The Unexpected Properties of Alkali Metal Iron Selenide Superconductors
TThe Unexpected Properties of Alkali Metal Iron Selenide Superconductors
Elbio Dagotto
1, 2 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996 Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 (Dated: October 31, 2018)
The iron-based superconductors that contain FeAs layers as the fundamental building block inthe crystal structures have been rationalized in the past using ideas based on the Fermi Surfacenesting of hole and electron pockets when in the presence of weak Hubbard U interactions. Thisapproach seemed appropriate considering the small values of the magnetic moments in the parentcompounds and the clear evidence based on photoemission experiments of the required electronand hole pockets. However, recent results in the context of alkali metal iron selenides, withgeneric chemical composition A x Fe − y Se ( A = alkali element), have drastically challenged thoseprevious ideas since at particular compositions y the low-temperature ground states are insulatingand display antiferromagnetic magnetic order with large iron magnetic moments. Moreover, angleresolved photoemission studies have revealed the absence of hole pockets at the Fermi level inthese materials. The present status of this exciting area of research, with the potential to alterconceptually our understanding of the iron-based superconductors, is here reviewed, covering bothexperimental and theoretical investigations. Other recent related developments are also brieflyreviewed, such as the study of selenide two-leg ladders and the discovery of superconductivityin a single layer of FeSe. The conceptual issues considered established for the alkali metal ironselenides, as well as the several issues that still require further work, are discussed in the text. I. INTRODUCTION
One of the most active areas of research in CondensedMatter Physics at present is the study of the high crit-ical temperature ( T c ) superconductors based on iron.This field started with the seminal discovery of super-conductivity at 26 K in F-doped LaFeAsO (Kamihara etal. , 2008). Several other superconductors with a simi-lar structure were synthesized since 2008 (for a reviewsee Johnston, 2010; Stewart, 2011). They all have FeAsor FeSe layers that are widely believed to be the keycomponent of these iron-based superconductors, similarlyas the CuO layers are the crucial ingredients of the fa-mous high T c cuprates (Dagotto, 1994; Scalapino, 1995).The many analogies between the iron-based supercon-ductors and the cuprates lie not only on the quasi two-dimensional characteristics of the active layers, but alsoin the proximity to magnetically ordered states that inmany theoretical approaches are believed to induce su-perconductivity via unconventional pairing mechanismsthat do not rely on phonons. However, at least for thecase of the iron-superconductors based on As, the par-ent magnetic compounds are metallic, as opposed to theMott insulators found in the cuprates, establishing animportant difference between cuprates and pnictides.The FeAs tetrahedra is the basic building block ofthe FeAs layers. Materials such as LaFeAsO belong tothe “1111” family, with a record critical temperature of55 K for SmFeAsO (Ren et al. , 2008). Subsequent effortsunveiled superconductivity also in the doped versions of“122” compounds such as BaFe As , “111” compoundssuch as LiFeAs, and others (Johnston, 2010; Paglioneand Greene, 2010; Stewart, 2011; Wang and Lee, 2011;Hirschfeld, Korshunov, and Mazin, 2011).It is important to remark that there are structurally related materials, known as the “11” family, that displayequally interesting properties. A typical example is FeSethat also superconducts, although at a lower T c of 8 K(Hsu et al. , 2008). FeSe has a simpler structure than thepnictides since there are no atoms in between the FeSelayers. Locally, the iron cations are tetrahedrally coordi-nated to Se, as it occurs in FeAs . The critical tempera-ture can dramatically increase by Te substitution or evenmore by pressure up to 37 K (Fang, M. H., et al. , 2008 ;Yeh et al. , 2008; Margadonna et al. , 2009). The normalstate of Fe(Se,Te) is electronically more correlated thanfor iron pnictides (Tamai et al. , 2010). The study of ironsuperconductors based on Se (the iron selenides) is lessadvanced than the similar studies in the case of As (theiron pnictides), and it is precisely the goal of this reviewto focus on the most recent developments in the area thatis often referred to as the “alkaline iron selenides,” withan alkali metal element intercalated in between the FeSelayers. Note that this set of compounds should be bettercalled “alkali metal iron selenides” to avoid a confusionwith the “alkaline earth metals” (Be, Mg, Ca, Sr, Ba, andRa). For this reason, in this review the more precise no-tation alkali metal iron selenides will be used. Also themore general term chalcogenides will not be used heresince our focus below is exclusively on compounds withFeSe layers, not with FeTe layers. At present, the fieldof alkali metal iron selenides is receiving considerable at-tention not only because the T c s are now comparableto those of the iron pnictides but also because some ofthese selenides are magnetic insulators, potentially bring-ing closer the fields of the iron-superconductors and thecopper-superconductors.One of the motivations for the use of alkali elements toseparate the FeSe layers is that the T c of the iron-basedsuperconductors appears to be regulated by the “anion a r X i v : . [ c ond - m a t . s up r- c on ] O c t height,” i.e. the height of the anion from the iron-squarelattice planes (Mizuguchi et al. , 2010). Alternatively, ithas been proposed that the closer the Fe Anion is tothe ideal tetrahedron, the higher T c becomes (Qiu et al. ,2008). Then, via chemical substitutions or intercalations T c could be further enhanced since that process will alter,and possibly optimize, the local structure.In this manuscript, this very active field of “alkalimetal iron selenides” will be reviewed. Before explainingthe organization of this article, it is important to remarkthat this is not a review of the full field of iron-basedsuperconductors, which would be a formidable task. In-stead the focus is on the recent developments for com-pounds with chemical formulas A x Fe − y Se ( A = alkalielement) that not only show superconductivity at tem-peratures comparable to those of the pnictides, but theyalso present insulating and magnetic properties at severalcompositions, establishing a closer link to the cuprates.In fact, many studies reviewed below suggest that aproper description of A x Fe − y Se requires at the mini-mum an intermediate value of the Hubbard repulsion U in units of the carriers’ bandwidth. This degree of elec-tronic correlation is needed, for instance, to explain thelarge magnetic moment per iron observed in these novelcompounds. Last but not least, the notorious absenceof Fermi Surface (FS) hole pockets in these materials,as also reviewed below, prevents the applicability of theideas widely discussed for the iron pnictides that rely onthe FS nesting between electron and hole pockets. Sincethere are no hole pockets, an alternative starting pointis needed to explain the physics of the iron selenides. Itis fair to express that pnictides and selenides may be indifferent classes of magnetic and superconducting materi-als, even if the pairing arises in both cases from magneticfluctuations. For instance, the former could be based onitinerant spin density wave states, while the latter couldarise from local moments. However, mere simplicity alsosuggests that pnictides and selenides may share a uniquemechanism to generate their magnetic and superconduct-ing states. If this is the case, then learning about thephysics of the A x Fe − y Se compounds may drasticallyalter the conceptual framework used for the entire fieldof research centered at the iron-based superconductors.The organization of the review is as follows: In SectionII, the early history of the alkali metal iron selenides isprovided, with information about the crystal structure,basic properties, and the ordered states of the iron vacan-cies. In Section III, investigations using angle-resolvedphotoemission are reviewed, with emphasis on the twomost important results: absence of hole pockets at the FSand isotropic superconducting gaps. Section IV containsthe neutron scattering results, showing the exotic mag-netic states in the presence of iron vacancies, particularlythe block antiferromagnetic state. Section V addressesthe existence of phase separation into superconductingand magnetic regions, and also the much debated issueof which states should be considered the parent statesfor superconductivity. Results obtained using a variety of other experimental techniques are in Section VI. Theo-retical calculations, using both first principles and modelHamiltonian approaches, are in Section VII. The exper-imentally observed phases are discussed from the theoryperspective, as well as a variety of competing states. Sec-tion VIII describes recent efforts focussed on two-leg lad-der selenides, which display several common aspects withthe layered iron selenides. Finally, in Section IX severalclosely related topics are discussed, including the discov-ery of superconductivity in a single layer of FeSe. Dueto length constrains some topics that would make thisreview self-contained, such as the crystallography of thematerials of focus here, cannot be included. However,recent reviews (Johnston, 2010; Stewart, 2011) can beconsulted by the readers to compensate for this missinginformation. A recent brief review about the alkali metaliron selenides (Mou, Zhao, and Zhou, 2011) can also beconsulted for a broader perspective on this topic. FIG. 1 (Color online) Crystal structure of A Fe x Se , from Bao et al. (2011b). All the other compounds described in thisreview have a similar structure. A is an alkali metal element(K in the figure). If x<
2, iron vacancies are present.
