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.
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