Hund Interaction, Spin-Orbit Coupling, and the Mechanism of Superconductivity in Strongly Hole-Doped Iron Pnictides

We present a novel mechanism of $s$-wave pairing in Fe-based superconductors. The mechanism involves holes near $d_{xz}/d_{yz}$ pockets only and is applicable primarily to strongly hole doped materials. We argue that as long as the renormalized Hund’s coupling $J$ exceeds the renormalized interorbital Hubbard repulsion $U^{\prime }$, any finite spin-orbit coupling gives rise to $s$-wave superconductivity. This holds even at weak coupling and regardless of the strength of the intraorbital Hubbard repulsion $U$. The transition temperature grows as the hole density decreases. The pairing gaps are fourfold symmetric, but anisotropic, with the possibility of eight accidental nodes along the larger pocket. The resulting state is consistent with the experiments on ${\mathrm{KFe}}_2{\mathrm{As}}_2$.

Figure 1

Left panel: Illustrative Fermi surfaces (FSs) for the $d_{xz}/d_{yz}$ hole pockets, where $k_0=\sqrt{2m\mu }$. In the SC state, the pairing amplitude on the outer Fermi surface is ${\mathrm{\Delta }}_{+}$ and on the inner ${\mathrm{\Delta }}_{-}$. Right panel: Schematic quasiparticle dispersion in the superconducting state (solid black lines). The gap away from the Fermi level is due to the $A_{2g}$ pairing and is present already without SOC. Once SOC is included, the gaps on the FS appear. The dashed lines are approximations which capture the gaps on the FS only.

Figure 2

The phase diagram at $T=0$ calculated at a fixed ratio of the SOC $\lambda$ to Fermi energy $\mu$. Displayed are the boundaries of the nodal region, which depend on the ratio of $A_{2g}$ (${\mathrm{\Delta }}_2$) and $A_{1g}$ (${\mathrm{\Delta }}_0$) components of the pairing gap at $T=0$. They also depend on $p_0$ and $p_1$, dimensionless parameters which enter into the angle dependence of the normal state band dispersion as in Eqs. (7) and (8). ($p_0$ is a measure of the off-diagonal orbital hopping, while $p_1$ is a measure of the anisotropy of the diagonal hopping). The pairing amplitudes on the larger and the smaller Fermi surfaces are ${\mathrm{\Delta }}_{+}={\mathrm{\Delta }}_0+(\lambda /|{\overrightarrow{B}}_{\mathbf{k}}|){\mathrm{\Delta }}_2$ and ${\mathrm{\Delta }}_{-}={\mathrm{\Delta }}_0-(\lambda /|{\overrightarrow{B}}_{\mathbf{k}}|){\mathrm{\Delta }}_2$, respectively; $2|{\overrightarrow{B}}_{\mathbf{k}}|$ is the energy of the band splitting, Eq. (7). Shaded area marks the appearance of the accidental nodes in ${\mathrm{\Delta }}_{+}$ for $p_1=0.25$. For a different value of $p_1$, the upper boundary of the shaded area shifts to the corresponding dashed line, while the lower boundary is $p_1$ independent. Below (above) the shaded region, the signs of ${\mathrm{\Delta }}_{+}$ and ${\mathrm{\Delta }}_{-}$ are opposite (same) and the pairing state can be viewed as $s^{+-}$ ($s^{++}$). Interestingly, numerical solutions of the self-consistency equations find that it is possible to start outside of the nodal region at $T_c$ (red and orange circles) and end up inside of it at $T=0$ (black and blue circles).

Figure 3

Angle dependence of the gap at $T=0$ on the inner (a) and the outer (b) hole Fermi surfaces (FSs) for parameters corresponding to the (black) end point of the down (red) arrow in Fig. 2. (c) and (d) show the same but for the parameters corresponding to the (blue) end point of the up (orange) arrow in Fig. 2. In both cases, there are eight nodal points on the outer FS.