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. Published by the American Physical Society 2024