Superconducting proximity effect to the block antiferromagnetism in K y Fe 2−x Se 2
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Superconducting proximity effect to the block antiferromagnetism in K y Fe − x Se Hong-Min Jiang,
1, 2
Wei-Qiang Chen,
3, 1
Zi-Jian Yao, and Fu-Chun Zhang
1, 4 Department of Physics and Center of Theoretical and Computational Physics,The University of Hong Kong, Hong Kong, China Department of Physics, Hangzhou Normal University, Hangzhou, China Department of Physics, South University of Science and Technology of China, Shenzhen, China Department of Physics, Zhejiang University, Hangzhou, China (Dated: November 2, 2018)Recent discovery of superconducting (SC) ternary iron selenides has block antiferromagentic(AFM) long range order. Many experiments show possible mesoscopic phase separation of thesuperconductivity and antiferromagnetism, while the neutron experiment reveals a sizable suppres-sion of magnetic moment due to the superconductivity indicating a possible phase coexistence.Here we propose that the observed suppression of the magnetic moment may be explained due tothe proximity effect within a phase separation scenario. We use a two-orbital model to study theproximity effect on a layer of block AFM state induced by neighboring SC layers via an interlayertunneling mechanism. We argue that the proximity effect in ternary Fe-selenides should be largebecause of the large interlayer coupling and weak electron correlation. The result of our meanfield theory is compared with the neutron experiments semi-quantitatively. The suppression of themagnetic moment due to the SC proximity effect is found to be more pronounced in the d -wavesuperconductivity and may be enhanced by the frustrated structure of the block AFM state. PACS numbers: 74.20.Mn, 74.25.Ha, 74.62.En, 74.25.nj
I. INTRODUCTION
The recent discovery of high- T c superconductivityin the ternary iron selenides A y Fe − x Se (A=K; Rb;Cs;...) has triggered a new surge of interest in study ofiron-based superconductors (Fe-SC). The fascinating as-pect of these material lies in the tunable Fe-vacancies inthese materials, which substantially modifies the normal-state metallic behavior and enhances the transition tem-perature T c to above 30K from 9K for the binary sys-tem FeSe at ambient pressure. Particular attentionhas been focused on the vacancy ordered 245 system,K . Fe . Se , as it introduces a novel magnetic struc-ture into the already rich magnetism of Fe-SC. Unlike thecollinear or bi-collinear AFM order observed in theparent compounds of other Fe-SC, the neutron diffractionexperiment has clearly shown that these materials havea block AFM (BAFM) order. Meanwhile, the AFM or-der with an unprecedentedly large magnetic moment of3 . µ B /Fe below the N´eel temperature is the largest oneamong all the known parent compounds of Fe-SC. Moreover, the carrier concentration is extremely low, in-dicating the parent compound to be a magnetic insu-lator/semiconductor, in comparison with a metallicspin-density-wave (SDW) state of the parent compoundin other Fe-SC.
The relation between the novel magnetism and super-conductivity in ternary Fe selenides is currently an in-teresting issue under debate. The question is whetherthe superconductivity and the BAFM order are phaseseparated or co-exist in certain region of the phase di-agram. The neutron experiment shows the suppressionof the AFM ordering below SC transition point, sug-gesting the coexistence. Some other experiments, such as two-magnon Raman-scattering, and muon-spin ro-tation and relaxation are consistent with this picture.On the other hand, the ARPES, NMR and TEM experiments indicate a mesoscopic phase separationbetween the superconductivity and the insulating AFMstate. Most recently, Li et al. showed the superconduc-tivity and the BAFM orders to occur at different layersof the Fe-selenide planes in the STM measurement. The vacancy in Fe-selenides is an interesting but com-plicated issue. The vacancy in the Fe-selenide carries anegative charge since the Fe-ion has a valence of 2+. Inthe equilibrium, we expect the vacancies to repel eachother at short distance for the Coulomb interaction andto attract to each other at a long distance for the elasticstrain. Such a scenario would be in favor of the phase sep-aration to form a vacancy rich and vacancy poor regionsin the compound. The challenge is then to explain the ob-served suppression of magnetic moment of the BAFM dueto the superconductivity. At the phenomenological level,the suppression of magnetism due to superconductivityhas been reported previously, and such phenomenonmay be explained by Ginzburg-Landau theory. In this paper, we propose that the proximity effectof superconductivity to the BAFM in a mesoscopicallyphase separated Fe-selenides may be large to accountthe suppression of the AFM moments observed in neu-tron experiment. More specifically, we use a microscopicmodel to study the proximity effect on a layer of theBAFM state induced by adjacent SC layer. The prox-imity effect in Fe-selenides is expected to be importantfor the two reasons. One is the weaker correlation ef-fect, and the other is the larger interlayer hopping am-plitude, compared with those in cuprates. Both of themmay enhance proximity effect on the magnetism from theneighboring SC layer. Our model calculations show theproximity effect in a mesoscopically phase separated stateof Fe-selenides may explain various seemingly conflictedexperiments.
