Theory of the bound state of $4f$ excitations and magnetic resonance in unconventional superconductors

We analyze the influence of unconventional superconductivity on crystalline electric-field (CEF) excitations of rare-earth ions in layered superconductors. We show that resonant magnetic excitations of the conduction electrons that have been observed in these systems below $T_c$ may result in the formation of the bound state in the $4f$-electron susceptibility at energies well below the CEF excitation energy. Our results are in agreement with the observed increase of the linewidth of CEF excitations below $T_c$ in superconducting ferropnictides and support the $s^{\pm }$ Cooper-pairing state in these compounds.

Figure 1

(a) Graphical illustration of the solution of Eq. (4). The dotted curve refers to the function ${\Delta }_{CEF}^2-{\omega }^2$ with ${\Delta }_{CEF}=2{\Delta }_0$ (black) and ${\Delta }_{CEF}=1.5{\Delta }_0$ (gray), while $2{\Delta }_{CEF}{I}_0^2{\chi }_{RPA}^{\left(d\right)}\left(\mathbf{q},{\omega }_{\mathbf{q}}\right)$ is shown for the normal (dashed curve) and the superconducting (solid curve) state assuming $I_0\approx {\Delta }_0$. (b) Calculated $\text{Im}{\chi }^f\left(\mathbf{Q},\omega \right)$ at $T=T_c$ (dashed curve) and in the superconducting state $\left(T=0.1T_c\right)$ (solid curve). The dispersion parameters and Coulomb repulsion for the electron gas on a square lattice have been used (Ref. 15).

Figure 2

(Color online) Calculated temperature dependence of the lowest CEF excitation in an unconventional superconductor for (a) strong coupling $\left(I_0\approx {\Delta }_0\right)$ and (b) weak coupling $\left(I_0<{\Delta }_0\right)$ coupling between conduction electrons and localized $4f$ electrons. The tight-binding parameters for the electron-doped cuprates (Ref. 15) and ferropnictides (Ref. 17) are used. In both cases we assume a temperature dependence of the superconducting order parameter of the form ${\Delta }_{\mathbf{k}}\left(T\right)={\Delta }_{\mathbf{k}}\text{tanh}\left(1.76\sqrt{\frac{T_c}{T}-1}\right)$.

Figure 3

Calculated temperature dependence of the frequency position, ${\omega }_q$ (a) and the linewidth (b) of the lowest CEF level assuming weak coupling between the conduction and the $4f$-shell. The insets show the corresponding experimental data for the superconducting CeFeAsOF system (Ref. 13).