Biomedical Engineering Reference
In-Depth Information
2
. The energy-
dependent diagonal and off-diagonal elements of the self-energy are interpreted here
as effective or dressed on-site energies and hoppings between the sites
l
and
r
,
respectively. Now assume that sites
l
and
r
are coupled to the rest of the molecule
through sites
l
,
l
,
r
and
r
through identical hoppings.
Then,
Δ
(
)=
(
−
)=(
−
ε
−
Σ
)(
−
ε
l
−
Σ
ll
)
−|
t
lr
+
Σ
lr
|
where
E
det
E
H
eff
E
E
r
rr
t
lr
+
Σ
lr
Δ
G
lr
=
,
(8.29)
where
t
lr
=
t
if
l
and
r
are adjacent to each other, like in the (1,2)-connection, and
zero otherwise.
In our four-QD ring with (1,3) connection, the
P
-space of the connected sites is
formed by the states
{|
1
,|
3
}
centered at QD1 and QD3. The
Q
-space of the rest
of the system is
{|
2
,|
4
}
. Thus, for the (1,3)-connection,
G
13
=
Σ
13
/
Δ
, with
t
2
Σ
13
=(
g
22
+
g
44
+
g
24
+
g
42
)
,
(8.30)
where
g
refers to the
Q
-block of the Green function
G
.Thisgives
Σ
13
as the sum
13
g
22
t
2
,below:
13
g
44
t
2
, and through the
of pathways defined from above:
Σ
=
Σ
=
13
2
g
24
t
2
, and the Green function has the form
coupling:
Σ
=
g
A
g
B
g
C
g
=
.
(8.31)
g
C
†
From (
8.27
)and(
8.30
), it can be seen that
Σ
13
=
0when
E
=(
ε
2
+
ε
4
)
/
2
−
V
, thus
explaining the antiresonance of Fig.
8.2
c.
When the molecule has two paths disconnected from each other, i.e., two disjoint
sets of sites, as in a ring, the Green function in the
Q
-space becomes blocked
g
A
0
g
B
0
g
=
,
(8.32)
so that,
g
24
=
0, and there are only two contributing pathways A (QD1-QD2-
QD3) and B (QD1-QD4-QD3):
g
42
=
t
2
Σ
13
=(
g
22
+
g
44
)
.
(8.33)
Then, a zero of transmission coming from
G
13
=
0 can be attributed to a cancellation
of the two contributions of the dynamical phase solely, one from each path:
g
22
+
g
44
=
0.