Solvable Two-dimensional Dirac Equation with Matrix Potential: Graphene in External Electromagnetic Field

It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different forms of external potentials but, as a rule, depending on one spatial variable only. Here, we shall find analytically the solutions for a wide class of combinations of matrix and scalar external potentials which physically correspond to applied mutually orthogonal magnetic and longitudinal electrostatic fields, both depending really on two spatial variables. The main tool for this progress was provided by supersymmetrical (SUSY) intertwining relations, namely, by their most general - asymmetrical - form proposed recently by the authors. Such SUSY-like method is applied in two steps similarly to the second order factorizable (reducible) SUSY transformations in ordinary Quantum Mechanics.