Acoustical and optical spin modes of multilayers with ferromagnetic and antiferromagnetic coupling

A study of the in-phase ‘‘acoustical’’ and of the out-of-phase ‘‘optical’’ spin modes in layered ferromagnetic structures with both ferromagnetic and antiferromagnetic coupling is presented, in particular for the system Fe~20 Å/Cr~20 Å/Fe~20 Å/Cr~9 Å/Fe~100 Å. A calculation of the dynamic magnetization, assumed constant in each film, gives an explanation of the behavior of the two kinds of modes at different magnetic fields. The ‘‘acoustical’’ mode is the lowest frequency one for low applied fields, but its frequency switches with that of the ‘‘optical’’ modes at higher fields. It is found that the system exhibits soft modes as a function of the applied field. One of these is a Goldstone mode