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.
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