Superconducting phase diagram of finite-layer nickelates Nd$_{n+1}$Ni$_n$O$_{2n+2}$

Following the successful prediction of the superconducting phase diagram for infinite-layer nickelates, here we calculate the superconducting $T_{\mathrm{c}}$ vs. the number of layers $n$ for finite-layer nickelates using the dynamical vertex approximation. To this end, we start with density functional theory, and include local correlations non-perturbatively by dynamical mean-field theory for $n=2$ to 7. For all $n$, the Ni $d_{x^2-y^2}$ orbital crosses the Fermi level, but for $n>4$ there are additional $(\pi, \pi)$ pockets or tubes that slightly enhance the layer-averaged hole doping of the $d_{x^2-y^2}$ orbitals beyond the leading $1/n$ contribution stemming from the valence electron count. We finally calculate $T_{\mathrm{c}}$ for the single-orbital $d_{x^2-y^2}$ Hubbard model by dynamical vertex approximation.