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

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Figure 6.8: The real (left) and reciprocal (right) two-dimensional hexagonal Bravais lattice,
containing the Wigner Seitz cell. The IBZ (marked in gray) is limited by the high symmetry
lines which connect the symmetry points Γ, K and M.
we take the q = 2π
a (0, 1/√ 3) point, which corresponds to two real space lattice sites
R1 = a(1/2, √ 3/2) and R2 = a(0, √ 3), i. e. nearest neighbor sites will be included. Using
equation (A-16), we get M1 = M and M2 = −M leading to a RW-AFM real space spin
structure. In the same procedure, any K-point will correspond to three real space spin
lattice sites. For example, the point q = 2π 2 ( a 3 , 0) corresponds to R1 = a(1, 0), R2 = a(2, 0)
and R3 = a(3, 0)), then the next nearest neighbor sites are included. Using equation
(A-16), we get M1 = M{cos( 4π 4π ), sin( 3 3 ), 0}, M2 = M{cos( 8π 8π ), sin( 3 3 ), 0} and M3 =
M{cos( 12π
12π ), sin( 3 3 ), 0}, which corresponds to the 120◦ Néel state. All high symmetry
points of the two-dimensional hexagonal lattice are presented in Cartesian coordinates in
units of 2π
a .
Table 6.3: Cartesian coordinates of the high symmetry points of the two-dimensional
hexagonal lattice in units of 2π/a.
symmetry point coordinates ( 2π
a )
Γ (0, 0)
M (0, 1/ √ 3), (0, −1/ √ 3)
(1/2, 1/(2 √ 3)), (−1/2, −1/(2 √ 3))
(1/2, −1/(2 √ 3)), (−1/2, 1/(2 √ 3))
K (2/3, 0), (1/3, 1/ √ 3), (1/3, −1/ √ 3)
(−2/3, 0), (−1/3, −1/ √ 3), (−1/3, 1/ √ 3)