4 Coulomb blockade

74 4 Coulomb blockade
The average number of electrons is
〈n〉 =
np(n) = 1
n exp −
Z
E∗
(n)
. (4.28)
T
n
The results of calculation at different temperatures are shown in Fig. 4.3.
We see, quite obviously, that the steps are smeared with temperature and
disappear almost completely at T ≈ EC, in agreement with the estimate
(4.6).
4.3 Single-electron transistor
Now we come back to a transport problem. Consider the system between two
leads coupled by two weak tunneling contacts (Fig. 4.4). When there is also
the third electrode (gate), the system is called single-electron transistor.
The electrostatic energy of the central region is given by the same formula
(4.24) with the gate charge Q ∗ determined by the combination of all voltages
applied to a system and the full capacitance is
n
C = CG + CL + CR. (4.29)
The addition energy to increase the number of electrons from n to n +1 is
∆E + n (n → n +1)=E ∗ (n +1)− E ∗ (n) = e2
C
3
2
1
0
n + 1 Q∗
+
2 e
-1 0 1 2 3
Q /|e|
*
-1
. (4.30)
Fig. 4.3. The average number of electrons 〈n〉 in a single-electron box as a function
of the gate voltage at different temperatures T = 0.01EC (steps), T = 0.1EC,
T =0.3EC, T = EC (nearly linear).