Information Technology Reference
In-Depth Information
The two mobile electrons are free to tunnel between adjacent dots, however,
tunneling out of the cell is assumed to be completely suppressed.
1.1 Full-Basis Quantum Mechanical Treatment
For the single four-dot cell considered in this chapter (with two electrons of oppo-
site spin), there are a total of sixteen underlying basis vectors. Modelling a single
cell within the full sixteen-dimensional Hilbert space makes it possible to capture
the full dynamics of the cell, and represents the most complete way of modelling
a QCA cell [
39
]. To maintain the full many-electron degrees of freedom of an
N
-cell system however (including the correlation and exchange effects between
cells), one must treat the entire
N
-cell system as a single quantum system by
constructing a new basis set consisting of the direct-product of all combinations
of the single-cell basis vectors. Thus, a complete basis set for an
N
-cell system
would require 16
N
basis vectors. A 16
N
16
N
Hubbard-type Hamiltonian for this
system can be constructed using the standard second-quantized notations [
39
],
×
H
=
i,σ,m
(
a
i
(
m
)
a
j
(
m
)+
a
j
(
m
)
a
i
(
m
))
E
0
n
i,σ
(
m
)
− t
i,j
i>j,σ,m
+
i,m
n
i,σ
(
m
)
n
j,σ
(
m
)
|r
i
(
m
)
E
Q
n
i,ⓦ
(
m
)
n
i,ₓ
(
m
)+
V
Q
− r
j
(
m
)
|
i>j,σ,σ
,m
n
i,σ
(
m
)
n
j,σ
(
k
)
|r
i
(
m
)
+
V
Q
,
(1)
− r
j
(
k
)
|
i>j,σ,σ
,k>m
where the operator
a
i,σ
(
m
)(
a
i,σ
(
m
)) annihilates (creates) an electron in the
i
th
site of cell
a
i,σ
(
m
)
a
i,σ
(
m
)isthe
number operator for the
i
th
site of cell
m
,and
V
Q
=
q
e
/
(4
ʾ˃
) is a constant,
where
q
e
is the charge of the electron and
˃
the electrical permittivity of the
medium. The first term in Eq. (
1
) represents the on-site energy of a dot. The
second term describes the electron tunnelling between neighbouring sites
i
and
j
within a cell,
m
. The third term in Eq. (
1
) accounts for the energetic cost of
putting two electrons of opposite spin at the same site, and the final two terms
are related to the Coulombic interactions between electrons in the same cell and
in neighbouring cells, respectively. The polarization of each cell can be found by
evaluating,
m
ʳ
, the operator
n
i,σ
(
m
)
↕
with spin
P
m
=
(
ʻ
1
+
ʻ
3
)
(
ʻ
2
− ʻ
4
)
ʻ
1
+
ʻ
2
+
ʻ
3
+
ʻ
4
−
,
(2)
where
ʻ
i
is the expectation value of the number operator on the
i
th
site of
cell
m
, i.e.,
ʻ
i
. Using this treatment, it has been shown that the
dynamics of small arrays can indeed be solved for directly while retaining the
full many-electron degrees of freedom [
39
]. However, the exponential growth in
=
≤
n
i
(
m
)
≥
Search WWH ::
Custom Search