Spin fluctuations and superconductivity in a three-dimensional tight-binding model for ${\text{BaFe}}_2{\text{As}}_2$

Despite the wealth of experimental data on the Fe-pnictide compounds of the $K{\text{Fe}}_2{\text{As}}_2$ type, $K=\text{Ba}$, Ca, or Sr, the main theoretical work based on multiorbital tight-binding models has been restricted so far to the study of the related 1111 compounds. This can be ascribed to the more three-dimensional electronic structure found by ab initio calculations for the 122 materials, making this system less amenable to model development. In addition, the more complicated Brillouin zone (BZ) of the body-centered tetragonal symmetry does not allow a straightforward unfolding of the electronic band structure into an effective 1Fe/unit cell BZ. Here we present an effective five-orbital tight-binding fit of the full density functional theory band structure for ${\text{BaFe}}_2{\text{As}}_2$ including the $k_z$ dispersions. We compare the five-orbital spin fluctuation model to one previously studied for LaOFeAs and calculate the random-phase approximation enhanced susceptibility. Using the fluctuation exchange approximation to determine the leading pairing instability, we then examine the differences between a strictly two-dimensional model calculation over a single $k_z$ cut of the BZ and a completely three-dimensional approach. We find pairing states quite similar to the 1111 materials, with generic quasi-isotropic pairing on the hole sheets and nodal states on the electron sheets at $k_z=0$, which however are gapped as the system is hole doped. On the other hand, a substantial $k_z$ dependence of the order parameter remains, with most of the pairing strength deriving from processes near $k_z=\pi$. These states exhibit a tendency for an enhanced anisotropy on the hole sheets and a reduced anisotropy on the electron sheets near the top of the BZ.

Figure 1

(Color online) Sketch of the Brillouin zone of the I4/mmm crystal symmetry (a) and of the large effective BZ corresponding to the 1Fe/unit cell (b). The blue line shows the two paths in the 1Fe/unit cell BZ that have to be folded by the reciprocal lattice vector $\mathbf{T}=\left(\pi ,\pi ,\pi \right)$ (red arrow) to give the corresponding path in the 2Fe/unit cell BZ of the P4/nmm symmetry.

Figure 2

(Color online) The paramagnetic DFT band structure (full line) and a Wannier fit (crosses) of the 10 bands in the vicinity of the Fermi surface onto the $\text{Fe-}3d$ orbitals (a). The five orbital tight-binding fit (colored points) of the ten-orbital Wannier fit (black points) with a color coding of the main orbital contributions (b). The colors correspond to $d_{xz}$ (red), $d_{yz}$ (green), $d_{xy}$ (blue), $d_{x^2-y^2}$ (orange), and $d_{3z^2-r^2}$ (magenta). All energies are measured from the Fermi energy $E_F=10.86\text{eV}$

Figure 3

(Color online) The partial density of states of the five-orbital tight-binding fit using the same color coding as in Fig. 2b.

Figure 4

(Color online) The main orbital contributions to the Fermi surfaces of the hole doped compound with $⟨n⟩=5.9$ at $k_z=0$ (a) and $k_z=\pi$ (b) using the same color coding as in Fig. 2b.

Figure 5

(Color online) The orbital composition of the Fermi surface sheets for a hole doped compound ($⟨n⟩=$5.9) as given by ${\left|a_{\nu }^t\left(k\right)\right|}^2$ for $k_z=0$ (a) and $k_z=\pi$ (b). The orbital contributions are shown as a function of the winding angle $\alpha$ with $\alpha =0$ corresponding to the rightmost point on each Fermi surface sheet.

Figure 6

(Color online) 2D susceptibility: the real part of the RPA enhanced susceptibility ${\chi }_{\text{RPA}}\left(q\right)$ as a function of the in-plane momentum transfer calculated in two dimensions for a single value of $k_z$, $k_z=0$ (a,c,e) and $k_z=\pi$ (b,d,f) for a hole doped compound with $⟨n⟩=5.9$. For (a) and (b) we have used $\overline{U}=0.65$ and $\overline{J}=0$, while for (c) and (d) we have used $\overline{U}=0.55$ and $\overline{J}=0.25\overline{U}$. In panels (e) and (f) the susceptibility is shown along the main symmetry lines with $\overline{U}=0.65$, $\overline{J}=0$ (red), and $\overline{U}=0.55$, $\overline{J}=0.25\overline{U}$ (blue).

Figure 7

(Color online) 3D susceptibility: the real part of the RPA enhanced susceptibility ${\chi }_{\text{RPA}}\left(q\right)$ as a function of the in-plane momentum transfer for two different values of $q_z$, $q_z=0$ (a,c,e) and $q_z=\pi$ (b,d,f) for a hole doped compound with $⟨n⟩=5.9$. For (a) and (b) we have used $\overline{U}=1.1$ and $\overline{J}=0$, while for (c) and (d) we have used $\overline{U}=0.8$ and $\overline{J}=0.25\overline{U}$. In panels (e) and (f) we use the same coloring scheme as in Fig. 6.

