Performance Comparison between Carbon Nanotube and Copper Interconnects for GSI
Azad Naeemi, Reza Sarvari, and James D. Meindl
791 Atlantic Dr. N.W. Atlanta, GA 30332, Georgia Institute of Technology, Email: azad@ece.gatech.edu
Abstract
The performances of minimum-size copper and carbon
nanotube interconnects are compared for various ITRS
generations. Results offer important guidance regarding the
nature of carbon nanotube technology development needed for
improving interconnect performance. Since wave propagation is
slow in a single nanotube, nanotube-bundles with larger wave
speeds must be used. At the 45nm node (year 2010), the
performance enhancement that can be achieved by using
nanotube-bundles is negligible, and at the 22nm node (year
2016) it can be as large as 30% and 80% if carbon nanotubes
with electron mean free paths of 5µm and 10µm, respectively,
can be fabricated.
Introduction
As interconnect feature sizes shrink, copper resistivity
increases due to surface and grain boundary scatterings, and also
surface roughness [1]. Furthermore, wires, especially power and
ground lines, are becoming more and more vulnerable to
electromigration because of rapid increases in current densities
[2]. In contrast, carbon nanotubes exhibit ballistic flow of
electrons with electron mean free paths of several microns, and
are capable of conducting very large current densities [3].
Carbon nanotubes are therefore proposed as potential candidates
for power and signal interconnection [4, 5]. In this paper, after
reviewing the circuit models for single wall carbon nanotubes
(SWCN), the performances of copper wires and nanotube
interconnects are compared. The results offer important guidance
regarding the nature of carbon nanotube technology development
needed for improving interconnect performance.
Circuit Models for Metallic Carbon Nanotubes
Neglecting electron spin and sub-lattice degeneracy, the dc
conductance of an ideal quantum wire is e2/h, where h is the
Plank constant, and e is electron charge [6]. For a nanotube
above a ground plane, its magnetic inductance is
rg
µ
(1)
lm = 0 ln( ) ≈ 0.2 pH / µ m
r0
2π
where rg and r0 dimensions are shown in Fig. 1. The current
carriers of a nanotube occupy the 1-D conduction bands with
very low density of states, and hence the kinetic energy stored in
current is so large that it results in a very large kinetic inductance
[6, 7, 8]. To explain this qualitatively, the energy levels and
wave vectors accessible to free electrons in a quantum wire are
shown in Fig. 2. Due to the degenerative approximation, all
states below the Fermi level are occupied and the states above
are empty. At zero current, the number of electrons moving from
left to right (with positive k vectors) is the same as those moving
in opposite direction (negative k vectors). To generate current,
ε=
h2k 2
2m
εF
kF
δk =
2π
L
k
Figure 2: Electron energy versus wave-vector in a quantum wire.
Allowable states are shown by circles. Wave-vectors of electrons
have to be integer multiplications of 2π/L where L is the length of the
quantum wire. Due to the degenerative approximation, all states
below the Fermi-level, εF, (gray area) are filled. For current equal to
zero, the numbers of right-mover electrons (positive k) and left-mover
electrons (negative k) are the same. The axes are not to scale since
kF >>> δk.
some of the left-movers should be converted to right-movers
or vice versa, and since only one electron can stay at each state
(two if spin of electrons is considered), the converted electrons
have to go to higher states (Fig. 3). In this manner, the kinetic
energy of the system increases as N2/D(εF) where N is the
number of converted electrons and D(εF) is the density of
states near the Fermi level. By equating the kinetic energy
stored in current I with LkI2/2, one can calculate kinetic
inductance per unit length as
h
(2)
lk = 2
2e v F
where vF is the Fermi velocity [8]. For graphene and hence
carbon nanotubes vF =8×105 m/s [6, 7]. The kinetic inductance
per unit length of carbon nanotubes is therefore around
16nH/µm, more than 4 orders of magnitude larger than its
magnetic counterpart (1).
