Nonlocal feedback in ferromagnetic resonance

Ferromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback effects. The equation of motion for the magnetization dynamics is nonlocal in both space and time and includes isotropic, anisotropic, and dipolar energy contributions as well as the conserved Gilbert and the nonconserved Bloch damping. We derive an analytical expression for the peak-to-peak linewidth. It consists of four separate parts originated by Gilbert damping, Bloch damping, a mixed Gilbert-Bloch component, and a contribution arising from retardation. In an intermediate frequency regime the results are comparable with the commonly used Landau-Lifshitz-Gilbert theory combined with two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth whose frequency dependence becomes strongly nonlinear. The relevance and the applicability of our approach to ferromagnetic resonance experiments are discussed.

Figure 1

The geometry of the film and the magnetization. Further description is given in the text.

Figure 2

Dependence of the magnetization angle ${\Theta }_{\mathrm{M}}$ on the angle ${\Theta }_{\mathrm{H}}$ under which the static external field is applied for ${\omega }_{\mathrm{r}}/\left(2\pi \right)=10$ GHz. The parameters are taken from Ref. 16: $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G.

Figure 3

Effect of varying retardation strength on the uniaxial anisotropy field for various frequencies and ${\Theta }_{\mathrm{M}}=\pi /3$. Other parameters are $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G; see Ref. 16.

Figure 4

Influence of the retardation strength ${\Gamma }_0$ (top) on the peak-to-peak linewidth $\Delta H_{\mathrm{T}}$ for various frequencies and (bottom) on the single contributions $\Delta H_{\eta }$ for $f=70$ GHz. ${\Delta }_{\mathrm{B}}=0$ is the frequency region. The parameters from this study are ${\Theta }_{\mathrm{H}}={\Theta }_{\mathrm{M}}=0$, $\beta =0.5$, $\alpha =0.01$, $T_2=5\times {10}^{-8}$ s, and $\tau =1.7\times {10}^{-14}$ s. The other parameters are $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G; see Ref. 16.

Figure 5

Influence of the dimensionless retardation length $\beta =\xi k_z$ (top) on the total peak-to-peak linewidth $\Delta H_{\mathrm{T}}$ for various frequencies and (bottom) on the single contributions $\Delta H_{\eta }$ for $f=70$ GHz; ${\Delta }_{\mathrm{B}}=0$ in this range. The parameters from this study are ${\Theta }_{\mathrm{H}}={\Theta }_{\mathrm{M}}=0$, ${\Gamma }_0=1.1$, $\alpha =0.01$, $T_2=5\times {10}^{-8}$ s, and $\tau =1.7\times {10}^{-14}$ s. The other parameters are $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G and are taken from Ref. 16.

Figure 6

Influence of the retardation time $\tau$ (top) on the total peak-to-peak linewidth $\Delta H_{\mathrm{T}}$ for various frequencies and (bottom) on the single contributions $\Delta H_{\eta }$ for $f=70$ GHz. ${\Delta }_{\mathrm{B}}=0$ in this region. The parameters from this study are ${\Theta }_{\mathrm{H}}={\Theta }_{\mathrm{M}}=0$, $\beta =0.5$, $\alpha =0.01$, $T_2=5\times {10}^{-8}$ s, and ${\Gamma }_0=1.1$; the other parameters are taken from Ref. 16: $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G.

Figure 7

Frequency dependence of all contributions to the peak-to-peak linewidth for ${\Theta }_{\mathrm{H}}={\Theta }_{\mathrm{M}}=0$, $\beta =0.5$, $\alpha =0.01$, $T_2=5\times {10}^{-8}$ s, $\tau =1.7\times {10}^{-14}$ s, and ${\Gamma }_0=1.2$. The parameters taken from Ref. 16 are $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G. The Bloch contribution $\Delta H_{\mathrm{B}}$ is shown in the inset.

Figure 8

Comparison of the retardation model with the two-magnon model. The frequency dependence of the total peak-to-peak linewidth $\Delta H_{\mathrm{T}}$ is shown for ${\Theta }_{\mathrm{H}}={\Theta }_{\mathrm{M}}=0$, $\beta =0.5$, ${\alpha }_1=0.003$, ${\alpha }_2=0.0075$, $T_2=5\times {10}^{-8}$ s, $\tau =1.22\times {10}^{-14}$ s, and ${\Gamma }_0=1.2$. The parameters taken from Ref. 7 are $4\pi M_{\mathrm{S}}=21000$ G and $H_{\mathrm{S}}=-15000$ G, and that from Ref. 40 is $\gamma =0.018$ GHz/G (derived from $g=2.09$ for bulk Fe). The dotted line is a superposition of Fig. 4 in Ref. 7 reflecting the two-magnon contribution and the Gilbert contribution (denoted as ${\alpha }_1$ in the text), which is linear in the frequency.

Figure 9

(top) Angular dependence of the total peak-to-peak linewidth $\Delta H_{\mathrm{T}}$ for various frequencies and (bottom) all contributions $\Delta H_{\eta }$ for $f=10$ GHz with $\beta =0.5$, $\alpha =0.01$, $T_2=5\times {10}^{-8}$ s, $\tau =1.7\times {10}^{-14}$ s, and ${\Gamma }_0=1.1$. The parameters are taken from Ref. 16: $4\pi M_{\mathrm{S}}=16980$ G, $H_{\mathrm{S}}=-3400$ G, and $\gamma =0.019$ GHz/G.