II. MODEL AND MEAN FIELD THEORY A y Fe − x Se is a layered material with FeSe layers sepa-rated by alkali atoms, similar to the 122 material in ironpnictides family. To investigate the proximity effect tothe BAFM layer, we consider a single BAFM layer nextto a SC layer as shown schematically in Fig. 1. The elec-tronic Hamiltonian describing the BAFM layer is givenby H = H + H inter , (1)where H describes the electron motion and spin cou-plings in the BAFM layer and H inter describes the cou-pling to the neighboring SC layer. We consider a two-orbital model to describe H , H = − X ij,αβ,σ t ij,αβ C † i,ασ C j,βσ − µ X i,ασ C † i,ασ C i,ασ + J X
13 eV; they are exchanged to the x di-rection; the NN interorbital hoppings are zero; the NNNintraorbital hopping integral t = − .
25 eV for both d xz and d yz orbitals, and the NNN interorbital hopping is t = 0 .
07 eV. The hopping integral t is taken as the en-ergy unit. We keep J : J ′ : J : J ′ = − − The doping level is given by δ = n − . III. RESULTS
To begin with, we present the energy band structureat half filling with n = 2 in Fig. 2(a), where J = 2 . ∼ For theelectron doping with n = 2 .
1, the Fermi level crosses anenergy band around the center of the Brillouin zone [Γpoint in Fig. 2(b)], while it intersects with an energy bandaround the zone corner at the hole doping with n = 1 . M point in Fig. 2(c)]. Although a simple two-orbitalmodel is adopted here, both the electron and hole dop-ing cases with δ = 0 . In the presence of theordered vacancies and BAFM order, the original two-band structures are splitting to sixteen subbands as aresult of the enlarged unit cell with 8 sites. At half fill-ing, 8 lower bands are occupied, i.e., 1 / δ = 0 .
1, the chemicalpotential crosses one subband which produce the char-acteristic features of the Fermi surface and the metallicBAFM state.
FIG. 2: (color online) Electronic band structures of H givenby Eq. (2). The parameters are given at the end of section IIof the text, and J = 2 .
0. (a): at half filling or n = 2 .
0; (b):at electron doping n = 2 .
1; and (c): at hole doping n = 1 . Motivated by the agreement of the self-consistentmean-field solutions with the mentioned first principlecalculations, we consider now the proximity effect inBAFM layer induced by the SC in SC layer. For ex-plicit reason, we choose two possible singlet pairing sym-metries in the SC layer, i.e., the NNN s ± -wave and theNN d -wave symmetries with their respective gap func-tions ∆ s ± = ∆ cos( k x ) cos( k y ) and ∆ d = ∆ [cos( k x ) − cos( k y )], where the former results in the NNN bond andthe latter the NN bond couplings in the BAFM layer.The interlayer hopping constant t τ is assumed to be siteindependent. Fig. 3 displays the moment of the BAFMorder as a function of the effective tunneling strength V τ,ij . At the half filling, both symmetries of the SC or-der in the SC layer introduce the decrease of the BAFMorder and simultaneously induce the SC correlation withthe same symmetries in the BAFM layer as the tunnel-ing strength increases. A main difference between the s ± - and the d -wave symmetries is the more pronouncedproximity effect in reducing the moment of the BAFMorder produced by the d -wave symmetry as the tunnel-ing strength increase, as shown in Figs. 3(a) and 3(b).In the case of electron doping with n = 2 .
