Structural, magnetic and electronic properties of the iron–chalcogenide A_xFe_2-ySe₂ (A=K, Cs, Rb, and Tl, etc.) superconductors

Abstract

The latest discovery of a new iron–chalcogenide superconductor A_xFe_2-ySe₂ (A =K, Cs, Rb, and Tl, etc.) has attracted much attention due to a number of its unique characteristics, such as the possible insulating state of the parent compound, the existence of Fe-vacancy and its ordering, a new form of magnetic structure and its interplay with superconductivity, and the peculiar electronic structures that are distinct from other Fe-based superconductors. In this paper, we present a brief review on the structural, magnetic and electronic properties of this new superconductor, with an emphasis on the electronic structure and superconducting gap. Issues and future perspectives are discussed at the end of the paper.

Graphical abstract

Keywords

superconductor / iron–chalcogenides / electronic structure / photoemission / Fermi science / superconducting gap / crystal structure / magnetic structure

1 Introduction

The discovery of superconductivity in iron-based compounds in 2008 by Hosono and collaborators [1] with a superconducting critical temperature up to ∼56 K [2–5] ushered in a second class of “high temperature superconductors” after the discovery of the first class of high temperature superconductors in copper–oxide compounds (cuprates) in 1986 [6]. So far, four main families of Fe-based superconductors have been found, denoted as “1111”-type ReFeAsO (Re = rare earth) (FeAs1111) [1–5], “122”-type BFe₂As₂ (B=Ba, Sr, or Ca) (FeAs122) [7], “111”-type AFeAs (A = alkali metal) (FeAs111) [8], and “11”-type tetragonal α-FeSe(Te) (FeCh11) [9]. It is instructive to compare and contrast the Fe-based superconductors to the copper–oxide superconductors in order to pinpoint some essential ingredients in realizing high temperature superconductivity [10–13]. There are similarities between the Fe-based superconductors and cuprate superconductors: (i) Structurally speaking, both the Fe-based compounds and cuprates have layered structures. The Fe-based compounds consist of a common FePn (Pn=As or Se) layers which are considered to be essential for the occurrence of superconductivity, similar to the CuO₂ planes in cuprates. (ii) Superconductivity in the Fe-based compounds is realized in a vicinity of antiferromagnetic state, a case that is similar to that in the cuprates. There are also some significant differences between the Fe-based and copper-based superconductors: (i) Usually the perfectness of CuO₂ plane is critical in sustaining superconductivity in cuprate superconductors. But the FeAs(Se) layers appear to be more tolerant to modifications of external perturbations. In fact, substitution of Fe by other ions like Co or Ni can even transform a non-superconductor into a superconductor [14, 15]. (ii) It is known that the parent compounds of cuprate superconductors are insulating Mott insulator [16], but most parent compounds of the Fe-based superconductors are bad metals. (iii) In the cuprate superconductors, the electronic structures are mainly dictated by a single \text{Cu}3{\text{d}}_{x^2}{}_{-y^2} band. But in the Fe-based compounds, all the five Fe 3d orbitals contribute to the formation of low-lying electronic states, thus forming a typical multi-band system.

The Fe-based superconductors discovered so far possess some common characteristics of electronic structure [17–20]. The low-lying electronic excitations are mainly dominated by five 3d orbitals which give rise to a couple of hole-like bands near the zone center \Gamma and electron-like bands near the zone corner M. Since the parent compound of the Fe-based superconductors is a bad metal, the electron correlation is believed to be not as strong as in cuprates. It remains under debate whether electrons in the Fe-based superconductors should be treated locally or itenerantly [10–13]. Moreover, while it is well-established that in the cuprate superconductors, the superconducting order parameter has predominantly d-wave symmetry, the pairing symmetry in the Fe-based superconductors remains unclear. It has been proposed that the interband scattering between the hole-like bands near \Gamma and the electron-like bands near M gives rise to electron pairing and superconductivity [18, 21–26]. On the other hand, another approach based on strong coupling suggested that the parent compound antiferromagnetism could be understood with frustrated Heisenberg model and the local spin coupling could give rise to superconductivity [27–31]. Both of these two approaches give an s^± pairing symmetry (nodeless gap with sign change between hole and electron Fermi surface sheets). In addition, the orbital degree of freedom has also been proposed [32, 33] to play an important role in this multiband system. Enhanced by electron–phonon coupling, orbital order is considered to be the driving force of antiferromagnetic transition and its fluctuation could lead to superconductivity with s⁺⁺ pairing symmetry (no gap sign change between two kinds of Fermi surface sheets). Although nodeless superconducting gap was revealed by Angle Resolved Photoemission Spectroscopy (ARPES) [34, 35] and gap sign change implied by some experimental techniques [36, 37], at this stage it remains to be investigated which of these candidates provide the best description of the Fe-based compounds.

Very recently, a new superconductor A_xFe_2-ySe₂ (A=K, Cs, Rb, and Tl, etc.) with T_{\mathrm{c}} around 30 K was reported [38–42]. This new superconductor triggered a new wave of excitement in the superconductivity community because it exhibits a couple of unique characteristics [43]. First, while the parent compounds of other Fe-based superconductors are bad metal, it is suggested that the parent compound of this superconductor could be an insulator [41]. Second, while the Fe-sites in FeAs(Se) layers of other Fe-based superconductors are filled, there could be Fe vacancies in this new superconductors [41]. The superconductor may show unique magnetic structure with high magnetic transition temperature and large magnetic moment on the Fe site [44, 45]. Particularly, the electronic structure of the new superconductor is distinct from other Fe-based superconductors in that no hole-like Fermi surface around \Gamma is present [46–50]. These characteristics will provide new perspectives on understanding the Fe-based superconductors. On the other hand, many issues remain unclear at this moment that need further experimental and theoretical efforts [51].

