Lifting of nodes by disorder in extended-$s$-state superconductors: Application to ferropnictides

We show, using a simple model, how ordinary disorder can gap an extended-$s$- $\left(A_{1g}\right)$ symmetry superconducting state with nodes. The concomitant crossover of thermodynamic properties, particularly the $T$ dependence of the superfluid density, from pure power-law behavior to an activated one is exhibited. We discuss applications of this scenario to experiments on the ferropnictide superconductors.

Figure 1

(Color online) Fermi surface and extended-$s$-gap eigenfunction from Fig. 15(d) of Ref. [29]. Red/blue color indicates sign of order parameter.

Figure 2

(Color online) Suppression of the critical temperature $T_c/T_{c0}$ vs normal-state scattering rate $\Gamma /T_{c0}$ for anisotropy parameter $r=1.1$ (dashed), 1.3 (solid). The inset shows the two curves at smaller values of the scattering rate. The $T_c$ suppression curve expected from Abrikosov-Gor’kov theory (Ref. [41]) (dotted) is included for comparison.

Figure 3

(Color online) Normalized spectral gap ${\Omega }_G\left(\varphi \right)/T_{c0}$ vs angle $\varphi$ on the Fermi surface for an extended $s$-wave state with $r=1.3$ and $\Gamma /T_{c0}=0$ (dashed), 0.3 (dotted), 1.0 (solid), and 3.1 (dashed-dotted).

Figure 4

(Color online) Normalized density of states $N\left(\omega \right)/N_0$ vs energy $\omega$ for the same parameters and line types as Fig. 3.

Figure 5

(Color online) Superfluid density ${\rho }_s/m^{\ast }$ vs $T/T_{c0}$ for $r=1.3$ and same disorder parameters and line types as in Figs. 3 and 4.

Figure 6

(Color online) ${\text{log}}_{10}\left(1-{\rho }_s\left(T\right)/{\rho }_s\left(0\right)\right)$ vs ${\text{log}}_{10}T/T_c$ for $r=1.3$ and various scattering rates. Circles: $\Gamma =0$, triangles: $\Gamma /T_{c0}=0.3$, crosses: $\Gamma /T_{c0}=1.$, and squares: $\Gamma /T_{c0}=3.1$. Solid line: $T$; dashed-dotted line: $T^2$.

Figure 7

(Color online) Spectral gaps ${\Omega }_G\left(\varphi \right)/T_{c0}$ vs $\varphi$ for the $\alpha$ (top) and $\beta$ (bottom) sheets for the two-band model. Here the pairing parameters $V_{11}=1.0,V_{12}=-0.6,V_{22}=1.5$ were chosen so that $r_{\text{eff}}=1.3$ is the same as in the one-band case. The densities of states ratio was taken as $N_2\left(0\right)/N_1\left(0\right)=1.25$. Scattering rates for intraband scattering correspond to $\Gamma \equiv n_i\pi N_1\left(0\right){\left|U_{11}\right|}^2=0$ (dashed), 0.3 (dashed-dotted), 0.6 (solid).

Figure 8

(Color online) Density of states $N\left(\omega \right)/N_{\text{Total}}$ vs $\omega /T_{c0}$ for two-band system with $r_{\text{eff}}=1.3$. Same parameters and line types as in Fig. 7. Inset: expanded low-energy region.

Figure 9

(Color online) Normalized spectral gap $\text{min}{\Omega }_G\left(\varphi \right)/T_{c0}$ vs $T_c$ suppression $1-T_c/T_{c0}$ for various impurity concentrations comparing one-band and two-band models. Parameters are chosen such that $r_{\text{eff}}=1.3$ on the one $\left(\beta \right)$ sheet for the pure superconducting state in all cases. Diamonds: one-band model with $T_c/T_{c0}$ and ${\Omega }_G/T_{c0}$ taken from Figs. 2 and 3, respectively. Squares: two-band model with $V_{11}=1.0,V_{12}=-0.6,V_{22}=1.5$, with densities of states ratio $N_2\left(0\right)/N_1\left(0\right)=1.25$ and one sheet anisotropy parameter $r=1.76$. Circles: two-band model with same $V_{11}=1.0,V_{12}=-0.6,V_{22}=1.5$, but with densities of states ratio $N_2\left(0\right)/N_1\left(0\right)=1.5$ and $r=1.88$.

Figure 10

(Color online) Spectral gap vs Fermi-surface arc length in arbitrary units on each of four Fermi-surface sheets (${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and ${\beta }_2$) for realistic spin-fluctuation model of Fig. 17 of Ref. [29]. Curves correspond to an arbitrary range of impurity concentrations, from red (clean) to blue (dirty). Bottom panel: detail of nodal region of $\beta$ sheets.

Figure 11

(Color online) $\text{min}{\Omega }_G/T_{c0}$ vs $1-T_c/T_{c0}$ for same disorder parameters as Fig. 10.