Biomedical Engineering Reference
In-Depth Information
of the z -projection of the quasi-angular momentum are the ones that diagonalize
all operators
R φ
with angles multiples of
π /
3. The corresponding eigenvalues are
phase factors e i φ where h
, the quasi-angular momentum of the state, is an integer
multiple of h . The discrete rotation operator of an angle
( C 2 symmetry
operation) is the one relevant for the para configuration. All orbitals are eigenstates
of the C 2 rotation with the eigenvalue
φ = π
1.
The relevant rotation operator for the meta configuration corresponds to an
angle
±
3( C 3 symmetry operation). Orbitals with an A or B symmetry are
eigenstates of this operator with the eigenvalue
φ =
2
π /
+
1 (angular momentum
=
0or
=
3). Hence we can already predict that there will be no difference based on
rotational symmetry between the para and the meta configuration for transitions
between states involving A -and B -type symmetries. Orbitals with E symmetry,
however, behave quite differently under the C 3 operation. They are the pairs of states
of angular momenta
2. The diagonal form of the rotation operator
on the twofold degenerate subspace of E -symmetry reads:
= ±
1or
= ±
e −||·
2
3
i
0
=
.
C 3
(7.33)
2 3
e ||·
i
0
For the twofold orbitally degenerate 7-particle ground states
2. This analysis
in terms of the quasi-angular momentum makes easier the calculation of the
fundamental interference condition ( 7.1 ) given in the introduction. In fact the
following relation holds between the transition amplitudes of the 6- and 7-particle
ground states:
|| =
d R τ |
φ
d R φ ,|
e i φ γ L
γ R
7 g τ |
6 g =
7 g τ |R
6 g =
(7.34)
and ( 7.1 ) follows directly.
7.4.3
Transport Calculations
With the knowledge of the eigenstates and eigenvalues of the Hamiltonian for the
isolated molecule, we implement Eq. ( 7.5 ) and look for a stationary solution. The
symmetries of the eigenstates are reflected in the transition amplitudes contained
in the generalized master equation. We find numerically its stationary solution and
calculate the current and the differential conductance of the device. In Fig. 7.4 we
present the stability diagram for the benzene I-SET contacted in the para (upper
panel) and meta position (lower panel). Bright ground state transition lines delimit
diamonds of zero differential conductance typical for the Coulomb blockade regime,
while a rich pattern of satellite lines represents the transitions between excited states.
Though several differences can be noticed, most striking are the suppression of the
linear conductance, the appearance of negative differential conductance (NDC), and
 
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