Fe3O4 THICK FILMS AS DESCRIBED BY SECOND ORDER PERTURBED HEISENBERG HAMILTONIAN P. SAMARASEKARA

Fe3O4 THICK FILMS AS DESCRIBED BY SECOND ORDER PERTURBED HEISENBERG HAMILTONIAN P. SAMARASEKARA Department of Physics, University of Peradeniya, Peradeniya, Sri Lanka Received 19th November 2019 / Accepted 20th February 2020 ABSTRACT Modified second order perturbed Heisenberg Hamiltonian has been used to explain magnetic properties including magnetic easy axis orientation of Fe3O4 thick films up to 1000 spin layers. The variation of magnetic easy and hard directions with the number of spin layers and stress induced anisotropy was investigated. 3-D plot of energy versus angle and number of spin layers was plotted to find the values of number of spin layers corresponding to energy minima and maxima. By plotting the curve of energy versus angle at one selected value of number of layers, magnetic easy and hard directions were determined. Similarly 3-D plot of energy versus angle and stress induced anisotropy was plotted to find the values of stress induced anisotropy corresponding to energy minima and maxima. Then the curve of energy versus angle at one selected value of stress induced anisotropy was plotted to find the magnetic easy and hard directions. MATLAB software was employed to plot all the graphs.


INTRODUCTION
Fe3O4 (magnetite) is a prime candidate in applications of magnetic storage, industrial catalysts, water purification and drug delivery. Fe3O4 is a non-uniaxial ferrite with inverse spinel structure. Magnetic easy axis of Fe3O4 is along one of the body diagonal of the cubic cell due to the spin arrangement parallel and antiparallel to the body diagonal. Spinel structure with tetrahedral and octahedral sites can be found in detail in some previous publications (Ahmed Farag et al., 2001a, Ahmed Farag et al., 2001b, Kahlenberg et al., 2001, John Zhang et al., 1998, Sickafus et al., 1999. Five of Fe 3+ ions occupy tetrahedral sites. Other five Fe 3+ ions and four Fe 2+ ions occupy octahedral sites. The spinel structure of this ferrite is represented by Fe 3+( Fe 2+ Fe 3+) O4. The magnetic moments of Fe 2+ and Fe 3+ are 4 B and 5 B, respectively. Because magnetic moments of Fe 3+ in tetrahedral and octahedral sites cancel each other, the net magnetic moment of Fe3O4 is completely due to the magnetic moments of four Fe 2+ ions. Therefore, the theoretical net magnetic moment of Fe3O4 is 4 Bohr magnetons.
Rietveld method has been employed to determine the cation distribution of ferrite like compounds (Ahmed Farag et al., 2001a, Ahmed Farag et al., 2001b. Surface spin waves in CsCl type ferrimagnet with a (001) surface has been studied by combining Green function theory with the transfer matrix method (Dai & Li, 1990). Anisotropy of ultrathin ferromagnetic films and the spin reorientation transition have been investigated using Heisenberg Hamiltonian with few terms (Usadel & Hucht, 2002). In addition, the surface magnetism of ferrimagnet thin films has been studied using Heisenberg method (Ding et al., 1993). The surface spin wave spectra of both the simple cubic and body centered ferrimagnets have been theoretically studied using Heisenberg Hamiltonian (Hung et al., 1975). The cation distribution and oxidation state of Mn-Fe spinel nanoparticles have been systematically studied at various temperatures by using neutron diffraction and electron Fe3O4 thick films as described by second order perturbed Heisenberg Hamiltonian 3 energy loss spectroscopy (John Zang et al., 1998). The crystal structure of spinel type compounds has been found using single crystal X-ray diffraction data (Kahlenberg et al., 2001). 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 (Zheng & Lin, 1988). The lattice parameter, anion parameter and the cation inversion parameter of spinel structures have been presented (Sickafus et al., 1999).
Ferromagnetic ultra thin and thick films have been studied previously using third order perturbed Heisenberg Hamiltonian (Samarasekara, 2008, Samarasekara & Mendoza, 2010. Ferromagnetic ultra-thin and thick films have been investigated using second order perturbed Heisenberg Hamiltonian by us (Samarasekara & Gunawardhane, 2011).
Furthermore, ferrite ultra-thin and thick films have been investigated using second order perturbed Heisenberg Hamiltonian by us (Samarasekara et al., 2009, Samarasekara, 2010.
Ferrite ultra-thin and thick films have been investigated using third order perturbed Heisenberg Hamiltonian (Samarasekara & Mendoza, 2011, Samarasekara, 2011. The magnetic properties of the spinel Fe3O4 thick films are described in this manuscript.

