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

5.2 Results of Fe monolayer on different hexagonal substrates from non-collinear cal. 83
E = −M 2 (+2J1 − 2J2 − 2J3 − 4J4)+2B1M 4 − 12K1M 4
if we use QMΓ/2 = 1
2 in eq. (A-21) for the single-QMΓ/2 .
If we use the double-Q biquadratic and 4-spin terms derived in Ref. [38]
Ebiquardr,2Q = −2B1M 4
and E4−spin,2Q = −4K1M 4
(5.10)
(5.11)
and then plug them in the uudd–model Hamiltonian up to the fourth nearest neighbor, we
get
for the uudd(3ΓK/4)-state and
E = −M 2 (−2J1 +2J2 − 2J3 − 4J4) − 2B1M 4 − 4K1M 4
E = −M 2 (+2J1 − 2J2 − 2J3 − 4J4) − 2B1M 4 − 4K1M 4
(5.12)
(5.13)
for the uudd(MΓ/2)-state.
A a result (5.12)-(5.9)=(5.13)-(5.10), which means that we will have the same energy
difference between the single- and double-Q-state in both Γ-K-M and Γ-M directions, i. e
Euudd − E 3ΓK/4 = Euudd − E MΓ/2 =4{2K1 − B1} (5.14)
for the difference between the uudd � 3ΓK/4 � (uudd � MΓ/2 � ) and the single-Q3ΓK/4 (QMΓ/2 )
along Γ-K-M (M-Γ) direction. By that, we are able to calculate B1 and K1 from equations
(5.14) and (5.8).
Using equations (A-20) and (A-21), different cuts through the zero temperature magnetic
phase diagrams can be calculated analytically. The magnetic phase diagram, shown
in figure 5.6, are calculated only for the first three exchange interaction terms J1, J2 and
J3. I. e., the exchange interactions taken into account are up to the third nearest neighbor.
From these phase diagrams, it can be seen that not only collinear but mainly non-collinear
magnetic structures exist on 2D hexagonal lattices.
If we calculate the phase diagram in arbitrary units of J1 (5.6(a)), the Néel ground
state is expected for any positive value of J2 if J1 is negative. A spin spiral structure
is predicted for large positive J1 if J2 is smaller than −J1. On the other hand, if the
third nearest neighbor interaction term J3 is included, the possibility of predicting more
complex structures increases (c. f. 5.6(b) and (c)). For example, the possibility to have
non-collinear magnetic structures is very large if both J2 and J3 values are negative and
larger than ≈ |J1|
2 .
In practice, these phase diagrams can be used as a starting point to predict the magnetic
order on triangular lattices by tuning the substrate or the overlayer. A very simple, and
computationally less expensive approach is to employ the virtual crystal approximation
(VCA) [148], in which one studies a crystal with the primitive periodicity, but composed