Biomedical Engineering Reference
In-Depth Information
Ug
Q
U
†
is the self-energy that contains the interactions involving the
orbitals not connected to the leads.
The Green function written in terms of the self-energy,
where
Σ
=
E
1
Δ
−
ε
n
−
Σ
nn
t
1
n
+
Σ
1
n
g
P
(
E
)=
,
(8.25)
t
n
1
+
Σ
n
1
E
−
ε
1
−
Σ
11
2
, contains all the
matrix elements needed for the calculation of the conductance between the sites 1
and
n
. Their poles are given by the zeros of the secular determinant
with
Δ
=
det
(
E
−
H
eff
)=(
E
−
ε
1
−
Σ
11
)(
E
−
ε
n
−
Σ
nn
)
−|
Σ
1
n
|
Δ
=
0, while
the antiresonances come from zeros in
g
1
n
=(
t
1
n
+
Σ
1
n
)
/
Δ
.When
t
1
n
=
0, as in
the (1,3) connection, the antiresonances arise from the zeros of
Σ
1
n
. Interestingly,
ij
ij
C
the self-energy
ij
becomes a sum of contributions throughout
the paths from above (A), from below (B), and through the interarm coupling
(C). The vanishing of the transmission occurs when the contributions from those
paths interfere destructively, thus cancelling the element
t
1
n
+
Σ
1
n
of the effective
Hamiltonian.
For the (1,3) connection, the contributions become
Σ
ij
=
Σ
+
Σ
+
Σ
t
41
g
44
t
34
e
−
2
i
ϕ
,
Σ
A
13
t
12
g
22
t
23
e
2
i
ϕ
,
Σ
B
13
C
13
Σ
=
=
=
g
24
(
t
41
t
23
+
t
12
t
34
)
,
(8.26)
where
g
22
=(
E
−
ε
4
)
/
D
,
g
44
=(
E
−
ε
2
)
/
D
,
g
24
=
V
/
D
,
(8.27)
V
2
and
D
, with
E
a
and
E
b
being the
energies of the bonding (
E
b
) and antibonding (
E
a
) orbitals of the molecule QD2-
QD4 described by
H
Q
.
=(
E
−
ε
2
)(
E
−
ε
4
)
−
=(
E
−
E
a
)(
E
−
E
b
)
8.2.8
Antiresonances from Interfering Pathways
As we discussed above antiresonances occur at the energy eigenvalues of the
molecule. Nevertheless, (
8.15
) shows that also the roots of
G
lr
can lead to zeros
of conductance that are termed
multipath zeros
[
48
].
The Green function of the isolated system can be written as the resolvent
(
E
−
H
eff
)
−
1
of an effective 2
×
2 Hamiltonian in the
P
-space, in terms of the self-energy
Σ
(
E
)
as [
48
,
50
]
E
1
Δ
−
ε
r
−
Σ
rr
t
lr
+
Σ
lr
G
(
E
)=
,
(8.28)
t
rl
+
Σ
rl
E
−
ε
l
−
Σ
ll
