Two types of $s$-wave pairing due to magnetic and orbital fluctuations in the two-dimensional 16-band $d\text{−}p$ model for iron-based superconductors

We study superconductivity in the two-dimensional 16-band $d\text{−}p$ model extracted from a tight-binding fit to the band structure of LaFeAsO, using the random phase approximation. When the intraorbital repulsion $U$ is larger than the interorbital one $U^{\prime }$, an extended $s$-wave ($s_{\pm }$-wave) pairing with sign reversal of order parameter is mediated by antiferromagnetic spin fluctuations, while when $U<U^{\prime }$ another kind of $s$-wave ($s_{++}$-wave) pairing without sign reversal is mediated by ferro-orbital fluctuations. The $s_{++}$-wave pairing is enhanced due to the electron-phonon coupling and then can be expanded over the realistic parameter region with $U>U^{\prime }$.

Figure 1

(Color online) Crystal structure of ${\text{Fe}}_2{\text{As}}_2$ layer. Small and large balls represent Fe and As atoms, respectively. The solid line represents the unit cell. It is noted that ${\text{As}}^1$ and ${\text{As}}^2$ denote the As atoms on the upper side and on the lower side of the ${\text{Fe}}_2{\text{As}}_2$ layer, respectively.

Figure 2

(Color online) (a) The band structure obtained from the $d\text{−}p$ model, Eq. (2), (solid line) and that obtained from the density-functional calculation (cross) for $x=0.1$. (b) Fermi surface obtained from the $d\text{−}p$ model for $x=0.1$. The solid and dashed lines show the Fermi surfaces which have mainly $d_{yz},d_{zx}$ and $d_{x^2-y^2}$ orbital character, respectively. (c) The DOS obtained from the $d\text{−}p$ model for $x=0.1$. Upper panel: total DOS, middle panel: partial DOS of the $\text{Fe}3d$ orbitals, and lower panel: partial DOS of the $\text{As}4p$ orbitals. The inset of the middle panel shows the DOS near the Fermi level. We note that the Fermi level set to 0 on the energy axis.

Figure 3

(Color online) (a) Several components of the spin susceptibility ${\widehat{\chi }}^s\left(\mathbf{q}\right)$, (b) the charge-orbital susceptibility ${\widehat{\chi }}^c\left(\mathbf{q}\right)$, and (c) the effective pairing interaction $\widehat{V}\left(\mathbf{q}\right)$ in $\mathbf{q}$ space for $U=1.71$, $U^{\prime }=1.4$, and $J=J^{\prime }=0.1\text{eV}$. It is noted that we number the orbitals as follows: (1) $d_{3z^2-r^2}$, (2) $d_{x^2-y^2}$, (3) $d_{xy}$, (4) $d_{yz}$, and (5) $d_{zx}$. (d) $U$ dependence of the eigenvalues of the gap equation for several symmetries.

Figure 4

(Color online) (a) Several components of the spin susceptibility ${\widehat{\chi }}^s\left(\mathbf{q}\right)$, (b) the charge-orbital susceptibility ${\widehat{\chi }}^c\left(\mathbf{q}\right)$, and (c) the effective pairing interaction $\widehat{V}\left(\mathbf{q}\right)$ in $\mathbf{q}$ space for $U=0.4$, $U^{\prime }=1.15$, and $J=J^{\prime }=0.1\text{eV}$. It is noted that (d) $U^{\prime }$ dependence of the eigenvalues of the gap equation for several symmetries. It is noted that the legends are the same as Fig. 3.

Figure 5

(Color online) [(a)–(d)] The diagonal components of the gap function $\widehat{\Delta }\left(\mathbf{k}\right)$ in the orbital representation and [(e)–(h)] those in the band representation for $U=1.71$, $U^{\prime }=1.4$, and $J=J^{\prime }=0.1\text{eV}$. The solid and dashed lines represent the Fermi surfaces and the nodes of the gap function, respectively.

Figure 6

(Color online) [(a)–(d)] The diagonal components of the gap function $\widehat{\Delta }\left(\mathbf{k}\right)$ in the orbital representation and [(e)–(h)] those in the band representation for $U=0.4$, $U^{\prime }=1.15$, and $J=J^{\prime }=0.1\text{eV}$. The solid and dashed lines represent the Fermi surfaces and the nodes of the gap function, respectively.

Figure 7

(Color online) [(a)–(d)] The diagonal components of the gap function $\widehat{\Delta }\left(\mathbf{k}\right)$ in the orbital representation and [(e)–(h)] those in the band representation for $U=0.4$, $U^{\prime }=1.18$ and $J=0.1$, $J^{\prime }=0\text{eV}$. The solid and dashed lines represent the Fermi surfaces and the nodes of the gap function, respectively.

Figure 8

(Color online) The phase diagram on $U^{\prime }\text{−}U$ plane for $J=J^{\prime }=0.1\text{eV}$ at $x=0.1$ and $T=0.02\text{eV}$. The solid and dotted lines show the $s_{++}$ wave and the $s_{\pm }$-wave superconducting instabilities, respectively. The dashed and dot-dashed lines show instabilities towards the incommensurate spin-density wave and the ferro-orbital order, respectively.