  We explore universal behaviors of magnetic domain wall driven by the
spin-transfer of an electrical current, in a ferromagnetic (Ga,Mn)(As,P) thin
film with perpendicular magnetic anisotropy. For a current direction transverse
to domain wall, the dynamics of the thermally activated creep regime and the
depinning transition are found to be compatible with a self-consistent
universal description of magnetic field induced domain wall dynamics. This
common universal behavior, characteristic of the so-called quenched
Edwards-Wilkinson universality class, is confirmed by a complementary and
independent analysis of domain wall roughness. However, the tilting of domain
walls and the formation of facets is produced by the directionality of
interaction with the current, which acts as a magnetic field only in the
direction transverse to domain wall.