II. EARLY DEVELOPMENTS
The report that started the area of research of alkalimetal iron selenides was published by Guo et al. (2010).In this publication, results for polycrystalline samples ofK . Fe Se (nominal composition) were presented. Thecrystal structure is in Fig. 1. It contains layers of an al-kali element, such as K, separating the FeSe layers. As inthe 122 pnictide structures based on, e.g., Ba, here theFeSe layers are the “conducting layers” while the K + ionsprovide charge carriers. The presence of the K layer in-creases the distance between FeSe layers, magnifying thereduced dimensionality characteristics of the material.The resistance versus temperature is in Fig. 2. Uponcooling, insulating behavior is first observed (a resistancethat grows with decreasing temperature), followed by abroad peak at 105 K where a metallic-like region starts.At ∼
30 K, the resistance abruptly drops leading to asuperconducting (SC) state. To explain the high value ofthe critical temperature as compared to the T c of FeSe(8 K) or FeSe doped with Te (15.2 K), Guo et al. (2010)argued that the Se-Fe-Se bond angle is close to the idealFeSe tetrahedral shape and also the interlayer distanceis large as compared to that of FeSe. FIG. 2 (Color online) Temperature dependence of the elec-trical resistance of polycrystalline K . Fe Se , from Guo etal. (2010). The dominant features include the SC transitiontemperature at ∼
30 K, with the lower inset containing betterresolution details of that transition. The peak slightly above100 K, that other efforts found to be located at higher tem-peratures when using single crystals (Mizuguchi et al. , 2011),is caused by the ordering of the iron vacancies (Wang, D. M., et al. , 2011). The coexistence of features related with iron va-cancies and superconductivity was later explained based onphase separation (Section V). The upper inset is the temper-ature dependence of the normal state Hall coefficient.
Subsequent work employing single crystals reportedthat the resistivity broad peak of K . Fe Se is actuallylocated above 200 K, i.e. at a higher temperature thanreported for polycrystals, and its SC critical temperatureis 33 K (Mizuguchi et al. , 2011). Related efforts showedthat the hump in the normal state resistivity was relatedto the iron vacancies ordering process (Wang, D. M., etal. , 2011) that was shown to exist in parts of the sample,as discussed in Section V devoted to phase separation(i.e. some of the early development samples were latershown to contain two phases, either at nanoscopic or mi-croscopic length-scale levels). There was no correlationbetween the hump and the SC critical temperatures.Similar properties were observed in other compounds.For instance, Krzton-Maziopa et al. (2011a) reporteda T c = 27 K for Cs . (FeSe . ) . Superconductivity at T c = 32 K was also found in Rb . Fe . Se (Wang, A.F., et al. , 2011), now including iron vacancies explicitly.Other studies using K and Cs as alkali elements were re-ported by Ying et al. (2011), superconductivity at 32 Kwas reported for (Tl,Rb)Fe x Se by Wang, Hangdong, etal. (2011), and using a mixture (Tl,K) by M. H. Fang et FIG. 3 (Color online) (left panel)
Iron-vacancy order corre-sponding to A Fe . Se . The blue solid circles are iron atoms.The green open circles are vacancies. Each iron atom has ei-ther two or three iron neighbors. This type of order is calledhere the 2 × (center panel) The case of A Fe . Se with its √ ×√ √ (right panel) State with no iron vacancies, corresponding to A Fe Se , be-lieved to be of relevance for the SC state. Reproduced fromFang, M. H. et al. (2011), where A =(Tl,K). al. (2011). The latter also contains an interesting phasediagram varying the amount of iron in (Tl,K)Fe x Se , con-structed from the temperature dependence of the resis-tivity. This phase diagram displays the evolution frominsulating to SC phases in the (Tl,K)Fe x Se system, re-sembling results in the cuprates. From anomalies in mag-netic susceptibilities, several of these efforts also reportedthe presence of antiferromagnetic (AFM) order in regimesthat are insulating at all temperatures (M. H. Fang et al. ,2011; Bao et al. , 2011b). Based on previous literature onmaterials such as TlFe x S , Fang, M. H, et al. (2011)concluded that there must be regularly arranged iron va-cancies similarly as when Se is replaced by S, and alsoa concomitant AFM order. The expected iron vacanciesorder is shown schematically in Fig. 3 for the cases of x = 1.5, 1.6, and 2.0 in the chemical formula (Tl,K)Fe x Se (Fang, M. H., et al. , 2011). In this context, Bao et al. (2011b) argued that decorating the lattice with vacanciesoffers a new route to high- T c superconductivity by modi-fying the FS and altering the balance between competingtendencies. Using x-ray diffraction and single crystals,the arrangement of iron vacancies sketched in the cen-tral panel of Fig. 3, i.e. the so-called √ ×√ et al. (2011) (and those samples have phase sepa-ration, see Section V). Transmission electron microscopyresults also provided evidence of this type of vacancy or-der (Wang, Z., et al. , 2011).All these early discoveries established the field of alkalimetal iron selenides, and the subsequent work reviewedbelow provided a microscopic perspective of the proper-ties of these compounds. III. ARPES
Several photoemission experiments have been carriedout for the alkali metal iron selenides. The main commonresult is the absence of hole pockets at the FS in materialsthat are nevertheless still SC. For instance, angle resolvedphotoemission (ARPES) studies of A x Fe Se ( A =K,Cs,nominal composition) by Zhang, Y., et al. (2011) re-vealed large electron-like pockets at the FS around thezone corners with wavevectors ( π ,0) and (0, π ) (in theiron sublattice notation), with an almost isotropic SCgap of value ∼ etal. (2011) remarked that FS nesting between hole andelectron pockets is not a necessary ingredient for the su-perconductivity of the iron-based superconductors. q δ Δ ( m e V ) FIG. 4 (Color online) Magnitude of the SC gap of K . Fe Se corresponding to the M-points electron pockets (there are nohole pockets in this compound) (from Zhang, Y., et al. , 2011).The radius represents the gap while the polar angle θ is mea-sured with respect to the M-Γ direction defined as θ =0. Theresults indicate that there are no nodes and also that thegap is fairly uniform, i.e. not strongly momentum dependent.Here the M=( π , π ) point is with regards to unit cells 45 o -rotated with respect to the Fe-Fe axes. In the iron sublatticeconvention, this point would be ( π ,0) or (0, π ). Similar ARPES results were presented for K . Fe . Se by Qian et al. (2011). This study reported the presenceof electron pockets at the zone boundary, nodeless super-conductivity, and a hole band at Γ with the top of theband at ∼
90 meV below the Fermi level (Fig. 5). Qian et al. , 2011) remarked that if the FS nesting theories areused, then nesting with wavevector ( π , π ) between theelectron pockets should dominate (as explained in sev-eral theoretical efforts summarized in Section VII) con-trary to what appears to occur in other iron-based su-perconductors. Also note that in principle FS nestingbetween electron- and hole-like pockets is required forthe magnetic susceptibility to be enhanced, so nestingbetween electron pockets may not be sufficient to ad-dress the magnetic states. The same group also studiedTl . K . Fe . Se arriving to similar conclusions withregards to the electron pockets at ( π ,0)-(0, π ) (iron sub-lattice convention), but in addition they also observedan unexpected electron-like pocket at Γ. This electron pocket has a SC gap of value comparable to that at thezone boundary pockets (Wang, X.-P., et al. , 2011). M -50-100-150 meV0 k x k y E - E F G XX M G FIG. 5 (Color online) Schematic diagram summarizing theelectronic band structure of K . Fe . Se obtained fromARPES, with the top of the hole band located below the FSat the Γ point. Reproduced from Qian et al. (2011). Studies of (Tl . Rb . )Fe . Se using ARPES alsoled to similar conclusions (Mou et al. (2011)), includingthe presence of small electron-like FS sheets around theΓ point (Fig. 6) and a nearly isotropic SC gap of value ∼
12 meV at the M points. While the SC gap at thelarger Γ point sheet is also nearly isotropic, for the innersmall Γ sheet pocket there is no SC gap. The same groupalso reported ARPES studies for K . Fe . Se ( T c = 32K) and (Tl . K . )Fe . Se ( T c = 28 K) (Zhao et al. ,2011). These results establish a universal picture withregards to the FS topology and SC gap in the A x Fe − y Se materials: there are no FS hole-like pockets at Γ (thusthere is no FS nesting as in some pnictides) and the SCgaps at the FS electron pockets are isotropic (nodeless). -2-1012 k y ( p / a ) -2 -1 0 1 2 k x ( p/ a) G M M h n =21.2eV G ' M M LowHigh E α β γ FIG. 6 (Color online) FS of (Tl . Rb . )Fe . Se , fromARPES studies (Mou et al. , 2011). Note the presence of asmall Γ pocket that has electron-like energy dispersion. Thelattice constant a is 3.896 ˚A. The M points are equivalent tothe ( π ,0) and (0, π ) points in the iron sublattice notation. Recent ARPES studies of K x Fe − y Se focused on theSC gap of the small electron Fermi pocket around the Z point. An isotropic SC gap ∼ et al. (2012) concluded that thesymmetry of the order parameter must be s -wave sincea d -wave should have nodes in that Z -centered pocket.Similar ARPES results were independently presented forTl . K . Fe . Se (Wang, X.