Electrostatically defined serial triple quantum dot charged with few electrons

A serial triple quantum dot (TQD) electrostatically defined in a $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ heterostructure is characterized by using a nearby quantum point contact as charge detector. Ground-state stability diagrams demonstrate control in the regime of few electrons charging the TQD. An electrostatic model is developed to determine the ground-state charge configurations of the TQD. Numerical calculations are compared with experimental results. In addition, the tunneling conductance through all three quantum dots in series is studied. Quantum cellular automata processes are identified, which are where charge reconfiguration between two dots occurs in response to the addition of an electron in the third dot.

Figure 1

(Color online) Scanning electron microscope micrograph of the sample structure. Gate electrodes (bright tone) are used to electrostatically define a TQD and three QPCs. The approximate position of the three QDs (A, B, and C) are depicted by black circles. Large (small) arrows mark possible tunneling current paths through QPCs (the TQD). Ohmic contacts are labeled with roman numbers. The gates marked with $\alpha$, $\beta$, and $\gamma$ are used as plunger gates of the three QDs A, B, and C. Gates marked as d1, d2, and d3 serve to define QPCs as charge sensors.

Figure 2

Equivalent circuit diagram for three tunnel-coupled QDs A, B, and C in serial configuration. QDs A and C are, in addition, tunnel coupled to leads II and III, respectively. Tunnel barriers are modeled as resistors and capacitors in parallel and electrostatic coupling to three plunger gates $\alpha$, $\beta$, and $\gamma$ as capacitors.

Figure 3

(Color online) Numerically calculated ground-state stability diagram of the TQD device shown in Fig. 1 for charging energies, electrostatic interdot coupling energies, and capacitances between QDs and plunger gates, similar to experimentally derived values. The color scale of the lines is chosen to simulate a possible measurement of the transconductance of the left QPC in Fig. 1 as a function of the plunger-gate voltages $V_{\alpha }$ and $V_{\gamma }$, where $V_{\gamma }$ is modulated. The background color denotes zero transconductance. Stable charge configurations are labeled by triples of numbers $(N_{\mathrm{A}},N_{\mathrm{B}},N_{\mathrm{C}})$.

Figure 4

(Color online) Measured charge stability diagram of the TQD device shown in Fig. 1 as a function of gate voltages $V_{\alpha }$ and $V_{\gamma }$. The color scale measures the transconductance of the left QPC in Fig. 1 as a function of the (modulated) plunger-gate voltage $V_{\gamma }$. Stable charge configurations are denoted by triples of numbers $(N_{\mathrm{A}},N_{\mathrm{B}},N_{\mathrm{C}})$. The graph is composed of several consecutive measurements explaining, e.g., the horizontal line at $V_{\alpha }\sim -0.22\mathrm{V}$.

Figure 5

(Color online) Expansion of a region of the TQD charge stability diagram in Fig. 4 for similar surface gate voltages. A black rectangle marks an area also marked by rectangles in Fig. 7.

Figure 6

(Color online) TQD charge stability diagram as in Fig. 5, but as a function of the plunger-gate voltages $V_{\beta }$ and $V_{\gamma }$. The graph is composed of three independent measurements. The white region was not investigated. Slight changes of the internal potential between the measurements lead to slightly imperfect seams between parts of the graph (e.g., at $V_{\beta }\simeq -0.37\mathrm{V}$; $V_{\gamma }\simeq -0.28\mathrm{V}$). Two dashed lines are a guide for the eyes and follow very faint charge reconfiguration lines of positive transconductance. The bistable region observed in Fig. 5 is also visible.

Figure 7

(Color online) The differential conductance $d{I}_{\mathrm{TQD}}∕d{V}_{\mathrm{SD}}$ measured through the TQD plotted with a logarithmic color scale as a function of the plunger-gate voltages $V_{\alpha }$ and $V_{\gamma }$. Voltages applied to other gates are the same as for Fig. 5. Black rectangles mark the same region as in Fig. 5. White color indicates a differential conductance exceeding the full range of the amplifier, small areas of the darkest blue color denote negative differential conductance. Applied bias voltages $V_{\mathrm{SD}}$ are indicated.

Figure 8

(Color online) Three-dimensional charging diagram, or beehive diagram, showing stable configurations of the triple-dot system as a function of the three plunger-gate biases, normalized by their capacitive couplings to the closest dot. The front and bottom regions of the structure corresponds to the configuration (0,0,0). Each of three visible end faces resembles a two-dimensional honeycomblike diagram, although more complicated diagrams can be seen at other slices, as shown in Fig. 9.

Figure 9

(Color online) Expansion of a region of the stability diagram of the TQD as a function of the plunger-gate voltages $V_{\alpha }$ and $V_{\gamma }$. Here, charging lines of all three QDs are close by. $V_{\beta }$ is increased in steps of $2\mathrm{mV}$ from (a) to (d) or from (e) to (i). The upper two rows show identical results from model calculations in two different representations. Triples of numbers denote stable charge configurations, where $X=1,2,3$ and $Y=2,1,4$. The third row shows transconductance measurements as, e.g., in Fig. 5. The two lowest rows display identical differential conductance measurements of the TQD with a linear (upper) and a logarithmic (lowest row) color scale (see main text for more explanations). The voltage ranges of $V_{\alpha }$ and $V_{\gamma }$ are identical for all subplots and shown in the left corner.