large values of Tc in these materials.

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FIG. 5. (Color online) (a) FSs in the 2-Fe-unit cell BZ are shownwith various nesting channels (arrows). (c) Bare static susceptibilityχ ′

0 is plotted in a 2-Fe-unit cell. (c) Corresponding RPA instability atU = 1.7 eV. The unitary transformation leads to both nesting vectorsof the 1-Fe-unit cell, shown in Fig. 2(a), coinciding around q ∼(π/2,π/2). (e) The dynamical spin suscebtibility χ ′′ is calculatedat U = 1.8 eV as shown along the zone boundary [green arrow in(b)] for dxy pairing in the 2-Fe-unit cell (dx2−y2 in 1-Fe-unit cell).(e) Computed spin susceptibility is plotted for nodal dx2−y2 pairingin the 2-Fe-unit cell (dxy in 2-Fe-unit cell). Here U is chosen to be1.5 eV.

VI. TEMPERATURE DEPENDENCE OF THE SPINRESONANCE AT Q

In pnictide and chalcogenide superconductors the tem-perature evolution of the spin resonance follows the BCSmean-field behavior of the SC gap (T ).20,24 We predict thata similar phenomenon also occurs in FeSe superconductors.To prove that, we calculate the (T ) for the two-band systemsolving the standard BCS gap equation with phenomenologicalpairing strength parameters V1,2 = 52 and 46 meV (neglectinginterband pairing). The parameters are adjusted to obtain anexperimental gap of 0 = 8.5 meV with a BCS ratio of2/kBTc = 6.7, in close agreement with the experimentalvalue of 7.5 (T ) is shown in Fig. 6(a) (cyan line) on top of thecalculated RPA-BCS spin resonance spectrum at q = (π,π )with U = 1.5 eV. The intensity at the resonance is plotted ininset to Fig. 6(a) in red line. Both the resonance energy andthe intensity shows a remarkable one-to-one correspondence

FIG. 6. (Color online) (a) Temperature dependence of the spinexcitation spectrum at q = (π,π ) for U = 1.5 eV is compared withsimilar observations by inelastic neutron scattering measurements(b) in pnictide20 and (c) in chalcogeniges.24 Inset: Computed value ofthe intensity of the resonance peak is compared with BCS (T ).

to the value of (T ). The results agree well with those forpnictide and chalcogenide, shown in Figs. 6(b) and 6(c),respectively. The T dependence of the resonance energy andits intensity follows the BCS form of (T ), suggesting thatthe itinerant RPA-BCS Fermi-liquid theory will be valid inFeSe-based superconductors, in particular, and in all iron-based superconductors studied to date, in general.

VII. FIVE-BAND MODEL

We repeat the calculation presented in Figs. 1–3 withina five-band model and find that the magnetic structure andmagnetic resonance do not depend on the overall band structurebut are essentially governed by the FS topology. The TB modelfor five bands is taken from Ref. 40, which includes the five d

orbitals of the Fe atoms in the 1-Fe-unit cell. The band structureis given in Fig. 7(a) and the corresponding FS is shown inFig. 7(b). The topology of the FS is qualitatively capturedhere, although the five-band model does not incorporate thenearly degenerate FSs as calculated in the LDA. Nevertheless,this minor technical drawback does not alter our conclusions.

The magnetic structure including the two nesting vectorsat q = (π,0.5π )/(0.5π,π ) and q = (0,0.46π )/(0.46π,0) iswell reproduced in the five-band calculation as shown inFigs. 7(b) and 7(c). Furthermore, the magnetic resonancespectrum is remarkably close to the one obtained in Fig. 3.The subtle differences between the two-band and the five-bandcalculation come from the inconsistency in the FS topology,not in the overall band dispersion. The remarkable consistencybetween the two-band and the five-band model suggests thatthe magnetic ground state as well as the SC properties can bestudied properly once all the FS pieces are modeled accurately.

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FIG. 7. (Color online) Calculations with the five-band model inthe 1-Fe-unit cell.40 (a) The calculated band structure is drawn alongthe high-symmetry lines for a chemical potential μ = 7.45 eV. (b) Thecorresponding electron-pocket FS is plotted in the first quadrant of theBrillouin zone. (c) The bare and (d) the RPA component of the staticspin susceptibility. (e) Computed spin excitation spectrum is shownalong the diagonal direction in a dx2−y2 -wave pairing symmetry. Theseresults should be compared with the results obtained with the two-band model in the 1-Fe-unit cell in Figs. 1–3. The value of U for allRPA calculations here is set to be 1.5 eV.

VIII. CONCLUSION

In summary, we have calculated the static and dynamicalspin susceptibility for an effective two-orbital TB modelwith d superconductivity. We find that the leading insta-bility in χ ′ is evolved around an incommensurate vectorq = (π,0.5π ) at some critical value of U , pinning stripe-like SDW or AFM order, in agreement with a number ofexperiments.11–16 Unlike in other iron-based superconductors,here the SDW order has the same q vector for the d pairingsymmetry to obtain the opposite sign at the “hot spot.” Ourchoice of parameters also predicts a comparatively weakCDW modulation with q = (0,0.46π ), similar to cuprates31,33

and pnictides.32 The d-wave gap yields a spin resonancebehavior with an “hourglass” or upward dispersion and a45◦ rotation of the resonance profile, in close agreementwith observations in chalcogenides. The present results alsodemonstrate that the weak to moderate pair-coupling the-ory of the spin fluctuation mechanism of superconductiv-ity is also appropriate in FeSe-based superconductors, inparticular, and in all iron-based superconductors studied todate, in general. A similar intermediate coupling theoryhas been successful in describing many saliant features incuprates.41,42

Recently, we became aware of the works by Wang et al.40

and Maier et al.43 which also predicted the d-wave gap in FeSebased systems.

ACKNOWLEDGMENTS

We are grateful to H. Ding, A. V. Chubukov, and W. Bao foruseful discussion. This work was funded by the US DOE, BES,and LDRD and benefited from the allocation of supercomputertime at the NERSC.

*tnmydas@gmail.com1J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He, andX. Chen, Phys. Rev. B 82, 180520(R) (2010).

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