Colloquium: The unexpected properties of alkali metal iron selenide superconductors

The iron-based superconductors that contain FeAs layers as the fundamental building block in the crystal structures have been rationalized in the past using ideas based on the Fermi surface nesting of hole and electron pockets when in the presence of weak Hubbard $U$ interactions. This approach seemed appropriate considering the small values of the magnetic moments in the parent compounds and the clear evidence based on photoemission experiments of the required electron and hole pockets. However, recent results in the context of alkali metal iron selenides, with generic chemical composition $A_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ ($A=\mathrm{\text{alkali}}$ metal element), have challenged those previous ideas since at particular compositions $y$ the low-temperature ground states are insulating and display antiferromagnetic order with large iron magnetic moments. Moreover, angle-resolved photoemission studies have revealed the absence of hole pockets at the Fermi level in these materials. The present status of this exciting area of research, with the potential to alter conceptually our understanding of the iron-based superconductors, is here reviewed, covering both experimental and theoretical investigations. Other recent related developments are also briefly reviewed, such as the study of selenide two-leg ladders and the discovery of superconductivity in a single layer of FeSe. The conceptual issues considered established for the alkali metal iron selenides, as well as several issues that still require further work, are discussed.

Figure 1

Crystal structure of $A{\mathrm{Fe}}_x{\mathrm{Se}}_2$. All the other compounds described here have a similar structure. $A$ is an alkali metal element (K in the figure). If $x<2$, iron vacancies are present. From 3.

Figure 2

Temperature dependence of the electrical resistance of polycrystalline ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_2{\mathrm{Se}}_2$. The dominant features include the SC transition temperature at $\sim 30\mathrm{K}$, with the inset containing better-resolution details of that transition. The peak slightly above 100 K, later found using single crystals at higher temperatures (92), is caused by the iron-vacancy ordering (135). The coexistence of features related to iron vacancies and superconductivity was later explained based on phase separation (see Sec. 3). From 39.

Figure 3

(Left panel) Iron-vacancy order corresponding to $A{\mathrm{Fe}}_{1.5}{\mathrm{Se}}_2$. The solid circles are iron atoms. The open circles are vacancies. Each iron atom has either two or three iron neighbors. This type of order is called the $2\times 4$ iron-vacancy order since along the horizontal (vertical) axis the vacancies are separated by 2 (4) Fe-Fe lattice spacings. (Center panel) The case of $A{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$ with its $\sqrt{5}\times \sqrt{5}$ iron-vacancy distribution. All iron atoms have three iron neighbors. The label refers to the distance between nearest-neighbor vacancies which is $\sqrt{5}$ in two perpendicular directions, in units of the Fe-Fe lattice spacing. (Right panel) State with no iron vacancies, corresponding to $A{\mathrm{Fe}}_2{\mathrm{Se}}_2$, where $A=(\mathrm{Tl},\mathrm{K})$, believed to be of relevance for the SC state. From 34.

Figure 4

Spatial distribution of the ratio of the compressed and the expanded phases in a region of size $22\times 55\mu {\mathrm{m}}^2$ of a ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$ crystal. Illustration of several length scales involved in the phase-separated state, resembling those found in cuprates and manganites. From 108.

Figure 5

Cartoon of the phase separation in superconducting ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$, obtained via photoemission and TEM techniques. The upper insets are the photoemission signals for the two regions: the left region corresponds to the $\sqrt{5}\times \sqrt{5}$ vacancy order, while the right region is the density of states of a superconductor. From 19.

Figure 6

(Left panel) STS results showing the DOS of a region of a ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ film that displays features compatible with a SC phase. (Right panel) As the left, but for another region of the film, with results this time compatible with an insulating phase, presumably with ordered iron vacancies. From 73.

Figure 7

Log-log plot of the spectral weight of the superfluid density $N_c$ vs the residual conductivity ${\sigma }_{\mathrm{dc}}$ times the critical temperature $T_c$. Results include those for cuprate superconductors and several iron-based superconductors, and the volume-average and effective-medium approximation (EMA) results for ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$. While the volume average signals a Josephson phase, the EMA result is now very close to the coherent regime. From 46.

Figure 8

Magnitude of the SC gap of ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_2{\mathrm{Se}}_2$ corresponding to the $M$point electron pockets (there are no hole pockets in this compound). The radius represents the gap while the polar angle $\theta$ is measured with respect to the $M\mathrm{\text{−}}\Gamma$ direction defined as $\theta =0$. The results indicate that there are no nodes and also that the gap is fairly uniform, i.e., not strongly momentum dependent. Here the $M=(\pi ,\pi )$ point is 45° rotated with respect to the Fe-Fe axes with regard to unit cells. In the iron-sublattice convention, this point would be $(\pi ,0)$ or $(0,\pi )$. From 173.

Figure 9

Schematic diagram summarizing the electronic band structure of ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_{1.7}{\mathrm{Se}}_2$ obtained from ARPES, with the top of the hole band at the $\Gamma$ point located below the FS. From 104.

