Rado-Weertman Boundary Equation Revisited in Terms of the Free-Energy Density of a Thin Film

Abstract

Historically, the first boundary conditions to be formulated and used in the theory of ferromagnetic thin films, the Rado–Weertman (RW) conditions, have a general advantage of being a simple differential equation, 2Aex ∂m ∂n − Ksurfm = 0. A key role in this equation is played by the phenomenological quantity Ksurf known as the surface anisotropy energy density; Aex denotes the exchange stiffness constant, and m is the amplitude of the transverse component of dynamic magnetization. In the present paper we use a microscopic theory to demonstrate that the surface anisotropy energy density of a thin film is directly related with its free-energy density, a fact not observed in the literature to date. Using two local free-energy densities F surf and F , defined separately on the surface and in the bulk, respectively, we prove that Ksurf = d ( F surf − F bulk ) , where d is the lattice constant. The above equation allows to determine the explicit configuration dependence of the surface anisotropy constant Ksurf on the direction cosines of the magnetization vector for any system with a known formula for the free energy. On the basis of this general formula the physical boundary conditions to be fulfilled for a fundamental uniform mode and surface modes to occur in a thin film are formulated as simple relations between the surface and bulk free-energy densities that apply under conditions of occurrence of specific modes.