4 Coulomb blockade

4.4 Coulomb blockade in quantum dots 83
much smaller than the energy relaxation time, and quasiparticle nonequilibrium
effects are usually neglected).
With these new assumptions, the theory of sequential tunneling is quite
the same, as was considered in the previous section. The master equation
(4.40) is replaced by
dp(n, ˆn, t)
=
dt
ˆn ′
nn−1
Γˆnˆn ′ p(n − 1, ˆn ′ ,t)+Γ nn+1
ˆnˆn ′ p(n +1, ˆn ′ ,t) −
n−1 n
Γˆn ′ n+1 n
ˆn + Γˆn ′
ˆn p(n, ˆn, t)+I {p(n, ˆn, t)} , (4.51)
ˆn ′
where p(n, ˆn, t) is now the probability to find the system in the state |n, ˆn〉,
Γ nn−1
is the transition rate from the state with n−1 electrons and single level
ˆnˆn ′
occupation ˆn ′ into the state with n electrons and single level occupation ˆn.
The sum is over all states ˆn ′ , which are different by one electron from the state
ˆn. The last term is included to describe possible inelastic processes inside the
system and relaxation to the equilibrium function peq(n, ˆn). In principle, it
is not necessary to introduce such type of dissipation in calculation, because
the current is in any case finite. But the dissipation may be important in
large systems and at finite temperatures. Besides, it is necessary to describe
the limit of classical single-electron transport, where the distribution function
of qausi-particles is assumed to be equilibrium. Below we shall not take into
account this term, assuming that tunneling is more important.
While all considered processes are, in fact, single-particle tunneling pro-
cesses, we arrive at
dp(ˆn, t)
=
dt
β
δnβ1Γ
δnβ1Γ nn−1
β
β
p(ˆn, nβ =0,t)+δnβ0Γ nn+1
β p(ˆn, nβ =1,t) −
n−1 n
β
+ δnβ0Γ
n+1 n
β
p(ˆn, t), (4.52)
where the sum is over single-particle states. The probability p(ˆn, nβ =0,t)
is the probability of the state equivalent to ˆn, but without the electron in
the state β. Consider, for example, the first term in the right part. Here the
delta-function δnβ1 shows, that this term should be taken into account only if
the single-particle state β in the many-particle state ˆn is occupied, Γ nn−1
β is
the probability of tunneling from the lead to this state, p(ˆn, nβ =0,t)isthe
probability of the state ˆn ′ , from which the system can come into the state ˆn.
The transitions rates are defined by the same golden rule expressions, as
before (4.31), (4.32), but with explicitly shown single-particle state α
n+1 n 2π
ΓLα = n +1,nα =1|
¯h
ˆ 2
HTL|n, nα =0 δ(Ei − Ef) =
2π
|Vkα|
¯h
2 fkδ(∆E + nα − Ek), (4.53)
k