Time-resolved investigation of magnetization dynamics of arrays of nonellipsoidal nanomagnets with nonuniform ground states

Time-resolved scanning Kerr microscopy measurements have been performed upon arrays of square ferromagnetic nanoelements of different sizes and for a range of bias fields. The experimental results were compared to micromagnetic simulations of model arrays in order to understand the nonuniform precessional dynamics within the elements. In the experimental spectra acquired from an element of length of 236 nm and thickness of 13.6 nm, two branches of excited modes were observed to coexist above a particular bias field. Below this so-called crossover field, the higher frequency branch was observed to vanish. Micromagnetic simulations and Fourier imaging revealed that modes from the higher frequency branch had large amplitude at the center of the element where the effective field was parallel to the bias field and the static magnetization. Modes from the lower frequency branch had large amplitude near the edges of the element perpendicular to the bias field. The simulations revealed significant canting of the static magnetization and effective field away from the direction of the bias field in the edge regions. For the smallest element sizes and/or at low bias field values, the effective field was found to become antiparallel to the static magnetization. The simulations revealed that the majority of the modes were delocalized with finite amplitude throughout the element while the spatial character of a mode was found to be correlated with the spatial variation in the total effective field and the static magnetization state. The simulations also revealed that the frequencies of the edge modes are strongly affected by the spatial distribution of the static magnetization state both within an element and within its nearest neighbors. Furthermore, the simulations suggest that collective modes may be supported in arrays of interacting nanomagnets, which act as magnonic crystals.

Figure 1

(Color online) (a) An optical micrograph of the sample is shown. (b) The calculated profile of the in-plane (open squares) and out-of-plane (open circles) components of the pulsed magnetic field is shown. (c) A SEM image of the array of 236 nm square elements is shown. (d) A $3\times 3$ model array corresponding to the SEM image in (c) and used in micromagnetic simulations is shown. The inset in (a) shows the coordinate system used in the paper.

Figure 2

(Color online) The static magnetization states of $3\times 3$ model arrays of 236 nm elements simulated for bias field values of 1 kOe and 770 Oe. The grayscale represents the component of magnetization that is parallel to the $y$ direction $\left(M_y\right)$ normalized to the saturation magnetization of $838\text{emu}/{\text{cm}}^3$. The white and black shades represent $+M_y$ and $-M_y$, respectively. In (a) the static state at each bias field (applied parallel to the $x$ axis) was prepared by allowing the magnetization to relax from the uniform state. In (b) the static state at a bias field of 1 kOe (applied $4°$ from the $x$ axis) was prepared by relaxing the magnetization from the uniform state. Subsequently, the static state at 770 Oe was prepared by relaxing the magnetization from the state at 1 kOe.

Figure 3

(Color online) The (a) experimental and simulated spectra of (b) the center element of the $3\times 3$ model array shown in Fig. 1d, (c) a single element from that array, (d) the center element of a $3\times 3$ array of perfect squares, and (e) a single perfect square are shown. The simulations were performed for a 236 nm element with a bias field of 590 Oe applied $4°$ from the $x$ direction. The static magnetization of all the simulated elements was found to be in the $S$ state. It is clear from the simulated spectra that both imperfections in element shape and different magnetic environments give rise to very different spectra.

Figure 4

(Color online) (a) A typical raw time-resolved signal (line with squares), the slowly varying background (solid line), and the signal with the background subtracted (line with circles) are shown for the array of 236 nm elements at a bias field of 270 Oe. (b) The FFT power spectra of the traces in (a) are shown.

Figure 5

(Color online) The dependence of the mode frequency upon the element size is shown for bias field values of (a) 1 kOe and (b) 150 Oe. The shaded spectra were obtained experimentally while simulated spectra are shown with a solid curve. For each element size, the spatial distribution of the FFT magnitude of the modes in the simulated spectra is shown in the insets.

Figure 6

(Color online) The simulated static magnetization states within the center element of $3\times 3$ model arrays are shown. The grayscale represents $M_y$ normalized to $M_s=838\text{emu}/{\text{cm}}^3$, where white and black represent $+M_y$ and $-M_y$, respectively. The bias field was applied $4°$ from the $x$ direction in order to stabilize the $S$ state.

Figure 7

(Color online) Simulated spectra for isolated 236 nm square elements with rounded corners are shown for (a) a square with no edge roughness, (b) a square with roughness along the edges of the element parallel to the bias field, and (c) a square with roughness along edges perpendicular to the bias field. The bias field strength was 770 Oe and was applied $4°$ from the $x$ direction. The static magnetization of all the simulated elements was found to be in the $S$ state. In (d)–(f) the spectra are shown for the same elements as those in (a)–(c) but the magnetization occupies the $C$ state instead. Again the bias field was 770 Oe but was applied along the $x$ direction in order to prevent the $S$ state from being stabilized.

