Biomedical Engineering Reference
In-Depth Information
which is almost equal to the current operator corresponding to the secular
approximation. The only difference is here the absence of the second projector
operator that allows contributions to the current coming from coherences between
different energy eigenstates.
7.7.2
Interference in a Reduced Symmetry I-SET
In this section we study the effect of reduced symmetry on the transport characteris-
tics of a benzene I-SET. For this purpose, we generalize the model Hamiltonian by
taking into account the perturbations on the molecule due to the contacts and the bias
voltage. The contact between molecule and leads is provided by different anchor
groups. These linkers are coupled to the contact carbon atoms over a
bond, thus
replacing the corresponding benzene hydrogen atoms. Due to the orthogonality of
σ
π
and
orbitals of
benzene. In particular the different electron affinity of the atoms in the linkers imply
a redistribution of the density of
σ
orbitals, the anchor groups affect in first approximation only the
σ
σ
electrons. Assuming that transport is carried by
π
electrons only, we model the effect of this redistribution as a change in the on-site
energy for the
p
z
orbitals of the contact carbon atoms:
H
ben
:
χσ
d
†
χσ
=
H
contact
=
ξ
c
d
χσ
,
χ
=
L
,
R
(7.64)
where
d
R
σ
=
d
5σ
, respectively, in the para and meta configuration, while
d
L
σ
=
d
1σ
in both setups.
We also study the effect of an external bias on the benzene I-SET. In particular
we release the strict condition of potential drop all concentrated at the lead-molecule
interface. Nevertheless, due to the weak coupling of the molecule to the leads, we
assume that only a fraction of the bias potential drops across the molecule. For
this residual potential we take the linear approximation
V
b
(
d
4σ
,
V
b
,
where we choose the center of the molecule as the origin and
r
sd
is the unity vector
directed along the source to drain direction.
a
0
=
r
)=
−
a
(
r
·
r
sd
/
a
0
)
43 A is the bond length between
two carbon atoms in benzene,
a
is the coefficient determining the intensity of the
potential drop over the molecule. Since the
p
z
orbitals are strongly localized, we
can assume that this potential will not affect the inter-site hopping, but only the
on-site term of the Hamiltonian:
1
.
H
ben
:
i
σ
ξ
b
i
d
i
σ
=
H
bias
=
e
d
i
σ
(7.65)
d
r
p
z
(
with
.
Under the influence of the contacts or the bias potential, the symmetry of
the molecule changes. Table
7.6
shows the point groups to which the molecule
ξ
b
i
=
r
−
R
i
)
V
b
(
r
)
p
z
(
r
−
R
i
)
