“…Films with 46 nm ≤ t ≤ 236 nm possess IP easy axes while the 28 nm film has an OOP easy axis of magnetization, which was confirmed via IP-magnetometry and OOP p -MOKE measurements (see Supplementary Figure 3e). The total magnetic anisotropy of a (111)-oriented TmIG film, neglecting growth and interfacial anisotropies, has contributions from shape anisotropy ( K shape ), cubic magnetocrystalline anisotropy ( K mc ), and magnetoelastic anisotropy ( K me ) ,, i.e. K e f f = K s h a p e + K m c + K m e = − 1 2 μ 0 M S 2 − K 1 12 − 9 4 λ 111 c 44 true( π 2 − β true) where K 1 is the magnetocrystalline anisotropy coefficient, λ 111 is the magnetostriction along the [111] direction, c 44 is the shear modulus, and β is the corner angle of the rhombohedrally distorted unit cell. For a negative magnetostriction (λ 111 = −5.2 × 10 –6 for bulk TmIG), the tensile IP strain, which results from the difference in lattice parameters ( a GSGG = 12.57 Å and a TmIG = 12.32 Å) promotes PMA ( K eff > 0). ,, PMA is expected for fully strained films (28 nm), but strain relaxation in thicker films reduces the magnetoelastic contribution, and the easy axis reorients to the IP direction …”