4 Coulomb blockade

90 4 Coulomb blockade
4.5.2 Elastic cotunneling
So-called elastic cotunneling is a result of the second-oder process excluded
in previous consideration, namely
Γ (2)n
2π
V
el =
¯h
∗ αqVβk
Eβ + ∆E + V
+
n − Ek
∗
αqVβk
− Eα
2
k∈L,q∈R
αβ
Eq − ∆E + n−1
× f 0 L(Ek) 1 − f 0 R(Eq) δ(Ek − Eq). (4.72)
For metallic dots the result of calculation is
Γ (2)n
el =2π
VβkV
¯h
∗ αkVqβV ∗
qαf 0 L(Ek) 1 − f 0 R(Eq)
kqαβ
× F (Eα,Ek,Eq)F (Eβ,Ek,Eq)δ(Ek − Eq), (4.73)
1 − f
F (E,Ek,Eq) =
0 (E)
E + ∆E + f
+
n − Ek
0 (E)
E + ∆E + . (4.74)
n−1 − Eq
We shall not go into further details here and refer to the original publications.
Elastic cotunneling though metallic dots is usually much smaller, than
inelastic, because the number of states available for elastic process is small.
It can be important only at very low temperatures and voltages, because it
is linear in voltage at zero temperature, while the inelastic cotunneling in
nonlinear (4.71).
Note once more, that the expressions for cotunneling current obtained
above are valid only in the Coulomb blockade regime far from the degeneracy
points (where the maximums of conductance are observed), when the sequential
tunneling is suppressed, voltage and temperature are not very large. The
systematic theory of high-order tunneling processes will be considered in other
chapter.
4.6 The effects of environment
To be written later.
4.7 Problems
4.7.1 The electrostatic energy of a single-electron transistor
4.7.2 Analytical formula for linear conductance
4.7.3 Elastic cotunneling through a single level