Theory of Multiband Superconductivity in Spin-Density-Wave Metals

We study the emergence of multiband superconductivity with $s$- and $d$-wave symmetry on the background of a spin density wave (SDW). We show that the SDW coherence factors renormalize the momentum dependence of the superconducting (SC) gap, yielding a SC state with an unconventional $s$-wave symmetry. Interband Cooper pair scattering stabilizes superconductivity in both symmetries. With increasing SDW order, the $s$-wave state is more strongly suppressed than the $d$-wave state. Our results are universally applicable to two-dimensional systems with a commensurate SDW.

Figure 1

FS (green solid lines) for the $n$-type (a) and $p$-type (b) cuprates ($x=0.12$) in the paramagnetic state. In the SDW state ($W_0=0.1\mathrm{eV}$), the $\alpha$ band possesses electron pockets (red long-dashed area) and the $\beta$-band hole pockets (blue dashed area). The RBZ is shown as a thin black line. The phase of the (a) unconventional $s$ and (b) $d_{x^2-y^2}$ SC OPs (of the $c$ electrons) are denoted by $+/-$. (c),(d) Angular dependence of the SC OP in the unconventional $s$-wave and $d_{x^2-y^2}$-wave channels for $n$-type doping close to $T_c$.

Figure 2

$T_c$ for unconventional $s$- and $d$-wave symmetries, as well as the DOS, as a function of $W_0$ at fixed chemical potential and for (a),(c),(e) $n$- and (b),(d),(f) $p$-type cuprates. We set $W_0=0.1\mathrm{eV}$, and for the $n$-type ($p$-type) cuprates $V_d=-2\mathrm{eV}$ ($V_d=-1.2\mathrm{eV}$) and $V_s=-0.6\mathrm{eV}$ ($V_s=-0.47\mathrm{eV}$).

Figure 3

The projected inter- and intraband effective interactions as a function of $W_0$.

Figure 4

Synopsis of our results shown as a schematic phase diagram. The system is an antiferromagnetic metal with electron pockets for $W_{\mathrm{cr}1}<W_0<W_{\mathrm{cr}2}$ ($e-\mathrm{FS}$), and with electron and hole pockets for $W_0<W_{\mathrm{cr}1}$ ($e+h-\mathrm{FS}$).