Tilted anisotropic Dirac cones in quinoid-type graphene and $\alpha \text{−}{\left(\text{BEDT-TTF}\right)}_2{\text{I}}_3$

We investigate a generalized two-dimensional Weyl Hamiltonian, which may describe the low-energy properties of mechanically deformed graphene and of the organic compound $\alpha \text{−}{\left(\text{BEDT-TTF}\right)}_2{\text{I}}_3$ $\left[\text{BEDT-TTF}=\text{bis}(\text{ethylene}\text{dithio)tetrathiafulvalene}\right]$ under pressure. The associated dispersion has generically the form of tilted anisotropic Dirac cones. The tilt arises due to next-nearest-neighbor hopping when the Dirac points, where the valence band touches the conduction band, do not coincide with crystallographic high-symmetry points within the first Brillouin zone. Within a semiclassical treatment, we describe the formation of Landau levels in a strong magnetic field, the relativistic form of which is reminiscent of that of graphene, with a renormalized Fermi velocity due to the tilt of the Dirac cones. These relativistic Landau levels, experimentally accessible via spectroscopy or even a quantum-Hall-effect measurement, may be used as a direct experimental verification of Dirac cones in $\alpha \text{−}{\left(\text{BEDT-TTF}\right)}_2{\text{I}}_3$.

Figure 1

(Color online) Energy dispersion (4) for the special choice of $w_x=w_y=1$ and ${\mathbf{w}}_0=\left(0,0.6\right)$, in natural units. The Dirac cone is tilted in the $y$ direction.

Figure 2

Renormalized Fermi velocity $v_F^{\ast }/\sqrt{w_x{w}_y}$ as a function of the effective tilt parameter ${\tilde{w}}_0\equiv \sqrt{{\left(w_{0x}/w_x\right)}^2+{\left(w_{0y}/w_y\right)}^2}$. The Fermi velocity vanishes for $\tilde{w}=1$, where the orbits change from ellipses to hyperbolas.

Figure 3

Quinoid-type deformation of the honeycomb lattice—the bonds parallel to the deformation axis (double arrow) are modified. The shaded region indicates the unit cell of the oblique lattice, spanned by the lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. The dashed and dashed-dotted lines indicate next-nearest neighbors, with characteristic hopping integrals $t_{\text{nnn}}$ and $t_{\text{nnn}}^{\prime }$, respectively, which are different due to the lattice deformation.

Figure 4

(Color online) Energy dispersion of the quinoid-type deformed honeycomb lattice for a lattice distortion of $\delta a/a=0.4$, with $t=3\text{eV}$, $t_{\text{nnn}}/t=0.1$, $\partial t/\partial a=-5\text{eV/Å}$, and $\partial t_{\text{nnn}}/\partial a=-0.7\text{eV/Å}$ The inset shows a zoom on one of the Dirac points, $D^{\prime }$.

Figure 5

Anisotropic triangular lattice model with four different nn hopping energies, $t_1,t_1^{\prime },t_2$, and $t_2^{\prime }$, and the nnn hopping energy $t_{\text{nnn}}$ (dashed lines). The unit cell with two inequivalent sites is represented by the shaded region. The sites of the $A$ and $B$ sublattices are depicted as filled and open circles, respectively.