Magnetic Surface Modes: Classification, New Types, and Simple Method of Solution

A physical interpretation of the eigenvalue equation is used to obtain the frequencies and spin configurations of ferromagnetic and antiferromagnetic surface modes and to provide a classification of these modes. A physical understanding of the features of surface modes also is provided. Since all spins precess at the same frequency in a normal mode, the frequency can be obtained from the equations of motion of one, or sometimes two or three, spins near the surface. One class of surface modes is the changed-surface-parameter class, in which the missing neighbors of a surface spin are compensated for by a changed parameter, such as an exchange constant, at the surface. Another is the equivalent-layer class, in which the crystal contains equivalent layers such that the net torque on each spin $\overrightarrow{\mathrm{S}}$ on an equivalent layer exerted by all spins on the surface side of $\overrightarrow{\mathrm{S}}$ is zero. Two new types of surface waves are studied. In the first, the spin precession changes phase as well as amplitude as a function of the distance from the surface. In the second, a surface layer of spins is antiparallel to the bulk spins.