The three dimensionality of triple quantum dot stability diagrams

We present the full three dimensionality of an electrostatically calculated stability diagram for triple quantum dots. The stability diagram maps out the favored charge configuration of the system as a function of potential shifts due to gate voltages. For triple dots only a three dimensional visualization allows for the complete identification of all its components. Those are most notably the so called quadruple points where four electronic configurations are degenerate, and quantum cellular automata (QCA) processes. The exact positions of these features within the stability diagram are now revealed. Furthermore the influence on transport is studied by comparing the model with a two path triple quantum dot made with local anodic oxidation. The two path setup allows to study the influence of the dots arrangement.


Introduction
The electronic properties of quantum dots have been studied for quite some decades by now [1]. Since quantum dots have been proposed as crucial elements of future quantum information technologies [2], these systems have gained even more attention. First single quantum dots were studied as they could be used to realize a quantum mechanical bit (qubit) based on the electronic spin. Soon also double quantum dots [3] came into focus. Due to technical and scientific improvements during recent years, the research on quantum dots finally reached a level allowing for the detailed analysis of coupled triple quantum dots (e.g. [4,5,6,7,9]). They represent the first step towards coupled qubit chains as they are needed for quantum computers. They can also work as single qubits themselves [10]. Other interesting applications can be built with triple quantum dots, e.g. charge rectifiers and ratchets [11,4] or spin entanglers [12]. They can be used to study other effects like Kondo and Aharonov-Bohm effect [13,14] or dark states [15].
The first and most crucial requirement for high level research on quantum dot systems is the understanding of the stability diagram. The stability diagram maps out the charge configuration of the system as a function of potential detuning and does not depend on the setup of transport leads (as long as there are no instabilities, see below). In particular it shows degeneracy points, where several charge configurations are degenerate, a precondition for transport in multiple dot systems. For triple quantum dots the full complexity of the stability diagram can only be revealed in three dimensions. The degeneracy points consist of at least four degenerate configurations (quadruple points). Although any researcher is reliant on this map for exact analysis, these diagrams are still not well known. Parts of it were presented by Vidan et al. [16] and also Stopa [11], but the presentaions remain incomplete showing only the first quadruple point. Gaudreau et al. [5] firstly discovered several quadruple points with four degenerate electronic configurations. Furthermore they found quantum cellular automata (QCA) processes [17,18,19] leading to charge rearrangements in two dots, when an electron is added to the third dot. However, the diagram is presented in a two dimensional visualization only, not capable to reveal the full properties. Schröer et al. [6] showed a three dimensional visualization, but due to the large number of electrons covered by the model, it is not possible to see the dynamics at triple dot resonances, namely the positions and properties of quadruple points. On vertical quantum dots Amaha et al. [20] mapped out the stability diagram and also presented it in three dimensions, but the quadruple points are not resolved due to high temperatures. So even though the mathematics is well established and stability diagrams were calculated, measured and visualized, a complete and detailed three dimensional stability diagram was never developed. This is a huge deficiency in the understanding of these systems. As a result false assumptions were made. For example Schröer et al. [6] stated that the number of quadruple points is four. Furthermore, quadruple points do not feature four intersecting charging or charge reconfiguration planes but six. And most important, transport in a serial triple dot is possible not only at these quadruple points. In this paper we develop and visualize an electrostatic model for stability diagrams of triple quantum dots. Although the underlying mathematics is straight forward and does not differ considerably from other ansatzes, the model and its visualization goes far beyond existing models as it covers the full three dimensionality of these diagrams. Thus it finally fills the former mentioned deficiencies. It allows for the complete analysis of its components especially the intensively discussed quadruple points and the interplay with QCA processes. It is found, that there are in general six quadruple points instead of only four. All six points can show sequential transport. Furthermore, the QCA processes appear not only at certain points or lines as implied until now, but at a full plane in the center of the diagram. Depending on the arrangement of the quantum dots and the leads, the complete plane can contribute to transport.
The parameters for the model are chosen such that it compares to a real triple quantum dot, measured with transport and charge detection. This allows to study the influence of the model's features on transport. The two path setup of the device allows to understand the role of the dots arrangement. From the two path transport a single path transport can be calculated that nicely fits to the model's predictions.