-P., et al. , 2012). In thiscase the Z -centered electron FS has an isotropic SC gapof ∼ d -wave super-conductivity appears to be ruled out in these materials.However, the doping effects of Co on a pnictide (not aselenide) such as KFe As have been interpreted via a d -wave SC state (Wang, A. F., et al. , 2012) since the criti-cal temperature rapidly decreases with increasing the Coconcentration, similarly as in cuprates. Thermal conduc-tivity also suggests d -wave symmetry for the same mate-rial (Reid et al. , 2012). Thus, if some pnictides appearto be d -wave superconductors, the symmetry of the SCstate in the alkali metal iron selenides of focus here stillneeds to be further investigated. K x Fe Se k // FIG. 7 (Color online) Sketch of the SC gap of K x Fe − y Se ,from Xu et al. (2012). This figure shows the energy gapversus wave vector parallel to the a - b plane passing throughthe Z =(0,0, π ) point. The presence of an isotropic gap at thecenter rules out d -wave superconductivity. How do all these ARPES results compare with similarpnictide investigations, namely with As instead of Se inthe chemical formulas? The ARPES pnictides effort issimply huge and will not be described here, but inter-ested readers can consult Richard et al. (2011) for a re-cent review. In fact, there are many similarities betweenpnictides and selenides if it is simply accepted that thechemical potential for the case of Se is above the entirehole pocket band located at Γ. Thus, a transition occursfrom a combination of hole and electron pockets for thepnictides, to only electron pockets for the selenides.These results are important for the FS nesting theo-ries that may work for pnictides but not for selenidesdue to the absence of hole pockets. Thus, alternativepairing mechanisms other than those based on weak cou-pling spin density wave scenarios are needed for a properdescription of the iron-based superconductors, such aspurely electronic theories where the Hubbard coupling U is not small or, alternatively, theories where the latticeis involved in the Cooper pair formation. Recent Lanc-zos investigations of the two-orbital Hubbard model in abroad range of Hubbard U and Hund J H couplings con-cluded that s -wave pairing induced by magnetism can not only be found at weak and intermediate couplings,but also in strong coupling where the parent compoundis an insulator (Nicholson et al. , 2011) and thus there isno simple visual representation of the paired state basedon a metallic FS. Then, although evidence is building upthat FS nesting is not needed in the iron-superconductors(Dai, Hu, and Dagotto, 2012) the pairing symmetry maystill be s -wave.Returning to ARPES, the nearly isotropic nature ofthe nodeless SC gaps is similar in both pnictides and se-lenides. However, in pnictides many bulk experimentssuggest the presence of nodes in the SC state (John-ston, 2010; Stewart, 2011). Since ARPES is a surface-sensitive technique, in these materials the surface andthe bulk could behave differently (Hirschfeld, Korshunov,and Mazin, 2011). Then, more work is needed to clarifythe symmetry of the SC state. IV. NEUTRON SCATTERING
Neutron scattering studies of the alkali metal iron se-lenides have revealed an unexpected and complex mag-netic state when in the presence of the ordered iron va-cancies. The details are as follows:
A. Elastic neutron scattering
The first powder neutron diffraction studies of the al-kali metal iron selenides were presented for K . Fe . Se (Bao et al. , 2011a), with Fe in a valence state 2+. Theseinvestigations confirmed the presence of the √ ×√ et al. , 2011). Other neutrondiffraction studies of Cs y Fe − x Se , A x Fe − y Se ( A =Rb, K), and Rb y Fe . x Se also concluded that there isa √ ×√ et al. , 2011a and2011b; Wang, Meng, et al. , 2011).More importantly, Bao et al. (2011a) reported a noveland exotic magnetic order in this compound, that is sta-ble in the iron-vacancies environment. This magneticorder contains 2 × µ B /Fe, thelargest observed in the family of iron-based supercon-ductors. These neutron results, particularly the largemagnetic moments, again challenge the view that thesecompounds are electronically weakly coupled and that FSnesting explains their behavior. While pnictides and se-lenides may have different Hubbard U coupling strengths,thus explaining their different properties, it could alsooccur that the prevailing view of the pnictides as weakor intermediate U materials is incorrect. More work isneeded to clarify these matters. Adding to the discrep-ancy with the weak coupling picture, an unprecedentedhigh N´eel temperature of T N =559 K was reported forthese iron-vacancy ordered compounds. The magneticordering temperature is 20 K smaller than the order-disorder transition temperature for the iron vacancies. FIG. 8 (Color online) In-plane crystal and magnetic structureof K . Fe . Se , reproduced from Bao et al. (2011a). Theopen squares are the iron vacancies and the red dark circleswith the “+” or “-” denote the occupied iron sites with theorientation of their spins. The green open circles correspondto Se, while the K atoms are in yellow as small open circles. Single-crystal neutron diffraction studies of A Fe Se ( A = Rb, Cs, (Tl,Rb), and (Tl,K)) by Ye et al. (2011)found the same iron vacancy order and magnetic block-AFM states as observed in K Fe Se . The order-disordertransition occurs at T S = 500-578 K, and the AFM transi-tion at T N = 471-559 K with a low-temperature magneticmoment ∼ µ B /Fe. Ye et al. (2011) showed that all 245iron selenides share a common crystalline and magneticstructure, which are very different from other iron-basedsuperconductors such as the pnictides.Neutron diffraction studies of TlFe . Se (May et al. ,2012; H. Cao et al. , 2012) have unveiled spin arrange-ments that may deviate from the block-AFM order, com-patible with theoretical calculations (Luo, Q., et al. et al. , 2011) and x-rays (Ricci et al. , 2011b) diffraction studies of the SCstate also provided evidence for phase separation betweenthe above mentioned regular distribution of iron vacan-cies and another state with a √ ×√ B. Inelastic neutron scattering
Inelastic neutron scattering studies (Wang, Miaoyin, et al. , 2011) showed that the spin waves of the insulatingantiferromagnet Rb . Fe . Se , with the block-AFM or-der and N´eel temperatures of ∼
500 K, can be accuratelydescribed by a local moment Heisenberg model withiron nearest-neighbors (NN), next-NN (NNN), and next-NNN (NNNN) interactions, as reviewed by Dai, Hu, andDagotto (2012). These results are contrary to the caseof the iron pnictides, with As instead of Se, where con-tributions from itinerant electrons are needed to under-stand their spin wave properties (Zhao, J., et al. , 2009).Moreover, Rb . Fe . Se has three spin-wave branches,while all the other materials studied with neutrons haveonly one. However, as the energy of the spin excitationsgrows the neutron results of Wang, Miaoyin, et al. (2011)also show (Fig. 9) an evolution from a low-energy statewith eight peaks, as expected from the block-AFM stateafter averaging the two chiralities of the iron vacanciesdistribution, to a high-energy state with spin waves verysimilar to those of pnictides such as BaFe As in spite oftheir very different N´eel temperatures. This observationreveals intriguing common aspects in the magnetism ofselenides and pnictides. In addition, a fitting analysis ofthe neutrons spin-wave spectra shows that in these mate-rials and others the effective NNN Heisenberg couplings(i.e. the coupling along the diagonal of an elementaryiron plaquette) are all of similar value. Since in the sameanalysis the effective NN couplings (i.e. at the short-est Fe-Fe distance) vary more from material to materialeven changing signs, the effective NNN coupling may becrucial to understand the common properties of the iron-based superconductors (Wang, Miaoyin, et al. , 2011). Infact, a robust real (as opposed to effective) NNN su-perexchange coupling comparable or larger in strengthto the real NN superexchange is needed for the stabil-ity of the magnetic state with magnetic wavector ( π ,0),in the iron-sublattice notation, that dominates in manyiron-based superconductors. Recent results for supercon-ducting Rb . Fe . Se (Wang, Miaoyin, et al. , 2012)also suggest that the magnetic excitations arise from lo-calized moments. For details see the recent review Dai,Hu, and Dagotto (2012). Note that the spin-wave spectrahave also been addressed using ab - initio linear responseby Ke, van Schilfgaarde, and Antropov (2012b).Since its discovery in the context of the high- T c Cu-oxide superconductors, an aspect of the inelastic neutronscattering data that is considered of much importanceis the neutron spin resonance (Scalapino, 2012). In su-perconducting A x Fe − y Se the presence of neutron spinresonances was reported in Park et al. (2011), Friemel et al. (2012a) and (2012b), and Taylor et al. (2012) (seealso Inosov et al. (2011)). The energies of the resonancesfor many compounds are summarized in Fig. 10, show-ing that the normalized resonance energy is similar in allof the iron-based superconductors. The neutron resultsshowing a resonance are compatible with the expectation
828 1.12.4 -2 -1 2
H (r.l.u) O K (r . l . u ) O -2-1012-101 -1 H (r.l.u) O a b E = 26 ± 2 meV E = 200 ± 20 meV
FIG. 9 (Color online) Wave-vector dependence of the spin-wave excitations of Rb . Fe . Se at two representative in-dicated energies (from Wang, Miaoyin, et al. , 2011). (a) showsthe eight peaks expected from the √ ×√ As . arising from FS nesting involving the electron pockets forthe case of a d -wave symmetric condensate (Scalapino,2012). However, the discussion is still open since FSnesting may not be sufficient to explain the properties ofthe iron-based superconductors, not even the pnictides(Dai, Hu, and Dagotto, 2012). Perhaps an intermediateHubbard U coupling is a more appropriate starting pointfor the pnictides while the selenides may require an evenstronger coupling. Also ARPES experiments reviewed inSection III tend to favor s -wave superconductivity due tothe absence of nodes in the small electron pocket at Γ.Thus, the d vs. s pairing symmetry of the alkali metaliron selenides remains an open and fascinating question. FIG. 10 (Color online) Normalized resonance energy of sev-eral iron-based superconductors, obtained via inelastic neu-tron scattering, reproduced from Park et al. (2011). RFSstands for Rb Fe Se , BFNA for Ba(Fe − x Ni x ) As , and therest of the abbreviations are for 122, 111, or 1111 materials(for details see Park et al. , 2011). V. TENDENCIES TO PHASE SEPARATION
