  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 $\phi(H)$ 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 $GMR(\phi)\propto (1-\cos\phi)$ 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.
