Biomedical Engineering Reference
In-Depth Information
a
b
c
E
1
E
2
E
3
E
4
1
10
-2
10
-4
10
-6
-2
-1
0
1
2
3
4
5
6
7
E
Fig. 8.2
Scheme of molecular orbitals for the 4-QDs ring: (
a
) without interarm coupling and
(
b
) with interarm coupling QD2-QD4. (
c
) Transmission functions through the ring without
coupling (
solid line
) and with coupling
V
=
2
.
5(
dashed line
). The
horizontal dot-dashed line
t
2
background of transmission. The
vertical dashed lines
show the position of the
energy eigenvalues
E
i
(
i
2
shows the
Γ
/
=
1
,...,
4) at which perfect transmission occurs. The antiresonance is
located at the energy
V
and is due to coupling of the quantum dot molecule QD2-
QD4. QD1 and QD3 are connected to the left and right leads and have on-site energies
(
ε
+
ε
)
/
2
−
2
4
ε
=
ε
=
0,
1
2
while
ε
=
2and
ε
=
4
2
4
eigenstate
ψ
k
of the isolated system have non-vanishing projection on the orbitals
|
l
and
|
r
:
l
|
ψ
k
and
r
|
ψ
k
. Reciprocally, if one of them equals zero, the electron
of energy
E
E
k
has a vanishing probability of being at both sites, and therefore no
transmission can occur. Since such a vanishing of transmission occurs at the energy
eigenvalues of the system, they have been named as
resonant zeros
.
Figure
8.2
depicts the scheme of the orbitals for the four-QD ring without
coupling between the arms (Fig.
8.2
a) and with an interarm coupling
V
=
5
(Fig.
8.2
b). In this representation, the spatial wave function for the orbital
k
is given
by
=
2
.
ϕ
i
are trial Gaussian functions centered at the QD locations.
The dark and light regions are zones of opposite signs and the dotted curves between
them are nodal lines. The transmission spectra are shown in Fig.
8.2
c with solid
(
V
i
c
ki
ϕ
i
(
r
)
,where
∑
. As discussed above, the peaks of perfect
transmission occurs at the energy eigenvalues of the ring
E
i
(
i
=
0) and dashed lines
(
V
=
2
.
5
)
=
1,...,4).Theorigin
of the antiresonance at
E
V
is the interference of the wave function
along the two arms and will be discussed at Sect.
8.2.8
. This allows one to change
the energy at which cancellation of transmission occurs by tuning the
V
coupling.
The energy of the second orbital is
E
=(
ε
2
+
ε
4
)
/
2
−
0 independently of
V
because of the QD2-
QD4 molecule is symmetrically coupled to terminal dots QD1 and QD3.
=
