4 Coulomb blockade

4.3.2 Master equation
4.3 Single-electron transistor 77
Now we can write the classical master equation for the probability p(n, t) to
find a state with n electrons
dp(n, t)
dt
= Γ nn−1
L + Γ nn−1
nn+1
R p(n − 1,t)+ ΓL + Γ nn+1
R p(n +1,t)−
n−1 n n+1 n n−1 n n+1 n
ΓL + ΓL + ΓR + ΓR p(n, t), (4.39)
where the transitions rates are calculated previously (4.35), (4.38). Introducing
the short-hand notations
we arrive at
dp(n, t)
dt
Γ nn−1 = Γ nn−1
L
Γ nn+1 = Γ nn+1
L
+ Γ nn−1
R ,
+ Γ nn+1
R ,
= Γ nn−1 p(n − 1,t)+Γ nn+1 p(n +1,t) − Γ n−1 n + Γ n+1 n p(n, t).
(4.40)
The structure of this equation is completely clear. The first two terms describe
transfer of one electron from the leads to the system (if the state before
tunneling is |n − 1〉) or from the system to the leads (if the initial state is
|n +1〉), in both cases the final state is |n〉, thus raising the probability p(n)
to find this state. The last term in (4.40) describes tunneling from the state
|n〉 into |n − 1〉 or |n +1〉.
The current (from the left or right lead to the system) is
Ji=L,R(t) =e
n
Γ n+1 n
i
− Γ
n−1 n
i
p(n, t). (4.41)
In the stationary case the solution of the master equation (4.40) is given
by the recursive relation (which is actually the stationary balance equation
between |n〉 and |n − 1〉 states).
Γ n−1 n p(n) =Γ nn−1 p(n − 1). (4.42)
It can be solved very easily numerically, one can start from an arbitrary value
for one of the p(n), then calculate all other p(n) going up and down in n to
some fixed limiting number, and, finally, normalize all the p(n) tosatisfythe
condition (4.27).
Below we present the numerical solution in two cases: for conductance in
the limit of low voltage, and for the voltage-current curves. The transport is
blocked at low temperature and low voltage because of the non-zero addition
energy (4.30).