Material-scale effective-band-edge theory of charged-domain-wall bound states in α-In2Se3

Charged ferroelectric domain walls in two-dimensional semiconductors provide a route to rewritable one-dimensional electronic channels, but the relation between atomic reconstruction and transverse confinement is not transparent from either macroscopic electrostatics or first-principles calculations alone. Motivated by recent atomic-resolution studies of α-In2Se3, where head-to-head walls contain a nonpolar β-like intercalated layer whereas tail-to-tail walls remain atomically abrupt, we develop a material-scale effective-band-edge theory for transverse domain-wall bound states. The material input is used as deliberately broad consistency windows, not as a direct fit to digitized density- functional or spectroscopic band-edge profiles. The smooth sector is exactly solvable: a lowest-order local band-edge ansatz reduces neutral walls to the Pöschl-Teller class and charged walls to the asymmetric Rosen-Morse class. This yields closed-form spectra, wave-function skewness, and exact delocalization boundaries. The same solution gives a useful constraint on smooth confinement: states near the localization boundary collapse onto the lower continuum edge, so smooth band bending alone does not, within the material windows considered here, simultaneously reproduce the HH depth, width, and center locking. We then add the minimal HH structural correction, represented by a central short-range β-layer lowering, and analyze both its δ-layer limit and finite- width implementations. Material-scale anchoring to effective masses, nanometer-scale wall widths, and structural energy scales relevant to α-In2Se3 supports a branch-selective picture: the abrupt TT wall is naturally described as a shallow hole-like effective band-bending channel, whereas the reconstructed HH wall is described by an electron-like smooth baseline supplemented by a central β-layer lowering. The theory provides a compact intermediate description connecting atomistic charged-wall reconstruction to mesoscopic one-dimensional confinement.