In analogy to the properties of a single quantum dot, one characterizes the system by a stability diagram. In the case of coupled quantum dots, however, the diagram will look quite differently since one now has to combine two Coulomb diamond pictures to just one common characteristical plot. In this section, we follow the review of van der Wiel et al. [66] to explain how the honeycomb diagram is formed and which regimes exist.
The transport properties of a double quantum dot can be understood by analyzing a network diagram (see Figure 2.9). Similar to the single quantum dot, one can define electrochemical potentialsµ1(N1, N2) andµ2(N1, N2) for dot 1 (left) and dot 2 (right) with N1 electrons on dot 1 and N2 electrons on dot 2. Again, these potentials are defined by the differences of electrostatic potentials U(N1, N2):
µ1(N1, N2) = U(N1, N2)−U(N1−1, N2) = µ N1− 1 2 ¶ EC1+N2ECm− 1 |e|(Cg1Vg1EC1+Cg2Vg2ECm) (2.10)
Figure 2.9: From Ref. [66]: Network of relevant capacitances and tunnel junctions for a double quantum dot structure. The tunnel barriers are modeled by a resistor and a capacitance in parallel as indicated in the upper right corner of this Figure.
µ2(N1, N2) = U(N1, N2)−U(N1, N2−1) = µ N2− 1 2 ¶ EC2+N1ECm− 1 |e|(Cg1Vg1ECm+Cg2Vg2EC2), (2.11) where EC1, EC2 and ECm are charging energies associated with the dots 1, 2 and with change in energy of one dot when an electron is added to the other dot. Cg1, Cg2 and Cm are the capacitances from gate 1 (with voltageVg1) to dot 1, from gate 2 (with voltage Vg2) to dot 2 and between the two quantum dots.
Again, as in the case of a single dot, two addition energies Eadd,1 and Eadd,2 in the quantum regime can be defined as
Eadd,1 =µ1,`(N1+ 1, N2)−µ1,k(N1, N2) =EC1+ (E`−Ek) = EC1 + ∆E, (2.12) where the (N1+1)th electron is an discrete state`and theN1th electron occupies the state k. ∆E is again the level spacing between the discrete levels for one charging state. ∆E can again also be zero, where one would then find the classical expression for the addition energy Eadd,1 =EC1, which is only determined by the charging energy for dot 1. For dot 2, one finds in a similar way Eadd,2 =EC2+ ∆E in the quantum case.
The charge stability diagram can then be determined by the expressions for the elec- trochemical energies µ1(N1, N2) and µ2(N1, N2) as depicted in Figure 2.10. One finds the equilibrium electron numbers on dots 1 and 2 asN1, N2 as a function of the gate voltages Vg1 and Vg2. The electrochemical potentials on the two leads S (source) and D (drain) have been defined to be zero, if no bias voltage VSD is applied. Therefore, the equilib- rium charges on the dots are the largest integers N1 and N2 for which both µ1(N1, N2) and µ2(N1, N2) are less than zero. If either would be larger than zero, electrons would escape to the leads. This creates the hexagonal shape of the regions of charge stability in Figure 2.10.
Now, we will have a look on the different coupling regimes between the two quantum dots. If the dots are completely decoupled [Figure 2.10 (a)], the gate voltages Vg1 and Vg2
Figure 2.10: From Ref. [66]: Stability diagrams of the double quantum dot system with different coupling strengths between the dots: (a) weak inter-dot coupling, (b) intermediate inter-dot coupling and (c) strong inter-dot coupling. The triple points and their charge transfer processes are depicted in (d). The electron transfer cycle (•) has a counterclockwise orientation in the stability diagram, the hole transfer cycle (◦), however, is oriented in a clockwise direction.
change the charge on dot 1 and 2 without affecting the charge on the other dot. For larger couplings between the dots, the domains of stability become hexagonal [Figure 2.10 (b)]. The vertices in this diagram are nowtriple points. If one increases the coupling even more, such that the capacitance Cm becomes the dominant capacitance of the system CCm1(2) →1, the double dot behaves as one large dot [Figure 2.10 (c)] with charge N1+N2.
