  The possibility of inducing topological superconductivity with cuprate
high-temperature superconductors (HTSC) is studied for various
heterostructures. We first consider a ballistic planar junction between a HTSC
and a metallic ferromagnet. We assume that inversion symmetry breaking at the
tunnel barrier gives rise to Rashba spin-orbit coupling in the barrier and
allows equal-spin triplet superconductivity to exist in the ferromagnet.
Bogoliubov-de Gennes equations are obtained by explicitly modeling the barrier,
and taking account of the transport anisotropy in the HTSC. By making use of
the self-consistent boundary conditions and solutions for the barrier and HTSC
regions, an effective equation of motion for the ferromagnet is obtained where
Andreev scattering at the barrier is incorporated as a boundary condition for
the ferromagnetic region. For a ferromagnet layer deposited on a (100) facet of
the HTSC, triplet p-wave superconductivity is induced. For the layer deposited
on a (110) facet, the induced gap does not have the p-wave orbital character,
but has an even orbital symmetry and an odd dependence on energy. For the layer
on the (001) facet, an exotic f-wave superconductivity is induced. We also
consider the induced triplet gap in a one-dimensional half-metallic nanowire
deposited on a (001) facet of a HTSC. We find that for a wire axis along the
a-axis, a robust triplet p-wave gap is induced. For a wire oriented 45 degrees
away from the a-axis the induced triplet p-wave gap vanishes. For the
appropriately oriented wire, the induced p-wave gap should give rise to
Majorana fermions at the ends of the half-metallic wire. Based on our result,
topological superconductivity in a semi-conductor nanowire may also be possible
given that it is oriented along the a-axis of the HTSC.
