Ferromagnetic Resonance Effect on Resistivity in Mesoscopic Structures

Abstract

Rapid advances in controlled preparation techniques have enabled us to fabricate submicron-scale magnetic structures. Considerable interest has been paid to the spin-excitation spectrum of a small magnetic structure. Several experiments using ferromagnetic resonance absorption or Brillouin light scattering have been performed to probe how the reduction of sample size affects the spin-excitation spectrum. The measurement in these experiments is carried out for arrays of small samples. Thus, strictly speaking, the results do not reveal the spectrum of individual samples. Besides the dynamical magnetic properties, electron transport properties in mesoscopic ferromagnetic metals have been the subject of intesive study. Particularly, the magnetoresistance of thin wires containing a magnetic domain wall (DW) has attracted attention with the aim of studying DW dynamics. It has been shown that we can detect DW depinning by measuring the resistance as a function of a magnetic field. Furthermore, it it shown recently that the dynamical property of a DW can be studied by measuring the magnetoresistance. These studies suggest that the magnetoresistance of various magnetic structures may provide us useful informations on their dynamical property. In this short note, we show that the ferromagnetic resonance in small structures can be detected by measuring the magnetoresistance. We consider the resistivity of disordered ferromagnetic metals in the presence of a static field H0 and a uniform rf field with frequency being perpendicular to H0. The rf field excits uniform spinprecession mode, which induces spin-flip scattering of conduction electrons. Although the momentum of conduction electrons is completely conserved in the scattering processes, the mixing of majority and minority spin channels results in an increase of the resistivity in the presence of spin asymmetry in the elastic mean free time. The resonance frequency !0 of the uniform precession mode depends on H0 as well as the demagnetizing field. We expect that the increase of the resistivity becomes large when the ferromagnetic resonance condition 1⁄4 !0 is nearly satisfied. It is shown that the correction to conductivity normalized by the Drude conductivity 0 is given by = 0 / ð hÞ=1⁄2ð!0 Þ þ 2 , where and h are the gyromagnetic ratio and the amplitude of the rf field, respectively, and is the loss parameter of the uniform precession mode. This indicates that we can observe the ferromagnetic resonance in individual ferromagnetic structures by measuring the resistivity as a function of H0. We set h 1⁄4 kB 1⁄4 1 in the following. As an example of magnetic structures in mesoscopic dimensions, we consider a thin cylindrical wire of ferromagnetic metal lying along the z direction. Its length is assumed to be much longer than its width. Although we treat only the thin wire structures, an extention to other structures is straightforward. Let MðrÞ be the magnetization vector in the wire. We consider the case where the static field is applied parallel to the wire, that is, H0 1⁄4 ð0; 0;H0Þ. Thus, the magnetization vector becomes MðrÞ 1⁄4 ð0; 0;M0Þ in the absence of an rf field (M0: saturation magnetization). We apply a uniform rf field hðr; tÞ 1⁄4 ðh cosð tÞ; h sinð tÞ; 0Þ circularly polarized in the xy plane. The magnetization vector obeys the Landau–Lifshitz equation @Mðr; tÞ=@t 1⁄4 Mðr; tÞ Hðr; tÞ, where is the gyromagnetic ratio and H is the total effective field H 1⁄4 H0 þ hþHd. Here, the demagnetizing field Hd is approximately expressed as Hd 1⁄4 ð 2 Mx; 2 My; 0Þ in our long cylindrical structure. We have neglected the skin effect for the rf field by assuming that the width of the wire is much smaller than the penetration depth. The typical penetration depth of ferromagnetic metals is longer than micrometer scale when is of the order of 10 GHz, so that the above assumption safely holds for wire structures of sub-micron scale diameter. We assume that conduction electrons are described by