1, where themetallic BAFM state results, although the proximity ef-fect is more pronounced, the magnetic and the inducedSC correlation remain the qualitatively unchanged, duepossibly to the very low total carrier concentration.The unique feature of the the effective tunneling inthe second order is it’s temperature dependence via theSC pairing ∆ ij,αα ′ ,σ,σ ′ , which differs from that in oneparticle tunneling process. The temperature depen-dence of the SC pairing parameter is modeled by aphenomenological form with ∆ = ∆ p − T /T c . We FIG. 3: (color online) Block AFM moment and the SC pair-ing correlation as functions of the effective tunneling strength V τ,ij . Black curves are for next nearest neighbor s ± -wavepairing, and red for nearest neighbor d -wave pairing. Up-per panel (a) and (b): n = 2 . n = 2 . present the temperature dependence of the magnetic mo-ment in Fig. 4(a) for the typical choice of the couplingconstants g τ = 2 V τ,ij ∆ = 0 .
25. As temperature de-crease, the magnetic order increases when temperature isabove T c , while it decreases when temperature is below T c , resulting in a broad peak around T c . We note that thetemperature dependence of the AFM moment is reminis-cent of the neutron diffraction and the two-magnon ex-periments [Fig. 4(b)]. There is another scenario thatthe competition between the AFM and the SC orders inthe microscopic coexistence of them may also producethe decrease of the AFM moment below T c . The studyof such possibility is currently under way and the resultswill be published elsewhere. It is worthwhile to noticethat the sizable proximity effect relies on the substantialinterlayer hoping constant t τ . Based on the first princi-ple calculation, the interlayer hopping t τ was estimatedto have a comparable magnitude with t possibly due tothe high values of electron mobility from the intercalatedalkaline atoms, and leads to the highly three dimen-sional Fermi surface. IV. SUMMARY AND DISCUSSIONS
In summary, we have proposed that various seeminglyconflict experiments on the phase separation or coexis-tence of superconductivity and BAFM may be explainedwithin a phase separation scenario by taking into ac-count of the proximity effect of superconductivity to theneighboring layer of BAFM. We have theoretically stud-ied the proximity effect to a BAFM layer induced byadjacent SC layers in a simplified two-orbital model forFe-selenides. The proximity effect in reducing the mo-
FIG. 4: (color online) (a) Temperature dependence of theblock AFM moment at n = 2 .
0. Black and red curves are fornext nearest neighbor s ± -wave and nearest neighbor d -wavepair couplings, respectively. (b): the re-plotted curve of theneutron data from Ref. 11. ment of the BAFM order highly depends on the couplingconstant V τ,ij . For realistic parameters of the interlayertunneling, our calculation shows that the superconduc-tivity proximity effect may result in substantial suppres-sion of the magnetic moment. This is in contrary to thatin the cuprate superconductor, where the coupling con-stant V τ,ij is very small because of small c-axis hoppingintegral due to the large anisotropy, and because of therenormalization of V τ,ij by a factor proportional to holeconcentrations due to the no double-occupation condi-tion. In iron-based superconductor, the anisotropy ofiron-based material is much smaller than in the cuprate,which lead to a relative larger t τ . And the moderatecorrelation effect in iron-based superconductor leads to amoderate renormalization factors. As a consequence, thecoupling constant V τ,ij in iron chalcogenide superconduc-tor should be moderate.We remark that we’d be careful in drawing a concreteconclusion to compare with the experiments. The ap-proximation that only d xz and d yz orbitals are impor-tant in the bands close to Fermi energy is good in termsof the band structures. But the maximum magneticmoment in two-orbital model is only 2 µ B, smaller thanthe moment of 3 . µ B measured in experiments.
