Ferromagnetic resonance force spectroscopy of individual submicron-size samples

We review how a magnetic-resonance force microscope (MRFM) can be applied to perform ferromagnetic resonance spectroscopy of individual submicron-size samples. We restrict our attention to a thorough study of the spin-wave eigenmodes excited in Permalloy (Py) disks patterned out of the same 43.3-nm-thin film. The disks have a diameter of either 1.0 or $0.5\mu \text{m}$ and are quasisaturated by a perpendicularly applied magnetic field. It is shown that quantitative spectroscopic information can be extracted from the MRFM measurements. In particular, the data are extensively compared with complementary approximate models of the dynamical susceptibility: (i) a two-dimensional analytical model, which assumes a homogeneous magnetization dynamics along the thickness, and ii) a full three-dimensional micromagnetic simulation, which assumes a homogeneous magnetization dynamics below a characteristic length scale $c$ and approximates the cylindrical sample volume by a discretized representation with regular cubic mesh of lateral size $c=3.9\text{nm}$. In our analysis, the distortions due to a breaking of the axial symmetry are taken into account; both models incorporating the possibility of a small misalignment between the applied field and the normal of the disks.

Figure 1

(Color online) Schematic representation of the mechanical-FMR setup: the resonance spectra of a small size sample are detected by a magnetic force microscope. The probe consists here of a magnetic sphere glued at the extremity of the cantilever. The excitation comes from a microwave antenna placed underneath the sample.

Figure 2

(Color online) (a) Photograph of the MRFM bottom base showing (1) the microwave antenna, (2) the mica substrate where the Py disks are, (3) the piezoelectric scanner, and (4) the magnetic coil used for a calibrated modulation of the applied field. (b) Section view of the above elements. The top part with the cantilever and the optical detection is not shown.

Figure 3

(Color online) Visualization of the dipolar coupling interaction between a spherical magnetic probe (diameter $\Psi$) and a cylindrical Py disk (diameter $D$) separated by a distance $s$. The magnetic configurations in both objects are assumed to be saturated along $z$. The color lines represent the iso-$B_z$ field lines. The bright and dark shades separate, respectively, the positive and negative coupling regions.

Figure 4

(Color online) Variation in the dipolar magnetic force between a sphere and a disk as a function of the ratio between their diameters, respectively, $\Psi$ and $D$. The continuous line shows the dependence when the two elements are brought into contact $\left(s=0\right)$ and only $\Psi$ is varied. The maximum force occurs when $\Psi =2D$. The dash line shows the behavior when the two elements are separated by $s=2D$ and only $D$ is varied (close to the experimental conditions); the vertical scale has been multiplied by a factor of 50 for clarity purpose.

Figure 5

(Color online) Scanning electron microscopy image of the CoFeSi magnetic sphere glued at the tip of an Olympus BioLever.

Figure 6

(Color online) Color code representation of the eigenmodes basis ${\mathcal{J}}_m^{\ell }\left(\mathit{r}\right)$, where $\ell$ and $m$ indicate, respectively, the number of nodes in the circumferential and radial direction. Only the modes $\ell =0$ couple to a uniform microwave excitation.

Figure 7

(Color online) Calculated spectral deformation induced by the magnetic tip placed at distances $s=3.1$ and $1.0\mu \text{m}$ above the disk samples $D_1=1.0\mu \text{m}$ and $D_2=0.5\mu \text{m}$, respectively. The graphs show the downward shift in field of each resonance mode (indexed by $m$) compared to the unperturbed FMR spectrum (dashed line).

Figure 8

(Color online) SEM images of the two Py disks samples. The images are being viewed at an angle of about $15°$ with respect to the normal of the substrate, thus distortions exist in the vertical direction.

Figure 9

(Color online) Mechanical-FMR spectra of the $D_1=1.0\mu \text{m}$ disk measured at 4.2, 5.6, 7.0, and 8.2 GHz and $s\approx 3.1\mu \text{m}$. The blue circles indicate the positions of the most intense peak. The lines are the analytical predictions of the locus of the $\left(\ell ,m\right)$ modes when ${\theta }_H=4.9°$ and $H_{\text{off}}=-70\text{Oe}$.

