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

3.4 Magnetic Anisotropy 49
This means that as long as R1 an out-of-plane magnetization easy axis is possible.
3.4.1 Magnetic anisotropy and critical temperature:
In ferromagnetic two dimensional systems, we can use the magnetic anisotropy energy
(EMAE) to estimate the two dimensional Néel critical temperature, T (2D)
c from . The simplest
statistical-mechanics method to estimate the magnetic transition temperature is the
mean-field approximation[111, 112, 113, 114]. This method is, however, unsatisfactory
in the case of 2D magnets since it neglects the long-wavelength fluctuations and, therefore,
gives a nonzero value of the magnetic transition temperature for isotropic Heisenberg
Hamiltonian contradicting to the Mermin-Wagner theorem[115]. Much better suited for
the study of the 2D systems is the random phase approximation (RPA)[116, 114]. According
to a renormalization group analysis[117], the two dimensional T (2D)
c scales with
the three dimensional critical temperature T (3D)
c obtained from the 3D-Heisenberg model,
which is normalized by logarithmic factor:
T (2D)
c =
(3D)
2T c
� � (3.59)
π2J1 ln
E⊥ MAE
This equation contains the strength of the uniaxial anisotropy energy E⊥ MAE . J1 is the next
nearest neighbor exchange interaction parameter in Heisenberg picture.