Mechanism for a pairing state with time-reversal symmetry breaking in iron-based superconductors

The multipocket Fermi surfaces of iron-based superconductors promote pairing states with both extended $s$-wave and $d$-wave symmetry. We argue that the competition between these two order parameters could lead to a time-reversal symmetry-breaking state with $s+id$ pairing symmetry in the iron-based superconductors, and propose several scenarios in which this phase may be found. To understand the emergence of such a pairing state on a more rigorous footing, we start from a microscopic five-orbital description representative of the pnictides. Using a combined approach of functional renormalization-group and mean-field analysis, we identify the microscopic parameters of the $s+id$ pairing state. There, we find the most promising region for $s+id$ pairing in the electron-doped regime with an enhanced pnictogen height.

Figure 1

Frustrating the $d$-wave limits of $\mathrm{K}{\mathrm{Fe}}_2{\mathrm{As}}_2$ (a) and ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ (b). Upon doping or differently induced band-structure effects, electron pockets appear (dashed red) in (a) and a hole pocket appears (dashed red) in (b) which populate the $q\sim (\pi ,0)/(0,\pi )$ scattering channels enhancing the $s_{\pm }$ symmetry. This leads to frustration providing the background for $s+id$ pairing.

Figure 2

Competing pairing orders and $s$-wave SC form factors for $U_1\left(d_{X^2-Y^2}\right)=U_1^{*}=2.5\mathit{eV}$ (a) and $U_1\left(d_{X^2-Y^2}\right)=1.6U^{*}$ (b) at electron-doped filling $n=6.13$. RG channel flow [(a1),(b1)] and $s_{\pm }$-gap form factor [(a2),(b2)]. $s/d$ transition from (a) to (b): Increasing $U_1\left(d_{X^2-Y^2}\right)$ enhances the gap anisotropy of the $s_{\pm }$ form factor on the electron pockets [$\mathbf{k}$ patching: points 33–64; see (a3)] shown in (a2) and (b2) until extended $d_{x^2-y^2}$ becomes competitive. The $d$-wave form factor (not shown) does not change from (a) to (b). [(a3),(b3)] Interactions mediated by $U_1$, inducing $s_{\pm }$-pairing tendency $(\Gamma \leftrightarrow X)$ and competing $d_{x^2-y^2}$-pairing symmetry due to $(X\leftrightarrow X)$. (c) Variation of the pnictogen height $h_p$ mostly affects the spread of the $d_{X^2-Y^2}$ orbital and thus $U_1\left(d_{X^2-Y^2}\right)$, as it is oriented to the planar projection of the pnictogen.

Figure 3

Preferred pairing as a function of electron doping and intraorbital Coulomb interaction $U_1\left(d_{X^2-Y^2}\right)$. The results are obtained by minimizing the mean-field free energy of the effective theory taken from fRG at $\Lambda \approx 0.001\mathit{eV}$. At $27\%$ electron doping, the $s+id$ pairing state occurs at $U_1\left(d_{X^2-Y^2}\right)=3\mathit{eV}$, which is comparable to the intraorbital repulsion in the remaining orbitals $U_1=2.5\mathit{eV}$.