We study theoretically the magic-angle twisted bilayer graphene with proximity-induced Ising and Rashba spin-orbit couplings on the top layer. Topological flat bands (with three distinct phases) are generically realized by the spin-orbit couplings. Using a mean field analysis, we find that (partial) valley polarization prevails for a wide range of doping, suppressing the usual superconductivity with a pairing between time-reversal partners. Remarkably, we uncover that observable unconventional intervalley interband phonon-mediated superconductivity (with the highest $T_c\approx 1.2\mathrm{K}$) can coexist with strong valley imbalance due to the approximate Fermi-surface nesting between two flat bands not related by time-reversal symmetry, and the dominant pairing is an intersublattice Ising pairing, corresponding to a mixture of $p$ and $d$ waves. In contrast, the intrasublattice Ising phonon-mediated superconductivity with $s$- and $f$-wave mixing emerges in the absence of valley imbalance. Our work reveals an unprecedented route of realizing unconventional superconductivity.

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• Spin-orbit coupling
• Superconductivity
• Topological materials
• Twisted bilayer graphene