Biomedical Engineering Reference
In-Depth Information
(SETs) in which, usually, the decoherence introduced by the leads dominates
the picture and the dynamics essentially consists of sequential tunnelling events
connecting the many-body eigenstates of the isolated system. Yet, interference is
achieved whenever two
energetically equivalent
paths involving degenerate states
contribute to the dynamics (see Fig.
7.1
)[
22
]. The associated fingerprints in the
transport characteristics are a strong negative differential conductance (NDC) and
eventually a current blocking in the case of fully destructive interference.
In the simplest case, NDC and current blocking triggered by interference
take place any time a SET presents an
N
-particle non-degenerate state and two
degenerate
N
+
1-particle states such that the ratio between the transition
amplitudes
γ
i
χ
1-particle states is different for
tunnelling at the source (
S
) and at the drain (
D
) lead:
(
i
=
1
,
2
,
χ
=
S
,
D
) between those
N
-and
N
+
γ
1
S
γ
2
S
=
γ
1
D
2
D
.
(7.1)
γ
2
,
is implied by Eq. (
7.1
). This fact excludes the interpretation of the physics of the
interference SET in terms of standard NDC with asymmetric couplings. Instead,
due to condition (
7.1
) there exist linear combinations of the degenerate
N
Notice that no asymmetry in the tunnelling
rates
, which are proportional to
|
γ
i
χ
|
1-
particle states which are connected to the
N
-particle state via a tunnelling event to
one of the leads but
not
to the other. The state which is decoupled from the drain lead
(i.e., the lead with the lower chemical potential) represents a blocking state which
prevents the current to flow since electrons can populate this state by tunnelling
from the source but cannot tunnel out towards the drain.
It should be noticed that several blocking states can be associated with the same
system. Let us consider again the example associated with (
7.1
) and analyze an
inversion of the bias polarity which interchanges the source and the drain lead. If
the state decoupled from the right lead blocks the current
L
+
R
, vice versa the state
decoupled from the left lead is a blocking state for the current
R
→
L
. Typically these
two different blocking states are not orthogonal and cannot form together a valid
basis set of the
N
→
1 particle space. The basis set that diagonalizes the stationary
density matrix (what we call in the manuscript the
physical basis
) contains at large
positive biases the
L
+
R
blocking state and is thus different from the physical basis
at large negative biases which necessarily contains the
R
→
L
blocking state. More
generally, the physical basis depends
continuously
on the bias. Thus only a treatment
that includes also coherences and not only populations of the density matrix can
capture the full picture at all biases.
It could be argued about the fragility of an effect which relies on the presence
of degeneracies in the many-body spectrum. Interference effects are instead rather
robust. The exact degeneracy condition can be in fact relaxed and interference
survives as far as the splitting between the many-body levels is smaller that the
tunnelling rate to the leads. In this limit, the system still does not distinguish between
the two energetically equivalent paths sketched in Fig.
7.1
. Summarizing, despite the
→
