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

Abstract

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.

Calculation of magnetic oscillations via the magnetic-field-containing relativistic tight-binding approximation method: Revisiting the de Haas–van Alphen effect

“…In this subsection, we briefly review the MFRTB method that has recently been developed [11]. The Dirac equation for an electron that moves in a uniform magnetic field and periodic potential of the crystal is given by…”

“…(4) is treated as the perturbation term [11]. We also neglect the integrals involving three different centres.…”

“…The vector k denotes the wave vector that is defined through the magnetic Bloch theorem [11], which is mentioned in Sec. II B.…”

“…However, unfortunately, it has been difficult to solve them directly [6,10]. Most recently, we have developed the magnetic-fieldcontaining relativistic tight-binding approximation (MFRTB) method that enables us to calculate electronic structures of materials immersed in a uniform magnetic field [11]. This method is the first-principles calculation method that is applicable to various kinds of realistic materials immersed in a uniform magnetic field [11].…”

“…Besides the description of the dHvA effect, the MFRTB method is also expected to be used for revealing the mechanism of the elastic softening of the boron-doped silicon immersed in the magnetic field [12][13][14][15], and for solving the Kohn-Sham equations of the current-density functional theories [16][17][18][19][20][21][22]. In the previous paper, we have applied this method to the crystalline silicon immersed in the magnetic field and revealed the specific structures of the energy spectrum [11].…”