Device and initial characterization

Figure 1a shows a scanning electron micrograph of a device nominally identical to the one used in the experiment. The gate layout has been adapted from previous triple and quadruple quantum dot devices.14,23 On one side, 17 gates with a pitch of 80 nm are fabricated to control the tunnel barriers and electrochemical potentials of the QDs. The upper part of the sample accommodates two sensing dots (SD) that are capacitively coupled to the linear QD array. The circles indicate the intended positions of the QDs. All measurements are carried out in a dilution refrigerator with a base temperature below 20 mK.

Fig. 1 Device design and tuning principle. a Scanning electron micrograph of a device nominally identical to the one used in the experiment. The scale bar measures 500 nm. The circles indicate the intended positions of eight quantum dots (QDs) that define a Qubyte register and of two additional dots that are used for charge sensing. For the linear array, the designed dot-to-dot pitch is 160 nm. The plunger gates connected to high-frequency lines are marked with blue triangles. The white squares indicate the position of the Fermi reservoirs. Two on-board tank circuits for RF reflectometry readout are connected to each of the sensing dots. b Charge stability diagram of a sextuple dot formed between barrier gates B 1 and B 7 . The sum of the differential demodulated voltages of both sensing dots is plotted. The dashed lines highlight charge transitions of each of the six QDs (the numbers refer to the labels in a). c Illustration of the potential landscape of a double QD. Gates P 4 and B 4 are used to form a third QD. Capacitive cross-talk, indicated by the capacitor symbols, has not been compensated for. Thus, these gates influence the potential of the other QDs as well (to avoid clutter, we did not draw any other capacitor symbols). d A double QD is extended to a triple QD using the virtual plunger VP 4 and barrier VB 4 . Due to cross-capacitance compensation, these parameters only act locally on the potential landscape Full size image

Initially, the device is characterized by DC transport measurements. The pinch-off characteristics of the channel between each of the plunger P i or barrier B i gates and the central gate D is measured (see schematics in Fig. 1a) and single QDs are formed by sweeping pairs of neighboring barrier gates. These measurements confirm that all QDs, including the sensing dots, can be formed (if not, the sample is considered unsuitable to host eight QDs). Moreover, the pinch-off values determined for each gate act as starting parameters for further tuning. In all subsequent measurements, we probe the linear QD array via the two sensing dots, which are sensitive to the number of electrons in the array, as well as to their position in the array. The charge sensors are probed using RF reflectometry (see Methods section).

To illustrate the difficulty of traditional tuning strategies, Fig. 1b shows a charge stability diagram for a linear six-dot array (sextuple dot) confined between the barrier gates B 1 to B 7 . The charge stability diagram has been recorded sweeping the voltages of gates P 2 and P 7 , i.e., the gates mostly coupled to the outer QDs. In the diagram, charge addition lines with different slopes can be identified. However, charge transitions with similar slopes can only be assigned unambiguously to specific dots, after also stepping other gate voltages (see e.g., the small difference in slope between the transitions for dots 6 and 7). Even then, the complex pattern of transitions in the center of the diagram makes it extremely difficult to determine the charge occupation at every point in this gate space. Moreover, cross-capacitances hinder local tuning of the electrochemical potential and tunnel rates.

n + 1 method

To tune up a multi-dot array dot by dot, we make use of virtual gates, which compensate for the cross-talk on the electrochemical potentials that occurs when sweeping actual gate voltages (see Fig. 1c, d). The virtual gates as used here do not compensate for cross-talk effects on tunnel barriers. The virtual plunger gate VP i directly corresponds to the electrochemical potential of QD i , up to a lever arm. The compensation is performed to first order, so that we can express the virtual gates as linear combinations of the physical gate voltages, summarized by a cross-capacitance matrix.5,22

The tuning procedure consists of the following steps, described in more detail below:

1. Tune up a double quantum dot (DQD) and one sensing dot with the traditional strategy. 2. Measure the cross-capacitance between all gates and the electrochemical potentials of these three QDs and record them in a cross-capacitance matrix. This matrix can now be used to generate virtual gates. 3. Use the virtual plunger and barrier gates adjacent to the existing dots to form the next QD without disturbing the former. 4. Measure the cross-capacitance between all gates and the electrochemical potential of the newly formed QD and place these values in the corresponding row of the matrix. 5. Re-measure the cross-capacitances to the previously formed QDs and update the matrix accordingly.

Steps 3–5 are repeated to extend the array, adding one QD at a time.

