Noncollinear Korringa-Kohn-Rostoker Green function method: Application to $3d$ nanostructures on $\mathrm{Ni}\left(001\right)$

Magnetic nanostructures on nonmagnetic or magnetic substrates have attracted strong attention due to the development of interesting experimental methods with atomic resolution. Motivated by this progress we have extended the full-potential Korringa-Kohn-Rostoker Green-function method to treat noncollinear magnetic nanostructures on surfaces. We focus on magnetic $3d$ impurity nanoclusters, sitting as adatoms on or in the first surface layer on $\mathrm{Ni}\left(001\right)$, and investigate the size and orientation of the local moments and, moreover, the stabilization of noncollinear magnetic solutions. While clusters of Fe, Co, Ni atoms are magnetically collinear, noncollinear magnetic coupling is expected for Cr and Mn clusters on surfaces of elemental ferromagnets. The origin of frustration is the competition of the antiferromagnetic exchange coupling among the Cr or Mn atoms with the antiferromagnetic (for Cr) or ferromagnetic (for Mn) exchange coupling between the impurities and the substrate. We find that Cr and Mn first-neighboring dimers and a Mn trimer on $\mathrm{Ni}\left(001\right)$ show noncollinear behavior nearly degenerate with the most stable collinear configuration. Increasing the distance between the dimer atoms leads to a collinear behavior, similar to the one of the single impurities. Finally, we compare some of the noncollinear ab initio results to those obtained within a classical Heisenberg model, where the exchange constants are fitted to total energies of the collinear states; the agreement is surprisingly good.

Figure 1

(Color online) $3d$ adatoms and inatoms on $\mathrm{Ni}\left(001\right)$: (a) Energy difference between the AF and FM coupling, the values related to adatom and inatoms are described by, respectively, full and empty black diamonds; (b) magnetic moments of the adatoms (triangles) and inatoms (circles) within the two possible magnetic configurations FM and AF; (c) the variation of the magnetic moments of Ni first nearest neighbors of the adatoms.

Figure 2

(Color online) Alexander-Anderson model for neighboring magnetic atoms: (a) early $3d$ transition elements in interaction with Ni surface atoms; (b) late $3d$ transition elements in interaction with Ni surface atoms; (c) Cr or Mn dimer.

Figure 3

(Color online) Different geometrical configurations considered for dimers at the surface of $\mathrm{Ni}\left(001\right)$. Dimer-1 type corresponds to the case where the atoms are first neighboring atoms, dimer-2 type where the atoms are $2^{\prime }\mathrm{NN}$ and finally dimer-3 type to $4^{\prime }\mathrm{NN}$. The collinear magnetic ground states are also shown for V, Cr, Mn, and Fe dimers.

Figure 4

(Color online) Most stable configurations of Cr dimer-1 type obtained with (a) the collinear KKR method and (b) the noncollinear KKR method. The rotation angle with respect to the $z$ axis is equal to $94.2^{\circ }$. The collinear state is the ground state, with the noncollinear state being a local minimum (see text).

Figure 5

(Color online) Most stable configurations of Mn dimer-1 type obtained with the collinear KKR method (a) and noncollinear KKR method (b). The rotation angle with respect to the $z$ axis is equal to $72.6^{\circ }$. The noncollinear state is the ground state.

Figure 6

(Color online) Noncollinear state of the Mn trimer on $\mathrm{Ni}\left(001\right)$ surface. Side view (a) and front view (b) are shown. This represents a local minimum in energy, with the collinear state being the ground state (see text).