  Cuprous oxide has been conceived as a potential alternative to traditional
organic hole transport layers in hybrid halide perovskite-based solar cells.
Device simulations predict record efficiencies using this semiconductor, but
experimental results do not yet show this trend. More detailed knowledge about
the Cu$_2$O/perovskite interface is mandatory to improve the photoconversion
efficiency. Using density functional theory calculations, here we study the
interfaces of CH$_3$NH$_3$PbI$_3$ with Cu$_2$O to assess their influence on
device performance. Several atomistic models of these interfaces are provided
for the first time, considering different compositions of the interface atomic
planes. The interface electronic properties are discussed on the basis of the
optimal theoretical situation, but in connection with the experimental
realizations and device simulations. It is shown that the formation of
vacancies in the Cu$_2$O terminating planes is essential to eliminate dangling
bonds and trap states. The four interface models that fulfill this condition
present a band alignment favorable for photovoltaic conversion. Energy of
adhesion, and charge transfer across the interfaces are also studied. The
termination of CH$_3$NH$_3$PbI$_3$ in PbI$_2$ atomic planes seems optimal to
maximize the photoconversion efficiency.
