Density of states of relativistic and nonrelativistic two-dimensional electron gases in a uniform magnetic and Aharonov-Bohm fields

We study the electronic properties of two-dimensional electron gas (2DEG) with quadratic dispersion and with relativistic dispersion as in graphene in the inhomogeneous magnetic field consisting of the Aharonov-Bohm flux and a constant background field. The total and local density of states (LDOS) are obtained on the base of the analytic solutions of the Schrödinger and Dirac equations in the inhomogeneous magnetic field. It is shown that as it was in the situation with a pure Aharonov-Bohm flux, in the case of graphene there is an excess of LDOS near the vortex, while in 2DEG the LDOS is depleted. This results in excess of the induced by the vortex DOS in graphene and in its depletion in 2DEG.

Figure 1

The normalized full LDOS $N_{\eta }^{\mathrm{S}}(r,E,B)/N_0^{\mathrm{S}}$ as a function of $E$ in the units of $\hslash {\omega }_c$. (a) $\eta =0$ (no vortex and LDOS is $r$ independent) and $\eta =1/2$ for $r=l$. (b) Both lines are for $\eta =1/2$, $r=0.5l$, and $r=5l$. In all cases, the width is $\Gamma =0.05\hslash {\omega }_c$.

Figure 2

The normalized full LDOS $N_{1/2}^{\mathrm{S}}(r,E,B)/N_0^{\mathrm{S}}$ as a function of the distance $r$ measured in the units of the magnetic length $l$ for four values of $E/\hslash {\omega }_c=0.5,1.5$ (usual LLs) and $E/\hslash {\omega }_c=1,2$ (vortex-like levels). The width is $\Gamma =0.05\hslash {\omega }_c$.

Figure 3

The normalized full LDOS $N_{\eta }^{\mathrm{D}}(r,E,B)/N_0^{\mathrm{D}}\left({\epsilon }_0\right)$ as a function of energy $E$ in the units of the relativistic Landau scale ${\epsilon }_0$. The LDOS is an even function of $E$, so only the positive-energy region is shown. (a) $\eta =0$ (no vortex and LDOS is $r$ independent) and $\eta =1/2$ for $r=l$. (b) Both lines are for $\eta =1/2$, $r=0.5l$, and $r=4l$. In all cases, the width is $\Gamma =0.05{\epsilon }_0$, $W/{\epsilon }_0=3.35$, and $\Delta =0$.

Figure 4

The normalized full LDOS $N_{1/2}^{\mathrm{D}(A,B)}(r,E,B)/N_0^{\mathrm{D}}\left({\epsilon }_0\right)$ as a function of energy $E$ in the units of the relativistic Landau scale ${\epsilon }_0$ for $r=l$. The gap is $\Delta ={\epsilon }_0$, the width is $\Gamma =0.05{\epsilon }_0$ and $W/{\epsilon }_0=3.35$.

Figure 5

The normalized full LDOS $N_{1/2}^{\mathrm{D}}(r,E,B)/N_0^{\mathrm{D}}\left({\epsilon }_0\right)$ as a function of distance $r$ measured in the units of the magnetic length $l$ from the vortex for four values of $E/{\epsilon }_0=\sqrt{2},2$ (usual LLs) and $E/{\epsilon }_0=\sqrt{3},\sqrt{5}$ (vortex-like levels). The width is $\Gamma =0.05{\epsilon }_0$, $W/{\epsilon }_0=3.35$, and $\Delta =0$.