Landau level collapse in graphene in the presence of in-plane radial electric and perpendicular magnetic fields

It is known that in two-dimensional relativistic Dirac systems placed in orthogonal uniform magnetic and electric fields, the Landau levels collapse as the applied in-plane electric field reaches a critical value $\pm E_c$. We study this phenomenon for a distinct field configuration with in-plane constant radial electric field. The Dirac equation for this configuration does not allow analytical solutions in terms of known special functions. The results are obtained by using both the WKB approximation and the exact diagonalization and shooting methods. It is shown that the collapse occurs for positive values of the total angular momentum quantum number, the hole (electron)-like Landau levels collapse as the electric field reaches the value $+(-)E_c/2$. The investigation of the Landau level collapse in the case of gapped graphene shows a number of distinctive features in comparison with the gapless case.

Figure 1

Schematic figure for the electric and magnetic field configuration. The radial electric field $E$ is in the plane of the graphene sheet and the uniform magnetic field $H$ is perpendicular to this plane.

Figure 2

The effective potential $U_{\mathrm{eff}}\left(\rho \right)$ versus $\rho$ for $\beta =1/2$ and different values of $\epsilon$ and $j$.

Figure 3

WKB and exact diagonalization (ED) spectra in units ${\epsilon }_0=\hslash v_F/l$ versus electric field in terms of $\beta =cE/\left(v_{F}H\right)$ for the gapless $\mathrm{\Delta }=0$ case. Panel (a) shows positive values of $j$ and panel (b) shows negative values of j. In both panels, solid lines show the results of the WKB approximation and crosses show the results calculated using ED. The Landau levels with $n=0,1,2,3,4$ are shown by the green, dark purple, purple lines, red and orange lines, respectively. The levels with $\left|j\right|=1/2,3/2,5/2,7/2$ are marked by the increasing thickness of the lines.

Figure 4

The WKB eigenenergies $\epsilon (n,j,\beta =1/2)$ versus the total angular momentum quantum number $j$ for $n=1$. Large blue dots correspond to the numerical solution of Eq. (B6) and small green dots are for the approximate solution Eqs. (41).

Figure 5

Spectra for the ${\mathbf{K}}_{+}$ point obtained by the shooting method in units ${\epsilon }_0=\hslash v_F/l$ versus electric field in terms of $\beta =cE/\left(v_{F}H\right)$ for the finite gap, $\delta =l\mathrm{\Delta }/\left(\hslash v_F\right)$, case. (a) Positive $j$: The solid lines are the solutions for $\delta =3$ and the dashed lines are for $\delta =0$. (b) Negative $j$: The gap $\delta =\sqrt{2}$. (c) The zoom of plot (b) in the vicinity of $\beta =1/2$. The same color and thickness scheme as the one displayed in Fig. 3 is used.