  We use symmetry analysis and density functional theory to characterize
antiferromagnetic (AFM) materials which have a finite equilibrium magnetization
density on particular surface terminations. A nonzero magnetic dipole moment
per unit area or "surface magnetization" can arise on particular surfaces of
many AFMs due to the bulk magnetic symmetries. Such surface magnetization plays
an essential role in numerous device applications, from random-access
magnetoelectric (ME) memory to exchange bias. However, at this point a
universal description of AFM surface magnetization is lacking. We first
introduce a classification system based on whether the surface magnetization is
sensitive or robust to roughness, and on whether the surface of interest is
magnetically compensated or uncompensated in the bulk magnetic ground state. We
show that uncompensated surface magnetization can be conveniently described in
terms of ME multipoles at the local-moment, unit cell level, and demonstrate
that the symmetry of the multivalued "multipolization lattice" distinguishes
between roughness-robust and roughness-sensitive surface magnetization. We then
demonstrate that magnetization on bulk-compensated surfaces arises due to ME
multipoles (in addition to higher-order magnetic terms) at the atomic site
level. These can further be understood in terms of bulk ME responses, arising
from the effective electric field resulting from the surface termination. We
also show with density functional calculations that nominally compensated
surfaces in Cr2O3 and FeF2 develop a finite magnetization density at the
surface, in agreement with our predictions based on both group theory and the
linear and higher-order ME response tensors. Our analysis provides a
comprehensive basis for understanding the surface magnetic properties in AFMs,
and has important implications for phenomena such as exchange bias coupling.