In this paper, we will present a brief review on the current status of research on the A_xFe_2-ySe₂ superconductors, with an emphasis on their unique electron structure. The paper is organized as follows: in Section 2, structural properties including Fe vacancy order and several proposed phase diagrams are first discussed. Then we summarize magnetic structures of several reported phases and their spin dynamics. In Section 3, electronic properties, including band structure and Fermi surface, gap structures and pairing symmetry are discussed. In Section 4, we provide a summary and discussion on the future issues.

2 Structural and magnetic properties

2.1 Crystal structure and phase diagram

The superconductivity with a T_{\mathrm{c}} at ∼30 K was first reported in a compound with a nominal chemical formula K_xFe₂Se₂, which was considered to be isostructural to FeAs122, as shown in Fig. 1 [38]. Later on Fang et al. [41] pointed out that Fe deficiency exists in (Tl_1-yK_y)Fe_xSe₂ compounds, as those discovered previously in the TlFe_xCh₂ (Ch=S, Se) compounds [52–54]. By tuning the Fe content x, different phases from an insulator to a superconductor can be obtained [Figs. 2(a) and (b)]. At low Fe content x, the compound is an antiferromagnetic insulator with a high Néel temperature. With increasing x, the antiferromagnetism is gradually suppressed and superconductivity emerges around x=1.7 with only a small fraction of superconducting volume. Bulk superconductivity can be found when x⩾1.78. Similar insulator-to-superconductor transition is also reported in K_xFe_2-ySe₂ by varying the potassium content, x [55]. A sign of possible T_{\mathrm{c}} at 40 K was reported in the A_xFe_2-ySe₂ system (A=\mathrm{K},(\mathrm{Tl},\mathrm{K})) [41, 55] but it remains hard to isolate a pure superconducting phase. T_{\mathrm{c}} of the A_xFe_2-ySe₂ (A=\mathrm{K},(\mathrm{Tl},\mathrm{Rb})) superconductor decreases from 31 K to zero with increasing pressure and then another phase with a T_{\mathrm{c}} as high as 48 K was reported under higher pressure [56].

It is apparent that the physical properties of A_xFe_2-ySe₂ rely on both the content of A (x) and the content of Fe (y). It is therefore essential to have a precise determination of the sample composition in order to build a clear correspondence between the composition, structure and physical properties. A phase diagram was constructed in Rb_xFe_2-ySe₂ based on composition determination and corresponding magnetic, conductivity and specific heat measurements [57]. As shown in Fig. 2(c), superconductivity was discovered with an Fe content between 1.53 and 1.6, while insulating and semiconducting behaviors were observed with Fe contents at and 2-y>1.6, respectively. Another phase diagram based on the Fe valence state (V_{\mathrm{Fe}}) was proposed in K_xFe_2-ySe₂ [58] which is divided into three regions [Fig. 2(d)]. These regions show structural and AFM transitions at similar temperatures. But superconductivity appears only in region II with and T_{\mathrm{c}} is nearly independent of V_{\mathrm{Fe}}. AFM insulating behavior is found on both sides of the superconducting region II, but they show different Fe vacancy ordering. In region I with V_{\mathrm{Fe}}⩾2.00, Fe vacancy order with a {\mathit{q}}_2=(1/4,3/4,0) is observed while in region III with , the Fe vacancy order has a wave vector of {\mathit{q}}_1=(1/5,3/5,0). It has been found that all the samples with a chemical formula K_1-xFe_1.5+x/2Se₂ are insulators; superconductivity can be obtained only by adding extra Fe content if keeping the potassium content x at 0.8. Therefore the stoichiometric A_0.8Fe_1.6Se₂ compound was proposed to be the parent compound of the A_xFe_2-ySe₂ superconductors which is an AFM insulator [41, 58].

It is worth noting that the A_xFe_2-ySe₂ compounds show peculiar resistivity-temperature dependence. As shown in Fig. 1(b), in addition to an abrupt superconducting transition at 31 K, the resistivity of K_0.8Fe₂Se₂ superconductor exhibits a broad hump around 140\sim150 K (T_{\mathrm{H}}) where there appears to be an insulator–metal transition [38]. Such a hump is common in other A_xFe_2-ySe₂ superconductors [39–41]. It was found that the hump maximum temperature (T_{\mathrm{H}}) depends sensitively on the type of A atom [59] and the Fe content [41, 55]. It shows little change with the applied magnetic field [60]. Pressure can gradually suppress the magnitude of the hump with a slight increase of T_{\mathrm{H}} [60–63]. It changes non-monotonically with isovalent substitution of Se with sulfur in K_xFe_2-y(Se_2-zS_z) [64]. When it is compared to the magnetic susceptibility, no corresponding anomaly is found at the temperature of the broad hump, suggesting it may not be a magnetic transition [65]. Moreover, structural analysis reveals no structural transition around T_{\mathrm{H}} [44, 60, 66]. It is possible that such a resistivity hump is related to the phase separation, as will be discussed below.

