Spin-wave excitation modes in thick vortex-state circular ferromagnetic nanodots

We study both experimentally and using micromagnetic simulations how the spin excitation spectra of the vortex-state circular dots made of soft magnetic material change with the dot thickness $t$ in the range $t=20–80\mathrm{nm}$. It is found that in addition to higher-order gyrotropic modes which are nonuniform along the dot thickness and were observed earlier, azimuthal spin-wave modes having curled structure at the dot top and bottom faces appear in the spectrum when increasing the dot thickness. For the dot thickness $t>50\mathrm{nm}$ these “curled” modes become the lowest ones in the spin-wave excitation spectrum. It is also shown that all spin-wave modes with azimuthal index $m=\pm 1$ are hybridized with the vortex gyrotropic modes. However, while “common” azimuthal $(0,\pm 1)$ modes are hybridized with the main gyrotropic ${\mathrm{G}}_0$ mode and reveal large frequency splitting of their doublet, the curled modes can be hybridized with higher-order gyrotropic modes and the doublet frequency splitting vanishes with the dot thickness increase.

Figure 1

Scanning electron microscopy image of fabricated dots with 80 nm thickness (a) and measured hysteresis loops of dots with different thickness (b).

Figure 2

Schematic diagram of the measurement setup and the field configuration.

Figure 3

Experimental (a) and simulated (b) microwave absorption spectra of the circular permalloy dots having 20 nm thickness and 150 nm radius. In (b) blue solid line corresponds to in-plane magnetic excitation field, red dashed to out-of-plane excitation field.

Figure 4

Azimuthal spin-wave mode profiles in a nanodot having 20 nm thickness. (a) and (b) Zeroth azimuthal modes running in the CW (${\mathrm{A}0}_{-}$ mode) and CCW (${\mathrm{A}0}_{+}$) directions, respectively. (c) and (d) First azimuthal modes ${\mathrm{A}1}_{-}$ and ${\mathrm{A}1}_{+}$. Arrows show the running spin-wave mode direction.

Figure 5

Experimental (a) and simulated (b) microwave absorption spectra of the dots having 40 nm thickness; in (b) blue solid line corresponds to in-plane magnetic excitation field, red dashed to out-of-plane excitation field.

Figure 6

Dynamical magnetization profiles of the curled azimuthal spin-wave modes ${\mathrm{C}1}_{-}$ (a) and ${\mathrm{C}1}_{+}$ (b) in 40 nm thick dot at the lower, central, and upper plane of the dot. Color scale is the same as in Fig. 4.

Figure 7

Microwave absorption spectra of 60 nm thick [(a), (b)] and 80 nm thick [(c), (d)] dots. (a) and (c) Experiment. (b) and (d) Simulations. In (b) and (d), blue solid line corresponds to in-plane magnetic excitation field, red dashed to out-of-plane excitation field.

Figure 8

Profiles of the azimuthal spin-wave modes in 80 nm thick dot at the bottom, central, and top planes of the dot. (a) Zeroth azimuthal mode ${\mathrm{A}0}_{+}$. (b) and (c) Curled modes ${\mathrm{C}1}_{-}$ and ${\mathrm{C}1}_{+}$, respectively. (d) Curled mode ${\mathrm{C}3}_{-}$. Color scale is the same as in Fig. 4; arrows show the running mode direction.

Figure 9

Dependencies of simulated spin excitation mode frequencies (a) and absorption peak amplitudes, normalized to the dot thickness (b), on the dot thickness $t$, obtained from micromagnetic simulations; open circles, squares, and diamonds correspond to the gyrotropic modes, up and down triangles to CW ($m=-1$) and CCW ($m=+1$) azimuthal spin-wave modes, respectively.

Figure 10

Static magnetization distribution in 80 nm thick nanodot at the bottom (a) and top (b) dot faces; color scale depicts the distribution of the normalized $m_{\rho }=M_{\rho }/M_s$ magnetization component.

Figure 11

Profiles of the dynamical magnetization of the azimuthal spin-wave modes across the dot thickness. (a) Zeroth azimuthal mode ${\mathrm{A}0}_{-}$. (b) Mode ${\mathrm{A}0}_{+}$. (c) and (d) Curled modes ${\mathrm{C}1}_{-}$ and ${\mathrm{C}1}_{+}$, respectively. Color scale is the same as in Fig. 4.