Comparison of first-principles methods to extract magnetic parameters in ultrathin films: Co/Pt(111)

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 noncollinear 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.

Figure 1

Side and top views onto two clockwise-rotating noncollinear spin configurations in a four-atom supercell (basis vectors are shown in red), representing (a) a flat spin spiral where the magnetic moments are fully contained in the $xz$ plane and (b) a coned spin spiral with small cone angle, where only a small component of the magnetic moments is contained in the $xz$ plane. In the latter case, the magnetic moments are almost pointing along the $y$ direction. The direction of the spin-spiral vector $\mathit{q}$ is indicated by a blue arrow.

Figure 2

Energy dispersion of spin spirals for a monolayer Co(fcc) on Pt(111). (a) Nonrelativistic dispersion curves $E_{\mathrm{SS}}\left(\mathit{q}\right)$ along the high-symmetry path of the first Brillouin zone. (b) Zoom into the parabolic region of (a) around the $\overline{\mathrm{\Gamma }}$ point to obtain the spin stiffness. (c) Spin-orbit induced antisymmetric corrections $E_{\mathrm{DM}}\left(\mathit{q}\right)$ to the energies along the high-symmetry path and (d) zoom onto the $\overline{\mathrm{\Gamma }M}$ direction. Full lines represent KKR-derived spin-spiral energies including all relevant interactions constants $J_{ij}$ (full calc.), exclude interactions between Co and Pt (excl. Pt) or consider only the nearest-neighbor Co interactions (only Co n.n.). Dashed lines indicate the slopes in the limit $\mathit{q}\to 0$ and represent the (b) spin stiffness and [(c) and (d)] DMI spiralization. gBT = generalized Bloch theorem, 1 shot SOC = SOC included in last iteration only. For better comparison, the energies of coned spin spirals have been scaled by $1/{sin}^2\theta$, where $\theta$ is the cone angle.

Figure 3

Magnetic moments (a) for Co atoms and (b) for induced Pt moments in the first substrate layer. The values from the ferromagnetic state of KKR calculations are shown as horizontal full lines. Induced Pt moments as modeled by Eq. (C1) (see Appendix pp3) are shown by the dashed line.

Figure 4

Calculation of the critical temperature of Co/Pt(111) for the different parametrization of the extended Heisenberg Hamiltonian from Monte Carlo simulations. (a) Averaged total energy, (b) magnetization, (c) specific heat, and (d) magnetization susceptibility as a function of temperature.

Figure 5

Power spectrum of the magnetization profile for a Néel-type domain wall. Vertical dotted lines indicate the interval that contains 90% of the components ${\tilde{m}}_x$.

Figure 6

Energy dispersions for spin spirals as evaluated with parameters as obtained by the KKR method for (a) the exchange interaction and (b) Dzyaloshinskii-Moriya interaction. The contributions from interactions between Co and induced Pt moments are included assuming either rigid Pt moments or a variation according to Eq. (C1).