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Recent experimental progress has brought about better, defect-free infinite-layer nickelates [1] and the first finite-layer nickelate superconductor [2]. The measured superconducting dome is in exceedingly good agreement with our theoretical prediction [3] based on single Ni-$d_{x^2-y^2}$-band Hubbard model plus a decoupled electron reservoir for the pockets around the $A$ momentum and at low doping $\Gamma$ momentum [4]. These pockets are essential for the obtaining the correct filling of the Ni-$d_{x^2-y^2}$-band for the infinite layer compound. In contrast, there are no such pockets for the pentalayer nickelate superconductor if electronic correlations included [5]; its $T_c$ not only well agrees with theory but also with that of the new infinite-layer films at the same $d_{x^2-y^2}$ doping. The absence and presence of pockets also explains the very different Hall coefficient in the infinite and finite-layer nickelates, respectively.
Nickelates are not optimal for superconductivity. In particular they have a too large Coulomb interaction, i.e., are too strongly correlated. For this reason, pressure, strain or replacing 3d nickelates by 4d palladates are viable alternatives for obtaining higher $T_c$'s [3]. This prediction has been spectacularly confirmed in a first experiment [6], showing an increase to a record high $T_c>30 K$ at 12 GPa in infinite-layer nickelates, and no indication yet of a saturation.
[1] K. Lee et al., arXiv:2203.02580.
[2] G. A. Pan et al., Nature Phys. 21, 160 (2022).
[3] M. Kitatani et al., npj Quantum Materials 5, 59 (2020).
[4] For a review of this perspective cf. K. Held et al., Frontiers in Physics 9, 810394 (2022).
[5] P. Worm et al., arXiv:2111.12697.
[6] N. N. Wang et al., arXiv:2109.12811.
