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
,
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