LAUSR

We theoretically study the de Haas--van Alphen (dHvA) oscillations in the system with changing the topology of the Fermi surface (the Lifshitz transition) by electron dopings. We employ the two-dimensional tight-binding model for $\ensuremath{\alpha}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{I}}_{3}$ under pressure which has two Dirac points in the first Brillouin zone. When this system is slightly doped, there exists two closed Fermi surfaces with the same area and the dHvA oscillations become saw-ooth pattern or inversed sawtooth pattern for both cases of fixed electron filling ($\ensuremath{\nu}$) or fixed chemical potential ($\ensuremath{\mu}$) with respect to the magnetic field, respectively. By increasing dopings, the system approaches the Lifshitz transition where two closed Fermi surfaces are close each other. Then, we find that the pattern of the dHvA oscillations changes. A jump of the magnetization appears at the center of the fundamental period, and its magnitude increases in the case of the fixed electron filling, whereas a jump is separated into a pair of jumps and its separation becomes large in the case of the fixed chemical potential. This is due to the lifting of double degeneracy in the Landau levels. Since this lifting is seen in the two-dimensional Dirac fermion system with two Dirac points, the obtained results in this paper can be applied to not only $\ensuremath{\alpha}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{I}}_{3}$, but also other materials with closely located Dirac points, such as graphene under the uniaxial strain, in black phosphorus, twisted bilayer graphene, and so on.