Recent investigations showed that the often puzzlingproperties of several alkali metal iron selenides can be understood by realizing that phase separation occurs inthese compounds. As it happens in manganites andcuprates, in the materials reviewed here several lengthscales are involved in the phase coexistence. The twocompeting (or maybe cooperating) states involved in theprocess are the SC and the magnetic states, the formerwith ordered iron vacancies. The coexistence of mag-netism, albeit free of vacancies, and superconductivityhas been reported in pnictides as well (Julien, 2009; John-ston 2010). Below, a summary of results on phase sepa-ration in selenides is presented, ordered by technique butalso approximately chronologically. A. µ SR The microscopic coexistence of magnetism and super-conductivity was reported via muon spin spectroscopyinvestigations of Cs . (FeSe . ) (Shermadini et al. ,2011) and A x Fe − y Se ( A = Rb, K) (Shermadini et al. ,2012). Additional evidence for phase separation was pro-vided by a simultaneous ARPES and µ SR analysis ofRb . Fe . Se with T c =32.6 K (Borisenko et al. , 2012).That study showed that the results can be rationalizedvia a macroscopic separation into metallic ( ∼ ∼ Se , and Borisenko et al. (2012) believe that the insulating component is a com-peting order, not relevant for superconductivity. In-stead, they argue that van Hove singularities are thekey ingredient for superconductivity. On the other hand,studies of the resistivity and magnetic susceptibility of A . Fe − y Se are also interpreted as coexisting supercon-ductivity and antiferromagnetism (Liu et al. , 2011) butnot simply competing with each other. While phase sepa-ration between magnetic and SC states is experimentallyproven, the implications are still under considerable de-bate. For the cases where antiferromagnetism and super-conductivity (SC) do coexist microscopically or at leastare so close in space that they can influence one another,does AFM induce or suppress SC? B. Raman scattering, TEM, x-rays
Phase separation with mutual exclusion between insu-lating and SC states, at the micrometer scale, was alsoproposed from the analysis of Raman scattering exper-iments on A . Fe . Se , where the intensity of a two-magnon peak decreases sharply on entering the SC phase(Zhang, A. M., et al. , 2012a and 2012b). Transmissionelectron microscopy (TEM) on K . Fe x Se and KFe x Se by Wang, Z., et al. (2011) also provided evidence of nano-scale phase separation (i.e. not a coexistence of the twostates but physical separation), including the formationof stripe patterns at the micrometer scale together withnanoscale phase coexistence between magnetic and SCphases (Wang, Z. W., et al. , 2012). Percolative scenariosinvolving weakly coupled SC islands were also discussedby Shen et al. (2011) and by Wang, Z. W., et al. (2012). FIG. 11 (Color online) Spatial distribution of the ratio ofthe compressed and the expanded phase in a region of size22 × µ m of a K . Fe . Se crystal, reproduced from Ricci et al. (2011a) where more details can be found. The figureillustrates the several lengths scales involved in the phase sep-arated state, resembling those found in other compounds suchas cuprates and manganites (Ricci et al. , 2011a). X-ray absorption and emission spectroscopy applied toK . Fe . Se also reported coexisting electronic phases,and found superconductivity to have glassy (granular)characteristics (Simonelli et al. , 2012). Using scanningnanofocus X-ray diffraction, studies of the same com-pound focusing down to a size of 300 nm collected thou-sands of diffraction patterns that allowed for the con-struction of a real-space imaging of the k-space resultsobtained by diffraction. These results provided explicitimages of the intrinsic phase separation below 520 K,and they contain an expanded lattice, compatible witha magnetic state in the presence of iron vacancies, anda compressed lattice with non-magnetic characteristics(Ricci et al. , 2011a) (see Fig. 11). Micrometer-sized re-gions with percolating magnetic or nonmagnetic domainsform a multiscale complex network of the two phases.Note that for phase separation at large length scales,x-ray diffraction techniques are sufficient to observe twostructurally distinct phases (Luo, X. G., et al. , 2011;Bosak et al. , 2011; Lazarevi´c et al. , 2012; Liu, Y., etal. , 2012; Pomjakushin et al. , 2012). This shows that theSC phase is a real bulk phase rather than an interfacialproperty. It is for shorter length scales that more mi-croscopic techniques are needed to clarify the interplaybetween the two phases. C. ARPES and phase separation
Using ARPES and high-resolution TEM applied toK x Fe − y Se , evidence was provided for a mesoscopicphase separation at the scale of several nanometers be-tween the SC and semiconducting phases and the AFMinsulating phases (Chen, F., et al. (2011)). One of the in-sulators has the √ ×√ et al. (2011) re-marked that the insulators are mesoscopically separated from the SC or semiconducting phases, and they believethat the semiconducting phase (free of magnetic and va-cancy order) is the parent compound that upon electrondoping leads to superconductivity. E F - . V q AFI1 e - - . V Superconductor e - -0.04 0 0.04
40K 5K
E-E F (eV) E F h υ FIG. 12 (Color online) Cartoon for the phase separation insuperconducting K x Fe − y Se , from Chen, F., et al. (2011),obtained via photoemission and TEM techniques. The upperinsets are the photoemission signals for the two regions: leftcorresponds to the √ ×√ D. STM and neutron diffraction
Using thin films of K x Fe − y Se grown using molecular-beam epitaxy techniques, Scanning Tunneling Mi-croscopy (STM) results were interpreted as caused bythe samples containing two phases: an insulating onewith the √ ×√ Se free of vacancies (Li, W., et al. ,2012a). The density of states (DOS) of the two phasesmeasured via Scanning Tunneling Spectroscopy (STS)are in Fig. 13. It is interesting that the SC phase is asso-ciated with the “122” rather than the “245” compositionthat contains the ordered iron vacancies, which naivelywas expected to be the parent compound.In related STS studies of K . Fe . Se (Cai et al. ,2012), a SC gap was found microscopically coexistingwith a so-called √ ×√ etal. (2012) argued that it is not a necessary ingredient forsuperconductivity. In fact, their results in the region ofthe charge modulation are compatible with the ferromag-netic block state but in the absence of the √ ×√ et al. (2012c)(Fig. 17). Other STM studies of K x Fe − y Se − z (Li,W., et al. , 2012b) concluded that KFe Se is the par-ent compound of superconductivity (with this state be-ing induced by Se vacancies or via the interaction withthe nearby “245” regions perhaps by modifying the dop-ing concentration). This STM study concluded that the d I/ d V ( a . u . ) Δ Δ d I/ d V ( a . u . ) FIG. 13 (Color online) ( left panel ) STS results from Li, W., et al. (2012a) showing the DOS of a region of a K x Fe − y Se film that displays features compatible with a SC phase. ( rightpanel ) Same as left, but for another region of the film, withresults this time compatible with an insulating phase, pre-sumably with ordered iron vacancies. phase with the √ ×√ √ ×√ et al. ,2012b). The “122” phase charge modulation is compat-ible with a block spin order without iron vacancies (Li,W., et al. , 2012c), since the distance between equivalentferromagnetic blocks (with spins pointing in the same di-rection) is 2 √ et al. (2012b) also reported an exotic √ ×√ et al. (2012b)).Recently, another possibility has been presented. Us-ing neutron diffraction techniques for K x Fe − y Se , Zhao et al. (2012) proposed the state in Fig. 3 (left panel),with a rhombus-type iron vacancy order, as the par-ent compound of the SC state. In this state the ironspins have parallel (antiparallel) orientations along thedirection where the iron vacancies are separated by four(two) lattice spacings. This state has ideal compositionKFe . Se , iron magnetic moments 2.8 µ B , and an AFMband semiconductor character, as in the first-principlescalculations by Yan, X.-W., et al. (2011a). FS nesting isnot applicable in this state and the large moments sug-gest that correlation effects cannot be neglected. Thesemiconducting nature of this state is also compatiblewith ARPES experiments (Chen, F., et al. (2011)) thatalso proposed a semiconductor as the parent compound. E. Optical spectroscopy
