We investigate extremal black holes with scalar hair in the generic Einstein-Euler-Heisenberg (EEH)-scalar theory with two scalar couplings to the Maxwell term. One is an exponential form with coupling constant $\alpha$ and the other is the polynomial coupling. We consider the cold horizon of Einstein–Euler–Heisenberg black holes described by mass $M$ and magnetic charge $P$. Fixing the Euler-Heisenberg parameter $\epsilon =0.03$ with $M=1$, we study spontaneous scalarization at the linear level, through the existence curves ${\alpha }_n(q)$ with $q=P/M$ and construct fully backreaction solutions. As extremality ($q=q_e$) is approached, all ${\alpha }_n(q)$ accumulate, following ${\alpha }_n-{\alpha }_c\sim 1/{\ln }^2(q_e-q)$, at the critical branch ${\alpha }_c$ fixed by the Breitenlohner-Freedman bound for the near-horizon throat. We obtain scalarized extremal black hole (SEBH) with constant scalar hair and it is recovered exactly from the entropy function approach on the near-horizon throat. Its entropy is an attractor invariant, being independent of the asymptotic modulus, so the scalar hair remains secondary. Exploiting this constant scalar, we seek SEBHs with scalar hair keeping its charge $Q_s$ for the zero asymptotic scalar (${\varphi }_{\infty }=0$). In the $(q,\alpha )$ plane, we observe that these extremal solutions occupy $q\ge q_e$. Hence, the existence curves ${\alpha }_n(q)$ existing for $q\le q_e$ and the extremal branches bound the scalarized domain from opposite sides and they meet only at $q=q_e$.

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• Alternative gravity theories
• Classical black holes
• General relativity