  In this work, the high entropy oxide A2CuO4 Ruddlesden-Popper
(La0.2Pr0.2Nd0.2Sm0.2Eu0.2)2CuO4 is explored by charge doping with Ce+4 and
Sr+2 at concentrations known to induce superconductivity in the simple parent
compounds, Nd2CuO4 and La2CuO4. Electron doped
(La0.185Pr0.185Nd0.185Sm0.185Eu0.185Ce0.075)2CuO4 and hole doped
(La0.18Pr0.18Nd0.18Sm0.18Eu0.18Sr0.1)2CuO4 are synthesized and shown to be
single crystal, epitaxially strained, and highly uniform. Transport
measurements demonstrate that all as-grown films are insulating regardless of
doping. Annealing studies show that resistivity can be tuned by modifying
oxygen stoichiometry and inducing metallicity but without superconductivity.
These results in turn are connected to extended x-ray absorption fine structure
(EXAFS) results indicating that the lack of superconductivity in the high
entropy cuprates likely originates from a large distortion within the Cu-O
plane ({\sigma}2>0.015 {\AA}2) due to A-site cation size variance, which drives
localization of charge carriers. These findings describe new opportunities for
controlling charge- and orbital-mediated functional responses in
Ruddlesden-Popper crystal structures, driven by balancing of cation size and
charge variances that may be exploited for functionally important behaviors
such as superconductivity, antiferromagnetism, and metal-insulator transitions,
while opening less understood phase spaces hosting doped Mott insulators,
strange metals, quantum criticality, pseudogaps, and ordered charge density
waves.
