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Multiferroics are novel classes of materials that exhibit cross-coupling of mutually excluding phenomena, i.e. magnetism and ferroelectricity. In recent years, the coexistence of ferroelectricity and magnetic orderings has become a hot issue and drawn considerable attentions due to the promising applications to these days technology and the fundamental science involved in these classes of materials. The microscopic origins of magnetism and ferroelectricity differ fundamentally, while the real mechanism of ferroelectricity is still under debate. In the present work, we have started from a simple method Heisengerg hamiltonian and an interaction term resulting from electric field coupling with the magnetic spins with anisotropic limit, demonstrated that magnetization can be manipulated by electric field and anisotropic field in agreement with results experimentally observed. In the multiferroic thin film system the magnetic field tends to play a role in stabilizing the spins in preferred orientations and induces a coupling of magnetism and ferroelectricity that opens a route to switch magnetization with electric polarazation and vice-versa.

Multiferroics are unique classes of materials whose fundamental state is both magnetic and ferroelectric [_{3} is found to be proportional to the photon energy which depends on the direction of magnetization [8-10].

We consider the spin Hamiltonian of the multiferroic thin film system that exhibits frustrated spin interactions [11,12]. The Hamiltonian of this anisotropic many body interaction system on the influence of external fields can be expressed by [13-15],

where S_{i}(S_{j}) is spin operator at sites i(j). J_{ij} is the exchange coupling that depends on the relative positions of neigbhoring spins and is defined by J_{ij} = J(R_{i} − R_{j}). The constant g is the gyromagnetic ratio, and the second term of Equation (1) is the effective Zeeman field with an external deriving field while in this model chosen to be parallel to the +z axis to line up all the spins in the same direction. Whereas, the third term includes an electric field E that interacts with the spins in the system. The constant μ describes the coupling strength between the spins and the radiation field that is either intrinsic or extrinsic [16,17]. The terms μ_{B} and D are the Bohr magneton and magnetic anisotropic (or spin wave stiffness) constants, respectively [17,18]. The total spin operators in the Hamiltonian are:

The ground state of N identical atoms of spin S is defined as

The total spin operator at any site i has components S_{iz}, S_{ix} and S_{iz} that should be treated independently with an identity S_{i}. S_{i} = S(S + 1). Transferring the spin operator problems into many body interaction systems upon replacing with bosonic creation and annihilation operators expressed as

where a_{i}^{†} and a_{i} are the creation and annihilation operators satisfying the commutation relation [a_{i}, a_{i}^{†}] = δ_{il} It is also possible to transform this bosonic operators to magnon variable operators b_{k} and b_{k}^{†} so that these operators can be related to the bosonic operators [19,20]

and

where the inverse transformation is

and

At low excitation i.e. where the interaction is predominated by bilinear spin variables, neglecting the interaction corresponding to higher order terms, the reduced Hamiltonian is of the form

Here, z is defined to be the number of nearest neighbor spins. In this Hamiltonian we consider only the term bilinear to the magnon variables

where,.

This simplified interaction term is equivalently expressed as, if we write

The dispersion equation on the approximation that can be reduced to

so that

The above expression is a dispersion relation with some additional term that tends to show how the dispersion term depends on the field components. Moreover, the constant.

The spin-wave dispersion function of the multiferroic thin film system is obtained as a function of the wave vector and the field components.

A significant change in the dispersion relation is observed as depicted by

and electric field values in some orientations. The inter-dependence of the magnetic and electric fields (their couplings) in multiferroic materials is also analyzed based on the spin-wave occupation number,.

The change in magnetization is obtained to be

where and and the change in magnetization versus temperature at some specific γ is shown in

Consequently, the magnetization is a function of the γ values at low temperature as indicated by

magnetization can be easily deduced from the expression γ = ζ + μE where ζ = −2DS + gμB H for some specific values of the parameters, D, S, and H at a particular temperature.

This work was supported by the research programm of Aaddis Ababa University, Ethiopia and Mizan-Tepi University, Ethiopia.