Electronic correlations, layer distinction, and electron doping in the alternating single-layer trilayer La$_{3}$Ni$_{2}$O$_{7}$ polymorph

We employ a density-functional theory plus dynamical mean-field theory framework to investigate the correlated electronic structure of the alternating single-layer trilayer (1313) polymorph of La$_3$Ni$_2$O$_7$ under pressure. At ambient pressure, the single-layer is in a Mott insulating regime and the low-energy physics is dominated by the trilayer block. Under pressure, the gap in the single-layer block closes due to orbital-selective physics, enabling charge transfer into the trilayer block. This change in effective doping of the trilayer block is likely linked to the higher T$_c$ obtained in La$_3$Ni$_2$O$_7$-1313 ($\sim$ 80 K) when compared to the nominal trilayer La$_4$Ni$_3$O$_{10}$ ($\sim$ 30 K). We conclude that correlation-driven layer differentiation is crucial in the La$_3$Ni$_2$O$_7$-1313 polymorph and that its low-energy physics aligns closely with the trilayer La$_4$Ni$_3$O$_{10}$ compound (in spite of the apparent differences in nominal filling) rather than with the conventional bilayer La$_3$Ni$_2$O$_7$.