  Employing $\textit{ab-initio}$ density functional theory (DFT), we performed
a systematic investigation of the electronic structures and the magnetic
properties of atomic bilayers composed of a 4$d$ transition-metal layer (Rh, Pd
and Ru) and a Co layer on the Re(0001) surface. Our DFT calculations reveal the
influence of the bilayer composition and their stacking sequence on the
magnetic ground state and magnetic interactions. We obtain the magnetic
interactions by mapping the total energies onto an effective spin Hamiltonian
which contains the exchange interaction and the Dzyaloshinskii-Moriya
interaction (DMI) as well as the magnetocrystalline anisotropy energy (MAE). We
observe noticeable changes in bilayer hybridization due to variation in bilayer
composition and overlayer symmetry, leading to significant variation in
magnetic interactions. In all considered systems, the nearest-neighbor exchange
interaction is ferromagnetic, however, the value varies by up to a factor of 5.
The effective, nearest-neighbor DMI constant exhibits variation in sign over
the films considered, favoring either right- (clockwise) or left-handed
(anticlockwise) cycloidal spin spirals. The value of the DMI changes by up to a
factor of 27 among the films. For most of the systems, the MAE favors an
out-of-plane easy magnetization axis, however, for hcp-Co/Rh and hcp-Co/Ru
bilayers on Re(0001), it prefers an in-plane magnetization axis. The magnitude
of the MAE varies from a small value of about 0.1 meV/Co atom up to about 2
meV/Co atom for Co/Pd bilayers. The spin spiral energy dispersion curve rises
quite quickly close to the ferromagnetic state for films in which the Co layer
is adjacent to the vacuum indicating a large effective exchange constant which
stabilizes a ferromagnetic ground state in Co/$4d$ bilayers on Re(0001).
