Quantum Hall conductance and de Haas van Alphen oscillation in a tight-binding model with electron and hole pockets for (TMTSF)$_2$NO$_3$

Abstract

Quantized Hall conductance and de Haas van Alphen (dHvA) oscillation are studied theoretically in the tight-binding model for (TMTSF)$_2$NO$_3$, in which there are small pockets of electron and hole due to the periodic potentials of anion ordering in the $a$-direction. The magnetic field is treated by hoppings as complex numbers due to the phase caused by the vector potential, i.e. Peierls substitution. In realistic values of parameters and the magnetic field, the energy as a function of a magnetic field (Hofstadter butterfly diagram) is obtained. It is shown that energy levels are broadened and the gaps are closed or almost closed periodically as a function of the inverse magnetic field, which are not seen in a semi-classical theory of the magnetic breakdown. Hall conductance is quantized with an integer obtained by Diophantine equation when the chemical potential lies in an energy gap. When electrons or holes are doped in this system, Hall conductance is quantized in some regions of a magnetic field but it is not quantized in other regions of a magnetic field due to the broadening of the Landau levels. The amplitude of the dHvA oscillation at zero temperature decreases as the magnetic field increases, while it is constant in the semi-classical Lifshitz Kosevich formula.