  In many growth processes particles are highly mobile in an active layer at
the surface, but are relatively immobile once incorporated in the bulk. We
study models in which atoms are allowed to interact, equilibrate, and order on
the surface, but are frozen in the bulk. Order parameter correlations in the
resulting bulk material are highly anisotropic, reflecting its growth history.
In a flat (layer by layer) growth mode, correlations perpendicular to the
growth direction are similar to a two dimensional system in equilibrium, while
parallel correlations reflect the dynamics of such a system. When the growing
film is rough, various couplings between height and order parameter
fluctuations are possible. Such couplings modify the dynamic scaling properties
of surface roughness, and may also change the critical behavior of the order
parameter. Even the deterministic growth of the surface profile can result in
interesting textures for the order parameter.
