Second order perturbed Heisenberg Hamiltonian of Fe3O4 ultra-thin films

Due to the wide range of applications, theoretical models of Fe3O4 films are found to be important. Ultra thin Fe3O4 films with ferrite structure have been theoretically investigated using second order perturbed modified Heisenberg Hamiltonian. Matrices for ultra thin films with two and three spin layers are presented in this manuscript. Total magnetic energy was expressed in terms of spin exchange interaction, magnetic dipole interaction, second order magnetic anisotropy and stress induced magnetic anisotropy. Magnetic properties were observed for films with two spin layers and variant second order magnetic anisotropy. For the film with three spin layers, second order anisotropy constant was fixed to avoid tedious derivations. Magnetic easy axis rotates toward the in plane direction as the number of spin layers is increased from two to three because the stress induced anisotropy energy dominates at higher number of spin layers. According to some other experimental data, the magnetic easy axis of thin films rotates toward the in plane direction as the thickness is increased. For ferrite film with two spin layers, magnetic easy and hard directions can be observed at 0.75 and 1.2 radians, respectively, when the ratio of stress induced anisotropy to the long range dipole interaction strength is 3.9. For ferrite film with three spin layers, magnetic easy and hard directions can be observed at 2.4 and 2.3 radians, respectively, when the ratio of stress induced anisotropy to the long range dipole interaction strength is 4.2.


Introduction:
Magnetite (Fe 3 O 4 ) and maghemite (γFe 2 O 3 ) are the most popular natural oxides. Fe 3 O 4 finds potential applications in magnetic storage, industrial catalysts, water purification and drug delivery. Fe 3 O 4 is a ferrite with inverse spinel structure. Spinel structure with tetrahedral and octahedral sites can be found in detail in some previous publications [1,2,3,4,5]. Five of Fe 3+ ions occupy tetrahedral sites. Other five Fe 3+ ions and four Fe 2+ ions occupy octahedral sites. Cation distribution of ferrite like compounds has been found using Rietveld method [1,2].
Surface spin waves in CsCl type ferrimagnet with a (001) surface has been studied by combining Green function theory with the transfer matrix method [6]. Anisotropy of ultrathin ferromagnetic films and the spin reorientation transition have been investigated using Heisenberg Hamiltonian with few terms [7]. In addition, the surface magnetism of ferrimagnet thin films has been studied using Heisenberg method [8]. The surface spin wave spectra of both the simple cubic and body centered ferrimagnets have been theoretically studied using Heisenberg Hamiltonian [9]. The cation distribution and oxidation state of Mn-Fe spinel nanoparticles have been systematically studied at various temperatures by using neutron diffraction and electron energy loss spectroscopy [4]. The crystal structure of spinel type compounds has been found using single crystal X-ray diffraction data [3]. The lattice parameter, anion parameter and the cation inversion parameter of spinel structures have been presented [5]. Surface spin waves on the (001) free surface of semi-infinite two lattice ferrimagnets on the Heisenberg model with nearest neighbor exchange interactions has been investigated [10].
Ferromagnetic ultra-thin and thick films have been investigated using second order perturbed Heisenberg Hamiltonian by us [11]. Previously ferromagnetic ultra thin and thick films have been studied using third order perturbed Heisenberg Hamiltonian [12,13]. Furthermore, ferrite ultra-thin and thick films have been investigated using second order perturbed Heisenberg Hamiltonian by us [14,15]. Ferrite ultra-thin and thick films have been investigated using third order perturbed Heisenberg Hamiltonian [16,17]. In this manuscript, the spinel structure of Fe 3 O 4 was used to find the magnetic properties of Fe 3 O 4 ultra thin films.

