Origin of gap anisotropy in spin fluctuation models of the iron pnictides

We discuss the large gap anisotropy found for the $A_{1g}$ ($s$ wave) state in random-phase approximation spin fluctuation and functional renormalization-group calculations and show how the simple arguments leading to isotropic sign-switched $s$-wave states in these systems need to be supplemented by a consideration of pair scattering within Fermi-surface sheets and between the individual electron sheets as well. In addition, accounting for the orbital makeup of the states on the Fermi surface is found to be crucial.

Figure 1

(Color online) The Fermi surface of the five-orbital tight-binding model (Ref. 15). The main orbital contributions are shown by the following colors/symbols: $d_{xz}$ (red/solid circles), $d_{yz}$ (green/open circles), and $d_{xy}$ (blue/diamonds).

Figure 2

(Color online) The orbital weights as a function of the winding angle $\varphi$ on the different Fermi-surface sheets (Ref. 15). The different colors/lines refer to $d_{xz}$ (red/solid lines), $d_{yz}$ (green/dashed lines), $d_{xy}$ (blue/dash-dotted line), and $d_{x^2-y^2}$ (yellow/short-dashed line).

Figure 3

(Color online) The strength ${\Gamma }_{ij}\left(k_0,k\right)$ associated with scattering a pair from the ${\alpha }_1$ Fermi surface with momenta (a) $k_0=\left(k_F\hat{ı},-k_F\hat{ı}\right)$ and (b) $k_0=\left(k_F\hat{j},-k_F\hat{j}\right)$ as indicated by the black circles as a function of momentum $k$. The orbital weight factors favor scattering from $d_{xz}$ to $d_{xz}$ and $d_{yz}$ to $d_{yz}$ orbital states.

Figure 4

(Color online) The RPA gap function (a) plotted on the Fermi surfaces (red/solid circles positive and blue/open circles negative), (b) $g\left(k\right)$ as a function of angle, and (c) bar graph of the various interaction components ${\lambda }_{ij}$. Note that the off-diagonal terms contribute to the total $\lambda$ [Eq. (5)] with weights of two or four as indicated.

Figure 5

(Color online) The sign-switched gap function (similar to Fig. 4).

Figure 6

(Color online) The phenomenological anisotropic gap function on the $\beta$-Fermi surface versus angle for several different values of $r$.

Figure 7

(Color online) The pairing strength and its components versus $r$ for the anisotropic gap shown in Fig. 6. Here, $\lambda$ is the total pairing strength. The intra- and inter-${\alpha }_1$ and ${\alpha }_2$ Fermi-surface contributions are given by ${\lambda }_{\alpha }={\lambda }_{{\alpha }_1{\alpha }_1}+{\lambda }_{{\alpha }_2{\alpha }_2}+2{\lambda }_{{\alpha }_1{\alpha }_2}$ and the intra-$\beta$ Fermi-surface contribution by ${\lambda }_{\beta }={\lambda }_{{\beta }_1{\beta }_1}+{\lambda }_{{\beta }_2{\beta }_2}$. The contribution to the pairing comes from ${\lambda }_{\alpha \text{−}\beta }=4{\lambda }_{{\alpha }_1{\beta }_1}+2{\lambda }_{{\alpha }_2{\beta }_1}$, while for this $A_{1g}$ gap, the inter-${\beta }_1\text{−}{\beta }_2$ contribution opposes the pairing for small $r$ and is neutralized for a large anisotropy. The red dashed line indicates the optimal degree of anisotropy $r=1.5$.