  We demonstrate computationally the existence of magnetoelectric multipoles,
arising from the second order term in the multipole expansion of a
magnetization density in a magnetic field, in noncentrosymmetric magnetic
metals. While magnetoelectric multipoles have long been discussed in the
context of the magnetoelectric effect in noncentrosymmetric magnetic
insulators, they have not previosuly been identified in metallic systems, in
which the mobile carriers screen any electrical polarization. Using
first-principles density functional calculations we explore three specific
systems: First, a conventional centrosymmetric magnetic metal, Fe, in which we
break inversion symmetry by introducing a surface, which both generates
magnetoelectric monopoles and allows a perpendicular magnetoelectric response.
Next, the hypothetical cation-ordered perovskite, SrCaRu$_2$O$_6$, in which we
study the interplay between the magnitude of the polar symmetry breaking and
that of the magnetic dipoles and multipoles, finding that both scale
proportionally to the structural distortion. Finally, we identify a hidden
antiferromultipolar order in the noncentrosymmetric, antiferromagnetic metal
Ca$_3$Ru$_2$O$_7$, and show that, while its competing magnetic phases have
similar magnetic dipolar structures, their magnetoelectric multipolar
structures are distinctly different, reflecting the strong differences in
transport properties.
