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By combined top- and backgating, we explore the correlation ofsuperconductivity with band filling and electron confinement at theLaAlO$_3$-SrTiO$_3$ interface. We find that the top- and backgate voltages havedistinctly different effects on the superconducting critical temperature,implying that the confining potential well has a profound effect onsuperconductivity. We investigate the origin of this behavior by comparing thegate-dependence of $T_c$ to the corresponding evolution of the band fillingwith gate voltage. For several backgate voltages, we observe maximum $T_c$ toconsistently coincide with a kink in tuning the band filling for high topgatevoltage. Self-consistent Schr\"odinger-Poisson calculations relate this kink toa Lifshitz transition of the second $d_{xy}$ subband. These results establish amajor role for confinement-induced subbands in the phase diagram of SrTiO$_3$surface states, and establish gating as a means to control the relative energyof these states.

Correlation between Superconductivity, Band Filling and Electron Confinement at the LaAlO$_{3}$-SrTiO$_{3}$ Interface