Electric Field-Controlled Magnetic Anisotropy …
15
with θ and ϕ the spherical angles describing the orientation of the ferromagnetic
background. It follows from perturbation theory arguments that K 2n ∝ t
(t
/J )
2n−1 .
Higher order anisotropy constants should decline rapidly in magnitude, as they are
proportional to higher powers of the ratio between the spin–orbit interaction strength
and the spin splitting, which is often small. The anisotropy constants can then be
determined by fitting the angular dependence of the internal energy. Keeping all
other parameters fixed, the internal energy given by (9) is an explicit function of the
angles describing the ferromagnetic orientation,U (θ, ϕ). Assuming that the model
form in (20) holds, evaluating the internal energy for three orientations is sufficient
to fix the anisotropy. The system will have PMA provided that both of the following
inequalities are satisfied:
U (π/2, 0) − U (0, 0) = K 2 + K 4 + K
4 ,
U (π/2, π/4) − U (0, 0) = K 2 + K 4 − K
4 .
> 0
(21)
Here, the magnetic anisotropy coefficients K i as functions of the macroscopic
parameters are determined by derivatives of the relation, U = U M AE , with respect to
the spherical angles, θ and ϕ. It can be shown that the anisotropy energy goes from
IMA → PMA → IMA as function of the band filling.
The first and second derivatives of the internal energy with respect to the
ferromagnetic moment orientation are described by the expressions
∂U
∂θ
= M ·
∂ B
∂θ
=
K 2 + 2
K 4 + K
4 cos 4ϕ
sin
2
θ
sin 2θ,
(22)
∂U
∂ϕ
= −M ·
∂ B
∂ϕ
= −K
4 sin 4ϕ sin
4
θ,
(23)
where M is the spin magnetic moment of the electrons defined in (8), and B is the
effective magnetic field produced by the local moments defined in (4c). From the
phenomenological expression for U M AE (θ, ϕ), it follows that the magnetic torque
[M × B] vanishes for the high-symmetry nearest- and next-nearest-neighbor directions (θ = π/2, ϕ = nπ/4) with n = {0, 1, . . . , 7}, and for magnetization normal to
the lattice plane (θ = 0, π). The second derivatives of the internal energy are particularly simple to evaluate for these high-symmetry directions, since cross derivatives
involving both polar and azimuthal angles vanish. Therefore, it is required only the
Cartesian component of the spin susceptibility tensor for the plane perpendicular
to a chosen magnetization direction (i.e., only the transverse spin susceptibility is
needed). For the high-symmetry directions, the net spin moment of the itinerant
electrons is aligned with the ferromagnetic background, M||B. For the in-plane
high-symmetry directions,
1
2
∂
2 U
∂θ 2
M||x
=
J
2
2
M
J
− χ
zz
= −K 2 + 2
K 4 + K
4
,
(24)
15
with θ and ϕ the spherical angles describing the orientation of the ferromagnetic
background. It follows from perturbation theory arguments that K 2n ∝ t
(t
/J )
2n−1 .
Higher order anisotropy constants should decline rapidly in magnitude, as they are
proportional to higher powers of the ratio between the spin–orbit interaction strength
and the spin splitting, which is often small. The anisotropy constants can then be
determined by fitting the angular dependence of the internal energy. Keeping all
other parameters fixed, the internal energy given by (9) is an explicit function of the
angles describing the ferromagnetic orientation,U (θ, ϕ). Assuming that the model
form in (20) holds, evaluating the internal energy for three orientations is sufficient
to fix the anisotropy. The system will have PMA provided that both of the following
inequalities are satisfied:
U (π/2, 0) − U (0, 0) = K 2 + K 4 + K
4 ,
U (π/2, π/4) − U (0, 0) = K 2 + K 4 − K
4 .
> 0
(21)
Here, the magnetic anisotropy coefficients K i as functions of the macroscopic
parameters are determined by derivatives of the relation, U = U M AE , with respect to
the spherical angles, θ and ϕ. It can be shown that the anisotropy energy goes from
IMA → PMA → IMA as function of the band filling.
The first and second derivatives of the internal energy with respect to the
ferromagnetic moment orientation are described by the expressions
∂U
∂θ
= M ·
∂ B
∂θ
=
K 2 + 2
K 4 + K
4 cos 4ϕ
sin
2
θ
sin 2θ,
(22)
∂U
∂ϕ
= −M ·
∂ B
∂ϕ
= −K
4 sin 4ϕ sin
4
θ,
(23)
where M is the spin magnetic moment of the electrons defined in (8), and B is the
effective magnetic field produced by the local moments defined in (4c). From the
phenomenological expression for U M AE (θ, ϕ), it follows that the magnetic torque
[M × B] vanishes for the high-symmetry nearest- and next-nearest-neighbor directions (θ = π/2, ϕ = nπ/4) with n = {0, 1, . . . , 7}, and for magnetization normal to
the lattice plane (θ = 0, π). The second derivatives of the internal energy are particularly simple to evaluate for these high-symmetry directions, since cross derivatives
involving both polar and azimuthal angles vanish. Therefore, it is required only the
Cartesian component of the spin susceptibility tensor for the plane perpendicular
to a chosen magnetization direction (i.e., only the transverse spin susceptibility is
needed). For the high-symmetry directions, the net spin moment of the itinerant
electrons is aligned with the ferromagnetic background, M||B. For the in-plane
high-symmetry directions,
1
2
∂
2 U
∂θ 2
M||x
=
J
2
2
M
J
− χ
zz
= −K 2 + 2
K 4 + K
4
,
(24)