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Compact silicon integrated lasers are of significant interest for various applications. We present a detailed investigation for realizing sub-mm long on-chip laser structures operating at λ = 1.533 µm on the silicon-on-insulator photonic platform by combining a multi-segment silicon waveguide structure and a recently demonstrated erbium-doped thin film deposition technology. Quarter-wave shifted distributed feedback structures (QWS-DFB) are designed and a detailed calculation of the lasing threshold conditions is quantitatively estimated and discussed. The results indicate that the requirements for efficient lasing can be obtained in various combinations of the designed waveguide DFB structures. Overall, the study proposes a path to the realization of compact (< 500 µm) on-chip lasers operating in the C-band through the hybrid integration of erbium-doped aluminum oxide processed by atomic layer deposition in the silicon photonic platform and operating under optical pumping powers of few mW at 1,470 nm.