Band structure for relativistic charge carriers submitted to a helical electric field

We study the relativistic dynamics of electrons submitted to a uniform harmonic electric fields longitudinally applied to the electron velocity. In order to obtain explicitly the set of time-dependent equations for the bi-spinors components, we write down the corresponding Dirac equation for this system. By performing a suitable transformation to this set of equations, we get a matrix equation whose eigenvalues can be found directly. We found analytically the band structure and show a break of degeneracy and the existence of three bands: two total and one partial. The thicker band preventing from the Schwinger mechanisms can be shortened by proper combination of the frequency and amplitude of the applied electric field.