$c$-axis lattice dynamics in Bi-based cuprate superconductors

We present the results of a systematic study of the c-axis lattice dynamics in single-layer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_6$ (Bi2201), bilayer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ (Bi2212), and trilayer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10}$ (Bi2223) cuprate superconductors. Our study is based on both experimental data obtained by spectral ellipsometry on single crystals and theoretical calculations. The calculations are carried out within the framework of a classical shell model, which includes long-range Coulomb interactions and short-range interactions of the Buckingham form in a system of polarizable ions. Using the same set of shell model parameters for Bi2201, Bi2212, and Bi2223, we calculate the frequencies of the Brillouin-zone center phonon modes of $A_{2u}$ symmetry and suggest the phonon mode eigenvector patterns. We achieve good agreement between the calculated $A_{2u}$ eigenfrequencies and the experimental values of the c-axis TO phonon frequencies which allows us to make a reliable phonon mode assignment for all three Bi-based cuprate superconductors. We also present the results of our shell model calculations for the $\Gamma$-point $A_{1g}$ symmetry modes in Bi2201, Bi2212, and Bi2223 and suggest an assignment that is based on published experimental Raman spectra. The superconductivity-induced phonon anomalies recently observed in the c-axis infrared and resonant Raman scattering spectra in trilayer Bi2223 are consistently explained with the suggested assignment.