4 Coulomb blockade

4.5 Cotunneling 89
In the metallic dots we can assume that the distribution over internal
single-particle states is equilibrium, and obtain
Γ (2)n
in
2π
=
¯h
k∈L,q∈R αβ
1
Eβ + ∆E + +
n − Ek
1
Eq − ∆E + n−1 − Eα
2
|Vαq| 2 |Vβk| 2
× f 0 L(Ek) 1 − f 0 R(Eq) 1 − f 0 S(Eβ) f 0 S(Eα)δ(Ek + Eα − Eq − Eβ). (4.67)
The two terms in this expression correspond to two physical processes. The
first one describes the jump (virtual!) of the extra electron from the state k in
the left lead into the state β in the system, and then the other electron jumps
through the second junction from the state α into the state q in the right
lead (Fig. 4.13b). In the second process (Fig. 4.13d) first the electron jumps
from the system into the right lead, and then the second electron jumps from
the left lead. All other matrix elements of the tunneling Hamiltonians in the
expression (4.63) are zero. The distribution functions describe as usually the
probabilities of necessary occupied or empty states.
In the wide-band limit with energy independent matrix elements we arrive
at
Γ (2)n
in
=¯hGLGR
2πe4
dEk dEq dEα dEβf 0 L(Ek) 1 − f 0 S(Eβ) f 0 S(Eα) 1 − f 0 R(Eq)
1
×
Eβ + ∆E + 1
+
n − Ek Eq − ∆E + 2 δ(Ek + Eα − Eq − Eβ).
n−1 − Eα
(4.68)
At zero temperature the integrals in this expression can be performed
analytically [31, 24].
The current is determined by the balance between forward and backward
tunneling rates
Jin = ¯hGLGR
12πe 3
J(V )=e
∆E + n
Γ (2)n
in
∆E + n−1
(V ) − Γ (2)n
in
(−V ) . (4.69)
At finite temperatures and small voltage V ≪ ∆E + n ,∆E + n−1 , the analytical
solution for the current is
1
−
1
2 (eV 2 2
) +(2πT) V. (4.70)
The linear conductance can be estimated as
T
2 . (4.71)
G ∼ h
GLGR
e2 EC