Biomedical Engineering Reference
In-Depth Information
|
d
ατ
|
+
Let us now consider the generic transition amplitude
N
N
1
,where
d
ατ
destroys an electron of spin
τ
on the contact atom closest to the
α
lead. It is useful
to rewrite this amplitude in the form
v
v
|
d
ατ
|
+
=
|
σ
σ
v
|
+
,
N
N
1
N
σ
v
d
ατ
σ
N
1
(7.29)
v
σ
v
=
where we have used the property
σ
1. Since in the para configuration both
v
contact atoms lie in the mirror plane
σ
v
, it follows
σ
v
d
α
σ
=
d
α
. If the participating
states are both symmetric under
σ
v
,Eq.(
7.29
) becomes
v
d
ατ
σ
v
|
,
|
σ
+
,
=
,
|
d
ατ
|
+
,
N
sym
N
1
sym
N
sym
N
1
sym
(7.30)
and analogously in the case that both states are antisymmetric. For states with
different symmetry it is
,
|
d
ατ
|
+
,
=
−
,
|
d
ατ
|
+
,
=
.
N
sym
N
1
antisym
N
sym
N
1
antisym
0
(7.31)
In other terms, there is a selection rule that forbids transitions between symmetric
and antisymmetric states. Further, since the ground state of the neutral molecule
is symmetric, for the transport calculations in the para configuration we select
the effective Hilbert space containing only states symmetric with respect to
σ
v
.
Correspondingly, when referring to the
N
particle ground state we mean the
energetically lowest
symmetric
state. For example, in the case of 4- and 8-particle
states it is the first excited state to be the
effective
ground state. In the para
configuration also the orbital degeneracy of the
E
-type states is effectively cancelled
due to the selection of the symmetric orbital (see Table
7.3
).
Small violations of this selection rule, due, e.g., to molecular vibrations or
coupling to an electromagnetic bath, result in the weak connection of different
metastable electronic subspaces. We suggest this mechanism as a possible ex-
planation for the switching and hysteretic behavior reported in various molecular
junctions. This effect is not addressed in this work.
For a simpler analysis of the different transport characteristics it is useful to
introduce a unified geometrical description of the two configurations. In both cases,
one lead is rotated by an angle
with respect to the position of the other lead. Hence
we can write the creator of an electron in the right contact atom
d
Rτ
φ
in terms of the
creation operator of the left contact atom and the rotation operator:
d
Rτ
=
R
†
φ
d
Lτ
R
φ
,
(7.32)
where
around
the axis perpendicular to the molecular plane and piercing the center of the benzene
ring;
R
φ
is the rotation operator for the anticlockwise rotation of an angle
φ
for the meta configuration.
The energy eigenstates of the interacting Hamiltonian of benzene can be
classified also in terms their quasi-angular momentum. In particular, the eigenstates
φ
=
π
for the para and
φ
=(
2
π
/
3
)
