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In-Depth Information
2.1
Geometry of Single and Bundled MWCNT
MWCNT consists of several concentric rolled up graphene sheets as shown in Fig.
1(a) where the distance between each shell is approximately equivalent to the Vander-
Waal's gap (
ʴ
) ≈ 0.34nm [12]. Inner and outer diameters are represented as
D
1
and
D
M
respectively, where
M
denotes the number of shells in an MWCNT. The distance
between the center of MWCNT and ground plane is denoted as
y
. Outer diameter of
MWCNT depends on the number of shells and can be expressed as [13]
(
)
=+×× −
(1)
DD
2
δ
M
1
M
1
Figure 1(b) presents an MWCNT bundle (height=
H
and width=
W
)
that consists of
different number of MWCNTs having similar number of shells. Therefore, the total
number of MWCNTs in a bundle can be expressed as [4]
(
)
[
(
)
]
(2)
n
=
n
n
−
n
2
=
n
n
−
n
−
1
2
H
H
MWCNT
(if
n
H
is an even and odd number respectively)
where
(
)
(
)
and
(3)
n
=
W
D
+
δ
n
=
H
D
+
δ
H
2.2
Equivalent
RLC
Model for MWCNT Bundle
MWCNT consists of numbers of cylindrical shells that can be viewed as several shells
in parallel. It is seemed to be similar to an SWCNT bundle that consists of several
nanotubes in parallel; but in fact, it has significant differences with SWCNT bundle.
For an MWCNT, the number of conducting channels depends on the shell diameters.
The interconnect parasitics such as resistance, inductance and capacitance are
modeled based on the total number of conducting channels that accounts for the effect
of spin degeneracy and sub-lattice degeneracy of carbon atoms. Using the assumption
of one third shells as metallic, the average number of conducting channels for a
particular shell in MWCNT can be expressed as [15]
ND
()
≈
kTD k D
+
,
>
d
/
T
i
T
i
1
i
2
i
D
(4)
≈
2 / 3
≤
/ T
T
i
where
D
i
represents the diameter of
i
th
shell in MWCNT,
k
1
and
k
2
are equivalent to
2.04 × 10
-4
nm
-1
K
-1
and 0.425 respectively [15]. The thermal energy of electrons and
gap between the sub-bands determines the quantitative value of
d
T
which is equivalent
to 1300nmK at room temperature (T=300K) [15]. Thus, the total numbers of
conducting channels in an MWCNT bundle can be calculated using the summation of
conducting channels (
N
i
) of each shell in an MWCNT [15]
p
;
p
= total number of shells in a MWCNT (5)
Nn
=
N
i
MWCNT
i
=
1
