4 Coulomb blockade

4.2 Single-electron box 73
E ∗ (n) = Q2L +
2CL
Q2G + QGVGL, (4.22)
2CG
where C = CL + CG. Substituting (4.20) and (4.21) in this expression, we
obtain
E ∗ (n) = (en)2 CGVGL
+
2C C (en)+CLCGV 2 GL
. (4.23)
2C
Neglecting the last irrelevant term independent on n, we get the free electrostatic
energy of a SEB with n = N − N0 excess electrons:
E ∗ (n) = e2 (n + Q∗ /e) 2
, (4.24)
2C
where Q ∗ = CGVGL is determined by the gate voltage and is non-integer in
the units of the elementary charge e.
This expression can be obtained alternatively from the simple argument,
that the full energy of the system is the sum of its own electrostatic energy and
the energy of the charge ne in the external potential. Indeed, the first term in
the formula (4.23) is the electrostatic energy of the isolated system, and the
second term is the electrostatic energy in the external potential (CG/C)VGL in
relation to the potential of the left lead. The full potential difference (voltage
VLS) between the system and the left lead is (from (4.20))
VLS = QL CGVGL
+ , (4.25)
CL
C
where the first term is the electrical potential produced by the charge of the
island with capacitance C, and the second term is produced by the external
voltage. Integrating it over the charge ne we obtain the free energy (4.23).
We plot the energy E∗ (n) as a function of the gate charge Q∗ at different n
(thenegativeelectronchargee = −|e| is taken into account explicitly) in the
Fig. 4.2. This picture shows, that the state with minimal energy is changed
with Q∗ . The neutral state n = 0 is stable at −0.5