Mode coupling effect in infrared spectra of Tl$_2$Ba$_2$Ca$_{n-1}$Cu$_n$O$_x$ (n=1, 2, 3)

We performed in-plane optical reflectance measurements on Tl$_2$Ba$_2$Ca$_{n-1}$Cu$_n$O$_x$ with n=1, 2, 3. The single crystals were successfully grown by flux method with maximum T$_c$=90 K, 109 K, and 119 K for n=1, 2, 3, respectively. For all three phases near optimal doping, the reflectance roughly has a linear-frequency dependence in the normal state, but displays a pronounced knee structure followed by a dip-like feature at higher frequency below T$_c$. Such characteristic features were commonly ascribed to the coupling of electrons with a bosonic mode. Very remarkably, we found that the energy levels of those features scale with T$_{c}$ for the three phases. The results suggest against a phonon origin for the bosonic mode. We also investigated the spectral evolution with doping for Tl-2201 crystals in the overdoped side. We found that the mode coupling effect weakens with doping and disappears in the heavily overdoped sample. Meanwhile, the optical scattering rate evolves from a linear- $\omega$ dependence to a shape with upward curvature in the normal state. Both the temperature and frequency dependence of the scattering rate can be described by a power law relation. Compared with ARPES results, we suggest that the overall decrease of the scattering rate may mainly originate from the increase of the quasiparticle life time near the ($\pi$,0) region in the Fermi surface.