4 Coulomb blockade

84 4 Coulomb blockade
n−1 n 2π
ΓLα = n − 1,nα =0|
¯h
ˆ
2
HTL|n, nα =1 δ(Ei − Ef) =
2π
|Vkα|
¯h
k
2 (1 − fk) δ(∆E + n−1 α − Ek), (4.54)
there is no occupation factors (1 − fα), fα because this state is assumed to
be empty in the sense of the master equation (4.52). The energy of the state
is now included into the addition energy.
Using again the level-width function
Γi=L,R α(E) = 2π
|Vik,α|
¯h
2 δ(E − Ek). (4.55)
we obtain
k
n+1 n Γα = ΓLαf 0 L (∆E+ nα)+ΓRαf 0 R (∆E+ nα), (4.56)
n−1 n
0 Γα = ΓLα 1 − fL (∆E + n−1 α )
0 + ΓRα 1 − fR (∆E + n−1 α ) . (4.57)
Finally, the current from the left or right contact to a system is
Ji=L,R = e
p(ˆn)Γiα δnα0f 0 i (∆E + nα) − δnα1(1 − f 0 i (∆E + nα)) . (4.58)
α
ˆn
The sum over α takes into account all possible single particle tunneling events,
the sum over states ˆn summarize probabilities p(ˆn) of these states.
4.4.1 Linear conductance
The linear conductance can be calculated analytically [25, 26]. Here we present
the final result:
G = e2
T
α
∞
ΓLαΓRα
ΓLα + ΓRα
n=1
Peq(n, nα =1) 1 − f 0 (∆E + n−1 α ) , (4.59)
where Peq(n, nα = 1) is the joint probability that the quantum dot contains
n electrons and the level α is occupied
Peq(n, nα =1)=
⎛
peq(ˆn)δ ⎝n −
⎞
⎠ δnα1, (4.60)
ˆn
and the equilibrium probability (distribution function) is determined by the
Gibbs distribution in the grand canonical ensemble:
peq(ˆn) = 1
Z exp
− 1
˜ɛα + E(n) . (4.61)
T
α
β
nβ