4 Coulomb blockade

68 4 Coulomb blockade
The minimal energy required to add one electron to the neutral system is
called charging energy
EC = ∆E +
N0 (N0 → N0 +1)= e2
. (4.3)
2C
If we want to add the second electron, the addition energy
∆E +
N0+1 (N0 +1→ N0 +2)= 3e2
, (4.4)
2C
is required, and so on. We can say that in this case the discrete energy levels
are formed by the Coulomb interaction, in spite of the fact that the initial
energy spectrum is dense (quasi-continuous).
The charging energy is an important energy scale. Let us estimate it in
some typical cases. The classical expression for the capacitance of the usual
oxide tunneling junction is C = εS/(4πd), where ε ≈ 10 is the dielectric constant
of the oxide, S ≈ (100nm) 2 is the area of the junction, and d ≈ 10˚A is
the thickness of the oxide layer. With these typical parameters the capacitance
is C ≈ 10 −15 F, which corresponds to the charging energy EC ≈ 10 −4 eV and
temperature T ∼ 1 K. This means that in metallic junctions the charging effects
can be observed at sub-Kelvin temperatures. However, for nanoscale
metallic particles, semiconductor quantum dots, and single molecules the
charging energy, and corresponding temperature, can be orders of magnitude
larger.
4.1.2 Discrete energy levels
The first important energy scale in small systems is the charging energy EC,
the second one is the energy level spacing ∆ɛ between the eigenstates of a
system. In metallic islands and semiconductor quantum dots the level spacing
is determined by the geometrical size, because the discrete energy levels are
formed as a result of spatial quantization of quasiparticles. It can be estimated
using the free-electron expression near the Fermi surface. For example, in a
3D system with characteristic size L
∆ɛ(ɛF ) ∼ π2 ¯h 2
, (4.5)
mkF L3 here m is the effective mass, kF is the Fermi wave-vector.
In smaller systems, like small quantum dots, multi quantum dot systems,
and single molecules, the energy spectrum is more complex and the level
spacing (or, more precisely, the energy spectrum ɛλ itself) should be calculated
from the microscopic model. In this case the interactions can not be neglected
and the level spacing ∆ɛ can not be unambiguously distinguished from the
charging energy EC. In the limit of small single molecules the charging energy
is, in fact, the same as the ionization energy, which in turn is related to