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The rich phase diagram of the two dimensional electron gas (2DEG) at the\STO/\LAO interface is probed using Hall and longitudinal resistivity. Thanksto a special bridge design we are able to tune through the superconductingtransition temperature T$_c$ and to mute superconductivity by either adding orremoving carriers in a gate bias range of a few volts. Hall signal measurementspinpoint the onset of population of a second mobile band right at the carrierconcentration where maximum superconducting T$_c$ and critical field H$_c$occur. These results emphasize the advantages of our design, which can beapplied to many other two dimensional systems assembled on top of a dielectricsubstrate with high permittivity.

Phase Diagram of Micron-Size Bridges of SrTiO$_3/$LaAlO$_3$ Interface: Link Between Multiple Band Structure and Superconductivity