Longitudinal resonance for thin film ferromagnets with random anisotropy

At the microscopic level, individual spins in ferromagnets with random anisotropy tip transversely with distinct local angles relative to the magnetization $\vec{M}$. When driven by an rf field along the equilibrium ${\vec{M}}_0$, which changes $dM={\widehat{M}}_0·d\vec{M}$, the transverse tippings rotate about ${\vec{M}}_0$, corresponding to a new macroscopic collective angle $\varphi$ about ${\vec{M}}_0$. The coupling of $dM$ and $\varphi$ leads to a new longitudinal mode that in bulk has a frequency that is largely independent of field $H$. A longitudinal mode has been observed in thin films of ferromagnets with random anisotropy, but its frequency is $H$ dependent; for $H$ at angle $\theta$ to the film normal and fixed resonator frequency $f$, $Hcos\theta$ was constant to angles of ${80}^{\circ }$, with $H$ saturating for larger angles. When the demagnetization field is included the theory yields such an $H$ vs $\theta$ for $\theta <{80}^{\circ }$, thus providing evidence for the predicted angle $\varphi$. However, lower frequency resonators are needed to manifest the predicted macroscopic anisotropy energy.

Figure 1

Relative intensity $I$ data from $I\left(\alpha \right)$ from Ref. [2], establishing longitudinal nature of new mode. Theoretical curve is ${cos}^2\alpha$. Reproduced from G. Suran, E. Boumaiz, and J. Ben Youssef, J. Appl. Phys. 79, 5381 (1996), with the permission of AIP Publishing.

Figure 2

Data points for $\omega$ vs $H$ measurements from Ref. [2] for $\alpha =0$ and $\theta ={90}^{\circ }$. Theoretical curve from Eq. (26) for $f_0=2$ GHz, $M=1.0\times {10}^6$ A/m, and ${\chi }_L=3.6\times {10}^{-3}$.

Figure 3

Data points from Ref. [24], for $\alpha =0$. Open circles correspond to data with perhaps 25% uncertainty. Theoretical curves from Eq. (14) (solid) and Eq. (15) (dashed) with $M={10}^6$ A/m and $sin\beta =0.092$. $\theta$ is measured relative to the normal to the plane.

Figure 4

Data points from Ref. [25]. Theoretical curves from Eq. (14) (solid) and Eq. (15) (dashed) with $M={10}^6$ A/m and $sin\beta =0.054$.

Figure 5

Geometry for field $H_{zy}$ with cooling field component and field $H_{zx}$ with no cooling field component.