Rado-Weertman Boundary Equation Revisited in Terms of the Free-Energy Density of a Thin Film |
H. Puszkarski
Surface Physics Division, Faculty of Physics, Adam Mickiewicz University,, Umultowska 85, 61-614 Poznań, Poland |
Received: June 7, 2016 |
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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, 2A_{ex}∂m/∂n - K_{surf}m = 0. A key role in this equation is played by the phenomenological quantity K_{surf} known as the surface anisotropy energy density; A_{ex} 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^{bulk}, defined separately on the surface and in the bulk, respectively, we prove that K_{surf} = 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 K_{surf} 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. |
DOI: 10.12693/APhysPolA.129.RK.129.6.1-1 PACS numbers: 75.70.-i, 75.40.Gb, 75.30.Ds, 75.30.Gw, 75.10.-b |