Biomedical Engineering Reference
In-Depth Information
7.4.1
Model Hamiltonian
For the description of the benzene molecule weakly coupled to source and drain
leads, we adopt the total Hamiltonian (
7.2
) introduced in Sect.
7.2
where the first
term is now the interacting Hamiltonian for isolated benzene [
44
-
46
] that we recall
here for clarity:
d
i
σ
H
ben
=
ξ
0
i
σ
d
i
σ
i
σ
d
i
+
1σ
d
i
σ
+
b
d
i
+
1σ
+
d
i
σ
n
i
↑
−
n
i
↓
−
1
2
1
2
∑
i
+
U
∑
i
n
i
↑
+
n
i
↓
−
1
n
i
+
1
↑
+
n
i
+
1
↓
−
1
,
+
V
(7.27)
where
d
i
σ
creates an electron of spin
σ
in the
p
z
orbital of carbon
i
,
i
=
1
,...,
6
d
i
σ
runs over the six carbon atoms of benzene and
n
i
σ
=
. Only the
p
z
orbitals
(one per carbon atom) are explicitly taken into account, while the core electrons
and the nuclei are combined into frozen ions, with the same spatial symmetry as the
relevant electrons. They contribute only to the constant terms of the Hamiltonian
and enforce particle-hole symmetry. Mechanical oscillations are neglected and all
atoms are considered at their equilibrium position.
This Hamiltonian for isolated benzene is respecting the
D
6
h
symmetry of the
molecule. Since for every site there are 4 different possible configurations (
d
i
σ
|
0
,
, the Fock space has the dimension 4
6
|↑
096, which requires
a numerical treatment. Though the diagonalization of the Hamiltonian is not a
numerical challenge, it turns out to be of benefit for the physical understanding
of the transport processes to divide
H
ben
into blocks, according to the number
N
of
p
z
electrons (from 0 to 12), the
z
projection
S
z
of the total spin and the orbital
symmetries of benzene (see Table
7.2
).
The parameters
b
,
U
,and
V
for isolated benzene are given in the literature [
48
]
and are chosen to fit optical excitation spectra. The presence, in the molecular I-
SET, of metallic electrodes and the dielectric is expected to cause a substantial
renormalization of
U
and
V
[
49
-
51
]. Nevertheless, we do not expect the main results
of this work to be affected by this change. We consider the benzene molecule weakly
coupled to the leads. Thus, to first approximation, we assume the symmetry of the
isolated molecule not to be changed by the screening. Perturbations due to the lead-
molecule contacts reduce the symmetry in the molecular junction. They are included
in
H
ben
(see Eqs. (
7.64
)and(
7.65
)) and will be treated in Sect.
7.7
.
The effect of the gate is included as a renormalization of the on-site energy
ξ
=
ξ
0
−
,
|↓
,
| ↑↓
)
=
4
,
0atthe
charge neutrality point. Source and drain leads are two reservoirs of non-interacting
electrons:
H
leads
=
∑
χ
k
σ
(
ε
k
−
μ
χ
)
eV
g
(
V
g
is the gate voltage) and we conventionally set
V
g
=
c
†
χ
c
,where
χ
=
L
,
R
stands for the left or
k
σ
χ
k
σ
right lead and the chemical potentials
μ
χ
of the leads depend on the applied bias
V
2
. In the following we will measure the energy starting from
voltage
μ
L
,
R
=
μ
0
±
