Magnetic-field-induced charge redistribution in disordered graphene double quantum dots

We have studied the transport properties of a large graphene double quantum dot under the influence of a background disorder potential and a magnetic field. At low temperatures, the evolution of the charge-stability diagram as a function of the $B$ field is investigated up to 10 T. Our results indicate that the charging energy of the quantum dot is reduced, and hence the effective size of the dot increases at a high magnetic field. We provide an explanation of our results using a tight-binding model, which describes the charge redistribution in a disordered graphene quantum dot via the formation of Landau levels and edge states. Our model suggests that the tunnel barriers separating different electron/hole puddles in a dot become transparent at high $B$ fields, resulting in the charge delocalization and reduced charging energy observed experimentally.

Figure 1

(a) Atomic force micrograph of the double quantum dot device measured in this paper. (b) Measurement of the differential conductance through the DQD for varied backgate voltages. Data collected at $V_{ac}=200\mu \mathrm{V}$ and $T=1.4\mathrm{K}$. Current through the DQD as a function of $V_{PG1}$ and $V_{PG2}$ measured in a dilution refrigerator at $T=100\mathrm{mK}$ with applied dc biases (c) $V_b=400\mu \mathrm{V}$, (d) $V_b=1\mathrm{mV}$, and (e) $V_b=2\mathrm{mV}$. The position of the triple points can be determined at low bias [see (c)], whereas at high bias the triple points evolve into triangles [see (d) and (e)].

Figure 2

(a) The evolution of the charge-stability diagram (taken at $T=100\mathrm{mK},V_{BG}=8.61\mathrm{V}$, and $V_b=-1\mathrm{mV}$) under the influence of a perpendicular magnetic field from 4 to 10 T. (b) Closeup of the triangle in the lowest position of each panel as highlighted by the dashed square in the leftmost panel of (a). Note that the voltage ranges on the $x-y$ axes of (a) and (b) are chosen to be the same in each panel. (c) The charging energies of the QDs as a function of the $B$ field. (d) The interdot coupling energy as a function of the $B$ field. Note that in order to consider the distortion of the triangle at certain $B$ fields (e.g., $B=7$ and 9 T), we use the right tip and the left tip of each triangle to doubly define the shape of the triangle. We take the average of both fittings (meaning the triangle shape determined by the left tip and the right tip) to extract the data points, and the error bar is determined by the difference between the two fittings.

Figure 3

(a) Example of potential distribution in a large disordered quantum dot. (b) Left panel: Schematic band structure of a GQD in zero magnetic field. Right panel: in a high magnetic field. The red lines denote the electronlike levels, and the blue lines denote the holelike levels. The black solid lines indicate the Dirac conelike dispersion. (c)–(e) Expected density of state (DOS) distribution in the dot at zero field, low magnetic field, and high magnetic field, respectively.

Figure 4

(a) Potential distribution in a quantum dot with $R=47.5\mathrm{nm}$. The black line in the potential profile indicates the region where the potential is $V=0$. (b)–(f) Calculated local DOS in the dot at different magnetic fields. (g)–(j) Calculated current distribution in the dot at different magnetic fields.

Figure 5

The charge-stability diagram measured at $V_{BG}=9\mathrm{V}$ and $V_b=1\mathrm{mV}$ in (a) $B=3.2\mathrm{T}$, (b) $B=3.8\mathrm{T}$, and (c) $B=4.4\mathrm{T}$ showing a formation of an additional dot under the influence of a magnetic field and strongly coupled to the original dots. (d) Graphic illustration of an effect of a localized state formed in magnetic fields. (e) Same as (d) but with a charge added into the localized state. It capacitively couples to the original dots and forces the DQD to reconstruct its wave function.