Biomedical Engineering Reference
In-Depth Information
where
S
and
D
indicate source and drain, respectively, and 1 and 2 label the two
degenerate states with
N
γ
S
(
D
)
±
1 particles.
γ
S
(
D
)
i
are the elements of the
matrices
introduced in Eqs. (
7.14
)and(
7.15
). The decoupled space reads:
ker
T
N
±
1
.
D
N
±
1
=
(7.23)
1 particles relevant Hilbert space has dimension 2 the only
possibility to find a blocking state is that
Since the
N
±
D
N
±
1
=
0. In other terms:
det
T
N
±
1
=
γ
S
1
γ
D
2
−
γ
D
1
γ
S
2
=
0
(7.24)
This condition is identical to Eq. (
7.1
). The blocking state can finally be calcu-
lated as:
B
N
+
1
=
P
N
+
1
ker
γ
S
1
γ
S
2
1
∩C
N
+
1
(7.25)
γ
D
1
γ
D
2
0
or
B
N
−
1
=
P
N
−
1
ker
γ
S
1
γ
S
2
0
∩C
N
−
1
,
(7.26)
γ
D
1
γ
D
2
1
where the
P
N
±
1
simply removes the last component of the vector that defines the one-dimensional
kernel.
C
N
±
1
is, in the relevant case, the entire space and the projector
7.4
The Benzene I-SET
The general ideas on interference blocking presented in the previous section apply
to a large class of devices. As a first example of interference SET based on quantum
dot molecules we consider a benzene single molecule transistor. We treat the
transport through the benzene I-SET in two different setups, the para and the meta
configuration, depending on the position of the leads with respect to the benzene
molecule (see Fig.
7.3
). Similar to [
47
], we start from an interacting Hamiltonian
of isolated benzene where only the localized
p
z
orbitals are considered and the
ions are assumed to have the same spatial symmetry as the relevant electrons. We
calculate the 4
6
096 energy eigenstates of the benzene Hamiltonian numerically.
Subsequently, with the help of group theory, we classify the eigenstates according
to their different symmetries and thus give a group-theoretical explanation to the
large degeneracies occurring between the electronic states. For example, while
the six-particle ground state (
A
1
g
symmetry) is non-degenerate, there exist four
seven-particle ground states due to spin and orbital (
E
2
u
symmetry) degeneracy.
Fingerprints of these orbital symmetries are clearly visible in the strong differences
in the stability diagrams obtained by coupling the benzene I-SET to the leads in the
=
4
,