Generalized equation of motion for a ferromagnet

Abstract

We derive microscopically an equation of motion for a ferromagnetic substance at nonzero temperatures allowing for both transverse and longitudinal relaxation and generalizing the Landau-Lifshitz equation. The consideration starts from the density matrix equation for a quantum spin interacting with the environment, which is within about 7% accuracy reduced to the closed equation for the first moment of the distribution function — the magnetization. The latter interpolates between the Landau-Lifshitz equation ( S ⪢ 1 and low temperatures) and the Bloch equation ( S = 1 2 or high temperatures). For condensed magnetic media (i.e. a ferromagnet) one can replace in the spirit of the mean field theory the magnetic field by the molecular one containing the exchange field acting on a given magnetic ion from its neighbours, which results in a Landau-Lifshitz type equation of motion with a longitudinal relaxation term providing the Curie-Weiss static solution. Further we consider the mobility of a domain wall (DW) in a uniaxial ferromagnet at nonzero temperatures where the magnetization in the middle of the domain boundary is smaller than in the domains (the elliptic DW transforms to the linear one near T c ). It is shown that longitudinal relaxation plays a crucial role in DW dynamics in a wide range of high temperatures.