Biomedical Engineering Reference
In-Depth Information
Fig. 7.20
Combination of the bias and contact perturbations.
Left panel
—Energy levels of the
7-particle ground and first excited state as functions of the bias voltage.
Right panel
—Stability
diagram around the 6
↔
7 resonance. The perturbation parameters are in this case
ξ
c
=
2
h
Γ
and
a
=
12. From [
20
]
d
i
σ
i
π
χσ
ij
f
χ
(
−
(
−
)
+
p
χ
(
−
)+
−
)
d
j
σ
Γ
H
sys
E
H
sys
E
H
sys
E
d
i
σ
i
π
d
j
σ
Γ
χσ
f
χ
(
−
(
H
sys
−
F
)
−
p
χ
(
H
sys
−
F
)+
H
sys
−
F
)
P
NF
(7.62)
ij
where
R
indicates the left or right contact. Nevertheless, within the limits
of derivation of the master equation, this formula can be simplified. Actually, if
E
χ
=
L
,
,then
F
can be safely substituted with
E
in the argument of the principal
values and of the Fermi functions, with an error of order
−
F
≤
h
Γ
Γ
k
B
T
which is
negligible (the generalized master equation that we are considering is valid for
h
E
F
k
B
T
−
h
<
Γ
k
B
T
). The approximation
E
∼
F
breaks down only if
E
−
F
∼
k
B
T
, but this
implies
E
which is the regime of validity of the secular approximation.
Consequently, in this regime, terms with
E
−
F
h
Γ
F
do not contribute to the average
current because they vanish in the stationary density matrix. Ultimately we can thus
reduce the current operators to the simpler form:
I
χ
=
NE
ij
σ
P
NE
d
j
σ
Γ
χσ
=
f
χ
(
d
i
σ
(
H
sys
−
E
)
H
sys
−
E
)
ij
d
i
σ
Γ
χσ
f
χ
(
−
(
E
−
H
sys
)
E
−
H
sys
)
d
j
σ
,
(7.63)
ij