We report on results of our analysis of the $c$-axis infrared conductivity, ${\sigma }_c\left(\omega \right)$, of bilayer ${\mathrm{LnBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ (Ln=La, Nd, Y) and trilayer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }$ high-$T_{\mathrm{c}}$ superconductors. The analysis employs the multilayer model involving the conductivity of the bilayer or trilayer unit, ${\sigma }_{\mathrm{bl}}\left(\omega \right)$, and that of the spacing layers separating the latter units, ${\sigma }_{\mathrm{int}}\left(\omega \right)$. For the ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ sample with concentration of holes $p=0.09$, our fitting of the data strongly suggests that at low temperatures, the conductivity ${\sigma }_{\mathrm{bl}}\left(\omega \right)$ possesses a pronounced and narrow Drude peak. For samples with $p\ge 0.115$ however, the fitting indicates that ${\sigma }_{\mathrm{bl}}\left(\omega \right)$ is, at low temperatures, dominated by a mode at a finite energy in the range from 30 to 60 meV. The properties of this resonance are in accord with those of a collective mode that appears in the spectra of ${\sigma }_{\mathrm{bl}}\left(\omega \right)$ calculated using a microscopic gauge-invariant theory of ${\sigma }_c\left(\omega \right)$ by J. Chaloupka and coworkers [Phys. Rev. B 79, 184513 (2009)]. The frequency and spectral weight of the latter mode are determined by the magnitude of the splitting between the bonding and the antibonding band of the bilayer or trilayer unit. Our results, in conjunction with the microscopic theory, thus demonstrate that in moderately underdoped bilayer and trilayer high-$T_{\mathrm{c}}$ cuprates the bilayer (or trilayer) splitting is already developed. The observed doping dependence is consistent with results from angular resolved photoemission spectroscopy.

• Received 16 September 2018
• Revised 28 January 2019

DOI:https://doi.org/10.1103/PhysRevB.99.054513