Spontaneous Magnetization of an Axion Domain Wall and a Primordial Magnetic Field

We show that axion domain walls gain spontaneous magnetization in early Universe by trapping either electrons or positrons with their spins polarized. The reason is that the walls produce an attractive potential for these particles. We argue that the wall bounded by an axionic superconducting string leaves a magnetic field after its decay. We obtain a field $\sim {10}^{-23\mathrm{}}\mathrm{G}$ on the scale of horizon at the recombination.