Sample fabrication

A wafer of epitaxially grown two-µm-thick film of BIG (Bi_0.7Gd_0.3Lu_2.0Ga_0.8Fe_4.2O₁₂) on gadolinium gallium garnet (GGG) was cut into 1 cm × 1 cm pieces via dicing saw. The film was grown by liquid-phase-epitaxy on (001)-oriented GGG substrate at II-VI, Inc. Samples were cleaned by sonication in acetone/IPA/DI water spray and blow-dried with nitrogen. The BIG film thickness was reduced from 2 µm to 300 nm by wet etching in ortho-phosphoric acid (85% from Sigma-Aldrich) for about an hour. The acid was heated to 130 °C and stirred using a magnetic stirrer for uniform etching of the film. The thickness of the film was measured using V-vase ellipsometry. Nano-patterns were then fabricated using a 100 KeV e-beam lithography system (VISTEC EBPG 5000+). A 250 nm-thick ZEP positive e-beam resist was spin-coated on the substrate together with a 30 nm-thick Au layer on top to suppress electrical charging of the dielectric garnet film during electron-beam exposure. 2D circular disk patterns were patterned on ZEP resist by uniform e-beam exposure at a dose of 150 µC/cm² under proximity effect correction (PEC). The Au layer was first removed by wet etching in a gold etchant solution and afterwards the resist was developed in an amyl acetate solution. The resist patterns were then transferred onto the BIG film by sputter-etching in an argon-ion milling system (Intlvac Nanoquest) with ion source parameters: beam voltage – 200 V, accelerating voltage – 24 V, beam current – 70 mA and plasma forward power – 72 Watt. The sample stage temperature was kept at 6 °C throughout the etching duration of about 90 min. The resist was then removed using resist remover N-methyl-2-pyrrolidine (NMP) by heating at 80 °C for about half an hour.

Finally, the sample was wet-etched in phosphoric acid at 130 °C for ~1 min to smoothen the side walls of the nanopillars. The higher etch rate towards the tip of the nanopillars due to stirred acid, produced the truncated-cone shape of the nanopillars. The roughness of the wet-etched surface is usually less than 1 nm for wet-etched film, however the trade-off here is the imperfect nanopillars. The AFM images are shown in Supplementary Note ³.

The film has in-plane anisotropy with in-plane saturation field of H_in = 30 Oe, out-of-plane saturation field of H_out = 2 kOe, and saturation magnetization 4πMs=860G.

Iron-garnet film was described by the following spectral dependence of permittivity and gyration constant:

and g=0.8⋅10−2−0.004λ+6.4⋅10−6λ2−5⋅10−9λ3+14⋅10−13λ4), (λ in µm) which was obtained previously for a similar smooth film³⁸.