  In this work, we discuss the nature of Dzyaloshinskii-Moriya interaction
(DMI) in transition metal heterostructures. We first derive the expression of
DMI in the small spatial gradient limit using Keldysh formalism. This
derivation provides us with a Green's function formula that is well adapted to
tight-binding Hamiltonians. With this tool, we first uncover the role of
orbital mixing: using both a toy model and a realistic multi-orbital
Hamiltonian representing transition metal heterostructures, we show that
symmetry breaking enables the onset of interfacial orbital momentum that is at
the origin of the DMI. We then investigate the contribution of the different
layers to the DMI and reveal that it can expand over several nonmagnetic metal
layers depending on the Fermi energy, thereby revealing the complex orbital
texture of the band structure. Finally, we examine the thickness dependence of
DMI on both ferromagnetic and nonmagnetic metal thicknesses and we find that
whereas the former remains very weak, the latter can be substantial.
