Dependence of the superconducting critical temperature on the number of layers in a homologous series of high- T c cuprates

We study a model of $n$-layer high-temperature cuprates of homologous series like $\mathrm{Hg}{\mathrm{Ba}}_{2}{\mathrm{Ca}}_{n\ensuremath{-}1}{\mathrm{Cu}}_{n}{\mathrm{O}}_{2+2n+\ensuremath{\delta}}$ to explain the dependence of the critical temperature ${T}_{c}(n)$ on the number $n$ of $\mathrm{Cu}\ensuremath{-}\mathrm{O}$ planes in the elementary cell. Focusing on the description of the high-temperature superconducting system in terms of the collective phase variables, we have considered a semimicroscopic anisotropic three-dimensional vector $XY$ model of stacked copper-oxide layers with adjustable parameters representing microscopic in-plane and out-of-plane phase stiffnesses. The model captures the layered composition and block structure along the $c$ axis of homologous series. Implementing the spherical closure relation for vector variables we have approximately solved the phase $XY$ model with the help of the transfer matrix method and calculated ${T}_{c}(n)$ for arbitrary block size $n$, elucidating the role of the $c$-axis anisotropy and its influence on the critical temperature. Furthermore, we accommodate inhomogeneous charge distribution among planes characterized by the charge imbalance coefficient $R$ being the function of number of layers $n$. By making a physically justified assumption regarding the doping dependence of the microscopic phase stiffnesses, we have calculated the values of the parameter $R$ as a function of block size $n$ in reasonable agreement with the nuclear magnetic resonance data of the carrier distribution in multilayered high-${T}_{c}$ cuprates.