  Topological superconductors are a class of unconventional superconducting
materials featuring sub-gap zero-energy Majorana bound modes that hold promise
as a building block for topological quantum computing. In this work, we study
the realization of second-order topology that defines anomalous gapless
boundary modes in a two-orbital superconductor with spin-orbital couplings. We
reveal a time-reversal symmetry-breaking second-order topological
superconducting phase with $d+id$-wave orbital-dependent paring without the
need for the external magnetic field. Remarkably, this orbital-active $d$-wave
paring gives rise to anomalous zero-energy Majorana corner modes, which is in
contrast to conventional chiral $d$-wave pairing, accommodating one-dimensional
Majorana edge modes. Our work not only reveals a unique mechanism of
time-reversal symmetry breaking second-order topological superconductors but
also bridges the gap between second-order topology and orbital-dependent
pairings.
