Spin waves in a strong tight-binding itinerant ferromagnet with a (100) surface

The surface spin-wave problem for a simple-cubic tight-binding ferromagnet is formulated in terms of a spin-wave Green’s function. The effect of the surface is treated as a perturbation to the bulk problem. The spin-wave Green’s function of a semi-infinite ferromagnet satisfies a Dyson equation with the bulk Green’s function as a kernel. Both the bulk spin-wave Green’s function and the surface perturbation are parametrized in terms of Heisenberg-like effective exchange integrals. The effective exchange integrals are expressed in terms of one-electron Hartree-Fock (HF) propagators and evaluated. The bulk effective exchange integrals and the surface perturbation for a strong ferromagnet are shown to be negligible beyond the range of electron hopping. The effect of the surface is separated into a geometric effect and a surface renormalizaton of the bulk exchange integrals due to the surface core shift, HF corrections, and Friedel oscillations. It is shown that the renormalization of the effective exchange integrals in the first two atomic planes is sufficient for a strong ferromagnet. The dependences of the renormalized surface exchange integrals on the occupation of the surface layer $n_s$ are computed. For a neutral (100) surface ($n_s$=n), the surface exchange integrals are very close to their bulk values. For $n_s$/n$n_s$/n>0.88. Acoustic surface modes can exist for other surfaces, e.g., (110).