Biomedical Engineering Reference
In-Depth Information
Γ
χσ
ij
The argument
E
N
of the
many-body states involved in the tunnelling process, sometimes written in Eq. (
7.6
)
in terms of the operator
H
sys
.
Until now we only concentrated on the sequential tunnelling processes in the
system. We still have to discuss the term in Eq. (
7.5
) which contains the effective
Hamiltonian
H
eff
. The latter is defined as:
Δ
E
of the rate
is the energy difference
E
N
+
1
−
π
NE
χσ
ij
P
NE
d
i
σ
Γ
χσ
1
2
H
eff
=
(
E
−
H
sys
)
p
χ
(
E
−
H
sys
)
d
j
σ
ij
d
j
σ
Γ
χσ
d
i
σ
+
(
H
sys
−
E
)
p
χ
(
H
sys
−
E
)
P
NE
,
(7.8)
ij
2
+
, has been
ı
2π
k
B
T
(
where the principal part function
p
χ
(
x
)=
−
Re
Ψ
x
−
μ
χ
)
introduced, with
T
being the temperature and
the digamma function. Eq. (
7.8
)
shows that the effective Hamiltonian is block diagonal in particle number and
energy, exactly as the density matrix in the secular approximation. Consequently,
it only influences the dynamics of the system in presence of degenerate states. The
effective Hamiltonian depends on the details of the system, yet in all cases it is bias
and gate voltage dependent and this property has important consequences on the
interference blocking phenomena that we are considering.
A natural expression for the current operators is obtained in terms of the time
derivative of the reduced density matrix:
Ψ
I
D
=
NE
Tr
N
ρ
NE
,
I
S
+
(7.9)
where
I
S
/
D
are the current operators calculated for the source and the drain
interfaces. Conventionally we assume the current to be positive when it increases
the charge on the molecule. Thus, in the stationary limit,
I
S
+
I
D
is zero. The
stationary current is obtained as the average:
I
S
=
Tr
{
ρ
stat
I
S
}
=
−
I
D
,
(7.10)
where
ρ
stat
=
lim
t
→
∞
ρ
(
t
)
is the stationary density operator that can be found from
ρ
stat
=
L
ρ
stat
=
˙
0
,
(7.11)
where
is the full Liouville operator defined in (
7.5
). Finally, by following, for
example, the procedure described in detail in [
20
], one finds the explicit expressions
for the current operators:
L
ij
P
NE
d
j
σ
Γ
χσ
∑
NE
f
χ
(
d
i
σ
+
I
χ
=
(
H
sys
−
E
)
H
sys
−
E
)
ij
σ
d
i
σ
Γ
χσ
f
χ
(
−
(
E
−
H
sys
)
E
−
H
sys
)
d
j
σ
P
NE
,
(7.12)
ij
where
the
energy
renormalization
terms,
present
in
the
generalized
master
equation (
7.5
), do not appear.
