Effect of pairing fluctuations on the spin resonance in Fe-based superconductors

The spin resonance observed in the inelastic neutron scattering data on Fe-based superconductors has played a prominent role in the quest for determining the symmetry of the order parameter in these compounds. Most theoretical studies of the resonance employ an RPA-type approach in the particle-hole channel and associate the resonance in the spin susceptibility ${\chi }_S(\mathbf{q},\omega )$ at momentum $\mathbf{Q}=(\pi ,\pi )$ with the spin-exciton of an $s^{+-}$ superconductor, pulled below $2\Delta$ by residual attraction associated with the sign change of the gap between Fermi points connected by $\mathbf{Q}$. Here we explore the effect of fluctuations in the particle-particle channel on the spin resonance. Particle-particle and particle-hole channels are coupled in a superconductor and to what extent the spin resonance can be viewed as a particle-hole exciton needs to be addressed. In the case of purely repulsive interactions, we find that the particle-particle channel at total momentum $\mathbf{Q}$ (the $\pi$ channel) contributes little to the resonance. However, if the interband density-density interaction is attractive and the $\pi$ resonance is possible on its own, along with spin-exciton, we find a much stronger shift of the resonance frequency from the position of the would-be spin-exciton resonance. We also show that the expected double-peak structure in this situation does not appear because of the strong coupling between particle-hole and particle-particle channels, and $\mathrm{Im}{\chi }_S(\mathbf{Q},\omega )$ displays only a single peak.

Figure 1

Some mixed-channel contributions to the spin susceptibility and Raman scattering. The solid lines represent $f$ fermions and the dashed lines $c$ fermions, corresponding to quasiparticles from the electron and hole pockets, respectively.

Figure 2

Real part of the bare spin susceptibilities ($T=0$, units of ${\Delta }^{-1}$) in the case of perfectly nested circular pockets ($m\equiv m_c=m_f=\frac{1}{100\Delta }$). In all numerical calculations, $\mu =10\Delta$ and we include a finite broadening $\gamma =\Delta /200$ when evaluating momentum integrals.

Figure 3

Imaginary part (main plot) and real part (insert) of the full spin susceptibility ($T=0$, units of ${\Delta }^{-1}$) in the case of repulsive interactions. In this plot, $u_S=26\Delta$ and $u_{\pi }=-13\Delta$. The solid, black line indicates the full calculation. For comparison, the dashed, blue line indicates a calculation that neglects particle-particle contributions.

Figure 4

Imaginary part (main plot) and real part (insert) of the full spin susceptibility ($T=0$, units of ${\Delta }^{-1}$) in the case of $u_S=30.5\Delta$ and $u_{\pi }=130\Delta$. In the absence of coupling we would observe resonance peaks in the spin and $\pi$ channels at frequencies ${\omega }_S\approx 0.57\Delta$ and ${\omega }_{\pi }\approx 1.0\Delta$, respectively, indicated on the plot.