Pnictogen height as a possible switch between high-$T_c$ nodeless and low-$T_c$ nodal pairings in the iron-based superconductors

We study the effect of the lattice structure on the spin-fluctuation-mediated superconductivity in the iron pnictides adopting the five-band models of several virtual lattice structures of LaFeAsO, as well as actual materials such as NdFeAsO and LaFePO obtained from the maximally localized Wannier orbitals. Random phase approximation is applied to the models to solve the Eliashberg equation. This reveals that the gap function and the strength of the superconducting instability are determined by the cooperation or competition among multiple spin-fluctuation modes arising from several nestings among disconnected pieces of the Fermi surface, which is affected by the lattice structure. Specifically, the appearance of the Fermi surface $\gamma$ around $\left(\pi ,\pi \right)$ in the unfolded Brillouin zone is sensitive to the pnictogen height $h_{\text{Pn}}$ measured from the Fe plane, where $h_{\text{Pn}}$ is shown to act as a switch between high-$T_c$ nodeless and low-$T_c$ nodal pairings. We also find that reduction in the lattice constants generally suppresses superconductivity. We can then combine these to obtain a generic superconducting phase diagram against the pnictogen height and lattice constant. This suggests that NdFeAsO is expected to exhibit a fully gapped, sign-reversing $s$-wave superconductivity with a higher $T_c$ than in LaFeAsO, while a nodal pairing with a low $T_c$ is expected for LaFePO, which is consistent with experiments.

Figure 1

(Color online) Lattice structure of one Fe-Pn layer, with the pnictogen height indicated.

Figure 2

(Color online) (a) The original (dashed lines) and reduced (solid) unit cells with ● (Fe), ▽ (As below the Fe plane) and △ (above Fe). (b) The band structure (left) of the five-band model for LaFeAsO, and the Fermi surface (right) at $k_z=0$ for $n=6.1$. The main orbital characters of some portions of the bands and the Fermi surface are indicated. The dashed horizontal line in the band structure indicates the Fermi level for $n=6.1$. The short arrow in the band structure indicates the position of the Dirac cone closest to the Fermi level. The gray areas in the Fermi surface around the zone corners represent the $\gamma$ Fermi surface. (c) The portion of the band that has mainly the $d_{X^2-Y^2}$ orbital character.

Figure 3

(Color online) Top panels: Diagonal components, ${\chi }_{s3333}$ and ${\chi }_{s4444}$, of the spin susceptibility matrix in the orbital representation $\left(3:YZ,4:X^2-Y^2\right)$ for the five-band model of LaFeAsO with $n=6.1$. Bottom: Nesting is shown for the Fermi surface for orbitals $XZ,YZ$ (left) and $X^2-Y^2$ (right). Here the thickness of the Fermi surface represents the strength of the respective orbital character.

Figure 4

(Color online) The fully gapped $s\pm$ wave (top panel), the nodal $s\pm$ wave (middle), and the $d$-wave gap (bottom) are schematically shown. The solid red (dashed blue) curves represent positive (negative) sign of the gap. The arrows indicate the dominating nesting vectors. $\gamma$ Fermi surface is present when the pnictogen height is large.

Figure 5

(Color online) The $s$-wave (left panels) and $d$-wave (right) gap functions for the second to fourth bands from top to bottom in the band representation for the five-band model of LaFeAsO with $n=6.1$. Solid lines represent the Fermi surface, and green dashed lines the nodes in the gap.

Figure 6

(Color online) (a) Eigenvalues of the Eliashberg equation for $s$ wave and $d$ wave, respectively, for the five-band model of LaFeAsO plotted against the band filling $n$. The light red (or gray) symbol for the $s$ wave at $n=6.3$ indicates that the gap is nearly nodal. (b) ${\chi }_{s3333}$ and ${\chi }_{s4444}$ for $n=6.3$. (c) The $s$-wave gap function for band 4 with $n=6.2$ (left) and $n=6.3$ (right).

Figure 7

(Color online) Band structure in the five-band model for (a) $z_{\text{As}}=0.658$ and (b) $z_{\text{As}}=0.6304$. The lattice constants are kept at the original values of LaFeAsO. $X^2-Y^2$ and $Z^2$ denote the main characters of the bands around $\left(\pi ,\pi ,k_z\right)$. Dashed lines represent the Fermi energy, and the Fermi surface at $k_z=0$ for $n=6.1$ is shown on the right panels.

