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

100 5 Fe monolayers on hexagonal nonmagnetic substrates
for the double-Q state.
Here IFe (Isub) is the Stoner parameter of Fe (substrate), M Fe 1Q (M sub 1Q) is the mag-
netic (induced) moment of Fe (substrate) calculated from single-Q state, δM Fe 1Q −Fe 2Q
1
(δMsub1Q 2Q
−sub ) is the magnetic local (induced) moments difference between Fe (substrate)
1
from single-Q and the first Fe (substrate) local (induced) moments from double-Q state,
and δM 2Q
Fe1 −Fe2Q (δM 2Q
2 sub1 −sub2Q)
is the difference between the first Fe (substrate) and the
2
second Fe (substrate) local (induced) moments.
Using equation (5.19), the difference between the single- and double-Q model Hamiltonian
(eq. 5.14) can be expressed as:
Euudd − E 3K/4 = Euudd − E M/2 =4{2K1 − B1}
+ IFe(δ 2 M Fe 1Q −Fe uudd
1
+ Isub(δ 2 M sub 1Q −sub uudd
1
+ δ2 M Fe uudd
1
−Feuudd 2
+ δ2 M sub uudd
1
)
−subuudd 2
). (5.19)
From this equation we see, that if the substrate unpolarized, like in uudd � 3ΓK/4 � case,
the substrate terms cancel, and almost no change or effect is expected on the Heisenberg
model Hamiltonian, by adding the substrate if the Fe moment remains constant. This can
bee seen in our results as mentioned above (Fig. 5.16), where adding the substrate to Fe
UML did not change the trend of the total energy differences. Where as, if Fe moments
change due substrate polarization, as in case of uudd � MΓ/2 � configuration, the model
Hamiltonian is modified and adding the substrate to Fe UML will affect the total energy
differences between signaler- and double-Q. If we use Fe and Rh(I) uudd � MΓ/2 � moments
(tab. 5.6) in equation (5.19), we find out that the maximum change in the total energy
differences, between single- and double-Q, will occur if we add Rh substrate to the Fe
UML, this is also consistent with our results (Fig. 5.16).
In conclusion of this chapter, we have studied the magnetism of Fe monolayer on 4d-
TM hexagonal substrates. Firstly, we performed structural relaxations for the Fe and
4d(I) layer to optimize Fe-Rh(I) interlayer distance. We performed total energy collinear
calculations to calculated Fe ground state on Rh, Pd, Tc substrates, and compared our
results with previous collinear calculations of Fe ground state on Ru, Re, Ir, Pt, Os, to have
a complete picture of the effect of the substrate on the Fe ground state. We found that
4d-TMs substrates produce the similar collinear results like 5d-TMs. Using total energy
differences, We found that Fe collinear ground state is FM on fcc hexagonal substrates
(Rh, Pd, Ir, Pt) and RW-AFM on hcp substrates (Tc, Ru, Re, Os). We found a small
total energy difference between FM and RW-AFM of Fe monolayer on Ru and Rh (Os and
Ir), which indicates that on these substrates Fe is very close to phase transition. From
experiments and non-collinear theoretical DFT calculations, Fe monolayer was found to
have a Néel ground state on Ru(0001) and a complex magnetic ground state on Ir(111)
surfaces. This encouraged us to see what happens to Fe magnetism on Rh(111) substrate
by performing non-collinear total energy calculations of flat spin spirals, which are the
general solution to the classical Heisenberg model, along the high symmetry line Γ-K-M-Γ