Spin Resonance in Three-Dimensional Superconductors: The Case of ${\mathrm{CeCoIn}}_5$

The recent observation of resonance spin excitation at ($1/2$, $1/2$, $1/2$) in the superconducting state of ${\mathrm{CeCoIn}}_5$ [C. Stock et al., Phys. Rev. Lett. 100, 087001 (2008)] was interpreted as evidence for $d_{x^2-y^2}$ gap symmetry, by analogy with cuprates. This is true if the resonance is a spin exciton. We argue that such a description is undermined by the three dimensionality of ${\mathrm{CeCoIn}}_5$. We show that in 3D systems the excitonic resonance only emerges at strong coupling, and is weak. We argue in favor of the alternative, magnon scenario, which does not require a $d_{x^2-y^2}$ gap.

Figure 1

Panels (a) and (b)—the imaginary and real parts of the polarization operator $\Pi (Q,\Omega )$ for 3D spherical FS with hot lines. At large frequencies, $\mathrm{Im}\Pi (\Omega )=\Omega /\Delta$, where $\Delta$ is the maximum of the $d_{x^2-y^2}$ gap. We set $r_0=1$. Panels (c) and (d)—the imaginary part of the full spin susceptibility, Eq. (1) (in arb. units) in the normal state (dashed) and $d$-wave superconducting state (solid) for ${\xi }^{-2}/({\gamma }_{3\mathrm{D}}\Delta )=0.1$ (c), and ${\xi }^{-2}/({\gamma }_{3\mathrm{D}}\Delta )=0.25$ (d). Observe that $\mathrm{Im}\chi (Q,\Omega )$ in a $d_{x^2-y^2}$ superconductor does has a “resonance” peak, but its intensity is not enhanced compared to the normal state.