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Evolution of the Fe-3$d$ impurity band state as the origin of high Curie temperature in p-type ferromagnetic semiconductor (Ga,Fe)Sb

(Ga$_{1-x}$,Fe$_x$)Sb is one of the promising ferromagnetic semiconductors for spintronic device applications because its Curie temperature ($T_{\rm C}$) is above 300 K when the Fe concentration $x$ is equal to or higher than ~0.20. However, the origin of the high $T_{\rm C}$ in (Ga,Fe)Sb remains to be elucidated. To address this issue, we use resonant photoemission spectroscopy (RPES) and first-principles calculations to investigate the $x$ dependence of the Fe 3$d$ states in (Ga$_{1-x}$,Fe$_x$)Sb ($x$ = 0.05, 0.15, and 0.25) thin films. The observed Fe 2$p$-3$d$ RPES spectra reveal that the Fe-3$d$ impurity band (IB) crossing the Fermi level becomes broader with increasing $x$, which is qualitatively consistent with the picture of double-exchange interaction. Comparison between the obtained Fe-3$d$ partial density of states and the first-principles calculations suggests that the Fe-3$d$ IB originates from the minority-spin ($\downarrow$) $e$ states. The results indicate that enhancement of the interaction between $e_\downarrow$ electrons with increasing $x$ is the origin of the high $T_{\rm C}$ in (Ga,Fe)Sb.