Eigen spectra and wave functions of the massless Dirac fermions under the nonuniform magnetic fields in graphene

In this research, firstly, by using the new form of Dirac-Weyl equation and the series method with submitting more suitable details, the energy spectrum and wave functions of the massless Dirac fermions are calculated under the inhomogeneous and q-deformed spatially magnetic fields. Although, we discussed about the results of the energy levels, further, we obtained the wave function as the Hessenberg determinant with calculating the elements of it as exact. On the other hand, by using the Mellin-Barnes integral representation and Hurwitz zeta function, we have achieved the thermodynamic physical quantities of the Dirac-Weyl fermions in the absence of a magnetic field for inside of the graphene quantum dot. Finally, our numerical results for the wave functions and probability densities are presented too.