Theother effect is that we only calculated the suppressionof the BAFM order of the surface layer of the BAFMdomain. According to TEM experiment, each BAFMdomain has around ten layers. And the suppression ofBAFM order of the layers in the middle of domain maybe more complicated. In brief, the suppression of theBAFM moment is sizable because of the moderate cou-pling constant V τ,ij , and our calculation may be viewedas a semi-quantitative result.We also investigated proximity effect for various pair-ing symmetry of the SC phase. It has shown that the SCpairing with NN d -wave symmetry resulted a more pro-nounced proximity effect in reducing the moment of theBAFM order than the NNN s ± -wave pairing. The secondorder process induced proximity effect has a temperaturedependent as the SC pairing, which may be relevant tothe experimental observations. More remarkable proxim-ity effect was found in the BAFM state by comparisonwith the conventional AFM state, which was the con-sequence of the frustrated structure and the associatedanisotropic exchange interactions. V. ACKNOWLEDGEMENT
We thank W. Bao, G. Aeppli, Y. Zhou, and T.M. Rice for helpful discussions. This work is sup-ported in part by Hong Kong’s RGC GRF HKU706809and HKUST3/CRF/09. HMJ is grateful to the NSFC(Grant No. 10904062), Hangzhou Normal University(HSKQ0043, HNUEYT).
VI. APPENDIX
In the following, we compare the above proximity effectwith that in the single band conventional AFM (CAFM)system. In order to make the comparison more con-vincing, we choose the dispersion ε k = − t [cos( k x ) +cos( k y )] − t ′ cos( k x ) cos( k y ) − µ with t = t and t ′ = t ,which gives rise to the similar energy band width withthat in the above two-orbital model and is close to thecase of the cuprates. The AFM order is introduced by theAFM exchange interaction J P h ij i S i · S j between the NNsites. At the half filling n = 1, we find that J = 1 . V τ,ij . The upper panel shows the results for the NNN s ± -wave pairing and the lower panel the results for the NN d -wave pairing. In the figure, the magnitude of the mag-netic order in both cases is renormalized. The proximityeffect in reducing the AFM order is more pronounced forthe BAFM state as shown in Figs. 5(a) and 5(c). As forthe induced pairing correlation, the larger correlation isfound in the BAFM state for the s ± -wave paring and inthe CAFM state for the d -wave pairing, as displayed inFigs. 5(b) and 5(d), respectively.We can understand the above results by consideringthe different spin configurations of the BAFM and CAFMorders, as shown in Fig. 6. In the BAFM state, when twoelectrons transfer from the BAFM layer to the SC one,the energy changes due to the bonds breaking for theNN bond coupling are ∆ E ↑↑ NN = | J | [A1 and A2 bondsin Fig. 6(a)] and ∆ E ↑↓ NN = 7 | J | / E ↑↑ NNN = 9 | J | / E ↑↓ NNN = 5 | J | / d -wave pairing is more remark-able than that by the s ± one. However, the energychanges due to the bonds breaking in the CAFM state are FIG. 5: (color online) Comparison of the proximity effectbetween the block and conventional AFM states. Left columnshows the moment of the AFM order, and right column theSC pairing correlation as functions of the effective tunnelingstrength V τ,ij . Upper panel: for the next nearest neighbor s ± -wave pairing and lower panel for the nearest neighbor d -wavepairing.FIG. 6: (color online) Comparison of the spin structures andtheir respective NN and NNN bonds. (a): block AFM state;(b): conventional AFM state. ∆ E NN = 7 | J | [C bond in Fig. 6(b)] and ∆ E NNN = 8 | J | [D bond in Fig. 6(b)] for the NN and NNN band cou-plings. Therefore, the proximity effect in reducing themoment of the AFM in the CAFM state is rather weakfor both the s ± - and d -wave pairing couplings. As forthe induced pairing correlation, the extent of the matchbetween the AFM order configuration and the singlet SCpairing largely determines the magnitude of the inducedpairing correlation. For example, the CAFM matcheswell with the NN d -wave pairing, so that one can ex-pect a large induced pairing correlation without the se-vere decrease of the AFM order as displayed in Figs. 5(c)and 5(d). J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M.He, and X. 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