2.2 Iron vacancy order and phase separation

As mentioned before, the stoichiometry of synthesized samples always deviate from the ideal AFe₂Se₂ due to the restriction of the Fe valence. Iron vacancy would form an ordered state in Fe-deficient A_xFe_2-ySe₂ as reported years ago [52–54]. Because of the discovery of superconductivity, many new experimental studies have been carried out recently which have revealed different types of iron ordering in A_xFe_2-ySe₂ by transmission electron microscope (TEM) [67–70], neutron scattering [44, 45, 66, 71–73], X-ray diffraction (XRD) [74, 75] and scanning tunneling microscopy (STM) [76, 77]. Up to now, five different phases were found (Fig. 3) which include vacancy free phase [Fig. 3(a)], \sqrt{5}\times \sqrt{5} superstructure phase [Fig. 3(b)], 2\times 2 superstructure phase [Fig. 3(c)], \sqrt{2}\times 2\sqrt{2} superstructure phase [Fig. 3(d)] and \sqrt{2}\times \sqrt{2} superstructure phase. From the iron ordering pattern, the Fe content of \sqrt{5}\times \sqrt{5} superstructure phase corresponds to 1.6 and its stoichiometry can be written as K_0.8Fe_1.6Se₂ or K₂Fe₄Se₅ (245 phase) if undoped. Similarly, both stoichiometries of 2\times 2 and \sqrt{2}\times 2\sqrt{2} superstructure phases can be written as K₂Fe₃Se₄ (234 phase I and 234 phase II respectively). Table 1 summarizes observed superstructures in different A_xFe_2-ySe₂ samples. The vacancy free phase has been observed only in superconducting samples while the \sqrt{2}\times 2\sqrt{2} superstructure is observed only in insulators. The other three superstructures were observed both in superconducting and insulating samples. Compared with other phases that are related to iron vacancy, the origin of the \sqrt{2}\times \sqrt{2} superstructure has not reached a consensus. It was observed by TEM [68], XRD [73] and neutron scattering [45], suggesting it is related to the lattice. But the STM results suggest it may have a magnetic origin [76, 77].

Temperature dependent neutron scattering studies on K_xFe_2-x/2Se₂ compounds with various iron content were reported by Bao et al. and a phase diagram was proposed as shown in Fig. 4 [66]. At high temperature, all samples show tetragonal ThCr₂Si₂ structure with I4/mmm group symmetry and both K and Fe sites are partially occupied in random. As temperature goes down, for insulator compounds with low iron content, two vacancy ordered state with I4/m(\sqrt{5}\times \sqrt{5} superstructure) and Pmna(\sqrt{2}\times 2\sqrt{2} superstructure) group symmetry appear and coexist with I4/mmm phase between 295 K and 500 K. Only \sqrt{5}\times \sqrt{5} superstructure phase exists when temperature is further lowered below 295 K. For superconductors with higher iron content, vacancy ordered state with I4/m group symmetry develops after the structure transition at 500 K and is maintained at the lowest temperature even below T_{\mathrm{c}}, indicating the superconductivity may coexist with the \sqrt{5}\times \sqrt{5} superstructure.

So far it is hard to prepare single phase A_xFe_2-ySe₂ samples, especially superconducting ones. For superconducting samples, it is common to observe two sets of c lattice constants, even for single crystal samples [45, 65, 78]. The coexistence of multiple phases has been observed in TEM [67, 70] and STM [76, 77] results (see Fig. 5). While phase separation is possible along the a–b plane, it is interesting to see that, in the superconducting samples, TEM revealed that the Fe-vacancy disorder state (DOS) and order state (OS) alternate along the c-axis direction [see Fig. 5(a)]. The temperature evolution of the phase separation was also investigated by XRD studies on a K_0.8Fe_1.6Se₂ superconductor [79]. At a high temperature above 600 K, no superstructure was found and it is a vacancy-disordered tetragonal phase. After formation of \sqrt{5}\times \sqrt{5} superlattice at 580 K, the (220) peak splits and another set of superstructure spots appears at 520 K, which can be assigned to \sqrt{2}\times \sqrt{2} superlattice. Using nanofocused XRD on different parts of the sample, the proportion of these two phases is found to vary from one part to another [80]. It remains to see whether a pure A_xFe_2-ySe₂ superconducting sample can be prepared or the phase separation is an intrinsic process for superconducting samples.

2.3 Magnetic structures

It has been well-established that the parent compounds of FeAs1111, FeAs122 and FeAs111 have collinear magnetic structure while the FeCh11 phase (FeTe) has a bi-collinear magnetic structure [81–85]. For the A_xFe_2-ySe₂ compound, the magnetic structures with different iron vacancy ordering are predicted from the theoretical calculations, as summarized in Fig. 3. For the hypothetical stoichiometric AFe₂Se₂ compound [86], the ground state shows a bi-collinear antiferromagnetic order, with the Fe moments having collinear antiferromagnetic order in each bipartite sublattice [Fig. 3(a)]. It results from Se 4p orbitals mediated super-exchange interactions of Fe moments, similar to the FeCh11 compound [84]. For the 245 phase with the \sqrt{5}\times \sqrt{5} superstructure [87–90], the ground magnetic state favors block AFM structure with all the magnetic moment parallel to c axis and four nearest iron atoms forming a cluster [Fig. 3(b)]. It is ferromagnetic (FM) within one cluster and antiferromagnetic between the adjacent clusters. Two different magnetic structures were predicted in the compound with the same 234 stoichiometry. For the square ordered 234 phase I [91], the magnetic structure is c-AFM order in which the next-nearest Fe moments are ordered in antiparallel if one just ignores the iron vacancy [Fig. 3(c)]. For the rhombus ordered 234 phase II [91], it has an A-collinear AFM order magnetic structure where the Fe moments are antiferromagnetically ordered along the line without vacancies [Fig. 3(d)].

