We report the effects of electron doping on the ground state of a diamagnetic semiconductor FeGa$_{3}$ with a band gap of 0.5 eV. By means of electrical resistivity, magnetization and specific heat measurements we have found that gradual substitution of Ge for Ga in FeGa$_{3-y}$Ge$_{y}$ yields metallic conduction at a very small level of $y = 0.006$, then induces weak ferromagnetic (FM) order at $y = 0.13$ with a spontaneous moment of 0.1 $mu_{B}$/Fe and a Curie temperature $T_{C}= 3.3$ K, which continues increasing to $T_{C} = 75$ K as doping reaches $y = 0.41$. The emergence of the FM state is accompanied by quantum critical behavior as observed in the specific heat, $C/T propto -$ln$T$, and in the magnetic susceptibility, $M/B propto T^{-4/3}$. At $y= 0.09$, the specific heat divided by temperature $C/T$ reaches a large value of 70 mJ/K$^{2}$molFe, twice as large as that reported on FeSi$_{1-x}$Ge$_{x}$ for $x_{c}= 0.37$ and Fe$_{1-x}$Co$_{x}$Sb$_{2}$ for $x_{c}=0.3$ at their respective FM quantum critical points. The critical concentration $y_{c}=0.13$ in FeGa$_{3-y}$Ge$_{y}$ is quite small, despite the fact that its band gap is one order of magnitude larger than those in FeSi and FeSb$_{2}$. In contrast, no FM state emerges by substituting Co for Fe in Fe$_{1-x}$Co$_{x}$Ga$_{3}$ in the whole range $0 leq x leq 1$, although both types of substitution should dope electrons into FeGa$_{3}$. The FM instability found in FeGa$_{3-y}$Ge$_{y}$ indicates that strong electron correlations are induced by the disturbance of the Fe 3d - Ga 4p hybridization.
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