The current-voltage characteristics have been calculated in figure 10 for a double dot system with specifications as indicated in the caption. Figure 11 shows the results for a system where the dot specifications have been swapped. Physically this simply amounts to varying the chemical potential in the right reservoir instead of the left reservoir.
At low temperatures two periods may be observed in the I-V characteristics.
The larger period is given by the Coulomb repulsion in the first dot plus a
single
particle spacing 
. The smaller period is simply given by
the single particle spacing 
. This behaviour is reminiscent of
the current through a single dot with 
(see section
iii).
In other words, the second dot with its surrounding barriers acts as a
single barrier with a reduced transparency. The electrons which have entered
the first dot will tunnel into a lower energy level in the second dot at a
rate of approximately 
since the levels in the dots will not
normally line up. At larger temperatures the current curves will lose some
features due to thermal smearing. This is clearly the case in figure
11.
However, the current curve of figure 10 has a more complicated form at high temperatures. It has a region of negative differential conductance which occurs in the bias range where the average occupancy of the first dot increases from one to two electrons with respect to the average occupation number at zero bias. This can be explained by considering the (unlikely) case where a pair of energy levels matches up for a given set of occupation numbers. This will result in a highly increased tunnelling rate between the dots. When the bias across the device is raised the average occupancy of the dot will increase. This means that the dots are less likely to contain the number of electrons for which the levels lined up. Consequently the tunnelling rate between the dots and hence the overall current will decrease, in spite of the fact that the total number of tunnelling paths is likely to have increased.
The temperature at which negative differential conductance can occur
(if at all) is set by the energy difference of the matching pair of
levels in the dots. Their energy separation has to be significantly less
than 
for the tunnelling rate to increase dramatically. A higher
temperature can cause some levels to match up which could not really be
considered energetically close at lower temperatures. However, a higher
temperature also means that there is a smaller probability that
the total inter-dot tunnelling is dominated by just a single pair of
levels. This will reduce the effect. The conclusion is that negative
differential conductance is more likely to occur at higher temperatures, but
when it occurs at a lower temperature the effect will be more pronounced.
Similar to the one dot case, one can extract information about the level
spacing when one considers the current through the system at fixed bias
while varying
the gate voltage of one of the dots, a measurement first performed by
van der Vaart et al. [12].
This is depicted in
figure 12. The Coulomb repulsion energies and the level
spacings are the same as before. The inelastic tunnelling coefficient
is small compared to the dot-reservoir coupling.
The empty levels are the levels that an excess
electron can occupy. Note that no more than a single extra electron can
be contained in the dot, since this is prevented by the Coulomb blockade.
Since electrons carry a negative charge, a rise in the gate potential
causes a downward shift of the energy levels.
The results are shown in figure 13 for a range of values of
. A few generic features can be noted which are also true
for tunnelling through a single dot.
Firstly, the current is periodic in the gate voltage with a period
. Only a single period is shown in the figure.
Secondly, a significant current is only allowed to flow when the
Coulomb blockade is overcome in both dots. This requires the lowest
available level in each dot to lie within the energy window .
This is clearly shown in figure 13 where a larger bias voltage
allows the current to flow at more values of the gate potential.
The most striking feature of figure 13 is the occurrence of
sharp current peaks. These happen at values of the gate voltage where one
of the occupied states of the first dot (containing the excess electron)
lines up with an empty level in the second dot. It is clear that this
will happen with a periodicity (in the figure 
). When the bias voltage is
such that several levels in the second dot are contained within the
energy window , then peaks will also occur at intervals
. This is the case in the third graph of figure 13,
where 
.
Finally, note that the current in the valleys between the peaks increases
more or less linearly with the number of peaks.
This reflects the fact that at a higher gate potential there are
more electrons in the first dot which are able to tunnel downwards in energy
into the second dot. Each tunnelling path has an associated
off-resonance tunnelling rate of which will contribute an
amount 
towards the total current (assuming
).
Therefore the current minima should increase by this amount every time
the gate potential is increased by an amount .
This seems to be a much more powerful method for determining the single particle
level spacing than the analogous experiment with a single dot.
In the model used in this chapter, the
line shape will be a multiple of the Bose-Einstein distribution. However,
at very small energy separations 
the intrinsic width of the levels becomes
significant
and the lineshape will be approximately a Lorentzian of width 
near
resonance but will be asymmetric off-resonance. This seems to
corresponds at least qualitatively to recent experiment [12].
In the case of large 
, the middle barrier is no longer the
current limiting barrier and the current will not peak anymore but simply
increase with until a saturation value of 
is
reached.
Further experiments will have
to show whether it is justified to assume that the inelastic tunnelling rate
can be approximated by the Bose-Einstein distribution. If this turns out
to be a false assumption, then the master equation can still be used to model
the effect of a more realistic inelastic tunnelling rate.