Environmental Engineering Reference
In-Depth Information
Next, we treat all nearest neighboring inter-layer blocks of
the dynamical matrix of the central conductor as a perturbation,
while the intra-layer blocks are treated nonperturbatively. For this
purpose, we consider the situation where these isolated layers are
connected with the adjacent layers one by one from left to right
of Fig. 2.3a: the 1st layer is added to the isolated left buffer layers
(0th layer) as a perturbation. As mentioned above, this process
is performed based on the Dyson's equation, which relates the
perturbedGreen'sfunctionmatrixtothenon-perturbedone.Forthe
firststep,thatis,combiningthe0thand1stlayers,wecanobtainthe
following key relations:
G
(1)
1,1
(
ω
)
=
g
1,1
(
ω
)
+
g
1,1
(
ω
)
D
1,0
G
(1)
0,1
(
ω
),
(2.15)
G
(1)
0,1
(
ω
)
=
g
0,0
(
ω
)
D
0,1
G
(1)
1,1
(
ω
),
(2.16)
G
(1)
1,0
(
ω
)
=
G
(1)
1,1
(
ω
)
D
1,0
g
0,0
(
ω
),
(2.17)
where
G
(
n
)
ω
)represents(
i
,
j
) block of the Green's function on
n
th
iteration step. The unperturbed Green's function isdefined as
g
i
,
i
(
ω
)
=
ω
i
,
j
(
+
i
0
+
−
D
i
,
i
−
1
.
2
(2.18)
Note that
G
(1)
1,1
(
ω
) is self-consistently determined from Eqs. 2.15
and 2.16, and then
G
(1)
1,0
(
ω
) can be obtained from Eq. 2.17. Next, the
second layer isadded to the combinedlayers, and wecan obtainthe
following relations:
G
(2)
2,2
(
ω
)
=
g
2,2
(
ω
)
+
g
2,2
(
ω
)
D
2,1
G
(2)
1,2
(
ω
),
(2.19)
G
(2)
1,2
(
ω
)
=
G
(1)
1,1
(
ω
)
D
1,2
G
(2)
2,2
(
ω
),
(2.20)
G
(2)
2,0
(
ω
)
=
G
(2)
2,2
(
ω
)
D
2,1
G
(1)
1,0
(
ω
).
(2.21)
In the same manner as Eqs. 2.15 and 2.16, Eqs. 2.19 and 2.20
are solved self-consistently. The same procedure is repeated for
the combined system and its adjacent isolated layer up to the
(
N
1)th iteration (see Fig. 2.3b), and eventually, the combined
system becomes the same as the original system of Fig. 2.3a. Here,
+