Electrostatic model
For the calculation of the three dimensional model of a triple dot charge diagram we determine the energetically favored electronic configuration around the transition from zero electrons in the system to one on each dot (equivalent to any other total electron numbers from N to N + 1 on each dot) for any given set of three gate voltages V G1 , V G2 and V G3 used for detuning the potential. The number of possible electronic configurations of a system containing M quantum dots is 2 M . The number of configurations for a total electron number N is M N with either zero or one electron on each dot. Thus a triple dot has 2 3 = 8 possible configurations (N A , N B , N C ) with N A , N B , N C being the electron numbers for the three dots named A, B, C. There is one configuration for zero electrons in the system: (0, 0, 0), three for one electron: (1, 0, 0), (0, 1, 0), (0, 0, 1), three for two electrons: (1, 1, 0), (1, 0, 1), (0, 1, 1) and one for three electrons: (1, 1, 1). The electrostatic energies for these configurations as a function of gate voltages are for zero electrons: for one electron on dot i: for two electrons, one on dot i, one on dot j: and for three electrons with one electron on dots i, j, k each: are the capacitances between dot i and gates 1, 2, 3 and C Σi is the total capacitance of dot i. The capacitances C ij reflect the capacitive interaction between dots i and j, that leads to anticrossings in stability diagrams. The energetically favored configuration of the system is the one with the lowest energy depending on gate voltages. The adjustment to the measured triple dot is done by adjusting the capacitances. An artistic view of the measured triple dot device is shown in Fig. 1. The sample is made by local anodic oxidation [21,22,23] on a GaAs/AlGaAs heterostructure. The oxide lines (black) define three dots A, B and C, which are connected to three leads Source (S), Drain1 (D1) and Drain2 (D2). Thus transport can be measured along two paths, from Source via dots A and B to Drain1 (blue arrows) and from Source via A and C to Drain2 (red arrows). A quantum point contact (QPC) is added for charge detection. The geometry of the device and the measurement technique is explained in full detail in Refs. [7,8].
The upper part of Fig. 2 shows measurements on this device recorded using charge detection. This method does not depend on the setup of the leads (again as long as there are no instabilities) and is thus capable to measure the stability diagram. As a function of gate voltages V G1 and V G3 several lines are visible denoting charge rearrangements of the system. Lines with a steep slope correspond to charging of dot C with an additional electron, those with a shallow slope correspond to charging of dot B. Intermediate slopes appear due to charging of dot A (see central measurement). The three shown measurements are cuts through the three dimensional stability diagram

Three dimensional Visualization
The three images shown at the bottom of Fig. 2 are as explained cuts through the three dimensional stability diagram. Even though the central image is a cut through the center of the diagram, right here no quadruple points appear. However, they are in direct vicinity, close enough to be visible in measurements due to finite peak widths. To completely understand the model, a full three dimensional visualization is indispensable. This visualization is presented in Fig. 3. The stability diagram is shown from four different perspectives as a function of all three gate voltages. It is in general composed of three planes colored red, blue and green. At each of these planes the total charge of the system is changed by one electron which is added to a certain dot via the leads. The green (blue, red) plane corresponds to charging of quantum dot A (B, C). These planes intersect as they have different gradients. At these intersections double dot resonances are established. Due to the finite interdot capacitances the dots interact and thus the planes shift. Therefore when two planes intersect, anticrossings are formed with three degenerate electronic configurations on each side. In two dimensions these degeneracies are called triple points. In three dimensions they are transformed into triple lines. A pair of triple lines is connected with a grey plane. At these planes an electron is moved from one dot to another without changing the total number of electrons in the system. The spaces between all the planes are characterized by stable electronic configurations.
At the center of the model the situation becomes more complex as now all three colored planes intersect. Here quadruple points are expected as well as QCA processes. Indeed the model shows a feature that cannot be explained with double dot physics. In the center of the model another plane appears (colored yellow). This is shown in more detail in Fig. 4 where the center of the model is highlighted. The yellow plane is different to all other planes of the model. It has a different gradient and does neither correspond to charging of a certain dot only nor to pure charge rearrangements within the system. Instead it features both. At the whole plane the configurations (1,0,0) and (0,1,1) are degenerate. Thus the following process is possible: when there is one electron on dot A (1,0,0), a second electron can enter via dot B (C). At the same time the electron on dot  Figure 4. Highlighted center of the three dimensional stability diagram. The central yellow plane is defined by four quadruple points qp2, qp3, qp4 and qp5. Another one is below the plane (qp1) and above (qp6).
A is shifted to dot C (B) leaving the system in the (0,1,1) configuration. This is a QCA process, one of the major discoveries in triple dot systems. It appears along the whole yellow plane.
This central plane is surrounded by points, where six planes (colored and grey) intersect. These are the so called quadruple points that feature four degenerate charge configurations. Together with the QCA process, these points are the manifestation of the triple dot. Six quadruple points exist, two more than expected according to former publications. There is one below the central plane (qp1), one above (qp6) and four more that actually define the central plane (qp2, qp3, qp4, qp5). Only under special conditions, the number of degeneracy points can be reduced. If for example the three dots do not interact, there is only one point, where all planes intersect with all eight charge configurations being degenerate. For the more general case of six quadruple points, four of them are already known. For those the following configurations are degenerate: all four charge configurations that meet at the quadruple point. For qp3 and qp4, only three configurations meet at one point in two dimensions. The fourth configuration touches this point coming from the third dimension. So this point does not appear as a quadruple point in two dimensions. qp3 and qp4 were therefore overlooked.

