Relativistic tight-binding approximation method for materials immersed in a uniform magnetic field: Application to crystalline silicon

We present a relativistic tight-binding (TB) approximation method that is applicable to actual crystalline materials immersed in a uniform magnetic field. The magnetic Bloch theorem is used to make the dimensions of the Hamiltonian matrix finite. In addition, by means of the perturbation theory, the magnetic hopping integral that appears in the Hamiltonian matrix is reasonably approximated as the relativistic hopping integral multiplied by the magnetic-field-dependent phase factor. In order to calculate the relativistic hopping integral, the relativistic version of the so-called Slater-Koster table is also given in an explicit form. We apply the present method to crystalline silicon immersed in a uniform magnetic field, and reveal its energy-band structure that is defined in the magnetic first Brillouin zone. It is found that the widths of energy-bands increase with increasing the magnetic field, which indicates the magnetic-field dependence of the appropriateness of the effective mass approximation. The recursive energy spectrum, which is the so-called butterfly diagram, can also be seen in the k-space plane perpendicular to the magnetic field