Block AFM structure of the 245 phase with the \sqrt{5}\times \sqrt{5} superstructure was established from neutron diffraction on both polycrystalline samples [44, 72, 73] and single crystal samples [45, 71]. However, no experimental results on magnetic structure of other ordered states are available yet. Figure 6 shows the development of the block AFM structure of the 245 phase in various A_xFe_2-ySe₂ (A = K, Rb, Cs, (Tl,K), (Tl,Rb)) samples [71]. The antiferromagnetic Néel temperature (T_{\mathrm{N}}), structural transition temperature (T_{\mathrm{S}}) and magnetic moment of different A_xFe_2-ySe₂ samples measured by different experimental techniques are summarized in Table 2. Because the establishment of the block AFM is based on the formation of the \sqrt{5}\times \sqrt{5} superstructure, T_{\mathrm{N}} is always slightly lower than T_{\mathrm{S}}, except for a K_xFe_2-ySe₂ insulator where the two transitions occur at nearly the same temperature [66]. The A_xFe_2-ySe₂ compounds have a rather high Néel temperature (above 500 K), and large magnetic moment (up to 3μ_B/Fe) which is the largest among all Fe-based compounds.

As mentioned above, due to the existence of multiple phases in A_xFe_2-ySe₂ samples, it remains unclear which phase is really superconducting. A related issue is whether antiferromagnetism and superconductivity can coexist in the system. On the one hand, coexistence of superconductivity and magnetism in A_xFe_2-ySe₂ was proposed from results of neutron scattering [44, 45, 71–73], transport measurements [57–59], μSR [92], Mössbauer [93] and Raman scattering [94]. As shown in the inset of Fig. 6, the magnetic Bragg intensity from the \sqrt{5}\times \sqrt{5} superstructure shows a kink around T_{\mathrm{c}}, indicating that the antiferromagnetism and superconductivity are coupled [44, 71]. On the other hand, Mössbauer spectroscopy studies on Rb_0.8Fe_1.6Se₂ superconductor revealed the presence of 88% magnetic and 12% non-magnetic Fe²⁺ species which could be attributed to the \sqrt{5}\times \sqrt{5} superstructure phase and vacancy free phase, respectively [95]. STM results also indicate that the superconductivity comes from the vacancy free phase and the \sqrt{5}\times \sqrt{5} superstructure phase is an insulator [Figs. 6(b)–(d)] [76, 77]. Further work are needed to reconcile these seemingly conflicting results. Recently, Hu et al. proposed [96] that the vacancy order free phase(also superconducting phase) may have block-AFM ground state as well, similar to that in \sqrt{5}\times \sqrt{5} superstructure phase. Since these two block-AFM states can be described by the same magnetic model, two separated phases can couple with each other in one sample.

2.4 Spin dynamics

The spin dynamics of the insulating Rb_0.89Fe_1.58Se₂ compound with a \sqrt{5}\times \sqrt{5} superstructure has been investigated by inelastic neutron scattering (INS) [97]. As shown in Fig. 7, the spin waves exist in three separated energy ranges. The lowest branch, which is an acoustic mode arising mostly from antiferromagnetic interactions of the FM blocked spins [89, 98], starts from 9 meV to 70 meV. The other two branches, which are optical spin waves associated with exchange interactions of iron spins within the FM blocks, are from 80 meV to 140 meV and 180 meV to 230 meV respectively. The magnetic exchange couplings, obtained by fitting the data with effective J_1\text{–}{J}_1^{\prime }\text{–}{J}_2\text{–}{J}_2^{\prime }\text{–}{J}_3\text{–}{J}_c Heisenberg model, are summarized in Table 3, together with those in Fe_1.05Te [99] and CaFe₂As₂ [100] for comparison. Although their static antiferromagnetic orders have completely different structures, these three iron-based compounds have comparable effective exchange couplings J₂, which is mainly determined by a local superexchange mediated by As or Se/Te [27]. This is consistent with the idea that J₂ is the leading parameter of ground magnetic state and closely related to superconductivity in the Fe-based superconductors [101].

It is predicted that, if there is a sign change in the superconducting order parameter, a spin resonance mode with an energy between one and two times of the superconducting gap would appear at the wave vector connecting two parts of the Fermi surface with opposite gap signs [102]. The spin resonance mode, generally taken as a hallmark of unconventional pairing symmetry of superconductivity, has been observed in cuperate superconductors [103–106], heavy Fermion superconductors [107, 108] and the Fe-based superconductors [36, 85]. Such a resonance mode has been also revealed in Rb_0.8Fe_1.6Se₂ superconductor [109]. As shown in Fig. 8, the intensity is obviously enhanced at 14 meV across T_{\mathrm{c}} at a wave vector (0.5, 0.3125, 0.5), in contrast to (0.5, 0.5, 0.5) where the resonance has been theoretically predicted in d-wave pairing symmetry [110, 111] and (0.5, 0, 0.5) where it is usually found in other Fe-based superconductors [36, 85]. Temperature dependence of the measured resonance intensity follows an order-parameter-like increase below T_{\mathrm{c}}, indicating it is related to the superconducting transition [Fig. 8(c)]. The ratio of \mathrm{\hslash }{\omega }_{\mathrm{Res}}/(k_{\mathrm{B}}{T}_{\mathrm{c}}) is 5.1±0.4, slightly above the nearly universal ratio of 4.3 estimated for 122-compounds, but is close to that in FeTe_1-xSe_x, LiFeAs, La-1111, and cuprate superconductorse [11, 85, 112, 113]. Another scaling parameter 2\Delta /(\mathrm{\hslash }{\omega }_{\mathrm{Res}}) would be 0.7±0.1, if we take the superconducting gap as 10 meV measured by ARPES [47–50]. It is slightly larger than the typical value of 0.64 for cuprates but smaller than 1, following the general trend found in all Fe-based superconductors [112, 113].

