Electric control of tunneling energy in graphene double dots

We theoretically investigate the spectrum of a single electron double quantum dot, defined by top gates in a graphene with a substrate-induced gap. We examine the effects of electric and magnetic fields on the spectrum of localized states, focusing on the tunability of the interdot coupling. We find that the substrate-induced gap allows for electrostatic control, with some limitations that for a fixed interdot distance, the interdot coupling cannot be made arbitrarily small due to the Klein tunneling. On the other hand, the proximity of the valence band in graphene allows for new regimes, such as an $npn$ double dot, which have no counterparts in GaAs.

Figure 1

Illustration of the electrostatically defined graphene double dot. The top gates (colored/shaded regions) define the area of confinement, which can be adjusted for the circular dots via $V_0$ (blue areas/circles) and the bridge between the dots via $V_1$ (red area/center) separately. In numerics, we terminate the grid at a distance $\delta$ beyond the structure potential boundary, and we introduce a slightly increased mass at the edge of the grid (the outer black line).

Figure 2

Illustrative plot of the potential landscape $V\left(\mathbf{r}\right)$ [Eq. (2)] for parameters $V_0=-V_1$ and $w=2R$.

Figure 3

Schematic energy diagram of a single dot. Within the diameter of $2R$, the energy bands are shifted by the confinement depth $V_0$ (this figure corresponds to $V_0<0$). The ground state is offset from the band bottom by ${\epsilon }_0$ (ground-state energy) from the nearest localized state by ${\epsilon }_{\mathrm{ex}}$ (excitation energy) and from the nearest extended state by ${\varepsilon }_{\text{ion}}=min({\varepsilon }^{\prime },{\varepsilon }^{\prime \prime })$ (ionization energy).

Figure 4

Calculated energy spectrum of a single dot as a function of the confinement depth $V_0$. For $V_0>0(V_0<0)$, the holes (electrons) are localized in the dot. The gray regions represent extended states. The characteristic energies are indicated.

Figure 5

Relative ground-state energy ${\varepsilon }_0$ (dash-dot line), excitation energy ${\varepsilon }_{\text{ex}}$ (dashed line), and ionization energy ${\varepsilon }_{\text{ion}}$ (dotted line) as a function of the confinement depth $V_0$.

Figure 6

Calculated energy spectrum of a single dot ($V_0=-60$ meV) as a function of a perpendicular magnetic field. The valley index of the lowest states is indicated by $K/K^{\prime }$.

Figure 7

Schematic energy diagram of a double dot. The energy is shifted locally by $V_0$ for the dots, and by $V_1$ for the barrier (this figure corresponds to $V_0<0<V_1$). The interband tunneling from the left to the right dot via the barrier is indicated by black arrows.

Figure 8

(a) Calculated energy spectrum of a double dot ($2d=110$ nm, $V_0=-60$ meV) as a function of the barrier potential $V_1$. The red (2) and blue (3) lines denote the dot ground state and the first excited state, respectively. Half of their energy difference is the tunneling energy $T$. The shaded region between those two states and the vertical line at $V_1=0$ are a guide for the eye. The green curve (1) denotes the highest localized state of the barrier. The dashed line is the top line for $V_0>0$ from Fig. 4. (b) The tunneling energy replotted from the data in (a). Line with symbols is a fit from the model described below Eq. (6).

Figure 9

Calculated energy spectrum of a double dot ($V_0=-60$ meV, $V_1=0$) as a function of the interdot distance $2d$. The red and blue curves give the localized ground and first excited state, respectively. The tunneling energy $T$ is indicated.

Figure 10

Calculated tunneling energy $T$ for a graphene double dot ($2d=110$ nm, $V_0=-60$ meV) as a function of perpendicular magnetic field for two values of the barrier potential.

Figure 11

Eigenenergies of the states 1, 2, and 3, taken from Fig. 8 (solid lines) and from the analytical model, described below Eq. (6) (lines with circles). A linear trend $\delta \epsilon =0.0593V_1$, fitted from the exact values of the level 2 energy, was added to all fitted values.