Modeling the magnetoresistance vs. field curves of GMR multilayers with antiferromagnetic and/or orthogonal coupling by assuming single-domain state and coherent rotation

In order to better understand the role of possible couplings in determining the giant magnetoresistance (GMR) behavior of multilayers, a knowledge of the dependence of the $GMR$ on magnetic field $H$ appears to be useful. Since a few specific cases have only been treated theoretically in the literature, it was decided to carry out a modeling of the $GMR(H)$ curves of ferromagnetic/non-magnetic (FM/NM) multilayers with various interlayer couplings. For simplicity, we focused on a trilayer structure (FM1/NM/FM2) corresponding fairly well to the case of a large number of FM/NM bilayers. To carry out the calculations, some fundamental assumptions were made: (i) single domain FM layer, in plane magnetization; (ii) the magnetization of each layer is the same; (iii) the magnetization vectors rotate in the plane of the layers in an external magnetic field. In order to calculate the $GMR(H)$ function, we need to know the magnetization process in the trilayer, i.e., the $M(H)$ function. Therefore, first we calculate the equilibrium angle $\varphi (H)$ between the two magnetization vectors as a function of the field by minimizing the total energy of the trilayer. According to most previous theoretical and experimental works, the angular dependence of the GMR is fairly well described by the relation $GMR(\varphi )\propto (1-\cos \varphi )$ and we used this relation to derive the $GMR(H)$ function. Along this line, the $M(H)$ and $GMR(H)$ curves were calculated for the following cases: (i) pure AF coupling; (ii) pure orthogonal coupling; (iii) AF coupling and orthogonal coupling simultaneously present. As to the calculation of the $GMR(H)$ curves, some of these configurations have not yet been treated formerly or for some specific parameter values only. For those cases for which calculations were reported in the literature for $M(H)$ and $GMR(H)$, our results agree with previous reports.