The electrostatic capacitance of a nanotube above a ground
plane is
2πε r ε 0
2πε r ε 0
(3)
cE =
≈
≈ 100aF / µ m
cosh −1 (rg / 2r0 ) ln(rg / r0 )
ε
εF-R
εF-L
k
rg
r0
Figure 1: A single-wall carbon nanotube above an ideal ground plane.
Figure 3: The Fermi levels for left and right movers (εF-L and εF-R,
respectively) in a quantum wire with a net current from left to right.
Some of the left-movers are converted to right-movers because of
which the kinetic energy of the system has increased. Due to small
density of states compared to the 3-D cases, the kinetic energy stored in
current is large which results in a large kinetic inductance. The axes are
not to scale since kF >>> δk.
0-7803-8684-1/04/$20.00 ©2004 IEEE
CL
h/2e2
lk
cQ
h/2e2
lk
lk
lk
cQ
cQ
cQ
h/2e2
Ground Plane
h/2e2
Figure 6: Single layer of carbon nanotubes above a ground plane.
Connecting these nanotubes in parallel lowers the overall contact
resistance, characteristic impedance, and inductance. The wave
propagation speed; however, does not change considerably because
overall capacitance increases.
cE
h/2e2
2
h/2e
h/2e2
h/2e2
Figure 4: The circuit model for a single wall carbon nanotube
considering spin and sub-lattice degenercies [6]. The magnetic
inductance is not shown because it is more than 4 orders of magnitude
smaller than the kinetic inductance.
where εrε0 is the dielectric permittivity. To add electric charge to
a quantum wire, one must add electrons to available states above
the Fermi level (Pauli exclusion principle); hence there is a
quantum capacitance of
2e 2
(4)
cQ =
hvF
in series with the electrostatic capacitance [6, 7, 8]. The
quantum capacitance of a carbon nanotube is 100aF/µm,
which is in the same order of magnitude as its electrostatic
counterpart (3).
Electron spin and sublattice degeneracy result in four parallel
channels shown in Fig. 4 through which three spin-modes and
one charge-mode can travel [6]. The effective circuit for the
charge-mode is shown in Fig. 5 [6]. The wave propagation speed
of nanotubes and their characteristic impedance are
1 1
1
(5)
v=
( + )
lK cE cQ
and
Z 0 = lK (
(6)
1
1
+ )
cE cQ
respectively. Assuming that the electrostatic and quantum
capacitances are equal, the wave speed is 5vF= 1.78×106, which
is 162 times smaller than the speed of light in free space. The
characteristic impedance also becomes numerically 7.1 KΩ. The
speed of spin-modes through which no charge is transferred is
always vF regardless of nanotube electrostatic capacitance [6].
Due to large values of contact resistance and characteristic
impedance of a single nanotube, many of them need to be used
in parallel for interconnect applications. Connecting nanotubes
above a ground plane in parallel as shown in Fig. 6 decreases the
inductance; however, the overall capacitance also increases
Contact
Resistance,
h/8e2
Contact
Resistance,
lk/4
h/8e2
4cQ
cE
CL
Ground Plane
Figure 7: A bundle of carbon nanotubes above a ground plane. The
wave propagation speed increases as n , where n is the number of
nanotubes in the bundle.
because of which the wave propagation speed remains
approximately constant. Latency of such a “single layer”
nanotube interconnect is
L
(7)
τ
= 0.7
+
nano
Rtr CL
where Rtr is driver resistance (equal to the line characteristic
impedance for impedance matching), CL is load capacitance
and L is interconnect length. In contrast, using a bundle of
carbon nanotubes above a ground plane as shown in Fig. 7
allows lowering inductance with no considerable change in
capacitance because of which the wave propagation speed
increases as n0.5, where n is the number of nanotubes in the
bundle. For n>100, the wave travel time becomes much
smaller than the RC charge up time of most typical
interconnects even at the end of the ITRS, and hence latency
would be equal to
(8)
τ bundle = Rtr (cbundle L + CL )
where cbundle is capacitance per unit length of the bundle of
nanotubes. Since the diameter of SWCNs can be less than 1
nm [9], a bundle of, for instance, 400 SWCNs can be as
narrow as 20nm.
