Exchange boson dynamics in the cuprates

The electron-boson spectral density function $I^{2}\chi(\Omega)$ responsible for carrier scattering of high temperature superconductors can be calculated from the optical scattering rate using a maximum entropy technique. Published data on a range of high temperature superconductors (with two notable exceptions) show a peak at an energy ($\Omega_{r}$) and a broad high frequency background. The energy of the peak is proportional to the superconducting transition temperature $\Omega_{r}\approx$ 6.3$k_{B}T_{c}$ for a large range of materials with $T_{c}$'s ranging from of 30 K for LSCO up to 130 K for HgBa$_{2}$Ca$_{2}$Cu$_{3}$O$_{8+\delta}$. This relationship in the charge channel compares well with a similar one in the spin channel where the frequency of the magnetic resonance seen by polarized neutron scattering is also found to be proportional to $T_c$, $\Omega_r^{neutron}\approx$ 5.4$k_{B}T_{c}$. The amplitude of both the optical resonance and the magnetic neutron peak decrease strongly with increasing temperature.