Ferromagnetic resonance linewidths in ultrathin structures: A theoretical study of spin pumping

We present theoretical studies of the spin pumping contribution to the ferromagnetic resonance linewidth for various ultrathin film ferromagnetic structures. We consider the isolated film on a substrate, with Fe on $\mathrm{Au}\left(100\right)$ and Fe on $\mathrm{W}\left(110\right)$ as examples. We explore as well the linewidth from this mechanism for the optical and acoustical collective modes of $\mathrm{FM}∕{\mathrm{Cu}}_{\mathrm{N}}∕\mathrm{FM}∕\mathrm{Cu}\left(100\right)$ structures. The calculations employ a realistic electronic structure, with self-consistent ground states generated from the empirical tight binding method, with nine bands for each material in the structure. The spin excitations are generated through use of the random phase approximation applied to the system, including the semi-infinite substrate on which the structure is grown. We calculate the frequency response of the system directly by examining the spectral density associated with collective modes whose wave vector parallel to the surface is zero. Linewidths with origin in leakage of spin angular momentum from the adsorbed structure to the semi-infinite substrate may be extracted from these results. We discuss a number of issues, including the relationship between the interfilm coupling calculated adiabatically for trilayers, and that extracted from the (dynamical) spin wave spectrum. We obtain excellent agreement with experimental data, within the framework of calculations with no adjustable parameters.

Figure 1

We plot the spectral density function $A({\vec{Q}}_{\parallel }=0,\Omega ;l_{\perp })$ for two cases: (a) a two layer Co film adsorbed on the $\mathrm{Cu}\left(100\right)$ surface and (b) a trilayer consisting of two Co films separated by two layers of Cu, with the complex adsorbed on the $\mathrm{Cu}\left(100\right)$ surface. Each of the Co films has two layers. The Zeeman energy has been taken to be ${\Omega }_0=2\times {10}^{-3}\mathrm{Ry}$, or $27.2\mathrm{meV}$. Here $l_{\perp }$ is chosen to be the innermost atomic layer of the magnetic structure.

Figure 2

For an Fe film on the $\mathrm{Au}\left(100\right)$ substrate, we plot the ration $\mathrm{\Delta }\Omega ∕{\Omega }_0$ as a function of $N_{\mathrm{Fe}}$, the number of Fe layers. The results of the calculations are given as open circles, and the solid line is the best power law fit to the data, which is $1∕(N_{\mathrm{Fe}}{)}^{0.98}$. The solid circles are taken from measurements reported in Ref. 6.

Figure 3

The ratio $\mathrm{\Delta }\Omega ∕{\Omega }_0$, for an Fe film adsorbed on the $\mathrm{W}\left(110\right)$ surface. The straight line is a best power law fit to the calculations, and gives a fall off with film thickness of $1∕(N_{\mathrm{Fe}}{)}^{0.81}$. Clearly quantum oscillations such as discussed in Ref. 10 are present for small film thicknesses, so the power law fit is an oversimplification.

Figure 4

For a two layer Fe film on $\mathrm{Cu}\left(110\right)$ (left panel) and for such a film on $\mathrm{W}\left(110\right)$ (right panel), we show how the linewidth is influenced by various coupling across the interface. In both cases, the solid curve shows the spectral density function generated by a complete calculation. The dashed curve has the $d$-$d$ hopping terms across the interface set to zero, whereas the dot dash curve has the $sp$-$sp$ hopping terms turned off.

Figure 5

For a ${\mathrm{Co}}_2{\mathrm{Cu}}_4{\mathrm{Co}}_2∕\mathrm{Cu}\left(100\right)$ structure, we show (a) the spectral density function $A(0,\Omega ;l_{\perp })$, for the case where $l_{\perp }$ is chosen to be the innermost layer of the inner film. Then in (b) we show the acoustical mode profile, where the mode has been isolated by the procedure described in the text. In (c), we display the optical mode.

Figure 6

For a ${\mathrm{Co}}_2{\mathrm{Cu}}_N{\mathrm{Co}}_2∕\mathrm{Cu}\left(100\right)$ structure, we compare interfilm couplings deduced from our dynamic calculations of the splitting between the optical and acoustic mode of the trilayer (open circles, dashed line) with that calculated by adiabatic theory (solid squares, solid line).

Figure 7

For a ${\mathrm{Co}}_2{\mathrm{Cu}}_N{\mathrm{Co}}_2∕\mathrm{Cu}\left(100\right)$ trilayer, we show the variation with $N$ of the linewidth for (a) the acoustic spin wave mode of the structure, and (b) the optical spin wave mode of the structure.

Figure 8

This figure gives information on the spectrum of a ${\mathrm{Co}}_2{\mathrm{Cu}}_{10}{\mathrm{Co}}_2∕\mathrm{Cu}\left(100\right)$ structure, where the frequency of the outer film is $\lambda {\Omega }_0$ and that of the inner film is ${\Omega }_0$. In (a), for the case where $\lambda$ is 1.2, we show with the solid line the spectral density function $A\left(\Omega \right)$ defined in the text. The dashed line is the spectral function $A_1\left(\Omega \right)$ and the dot dash line is $A_2\left(\Omega \right)$. In (b) we give the linewidth of the acoustical and optical modes of the trilayer as a function of the parameter $\lambda$.