Before we describe these five steps in more detail, we first explain how the cross-capacitance matrix A = (α ij ) is used to create virtual gates. For illustration purposes, we do this for the first three dots, leaving out the outer barrier gates and the gates of the sensing dots for simplicity. The following relationship expresses how much each physical gate affects each virtual gate:

$$\left( {\begin{array}{*{20}{c}} {\Delta {\mathrm{VP}}_1} \\ {\Delta {\mathrm{VB}}_1} \\ {\Delta {\mathrm{VP}}_2} \\ {\Delta {\mathrm{VB}}_2} \\ {\Delta {\mathrm{VP}}_3} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 & {\alpha _{12}} & {\alpha _{13}} & {\alpha _{14}} & {\alpha _{15}} \\ 0 & 1 & 0 & 0 & 0 \\ {\alpha _{31}} & {\alpha _{32}} & 1 & {\alpha _{34}} & {\alpha _{35}} \\ 0 & 0 & 0 & 1 & 0 \\ {\alpha _{51}} & {\alpha _{52}} & {\alpha _{53}} & {\alpha _{54}} & 1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\Delta P_1} \\ {\Delta B_1} \\ {\Delta P_2} \\ {\Delta B_2} \\ {\Delta P_3} \end{array}} \right)$$ (1)

For convenience, we set the diagonal entries to 1 (dimensionless), disregarding the lever arm. This implies we express virtual gates in units of Volt, similar to the physical gates. Furthermore, since we do not include cross-talk effects on tunnel barriers, the off-diagonal matrix elements relating the physical gate voltages to virtual barrier gates are set to zero. The inverse matrix expresses the linear combination of physical gate voltages that is needed to sweep a virtual gate. We note that the diagonal entries of the inverse matrix do not need to be equal to 1.

We now turn to the five steps in the n + 1 method. In step 1, we form QD 2 , QD 3 and the left sensing dot using conventional methods. The resulting charge stability diagram is shown in Fig. 2a. The matrix A 1 at this point is simply the identity matrix (this matrix and the matrices produced in subsequent steps are shown in the Supplementary information). For step 2, the matrix entries for QD 2 , QD 3 and the left sensing dot are determined by recording how much an addition line for QD i in a P i scan is displaced when stepping any of the other plunger (barrier) gates P j (B j ) by an amount δV (see Supplementary Fig. S1). The ratio of the shift of the charge transition line of QD i in the P i scan and δV yields the corresponding entry in the cross-capacitance matrix. We do this for all eight plunger and nine barrier gates of the linear array, as well as for the plunger gates of both sensing dots. The resulting matrix is A 2 .

Fig. 2 Tuning method. a Charge stability diagram of a DQD in the single electron regime. The charge sensor response (the differential demodulated voltage, in arbitrary units, is plotted here and in similar plots below) is plotted as a function of the plunger gate voltages P 2 and P 3 . b Charge stability diagram of the same DQD recorded as a function of the virtual plungers VP 2 and VP 3 . c Charge stability diagram where an additional QD has been formed to the right of the DQD by raising the relevant tunnel barrier. The arrows indicate the position of the addition lines of the newly formed dot. d Differentiated demodulated charge sensor signal as a function of virtual plunger VP 4 and virtual barrier VB 4 . The charge addition lines corresponding to the newly formed QD are clearly visible. No transitions of the pre-existing dots are observed, due to the use of virtual gates. e Visualization of the cross-capacitance matrix of the eight-dot array. The entries of each row show how the virtual plunger value (and hence the electrochemical potential) of a QD is influenced by other gate voltages. The rows for virtual barrier gates are omitted for simplicity. The plungers of both sensing dots are included in the matrix Full size image

The effectiveness of the cross-talk compensation can be seen by recording a charge stability diagram in the virtual gate space, i.e., using VP 2 and VP 3 as sweep parameters (see Fig. 2b). Ideally, addition lines of QD 2 and QD 3 appear as orthogonal (horizontal and vertical) lines. In practice, the compensation is not always perfect because we extrapolate each cross-capacitance from just two data points (see Supplementary Fig. S1), but it is usually good enough.

To add the next QD (step 3), here QD 4 , we form a new tunnel barrier using the neighboring virtual barrier gate, VB 4 . The pinch-off values determined in DC transport indicate a suitable voltage range to scan with the barrier gate. Optionally, we then monitor the charge stability diagram VP 2 −VP 3 while stepping VB 4 . Once the barrier is raised sufficiently to form an additional QD, new addition lines appear in the charge stability diagram (see arrows in Fig. 2c). The charge transitions of the previously tuned QDs are only slightly affected, indicating the effectiveness of the virtual gate concept.

We complete the tuning of the newly formed dot to the single electron regime by measuring a charge stability diagram, here sweeping virtual plunger VP 4 versus virtual barrier VB 4 . A set of diagonal lines indicates charge transitions of the newly formed QD (see Fig. 2d). We can identify the last charge transition in the bottom left of the figure. The cross-capacitance matrix is unchanged, A 3 = A 2 .

