Spin-orbit coupling and spin waves in ultrathin ferromagnets: The spin-wave Rashba effect

We present theoretical studies of the influence of spin-orbit coupling on the spin-wave excitations of the Fe monolayer and bilayer on the W(110) surface. The Dzyaloshinskii-Moriya interaction is active in such films by virtue of the absence of reflection symmetry in the plane of the film. When the magnetization is in plane, this leads to a linear term in the spin-wave dispersion relation for propagation perpendicular to the magnetization, to produce a dispersion curve similar in nature to that found for electrons on semiconducting surfaces when the Rashba coupling is active. We also show spin-polarized electron-loss spectroscopy response functions that illustrate the role of spin-orbit coupling in such measurements. In addition to the modifications of the dispersion relations for spin waves, the presence of spin-orbit coupling in the W substrate leads to a substantial increase in the linewidth of the spin-wave modes. The formalism we have developed applies to a wide range of systems, and the particular system explored in the numerical calculations provides us with an illustration of phenomena which will be present in other ultrathin ferromagnet/substrate combinations.

Figure 1

(Color online) For the Fe monolayer adsorbed on W(110), we show the local density of states in the Fe monolayer. The majority-spin density of states is shown positive and the minority-spin density of states is negative. The zero of energy is at the Fermi energy. In (a), bulk electronic structure parameters are used as in the second of the two papers cited in Ref. 5 (case A). In (b), we have the density of states generated by method B. The black curve is found by fitting the KKR electronic energy band structure to tight-binding parameters as described in the text and the red dashed curve is calculated directly from the KKR calculation. In (c) we have the local density of states generated by method C.

Figure 2

Schematic representation of in-plane lattice points and two-dimensional Brillouin zone geometry. The easy axis for the monolayer is parallel to $\left[1\overline{1}0\right]$ in both cases B and C. For the bilayer, the easy axis is perpendicular to the film in case B and parallel to $\left[1\overline{1}0\right]$ in case C.

Figure 3

(Color online) The spectral density functions $A\left({\vec{Q}}_{\parallel },\Omega ;l_{\perp }\right)$ evaluated in the Fe monolayer for three values of the wave vector in the direction perpendicular to the magnetization. We have (a) $\left|{\vec{Q}}_{\parallel }\right|=0.4{\text{Å}}^{-1}$, (b) $\left|{\vec{Q}}_{\parallel }\right|=1.0{\text{Å}}^{-1}$, and (c) $\left|{\vec{Q}}_{\parallel }\right|=1.4{\text{Å}}^{-1}$. The green curve (dashed) is calculated with spin-orbit coupling set to zero; the spectral density here is independent of the sign of ${\vec{Q}}_{\parallel }$. The red and black curves are calculated with spin-orbit coupling turned on. Now we see asymmetries for propagation perpendicular to the magnetization with the red curve ${\vec{Q}}_{\parallel }$ directed from left to right and the black curve from right to left.

Figure 4

(Color online) Spin-wave dispersion relations constructed from peaks in the spectral density for the Fe monolayer on W(110). The wave vector is in the direction perpendicular to the magnetization. The red curve is constructed in the absence of spin-orbit coupling, which is included in the black curve. Panel (b) is a zoom of panel (a) at the small wave-vector region.

Figure 5

(Color online) The spectral density in the outermost layer, in the acoustic spin-wave regime, for wave vectors of $Q_{\parallel }=+0.5{\text{Å}}^{-1}$ (black curve) and $Q_{\parallel }=-0.5{\text{Å}}^{-1}$ (red curve). Model C has been used for the calculation. In the ground state, the magnetization lies in plane along the long axis.

Figure 6

(Color online) For the wave vector $Q_{\parallel }=0.5{\text{Å}}^{-1}$ we show the spectral densities in the innermost Fe layer (upper panel) and in the outermost layer (lower panel) for the Fe bilayer on W(110). The figure includes the optical spin-wave feature. As in Fig. 5, the black curve is calculated for $Q_{\parallel }$ positive and the red curves are for $Q_{\parallel }$ negative. The calculations employ model C.

Figure 7

(Color online) For the Fe bilayer and for several values of the wave vector (all positive), we show the spectral density functions for the innermost layer adjacent to the substrate (black curve) and those for the outer layer of the film. The calculations employ model C.

Figure 8

(Color online) For the Fe bilayer with magnetization in plane we show, in panel (a), the spin-wave dispersion curves calculated with spin-orbit coupling (black symbols) and without spin-orbit coupling (red symbols). Panel (b) is a zoom of panel (a) at the small wave-vector region, together with fittings to quadratic functions (see text). Model C has been employed for these calculations.

Figure 9

(Color online) For the bilayer and the case where the magnetization is perpendicular to the surface (model B), and for $Q_{\parallel }=0.6{\text{Å}}^{-1}$, we show spectral density function calculated for positive values of $Q_{\parallel }$ (continuous lines) and negative values of $Q_{\parallel }$ (symbols). The black curve is the spectral density for the inner layer and the red curve is the outermost Fe layer.