An Analytical Theory of FMR in Bulk Metals (Parallel Configuration). I. Dispersion Relations

Abstract

A linear solntion of Maxwell's equation for the propagation of electromagnetic waves in a ferromagnetic metal in the region of ferromagnetic resonance is presented; the motion of the magnetization vector is described by the Landau-Lifshits equation (including the exchange effect in the damping term). Dispersion relations (DR) are derived for the roots of the bicnbic secular equation, representing three waves of generally combined electromagnetic-spin-wave character. They are compared with the DR for the case of low-conductivity ferromagnets; the crosaover effect is discussed.Eine lineare Losung der Ma.xwellgleichungen fur die Ausbreitung von elektromagnetischen Wellen in ferromagnetischen Metallen im Gebiet der ferromagnetischen Resonanz wird vorgelegt ; die Bewegung des Magnetisierungsvektors wird durch die Landau-Lifschitz-Gleichung beschrieben (einschlieWlich des Austauscheffektes im Ausdruck fur die Diimpfung). Die Dispersionsbeziehungen (DR) fur die Wurzeln der bikubischen Sekulargleichung werden abgeleitet ; sie stellen drei Wellen des allgemein kombinierten elektromognetischen-spinwellen Charokters dar. Diese Beziehungen werden mit den DR fur den Fall eines schwachleitenden Ferromagneten verglichen; der ,,crossover"-Effekt wird besprochen.

A Theoretical Study of the FMR Surface Mode Excitation in Metals (Parallel Configuration)

“…[19] for the surface impedance [ 2 ] where Q is the resistivity, M , the saturation magnetization, A the exchange constant, po the SI system constant; the coordinate system is defined in [17]; the static mag- (in the SI unit system).…”

“…This situation is illustrated in Fig. 2 a, With rising q after passing the cross-over region (for q near ?pa) also a n interchange of the character of both individual waves i = 1 and i = 3 occurs (for more details see [17]). …”

“…It contains a cross-over point [17], in which the two roots Kt of the bicubic secular equation [19] (KF represents the wave of originally electromagnetic behaviour, Kg is the Larmor spin wave) would cross in the complex K 2 space, in reality they avoid a contact, and their character becomes mixed. This situation is illustrated in Fig.…”

“…For a further discussion we need to quote the conditions, which specify the FMR cross-over region [17]. There the frequency 52 is given by where wid 3 L2)) is Kittel's main mode resonance point; the magnetic field distance 6q = q -T~M R (from Kittel's MM) is given by By analysing the cross-over conditions (a), (3) for the Fe40Ni40B20 alloy, one can conclude (because of the small exchange and electrical conductivity parameters and large L.L.…”

“…the quantity measured in the experiment) will be investigated as dependent on the static magnetic field intensity and for different values of the surface anisotropy constant K , and a t frequencies sufficient for the observation of the surface mode. The use of the analytical FMR theory [3,4,9,15,17,18] is appropriate in this case, because only K, values corresponding t o 6 up to 0.5 (see below and e.g, [9]) are considered.…”