10.5. MESOSCOPIC STRUCTURES 503
systems, electron charging energy effects arising from Coulomb interactions between electrons
can become significant. This phenomenon is called the Coulomb blockade effect. We are fa-
miliar with the parallel plate capacitor with capacitanceCand the relation between a charge
incrementΔQand the potential variationΔV
C=
ΔQ
ΔV
orΔV=ΔQ
C
(10.5.9)
The capacitance is given by the spacing of the plates(d)and the area(A)
C=
A
d(10.5.10)
Now consider a case where the capacitance decreases until a single electron on the capacitor
causes a significant change in the voltage. The charging energy to place a single electron on a
capacitor is
ΔE=e^2
2 C(10.5.11)
and the voltage needed is
e
2 C
=
80 mV
C(aF)(10.5.12)
where the capacitance is in units of 10 −^18 F(aF). If we write the charging energy as a thermal
energy,kBT 0 , the temperature associated with the charging energy is
T 0 =
e^2
2 kBC=
928. 5 K
C(aF)(10.5.13)
Coulomb blockade effects will manifest themselves if the sample temperatureTis smaller than
this effective charging temperatureT 0 and we expect the following to occur:
- When the capacitance reaches values approaching∼ 10 −^18 F, each electron causes a shift in
voltage of several 10s of millivolts. - The charging energy of the capacitor, i.e., the energy needed to place a single extra electron
becomes comparable to or larger thankBTwithTreaching 10 K or even 100 K if the capacitance
becomes comparable to 10 −^18 F.
To get the small capacitors needed to generate Coulomb blockade effects at reasonable tem-
peratures one has to use areal dimensions of<∼ 1000 A ̊× 1000 A with spacing between the ̊
contacts reaching∼50–100A. With such dimensions (using a relative dielectric constant of ̊ ∼
- we get capacitors with capacitances of the order of∼ 10 −^16 F. The charging voltages are
then∼1mVandT 0 ∼10 K. If the area of the capacitor is reduced further these values increase.
It is possible to fabricate small capacitors with capacitance approaching 10 −^18 F.
In figure 10.12 we show the band profile of a typical tunnel junction capacitor which consists
of two metal contacts separated by a thin tunneling barrier. In the absence of any Coulomb
blockade we observe a monotonic increase in current with applied bias as shown in figure 10.12a.
In case the Coulomb blockade is significant we get a very different device behavior. In fig-
ure 10.12b we show the behavior for a structures where the charging energy is large enough to