  The longitudinal spin-Seebeck effect (SSE) in magnetic
insulator$|$non-magnetic metal heterostructures has been theoretically studied
primarily with the assumption of an isotropic interfacial exchange coupling.
Here, we present a general theory of the SSE in the case of an antisymmetric
Dzyaloshinskii-Moriya interaction (DMI) at the interface, in addition to the
usual Heisenberg form. We numerically evaluate the dependence of the spin
current on the temperature and bulk DMI using a pyrochlore iridate as a model
insulator with all-in all-out (AIAO) ground state configuration. We also
compare the results of different crystalline surfaces arising from different
crystalline orientations and conclude that the relative angles between the
interfacial moments and Dzyaloshinskii-Moriya vectors play a significant role
in the spin transfer. Our work extends the theory of the SSE by including the
anisotropic nature of the interfacial Dzyaloshinskii-Moriya exchange
interaction in magnetic insulator$|$non-magnetic metal heterostructures and can
suggest possible materials to optimize the interfacial spin transfer in
spintronic devices.