Next, we update the cross-capacitance matrix (step 4). First, we fill the row corresponding to VP 4 in an otherwise unity matrix, which gives \(A_4^\prime\). The effect of all VP j and VB j on VP 4 is determined, as described for the first double dot, with the distinction that we now start from a set of virtual gates, expressed by matrix A 2 . We then consolidate the virtual gates for dots 2–4 (and the first sensing dot) into the matrix \(A_4 = A_4^\prime A_3\).

Finally, in step 5, the matrix entries in A 4 are updated to account for reduced screening of the gate potentials when the two-dimensional electron gas at the location of QD 4 is depleted. To do so, we remeasure the cross-talk from all the virtual plunger and barrier gates to all the virtual plunger gates. This results in the matrix \(A_5^\prime\), which represents a correction on A 4 and is constructed using the virtual gates expressed by A 4 . Therefore, the updated cross-capacitance matrix A 5 is found by matrix multiplication, \(A_5 = A_5^\prime A_4\). Additional dots, including the second sensing dot, are formed by repeating steps 3 to 5.

Qubyte in the single electron regime

We apply the n + 1 method to form a sextuple QD and octuple QD. We start with a DQD confined between the barrier gates B 1 and B 3 and initially extend the array to the right. The electrochemical potentials and thus the number of electrons residing on all QDs can be independently controlled. The results are verified by charge stability diagrams of neighboring pairs of virtual plunger gates, see Fig. 3b–f, where the sextuple dot has been initialized with one electron in each of the QDs. The gate voltages at the center of all of these plots are identical. All data sets have been acquired by fast voltage sweeps. Each plot can be interpreted as a charge stability diagram of a DQD, independent of the neighboring QDs. The virtual gates control the electrochemical potential of the DQD and the number of electrons can be determined easily from the measurements. This set of measurements contains the full information of the charge state of the sextuple QD and is much easier to interpret and work with than conventional charge stability diagrams, where multiple charge addition lines, as well as interdot transitions, are visible in a single plot. In fact, the data of Fig. 1b was taken for illustration purposes only, after forming the sextuple dot using measurements such as those in Fig. 3.

Fig. 3 Qubyte in the single electron regime. a–g Charge stability diagrams of pairs of neighboring QDs within the array. The differential demodulated voltage is plotted as a function of the virtual plunger gates. For panels b–f, the charge stability diagrams were measured with a sextuple QD defined between barrier gates B 1 and B 7 . The measurements are centered around the single electron occupation. Panels a, g show charge stability diagrams of the two outermost DQDs of an octuple QD array. These panels took about 2 h each, compared to 10 s for the other panels, and may thus be more susceptible to background charge rearrangements. h Scan along the interdot detuning axis of QD 3 and QD 4 with one electron in those two dots (see arrow in c). The data has been fit according to the model described in refs 22,27 Full size image

Following the same n + 1 method, the sextuple QD is further extended to an octuple QD array by adding another QD on each side. Due to limitations of the experimental setup (see Methods section for details), the plunger gates P 1 and P 8 are not connected to high-frequency lines necessary to apply fast gate voltage sweeps. Therefore, any measurement using virtual gates involving these gates must rely on slow gate voltage sweeps. For this reason, we first formed a sextuple dot in the center and only then extended it to an octuple dot. Figure 3a, g show charge stability diagrams as a function of VP 1 , VP 2 and VP 7 , VP 8 , respectively, completing the formation of the Qubyte register.

The cross-capacitance matrix for the octuple QD configuration of Fig. 3 is shown in Fig. 2e. It visualizes the effect of plunger and barrier gates on the electrochemical potential of all QDs. As discussed, each row has been normalized such that the diagonal elements are 1. In these units, the effect of the closest barrier gates on the electrochemical potential of a QD is typically between 0.9 and 1.1. This is in agreement with the device geometry (see Fig. 1a) where the barrier gates are 30 nm longer than the plunger gates, bringing them close to the expected QD position. The influence of a neighboring plunger gate on a dot potential is on the order of 0.4–0.5 and the one of the next-nearest neighbor 0.15–0.2, so the coupling diminishes with distance, as expected (Supplementary Fig. S2 plots the cross-capacitance versus distance). The cross-capacitance to the sensing dots is small (typically below 0.1), but nevertheless it is relevant to correct for, as the sensing dots are operated at a steep slope of a Coulomb peak to maximize the charge detection sensitivity.

