Fig. 3: Micromagnetic simulations.

a The static magnetization at 0 T after a magnetic field scan from a saturation field along the +X axis to zero. The orange dot is the detection position. (The scale bar in Fig. 3a is 5 μm) (b) The FFT spectrum of the computed magnetization time series at the orange dot marked in Fig. 3a under a microwave field with frequency f₀ = 2.87/3 GHz. The errors margins of the FFT spectrum are plotted as the gray region. c The dispersion relation of spin waves along the +X axis under a zero magnetic field. The green color and black curve are obtained from the simulations and Eq. (1), respectively. The blue curve depicts the dispersion of the surface spin waves⁴⁷. d The mode profiles of spin waves with the frequencies f₀, 2f₀ and 3f₀. f₀ is the frequency of an external RF magnetic field far below the spin wave gap of the bulk material (Fig. 3c) that resonates with the low-frequency edge modes. Since it penetrates the entire sample, it also reactively (non-dissipatively) drives the bulk magnetization as seen in the upper panel of Fig. 3d. The middle panel of Fig. 3d illustrates that the waves with frequency 2f₀ below the band gap decay away from the edge. Those at 3f₀ belong to the continuous spin-wave spectrum above the band gap and may propagate, as shown in the lower panel of Fig. 3d.