Dirac gap-induced graphene quantum dot in an electrostatic potential

A spatially modulated Dirac gap in a graphene sheet leads to charge confinement, thus enabling a graphene quantum dot to be formed without the application of external electric and magnetic fields [G. Giavaras and F. Nori, Appl. Phys. Lett. 97, 243106 (2010)]. This can be achieved provided the Dirac gap has a local minimum in which the states become localized. In this work, the physics of such a gap-induced dot is investigated in the continuum limit by solving the Dirac equation. It is shown that gap-induced confined states couple to the states introduced by an electrostatic quantum well potential. Hence the region in which the resulting hybridized states are localized can be tuned with the potential strength, an effect which involves Klein tunneling. The proposed quantum dot may be used to probe quasirelativistic effects in graphene, while the induced confined states may be useful for graphene-based nanostructures.

Figure 1

(a) Quantum states for $m=2$, $\tau =+1$ and different potential strengths $V_0$. The mass term is modelled by $\Delta ={\Delta }_0{r}^2$, with ${\Delta }_0=1$ $\mu$eV nm${}^{-2}$, and the electrostatic potential is modelled by $V=V_0{r}^2$. The potential strength $V_0$ is from top to bottom: $V_0/{\Delta }_0=0$, 0.9, and 1.5. The vertical axes of the insets to the top and middle panels are on logarithmic scale. (b) As in (a) but for $\tau =-1$.

Figure 2

Energy levels of confined states for the two valleys: $\tau =+1$ (solid lines) and $\tau =-1$ (dashed lines). The angular momentum number is $m=2$, the electrostatic potential is modelled by $V=V_0{r}^2$ and the mass term is modelled by $\Delta ={\Delta }_0{r}^2$. Here ${\Delta }_0=1$ $\mu$eV nm${}^{-2}$, whereas $V_0$ is tuned.

Figure 3

Quantum states for a piecewise-constant mass term $\Delta$ given by Eq. (17). The parameters are $m=2$, $R=100$ nm, and ${\delta }_0=80$ meV. Results for both valleys, $\tau =\pm 1$, are shown. The states correspond to the lowest positive eigenenergy.

Figure 4

(a), (b) Energy levels, for $\tau =+1$, as a function of the asymptotic value of the mass term ${\delta }_0$ which generates a Dirac gap of $2{\delta }_0$. The mass term $\Delta$ is modelled by Eq. (23). (c) The contour plot shows the density of states; the maximum number of states is restricted to five.

Figure 5

(a) Energy levels versus the electrostatic potential depth $V_0$, for $m=6$, $\tau =+1$ and a constant Dirac gap of $2\Delta =50$ meV. (b) As in (a) but for a spatially modulated Dirac gap with an asymptotic value of $2{\delta }_0=50$ meV. (c) As in (b) but for $m=2$. The states of the energies marked by circles are shown in Fig. 6.

Figure 6

(a) Quantum states for $m=2$ and for different electrostatic potential depths $V_0$; from top to bottom: $V_0=0$, 30, 50, and 70 meV for the left panels, and $V_0=0$, 30, 70, and 100 meV for the right panels. The Dirac gap is spatially modulated with an asymptotic value of $2{\delta }_0=50$ meV. Left (right) panels show states with energies marked by $•$ ($ˆ$) in Fig. 5c. (b) As in (a) but for $m=6$ and $V_0=0$, 40, 85, and 100 meV for the left panels, and $V_0=0$, 40, 85, and 120 meV for the right panels. Left (right) panels show states with energies marked by $•$ ($ˆ$) in Fig. 5b. The vertical axes of the insets range from 0 to $1\times {10}^{-3}$.

Figure 7

Density of states versus energy $E$ ($\left|E\right|<24.8$ meV) and electrostatic potential depth $V_0$, for $\tau =+1$, and an asymptotic Dirac gap of $2{\delta }_0=50$ meV.