  Domain wall morphologies in ferroelectrics are believed to be largely shaped
by electrostatic forces. Here, we show that for conducting domain walls, the
morphology also depends on the details of the charge-carrier band structure.
For concreteness, we focus on transition-metal perovskites like BaTiO$_3$ and
SrTiO$_3$. These have a triplet of $t_{2g}$ orbitals attached to the Ti atoms
that form the conduction bands when electron doped. We solve a set of coupled
equations -- Landau-Ginzburg-Devonshire (LGD) equations for the polarization,
tight-binding Schr\"odinger equations for the electron bands, and Gauss' law
for the electric potential -- to obtain polarization and electron density
profiles as a function of electron density. We find that at low electron
densities, the electron gas is pinned to the surfaces of the ferroelectric by a
Kittel-like domain structure. As the electron density increases, the domain
wall evolves smoothly through a zigzag head-to-head structure, eventually
becoming a flat head-to-head domain wall at high density. We find that the
Kittel-like morphology is protected by orbital asymmetry at low electron
densities, while at large electron densities the high density of states of the
multiorbital band structure provides effective screening of depolarizing fields
and flattens the domain wall relative to single-orbital models. Finally, we
show that in the zigzag phase, the electron gas develops tails that extend away
from the domain wall, in contrast to na\"{i}ve expectations.
