Biomedical Engineering Reference
In-Depth Information
is the above discussed Fano resonance due to the path interference; the application
of a flux 0
4 suppresses the antiresonance leaving only a dip in the
transmission which also disappears at
<
Φ
<
Φ
0
/
4 leaving only the resonant peak,
as seen in Fig.
8.5
b. Further increase of the flux in the range
Φ
=
Φ
0
/
Φ
0
/
4
<
Φ
<
Φ
0
/
2
produces a new dip at an energy slightly smaller than
ε
2
moving the peak to slightly
higher energies. At
2 the dip in the transmission of the ring becomes an
antiresonance, with the resonant peak tuned with that of the upper chain as seen in
the inset of Fig.
8.5
c. Between
Φ
=
Φ
0
/
Φ
0
/
2and
Φ
0
, the behavior of
T
(
E
)
is reversed, such
that a cycle is completed in a period of
Φ
0
. Figure
8.5
d depicts the transmission
through the ring with an interarm coupling
V
5. Now the transmission through
the lower (dashed line) and upper (dotted line) arms, including the site laterally
coupled by
V
, shows Fano-like resonances at
=
0
.
ε
=
ε
4
, respectively.
This can be understood as the electron propagation along a linear chain between
macroscopic leads (continuous state) with a laterally coupled dot (discrete state).
The transmission through the ring (solid line) still remains close to that of the
lower arm with a lateral connection to site 2. Then, the overall picture for the
transmission through a ring with different connection strengths along each arm is
that of the transmission throughout the stronger pathway (i.e., the one with larger
hoppings) at almost every energy, except at the one resonant with the energy of the
site connecting the arms where a Fano interference occurs.
In the presence of a magnetic flux, the self-energy is a complex quantity and its
modulus should be considered. The self-energy contributions (
8.26
) show that the
flux
ε
=
ε
2
and at
in the paths throughout the arms while no phase
change occurs in the self-energy corresponding to the interarm coupling. Let us call
Σ
13
(
ϕ
)
Φ
introduces a phase
±
2
ϕ
13
e
2
i
ϕ
+
Σ
13
e
−
2
i
ϕ
+
the self-energy with magnetic field. Then,
Σ
13
(
ϕ
)=
Σ
A
,
B
,
C
13
,where
Σ
Σ
are the real self-energies at zero magnetic field, such that
13
2
A
13
2
B
13
2
C
13
2
A
13
B
13
cos 4
A
13
B
13
C
13
cos 2
|
Σ
13
(
ϕ
)
|
=(
Σ
)
+(
Σ
)
+(
Σ
)
+
2
Σ
Σ
ϕ
+
2
(
Σ
+
Σ
)
Σ
ϕ
,
(8.35)
where the first three terms represent the non-interfering transmission along the paths
A, B, and C. The last two terms contain the effect of the interference due to the
quantum and magnetic phases. Even for
0 there is an interference between
the path contributions to the self-energy. Interestingly, there are two periods in the
magnetic phase; a period
ϕ
=
Φ
=
Φ
0
(associated with cos 4
ϕ
)andaperiod
Φ
=
2
Φ
0
13
(associated with cos 2
0, the latter is
not present. On the other hand, as soon as a finite
V
exists, the self-energy acquires
the longer period modulated by the shorter one. Such a behavior has been observed
in experiments [
6
] and were termed as Fano resonances of the
big
and
small orbits
.
Figure
8.6
a shows the conductance in the (1,3) connection, with the hopping
t
23
set equal to zero, such that there is a single small orbit. By setting
t
23
=
ϕ
). When there is no interarm coupling,
Σ
=
1and
V
0, one has a big orbit enclosing twice the flux threading the small one, whose
transmission is shown in Fig
8.6
b. For a given energy
E
, the passage from the former
to the latter can be realized by decreasing the interarm coupling at the time that the
=
