14
A. M. Korostil and M. M. Krupa
−
1
π
ImTr
E F
−∞
d E
1
(E − E n (k)) 2 =
∂ f n (k)
∂ E n (k)
,
(17)
which is the partial derivative of (16) with respect to E n (k).
Taking into account (16) and (17), the intra- and interband magnetization
susceptibilities can be represented as
χ
αβ
intra =
d k
(2π) 2
ˆ
b α (k) ˆ
b β (k)
n=±
δ(E F − E n (k))
(18)
and
χ
αβ
inter =
d k
(2π) 2
ˆ
b α (k) ˆ
b β (k) − δ αβ
f − (k) − f + (k)
E − (k) − E + (k)
,
(19)
respectively. The intraband term collects the contributions from the Fermi surface,
while the interband term collects those from the Fermi sea.
Band dispersions given by (5a) and respective densities of states essentially
depend on the parameters t
and J of the Rashba spin–orbit coupling strength and
the exchange coupling to the ferromagnetic order parameter, respectively, as it is
represented in Fig. 1 [1].
None of the interfacial Rashbs SOC (t
= 0) and finite exchange coupling (J )
to the background magnetization leads to a constant vertical splitting of the bands
(Fig. 1a). Finite Rashba SOC (t
= 0) and no background magnetization (J = 0)
lead to k-dependent horizontal splitting of the bands (Fig. 1b). The finite Rashba
SOC and background magnetization lead to dependence of the dispersion on the
orientation of the magnetization with respect to the lattice. When the magnetization
is normal to the plane (B||z), the system has fourfold rotational symmetry. When
the magnetization is along a nearest-neighbor direction (B||x), the bands have a
unidirectional shift in the perpendicular direction ( y).
3 Magnetic Anisotropy Energy
The MAE is due to the variation of the internal energy of the itinerant electrons as the
ferromagnetic background orientation rotates. The MAE vanishes if there is no spin–
orbit coupling, i.e., in the given model if there is no Rashba coupling (φ R = t
= 0).
Phenomenologically, the MAE is expanded in angular functions with respect to the
symmetry of the system (7). For the square lattice (effectively tetragonal symmetry),
U MAE (θ, ϕ) ≈ K 2 sin
2
θ + (K 4 + K
4 cos 4ϕ) sin
4
θ,
(20)
A. M. Korostil and M. M. Krupa
−
1
π
ImTr
E F
−∞
d E
1
(E − E n (k)) 2 =
∂ f n (k)
∂ E n (k)
,
(17)
which is the partial derivative of (16) with respect to E n (k).
Taking into account (16) and (17), the intra- and interband magnetization
susceptibilities can be represented as
χ
αβ
intra =
d k
(2π) 2
ˆ
b α (k) ˆ
b β (k)
n=±
δ(E F − E n (k))
(18)
and
χ
αβ
inter =
d k
(2π) 2
ˆ
b α (k) ˆ
b β (k) − δ αβ
f − (k) − f + (k)
E − (k) − E + (k)
,
(19)
respectively. The intraband term collects the contributions from the Fermi surface,
while the interband term collects those from the Fermi sea.
Band dispersions given by (5a) and respective densities of states essentially
depend on the parameters t
and J of the Rashba spin–orbit coupling strength and
the exchange coupling to the ferromagnetic order parameter, respectively, as it is
represented in Fig. 1 [1].
None of the interfacial Rashbs SOC (t
= 0) and finite exchange coupling (J )
to the background magnetization leads to a constant vertical splitting of the bands
(Fig. 1a). Finite Rashba SOC (t
= 0) and no background magnetization (J = 0)
lead to k-dependent horizontal splitting of the bands (Fig. 1b). The finite Rashba
SOC and background magnetization lead to dependence of the dispersion on the
orientation of the magnetization with respect to the lattice. When the magnetization
is normal to the plane (B||z), the system has fourfold rotational symmetry. When
the magnetization is along a nearest-neighbor direction (B||x), the bands have a
unidirectional shift in the perpendicular direction ( y).
3 Magnetic Anisotropy Energy
The MAE is due to the variation of the internal energy of the itinerant electrons as the
ferromagnetic background orientation rotates. The MAE vanishes if there is no spin–
orbit coupling, i.e., in the given model if there is no Rashba coupling (φ R = t
= 0).
Phenomenologically, the MAE is expanded in angular functions with respect to the
symmetry of the system (7). For the square lattice (effectively tetragonal symmetry),
U MAE (θ, ϕ) ≈ K 2 sin
2
θ + (K 4 + K
4 cos 4ϕ) sin
4
θ,
(20)