Figure 10

(Color online) Mechanical-FMR spectra of the $D_2=0.5\mu \text{m}$ disk measured at 4.2, 5.6, 7.0, and 8.2 GHz and $s\approx 1.0\mu \text{m}$. The lines are the analytical predictions of the locus of the $\left(\ell ,m\right)$ modes when ${\theta }_H=9.5°$ and $H_{\text{off}}=-760\text{Oe}$.

Figure 11

(Color online) Graphic representation of the Cartesian frame $\left(\xi ,\zeta \right)$ (blue) rotated in the direction of the equilibrium magnetization when the applied field makes an angle ${\theta }_H$ with the normal of the disk ($z$ axis).

Figure 12

(Color online) Comparison between the 2D analytical model (oblique lines) and the 3D micromagnetic simulation (horizontal spectra) for the prediction of the resonance fields of a $D_2=0.5\mu \text{m}$ Py disk magnetized at ${\theta }_H=0°$. The crosses indicate the positions of the most intense peaks in the simulated spectra. The continuous lines are the locus of the resonance field of the radial modes $\left(\ell =0,m\right)$ predicted analytically. The dashed lines correspond to the hidden modes $\left(\ell =1,m\right)$.

Figure 13

(Color online) (a) Images of the transverse susceptibility obtained through the 3D micromagnetic simulation for the first three modes (red crosses of Fig. 12) at 8.2 GHz of a $D_2=0.5\mu \text{m}$ Py disk magnetized at ${\theta }_H=0°$. The thick lines indicate the nodes of precession. (b) Image of the transverse susceptibility for the third mode (at $H_{\text{ext}}=8.8\text{kOe}$) of the 5.6 GHz spectrum showing that the magnetization dynamics is not uniform across the thickness of the disk.

Figure 14

(Color online) Comparison between the 2D analytical model (oblique lines) and the 3D micromagnetic simulation (horizontal spectra) for the prediction of the resonance fields of a $D_2=0.5\mu \text{m}$ Py disk magnetized at ${\theta }_H=5°$. The continuous lines are the locus of the resonance field of the radial modes $\left(\ell =0,m\right)$ predicted analytically [dashed lines correspond to the hidden modes $\left(\ell =1,m\right)$]. The important disagreement between the 3D and 2D models originate from the two simplifications in the analytical model (see text).

Figure 15

(Color online) (a) Image of the spatial distribution of the resonant profile for the second peak at 9.2 kOe on the 8.2 GHz simulated spectrum of Fig. 14. The cartography is shown both in the axial and radial middle sections. (b) The mode shown in (a) is now decomposed in the ${\mathcal{J}}_m^{\ell }$ basis. The figure reveals here the spectral weights of the different eigenmodes.

Figure 16

(Color online) Resonance field of the fundamental mode. Blue dots (see Fig. 9) indicate the measured field on the $D_1=1.0\mu \text{m}$ disk. Continuous lines are the analytically predicted angle dependence of $H_{\text{res}}\left({\omega }_s\right)$ for different values of ${\theta }_H$ (lines) with $H_{\text{off}}=-70\text{Oe}$. Also indicated is the measured resonance position in the extended film (red triangle). The dashed line represents the asymptote crossing this point and whose slope is the measured gyromagnetic ratio: the shift with the ${\theta }_H=0°$ line is mainly due to finite-size effects in the disk.

Figure 17

(Color online) Comparison between experiment and analytical model of the amplitude of the peaks for the disk of diameter $1.0\mu \text{m}$ at 5.6 GHz.

Figure 18

(Color online) Frequency dependence of the linewidth measured on both the $D_1=1.0\mu \text{m}$ disk (full circles) and the $D_2=0.5\mu \text{m}$ disk (open boxes). The red triangle indicates the value measured by cavity FMR on the extended film on Si. The mechanical-FMR data are renormalized by ${\gamma }_{\text{eff}}$. The dashed line corresponds to a damping coefficient $\alpha =6\times {10}^{-3}$.