A contribution to the problem of the spatial distribution of the spontaneous magnetization in thin films

Abstract

This report summarizes some of the calculations based on the Heisenberg model using the molecular field theory approximation and the Green's fnnct.ion technique. The results are compared with the corresponding ones of other authors. Some consequences are pointed out for the experiment e.g. application of the Mijssbauer effect.In dieaer Arbeit wird uber einige Berechnungen der riiumlichon Magnetisieriingsverteilung in dlnnen Schichten auf der Grundlage dea HeisenbergModells nnter Benutzung der Molekularfeldtheorie und der Technik der Greenechen Funlitionen berichtet. Die erhaltenen Ergebnisse werden mit den entsprechenden Ergebnissen anderer Aiitoren verglichen. Ferner werden einige Konsequenzen ftir das Experiment, z. B. fur den MoBbauer-Effekt, aufgezeigt. IntroductionThere are some phenomena, as e.g. the spin wave resonance, the understanding of which necessitates a detailed knowledge of the spatial distribution of the magnetization in magnetic thin films (see e.g. [l]). Thanks t o the recent progress in the experimental technique using the M6ssbeuer effect it, seems to be possible to investigate the local distribution of the magnetization (see e.g. [2]). Itt is, therefore, desirable t o see what may be expected from the theory starting with some simple theoretical model. I n this paper we report on some older calculations not yet published till now and on some recent calculations using the molecular field treatment and the Green's function technique. Calculating the local magnetization of a film we use the subdivision of the film into atomic layers parallel to the surface of the film as introduced in [3] (see also [4]). We treat then the layers aa some generalized NBel sublattices and evaluate their mean magnetization. Though the magnetization of a layer is not a good quantum number, there are some reasons, given in [3] and recently by Koppe and Jelitto in [5] in favour of such a treatment. We start with a survey of our results of the molecular field treatment,

A Status Report of Green's Function Theory in Uniaxial Ferromagnetic Unpinned Thin Films

“…I n order to make the connection with known numerical computations of I ----- It will be mentioned that contrary to the papers [5] to [7] collapse near the surfaces. The GFA shows that the increasing collapse of the magnetization near the unpinned surfaces is replaced a t high temperatures by a progressive smoothing of the spatial magnetization distribution, so that a t the Curie temperature the ferromagnetism of all layer magnetizations disappears simultaneously, as predicted in Section 4.1.…”

“…Furthermore, we demand periodic boundary conditions in the plane of the film and assume a quadratic form of the surface. For that reason, the quantities (Si ,) and A,,,, are independent of the index m, and we can expand the Fourier transformed quantity gvwt.nrel(w) of equation (1) for the Green's function (7) into a two-dimensional Fourier series : 1 gp,gz(w) = -2 eik(m -QEe(w) 7 (111 N h where h = (ha, h,) denotes a two-dimensional wave vector lying in the plane of the film which obeys the periodic boundary conditions. N is the number of spin-carrying lattice points belonging to one single-atom layer having quadratic form, and the summation is performed over the two-dimensional first Brillouin zone.…”

“…sPv means the number of nearest neighbours in the v-th layer to an atom in the p t h layer. I n [ 7 ] , in the absence of a dc magnetic field, the spatial distribution of the magnetization for Fe( loo), Co(OOOl), and Ni( 111) films has been calculated numerically as a function of film thickness starting from the system of equations (21). I n order to solve the system of equations (21) analytically we consider the temperature range where the Brillouin function can be developed in a Taylor series and restrict ourselves to the linear approximation with respect to B.…”

“…connected with the Green's function (7) can be written as taking into consideration the relations ( l ) , ( 2 ) , ( l l ) , (12), and According to Callen [17] we can derive from the equilibrium correlation function (14) a differential equatioil of second order for the quantity (e'"} if we put t = t', el = pm, and B,, = e q m SG, ( r is a parameter). By using Callen's momentum generating function method we get for <e'") the solution (XE} (rp = 1, 2 , .…”

“…I n order to make the connection with known numerical computations of It will be mentioned that contrary to the papers [5] to [7] films has been calculated numerically and is represented in Fig. 6 t b 10.…”