4 Coulomb blockade

78 4 Coulomb blockade
G [arb. u.]
-1 0 1 2 3
Q /|e|
*
Fig. 4.5. Conductance oscillations as a function of the gate voltage at different
temperatures T =0.01EC, T =0.1EC, T =0.3EC, T =0.5EC (upper curve).
4.3.3 Conductance: CB oscillations
Consider first the conductance of a system as a function of the gate voltage
VG (gate charge Q ∗ ). The result is shown in Fig. 4.5. The conductance has
maximum in the degeneracy points Q ∗ /|e| = n+0.5, when the addition energy
(4.30) vanishes. Between these points the conductance is very small at low
temperatures, because the electron can not overcome the Coulomb energy,
the effect was nick-named ”Coulomb blockade”.
The phenomena can be described within the noninteracting particle picture,
if we notice, that the charging energy produce, in fact, discrete energy
levels, while the gate voltage works as a potential changing the position of
the levels up and down in energy. The maximum of the conductance is observed
when this induced discrete level is in resonance with the Fermi levels
in the leads. This single-particle picture (or better to say analogy), however,
should be used with care. The problem is that this single-particle level is
fictional and corresponds, in fact, to a superposition of two many-particle
states with different number of electrons. In the degeneracy point the probabilities
p(n) andp(n + 1) of the states with different number of electrons
are equal. This circumstance shows that the Coulomb blockade is essentially
many-particle phenomena, and the mean-field (Hartree-Fock) approximation
can not be used to describe it. Later we shall see, that it is a consequence of
the degeneracy of a single-particle spectrum.
At higher temperatures, the oscillations are smeared and completely disappear
at T ∼ EC, the effect is analogous to the charge quantization in a
single-electron box.