Figure 10

FS of $({\mathrm{Tl}}_{0.58}{\mathrm{Rb}}_{0.42}){\mathrm{Fe}}_{1.72}{\mathrm{Se}}_2$, from ARPES studies (93). Note the presence of a small $\Gamma$ pocket that has electronlike energy dispersion. The lattice constant $a$ is 3.896 Å. The $M$ points are equivalent to the $(\pi ,0)$ and $(0,\pi )$ points in the iron-sublattice notation.

Figure 11

Sketch of the SC gap of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{26}$. The energy gap vs wave vector parallel to the $a\mathrm{\text{−}}b$ plane passing through the $Z=(0,0,\pi )$ point. The presence of an isotropic gap at the center rules out $d$-wave superconductivity. From 149.

Figure 12

In-plane crystal and magnetic structure of ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$. The open squares are the iron vacancies and the dark circles with the “$+$” or “$-$” denote the occupied iron sites with the orientation of their spins. The open circles correspond to Se, while the K atoms are small open circles. From 2.

Figure 13

Wave vector dependence of the spin-wave excitations of ${\mathrm{Rb}}_{0.89}{\mathrm{Fe}}_{1.58}{\mathrm{Se}}_2$ at two representative energies (142). (a) Eight peaks of the $\sqrt{5}\times \sqrt{5}$ iron distribution when the two chiralities are averaged; (b) is similar to results for ${\mathrm{BaFe}}_2{\mathrm{As}}_2$.

Figure 14

Normalized resonance energy of several iron-based superconductors, from inelastic neutron scattering. RFS stands for ${\mathrm{Rb}}_2{\mathrm{Fe}}_4{\mathrm{Se}}_5$, BFNA for $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Ni}}_x{)}_2{\mathrm{As}}_2$, and the rest of the abbreviations are for 122, 111, or 1111 materials. From 100.

Figure 15

(a) Electronic band structure of ${\mathrm{K}}_{0.8}{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$ in the ground state with the $2\times 2$ block-AFM order. The top of the valence band is set to zero. (b) Explanation of the convention followed to label points of the Brillouin zone. These theoretical calculations are carried out in a tetragonal structure with lattice parameters in excellent agreement with experiments. From 152.

Figure 16

(a)–(c) Some of the competing states in the presence of a $\sqrt{5}\times \sqrt{5}$ distribution of iron vacancies at $n=6$. From 81. (a) The experimentally dominant $2\times 2$ block-AFM state, (b) a competing state found by 81 by reducing $J_H/U$, and (c) the C-type AFM state described by 21 and 81 that could be stabilized by increasing pressure. (d) Phase diagram of the five-orbital Hubbard model in the presence of the $\sqrt{5}\times \sqrt{5}$ iron-vacancy order, using Hartree-Fock (HF) techniques (81). A variety of phases are stable, including the state of (a) called “AF1” (often also called the plaquette state), the state of (b) called “AF4,” and the state of (c) called “C.” The FM phase is not shown because it is obvious, and for the remaining states “AF2,” “AF5,” and “E” see 11, 81, and 162.

Figure 17

The block-AFM spin order predicted for ${\mathrm{KFe}}_2{\mathrm{Se}}_2$ (no iron vacancies) based on spin model calculations. From 72, where details can be found about the several Heisenberg couplings shown.

Figure 18

The two-leg ladder substructures of ${\mathrm{BaFe}}_2{\mathrm{Se}}_3$, with their legs oriented perpendicular to the figure. From 65.

Figure 19

Relation between a complete FeSe layer and the structure of the ladders. The dark spheres are the Se atoms and the light spheres are the Fe atoms. The ladders simply amount to the removal of every third iron atom from the layers. From 113.

Figure 20

Magnetic order of the two-leg ladders for the cases of ${\mathrm{KFe}}_2{\mathrm{Se}}_3$ and ${\mathrm{BaFe}}_2{\mathrm{Se}}_3$ obtained using neutron diffraction. From 15.

Figure 21

Electronic band structure of the block-AFM state of the two-leg ladder ${\mathrm{BaFe}}_2{\mathrm{Se}}_3$. The gap is 0.24 eV. From 71.

Figure 22

Phase diagram of the five-orbital Hubbard model in the real-space HF approximation, at electronic density $n=5.75$ ($n$ is the number of electrons per iron), using a $2\times 16$ lattice. $J_H$ in units of $U$ and $U$ in units of the bandwidth $W$ are varied. PM stands for paramagnetic and FM for ferromagnetic. The other magnetic states are schematically shown at the bottom. The hoppings used are from band structure calculations corresponding to ${\mathrm{BaFe}}_2{\mathrm{Se}}_3$. From 83.

Figure 23

Fermi surface of a single-layer FeSe superconductor using ARPES techniques. Only electronlike pockets are present. From 77.

Figure 24

The phase diagram of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ vs the iron valence. The SC phase appears sandwiched between AFM insulators. The Fe valence state was systematically controlled by varying the $x$ and $y$ concentrations in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$. From 153.

Figure 25

Superconducting $T_c$ vs pressure for the compounds indicated. Two SC phases were found. SC-II has a $T_c\sim 48.7\mathrm{K}$. From 124.