Biomedical Engineering Reference
In-Depth Information
where one appreciates the linear dependence of the spin splitting on the lead
polarization
P
α
. The first and the third terms of the sum would cancel each other if
the energy of the singlet and triplet 8-particle states would coincide. An analogous
condition, but this time on the 6-particle states, concerns the second and the
fourth terms. For this reason the exchange interaction on the system is a necessary
condition to obtain spin splitting of the renormalization frequencies and thus the full
all-electric spin control.
In Fig.
7.17
we show the frequencies
0 vs. bias voltage also for a finite
values of the polarization
P
calculated for the benzene I-SET, where exchange
splitting is ensured by the strong Coulomb interaction on the system. The inter-
ference blocking conditions
ω
L
σ
=
R
current are satisfied at different
biases for the different spin species. The dotted and dashed lines in Fig.
7.16
are
the representation of the relations
ω
L
σ
=
0forthe
L
→
ω
L
↑
=
0
,
ω
L
↓
=
0 as a function of the bias and
polarization, respectively.
7.7
Robustness
One could argue about the fragility of an effect which relies on the degeneracy of
the many-body spectrum. Interference effects are instead rather robust. The exact
degeneracy condition can in fact be relaxed and interference survives also for a
quasi-degeneracy
condition: i.e., as far as the splitting between the many-body
levels is smaller than 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
.
To quantify the robustness of the effect we will address, in this section, two
issues: the first is the modification of the master equation, Eq. (
7.5
), necessary to
capture the interference between quasi-degenerate states, the second is the detailed
study of an example of I-SET (the benzene single molecule junction) under several
perturbations that lower the symmetry of the system.
7.7.1
GME and Current in the Non-secular Approximation
The bias and the contact perturbations in our model for a benzene I-SET lower the
symmetry of the active part of the junction and consequently lift the degeneracy that
appeared so crucial for the interference effects. The robustness of the latter relies
on the fact that the necessary condition is rather quasi-degeneracy, expressed by the
relation
.
Nevertheless, if the perfect degeneracy is violated, the secular approximation ap-
plied to obtain Eqs. (
7.5
)-(
7.8
) does not capture this softer condition. We report here
the general expression for the generalized master equation and the associated current
operator in the Born-Markov approximation and under the only further condition
δ
E
h
Γ