MODEL
The modified Heisenberg Hamiltonian of any thin film with N spin layers can be given as Here J is spin exchange interaction,is the strength of long range dipole interaction,is azimuthal angle of spin The spinel cubic cell can be divided into 8 spin layers with alternative Fe 2+ and Fe 3+ spins layers (Sickafus et al., 1999). The spins in one layer and adjacent layers point in one direction and opposite directions, respectively. The spins of Fe 2+ and Fe 3+ will be taken as 1 and p, respectively. A cubic unit cell with length a will be considered. Due to the super exchange interaction between spins, the spins are parallel or antiparallel to each other within the cell. Therefore the results proven for oriented case in one of our early report (Samarasekara et al., 2009) will be used for following equations. But the angle  will vary from m to m+1 at the interface between two cells.
Following equations will be proven for a thin film with thickness Na.

Spin exchange interaction energy =
The first and second term in each above equation represent the variation of energy within the cell (Samarasekara et al., 2009) and the interface of the cell, respectively.

Total energy
Fe3O4 thick films as described by second order perturbed Heisenberg Hamiltonian The anisotropy energy term and the last term given in above equations have been explained in our previous report for oriented spinel ferrite (Samarasekara et al., 2009). The angle can be given as m = +m with perturbation m.

The total energy can be given as E() = E0+E()+E( 2 )
Here the energy is given only up to the second order perturbation of . Then after expanding sin and cosine terms, where where the elements of matrix C can be given as following, For m = 2, 3, ----, N-1 Finally the total energy can be given as Here C + is the pseudo-inverse given by .
For a thick film with N = 10000, the second term in equation number 9 can be neglected.
Then C + will be the normal inverse of matrix C. C m,m+1 is zero, then the matrix C will be diagonal. Then the elements of inverse matrix (C + ) is given by

If anisotropy constants
To avoid tedious calculations, the solution will be found under assumption Cm,m+1 = 0.  According to this graph, the magnetic easy direction depends on the number of spin layers or the thickness of the film. In addition, the variation of magnetic easy direction with the deposition temperature has been explained using Heisenberg Hamiltonian coupled with spin reorientation by us (Samarasekara & Gunawardhane, 2011, Samarasekara & Saparamadu, 2012, Samarasekara & Saparamadu, 2013. This graph is similar to the same graph plotted for thick ferromagnetic films using third order perturbed Heisenberg Hamiltonian, and the maximum value of energy is also the same in the both cases (Samarasekara, 2008). The same graph plotted for thick nickel ferrite films using second order perturbed Heisenberg Hamiltonian is similar to this graph (Samarasekara, 2010).  Heisenberg Hamiltonian by the authors (Samarasekara et al., 2009). Broad peaks could be observed for films with 2 spin layers (Samarasekara et al., 2009). However, the same graph plotted for ferromagnetic ultra thin films with three spin layers using third order perturbed Heisenberg Hamiltonian is somewhat similar to the plot given in this case (Samarasekara & Mendoza, 2010).  (Samarasekara & Mendoza, 2011). However, this graph is similar to the same kind of 3-D graph plotted for nickel ferrite thick films using third order perturbed Heisenberg Hamiltonian (Samarasekara, 2011). Figure 5 shows the graph of   (Samarasekara & Cadieu, 2001).