Spin waves and their damping in itinerant ultrathin ferromagnets: Intermediate wave vectors

Our earlier papers explore the nature of large wave vector spin waves in ultrathin ferromagnets, and also the properties and damping of spin waves of zero wave vector, at the center of the two-dimensional Brillouin zone, with application to ferromagnetic resonance (FMR) studies. The present paper explores the behavior of spin waves in such films at intermediate wave vectors, which connect the two regimes. For the case of Fe films on Au(100), we study the wave vector dependence of the linewidth of the lowest frequency mode to find that it contains a term which varies as the fourth power of the wave vector. It is argued that this behavior is expected quite generally. We also explore the nature of the eigenvectors of the two lowest lying modes of the film, as a function of wave vector. Interestingly, as wave vector increases, the lowest mode localizes onto the interface between the film and the substrate, while the second mode evolves into a surface spin wave, localized on the outer layer. We infer similar behavior for a Co film on Cu(100), though this evolution occurs at rather larger wave vectors where, as we have shown previously, the modes are heavily damped with the consequence that identification of distinct eigenmodes is problematical.

Figure 1

(Color online) For the (a) four layer and (b) eight layer Fe film on Au(100), we show the dispersion relation of the two lowest lying modes of the film as a function of reduced wave vector, along the [11] direction in the Brillouin zone. As discussed in the text, a Zeeman field has been imposed so the lowest mode has finite frequency at the zone center.

Figure 2

We plot the frequency variation of the spectral density $A({\vec{Q}}_{\Vert },\Omega ;l_{\perp })$ as a function of layer index $l_{\perp }$, for the eight layer Fe film on Au(100) and for two selected wave vectors. In the left panel, we have zero wave vector, and in the right panel the reduced wave vector is $0.25\sqrt{2}$ along the [11] direction. The top entries, labeled I, are the layers adjacent to the Au substrate, and the lowest entry labeled S is the outer surface layer of the film.

Figure 3

(Color online) Linewidths of the lowest mode as a function of wave vector along the [11] direction in a four layer (a) and an eight layer (b) Fe film on Au(100). The solid curves are fittings to $Q^4$ functions.

Figure 4

The amplitude and phase of the eigenvector associated with the lowest frequency mode of the eight layer Fe film on Au(100). The top two panels show the eigenvector of the mode at ${\vec{Q}}_{\Vert }=0$, and the bottom two panels give the same for a reduced wave vector of 0.25 along the [11] direction in the two-dimensional Brillouin zone. The layer labeled I is the Fe layer against the substrate, and the layer labeled S is the outermost surface layer of the film.

Figure 5

The same as Fig. 4, but now for the second mode of the film.

Figure 6

(Color online) We show eigenvectors for Co on Cu(100) calculated using the Heisenberg model, as described in the text. The left panel shows the behavior of the lowest mode at the center of the Brilluoin zone (open circles) and at the reduced wave vector of 0.6 along the [11] direction in the Brillouin zone (triangles). The right panel gives the same information for the second mode. The layer labeled I is the layer of Co spins adjacent to the Cu(100) substrate, and the layer labeled S is the outer surface layer.

Figure 7

(Color online) The dispersion relation of the two lowest lying modes in the Co film on Cu(100). The wave vector is directed along the [11] direction in the two-dimensional Brillouin zone.