Angular dependence of spin-wave resonances and surface spin pinning in ferromagnetic (Ga,Mn) As films

We present a study of exchange-dominated surface and bulk spin-wave modes in (Ga,Mn)As films grown by molecular beam epitaxy. Multimode spin-wave spectra were observed using the ferromagnetic resonance technique, as the direction of the magnetic field H was varied within the plane of the layer, as well as relative to the plane. The field corresponding to the main (strongest) resonance peak at each angle is used to calculate bulk magnetic anisotropy parameters and the g factor of the magnetic films. The dependence of spin-wave (SW) modes on the orientation of H is analyzed in terms of two fundamental models: the surface inhomogeneity model and the volume inhomogeneity model. Typically, when H is normal or near normal to the sample plane (i.e., H[001]), a spectrum consisting of a series of well-resolved SW modes is observed. However, as H is rotated significantly away from the normal, at some orientation, a ``critical angle'' is found, at which only the single uniform mode is observed. As H is rotated further and approaches the in-plane direction (such as the [110] direction), a multiple-mode spectrum reemerges, ascribed in part to the appearance of a surface spin excitation due to unpinned surface spins. The analysis of spin-wave resonance spectra in terms of the dynamic surface spin pinning (derived from the surface anisotropy) allows us to determine the value of the exchange stiffness constant D.