Biomedical Engineering Reference
In-Depth Information
second approach employs phenomenological models capturing the main physical
mechanisms and providing qualitative physical insight of the underlying effects.
While the former is more precise and rigorously grounded, the latter gives results of
a wider generality, beyond the details or the peculiarities of the system under study.
Therefore, both approaches are complementary and a number of methods lie in be-
tween these pure extremes. Here, we shall consider the second approach, neglecting
the spatial extension of the wave function, and considering a discretization of the
Hamiltonian. The electron is assumed to be in states of given sites (QD levels) with
a probability of transition between them (hopping term).
8.2.4
A Quantum Dot Molecule in an AB Interferometer
As a model for a quantum dot molecule embedded into a Aharonov-Bohm
interferometer, we consider four quantum dots forming a ring and coupled to two
leads L and R. Sites 2 and 4 of the ring are connected to each other forming the
artificial molecule, as shown in Fig.
8.1
b. We consider only one energy level in each
dot and both the intradot and interdot electron-electron interactions are neglected.
The system shall be described by a Hamiltonian
H
=
H
ring
+
H
lead
+
H
tunnel
,
(8.11)
where
H
ring
is the Hamiltonian of the isolated bicyclic ring,
4
i
=
1
ε
i
d
i
d
i
+
4
i
=
1
t
i
,
i
+
1
(
d
i
d
i
+
1
e
i
ϕ
+
d
i
+
1
d
i
e
−
i
ϕ
)+
V
(
d
2
d
4
+
d
4
d
2
)
=
H
ring
(8.12)
H
lead
is the Hamiltonian of the leads
R
ε
k
α
c
k
α
∑
H
lead
=
c
k
α
,
(8.13)
k
,
α
∈
L
,
and
H
tunnel
is the Hamiltonian describing the tunneling between the leads and the
ring
H
tunnel
=
∑
k
(
V
L
c
kL
d
1
+
V
R
c
kR
d
n
)+
H
.
c
.,
(8.14)
where
i
are the on-site energy at the dots,
t
i
,
i
+
1
are the nearest-neighbor hopping
parameters (where
t
45
ε
t
41
should be understood),
V
is the interdot hopping that
couples the upper and lower arms of the interferometer,
=
ϕ
=
2
π Φ
/
4
Φ
=
π Φ
/
2
Φ
0
0
is the phase
2 acquired due to interdot hopping in a magnetic field threading
the ring with a reduced flux
ϕ
=
πφ
/
e
),
and
n
is the site of contact to the right lead. The left lead is always attached to
φ
=
Φ
/
Φ
0
(i.e., in units of the quantum
Φ
=
h
/
0
