Evolution of symmetry and structure of the gap in iron-based superconductors with doping and interactions

We present a detailed study of the symmetry and structure of the pairing gap in Fe-based superconductors (FeSCs). We treat FeSCs as quasi-2D, decompose the pairing interaction in the $XY$ plane in $s$-wave and $d$-wave channels into contributions from scattering between different Fermi surfaces, and analyze how each scattering evolves with doping and input parameters. We verify that each interaction is well approximated by the lowest angular harmonics. We use this simplification to analyze the interplay between the interaction with and without spin-fluctuation components, the origin of the attraction in the $s^{\pm }$ and $d_{x^2-y^2}$ channels, the competition between them, the angular dependence of the $s^{\pm }$ gaps along the electron Fermi surface, the conditions under which the $s^{\pm }$ gap develops nodes, and the origin of superconductivity in heavily electron- or hole-doped systems, when only Fermi surfaces of one type are present. We also discuss the relation between RPA and RG approaches for FeSCs.

Figure 1

Fits of the actual interactions ${\Gamma }_{ij}({\mathbf{k}}_F,{\mathbf{k}}_F^{\prime })$ by LAHA. Symbols represent interactions computed numerically for the 5-orbital model using LDA band structure; solid lines are fits using Eq. (10). The fit is for set 1 for bare interactions (no SF component). The chemical potential is $\mu =0.08$ which corresponds to electron doping. We set ${\mathbf{k}}_F$ in ${\Gamma }_{ij}({\mathbf{k}}_F,{\mathbf{k}}_F^{\prime })$ to be either along the $x$ or along the $y$ direction on a given FS (its location is specified in the title on top of each figure) and varied ${\mathbf{k}}_F^{\prime }$ along each of the FSs. The angle ${\theta }^{\prime }$ is measured relative to $k_x$. The fit is reasonably good.

Figure 2

The same as in Fig. 1 but for the full interaction, with SF component. The fit is again reasonably good. We verified that the slight discrepancies are removed if in LAHA we add a $4\theta$ harmonics to the interaction.

Figure 3

Top: $s$-wave solutions (left pair) and $d$-wave solutions (right pair) obtained by applying LAHA for the gap structure for set 1 (NSF) with $\mu =0.08$ and the ones obtained numerically from the 5-orbital model based on the LDA band structure, respectively. There are no solutions with positive $\lambda$, so we show the solutions with the smallest negative $\lambda$ in both channels. The agreement of the gap structure with LAHA results is quite good. For LAHA, the $s$-wave solution has ${\lambda }_s=-1.02$ and the $d$-wave solution has ${\lambda }_d=-0.99$ (${\lambda }_s/{\lambda }_d\approx 1.0$). ${\lambda }_s$ and ${\lambda }_d$ obtained from the numerical approach are negative, and their ratio is ${\lambda }_s/{\lambda }_d=0.6$. Bottom: The same, but now with the SF component of interaction included. The couplings are now positive: ${\lambda }_s=4.6$ and ${\lambda }_d=4.8$ (${\lambda }_s/{\lambda }_d\approx 0.96$) for LAHA and $\approx$1.1 for RPA-SF calculations for the 5-orbital model (Ref. 1).

Figure 4

Effect of electron-hole interactions: $s$-wave solutions for set 1 (NSF) for $\mu =0.08$ with the angular parts of electron-hole interactions (${\alpha }_{h_{i}e}$) set to zero and increased by a factor of 2, respectively (left and center). The values of ${\lambda }_s$ are $-$0.16 and 0.226; positive ${\lambda }_s$ is for enhanced ${\alpha }_{h_{i}e}$. In the right figure, we introduce additional enhancement of the overall factors of the electron-hole interactions by $\sim$4 and this causes nodes in the gap to disappear.