Figure 8

(Color online) 2D susceptibility: the real part of the RPA enhanced susceptibility ${\chi }_{\text{RPA}}\left(q\right)$ as a function of the in-plane momentum transfer calculated in two dimensions for a single value of $k_z$, $k_z=0$ (a,c,e), and $k_z=\pi$ (b,d,f) for an undoped compound with $⟨n⟩=6$. For (a) and (b) we have used $\overline{U}=0.7$ and $\overline{J}=0$, while for (c) and (d) we have used $\overline{U}=0.6$ and $\overline{J}=0.25\overline{U}$. In panels (e) and (f) we use the same coloring scheme as in Fig. 6.

Figure 9

(Color online) 3D susceptibility: the real part of the RPA enhanced susceptibility ${\chi }_{\text{RPA}}\left(q\right)$ as a function of the in-plane momentum transfer for two different values of $q_z$, $q_z=0$ (a,c,e) and $q_z=\pi$ (b,d,f) for an undoped compound with $⟨n⟩=6$. For (a) and (b) we have used $\overline{U}=1.1$ and $\overline{J}=0$, while for (c) and (d) we have used $\overline{U}=0.9$ and $\overline{J}=0.25\overline{U}$. In panels (e) and (f) we use the same coloring scheme as in Fig. 6.

Figure 10

(Color online) Fermi surface mesh of the hole doped compound applied to the calculation of the pairing functions. Here we used $24\times 10$ $\mathbf{k}$-points for every Fermi surface sheet with ${\alpha }_1$ (red), ${\alpha }_2$ (blue), ${\beta }_1$, ${\beta }_2$ (green), and $\gamma$ (yellow).

Figure 11

(Color online) 2D pairing functions, $\overline{J}=0$: the leading (upper row) and subleading (lower row) pairing function for the hole doped compound $\left(⟨n⟩=5.9\right)$ plotted along the Fermi surfaces at two different $k_z$ cuts. The pairing functions are shown in the order ${\alpha }_1$, ${\alpha }_2$, $\gamma$, ${\beta }_1$, and ${\beta }_2$ running counterclockwise around each Fermi surface sheet with the rightmost point as the starting point on each sheet, except the ${\beta }_2$ pocket where the plots start with the uppermost point. The calculations were performed for $\overline{U}=0.65$ and $\overline{J}=0$ and the eigenvalues are $\lambda =0.022$ ($s$ wave) and $\lambda =0.015$ ($d$ wave) for $k_z=0$, and $\lambda =1.615$ ($s$ wave) and $\lambda =0.377$ ($d$ wave) for $k_z=\pi$.

Figure 12

(Color online) 2D pairing functions, $\overline{J}>0$: the leading (upper row) and subleading (lower row) pairing functions for the hole doped compound $\left(⟨n⟩=5.9\right)$ plotted as before for two different values of $k_z$, calculated for $\overline{U}=0.55$ and $\overline{J}=0.25\overline{U}$. The eigenvalues are $\lambda =0.027$ ($s$ wave) and $\lambda =0.015$ ($d$ wave) for $k_z=0$, and $\lambda =0.638$ [$s$(1) wave] and $\lambda =0.381$ [$s$(2) wave] for $k_z=\pi$.

Figure 13

(Color online) 3D pairing functions, hole doped: The leading pairing functions for the hole doped compound $\left(⟨n⟩=5.9\right)$ plotted as before for two different values of $k_z$, calculated for $\overline{U}=1.0$ and $\overline{J}=0$ (upper row) and $\overline{U}=0.8$ and $\overline{J}=0.25\overline{U}$ (lower row). Here the maximum eigenvalues are $\lambda =0.956$ and $\lambda =1.077$, respectively.

Figure 14

(Color online) 3D pairing functions, $\overline{J}=0$: the leading $A1g$ pairing function (extended $s$-wave) on the ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and $\gamma$ Fermi surface sheet for the hole doped compound $\left(⟨n⟩=5.9\right)$. The semitransparent color mesh visualizes the gap on each of the FS sheets that is also shown without the gap for comparison. The calculations were performed for $\overline{U}=1.0$ and $\overline{J}=0$. In this figure we have changed the overall phase of the gap by $-1$ from that was used in Fig. 13 in order to highlight the nodal structure.

Figure 15

(Color online) 3D pairing functions, undoped: the leading and subleading pairing functions ($d$ wave and $s$ wave) for the undoped compound $\left(⟨n⟩=6\right)$ plotted as before for two different values of $k_z$, calculated for $\overline{U}=0.9$ and $\overline{J}=0.25\overline{U}$. Here the eigenvalues are $\lambda =1.14$ and $\lambda =0.617$, respectively.