  We explore the emergence of spin-polarised flat-bands at head-to-head domain
walls in a recently predicted class of antiferromagnetic topological insulators
hosting planar magnetisation. We show, in the framework of quantum well
physics, that by tuning the width of a domain wall one can control the
functional form of the bound states appearing across it. Furthermore, we
demonstrate the effect that the parity of the number of layers in a multilayer
sample has on the electronic dispersion. In particular, the alignment of the
magnetisation vectors on the terminating surfaces of odd layer samples affords
particle-hole symmetry leading to the presence of linearly dispersing
topologically non-trivial states around $E = 0$. By contrast, the lack of
particle-hole symmetry in even layer samples results in a gapped system, with
spin-polarised flat-bands appearing either side of a band gap, with
characteristic energy well within terahertz energy scales. In addition to being
a versatile platform for the development of spintronic devices, when many-body
interactions are accounted for we predict that these flat-bands will host
strong correlations capable of driving the system into novel topological
phases.
