Electronic polarons and bipolarons in iron-based superconductors: The role of anions

We introduce a quantum-mechanical model to describe the low-energy scale electronic structure of the Fe-based pnictides, placing a special importance on the large polarizability of the large anions, such as ${\text{As}}^{3-}$. These are modeled in terms of an atomic two-level system coupled strongly to charge fluctuations on the nearby Fe ions. This strong coupling results in electronic polaron formation, a strong effective reduction in the Fe on-site Coulomb repulsion, and the appearance of a strong attractive interaction for $d$ electrons on nearest-neighbor Fe sites, which could be an essential component of the pairing mechanism. This quantum model allows us to investigate this phenomenology away from the linear regime and to also study the dynamic properties of these polarons, as well as the binding of polaron pairs and their dynamics.

Figure 1

(Color online) Three-dimensional sketch of the FeAs layer. The small (green) spheres indicate Fe and the large (red) ones indicate As locations. The lattice constant is $a$ and the angle $\theta$ between the Fe-As direction and the $z$-axis is indicated.

Figure 2

(Color online) View from above of the Fe-As structure. Several sites are indexed. The orientations of the in-plane axes 1,2 and $x,y$ is also indicated. The Fe and As layers have different depths in the $z$ direction.

Figure 3

(Color online) Ratio $g/\Omega$ vs $\Omega$, for a fixed polarizability ${\alpha }_p=10{\text{Å}}^3$ (full line) and $12{\text{Å}}^3$ (dashed line). The dot indicates our typical choice of parameters.

Figure 4

(Color online) Pictorial illustration of a static electronic polaron. The electron residing on the central Fe atom polarizes the holes on the nn As sites into the $4p$ orbital pointing toward the Fe. More distant As are assumed to remain in the $5s$ ground state.

Figure 5

(Color online) Polaron hopping in the $x$ direction. The hopping integral is renormalized by the overlap between the polarization clouds in the initial and final positions. More distant As are assumed to be in the $5s$ ground state.

Figure 6

(Color online) $t_{\text{eff}}/t$ (left) and $t_{\text{eff}}^{\prime }/t^{\prime }$ (right) vs $\Omega$, for different As polarizabilities. The dots indicate the main values used here.

Figure 7

(Color online) Polarization clouds for a second-nearest-neighbor electronic bipolaron. The central “shared” As atom has a polarization different from that of the usual polaron clouds.

Figure 8

(Color online) (a) Renormalization of on-site interaction, $U_0-U_H$; (b) nn energy $U_1$ and (c) second-nn energy $U_2$ vs $\Omega$ for various polarizabilities. (d) $U_1$ vs $g/\Omega$ when $\Omega =4,6,8\text{eV}$. The dots show our typical values.

Figure 9

(Color online) Polarization clouds for a nearest-neighbor electronic bipolaron. The two central “shared” As atoms have a polarization different from that of the usual polaron clouds.

Figure 10

(Color online) (a) Bound eigenstates measured from the two-polaron continuum, $\Delta E_{BP}=E_{BP}\left(0\right)+8t_{\text{eff}}$ vs $U_H$. The symmetry of the eigenstates is labeled in the inset. (b) Probability for on-site, first-, second-, and third-nn separations in the ground state, vs $U_H$. The dashed lines show the same quantities for the $d$ state. Here $t^{\prime }=0,{\alpha }_p=10{\text{Å}}^3$, $\Omega =6\text{eV}$ (similar results are found for all ${\alpha }_p=7–12{\text{Å}}^3,\Omega =4–8\text{eV}$).

Figure 11

(Color online) Dispersion of bound bipolaron states along high-symmetry axes in the Brillouin zone, for (a) $t^{\prime }=0$ and (b) $t^{\prime }=-t/2$. The two-polaron continuum is also shown. Parameters are $U_H=10\text{eV}$, ${\alpha }_p=10{\text{Å}}^3,\Omega =6\text{eV}$ (similar results are found for all ${\alpha }_p=7–12{\text{Å}}^3,\Omega =4–8\text{eV}$). The symmetry of the ground state changes from $s$ to $d$ if $t^{\prime }\ne 0$.

Figure 12

(Color online) Ground-state bipolaron (a) binding energy, and (b) effective mass in units of the free-carrier mass vs $\Omega$, for various polarizabilities. The full lines correspond to $t^{\prime }=0$, dashed lines to $t^{\prime }=-t/2$. Here $U_H=10\text{eV}$.

Figure 13

(Color online) Effective hoppings for closely spaced bipolarons. For these parameters, the $t_2$ and $t_3$ curves are essentially superimposed, therefore we only show $t_2/t$.