Orbital fluctuation theory in iron pnictides: Effects of As-Fe-As bond angle, isotope substitution, and $Z^2$-orbital pocket on superconductivity

We study the pairing mechanism in iron pnictide superconductors based on the five-orbital Hubbard-Holstein model. Due to Fe-ion oscillations, the $s$-wave superconducting (SC) state without sign reversal ($s_{++}$-wave state) is induced by orbital fluctuations by using realistic model parameters. The virtue of the present theory is that the famous empirical relation between $T_c$ and the As-Fe-As bond angle is automatically explained since the electron-phonon coupling that creates the orbital fluctuations is the strongest when the ${\text{As}}_4$ tetrahedron is regular. The negative iron isotope effect is also reproduced. In addition, the magnitude of the SC gap on hole pockets is predicted to be rather insensitive to the corresponding $d$ orbital ($xz/yz$ or $z^2$ orbital), which is consistent with the recent bulk-sensitive angle-resolved photoemission spectroscopy (ARPES) measurement for $\left(\text{Ba},\text{K}\right){\text{Fe}}_2{\text{As}}_2$ and ${\text{BaFe}}_2{\left(\text{As},\text{P}\right)}_2$. These obtained results indicate that the orbital fluctuation mediated $s_{++}$-wave state is a plausible candidate for iron pnictides.

Figure 1

(Color online) (a) ${\text{As}}_4$ tetrahedron in iron pnictides shown along the $z$ axis. Here, we put $R_{\text{Fe-As}}=1$. The As-Fe-As bond angle $\alpha$ is illustrated in Fig. 6a. (b) The diagrammatic expression for ${\chi }^{s\left(c\right)}$.

Figure 2

(Color online) Obtained $U\text{−}g\left(0\right)$ phase diagram for $n=6.1$. Near the orbital-density-wave boundary, $s_{++}$-wave SC state is realized by orbital fluctuations.

Figure 3

(Color online) Obtained (a) ${\chi }_{24,24}^c\left(\mathit{q},0\right)$ and (b) ${\chi }_{22,22}^c\left(\mathit{q},0\right)$, for $n=6.1$, $U=1$, $T=0.02$, and $g\left(0\right)=0.21$. Note that ${\chi }_{24,24}^c\left(\mathit{q},0\right)={\chi }_{34,34}^c\left({\mathit{q}}^{\prime },0\right)$ and ${\chi }_{22,22}^c\left(\mathit{q},0\right)={\chi }_{33,33}^c\left({\mathit{q}}^{\prime },0\right)$, where ${\mathit{q}}^{\prime }$ is given by the rotation of $\mathit{q}$ by $\pi /2$. In (c), charge susceptibility ${\chi }^c\left(\mathit{q},0\right)\equiv {\sum }_{l,m}{\chi }_{ll,mm}^c\left(\mathit{q},0\right)$ is shown. We use 2048 Matsubara frequencies.

Figure 4

(Color online) [(a) and (b)] Obtained SC gap functions for (a) $U=1.09$ and (b) $U=1.10$, respectively. We put $g\left(0\right)=0.21$ $\left({\alpha }_c=0.98\right)$, $T=0.02$, and ${\omega }_D=0.02$. They are normalized as $N^{-1}{\sum }_{\mathit{k},lm}{\left|{\Delta }_{lm}\left(\mathit{k}\right)\right|}^2=1$. We use 2048 Matsubara frequencies. (c) FSs in the unfolded Brillouin zone. FS1,2 (FS3,4) are composed of 2,3-orbitals (2,3,4-orbitals). FS5 is composed of 1-orbital in $\left(\text{Ba},\text{K}\right){\text{Fe}}_2{\text{As}}_2$ and ${\text{BaFe}}_2{\left(\text{As},\text{P}\right)}_2$, which will be discussed in Sec. 5A. Note that FS5 moves to (0,0) in the folded zone.

Figure 5

(Color online) Obtained ${\lambda }_E$ as function of $1-{\alpha }_c$ for (a) $U=0$ and (b) $U=1$, respectively, at $T=0.02$. The former (latter) corresponds to the positive (negative) isotope effect. We use 2048 Matsubara frequencies. Inset in (a): $T$ dependence of ${\lambda }_E$ for ${\alpha }_c=0.98$.

Figure 6

(Color online) (a) The definitions of the As-Fe-As bond angle $\alpha$. (b) $g\left(0,\alpha \right)/g\left(0\right)$ as a function of $\alpha$.

Figure 7

(Color online) Obtained (a) ${\chi }_{24,24}^c\left(\mathit{q},0\right)$ and (b) ${\chi }_{22,22}^c\left(\mathit{q},0\right)$ for $\alpha =120°$, $g\left(0\right)=0.21$, $U=1$, and $T=0.02$. We use 1024 Matsubara frequencies.

Figure 8

(Color online) $\alpha$ dependence of ${\lambda }_E$ for (a) $U=0$ and $g\left(0\right)=0.23$, and (b) $U=1$ and $g\left(0\right)=0.21$, respectively. We put $T=0.02$, and ${\omega }_D=0.02$. In (a), $s_{++}$-wave state is always realized. We use 1024 Matsubara frequencies.

Figure 9

(Color online) (a) Band structure (in the unfolded Brillouin zone) given by shifting the $Z^2$-orbital level by $+0.32\text{eV}$. The hole pocket around $\left(\pi ,\pi \right)$, FS5 in Fig. 4c, is composed of $Z^2$ orbital. (b) $s_{\pm }$-wave SC gap function for $U=1.0$ and $g\left(0\right)=0$. (c) $s_{++}$-wave SC gap function for $U=0$ and $g\left(0\right)=0.19$.

Figure 10

(Color online) (a) Obtained $U\text{−}g\left(0\right)$ phase diagram for $J/U=1/4$, 1/5, and 1/6. (b) Obtained ${\chi }_{24,24}^c\left(\mathit{q},0\right)$ for $J/U=1/6$ and $n=6.1$. (c) Obtained ${\chi }_{22,22}^c\left(\mathit{q},0\right)$ for $J/U=1/6$ and $n=6.1$.