Biomedical Engineering Reference
In-Depth Information
the dot 1, as shown in Fig. 8.1 b; for brevity we refer to it as the connection (1,3).
In experimental setup, the quantum dots 2 and 4 have a molecular coupling
controlled by a plunger gate; this allows to open or to pinch-off gradually the direct
connection between them. In our model, the interdot hopping V parameter is allowed
to vary continuously between zero (pinch-off) and a finite value (open).
8.2.5
Properties of the Transmission Function T(E)
Some characteristics of the transmission spectrum can be derived in general terms,
from the relation between the Green functions for the isolated system, G , and for the
system connected to the terminals,
. We will assume that the ring is connected to
each terminal through single sites, l and r , attached to leads L and R, respectively.
We assume the wide band approximation, in which the spectral densities at the
leads are energy-independent, and a symmetric coupling to the leads (
G
Γ l = Γ r = Γ
).
Using Dyson equation ( 8.7 ), the Green function of the connected system,
G
, can be
written in terms of the Green function of the isolated molecule, G ,as
G lr
G lr =
) ,
(8.15)
1
Γ
2
(
G ll G rr
−|
G lr |
2
)
i
Γ (
G ll +
G rr
from which the transmission T lr can be obtained:
2
2
T lr =
4
Γ
|G lr |
.
(8.16)
In the present tight-binding treatment of the electronic structure of QD arrays, the
charge transport results from a competition between the on-site energy
that tends
to localize the electron in the dot positions, and the hopping energy t that favors the
motion from a site to its nearest neighbor. As a result, the spectrum of transmission
for a weakly coupled QD molecule (
ε
t ) typically consists of narrow peaks of
high (eventually perfect) conductance and narrower antiresonances (i.e., complete
suppression of transmission) or dips at a discrete set of energy values, on top of
a background of a smooth transmission function. The system is characterized by
a set of energy eigenvalues, roughly in the range
Γ
|
E
ε |
2 t , where the Green
function behaves approximately as
G lr
1
/
2 t , so the transmission becomes of order
t 2 . The expression for the connected Green function, ( 8.15 ),
and its corresponding transmission, ( 8.16 ), shows that the transmission through the
connected molecule depends on the electronic structure of the isolated molecule
both through the diagonal Green functions ( G ll and G rr ) at the connecting sites and
through the off-diagonal function G lr between them. The choice of the pair
2
2
2
T lr =
4
Γ
|G lr |
Γ
/
)
corresponds to the dependence on the topology of the connection, which we will
refer to as the
(
l
,
r
(
l
,
r
)
connection.
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