Optical spectroscopy studies of K . Fe . Se by Yuan et al. (2012) revealed a sharp reflectance edge below T c at a frequency much smaller than the SC gap, on an inco-herent electronic background. This edge was interpretedas caused by a Josephson-coupling plasmon in the SCcondensate. This study provided evidence for nanoscalephase separation between superconductivity and mag-netism. The coupling between the two states can be understood if it occurs at the nanometer scale, since atthis scale there is a large fraction of phase boundary inthe sample, while at a longer length scale a very weakcoupling between the states would exist (Yuan et al. ,2012). Infrared spectroscopy studies of K . Fe . Se were also presented (Chen, Z. G., et al. , 2011), reveal-ing abundant phonon modes that could be explained bythe iron vacancy ordering. Studies of the complex di-electric function of Rb Fe Se (Charnukha et al. , 2012b)also concluded that there are separated SC and mag-netic regions in this compound. Investigations via opticalmicroscopy and muon spin rotation reported an intrigu-ing self-organization of this phase-separated state into aquasiregular heterostructure (Charnukha et al. , 2012a). FIG. 14 (Color online) Log-log plot of the spectral weight ofthe superfluid density N c vs. the residual conductivity σ dc times the critical temperature T c , reproduced from Homes etal. (2012b). Results include cuprate superconductors, severaliron based superconductors, and the volume average and effec-tive medium approximation (EMA) results for K . Fe − y Se .While the volume average signal a Josephson phase, the EMAresult is now very close to the coherent regime. Other optical studies (Homes et al. , 2012a) initiallycharacterized K . Fe − y Se as a phase-separated Joseph-son phase, with inhomogeneous characteristics. However,more recent studies (Homes et al. , 2012b) distinguishedbetween the volume average measurements of the orig-inal report (Homes et al. , 2012a) and the results aris-ing from an effective medium analysis (EMA) to deter-mine which fraction of the material is actually metal-lic/superconducting. The volume average case has a nor-mal resistance too high for coherent transport, locatingthis case in the Josephson coupling region, as shown inFig. 14 that contains a scaling plot previously used todiscuss cuprates and other iron-based superconductors.However, the material is not homogeneous and the EMAshows that only 10% is metallic/SC. Homes et al. (2012b)then concluded that if a sample could be constructedcomposed of just this phase, then it would be a coherent0metal, falling closer to the other iron-based materials asshown also in Fig. 14. This is in agreement with the con-clusions by Wang, C. N., et al. (2012) using muon spinrotation and infrared spectroscopy. The use of the EMAto rationalize results in phase separated systems was alsosuggested by Charnukha et al. (2012a, 2012b).In summary, the discussion regarding the character-istics of the parent compound of the superconductingKFe Se state is still very fluid, defining an intriguingand exciting area of research of much importance. Sev-eral candidate states have been proposed for the parentcomposition of the SC state. VI. RESULTS USING NMR, TEM, M ¨OSSBAUER, ANDSPECIFIC HEAT TECHNIQUES Se Nuclear Magnetic Resonance (NMR) studies andKnight-shift studies of K . Fe . Se and K . Fe . Se below T c have demonstrated that the superconductivityis in the spin singlet channel, although without coherencepeaks in the nuclear spin-lattice relaxation rate below T c suggesting that the state is probably non-conventional(Yu, W., et al. , 2011). These results are similar to thoseknown from the pnictides. Moreover, above T c the tem-perature dependence of 1/ T indicates that the systembehaves as a Fermi liquid, suggesting the absence ofstrong low-energy spin fluctuations at the Se site (Yu,W., et al. , 2011). Other Se NMR measurements ofK . Fe . Se (Torchetti et al. , 2011) and Se and RbNMR studies of Tl . Rb . Fe . Se (Ma et al. , 2011)arrived to similar conclusions. Torchetti et al. (2011)also suggested that the K vacancies may have a super-structure and the symmetry of the Se sites is lower thanthe tetragonal fourfold symmetry of the average struc-ture. In addition, transmission electron microscopy ex-periments on K x Fe − y Se suggested the ordering of theK ions in the a - b plane, and also addressed the resistiv-ity hump anomaly in the iron-vacancy ordering (J. Q. Li et al. , 2011, and Song et al. , 2011). Using Se NMR,the absence of strong AFM spin correlations was alsoreported for superconducting K . Fe Se , with a nonex-ponential behavior in the nuclear spin lattice relaxationrate 1/ T indicating disagreement with a single isotropicgap (Kotegawa et al. , 2011 and 2012). Se and RbNMR studies of Rb . Fe . Se also reported two coex-isting phases (Texier et al. , 2012), and the SC regions donot have iron vacancies nor magnetic order.M¨ossbauer spectroscopy studies of superconductingRb . Fe . Se also report the presence of 88% magneticand 12% nonmagnetic Fe regions (Ksenofontov et al. ,2011), compatible with previously discussed reports. Themagnetic properties of superconducting K . Fe . Se were also studied using M¨ossbauer spectroscopy (Ryan et al. (2011)). Magnetic order involving large iron mag-netic moments is observed from well below the T c ∼
30 Kto the N´eel temperature T N =532 K.Via the study of the low-temperature specific heat, nodeless superconductivity and strong coupling charac-teristics were reported by Zeng et al. (2011) for singlecrystals of K x Fe − y Se , compatible with results foundusing ARPES techniques. On the other hand, ther-mal transport results for superconducting K . Fe . Se were interpreted as corresponding to a weakly or inter-mediately correlated superconductor by Wang, Lei, andPetrovic (2011a) and (2011b). A numerical study of thethermal conductivity and specific heat angle-resolved os-cillations in a magnetic field for A y Fe Se superconduc-tors addressed the gap structure and presence of nodes(Das et al. , 2012), concluding that care must be taken inthe interpretation of results using these techniques sinceeven for isotropic pairing over an anisotropic FS, ther-modynamic quantities can exhibit oscillatory behavior. VII. THEORYA. Band structure in the presence of iron vacancies
The magnetic state of the alkali metal iron selenideshas been investigated from the perspective of theory us-ing a variety of techniques. For example, employingfirst-principles calculations and comparing several mag-netic configurations, the ground state of (K,Tl) y Fe . Se was found to be the magnetic configuration with anti-ferromagnetically coupled 2 × y =0.8 and K as the alkali element, a band gap ∼
600 meV opens leading to an AFM insulator (Cao andDai, 2011a). For y =1, the Fermi level is near the topof the band gap of y =0.8, leading to a metallic statewith a ∼ ab - initio calculations by Yan, X.-W., et al. (2011b) agree with these results, and band structure cal-culations for K x Fe Se can also be found in Shein andIvanovskii (2010) and Yan, X.-W., et al. (2011c). Theblock-AFM ground state band structure is in Fig. 15. Inaddition, via studies of K . Fe . Se and K . Fe . Se ,i.e. varying the concentration of K to affect the va-lence of iron and the associated carrier concentration,it was found that the band structure and magnetic or-der almost do not change in that range of doping. Then,K . Fe . Se could be considered as a parent compoundwhich becomes superconducting upon electron or holedoping (Yan, X.-W., et al. , 2011b). This is relevantsince in (Tl,K)Fe x Se , superconductivity already occursat x =1.7 or higher (M. H. Fang et al. , 2011). However,the issue of phase separation discussed in Sec. V rendersthe identification of the parent compound far more com-plicated than naively anticipated. B. Influence of electron-electron correlations
First-principles calculations for the related materialTlFe . Se (i.e. with Fe . instead of Fe . , and thuswith a different distribution of iron vacancies) using the1 FIG. 15 (Color online) (a) Electronic band structure ofK . Fe . Se in the ground state with the 2 × et al. (2011b). The top of the va-lence band is set to zero. (b) Explanation of the conventionfollowed to label points of the Brillouin Zone. These theoreti-cal calculations are carried out in a tetragonal structure withlattice parameters in excellent agreement with experiments. GGA + U method were also reported by Cao and Dai(2011b). The conclusion is that the magnetic state, aspin density wave, becomes stable because of an effectiveincrease of U/W due to the reduction in W caused bythe loss of kinetic energy of the electrons in a backgroundwith iron vacancies (Cao and Dai, 2011b; Chen, Cao, andDai, 2011). This is similar to the conclusion of model cal-culations that addressed the stability of the block-AFMstate for the case Fe . (Luo, Q., et al. U ∼ U ∼ et al. U/W strength for the 1111 and 122 pnictides see Vilmercati etal. (2012)). The relevance of Mott physics, as opposedto an insulator caused by band structure effects, was alsoremarked by Craco, Laad, and Leoni (2011) using bandstructure plus dynamical mean-field theory. In fact, amore general study of the influence of correlations, notonly in selenides but in pnictides as well, arrives to theconclusion that the weak coupling Fermi Surface nestingpicture is incomplete and the intermediate U couplingregime is more realistic (Yin, Haule, and Kotliar, 2011;Dai, Hu, and Dagotto, 2012).Model calculations using a three-orbital Hubbardmodel in the random phase approximation (RPA)(Huang and Mou, 2011) also concluded that for Fe . theblock-AFM spin state is caused by electron correlationeffects, although at a smaller U ∼ U to representthe same physics as a five-orbital model, due to the re-duction in the bandwidths when reducing the number oforbitals. This value of U is also compatible with resultsby Luo et al. (2010) using also a three-orbital model, butin the context of pnictides. Note that in Huang and Mou (2011) the ratio J H /U is 0.2, similar to the 0.25 found byLuo, Q., et al. (2011). Studies for pnictides also suggesta similar ratio for J H /U (Luo et al. , 2010). Moreover,the importance of a robust J H has been remarked fromthe dynamical mean-field theory perspective (Georges,de’ Medici, and Mravlje, 2012, and references therein) aswell as from the orbital differentiation perspective (seeBascones, Valenzuela, and Calder´on, 2012, and referencestherein; for recent experimental results see Yi, M., et al. ,2012) where some orbitals develop a gap with increasing U while others remain gapless. In addition, in the workby Luo, Q., et al. (2011), and also via mean-field approx-imations and the three-orbital model by Lv, Lee, andPhillips (2011), it was concluded that for a sufficientlylarge U an orbitally ordered state should be stabilizedfor the iron-vacancies ordered state, with the populationof the d xz and d yz orbitals different at every iron site. C. Competing states
