  We compare three distinct computational approaches based on first-principles
calculations within density functional theory to explore the magnetic exchange
and the Dzyaloshinskii-Moriya interactions (DMI) of a Co monolayer on Pt(111),
namely (i) the method of infinitesimal rotations of magnetic moments based on
the Korringa-Kohn-Rostoker (KKR) Green function method, (ii) the generalized
Bloch theorem applied to spiraling magnetic structures and (iii) supercell
calculations with non-collinear magnetic moments, the latter two being based on
the full-potential linearized augmented plane wave (FLAPW) method. In
particular, we show that the magnetic interaction parameters entering
micromagnetic models describing the long-wavelength deviations from the
ferromagnetic state might be different from those calculated for fast rotating
magnetic structures, as they are obtained by using (necessarily rather small)
supercell or large spin-spiral wave-vectors. In the micromagnetic limit, which
we motivate to use by an analysis of the Fourier components of the domain-wall
profile, we obtain consistent results for the spin stiffness and DMI
spiralization using methods (i) and (ii). The calculated spin stiffness and
Curie temperature determined by subsequent Monte Carlo simulations are
considerably higher than estimated from the bulk properties of Co, a
consequence of a significantly increased nearest-neighbor exchange interaction
in the Co-monolayer (+50%). The calculated results are carefully compared with
the literature.