Ideal Carbon Nanotubes versus Copper Wires
For copper interconnects, the RC distributed model
τ wire = 0.7 Rtr C L + 0.7( rint CL + cint Rtr ) L + 0.4rint cint L2 (9)
can be used where rint and cint are resistance and capacitance
per unit length of copper wires [10]. Resistance per unit
length, rint can be calculated by the combination of FuchsSondheimer and Mayadas-Shatzkes models for surface
scattering and grain boundary scattering [11, 12], respectively.
For the 22nm node (year 2016), latencies of carbon
nanotube and copper interconnects are plotted versus length in
Fig. 8. It is evident that for all practical lengths, a single level
of nanotubes above a ground plane has a latency larger than
that of a copper wire because of its very small wave
propagation speed. A bundle of nanotubes, however, has a
smaller latency especially for long lengths. By comparing (8)
and (9), and assuming that cbundle≈ cint, the ratio of latencies of
nanotube bundles and copper wires can be written as
0.57 +
Figure 5: The effective circuit model for the charge mode propagation
in a single wall carbon nanotube [6].
2v F
CL
τ copper
cint L rint L
r L
= 1+
≈ 1 + int
CL
R
Rtr
τ Bundle
tr
1+
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cint L
(10)
Figure 8: Latency versus interconnect length for ideal single-layered
carbon nanotubes above a ground plane, minimum sized copper wires
implemented at the 22 nm technology node (year 2016) and bundles of
ideal SWCNs for n>100. For copper interconnects, surface scattering
co-efficient, p, and grain boundary reflection coefficient, R, are
pessimistically assumed to be 0 and 0.5, respectively.
Figure 9: Latency versus interconnect length for minimum sized
copper wires implemented at the 22 nm node and bundles of SWCNs
with electron mean free paths of 5µm and 10µm. For copper
interconnects p=0 and R=0.5 are assumed.
suitable for lengths larger than 10 times the electron mean free
path.
The performance enhancement that can be achieved by
using carbon nanotubes is a function of length as shown in
Fig. 11. The maximum performance enhancement and the
which shows that performance enhancement is larger for longer length at which it occurs are given by
interconnects and larger drivers.
⎧⎪
⎫⎪
τ copper
⎡L r
⎤
2R
(14)
≈ max ⎨0.76 ⎢ 0 int (ln tr − 1) + 1⎥ , 1⎬
R
R
τ Bundle max
⎪
⎪
tr
0
⎣
⎦
⎩
⎭
Non-Ideal Carbon Nanotubes versus Copper wires
Even initially ideal carbon nanotubes become disordered and
once they are physisorbed on a surface [13]. Electron mean free
⎡
2R
R ⎤
(15)
Lmax ≈ 2 L0 ⎢0.8ln( tr ) + 0.25 − 0.25 tr ⎥ ,
path in real carbon nanotubes is hence finite. Electron-phonon
R
L
r
0
0 int ⎦
⎣
scattering that is dominant in copper wires is inelastic. In
respectively. In Fig. 12, the maximum performance
contrast, electron-defect scattering, which is the major scattering
enhancement that can be achieved by carbon nanotubes is
mechanism in carbon nanotubes, is elastic because of which
plotted versus electron mean free path in carbon nanotubes for
electron waves in nanotubes do not loose their phase coherency.
four various technology generations. It can be seen that the
The incident electron wave and reflected waves therefore
performance enhancement at the 45nm node (year 2010) is
interfere and resistance increases exponentially with length [13,
14]. Resistance of a bundle of nanotubes can be written as [13, negligible even if mean free paths as large as 10µm or 20µm
are achieved. This is mainly due to small resistance per unit
15]
L
(11)
R = R0 e 2 L0
where R0 is the resistance of the bundle of defect free nanotubes
(ideally equal to h /(4e2n)), and L0 is the electron mean free path.
