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.