Model:
Classical Heisenberg Hamiltonian of a thin film can be written as following. Here J, ω, θ,

, ,
The cubic cell was divided into 8 spin layers with alternative Fe 2+ and Fe 3+ spins layers (Sickafus et al. 1999). The spins of Fe 2+ and Fe 3+ will be taken as 1 and p, respectively. While the spins in one layer point in one direction, spins in adjacent layers point in opposite directions.
A thin film with (001) spinel cubic cell orientation will be considered. The length of one side of unit cell will be taken as "a". Within the cell the spins orient in one direction due to the super exchange interaction between spins (or magnetic moments). Therefore, the results proven for oriented case in one of our early report [15] will be used for following equations. But the angle θ will vary from θ m to θ m+1 at the interface between two cells.
Here the anisotropy energy term and the last term have been explained in our previous report for oriented spinel ferrite [15]. If the angle is given by θ m =θ+ε m with perturbation ε m , after taking the terms up to second order perturbation of ε only, The total energy can be given as The sin and cosine terms in equation number 2 have been expanded to obtain above equations. Therefore, E(ε)= ε α r r .
, and matrix C is assumed to be symmetric (C mn =C nm ).
Here the elements of matrix C can be given as following, Therefore, the total energy can be given as Here C + is the pseudo-inverse given by .

(9)
Here E is the matrix with all elements E mn =1.

Results and discussion:
A film with two spin layers (N=2) will be considered first. If the anisotropy constants vary within the film, then C 12 =C 21 and  to thick ferromagnetic films [12]. However, the spikes observed in energy versus angle graph of thin ferromagnetic films with two layers don't appear in this graph [13]. For N=3, the each C + nm element found using equation 9 is consist of more than 20 terms. To avoid this problem, matrix elements were found using C.C + =1. Then C + mn is given by , and the average value of first order perturbation is zero. The second order anisotropy constant is assumed to be an invariant for the convenience.
Then C 11 =C 33 , C 12 =C 21 =C 23 =C 32 , C 13 =C 31 =0, α 1 =α 2 =α 3 . third order perturbation is comparable to the total energy in this case [16]. The total energy of thick ferrite films obtained using third order perturbed Heisenberg Hamiltonian is higher than that of this ultra thin ferrite film [17]. Figure 4 shows the graph of of energy versus angle graph of ferrite thin films with three layers obtained using third order perturbation is different from the same graph in this case [16]. Some spikes observed in energy versus angle graph of ferromagnetic films with five layers don't appear in this graph. According to our previous experimental data of ferrite thin films, the coercivity and the magnetic anisotropy depend on the stress of the film [18]. According to figures 2 and 4, the first magnetic easy direction can be observed at 0.25 and 0.75 radians for two and three spin layers, respectively. Therefore, the magnetic easy axis rotates toward the in plane direction as the number of layers is increased. According to the equation of total energy, only the E 0 term depends on the number of layers (N). Only term with N is the stress induced anisotropy. Because 1-p is negative for most of the spinel ferrites, the energy of stress induced anisotropy increases with the number of layers. As the thickness is increased, stress induced anisotropy dominates spin exchange interaction, second order magnetic anisotropy and magnetic dipole interaction. The domination of stress induced anisotropy is the possible reason for the rotation of easy direction with the number of layers. The same phenomenon was observed for thick ferromagnetic films using second order and third order perturbed Heisenberg Hamiltonian by us. According to some experimental data, the easy axis of magnetic thin films rotates toward the in plane direction, as the number if layers is increased [19,20]. In addition, the magnetic easy direction of thin films depends on the deposition temperature. The variation of easy axis with deposition temperature can be explained using spin reorientation coupled with Heisenberg Hamiltonian [11,21,22].

Conclusion:
Total magnetic energy, magnetic easy direction and magnetic hard direction were determined for films with two and three spin layers by plotting 3-D graphs of energy versus stress induced anisotropy and angle, and graphs of energy versus angle. Minimum of energy of 3-D plot can be observed at several values of stress induced anisotropy. For two spin layers, consecutive minimum and maximum can be observed at 0.75 and 1.2 radians, respectively for ω s K =3.9. For three spin layers, consecutive minimum and maximum can be observed at 2.4 and 2.3 radians, respectively for ω s K =4.2. According to the first minima of 2-D plots of two and three layers, the magnetic easy axis gradually rotates toward the in plane direction of the film as the number of spin layers is increased. The possible reason for the rotation of magnetic easy axis is the domination of stress induced magnetic anisotropy. This theoretical data qualitatively agree with some experimental data of magnetic thin films.