Environmental Engineering Reference
In-Depth Information
Figure 2.3
(a) A central conductor with a square lattice structure. The 0th
and (
N
+
1)th columns are respectively the left and right buffer layers.
Here
N
is the total number of layers of the scattering region in the central
conductor.(b)
n
thiterationstepintherecursiveGreen'sfunctiontechnique.
n
th layer is added into the subsystem constructed in the previous iteration.
each other, as shown in Fig. 2.3a. In this case, it is convenient to
choose a column in the lattice as a layer. In addition, we suppose
thatthecentralconductorincludesnotonlythescatteringregionbut
also portions of the left and right leads (0th and (
N
+
1)th layers),
namelybufferlayers.Here
N
isthenumberofthelayersconstituting
the scattering region. Note that by introducing the buffer layers, all
the other atoms in the leads can be regarded as the same as those
in infinite leads, which makes the formulation easier. At this time,
Green's functions in Eqs. 2.11 and 2.12, and the effective dynamical
matrix of the central conductor can be represented by (
N
+
2)
×
(
N
+
2) block matrices. Then, the 0th and (
N
+
1)th intra-layer
blocks of the effective dynamical matrix of the conductor
D
are
respectively described as
D
0,0
=
D
0,0
+
L
(
ω
),
D
N
+
1,
N
+
1
=
D
N
+
1,
N
+
1
+
R
(
ω
), (2.13)
where
A
i
,
j
is a block element of a matrix
A
between the
i
th and
j
th
layers.
D
denotes the dynamical matrix of the isolated conductor,
and
L/R
(
ω
)is the intra-buffer-layer block of the self energy due to
the left/right lead
L/R
(
ω
). All elements in the other blocks of the
selfenergymatricesarezero,andsotheotherblocksoftheeffective
dynamical satisfy the relation
D
i
,
j
=
D
i
,
j
. (2.14)
Itshouldbenotedthattheeffectivedynamicalmatrixhasablock
tri-diagonal structure with the diagonal blocks of
D
i
,
i
and the off-
diagonalblocks of
D
i
,
i
−
1
and
D
i
−
1,
i
.
