Spontaneous scalarization of charged black holes at the approach to extremality

We study static, spherically symmetric and electrically charged black hole solutions in a quadratic Einstein-scalar-Gauss-Bonnet gravity model. Very similar to the uncharged case, black holes undergo spontaneous scalarization for sufficiently large scalar-tensor coupling $gamma$ - a phenomenon attributed to a tachyonic instability of the scalar field system. While in the uncharged case, this effect is only possible for positive values of $gamma$, we show that for sufficiently large values of the electric charge $Q$ two independent domains of existence in the $gamma$-$Q$-plane appear: one for positive $gamma$ and one for negative $gamma$. We demonstrate that this new domain for negative $gamma$ exists because of the fact that the near-horizon geometry of a nearly extremally charged black hole is $AdS_2times S^2$.This new domain appears for electric charges larger than approximately 74$%$ of the extremal charge. For positive $gamma$ we observe that a singularity with diverging curvature invariants forms outside the horizon when approaching extremality.