The issue of the magnetic states that compete with the2 × π ,0)with regards to the iron sublattice) was found to becomestable if a pressure of 12 GPa is applied (Chen, Lei, etal. , 2011). This state corresponds to the same ( π ,0) mag-netic order (C-AFM) of the “122” and “1111” families,simply removing the spins corresponding to the locationof the iron vacancies (Fig. 16 (c)). Further increasingthe pressure to 25 GPa a non-magnetic metallic stateis reached (Chen, Lei, et al. , 2011). These results arequalitatively compatible to those found via Hartree-Fock(HF) approximations to the five-orbital Hubbard model(Luo, Q., et al. W . Since the Hubbard U is local, it should notbe affected as severely as W by these effects. Thus,a pressure increase amounts to a decrease in U/W inHubbard model calculations. Indeed, working at a fixed J H /U =0.25, Luo, Q., et al. (2011) found that by re-ducing U/W then transitions occur from the block-AFMstate Fig. 16 (a) to the C-AFM state Fig. 16 (c), andthen eventually to a non-magnetic state, if at a constant J H /U . If J H /U is reduced, then the state Fig. 16 (b)could also be reached, with staggered order within the2 × et al. (2011)show that the stabilization of the block-AFM state iscaused by a lattice tetramer distortion, otherwise the C-2 FIG. 16 (Color online) (a-c) Some of the competing statesin the presence of a √ ×√ et al. (2011). Shown are (a) theexperimentally dominant 2 × et al. (2011) by reducing J H /U , and (c) the C-type AFM state described by Luo, Q., et al. , (2011) and Chen, Lei, et al. (2011) that could be sta-bilized by increasing pressure. (d) Phase diagram of the five-orbital Hubbard model in the presence of the √ ×√ et al. et al. (2011). Competing states can also be found in Caoand Dai (2011b) and Yu, Goswami, and Si (2011). AFM state would be stable. This effect is not consideredin the Hubbard model calculations where the block-AFMstate is stabilized by an increase in
U/W (Luo, Q., et al. A . Fe . Se reported that pressure induces a transi-tion from the block-AFM state to the metallic “N´eel-FM”state where each 2 × et al. (2011)and Cao, Fang, and Dai (2011) (C. Cao, private commu-nication). As already remarked, note also that the modelHamiltonian calculations (Luo, Q., et al. J H /U - U phase diagram (Fig. 16, lower panel), thussmall variations in the first-principles calculations maylead to different states. These differences highlight thecomplexity of the phase diagram of various materials,displaying several competing phases when in the pres-ence of iron vacancies. From the strong coupling limitperspective, calculations based on localized spin modelsfor A . Fe . Se also revealed many competing states, in-cluding the magnetic arrangement found in neutron ex-periments (Yu, Goswami, and Si, 2011; Fang et al. , 2012).Similar competition of states was found for A . Fe . Se ,i.e. with Fe . instead of Fe . (Yu, Goswami, and Si,2011). Note also that Li, W., et al. (2012c) predictedan insulating block-AFM spin state even in the absenceof iron vacancies, for instance for KFe Se . This state issketched in Fig. 17. The dominant magnetic instabilityof vacancies-free KFe Se was also studied by Cao andDai (2011c), reporting a state similar to that of pnictidesand a FS with only electron-like pockets without nesting,and by Liu, D.-Y., et al. (2012). X' X J J 'J J J J ' FIG. 17 (Color online) The block-AFM spin order predictedfor KFe Se (no iron vacancies) based on spin model calcula-tions (from Li, Wei, et al. (2012c) where details can be foundabout the several Heisenberg couplings shown). D. Pairing symmetry
As remarked before, the states with chemical composi-tion A . Fe . Se , A Fe . Se , and A Fe Se have receivedconsiderable attention both experimentally and theoret-ically. Predicting the pairing symmetry of the SC statein these materials has been one of the areas of focus. Us-ing a slave-spin technique to study the Mott transitionof a two-orbital Hubbard model, and an effective per-turbation theory once the system is in the Mott state,the superconductivity of slightly doped (Tl,K)Fe . Se was studied, unveiling a competition regulated by J H be-tween a d -wave state (with a positive order parameter intwo of the electron-like pockets and negative in the othertwo) and an s -wave state with the same sign of the orderparameter in all the electron pockets (there are no holepockets in these materials) (Zhou, Yi, et al. , 2011). The3importance of superconductivity mediated by spin fluc-tuations was also analyzed using spin fermion models, i.e.mixing itinerant and localized degrees of freedom as op-posed to using directly a Hubbard model (Zhang, G. M., et al. , 2011). For K x Fe − y Se , the fluctuation exchangeapproximation applied to a five-orbital Hubbard model(Maier et al. , 2011) leads to d -wave superconductivitydue to pair scattering between the electron pockets. TheRPA enhanced static susceptibility has a broad peak at( π , π ) in the Fe sublattice notation. A similar d -wave pair-ing was found using the two-orbital model within RPA(Das and Balatsky, 2011), and a possible s +i d pairingwas also discussed by Yu, R., et al. (2011). The resultsof Maier et al. (2011) contain a robust dependence of theSC gap with wavevector along the electron pockets.However, ARPES results seem in disagreement with d -wave pairing (Xu et al. , 2012; Wang, X.-P., et al. , 2012).In addition, the calculations that lead to d -wave super-conductivity have been criticized because they are basedon the “unfolded” Brillouin zone, neglecting the symme-try lowering of the staggered Se atom positions (Mazin,2011). Based on this consideration, Mazin (2011) arguedthat the d -wave states should develop nodal lines at thefolded BZ electron pockets, which are not observed exper-imentally. It is then concluded that either a conventionalsame-sign s -wave state, with the same sign for the SCorder parameter in all the FS pockets, or another formof the s + − state, different from the one proposed for thepnictides, should be the dominant symmetries (Mazin,2011; for details and references on the possible pairingchannels discussed in the literature see Johnston, 2010;for another form of s + − pairing for A Fe Se see Khodasand Chubukov, 2012). The dominance of s -wave pair-ing was also concluded from mean-field studies based onmagnetic exchange couplings (Fang, Chen, et al. , 2011).Those authors remarked that in strong coupling s -wavepairing can exist even without the electron and hole pock-ets needed in weak coupling. Lanczos calculations byNicholson et al. (2011) reached similar conclusions. The d - vs s -wave competition, the latter with the same signin all pockets, was also studied by Saito, Onari, and Kon-tani (2011) via orbital and spin fluctuations in models forKFe Se . For the orbital fluctuations a small electron-phonon coupling is needed. In the phase separation con-text, the differences between d - and s -wave pairing forthe superconducting proximity effect into the magneticstate and the suppression of the magnetic moments werealso addressed via two-orbital models and mean-field ap-proximations (see Jiang et al. , 2012; a related work totest the pairing symmetry via nonmagnetic impuritieswas proposed by Wang, Yao, and Zhang, 2012). E. Other topics addressed by theory
Several other topics have been addressed using theoret-ical techniques. For example, (i) the effect of disorderedvacancies on the electronic structure of K x Fe − y Se was studied using new Wannier function methods (Berlijn,Hirschfeld, and Ku, 2012) and also via the two-orbitalHubbard model in the mean-field approximation (Tai et al. , 2012). Also in this context and to distinguishbetween the d - and s -wave pairing channels in the ab-sence of hole pockets it was argued that the influence ofnonmagnetic impurity scattering needs to be considered(Zhu and Bishop, 2011). Similar issues were addressedby Zhu et al. (2011). In addition, it has been argued thatadding Fe atoms to K Fe x Se creates impurity bandswith common features to iron-pnictides, thus addressingthe coexistence of superconductivity and magnetic states(Ke, van Schilfgaarde, and Antropov, 2012a); (ii) Bandstructure calculations have shown that the stoichiomet-ric KFe Se has a rather different FS than Ba122, butstill the d xz , d yz , and d xy orbitals dominate at the Fermienergy (Nekrasov and Sadovskii, 2011). VIII. TWO-LEG LADDERSA. Introduction and experiments
Considering the vast interest in the alkali metal ironselenides summarized in the previous sections, and alsoconsidering that deviations from an iron square lattice,as in the presence of the iron vacancies order, lead to in-teresting physics, then other crystal geometries are worthexploring. In this subsection, recent experimental efforts(Caron et al. , 2011; Saparov et al. , 2011; Lei et al. , 2011d;Krzton-Maziopa et al. , 2011b; Caron et al. , 2012; Nambu et al. , 2012) in the study of selenides with the geometryof two-leg ladders (sometimes also referred to as doublechains) will be reviewed, while a description of the statusof the theoretical work will be presented in the next sub-section. A typical compound in this context is BaFe Se that contains building blocks made of [Fe Se ] − thatwhen assembled along a particular direction leads to anarray of two-leg ladder structures, as sketched in Fig. 18. FIG. 18 (Color online) The two-leg ladder substructures ofBaFe Se , with their legs oriented perpendicular to the figure,from Lei et al. (2011d). The ladders in this compound can be considered as4cut-outs of the layers of edge-sharing FeSe tetrahedraof the two-dimensional selenides (Fig. 19). Each lad-der has a long direction (“legs”) and a short directioninvolving two Fe atoms (“rungs”). Note that the fieldof research involving similar ladder structures but withspin 1/2 copper instead of iron, is also very active sincein that context two interesting effects were found: a spingap and superconductivity upon doping (Dagotto, Ri-era, and Scalapino, 1992; Dagotto and Rice, 1996). Forinstance, SrCu O is a material analogous to BaFe Se (Dagotto, 1999). b-1/3 FeFeSe Fe Se FIG. 19 (Color online) Relation between a complete FeSelayer, and the structure of the ladders. The magenta (darkin the black and white version) spheres are the Se atoms andthe light blue (grey if in black and white) the Fe atoms. Theladders simply amount to the removal of every third iron atomfrom the layers. From Saparov et al. (2011).