It is expected that antiferromagnetic spin fluctuations, which is proposed to be a strong candidate of the pairing glue in the Fe-based superconductors and has been observed in 11-type iron–chalcogenide and iron–pnictide superconductors, would introduce a Curie–Weiss upturn near T_{\mathrm{c}} in 1/(T_{1}T) in the nuclear magnetic resonance measurements [114, 115]. However, no indication of such an enhancement is detected in several A_xFe_2-ySe₂ (A = K, (Tl,Rb)) superconductors so far [116–119]. Instead, the results are more similar to the overdoped non-supercondcting 122-type iron–pnictide superconductors where the absence of Curie–Weiss upturn was interpreted as due to the lack of nesting between the hole and electron Fermi surface sheets because the hole band sinks below the Fermi level by electron doping. Similar band structure was observed by ARPES in various A_xFe_2-ySe₂ superconductors, as described below.

3 Electronic properties

3.1 Band structure calculations

In A_xFe_2-ySe₂, the intercalated A can donate electrons into the Fe_2-ySe₂ layers. The electronic structure of the iron vacancy free phase, A_xFe₂Se₂ (A=K, Cs, Rb, Tl), can be treated as electron-doped FeSe (doping level is x) with the chemical potential raising up, or hole-doped AFe₂Se₂ (doping level 1-x) with the chemical potential moving down, as confirmed by LDA calculations [86, 120–122]. The calculated electronic structures have little dependence on the A element because the density of states (DOS) around the Fermi level are mainly associated with Fe-3d and Se-4p orbitals [121]. The calculated band structure of AFe₂Se₂ (A=K, Cs) is shown in Fig. 9(b), together with that of BaFe₂As₂ for comparison [122]. The band structure of A-deficient A_xFe₂Se₂ looks similar to that of BaFe₂As₂ if the chemical potential of the A-filled AFe₂Se₂ band is lowered, as shown in Fig. 9(b). Compared to BaFe₂As₂ where both the two e_g orbitals (3z^2-r^2 and x^2-y^2) and the three t_{2g} orbitals (xz, yz and xy orbitals) are at the Fermi level, the e_g states in A_xFe₂Se₂ are sinked below E_{\mathrm{F}} and the states at E_{\mathrm{F}} are only contributed by the t_{2g} states. As a result, the Fermi surface of AFe₂Se₂ [Fig. 9(c)] consists of two quasi-two-dimensional electron-like Fermi surface sheets near X and a small electron-like Fermi pocket near Z which is more three-dimensional. Note that in this case all the Fermi surface sheets are electron-like.

The electronic structure of various Fe-deficient A_xFe_2-ySe₂ phases has also been calculated, such as vacancy free phase with bi-collinear AFM [86], 245 phase with block AFM [87, 88], 245 phase in non-magenetic state [87, 88], and 234 phase II with A-collinear AFM [91]. Figure 10 shows the band structure of two typical Fe-vacancy-ordered phases. Of particular interest is the electronic structure of the A_0.8Fe_1.6Se₂ (245 phase) that has been commonly revealed in structural analysis. The calculated band structure for the 245 phase with block AFM [Fig. 10(a)] indicates that it is a semiconductor with band gap as large as 400–600 meV [87, 88]. This is totally different from the vacancy free A_xFe₂Se₂ phase or any other Fe-based superconductors. Detailed analysis of the orbital contribution indicates that the top valence bands are composed of both Fe-3d and Se-4p orbitals and the lower conduction bands are mainly contributed by the Fe-3d orbital. All the five up-spin orbitals are almost fully filled, while the down-spin orbitals are partially filled by d{}_{z^2}, d_xy, and d_yz orbitals, suggesting the Hund coupling is larger than the crystal-field splitting induced by Se atoms. The 234 phase II in magnetic ground state was also indicated by LDA calculations as an semiconductor with a smaller band gap [91], as shown in Fig. 10(b). It varies from tens to more than a hundred meV between different compounds, which is comparable to the gap value obtained from resistivity measurements [41]. It has been predicted that Mott insulator behavior can exist in both the 234 and 245 vacancy-ordered phases [123–126]. The iron vacancy is expected to reduce the neighboring iron number in 234 or 245 phases, thus making the electron hoping term between neighboring irons smaller and then the bandwidth narrower. This reduction of bandwidth tends to push the vacancy ordered phase of A_xFe_2-ySe₂ into the Mott insulator regime.