Comparison with transport measurements
As mentioned before, the properties of the stability diagram (and of charge detection) do not depend on how the dots are connected to the leads. In particular they do not depend on the transport paths. This is not true for the conductance properties. They strongly depend on the dots' arrangement. With the model now well known the transport properties of our two path triple quantum dot can be analyzed. The upper part of Fig. 5 shows two transport measurements as a function of V G1 and V G3 . The conductance of both transport paths is plotted simultaneously, path1 in blue, path2 in red (results from charge detection are added as dashed lines). The left plot at V G2 = 0 mV corresponds to the back side of the three dimensional model. There is no resonance of all three dots. Nevertheless features are visible in both paths showing double dot resonances. The red features (marked with a red arrow) appear due to a resonance of dots A and C. They correspond to the triple lines on both sides of the anticrossing when the model's red and green planes intersect (see Fig. 2). Indeed one can see that the two involved triple lines form a double feature of two triple points in the measurement. The blue feature (blue arrow) corresponds to the intersection of the blue and the green planes from the model due to resonance of dots A and B. Although there are two triple lines involved as well, there is only one spot visible as the anticrossing is not large enough and cannot be resolved in transport. Assuming the two paths to be ohmic with conductances G1 and G2, one can calculate a hypothetic transport from Drain1 via dots B, A, C to Drain2. Thus the arrangement of the dots can be mathematically changed to form three dots in series with a combined conductance G = 1 1/G1+1/G2 . Such arrangement can only show transport, if all three dots are resonant. Therefore, double dot features should be wiped out. Of course this method does not include nonohmic effects. Furthermore the tunnel resistance to the Source is still included which is not the case in a real serial device. And finally the real arrangement is of course not changed. The observed features must still be analyzed in terms of the two path setup.
The result of this method is presented in the lower part of Fig. 5 for both voltages V G2 . Indeed the double dot features are strongly suppressed. For V G2 = 0 mV almost no conductance is visible. All the triple points have disappeared. For V G2 = −15 mV transport is suppressed as well. However, two spots are left (dashed circle). These are the features from the triple dot, that appear now well separated.
As the conductance properties depend on the dots' arrangement, it is not guaranteed that all quadruple points can contribute to transport. However, the quadruple points qp1 and qp6 are always visible as long as there are at least two leads connected to three coupled dots. At qp1 an electron can enter the empty system at any dot that is connected to a lead. It can then travel from dot to dot until it reaches the exit lead. The same accounts at qp6 for holes. Quadruple points qp2 to qp5 are not that trivial. For the two path setup, transport means charge exchange between the Source lead and one or both Drain leads. This is possible at qp2 and qp5. At qp2, transport in path1 can occur via the transition (1,0,0) → (1,1,0) → (0,1,0) → (1,0,0). An electron can enter from Drain1 to dot B, another electron then leaves via dot A to Source and the remaining electron shifts from B to A. The analoge process is possible for a hole in path2 with the transition (0,1,1) → (0,1,0) → (1,1,0) → (0,1,1). Similarly transport is possible at qp5 in both paths via (0,1,1) → (0,0,1) → (1,0,1) → (0,1,1) in path1 and via (1,0,0) → (1,0,1) → (0,0,1) → (1,0,0) in path2. At qp3 and qp4 no transport is possible for the two path setup. At both quadruple points neither electrons nor holes can enter the system via Source. So the observed features in Fig. 5 must be quadruple points qp1, qp2, qp5 and qp6. According to the model, the quadruple points qp1 and qp2 are much closer than e.g. qp1 and qp5. The same accounts for qp5 and qp6. So there are two pairs of quadruple points. Indeed the measurements shows only two spots instead of four. They correspond to these two pairs. The quadruple points within a pair are again too close to be resolved. The left spot contains qp1 and qp2, the right spot contains qp5 and qp6. The observation of these pairs confirms the model nicely.
The gap between the two pairs also confirms that the real dot arrangement is of course not changed by the calculation of a hypothetic serial transport. For a real serial system with Source and Drain at dots B and C, transport would occur along the whole central plane via the QCA process (and via sequential tunneling at qp3 and qp4). As this process does not involve charge exchange with the Source lead, it does not produce transport in our device. Even these stability diagrams can have instabilities. If for example there is only one lead attached to dot A at one end of a triple dot chain A, B, C, the configuration (1,0,0) could not be transferred directly to (0,0,1) even if both were degenerate. One had to lift this degeneracy to transfer the electron from A to B and from B to C. Then the degeneracy could be established again with the system in (0,0,1). So the actual charge configuration depends on the path through the stability diagram. Such instabilities could appear in quadruple quantum dots even when two leads are attached.

Conclusion
In conclusion we have visualized the full three dimensionality of triple dot stability diagrams. We have developed an electrostatic model that was fit to match the properties of a triple dot device with a two path setup. Due to the full visualization we were able to completely analyze all the model's features for triple dots. We found the maximum number of quadruple points (and degeneracy points in general) to be six instead of four as reported in other publications. We identified quantum cellular automata processes that occur along a whole plane in the center of the model. The comparison with the two path triple dot confirms the model and is capable to explain all features visible in transport. Due to the two path setup we were able to separate double dot features from triple dot features. As a result, the transport properties in triple dots strongly depend on the dot arrangement. Under certain circumstances transport is possible not only at quadruple points but along the whole central plane using the quantum cellular automata process.