Figure 8

(Color online) ${\chi }_{s3333}$ (left panels) and ${\chi }_{s4444}$ (right) for the model with (a) $z_{\text{As}}=0.658$ and (b) $z_{\text{As}}=0.6304$.

Figure 9

(Color online) $s$-wave and $d$-wave eigenvalues of the Eliashberg equation plotted against $z_{\text{As}}$ (lower scale) or $h_{\text{As}}$ (upper scale) for $n=6.1$. The lattice constants are fixed at the original values for LaFeAsO. For the $s$ wave, the open (solid) circles indicate that the gap is nodeless (nodal).

Figure 10

(Color online) The gap functions for the model with $z_{\text{As}}=0.6304$ (left panels) or $z_{\text{As}}=0.658$ (right). From top to bottom: $s$ wave in band 3, $s$ wave in band 4, $d$ wave in band 3, and $d$ wave in band 4.

Figure 11

(Color online) Upper panels: $s$-wave and $d$-wave eigenvalues of the Eliashberg equation plotted against $a$ (left) or $c$ (right), where $h_{\text{As}}$ is fixed at the original value for LaFeAsO. Lower panels: ${\chi }_{s3333}$ and ${\chi }_{s4444}$ for $a=3.88\text{Å}$ with $c$ fixed at the original value (left) and $c=8.0\text{Å}$ with $a$ fixed at the original value (right).

Figure 12

(Color online) $s$-wave and $d$-wave eigenvalues of the Eliashberg equation calculated for orbital-dependent interactions, plotted against (a) $n$ for the model of LaFeAsO, (b) $z_{\text{As}}$ or $h_{\text{As}}$ for $n=6.1$ with the lattice constants fixed at the original values for LaFeAsO, (c) $a$, and (d) $c$ with $h_{\text{As}}$ fixed at the original value for LaFeAsO. In (a), the light red (or gray) symbol for $s$ wave at $n=6.2$ indicates that the gap is nearly nodal, while the red (or solid) symbol at $n=6.3$ stands for the nodal $s$ wave.

Figure 13

(Color online) The gap function for band 4 calculated with the orbital-dependent interactions for (a) LaFeAsO with $n=6.1$ (left) and $n=6.3$ (right), and (b) $z_{\text{As}}=0.658$ (left) and $z_{\text{As}}=0.6304$ (right) with $n=6.1$. (c) ${\chi }_{s3333}$ and ${\chi }_{s4444}$ for $z_{\text{As}}=0.658$ and $n=6.1$.

Figure 14

(Color online) The band structure of the five-band model of the optimally doped NdFeAsO (upper panels) and LaFePO (lower). The Fermi surface at $k_z=0$ for $n=6.1$ is shown on the right.

Figure 15

(Color online) Upper panels: ${\chi }_{s3333}$ and ${\chi }_{s4444}$ for the five-band model for LaFePO with $n=6.1$, $U=1.7$, $U^{\prime }=1.4$, and $J=J^{\prime }=0.15$ (orbital-independent interactions). Lower panels: $s$-wave (left) and $d$-wave (right) gap functions for bands 3 and 4 for the same parameter values.

Figure 16

(Color online) $s$-wave and $d$-wave eigenvalues of the Eliashberg equation for LaFePO plotted against (a) $U$ with orbital-independent interactions with $U-U^{\prime }=2J=0.3$ fixed, and (b) the multiplication factor $f$ with orbital-dependent interactions. The $s$-wave gap here always has nodes intersecting the $\beta$ Fermi surface.

Figure 17

(Color online) ${\chi }_{s3333}$ (left panels) and ${\chi }_{s4444}$ (right) in the five-band model for (a) the optimally doped NdFeAsO with $n=6.1$, and (b) the underdoped NdFeAsO with $n=6.03$.

Figure 18

(Color online) The $s$-wave gap function for band 3 (left panels) and band 4 (right) for the optimally doped NdFeAsO. Orbital-independent (a) and dependent (b) interactions are adopted.

Figure 19

(Color online) Schematic superconducting phase diagram on the $h_{\text{Pn}}\text{−}\left[a,c\right]$ (upper) and $\alpha \text{−}\left[a,c\right]$ (lower) planes. The dashed lines are schematic contours of $T_c$. The curved arrow in the lower panel schematically indicates how the lattice parameters vary as the bond angle $\alpha$ is decreased in Ref. 50. The shorthands for materials are the same as in Table .