Surface spin waves in the Hubbard model

Abstract

A general random-phase-approximation (RPA) equation for the surface susceptibility of an itinerant ferromagnet is derived. The surface is treated as a perturbation to the bulk problem. It is shown that the resulting secular equation for surface spin waves has the same structure as for magnetic insulators. It is demonstrated that the spin-rotational invariance of the Hubbard Hamiltonian imposes an important self-consistency condition on the surface perturbation to the bulk susceptibility. The previous calculations which do not satisfy this self-consistency condition are critically reviewed. The secular equation for surface spin waves is solved explicitly for a strong itinerant ferromagnet containing a plane of impurities with an excess intra-atomic repulsion $\ensuremath{\Delta}U$ modeling the surface. The unenhanced surface susceptibility is treated exactly in the surface plane and approximated by the bulk susceptibility outside the plane. This approximation is shown to be asymptotically exact within RPA for a ferromagnet with exchange splitting much larger than the bandwidth. Surface spin waves in this model split off the bottom of the bulk spin-wave band for a magnetically weaker surface ($\ensuremath{\Delta}U<0$) and they deviate downward from the bulk band only to the order ${q}_{\ensuremath{\parallel}}^{4}$ (${q}_{\ensuremath{\parallel}}$ is the wave vector parallel to the surface). The attenuation of long-wavelength surface spin waves is exponential and their attenuation length is proportional to the square of the wavelength. All these properties are in qualitative agreement with the properties of surface spin waves in magnetic insulators. The similarities and differences between surface spin waves in metals and insulators are discussed in the light of the modern approach to magnetic excitations in bulk itinerant ferromagnets.