Effect of the Dirac-cone tilt on the disorder-broadened Landau levels in a two-dimensional Dirac nodal system

Abstract

By means of the kernel polynomial method (KPM), a numerically exact theoretical approach, we calculate the density of states (DOS) and diagonal conductivity of a two-dimensional Dirac nodal system in the presence of a high magnetic field. The effect of Dirac cone tilt on the profiles of disorder-broadened Landau level (LL) peaks in both the DOS and diagonal conductivity spectra is our main concern. Our numerical results show that the profile of an isolated LL peak in DOS, especially the $n=0$ one, is tilt independent. On the other hand, the Dirac cone tilt enhances/reduces the diagonal conductivity peaks in the direction perpendicular/parallel to the cone tilt direction ($y$ direction). In particular, at the Dirac point, i.e., zero energy, the ratio of the former to the latter is ${\ensuremath{\sigma}}_{xx}(0)/{\ensuremath{\sigma}}_{yy}(0)=1/(1\ensuremath{-}{\ensuremath{\beta}}^{2})$, with $\ensuremath{\beta}$ being the tilt parameter. In addition, in comparison with the results obtained by KPM, we check the validity of the self-consistent Born approximation (SCBA) which has thus far been widely exploited for studying the DOS and quantum transport properties of Dirac or Weyl systems with disorder. We find that the SCBA fails to describe the detail of the LL peak around zero energy. At zero energy, the KPM result of the diagonal conductivity increases with the disorder strength before saturation, rather than a disorder-independent constant as reported previously by SCBA.