Biomedical Engineering Reference
In-Depth Information
the strong suppression of the current at the right(left) border of the 7 (5) particle
diamond when passing from the para to the meta configuration. All these features
are different manifestations of the interference between orbitally degenerate states
and ultimately reveal the specific symmetry of benzene.
7.4.3.1
Linear Conductance
We study the linear transport regime both numerically and analytically. For the
analytical calculation of the conductance we consider the low temperature limit
where only ground states with
N
and
N
1 particles have considerable occupation
probabilities, with
N
fixed by the gate voltage. Therefore only transitions between
these states are relevant and we can treat just the terms of (
7.5
) with
N
and
N
+
1
particles and the ground state energies
E
g
,
N
and
E
g
,
N
+
1
, respectively. A closer look
at (
7.5
) reveals that the spin coherences are decoupled from the other elements of
the density matrix. Thus we can set them to zero, and write (
7.5
) in a block diagonal
form in the basis of the ground states of
N
and
N
+
1 particles. Additionally, since
the total Hamiltonian
H
is symmetric in spin, the blocks of the GME with the same
particle but different spin quantum number
+
must be identical. Finally, since around
the resonance the only populated states are the
N
and
N
τ
+
1 particle states, the
conservation of probability implies that:
=
n
ρ
N
nn
+
m
ρ
N
+
1
1
,
(7.35)
mm
nn
is the population of the
N
-particle ground state and
n
contains the
orbital and spin quantum numbers. With all these observations we can reduce
(
7.5
) to a much smaller set of coupled differential equations that can be treated
analytically. The stationary solution of this set of equations can be derived more
easily by restricting in (
7.5
) to the dynamics generated by the sequential tunnelling
Liouvillean
where
ρ
L
tun
. With this simplification we derive an analytical formula for the
conductance close to the resonance between
N
and
N
1 particle states as the first
order coefficient of the Taylor series of the current in the bias:
+
1
S
N
S
N
+
1
f
(
Δ
Γ
L
Γ
R
Γ
L
+
Γ
R
Λ
E
)
2
e
2
G
N
,
N
+
1
(
Δ
E
)=
−
(7.36)
N
,
N
+
(
S
N
+
1
−
S
N
)
f
(
Δ
E
)+
S
N
where
Δ
E
=
E
g
,
N
−
E
g
,
N
+
1
+
eV
g
is the energy difference between the benzene
ground states with
N
and
N
1 electrons diminished by a term linear in the gate
voltage. Interference effects are contained in the overlap factor
+
Λ
N
,
N
+
1
:
nm
τ
2
d
Rτ
|
N
,
n
|
d
Lτ
|
N
+
1
,
m
N
+
1
,
m
|
N
,
n
Λ
N
,
N
+
1
=
,
(7.37)
2
S
N
S
N
+
1
N
,
n
|
d
ατ
|
N
+
1
,
m
∑
nm
ατ
