In this work, we patterned a two-dimensional electron gas (2DEG) on the surface of a ${\mathrm{SrTiO}}_3$ thin film grown homoepitaxially on ${\mathrm{SrTiO}}_3$ by hybrid molecular beam epitaxy (hMBE). We explored the superconducting dome in this material system by tuning electron density with ionic liquid gating. We found superconducting transitions up to 503 mK near an optimal electron density of approximately $3\times$ ${10}^{13}$ ${\mathrm{cm}}^{-2}$. This is a meaningful increase from the typical optimal transition near 350 mK in similar 2DEGs on ${\mathrm{SrTiO}}_3$ single crystal substrate surfaces. Systematic tuning of 2DEG electron density revealed a consistent BCS scaling between superconducting critical temperature, coherence length, and electron mean free path. Substantial variation of transition width across the dome was described by a paraconductivity model combining Aslamazov-Larkin and Maki-Thompson contributions.

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• Transport phenomena
• Superconductors
• Thin films
• Transition metal oxides
• Two-dimensional electron gas
• Molecular beam epitaxy