Such an analysis is only valid in the linear conductance or equilibrium regime, where the bias voltage over the double quantum dot is very small or even vanishesVSD =µS−µD ≈0. To obtain a measurable current, the tunnel barriers should be sufficiently transparent. But they also need to be sufficiently opaque to ensure a fixed, well-defined electron number on each dot. A conductance resonance can be found, when electrons can tunnel through both dots. This happens, if three charge states are degenerate, i.e. their boundaries meet at one point, the triple point. One can distinguish two of such triple points depending on the charge transfer processes [Figure 2.10 (d)]. There is an electron transfer cycle [full circle (•) in Figure 2.10 (d)]
(N1, N2)→(N1+ 1, N2)→(N1, N2+ 1)→(N1, N2), (2.13) which shuttles an electron through the double quantum dot and a hole transfer cycle [open circle (◦) in Figure 2.10 (d)]
(N1+ 1, N2 + 1)→(N1+ 1, N2)→(N1, N2+ 1)→(N1+ 1, N2+ 1), (2.14) where a hole is tunneling through the double dot.
The distances in a single stability cell [Figure 2.11] of the full charge stability diagram can be deduced by considering the electrochemical potentials of the two dots at the borders between two charge domains as
∆Vg1(2) = | e| Cg1(2) µ 1 + ∆E EC1(2) ¶ (2.15)
Figure 2.11: From Ref. [66]: one stable charge configuration in detail. By connecting the triple points in an experimentally determined stability diagram, one can deduce the spacings between the lines (see text) and thus, get more information on the capacitances in the electrostatic network of this structure.
∆Vgm1(2) = |e|Cm Cg1(2)C2(1) µ 1 + ∆E ECm ¶ , (2.16)
whereC1(2)is the sum of all capacitances attached to dot 1(2). Here, the level spacing ∆E is the expression from the addition energies Eq. (2.12).
A more detailed picture of the hexagonal pattern of a single charge stability cell is shown in Figure 2.12 (a). There one can see, where the dot levels of the two dots align or how their positions are related with the electrochemical potentials of the leads.
If one now turns to the case of a finite bias voltageVSDapplied over the double quantum dot, the triple points are reshaped to triangular patterns, see Figure 2.12 (b).
We assume here that the right lead is grounded µD = 0 and that the bias voltageVSD is only applied to the left lead µS = −|e|VSD. Now the condition −|e|VSD = µS ≥ µ1 ≥ µ2 ≥µD = 0 gives rise to the boundaries of the trangular pattern in Figure 2.12 (b)
α1δVg1 = Cg1 C1 | e|δVg1 =|eVSD| (2.17) α2δVg2 = Cg2 C2 | e|δVg2 =|eVSD|, (2.18)
whereα1 andα2 are the conversion factors between gate voltages and energies. In addition to the new triangular shape of the original triple points, the triangular region has some lines in it, if one considers the discreteness of the levels and if excited states can contribute to the nonlinear transport. The details can be found in Ref. [66]. Also more information about resonant tunneling, magnetic field spectroscopy and microwave spectroscopy of the double quantum dot system can be found in this review article. Since we are interested in a charge qubit in a double quantum dot system, we will not need all of these insights into the system.
(a) (b)
Figure 2.12: From Ref. [66]: (a) a “unit cell” of the stability diagram [dotted square in Figure 2.10 (b)]. One can distinguish four different charge configurations, separated by solid lines. At most solid lines, the electrochemical potential of at least one dot is zero, whereas at the solid line between the triple points, the two charge states (0,1) and (1,0) are degenerate. The dashed lines are extensions of the solid lines within the hexagonal patterns. (b) nonlinear (finite bias) transport in a “unit cell” of the stability diagram. The triple points are then reshaped to triangular regions. The solid lines again separate the different charge configurations. In the case of only one discrete level per dot, resonant tunneling (aligning the two dot states) is only possible along the side of the triangles that coincides with the line connecting the original triple points (• and ◦). Inelastic tunneling and cotunneling processes can still contribute to finite current within the triangles. Additional lines within the gray triangles could be observed, if one included more discrete levels for both dots.
Another double quantum dot setup has been realized in Ref. [103]. There, the two quantum dots were arranged in a parallel geometry, such that both dots were coupled to both leads. Figure 2.13 shows the layout and an atomic force microcopy (AFM) picture of the sample. The characteristic honeycomb pattern from Figure 2.10 (b) can be found for this sample, but the main focus of the article is the probing of the bonding between the two dots in the cotunneling regime. More details on the sample design can also be found in Ref. [78].