4 Coulomb blockade

88 4 Coulomb blockade
Before considering the particular cases, let us perform all possible general
calculations in (4.62). First of all, the tunneling Hamiltonian is the sum of
left and right parts, we are interested only in the processes, which transfer
electrons from the left to the right, and include one tunneling through the
right junction and one tunneling through the left junction. Then, the initial
state is a mixed state with different probabilities for the states |k〉 and |q〉 to be
occupied or empty, following the tunneling theory we introduce the sum over
all possible configurations of the initial state with corresponding occupation
probabilities (distribution functions) f 0 L (Ek) andf 0 R (Eq). Thus, we obtain
Γ (2) 2π
L→R =
¯h
×
λ
fq
k∈L,q∈R
f 0 L(Ek) 1 − f 0 R(Eq) δ(Ei − Ef )
ˆ
HTR
λ λ ˆ
HTL
ik
+
fq
Eλ − Ei
ˆ
HTL
λ λ ˆ
HTR
ik
2
. (4.63)
The intermediate states |λ〉 are the states with one extra electron (or
without one electron, e.g. with one extra hole) in the central system. Note,
that the only thing we assume is that the charge of the central system is the
same in the initial and the final states (and ±e in the virtual intermediate
state). However the state and energy of the central system can change after
tunneling event. In the last case the process is called inelastic cotunneling.
Up to now, all our calculation is general, now it is reasonable to consider
separately the contributions of inelastic and elastic cotunneling.
4.5.1 Inelastic cotunneling
Consider the final state as the state with one electron transferred from the
state k in the left lead to the state q in the right lead with the change of the
state of the system from |λi〉 with electron in the state α and empty state β
to |λf 〉 with electron removed from α to β
f = c † qckd †
i , (4.64)
the energy conservation implies that
β dα
Ei = Ek + Eα = Ef = Eq + Eβ. (4.65)
Due to mutual incoherence of these states, not the amplitudes, but the
probabilities of different events should be summarized. Below we neglect all
cross-terms with different indices λ, and apply
Xλ
2
= |Xλ| 2 . (4.66)