Third order perturbed energy of fcc ferromagnetic thin films as described by Heisenberg Hamiltonian

Abstract: Magnetic energy of fcc structured ferromagnetic thin films was investigated using third order perturbed Heisenberg Hamiltonian with seven magnetic energy parameters by a MATLAB computer program for the first time. Spin exchange interaction, magnetic dipole interaction, second order anisotropy, fourth order anisotropy, applied field, demagnetization factor and stress induced anisotropy were taken into account. Ultra-thin films with three layers (N=3) were considered. Several magnetic easy and hard directions were found in all the 3-D plots. According to some graphs, the angle between easy and hard directions was 90 degrees. Previously, the third order perturbed Heisenberg Hamiltonian with spin exchange interaction, second order magnetic anisotropy and stress induced anisotropy terms only was solved manually for ferromagnetic films with two and three layers by us. Due to the unavailability of the experimental values of these seven magnetic parameters, some reasonable values were used for this simulation.


INTRODUCTION
Previously the Heisenberg Hamiltonian up to 2 nd order perturbation with few magnetic energy parameters has been solved by some other researchers.Ferromagnetic thin films have been studied using the Heisenberg Hamiltonian with spin exchange interaction, magnetic dipole interaction, applied magnetic field, second and fourth order magnetic anisotropy terms (Hucht and Usadel, 1997;Hucht and Usadel, 1999;Usadel and Hucht, 2002).Domain structure and Magnetization reversal in thin magnetic films have been theoretically investigated (Nowak, 1995).In-plane dipole coupling anisotropy of a square ferromagnetic Heisenberg monolayer has been explained using Heisenberg Hamiltonian (Dantziger et al., 2002).Effect of the interracial coupling on the magnetic ordering in ferroantiferromagntic bilayers has been studied using the Heisenberg Hamiltonian (Tsai et al., 2003).
Due to the applications of magnetic thin films in magnetic and microwave devices, magnetic thin films are synthesized using many different techniques.Previously strontium ferrite (Hegde et al., 1994) and nickel ferrite (Samarasekara, 2003) films were synthesized using sputtering by us.In addition, lithium mixed ferrite films were fabricated using pulsed laser deposition (Samarasekara, 2002).For all these films, the coercivity of film increased due to the stress induced anisotropy.The change of coercivity due to the stress induced anisotropy was qualitatively calculated for all these films.The calculated values of the change of coercivity agreed with the experimentally found values.So the stress induced anisotropy plays a major role in magnetic thin fabrications.Previously the Heisenberg Hamiltonian was employed to investigate the second order perturbed energy of ultrathin ferromagnetic films (Samarasekara, 2006a), unperturbed energy of thick ferromagnetic films (Samarasekara, 2006b), unperturbed energy of spinel ferrite films (Samarasekara, 2007), second order perturbed energy of thick ferromagnetic films (Samarasekara and De Silva, 2007), third order perturbed energy of thick spinel ferrite (Samarasekara, 2011), third order perturbed energy of thin spinel ferrite (Samarasekara and Mendoza, 2011), second order perturbed energy of thick spinel ferrite films (Samarasekara, 2010) and spin reorientation of barium ferrite (Samarasekara and Saparamadu, 2013).The magnetic dipole interaction and demagnetization factor are microscopic and macroscopic effects, respectively.Therefore, both these terms were taken into consideration in our model.
Other theoretical work of magnetic thin films can be summarized as following.EuTe films with surface elastic stresses have been theoretically studied using Heisenberg Hamiltonian (Radomska and Balcerzak, 2003).Magnetostriction of dc magnetron sputtered FeTaN thin films has been theoretically studied using the theory of De Vries (Cates and Alexander, 1994).Magnetic layers of Ni on Cu have been theoretically investigated using the Korringa-Kohn-Rostoker Green's function method (Ernst et al., 2000).Electric and magnetic properties of multiferroic thin films have been theoretically explained by modified Heisenberg and transverse Ising model using Green's function technique (Kovachev and Wesselinowa, 2009).The quasistatic magnetic hysteresis of ferromagnetic thin films grown on a vicinal substrate has been theoretically investigated by Monte Carlo simulations within a 2D model (Zhao et al., 2002).Structural and magnetic properties of two dimensional FeCo orders alloys deposited on W(110) substrates have been studied using first principles band structure theory (Spisak and Hafner, 2005).

MODEL
The Heisenberg Hamiltonian with all seven magnetic parameters can be expressed as below.

H=
Here J is spin exchange interaction,is the strength of long range dipole interaction m is azimuthal angle of spin, is the applied magnetic field, s K is stress induced anisotropy constant, and n and m are spin plane indices.When the stress applies normal to the film plane, the angle between m th spin and the stress is  m .In this 2-D model, only the x and y components of the spin are considered.N d is the demagnetization factor changing from 0 (in film plane) to 1 (perpendicular to film plane) in SI units.
The total energy per unit spin can be deduced to the following equation.
, N, H in and H out being indices of layers, number of nearest spin neighbors, constant arising from summation of dipole interactions, azimuthal angles of spins, total number of layers, in plane applied field and out of plane applied field, respectively.
With some perturbation, above angles θ m and θ n measured with film normal can be expressed in forms of , and above energy can be expanded up to the third order of as following.Here  m (or  n ) is a small perturbation of the angle. Here Here the elements of matrix C can be given as following, In addition, third order can be expressed as the Here elements of matrix β can be given as Also  nm = mn , implying that matrix is symmetric.
After substituting equations ( 8) and ( 6) in equation ( 3), total energy can be expressed as At the energetically favorable state, the derivative of above E( with respect to  will be zero.Using that condition,  can be found.After substituting that in above equation of E(), following equation can be derived.
Following equation has been used to calculate the elements of matrix C + .
Here E is the matrix with all elements given by E mn =1, and C + is a pseudo inverse.

Because the experimental values of
, H in , H out , N d and K s are not available, these simulations were performed for some reasonable values of the ratios between those parameters., easy and hard directions appear at 4 and 7 radians, respectively.Stress is induced in the thin films during annealing or subsequent cooling process due to the difference between thermal expansion coefficients of the film and the substrate (Samarasekara 2002).4, easy and hard directions can be seen at 1.43 and 3 radians, respectively.The angle between easy and hard directions is 90 degrees according to Figures 2 and 4. In some graphs, two hard or easy directions corresponding to two different energies could be found.Although this simulation was performed for some selected values of the seven magnetic parameters, this simulation can be carried out for any values of the seven magnetic parameters.

Here
the easy directions with smaller energies.Hard directions can be observed at  s K =6, 16, 26.---etc.Two hard directions with different energies can be observed.The  s K values are given for the hard directions with higher energies.For each  s K

Figure
Figure 2 shows the 3-D plot of   ) ( E

Figure.
Figure 3 shows the 3-D plot of   ) ( E