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Effect of interfacial spin mixing conductance on gyromagnetic ratio of Gd substituted Y$_{3}$Fe$_{5}$O$_{12}$

Due to its low intrinsic damping, Y$_3$Fe$_5$O$_{12}$ and its substituted variations are often used for ferromagnetic layer at spin pumping experiment. Spin pumping is an interfacial spin current generation in the interface of ferromagnet and non-magnetic metal, governed by spin mixing conductance parameter $G^{\uparrow\downarrow}$. $G^{\uparrow\downarrow}$ has been shown to enhance the damping of the ferromagnetic layer. The theory suggested that the effect of $G^{\uparrow\downarrow}$ on gyromagnetic ratio only come from its negligible imaginary part. In this article, we show that the different damping of ferrimagnetic lattices induced by $G^{\uparrow\downarrow}$ can affect the gyromagnetic ratio of Gd-substituted Y$_3$Fe$_5$O$_{12}$.