Lorentzian Quantum Wells in Graphene: The Role of Shape Invariance in Zero-Energy-State Trapping

Confining Dirac fermions in graphene using electrostatic fields is a challenging task. Electric quantum dots created by a scanning tunneling microscope tip can trap zero-energy quasiparticles. The Lorentzian quantum well provides a faithful, exactly solvable approximation for such a potential, hosting zero-energy bound states for certain values of the coupling constant. We show that in this critical configuration, the system can be related to the free-particle model by means of a supersymmetric transformation. The revealed shape invariance of the model greatly simplifies the calculation of the zero modes and naturally explains the degeneracy of the zero energy. © 2024 authors. Published by the American Physical Society.