Origin of orthorhombic transition, magnetic transition, and shear-modulus softening in iron pnictide superconductors: Analysis based on the orbital fluctuations theory

The main features in iron pnictide superconductors are summarized as (i) the orthorhombic transition accompanied by a remarkable softening of the shear modulus, (ii) high-$T_c$ superconductivity close to the orthorhombic phase, and (iii) stripe-type magnetic order induced by orthorhombicity. To present a unified explanation for these features, we analyze the multi-orbital Hubbard-Holstein model with Fe-ion optical phonons based on the orbital fluctuation theory. In the random-phase approximation (RPA), a small electron-phonon coupling constant ($\lambda ~0.2$) is enough to produce large orbital (charge quadrupole) fluctuations. The most divergent susceptibility is the $O_{\mathit{xz}}$-antiferroquadrupole (AFQ) susceptibility, which causes $s$-wave superconductivity without sign reversal ($s_{++}$-wave state). At the same time, divergent development of $O_{x^2-y^2}$-ferroquadrupole (FQ) susceptibility is brought about by the “two-orbiton process” with respect to the AFQ fluctuations, which is absent in the RPA. The derived FQ fluctuations cause the softening of the $C_{66}$ shear modulus, and its long-range order not only triggers the orthorhombic structure transition, but also induces the instability of the stripe-type antiferromagnetic state. In other words, the condensation of composite bosons made of two orbitons gives rise to the FQ order and structure transition. Therefore, the theoretically predicted multi-orbital criticality presents a unified explanation for the above-mentioned features of iron pnictide superconductors.

Figure 1

Crystal structure of the FeAs layer, in which the unit cell is given by the two-iron unit cell composed of Fe-A, Fe-B, As-A, and As-B. The single-iron unit cell is realized by applying the “unfold gauge transformation.”

Figure 2

(a) FS’s for $n=6.0$ in the unfolded model. The colors correspond to $2$ (green), $3$ (red), and $4$ (blue), respectively. The interorbital nesting between FS ${\alpha }_2$ (green) and FS ${\beta }_2$ (blue) causes the AFQ fluctuations. (b) FS’s for the original ten-orbital model.

Figure 3

Interorbital scattering processes due to the $e$-ph interaction by $u_{\mu }$ ($\mu =x,y,z$) within the $t_{2g}$ orbitals ($l=2–4$).

Figure 4

Quadrupole susceptibilities in the five-orbital model for (a) ${\chi }_{\mathit{xz}}^Q(\mathit{q})$, (b) ${\chi }_{\mathit{yz}}^Q(\mathit{q})$, and (c) ${\chi }_{\mathit{xy}}^Q(\mathit{q})$, respectively. The used model parameters are $n=6.05$, $T=0.05$, and $g=0.22$ (${\alpha }_c=0.98$). The correlation length in (a) or (b) is derived as $\xi =\pi /\Delta q~3$, where $\Delta q$ is the half-width of the peak. Therefore, we obtain the relations $6{\xi }^2~(1-{\alpha }_c){}^{-1}$ and $c{\xi }^2~2(1-{\alpha }_c){}^{-1}~12{\xi }^2$.

Figure 5

Quadrupole susceptibilities as a function of ${\alpha }_c$ for $n=6.0$ and $6.05$ given by the RPA. Model parameters are $U=0.8$ and $T=0.05$. We can recognize the relation ${\chi }_{\mathit{xz}}^Q(\mathit{Q})\propto (1-{\alpha }_c){}^{-1}\propto (g_c-g){}^{-1}$, which diverges at ${\alpha }_c=1$ or $g=g_c$.

Figure 6

(a) Fe-ion in-plane optical phonons with momentum $\mathit{q}=(\pi ,0)$ and $(\pi ,\pi )$. (b) Fe-ion out-of-plane optical phonons with momentum $\mathit{q}=(\pi ,0)$ and $(0,0)$.

Figure 7

Displacement vectors ${\mathit{u}}_{\mathrm{As}}$ and ${\mathit{u}}_{\mathrm{Fe}}$ in the transverse acoustic modes that couple with (a) $C_{44}$, (b) $C_E$, and (c) $C_{66}$. The $C_{66}$ mode corresponds to the orthorhombic structure transition.

Figure 8

(a) Diagrammatic expression for the shear modulus susceptibilities ${\chi }_{\varphi }$ ($\varphi =44,66,E$) in the RPA. (b) The second-order term with respect to the third-order anharmonic phonon-phonon interaction $A_{(3)}{u}_x^2{u}_{66}$. This diagram represents the virtual process in which an acoustic phonon with $\mathit{q}=0$ breaks into two optical phonons conserving the total momentum. (c) Two-orbiton term for the shear modulus susceptibilities ${\chi }_{66}^{\mathrm{TO}}$, which is the irreducible susceptibility of ${\chi }_{x^2-y^2}(0,0)$. This term gives the softening of $C_{66}$. (d) Dominant contribution for the three-point vertex $A_{x^2-y^2}(\Gamma ,\Gamma ;\mathit{q})$ for $\Gamma =\mathit{xz}$ and $\mathit{yz}$. (e) Self-energy correction due to ${\chi }_{\mathit{xz}}^Q(q)$. (f) Second-order Maki-Thompson-type vertex correction with respect to ${\chi }_{\mathit{xz}}^Q(q)$.

