Ab initio investigations of magnetic properties of ultrathin transition ...

86 5 Fe monolayers on hexagonal nonmagnetic substrates
structures the Mn 3d-band filling is increased. This supports our idea that the exchange
interaction parameters can be controlled by tuning the substrate, which might lead to
unexpected complex magnetic structures.
5.2.2 Results of Fe monolayer on Rh(111):
In this section, the search for the magnetic ground state of Fe on Rh and Tc will be shown
by performing non-collinear spin-spiral calculations along the high symmetry line, Γ-K-M-
Γ, in the hexagonal BZ. The non-collinear magnetic states have been studied employing an
asymmetric film consisting of six substrate layers and an Fe monolayer on one side of the
film at the distance optimized for the collinear (FM or AFM) state of lowest energy. The
spin spirals have been calculated exploiting the generalized Bloch theorem [83]. We have
used about 120 basis functions per atom for all calculations and at least 1024 k� points in
the 2D-BZ for the spin-spiral calculations, 48 k� points in one quarter of the 2D-BZ for
the uudd configurations and 32 k� points in the 2D-BZ for the 3Q-state requiring a surface
unit-cell comprising 4 atoms. All other numerical parameters were kept to be the same as
for the collinear calculations in sec. (4.3.1).
The total energy relative to the FM spin configuration of Fe monolayer on Rh(111), is
shown in fig. 5.7 as a function of the spin-spiral q-vector, using GGA and LDA exchange
correlation potentials, along the high symmetry line Γ-K-M-Γ, and for hcp or fcc Fe-Rh(I)
stacking. We see that both stacking sequences share the same global minimum energy. The
fcc Fe-Rh(I) is lower than the hcp stacking minimum energy by ∼−5 meV/Fe atom. But
the Fe prefers hcp stacking by −9 meV/Fe atom (see subsec. 5.1.2) from the independently
FM relaxed solution of each stacking, this means that the real spin spiral curve of the hcp
Fe-Rh(I) stacking is lower in energy by −9meV, i. e. the hcp is lower than the fcc stacking
global minimum energy by −4 meV/Fe atom, provided that the difference between fcc and
hcp Fe-Rh(I) optimized d12 was 0.01˚A. We checked that, and found the our analogy is true,
the hcp at the minimum q-point is always lower than the fcc stacking total energy by −4
meV/Fe atom. Because of that, we always use the Fe-Rh(I) hcp stacking results. On the
other hand, we compare GGA and LDA results, and we see that both confirm the same
result, where Fe monolayer might have a possible spin spiral ground state along Γ-K, with
−5 (−10) meV/Fe atom in case of GGA (LDA).
A simple approach to estimate the J’s, is to exclude the higher order terms in equation
A-20, and calculate J1, J2 and J3 from the high symmetry points total energies, Γ, M
and K, where they are presented in terms of J1, J2 and J3 from equation A-20 as following:
EΓ = −M2 (6J1 +6J2 +6J3)
EM = −M2 (−2J1 − 2J2 +6J3)
EK = −M2 (−3J1 +6J2− 3J3) (5.15)
As a simple approximation, if the variation of Fe magnetic moments is considerably small
at the high symmetry points, , we can assume an arbitrary value of M=1. We have three