Magnetism, superconductivity, and pairing symmetry in iron-based superconductors

We analyze antiferromagnetism and superconductivity in novel Fe-based superconductors within the itinerant model of small electron and hole pockets near (0,0) and $\left(\pi ,\pi \right)$. We argue that the effective interactions in both channels logarithmically flow toward the same values at low energies; i.e., antiferromagnetism and superconductivity must be treated on equal footing. The magnetic instability comes first for equal sizes of the two pockets, but loses to superconductivity upon doping. The superconducting gap has no nodes, but changes sign between the two Fermi surfaces (extended $s$-wave symmetry). We argue that the $T$ dependencies of the spin susceptibility and NMR relaxation rate for such a state are exponential only at very low $T$, and can be well fitted by power laws over a wide $T$ range below $T_c$.

Figure 1

(Color online) A simplified FS geometry of doped Fe-based superconductors, used in the present work. At zero doping, the Fermi surface consists of an electron pocket around $\left(\pi ,\pi \right)$ (blue dashed curve), and a hole pocket of roughly equal size around (0,0) (blue solid curve). We neglect in this work the fact that there are two hole and two electron pockets. In this system, there is a near-perfect nesting between hole and electron pockets [moving a hole FS by $\left(\pi ,\pi \right)$ one obtains a near-perfect match with an electron FS]. Upon electron doping, the size of the electron pocket increases (dashed $\text{blue}\to \text{solid black}$), and this breaks the nesting. $+\Delta$ and $-\Delta$ are the values of the superconducting gaps on the two FS for $s^{+}$ superconducting state.

Figure 2

Five relevant bare vertices. Solid and dashed lines represent fermions from $c$ band (near $\mathbf{k}=0$) and $f$ band [near $\mathbf{k}=\left(\pi ,\pi \right)$].

Figure 3

Diagrammatic representation of the equations for SDW, CDW, and SC instability temperatures. The equations for density-wave $T_{\text{SDW}}^{\left(r,i\right)}$ and $T_{\text{CDW}}^{\left(r,i\right)}$ are obtained by adding and subtracting equations for ${\Delta }_{\text{SDW}}$ and ${\Delta }_{\text{SDW}}^{\ast }$, and for ${\Delta }_{\text{CDW}}$ and ${\Delta }_{\text{CDW}}^{\ast }$, respectively. The equations for $T_{\text{SC}}^{\left(s\right)}$ and $T_{\text{SC}}^{\left(s^{+}\right)}$ obtained by adding and subtracting equations for ${\Delta }^c$ and ${\Delta }^f$.

Figure 4

Diagrams for one-loop vertex renormalizations. The renormalizations of $u_1$ and $u_3$ are shown; others are obtained in a similar way.

Figure 5

(Color online) The RG flow of Eq. (13) in variables $u_4/u_3$ and $u_1/u_3$. The fixed point is $u_4/u_3=-1/\sqrt{5}$, $u_1/u_3=1/\sqrt{5}$. The fourth variable, $u_2$, becomes small near the fixed point compared to the other $u_i$. We used boundary conditions $u_1^0=u_4^0$, $u_3^0=0.1u_1^0$, and for simplicity set $u_0^2=0$.

Figure 6

(Color online) The RG flow of the effective couplings in various density-wave and superconducting channels vs $L=u^0\text{log}W/E$. The boundary conditions are $u_1^0=u_4^0=u_0$ and $u_2^0=u_3^0=\pm 0.1u_0$. The running couplings are in units of $u_0$. (a) The RG flow for $u_3^0>0$. The extended $s$-wave $\left(s^{+}\right)$ superconducting channel becomes attractive above some $L$. The strongest density-wave instability is in SDW channel, for real order parameter. (b) The same for $u_3^0<0$. The conventional $s$-wave superconducting channel becomes attractive above some $L$. The strongest density-wave instability is in CDW channel, again for real order parameter.

Figure 7

(Color online) $T_c$ (a) and $b_0=2U_i\left(\pi \right)/{\Delta }_0$ (b) as functions of $b=2U_i\left(\pi \right)/\Delta$, where $\Delta$ is the order parameter, and ${\Delta }_0$ is the gap in the absence of impurities. The inset of (a) shows the dependence of $T_c$ on $b_0$.

Figure 8

(Color online) (a) Calculated temperature dependence of $1/T_1$ for an $s^{+}$ superconductor with nonmagnetic impurities. The normalization is chosen such that $1/T_1=1$ at $T=T_c$. The theoretical curves are for various values of the parameter $b=2U_i\left(\pi \right)/\Delta \left(T=0\right)$ which measures the strength of the pair-breaking component of nonmagnetic impurities. Gapless superconductivity occurs for $b>1$. The experimental data are taken from Refs. 31, 33. (b) Theoretical $1/T_1$ for different $b$ vs power-law forms. All theoretical dependencies are exponential in $T$ at very low $T$, but are described by power laws $T^{\alpha }$ over a wide $T$ range below $T_c$. The exponent $\alpha$ decreases as $b$ increases from $\alpha \approx 3$ for $b=0.3$ to $\alpha \approx 2.3$ for $b=0.7$.

Figure 9

(Color online) Calculated temperature dependence of the uniform susceptibility for various values of $b$. The experimental data are taken from Ref. 33.