4 Coulomb blockade

76 4 Coulomb blockade
for the state |k〉 to be occupied, and for the state |α〉 to be unoccupied. In such
a way we take into account that the state denoted as |n〉 is, in fact, a mixed
state of many single-particle states. We assume also for simplicity that the
energy relaxation time is fast enough and the distribution of particles inside
the system is equilibrium (but not the distribution p(n) of the different charge
states!).
Analogously, the transition from the state |n〉 to the state |n − 1〉 is determined
by the probability of tunneling of one electron from the state with n
electrons to the left lead
n−1 n
ΓL = 2π
¯h
2π
¯h
n − 1| ˆ
2
HTL|n δ(Ei − Ef )=
|Vkα| 2 fα (1 − fk) δ(Eα + ∆E + n−1 − Ek). (4.32)
kα
Changing to the energy integration, we obtain
dE1 dE2ΓL(E1,E2)f 0 L(E1) 1 − f 0 S(E2) δ(E2 + ∆E + n − E1),
(4.33)
n+1 n
ΓL =
where
Γi=L,R(E1,E2) = 2π
¯h
|Vα,ik| 2 δ(E1 − Eα)δ(E2 − Ek). (4.34)
kα
Further simplification is possible in the wide-band limit, assuming that
Γi=L,R(E1,E2) is energy-independent. Finally, we arrive at
where Gi is the tunneling conductance
n+1 n
Γi=L,R =(Gi/e 2 )f(∆E + n + eViS), (4.35)
Gi=L,R = 4πe2
¯h Ni(0)NS(0)|V | 2 , (4.36)
with the densities of states Ni(0) and NS(0), and f(E) is
E
f(E) =
exp . (4.37)
E
T − 1
In the same way for the transition from n to n − 1weget
Γ n−1 n
i=L,R =(Gi/e 2 )f(−∆E + n−1 − eViS). (4.38)