Figure 5

The dependence of the gap structure for $s$-wave solution on the strength of angle-dependent parts of the interactions. We used set 1 (SF) for $\mu =0.08$. The solution for the original ${\alpha }_{h_{i}e}$ is shown in Fig. 3. Top panel: The gap structure for the cases when the angular parts of the electron-hole interaction (${\alpha }_{h_{i}e}$) are increased by factor of 2, reduced by factor of 2, and set to zero, while the angular parts of the electron-electron interaction (${\alpha }_{ee}$ and ${\beta }_{ee}$) are kept at the original values. The values of ${\lambda }_s$ are 7.3, 3.4, and 2.5, respectively. The angular dependence does change, and, in particular, the nodes along the hole FS disappear when we set ${\alpha }_{h_{i}e}=0$. Bottom panel: The same but now we keep ${\alpha }_{h_{i}e}$ at the original value and change the angular parts of the electron-electron interactions (${\alpha }_{ee}$ and ${\beta }_{ee}$ terms): increase them by 2, reduce by 2, and set to zero. The $\lambda$'s are 4.5, 4.7, and 4.0, respectively. We see that this has very little effect on the structure of the gap.

Figure 6

Top, left to right: FSs for increasing electron doping with $\mu =0.05,0.18,0.30$, respectively. $\mu =0.18$ represents the electron doping at which the hole FS almost disappears, while $\mu =0.30$ represents extreme electron doping where the hole FS completely disappears. Bottom, left to right: FSs for increasing hole doping with $\mu =-0.05,-0.18,-0.20$, respectively. Here electron FSs almost disappear at $\mu =-0.18$ and completely disappear at $\mu =-0.20$.

Figure 7

Top panel: The best and worst LAHA fits of the interactions $\Gamma ({\mathbf{k}}_F,{\mathbf{k}}_F^{\prime })$ obtained from RPA-SF calculations for parameter set 2 (SF) for $\mu =0.05$. Bottom panel: Gap functions in $s$ and $d$ channels obtained in the LAHA. ${\lambda }_s=0.25$, ${\lambda }_d=0.37$. Note that ${\lambda }_d$ is somewhat larger.

Figure 8

The same as in Fig. 7, but for different $\mu =0.18$. ${\lambda }_s=0.21$, ${\lambda }_d=0.35$. Again, ${\lambda }_d$ is larger, and the difference between ${\lambda }_d$ and ${\lambda }_s$ is larger than for $\mu =0.05$.

Figure 9

The same as in Fig. 7, but for negative $\mu =-0.05$, when extra hole FS appears. ${\lambda }_s=0.58$, ${\lambda }_d=0.31$. Observe that now ${\lambda }_s>{\lambda }_d$ and the $s^{\pm }$ gap has no nodes.

Figure 10

The same as in Fig. 9, but for $\mu =-0.18$ and $U=0.9,J=0.0,V=0.9$. We obtain ${\lambda }_s=1.8$, ${\lambda }_d=1.2$. Observe that, still, ${\lambda }_s>{\lambda }_d$ and the $s^{\pm }$ gap has no nodes.

Figure 11

$s^{\pm }$ gap structure obtained within LAHA for parameter set 3 (SF) for $\mu =-0.05$. The coupling ${\lambda }_s=0.79$. The gap has nodes on electron FS despite that there is the third Fermi surface at $(\pi ,\pi )$. This nodal solution has a strong $d$-wave competitor for which ${\lambda }_d=0.8$.

Figure 12

Top: The fits of the interactions and the structure of $s$-wave and $d$-wave gaps in LAHA for the case of heavy electron doping, when only electron FSs are present. The $\lambda$'s are ${\lambda }_s=-0.12$ and ${\lambda }_d=0.13$ ($\mu =0.30$). Bottom: The same, but for a different band structure, used in Ref. 26. The couplings are ${\lambda }_s=0.1$ and ${\lambda }_d=5.9$.

Figure 13

The fits of the interactions by LAHA and the structure of $s$-wave and $d$-wave gaps for the case of heavy hole doping, $\mu =-0.20$, when only hole FSs are present. The parameters are $U=1,J=0.25,V=0.69$ (see text). The eigenvalues are ${\lambda }_s=-.05$ and ${\lambda }_d=0.05$.