Biomedical Engineering Reference
In-Depth Information
nonlinear current characteristics, we have to take into account energy conservation,
the Pauli exclusion principle and the interference between participating states. For
the visualization of the interference effects, we introduce the transition probability
(averaged over the
z
coordinate and the spin
σ
):
L
/
2
1
2
L
→
∞
σ
†
σ
(
2
P
(
x
,
y
;
n
,
τ
)=
lim
L
d
z
|
7
g
n
τ
|
ψ
r
)
|
6
g
|
(7.40)
−
L
/
2
for the
physical
7-particle basis, i.e., the 7-particle basis that diagonalizes the
stationary density matrix at a fixed bias. Here
2
labels the two states of the physical basis which are linear combinations of the
orbitally degenerate states
τ
is the spin quantum number,
n
=
1
,
and can be interpreted as conduction channels.
Each of the central panels of Fig.
7.6
are surface plots of (
7.40
) at the different bias
voltages
a
-
c
. The 7-particle ground states can interfere and thus generate nodes in
the transition probability at the contact atom close to one or the other lead, but, in
the meta configuration, never at both contact atoms at the same time.
Energetic considerations are illustrated in the lower panels of Fig.
7.6
for two
key points of the current curve at positive biases. The left panel corresponds to
the resonance peak of the current. Due to energy conservation, electrons can enter
the molecule only from the left lead. On the contrary the exit is allowed at both
leads. The current is suppressed when transitions occur to a state which cannot be
depopulated (a blocking state). Since, energetically, transmissions to the 6-particle
state are allowed at both leads, each 7-particle state can always be depopulated and
no blocking occurs.
The current blocking scenario is depicted in the lower right panel of Fig.
7.6
.
For large positive bias the transition from a 7-particle ground state to the 6-particle
ground state is energetically forbidden at the left lead. Thus, for example, the
c
panel in Fig.
7.6
visualizes the current blocking situation yielding NDC: while for
both channels there is a non-vanishing transition probability from the source lead to
the molecule, for the upper channel a node prevents an electron from exiting to the
drain lead. In the long time limit the blocking state gets fully populated while the
non-blocking state is empty. At large negative bias the blocking scenario is depicted
in the panel
a
that shows the left-right symmetry obtained by a reflection through
a plane perpendicular to the molecule and passing through the carbon atoms 6 and
3. The temperature sets the scale of the large bias condition and, correspondingly,
the width of the current peak presented in Fig.
7.6
grows with it. The peak is not
symmetric, though, its shape depends also on the energy renormalization introduced
by the coupling to the leads [
21
] and described by the effective Hamiltonian (
7.8
).
In fact the interference blocking is not a threshold effect in the bias. The complete
blocking corresponds to a very precise bias which is determined by the form of
H
eff
. We will return to this point in Sect.
7.6
, while discussing the spin-dependent
transport. Moreover, we remark that only a description that retains coherences
between the degenerate 7-particle ground states correctly captures NDC at both
positive and negative bias.
|
7
g
τ
