Biomedical Engineering Reference
In-Depth Information
8.3
Transmission Through Laterally Coupled QDs
Let us analyze in this section some numerical calculations for a four-QD ring
with on-site energies
4 in arbitrary units. The
connecting dots are assumed to be weakly coupled to the leads by
ε 1 = ε 3 =
0,
ε 2 =
2and
ε 4 =
Γ =
0
.
05.
Figure 8.3 shows the dimensionless conductance T
for the ring connected to
the leads in the configuration (1,3) (solid line), and for the three-site chains forming
the upper (dotted line) and lower (dashed line) arms with the side-dot QD4 and
QD2, respectively, coupled by V . The transmission with an applied magnetic flux
Φ =
(
E
)
Φ 0 is shown in dot-dashed line. The transmission shows four peaks at the
energy eigenvalues of the rings. The ring with disconnected arms, Fig. 8.3 a, shows
also an antiresonance, not present in the chains because, for V
0
.
1
0, there is no
side-coupled dot. This suppression of the transmission in the ring is due to the
cancellation of the contributions to the self-energy throughout the upper and lower
paths (
=
13
13
Σ 13 = Σ
+ Σ
=
0). In general, ( 8.26 ) shows that the self-energy vanishes if
D (
2 V =
t 2
t 2
g 22 e 2 i ϕ +
g 44 e 2 i ϕ +
e 2 i ϕ +(
e 2 i ϕ +
Σ
=
(
2 g 24
)=
E
ε
)
E
ε
)
0
13
4
2
(8.34)
Figure 8.3 a and b depict the tuning of the antiresonance with V
=
0and V
=
( ε 2 ε 4 ) /
2, respectively. Since, as mentioned above, in absence of magnetic flux
(
ϕ =
0),
Σ 13 vanishes at the energy E
=
¯
ε
V
=( ε 2 + ε 4 ) /
2
V . When there is a
finite magnetic flux,
Σ 13 is complex and its cancellation requires vanishing its real
and imaginary parts, i.e.,
(
2 E
ε 2 ε 4 )
cos 2
ϕ +
2 V
=
0, and
( ε 2 ε 4 )
sin 2
ϕ =
0.
Both equations cannot be satisfied simultaneously, except when
ε 2 = ε 4 ,which
presents an antiresonance at
. Therefore, the magnetic field eliminates
the antiresonance for a ring with different site energies
ε 2
V
/
cos 2
ϕ
ε 2 = ε 4 for arbitrary V .This
suppression of the antiresonance is shown with dot-dashed lines in Fig. 8.3 for a
flux
Φ =
0
.
1
Φ 0 .
a
b
T ( E )
T ( E )
1
10 -2
10 -2
10 -4
10 -4
10 -6
10 -6
E
E
-2
0
2
4
6
-2
0
2
4
6
Fig. 8.3 Transmission function of a four-QD ring in the connection (1,3) in absence of magnetic
field ( solid line ) and with an applied magnetic flux Φ = 0 . 0 ( dot-dashed line ). For comparison,
transmission for the three-site chains QD1-QD2-QD3 ( upper arm ) and QD1-QD4-QD3 ( lower
arm )isshownin dotted and dashed lines , respectively. ( a ) Without interarm coupling and ( b ) with
coupling V = 1 between the arms through QD2-QD4. The insets show the scheme of the chains
pointing to the respective curve
 
Search WWH ::




Custom Search