3.2 Band structure and Fermi surface

Angle-resolved photoemission (ARPES) experiments have been successfully carried out on the A_xFe_2-ySe₂ (A=K, Cs, (Tl,K), (Tl,Rb)) compounds [46–50, 127, 128]. Figure 11(a) shows a wide energy scan of a photoemission spectrum which includes shallow core levels and the valence band of the K_0.8Fe_1.7Se₂ superconductor [46]. Besides a sharp peak of K 3p core level at a binding energy of 17.55 eV and another peak at 12 eV, a weak feature at 0.9 eV is also revealed. A weakly-dispersive broad band is observed around this energy from the ARPES measurements [Figs. 11(c) and (d)] [46]. This feature shifts to lower binding energy with increasing temperature [50]. By systematically analyzing the electronic structures of superconducting, semiconducting and insulating K_xFe_2-ySe₂ samples, it is suggested that it may come from an insulating phase in the samples [127].

Detailed band structure in the vicinity of Fermi level in several A_xFe_2-ySe₂ superconductors have been investigated and typical results are shown in Fig. 12. The band structure of (Tl,Rb)_xFe_2-ySe₂ superconductor (T_{\mathrm{c}}=32 K) around \Gamma shows two electron bands [α and β in Fig. 12(a)] with the α band bottom barely touching the Fermi level. There is also a hole-like band sinking below the Fermi level [Fig. 12(b)] [48]. In (Tl,K)_xFe_2-ySe₂ and K_xFe_2-ySe₂ superconductors [49, 50], in addition to two electron bands, two hole bands [denoted as GA and GB in Fig. 12(j)] below the Fermi level are observed. Around M, one electron-like band is clearly observed [Figs. 12(c) and (d)]. But a more detailed measurements [Figs. 12(e) and (f)] indicate that the band is composed of two bands with similar Fermi momenta but with different band bottoms, one at ∼ 40 meV and the other at ∼ 60 meV [Fig. 12(f)]. The k_Z dependence of these bands are obtained by ARPES using different photon energies [47]. As shown in Fig. 13, the band crossings around M do not change much at different k_Z, indicative of their quasi-two-dimensional nature. However, the electron-like α band near \Gamma exhibits obvious k_Z dependence: an electron-like band crosses the Fermi level near Z but it is above the Fermi level near \Gamma [47].

The A_xFe_2-ySe₂ superconducting compounds exhibit similar Fermi surface topology, as summarized in Fig. 14. They all show two electron-like pockets around \Gamma (denoted as α and β) and one electron-like pocket around M (γ) which is in fact composed of two nearly degenerate Fermi surface sheets. Around \Gamma point, no Fermi surface sheet or only one small sheet were first reported in K_xFe_2-ySe₂ and Cs_xFe_2-ySe₂ [46, 47]. Later results on (Tl,Rb)_xFe_2-ySe₂ with improved data quality revealed two electron-like Fermi surface sheets around \Gamma [48]. This distinct Fermi surface topology was further confirmed in other measurements of (Tl,K)_xFe_2-ySe₂ and K_xFe_2-ySe₂ [49, 50], making it a common Fermi-surface topology in the A_xFe_2-ySe₂ superconductor. By calculating the enclosed area of all measured Fermi pockets, the doping concentration can be estimated to be 0.27 e/Fe for (Tl_0.58Rb_0.42)Fe_1.72Se₂ [48] and 0.32 e/Fe for (Tl_0.63K_0.37)Fe_1.78Se₂ [50]. It is 0.18 e/Fe for both compounds, if only the two degenerate pockets around M are counted.

It is noted that the measured Fermi surface and band structure of the A_xFe_2-ySe₂ superconductors are qualitatively consistent with band structure calculations on the vacancy free AFe₂Se₂ phase (Fig. 9). One outstanding difference is the β Fermi surface sheet around \Gamma (Fig. 13) that is absent in the band calculations. We note here that, since the exact superconducting phase in A_xFe_2-ySe₂ remains unclear, one should be cautious in make such a direct comparison to jump on some conclusions. An immediate issue is on the origin of this β Fermi surface. The first possibility is a surface state. While surface state on some Fe-based compounds like the FeAs1111 system was reported before [129], it has not been observed in the “11”-type Fe(Se,Te) system [130]. The second possibility is whether the β band can be caused by the folding of the electron-like γ Fermi surface near M. It is noted that the Fermi surface size, the band dispersion, and the band width of the β band at \Gamma is similar to that of the γ band near M. A band folding picture would give a reasonable account for such a similarity if there exists a (π, π) modulation in the system that can be either structural or magnetic. An obvious issue with this scenario is that, in this case, one should also expect the folding of the α band near \Gamma onto the M point; but such a folding is not observed at the M point [Figs. 12(c) and (d)]. The third possibility is whether the measured β sheet is a Fermi surface at a special k_Z cut. Due to its weak intensity, this β Fermi surface is not revealed in the k_Z dependence measurements (Fig. 13). But almost the same Fermi crossing was observed with three different energies (6.994 eV, 21.2 eV and 40.8 eV), implying it is nearly two-dimensional-like [48, 128]. Note that this β Fermi surface should not be confused with the α pocket at Z because they have very different size, particularly they are observed simultaneously near \Gamma point. The origin of the β Fermi surface is still an open question to be further addressed.

