Biomedical Engineering Reference
In-Depth Information
where
T
S
reads
⎛
⎞
T
10000000
01000000
00100000
00010000
⎝
⎠
T
S
=
,
(7.50)
in accordance with its general definition given in Eq. (
7.17
), and the projector
P
2
1
T
2
,
2
1
1
0
,
removes the last four components from the vectors that span ker
(
T
S
)
.Itis
then straightforward to calculate the vectors that span the blocking space
B
1
0
:
2
,
2
1
,
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
e
−
i
6
√
2
0
0
e
+
i
6
√
2
0
0
0
e
−
i
6
√
2
0
0
e
+
i
6
√
2
0
0
0
e
−
i
6
√
2
0
0
e
+
i
6
√
2
v
1
=
,
v
2
=
,
v
3
=
.
(7.51)
The vectors
v
1
,
v
2
,and
v
3
are the components of the blocking states written in the 2
1
basis set presented in (
7.45
). Thus, the three blocking states correspond to the three
different projectors of the total spin
S
z
=
h
, respectively. Essentially,
there is a blocking state for each of the three projection of the spin
S
z
. This result is
natural since, for unpolarized or parallel polarized leads, coherences between states
of different spin projection along the common lead quantization axis do not survive
in the stationary limit.
h
,0,and
−
7.6
Spin-Dependent Transport
In the previous sections we have shown different types of interference blocking,
involving both ground and excited many-body states. All of them were essentially
described in terms of the sequential tunnelling dynamics generated by
L
tun
(see
Eq. (
7.6
)). We neglected
H
eff
in the analysis of the numerical results and correspond-
ingly the role of the third condition (
0) in the definition of a blocking
state. Indeed, the consequences of the dynamics generated by
H
eff
on the transport
characteristics of the benzene and triple dot I-SETs are marginal for unpolarized
leads.
The scenario changes completely for the case of spin polarized leads (Fig.
7.13
).
Thanks to (
7.21
) the destructive interference between orbitally degenerate electronic
states typical of I-SETs produces current blocking at
specific
bias voltages (see
Figs.
7.16
and
7.11
). In the presence of parallel polarized ferromagnetic leads the
interplay between interference and the exchange coupling on the system generates
an effective energy renormalization yielding different blocking biases for majority
and minority spins. Hence, by tuning the bias voltage full control over the spin of
[
H
eff
,
ρ
block
]=