Glide-Plane Symmetry and Superconducting Gap Structure of Iron-Based Superconductors

We consider the effect of glide-plane symmetry of the Fe-pnictogen/chalcogen layer in Fe-based superconductors on pairing in spin fluctuation models. Recent theories have proposed that so-called $\eta$-pairing states with nonzero total momentum can be realized and possess exotic properties such as odd parity spin singlet symmetry and time-reversal symmetry breaking. Here we show that $\eta$ pairing is inevitable when there is orbital weight at the Fermi level from orbitals with even and odd mirror reflection symmetry in $z$; however, by explicit calculation, we conclude that the gap function that appears in observable quantities is identical to that found in earlier, 1 Fe per unit cell pseudocrystal momentum calculations.

Figure 1

(a) Fermi surfaces and (b) the leading gap function for the five-orbital model in the zone of the pseudocrystal momentum $\tilde{\mathbf{k}}$. The Fermi surface is colored to show the dominant orbital weight [$d_{xz}$ red (gray), $d_{yz}$ green (light gray), $d_{xy}$ blue (dark gray)]. (c) The unfolded Fermi surface in the physical crystal momentum $\mathbf{k}$-space. The size of the dots is proportional to the sum of the orbital weights of the spectral function and the color shows the dominant orbital weight. The red line denotes the boundary of the 2 Fe per unit cell Brillouin zone. An “$\eta$” pair ($\mathbf{k}\uparrow$, $-\mathbf{k}+\mathbf{Q}\downarrow$) is shown. (d) Comparison of the angle dependence of the leading gap function calculated from the five-orbital model (orange line) and the ten-orbital model (blue dots) on the various Fermi pockets. Here we denote the two crossed electron pockets in the ten-orbital model at the $X$ point as ${\beta }_1$ and ${\beta }_1^{\prime }$ and at the $Y$ point ${\beta }_2$ and ${\beta }_2^{\prime }$. The gaps along the ${\beta }_{1,2}^{\prime }$ pockets are not plotted since ${\beta }_1^{\prime }={\beta }_2$ and ${\beta }_2^{\prime }={\beta }_1$ by symmetry.

Figure 2

Gap functions ${\mathrm{\Delta }}_{{\ell }_1{\ell }_2}(\mathbf{k})$ in the orbital basis plotted on the Fermi surface in the physical momentum $BZ$. (a)-(b) When ${\ell }_1$ and ${\ell }_2$ have the same $z$-reflection symmetry, one has a normal ($\mathbf{k}$, $-\mathbf{k}$) pairing and the gap function ${\mathrm{\Delta }}_{{\ell }_1{\ell }_2}^N(\mathbf{k})$ is real and has even parity. (c)-(d) When ${\ell }_1$ and ${\ell }_2$ have different $z$-reflection symmetry, one has a ($\mathbf{k}$,$-\mathbf{k}+\mathbf{Q}$) $\eta$ pairing and ${\mathrm{\Delta }}_{{\ell }_1{\ell }_2}^{\eta }(\mathbf{k})$ is purely imaginary and has odd parity.

Figure 3

(a) The leading gap function $\tilde{\mathrm{\Delta }}(\tilde{\mathbf{k}})$ in the band representation calculated in the five-orbital model in pseudocrystal momentum space [$\text{red}=\text{positive}$ (gap along pockets at $X$ and $Y$ points), $\text{blue}=\text{negative}$ (gap along pockets at $\mathrm{\Gamma }$ and $M$ points)]. When transformed to physical crystal momentum $\mathbf{k}$ space, the gap splits into the normal even-even and odd-odd contributions ${\mathrm{\Delta }}_{\mathrm{odd}}^N(\mathbf{k})+{\mathrm{\Delta }}_{\text{even}}^N(\mathbf{k}+\mathbf{Q})$ plotted in (b) and the even-odd and odd-even $\eta$ contributions ${\mathrm{\Delta }}_{\text{odd-even}}^{\eta }(\mathbf{k})+{\mathrm{\Delta }}_{\text{even-odd}}^{\eta }(\mathbf{k}+\mathbf{Q})$ shown in (c). In the band representation, all of the contributions have even parity.