It is under debate whether the superconducting phase in A_xFe_2-ySe₂ coexists with the Fe vacancy ordered \sqrt{5}\times \sqrt{5} phase [71]. Under such a circumstance, one would expect the Brillouin zone for the \sqrt{5}\times \sqrt{5} phase to be 1/5 of the original 1×1 phase [Fig. 3(e)]. The \sqrt{5}\times \sqrt{5} superstructure would produce multiple folded Fermi surface sheets from the original one pocket near M or near \Gamma. However, no indication of such folded Fermi surface sheets is observed in the measured results (Fig. 14). While one cannot fully rule out that the folded Fermi surface is not seen because they may be too weak, their absence is not compatible with the scenario of coexistence of superconductivity and the \sqrt{5}\times \sqrt{5} superstructure. The same argument is also applicable to other possible ordered superstructures like the 2\sqrt{2}\times \sqrt{2} and 2\times 2 phases.

Study on the electronic structure evolution from an insulator to a superconductor is helpful in addressing the key issue on what the parent compound is for the A_xFe_2-ySe₂ superconductors. Valence bands were measured on three types of K_xFe_2-ySe₂ compounds (insulator, semiconductor and superconductor) [113]. In comparison with a superconductor, no bands are observed within the 0.5 eV energy range below E_{\mathrm{F}} in an antiferromagnetically insulating sample. Two high energy features around 0.7 eV and 1.6 eV binding energies were observed in all three type of samples. It was found that, by increasing the temperature or reducing the photon intensity, these two features shift toward low binding energy, which is a typical behavior called charging effect in photoemission measurements on insulating samples. On the other hand, no such charging effect was observed in low energy feature of a superconductor. Therefore, the high energy features (0.7 eV and 1.6 eV) and low energy near-E_{\mathrm{F}} features may not come from the same phase in the sample: the former may come from the insulating phase while the latter from metallic or superconducting phase. Similar phase separation behavior was also found in the semiconducting sample. Such results are consistent with the phase separation picture also revealed by TEM [67] and STM [76, 77].

In contrast to the dramatic difference between the electronic structure of the insulating and superconducting phases, the low energy features of the semiconductor are reminiscent of those in the superconducting sample. As illustrated in Fig. 15(b), only hole-like bands with a band gap of \sim20 meV is observed around \Gamma. The band structure of the semiconductor seems to resemble that of the superconductor by shifting the chemical potential of the superconductor downwards by 55 meV and coincidentally the chemical potential lies in the gap between the electron and hole bands. No shadow band possibly caused by magnetic order or lattice superstructure was observed in this semiconductor. These observations led the authors [127] to propose that the actual parent compound of K_xFe_2-ySe₂ superconductor is not the insulating phases, but the semiconducting phase; superconductivity can be obtained by doping the semiconductor. More work needs to be done to confirm this interesting scenario.

3.3 Superconducting gap and pairing symmetry

ARPES can directly measure superconducting gap and its momentum dependence, thus providing key information on the pairing symmetry of superconductivity. Figure 16 first shows typical ARPES results on measuring temperature dependence of superconducting gap in (Tl,Rb)_xFe_2-ySe₂ superconductor [48, 128]. The photoemission data are taken on different Fermi surface sheets at different temperatures [Figs. 16(a) and (e)] and the photoemission spectra (energy distribution curves, EDCs) on the Fermi surface are symmetrized [Figs. 16(c) and (f)] to visualize the superconducting gap opening and obtain the gap size. It is clear that for both the β and γ Fermi surface sheets the superconducting gap opens right below T_{\mathrm{c}}=32 K. Also the quasiparticle peak sharpens up upon entering the superconducting state. As seen in Fig. 16(b), a sharp quasiparticle peak with a narrow width of only 9 meV (Full-width-at-half-maximum) is observed at low temperature for the γ Fermi surface near M. The superconducting gap size can be determined by taking the EDC peak position. As seen in Fig. 16(d), the gap size on the γ Fermi surface near M follows the standard BCS form with a {\Delta }_0\sim9.7 meV [128]. Similar temperature dependence of the superconducting gap is also reported in other A_xFe_2-ySe₂ superconductors, but with different gap sizes [47, 49, 50]. With respect to the small α pocket, no gap opening was detected in (Tl,Rb)_xFe_2-ySe₂ because the band is slightly above the Fermi level near \Gamma [48]. A superconducting gap of \sim7 meV was reported below T_{\mathrm{c}} in K_xFe_2-ySe₂ on the α Fermi surface near Z with a clear Fermi crossing [47].

The ARPES measurements on different A_xFe_2-ySe₂ (A=K, (Tl,K), (Tl,Rb)) superconductors give nearly isotropic superconducting gap on the γ Fermi surface around M [47–50, 131] [Fig. 17(e)]. In order to measure the momentum-dependent superconducting gap on the weak β Fermi surface around \Gamma, high resolution laser-based ARPES measurements have been performed which also give a nearly isotropic superconducting gap [Fig. 17(d)] [128]. It was also found that the gap sizes on both α and γ Fermi surface sheets show little variation at different Fermi crossing along k_Z [47]. So far no evidence of gap node is observed in the A_xFe_2-ySe₂ superconductors.

The low-temperature specific heat measurement [132] and NMR measurement [118] also suggest a nodeless gap. The scanning tunneling spectroscopy measurements, which reveal the averaged electronic structure of the entire Brillion Zone, show a 7 meV superconducting gap in the cleaved K_xFe_2-ySe₂ sample [77]. For thin film sample grown by MBE, however, two superconducting gaps with much smaller size, 1 meV and 4 meV, were reported [Fig. 5(c)] [76]. Table 4 summarizes the superconducting gap size of different A_xFe_2-ySe₂ samples measured by different techniques. The ratio of 2\Delta /(k_{\mathrm{B}}{T}_{\mathrm{c}}) obtained from single crystal samples is larger than the traditional BCS weak-coupling value of 3.54, which indicates that A_xFe_2-ySe₂ superconductor is in a strong-coupling regime in the BCS picture. On the other hand, the small ratio of 0.9 and 3.5 obtained in MBE-grown film samples puts it in a weak-coupling regime. This difference between the bulk samples and film samples needs to be further clarified.

