Longitudinal conductivity of massless fermions with tilted Dirac cone in magnetic field

We investigate a longitudinal conductivity of a two-dimensional relativistic electron gas with a tilted Dirac cone in magnetic field. It is demonstrated that the conductivity behaves differently in the directions parallel and perpendicular to the tilting of the cone. At high magnetic fields, the conductivity at nonzero Landau levels in the direction perpendicular to the tilting modifies nontrivially, in contrast to the parallel case. At zero temperature, the crossover of the conductivity at the Dirac point from high to low magnetic field is studied numerically. It is found that the tilting produces anisotropy of the conductivity which changes with the magnetic field which is different from the anisotropy coming from the Fermi velocity. We also discuss the conductivity at finite temperatures and finite magnetic fields which can be directly compared with the experiments in $\alpha$-(BEDT-TTF)${}_2{\mathrm{I}}_3$ organic conductor. We find that the tilting does not affect so much the magnetic-field dependence of the conductivity except for the prefactor. We discuss the interpretation of recent experimental data and make some proposals to detect the effect of the tilting in future experiments.

Figure 1

Diagrammatic representation of Eq. (23). Each dashed line on the right-hand side stands for $u\delta (\mathit{r}-{\mathit{R}}_j)$ and the star means the average over ${\mathit{R}}_j$ and summation over $j$.

Figure 2

$\mathrm{Im}\Sigma \left(\epsilon \right)$ at $\epsilon =0$ as a function of $\hslash {\omega }_c$ for several $\varkappa$ with $E_c=1$ and $\eta =0$.

Figure 3

Chemical potential dependence of the normalized conductivities ${\tilde{\sigma }}_{\parallel }={\lambda }^{-1}{\sigma }_{\parallel }$ and ${\tilde{\sigma }}_{\perp }=\lambda {\sigma }_{\perp }$ in SCBA with $\eta =0.5, \varkappa =0.005$, and $E_c/\hslash {\omega }_c^{*}=100$. The dotted-dashed (dotted) horizontal line is a guide for eyes at the value $2N$ ($2N-N{\eta }^2$).

Figure 4

Semiclassical trajectories of a massless Dirac electron with the tilted cone in the $k$ space with $\eta =0.6$ for $n=-5,...,5$. The units of $k_x$ and $k_y$ are ${\ell }^{-1}$. The blue (purple) ellipses correspond to the upper (lower) band. The dots show centers of the ellipses. Each ellipse has its focus at the origin and its center displaced by $x_n{\ell }^{-2}$ in the $y$ direction. The eccentricity of each ellipse is $\eta$.

Figure 5

Normalized conductivities ${\tilde{\sigma }}_{\parallel }={\lambda }^{-1}{\sigma }_{\parallel }$ and ${\tilde{\sigma }}_{\perp }=\lambda {\sigma }_{\perp }$ at $\mu =0$ with $\eta =0.4$ and $0.5$ calculated from Eqs. (14) and (15) numerically using $N=150$ Landau levels. (a) The conductivities in the constant broadening approximation as functions of $\Gamma /\hslash {\omega }_c^{*}$; $\eta =0$ case is indicated by the dotted-dashed line. (b) The conductivities in SCBA as functions of $\varkappa$ with $E_c/\hslash {\omega }_c^{*}=100$. In each figure, the upper (lower) dashed line corresponds to the limiting value ${sin}^{-1}\eta /\left(\lambda \eta \right)$ at zero field for $\eta =0.5$ ($\eta =0.4$) [30].

Figure 6

Magnetic-field dependence of the conductivity at $\mu =0$ in the constant broadening approximation for several temperatures without the Zeeman interaction (a) and with the Zeeman interaction included (b).

Figure 7

Temperature dependence of the resistivity at $\mu =0$ for several magnetic fields. The dashed line shows the resistivity at zero magnetic field. The inset shows the position of the maximum $k_B{T}_{\max }$ as a function of $\hslash {\omega }_c$ with $g=0$.

Figure 8

(a) Magnetic-field dependence of the conductivity at $\mu =0$ in SCBA for several temperatures with $\varkappa =0.02$ and $E_c=0.05$ eV. The inset shows $B_{\min }^{1/2}$ vs temperature. (b) Magnetic-field dependence of $\hslash {\omega }_c, \Gamma \left(B\right), {\mu }_{B}B$, and the conductivity in the unit of $e^2/\pi h$ at $T=2.5$ K (dashed line) in the same plot. The horizontal dotted-dashed line corresponds to $T=2.5$ K.

Figure 9

Magnetic-field dependence of the conductivity at $\mu =0$ in SCBA for $T=4.5$ K, $\varkappa =0.02$, and $E_c=0.05$ eV calculated from Eqs. (18) and (30) (solid line). Closed squares (triangles) show ${\tilde{\sigma }}_{\perp }$ (${\tilde{\sigma }}_{\parallel }$) with $\eta =0.5$ calculated from Eqs. (7) and (14) [Eq. (15)] using summation over 30 Landau levels. The horizontal axis is renormalized as $\tilde{B}=B{(1-{\eta }^2)}^{-3/2}$.

Figure 10

Resistivity at $\mu =0$ as a function of temperature for several magnetic fields calculated in SCBA with $\varkappa =0.02$ and $E_c=0.05$ eV.