BaFe Se is an insulator with a gap 0.14-0.18 eV (Lei et al. , 2011d ; Nambu et al. , 2012). This material haslong-range AFM order at ∼
250 K, low-temperature mag-netic moments ∼ µ B , and short-range AFM order(presumably along the leg directions) at higher temper-atures (Caron et al. , 2011; Saparov et al. , 2011; Lei etal. , 2011d). Establishing an interesting analogy withthe alkali metal iron selenides, neutron diffraction stud-ies (Caron et al. , 2011; Nambu et al. , 2012) reported adominant order involving 2 × √ ×√ Se are replaced by K, even-tually arriving to KFe Se , the magnetic order changesto that in Fig. 20 (upper panel), with spins along therungs coupled ferromagnetically, and with an AFM cou-pling along the legs (Caron et al. , 2012). B. Theory
The theoretical study of selenide ladders is only at anearly stage. First-principles calculations and spin modelstudies (W. Li et al. , 2012d) showed the dominance ofthe block-AFM state found experimentally. The bandstructure calculation in this magnetic state was presentedby Li, W., et al. (2012d) (see also Saparov et al. , 2011)and it contains a gap of 0.24 eV (Fig. 21).
FIG. 20 (Color online) Magnetic order of the two-leg laddersfor the cases of KFe Se and BaFe Se obtained using neutrondiffraction. From Caron et al. (2012). -0.500.5 E n e r gy ( e V ) (b) Γ N M N’ Γ Z R A R’ Z
FIG. 21 (Color online) Electronic band structure of the block-AFM state of the two-leg ladder BaFe Se , from Li, W., etal. (2012d). The gap is 0.24 eV. With regards to model Hamiltonians, calculations us-ing the five-orbital Hubbard model in the HF approx-imation have been reported by Luo et al. (2012).Varying U and J H , the phase diagram in Fig. 22 wasfound. The block-AFM phase, called the plaquette phase(P) in the figure, is stable in a robust portion of thephase diagram. This includes the regime with the ra-tio J H /U =0.25 widely believed to be realistic for thesecompounds (Fig. 22, upper panel). Moreover, the otherphase of ladders that was recently reported in neutronexperiments (Caron et al. , 2012), denoted as CX in thefigure, is also part of the phase diagram. In addition, sev-eral other phases not yet observed experimentally are alsostable varying the couplings, suggesting that many statesare close in energy and likely competing. Figure 22 (lowerpanel) contains a sketch of those states. Note also thatthe ratio U/W starts at ∼ et al. , 2012). FIG. 22 (Color online) Phase diagram of the five-orbital Hub-bard model in the real-space HF approximation, at electronicdensity n =5.75 ( n are electrons per iron), using a 2 ×
16 lattice(from Luo et al. (2012)). J H in units of U and U in units ofthe bandwidth W are varied. PM stands for paramagnetic,and FM for ferromagnetic. The other magnetic states areschematically shown at the bottom. The hoppings used arefrom band structure calculations corresponding to BaFe Se . Our understanding of ladder iron selenides is still prim-itive and more work should be carried out in this context.The main advantage of studying ladders is that the quasione dimensionality of these systems allows for more ac-curate theoretical calculations than those routinely per-formed for two-dimensional systems, thus improving theback-and-forth iterative process between theory and ex-periments to understand these materials.
IX. RELATED AND RECENT DEVELOPMENTS
An exciting recent result is the report of superconduc-tivity in a single unit-cell FeSe film grown on SrTiO (Wang, Qing-Yan, et al. , 2012), displaying signatures ofthe SC transition above 50 K, and a SC gap as largeas 20 meV. The electronic structure of this single-layerFeSe superconductor was studied via ARPES techniquesby Liu, Defa, et al. (2012). The FS is in Fig. 23 andit consists only of electron pockets near the zone corner,without any indication of even a small pocket at the zonecenter. Thus, there are no scattering channels betweenthe Γ and M points of the Brillouin zone. The top ofthe hole-like band at Γ is 80 meV below the Fermi level.The critical temperature is ∼
55 K and the SC gap wasfound to be large and nearly isotropic, and since this isa strictly two-dimensional system, then the presence ofnodes along the z -axis is ruled out. From first principlescalculations Liu, Lu, and Xiang (2012) concluded that the single and double layer FeSe films are weakly dopedAFM semiconductors, i.e. for the mono layer FeSe to besuperconducting doped electrons may be needed via Oor Se vacancies. Clearly, the in-depth study of this singlelayer system will contribute significantly to the under-standing of the SC state of the iron superconductors. -2-1012 k y ( p / a ) -2 -1 0 1 2 k x ( p/ a) G g E FIG. 23 (Color online) Fermi Surface of a single-layer FeSesuperconductor using ARPES techniques. Only electron-likepockets are present. From Liu, Defa, et al. (2012).
