4 Coulomb blockade

82 4 Coulomb blockade
4.4 Coulomb blockade in quantum dots
Here we want to consider the Coulomb blockade in intermediate-size quantum
dots, where the typical energy level spacing ∆ɛ is not too small to neglect it
completely, but the number of levels is large enough, so that one can use the
constant-interaction model (4.17), which we write in the eigenstate basis as
ˆHCIM =
α
˜ɛαd † αdα + E(n), (4.46)
where the charging energy E(n) is determined in the same way as previously,
for example by the expression (4.16). Note, that for quantum dots it is not
convenient to incorporate the gate voltage into the free electrostatic energy
(4.24), as we did before for metallic islands, also the usage of classical capacitance
is not well established, although for large quantum dots it is possible.
Instead, we shift the energy levels in the dot ˜ɛα = ɛα + eϕα by the electrical
potential
ϕα = VG + VR + ηα(VL − VR), (4.47)
where ηα are some coefficients, dependent on geometry. This method can be
easily extended to include any self-consistent effects on the mean-field level by
the help of the Poisson equation (instead of classical capacitances). Besides,
if all ηα are the same, our approach reproduce again the relevant part of the
classical expressions (4.23), (4.24)
ÊCIM =
ɛαnα + E(n)+enϕext. (4.48)
α
The addition energy now depends not only on the charge of the molecule,
but also on the state |α〉, in which the electron is added
∆E + nα(n, nα =0→ n +1,nα =1)=E(n +1)− E(n)+ɛα, (4.49)
we can assume in this case, that the single particle energies are additive to the
charging energy, so that the full quantum eigenstate of the system is |n, ˆn〉,
where the set ˆn ≡{nα} shows weather the particular single-particle state |α〉
is empty or occupied. Some arbitrary state ˆn looks like
ˆn ≡{nα} ≡ n1,n2,n3,n4,n5, ... = 1, 1, 0, 1, 0, ... . (4.50)
Note, that the distribution ˆn defines also n =
α nα. It is convenient, however,
to keep notation n to remember about the charge state of a system,
below we use both notations |n, ˆn〉 and short one |ˆn〉 as equivalent.
The other important point is that the distribution function fn(α) inthe
charge state |n〉 is not assumed to be equilibrium, as previously (this condition
is not specific to quantum dots with discrete energy levels, the distribution
function in metallic islands can also be nonequilibrium. However, in the parameter
range, typical for classical Coulomb blockade, the tunneling time is