Origin of the superconducting dome at $\mathrm{LaAlO_3}$/$\mathrm{SrTiO_3}$ interfaces
In plain words
Where $\mathrm{LaAlO_3}$ meets $\mathrm{SrTiO_3}$, a thin conducting sheet forms that superconducts below about $0.3\,\mathrm{K}$, and a gate voltage tunes its transition temperature up and then down. What causes this dome-shaped curve is unresolved.
Precise statement
The two-dimensional electron gas at (001) $\mathrm{LaAlO_3}$/$\mathrm{SrTiO_3}$, with sheet density $n_{s}$ of order $10^{13}\,\mathrm{cm}^{-2}$, shows a gate-tuned dome $T_{c}(V_{g})$ with maximum about $0.2\ \text{to}\ 0.3\,\mathrm{K}$ near a Lifshitz transition where $d_{xz}/d_{yz}$ subbands fill and Rashba spin-orbit coupling grows. Determine the cause of the decrease of $T_{c}$ on both sides of the maximum (multiband pairing, loss of superfluid stiffness and phase fluctuations, change of dielectric screening) from a calculation matching $T_{c}(n_{s})$, superfluid stiffness and tunneling gap data. An answer is a quantitative model of the dome.
What would settle it
Simultaneous measurement of gap, superfluid stiffness and $T_{c}$ versus gate voltage on one device, reproduced by a model.
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