20
A. M. Korostil and M. M. Krupa
Fig. 2 MAE as the function of N e for the strong exchange case, J >> t >> t . a The MAE
from the second derivatives of the band energy, from the connection to its phenomenological form.
b Internal energy differences decomposed using (10)
term remains finite and it favors PMA. When M||x, the volume susceptibility (28) is
the only nonzero contribution and reproduces essentially the same MAE as found for
M||z. This agreement shows that the higher order anisotropy constants (K 4 and K
4 )
are very small when compared with K 2 . The MAE from the band energy difference
between M||x and M||z is in perfect agreement with the one extracted from the
susceptibility (Fig. 2b). The PMA is realized near half-filling. Moving from electron
per site N e = 1 to N e = 0, the interband contribution is accurately proportional to
M, which decreases monotonically to zero. The intraband contribution qualitatively
follows ρ(E F ), which increases up to the Van Hove singularity and then decreases
again.
In the second case (Fig. 3), where the exchange energy is comparable to the
non-relativistic bandwidth, by setting J = t
the two bands overlap with minority
band occupation beginning for N e > 0.5, and the lower band being completely full
for N e > 1.5. This intermediate exchange coupling strength case is applicable to
many ferromagnetic metals. The MAE from the spin susceptibility is characterized
Fig. 3 MAE as the function of N e for the intermediate exchange case, J ~ t’ > > t”. a The MAE
from the second derivatives of the band energy, from the connection to its phenomenological form.
It is shown the dependence of the spin susceptibility contributions on the magnetization direction. b Internal energy differences, decomposed using (10), which agrees with the results of the
susceptibility calculations
A. M. Korostil and M. M. Krupa
Fig. 2 MAE as the function of N e for the strong exchange case, J >> t >> t . a The MAE
from the second derivatives of the band energy, from the connection to its phenomenological form.
b Internal energy differences decomposed using (10)
term remains finite and it favors PMA. When M||x, the volume susceptibility (28) is
the only nonzero contribution and reproduces essentially the same MAE as found for
M||z. This agreement shows that the higher order anisotropy constants (K 4 and K
4 )
are very small when compared with K 2 . The MAE from the band energy difference
between M||x and M||z is in perfect agreement with the one extracted from the
susceptibility (Fig. 2b). The PMA is realized near half-filling. Moving from electron
per site N e = 1 to N e = 0, the interband contribution is accurately proportional to
M, which decreases monotonically to zero. The intraband contribution qualitatively
follows ρ(E F ), which increases up to the Van Hove singularity and then decreases
again.
In the second case (Fig. 3), where the exchange energy is comparable to the
non-relativistic bandwidth, by setting J = t
the two bands overlap with minority
band occupation beginning for N e > 0.5, and the lower band being completely full
for N e > 1.5. This intermediate exchange coupling strength case is applicable to
many ferromagnetic metals. The MAE from the spin susceptibility is characterized
Fig. 3 MAE as the function of N e for the intermediate exchange case, J ~ t’ > > t”. a The MAE
from the second derivatives of the band energy, from the connection to its phenomenological form.
It is shown the dependence of the spin susceptibility contributions on the magnetization direction. b Internal energy differences, decomposed using (10), which agrees with the results of the
susceptibility calculations