In the Fe-based superconductors, it has been proposed that the interband scattering between the hole-like bands near \Gamma and the electron-like bands near M gives rise to electron pairing and superconductivity with a s^± symmetry (Fig. 18, left panel) [18, 21–26]. With the absence of hole-like Fermi surface around \Gamma, this picture is no longer applicable in the A_xFe_2-ySe₂ superconductors. In this case, from a weak coupling point of view, the repulsive inter-electron–pocket pair scattering between two M pockets (Fig. 18, right panel) would give a {\text{d}}_{x^2}{}_{-y^2} pairing symmetry [18, 110, 111, 133]. On the other hand, in the viewpoint of a doped Mott insulator, by calculating the t–J model, both s-wave and d-wave pairing are possible at different regions of phase diagram [125, 134]. But s-wave pairing symmetry is robust if the antiferromagnetic J₂ is the main factor for pairing and the J₁ is ferromagnetic [135]. It is also proposed that orbital order and its fluctuations would give rise to a s⁺⁺ wave pairing with the presence of a moderate electron–phonon interaction [136].

The observation of a nearly isotropic, nodeless superconducting gap in A_xFe_2-ySe₂ superconductors favors an s-wave pairing symmetry. It is also considered to be consistent with a nodeless d-wave pairing because the d-wave node is along the diagonal of 1 Fe Brillouin zone (Fig. 18, right panel); the γ Fermi surface sheet may show a nearly isotropic gap because it does not cross this node line. However, as pointed out by Mazin [137], if there is a sign change between the (0, -\pi) Fermi pocket and (π, 0) pocket, gap node would still be expected at the crossing points of two Fermi pockets around the same M point due to the folding from 1 Fe Brillouin zone to 2 Fe Brillouin zone, which is then inconsistent with existing experimental results. Up to now no conclusive answer to the pairing symmetry of the A_xFe_2-ySe₂ superconductors has been reached yet. We note that, the weak β Fermi pocket around \Gamma crosses the node line in d-wave pairing, and if it is proven to be intrinsic, its nearly isotropic gap [Fig. 17(d)] [48, 128] could rule out the d-wave pairing scenario.

3.4 Electron dynamics

The coupling between electron and boson modes plays an important role in giving rise to the electron pairing. Such a coupling can be probed directly by ARPES by measuring the electron self-energy which manifest itself as a kink structure on a band dispersion [138, 139]. A number of phonon modes are observed in the A_xFe_2-ySe₂ (A= K, Rb, Tl) superconductors by Raman scattering and optical studies [140–142]. A magnetic resonance mode has been predicted [110, 111] and observed (Fig. 8) in the superconducting state of A_xFe_2-ySe₂ at an energy of 14 meV [109]. Therefore, it is interesting to investigate the electronic dynamics in A_xFe_2-ySe₂ superconductors. Figure 19 shows the measured band dispersion of the (Tl,Rb)_xFe_2-ySe₂ superconductor around M point at different temperatures, together with the extracted imaginary and real parts of the electron self-energy [128]. The transition near 7 meV in the real part of electron self-energy [Fig. 19(c)] is caused by the superconducting gap opening and is not the effect of electron–boson coupling. Except for this feature, no obvious dispersion kink is observed in the real part of self-energy at both normal and superconducting states. These observations indicate that the electron-boson coupling is weak in the A_xFe_2-ySe₂ superconductors.

4 Summary and perspective

Although it has been only one year since the discovery of the A_xFe_2-ySe₂ superconductors, significant progress has been made in studying their unique structural, magnetic and electronic properties. Unique characteristics have been revealed and studied, such as the existence of Fe vacancy and its ordering, the novel magnetic structure of block antiferromagnetism with large magnetic moment and high Néel temperature, and the possible insulating nature of its parent compound. In particular, key insights have been drawn in understanding the pairing mechanism of the Fe-based superconductors with its distinct electronic structure, i.e., with the absence of hole-like Fermi surface around \Gamma, high T_{\mathrm{c}} (∼30 K at ambient pressure and 48 K under pressure) is achieved in this system.

On the other hand, there are complications and open questions around this new system. First and foremost, what is the exact superconducting phase in the A_xFe_2-ySe₂ superconductors? STM and ARPES data favor Fe-vacancy-free A_xFe₂Se₂ phase, but neutron scattering and Raman studies indicate that superconductivity is associated with the Fe-vacancy ordered \sqrt{5}\times \sqrt{5} superstructure phase. The answer becomes complicated due to the existence of multiple phases in the superconducting samples which leads to another question of phase separation. Is the phase separation in the A_xFe_2-ySe₂ superconductors inevitable or a problem of sample preparation? Can the phase separation properly account for many unusual properties observed so far? The final answer to the superconducting phase will help in answering the question of its parent compound: whether it is an insulator, a semiconductor, or a poor metal. Of course, the ultimately important issue is to reveal the pairing mechanism of superconductivity. To conclude, the A_xFe_2-ySe₂ superconductors provide a new platform for investigating the properties and mechanism of superconductivity. The storm caused by this new superconducting family is continuing and more exciting results are expected under the future effort.

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