Figure 8

(Color online) The dependence of the mode spectra and spatial character of the different modes upon the bias field is shown for the 637 nm element. The shaded spectra with symbols show the experimental data while the solid red line shows spectra obtained from micromagnetic simulations. The vertical dashed blue lines indicate the frequencies of the modes whose spatial character is shown in the images of the magnitude of the Fourier transform. The images are ordered in terms of increasing frequency.

Figure 9

(Color online) The dependence of the mode spectra and spatial character of the different modes upon the bias field is shown for the 428 nm element. The shaded spectra with symbols show the experimental data while the solid red line shows spectra obtained from micromagnetic simulations. The vertical dashed blue lines indicate the frequencies of the modes whose spatial character is shown in the images of the magnitude of the Fourier transform. The images are ordered in terms of increasing frequency.

Figure 10

(Color online) The dependence of the mode spectra and spatial character of the different modes upon the bias field is shown for the 236 nm element. The shaded spectra with symbols show the experimental data while the solid red line shows spectra obtained from micromagnetic simulations. The vertical dashed blue lines indicate the frequencies of the modes whose spatial character is shown in the images of the magnitude of the Fourier transform. The images are ordered in terms of increasing frequency.

Figure 11

(Color online) The dependence of the mode spectra and spatial character of the different modes upon the bias field is shown for the 124 nm element. The shaded spectra with symbols show the experimental data while the solid red line shows spectra obtained from micromagnetic simulations. The vertical dashed blue lines indicate the frequencies of the modes whose spatial character is shown in the images of the magnitude of the Fourier transform. The images are ordered in terms of increasing frequency.

Figure 12

(Color online) The dependence of the mode spectra and spatial character of the different modes upon the bias field is shown for the 70 nm element. The shaded spectra with symbols show the experimental data while the solid red line shows spectra obtained from micromagnetic simulations. The vertical dashed blue lines indicate the frequencies of the modes whose spatial character is shown in the images of the magnitude of the Fourier transform. The images are ordered in terms of increasing frequency.

Figure 13

(Color online) The experimental spectrum for the 236 nm element array at a bias field of 770 Oe is shown in (a). In (b)–(e) the simulated spectra derived from the average time-resolved response of the center element of $3\times 3$ arrays are shown for different ground states. In (b) and (d) the simulated spectra correspond to the ground state shown for 770 Oe in Figs. 2a, 2b, respectively. In (c) the ground state was similar to that in (b) except that the center element occupied the $C$ state. In (e) all elements occupied the $C$ state. For each case, illustrations and Fourier images (inset) show, respectively, the ground-state magnetization of the array and the center element spatial character for the mode at 10.4 GHz.

Figure 14

(Color online) The averaged spectra for simulated elements that occupy the $C$ and $S$ states are shown in (a) and (b), respectively. In both cases the element size was 236 nm and the bias field was 770 Oe. The corresponding experimental spectrum is shown in (c). The spectrum in (a) is the average of equal contributions from the spectra shown in Figs. 7d, 7e, 7f, 13c, 13e while the spectra in (b) is the average of equal contributions from the spectra shown in Figs. 7a, 7b, 7c, 13b, 13d.

Figure 15

The simulated total effective field within the center element of the $3\times 3$ model arrays is shown. The effective field includes contributions from the applied, demagnetizing, anisotropy, and exchange fields. The grayscale represents the magnitude of the effective field normalized to the bias field except at remanence where the field was normalized to 75 Oe.

Figure 16

(Color online) Cross sections of the simulated total effective field within the center element of the $3\times 3$ model arrays at a bias field of 1 kOe. The sections show the $x$ component of the effective field. Regions of negative effective field exist for all element sizes. The width of the regions, labeled as $\Delta x$, is found to occupy a larger proportion of the total element length as the element size is reduced.

Figure 17

(Color online) Cross sections are shown of (a) the $x$ component of the simulated total effective field and (b) the FFT magnitude within the center element of the $3\times 3$ model array of 637 nm elements. The sections are shown for four different values of the bias field.

Figure 18

(Color online) Cross sections are shown of (a) the $x$ component of the simulated total effective field and (b) the FFT magnitude within the center element of the $3\times 3$ model array of 236 nm elements. The sections are shown for four different values of the bias field.

Figure 19

(Color online) Cross sections are shown of (a) the $x$ component of the simulated total effective field and (b) the FFT magnitude within the center element of the $3\times 3$ model array of 70 nm elements. The sections are shown for four different values of the bias field.

Figure 20

(Color online) Simulated Fourier images of FFT magnitude and phase are shown in (a) for two modes with frequencies of 6.75 and 7.0 GHz excited in the center element of a $3\times 3$ array of 236 nm elements at a bias field of 270 Oe. In (b) the FFT magnitude is shown for all elements in the $3\times 3$ array.