Biomedical Engineering Reference
In-Depth Information
where
n
and
m
label the
S
N
-fold and
S
N
+
1
-fold degenerate ground states with
N
and
N
1 particles, respectively. In order to make the interference effects more visible
we remind that
d
Rτ
=
R
+
†
φ
d
Lτ
R
φ
, with
φ
=
π
for the para while
φ
=
2
π
/
3forthe
meta configuration. Due to the behavior of all eigenstates of
H
ben
under discrete
rotation operators with angles multiples of
π
/
3, we can rewrite the overlap factor:
nm
τ
|
2
e
i
φ
nm
2
N
,
n
|
d
Lτ
|
N
+
1
,
m
|
Λ
N
,
N
+
1
=
2
,
(7.38)
,
|
d
Lτ
|
+
,
S
N
S
N
+
1
2
nm
τ
N
n
N
1
m
where
φ
nm
encloses the phase factors coming from the rotation of the states
|
N
,
n
and
.
The effective Hamiltonian
H
eff
neglected in (
7.36
) only influences the dynamics
of the coherences between orbitally degenerate states. Thus, Eq. (
7.36
) provides an
exact description of transport for the para configuration, where orbital degeneracy
is cancelled. Even if Eq. (
7.36
) captures the essential mechanism responsible for the
conductance suppression, we have derived an exact analytical formula also for the
meta configuration which can be found in Appendix B of [
20
].
In Fig.
7.5
we present an overview of the results of both the para and the meta
configuration. A direct comparison of the conductance (including the
H
eff
term of
(
7.5
)) in the two configurations is displayed in the upper panel. The lower panel
illustrates the effect of the energy non-conserving terms on the conductance in the
meta configuration. The number of
p
z
electrons on the molecule and the symmetry
of the lowest energy states corresponding to the conductance valleys are reported.
The symmetries displayed in the upper panel belong to the (effective) ground states
in the para configuration, the corresponding symmetries for the meta configuration
are shown in the lower panel.
Figure
7.5
shows that the results for the para and the meta configuration coincide
for the 10
|
N
+
1
,
m
12
particles have
A
-or
B
-type symmetries, they are therefore orbitally non-degenerate,
no interference can occur and thus the transitions are invariant under configuration
change. For every other transition we see a noticeable difference between the results
of the two configurations (Fig.
7.5
). In all these transitions one of the participating
states is orbitally degenerate. First we notice that the linear conductance peaks for
the 7
↔
11 and 11
↔
12 transitions. The ground states with
N
=
10
,
11
,
9 transitions in the para configuration are
shifted
with respect
to the corresponding peaks in the meta configuration. The selection of an effective
symmetric Hilbert space associated with the para configuration reduces the total
degeneracy by cancelling the orbital degeneracy. In addition, the ground state energy
of the 4- and 8-particle states is different in the two configurations, since in the
para configuration the
effective
ground state is in reality the first excited state. The
degeneracies
S
N
,
↔
8and8
↔
S
N
+
1
of the participating states as well as the ground state energy
are both entering the degeneracy term of Eq. (
7.36
)
