Minimal two-band model of the superconducting iron oxypnictides

Following the discovery of the Fe-pnictide superconductors, local-density approximation (LDA) band structure calculations showed that the dominant contributions to the spectral weight near the Fermi energy came from the $\text{Fe}3d$ orbitals. The Fermi surface is characterized by two hole surfaces around the $\Gamma$ point and two electron surfaces around the $M$ point of the two Fe/cell Brillouin zone. Here, we describe a two-band model that reproduces the topology of the LDA Fermi surface and exhibits both ferromagnetic and $q=\left(\pi ,0\right)$ spin-density wave fluctuations. We argue that this minimal model contains the essential low energy physics of these materials.

Figure 1

(Color online) (a) The Fe ions form a square lattice and the crystallographic unit cell contains two Fe and two As ions. The As ions are located either directly above (solid circles) or below (dashed circles) the faces of the Fe square array. (b) A schematic showing the hopping parameters of the two-orbital $d_{xz}$,$d_{yz}$ model on a square lattice. The projections of the $d_{xz}$ $\left(d_{yz}\right)$ orbitals onto the xy plane are depicted in green (white). Here $t_1$ is a near-neighbor hopping between $\sigma$ -orbitals and $t_2$ is a near-neighbor hopping between $\pi$ -orbitals. We also include a second-neighbor hopping $t_4$ between different orbitals and a second-neighbor hopping $t_3$ between similar orbits.

Figure 2

(Color online) (a) The band structure of the two-band model with $t_1=-1$, $t_2=1.3$, $t_3=t_4=-0.85$, and $\mu =1.45°\text{C}$, plotted along the path $\left(0,0\right)\to \left(\pi ,0\right)\to \left(\pi ,\pi \right)\to \left(0,0\right)$ as shown in Fig. 3a by the black dashed lines. (b) The band structure folded to the small BZ, with the $\Gamma ,X,M$ defined in the small BZ as shown in Fig. 3b. (c) The two-$d$ band structure for $k_x,k_y∊\left[0,\pi \right]$. A saddle point exists for each band. (d) The density of states of the two-band model, with two Van Hove singularities. The dashed line shows the Fermi level corresponding to our choice of $\mu =1.45$.

Figure 3

(Color online) (a) The Fermi surface of the two-orbital model on the large one Fe/cell BZ. Here, ${\alpha }_{1,2}$ surfaces are hole Fermi pockets given by $E_{-}\left({\mathit{k}}_f\right)=0$ and ${\beta }_{1,2}$ are electron Fermi pockets by $E_{+}\left({\mathit{k}}_f\right)=0$. The dashed square indicates the BZ of the two Fe/cell. (b) The Fermi surface folded down into the two Fe/cell BZ consists of two $\alpha$ surfaces around $\Gamma$ and two elliptically deformed $\beta$ surfaces around the $M$ point. Here the parameters are the same as in Fig. 2.

Figure 4

(Color online) (a) The $\omega =0$ bare spin susceptibility ${\chi }_S\left(\mathbf{q}\right)$ versus $\mathbf{q}$ for the same tight-binding parameters as used in Fig. 3. (b) The bare spin susceptibility ${\chi }_S\left(\mathbf{q}\right)$ (red solid line) and the RPA spin susceptibility ${\chi }_S^{\text{RPA}}\left(\mathbf{q}\right)$ for $U=V=3$, $J=0$ (blue dashed line) along the $\left(0,0\right)\to \left(\pi ,0\right)\to \left(\pi ,\pi \right)\to \left(0,0\right)$ path in BZ, as shown by the dashed line in (a).

Figure 5

(Color online) (a) The chemical potential dependence of $\text{Re}\chi \left(\mathbf{q},\omega =0\right)$ with $\mathbf{q}=\left(0,0\right)$, $\left(0,\pi \right)$, and $\left(\pi ,\pi \right)$. The vertical dashed line shows the chemical potential $\mu =1.45$ which we are working on. (b) The three-$d$ color plot of spin susceptibility as a function of $k_x,k_y$ and chemical potential $\mu$. The hot points in BZ shifts from $\left(\pi ,0\right)$ to $\left(0,0\right)$ and $\left(\pi ,\pi \right)$ with electron doping as $\mu$ increases. The deepest red (dark) region shows the Van Hove singularities.