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
 
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