Three-quarter Dirac points, Landau levels and magnetization in $\alpha$-(BEDT-TTF)${}_2$I${}_3$

The energies as a function of the magnetic field ($H$) and the pressure are studied theoretically in the tight-binding model for the two-dimensional organic conductor, $\alpha$-(BEDT-TTF)${}_2$I${}_3$, in which massless Dirac fermions are realized. The effects of the uniaxial pressure ($P$) are studied by using the pressure-dependent hopping parameters. The system is semi-metallic with the same area of an electron pocket and a hole pocket at $P<3.0$~kbar, where the energies $({\epsilon }_{\mathrm{D}}^0$) at the Dirac points locate below the Fermi energy $({\epsilon }_{\mathrm{F}}^0$) when $H=0$. We find that at $P=2.3$~kbar the Dirac cones are critically tilted. In that case a new type of band crossing occurs at “three-quarter”-Dirac points, i.e., the dispersion is quadratic in one direction and linear in the other three directions. We obtain new magnetic-field-dependences of the Landau levels $({\epsilon }_n)$; ${\epsilon }_n-{\epsilon }_{\text{D}}^0\propto (nH{)}^{4\mathrm{/}5}$ at $P=2.3$~kbar (“three-quarter”-Dirac points) and $|\varepsilon_n-\varepsilon_{\rm F}^0| \propto (n H)^2$at$P=3.0$~kbar (the critical pressure for the semi-metallic state). We also study the magnetization as a function of the inverse magnetic field. We obtain two types of quantum oscillations. One is the usual de Haas van Alphen (dHvA) oscillation, and the other is the unusual dHvA-like oscillation which is seen even in the system without the Fermi surface.