Magnetic breakdown investigated by means of the magnetic-field-containing relativistic tight-binding approximation method

We investigate the magnetic breakdown (MB) phenomena by means of the recently proposed magnetic-field-containing relativistic tight-binding approximation (MFRTB) method [Phys. Rev. B 91, 075122 (2015)]. In the MFRTB method, the MB phenomena can be described as the electron hopping between adjacent semiclassical orbits. It is shown that a set of magnetic energy bands is generated by the MB in the cluster that corresponds to the semiclassical energy level. It is also found that magnetic energy bands originating from the MB and those originating from the semiclassical orbit lying on the constant energy surface are hybridized to each other. Such hybridization leads to various subclusters that correspond to energy states of the so-called forbidden orbits. Magnetic oscillations related to the MB occur when the subcluster changes from the occupied state to unoccupied one and vice versa.

Figure 1

Schematic view of the Fermi surface (dots line) and electronic orbits containing the MB points (solid lines denoted by MB1, MB2, MB3, and MB4). The arrows denote the direction of the orbital motion for electrons, which is predicted by the semiclassical approach. Note that the MB2, MB3, and MB4 orbits are so-called forbidden orbits, parts of which are incompatible with the direction of the orbital motion predicted by the semiclassical approach.

Figure 2

Fourier transformation of the magnetization for the case of $\eta =0.52$.

Figure 3

Dependence of the Fourier transformation of the magnetization on the TB parameter.

Figure 4

Dependence of the magnetization and the Fermi energy on the magnitude of the magnetic field ($p/q$). The vertical dashed lines indicate the positions of the dHvA and MB oscillations.

Figure 5

Overall view of the magnetic energy-band structure for the case of $p/q=17/151\approx 1/8.882$. The position of the Fermi energy ($E_F$) for the case of $\eta =0.52$ is indicated by the red line and arrow.

Figure 6

Dependence of magnetic energy bands near the center on $p/q$, the values of which are (a) $61/487(\approx 1/7.983)$, (b) $25/199(\approx 1/7.960)$, (c) $24/191(\approx 1/7.958)$, and (d) $21/167(\approx 1/7.952)$, respectively. The position of the Fermi energy ($E_F$) for the case of $\eta =0.52$ is indicated by the red line and arrow.

Figure 7

Magnetic-field dependence of magnetic energy bands for the higher magnetic-field case, the $p/q$ values of which are (a) $59/457(\approx 1/7.746)$, (b) $25/193(\approx 1/7.720)$, (c) $26/199(\approx 1/7.654)$, (d) $25/191(\approx 1/7.640)$, (e) $50/379(\approx 1/7.580)$, (f) $24/179(\approx 1/7.458)$, (g) $29/211(\approx 1/7.276)$, and (h) $48/349(\approx 1/7.270)$, respectively. The position of the Fermi energy ($E_F$) for the case of $\eta =0.52$ is indicated by the red line and arrow.