As an example, in Fig. 9, latencies of nanotube-bundles and
copper interconnects implemented at the 22nm node are plotted
versus interconnect length for L0 =5µm and L0 =10µm. There is a
length beyond which resistance of nanotube bundles become
larger than that of copper wires. This length is equal to
R
(12)
Lcrit = −2 LambertW (−1, − 0 )
rint L0
where LambertW(-1, x) is the solution of equation exp(x)=x.
Equation (12) can be approximated by
⎡
2r L ⎤
(13)
Lcrit ≈ 2 L0 ⎢ 0.9 + 1.15ln( int 0 ) ⎥
R0 ⎦
⎣
with less than 5% error for 2rL0 / R0 > 10 , which is valid for
virtually all practical cases. The critical length is plotted versus
electron mean free path in Figure 10. As a rule of thumb, it can
be said that the critical length is roughly 10 times the electron
mean free path. This means that carbon nanotubes are not
Figure 10: The critical length beyond which resistance of nanotubebundles becomes larger than that of minimum size copper
interconnects versus electron mean free path in carbon nanotubes for
three generations of technology.
0-7803-8684-1/04/$20.00 ©2004 IEEE
Figure 12: Maximum performance enhancement that can be achieved
by using carbon nanotubes versus electron mean free path in
nanotubes for four various generations of technology. At the 22nm
node (year 2016) performance enhancement is 30% for mean free path
of 5µm, and it can be as large as 80% if achieving a mean free path of
10µm is feasible
Figure 11: Relative performance of nanotube-bundles to minimum
sized copper interconnects implemented at the 22nm node (year 2016)
versus interconnect length for three various values of electron mean
free path in carbon nanotubes.
length of copper wires. At the 22nm node (year 2016), however,
performance enhancement is 30% for mean free path of 5µm,
and it can be as large as 80% if achieving a mean free path of
10µm is feasible. It is also evident from Fig. 12 that the
maximum performance enhancement increases linearly as
electron mean free path in nanotubes increases. As (14) shows,
the key factor determining the maximum performance
enhancement is the product of resistance of a copper
interconnect whose length is equal to the mean free path of
electrons in carbon nanotubes and reciprocal driver resistance
(rintL0/Rtr ). In other words, having a larger resistance per unit
length of copper interconnects, a larger electron mean free path
of electrons in carbon nanotubes, or a smaller driver resistance
augment the advantage of using carbon nanotube interconnects.
The electron mean free path in a carbon nanotube is a
function of the amount of disorder as well as the nanotube
diameter. Since electron wave has a doughnut-like wave packet
extended around its circumference, the effective disorder that an
electron experiences is smaller for larger diameters because of
which electron mean free path is linearly proportional to
nanotube diameter [13]. Nanotubes with very large diameters,
however, may not be mechanically robust.
Conclusions
For the first time, the performances of carbon nanotube and
minimum-size copper interconnects are compared for various
generations of technology. The wave propagation speed in a
single carbon nanotube above a ground plane is more than two
orders of magnitude smaller than the speed of light in free space
because of extremely large kinetic inductance values. It is
therefore critical to use bundles of nanotubes to lower the overall
inductance value without increasing capacitance to improve the
wave speed such that the RC charge-up time becomes dominant.
Otherwise, latency of minimum-sized copper interconnects will
be smaller than that of nanotubes for all practical lengths.
Resistance of carbon nanotubes is exponentially proportional to
the ratio of nanotube length to electron mean free path. Hence
there is a critical length beyond which minimum size copper
wires have smaller latencies compared to nanotube-bundles. As a
rule of thumb, this critical length is roughly 10 times the electron
mean free path in nanotubes. It is also shown that the maximum
performance enhancement that can be achieved using nanotubebundles is proportional to the product of reciprocal driver
resistance and resistance of a copper interconnect whose
length is equal to the mean free path of electrons in carbon
nanotubes. The performance enhancement at the 45nm node
(year 2010) is negligible even if mean free paths as large as
10µm or 20µm are achieved. At the 22nm node (year 2016),
however, maximum performance enhancement is 30% for a
mean free path of 5µm, and it can be as large as 80% if
achieving a mean free path of 10µm is feasible.
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