While completing this review two remarkable new re-sults were reported: (1) the SC T c of the single-layerFeSe film grown on a SrTiO substrate was optimized to T c =65 ± et al. , 2012),establishing a new T c record for the iron superconduc-tors. Photoemission studies indicate a FS with electronpockets at the M points (He et al. , 2012), as in the previ-ous report Liu, Defa, et al. (2012). (2) A single layer ofalkali-doped FeSe with the geometry of weakly coupledtwo-leg ladders was prepared by Li, Wei, et al. (2012e)and shown to become superconducting based on the pres-ence of a gap in the local DOS. This suggests that thepairing is likely local and establishes stronger analogieswith the Cu oxide ladders (Dagotto and Rice, 1996).There are several other recent exciting topics of re-search in these materials. As discussed before, the in-sulating characteristics of some of the alkali metal ironselenides suggests that Mott physics may be importantto understand their properties. Mott localization closeto iron-based superconductors has also been addressedin other contexts as well. For instance, the iron oxy-chalcogenides La O Fe O(Se,S) have been studied the-oretically and the conclusion is that they are Mott in-sulators because of enhanced correlation effects causedby band narrowing (Zhu et al. , 2010). The importanceof Mott localization in materials related to the iron-superconductors was also addressed for K . Fe . S andK . Fe . SeS (Guo et al. , 2011), and also for BaFe Se O(Han et al. , 2012; Lei et al. , 2012). Lei et al. (2011a)studied the phase diagram of K x Fe − y Se − z S z , showingthat T c is suppressed as the S concentration increases (seealso Lei et al. , 2011b and 2011c).6In a related context, the K . Fe − x Co x Se phase dia-gram was discussed by Zhou, T.T., et al. (2011). A smallamount of Co is sufficient to suppress the superconduc-tivity of the undoped material, and at x =0.03 there is nolonger a zero resistivity state. Zhou, T.T., et al. (2011)argue that this behavior is similar to that in the Cu-oxidesuperconductors and for this reason the alkali metal ironselenides are better described by localized 3 d spins thanby itinerant electrons. PM AFM II AFMI
IAFMI III SC T ( K ) Valence of iron
FIG. 24 (Color online) The phase diagram of K x Fe − y Se versus the iron valence, from Yan et al. (2012). The SCphase appears sandwiched between AFM insulators. The Fevalence state was systematically controlled by varying the x and y concentrations in K x Fe − y Se . Also among the most recent developments is the studyof the phase diagram of A x Fe − y Se ( A = K, Rb, andCs) versus the valence of iron (Yan et al. , 2012). Thisiron valence was controlled by varying systematically x and y . The resulting phase diagram is in Fig. 24 and itcontains three AFM insulating states (characterized bydifferent iron vacancy superstructures) and a SC state.Since the SC phase is surrounded by insulators, Yan etal. (2012) concluded that the SC phase must have thoseinsulating states as parent compounds.Another interesting result is the discovery of a sec-ond “re-emerging” SC phase (Sun et al. , 2012) forTl . Rb . Fe . Se , K . Fe . Se , and K . Fe . Se ,with critical temperatures T c ∼ T c with increasing pressure may be caused by structuralvariances within the basic tetragonal unit cell, and the √ ×√ et al. (2012).Along similar lines with regards to further increases in T c , superconductivity at 30 K-46 K in A x Fe Se was re-cently observed by Ying et al. (2012). Compatible withthese results, superconductivity at 44 K in A x Fe − y Se was also recently reported (A.M. Zhang et al. , 2012b).At these temperatures a sharp drop in resistivity and (cid:48) (cid:51) (cid:54) (cid:57) (cid:49)(cid:50) (cid:49)(cid:53)(cid:48)(cid:49)(cid:48)(cid:50)(cid:48)(cid:51)(cid:48)(cid:52)(cid:48)(cid:53)(cid:48)(cid:54)(cid:48) (cid:84) (cid:67) (cid:40) (cid:75) (cid:41) (cid:83)(cid:67)(cid:45)(cid:73) (cid:32) (cid:32) (cid:32) (cid:75) (cid:48)(cid:46)(cid:56) (cid:70)(cid:101) (cid:49)(cid:46)(cid:55) (cid:83)(cid:101) (cid:50) (cid:32) (cid:40)(cid:82)(cid:41)(cid:32) (cid:32) (cid:75) (cid:48)(cid:46)(cid:56) (cid:70)(cid:101) (cid:49)(cid:46)(cid:55)(cid:56) (cid:83)(cid:101) (cid:50) (cid:32) (cid:40)(cid:82)(cid:41) (cid:32) (cid:80)(cid:114)(cid:101)(cid:115)(cid:115)(cid:117)(cid:114)(cid:101)(cid:32) (cid:40)(cid:71)(cid:80)(cid:97)(cid:41) (cid:83)(cid:67)(cid:45)(cid:73)(cid:73) (cid:78) (cid:83) (cid:67) (cid:32) (cid:84)(cid:108) (cid:48)(cid:46)(cid:54) (cid:82)(cid:98) (cid:48)(cid:46)(cid:52) (cid:70)(cid:101) (cid:49)(cid:46)(cid:54)(cid:55) (cid:83)(cid:101) (cid:50) (cid:32) (cid:40)(cid:82)(cid:41)(cid:32) (cid:84)(cid:108) (cid:48)(cid:46)(cid:54) (cid:82)(cid:98) (cid:48)(cid:46)(cid:52) (cid:70)(cid:101) (cid:49)(cid:46)(cid:54)(cid:55) (cid:83)(cid:101) (cid:50) (cid:32) (cid:40)(cid:97)(cid:99)(cid:41) FIG. 25 (Color online) Superconducting T c vs pressure forthe compounds indicated, from Sun et al. (2012). Two SCphases were found. SC-II has a T c ∼ susceptibility were observed. The 44 K SC phase isclose to an ideal 122 structure, but with an unexpect-edly large c -axis lattice parameter 18.10 ˚A. In Zhang,A. M., et al. (2012c), a plot shows that T c increaseswith the distance between neighboring FeSe layers. Re-lated with these results, superconductivity at 44 K inLi x Fe Se (NH3) y (Scheidt et al. , 2012) and at 45 Kin Li x (C H N) y Fe − z Se (Krzton-Maziopa et al. , 2012)were also recently observed. X. CONCLUSIONS
In this publication the “hot” topic of alkali metal ironselenides has been reviewed. The main reasons for thecurrent excitement in this area of research includes therealization that these materials do not have hole pock-ets at the Γ point, altering conceptually the dominantperception that originated in the pnictides with regardsto the importance of Fermi Surface nesting between elec-tron and hole pockets to understand the magnetic andSC states. This conclusion is compatible with the recentaccumulation of evidence that Fermi Surface nesting anda weak coupling perspective are actually not sufficient forthe the pnictides (Dai, Hu, and Dagotto, 2012). More-over, via ARPES techniques applied to some alkali metaliron selenides, the small electron (not hole) pocket at Γwas investigated and in the SC state this pocket doesnot present nodes, removing the d -wave state as a pos-sibility (although this issue is still under discussion asexplained before). Thus, the menu of options for thesymmetry of the SC order parameter in these selenidesappears reduced to a conventional same-sign s -wave state(realized via a coupling of the electrons to the lattice),or a more exotic form of the s + − state (Mazin, 2011),different from the s + − state proposed for the pnictides(Johnston, 2010). Also note that the same-sign s -wavemay not explain the neutron spin resonances in the al-kali metal iron selenides (Scalapino, 2012). Thus, onlyfurther work can clarify entirely this subtle matter.7Another reason for the excitement in this area of re-search is the possibility of having an insulating parentcompound of the SC state, perhaps a Mott insulator.Candidate states with an ordered distribution of iron va-cancies have been identified at particular compositionsof iron. Some of these states display an exotic magneticstate that contains 2 × As , the readercan consult Johnston et al. (2011) and Pandey et al. (2012), and literature cited therein. This line of explo-ration is promising and it should be further pursued.Finally, the presence of phase separation has also at-tracted considerable attention. Are the magnetic and SCstates competing or cooperating? This is also a recurrentopen question for the SC copper oxides as well. Notethat such competition or cooperation is only relevant ifthe states can influence one another by either sharingthe same volume element, i.e. microscopically coexist-ing, or by forming an inhomogeneous state at such shortlength scales that one state can still affect the other andviceversa. In fact, in several FeAs-based materials thereis evidence that the two competing states do share thesame volume element (Johnston, 2010), while in the se-lenides the situation is still evolving with regards to thelength-scales involved in the phase separation process.In summary, the young subfield of alkali metal ironselenides is challenging the prevailing ideas for the pnic-tides. It could occur that selenides and pnictides mayharbor different pairing mechanisms, or they may havedifferent strengths in their Hubbard U couplings. Af-ter all, the pnictides have AFM metallic states as par-ent compounds of superconductivity, while the selenidesmay have AFM insulators as parent compounds basedon the discussion presented in this review. However, bymere simplicity it is also reasonable to assume that aunique qualitative mechanism could be at work simulta-neously in both families of compounds. Perhaps short-range AFM fluctuations may be similarly operative as thepairing mechanism in the context of both metallic andinsulating parent states. All these important issues arestill under much discussion, and by focusing on the newalkali metal iron selenides the several intriguing concep-tual questions raised by the discovery of the iron-basedsuperconductors may soon converge to an answer. XI. ACKNOWLEDGMENTS
The author thanks D.C. Johnston and A. Moreo for acareful reading of this manuscript and for making valu- able suggestions to improve the quality of the presen-tation. The author also thanks W. Bao, A. Bianconi,A. V. Boris, M. J. Calder´on, Chao Cao, A. Charnukha,Xi Chen, Xialong Chen, Xianhui Chen, Jianhui Dai,Pengcheng Dai, Hong Ding, Shuai Dong, M. H. Fang,D. L. Feng, P. J. Hirschfeld, C. Homes, J. P. Hu, D.S. Inosov, B. Keimer, Z.-Y. Lu, Qinlong Luo, T. A.Maier, G. Martins, T. M. McQueen, Tian Qian, C. Petro-vic, A. Ricci, A. Safa-Sefat, D.J. Scalapino, Gang Wang,Miaoyin Wang, Tao Xiang, Q. K. Xue, Yajun Yan, FengYe, Rong Yu, H.Q. Yuan, Fuchun Zhang, and X.J. Zhoufor many useful comments. The author is supported bythe U.S. DOE, Office of Basic Energy Sciences, MaterialsSciences and Engineering Division, and by the NationalScience Foundation under Grant No. DMR-1104386.
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