Biomedical Engineering Reference
In-Depth Information
Fig. 7.10
Spectrum of the
triple dot system for the
specific gate voltage
eV
g
=
8
b
chosen to favor a
configuration with two
electrons. The other
parameters in the system are
U
4
.
,where
b
is the hopping integral
between the different dots.
From [
22
]
=
5
|
b
|
and
V
=
2
|
b
|
7.5.1
The Model
The total Hamiltonian of the I-SET is in the generic form (
7.2
). We describe the
system with an Hamiltonian in the extended Hubbard form
3
:
d
i
σ
d
i
σ
d
i
+
1σ
i
σ
i
σ
H
3d
=
ξ
d
i
σ
+
b
d
i
+
1σ
+
d
i
σ
0
n
i
↑
−
n
i
↓
−
1
2
1
2
∑
i
+
U
(7.42)
∑
i
n
i
↑
+
n
i
↓
−
1
n
i
+
1
↑
+
n
i
+
1
↓
−
1
,
+
V
(7.43)
where
d
i
σ
creates an electron of spin
σ
in the ground state of the quantum dot
i
.
Here
i
3 runs over the three quantum dots of the system and we impose
the periodic condition
d
4σ
=
=
1
,...,
d
i
σ
d
1σ
. Moreover
n
i
σ
=
d
i
σ
. The effect of the gate is
included as a renormalization of the on-site energy
eV
g
where
V
g
is the gate
voltage. We measure the energies in units of the modulus of the (negative) hopping
integral
b
. The parameters that we use are
ξ
=
ξ
0
−
.
H
leads
in (
7.2
) describes two reservoirs of non-interacting electrons with a
difference
eV
b
between their electrochemical potentials. Finally,
H
tun
accounts for
the weak tunnelling coupling between the system and the leads, characteristic of
SETs, and we consider the tunnelling events restricted to the atoms or to the dots
closest to the corresponding lead.
The number of electrons considered for the triple dot structure goes from 0 to
6. Thus the entire Fock space of the system contains 4
3
ξ
0
=
0
,
U
=
5
|
b
|,
V
=
2
|
b
|
64 states. By exact
diagonalization we obtain the many-body eigenstates and the corresponding eigen-
values that we present in Fig.
7.10
for a gate voltage of
V
g
=
=
4
.
8
b
/
e
.InTable
7.5
3
This denomination of the Pariser-Parr-Pople Hamiltonian is more common in the solid state
community.
