Position-dependent mass effects on the electronic states of a quantum torus

Geometric effects affect the electronic properties of charged particles in nanostructured semiconductor systems. It is well known that regions of high curvature can induce additional electronic confinement. Motivated by these curvature-induced position-dependent mass effects, we investigate the bound states in a bilayer-graphene toroidal nanostructure with a position-dependent effective mass. Using the effective-mass approximation within a single-band approach, we solve the stationary Schrödinger-like equation in toroidal coordinates, including the geometry-induced da Costa potential and a curvature- and position-dependent effective mass. The mass profile is constructed from a scalar invariant formed by the mean and Gaussian curvatures of the surface, inspired by the local curvature density entering Willmore-type elastic energies. This allows us to reduce the problem to a one-dimensional Hermitian Schrödinger-like equation with a mass-dependent effective potential. Our results show that the curvature-mass coupling controls both the depth and topology of the effective confinement, that negative values of the coupling can split the inner-equator geometric well into two lateral wells separated by a repulsive barrier, and that finite azimuthal angular momentum can shift the dominant localization region toward the outer equator of the torus.