By means of the virtual barrier gates VB i , we can adjust the interdot tunnel couplings while cross-capacitance correction compensates the influence on the electrochemical potentials. To determine the interdot tunnel coupling, we measure the charge sensor response along the detuning axis across a single-electron transition. Figure 3h shows an example for the (1, 1, 0, 1, 1, 1)–(1, 0, 1, 1, 1, 1) transition, where the numbers in brackets indicate the number of charges on each of the six dots, from QD 2 to QD 7 . The data is fit according to a simple model considering broadening of the transition due to tunnel coupling and thermal excitation,22,27 using a measured effective electron temperature of T e = 90 mK. The tunnel coupling for all pairs of neighboring dots has been tuned to a range of 5 to 15 GHz (see Supplementary Fig. S3).

To further verify the validity of the n + 1 method implemented via the use of virtual gates, we record the charge stability diagram of two neighboring dots, while all other dots are kept in Coulomb blockade. One by one we step the virtual plunger gates of the neighboring dots, which ideally should not affect the measured charge stability diagram. Figure 4 depicts such a test for QD 5 and QD 6 . In panel (a) VP 4 has been increased compared to panel (b) and in panel (c) VP 7 has been increased. The charge stability diagram is not affected by small changes in neighboring electrochemical potentials, which implies that the virtual gates behave as expected and verifies that the charge stability diagram indeed shows addition lines of the expected dots. The 5 mV step size was chosen large enough such that the functioning (or not) of the virtual gates can be properly tested, while remaining well below half the addition energy. A step (positive or negative) of more than half the addition energy would change the electron occupation. The same measurements are repeated for all QDs; charge stability diagrams of neighboring QDs were measured while the electrochemical potential of all other QDs has been altered one by one. Data sets for all gate combinations are shown in Supplementary Fig. S4, showing similar results as presented in Fig. 4.

Fig. 4 Verifying independent control of the electrochemical potentials. Charge stability diagrams of QD 5 and QD 6 measured as function of the virtual plunger gates with all other QDs near the center of a charge stability region. Panel b is a reference measurement. Panel a, c are taken for identical gate voltages as b except that VP 4 (VP 7 ) is increased by 5 mV. We note that stepping VP 4 (VP 7 ) has almost no influence on the charge stability diagram of QD 5 and QD 6 . This figure shows a subset of the data shown in Supplementary Fig. S4 Full size image

We note that it is not trivial that this method works flawlessly and care has to be taken to ensure the electron occupation of each dot is as intended. Specifically, it is important that the neighboring QDs remain sufficiently far from any charge transitions. This requires that the cross-capacitances are measured with a reasonable accuracy, and that the neighboring QDs be detuned from the Fermi level by more than the interdot capacitive coupling energy. To illustrate this point, a set of charge stability diagrams for QD 4 and QD 5 is shown in Fig. 5a–c, with increasing values for VP 6 per panel (The Supplementary video shows a similar series of charge stability diagrams in steps of 0.5 mV in VP 6 .). Figure 5a shows a reference plot of a clean charge stability diagram. In Fig. 5b, the same gate voltages are scanned but VP 6 has been changed by 10 mV. Extra lines appear, which disappear again when increasing VP 6 further (Fig. 5c). The extra lines can be understood if we inspect the charge stability diagram for QD 5 and QD 6 , which is depicted in Fig. 5d with arrows indicating the values of QD 6 used in panels (a–c). We see that arrow b, which corresponds to the case of Fig. 5b, passes through an interdot transition of QD 5 and QD 6 , then intersects an addition line for QD 6 (since the virtual gates are not perfect, this addition line is slightly tilted) and finally cuts through another interdot transition of QD 5 and QD 6 . These three crossings occur at the positions of the red circles in Fig. 5b. By comparison, arrows a and c do not pass through any charge transitions involving QD 6 . This set of data makes clear how to avoid ambiguity in controlling and verifying the number of electrons on each dot.

Fig. 5 Possible pitfalls when using virtual gates. a–c Charge stability diagrams of QD 5 and QD 4 . The virtual plunger VP 6 is stepped by 10 mV from a to b and by another 15 mV from b to c. In b, additional lines appear, which arise both from interdot and dot-reservoir transitions, as explained in the text. d Charge stability diagram of QD 6 and QD 5 . The arrows indicating cuts along the VP 5 axis for three different values of VP 6 correspond to the arrows in a–c. e Simulated double dot charge stability diagram computed for a triple quantum dot system. The simulation assumes the constant interaction model, taking into account the capacitive interdot coupling and using on purpose an imperfect cross-capacitance matrix. The simulation results exhibit similar features as the measurements in d. The three arrows are drawn at the corresponding locations as in d as well. f Simulated charge stability diagram computed for the same triple quantum dot as in e, but for the other DQD. Similar to the data in b, we observe several additional interdot and dot-reservoir charge transition lines Full size image