Biomedical Engineering Reference
In-Depth Information
the quantization axis of the lead
. It is interesting to note that in that case, no
blocking states can be found unless the polarization of one of the leads is
P
χ
1.
The spin channel can in fact be closed only one at the time via linear combination
of different spin states.
A last comment on the definition of the blocking conditions is necessary.
A blocking state is a stationary solution of the Eq. (
7.5
) since by definition it does
not evolve in time. The density matrix associated with one of the blocking states
discussed so far (i) commutes with the system Hamiltonian since it is a state with
given particle number and energy; (ii) it is the solution of the equation
=
0
since the probability of tunnelling out from a blocking state vanishes, independent
of the final state. Nevertheless, a third condition is needed to satisfy the condition of
stationarity:
L
tun
ρ
=
3. The density matrix
ρ
block
associated with the blocking state should commute with
the effective Hamiltonian
H
eff
which renormalizes the coherent dynamics of the
system to the lowest non-vanishing order in the coupling to the leads:
[
ρ
block
,
H
eff
]=
0
.
(7.21)
The specific form of
H
eff
varies with the details of the system. Yet its generic bias
and gate voltage dependence implies that, if present, the current blocking occurs
only at specific values of the bias for each gate voltage. Further, if an energy level
has multiple blocking states and the effective Hamiltonian distinguishes between
them, selective current blocking, and correspondingly all electrical preparation of
the system in one specific degenerate state, can be achieved. In particular, for spin
polarized leads, the system can be prepared in a particular spin state without the
application of any external magnetic field as we will show explicitly in Sect.
7.6
.
Before continuing with the discussion, in the following sections, we derive here
the Eq. (
7.1
) as a specific example of the general theory presented so far. That
equation represents the interference blocking condition for the simplest possible
configuration involving only a non-degenerate and a doubly degenerate state. Let
us consider for simplicity a spinless
2
system and a gate and bias condition that
restricts the set of relevant many-body states to three: one with
N
particles and two
(degenerate) with
N
+
1or
N
−
1 particles. The interference blocking state, if it
exists, belongs to the
N
1 level. There is only one interesting tunnelling matrix to
be analyzed, namely
T
N
±
1
. Let us take for it the generic form:
±
γ
S
1
γ
S
2
T
N
±
1
=
(7.22)
γ
D
1
γ
D
2
2
The assumption of a spinless system is not restrictive for parallel polarized leads and transitions
between a spin singlet and a doublet since the different spin sectors decouple from each other.