Figure 9

(a) Obtained $A_{x^2-y^2}(\mathit{xz},\mathit{xz};\mathit{q})$ for $U=0$. We set $n=6.05$ and $T=0.05$. (b) ${\chi }_{66}^{\mathrm{TO}}/T$ given by Eq. (49) for $U=0$ and $0.8$ as a function of ${\alpha }_c$. Using the relation $6{\xi }^2~(1-{\alpha }_c){}^{-1}$ given in the caption of Fig. 4, we obtain ${\chi }_{66}^{\mathrm{TO}}/T~0.1(1-{\alpha }_c){}^{-1}~0.6{\xi }^2$.

Figure 10

(a) ${\chi }_{66}^{\mathrm{TO}}$ for $T_{\mathrm{AFQ}}=100$, $50$, $10$, and $-10$ K given by Eq. (61). (b) $C_{66}/C_{66,0}$ for $T_{\mathrm{AFQ}}=100$, $50$, $10$, and $-10$ K given by Eqs. (51) and (52). We set $X=5.7$, $l=0.086$, $l^{\prime }=2$, $g_{66}=0.17$, and $a_{66}=2.5$. $C_{66}=0$ is realized when ${\chi }_{66}^{\mathrm{TO}}=g_{66}^{-1}-a_{66}$, which is $3.38$ in the present parameters. Using these same parameters, we can fit the recent experimental data by Yoshizawa et al.[32] for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ for $x=0–0.1$ just by changing $T_{\mathrm{AFQ}}$.

Figure 11

(a) $O_{x^2-y^2}$-FQ order given by the divergence of $C_{66}^{-1}$. Be careful not to confuse ${\mathrm{Ô}}_{x^2-y^2}$ with the $(x^2-y^2)$-orbital operator. (b) $O_{\mathit{xz}}$-AFQ order brought by the divergence of ${\chi }_{\mathit{xz}}^Q(\mathit{Q})$. (c) The correspondence between $O_{x^2-y^2}$-quadrupole order ($O_{\mathit{xz}}$-quadrupole order) and the $d$ orbital with larger occupation number.

Figure 12

(a) ${\chi }^s(\mathit{q};\Delta E)$ for $\Delta E=0$. Used model parameters are $n=6.05$, $U=1.1$, $g=0$, and $T=0.05$. (b) ${\chi }^s(\mathit{q};\Delta E)-{\chi }^s(\mathit{q};0)$ for $\Delta E=0.04$. Model parameters are the same as those in (a). (c) $U_c$ as a function of $\Delta E$ given by the RPA for $m^{*}/m=1$. (d) $\Delta n_2$ and $\Delta n_3$ as a function of $\Delta E$ for $m^{*}/m=1$. Note that $\Delta n\equiv \Delta n_2-\Delta n_3$. (e) FS’s for $\Delta E=-0.04$ for $n=6.05$. We use the same color coding as in Fig. 2. (f) FS’s for $\Delta E=-0.08$ for $n=6.05$. In both cases, the best nesting vector is $\mathit{q}=(\pi ,0)$.

Figure 13

The phase diagram for iron-pnictide superconductors obtained by the present orbital fluctuation theory. $T_S$ is the orthorhombic transition temperature (= FQ order temperature), and $T_{\mathrm{N}}$ is the SDW transition temperature. The fact that two QCP’s at $T_S=0$ and $T_{\mathrm{AFQ}}=0$ almost coincide means that novel “multi-orbital QCP’s” are realized in iron pnictides. The left-hand (right-hand) side of the vertical dotted line corresponds to $T_{\mathrm{AFQ}}>0$ ($T_{\mathrm{AFQ}}<0$), in which the two-orbiton process is relevant (irrelevant). At $T_{\mathrm{AFQ}}$, the AFQ order does not occur since it is prevented by the FQ order at $T_S$.

Figure 14

Quadrupole susceptibilities in the ten-orbital model for (a) ${\chi }_{\mathit{xz}}^Q(\mathit{q})$, (b) ${\chi }_{\mathit{yz}}^Q(\mathit{q})$, and (c) ${\chi }_{\mathit{xy}}^Q(\mathit{q})$, respectively. We set $n=6.05$, $T=0.05$, and ${\alpha }_c=0.98$.

Figure 15

Quadrupole susceptibilities in the ten-orbital model for (a) ${\chi }_{\mathit{xz}}^{Q,\mathit{AA}}(\mathit{q})$ and ${\chi }_{\mathit{xz}}^{Q,\mathit{AB}}(\mathit{q})$, (b) ${\chi }_{\mathit{yz}}^{Q,\mathit{AA}}(\mathit{q})$ and ${\chi }_{\mathit{yz}}^{Q,\mathit{AB}}(\mathit{q})$, and (c) ${\chi }_{\mathit{xy}}^{Q,\mathit{AA}}(\mathit{q})$ and ${\chi }_{\mathit{xy}}^{Q,\mathit{AB}}(\mathit{q})$, respectively. We set $n=6.05$, $T=0.05$, and ${\alpha }_c=0.98$.