Modulated superconductivity due to vacancy and magnetic order in $A_y$Fe${}_{2-x/2}$Se${}_2$ [$A=\text{Cs}$, K, (Tl,Rb), (Tl,K)] iron-selenide superconductors

We present a calculation of a “modulated” superconducting state in iron-selenide superconductors. The zero-momentum $d$-wave pairing breaks the translational symmetry of the conventional BaFe${}_2$Se${}_2$-like crystal of the $I4/mmm$ space group. This pairing state becomes compatible when the Fe vacancies form an ordered state and the crystal symmetry changes to a low-temperature $I4/m$ one. For the specific case of an incommensurate vacancy order at $Q_v=(\frac{1}{5},\frac{3}{5})$ in K${}_{0.82\left(2\right)}$Fe${}_{1.626\left(3\right)}$Se${}_2$, we find that it induces a block checkerboard antiferromagnetic phase at wave vector ${\mathbf{Q}}_m=4{\mathbf{Q}}_v$. The coexistence of vacancy order and magnetic order leads to a reconstructed ground state which naturally couples to the $d$-wave superconductivity in a uniform phase in what we propose will be a general coupling for all iron-selenide superconductors. Our results agree with numerous experimental data available to date. We thus suggest that the incommensurability leads to a uniform coexistence of multiple phases as a viable alternative to a nanoscale phase separation in high-$T_c$ superconductors and play an important role in the enhancement of superconductivity.

Figure 1

(a) Experimental phase diagram of A${}_y$Fe${}_{2-x/2}$Se${}_2$ compound taken from Ref. 1 is also shown to be a common phase diagram for this class of materials $A_y$Fe${}_{2-x/2}$Se${}_2$ in Ref. 12. At high temperature, the system is in the Ba${}_2$Fe${}_2$As${}_2$-like tetragonal structure of $I4/mmm$ space group at all Fe vacancies. A tetragonal structure with an $I4/m$ unit cell is observed to stabilize at low temperature in the $\sqrt{5}\times \sqrt{5}\times 1$ superlattice. The metal-insulator transition is observed at $x\sim 0.865$ in numerous experiments albeit a smooth behavior of the Néel temperature $T_N$ (diamond) throughout the phase diagram. The open diamond symbol marks the transition temperature at which the disorder-to-order state of the Fe vacancy is measured. Solid squares represent $T_c\sim 30$ K, which seems to be symmetric around $x=0.8$ at which full occupancy of Fe2 sublattices and vacant Fe1 sites is realized. (b) Corresponding common crystal and magnetic structure in the real space tetragonal $I4/m$ unit cell (top view) in which only Fe atoms are shown (Refs. 1, 2, and 12). Magenta and cyan color circles give spin-up and spin-down configurations, while black and white dumbbells represent positive and negative phase of the pairing symmetry. Boxes of different colors and sizes stand for different unit cells (see text), while the gold box is the true unit cell for this system at low temperature.

Figure 2

The band dispersion along high-symmetry lines [(a1),(b1),(c1),(d1)], corresponding FSs [(a2),(b2),(c2),(d2)] and bare susceptibility [(a3),(b3),(c3),(d3)] maps at zero energy are presented for four combinations of interactions: (a) $\to$ all $V_i=0$, (b) $\to$ only $V_v=-0.5$ eV, (c) $\to$ only $V_m=0.02$ eV, and (d) $\to$ $V_v=-0.5,V_2=0.2,V_3=-0.1,V_m=0.01$ eV. FS and susceptibility are plotted only in the first quadrant of the BZ which are symmetric in the other quadrants. Note that opposite phases between consecutive interactions give maximal weight on main bands. The electronic structure consists of 20 split bands while the spectral weight is largest in the main band if coherence factors of the modulated order are included. Here we have not included the structure factor of the modulated orders which may further enhance the distribution of the spectral weight among the modulated bands.

Figure 3

Schematic behavior of the FS and pairing symmetry in 1 Fe per unit cell and 2 Fe per unit cell Brillouin zone for BaFe${}_2$As${}_2$ are shown in (a) and (b), respectively. The corresponding real-space view of the gap symmetry is depicted in (c) in which the SC phase is isotropic and nodeless on each Fe atom but changes phase between Fe1 and Fe2 sublattices. Black to white color scale depicts the pairing symmetry from negative to positive in momentum space while black and white dumbbells represent the negative and positive phase of the pairing symmetry in real space (exponential form of cosine and sine functions). Similar results for $A$Fe${}_2$Se${}_2$ compounds are given in (d)–(f). In momentum space FSs are isotropic and nodeless, but in real space the gap nodes pass through both of the Fe atoms. The phases of the pairing symmetries are $s_{xy}=2\cos \left(k_{x}a\right)\cos \left(k_{y}a\right)$, $s^{\pm }/d_{x^2-y^2}=\cos \left(k_{x}a\right)\pm \cos \left(k_{y}a\right)$, and $d_{xy}^{\prime }=2\sin \left(k_{x}a\right)\sin \left(k_{y}a\right)$.