Modeling of the Influence of Oxidation on the Energy of Interfacial Exchange Interaction in Co/CoO Films

The interphase exchange interaction energy of ferromagnetic/antiferromagnetic films is determined through the mean spin theory. The thickness of the oxidized layer affects the exchange field, as shown by the Co/CoO film.


Introduction
The controllability of ferromagnetic and antiferromagnetic (F/AF) thin film thickness has prompted a comprehensive study of the exchange bias, which is defined by the bias of the center of the magnetic hysteresis loop and characterized by the bias field H E [1][2][3].It was discovered that the H E field increases as the thickness of antiferromagnetic layers d AF in two-phase F/AF films grows, reaching its maximum value.This value remains consistent even as d AF continues to increase [3][4][5].
This observation of the bias field H E dependence on the thickness d AF was noted in core/shell nanoparticle systems [6][7][8][9][10].Note that the well-known relationship (see, for example, [3]) between the energy of the interphase exchange interaction and the field E in = H eb M F M V F M , where M F M and V F M are saturation magnetization and ferromagnetic volume, allows experimental energy estimate E in .
The thickness of the antiferromagnetic layer may be adjusted by controlling the duration of post-synthetic annealing or the partial pressure of oxygen during spraying.Through oxidation, it is feasible to establish the value of the H E field and, therefore, the interphase exchange interaction energy [9,10].
In this paper, we use the mean spin formalism [11] to model the impact of cobalt oxidation on the energy of interphase exchange interaction in ultrathin Co/CoO films.

Model
To calculate the temperature-dependent energy of interphase exchange interaction, we utilized the model as described in [12,13].
(i) Consider a film consisting of N monolayers in which the first α are antiferromagnetic and the subsequent ones are ferromagnetic.(ii) We will assume that the atoms located in the selected layer have the same average magnetic moments, which differ from the average magnetic moments in neighboring layers.(iii) The exchange interaction fields H among atoms are randomly distributed and are only realized between adjacent neighbors, whose amount is determined by the crystal lattice structure.(iv) The magnetic moments of atoms can only be in two states µ i = µ i σ i , where σ i = {1, −1}, with orientation on or against some Oz axis, which corresponds to the approximation of the Ising model.

Energy of Interphase Exchange Interaction
Consider only two boundary layers in the described system: antiferromagnetic with the number α and ferromagnetic with the number α + 1, and choose in them an arbitrary pair of atoms interacting through the interface with magnetic moments µ α and µ α+1 .To estimate the energy of the interfacial exchange interaction E in , per this pair of atoms, we write down the Hamiltonian of the interaction between µ α = µ α σ α and µ α+1 = µ α+1 σ α+1 as follows: where J int is constant of the interphase exchange interaction between µ α and µ α+1 , H α and H α+1 are random exchange interaction fields created by the magnetic moments of neighboring atoms on µ α and µ α+1 , respectively.The interaction energy E in will be considered as the value -J int σ α σ α+1 averaged using the Hamiltonian (1) over the states σ α and σ α+1 at temperature T , and the interaction fields H α and H α+1 : where W α (H α ) and W α+1 (H α+1 ) are density functions of the distribution over the interaction fields on the interface atoms µ α and µ α+1 , defined in [11] by the relations (19) -(24).Equilibrium value ⟨σ α σ α+1 ⟩ T is: Considering (3) and the relations (19) -( 24) defining W F (H 1 ) and W AF (H 2 ) in [11], the expression for E in takes the form: where z n,k is number of nearest neighbors of an atom from a layer numbered n in a layer numbered k, i int = J int /J F M is relative constant of interphase exchange interaction, J F M exchange coupling constant of a ferromagnet, is constant of the exchange interaction between atoms in the n and k layers, and in ferromagnetic layers i n,k = 1, which corresponds to the approximation of a non-deformable lattice.Expression (4), along with relations (19)-( 24) from [11], enables examination of the temperature-dependent energy dependence of interphase exchange interaction.

Dependence of Interphase Exchange Interaction Energy on Oxidized Layer Thickness
For comparison with the findings of previous experiments [14,15], it can be postulated that oxidation of a ferromagnetic cobalt film grown on palladium leads to the formation of an antiferromagnetic CoO oxide layer on its free surface.Consequently, a crystal structure of Co(111)/Co(111) is produced, as presented in Table 1 which includes all modeling parameters.Figure 1 displays calculations illustrating how the thickness of the oxidized layer, d AF , affects the interfacial exchange interaction energy, E in (T ), as it varies with temperature.It is evident that, within a set temperature range, the interfacial exchange interaction energy E in (T ) increases as d AF thickness increases.Furthermore, an increase in the exchange coupling constant J int expands the temperature range in which the energy E in (T ) relies on the thickness d AF .The increase in energy input, E in , that corresponds to an increase in the thickness of the oxidized layer, d AF , at a characteristic temperature of T m = 65 K is illustrated in Fig. 2. It is noteworthy that the result obtained is in excellent agreement with the experimental findings [14,15].

Conclusion
Based on the method of random exchange interaction fields, we derived an equation for the interfacial exchange interaction energy E in , which confirms the experimental finding of an augmented exchange bias field H E with increased thickness of the oxidized layer.

Figure 1 .
Figure 1.The results of modeling the temperature dependence of the energy of the interphase exchange interaction E in (T ) for different values of the exchange coupling constant (J int = 1.5 * 10 −15 erg -black, J int = 1.0 * 10 −15 erg -purple, J int = 0.5 * 10 −15 erg -red), as well as the thicknesses of the antiferromagnet d AF = 2 and 20 ML.The insert shows the temperature dependence of the energy E in (T ) at J int = 0.5 * 10 −15 erg and the thickness of the antiferromagnet d AF = 2 ÷ 20 ML.

Table 1 .
The number of nearest neighbors z n,k and the exchange coupling constant J in the film interface Co(111)/Co(111).